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\begin{document}
\title{Minimal $4$-colored graphs representing an infinite family of hyperbolic $3$-manifolds}
\author{P. CRISTOFORI, E. FOMINYKH, M. MULAZZANI, V. TARKAEV}
\address{Paola CRISTOFORI - Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit\`a di Modena e Reggio Emilia, Italy}
\email{[email protected]}
\address{Evgeny FOMINYKH, Vladimir TARKAEV - Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia and Department of Mathematics, Chelyabinsk State University, Chelyabinsk, Russia.}
\email{[email protected]\quad [email protected]}
\address{Michele MULAZZANI - Dipartimento di Matematica and ARCES, Universit\`a di Bologna, Italy}
\email{[email protected]}
\maketitle
\centerline{\textit{To Professor Maria Teresa Lozano on the occasion of her 70th birthday}}
\begin{abstract} The graph complexity of a compact $3$-manifold is defined as the minimum order among all $4$-colored graphs representing it.
Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity $32$) and for compact orientable 3-manifolds with toric boundary (up to graph complexity $12$) and for infinite families of lens spaces.
In this paper we extend to graph complexity $14$ the computations for orientable manifolds with toric boundary and we give two-sided bounds for the graph complexity of tetrahedral manifolds. As a consequence, we compute the exact value of this invariant for an infinite family of such manifolds.
\noindent {\it 2010 Mathematics Subject Classification:} 57N10, \ 57Q15, \ 57M15.
\noindent {\it Key words and phrases:} $3$-manifolds, \ colored graphs, \ graph complexity, \ tetrahedral manifolds.
\end{abstract}
\section{Introduction}
Representation tecniques have long been used as an important tool in the study of PL manifolds. The theory of {\it crystallizations}, or more generally of {\it gems}, was introduced as a combinatorial representation of closed PL manifolds of arbitrary dimension by means of a particular class of edge-colored graphs (see \cite{[FGG]}). This tool has been proved to be particularly effective in dimension three adding to classical representation methods such as Heegaard diagrams, spines, framed knots and links, branched coverings, etc...
More recently, the representation by edge-colored graphs has been extended in \cite{[CM]} to non-closed compact $3$-manifolds.
More precisely, it has been proved that there is a well-defined surjective map from the whole set of {\it $4$-colored graphs} -- i.e., $4$-regular graphs equipped with an {\it edge-coloration}
(see Subsection \ref{colgraph}) -- to the set of $3$-manifolds that are either closed or have non-empty boundary with no spherical components.
In this context, it is natural to pose the problem of determining and listing
minimal (with respect to the number of vertices) $4$-colored graphs representing $3$-manifolds.
The order of a minimal graph $\Gammaamma$ is called the {\it graph complexity} of the represented manifold $M_\Gammaamma$.
By the duality between $4$-colored graphs and a particular kind of vertex-labeled pseudotriangulations (called {\it colored triangulations}), graph complexity of manifolds turns out to be also the number of tetrahedra in a minimal triangulation of this type (see details in Subsection \ref{colgraph}).
The graph complexity of a manifold is an important invariant in the theory of $3$-manifolds and the problem of its computation is usually very difficult.
Exact values of graph complexity can be obviously computed by enumerating $4$-colored graphs with increasing number of vertices and identifying the represented manifolds. This has been done first in the closed case and more recently
in the case of non-empty boundary. In particular, there exist tables of
\begin{itemize}
\item [(a)] closed orientable $3$-manifolds up to graph complexity $32$ (\cite{[CC],[CC1],[Li]});
\item [(b)] closed non-orientable $3$-manifolds up to graph complexity $30$ (\cite{[BCG]}, \cite{[C1]});
\item [(c)] compact orientable $3$-manifolds with toric boundary up to graph complexity $12$ (\cite{[CFMT]}).
\end{itemize}
As regards the computation of graph complexity for infinite families of $3$-manifolds, few results have been obtained up to now. It is proved in \cite{[CC1]} that lens spaces of the form
$L(qr + 1,q)$, with $q,r\gammae 1$ odd, have graph complexity $4(q + r)$,
while concrete examples of minimal graphs for the same family are constructed in \cite{[BD]}.
In Section \ref{sec 4} of this paper we extend table (c) to graph complexity $14$. Moreover, in Section \ref{sect: exact values} we give two-sided bounds for the graph complexity of compact tetrahedral manifolds (i.e., manifolds admitting a triangulation by regular ideal hyperbolic tetrahedra).
On the basis of this result we construct an infinite family of minimal $4$-colored graphs representing tetrahedral manifolds and, hence, compute the exact value of graph complexity for these manifolds.
\section{Preliminaries}
\subsection{Triangulations}\label{triang}
Let $\mathcal D = \{\tilde\Deltaelta_1, \ldots, \tilde\Deltaelta_n\}$ be a collection of pairwise-disjoint tetrahedra and suppose $\Phi = \{\varphi_1, \ldots, \varphi_{2n}\}$ is a family of affine homeomorphisms pairing faces of the tetrahedra in $\mathcal D$ so that every face has a unique counterpart. It is allowed that faces in each pair belong either to different tetrahedra or to the same tetrahedron.
We use $\mathcal D/ \Phi$ to denote the space obtained from the disjoint union of the tetrahedra of $\mathcal D$ by identifying all the faces via the homeomorphisms of~$\Phi$.
It is well known that, by the previous assumptions, the identification space $\mathcal D/ \Phi$ is a $3$-manifold except possibly at the images of some vertices and at the center of some edges of the tetrahedra $\tilde\Deltaelta_i$ under the projection $p: \cup_i \tilde\Deltaelta_i \to \mathcal D/ \Phi.$
In the following we restrict our attention to the cases where the singularities of $\mathcal D/ \Phi$ only appear at the images of the vertices.
This happens, for example, when all homeomorphisms of $\Phi$ are orientation-reversing with respect to a fixed orientation of the tetrahedra of $\mathcal D$, and therefore the complement of the singularities is an orientable $3$-manifold.
We collect all these information into a single symbol $\mathcal T$ and call $\mathcal T$ a triangulation of $\mathcal D/ \Phi$; moreover, we also use $|\mathcal T|$ to denote the space $\mathcal D/ \Phi$.
In the literature this kind of triangulation is often called pseudo- or singular triangulation.
A tetrahedron, face, edge, or vertex of this triangulation is, respectively, the image of a tetrahedron, face, edge, or vertex of the tetrahedra of $\mathcal D $.
We will denote the image of the vertices by $\mathcal T^{(0)}$.
The link of each vertex of $\mathcal T$ is either a $2$-sphere (such a vertex is called {\it regular}) or a closed surface distinct from the $2$-sphere (such a vertex is called {\it singular}).
Denote by $\mathcal T^{(0)}_s \subseteq \mathcal T^{(0)}$ the set of the singular vertices of $\mathcal T$. If $\mathcal T^{(0)}_s = \emptyset$, then $\mathcal T$ is a triangulation of the closed orientable $3$-manifold $M = |\mathcal T|$.
If $\mathcal T^{(0)}_s \neq \emptyset$, we say $\mathcal T\setminus \mathcal T^{(0)}_s$ is a triangulation of the noncompact $3$-manifold $\hat M = |\mathcal T| \setminus \mathcal |\mathcal T^{(0)}_s|$.
In some cases when $\mathcal T^{(0)}_s = \mathcal T^{(0)}$, then $\mathcal T\setminus \mathcal T^{(0)}_s$ is an ideal triangulation of $\hat M$ (an example are the tetrahedral manifolds in Subsection~\ref{tetrahedral}).
Assume that $\mathcal T^{(0)}_s \neq \emptyset$. Let us replace every tetrahedron of $\mathcal T$ by the corresponding partially truncated one, by removing open regular neighborhoods of all singular vertices of $\mathcal T$. In this way we get a compact $3$-manifold $M$ with nonempty boundary.
It is obvious that we can identify $\text{Int } M = M\setminus\partial M$ with the noncompact $3$-manifold $\hat M = |\mathcal T| \setminus \mathcal |\mathcal T^{(0)}_s|$.
In this situation, we also say that $\mathcal T\setminus \mathcal T^{(0)}_s$ is a triangulation of the compact $3$-manifold $M$ with nonempty boundary.
\subsection{From $4$-colored graphs to triangulated compact $3$-manifolds}\label{colgraph}
\begin{definition} \label{$4$-colored graph}
{\em A {\it $4$-colored graph} is a regular $4$-valent multigraph (i.e., multiple edges are allowed, but loops are forbidden) $\Gammaamma=(V(\Gammaamma), E(\Gammaamma))$ endowed with a map $\gammaamma: E(\Gammaamma) \rightarrow \mathcal C=\{0,1,2,3\}$ that is injective on adjacent edges.
\varphiootnote{Note that there exist (non-bipartite) $4$-regular multigraphs admitting no coloration of this type.}}
\end{definition}
A $3$-dimensional compact manifold $M_\Gammaamma$, possibly with non-empty non-spherical boundary, can be associated to any $4$-colored graph $\Gammaamma$ in the following way:
\begin{itemize}
\item consider a collection $\mathcal D(\Gammaamma)=\{\tilde\Deltaelta_1, \ldots, \tilde\Deltaelta_n\}$ of tetrahedra in bijective correspondence with $V(\Gammaamma)$ and label the vertices of each tetrahedron by different elements of $\mathcal C$;
\item for each pair of $c$-adjacent vertices of $\Gammaamma$ ($c\in\mathcal C$), glue the faces of the corresponding tetrahedra that are opposite to the $c$-labeled vertices, so that equally labeled vertices are identified;
\item remove from the resulting $3$-pseudocomplex $K(\Gamma)$ small open neighborhoods of the singular vertices.
\end{itemize}
As a consequence of the construction the pseudocomplex $K(\Gamma)$ inherits a natural vertex-labeling by $\mathcal C$ that is injective on each simplex.
We remark that the above construction is dual to the one introduced in \cite{[CM]}, where it is proved that any compact $3$-manifold without spherical boundary components admits a representation by $4$-colored graphs and that the manifold is orientable if and only if the representing graph is bipartite.
\begin{remark} {\em Note that any $4$-colored graph encodes a triangulation in the sense of Subsection \ref{triang}.
In fact, given the collection of tetrahedra of $\mathcal D(\Gamma)$, the affine homeomorphisms of the triangulation are defined naturally by the gluings of their faces induced by the vertex-labeling.
Therefore, the construction of the pseudocomplex $K(\Gamma)$ is a particular case of the one described in Subsection \ref{triang}.
Note also that in this case no singularities can arise at the images of the centres of the edges.
When the graph is bipartite the tetrahedra of $\mathcal D(\Gamma)$ can be subdivided into two classes according to the bipartition classes of the corresponding vertices of $\Gamma$ and, by giving to the tetrahedra of one class the orientation induced by the cyclic permutation $(0\ 1\ 2\ 3)$ of the labels of their vertices, and to the tetrahedra of the other class the opposite orientation, all the affine homeomorphisms of the triangulation turn out to be orientation-reversing; as a consequence the resulting manifold is orientable.}
\end{remark}
\subsection{Graph and tetrahedral complexities of $3$-manifolds}
A $4$-colored graph $\Gammaamma$ is called {\it minimal} if there exists no graph representing $M_{\Gammaamma}$ with less vertices than $\Gammaamma$.
\begin{definition} {\em The {\it graph complexity} of a compact $3$-manifold $M$, denoted by $c_{g}(M)$, is the number of vertices in a minimal $4$-colored graph representing $M$.}\end{definition}
In case $M$ is a closed manifold a notion of complexity in terms of colored graphs has been already introduced in \cite{[Li]}: it is called gem-complexity, denoted by $k(M)$, and the relation between the two invariants is $c_g(M) = 2k(M)+ 2.$
A triangulation of a compact $3$-manifold $M$ into tetrahedra is {\it minimal} if there is no triangulation of $M$ into fewer tetrahedra. The {\it tetrahedral complexity} $c_{tet}(M)$ of $M$ is the number of tetrahedra in a minimal triangulation.
The next result gives an inequality relating the complexities $c_{tet}$ and $c_{g}$.
\begin{lemma}
\label{2complexities}
For every compact $3$-manifold $M$ we have $c_{tet}(M) \leq c_{g}(M)$.
\end{lemma}
\begin{proof}
Consider a minimal $4$-colored graph $\Gamma$ representing the manifold $M$.
By definition, $\Gamma$ has $c_{g}(M)$ vertices. Therefore, the graph $\Gamma$ determines a triangulation of $M$ with $c_{g}(M)$ tetrahedra.
This implies that $c_{tet}(M) \leq c_{g}(M)$.
\end{proof}
In Section \ref{sect: exact values} we will apply Lemma \ref{2complexities} in order to find lower bounds for the graph complexity of the
so-called tetrahedral manifolds.
\subsection{Tetrahedral manifolds}\label{tetrahedral}
Let $M$ be a compact $3$-manifold with boundary consisting of tori. Suppose that the interior of $M$, denoted by $Q$, possesses a complete Riemannian metric with finite volume and constant sectional curvature $-1$.
Following \cite{[FGGTV]}, we say that $M$ is {\it tetrahedral} if there exists a decomposition of $Q$ into ideal regular hyperbolic tetrahedra. Equivalently, there exists an ideal triangulation of $M$ such that each edge class contains exactly six edges of the tetrahedra of~$\mathcal D$.
As mentioned in \cite{[A],[FGGTV],[VTF]}, coverings of tetrahedral manifolds yield infinite families of finite volume hyperbolic $3$-manifolds whose tetrahedral complexity can be calculated exactly. More precisely the following statement holds.
\begin{lemma}
\label{complexityoftetrahedral}
Let $M$ be a compact tetrahedral manifold such that the interior of $M$ is obtained by gluing together $k$ regular ideal tetrahedra, and let $N$ be an $n$-fold covering of $M$. Then
$$ c_{tet}(M) = k \text{\ \ and\ \ } c_{tet}(N) = nk.$$
\end{lemma}
\begin{proof}
Let us denote by $Q$ the interior of $M$. Recall that the volume of the regular ideal tetrahedron, that is $v_{tet} = 1.01494\dots$, is maximal among the volumes of all tetrahedra in $\mathbb H^3$. On this property the relation $c_{tet}(M) \gammaeq \operatorname{vol}(Q)/v_{tet}$ mentioned in \cite{[A]} is based. Since $Q$ is obtained by gluing $k$ regular ideal tetrahedra together, its volume $\operatorname{vol}(Q)$ is $kv_{tet}$. Hence, $c_{tet}(M) = k$.
Since the class of tetrahedral manifolds is closed under finite coverings, $N$ is a tetrahedral manifold such that the interior of $N$ is obtained by gluing together $nk$ regular ideal tetrahedra. Hence, $c_{tet}(N) = nk$.
\end{proof}
\section{Exact values and two-sided bounds for the graph complexity of tetrahedral manifolds}
\label{sect: exact values}
An {\it $n$-fold covering} between two $4$-colored graphs $G$ and $\Gamma$, where $n=\#V(G)/\#V(\Gamma)$, is a map $f : V(G)\to V(\Gamma)$ that preserves $c$-adjacency of vertices for all $c\in\mathcal C$ (i.e., for each pair of $c$-adjacent vertices $a,b\in V(G)$ the vertices $f(a), f(b)$ are $c$-adjacent in $\Gamma$).
We call a covering {\it admissible} if it is bijective when restricted to the bicolored cycles of the graphs.
The $n$-fold covering $f$ naturally induces a topological $n$-fold (possibly branched) covering $\vert f\vert\ :M_G\to M_{\Gamma}$. Moreover, $\vert f\vert$ is unbranched if and only if $f$ is admissible. Note also that the triangulation associated to $G$ is the lifting of the one associated to $\Gamma$.
The next result gives two-sided bounds for the graph complexity of compact tetrahedral manifolds.
\begin{theorem}
\label{twosidedbounds}
Let $\Gamma$ be a $4$-colored graph with $k$ vertices representing a compact tetrahedral manifold $M_\Gamma$ such that the interior of $M_\Gamma$ is obtained by gluing together $d$ regular ideal tetrahedra. Let $G$ be an admissible $n$-fold covering of $\Gamma$. Then
$$nd \leq c_{g}(M_{G}) \leq nk.$$
\end{theorem}
\begin{proof}
Since $G$ is an $n$-fold covering of $\Gamma$, $G$ has $nk$ vertices. This implies that $c_{g}(M_{G}) \leq nk$.
On the other hand, it follows from Lemma \ref{2complexities} that $c_{tet}(M_{G}) \leq c_{g}(M_{G})$. Since $G$ is an admissible $n$-fold covering of $\Gamma$, $M_{G}$ is an $n$-fold covering of $M_\Gamma$. Thus, by Lemma \ref{complexityoftetrahedral}, we have $c_{tet}(M_{G}) = nd$.
\end{proof}
Now we give examples of $4$-colored graphs satisfying the assumptions of Theorem \ref{twosidedbounds}.
They allow us to find either the exact values or two-sided bounds for the graph complexity of infinite families of compact tetrahedral manifolds.
\begin{theorem}
Let $\Gamma$ be the bipartite $4$-colored graph with $12$ vertices of Figure \ref{Gamma_1}. If $G$ is an admissible $n$-fold covering of $\Gamma$, then
$$c_{g}(M_{G}) = 12n.$$
\end{theorem}
\begin{proof}
It follows from \cite[Table 3]{[CFMT]} that $\Gamma$ represents the tetrahedral manifold \verb'otet12_00009', which is obtained by gluing together $12$ regular ideal tetrahedra (see details in \cite{[CFMT]}). The conclusion $c_{g}(M_{G}) = 12n$ now follows from Theorem \ref{twosidedbounds}.
\end{proof}
In Figure \ref{Gamma_2} we give a concrete example of such a graph $G$. As pointed out in \cite{[OVC]}, $M_{G}$ is the complement of the link in $S^3$ composed by the weaving knot ${\mathcal W}(3,3n)$ and its braid axis (see Figure \ref{Link_n}). As in \cite{[CKP]}, the weaving knot ${\mathcal W}(p,q)$ is the alternating knot or link with the same projection as the standard $p$-braid $(\sigma_1\cdots\sigma_{p-1})^q$ projection of the torus knot or link $T(p,q)$.
\begin{figure}
\caption{The 4-colored graph $\Gamma$.}
\label{Gamma_1}
\end{figure}
\begin{figure}
\caption{An admissible $n$-fold covering of the graph $\Gamma$ depicted in Figure \ref{Gamma_1}
\label{Gamma_2}
\end{figure}
\begin{figure}
\caption{The weaving knot ${\mathcal W}
\label{Link_n}
\end{figure}
The {\it code} of a bipartite $4$-colored graph $\Gamma$ with $2p$ vertices is a numerical ``string'' of length $3p$ which completely describes both combinatorial structure and coloration of $\Gamma$.
More precisely, the vertices of $\Gamma$ are divided into the two bipartition classes and labelled by the integers $\{-p,\ldots,-1\}$ and $\{+1,\ldots,+p\}$ respectively. Then, for each $i\in\{1,\dots,p\}$ and $c\in\{1,2,3\}$, the label of the vertex that is $c$-adjacent to $-i$ appears as the $(c-1)p+i$-th character of the string, while $-i$ and $+i$ are assumed to be $0$-adjacent.
Although there are obviously many ways of labeling the vertices and also of permuting the elements of the color set, there exists an algorithm to compute the string such that it uniquely determines $\Gamma$ up to relabeling of the vertices and permutations of the color set (see \cite{[Li]} for details).
When the vertices are few, the code is often displayed by using small letters for negative integers and capital ones for positive integers.
\begin{theorem}
Let $\Gamma_1, \Gamma_2, \Gamma_3$ be bipartite $4$-colored graphs represented by the following codes: \\
$\Gamma_1 : DABCFEFEABDCCDEFAB$; \\
$\Gamma_2 : FABCDEDEFABCCDEFAB$; \\
$\Gamma_3 : DABCFEFEDABCBCFEDA$. \\
If $G_i$, $1\leq i\leq 3$, is an admissible $n$-fold covering of $\Gamma_i$, then
$$10n \leq c_{g}(M_{G_i}) \leq 12n.$$
\end{theorem}
\begin{proof}
It follows from \cite[Table 3]{[CFMT]} that $\Gamma_1, \Gamma_2$ and $\Gamma_3$ represent the tetrahedral manifolds \verb'otet10_00014', \verb'otet10_00028' and \verb'otet10_00027' respectively, which are obtained by gluing together $10$ regular ideal tetrahedra (see details in \cite{[CFMT]}). The double inequality $\ \ 10n \leq c_{g}(M_{G_i}) \leq 12n\ \ $ now follows from Theorem \ref{twosidedbounds}.
\end{proof}
\section{Manifolds of graph complexity $14$}\label{sec 4}
The previous section shows how it could be useful to have a census of (prime) $3$-manifolds represented by $4$-colored graphs.
In \cite{[CFMT]}, all prime orientable $3$-manifolds with toric boundary representable by (bipartite) $4$-colored graphs with order $\leq 12$ have been classified.
In this section we extend the classification up to $14$ vertices of the associated graphs.
Moreover, we show that all manifolds appearing in this census, except four, are complements of links in the $3$-sphere whose diagrams are also determined.
The classification has been obtained starting from the catalogues of graphs described in \cite{[CFMT]} by using the programs \verb"3-Manifold" \verb" Recognizer" \cite{[Recognizer]} and \verb"SnapPy" \cite{[SnapPy]} and following the procedure described in the same paper.
\begin{theorem}
There exist exactly $34$ non-homeomorphic compact orientable prime $3$-manifolds with (possibly disconnected) toric boundary of graph complexity $14$, and exactly $30$ of them are complements of links in the $3$-sphere (see Table~\ref{tab:1}).
\end{theorem}
In order to refer with precision to each manifold in our census, we use a notational system analogous to that used in the knot and link tables.
For each $1\leq k\leq 5$, we sort in arbitrary order all $3$-manifolds with $k$ boundary components represented by a minimal $4$-colored graph with $14$ vertices, and we denote by $14_n^k$ the $n$-th manifold of this list.
Let us describe which kind of $3$-manifolds can be found in Table~\ref{tab:1}.
\textbf{Seifert manifolds.} A Seifert manifold will be denoted by $(F,(p_1,q_1),\ldots,(p_k,q_k))$, where $F$ is a compact surface with non-empty boundary, $k\gammaeq 0$ and the coprime pairs of integers $(p_i,q_i)$, with $p_i\gammaeq 2$, are the Seifert invariants of the exceptional fibers.
We point out that, by construction, any Seifert manifold with non-empty boundary is endowed with a coordinate system for each of its boundary tori, made by a pair of meridian/longitude suitably oriented.
All Seifert manifolds appearing in our census, either as single manifolds or as components of a graph manifold, have either disks or M\"obius strips, possibly with holes, as base spaces and at most two exceptional fibers.
In Table~\ref{tab:1}, we denote by $D^2_i$ and $M^2_i$ the disc and the M\"obius strip with $i>0$ holes respectively.
\textbf{Graph manifolds.} Graph manifolds of Waldhausen are obtained from Seifert manifolds by gluing them along boundary components.
The structure of the $14$ graph manifolds arising in our census is very simple: each of them is obtained by gluing together either two or three Seifert manifolds as follows.
\begin{itemize}
\item Let $M, M'$ be two Seifert manifolds with non-empty boundaries equipped with fixed coordinate systems. Chosen arbitrary tori $T$ and $T'$ of $\partial M$ and $\partial M'$, respectively, let $f_A: T\to T'$, with $A=(a_{ij})\in GL_2(Z)$, be a homeomorphism that takes any curve of type $(m, n)$ on $T$ to a curve of type $(a_{11}m+a_{12}n, a_{21}m+a_{22}n)$ on $T'$. So we define $M\cup_A M'= M\cup_{f_A} M'$.
\item Let $M, M', M''$ be three Seifert manifolds with non-empty boundaries equipped with fixed coordinate systems. Chosen arbitrary tori: $T$ of $\partial M$, $T'_1$ and $T'_2$ of $\partial M'$ and $T''$ of $\partial M''$, let $f_A: T\to T'_1$, $f_B: T''\to T'_2$ be homeomorphisms corresponding to the matrices $A, B\in GL_2(Z)$ as above, then we define $M\cup_A M'\cup_B M''=M\cup_{f_A} M'\cup_{f_B} M''$.
\end{itemize}
\textbf{Hyperbolic manifolds.} Of the seven hyperbolic manifolds in our census, three ($14^3_9,\ 14^3_{10}$ and $14^4_{14}$), by removing their boundary, give rise to cusped hyperbolic $3$-manifolds that are contained in the orientable cusped census \cite{[CHW]} or in the censuses of Platonic manifolds of \verb"SnapPy".
Therefore they are identified, in Table~\ref{tab:1}, by the notations of their corresponding cusped manifolds.
\textbf{Composite manifolds.} We call a $3$-manifold {\it composite} if its JSJ decomposition is non-trivial and contains a hyperbolic manifold.
Each of the $10$ composite manifolds arising in our census is obtained by gluing together one hyperbolic manifold and either one or two Seifert manifolds as follows.
\begin{itemize}
\item Let $M$ be a Seifert manifold with non-empty boundary equipped with fixed coordinate systems as remarked above.
Let $M_L$ be a hyperbolic manifold, which is the complement of an open regular neighbourhood of a link $L = L_1\sqcup\ldots\sqcup L_r$ in $S^3$.
A preferred coordinate system for $\partial M_L$ can be also chosen in the following way.
On the regular neighbourhood of each $L_i$, considered as a knot in $S^3$, we choose a standard coordinate system formed, as usual, by the boundary of a meridian disk and a homologically trivial curve in the complement of $L_i$.
Therefore, once a boundary torus $T$ of $M$ and an $i$-th component $(\partial M_L)_i$ of $\partial M_L$ corresponding to $L_i$ are chosen, a homeomorphism $f_{A, i}\ :\ T\to (\partial M_L)_i$ can be described by means of a matrix $A\in GL_2(Z)$ as in the case of graph manifolds.
Finally, we denote by $M\cup_{A, i} M_L$ the manifold obtained by gluing $M$ and $M_L$ through the homeomorphism $f_{A, i}$. Since in Table~\ref{tab:1} each manifold $M_L$ is represented by a link with up to $8$ crossings, we numerate the components $L_1, \ldots , L_r$ of $L$ as they appear in its Gauss code displayed in the corresponding page of \cite{[KnotAtlas]}.
\item {Given two Seifert manifolds with non-empty boundaries $M'$ and $M''$ and a hyperbolic manifold $M_L$ as above, we denote by $M\cup_{A, i} M_L\cup_{B, j} M''$ the manifold obtained by identifying two bondary tori of $M'$ and $M''$ with $(\partial M_L)_i$ and $(\partial M_L)_j$ respectively by the homeomorphisms $f_A$ and $f_B$ similarly to the previous case.}
\end{itemize}
All prime links appearing in Table \ref{tab:1} are contained in the Thistlethwaite link table up to $14$ crossings distributed with \verb"SnapPy"; they are identified through their Thistlethwaite name, that is of the form $L[k]a[j_1]$ or $L[k]n[j_2]$, depending on whether the link is alternating or not. Here $k$ is the crossing number and $j_1, j_2$ are archive numbers assigned to each $(a, k)$, $(n, k)$ pair, respectively.
All other links of Table \ref{tab:1} are not prime and their diagrams are depicted in Figure \ref{links}.
\begin{longtable}{|>{\tiny}c||>{\tiny}c|>{\tiny}c|>{\tiny}c|>{\tiny}c|}
\caption{Orientable prime $3$-manifolds with toric boundary of graph complexity $14$} \label{tab:1} \\
\hline
Name & Code & Manifold & Link \\
\hline \hline \endhead
\hline
$14^2_1$ & EABCDGFGDFEBCADGEFBAC & $\seifuno {D^2_1}31$ & L6a3 \\
\hline
$14^2_2$ & DABCGEFGFECDBABGDFACE & $\seifdue {D^2}2131 \bigu 1110 (D^2_2\times S^1)$ & see fig. \ref{links} \\
\hline
$14^2_3$ & GABCDEFEDGFABCDEFAGCB & $\seifuno {D^2_1}21 \bigu 0110 \seifuno {D^2_1}21$ & L11n204 \\
\hline
$14^3_1$ & EABCDGFGEFCADBCEGAFBD & $\seifuno {D^2_2}21$ & L12n1998 \\
\hline
$14^3_2$ & DABCGEFGFBADCEFCEAGDB & $\seifuno {M^2_2}10$ & -- \\
\hline
$14^3_3$ & DABCGEFFDBECGAEDGCFAB & $\seifdue {D^2}2131 \bigu 1110 (D^2_3\times S^1)$ & see fig. \ref{links} \\
\hline
$14^3_4$ & EABCDGFGDFEBCABDGAFEC & $\seifuno {D^2_1}21 \bigu 1211 (D^2_2\times S^1)$ & L8n6 \\
\hline
$14^3_5$ & DABCGEFGEFBDACFGEBACD & $\seifuno {D^2_1}31 \bigu 0110 (D^2_2\times S^1)$ & see fig. \ref{links} \\
\hline
$14^3_6$ & EABCDGFGFDABECCEFAGDB & $\seifuno {M^2_1}10 \bigu 0110 (D^2_2\times S^1)$ & -- \\
\hline
$14^3_7$ & EABCDGFGFDABECFDGBACE & $\seifuno {D^2_1}21 \bigu 0110 (D^2_2\times S^1) \bigu 0110 \seifuno {D^2_1}21$ & see fig. \ref{links} \\
\hline
$14^3_8$ & DABCGEFGDFCABEBFDECGA & $(D^2_2\times S^1) \bigcup\nolimits_{{\tiny{\matr {0} {1} {1} {0}}}, 1} M_{L5a1}$ & L13n9356 \\
\hline
$14^3_9$ & DABCGEFGEFBDACFCGABDE & \verb't12066', \verb'ooct02_00003' & L8n5 \\
\hline
$14^3_{10}$ & DABCGEFGDFBACECFAEDGB & \verb't12067', \verb'ooct02_00005' & L6a4 \\
\hline
$14^4_1$ & EABCDGFGFBACEDBCFDGAE & $(D^2_2\times S^1) \bigu 0110 \seifuno {D^2_2}21$ & see fig. \ref{links} \\
\hline
$14^4_2$ & EABCDGFGBEDFACEFAGCDB & $(D^2_2\times S^1) \bigu 0110 \seifuno {M^2_2}10$ & -- \\
\hline
$14^4_3$ & DABCGEFGCFADBEEGABCFD & $\seifuno {D^2_1}21 \bigu 0110 (D^2_3\times S^1) \bigu 0110 \seifuno {D^2_1}21$ & see fig. \ref{links} \\
\hline
$14^4_4$ & EABCDGFGEFCADBBFDGEAC & $(D^2_2\times S^1) \bigu 0110 \seifuno {D^2_1}21 \bigu {-1}21{-1} (D^2_2\times S^1)$ & L11n379 \\
\hline
$14^4_5$ & EABCDGFGFECABDCGDAFEB & $\seifuno {D^2_1}21 \bigu 1211 (D^2_2\times S^1) \bigu 0110 (D^2_2\times S^1)$ & see fig. \ref{links} \\
\hline
$14^4_6$ & EABCDGFGDFACEBBFEDGAC & $\seifuno {M^2_1}10 \bigu 0110 (D^2_2\times S^1) \bigu 0110 (D^2_2\times S^1)$ & -- \\
\hline
$14^4_7$ & EABCDGFGFDEBCAFDEGCAB & $\seifuno {D^2_1}21 \bigcup\nolimits_{{\tiny{\matr {1} {0} {0} {-1}}}, 1} M_{L8n7}$ & L14n63157 \\
\hline
$14^4_8$ & EABCDGFGFEBCDACGFEBAD & $\seifuno {D^2_1}21 \bigcup\nolimits_{{\tiny{\matr {0} {1} {1} {0}}}, 1} M_{L8n7}$ & L14n61549 \\
\hline
$14^4_9$ & EABCDGFGDFEBCAFCGADEB & $(D^2_2\times S^1) \bigcup\nolimits_{{\tiny{\matr {0} {1} {1} {1}}}, 1} M_{L6a5}$ & L14n62850 \\
\hline
$14^4_{10}$ & EABCDGFGFBEACDDCGAFEB & $(D^2_2\times S^1) \bigcup\nolimits_{{\tiny{\matr {0} {1} {1} {0}}}, 3} M_{L8n5}$ & see fig. \ref{links} \\
\hline
$14^4_{11}$ & DABCGEFGEFBDACFGCABDE & $(D^2_2\times S^1) \bigcup\nolimits_{{\tiny{\matr {0} {1} {1} {0}}}, 1} M_{L8n5}$ & L14n62541 \\
\hline
$14^4_{12}$ & EABCDGFGEFBDACCGAEFDB & $(D^2_2\times S^1) \bigcup\nolimits_{{\tiny{\matr {0} {1} {1} {0}}}, 1} M_{L6a4}$ & see fig. \ref{links} \\
\hline
$14^4_{13}$ & EABCDGFGEFBDACBGCEFDA & $(D^2_2\times S^1) \bigcup\nolimits_{{\tiny{\matr {0} {1} {1} {0}}}, 1} M_{L5a1} \bigcup\nolimits_{{\tiny{\matr {0} {1} {1} {0}}}, 2} (D^2_2\times S^1)$ & see fig. \ref{links} \\
\hline
$14^4_{14}$ & EABCDGFGDFACEBFCEGBAD & \verb'otet10_00011', \verb'ocube02_00044' & L8a21 \\
\hline
$14^4_{15}$ & EABCDGFGFDABECFDEGCAB & hyperbolic manifold with $\operatorname{Vol} = 10.6669791338$ & L14n60227 \\
\hline
$14^4_{16}$ & EABCDGFGFEACBDCDFGAEB & hyperbolic manifold with $\operatorname{Vol} = 11.202941612$ & L10n96 \\
\hline
$14^4_{17}$ & DABCGEFGEFBDACCGAFBDE & hyperbolic manifold with $\operatorname{Vol} = 12.8448530047$ & L11n456 \\
\hline
$14^4_{18}$ & DABCGEFGEFBDACFGEACDB & hyperbolic manifold with $\operatorname{Vol} = 12.3173273072$ & L14n63000 \\
\hline
$14^5_1$ & EABCDFGGFEBADCCDEGFAB & $(D^2_2\times S^1) \bigu 0110 (D^2_3\times S^1)$ & see fig. \ref{links} \\
\hline
$14^5_2$ & DABCGEFGFBADCEECFGABD & $(D^2_2\times S^1) \bigcup\nolimits_{{\tiny{\matr {1} {0} {0} {-1}}}, 1} M_{L8n7}$ & L12n2249 \\
\hline
$14^5_3$ & DABCGEFGCFADBECDEGAFB & $(D^2_2\times S^1) \bigcup\nolimits_{{\tiny{\matr {0} {1} {1} {0}}}, 1} M_{L8n7}$ & L14n63769 \\
\hline
\end{longtable}
\begin{figure*}
\caption{Non-prime links with complements represented by $4$-colored graphs of order $14.$}
\label{links}
\end{figure*}
\textbf{Acknowledgements:} P. Cristofori and M. Mulazzani have been supported by the National Group for
Algebraic and Geometric Structures, and their Applications'' (GNSAGA-INdAM), the University of Modena and Reggio Emilia and the University of Bologna, funds for selected research topics.
E. Fominykh and V. Tarkaev have been supported by RFBR (grant number 16-01-00609).
\end{document} |
\betagin{document}
\tauitle{A construction of 2-cofiltered bilimits of topoi}
\alphauthor{Eduardo J. Dubuc, \; Sergio Yuhjtman}
\muaketitle
{\sigmac introduction}
We show the existence of bilimits of 2-cofiltered diagrams of
topoi, generalizing the construction of cofiltered bilimits developed in
\cite{G2}. For any given such diagram, we show that it can be
represented by a 2-cofiltered diagram of small sites with finite
limits, and we construct a small site for the inverse limit
topos. This is done by taking the 2-filtered bicolimit of the
underlying categories and inverse image functors. We use the
construction of this bicolimit developed in \cite{DS}, where it is
proved that if the categories in the diagram have finite limits and
the transition functors are exact, then the bicolimit category has
finite limits and the pseudocone functors are exact. An application of our result here is the fact that every Galois topos \mubox{has points \cite{D}.}
\sigmaection{Background, terminology and notation} \lambdaabel{background}
In this section we recall some $2$-category and topos
theory that we shall explicitly need, and in this way fix notation
and terminology. We also include some in-edit proofs when it seems necessary. We distinguish between \epsilonmph{small} and \epsilonmph{large}
sets. Categories are supposed to have small hom-sets. A category with
large hom-sets is called \epsilonmph{illegitimate}.
{{\bf Bicolimits}}
By a \epsilonmph{2-category} we mean a $\muathcal{C}at$ enriched category, and
\mubox{\epsilonmph{2-functors}}
are $\muathcal{C}at$ functors, where $\muathcal{C}at$ is the category of small categories. Given a 2-category, as usual, we denote
horizontal composition by juxtaposition, and vertical composition
by a $''\circ''$. We consider juxtaposition more binding than
$''\circ''$ (thus $xy\circ z$ means $(xy)\circ z$). If $\muathcal{A},\;
\muathcal{B}$ are $2$-categories ($\muathcal{A}$ small), we will
denote by $[[\muathcal{A}, \muathcal{B}]]$ the $2$-category which has as objects
the $2$-functors, as arrows the \epsilonmph{pseudonatural transformations},
and as $2$-cells the \epsilonmph{modifications} (see \cite{G}
I,2.4.). Given $F,\,G,\,H\,: \muathcal{A} \mur{} \muathcal{B}$,
there is a functor:
\betagin{equation} \lambdaabel{3cat}
[[\muathcal{A},\, \muathcal{B}]](G,\,H) \tauimes [[\muathcal{A},\, \muathcal{B}]](F,\,G)
\mur{} [[\muathcal{A},\, \muathcal{B}]](F,\,H)
\epsilonnd{equation}
To have a handy reference
we will explicitly describe these data in the particular cases
we use.
A \epsilonmph{pseudocone} of a diagram given by a 2-functor $\mathcal A \xto F \mathcal B$
to an object \mubox{$X \in \muathcal{B}$} is a pseudonatural
transformation \mubox{$F \mur{h} X$} from $F$ to the 2-functor
which is constant at $X$. It consists of a family of
arrows \mubox{$(h_A: FA \tauo X)_{A \, \in \muathcal{A}}$,} and a family
of invertible $2$-cells \mubox{$(h_u: h_A \tauo h_B \circ Fu)_{(A \mur{u}
B) \, \in \muathcal{A}}$.}
A morphism $g \mathcal Mr{\varphi} h$ of pseudocones (with same vertex)
is a modification, as such, it consists of a
family of $2$-cells
$(g_A \mathcal Mr{\varphi_A} h_A)_{A \in \muathcal{A}}$. These data is subject to
the following:
\betagin{sinnadaitalica} [{\it Pseudocone and morphism of pseudocone
equations}] \lambdaabel{PCequations} ${}$
pc0. $h_{id_A} = id_{h_A}$,
for each object $A$ \hspace{3ex}
pc1. $h_v Fu \circ h_u = h_{vu}$,
for each pair
of arrows \; $A \mur{u} B \mur{v} C$ \hspace{3ex}
pc2. $h_B F\gamma \circ h_v = h_u$,
for each 2-cell \;
$A \cellr{u}{\gammaamma}{v} B$ \hspace{3ex}
pcM. $h_u \circ \varphi_A = \varphi_B Fu \circ g_u$,
for each
arrow \; $A \mur{u} B$ \hspace{3ex}
\epsilonnd{sinnadaitalica}
We state and prove now a lemma which, although expected, needs
nevertheless a proof, and for which we do not have a reference in the
literature. As the reader will realize, the statement concerns general
pseudonatural transformations, but we treat here the particular case
of pseudocones.
\betagin{lemma} \lambdaabel{translacion}
Let $\mathcal A \xto F \mathcal B$ be a 2-functor and $F \mur{g} X$ a
pseudocone. Let $FA \mur{h_A} X$ be a family of morphisms
together with
invertible $2$-cells $g_A \mathcal Mr{\varphi_A} h_A$.
Then, conjugating by $\varphi$ determines a pseudocone structure
for $h$, unique such that $\varphi$ becomes an isomorphism of
pseudocones.
\epsilonnd{lemma}
\betagin{proof}
If $\varphi$ is to become a pseudocone morphism, the equation pcM.
\mubox{$\varphi_B Fu \,\circ\, g_u = h_u \circ \varphi_A$} must hold.
Thus, $h_u = \varphi_B Fu \circ g_u \circ \varphi_A^{-1}$
determines and \mubox{defines $h$.} The pseudocone equations
\rhoef{PCequations} for $h$ follow from the respective equations
for $g$:
pc0.
$h_{id_A}\;=\;\varphi_A \circ g_{id_A} \circ \varphi_A^{-1} \;=\;
\varphi_A \circ id_{g_A} \circ \varphi_A^{-1} \;=\; id_{h_A} $
pc1. $A \xto u B \xto v C$:
$h_v Fu \circ h_u \;\; = $
$(\varphi_C Fv \circ g_v \circ \varphi_B^{-1})Fu \circ \varphi_B Fu
\circ g_u \circ \varphi_A^{-1} \;\;\;=$ \hspace{9ex}
$\varphi_C F(vu) \circ g_v Fu \circ \varphi_B^{-1} Fu
\circ \varphi_B Fu \circ g_u \circ \varphi_A^{-1} \;\;\;=$
\hspace{9ex}
$\varphi_C F(vu) \circ g_v Fu \circ g_u \circ\varphi_A^{-1}
\;\;\;=$ \hspace{9ex}
$\varphi_C F(vu) \circ g_{vu} \circ \varphi_A^{-1} \;\;\;=$
\hspace{2ex} $h_{vu}$ \hspace{2ex}
pc2. For $\xymatrix{A \alphar@<1ex>[r]^u \alphar@{}[r]|{\mathcal U} \def\UU{\-{\U}parrow \gamma}
\alphar@<-1ex>[r]_v & B}$ we must see
$h_B F\gamma \circ h_v = h_u$. This is the same as
$h_B F\gamma \circ \varphi_B Fv \circ g_v \circ
\varphi_A^{-1}=\varphi_B Fu \circ g_u
\circ\varphi_A^{-1}$.
Canceling $\varphi_A^{-1}$ and composing with
$(\varphi_B Fu)^{-1}$ yields
(1) $(\varphi_B Fu)^{-1} \circ h_B F\gamma \circ \varphi_B Fv
\circ g_v = g_u$.
From the compatibility between vertical and horizontal
composition it \mubox{follows} \mubox{$(\varphi_B Fu)^{-1} \circ h_BF\gamma \circ \varphi_B Fv \;=$}
$(\varphi_B^{-1} \circ h_B \circ \varphi_B)(Fu \circ F\gamma \circ Fv) = g_B
F\gamma$. Thus, after replacing,
(1) becomes $g_B F\gamma \circ g_v = g_u$.
\epsilonnd{proof}
Given a small 2-diagram $\muathcal{A} \mur{F} \muathcal{B}$, the category of pseudocones and its
morphisms is, by definition, \mubox{$pc \muathcal{B}(F, X) =
[[\muathcal{A},\,
\muathcal{B}]](F,\, X)$.} Given a pseudocone $F \mur{f} Z$ and a
$2$-cell
$Z \cellr{s}{\xi}{t} X$, it is clear and straightforward how to define
a morphism of pseudocones $F \cellr{sf}{\xi f}{tf} X$ which is the
composite
$F \mur{f} Z \cellr{s}{\xi}{t} X$. This is a particular case of
\rhoef{3cat}, thus composing with $f$
determines a functor (denoted $\rhoho_f$)
\mubox{$\muathcal{B}(Z,\, X) \mur{\rhoho_f} pc \muathcal{B}(F,\, X)$.}
\betagin{definition} \lambdaabel{bicolimit}
A pseudocone $F \mur{\lambdaambda} L$ is a \epsilonmph{bicolimit} of $F$ if for
every object $X \in \muathcal{B}$, the
functor $\muathcal{B}(L,\, X) \mur{\rhoho_\lambdaambda} pc\muathcal{B}(F,\, X)$ is
an
equivalence of categories. This amounts to the following:
{\it bl}) Given any pseudocone $F \mur{h} X$, there exists an arrow
$L \xto{\epsilonll} X$ and an invertible morphism of pseudocones
$\;h \mathcal Mr{\tauheta} \epsilonll \lambdaambda$. Furthermore, given any other
$L \xto{t} X$ and $\;h \mathcal Mr{\varphi} t \lambdaambda$,
there exists a \epsilonmph{unique}
$2$-cell $\;\epsilonll \mathcal Mr{\xi} t$ such that
$\varphi = (\xi \lambdaambda) \circ \tauheta$ (if $\varphi$ is invertible,
then so it is $\xi$).
\epsilonnd{definition}
\betagin{definition}
When the functor $\muathcal{B}(L,\, X) \mur{\rhoho_\lambdaambda} pc\muathcal{B}(F,\,
X)$
is an isomorphism of categories, the bicolimit is said to be a
\epsilonmph{pseudocolimit}.
\epsilonnd{definition}
It is known that the $2$-category $\muathcal C at$ of small categories has all
small pseudocolimits, then a ``fortiori'' all small
bicolimits (see for example \cite{S}). Given a 2-functor $\muathcal{A}
\mur{F} \muathcal{C}at$ we denote by $\Colim{F}$ the vertex of a bicolimit
cone.
In \cite{DS} a special
construction
of the pseudocolimit of a 2-filtered diagram of categories
(not necessarily small) is made, and using this construction it is
proved a result
(\mubox{theorem \rhoef{key} below}) which is the key to our
construction of small
$2$-filtered bilimits of topoi. Notice that even if the categories
of the system are large, condition {\it bl)} in definition
\rhoef{bicolimit} makes sense and it defines the bicolimit of large
categories.
We denote by $\muathcal{CAT}_{fl}$ the \epsilonmph{illegitimate} (in the sense
that its hom-sets are large) 2-category of
finitely complete
categories and exact (that is, finite limit preserving) functors.
\betagin{theorem}[\cite{DS} Theorem 2.5] \lambdaabel{key}
$\muathcal{CAT}_{fl} \sigmaubset
\muathcal{CAT}$ is closed under 2-filtered
pseudocolimits. Namely, given any 2-filtered diagram $\mathcal A \xto F
\muathcal{CAT}_{fl}$, the
pseudocolimit pseudocone $FA \mur{\lambdaambda_A}
\Colim{F}$ taken in $\muathcal{CAT}$ is a
pseudocolimit cone in $\muathcal{CAT}_{fl}$. If the index 2-category
$\muathcal{A}$ as well as all the categories $FA$ are small, then
$\Colim{F}$ is a small category.
$\mathcal Box$
\epsilonnd{theorem}
{\bf Topoi}
By a \epsilonmph{site} we mean a category furnished with a (Grothendieck)
topology, and a small set of objects capable of covering any
object (called \epsilonmph{topological generators} in \cite{G1}). \epsilonmph{To
simplify we will consider only sites with finite
limits.}
A \epsilonmph{morphism} of sites with finite limits $\mathcal D \mur{f} \C$ is a
\epsilonmph{continous} (that is, cover preserving) and exact functor in the
other direction $\C \mur{f^*} \mathcal D$.
A $2$-cell $\muathcal{D} \cellr{f}{\gammaamma}{g} \muathcal{C}$ is a natural
transformation $\muathcal{C} \cellr{g^*}{\gammaamma}{f^*}
\muathcal{D}$ \footnote{Notice that $2$-cells are also taken in the
opposite
direction. This is Grothendieck original convention, later changed
by some authors.}. Under the presence of topological generators it
can be easily seen there is only a small set of
natural
transformations between any two continous functors. We denote by
$\muathcal S it$ the resulting 2-category of sites with
finite
limits. We denote by $\muathcal{S}it^*$ the $2$-category whose objects
are the sites, but taking as arrows and $2$-cells the functors $f^*$
and natural transformations respectively. Thus $\muathcal{S}it$ is obtained
by formally inverting the arrows and the $2$-cells of $\muathcal{S}it^*$. We
have by definition
$\muathcal{S}it(\muathcal{D},\muathcal{C}) = \muathcal{S}it^*(\muathcal{C},\muathcal{D})^{op}$.
A topos (also ``Grothendieck topos'') is a category equivalent
to the category of sheaves on a site. Topoi are
considered as
sites furnishing them with the canonical topology. This determines a
full subcategory \mubox{$\muathcal{T}op^* \sigmaubset \muathcal{S}it^*$,}
$\muathcal{T}op^*(\muathcal{F},\,\muathcal{E})\;=\; \muathcal{S}it^*(\muathcal{F},\,\muathcal{E})$.
A morphism of topoi (also ``geometric morphism'') $\mathcal E \xto f \mathcal F$ is a
pair of adjoint functors ${f^* \deltaashv f_*}$
(called inverse and direct image respectively)
$\xymatrix{\mathcal E \alphar@<.5ex>[r]^{f_*} & \mathcal F \alphar@<.5ex>[l]^{f^*}}$
together with an adjunction isomorphism $[f^*C,D] \xto \cong
[C,f_*D]$. Furthermore, $f^*$ is required to preserve finite
limits.
Let $\muathcal{T}op$ be the \mubox{2-category} of topos with geometric
morphisms. 2-arrows are pairs of natural transformations ($f^*
\Rightarrow g^*$,
$g_* \Rightarrow f_*$) compatible with the adjunction (one of the
natural transformations completely determines the other).
The inverse image $f^*$ of a morphism is an arrow in $\muathcal{T}op^*
\sigmaubset \muathcal{S}it^*$. This determines a forgetful 2-functor (identity
on the objects)
$\muathcal{T}op \mur{} \muathcal{S}it$ which establish an equivalence of categories
\mubox{$\muathcal{T}op(\muathcal{E},\,\muathcal{F}) \;\cong\;
\muathcal{S}it(\muathcal{E},\,\muathcal{F})$}. Notice that
$\muathcal{T}op(\muathcal{E},\muathcal{F}) \cong \muathcal{T}op^*(\muathcal{F},\muathcal{E})^{op}$, not an
equality.
We recall a basic result in the theory of morphisms of Grothendieck
topoi \cite{G1} expose IV, 4.9.4. (see for example \cite{MM} Chapter VII, section 7).
\betagin{lemma} \lambdaabel{Diaconescu}
Let $\C$ be a site with finite limits, and
$\C \xto {\epsilon^*} \omegaidetilde{\C}$ the canonical morphism of sites to
the topos of sheaves $\omegaidetilde{\C}$. Then
for any topos $\mathcal F$, composing
with $\epsilon^*$ determines a functor
$\muathcal{T}op^*(\omegaidetilde{\C},\, \mathcal F) \xto {\cong}
\muathcal{S}it^*(\C,\, \mathcal F)$
which is an equivalence of categories. Thus,
$\muathcal{T}op(\muathcal{F},\, \omegaidetilde{\C}) \xto {\cong}
\muathcal{S}it(\mathcal F,\, \C)$.
\epsilonnd{lemma}
By the comparison lemma \cite{G1} Ex. III 4.1 we can state
it in the following form, to be used in the proof of lemma
\rhoef{pseudodiaconescu}.
\betagin{lemma} \lambdaabel{Diaconescubis}
Let $\mathcal E$ be any topos and $\muathcal{C}$ any small set of generators
closed under finite limits (considered as a site with the
canonical topology). Then,
for any topos $\mathcal F$, the
inclusion $\muathcal{C} \sigmaubset \muathcal{E}$ induce a restriction
functor
\mubox{$\muathcal{T}op^*(\mathcal E,\mathcal F) \mur{\rhoho} \muathcal{S}it^*(\C,\mathcal F)$} which
is an equivalence of categories.
\epsilonnd{lemma}
\sigmaection{2-cofiltered bilimits of topoi}
Our work with sites is auxiliary to prove our results for topoi, and
for this all we need are sites with finite limits. The 2-category
$\muathcal S it$ has all small \mubox{2-cofiltered} pseudolimits, which
are obtained by furnishing the 2-filtered pseudocolimit in
$\muathcal{CAT}_{fl}$ (\rhoef{key}) of the
underlying categories with the coarsest topology making the cone
injections site morphisms. Explicitly:
\betagin{theorem}\lambdaabel{sitelimit}
Let $\muathcal{A}$ be a small 2-filtered 2-category, and \mubox{$\muathcal{A}^{op} \mur{F}
\muathcal{S}it$} \mubox{($\muathcal{A} \mur{F} \muathcal{S}it^*$)} a \mubox{2-functor.}
Then, the
category $\Colim{F}$ is
furnished with a topology such that the pseudocone functors $FA
\mur{\lambdaambda_A^*} \Colim{F}$ become continuous and induce an isomorphism
of categories
\mubox{$\muathcal{S}it^*[\Colim{F},\, \muathcal{X}] \mur{\rhoho_{\lambdaambda}}
\muathcal{P}\muathcal{C}{\muathcal{S}it^*}[F,\, \muathcal{X}]$.} The corresponding site
is then a
pseudocolimit of $F$ in the 2-category $\muathcal{S}it^*$. If each $FA$ is a
small category, then so it is $\Colim{F}$.
\epsilonnd{theorem}
\betagin{proof}
Let $FA \mur{\lambdaambda_A} \Colim{F}$ be the colimit pseudocone in
$\muathcal{CAT}_{fl}$. We give $\Colim{F}$ the topology generated by the
families
$\lambdaambda_A c_\alphalpha \mur{} \lambdaambda_A c$, where $c_\alpha \mur{} c$
is a covering in some $FA$, $A \in \mathcal A$. With this topology, the
functors $\lambdaambda_A$ become continuous, thus they correspond to
site morphisms. This determines the upper horizontal arrow in the
following diagram (where the vertical arrows are full subcategories
and the lower horizontal arrow is an isomorphism):
$$
\xymatrix
{
\muathcal{S}it[\Colim{F},\, \muathcal{X}]
\alphar[r] \alphar[d]
& pc\muathcal{S}it[F,\, \muathcal{X}]
\alphar[d]
\\
\muathcal{C}at_{fl}[\Colim{F},\, \muathcal{X}]
\alphar[r]^\cong
& pc\muathcal{C}at_{fl}[F,\, \muathcal{X}]
}
$$
To show that the upper horizontal arrow is an isomorphism we have to
check that given a pseudocone $h \in pc\muathcal{S}it[F,\, \muathcal{X}]$, the
unique functor \mubox{$f \in \muathcal{C}at_{fl}[\Colim{F},\, \muathcal{X}]$},
corresponding to
$h$ under the lower arrow, is continuous. But this is clear
since from the equation $f\lambdaambda = h$ it follows that it preserves
the generating covers, and thus all covers as well. Finally, by the
construction of $\Colim{F}$ in \cite{DS} we know that every object in
$\Colim{F}$ is of the form $\lambdaambda_A c$ for some $A \in \muathcal{A}$,
$c \in FA$. It follows then that the collection of objects of the form
$\lambdaambda_A c$, with $c$ varying on the set of topological generators
of each $FA$, is a
set of topological generators for $\Colim{F}$.
\epsilonnd{proof}
In the next proposition we show that any 2-diagram of topoi
restricts to a 2-diagram of small sites with finite limits by means of a 2-natural (thus a
fortiori pseudonatural) transformation.
\betagin{proposition}\lambdaabel{res}
Given a 2-functor $\muathcal{A}^{op} \mur{\muathcal{E}} \muathcal{T}op$ there exists a
2-functor $\muathcal{A}^{op} \mur{\muathcal{C}} \muathcal{S}it$ such that:
i) For any $A\in \mathcal A$, $\C_A$ is a \epsilonmph{small} full generating
subcategory of
$\mathcal E_A$
closed under finite limits, considered as a site with the canonical
topology.
ii) The
arrows and the $2$-cells in the
$\muathcal{C}$ diagram are the restrictions of those in the $\muathcal{E}$
diagram: For any $2$ cell $A \cellr{u}{\gammaamma}{v} B$ in $\muathcal{A}$, the
following diagram commutes (where we omit notation for the action of
the $2$ functors on arrows and $2$-cells):
$$
\xymatrix@C=8ex
{
\muathcal{E}_A \alphar@<1ex>[r]^{u^*}
\alphar@<-1.6ex>[r]_(0.5){v^*}^(0.5){\gammaamma \:\mathcal Downarrow}
& \muathcal{E}_B
\\
\muathcal{C}_A \alphar@<1ex>[r]^{u^*}
\alphar@<-1.6ex>[r]_(0.5){v^*}^(0.5){\gammaamma \:\mathcal Downarrow} \alphar@{^(->}^{i_A}[u]
& \muathcal{C}_B \alphar@{^(->}^{i_B}[u]
}
$$
\epsilonnd{proposition}
\betagin{proof}
It is well known that any small set $\muathcal{C}$ of generators in a topos
can be enlarged so as to determine a (non canonical) small full subcategory
$\overline{\muathcal{C}} \sigmaupset \muathcal{C}$ closed under finite
limits: Choose a limit cone for each finite diagram, and repeat
this in a denumarable process. On the other hand, for the validity of
condition ii) it is enough that for each transition functor $\muathcal{E}_A
\mur{u^*} \muathcal{E}_B$ and object $c \in \muathcal{C}_A$, we have $u^*(c) \in \muathcal{C}_B$
(with this, natural transformations restrict automatically).
Let's start with any set of generators $\muathcal{R}_A \sigmaubset \mathcal E_A$ for all
$A\in \muathcal{A}$. We
will naively add
objects to these sets to remedy the failure of each condition
alternatively. In this way we achieve simultaneously the two
conditions:
Define $\muathcal{C}^0_A = \overline{\muathcal{R}}_A \sigmaupset
\muathcal{R}_A$.
Define $\muathcal{R}_A^{n+1} =
\bigcup\muathcal Lmits_{X \xto u A}
u^*(\muathcal{C}^n_X)$.
$\muathcal{R}_A^{n+1}$ is small because $\mathcal A$ is small.
$\muathcal{C}^n_X \sigmaubset \muathcal{R}_A^{n+1}$ due to $id_A$.
Suppose now $c\in \muathcal{R}_A^{n+1}$, $c=u^*(d)$ with $d\in
\muathcal{C}^n_X$, and
let $A
\mur{v} B$ in $\muathcal{A}$.
We have
$v^*(c)=v^*u^*(d) = (vu)^*(d)$,
thus $v^*(c) \in \muathcal{R}_B^{n+1}$.
Define $\C_A^{n+1} = \overline{\muathcal{R}_A^{n+1}} \sigmaupset
\muathcal{R}_A^{n+1}$.
Then, it is straightforward to check that
$\C_A = \bigcup\muathcal Lmits_{n\in \muathbb{N}}\C_A^n$ satisfy the two
conditions.
\epsilonnd{proof}
A generalization of lemma \rhoef{Diaconescubis} to pseudocones
holds.
\betagin{lemma} \lambdaabel{pseudodiaconescu}
Given any 2-diagram of topoi $\mathcal A^{op} \xto \mathcal E
\muathcal{T}op$, a restriction
$\mathcal A^{op} \xto \C \muathcal S it$ as before, and any topos $\muathcal{F}$, the
inclusions
$\muathcal{C}_A \sigmaubset \muathcal{E}_A$ induce a restriction
functor
\mubox{$pc\muathcal{T}op^*(\mathcal E,\mathcal F) \xto {\rhoho} pc\muathcal{S}it^*(\C,\mathcal F)$} which
is an equivalence of categories.
\epsilonnd{lemma}
\betagin{proof}
The restriction functor $\rhoho$ is just a particular case of \rhoef{3cat},
so it is well defined. We will check that it is essentially surjective
and fully-faithful.
The
following diagram
illustrates the situation:
$$
\xymatrix@R=1ex
{
{} & {} & {} & {} & {} & {}
\\
\muathcal{C}_A \alphar@{^(->}^{i_A}[r] \alphar[dd]^{u^*}
\alphar@(u,u)[rrrd]^{g^*_A}
& \muathcal{E}_A \alphar[dd]^{u^*} \alphar[rrd]^{h_A^*}
\alphar@{}[rrrru]^<{\hspace{5ex}\cong\, \varphi_A}
\\
{} & {} & {{}^{\hspace{-3ex}\mathcal Downarrow h_u}} & \muathcal{F}
\\
\muathcal{C}_B \alphar@{^(->}^{i_B}[r]
\alphar@{}[uur]|\epsilonquiv
\alphar@(d,d)[rrru]_{g^*_B}
& \muathcal{E}_B \alphar[rru]_{h_B^*}
\alphar@{}[rrrd]_<{\hspace{5ex}\cong\,\varphi_B}
\\
{} & {} & {} & {} & {}
}
$$
\epsilonmph{essentially surjective}:
Let \mubox{$g \in pc\muathcal{S}it^*(\C,\mathcal F)$.} For each $A \in
\mathcal A$, take by lemma \rhoef{Diaconescubis} $\mathcal E_A \xto {h^*_A} \mathcal F$,
$\varphi_A$,
\mubox{$h_A^* i_A \sigmatackrel{\varphi_A}{\sigmaimeq} g^*_A$}.
By lemma \rhoef{translacion}, $h^*i$ inherits a pseudocone structure
such that $\varphi$ becomes a
pseudocone isomorphism.
For each arrow $A \xto u B$ we have $(h^*i)_A
\sigmatackrel{(h^*i)_u}{\Rightarrow} (h^*i)_B u^*$. Since $\rhoho_A$ is fully-faithful, there exists a unique $h_A^*
\sigmatackrel{h_u}{\Rightarrow} h_B^* u^*$ extending $(h^*i)_u$. In this way we obtain
data \mubox{$h^* = (h^*_A,\, h_u)$} that restricts to a
pseudocone. Again from
the fully-faithfulness of each $\rhoho_A$ it is straightforward to check
that it satisfies the pseudocone equations \rhoef{PCequations}.
\epsilonmph{fully-faithful:}
Let \mubox{$h^*, l^* \in pc\muathcal{T}op^*(\mathcal E,\mathcal F)$} be two pseudocones, and
let
$\omegaidetilde{\epsilonta}$ be a morphism between the pseudocones $h^*i$ and
$l^*i$.
We have natural transformations $\xymatrix{h^*_A i_A
\alphar@{=>}[r]^{\omegaidetilde{\epsilonta_A}} & l^*_A i_A}$.
Since the inclusions $i_A$ are dense, we can extend
$\omegaidetilde{\epsilonta_A}$
uniquely to $\xymatrix{h^*_A \alphar@{=>}[r]^{\epsilonta_A} & l^*_A}$ such
that $\omegaidetilde{\epsilonta} = \epsilonta \,i$. As before, from
the fully-faithfulness of each $\rhoho_A$ it is straightforward to check
that $\epsilonta = (\epsilonta_A)$ satisfies the morphism of pseudocone equation \rhoef{PCequations}.
\epsilonnd{proof}
\betagin{theorem} \lambdaabel{main}
Let $\mathcal A^{op}$ be a small 2-filtered 2-category, and $\mathcal A^{op}
\xto \mathcal E \muathcal{T}op$ be a 2-functor. Let \mubox{$\mathcal A^{op}
\xto \C \muathcal{S}it$} be a restriction to small sites as in
\rhoef{res}. Then, the topos of sheaves $\omegaidetilde{\Colim{\muathcal{C}}}$ on
the site $\Colim{\muathcal{C}}$ of \rhoef{sitelimit} is a bilimit of
$\muathcal{E}$ in $\muathcal{T}op$, or, equivalently, a bicolimit in
$\muathcal{T}op^*$.
\epsilonnd{theorem}
\betagin{proof}
Let $\lambda^*$ be the pseudocolimit pseudocone $\C_A \xto {\lambda_A^*}
\Colim{\muathcal{C}}$ in the \mubox{2-category $\muathcal{S}it^*$}
(\rhoef{sitelimit}). Consider the composite pseudocone
\mubox{$\C_A \xto {\lambda_A^*} \Colim{\muathcal{C}} \mur{\varepsilon}
\omegaidetilde{\Colim{\muathcal{C}}}$} and
let $l^*$ be a
pseudocone from $\mathcal E$ to $\omegaidetilde{\Colim{\muathcal{C}}}$ such that $l^*i \sigmaimeq
\epsilon^*\lambda^*$
given by lemma \rhoef{pseudodiaconescu}. We have the following diagrams
commuting up to an isomorphism:
$$
\xymatrix
{
\muathcal{F}
& \omegaidetilde{\Colim{\muathcal{C}}}
\alphar[l]
\alphar@{}@<-1.3ex>[rd]^{\cong}
& \Colim{\muathcal{C}} \alphar[l]_{\varepsilon^*}
\\
{}
& \muathcal{E} \alphar[u]_{l^*}
& \muathcal{C} \alphar[u]_{\lambdaambda^*} \alphar[l]_{i}
}
\hspace{5ex}
\xymatrix
{
\muathcal{T}op^*(\omegaidetilde{\Colim{\muathcal{C}}},\, \muathcal{F})
\alphar[d]^{\rhoho_l}
\alphar[r]^{\rhoho_\varepsilon}
\alphar@{}@<-1.3ex>[rd]^{\cong}
& \muathcal{S}it^*(\Colim{\muathcal{C}},\, \muathcal{F})
\alphar[d]^{\rhoho_\lambdaambda}
\\
pc\muathcal{T}op^*(\muathcal{E},\,\muathcal{F})
\alphar[r]^{\rhoho}
& pc\muathcal{S}it^*(\muathcal{C},\,\muathcal{F})
}
$$
In the diagram on the right the arrows $\rhoho_\varepsilon$,
$\rhoho_\lambdaambda$ and $\rhoho$ are equivalences of categories
(\rhoef{Diaconescu}, \rhoef{sitelimit} and \rhoef{pseudodiaconescu}
respectively), so it follows that $\rhoho_l$ is an equivalence. This
finishes the proof.
\epsilonnd{proof}
This theorem shows the existence of small 2-cofiltered bilimits in the
\mubox{2-category} of topoi and geometric morphisms. But, it shows more,
namely, that given any small 2-filtered diagram of topoi, without loss of
generality, we can construct a
small site with finite limits for the bilimit topos out of a 2-cofiltered sub-diagram
of small sites with finite limits. However, this depends on the
\epsilonmph{axiom of choice} (needed for Proposition \rhoef{res}). We notice
for the interested reader that if we allow large sites (as in Theorem
\rhoef{sitelimit}), we can take the topoi themselves as sites, and the
proof of theorem \rhoef{main} with $\muathcal{C} = \muathcal{E}$ does not use
Proposition \rhoef{res}. Thus, without the use of choice we have:
\betagin{theorem} \lambdaabel{main2}
Let $\mathcal A^{op}$ be a small 2-filtered 2-category, and $\mathcal A^{op}
\xto \mathcal E \muathcal{T}op$ be a 2-functor. Then, the topos of sheaves $\omegaidetilde{\Colim{\muathcal{E}}}$ on
the site $\Colim{\muathcal{E}}$ of \rhoef{sitelimit} is a bilimit of
$\muathcal{E}$ in $\muathcal{T}op$, or, equivalently, a bicolimit in
$\muathcal{T}op^*$.
\epsilonnd{theorem}
\betagin{thebibliography}{00}
\bibitem{G1} Artin M, Grothendieck A, Verdier J., \tauextsl{SGA 4 ,
(1963-64)}, Lecture Notes in Mathematics 269 Springer, (1972).
\bibitem{G2} Artin M, Grothendieck A, Verdier J., \tauextsl{SGA 4 ,
(1963-64)}, Springer Lecture Notes in Mathematics 270 (1972).
\bibitem{D} Dubuc, E. J., \tauextsl{2-Filteredness and the point of
every Galois topos}, Proceedings of CT2007, Applied Categorical
Structures, Volume 18, Issue 2, Springer Verlag (2010).
\bibitem{DS} Dubuc, E. J., Street, R., \tauextsl{A construction of
2-filtered bicolimits of categories}, Cahiers de Topologie et
Geometrie Differentielle, (2005).
\bibitem{G} Gray J. W., \tauextsl{Formal Category Theory: Adjointness
for $2$-Categories}, Springer Lecture Notes in Mathematics 391
(1974).
\bibitem{MM} Mac Lane S., Moerdijk I., \tauextsl{Sheaves in Geometry
and Logic}, Springer Verlag, (1992).
\bibitem{S} Street R.,\tauextsl{Limits indexed by category-valued $2$-functors}
J. Pure Appl. Alg. 8 (1976).
\epsilonnd{thebibliography}
\epsilonnd{document} |
\begin{document}
\title{Components and Exit Times of Brownian Motion in two or more $p$-Adic Dimensions}
\author{Rahul Rajkumar$^1$ \and David Weisbart$^2$}
\address{
\begin{tabular}[h]{cc}
$^{1,2}$Department of Mathematics\\
University of California, Riverside
{\rm e}nd{tabular}
}
{\rm e}mail{$^[email protected]} {\rm e}mail{$^[email protected]}
\maketitle
\pagestyle{plain}
\begin{abstract}
The fundamental solution of a pseudo-differential equation for functions defined on the $d$-fold product of the $p$-adic numbers, $\mathds{Q}_p$, induces an analogue of the Wiener process in $\mathds{Q}_p^d$. As in the real setting, the components are $1$-dimensional $p$-adic Brownian motions with the same diffusion constant and exponent as the original process. Asymptotic analysis of the conditional probabilities shows that the vector components are dependent for all time. Exit time probabilities for the higher dimensional processes reveal a concrete effect of the component dependency.
{\rm e}nd{abstract}
\tableofcontents
\tilde{S}ection{Introduction}\label{sec:intro}
Two main ideas initially motivated the study of non-Archimedean diffusion: the idea that non-Archimedean physical models describe the observed ultrametricity in certain complex systems, and the idea that the extremely small scale structure of spacetime could be non-Archimedean. Ultrametric structures in spin glasses were already implicit in Parisi's investigations in \cite{Parisi:PRL:1979} and \cite{Parisi:JPA:1980}. In \cite{Volovich:1987}, Volovich proposed the idea that spacetime could have a non-Archimedean structure at small enough distance and time scales and he initiated a program to study analogues of physical theories with $p$-adic state spaces for this reason. Varadarajan discussed this contribution of Volovich in \cite[Chapter 6]{VSV:Reflections:2011}. The study of diffusion in non-Archimedean local fields goes back more than 30 years, and the study of diffusion processes in vector spaces over such fields goes back nearly as far. Seminal works in this area include \cite{koch92} and \cite{alb}. In \cite{koch92}, Kochubei gave the fundamental solution to the $p$-adic analogue of the diffusion equation, developed a theory of $p$-adic diffusion equations, and proved a Feynman-Kac formula for the operator semigroup with a $p$-adic Schr\"{o}dinger operator as its infinitesimal generator. Albeverio and Karwowski constructed in \cite{alb} a continuous time random walk on $\mathds Q_p$, computed its transition semigroup and infinitesimal generator, and showed among other things that the associated Dirichlet form is of jump type.
For any finite dimensional vector space $\mathcal S$ with coefficients in a division algebra that is finite dimensional over a non-Archimedean local field of arbitrary characteristic, Varadarajan constructed in \cite{var97} a general class of diffusion processes with sample paths in the Skorohod space $D([0, \infty)\colon\mathcal S)$ of c\`adl\`ag paths that take values in $\mathcal S$. The current work follows the approach of \cite{var97} and takes it as a starting point, but specializes to the setting where, for any prime number $p$ and any natural number $d$, paths take values in the $d$-fold Cartesian product of the $p$-adic numbers, $\mathds Q_p^d$. The results of \cite{var97} show that there is a triple $\big(D([0, \infty)\colon\mathds Q_p^d), P^d, \vec{X}\big)$ so that $P^d$ is a probability measure on $D([0, \infty)\colon\mathds Q_p^d)$ and, for any $\omega$ in $D([0, \infty):\mathds Q_p^d)$ and any positive $t$, \[\vec{X}(t, \omega) = \omega(t).\] Furthermore, the probability measure on $\mathds Q_p^d$ that for any Borel set $B$ of $\mathds Q_p^d$ is given by \[B\mapsto P^d(\vec{X}(t, \omega) \in B)\] has a density function that is a solution to a pseudo-differential equation that is analogous to the real diffusion equation. Refer to any process of this type as a {\rm e}mph{Brownian motion in $\mathds Q_p^d$}.
For any Brownian motion $\vec{X}$ in $\mathds Q_p^d$ with sample paths in $(D([0, \infty)\colon\mathds Q_p^d), P^d)$, the current paper establishes that the component processes of $\vec{X}$ are each Brownian motions in $\mathds Q_p$ with the same parameters (diffusion constant and exponent) as $\vec{X}$, and that for no positive real $t$ are the components of $\vec{X}_t$ independent. Section~\ref{sec:NormProd} briefly reviews some necessary results in $p$-adic analysis, discusses the max-norm process that is a special case of the more general process that Varadarajan discusses in \cite{var97}, and determines the infinitesimal generator of the {\rm e}mph{product process} in $\mathds Q_p^d$. Section~\ref{sec:components} studies the component processes of the max-norm process. The main results are Theorems~\ref{thm:components:marginal_distributions} and \ref{theorem:components:epsilon}. Theorem~\ref{thm:components:marginal_distributions} establishes that the component processes are Brownian motions in $\mathds Q_p$. Theorem~\ref{theorem:components:epsilon} gives precise estimates on certain conditional probabilities that establish the dependency of the component processes. The effect of the dependency of the components becomes strikingly apparent in the calculation in Section~\ref{sec:exit} of the exit probabilities for the product and max-norm process that generalize \cite[Theorem 3.1]{Weisbart:2021}.
Dragovich, Khrennikov, Kozyrev, Volovich, and Zelenov give a detailed review of the history of the research in non-Archimedean mathematical physics in \cite{DKKVZ:2017} that updates the earlier review \cite{DKKV:2009} by the first four authors. This review helps to put the current paper in context and discusses many areas where the current paper could find application. In their recent book \cite{KKZ:2018}, Khrennikov, Kozyrev, and Z\'{u}\~{n}iga-Galindo discuss many applications of ultrametric pseudodifferential equations, including many interesting recent developments. This work and the references therein also present many areas where the current paper could be useful. The works \cite{ave3, ave4, ave} of Avetisov, Bikulov, Kozyrev, and Osipov that deal with $p$-adic models for complex systems seem to be particularly relevant to this current paper, as is the work \cite{Avetisov-Bikulov:PSIM:2009} of Avetisov and Bikulov that involve biological applications. Ultrametricity can be found in data structures, as Bradley discusses in \cite{Bradley:2017}. The current paper may find application in the study of data structures and, in particular, in the recent work \cite{Bradley-Keller-Weinmann: 2018} of Bradley, Keller, Weinmann, as well as in Bradley's work, \cite{Bradley: 2019}, and in the work \cite{Bradley-Jahn: 2022}, of Bradley and Jahn.
\tilde{S}ection{The Norm and Product Processes}\label{sec:NormProd}
See Gouv\^{e}a's book \cite{Gov} for an accessible supplement to the cursory review of $p$-adic analysis that the current section presents. For more detail, see the book \cite{Ram} of Ramakrishnan and Valenza, and Weil's book \cite{Weil}. Vladimirov, Volovich, and Zelenov give a self-contained introduction to $p$-adic analysis and mathematical physics in their now classic book \cite{vvz}. Z\'{u}\~{n}iga-Galindo's recent book \cite{Zuniga:PDiffEBook} is an accessible and current reference that, among other things, investigates diffusion processes of a type that includes the max-norm processes that appear shortly.
This section follows the presentation of \cite{Weisbart:2021}, but generalizes it to the higher dimensional $p$-adic setting that is necessary for the sections that follow.
\tilde{S}ubsection{Analysis in $\mathds Q_p^d$}\label{sec:NormProd:sub:analysis}
For any prime $p$, denote by $|\cdot|$ the absolute value on $\mathds Q_p$. For any natural number $d$ and any $d$-tuple $(x_1, {\rm d}ots, x_d)$ in $\mathds Q_p^d$, write \[\vec{x} = (x_1, {\rm d}ots, x_d).\] Denote by $\|\cdot\|$ the {\rm e}mph{max-norm} on $\mathds Q_p$ that takes any $\vec{x}$ in ${\mathds Q}_p^d$ to $\|\vec{x}\|$, where \[\|\vec{x}\| = \max_{i\in\{1, {\rm d}ots, d\}}|x_i|.\] The max-norm induces an ultrametric on $\mathds Q_p^d$. The general linear group in $d$ dimensions with coefficients in $\mathds Z\tilde{S}lash p\mathds Z$, $GL_d(\mathds Z\tilde{S}lash p\mathds Z)$, is the maximal compact subgroup of the $p$-adic general linear group in $d$ dimensions, $GL_d(\mathds Q_p)$. Since the real orthogonal group in $d$ dimensions, $O_d(\mathds R)$, is the maximal compact subgroup of the real general linear group $GL_d(\mathds R)$, the group $GL_d(\mathds Z\tilde{S}lash p\mathds Z)$ is the natural analogue in the $\mathds Q_p^d$ setting of the real orthogonal group $O_d(\mathds R)$. The max-norm is $GL_d(\mathds Z\tilde{S}lash p\mathds Z)$ invariant, and so the max-norm is a natural analogue in the $\mathds Q_p^d$ setting of the Euclidean norm on $\mathds R^d$.
For each $\vec{x}$ in $\mathds Q_p^d$, denote respectively by $B_d(k,\vec{x})$ and $S_d(k,\vec{x})$ the ball and the circle of radius $p^k$ with center at $\vec{x}$, the compact open sets \[B_d(k, \vec{x}) = \big\{\vec{y}\in \mathds Q_{p}^d\colon \|\vec{y}-\vec{x}\| \leq p^k\big\} \quad {\rm and}\quad {S}_d(k, \vec{x}) = \big\{\vec{y}\in \mathds Q_{p}^d\colon \|\vec{y}-\vec{x}\| = p^k\big\}.\] The unit ball in $\mathds Q_{p}^d$ is the $d$-fold Cartesian product, $\mathds Z_{p}^d$, of the ring of integers in $\mathds Q_p$. For balls and circles with centers at the origin, simplify notation by respectively denoting by $B_d(k)$ and $S_d(k)$ the ball and the circle of radius $p^k$ in $\mathds Q_p^d$ with center $\vec{0}$. To further simplify notation, suppress $d$ in the notation for balls and circles to indicate that $d$ is equal to 1.
For any $x$ in $\mathds Q_{p}$, there is a unique function \begin{align}\label{theafunction}a_{x}\colon \mathds Z\to \{0,1,{\rm d}ots, p-1\}{\rm e}nd{align} with the property that \[x = \tilde{S}um_{k\in\mathds Z}a_x(k)p^k.\] Denote by $\{x\}$ the {\rm e}mph{fractional part of $x$}, the sum
\begin{equation}\label{eq:NormProd:def:fracpart}
\{x\} = \tilde{S}um_{k<0} a_{x}(k)p^k.
{\rm e}nd{equation}
For any $x$ in $\mathds Q_p$, the support of $a_x$ is bounded below and so the sum that defines $\{x\}$ in {\rm e}qref{eq:NormProd:def:fracpart} is a finite sum. Take $\chi$ to be the additive character on $\mathds Q_p$ that is given by \[\chi(x) = {\rm e}^{2\pi{\tilde{S}qrt{-1}}\{x\}}.\] For any natural number $d$, $\mathds Q_p^d$ is a totally disconnected, self-dual, locally compact, Hausdorff abelian group. The additive character $\chi$ induces an isomorphism between $\mathds Q_p^d$ and its Pontryagin dual, $\big(\mathds Q_p^d\big)^\ast$. Namely, for any additive character $\phi$ in $\big(\mathds Q_p^d\big)^\ast$, there is a $\vec{y}$ in $\mathds Q_p^d$ so that for any $\vec{x}$ in $\mathds Q_p^d$, \[\phi(\vec{x}) = \chi(\vec{x}\cdot \vec{y}) \quad \text{where}\quad \vec{x}\cdot \vec{y} = x_1y_1 +\cdots + x_dy_d.\] Take $\mu_d$ to be the unique Haar measure on $\mathds Q_{p}^d$ for which $\mathds Z_{p}^d$ has unit measure. For any integer $k$, the substitution formula for $p$-adic integrals implies that $\mu_1(B(k))$ is equal to $p^k$. The measure $\mu_d$ is a product measure and $B_d(k)$ is the $d$-fold Cartesian product of the ball $B(k)$. Furthermore, the circle $S(k)$ is the set $B(k)\tilde{S}etminus B(k-1)$. Translation invariance of $\mu_d$ therefore implies that for any $\vec{x}$ in $\mathds Q_p^d$,
\begin{equation}\label{EQ:NormProd:Ball_Sphere_measure_d}
\mu_d(B_d(k, \vec{x})) = p^{kd} \quad {\rm and}\quad \mu_d(S_d(k, \vec{x})) = p^{kd}- p^{(k-1)d} = p^{kd}\left(1 - \tfrac{1}{p^d}\right).
{\rm e}nd{equation}
Initially define $\mathcal F_d$ and $\mathcal F_d^{-1}$ on $L^1(\mathds Q_{p}^d)$ by \[(\mathcal F_df)(\vec{y}) = \int_{\mathds Q_{p}^d}\chi(-\vec{x}\cdot \vec{y})f(\vec{x})\,{\rm d}\mu_d\!\left(\vec{x}\right) \quad {\rm and}\quad (\mathcal F_d^{-1}f)(\vec{y}) = \int_{\mathds Q_{p}^d}\chi(\vec{x}\cdot \vec{y})f(\vec{x})\,{\rm d}\mu_d\!\left(\vec{x}\right).\] These operators are unitary on $L^1(\mathds Q_{p}^d)\cap L^2(\mathds Q_{p}^d)$ and extend to unitary operators on $L^2(\mathds Q_{p}^d)$. The extensions of these operators, again denoted by $\mathcal F_d$ and $\mathcal F_d^{-1}$, are the Fourier and inverse Fourier transforms on $L^2(\mathds Q_{p}^d)$, respectively. To simplify notation, suppress the measure in the notation for integrals by writing ${\rm d}\vec{x}$ to mean ${\rm d}\mu_d\big(\vec{x}\,\big)$. Furthermore, write ${\rm d}x$ to mean ${\rm d}\mu_1(x)$ in the case when $d$ is equal to $1$ and the integral is taken over a subset of $\mathds Q_p$.
The following lemma is helpful for performing calculations that involve the integration of characters.
\begin{lemma}\label{lem:NormProd:CharInt}
For any $m$ and $n$ in ${\mathds Z}$,
\begin{equation*}
\int_{B_d(m)} \int_{B_d(n)} \chi(\vec{x}\cdot \vec{y}\,) \,{\rm d}\vec{x} \,{\rm d}\vec{y} = p^{d(n + \min(-n,m))}.
{\rm e}nd{equation*}
{\rm e}nd{lemma}
\begin{proof}
Since the character $\chi$ is identically equal to 1 on $\mathds Z_p$ and the sum of the $p^{\rm th}$ roots of unity is equal to $0$,
\begin{equation}\label{lemma:NormProd:BasicCharInt}
\int_{B(n)} \chi(x)\,{\rm d}x = \begin{cases}p^{n}&\mbox{if }p^{n}\leq 1\\0&\mbox{if }p^{n}> 1.{\rm e}nd{cases}
{\rm e}nd{equation}
For any $y$ in $\mathds Q_p$, the substitution formula for $p$-adic integration implies that
\begin{equation}\label{lemma:NormProd:BasicCharInt}
\int_{B(n)} \chi(xy)\,{\rm d}x = \begin{cases}p^{n}&\mbox{if }p^{n}\leq \frac{1}{|y|}\\0&\mbox{if }p^{n}> \frac{1}{|y|}.{\rm e}nd{cases}
{\rm e}nd{equation}
For any subset $S$ of $\mathds Q_p^d$, take ${\mathds 1}_S$ to be the indicator function on $S$. Equation~\ref{lemma:NormProd:BasicCharInt} implies that
\begin{equation}\label{lemma:NormProd:BasicCharInt:Indicator}
\int_{B_d(n)} \chi(\vec{x}\cdot\vec{y}\,)\,{\rm d}\vec{x} = p^{dn}{\mathds 1}_{B_d(-n)}(\vec{y}\,),
{\rm e}nd{equation}
and so
\begin{align*}
\int_{B_d(m)} \int_{B_d(n)} \chi(\vec{x}\cdot\vec{y}\,) \,{\rm d}\vec{x}\, {\rm d}\vec{y} &= \int_{B_d(-m)} p^{dn}{\mathds 1}_{B_{-n}}(\vec{y}\,) \,{\rm d}\vec{y} \\&= p^{d(n + \min(-n,m))}.
{\rm e}nd{align*}
{\rm e}nd{proof}
\tilde{S}ubsection{The Max-Norm Process}\label{sec:NormProd:sub:1-d}
Denote by $SB(\mathds Q_{p}^d)$ the {\rm e}mph{Schwartz-Bruhat} space of complex valued, compactly supported, locally constant functions on $\mathds Q_{p}^d$. This space is the $\mathds Q_p^d$ analogue of the space of complex valued, compactly supported, smooth functions on $\mathds R^d$. Unlike its real analogue, $SB(\mathds Q_{p}^d)$ is closed under the Fourier transform. For any positive real number $b$, take ${\mathcal M}_b$ to be the multiplication operator that acts on $SB(\mathds Q_{p}^d)$ by \[({\mathcal M}_bf)(\vec{x}) = \|\vec{x}\|^bf(\vec{x}).\] Take $\Delta_{b,d}$ to be the self-adjoint closure of the densely defined operator that acts on $SB(\mathds Q_{p}^d)$ by \begin{equation}\label{EQ:NormProd:pseudoDelta}\big(\Delta_{b,d} f\big)(\vec{x}) = \big(\mathcal F^{-1}_d{\mathcal M_b}\mathcal F_df\big)\!(\vec{x}).{\rm e}nd{equation} Extend $\Delta_{b,d}$ to act on a complex valued function $f$ on $\mathds R_+\times \mathds Q_{p}^d$ by currying variables, as in \cite{Weisbart:2021}. This is to say that for each $t$ in $\mathds R_+$, if the function $f(t, \cdot)$ is in the domain of $\Delta_{b,d}$, then $f$ is in the domain of the extended operator, again denoted by $\Delta_{b,d}$, and that \[(\Delta_{b,d}f)(t,\vec{x}) := (\Delta_{b,d}f(t, \cdot))(\vec{x}).\] This extension is the {\rm e}mph{Taibleson-Vladimirov operator with exponent} $b$. To simplify notation, suppress $d$ to denote that $d$ is equal to $1$. Again follow \cite{Weisbart:2021} by currying variables to extend the Fourier and inverse Fourier transforms to act on functions on $\mathds R_+\times \mathds Q_{p}^d$. For any positive real number $\tilde{S}igma$, the pseudo-differential equation \begin{align}\label{EQ:NormProd:DiffusionEQ} {\rm d}frac{{\rm d}f(t,\vec{x})}{{\rm d}t} = -\tilde{S}igma\Delta_{b,d} f(t,\vec{x}){\rm e}nd{align} is a {\rm e}mph{$d$-dimensional $p$-adic diffusion equation} and has fundamental solution $\rho_d$, where for each $t$ in $(0,\infty)$ and for each $\vec{x}$ in $\mathds Q_p^d$, \[\rho_d(t,\vec{x}) = \left(\mathcal F_d^{-1}{\rm e}^{-\tilde{S}igma t\|\cdot\|^b}\right)\!(\vec{x}).\] Once again, suppress $d$ in the notation for $\rho_d$ when $d$ is equal to $1$.
With the necessary modifications to include the diffusion constant $\tilde{S}igma$, follow the arguments in \cite{var97} to see that $\rho_d(t,\cdot)$ is a probability density function that gives rise to a probability measure $P^d$ on $D([0,\infty) \colon \mathds Q_{p}^d)$ that is concentrated on the set of paths originating at 0. The inclusion of a diffusion constant that is not equal to 1 amounts to a rescaling of the real time parameter. The {\rm e}mph{max-norm process} $\vec{X}$ is the stochastic process that, for any pair $(t, \omega)$ in $[0,\infty)\times D([0,\infty) \colon \mathds Q_{p}^d)$, is defined by \[\vec{X}(t, \omega) = \omega(t).\] This stochastic process specializes the process that Varadarajan constructed in \cite{var97} to the setting of $\mathds Q_p^d$. If $d$ is equal to $1$, write $X$ rather than $\vec{X}$. The process $X$ is the process discussed in \cite{Weisbart:2021} and is the {\rm e}mph{Brownian motion} in $\mathds Q_p$ with {\rm e}mph{diffusion constant} equal to $\tilde{S}igma$ and {\rm e}mph{diffusion exponent} equal to $b$.
Denote by $\vec{X}_t$ the random variable $\vec{X}(t, \cdot)$ that takes any path $\omega$ in $D([0,\infty) \colon \mathds Q_{p}^d)$ to the value $\omega(t)$. The density function $\rho_d(t,\cdot)$ for $\vec{X}_t$ satisfies the equality
\begin{align}\label{pdf:NormProcess:d}
\rho_d(t,\vec{x}) & = \int_{\mathds Q_p^d} \chi(\vec{x}\cdot \vec{y}){\rm e}^{-\tilde{S}igma t\|\vec{x}\|^b}\,{\rm d}\vec{y}\notag\\&= \tilde{S}um_{r\in \mathds Z} {\rm e}^{-\tilde{S}igma tp^{rb}}\int_{S_d(r)} \chi(\vec{x}\cdot \vec{y})\,{\rm d}\vec{y}\notag\\ &= \tilde{S}um_{r\in \mathds Z} \Big({\rm e}^{-\tilde{S}igma tp^{rb}} - {\rm e}^{-\tilde{S}igma tp^{(r+1)b}}\Big)\int_{B_d(r)} \chi(\vec{x}\cdot \vec{y})\,{\rm d}\vec{y}\notag\\& = \tilde{S}um_{r\in\mathds Z}\Big({\rm e}^{-\tilde{S}igma t p^{rb}} - {\rm e}^{-\tilde{S}igma t p^{(r+1)b}}\Big)p^{dr}{\mathds 1}_{B_d(-r)}(\vec{x}).
{\rm e}nd{align}
\tilde{S}ubsection{The Product Process}\label{sec:NormProd:higher-d}
A function $f$ is a {\rm e}mph{Schwartz-Bruhat monomial} with domain ${\mathds Q}_p^d$ if for every $i$ in $\{1, {\rm d}ots, d\}$ there is a Schwartz-Bruhat function $f_i$ on $\mathds Q_p$ so that for every $\vec{x}$ in ${\mathds Q}_p^d$,
\begin{equation}
\vec{x} = (x_1, {\rm d}ots, x_d) \quad \text{implies that}\quad f(\vec{x}) = \prod_{i=1}^d f_i(x_i).
{\rm e}nd{equation}
The space $SB_0(\mathds Q_{p}^d)$ of {\rm e}mph{simple Schwartz-Bruhat} functions on $\mathds Q_p^d$ is the $d$-fold algebraic tensor product of the space of functions $SB({\mathds Q}_p)$---It is the complex vector space of functions that are finite linear combinations of Schwartz-Bruhat monomials. Each simple Schwartz-Bruhat function is a function in $L^2({\mathds Q}_p^d)$, and $L^2({\mathds Q}_p^d)$ is the analytic completion of $SB_0(\mathds Q_{p}^d)$ under the $L^2$-norm on $L^2({\mathds Q}_p^d)$.
For each $i$ in $\{1, {\rm d}ots, d\}$, take $H_i$ to be the linear extension to $SB_0(\mathds Q_{p}^d)$ of the operator that is initially defined for any Schwartz-Bruhat monomial $f$ by \[H_i(f) = f_1 \otimes \cdots \otimes f_{i-1} \otimes \tilde{S}igma\Delta_b f_i\otimes f_{i+1} \otimes \cdots \otimes f_d.\] Take $H_0$ to be the sum \[H_0 = H_1 + \cdots + H_d.\] The operator $H_0$ is the Fourier transform of real valued multiplication operator on a dense subset of $L^2({\mathds Q}_p^d)$, and so it is essentially self adjoint on $SB_0(\mathds Q_{p}^d)$. Take $H$ to be the self adjoint closure of $H_0$ on $L^2({\mathds Q}_p^d)$.
The max-norm process $\vec{X}$ has a radially symmetric law with respect to the max-norm, but it is not the only possible choice for a Brownian motion in ${\mathds Q}_p^d$. Take $P$ to be the probability measure on $D([0,\infty) \colon {\mathds Q}_p)$ such that the triple $(D([0,\infty) \colon {\mathds Q}_p), P, X)$ is the 1-dimensional Brownian motion studied in \cite{Weisbart:2021} with the property that, for each positive $t$, the probability density function $\rho(t,\cdot)$ for $X_t$ satisfies {\rm e}qref{EQ:NormProd:DiffusionEQ} with $d$ equal to $1$. For each positive real number $t$, take $\vec{Y}_t$ to be the random variable that, for any path $\omega$ in the $d$-fold Cartesian product $D([0,\infty) \colon \mathds Q_{p})^d$, is given by \[\vec{Y}_t(\omega) = \omega(t) = (\omega_1(t), {\rm d}ots, \omega_d(t)).\] Take $\vec{Y}$ to be the function that is defined for all $t$ in $(0,\infty)$ by \[\vec{Y}(t) = \vec{Y}_t.\] The process that is given by the triple $\big(D([0,\infty) \colon {\mathds Q}_p^d), \otimes_{i=1}^d P, \vec{Y}\big)$ is the {\rm e}mph{product process}, which is radially symmetric if and only if $d$ is equal to $1$. Simplify notation by henceforth writing $\otimes^dP$ rather than $\otimes_{i=1}^dP$.
For any positive real $t$, take $g_d(t, \cdot)$ to be the probability density function for $\vec{Y}_t$. The independence of the components of $\vec{Y}$ together with the product rule for differentiation implies that, for any $\vec{x}$ in ${\mathds Q}_p^d$,
\begin{align*}
\frac{{\rm d}}{{\rm d}t}g_d(t,\vec{x}) & = \frac{{\rm d}}{{\rm d}t}\big(\rho(t,x_1)\rho(t,x_2)\cdots \rho(t,x_d)\big)\\ & = \tilde{S}igma H g_d(t, \vec{x}),
{\rm e}nd{align*}
and so $H$ is the infinitesimal generator of the product process on ${\mathds Q}_p$.
In the setting of Brownian motion in ${\mathds R}^d$, where the norm is the usual Euclidean ${\rm e}ll^2$-norm, the analogous norm process is that which is given by the product of $d$ independent Brownian motions in ${\mathds R}$ that have the same diffusion constant. The next section will show that this is not the case in the ${\mathds Q}_p$ setting, and the goal of the present work is to precisely understand some differences between the stochastic processes $\vec{X}$ and $\vec{Y}$ when $d$ is greater than 1.
\tilde{S}ection{The Component Processes and their Dependence}\label{sec:components}
Section~\ref{sec:components:sub:marginals} establishes that the processes $\vec{X}$ and $\vec{Y}$ are similar in that the components of each are themselves Brownian motions with the same diffusion constant, $\tilde{S}igma$, and the same diffusion exponent, $b$. Section~\ref{sec:components:sub:dependency} establishes that the max-norm process, $\vec{X}$, has dependent components, where the product process, $\vec{Y}$, has independent components, and so $\vec{X}$ and $\vec{Y}$ are qualitatively different.
\tilde{S}ubsection{Calculation of the Marginals}\label{sec:components:sub:marginals}
For any natural number $i$ in $\{1, {\rm d}ots, d\}$, take $X^{(i)}$ to be the stochastic process that is given by the $i^{\rm th}$ component of the max-norm process $\vec{X}$. The goal of this subsection is to prove Theorem~\ref{thm:components:marginal_distributions}.
Some additional notation facilitates the presentation. For each $\vec{x}$ in $\mathds Q_p^d$, denote by $\vec{x}_i$ the vector in $\mathds Q_p^{d-1}$ that is given by \[\vec{x}_i = (x_1, {\rm d}ots, x_{i-1}, x_{i+1}, {\rm d}ots, x_d).\] To simplify notation, identify any ordered pair $(x_1, (x_2, {\rm d}ots, x_d))$ in $\mathds Q_p\times \mathds Q_p^{d-1}$ with the $d$-tuple $(x_1, {\rm d}ots, x_d)$ in $\mathds Q_p^d$. For each positive real number $t$, take $\rho^{(i)}(t, \cdot)$ to be the probability density function for the $i^{\rm th}$ component function of $\vec{X}_t$, the random variable $X_t^{(i)}$. For each $x$ in $\mathds Q_p$, the marginal $\rho^{(i)}(t, x)$ is given by the integral
\begin{align}\label{EQ:components_marginals:motivation}
\rho^{(i)}(t, x) & = \int_{{\mathds Q}_p^{d-1}} \rho_d(t, (x_1, {\rm d}ots, x_{i-1}, x, x_{i+1}, {\rm d}ots, x_d)) \,{\rm d}\vec{x}_i \notag\\
& = \int_{{\mathds Q}_p^{d-1}} \rho_d(t, (x, \vec{x}_1)) \,{\rm d}\vec{x}_1\notag\\
&= \int_{{\mathds Q}_p^{d-1}} \int_{{\mathds Q}_p^d} \chi((x, \vec{x}_1)\cdot\vec{y}\,){\rm e}^{-\tilde{S}igma t\|\vec{y}\|^b} \,{\rm d}\vec{y} \,{\rm d}\vec{x}_i,
{\rm e}nd{align}
where the change of variables formula and the symmetry of the integrand together imply the penultimate equality.
Switching the order of integration in {\rm e}qref{EQ:components_marginals:motivation} that determines the marginals facilitates calculation of the law for the component processes. The failure of the integrand to be absolutely integrable with respect to the product measure on ${\mathds Q}_p^{d-1} \times {\mathds Q}_p^d$ precludes a naive application of the Fubini-Tonelli theorem that would quickly verify Theorem~\ref{thm:components:marginal_distributions}. The technical complication necessitates the following more involved argument and implicitly makes use of the fact that the given integral is an oscillatory integral.
\begin{theorem}\label{thm:components:marginal_distributions}
Each $X^{(i)}$ is a Brownian motion in $\mathds Q_p$ with diffusion constant equal to $\tilde{S}igma$ and diffusion exponent equal to $b$.
{\rm e}nd{theorem}
\begin{proof}
For any $i$ in $\{1, {\rm d}ots, d\}$, the radial symmetry of $\rho_d$ and {\rm e}qref{EQ:components_marginals:motivation} together imply that $X^{(i)}$ and $X^{(1)}$ have the same law. For any positive $t$, the characteristic function $\phi_d(t, \cdot)$ of $\vec{X}_t$ is given for each $\vec{y}$ in $\mathds Q_p^d$ by \[\phi_d(t, \vec{y}\,) = {\rm e}^{-\tilde{S}igma t\|\vec{y}\,\|^b}.\] The function $\phi_d(t, \cdot)$ is bounded and integrable, and so the Fubini-Tonelli theorem guarantees that
\begin{align*}
\rho^{(1)}(t, x_1) & = \int_{{\mathds Q}_p^{d-1}} \rho_d(t, (x_1, \vec{x}_1))\,{\rm d}\vec{x}_1\\
& = \int_{{\mathds Q}_p^{d-1}}\left\{\int_{{\mathds Q}_p^{d}} \chi((x_1, \vec{x}_1)\cdot \vec{y}\,){\rm e}^{-\tilde{S}igma t\|\vec{y}\,\|^b}\,{\rm d}\vec{y}\right\}{\rm d}\vec{x}_1\\%
& = \int_{{\mathds Q}_p^{d-1}}\left\{\int_{{\mathds Q}_p} \int_{{\mathds Q}_p^{d-1}} \chi(x_1y_1)\chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t\|(y_1, \vec{y}_1)\|^b}\,{\rm d}\vec{y}_1{\rm d}y_1\right\}{\rm d}\vec{x}_1.
{\rm e}nd{align*}
For each $y_1$, decompose the innermost integral into a sum of integrals over the ball of radius $|y_1|$ and its complement to obtain the equalities
\begin{align*}
\rho^{(1)}(t, x_1) & = \int_{{\mathds Q}_p^{d-1}}\left\{\int_{\mathds Q_p}\int_{\|\vec{y}_1\| \leq |y_1|} \chi(x_1y_1)\chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t\|(y_1, \vec{y}_1)\|^b}\,{\rm d}\vec{y}_1\,{\rm d}y_1\right\}{\rm d}\vec{x}_1 \\&\qquad\qquad\qquad+ \int_{{\mathds Q}_p^{d-1}}\left\{\int_{\mathds Q_p}\int_{\|\vec{y}_1\|>|y_1|} \chi(x_1y_1)\chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t\|(y_1, \vec{y}_1)\|^b}\,{\rm d}\vec{y}_1\,{\rm d}y_1\right\}{\rm d}\vec{x}_1\\
& = \int_{{\mathds Q}_p^{d-1}}\Bigg\{\int_{\mathds Q_p}\chi(x_1y_1)\Bigg[{\rm e}^{-\tilde{S}igma t|y_1|^b}\int_{\|\vec{y}_1\| \leq |y_1|} \chi(\vec{x}_1\cdot \vec{y}_1)\,{\rm d}\vec{y}_1 \\&\qquad\qquad\qquad+ \int_{\|\vec{y}_1\|>|y_1|} \chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t\|\vec{y}_1\|^b}\,{\rm d}\vec{y}_1\Bigg]\,{\rm d}y_1\Bigg\}{\rm d}\vec{x}_1.
{\rm e}nd{align*}
Since $\rho_d(t, \cdot)$ is integrable,
\begin{equation}\label{EQ:components_marginals:Rhod_as_a_Limit}\rho^{(1)}(t, x_1) = \lim_{n\to\infty}\int_{B_{d-1}(n)} \rho_d(t, (x_1, \vec{x}_1))\,{\rm d}\vec{x}_1.{\rm e}nd{equation}
Take $I_1(x_1)$ and $I_2(x_1)$ to be the quantities
\begin{equation}\label{EQ:components_marginals:I1a}I_1(x_1) = \lim_{n\to\infty}\int_{B_{d-1}(n)} \left\{\int_{\mathds Q_p}\chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}\int_{\|\vec{y}_1\| \leq |y_1|} \chi(\vec{x}_1\cdot \vec{y}_1)\,{\rm d}\vec{y}_1\,{\rm d}y_1\right\}{\rm d}\vec{x}_1{\rm e}nd{equation} and
\begin{equation}\label{EQ:components_marginals:I2a}I_2(x_1) = \lim_{n\to \infty}\int_{B_{d-1}(n)}\left\{\int_{\mathds Q_p}\chi(x_1y_1)\int_{\|\vec{y}_1\|>|y_1|} \chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t\|\vec{y}_1\|^b}\,{\rm d}\vec{y}_1\,{\rm d}y_1\right\}{\rm d}\vec{x}_1,{\rm e}nd{equation} so that if both limits exist, then
\begin{equation}\label{EQ:components_marginals:I1+I2}
\rho^{(1)}(t, x_1) = I_1(x_1) + I_2(x_1).
{\rm e}nd{equation}
The domain of integration of the outermost integral of both {\rm e}qref{EQ:components_marginals:I1a} and {\rm e}qref{EQ:components_marginals:I2a} is the $\mu_{d-1}$--finite measure space $B_{d-1}(n)$.
The integrand \[ \chi(x_1y_1)\int_{\|\vec{y}_1\|>|y_1|} \chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t\|\vec{y}_1\|^b}\,{\rm d}\vec{y}_1 = \chi(x_1y_1)\int_{\mathds \mathbb{Q}_p^{d-1}} \chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t\|\vec{y}_1\|^b}\mathds{1}_{\|\vec{y}_1\|>|y_1|}(\vec{y})\,{\rm d}\vec{y}_1\] is bounded with compact support in $\mathbb{Q}_p$, so both \[ \chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}\int_{\|\vec{y}_1\| \leq |y_1|} \chi(\vec{x}_1\cdot \vec{y}_1)\,{\rm d}\vec{y}_1 \quad \text{and} \quad \chi(x_1y_1)\int_{\|\vec{y}_1\|>|y_1|} \chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t\|\vec{y}_1\|^b}\,{\rm d}\vec{y}_1\] are $L^1(B_{d-1}(n) \times \mathds \mathbb{Q}_p)$. The Fubini-Tonelli theorem implies that
\begin{equation}\label{EQ:components_marginals:I1b}I_1(x_1) = \lim_{n\to\infty}\int_{\mathds Q_p}\left\{\int_{B_{d-1}(n)}\chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}\int_{\|\vec{y}_1\| \leq |y_1|} \chi(\vec{x}_1\cdot \vec{y}_1)\,{\rm d}\vec{y}_1\,{\rm d}\vec{x}_1\right\}{\rm d}y_1{\rm e}nd{equation} and
\begin{equation}\label{EQ:components_marginals:I2b}I_2(x_1) = \lim_{n\to \infty}\int_{\mathds Q_p}\left\{\int_{B_{d-1}(n)}\chi(x_1y_1)\int_{\|\vec{y}_1\|>|y_1|} \chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t\|\vec{y}_1\|^b}\,{\rm d}\vec{y}_1\,{\rm d}\vec{x}_1\right\}{\rm d}y_1.{\rm e}nd{equation}
Lemma~\ref{lem:NormProd:CharInt} and {\rm e}qref{EQ:components_marginals:I1b} together imply that
\begin{equation*}
I_1(x_1) = \lim_{n\to\infty}\int_{\mathds Q_p} \chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}p^{\log_p|y_1| +\min(-\log_p|y_1|, n)}{\rm d}y_1.
{\rm e}nd{equation*}
For any natural number $M$, decompose the integral over $\mathds Q_p$ into a sum of integrals over $B(-M)$ and $B(-M)^c$ to obtain the equalities
\begin{align}\label{EQ:components_marginals:I1c}
I_1(x_1) &= \lim_{n\to\infty}\left\{\int_{B(-M)^c} \chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}p^{\log_p|y_1| +\min(-\log_p|y_1|, n)}{\rm d}y_1\right.\notag\\&\left.\qquad\qquad\qquad\qquad+ \int_{B(-M)} \chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}p^{\log_p|y_1| +\min(-\log_p|y_1|, n)}{\rm d}y_1\right\}\notag\\%
&= \int_{B(-M)^c} \chi(x_1y_1){\rm e}^{-t|y_1|^b}\,{\rm d}y_1\notag\\&\qquad\qquad\qquad\qquad+ \lim_{n\to\infty}\int_{B(-M)} \chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}p^{\log_p|y_1| +\min(-\log_p|y_1|, n)}{\rm d}y_1\notag\\
&= \int_{{\mathds Q}_p} \chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}\,{\rm d}y_1 + E_M(x_1),
{\rm e}nd{align}
where
\begin{align*}
E_M(x_1) &= - \int_{B(-M)} \chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}\,{\rm d}y_1\\& \qquad\qquad\qquad\qquad+ \lim_{n\to\infty}\int_{B(-M)} \chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}p^{\log_p|y_1| +\min(-\log_p|y_1|, n)}{\rm d}y_1.
{\rm e}nd{align*}
The inequality \[|E_M(x_1)| \leq 2p^{-M}\] and {\rm e}qref{EQ:components_marginals:I1c} together imply that
\begin{equation}\label{EQ:components_marginals:I1d}
I_1(x_1) = \int_{{\mathds Q}_p} \chi(x_1y_1){\rm e}^{-\tilde{S}igma t|y_1|^b}\,{\rm d}y_1.
{\rm e}nd{equation}
Take $F$ to be the function that is given for each pair $(y_1, \vec{y}_1)$ in ${\mathds Q}_p\times{\mathds Q}_p^{d-1}$ by
\begin{equation}\label{EQ:components_marginals:F}
F(y_1, \vec{y}_1) = {\rm e}^{-\tilde{S}igma t\|\vec{y}_1\|^b}{\mathds 1}_{B_{d-1}(\log_p|y_1|)^c}(\vec{y}_1)
{\rm e}nd{equation}
and for each $y_1$, take $\tilde{F}(y_1, \cdot)$ to be the Fourier transform, taken over ${\mathds Q}_p^{d-1}$, of $F(y_1, \cdot)$. Use {\rm e}qref{EQ:components_marginals:I2b} and {\rm e}qref{EQ:components_marginals:F} to rewrite $I_2(x_1)$ in terms of $F$ and obtain the equalities
\begin{align*}
I_2(x_1) & = \lim_{n\to \infty}\int_{B_{d-1}(n)}\left\{\int_{\mathds Q_p}\chi(x_1y_1)\int_{\mathds{Q}_p^{d-1}} \chi(\vec{x}_1\cdot \vec{y}_1)F(y_1, \vec{y}_1)\,{\rm d}\vec{y}_1\,{\rm d}y_1\right\}{\rm d}\vec{x}_1\\
& = \lim_{n\to \infty}\int_{B_{d-1}(n)}\left\{\int_{\mathds Q_p}\chi(x_1y_1)\tilde{F}(y_1, \vec{x}_1)\,{\rm d}y_1\right\}{\rm d}\vec{x}_1.
{\rm e}nd{align*}
Since $B_{d-1}(n)$ has finite measure, the Fubini-Tonelli theorem together with the equality \[F(y_1, \vec{0}) = 0\] implies that
\begin{align}\label{EQ:components_marginals:I2PreSplit}
I_2(x_1) & = \lim_{n\to \infty}\int_{\mathds Q_p}\left\{\int_{B_{d-1}(n)}\chi(x_1y_1)\tilde{F}(y_1, \vec{x}_1){\rm d}\vec{x}_1\right\}\,{\rm d}y_1\notag\\%
& = \lim_{n\to \infty}\int_{\mathds Q_p}\left\{\int_{{\mathds Q}_p^{d-1}}\chi(x_1y_1)\tilde{F}(y_1, \vec{x}_1){\rm d}\vec{x}_1 - \int_{B_{d-1}(n)^c}\chi(x_1y_1)\tilde{F}(y_1, \vec{x}_1){\rm d}\vec{x}_1\right\}\,{\rm d}y_1\notag\\
& = \lim_{n\to \infty}\int_{\mathds Q_p}\left\{F(y_1, \vec{0}) - \int_{B_{d-1}(n)^c}\chi(x_1y_1)\tilde{F}(y_1, \vec{x}_1){\rm d}\vec{x}_1\right\}\,{\rm d}y_1\notag\\
& = -\lim_{n\to \infty}\int_{\mathds Q_p}\left\{\int_{B_{d-1}(n)^c}\chi(x_1y_1)\tilde{F}(y_1, \vec{x}_1){\rm d}\vec{x}_1\right\}\,{\rm d}y_1.
{\rm e}nd{align}
For any natural number $M$, decompose the integral over $\mathds Q_p$ in {\rm e}qref{EQ:components_marginals:I2PreSplit} into a sum of integrals over $B(-M)$ and $B(-M)^c$ to obtain the equality
\begin{align}\label{EQ:components_marginals:I2Split}
I_2(x_1) & = -\lim_{n\to \infty}\left\{\int_{B(-M)}\left\{\int_{B_{d-1}(n)^c}\chi(x_1y_1)\tilde{F}(y_1, \vec{x}_1){\rm d}\vec{x}_1\right\}\,{\rm d}y_1 \right.\notag\\&\left.\qquad\qquad\qquad\qquad+ \int_{B(-M)^c}\left\{\int_{B_{d-1}(n)^c}\chi(x_1y_1)\tilde{F}(y_1, \vec{x}_1){\rm d}\vec{x}_1\right\}\,{\rm d}y_1\right\}.
{\rm e}nd{align}
For any non-zero $y_1$ in ${\mathds Q}_p$, $F(y_1, \cdot)$ is locally constant with a radius of local constancy equal to $|y_1|$. The function $\tilde{F}(y_1, \cdot)$ is therefore supported on the ball of radius $\frac{1}{|y_1|}$ that is centered at the origin, and so the second summand in {\rm e}qref{EQ:components_marginals:I2Split} is the integral of the zero function for any $n$ that is greater than $M$. Since the innermost integral of the first term is bounded, there is a constant $C$ so that for any natural number $M$, \[|I_2(x_1)| \leq \tfrac{C}{p^M},\] and so
\begin{equation}\label{EQ:components_marginals:I20}
I_2(x_1) = 0.
{\rm e}nd{equation}
Equations~{\rm e}qref{EQ:components_marginals:I1+I2}, {\rm e}qref{EQ:components_marginals:I1d}, and {\rm e}qref{EQ:components_marginals:I20} together imply that
\begin{equation*}
\rho^{(1)}(t, x_1) = \int_{{\mathds Q}_p} \chi(\vec{x}_1\cdot \vec{y}_1){\rm e}^{-\tilde{S}igma t|y_1|^b}\,{\rm d}y_1.
{\rm e}nd{equation*}
{\rm e}nd{proof}
\tilde{S}ubsection{Probabilities for the Conditioned Components}\label{sec:components:sub:probabilities} For each $i$ in $\{1, {\rm d}ots, d\}$, denote by $\vec{X}_{t, i}$ the $d-1$ tuple \[\vec{X}_{t, i} = \Big(X_t^{(1)}, {\rm d}ots, X_t^{(i-1)}, X_t^{(i+1)}, {\rm d}ots, X_t^{(d)}\Big), \quad \text{where}\quad \vec{X}_t = \Big(X_t^{(1)}, {\rm d}ots, X_t^{(d)}\Big).\] The components of the max-norm process fail to be independent because the spatial dependence of the law for the max-norm process involves only the component with the largest $p$-adic absolute value. For any integers $r$ and $R$, for any $i$ in $\{1, {\rm d}ots, d\}$, and for any $a$ in $B(R)$, if $r$ is less than $R$, then \[P^d\!\left(X_t^{(1)} \in B(r, a) \Big\vert \vec{X}_{t,1}\in S_{d-1}(R)\right) \leq p^{r-R}<\tfrac{1}{p}\] because this conditional probability is independent of $a$. However, as long as $t$ is small enough, \[P^d\!\left(X_t^{(1)} \in B(r, 0)\right) > \tfrac{1}{p},\] and so $\vec{X}_t$ does not have independent components as long as $t$ is small enough. Lemma~\ref{lemma:components:conditional_calculation} provides an explicit calculation of certain conditional probabilities that lead not only to a proof that the components of $\vec{X}_t$ are dependent for any positive $t$, but also to an explicit description of certain local (small time) behaviors of the conditioned component processes.
\begin{lemma}\label{lemma:components:conditional_calculation}
For any integers $r$ and $R$, for any $i$ in $\{1, {\rm d}ots, d\}$, and for any $a$ in $B(R)$, if $r$ is less than or equal to $R$, then
\begin{equation*}
P^d\!\left(X_t^{(i)} \in B(r, a) \Big\vert \vec{X}_{t,i}\in S_{d-1}(R)\right) = \frac{p^r\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}}{\tilde{S}um_{j \leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{(d-1)j}}.
{\rm e}nd{equation*}
{\rm e}nd{lemma}
\begin{proof}
Without loss in generality, take $i$ to be equal to $1$. For any $\vec{x}$ in $B(r, a)\times S_{d-1}(R)$ and for any $\vec{x}$ in $B(R, a)\times S_{d-1}(R)$, \[\|\vec{x}\| = p^R,\] and so if $-j$ is less than $R$, then
\begin{equation}\label{EQ:components:conditional_calculation:DomAZero}
\int_{B(r, a)\times S_{d-1}(R)} {\mathds 1}_{B_d(-j)}(\vec{x})\,{\rm d}\vec{x} = \int_{B(R, a)\times S_{d-1}(R)} {\mathds 1}_{B_d(-j)}(\vec{x})\,{\rm d}\vec{x} = 0.
{\rm e}nd{equation}
For any $\vec{x}$ in $B(R, a)^c\times S_{d-1}(R)$, \[\|\vec{x}\| > p^R,\] and so if $-j$ is less than or equal to $R$, then
\begin{equation}\label{EQ:components:conditional_calculation:DomBZero}
\int_{B(R, a)^c\times S_{d-1}(R)} {\mathds 1}_{B_d(-j)}(\vec{x})\,{\rm d}\vec{x} = 0.
{\rm e}nd{equation}
The Fubini-Tonelli theorem and {\rm e}qref{EQ:NormProd:Ball_Sphere_measure_d} together imply that if \[j \leq -R,\] then
\begin{align}\label{EQ:components:conditional_calculation:DomA}
\int_{B(r, a)\times S_{d-1}(R)} {\mathds 1}_{B_d(-j)}(\vec{x})\,{\rm d}\vec{x} & = \int_{B(r, a)} {\mathds 1}_{B(-j)}(x_1)\,{\rm d}x_1 \int_{S_{d-1}(R)} {\mathds 1}_{B_{d-1}(-j)}(\vec{x}_1)\,{\rm d}\vec{x}_1\notag\\&=p^rp^{(d-1)R}\left(1-\tfrac{1}{p^{d-1}}\right),
{\rm e}nd{align}
and
\begin{align}\label{EQ:components:conditional_calculation:DomB}
\int_{B(R,a)\times S_{d-1}(R)} {\mathds 1}_{B_d(-j)}(\vec{x})\,{\rm d}\vec{x} & = \int_{B(R, a)} {\mathds 1}_{B(-j)}(x_1)\,{\rm d}x_1 \int_{S_{d-1}(R)} {\mathds 1}_{B_{d-1}(-j)}(\vec{x}_1)\,{\rm d}\vec{x}_1\notag\\&=p^{dR}\left(1-\tfrac{1}{p^{d-1}}\right),
{\rm e}nd{align}
and if \[j <-R,\] then
\begin{align}\label{EQ:components:conditional_calculation:DomC}
\int_{B(R, a)^c\times S_{d-1}(R)} {\mathds 1}_{B_d(-j)}(\vec{x})\,{\rm d}\vec{x} & = \int_{B(R, a)^c} {\mathds 1}_{B(-j)}(x_1)\,{\rm d}x_1 \int_{S_{d-1}(R)} {\mathds 1}_{B_{d-1}(-j)}(\vec{x}_1)\,{\rm d}\vec{x}_1\notag\\&=\left(p^{-j} - p^R\right)p^{(d-1)R}\left(1-\tfrac{1}{p^{d-1}}\right).
{\rm e}nd{align}
The equality {\rm e}qref{pdf:NormProcess:d} implies that for any positive real $t$ and any Borel subset $U$ of $\mathds Q_p^d$,
\begin{equation}\label{EQ:components:probU}
P^d(X_t\in U) = \tilde{S}um_{j\in \mathds Z} \left({\rm e}^{-tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{j+1}b}\right)p^{dj}\int_U {\mathds 1}_{B_d(-j)}(\vec{x})\,{\rm d}\vec{x}.
{\rm e}nd{equation}
Together with {\rm e}qref{EQ:components:conditional_calculation:DomAZero}, {\rm e}qref{EQ:components:conditional_calculation:DomBZero}, {\rm e}qref{EQ:components:conditional_calculation:DomA}, {\rm e}qref{EQ:components:conditional_calculation:DomB}, and {\rm e}qref{EQ:components:conditional_calculation:DomC}, {\rm e}qref{EQ:components:probU} implies that
\begin{equation}\label{EQ:components:CondNum}
P^d\!\left(\vec{X}_t \in B(r, a)\times S_{d-1}(R)\right) = p^rp^{(d-1)R}\left(1-\tfrac{1}{p^{d-1}}\right)\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj},
{\rm e}nd{equation}
\begin{equation}\label{EQ:components:CondDenomA}
P^d\!\left(\vec{X}_t \in B(R, a)\times S_{d-1}(R)\right) = p^{dR}\left(1-\tfrac{1}{p^{d-1}}\right)\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj},
{\rm e}nd{equation}
and
\begin{align}\label{EQ:components:CondDenomB}
&P^d\!\left(\vec{X}_t \in B(R, a)^c\times S_{d-1}(R)\right) \notag\\& \hspace{1in} = p^{(d-1)R}\left(1-\tfrac{1}{p^{d-1}}\right)\tilde{S}um_{j < -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}\left(p^{-j} - p^R\right).
{\rm e}nd{align}
The equality
\begin{align*}P\!\left(\vec{X}_{t,1}\in S_{d-1}(R)\right) &= P\!\left(\big(X_t^{(1)} \in B(R)\big)\cap\big(\vec{X}_{t,1}\in S_{d-1}(R)\big)\right) \\&\qquad\qquad+ P\!\left(\big(X_t^{(1)} \in B(R)^c\big)\cap\big(\vec{X}_{t,1}\in S_{d-1}(R)\big)\right)
{\rm e}nd{align*}
implies that
\begin{align*}
&P^d\!\left(X_t^{(1)} \in B(r, a) \Big\vert \vec{X}_{t,1}\in S_{d-1}(R)\right) \\&\qquad = \frac{P^d\!\left(\big(X_t^{(1)} \in B(r, a)\big) \cap \big(\vec{X}_{t,1}\in S_{d-1}(R)\big)\right)}{P^d\!\left(\big(X_t^{(1)} \in B(R)\big)\cap\big(\vec{X}_{t,1}\in S_{d-1}(R)\big)\right) + P^d\!\left(\big(X_t^{(1)} \in B(R)^c\big)\cap\big(\vec{X}_{t,1}\in S_{d-1}(R)\big)\right)},
{\rm e}nd{align*}
and so {\rm e}qref{EQ:components:CondNum}, {\rm e}qref{EQ:components:CondDenomA}, and {\rm e}qref{EQ:components:CondDenomB} together imply that
\begin{align*}
&P^d\!\left(X_t^{(1)} \in B(r, a) \Big\vert \vec{X}_{t,1}\in S_{d-1}(R)\right)\\&\quad = \frac{p^r\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}}{p^{R}\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj} + \tilde{S}um_{j < -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}\left(p^{-j} - p^R\right)}
\\&\quad = \frac{p^r\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}}{p^{R}\left({\rm e}^{-\tilde{S}igma tp^{-Rb}} - {\rm e}^{-\tilde{S}igma tp^{(-R+1)b}}\right)p^{-dR} + \tilde{S}um_{j < -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{(d-1)j}}
\\&\quad = \frac{p^r\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}}{\tilde{S}um_{j \leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{(d-1)j}}.
{\rm e}nd{align*}
{\rm e}nd{proof}
\tilde{S}ubsection{Component Dependency}\label{sec:components:sub:dependency} The formula that Lemma~\ref{lemma:components:conditional_calculation} provides for the conditional probabilities is rather complicated. Lemma~\ref{lemma:components:conditional_calculation_Oest} gives a description of the local behavior of these conditional probabilities that is especially useful for understanding the effect of conditioning on the component processes. Denote by $\Gamma(p, b, d)$ the quantity
\begin{equation}\label{EQ:components:FDef}
\Gamma(p, b, d) = \frac{p^{b+d}-p}{p^{b+d+1} - p}.
{\rm e}nd{equation}
Use the standard ``little oh'' and ``big oh'' Landau notation to simplify the statements and proofs of the statements below.
\begin{lemma}\label{lemma:components:conditional_calculation_Oest}
For any integers $r$ and $R$, for any $i$ in $\{1, {\rm d}ots, d\}$, and for any $a$ in $B(R)$, if $r$ is less than or equal to $R$, then
\begin{equation*}
P^d\left(X_t^{(i)} \in B(r, a) \Big\vert \vec{X}_{t,i}\in S_{d-1}(R)\right) = \left(\Gamma(p, b, d)p^{-R} + {\rm o}(t)\right)p^r.
{\rm e}nd{equation*}
{\rm e}nd{lemma}
\begin{proof}
Take $G(R, d, \cdot)$ to be the function that is given for any $t$ in $[0, 1]$ by
\begin{equation}\label{EQ:components:Gdef}
G(R, d, t) = \tilde{S}um_{j\leq -R} \big({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\big)p^{dj}.
{\rm e}nd{equation}
Differentiate $G(R, d, \cdot)$ to obtain for each $t$ in $[0, \infty)$ the equality
\begin{align}\label{EQ:components:NumDenom}
G^\prime(R, d, 0) = \tilde{S}um_{j\leq -R} \big(\tilde{S}igma p^{(j+1)b} - \tilde{S}igma p^{jb}\big)p^{dj} = \frac{\tilde{S}igma\big(p^b-1\big)p^{(b+d)(1-R)}}{p^{(b+d)} - 1}.
{\rm e}nd{align}
The twice continuous differentiability of $G(R, d, \cdot)$ implies that
\begin{equation}
G(R, d, t) = G^\prime(R, d, 0)t + {\rm O}(t^2)\quad {\rm and}\quad G(R, d-1, t) = G^\prime(R, d-1, 0)t + {\rm O}(t^2),
{\rm e}nd{equation}
and so if $t$ is positive, then
\begin{align}\label{EQ:components:CondbigOestimate}
\frac{G(R,d,t)}{G(R,d-1,t)} &= \frac{G^\prime(R, d, 0)}{G^\prime(R, d-1, 0)} + {\rm o}(t)\notag\\& = p^{-R}\frac{p^{b+d}-p}{p^{b+d+1} - p} + {\rm o}(t).
{\rm e}nd{align}
Use {\rm e}qref{EQ:components:CondbigOestimate} to rewrite the righthand side of the equality that is given by Lemma~\ref{lemma:components:conditional_calculation} and obtain the desired description of the local behavior of the conditional probabilities.
{\rm e}nd{proof}
For any positive real number $t$ and any integer $R$, take $U(R)_t$ to be the random variable with the following law: For any Borel subset $V$ of $\mathds Q_p$,
\begin{equation}\label{EQ:uniformBorelDef}
{\rm Prob}(U(R)_t\in V) = P^d\!\left(X_t^{(1)} \in V \Big\vert \vec{X}_{t,1}\in S_{d-1}(R)\right).
{\rm e}nd{equation}
\begin{proposition}\label{prop:components:epsilon}
The random variable $U(R)_t$ is asymptotically uniformly distributed in $B(R)$. Furthermore, for any Borel subset $V$ of $B(R)$, \[\lim_{t\to 0^+} {\rm Prob}(U(R)_t\in V) = \mu(V)p^{-R}\Gamma(p, b, d).\]
{\rm e}nd{proposition}
\begin{proof}
For any Borel subset $V$ of $B(R)$, there is an at most countable index set $J$ and sequences $(a_j)$ in $B(R)$ and $(r_j)$ in $\mathds Z\cap(-\infty, R]$, both indexed in $J$, so that $(B(r_j, a_j))$ is a sequence of disjoint balls whose union is $V$. Lemma~\ref{lemma:components:conditional_calculation} and {\rm e}qref{EQ:components:CondbigOestimate} together imply that for each $j$,
\begin{align}\label{EQ:components:epsilonA}
{\rm Prob}(U(R)_t\in B(r_j, a_j)) &= p^{r_j}\frac{G(R, d, t)}{G(R, d-1, t)}\notag\\
& = p^{r_j}\left(p^{-R}F(p,b,d) + {\rm o}(t)\right).
{\rm e}nd{align}
The countable additivity of the conditioned measure implies that
\begin{align}\label{EQ:components:epsilonA}
{\rm Prob}(U(R)_t\in V) &= \tilde{S}um_{j\in J}p^{r_j}\left(p^{-R}\Gamma(p,b,d) + {\rm o}(t)\right)\notag\\
&= \left(\Gamma(p, b, d)p^{-R} + {\rm o}(t)\right)\tilde{S}um_{j\in J}\mu(B(r_j,a_j)) = \left(\Gamma(p, b, d)p^{-R} + {\rm o}(t)\right)\mu(V).
{\rm e}nd{align}
Lemma~\ref{lemma:components:conditional_calculation_Oest} and {\rm e}qref{EQ:components:epsilonA} together imply that
\begin{equation}
{\rm Prob}(U(R)_t\in V) = \left(\Gamma(p, b, d)p^{-R} + {\rm o}(t)\right)\mu(V) \to \mu(V)p^{-R}\Gamma(p, b, d)
{\rm e}nd{equation}
as $t$ tends to $0$ from the right, and so $U(R)_t$ is asymptotically uniformly distributed in $B(R)$.
{\rm e}nd{proof}
The symmetry between spatial and temporal scalings for the law for a $p$-adic Brownian motion suggests the following significant extension of Proposition~\ref{prop:components:epsilon}.
\begin{theorem}\label{theorem:components:epsilon}
For any positive real numbers $t$ and $\varepsilon$, there is an integer $M$ so that for any integer $N$ and for any Borel subset $V$ of $B(N)$,
\[N \ge M \quad \text{implies that} \quad \left|{\rm Prob}(U(N)_t\in V) - \mu(V)\Gamma(p, b, d)p^{-N}\right| < \varepsilon.\]
{\rm e}nd{theorem}
\begin{proof}
For any integer $R$ and any natural number $K$, take $G(R+K, d, t)$ and $G(R+K, d-1, t)$ to be given by {\rm e}qref{EQ:components:Gdef} so that
\begin{equation}\label{EQ:theorem:components:epsilonA}
\frac{G(R+K,d,t)}{G(R+K,d-1,t)} = \frac{\tilde{S}um_{j\leq -R-K} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}}{\tilde{S}um_{j \leq -R-K} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{(d-1)j}}.
{\rm e}nd{equation}
Reindex the sums in {\rm e}qref{EQ:theorem:components:epsilonA} to obtain the equalities
\begin{align}\label{EQ:theorem:components:epsilonB}
\frac{G(R+K,d,t)}{G(R+K,d-1,t)} & = \frac{\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma tp^{(j-K)b}} - {\rm e}^{-\tilde{S}igma tp^{(j-K+1)b}}\right)p^{d(j-K)}}{\tilde{S}um_{j \leq -R} \left({\rm e}^{-\tilde{S}igma tp^{(j-K)b}} - {\rm e}^{-\tilde{S}igma tp^{((j-K)+1)b}}\right)p^{(d-1)(j-K)}}\notag\\
& = \frac{\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma (tp^{-Kb})p^{jb}} - {\rm e}^{-\tilde{S}igma (tp^{-Kb})p^{(j+1)b}}\right)p^{dj}p^{-Kd}}{\tilde{S}um_{j \leq -R} \left({\rm e}^{-\tilde{S}igma (tp^{-Kb})p^{jb}} - {\rm e}^{-\tilde{S}igma (tp^{-Kb})p^{(j+1)b}}\right)p^{(d-1)j}p^{-K(d-1)}}\notag\\
& = p^{-K}\frac{G(R,d,tp^{-Kb})}{G(R,d-1,tp^{-Kb})}.
{\rm e}nd{align}
For any positive real number $t$, Lemma~\ref{lemma:components:conditional_calculation_Oest} implies that
\[
\frac{G(R,d,tp^{-Kb})}{G(R,d-1,tp^{-Kb})} = \left(\Gamma(p, b, d)p^{-R} + {\rm o}(tp^{-Kb})\right),
\]
and so {\rm e}qref{EQ:theorem:components:epsilonB} implies that
\begin{equation}\label{EQ:theorem:components:epsilonC}
\frac{G(R+K,d,t)}{G(R+K,d-1,t)} = p^{-K}\left(\Gamma(p, b, d)p^{-R} + {\rm o}(tp^{-Kb})\right).
{\rm e}nd{equation}
Lemma~\ref{lemma:components:conditional_calculation_Oest}, {\rm e}qref{EQ:uniformBorelDef}, and {\rm e}qref{EQ:theorem:components:epsilonC} together imply that for any $r$ that is less than $R+K$ and any $a$ in $B(R+K)$,
\begin{equation*}
{\rm Prob}(U(R+K)_t\in B(r,a)) = p^{r-K}\left(\Gamma(p, b, d)p^{-R} + {\rm o}(tp^{-Kb})\right).
{\rm e}nd{equation*}
Follow the proof of Proposition~\ref{prop:components:epsilon} to generalize to the case where $V$ is a Borel set that is not a ball and obtain for any Borel set $V$ in $B(R+K)$ the equality
\begin{equation*}
{\rm Prob}(U(R+K)_t\in V) = \mu(V)\Gamma(p, b, d)p^{-R-K} + \mu(V)p^{-K}{\rm o}(tp^{-Kb}).
{\rm e}nd{equation*}
Since $V$ is a subset of $B(R+K)$, $\mu(V)$ is no greater than $p^{R+K}$, and so Proposition~\ref{prop:components:epsilon} implies that \[\left|\mu(V)p^{-K}{\rm o}(tp^{-Kb})\right| \leq p^R\left|{\rm o}(tp^{-Kb})\right| < \varepsilon\] as long as $K$ is large enough.
{\rm e}nd{proof}
Note that Theorem~\ref{theorem:components:epsilon} implies that for any fixed positive $t$, $U(R)_t$ is asymptotically uniformly distributed in $R$ for large values of $R$.
\begin{corollary}\label{thm:components:anyt}
For any positive real number $t$, the components of $\vec{X}_t$ are not independent.
{\rm e}nd{corollary}
\begin{proof}
For any $i$ in $\{1, {\rm d}ots, d\}$ and for any positive real number $t$, Theorem~\ref{thm:components:marginal_distributions} implies that
\[
\lim_{r\to \infty} P^d\!\left(X_t^{(i)} \in B(r)\right) = \lim_{r\to \infty} P(X_t \in B(r)) = 1,
\]
and so for any positive real number $\varepsilon$ there is an integer $R$ so that for any $r$ that is greater than or equal to $R$,
\[
P^d\!\left(X_t^{(i)} \in B(r)\right) > 1 - \varepsilon.
\]
Since $\Gamma(p,b,d)$ is less than 1, Theorem~\ref{theorem:components:epsilon} implies that there is a natural number $K$ and a real number $e$ in $(-\varepsilon, \varepsilon)$ so that
\begin{align*}
P^d\!\left(X_t^{(i)} \in B(R+K) \Big\vert \vec{X}_{t,i}\in S_{d-1}(R+1+K)\right) &= \mu(B(R+K))p^{-R-K-1}\Gamma(p, b, d)+e\\
& = \tfrac{1}{p}\Gamma(p, b, d)+e < \tfrac{1}{p} +\varepsilon.
{\rm e}nd{align*}
As long as $\varepsilon$ is small enough, \[P^d\!\left(X_t^{(i)} \in B(R+K) \Big\vert \vec{X}_{t,i}\in S_{d-1}(R+1+K)\right) < P^d\!\left(X_t^{(i)} \in B(R+K)\right).\]
{\rm e}nd{proof}
\tilde{S}ection{First Exit Probabilities}\label{sec:exit}
Each component process of the product and max-norm process is a $p$-adic Brownian motion with diffusion constant $\tilde{S}igma$ and diffusion exponent $b$. The dependency of the components of these processes impacts the first exit times from balls. Namely, dimension has a large impact on the first exit time probabilities for the product process, but it has a rather small effect on these probabilities for the max-norm process.
\tilde{S}ubsection{Exit Times for the Components}\label{sec:exit:sub:1D}
Take $\alpha$ to be the positive real number that is given by \[\alpha = 1- \tfrac{p^b-1}{p^{b+1}-1}\] and take $X$ to be a one dimensional $p$-adic Brownian motion with diffusion constant $\tilde{S}igma$ and diffusion exponent $b$. For sake of clarity, slightly modify the notation in \cite{Weisbart:2021} and for any positive real number $T$ denote by $\vertiii{X}_T$ the quantity \[\vertiii{X}_T=\tilde{S}up_{0\leq t\leq T}|X_t|.\] The probability that a sample path for $X$ remains in $B(R)$ until time $T$, $P\!\left(\vertiii{X}_T\leq p^R\right)$, is a {\rm e}mph{survival probability} for $X$. The complement of this probability is a {\rm e}mph{first exit probability}, the probability that a sample path for $X$ has first exit from $B(R)$ before time $T$. Since every point in $B(R)$ is the center of $B(R)$, the exit times for $X$ from $B(R)$ do not depend on starting points, and so \[P\!\left(\vertiii{X}_{T+S}\leq p^R \big\vert{X}_{S}\leq p^R \right) = P\!\left(\vertiii{X}_{T}\leq p^R\right).\] The survival probabilities for $X$ are continuous from the right at $0$ and satisfy Cauchy's multiplicative functional equation, and so the first exit time for $X$ is an exponentially distributed random variable. Theorem~3.1 of \cite{Weisbart:2021} determines the parameter of this exponential distribution by establishing the equality
\begin{equation}\label{ExitProbability:Theorem_3.1}
P\!\left(\vertiii{X}_T \leq p^R\right) = {\rm e}^{-\tilde{S}igma \alpha T p^{-Rb}}.
{\rm e}nd{equation}
\tilde{S}ubsection{Exit Times for the Processes}\label{sec:exit:sub:exit}
Take $\alpha_d$ to be the quantity \[\alpha_d = 1- \tfrac{p^b-1}{p^{b+d}-1},\] and for any $\mathds Q_p^d$--valued stochastic process $\vec{Z}$ and any positive real number $T$, take $\vertiii{\vec{Z}}_T$ to be the quantity \[\vertiii{\vec{Z}}_T=\tilde{S}up_{0\leq t\leq T}\|\vec{Z}_t\|.\]
\begin{theorem}\label{theorem:exit:Prod}
For any integer $R$,
\begin{equation*}
\otimes^dP\!\left(\vertiii{\vec{Y}}_T \leq p^R\right) = {\rm e}^{-d\tilde{S}igma \alpha_1 T p^{-Rb}}.
{\rm e}nd{equation*}
{\rm e}nd{theorem}
\begin{proof}
Since $\vec{Y}_t$ lies outside $B_d(R)$ if and only if at least one component of $\vec{Y}_t$ lies outside $B_d(R)$, \[\left(\vertiii{\vec{Y}}_T\leq p^R\right) = \bigcap_{i\in\{1, {\rm d}ots, d\}}\left(\vertiii{Y^{(i)}}_T\leq p^R\right).\] Independence of the components of $\vec{Y}$ implies that \[\otimes^dP\!\left(\vertiii{\vec{Y}}_T\leq p^R\right) = \prod_{i\in\{1, {\rm d}ots, d\}}P\!\left(\vertiii{Y^{(i)}}_T\leq p^R\right) = {\rm e}^{-d\alpha\tilde{S}igma T p^{-Rb}}.\]
{\rm e}nd{proof}
With only minor modification, the arguments of \cite{Weisbart:2021} extend to the more general max-norm setting and determine the survival probabilities for the max-norm process $\vec{X}$. For this reason, the proof below for Theorem~\ref{theorem:exit:max-norm} will omit certain details that the proof of Theorem~3.1 in \cite{Weisbart:2021} includes.
\begin{theorem}\label{theorem:exit:max-norm}
For any integer $R$, \[P^d\!\left(\vertiii{\vec{X}}_T\leq p^R\right) = {\rm e}^{-\tilde{S}igma\alpha_d Tp^{-Rb}}.\]
{\rm e}nd{theorem}
\begin{proof}
Take $U$ in {\rm e}qref{EQ:components:probU} to be $B(R)$ in order to obtain the equality
\begin{align}\label{eq:components:intoverball}
P^d\!\left(\vec{X}_t\in B_d(R)\right) & = \tilde{S}um_{j\in \mathds Z} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}\int_{B_d(R)} {\mathds 1}_{B_d(-j)}(\vec{x})\,{\rm d}\vec{x}\notag\\
& = p^{dR}\tilde{S}um_{j \leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj} + \tilde{S}um_{j>-R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right).
{\rm e}nd{align}
The summands of the second sum in {\rm e}qref{eq:components:intoverball} telescope, and so
\begin{equation}\label{eq:components:intoverballSimp}
P^d\!\left(\vec{X}_t\in B_d(R)\right) = {\rm e}^{-\tilde{S}igma tp^{(-R+1)b}} + p^{dR}\tilde{S}um_{j\leq -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}.
{\rm e}nd{equation}
For any natural numbers $N$ and $j$, where $j$ is less than or equal to $N$, take $t_j$ to be given by \[t_j = \tfrac{jT}{N}.\] Since the max-norm on $\mathds Q_p^d$ satisfies the ultra-metric inequality,
\begin{align}\label{sec3:maxtoprodequality}&P^d\!\left(\max_{t_j} \big(\|\vec{X}_{t_1}\|, {\rm d}ots, \|\vec{X}_{t_n}\|\big) \leq p^R\right)\notag\\&\hspace{.5in} = P^d\!\left(\max_{t_j} \big(\|\vec{X}_{t_1}\|, \|\vec{X}_{t_1}- \vec{X}_{t_2}\|, {\rm d}ots, \|\vec{X}_{t_n}- \vec{X}_{t_{n-1}}\|\big) \leq p^R\right).{\rm e}nd{align} Take the random variable $X_0$ to be the zero function. The independence of the set of increments $\big\{\vec{X}_{t_j} - \vec{X}_{t_{j-1}}\colon i\in\{1, {\rm d}ots, N\}\big\}$ implies that \begin{align}\label{eq:exit:ProdInc}P^d\!\left(\max_{t_j} \big(\|\vec{X}_{t_1}\|, {\rm d}ots, \|\vec{X}_{t_n}\|\big) \leq p^R\right) &= P^d\!\left(\|\vec{X}_{t_1}\| \leq p^R\cap {\rm d}ots\cap \|\vec{X}_{t_n}\| \leq p^R\right)\notag\\& = \prod_{1\leq j\leq N}P^d\!\left(\|\vec{X}_{t_i}-\vec{X}_{t_{i-1}}\| \leq p^R\right)^N.{\rm e}nd{align} The increments are identically distributed, and so
\begin{equation}
P^d\!\left(\max_{t_j} \big(\|\vec{X}_{t_1}\|, {\rm d}ots, \|\vec{X}_{t_n}\|\big) \leq p^R\right)= P^d\!\left(\|\vec{X}_{\frac{T}{N}}\| \leq p^R\right)^N.
{\rm e}nd{equation}
Take $B$ to be the twice continuously differentiable function that is defined for any $t$ in $[0,\infty)$ by \[B(t) = P^d\!\left(\|\vec{X}_t\| \leq p^R\right).\] The twice continuous differentiability of $B$ implies that the equality \[B\big(\tfrac{T}{n}\big) = 1 + \tfrac{B^\prime(0)T}{n} + {\rm O}\big(\tfrac{1}{n^2}\big),\quad \text{hence} \quad B\big(\tfrac{T}{n}\big)^n = {\rm e}^{TB^\prime(0)}.\] The right continuity of the sample paths of $\vec{X}$ implies that
\begin{align}\label{eq:exit:EqforSurvivalProb}P^d\!\left(\vertiii{\vec{X}}_T\leq p^R\right) &= \lim_{n\to \infty}P^d\!\left(\max_{j} \big(\|\vec{X}_{\frac{T}{n}}\|, \|\vec{X}_{\frac{2T}{n}}\|, {\rm d}ots, \|\vec{X}_{\frac{jT}{n}}\|, {\rm d}ots, \|\vec{X}_{T}\|\big) \leq p^R\right)\notag\\ &= \lim_{n\to \infty} B\big(\tfrac{T}{n}\big)^n = {\rm e}^{TB^\prime(0)}.{\rm e}nd{align}
Rewrite {\rm e}qref{eq:components:intoverballSimp} to obtain the equality
\begin{equation}\label{eq:exit:intoverballSimp}
B(t) = {\rm e}^{-\tilde{S}igma tp^{-Rb}} + p^{dR}\tilde{S}um_{j < -R} \left({\rm e}^{-\tilde{S}igma tp^{jb}} - {\rm e}^{-\tilde{S}igma tp^{(j+1)b}}\right)p^{dj}.
{\rm e}nd{equation}
Differentiate both sides of {\rm e}qref{eq:exit:intoverballSimp} to obtain the equality
\begin{equation}\label{eq:exit:Bprime}
B^\prime(0) = -\tilde{S}igma p^{-Rb} + \tilde{S}igma \tilde{S}um_{j\leq -R-1}p^{d(R+j)}\Big(p^{(j+1)b} - p^{jb}\Big).
{\rm e}nd{equation}
Simplify the righthand side of {\rm e}qref{eq:exit:Bprime} to obtain the equality
\begin{equation}\label{eq:exit:Bprimeat0}
B^\prime(0) = -\tilde{S}igma \alpha_d p^{-Rb}
{\rm e}nd{equation} which, together with {\rm e}qref{eq:exit:EqforSurvivalProb}, implies that \[P^d\!\left(\vertiii{\vec{X}}_T\leq p^R\right) = {\rm e}^{-\tilde{S}igma \alpha_d Tp^{-Rb}}.\]
{\rm e}nd{proof}
Together with Theorem~\ref{theorem:exit:max-norm}, the equality
\begin{equation*}
\lim_{d\to \infty} \alpha_d = \lim_{d\to \infty} \left(1- \tfrac{p^b-1}{p^{b+d}-1}\right) = 1
{\rm e}nd{equation*}
implies that for any integer $R$,
\begin{equation}\label{EQ:exit:normLim}
\lim_{d\to \infty} P^d\!\left(\vertiii{\vec{X}}_T\leq p^R\right) = {\rm e}^{-\tilde{S}igma Tp^{-Rb}}.
{\rm e}nd{equation}
In contrast, Theorem~\ref{theorem:exit:Prod} implies that
\begin{equation}\label{EQ:exit:ProdLim}
\lim_{d\to \infty} \otimes^dP\!\left(\vertiii{\vec{Y}}_T\leq p^R\right) = \lim_{d\to \infty}{\rm e}^{-d\tilde{S}igma \alpha_1Tp^{-Rb}} = 0.
{\rm e}nd{equation}
This marked contrast between {\rm e}qref{EQ:exit:normLim} and {\rm e}qref{EQ:exit:ProdLim} demonstrates that exit probabilities from a fixed ball depend only to a small degree on dimension for the max-norm process, but are very sensitive to changes in dimension for the product process.
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{\rm e}nd{document} |
\begin{document}
\begin{abstract} We provide a local classification of
self-dual Einstein Riemannian
four manifolds admitting a positively oriented Hermitian structure and
characterize those
which carry a hyperhermitian, non-hyperk{\"a}hler
structure compatible with the negative
orientation. We finally show that self-dual Einstein 4-manifolds obtained as
quaternionic quotients of the Wolf spaces ${\mathbb H}P^2$, ${\mathbb H}H^2$,
$SU(4)/S(U(2)U(2))$, and $SU(2,2)/S(U(2)U(2))$ are always Hermitian.
\noindent
2000 {\it Mathematics Subject Classification}. Primary 53B35, 53C55\\
{\it Keywords:} Einstein metrics; complex structures; hypercomplex structures;
quaternionic K{\"a}hler manifolds.
\end{abstract}
\maketitle
\section*{Introduction}
The main goal of this paper is to give a local description of
all self-dual Einstein
$4$-manifolds $(M,g)$ which admit a positive Hermitian structure.
It follows from a (weak) Riemannian version of the Goldberg-Sachs theorem
\cite{PB,Bo4,Nu,AG} that a Riemannian Einstein $4$-manifold locally admits a
positive Hermitian structure if and only if the {\it self-dual Weyl
tensor} $W^+$ is {\it
degenerate}.
This means that
at any point of $M$ at least two of the three eigenvalues of
$W^+$ coincide, when $W ^+$ is viewed as a symmetric traceless
operator acting on
the three-dimensional space of self-dual 2-forms.
Riemannian Einstein 4-manifolds with
degenerate self-dual Weyl tensor have been much studied by
A. Derdzi\'nski; we here recall the following facts taken from \cite{De}:
\begin{enumerate}
\item[{\rm (i)}] $W^+$ either vanishes identically or else has no
zero, i.e. has exactly two distinct eigenvalues at any point (one of them, say $\lambdambda$,
is simple; the
other one is
of multiplicity $2$, and therefore equals $- \frac{\lambdambda}{2}$ as
$W ^+$ is trace-free).
\item[{\rm (ii)}] In the latter case, the K{\"a}hler form of the
Hermitian structure $J$ is a generator of the simple eigenspace
of $W^+$ --- in particular, the conjugacy class of $J$ is uniquely
defined by the metric ---
and the conformal
metric ${\bar g} = |W^+|^{\frac{2}{3}}g$ is {\it K{\"a}hler}
with respect to $J$.
\item[{\rm (iii)}] If, moreover, $g$ is assumed to be {\it
self-dual} --- meaning that the {\it anti-self-dual} Weyl tensor,
$W ^-$, vanishes identically ---
the simple eigenvalue $\lambdambda$ of $W ^+$ is constant (equivalently, the norm $|W ^+|$
is constant) if and only if $(M, g)$ is
locally symmetric, i.e., a real
or complex space form.
\end{enumerate}
We then have a natural bijection between the following three classes of
Riemannian $4$-manifolds (see Lemma \ref{de-ga} below):
\begin{enumerate}
\item Self-dual Einstein $4$-manifolds with degenerate self-dual Weyl
tensor $W ^+$, such that $|W ^+|$ is not constant.
\item Self-dual Einstein Hermitian $4$-manifolds which are neither
conformally-flat nor K{\"a}hler.
\item Self-dual K{\"a}hler manifolds with nowhere vanishing and
non-constant scalar curvature.
\end{enumerate}
In this correspondence, the Riemannian metrics are defined on the
same manifold and
belong to the same conformal class.
Observe that each class is defined by an algebraic closed condition (the
vanishing of some tensors) and an open genericity condition.
Since the compact case is completely understood, see e.g.
\cite{BYC} or \cite{De,Bu,It,Bo4, ADM} for a classification,
the paper will concentrate on the
local situation.
The first known examples of (non-locally-symmetric)
self-dual Einstein Hermitian metrics have been metrics of
cohomogeneity one
under the isometric action of a four-dimensional Lie
group. Einstein metrics which are of cohomogeneity one under the action of a
four-dimensional Lie group are automatically Hermitian \cite{De}. By
using this remark, A. Derdzi\'nski constructed \cite{de}
a family of cohomogeneity-one
self-dual Einstein Hermitian metrics under the action of
${\mathbb R}\times {\rm Isom}({\Bbb R}^2)$, U(1,1) and U(2);
this family actually includes (in a rather implicit way)
the well-known {\it Pedersen-LeBrun metrics}
\cite{P,Le} which play an important r{o}le in
Section 3 of this paper.
It is {\it a priori} far from obvious that there are any
other examples of self-dual Einstein Hermitian 4-manifolds, since the conditions of being self-dual, Einstein and
Hermitian constitute an over-determined second order PDE system for the metric
$g$. We show however that there
are actually many other examples; more precisely, we
classify all local solutions of this system
and provide a simple, {\it explicit} (local) Ansatz for self-dual Einstein
Hermitian 4-manifolds (see Theorem \ref{th3} and Lemma \ref{integrate} for a precise
statement).
An amazing, a priori unexpected fact comes out from the argument and
explains a posteriori the integrability of the above mentioned
Frobenius system~: {\it
all self-dual Einstein Hermitian
metrics admit
a local isometric action
of ${\Bbb R}^2$ with two-dimensional orbits} (Theorem \ref{th3} and Remark 3).
In particular, these metrics locally fall into the more general
context of
self-dual metrics with torus action considered in \cite{joyce}
and, more recently, in \cite{Ca0,Ca1} (see Remark \ref{rem3} (ii)).
It turns out that this property of having more
(local)
symmetries than expected is actually shared by K{\"a}hler metrics with
vanishing Bochner tensor in all
dimensions, as shown in the recent work of R. Bryant
\cite{Br} (see \cite{Br} for precise statements).
Since the Bochner tensor of a K{\"a}hler
manifold of real dimension four is the same as the anti-self-dual tensor $W ^-$ --- so that
Bochner-flat K{\"a}hler metrics are a natural generalization of
self-dual K{\"a}hler metrics in higher dimensions --- by using the
correspondence given by Lemma 2, Bryant's work provides an
alternative approach to our classification in Section 2.
Moreover, Bryant's work includes a large section devoted to complete
metrics; in particular, by specifying his general techniques to dimension four,
he has been able (again via Lemma 2) to give {\it complete} examples
of self-dual Einstein
Hermitian 4-manifolds, corresponding to the generic case considered
in Theorem \ref{th3}.
The paper is organized as follows:
Section 1 displays the background material; the notation closely
follows our previous work \cite{AG} --- with the exception of the Lee
form, whose definition here is slightly different --- and we send
back the reader to
\cite{AG} for more details and references.
Section 2.1 provides a complete description of (locally defined)
cohomogeneity-one
self-dual Einstein Hermitian metrics (Theorem \ref{th2}).
It turns out that they all
admit
a local isometric action (with three-dimensional orbits)
of certain four-dimensional Lie groups,
such that the metrics
can be put in a {\it diagonal} form; in other
words, they are
{\it biaxial diagonal Bianchi metrics of type A}, see {e.g.} \cite{Tod1,CP}.
Theorem \ref{th2} relies on the fact
that
every (non-locally-symmetric)
self-dual Einstein Hermitian metric $(g,J)$
has a distinguished non-trivial Killing field, namely
$K= J{\rm grad}_g(|W^+|^{-\frac{1}{3}})$, \cite{De}. Then, the
Jones-Tod reduction with respect to $K$ \cite{Tod2} provides
a three-dimensional space of {\it constant curvature}.
The diagonal form of the metrics follows from \cite{Tod2} and
\cite{Tod1} (a unified presentation for these
cohomogeneity-one metrics also appears in \cite{CP}).
To the best of our knowledge, apart from these metrics no other examples of
self-dual Einstein Hermitian metrics were known in the literature (see
however Section 4).
Section 2.2 is devoted to the generic case, when the
metric is neither locally-symmetric nor of cohomogeneity one.
Our approach is similar to Armstrong's one in \cite{Arm}:
When considering the Einstein condition alone,
the Riemannian Goldberg-Sachs theorem together
with Derdzi\'nski's results reported above imply a number of
relations for
the 4-jet of an Einstein Hermitian metric
(Sec. 2.1, Proposition \ref{prop2}); these happen to be the
only obstructions for prolonging the 3-jet solutions of the problem to
4-jet and no further obstructions appear when
reducing the equations for non-K{\"a}hler,
non-anti-self-dual Hermitian
Einstein
4-manifolds to a (simple) perturbated
${\rm SU}(\infty)$-Toda field equation \cite{Arm, Pl-Pr}.
If, moreover, we insist that $g$ be also {\it self-dual}, we find
further relations for the 5-jet of the metric and we show
that they have the form of an integrable closed
Frobenius system of PDE's for
the parameter space of the 4-jet of the metric. We thus prove the
local existence of non-locally symmetric and non-cohomogeneity-one
self-dual Einstein Hermitian
metrics (Theorem \ref{th3}). It turns out that this
Frobenius system can be explicitly integrated
(Lemma 3). We thus obtain a uniform local description
for {\it all}
self-dual Einstein Hermitian
metrics in an explicit way.
Section 3 is devoted to the subclass of self-dual Einstein Hermitian
metrics which admit a compatible, non-closed, anti-self-dual hypercomplex
structure. This is the same, locally, as the class of self-dual Einstein Hermitian
metrics which admit a non-closed Einstein-Weyl connection (see
Section 1.2). From this viewpoint, it is a particular case of
four-dimensional conformal metrics which admit two distinct
Einstein-Weyl connections. In our case, one of them is the
Levi-Civita connection of the Einstein metric, whereas the other one
is {\it non-closed}, hence, because of Proposition \ref{prop4},
attached to a non-closed hyperhermitian structure.
(Recall that a conformal 4-manifold admitting two distinct
{\it closed} Einstein-Weyl structures is necessarily conformally flat
(folklore), and that, conversely,
every conformally flat 4-manifold only admits closed Einstein-Weyl
structures \cite{ET}, see also Proposition \ref{prop4} and
Corollary \ref{cor1} below).
It turns out that self-dual Einstein Hermitian
metrics which admit a compatible, non-closed, anti-self-dual hypercomplex
structure, actually admit a second one and thus fall
in the {\it bi-hypercomplex} situation described by Madsen in
\cite{Mad}; in particular, these metrics admit a local action of
${\rm U(2)}$, with three-dimensional orbits, and are diagonal Bianchi XI
metrics, see Theorem \ref{th1} below.
Notice that a general description of
(anti-self-dual) metrics admitting two distinct compatible hypercomplex
structures appears in \cite{Ca2}, see also \cite{BCM}, whereas a
family of self-dual Einstein metrics with compatible non-closed hyperhermitian
structures, parameterized by
holomorphic functions of one variable,
has been constructed in \cite{CT}.
In Section 4, we show that all anti-self-dual,
Einstein four dimensional {\it orbifolds} obtained by quaternionic
K{\"a}hler reduction from the eight dimensional quaternionic K{\"a}hler
Wolf spaces ${\mathbb H}{P}^2$, $SU(4)/S(U(2)U(2))$
and their non-compact duals
(see \cite{galicki1,galicki2} and \cite{G-L}) are
actually Hermitian with respect to the opposite orientation,
hence locally isomorphic to metrics described in Section 2.
These orbifolds include the {\it weighted projective planes}
${\mathbb C} P^{[p_1,p_2,p_3]}$ for
integers $0<p_1\le p_2\le p_3$ satisfying $p_3<p_1+p_2$,
cf. \cite[Sec. 4]{G-L}. On these orbifolds,
Bryant has constructed Bochner-flat
K{\"a}hler metrics with everywhere positive scalar curvature,
hence also self-dual, Einstein Hermitian metrics
according to Lemma 2 below, \cite[Sec. 4.3]{Br};
in view of the results of Section 2,
Galicki-Lawson's and Bryant's metrics agree locally,
but the issue as to whether they agree globally remains unclear.
\noindent
{\bf Acknowledgments.} The first-named author thanks the
Dipartimento di Matematica, Universit{\`a} di Rome Tre and the
Max-Planck-Institut in Bonn
for hospitality during the preparation
of this paper. He would like to express his gratitude to J. Armstrong
for explaining his approach to Einstein Hermitian metrics
and many illuminating discussions.
The authors warmly
thank S. Salamon for being an initiator of this work and for gently
sharing his expertise, and
C. LeBrun, C. Boyer, K. Galicki, whose comments
are at the origin of the last section of the paper.
It is also a pleasure for us to thank N. Hitchin, S. Marchiafava,
H. Pedersen, P. Piccinni, M. Pontecorvo and K.P. Tod for their interest
and stimulating conversations, and
R. Bryant for his interest and remarks.
Finally, a special aknowledgment is due to D. Calderbank for his friendly assistance
in carefully reading the manuscript, checking computations, correcting mistakes and suggesting
improvements; he in particular decisively contributed to improving
the paper by pointing out a mistake in a former
version and thus revealing the rational
character of the metrics described in Section 2.2.
\section{Einstein metrics, Hermitian structures and Einstein-Weyl geometry
in dimension 4}
\subsection{Einstein metrics and compatible Hermitian structures}
In the whole paper $(M, g)$ denotes an oriented Riemannian
four-dimensional manifold.
A specific feature of the four-dimensional Riemannian geometry is
the splitting
\begin{equation} \lambdabel{split1} AM = A ^+M \oplus A ^-M, \end{equation}
of the Lie algebra bundle, $AM$, of skew-symmetric endomorphisms of the
tangent bundle, $TM$, into the direct sum of two Lie algebra subbundles, $A
^{\pm}M$, derived from the Lie algebra splitting $\mathfrak{so}
(4) = \mathfrak{so}
(3) \oplus \mathfrak{so}(3)$ of the orthogonal Lie algebra $\mathfrak{so}
(4)$ into the direct sum of two copies of $\mathfrak{so}
(3)$.
A similar decomposition occurs for the bundle $\Lambdambda ^2M$ of
$2$-forms
\begin{equation} \lambdabel{split2} \Lambdambda ^2 M = \Lambdambda ^+M \oplus
\Lambdambda ^-M, \end{equation}
given by the spectral decomposition of
the Hodge-star operator, $*$, whose restriction to $\Lambdambda ^2 M$ is an
involution; here,
$\Lambdambda ^{\pm} M$ is the eigen-subbundle for the eigenvalue
$\pm$ of $*$.
Both decompositions are actually determined by the
conformal metric $[g]$ only. When $g$ is fixed,
$\Lambdambda ^2 M$ is identified to $AM$ by
setting: $\psi (X, Y) = g(\Psi (X), Y)$, for any $\Psi$ in $AM$ and any
vector fields $X, Y$; then, we can arrange signs in (\ref{split1}) so
that (\ref{split1}) and
(\ref{split2}) are identified to each other. A similar
decomposition and a similar identification
occur for the bundle $\Lambdambda ^2 (TM)$ of bivectors.
Sections of $\Lambdambda ^+M$, resp. $\Lambdambda ^-M$, are called {\it
self-dual}, resp. {\it anti-self-dual}, and similarly for
sections of $AM$ or $\Lambdambda ^2 (TM)$.
In the sequel, the vector bundles $AM$, $\Lambdambda ^2 M$ and
$\Lambdambda ^2 (TM)$ will be freely identified to
each other; similarly, the cotangent bundle $T ^*M$ will be freely
identified to $TM$; when no confusion can arise, the inner product determined by $g$ will be
simply
denoted by $(\cdot, \cdot)$; we adopt the convention that $(\Psi _1 , \Psi
_2) = - \frac{1}{2} {\rm tr} \, (\Psi _1 \circ \Psi _2)$, for sections
of $AM$, and the corresponding convention for $\Lambdambda ^2 M$ and $\Lambdambda ^2 (TM)$.
The Riemannian curvature, $R$, is defined by
$R _{X, Y} = D ^g _{[X, Y]} - [D ^g _X, D ^g _Y],$
where $D ^g$ denotes the Levi-Civita connection of $g$; $R$ is thus a
$AM$-values $2$-form, but will be rather considered as a section of
the bundle $S^2 (\Lambdambda ^2M)$ of symmetric
endomorphisms of $\Lambdambda ^2 M$.
The Weyl tensor, $W$, commutes with $*$ and, accordingly, splits as $W = W ^+ + W ^-$, where $W ^{\pm} = \frac{1}{2} (W \pm W \circ *)$; $W ^+$
is called the {\it self-dual Weyl tensor}; it acts trivially on
$\Lambdambda ^- M$ and will be considered in the sequel as a
field of
(symmetric, trace-free) endomorphisms of $\Lambdambda ^+ M$; similarly,
the {\it anti-self-dual Weyl tensor} $W ^-$ will be considered as a
field of endomorphismes of $\Lambdambda ^ - M$.
The Ricci tensor, ${\rm Ric}$, is the symmetric bilinear
form defined by ${\rm Ric} (X, Y) = {\rm tr} \, \{ Z \to R _{X, Z} Y \}$;
alternatively, ${\rm Ric} (X, Y) =
\sum _{i = 1}^4 (R _{X, e _i} Y, e _i)$ for any $g$-orthonormal
basis $\{ e _i \}$. We then have ${\rm Ric} = \frac{s}{4} \, g + {\rm Ric} _0$,
where $s$ is the scalar curvature (= the trace of ${\rm Ric}$ with respect
to $g$) and ${\rm Ric} _0$ is the {\it trace-free Ricci tensor}. The latter can be
made into a section of $S ^2 (\Lambdambda ^2 M)$, then denoted by
$\widetilde{{\rm Ric} _0}$, by putting
$ \widetilde{{\rm Ric} _0} (X \wedge Y) = {\rm Ric} _0 (X) \wedge Y + X
\wedge {\rm Ric} _0 (Y). $
It is readily checked that $\widetilde{{\rm Ric} _0}$ satisfies the first Bianchi
identity, i.e. $\widetilde{{\rm Ric} _0}$ is a tensor of the same kind as $R$ itself, as well as $W
^+$ and $W ^-$; moreover, $ \widetilde{{\rm Ric} _0}$ anti-commutes with $*$,
so that it can be viewed as a field of homomorphisms from $\Lambdambda ^+
M$ into $\Lambdambda ^- M$, or from $\Lambdambda ^-
M$ into $\Lambdambda ^+ M$ (adjoint to each other);
we eventually get the well-known Singer-Thorpe decomposition of $R$, see
e.g. \cite{besse}:
\begin{equation}\lambdabel{SO(4)} R = \frac{s}{12} \, {\rm Id} _{| \Lambdambda ^2 M} +
\frac{1}{2} \widetilde{{\rm Ric}} _0 +
W ^+ + W ^-, \end{equation}
or, in a more pictorial way
\begin{equation*} R = \begin{pmatrix} & W ^+ + \frac{s}{12} \, {\rm
Id} _{| \Lambdambda ^+ M} & \frac{1}{2} \widetilde{{\rm Ric} _0} _{| \Lambdambda ^- M}
\\ \\ & \frac{1}{2} \widetilde{{\rm Ric} _0} _{| \Lambdambda ^+ M} & W ^- + \frac{s}{12} \, {\rm
Id} _{| \Lambdambda ^- M} \end{pmatrix} \end{equation*}
The metric $g$ is {\it Einstein} if ${\rm Ric} _0 = 0$ (equivalently,
$g$ is Einstein if $R$
commutes with $*$).
The metric $g$ (or rather the conformal class $[g]$) is {\it
self-dual} if $W ^- = 0$; {\it anti-self-dual} if $W ^+ = 0$.
An {\it almost-complex structure} $J$ is a field of automorphisms of $TM$ of
square $ - {\rm Id}|_{TM}$. An {\it integrable} almost-complex structure is simply called a
{\it complex structure}.
In this paper,
the metric $g$, or its conformal class $[g]$, is fixed and we only consider
$g$-orthogonal almost-complex structures, i.e. almost-complex
structure $J$ satisfying the identity $g (JX, JY) = g (X, Y)$, so
that the pair $(g, J)$ is an {\it almost-Hermitian structure}; then, the associated bilinear form,
$F$, defined by $F (X, Y) = g(JX, Y)$ is a $2$-form, called the {\it
K{\"a}hler form}.
The pair $(g, J)$ is {\it Hermitian} if $J$ is integrable; {\it
K{\"a}hler} if $J$ is parallel with respect to the Levi-Civita
connection $D ^g$; if $(g, J)$ is K{\"a}hler then $J$ is integrable and $F$
is closed; conversely, these two conditions together imply that $(g, J)$ is
K{\"a}hler.
A $g$-compatible almost-complex structure $J$ is either a section of
$A ^+M$ or a section of $A ^-M$; it is called {\it positive}, or
{\it self-dual}, in
the former case, {\it negative}, or {\it anti-self-dual} in the latter case. Alternatively, the
K{\"a}hler form {\rm is} either self-dual or
anti-self-dual.
Conversely, any
section $\Psi$ of $A ^+M$, resp. $A ^- M$, such that $|\Psi| ^2 = 2$,
is a positive, resp. negative, $g$-orthogonal almost-complex
structure. It follows that any non-vanishing section, $\Psi$, of $A
^+M$ --- if any --- determines a (positive) almost-complex structure
$J$, defined by $J = \sqrt{2} \frac{\Psi}{|\Psi|}$ (similarly for
non-vanishing sections of $A ^- M$).
Whereas the existence of a (positive) $g$-orthogonal almost-complex
structure is a purely topological problem, the similar issue for
{\it complex} structures heavily depends on the
geometry of $g$, and this dependence is essentially measured by the self-dual Weyl
tensor $W ^+$.
This assertion can be made more precise in the following way. We denote by $\lambdambda _ +
\geq \lambdambda _0 \geq \lambdambda _-$ the eigenvalues of $W ^+$ at some
point, $x$, of $M$, and we assume that $W ^+$ does not vanish at $x$;
equivalently, since $W ^+$ is trace-free, we assume that $\lambdambda _+ -
\lambdambda _-$ is positive; we denote by $F
_{+}$ an eigenform of $W ^+$ with respect to $\lambdambda _+$,
normalized by $|F _+| ^2 = 2$; similarly, $F _-$ denotes an eigenform
of $W ^+$ for $\lambdambda _-$, again normalized by $|F _-| ^2 = 2$; the
{\it roots}, $P$, of $W ^+$ at $x$ are then defined by
$P = \frac{(\lambdambda_+ - \lambdambda_0 )^{\frac{1}{2}}}{(\lambdambda_+ -
\lambdambda_- )^{\frac{1}{2}}}F_- +
\frac{(\lambdambda_0 - \lambdambda_- )^{\frac{1}{2}}}{(\lambdambda_+ -
\lambdambda_-)^{\frac{1}{2}}}F_+;$
it is easily checked that this expression actually determine {\it
two}
distinct pairs of opposite roots in the generic case, when the
eigenvalues are all distinct, and {\it one} pair in the degenerate case,
when $\lambdambda _0$ is
equal to either $\lambdambda _+$ or $\lambdambda _-$.
It is a basic fact that when $J$ is a positive, $g$-orthogonal
{\it complex} structure defined on $M$, the value of $J$ at any point $x$
where $W ^+$ does not vanish must be equal to a root of $W ^+$ at that
point. This means that on the open subset of $M$ where $W ^+$ does not
vanish, the conjugacy class of a positive, $g$-orthogonal
complex structure --- if any --- is almost entirely determined by $g$
(in fact by $[g]$), with at most a $2$-fold ambiguity.
On the other hand, it is an easy
consequence of the integrability theorem in \cite{AHS} that $A ^+M$
can be locally trivialized by integrable (positive, $g$-orthogonal)
almost-complex structures if and only if $[g]$ is anti-self-dual.
In the sequel, $W ^+$ will be called {\it degenerate} at some point $x$ if it has at
most two distinct eigenvalues at that point. The terms {\it
anti-self-dual} and {\it non-anti-self-dual} will be abbreviated as
ASD and non-ASD respectively.
For a given non-ASD metric $g$ it is a subtle question to decide whether
the roots of $W ^+$ actually provide complex structures (this is of
course not true in general). The situation is quite different if $g$
is Einstein. It is then settled by the following (weak) Riemannian version
of the Goldberg-Sachs theorem, cf. \cite{De,PB,Nu,AG}:
\begin{prop}\lambdabel{prop1} Let $(M,g)$ be an oriented Einstein
4-manifold; then the following three conditions are equivalent:
\begin{enumerate}
\item[(i)] $W^+$ is everywhere degenerate;
\item[(ii)] there exists a positive $g$-orthogonal complex structure in a neighbourhood
of each point of $M$;
\item[(iii)] $(M,g)$ is either ASD or $W^+$ has
two distinct eigenvalues at each point.
\end{enumerate}
\end{prop}
A consequence of this proposition is that the self-dual Weyl tensor $W
^+$ of a non-ASD Einstein Hermitian $4$-manifold
nowhere vanishes and has two distinct eigenvalues at any point, one
simple, the other one of multiplicity $2$; moreover, the K{\"a}hler form $F$ is an
eigenform of $W ^+$ for the simple eigenvalue. Conversely,
for any oriented, Einstein $4$-manifold
whose $W ^+$ has two distinct eigenvalues,
the generator of the simple eigenspace of
$W^+$ determines a (positive) Hermitian structure.
For any positive $g$-orthogonal almost-complex structure $J$, $A ^+ M$
splits as follows:
\begin{equation} \lambdabel{splitJ1} A ^+ M = {\mathbb R}\cdot {J} \oplus A ^{+, 0} M,
\end{equation}
where ${\mathbb R}\cdot{J}$ is the trivial subbundle generated by
$J$ and $A ^{+, 0} M$ is the orthogonal complement (equivalently, $A
^{+, 0} M$ is the subbundle of elements of $A ^+ M$ that anticommute with $J$); $A
^{+, 0} M$ is a rank $2$ vector bundle and will be also considered as
a complex line bundle by putting $J \Phi = J \circ \Phi$. We have the
corresponding decomposition
\begin{equation} \lambdabel{splitJ2} \Lambdambda ^+ M = {\mathbb R}\cdot{F} \oplus \Lambdambda
^{+, 0} M,
\end{equation}
where $\Lambdambda
^{+, 0} M$ is the subbundle of $J$-anti-invariant $2$-forms,
i.e. $2$-forms satisfying $\phi (JX, JY) = - \phi (X, Y)$; again, $\Lambdambda
^{+, 0} M$ is considered as a complex line bundle by putting $(J
\phi) (X, Y) = - \phi (JX, Y) = - \phi (X, JY)$. As complex line
bundles, both $A ^{+, 0} M$ and $\Lambdambda
^{+, 0} M$ are identified to the {\it anti-canonical bundle} $K
^{-1} M = \Lambdambda ^{0, 2} M$ of the (almost-complex) manifold $(M,
J)$.
For an Einstein, Hermitian $4$-manifold, the action of $W ^+$
preserves the decompositions
(\ref{splitJ1}) and (\ref{splitJ2}).
The {\it Lee form} of an almost-Hermitian structure
$(g, J)$ is the real $1$-form, $\theta$, defined by
\begin{equation} \lambdabel{lee} {\rm d} F = - 2 \theta \wedge F; \end{equation}
equivalently, $\theta = - \frac{1}{2} J \, \delta F$, where $\delta$
denotes the co-differential with respect to $g$ (here, and henceforth,
the action of $J$ on $1$-forms is defined via the identification $T
^*M \simeq TM$ given by the metric; we thus have $(J \alpha ) (X) =
- \alpha (JX)$, for any $1$-form $\alpha$). The reason
for the choice of the factor
$-2$ in (\ref{lee}) will be clear in the next subsection (notice that a
different
normalization is used in our previous work \cite{AG}).
When $(g, J)$ is Hermitian, it is K{\"a}hler if and only if $\theta$
vanishes identically; it is conformally K{\"a}hler if and only
if $\theta$ is exact, i.e. $\theta = -
{\rm d} \ln{f}$ for a positive smooth real function $f$ (then, $J$
is K{\"a}hler with respect to the
conformal metric $g' = f ^{-2} \, g)$; it is locally conformally
K{\"a}hler ---
lcK for short --- if and only if $\theta$ is closed, hence locally of
the above type.
The Lee form clearly satisfies $({\rm d} \theta, F) = 0$; this means
that the self-dual part, ${\rm d}
\theta ^+$, of ${\rm d}
\theta $ is a section of the rank $2$ subbundle,
$\Lambdambda ^{+,0} M$.
In the Hermitian case, ${\rm d}
\theta ^+$ is an eigenform of $W ^+$ for the mid-eigenvalue $\lambdambda
_0$; moreover, $\lambdambda _0 = - \frac{K{\"a}hler ppa}{12}$, where $K{\"a}hler ppa$ is the {\it
conformal scalar curvature}, of which a more direct definition is
given in the next subsection; ${K{\"a}hler ppa}$ is related to the (Riemannian) scalar curvature $s$ by
\begin{equation} K{\"a}hler ppa = s +
6 \, (\delta \theta - |\theta| ^2), \end{equation}
and we also have
\begin{equation} K{\"a}hler ppa = 3 \, (W ^+ (F), F), \end{equation}
see \cite{Va2,Ga1}. Notice that, in the Hermitian case, the mid-eigenvalue
$\lambdambda _0$ of $W ^+$ is always a {\it smooth} function (this,
however, is not
true in general for the remaining two eigenvalues of $W ^+$,
$\lambdambda _+$ and $ \lambdambda _-$, which are given by:
$$\lambda_{\pm}=\frac{1}{24}K{\"a}hler ppa \pm \frac{1}{8}(K{\"a}hler ppa^2 +
32|{\rm d} \theta ^+|^2)^{\frac{1}{2}},$$
cf. \cite{AG}).
It follows that for Hermitian 4-manifolds
the following
three conditions are equivalent (cf. \cite{Bo2,AG}):
\begin{enumerate}
\item[(i)] ${\rm d} \theta ^+=0$;
\item[(ii)] $W^+$ is degenerate;
\item[(iii)] $F$ is an eigenform of $W^+$.
\end{enumerate}
\noindent
(In the latter case $F$ is actually an eigenform for
the simple eigenvalue of $W ^+$, which is then equal to
$\frac{K{\"a}hler ppa}{6}$, {\rm also} equal to $\lambdambda _+$ or $\lambdambda _-$
according as $K{\"a}hler ppa$ is
positive or negative).
If, moreover, $M$ is compact, any one of the above three conditions is equivalent to
$(g,J)$ being locally conformally K{\"a}hler; if, in addition, the first
Betti number of $M$ is even, $(g, J)$ is then globally conformally K{\"a}hler \cite{Va1}.
By Proposition \ref{prop1} we conclude
that for every Einstein Hermitian $4$-manifold, we have
${\rm d} \theta ^+=0$, i.e. ${\rm d} \theta$ is self-dual.
In fact, a
stronger statement is true, see \cite[Prop.1]{AG} and
\cite[Prop.4]{De}:
\begin{prop}\lambdabel{prop2}
Let $(M,g,J)$ be an Einstein, non-ASD Hermitian 4-manifold.
Then the conformal scalar curvature $K{\"a}hler ppa$ nowhere vanishes and the Lee form $\theta$
is given by \ $\theta = \frac{1}{3}{\rm d}\ln{|K{\"a}hler ppa|}$ {\rm (}in particular,
$(g,J)$ is conformally K{\"a}hler{\rm )}.
If, moreover, $K{\"a}hler ppa$
is not constant, i.e. if $(g,J)$ is not K{\"a}hler, then
$K=J\rm{grad}_g(K{\"a}hler ppa^{-\frac{1}{3}})$ is a non-trivial Killing
vector field
with respect to $g$, holomorphic
with respect to $J$.
\end{prop}
\subsection{Einstein-Weyl structures and anti-self-dual conformal
metrics}
Another specific feature of the four-dimensional geometry is that to
each conformal Hermitian structure $([g], J)$ is canonically attached a unique
{\it Weyl connection} $D$ such that $J$ is parallel with respect to
$D$; in other words, any Hermitian structure is ``K{\"a}hler'' in the
extended context of
Weyl structures (of course, $(g, J)$ is K{\"a}hler in the usuel sense
--- the only one used in this paper --- if and only if $D$ is the
Levi-Civita connection of some metric in the conformal class $[g]$).
Recall that, given a conformal metric $[g]$, a Weyl connection (with
respect to $[g]$) is a torsion-free linear connection, $D$, on $M$
which preserves $[g]$; the latter condition can be reformulated as
follows: for any metric $g$ in $[g]$, there exists a real $1$-form
$\theta _g$ such that $D g = - 2 \theta _g \otimes g$; $\theta _g$ is
called the {\it Lee form} of $D$ with respect to $g$; then, the Weyl
connection $D$ and the Levi-Civita connection $D ^g$ are related by $D
= D ^g + \tilde{\theta} _g$, meaning
\begin{equation}\lambdabel{D^J} D _X Y = D ^g _X Y + \theta _g (X) Y + \theta _g (Y)
X - g(X, Y) \, \theta _g ^{\sharp _g}, \end{equation}
where $\theta _g ^{\sharp _g}$ is the Riemannian dual of $\theta _g$ with
respect to $g$.
If $g' = f ^{-2} g$
is another metric in $[g]$, the Lee form, $\theta
_{g'}$, of $D$ with respect to $g'$ is related to $\theta _g$ by
$\theta _{g'} = \theta _g + {\rm d} \ln{f}$.
A Weyl connection $D$ is the Levi-Civita connection of some metric in
the conformal class $[g]$ if and only if its Lee form with respect to
any metric $g$ in $[g]$ is exact, i.e. $\theta _g = - {\rm
d} \ln{f}$; then, $D = D ^{f ^{-2} g}$; such a Weyl connection is
called {\it exact}. More generally, a Weyl connection is said to be
{\it closed} if its Lee form with respect to any metric in $[g]$ is
closed; then, $D$ is locally of the above type, i.e. locally the
Levi-Civita connection of a (local) metric in $[g]$.
The definitions of the curvature $R ^D$ and the Ricci tensor ${\rm Ric} ^D$
of a Weyl
connection $D$ are formally identical as the ones we gave for $D ^g$
(notice that the derivation
of ${\rm Ric} ^D$ from $R ^D$ requires no metric); however, $R ^D$ is
now a $AM \oplus {\mathbb R} \, {{\rm Id}|_{TM}}$-valued $2$-form, i.e. has a
{\it scalar
part} equal to $F ^D \otimes{{\rm Id}|_{TM}}$, where the real $2 $-form $F
^D$,
the so-called {\it Faraday tensor} of the Weyl connection, is equal to $- {\rm d} \theta _g$ for any
metric $g$ in $[g]$; moreover, ${\rm Ric} ^D$ is not symmetric in general:
its skew-symmetric part
is equal to $\frac{1}{2} F ^D$; ${\rm Ric}^D$ is thus symmetric if and only if $D$ is closed.
A Weyl connection $D$ is called {\it Einstein-Weyl} if the symmetric,
trace-free part of ${\rm Ric} ^D$ vanishes; with respect to any metric
$g$
in $[g]$, and by writing $\theta$ instead of $\theta _g$, this conditions reads
\begin{equation} \lambdabel{EW} D ^g \theta - \theta \otimes \theta + \frac{1}{4}
(\delta \theta + |\theta| ^2) \, g - \frac{1}{2} {\rm d} \theta -
\frac{1}{2} {\rm Ric} _0 = 0, \end{equation}
see {e.g.} \cite{Ga2};
for a fixed metric $g$, (\ref{EW}) should be considered as an equation for an
unknown $1$-form $\theta$.
The {\it conformal scalar curvature} of $D$ with respect to $g$,
denoted by $K{\"a}hler ppa _g$, is
the trace of ${\rm
Ric}^D$ with respect to $g$; it is related to the (Riemannian) scalar curvature $s$ by:
\begin{equation}\lambdabel{kappag}
K{\"a}hler ppa _g = s + 6 \, (\delta \theta - |\theta |^2),
\end{equation}
see { e.g.} \cite{Ga2}.
A key observation is that the Lee form, $\theta$, of an
almost-Hermitian structure $(g, J)$ is also the Lee form with respect
to $g$ of the Weyl
connection canonically attached to the conformal
almost-Hermitian structure $([g], J)$; in other words, the Weyl
connection $D$ defined by $ D = D ^g + \tilde{\theta}$ is actually
independent of $g$ in its conformal class $[g]$. The Weyl connection
$D$ defined in this way is called the {\it canonical Weyl connection} of the
(conformal) almost-Hermitian structure $([g], J)$.
The scalar curvature $K{\"a}hler ppa _g$ of $D$ with respect to $g$ is called
the {\it conformal scalar curvature} of $(g, J)$; it coincides with
the function $K{\"a}hler ppa$ introduced in the previous paragraph.
The canonical Weyl connection is an especially interesting object when
$J$ is integrable, because of the following lemma:
\begin{Lemma} \lambdabel{weyl} {\rm (i)} $J$ is integrable if and only if $DJ = 0$.
{\rm (ii)} If $J _1$ and $J _2$ are two $g$-orthogonal complex
structures, the corresponding canonical connections $D ^1$ and $D ^2$
coincide if and only if the scalar product $(J _1, J_2)$ is constant.
\end{Lemma}
\begin{proof} (i) The condition $DJ = 0$ reads
\begin{equation} \lambdabel{integrable} D ^g _X J = [X \wedge \theta, J]; \end{equation}
this identity is proved e.g. in \cite{Ga1,Va2}.
(ii) Let $p$ denote the
{\it angle function} of $J _1$ and $J_2$, defined by $p = -
\frac{1}{4} {\rm tr} \, (J _1 \circ J _2) = \frac{1}{2} (J _1,
J_2)$; we then have
\begin{equation} J_1 \circ J _2 + J _2 \circ J_2 = - 2p \, {{\rm Id}|_{TM}}. \end{equation}
Let $\theta _1$ and $\theta _2$ be the Lee forms of $D ^1$, $D ^2$;
from (\ref{integrable}) applied to $J _1$, we infer $(D ^g J _1, J_2)
= ([J_1, J_2] X, \theta _1)$; similarly, we have $(D ^g J _2, J_1)
= ([J_2, J_1] X, \theta _2)$; putting together these two
identities, we get
\begin{equation} {\rm d} p = - \frac{1}{2} [J_1, J_2] (\theta _1 -
\theta _2). \end{equation}
This obviously implies ${\rm d} p = 0$ if $D ^1 = D ^2$; the converse
is also true, as the commutator $[J _1, J_2]$ is invertible at each point where $J _2 \neq \pm J_1$.
\end{proof}
An {\it almost-hypercomplex structure} is the datum of three
almost-complex structures, $I _1, I_2, I_3$, such that
$$ I _1 \circ I _2 = - I _2 \circ I _1 = I _3.$$
Since $M$ is a four-dimensional manifold, any almost-hypercomplex
structure $I _1, I_2, I_3$ determines a conformal class $[g]$ with
respect to which each $I_i$ is orthogonal: $[g]$ is defined by
decreeing
that, for any non-vanishing (local) vector field $X$, the frame $X, I_1X,
I_2X, I_3X$ is (conformally) orthonormal; for any $g$ in the
conformal class defined in this way, we thus get an {\it
almost-hyperhermitian structure} $(g, I_1, I_2,I_3)$; notice that
the $I_i$'s are pairwise orthogonal with respect to $g$, so that $I
_1, I_2, I_3$ is a (normalized) orthonormal frame of $A ^+ M$;
conversely, for a given Riemannian metric $g$ any (normalized)
orthonormal frame of $A ^+ M$ is an almost-hypercomplex structure and,
together with $g$ form an almost-hyperhermitian structure.
An almost-hyperhermitian structure $(g, I_1, I_2,I_3)$ is called
{\it hyperhermitian} if all $I_i$'s are integrable; it is called {\it
hyperk{\"a}hlerian} if the $I_i$'s are all parallel with respect to
the Levi-Civita connection $D ^g$.
In the hyperhermitian case the canonical Weyl connections, $D ^1, D
^2, D ^3$, of the
almost-Hermitian structures $(g, I_1)$, $(g, I_2)$, $(g, I_3)$ are the
same by
Lemma \ref{weyl}; the common Weyl connection, $D$, is
called the {\it canonical Weyl connection}
of the hyperhermitian structure.
Conversely, the
condition $D ^1 = D ^2 = D ^3$ implies that $(g, I_1, I_2,I_3)$ is
hyperhermitian (this observation is due to S. Salamon and F. Battaglia,
see e.g. \cite{GT}).
The canonical Weyl connection of a hyperhermitian structure $(g, I_1,
I_2,I_3)$ is
closed if and only if $I_1, I_2,I_3$ is locally hyperk{\"a}hler with
respect to some (local) metric belonging to the conformal class $[g]$; for
brevity, a hyperhermitian structure will be called {\it closed} or
{\it non-closed} according as its canonical Weyl connection being
closed or non-closed.
\begin{rem} {\rm In general, for any given hypercomplex structure $I _1,
I_2, I _3$ on a $n$-dimensional manifold, there exists a {\it unique}
torsion--free linear connection on $M$ that preserves the $I
_i$'s, called the {\it Obata connection}; the canonical connection thus coincides with the
Obata connection; for $n > 4$ however, there is no conformal
metric canonically attached to $I _1,
I_2, I _3$ and, in general, the Obata connection is
not a Weyl connection.}
\end{rem}
If $(g, I_1, I_2,I_3)$ is
hyperhermitian, we have $D I_1 = D I_2 = D I_3 = 0$, where $D$ is
the canonical Weyl connection acting on sections of $A ^+ M$; it
follows that the connection of $A ^+M$ induced by $D$
is {\it flat}; conversely, if $D$ is a Weyl connection, whose induced
connection on $A ^+ M$ is flat, then $A ^+ M$ can be locally trivialized by
a $D$-parallel (normalized) orthonormal frame $I_1, I_2,I_3$, which,
together with $g$, constitute a hyperhermitian structure.
The curvature, $R ^{D, A ^+M} $, of the induced connection is given by $R ^{D, A ^+M} _{X, Y} \Psi = [R ^D _{X, Y},
\Psi]$, where $R ^D _{X, Y}$ is understood as a field of
endomorphisms of $TM$ --- more precisely a section of $A M \oplus {\mathbb R}
\, {{\rm Id}|_{TM}}$ --- and $[R ^D _{X, Y},
\Psi]$ is the commutator of $R ^D _{X, Y}$ and $\Psi$; we easily
infer that the vanishing of $R ^{D, A ^+M} $ is equivalent to the
following four conditions:
\begin{enumerate} \lambdabel{swann}
\item $W ^+ = 0$;
\item $(F ^D) ^+ = 0$; if $\theta$ denotes the Lee form of $D$, this
also reads ${\rm d} \theta ^+ = 0$;
\item $D$ is Einstein-Weyl, i.e. the Lee form $\theta$ is solution of
(\ref{EW});
\item The scalar curvature of $D$ vanishes identically; in view of
(\ref{kappag}), this condition reads
\begin{equation}\lambdabel{hypherm}
s = 6 \, (-\delta \theta + |\theta|^2).
\end{equation}
\end{enumerate}
It follows from this discussion that, for an ASD Riemannian
4-manifold, the existence of a compatible hypercomplex structure is
locally equivalent to the existence of an Einstein-Weyl connection
satisfying the above conditions 2 and 4 (cf. \cite{PS} or \cite{GT}).
In this correspondence, conformally hyperk{\"a}hler
structures correspond to closed Einstein-Weyl structures. The existence of a
non locally hyperk{\"a}hler, hyperhermitian structure is actually (locally)
equivalent to the existence of a non-closed Einstein-Weyl connection,
in view of the following result of D. Calderbank:
\begin{prop}\lambdabel{prop4}{\rm (\cite{Ca})}
Let $(M,[g],D)$ be an anti-self-dual Einstein-Weyl 4-manifold. Then either
$D$ is closed, or else $D$ satisfies conditions 2 and 4 above, i.e. is the canonical Weyl connection of a
hyperhermitian structure.
\end{prop}
Notice that in the case when $M$ is
compact, $d \theta ^+ = 0$ implies ${\rm d}\theta=0$, hence {\it any} hyperhermitian
structure is locally
conformally hyperk{\"a}hler; a complete classification appears in \cite{Bo3}.
\section{Self-dual Einstein Hermitian 4-manifolds}
By Proposition \ref{prop2}, a Hermitian, Einstein 4-manifold, whose
self-dual Weyl tensor $W ^+$ has constant eigenvalues is either anti-self-dual or K{\"a}hler-Einstein, \cite{De}.
If, moreover, the metric $g$ is self-dual, this happens precisely when
$g$ is locally-symmetric, i.e. when $(M, g)$ is a real or a complex space form, see
\cite{TV}. More generally, a self-dual Einstein 4-manifold is
locally-symmetric if and only if $W ^+$ is degenerate, with constant
eigenvalues, \cite{De}.
In the opposite case, we have the following lemma:
\begin{Lemma} \lambdabel{de-ga} Non-locally-symmetric
self-dual Einstein Hermitian metrics are
in one-to-one correspondence with self-dual K{\"a}hler metrics of nowhere
vanishing and non-constant scalar curvature.
\end{Lemma}
\begin{proof} Every self-dual
Einstein Hermitian 4-manifold $(M,g,J)$ of
non-constant curvature is conformally related (via Proposition \ref{prop2})
to a
self-dual K{\"a}hler metric ${\bar g}$ of nowhere vanishing scalar curvature.
A self-dual K{\"a}hler metric is locally-symmetric if and only if its scalar
curvature is constant \cite{De}; thus, the one direction in the correspondence
stated in the lemma follows by observing that
${\bar g}$ is locally-symmetric as soon as $g$ is.
Since the Bach tensor of
a self-dual metric vanishes \cite{Gau3},
it follows from \cite[Prop.4]{De}
that any self-dual K{\"a}hler metric of nowhere vanishing scalar curvature
gives rise to an
Einstein Hermitian metric in the same conformal class.
\end{proof}
In the remainder of this section, $(M, g, J)$ is an Einstein,
self-dual Hermitian $4$-manifold, and we assume that $g$ is {\it not}
locally-symmetric; in particular, $W ^+$ is degenerate, but its
eigenvalues, $\lambdambda, - \frac{\lambdambda}{2}$, or, equivalently, its
norm $|W^+| = \sqrt{\frac{3}{2}} \, |\lambdambda|$, are not constant.
Since $(M,g,J)$ is not K{\"a}hler (Proposition \ref{prop2}), by
substituting to $M$ the dense open subset where the Lee form $\theta$
does not vanish, we shall assume throughout this section that $D^g J$ nowhere vanishes, see
(\ref{integrable}).
For convenience, we choose a (local, normalized) orthonormal frame of
$\Lambdambda ^{+, 0} M$ of the form $\{ \phi, J \phi \}$, where $|\phi| =
\sqrt{2}$; such a frame will be called a {\it gauge}. Then, the
triple $\{ F, \phi, J \phi \}$ is a (local, normalized) orthonormal
frame of $\Lambdambda ^ + M$.
Recall that by Proposition \ref{prop1} we have
\begin{equation} \lambdabel{lambda0} W
^+ (\psi) = - \frac{K{\"a}hler ppa}{12} \psi, \end{equation}
for any section $\psi$ of $\Lambdambda ^{+, 0} M$, whereas
\begin{equation} \lambdabel{lambda+} W ^+ (F) =
\frac{K{\"a}hler ppa}{6} F. \end{equation}
With respect to the gauge $\{ \phi, J \phi \}$, the covariant
derivative $ D^g F$ is written as
\begin{equation}\lambdabel{DF}
D^g F = \alpha\otimes \phi + J\alpha\otimes J\phi,
\end{equation}
where
\begin{equation} \alpha = \phi (J \theta); \end{equation}
equivalently,
\begin{equation} \lambdabel{phialpha} \phi = -\frac{1}{|\theta|^2}\big(\alpha\wedge J\theta + J\alpha\wedge \theta \big); \ \
J\phi = \frac{1}{|\theta|^2}\big(\alpha\wedge \theta - J\alpha\wedge
J\theta \big). \end{equation}
We also have
\begin{equation} \lambdabel{Dphi}
D^g \phi = - \alpha\otimes F + \beta\otimes J\phi; \ \
D^g (J\phi)= -J\alpha\otimes F - \beta\otimes \phi,
\end{equation}
for some 1-form $\beta$.
From (\ref{DF}), we infer
\begin{eqnarray}\nonumber
(D^g)^2|_{\Lambda^2M} F &=& ({\rm d}\alpha + J\alpha\wedge \beta)\otimes \phi + ({\rm d}(J\alpha) -\alpha\wedge \beta)\otimes J\phi \\ \nonumber
& =& -R(J\phi)\otimes \phi + R(\phi)\otimes J\phi.
\end{eqnarray}
Because of (\ref{lambda0}), this reduces to
\begin{equation}\lambdabel{ricci1}
\left\{
\begin{array}{c@{ = }c}
{\rm d}\alpha - \beta\wedge J\alpha \ & \frac{(K{\"a}hler ppa -s)}{12}J\phi\\
{\rm d}(J\alpha) + \beta\wedge \alpha \ & -\frac{(K{\"a}hler ppa -s)}{12}\phi.
\end{array}
\right.
\end{equation}
Similarly, because of (\ref{lambda+}), we infer
the following additional relation from (\ref{Dphi}):
\begin{equation}\lambdabel{ricci2}
{\rm d}\beta + \alpha \wedge J\alpha = - \frac{(s + 2K{\"a}hler ppa)}{12}F.
\end{equation}
Notice that 1-forms $\alpha$ and $\beta$ are both {\it gauge dependent}; if
$$\phi' = (\cos\varphi ) \phi + (\sin\varphi )J\phi $$
they transform to
$$\alpha'= (\cos\varphi) \alpha + (\sin\varphi) J\alpha; \ \
\beta'= \beta + {\rm d}\varphi.$$
We next introduce 1-forms $n_i, m_i, i=1,2$ by
\begin{equation}\lambdabel{Dtheta}
D^g \theta = m_1\otimes \theta + n_1\otimes J\theta + m_2 \otimes \alpha +
n_2\otimes J\alpha.
\end{equation}
By (\ref{DF}) and (\ref{phialpha}) we derive
\begin{equation}\lambdabel{DJtheta}
\begin{array}{c@{}c}
&D^g (J\theta) = -n_1\otimes \theta + m_1\otimes J\theta -(n_2+J\alpha)\otimes\alpha +(m_2 + \alpha)\otimes J\alpha; \\
& \ \ \ \ \ D^g \alpha \ = -m_2\otimes \theta + (n_2 + J\alpha)\otimes J\theta +
m_1\otimes\alpha - (n_1 -\beta)\otimes J\alpha; \\
& D^g(J\alpha) = - n_2\otimes \theta -(m_2 + \alpha)\otimes J\theta +
(n_1-\beta)\otimes \alpha + m_1\otimes J\alpha.
\end{array}
\end{equation}
A straightforward computation, using identities (\ref{ricci1}) and the fact that
the vector field $K= ({K{\"a}hler ppa}^{-\frac{1}{3}}J\theta)^{\sharp_g}$, the
dual of
${K{\"a}hler ppa}^{-\frac{1}{3}}J\theta$, is
Killing (see Proposition \ref{prop2}), gives the following
expressions
for $m_i$ and $n_i$:
\begin{equation}\lambdabel{mn}
\left\{
\begin{array}{c@{ = }c}
m_1 & m_0 + (p-\frac{(K{\"a}hler ppa -s)}{24|\theta|^2} +\frac{1}{2}) \theta \\
n_1 & Jm_0 +(p-\frac{(K{\"a}hler ppa -s)}{24|\theta|^2} -\frac{1}{2}) J\theta \\
m_2 & J\phi(m_0) -(p +\frac{(K{\"a}hler ppa -s)}{24|\theta|^2} +\frac{1}{2})\alpha \\
n_2 & -\phi(m_0) -(p +\frac{(K{\"a}hler ppa -s)}{24|\theta|^2} +\frac{1}{2})J\alpha,
\end{array}
\right.
\end{equation}
where $p$ is a smooth function, and $m_0$ is a 1-form which
belongs to the distribution
${\mathcal D}^{\perp}= {\rm span} \{ \alpha , J\alpha \}$,
the orthogonal
complement of
${\mathcal D} = {\rm span} \{ \theta , J\theta \}$.
Since
$m_1 = {\rm d}\ln |\theta|$, the 1-form $m_0$ is nothing else
than the projection of
${\rm d}\ln |\theta|$ to the subbundle ${\mathcal D}^{\perp}$. Moreover,
with respect to any gauge $\phi$, we write
\begin{equation}\lambdabel{m0}
m_0 = q\alpha + r J\alpha,
\end{equation}
for some smooth functions $q$ and $r$.
In view of (\ref{integrable}),
identities (\ref{Dtheta}) and (\ref{mn}) are
conditions on the 2-jet of $J$. Since
$J$ is completely determined by $W^+$ (see Proposition \ref{prop1}), these
are the conditions on the 4-jet of
the metric referred to in the introduction.
This completes the analysis of the Einstein condition and we are now going to
see how the vanishing of $W^-$ interacts on further jets of $g$.
For that,
we introduce the ``mirror frame'' of $\Lambda^-M$:
$${\bar F}= -F + \frac{2}{|\theta|^2}\theta\wedge J\theta; \ \
{\bar \phi} = \phi + \frac{2}{|\theta|^2} J\alpha\wedge \theta;$$
$$I{\bar \phi}= J\phi + \frac{2}{|\theta|^2} J\alpha\wedge J\theta, $$
where the {\it negative} almost Hermitian structure $I$, of which the
anti-self-dual 2-form ${\bar F}$ is the K{\"a}hler form, is
equal to $J$ on ${\mathcal D}$ and $-J$ on ${\mathcal D}^{\perp}$.
By (\ref{DJtheta}) and the fact that $\theta= \frac{{\rm d}K{\"a}hler ppa}{3K{\"a}hler ppa}$,
we obtain the following expression for the covariant derivative of the
Killing vector field
$K=(K{\"a}hler ppa^{-\frac{1}{3}}J\theta)^{{\sharp_g}}$
\begin{equation}\lambdabel{DK}
D^g K = K{\"a}hler ppa^{-\frac{1}{3}}|\theta|^2\big(q{\bar \phi} - rI{\bar\phi} - (p-\frac{1}{2}){\bar F} + \frac{(K{\"a}hler ppa -s)}{24|\theta|^2}F\big).
\end{equation}
Moreover, since $K$ is Killing, we have
\begin{equation} \lambdabel{killing} D^g_X \Psi = R(K,X), \end{equation}
where $\Psi = D^g K$.
Considering the ASD parts of both sides of (\ref{killing}), we infer
that the condition $W^-=0$ is equivalent to
\begin{equation}\lambdabel{W^-=0}
D^g(\Psi^-) = \frac{s}{24}(\bar{\phi}(K)\otimes {\bar \phi} + I\bar{\phi}(K)\otimes I\bar{\phi} + IK\otimes {\bar F}),
\end{equation}
where
$$\Psi^-= K{\"a}hler ppa^{-\frac{1}{3}}|\theta|^2\big(q{\bar \phi} - rI{\bar\phi} - (p-\frac{1}{2}){\bar F}\big)$$ is the ASD part of $\Psi=D^gK$,
see (\ref{DK}).
Furthermore, by (\ref{Dtheta}) and (\ref{DJtheta}) one gets
\begin{eqnarray}\nonumber
D^g {\bar F} &=& -(2m_2 + \alpha)\otimes {\bar \phi} + (2Jm_2 +J\alpha) \otimes I{\bar \phi};\\
\lambdabel{I}
D^g {\bar \phi} &=& \ \ \ (2m_2 +\alpha) \otimes {\bar F} + (2n_1- \beta)\otimes I{\bar \phi}; \\
\nonumber
D^g I{\bar \phi} &=& -(2Jm_2+ J\alpha)\otimes {\bar F} - (2n_1 -\beta)\otimes {\bar \phi}.
\end{eqnarray}
Keeping in mind that
$\theta=\frac{{\rm d}K{\"a}hler ppa}{3K{\"a}hler ppa}$ and $m_1={\rm d}\ln|\theta|$,
(\ref{W^-=0}) then reduces to
\begin{eqnarray}\lambdabel{system1}
{\rm d}p &=& -(p-\frac{1}{2})(2m_1 - \theta) + q(m_2+ \alpha) \\ \nonumber
& & + r(Jm_2 + J\alpha) -
\frac{s}{24|\theta|^2} \theta \\ \lambdabel{system2}
{\rm d}q &=& -(p-\frac{1}{2})(m_2+ \alpha) - q(2m_1 - \theta) \\\nonumber
& & - r(2n_1 - \beta) - \frac{s}{24|\theta|^2} \alpha \\\lambdabel{system3}
{\rm d}r &=& -(p-\frac{1}{2})(Jm_2+ J\alpha) + q(2n_1 - \beta) \\\nonumber
& & - r(2m_1 - \theta)-
\frac{s}{24|\theta|^2} J\alpha.
\end{eqnarray}
Now, taking into account (\ref{ricci1}) and (\ref{ricci2}), (\ref{system1})--(\ref{system3}) constitute
a closed differential system that a self-dual Einstein Hermitian metric
must satisfy; by (\ref{ricci1}), (\ref{ricci2}), (\ref{DJtheta})
and (\ref{mn})
one can directly check
that the integrability conditions ${\rm d(d}p)={\rm d(d}q)={\rm d(d}r)=0$ are satisfied. This
is a first evidence that the existence of self-dual
Einstein Hermitian metrics with prescribed 4-jet at a given point
can be expected.
To carry out this program explicitly, we first consider the case when
$q\equiv 0, r\equiv 0$ and show that it precisely corresponds
to {\it cohomogeneity-one} self-dual Einstein Hermitian metrics.
\subsection{Self-dual Einstein Hermitian metrics of cohomogeneity one}
A Riemannian 4-manifold $(M,g)$ is said to be (locally) {\it of
cohomogeneity one}, if it admits a (local)
isometric action of a Lie group $G$,
with three-dimensional orbits.
The manifold $M$ is then locally a product
$$ M \cong (t_1,t_2)\times G/H.$$
The metric $g$ descends to a left invariant metric $h(t)$
on each orbit $\{ t \} \times G/H$, and, by an appropriate choice of
the parameter $t$, can be written as
$$g= dt^2 + h(t).$$
If, moreover, $(M,g)$ is Einstein and self-dual, and $G$ is at least
of dimension four, then, according to a result of A. Derdzi\'nski \cite{De},
the spectrum of the self-dual Weyl tensor
of $g$ is everywhere degenerate, and
$g$ is Hermitian with respect
some invariant complex structure.
Here is a way of constructing such metrics, all belonging to the class of {\it diagonal
Bianchi metrics of type A} (see e.g. \cite{Tod1}). Let
$\widetilde{G}$ be one of
the following six three-dimensional Lie groups:
${\Bbb R}^3$,
${\rm Nil}^3, {\rm Sol}^3$, Isom(${\Bbb R}^2$), SU(1,1) or
SU(2); let $H$ be a discrete subgroup of $\widetilde{G}$ and
consider, on $\widetilde{G} / H$, the family of diagonal metrics $h (t)$ of the form
\begin{equation}\lambdabel{diagonal}
h(t) = A(t)\sigma_1^2 + B(t)\sigma_2^2 + C(t)\sigma_3^2,
\end{equation}
where $A,B,C$ are positive smooth functions,
and $\sigma_i$ are the standard left invariant
generators of the corresponding Lie algebras; we thus have
$$d\sigma^1 = n_1 \sigma_2\wedge \sigma_3; \ d\sigma_2=-n_2\sigma_1\wedge\sigma_3; \ d\sigma_3 = n_3\sigma_1\wedge \sigma_2$$
for a triple $(n_1,n_2,n_3)$, $n_i \in \{-1,0,1\}$, depending on the
chosen group, according to the following table:
\begin{center}
\begin{tabular}{|c|c|c|} \hline
{\rm class} & $ n_1 \ \ \ n_2 \ \ \ n_3 $ & ${\widetilde G}$ \\
\hline \hline
{\rm I} & $0 \ \ \ \ 0 \ \ \ \ 0\ $ & ${\Bbb R}^3$ \\ \hline
{\rm II} & $0 \ \ \ \ 0 \ \ \ \ 1\ $ & ${\rm {Nil}^3}$\\ \hline
${\rm VI}_0$ & $1 \ \ {-1} \ \ \ 0\ $ & ${\rm {Sol}^3}$ \\ \hline
${\rm VII}_0$& $1 \ \ \ \ 1 \ \ \ \ 0\ $ & ${\rm Isom}({\Bbb R}^2)$ \\ \hline
{\rm VIII} & $1 \ \ \ \ 1 \ {-1}\ $ & ${\rm SU}(1,1)$ \\ \hline
{\rm IX} & $1 \ \ \ \ 1 \ \ \ \ 1\ $ & ${\rm SU(2)}$ \\ \hline
\end{tabular}
\end{center}
Except for Class ${\rm VI}_0$, when $A = B$ all these metrics admit a further (local)
symmetry which rotates the $\{\sigma_1, \sigma_2 \}$-plane, i.e. we get
the so-called {\it biaxial} Bianchi metrics, see e.g. \cite{CP}.
We thus obtain
diagonal Bianchi metrics of Class A,
admitting a local isometric action of a four-dimensional Lee group $G$, where
$G$ is
${\Bbb R}\times{\rm Isom({\Bbb R}^2)}$, {\rm U(1,1)}, {\rm U(2)}, or the non-trivial
central extension of ${\rm Isom}({\Bbb R}^2)$ corresponding to
biaxial Class II metrics.
Clearly, any such metric admits
a positive {\it and} a negative invariant
Hermitian structure, $J$ and $I$, whose K{\"a}hler forms are given by
$$F= \sqrt{C}dt\wedge \sigma_3 + A\sigma_1\wedge \sigma_2,$$
and
$${\bar F} =\sqrt{C}dt\wedge \sigma_3 - A\sigma_1\wedge \sigma_2,$$
respectively.
When imposing the Einstein and the self-duality conditions,
we obtain an ODE system
for the unknown functions $A$ and $C$, which can be explicitly
solved, cf. {e.g.}
\cite{P}, \cite{Le}, \cite{DS}, \cite{Tod1}, \cite{CP},
\cite{bergery}.
In the sequel, we shall simply refer to these
(self-dual, Einstein, Hermitian) metrics as {\it diagonal Bianchi}
metrics.
Notice that $4$-dimensional locally symmetric metrics,
i.e. real and complex space forms, can also be put (in several ways) as
diagonal Bianchi metrics. For example, self-dual Einstein Hermitian metrics in
Class I are all flat \cite{Tod1}.
Our next result shows that, apart from locally symmetric
spaces, diagonal Bianchi metrics in the above sense are actually {\it
all}
(non-locally symmetric) cohomogeneity-one self-dual Einstein Hermitian
metrics, and, in fact,
can be characterized by the property
$m_0\equiv 0$ in the notation of the preceding section. More
precisely, we have:
\begin{theo}\lambdabel{th2}
Let $(M,g)$ be a self-dual
Einstein 4-manifold. Suppose that $(M,g)$ is not locally symmetric. Then the following three conditions are equivalent:
\begin{enumerate}
\item[(i)] $(M,g)$ is of cohomogeneity one and
the spectrum of $W^+$ is degenerate.
\item[(ii)] $(M,g)$ admits a local isometric action of a Lie group of
dimension at least four, with three-dimensional orbits, and is
locally isometric to a diagonal Bianchi self-dual Einstein Hermitian
metric belonging to one of the classes ${\rm II}$, ${\rm VII}_0$, ${\rm VIII}$ or
${\rm IX}$.
\item[(iii)] $(M,g)$ admits a positive, non-K{\"a}hler
Hermitian structure $J$, and a negative
Hermitian structure $I$ such that $I$ is equal to $J$ on
${\mathcal D}={\rm span} \{\theta , J\theta \}$ and to $-J$ on the orthogonal
complement ${\mathcal D}^{\perp}$ ; equivalently, the 1-form $m_0$
of $(g,J)$ vanishes identically.
\end{enumerate}
\end{theo}
\begin{proof}
${\rm (i)} \Rightarrow {\rm (iii)}$. By Propositions \ref{prop1}
and \ref{prop2}, $W^+$ has two distinct, non-constant eigenvalues at
any point and there exists a positive, non-K{\"a}hler Hermitian structure
$J$ whose K{\"a}hler form $F$
generates the eigenspace of $W^+$ corresponding to the
simple eigenvalue. It follows that the Hermitian structure is preserved by the
action of $G$, and therefore both functions
$|D^g F|^2 = 2|\theta|^2$ and
$|W^+|^2=\frac{{K{\"a}hler ppa}^2}{24}$ are constant along the orbits of $G$; in particular,
${\rm d}\ln|\theta|$ is colinear to $\theta = \frac{{\rm
d}K{\"a}hler ppa}{3K{\"a}hler ppa}$, at any point; this means that
$m_0=0$; by (\ref{I}) and (\ref{mn}),
the vanishing of $m_0$ is equivalent to the integrability of the
negative almost Hermitian structure $I$.
${\rm (iii)} \Rightarrow {\rm (ii)}$.
If $m_0\equiv 0$ or, equivalently, if the negative almost Hermitian
structure $I$ is integrable, then, by (\ref{I}), the Lie form
$\theta_I$ of $(g,I)$ reads:
\begin{equation}\lambdabel{thetaI}
\theta_I =(2p +\frac{(K{\"a}hler ppa -s)}{12|\theta|^2})\theta.
\end{equation}
According to (\ref{mn}) we also have
$m_1={\rm d}\ln|\theta| =(p-\frac{(K{\"a}hler ppa -s)}{24|\theta|^2} +
\frac{1}{2}) \theta$ and $\theta= \frac{1}{3}{\rm d}\ln|K{\"a}hler ppa|$;
it follows that ${\rm d}\theta_I=0$; then, locally,
$\theta_I = {\rm d}f$ for a positive function $f$, i.e., $g$ is
conformal to a K{\"a}hler metric ${g'} = f^2g$.
Since $W^-=0$, the K{\"a}hler metric $g'$ is
of zero scalar curvature. Clearly, the Killing
field $K$ preserves both $J$ and $g$, hence, also,
the K{\"a}hler structure $(g',I)$. Two cases occur, according as $g '$ is
homothetic or not to $g$.
(a) Suppose ${g'}$ is {\it not} homothetic to $g$; equivalently, the scalar
curvature $s$ of $g$ does not vanishes; then, by \cite{De},
$K'=I{\rm grad}_g(f^{-1})$ is a Killing vector field for $g$ and $g'$
and is
holomorphic with respect $I$. By the very definition of $I$ we have
that
$J|_{\mathcal D} = I|_{\mathcal D}$; the Killing
vector fields $K'$ and $K$ are thus colinear everywhere (see
(\ref{thetaI}));
it follows that
$K'$ is a
constant multiple of $K$. By considering $z=f^2$ as a local
coordinate on $M$ and, by introducing a holomorphic
coordinate $x+iy$ on the (locally defined)
orbit-space for the holomorphic action of $K+\sqrt{-1}IK$ on
$(M,I)$,
the metric $g$ can be written in the following form:
\begin{equation}\lambdabel{g}
g = \frac{1}{z^2}[e^{u}w({\rm d}x^2 + {\rm d}y^2) + w {\rm d}z^2 + w^{-1}\omega^2],
\end{equation}
where $u(x,y,z)$ is a smooth function satisfying the
${\rm SU(\infty )}$ Toda field equation:
$$u_{xx} + u_{yy} + (e^u)_{zz}=0,$$
$w$ is a positive function given by
$$w= \frac{6(zu_z -2)}{s},$$
and $\omega$ is a
connection 1-form of the ${\Bbb R}$-bundle
$M \mapsto N=\{(x,y,z)\} \subset {\Bbb R}^3$,
whose curvature is given by
\begin{equation}\lambdabel{domega}
{\rm d}\omega = -w_x{\rm d}y\wedge {\rm d}z - w_y {\rm d}z\wedge {\rm d}x - (we^u)_z
{\rm d}x\wedge {\rm d}y,
\end{equation}
(see, e.g. \cite{Tod2}).
Moreover, the Killing field $K$ is dual to $\frac{1}{wz^2}\omega$, and
the (anti-self-dual) K{\"a}hler form of
the negative Hermitian structure $I$ is given by
\begin{equation}\lambdabel{barF}
{\bar F} =\frac{1}{z^2}\big(we^u {\rm d}x\wedge {\rm d}y - {\rm d}z\wedge \omegaega \big).
\end{equation}
By (\ref{thetaI}) we have that
${\mathcal D}={\rm span}\{ \theta , J\theta \}=
{\rm span}\{\theta_I, I\theta_I \}={\rm span}\{K^{\sharp_g},IK^{\sharp_g}\}$, so that
the K{\"a}hler form $F$ of the positive Hermitian structure $J$ is given by
\begin{equation}\lambdabel{F}
F = \frac{1}{z^2}\big(we^u {\rm d}x\wedge {\rm d}y + {\rm d}z\wedge \omegaega \big).
\end{equation}
It is now easily seen that (\ref{barF}) and (\ref{F})
simultaneously define integrable almost complex structures if and only if
$w_x=w_y=0$, or equivalently if and only if
$u(x,y,z)=u_1(x,y) + u_2(z)$. This means that $u$ is a {\it separable} solution to
the ${\rm SU(\infty )}$ Toda field equation. Up to a change of the
holomorphic coordinate $x+ iy$, it
is explicitly given by \cite{Tod2}
$$e^u = \frac{4(c + bz + az^2)}{(1 + a(x^2 + y^2))^2},$$
for properly chosen constants $a,b,c$.
Any such solution gives rise to a
{diagonal Bianchi} self-dual Einstein Hermitian metric pertaining to
one of classes II, ${\rm VII}_0$, VIII and IX,
depending on the choice of the constants
$a,b,c$ (see e.g. \cite[Sec. 8]{CP}) for a common case of these
metrics in the Bianchi IX case).
(b) If $g ' $ is homothetic to $g$, i.e. $(g,I)$ is itself a K{\"a}hler structure of zero scalar curvature,
then $g$ is locally hyperk{\"a}hler and $K$
is a Killing vector field preserving the K{\"a}hler structure $I$. Then,
one of the two
following situations occurs:
(b1) {\it $K$ is {\it triholomorphic}}, i.e. $K$
preserves each K{\"a}hler structure in the hyperk{\"a}hler family: Then
the quotient space, $N$, for the (real) action of $K$ is flat and is
endowed with a field of parallel straight lines.
This situation is described by the Gibbons-Hawking Ansatz \cite{GH},
and the metric $g$ has the form:
$$g = w({\rm d}x^2 + {\rm d}y^2 + {\rm d}z^2) + \frac{1}{w}\omegaega^2,$$
for a positive harmonic function $w(x,y,z)$ on $N$ and a 1-form $\omegaega$
on $M$ satisfying
$${\rm d}\omega = -w_x{\rm d}y\wedge {\rm d}z - w_y {\rm d}z\wedge {\rm d}x - w_z
{\rm d}x\wedge {\rm d}y.$$
The Killing field $K$ is dual to $\frac{1}{w}\omega$
and one may consider
that
the positive and negative Hermitian structures, $J$ and $I$,
correspond to the 2-forms
$$F= w{\rm d}x\wedge {\rm d}y + {\rm d}z\wedge \omegaega; \ \
{\bar F}= w{\rm d}x\wedge {\rm d}y - {\rm d}z\wedge \omegaega,$$
respectively. We again conclude $w_x=0, w_y=0$, and therefore
$w=az +b$. The case $a=0$ corresponds to flat metrics in Class I,
whereas, when
$a\neq 0$, by putting
$at = az + b, \sigma_1 = {\rm d}x, \sigma_2 = {\rm d}y, \sigma_3 = \omegaega$,
the metric becomes a diagonal Bianchi metric of Class II.
(b2) {\it $K$ is {\it not} triholomorphic}:
Since, nevertheless, $K$ preserves $(g,I)$,
the metric $g$ takes the form \cite{BoF}
$$ g = e^{u}w({\rm d}x^2 + {\rm d}y^2) + w {\rm d}z^2 + w^{-1}\omega^2, $$
where $u(x,y,z)$ is a solution to the
${\rm SU(\infty)}$ Toda field equation, $w=au_z$,
$\omegaega$ satisfies (\ref{domega}) and
$a$ is a constant. Moreover, $K$ is dual to $\frac{1}{w}\omegaega$, and $I$ is defined by the anti-self-dual form
$${\bar F} = we^u {\rm d}x\wedge {\rm d}y - {\rm d}z\wedge \omegaega.$$
Similar arguments as above show that
$w_x=w_y=0$, i.e., $u$ is a separable solution to
the ${\rm SU(\infty )}$ Toda field equation, and therefore
our metric is again a diagonal Bianchi
metric in one of the classes II, ${\rm VII}_0$, VIII or IX, cf.
\cite{CP}.
The implication ${\rm (ii)} \Rightarrow {\rm (i)}$ is clear.
\end{proof}
\begin{rem}{\rm
A weaker version of Theorem \ref{th2} was announced in \cite{de}
(see \cite[Rem.~1.3]{de} and Lemma 2 above).}
\end{rem}
\subsection{The generic case} We now consider the generic case, when $m_0$ a {\it non-vanishing} section of
${\mathcal D}^\perp$, hence determines a gauge $\phi$ such that
$r\equiv 0, q\neq 0$ in (\ref{mn}). According to
(\ref{mn}), the 1-form $\alpha$ is then given by
\begin{equation}\lambdabel{alpha}
m_1 = {\rm d}\ln|\theta| = q\alpha + (p- \frac{(K{\"a}hler ppa -s)}{24|\theta|^2} + \frac{1}{2})\theta;
\end{equation}
moreover, by (\ref{system1})--(\ref{system3}), we have that
\begin{eqnarray}\lambdabel{beta}
\beta &= & \frac{1}{q}\Big(p(2p + \frac{(K{\"a}hler ppa -s)}{12|\theta|^2} - 1) - \frac{K{\"a}hler ppa}{24|\theta|^2} + 2q^2\Big)J\alpha \\\nonumber
& &-\frac{(K{\"a}hler ppa -s)}{12|\theta|^2}J\theta,
\end{eqnarray}
\begin{eqnarray}\lambdabel{frobenius1}
{\rm d}p &=&\Big(2q^2 - p(2p - \frac{(K{\"a}hler ppa -s)}{12|\theta|^2} - 1)
- \frac{K{\"a}hler ppa}{24|\theta|^2}\Big)\theta \\ \nonumber
& & - q\Big(4p +\frac{(K{\"a}hler ppa -s)}{12|\theta|^2} - 1\Big)\alpha,
\\\lambdabel{frobenius2}
{\rm d}q &=& - q\Big(4p -\frac{(K{\"a}hler ppa -s)}{12|\theta|^2} - 1\Big)\theta
\\ \nonumber
& & - \Big(2q^2 -p(2p + \frac{(K{\"a}hler ppa -s)}{12|\theta|^2} - 1)
+ \frac{K{\"a}hler ppa}{24|\theta|^2}\Big)\alpha.
\end{eqnarray}
By differentiating (\ref{alpha}) and by making use of
(\ref{frobenius1})--(\ref{frobenius2}),
we get
\begin{equation}\lambdabel{dalpha}
{\rm d}\alpha = \frac{(K{\"a}hler ppa -s)}{12|\theta|^2}\alpha \wedge \theta =
\alpha\wedge J\beta;
\end{equation}
this is nothing else than the first relation in (\ref{ricci1}), when
$\beta$ is given by (\ref{beta});
by substituting the expression (\ref{beta}) for $\beta$
into the second relation of (\ref{ricci1}), we obtain
\begin{equation}\lambdabel{dJalpha}
{\rm d}(J\alpha) = J\alpha \wedge J\beta.
\end{equation}
In view of (\ref{alpha}) and (\ref{frobenius1})--(\ref{frobenius2}),
it is not hard to check that the 1-form $J\beta$ is equivalently given by
\begin{equation}\lambdabel{dJbeta}
J\beta = {\rm d}\ln(\frac{|K{\"a}hler ppa|}{|q||\theta|^4}),
\end{equation}
so that (\ref{dJalpha}) becomes
\begin{equation}\lambdabel{dJa}
{\rm d}(\frac{K{\"a}hler ppa}{q|\theta|^4}J\alpha)=0;
\end{equation}
from (\ref{DJtheta}) we get
\begin{equation}\lambdabel{dJthe}
{\rm d}(J\theta)= J\theta\wedge\big(\frac{1}{3}{\rm d}\ln|K{\"a}hler ppa| -2{\rm d}\ln|\theta|\big) + J\alpha \wedge \eta,
\end{equation}
or, equivalently,
\begin{equation}\lambdabel{dJtheta}
{\rm d}(\frac{K{\"a}hler ppa^{\frac{1}{3}}}{|\theta|^2} J\theta)=
\frac{K{\"a}hler ppa^{\frac{1}{3}}}{|\theta|^2}J\alpha\wedge \eta,
\end{equation}
where
$$\eta = -2q\theta + (2p + \frac{(K{\"a}hler ppa -s)}{12|\theta|^2} -1)\alpha.$$
We are now rea{\rm d}y to prove the existence of self-dual Einstein Hermitian metrics
with $m_0\neq 0$. More precisely, we exhibit a 1--1-correspondence
between these metrics and the set of solutions of the integrable Frobenius system
(\ref{frobenius1})--(\ref{frobenius2}).
We start with the data
$(s, K{\"a}hler ppa, |\theta|)$ consisting
of a constant $s$ (the scalar curvature), a
nowhere vanishing smooth function $K{\"a}hler ppa$ (the conformal scalar curvature),
and a positive smooth function $|\theta|$ (the norm of the Lie form
$\theta= \frac{{\rm d}K{\"a}hler ppa}{3K{\"a}hler ppa}$), defined on an open subset ${U}$
of $M$,
such that $\theta\wedge {\rm d}|\theta|^2$ has no zero on ${U}$
(equivalently, $m_0$ does not vanish on ${U}$).
We then introduce local
coordinates $x= K{\"a}hler ppa^{\frac{1}{3}} \neq 0$ and $y=|\theta|^2 >0$.
Observe that $x$ is a {\it momentum map} for the Killing field $K$
with respect to the self-dual K{\"a}hler metric
${\bar g}={K{\"a}hler ppa}^{\frac{2}{3}}g$ while $y=|K|_{\bar g}^2$ is the
square-norm of $K$ with respect to ${\bar g}$ (see Proposition \ref{prop2}).
The Lee form $\theta$ is then given by
\begin{equation}\lambdabel{deftheta}
\theta= \frac{{\rm d}x}{x},
\end{equation}
and the 1-form $\alpha$ is
given by (\ref{alpha}) for some smooth functions $p(x,y)$ and
$q(x,y)\neq 0$ of $x,y$, i.e.
\begin{equation}\lambdabel{defalpha}
\alpha = \frac{1}{q}\Big( \frac{{\rm d}y}{2y} - \frac{1}{x}(p -\frac{(x^3-s)}{24y} + \frac{1}{2}){\rm d}x\Big).
\end{equation}
Then, (\ref{frobenius1})--(\ref{frobenius2}) can be made into the
following
Frobenius system for the
(unknown) functions $p$ and $q^2$:
\begin{eqnarray}\lambdabel{defp}
{\rm d}p &=& \frac{1}{x}\Big[ 2q^2 + 2(p+\frac{(x^3 -s)}{24 y})(p-\frac{(x^3 -s)}{24 y} +1) -\frac{1}{2} - \frac{x^3}{24y}\Big]{\rm d}x \\ \nonumber
& & - \frac{1}{y}\Big[2p + \frac{(x^3 -s)}{24y} - \frac{1}{2}\Big]{\rm d}y
\end{eqnarray}
\begin{eqnarray}\lambdabel{defq}
{\rm d}(q^2) &=& -\frac{1}{y}\Big[ 2q^2 -2p(p+\frac{(x^3 -s)}{24y} - \frac{1}{2}) +\frac{x^3}{24y}\Big]{\rm d}y\\ \nonumber
& &- \frac{2}{x}\Big[\Big(p-\frac{(x^3 -s)}{24y} + \frac{1}{2}\Big)\Big(2p(p +\frac{(x^3-s)}{24y} -\frac{1}{2}) - \frac{x^3}{24y}\Big)\\ \nonumber
& & \ \ \ \ \ \ \ - 2q^2(1-p)\Big]{\rm d}x
\end{eqnarray}
A straightforward computation shows
that the integrability condition ${\rm d}({\rm d}p)={\rm d}({\rm
d}q^2)=0$ is satisfied (as a matter of fact, the explicit solutions are given in Lemma 3
below). The above mentioned
correspondence
between solutions to (\ref{defp})--(\ref{defq}) and
self-dual Einstein Hermitian metrics with $m_0\neq 0$ now goes as
follows.
Since (\ref{defp})--({\ref{defq}) is integrable,
each value of $(p,q)$ at a given point $(x_0,y_0)$ can be extended
to a solution of (\ref{defp})--(\ref{defq})
in some neighborhood
$V$ of
$(x_0,y_0)$; moreover, by choosing $q(x_0,y_0) \neq 0$, we may assume
that $q$ has no zero
on $V$;
by (\ref{defalpha}) and (\ref{defp})--(\ref{defq}), one
immediately obtains (\ref{dalpha}) for the corresponding 1-form
$\alpha$. We then
introduce a third local coordinate, $z$, such that
\begin{equation}\lambdabel{defJalpha}
J\alpha =
\frac{qy^2}{x^3}{\rm d}z,
\end{equation}
see (\ref{dJa}). Finally, since the 1-form $J\theta$
satisfies
(\ref{dJthe}) or, equivalently, (\ref{dJtheta}),
the integrability condition reads as follows:
$${\rm d}(\frac{qy}{x^2}\eta)=0,$$
see (\ref{dJa}) and (\ref{dJthe}); by using
(\ref{frobenius1})--(\ref{dJalpha}), one easily checks that
the integrability condition is actually satisfied, so that
\begin{equation}\lambdabel{defJtheta}
J\theta= \frac{y}{x}({\rm d}t + h{\rm d}z),
\end{equation} where $t$ is a suitable
transversal coordinate to $(x,y,z)$,
and $h(x,y)$ is a smooth function on $V$, defined by
$${\rm d}h =-\frac{qy}{x^2}\eta.$$
It is an easy consequence of (\ref{defp}) that the above equation
is solved by
\begin{equation}\lambdabel{defh}
h = \frac{yp}{x^2} + \frac{x}{24}.
\end{equation}
The metric $g$ and the orthogonal
almost complex structure $J$ are then given by
$$g = \frac{1}{|\theta|^2}(\theta\otimes\theta + J\theta\otimes
J\theta + \alpha\otimes \alpha + J\alpha\otimes J\alpha);$$
according to (\ref{deftheta}),(\ref{defalpha}),(\ref{defJalpha}) and
(\ref{defJtheta}), and by using the coordinates
$(x,y,z,t)$, the metric $g$ takes the form
\begin{equation}\lambdabel{canonic}
g = \frac{1}{y}\Big[\frac{{\rm d}x^2}{x^2} + \frac{1}{q^2}\Big(\frac{{\rm d}y}{2y} - \frac{1}{x}(p -\frac{(x^3 -s)}{24y} + \frac{1}{2}){\rm d}x\Big)^2 + \frac{q^2y^4}{x^6}{\rm d}z^2 +
\frac{y^2}{x^2}({\rm d}t + h{\rm d}z)^2\Big];
\end{equation}
this shows that any self-dual Einstein Hermitian metric
with $m_0\neq 0$ is
locally isometric to a metric of the above form for some solution $(p,q)$ to
(\ref{defp})--(\ref{defq}).
Conversely, for any solution to
(\ref{defp})--(\ref{defq}), the corresponding almost-Hermitian metric
$(g,J)$ is
self-dual Einstein Hermitian metric with $m_0\neq 0$. Indeed,
by (\ref{dalpha}), (\ref{dJalpha}) and (\ref{dJtheta}),
$J$ is integrable and it is easily checked that $\theta=\frac{{\rm d}x}{x}$ is the Lee form for $(g,J)$,
i.e.,
$${\rm d}F = -2\theta\wedge F;$$
moreover, the 1-form $\alpha$ corresponds to
the gauge
$$ \phi = -\frac{1}{y}\big(\alpha\wedge J\theta + J\alpha\wedge \theta \big),$$
meaning that $\alpha = \phi(J\theta)$; one directly computes
$${\rm d}\phi = (\theta + J\beta)\wedge \phi,$$
where the 1-form $\beta$ is given by (\ref{beta}); it follows that
$\beta$ is precisely the {\rm 1-form} defined by (\ref{Dphi}) and that
(\ref{dalpha})--(\ref{dJalpha}) are nothing else than the
Ricci identities (\ref{ricci1}); this allows us to recognize the curvature:
By (\ref{ricci1}), the Ricci tensor of $(g,J)$ is
$J$-invariant, and, since $\theta= \frac{{\rm d}x}{x}$,
the dual vector field $K$ of
$K{\"a}hler ppa^{-\frac{1}{3}}J\theta=\frac{1}{x}J\theta$
is Killing, cf. e.g. \cite{AG}; by (\ref{dJtheta}) and (\ref{DF}),
the covariant derivative of
$\theta$ is given by (\ref{Dtheta}) for $p$ and $q$ constructed as above,
and $r\equiv 0$; hence, (\ref{beta}) and (\ref{frobenius1})--(\ref{frobenius2})
(equivalently, (\ref{defp})--(\ref{defq}))
are the same as relations
(\ref{system1})--(\ref{system3}); these, in turn, are a way of
re-writing (\ref{W^-=0}); it follows that
the projection of the curvature to $\Lambda^-M$ reduces to
$\frac{s}{12}{\rm Id}|_{\Lambdambda^-M}$, i.e. the Hermitian metric $g$
is Einstein and self-dual, with scalar curvature equal to $s$,
see (\ref{SO(4)}); turning back to (\ref{dalpha}),
we conclude that the conformal scalar curvature is $K{\"a}hler ppa=x^3$,
see (\ref{ricci1}); the metric constructed in this way is not of
cohomogeneity one, as
$m_0\neq 0$, see
Theorem \ref{th2}. Finally,
different solutions $(p,q)$ of (\ref{defp})--(\ref{defq}) give rise
to non-isometric metrics,
as $p$ and $q$ are completely determined by $|W^+|, {\rm d}|W^+|$ and ${\rm d}|D^gW^+|$,
see Sec.~ 2 and (\ref{alpha}).
We finally observe that the metric (\ref{canonic}) admits two commuting vector fields,
$\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial z}$.
We summarize the results obtained so far as follows:
\begin{theo}\lambdabel{th3} Let $(M,g,J)$ be a self-dual Einstein Hermitian
4-manifold. Suppose that $(M,g,J)$ is neither locally-symmetric nor
of cohomogeneity one.
Then, on an open dense subset of $M$, $g$ is locally given by (\ref{canonic}). In particular, $(M,g)$ admits a
local isometric action
of ${\mathbb R}^2$ almost-everywhere.
\end{theo}
\begin{rem} \lambdabel{rem3} {\rm (i) It is easily seen that the metrics (\ref{canonic})
have only
2-dimensional continuous symmetries. Moreover, as we already observed,
the coordinate $x=K{\"a}hler ppa^{\frac{1}{3}}$
is a {momentum map} of the Killing vector field
$\frac{\partial}{\partial t}$ with respect to the
K{\"a}hler metric
${\bar g}= x^2 g$ while, by (\ref{defp}) and (\ref{defh}),
a momentum map $\tilde{\mu}$ of the second Killing field,
$\frac{\partial}{\partial z}$,
is given by
$$2x{\tilde \mu} = y + \frac{x^3 + s}{12},$$
where $\frac{x^3 + s}{12} =
\frac{K{\"a}hler ppa + s}{12}$ is the (pointwise constant)
holomorphic sectional curvature of $(g,J)$.
The momentum map $x$ is also equal to the scalar curvature
of the K{\"a}hler metric ${\bar g}$.
A straighforward computation shows that the second momentum map $\tilde{\mu}$
defined above is related to the Pfaffian of the {\it normalized Ricci form}
${\bar \sigma}$ of the K{\"a}hler metric ${\bar g}$ by
$$ \tilde{\mu} = 12 \, ( {\rm Pfaff} \, {\bar \sigma} + b),$$
where $b$ is the constant appearing in (\ref{explicitef}) below. This
fits with an observation of R. Bryant in \cite{Br}.
({\rm Recall that for any
$2$-form $\psi$, the Pfaffian of $\psi$ with respect to ${\bar g}$
is defined by: $\psi \wedge \psi = 2\, {\rm Pfaff}\, \psi \, v _{{\bar
g}}$, where $v _{{\bar
g}}$ is the volume form of ${\bar g}$; the
normalized Ricci form ${\bar \sigma}$ is the $(1,1)$-form associated
to the normalized Ricci tensor, ${\bar S}$, appearing in the
usual decomposition ${\bar R} = {\bar S} \wedge {\bar g} + W$ of the
curvature operator of ${\bar g}$~; it is related to the usual Ricci
form ${\bar \rho}$ by ${\bar \sigma} = \frac{1}{2} \, ({\bar \rho} _0 +
\frac{x}{12} \, {\bar \omegaega})$, where ${\bar \rho} _0$ is the
trace-free part of ${\bar \rho}$; since $g = x ^{-2} {\bar g}$ is
Einstein and ${\rm d}^c x$ is the dual of a Killing vector field,
we have that ${\bar \rho} _0 = - \frac{1}{x} \, ({\rm d}
{\rm d}^c x) _0$; the result follows easily}).
(ii) It follows from Theorems \ref{th2} and \ref{th3} that
every self-dual Einstein
Hermitian 4-manifold admits a (local) isometric ${\Bbb R}^2$-action
compatible
with a {\it product structure} in the sense of \cite{joyce};
the general considerations in
\cite[Sec.2]{joyce} therefore apply to the present situation;
a detailed analysis
of self-dual Einstein 4-manifolds admitting ${\Bbb R}^2$-continuous symmetry
has been carried out by D. Calderbank \cite{Ca0}, based on results of \cite{Ca1}. }
\end{rem}
We end this section by providing an explicit form for the metric
(\ref{canonic}), in view of the following
\begin{Lemma}\lambdabel{integrate}
The solutions $p(x,y)$ and $q(x,y)$ of the system (\ref{defp})--(\ref{defq}) are explicitly given by
\begin{equation}\lambdabel{explicitep}
p = \frac{f}{y^2} - \frac{(x^3 -s)}{24y} + \frac{1}{4};
\end{equation}
\begin{equation}\lambdabel{expliciteq}
q^2= \frac{1}{y^2}\Big[\frac{x}{2}f' -f + \big(\frac{x^3-s}{24}\Big)^2\Big] - \frac{x^3}{24y} - p^2,
\end{equation}
where
\begin{equation}\lambdabel{explicitef}
f(x)= ax^2 + bx^4 - \frac{(x^6 - s^2)}{576},
\end{equation}
$a$ and $b$ are constants defined by positivity in (\ref{expliciteq}),
and $f'$ stands for the first derivative of $f$.
\end{Lemma}
\begin{proof}
We first observe that (\ref{defp}) can be equivalently written as
$${\rm d}\Big(y^2(p + \frac{(x^3 -s)}{24y} - \frac{1}{4})\Big) = $$
$$\frac{y^2}{x}\Big[2q^2 + 2(p + \frac{(x^3 -s)}{24y})(p -
\frac{(x^3 -s)}{24y}) + 2(p + \frac{(x^3 -s)}{24y} - \frac{1}{4}) + \frac{x^3}{12y}\Big]{\rm d}x;$$
this shows that $y^2(p + \frac{(x^3 -s)}{24y} - \frac{1}{4})$ is
function of $x$, say $f$; from the above equality, we get (\ref{explicitep}) and
(\ref{expliciteq}), where $f$ is a (still unknown) smooth function;
in order to determine $f$, we differentiate (\ref{expliciteq}) by
using (\ref{explicitep}) and
substitute into (\ref{defq}); then, cancellations occur and (\ref{defq})
eventually reduces to
\begin{equation}\lambdabel{ODE}
x^2f'' -5xf' + 8f + \frac{(x^6 -s^2)}{72}=0;
\end{equation}
the solutions of (\ref{ODE}) are given
by (\ref{explicitef}). \end{proof}
\section{Self-dual Einstein Hermitian metrics with hyperhermitian structures}
In this section, we consider self-dual, Einstein, Hermitian metrics
which in addition admit a {\it non-closed} hyperhermitian structure compatible
with the negative orientation. It is well-known that
LeBrun-Pedersen metrics, which are of cohomogeneity one under the action of the
unitary group ${\rm U(2)}$, carry such hyperhermitian structures;
in LeBrun's coordinates \cite{Le} these metrics read as follows:
\begin{equation}\lambdabel{g1}
g = \frac{1}{(b t^2 + 4c)^2}
\Big((1 + \frac{8b}{t^2} + \frac{16c}{t^4})^{-1}{\rm d}t^2 + \frac{t^2}{4}
\big[\sigma_1^2 + \sigma_2^2 + (1 + \frac{8b}{t^2} + \frac{16c}{t^4})
\sigma_3^2 \ \big]\Big),
\end{equation}
where $b$ and $c$ are properly chosen constants \cite{Mad}; more
precisely, we have the following
\begin{prop}\lambdabel{prop5}{\rm (\cite{Mad})}
Let $(M,g)$ be an oriented self-dual
Einstein 4-manifold. Assume that $(M,g)$ admits a $\rm{U}(2)$ isometric action with
generically three-dimensional $\rm{SU}(2)$-orbits.
If $g$ admits a non-closed, $\rm{U}(2)$-invariant
negative hyperhermitian structure, then
$g$ is isometric to (\ref{g1}) with $c > b^2$,
and actually admits exactly two distinct invariant hyperhermitian
structures.
\end{prop}
We here prove the following more general result:
\begin{theo}\lambdabel{th1} A self-dual Einstein Hermitian 4-manifold
$(M,g,J)$ locally admits a non-closed, negative hyperhermitian structure
if and only if
$g$ is locally isometric to one of the $\rm{U}(2)$-invariant metrics
(\ref{g1}) with $c > b^2$; then, $(M, g)$ actually carries
exactly two distinct hyperhermitian structures, each of them $\rm{U}(2)$-invariant.
\end{theo}
We first establish general facts concerning self-dual Einstein 4-manifolds
which carry a {\it non-closed} hyperhermitian structure compatible with the negative orientation.
As already observed in Sec.2, a (negative)
hyperhermitian structure $(g, I_1,I_2,I_3)$ is determined by a real $1$-form
$\theta$ --- the common Lee form of $(g,I_i)$, also the Lee form of
the Obata connection --- satisfying conditions
(\ref{EW}) and (\ref{hypherm}), and such that $\Phi := {\rm d} \theta$
is self-dual; in particular, the 2-form $\Phi$ is harmonic.
The next Lemma
shows that the self-dual Weyl tensor of $g$ is completely determined by
$\theta$, $\Phi$ and the
first covariant derivative $D^g\Phi$ of $\Phi$.
\begin{Lemma}\lambdabel{gauduchon1} Let $(M,g)$ be an oriented self-dual
Einstein 4-manifold and assume that $(M, g)$ carries a
negative hyperhermitian structure. Then, as a symmetric operator
acting on $\Lambdambda ^ + M$, the self-dual Weyl tensor $W^+$ is
given by
\begin{equation}\lambdabel{gau1}
W^+(\psi) = \frac{1}{2}[\psi, \Phi] + \frac{1}{|\theta|^2} D^g_{\psi(\theta)} \Phi,
\end{equation}
where $\psi$ is any self-dual 2-form, $\theta$ is viewed
as a vector field by Riemannian duality,
and $[\cdot, \cdot]$ denotes the commutator of
2-forms, viewed as skew-symmetric endomorphisms of the tangent bundle.
Moreover, $\theta$ and $\Phi$ are related by
\begin{equation}\lambdabel{gau3}
D^g_{\theta} \Phi = 2|\theta|^2\Phi.
\end{equation}
\begin{equation}\lambdabel{gau2}
{\rm d}|\theta|^2 -(\frac{s}{12} + |\theta|^2)\theta + \Phi(\theta)=0,
\end{equation}
\end{Lemma}
\begin{proof} By using (\ref{EW}), the right-hand side of
$$R_{X,Y}\theta= (D^g)^2_{Y,X}\theta - (D^g)^2_{X,Y}\theta $$
is easily computed; we thus obtain:
\begin{eqnarray}\lambdabel{util1}
R(\theta\wedge Z) &=& -\frac{1}{2}{\rm d}|\theta|^2\wedge Z - \frac{1}{2}(\frac{s}{12} -|\theta|^2)\theta \wedge Z\\ \nonumber
& &
-\frac{1}{2}\Phi(Z)\wedge \theta - \frac{1}{2}D^g_Z\Phi + \theta(Z)\Phi.
\end{eqnarray}
Since $g$ is self-dual and Einstein, $R = \frac{s}{12}{\rm Id}|_{\Lambdambda^2M} + W^+$,
see (\ref{SO(4)}).
Then, by projecting (\ref{util1}) to $\Lambda^-M$, we get
(\ref{gau2}), whereas
the projection of (\ref{util1}) to $\Lambda^+M$ gives (\ref{gau1}) and (\ref{gau3}).
\end{proof}
\begin{cor}\lambdabel{cor1} {\rm (\cite{ET,Ca})} Every hyperhermitian
structure on a conformally flat
4-manifold is closed.
\end{cor}
\begin{proof} If we assume that
$\Phi \neq 0$ somewhere on $M$ and that the anti-self-dual Weyl tensor
is identically zero, then,
after contracting (\ref{gau1}) and (\ref{gau3}) with $\Phi$, we obtain
$\theta = \frac{1}{4} {\rm d}\ln|\Phi|^2$,
which contradicts $\Phi = {\rm d}\theta \neq 0$. \end{proof}
We can compute the covariant derivative $D^g_{\theta}W^+$ of $W^+$
along the dual vector field of $\theta$ (still denoted by $\theta$), by using (\ref{gau1}) together with
(\ref{gau3}) and (\ref{gau2}) (the latter are used for evaluating the term
$(D^g)^2_{\theta,\psi(\theta)} \Phi$ which appears in the
calculation); we thus get
\begin{Lemma}\lambdabel{gauduchon2} Let $(M,g)$ be an oriented self-dual
Einstein 4-manifold, admitting a
negative hyperhermitian structure; then, the covariant derivative $D^g_{\theta} W^+$ of the
self-dual Weyl tensor $W^+$ along the dual vector field of the
Lee form $\theta$ is given by
\begin{eqnarray}\nonumber
\big( (D^g_{\theta} W^+)(\psi), \phi \big) &=& \big( [W^+(\phi),\psi] + [W^+(\psi),\phi], \Phi \big) \\ \nonumber
& & + (4|\theta|^2 - \frac{s}{6})\big( W^+(\psi), \phi \big) \\ \lambdabel{gau4}
& & + |\Phi|^2\big( \psi, \phi \big) -
3\big( \Phi, \psi \big) \big( \Phi, \phi \big) ,
\end{eqnarray}
for any sections, $\phi$ and $\psi$, of $\Lambda^+M$.
\end{Lemma}
From Lemma \ref{gauduchon2} and Propositions \ref{prop1} and
\ref{prop2}, we infer
\begin{prop}\lambdabel{prop6} Let $(M,g)$ be an oriented self-dual
Einstein 4-manifold, admitting a non-closed
hyperhermitian structure compatible with the negative orientation.
Then the following three conditions are equivalent:
\begin{enumerate}
\item[{\rm(i)}] the spectrum of $W^+$ is everywhere degenerate;
\item[{\rm(ii)}] $W^+$ has two distinct eigenvalues at any point;
\item[{\rm(iii)}] the self-dual 2-form $\Phi$ is a nowhere vanishing
eigenform for $W^+$ with respect to the simple eigenvalue,
and is proportional to a positive Hermitian structure $J$.
\end{enumerate}\
\end{prop}
\begin{proof}
${\rm (i)} \Rightarrow {\rm (ii)}$.
According to Proposition \ref{prop1}, if the spectrum of $W^+$ is
everywhere degenerate, then either $W^+$ vanishes identically (and therefore
the hyperhermitian
structure is closed by Corollary \ref{cor1}) or
$W^+$ has two distinct eigenvalues $\lambda$ and $-\frac{\lambda}{2}$ at any
point.
${\rm (ii)} \Rightarrow {\rm (iii)}$.
By Proposition \ref{prop1}, we know that
a normalized generator $F$ of the $\lambda$-eigenspace of $W^+$ is
the K{\"a}hler form of a positive Hermitian structure $J$.
Let $\phi$ be any self-dual 2-form orthogonal to
$F$, with $|\phi | ^2 = 2$; then, $\phi$ and $\psi = (J\circ \phi)$ are orthogonal,
$(-\frac{\lambda}{2})$-eigenforms of $W^+$; by substituting into
(\ref{gau4}), we get
$$ 0= \big( (D^g_{\theta} W^+)(\phi), \psi \big) = - 3\big( \Phi, \psi \big) \big( \Phi, \phi \big),$$
$$ -{\rm d}\lambda (\theta)= \big( (D^g_{\theta} W^+)(\phi), \phi \big) = -(4|\theta|^2 -\frac{s}{6})\lambda +2|\Phi|^2 - 3\big( \Phi, \phi \big)^2,$$
$$ -{\rm d}\lambda (\theta)= \big( (D^g_{\theta} W^+)(\psi), \psi \big) = -(4|\theta|^2 -\frac{s}{6})\lambda + 2|\Phi|^2 - 3\big( \Phi, \psi \big)^2.$$
From the last two equalities, we get
$\big( \Phi, \psi \big) = \pm \big( \Phi, \phi \big) $, and
by the first one we
conclude that $\big( \Phi, \psi \big) = \big( \Phi, \phi \big) =0$.
This shows that $\Phi$ is a multiple of $F$. It remains to prove that
$\Phi$ does not vanish on $M$; by taking a two-fold cover of $M$ if
necessary, we may assume that the Hermitian structure $J$ is globally
defined on $M$; by
Proposition \ref{prop2}, $(g,J)$ is conformally K{\"a}hler
and $\lambda^{\frac{2}{3}}F$
is the corresponding
closed K{\"a}hler form; but $\Phi$ is also a closed, self-dual 2-form,
and a multiple of ${F}$, hence a
constant (non zero) multiple of
$\lambda^{\frac{2}{3}}F$.
${\rm (iii)} \Rightarrow {\rm (i)}$.
This is an immediate consequence of Proposition \ref{prop1}.
\end{proof}
\noindent
{\bf Convention:}
From now on, we assume that $(M,g)$ is an oriented self-dual
Einstein 4-manifold
whose self-dual Weyl $W^+$ has degenerate spectrum, and which
admits a {\it non-closed} hyperhermitian structure compatible with the
negative orientation of $M$. According to Proposition \ref{prop6},
$W^+$ has two distinct eigenvalues which we denote by $\lambda$ and $-\frac{\lambda}{2}$,
and the harmonic self-dual 2-form $\Phi$
defines a positive Hermitian structure $J$ on
$(M,g)$ whose K{\"a}hler form, $F$, is an $\lambda$-eigenform for $W^+$. Moreover,
it follows from Proposition \ref{prop2} that,
after rescaling the metric if necessary, we may assume:
\begin{equation}\lambdabel{util2}
\Phi = \frac{1}{2}\lambda^{\frac{2}{3}}F.
\end{equation}
In the notation of Sec.2.1, the conformal
scalar curvature $K{\"a}hler ppa$ of $(g,J)$ is thus equal to $6\lambda$;
the Lee form
$\theta_J$ and the Killing vector field $K$, rescaled by an appropriate
positive constant, are therefore given by:
\begin{equation}\lambdabel{theta-X}
\theta_J= \frac{{\rm d}\lambda}{3\lambda}; \ \ K=J\rm{grad_g}(\lambda^{-\frac{1}{3}}),
\end{equation} (see Proposition \ref{prop2}).
At this point, our main technical result reads as follows:
\begin{prop}\lambdabel{prop7} A self-dual
Einstein Hermitian 4-manifold $(M,g,J)$ admits a
non-closed, hyperhermitian structure compatible with the negative orientation
if and only if
the Lee form $\theta_J$ satisfies
\begin{equation} \lambdabel{gau5} \begin{split}
D^g \theta_J &= \frac{(1+ \lambda^{\frac{2}{3}})(s +
3\lambda^{\frac{1}{3}})}{12}g \\ & \ \ \ +
\frac{(1 + 2\lambda^{\frac{2}{3}})}{(1 + \lambda^{\frac{2}{3}})} \theta_J\otimes \theta_J +
\frac{\lambda^{\frac{2}{3}}}{(1+\lambda^{\frac{2}{3}})}J\theta_J\otimes J\theta_J.
\end{split} \end{equation}
In this case,
$(M,g)$ actually admits exactly two non-closed hyperhermitian structures
$\{ I_1',I_2',I_3' \}$ and $\{ I_1'', I_2'', I_3'' \}$ whose Lee forms,
$\theta '$ and $\theta ''$, are given by
$$\theta ' = \frac{1}{(1 + \lambda^{\frac{2}{3}})}\big( \theta_J - \lambda^{\frac{1}{3}}
J\theta_J \big),$$
$$\theta '' = \frac{1}{(1 + \lambda^{\frac{2}{3}})}\big( \theta_J + \lambda^{\frac{1}{3}}
J\theta_J \big)$$
respectively.
Moreover, the Killing vector field $K$ is triholomorphic
for both hyperhermitian structures, i.e.,
$K$ preserves all complex structures $I_i'$ and $I_i''$, $i=1,2,3$.
\end{prop}
\begin{proof}
We first show that
if $(M,g,J)$ admits a non-closed hyperhermitian structure compatible with
the negative
orientation, then the corresponding Lee form $\theta$ must be one of the forms $\theta'$ and $\theta''$
given in Proposition \ref{prop7}.
From (\ref{gau3}) and the fact
that $\Phi$ is an $\lambda$-eigenform of $W^+$, we infer
\begin{equation}\lambdabel{util3}
{\rm d}|\Phi|^2 =4|\Phi|^2\theta + 4\lambda \Phi(\theta).
\end{equation}
By differentiating (\ref{util3}) and by using (\ref{gau2}) in order to
compute
${\rm d}(\Phi(\theta ))$, we obtain
$$({\rm d}\lambda - 3\lambda \theta)\wedge \Phi(\theta ) + \big(|\Phi|^2 + \lambda(\frac{s}{12} +
|\theta|^2)\big)\Phi =0; $$
we infer:
\begin{equation}\lambdabel{util4}
|\Phi|^2 = - \lambda(\frac{s}{12}+ |\theta|^2).
\end{equation}
By substituting the above expression of $|\Phi|^2$ in (\ref{util3}),
and by using (\ref{gau2}) again, we get
\begin{equation}\lambdabel{util5}
{\rm d}\lambda - 3\lambda\theta = \frac{3\lambda^2}{|\Phi|^2}\Phi(\theta).
\end{equation}
Now, according to the above convention,
by (\ref{theta-X}) and (\ref{util2}) we end up with the following
expression for $\theta$:
\begin{equation}\lambdabel{util7}
\theta = \frac{1}{(1 + \lambda^{\frac{2}{3}})}\big( \theta_J - \lambda^{\frac{1}{3}}
J\theta_J \big).
\end{equation}
This shows that every non-closed hyperhermitian structure
is completely determined by the self-dual harmonic 2-form $\Phi$.
It remains to prove that
$\Phi$ itself is
determined, up to sign, by the metric $g$; then, the two possible values of
$\theta$ appearing in Proposition \ref{prop7} will only differ by conjugation of
$J$ or, equivalently, by substituting $ - \Phi$ to $\Phi$.
Notice that, according to our convention, at this stage we have
the freedom to rescal the $2$-form $\Phi$ by a non-zero constant.
In other words, by
fixing one non-closed hyperhermitian structure and by following our
convention, we know that
any other non-closed hyperhermitian structure corresponds
to a harmonic 2-form of the form
$a \Phi = \frac{a}{2}\lambda^{\frac{2}{3}}F$, where $a$ is a non-zero
constant. Our claim is that $a=\pm 1$; to see this,
by using (\ref{gau1}) and (\ref{gau3}), we calculate
$$|D^g \Phi |^2 = 2|\theta|^2(3|\Phi|^2 + |W^+|^2);$$
in the present situation, when $W^+$ has degenerate spectrum,
the norm of $W^+$ is given by $|W^+|^2=\frac{3}{2}\lambda^2$;
then, by (\ref{util4}),
the above equality reduces itself to
\begin{equation}\lambdabel{util6}
|D^g \Phi |^2 = -(\frac{|\Phi|^2}{\lambda} + \frac{s}{12})(6|\Phi|^2 + 3\lambda^2);
\end{equation}
it is readily checked that if the $2$-forms
$\Phi$ and $a \Phi$ simultaneously satisfy (\ref{util6}), then $a=\pm 1$.
We now check that the conditions (\ref{EW})\&(\ref{hypherm}) for either
$\theta'$ or $\theta''$
are equivalent
to (\ref{gau5}).
Keeping (\ref{util2}) in mind, we see that (\ref{util5}) can be equivalently
re-written as
\begin{equation}\lambdabel{util8}
\theta_J= \theta + \lambda^{\frac{1}{3}}J\theta;
\end{equation}
then, the equivalence
``(\ref{gau5}) $\Leftrightarrow$ (\ref{EW})\&(\ref{hypherm})'' follows by
a straightforward computation involving the expressions
(\ref{util7}) and (\ref{util8}), and using formula (\ref{integrable});
the 1-forms
$\theta'$ and $\theta''$ thus
correspond to two distinct, non-closed hyperhermitian structures
$\{ I_1',I_2',I_3' \}$ and $\{ I_1'', I_2'', I_3'' \}$ provided that
(\ref{gau5}) holds, see Sec. 1.2.
As a final step, we have to prove that $K$ is triholomorphic with
respect to both
hyperhermitian structures. For a general hyperhermitian
structure $I_i, i=1,2,3$, with Lee form $\theta$, and
for any Killing field $K$, we have
$${\mathcal L}_K I_i = D_K I_i -[DK,I_i],$$
where $D$ is the
Weyl derivative given by (\ref{D^J}); we thus only need to check
that in our specific situation $D K$ commutes with $I_i$; by using (\ref{D^J}), (\ref{theta-X}),
(\ref{integrable}) and
(\ref{gau5}), we get
$$DK = \theta(K){{\rm Id}|_{TM}} + \frac{(1+ \lambda^{\frac{2}{3}})}{4} J;$$
the claim follows immediately. \end{proof}
\begin{cor}\lambdabel{cor2} {\rm (\cite{ET})} A locally-symmetric self-dual
Einstein 4-manifold does not admit non-closed hyperhermitian structures.
\end{cor}
\begin{proof} Any such manifold
is either a space of constant curvature, hence conformally flat, or
a K{\"a}hler manifold of constant holomorphic sectional curvature (see
Propositions \ref{prop1} and \ref{prop2}). In the former case, the claim follows
by Corollary \ref{cor1}, whereas in the latter case $\theta_J=0$; we
then
conclude by using
Proposition \ref{prop7}. \end{proof}
\begin{rem} {\rm D. Calderbank proved that
any conformal selfdual 4-manifold admitting two distinct Einstein-Weyl structures
is equipped with a canonical conformal submersion to an Einstein-Weyl
3-manifold \cite{Ca2}.
In the situation described by Proposition \ref{prop7}, this conformal
submersion is seen as follows: the hyperhermitian structures $\{
I_1',I_2',I_3' \}$ and $\{ I_1'', I_2'', I_3'' \}$ determine a SO(3)-valued
function, $p$, on $M$ defined by:
$$I_i'' = \sum_{j=1}^3 a_{ij}I_j'; \ A=(a_{ij})\in {\rm SO(3)};$$
we claim that $p$ is a conformal submersion of $(M,g)$ to
SO(3)=${\Bbb RP}^4$: The
differential of $p$ is easily computed by using the fact that $I_i''$ and $I_j'$ are both integrable; we thus obtain:
\begin{equation}\lambdabel{dA}
{\rm d}(a_{ij}) + \frac{\lambda^{\frac{2}{3}}}{2(1+ \lambda^{\frac{2}{3}})}\Sigma_{k=1}^3
a_{ik}\big([I'_k,I'_j] K\big)^{\sharp_g}=0;
\end{equation}
here, $[\cdot, \cdot]$ denotes the commutator of endomorphisms of $TM$
and $^{{\sharp}_g}$ stands for the Riemannian duality; from
(\ref{dA}), we infer:
$${\mathcal L}_{K} a_{ij} =0,$$
$$\sum_{i,j} \big(da_{ij}(X)\big)^2 =
\frac{\lambda^{\frac{4}{3}}}{2(1+\lambda^{\frac{2}{3}})^2}g(X,X), \ \forall
X\in K^{\perp};$$
The first equality shows
that $p$ coincides with the projection of $M$ to
the space, $N$, of orbits of $K$, whereas
the second equality means that the $K$-invariant metric
${\bar g}=\frac{\lambda^{\frac{2}{3}}}{(1+\lambda^{\frac{2}{3}})}g$ descends
to the round
metric of ${\rm SO(3)} = {\mathbb R} P^3$; in other words, $K$ defines a Riemannian submersion from $(M,{\bar g})$
to ${\rm SO(3)}$.}
\end{rem}
\noindent
{\bf Proof of Theorem \ref{th1}.}
We first notice that
the Killing vector field $K$ is trivial
if and only if $\lambda$ is constant (see (\ref{theta-X})), or, equivalently,
$\theta_J=0$.
Thus, according to Propositions \ref{prop6} and \ref{prop7},
if $(M,g,J)$ is a self-dual Einstein Hermitian 4-manifold
admitting a {\it non-closed} hyperhermitian structure,
the Killing vector field
$K$ does not vanish
on an open, dense subset of $M$.
It then follows from \cite{GT,CT,CP} that self-dual Einstein 4-manifolds
admitting two distinct hyperhermitian structures and a non-trivial
triholomorphic Killing
vector field are locally given by
Proposition \ref{prop5}.
For completeness, however, we here give a different and
more direct argument adapted to our ``Hermitian'' situation.
By Proposition \ref{prop5} it is sufficient to show that our metric can be
written
in the diagonal form (\ref{diagonal}). Since the eigenvalues of $W^+$ are
not constant, i.e., $\theta_J\neq 0$ (Proposition \ref{prop7}),
we introduce the variable $t=\lambda^{\frac{1}{3}}$;
the Lee form $\theta_J$ is then equal to $\frac{{\rm d}t}{t}$, whereas
the dual $1$-form of the Killing vector field is given by $-\frac{1}{t^2}J{\rm d}t$.
We set: $\sigma_3 = f(t)J{\rm d}t$,
for some smooth function $f$ of $t$,
and we insist that
\begin{equation}\lambdabel{dsigma3}
{\rm d}\sigma_3 = \sigma_1\wedge \sigma_2,
\end{equation}
where the 1-forms $\sigma_1$ and $\sigma_2=J\sigma_1$ are
both orthogonal to ${\rm d}t$ and satisfy
\begin{equation}\lambdabel{dsigma1}
{\rm d}\sigma_1 =\sigma_2\wedge \sigma_3; \ \ {\rm d}\sigma_2 = \sigma_3\wedge \sigma_1.
\end{equation}
We then derive $f$ from (\ref{dsigma3}):
By differentiating (\ref{util7}) and by making use of (\ref{util2}), we obtain
\begin{equation}\lambdabel{dJr}
{\rm d}(J{\rm d}t) = -\frac{(1+t^2)t^2}{2}F + \frac{2t}{(1+t^2)}{\rm d}t\wedge J{\rm d}t.
\end{equation}
By (\ref{util8}), (\ref{util4}) and (\ref{util2}), we also get
$$|{\rm d}t|^2 =-(\frac{t}{2} + \frac{s}{12})(t^4 + t^2);$$
it follows that $\big({\rm d}\sigma_3, {\rm d}t\wedge J{\rm d}t \big) =0$ if and only
if $(\ln f)'= -\frac{2t}{(1+t^2)} - \frac{1}{(t + \frac{s}{6})}$, where the
prime
stands for $\frac{{\rm d}}{{\rm d}t}$; we then have
$f= \frac{a}{(1+t^2)(t+ \frac{s}{6})}$, hence
\begin{equation}\lambdabel{sigma3}
\sigma_3 = \frac{a}{(1+t^2)(t + \frac{s}{6})} J{\rm d}t
\end{equation}
for a positive constant $a$.
In order to determine the 1-forms $\sigma_1$ and $\sigma_2$, we choose
a gauge
$\phi$ or, equivalently, a 1-form $\alpha =\phi(J\theta_J) \in {\mathcal
D}^{\perp}$; since $\sigma_1$ and $\sigma_2=J\sigma_1$ are orthogonal to ${\rm d}t$,
there certainly
exists
a smooth function $h$ of $t$ and a smooth function $\varphi$ on $M$, such that
$$\sigma_1 = h(\cos\varphi \alpha + \sin\varphi J\alpha);
\sigma_2=h(-\sin\varphi \alpha + \cos\varphi J\alpha);$$
by (\ref{sigma3}) and (\ref{dsigma3}), we obtain the following
expression for $h$:
\begin{equation}\lambdabel{h}
h^2 = \frac{at^2}{(t + \frac{s}{6})^2(1+t^2)};
\end{equation}
by using (\ref{sigma3}) and (\ref{ricci1}),
we now see that the conditions (\ref{dsigma1})
are equivalent to
\begin{equation}\lambdabel{varphi}
{\rm d}\varphi + \beta + \frac{(\frac{s}{6}-t^3 +at)}{t(1+t^2)(\frac{s}{6}+ t)}J{\rm d}t =0;
\end{equation}
therefore, the existence of a smooth function $\varphi$ satisfying
(\ref{varphi}) is equivalent to the following condition:
$${\rm d}(\beta + \frac{(\frac{s}{6}-t^3 +at)}{t(1+t^2)(\frac{s}{6}+ t)}J{\rm d}t)=0;$$
a straightforward computation involving (\ref{ricci2}) and (\ref{dJr}) shows
that the above equality holds whenever the constant $a$ is chosen
equal to $1 + \frac{s^2}{36}.$ \ \
\section{Hermitian structures on quaternionic quotients}
Let $(N,g)$ be a quaternionic K{\"a}hler manifold of real dimension $4n$,
endowed with a non-trivial Killing field $K$ which preserves the
quaternionic structure. According to
Galicki \cite{galicki1, galicki2} and
Galicki-Lawson \cite{G-L}, under some ``non-degeneracy'' condition
for $K$ one can define
a $4(n-1)$-dimensional quaternionic orbifold $(M,g^*)$
via the so-called {\it quaternionic reduction construction}. This
can be described as follows.
We first consider the following orthogonal
splitting
of the bundle of 2-forms:
\begin{equation}\lambdabel{quaternion-split}
\Lambda^2N = \Lambda^+N \oplus \Lambda^{1,1}N \oplus \Lambda^{\perp}N,
\end{equation}
where:
\begin{enumerate}
\item[$\bullet$] $\Lambda^+ N$ is the 3-dimensional
sub-bundle of ``self-dual''
2-forms which determines the {quaternionic structure} (also identified
to
a sub-bundle $A^+N$ of skew-symmetric endomorphism
of $TN$): both $A^+N$ and $\Lambda^+N$ are preserved by
the Levi-Civita connection, $D ^g$, and at each point $x$ of $N$
there is an orthonormal basis $\{ I_1, I_2, I_3 \}$ of $A^+N \subset {\rm End}(T_x N)$ with
the property that: $I_i\circ I_j = -\delta_{ij}{\rm Id}|_{TN} +
\epsilon_{ijk} I_k$
(resp. $\Lambda^+N = {\rm span}(\omegaega_1,\omegaega_2,\omegaega_3)$,
where $\omegaega_i$ are the fundamental 2-forms of the almost Hermitian
structures $(g,I_l)$. In the sequel, we refer to any such choice of
$I_l$'s (resp. $\omegaega_l$'s) as a {\it trivialization} of $A^+ N$
(resp. $\Lambda^+ N$);
\item[$\bullet$] $\Lambda^{1,1}N$ is the sub-bundle of 2-forms which are
$I_i$-invariant for any section of $A^+N$;
\item[$\bullet$] $\Lambda^{\perp}N$ denotes the orthogonal complement of
$\Lambda^+N \oplus \Lambda^{1,1}N$ in $\Lambda^2N$.
\end{enumerate}
We denote by ${\Pi^+}$ the projection of $\Lambda^2N$ to $\Lambda^+N$;
for any trivialization $\{ \omegaega_1, \omegaega_2, \omegaega_3 \}$ of
$\Lambda^+ N$ we then have
$$\Pi^+ = \frac{1}{2n}\sum_{l} \omegaega_l\otimes \omegaega_l,$$ and
$\Pi^+_{K} :=
\frac{1}{2n}\sum_{l} (i_K\omegaega_l \otimes \omegaega_l)$ is a section of
$T^*N\otimes \Lambda^+ N$. Then, Galicki-Lawson showed \cite[Th.~2.4]{G-L}.
that there exists
a section $f_K$ of $\Lambda^+N$ such that
$${\rm d}^{D^g} f_K = D ^g f_K = \Pi^+_K.$$
The section $f_K$ is called {\it the momentum map} associated to
$(N,g,K)$ and it is easily seen that the ``level set''
\begin{equation}\lambdabel{LK}\nonumber
L_{K} := \{ x\in N: f_K(x)=0 \}
\end{equation}
is $K$-invariant.
Assuming that $K_x \neq 0$ at $x\in L_{K}$,
Galicki-Lawson proved that $L_{K}$ is regular,
i.e. $L_K$ is a smooth submanifold of $N$. If moreover
the quotient space $M:= L_{K}/K$ is (locally) a $(4n-4)$-dimensional
manifold (or just an orbifold), then it becomes a
quaternionic K{\"a}hler manifold with respect to the ``projected''
quaternionic structure, $g^*$, of $N$.
Thus, when $N$ is 8-dimensional, the quaternonic reduction gives rise to a
four dimensional {\it anti-self-dual} Einstein orbifold
(with respect to the canonical orientation induced by $N$).
Note that when $K$ is the generator of a
$S^1$-quaternionic action on $N$, under
the non-degeneracy condition as above $M$ always inherits an orbifold
structure, cf. \cite[Th. 3.1 \& Cor. 3.2]{G-L}.
The above construction applies in particular to
$N = {\mathbb H}{P}^2$ endowed with certain {\it weighted} $S^1$-actions;
one thus obtains a wealth of examples of
{\it compact} anti-self-dual Einstein orbifolds; as shown by
Galicki-Lawson, the corresponding orbifolds are all
weighted projective planes ${\mathbb C} P^{[p_1,p_2,p_3]}$ for
some integers $0<p_1\le p_2\le p_3$ satisfying $p_3<p_1+p_2$,
\cite[Sec. 4]{G-L}.
Notice that, with respect
to the orientation induced by the canonical complex structure, the metric
becomes {\it self-dual}. (In the case when $p_1=p_2=p_3$ one obtains
the Fubini-Study metric on ${\mathbb C}{P}^2$).
On the other hand, R. Bryant showed \cite[Sec. 4.2]{Br}
that each weighted projective plane admits a
self-dual K{\"a}hler metric which under the above assumption
for the weights
has everywhere positive scalar curvature. Therefore, according to
\cite[Lemma \ref{de-ga}]{AG},
Bryant's metric
gives rise to
a self-dual {\it Einstein} Hermitian metric on
${\mathbb C} P^{[p_1,p_2,p_3]}$, $p_3<p_1+p_2$.
When considering both results together, a natural question arises:
\noindent
{\bf Question.} \cite{LeBrun}
Are the Galicki-Lawson metrics on ${\mathbb C} P^{[p_1,p_2,p_3]}$
Hermitian
with respect to some anti-self-dual complex structure?
In this section we show that this is indeed the case, at least on a
dense open subset; more generally,
we show that
the answer to the above question is essentially yes for any anti-self-dual
Einstein 4-orbifold obtained
by quaternionic reduction from the 8-dimensional Wolf spaces
${\mathbb H}P^2$, $SU(4)/S(U(2)U(2))$ and the corresponding
non-compact dual spaces (but according to \cite{kris}
the argument fails for quaternionic quotients of the exeptional 8-spaces
$G_2/SO(4)$ and $G^2_2/SO(4)$). More precisely, we have the following
\begin{prop}\lambdabel{quat-quot} Let $(N,g)$ be
${\mathbb H}{P}^2, SU(4)/S(U(2)U(2))$,
or one of the corresponding non-compact dual spaces.
Then, any anti-self-dual, Einstein
4-orbifold $(M,g^*)$ which is obtained as a quaternionic reduction
of $(N,g)$ by a quaternionic Killing field $K$
locally admits (a negatively oriented)
Hermitian structure $J$. In particular, the metric $g^*$ is locally given
by the explicit constructions in Sec. 2.
\end{prop}
The proof is based on the following simple observation.
\begin{Lemma} \lambdabel{Phi}
Let $(N,g)$ be a quaternionic K{\"a}hler manifold
of non-zero scalar curvature and $K$ be a Killing
field on $N$. Denote by $\Psi(X,Y)=(D ^g_X K, Y)$ the
2-form corresponding to $D ^g K$ and let $\Psi^+ = \Pi^+(\Psi)$
be the projection of $\Psi$ to
$\Lambda^+ N$. Then, up to multiplication by a constant,
the momentum map $f_K$ of $K$ is given by $\Psi^+$.
\end{Lemma}
\begin{proof} Since $K$ is Killing, equality (\ref{killing})
$$D ^g_X \Psi = R(K\wedge X)$$
holds.
For a quaternionic K{\"a}hler manifold the curvature operator $R$
acts on $\Lambda^+ N$ by $\lambdambda {\rm Id}|_{\Lambda^+N}$,
where $\lambdambda$ is a positive multiple of the scalar curvature,
cf. e.g. \cite{salamon}. Thus, projecting (\ref{killing}) to
$\Lambda^+N$ we get $D ^g_X \Psi^+ = \lambdambda \Pi^+_K. $
\end{proof}
By Lemma \ref{Phi}
the ``level set'' $L_K$ of $K$ is the same as the set of points
$x\in N$ where $\Psi^+_x =0$. Thus, at any point $x\in L_K$
the tangent space $T_xL_K$ is given by
$T_xL_K = \{ T_xN \ni X : D ^g_X \Psi^+ =0 \}.$
Since by assumption $K$ does not vanish on $L_K$, we conclude
by (\ref{killing}) and the fact that
$R|_{\Lambda^+N}= \lambdambda {\rm Id}|_{\Lambda^+N}$
$$T_xL_K = {\rm span}(I_1K,I_2K,I_3K)^{\perp},$$
where $\{I_1, I_2,I_3\}$ is any trivialization of $A^+N$.
We also observe that the 2-form $\Psi$ is a section
of $\Lambda^+N \oplus \Lambda^{1,1}N$, provided that
$K$ preserves the quaternionic structure. Indeed,
$$[D ^g K, I_l] = D ^g _K I_l -{\mathcal L}_K I_l, $$
where $[\cdot ,\cdot]$ stands for the commutator of ${\rm End}(TN)$.
Since $K$ is quaternionic, the left-hand-side of the above equality
is a section of $\Lambda^+ N$. By summing over $l$ in the above relation
we get
\begin{equation}\lambdabel{la11}
\Psi + 2\Pi^{1,1}(\Psi) \in \Lambda^+N,
\end{equation}
where
$\Pi^{1,1}$ denotes the projection to $\Lambda^{1,1}N$:
\begin{equation}\lambdabel{pi11}
\Pi^{1,1}(\psi)(\cdot,\cdot) = \frac{1}{4} \Big[(\psi(\cdot, \cdot) + \sum_l\psi(I_l\cdot ,I_l\cdot)\Big], \ \forall \psi \in \Lambda^2N.
\end{equation}
Thus, $\Psi$ is a section of
$\Lambda^+N \oplus \Lambda^{1,1}N$, and at $x\in L_K$, $\Psi_x$
actually belongs to $\Lambda_x^{1,1}N$.
Since $\Psi = \frac{1}{2} {\rm d} K^{\sharp}$,
where $K^{\sharp}$ is the $g$-dual 1-form of $K$, we conclude that
$${\mathcal L}_K \Psi = {\rm d}(i_K(\Psi)) = -\frac{1}{2} {\rm d}({\rm d}|K|^2)=0, $$
i.e. $\Psi$ is a closed $K$-invariant 2-form.
This shows that $\Psi$ projects to $M= L_K/K$ to define an
{\it anti-self-dual} form on $(M,g^*)$, then denoted by $\Psi^*$.
Considering the Riemannian submersion
$$\pi: L_K \longmapsto M = L_K/K,$$
the {\it horizontal} space, $H$, of $TL_K$ is given by
$$H = {\rm span}(K,I_1K,I_2K,I_3K)^{\perp}.$$
Note that $H$ is $I_l$-invariant for any section $I_l$ of $A^+N$.
Using the above remarks we calculate:
\begin{equation}\lambdabel{important}
(D ^{g^*}_{U^*} \Psi^*)(V^*, T^*) = (D ^g_U \Psi)(V,T)
-\frac{4}{|K|^2_g}\Pi^{1,1}(i_U\Psi \wedge i_K\Psi)(V,T),
\end{equation}
where $D ^{g^*}$ is the Levi-Civita connection of $g^*$, $U^*,V^*,T^*$
are any vectors on $M$, and $U,V,T$ are the corresponding horizontal lifts.
By assumption, $K$ has no zero on $L_K$;
it then follows from (\ref{important}) and (\ref{killing}) that
$\Psi^*$ does not vanish identically on $M$. Thus,
on the open subset of $(M,g^*)$ where $\Psi^* \neq 0$ the
normalised ASD form $\frac{{\sqrt 2} \Psi^*}{|\Psi^*|_{g^*}}$
determines a
{\it negative} almost Hermitian structure $J$.
By virtue of the Riemannian Goldberg-Sachs (\cite[Prop. 1]{AG}),
Proposition \ref{quat-quot} follows from the following
\begin{Lemma}\lambdabel{integrab} The almost-complex structure $J$ is integrable.
\end{Lemma}
\begin{proof}
We denote $Z^*_i$ any complex (1,0)-vector field of $(M,J)$ and $Z_i$
the corresponding horizontal lift (considered as complex vector in
$T_x^{\mathbb C} N$);
then, $J$ is integrable if and only if the following identity holds:
\begin{equation}\lambdabel{integrability0}
D ^{g ^*}_{Z^*_i} (\frac{{\sqrt 2}\Psi^*}{|\Psi^*|_{g^*}})(Z^*_j,Z^*_k) =
(D ^{g ^*}_{Z^*_i} \Psi^*)(Z^*_j, Z^*_k) =0 \ \forall i,j,k ;
\end{equation}
by the very definition of $J$ we have
$\Psi(Z_i,Z_j)=0$; moreover, since $\Psi$ belongs to $\Lambda^{1,1}N$ on $L_K$,
the almost complex structure $J$ (defined on $H$)
commutes with $I_l$'s for any trivialization $\{I_1,I_2,I_3\}$ of $A^+N$.
Then, by (\ref{important}) and (\ref{killing}) it is easily seen
that
the integrability condition (\ref{integrability0}) for $J$ is the
same as
\begin{equation}\lambdabel{integrability}
(D ^{g ^*}_{Z^*_i} \Psi^*)(Z^*_j, Z^*_k)=
({D^g}_{Z_i} \Psi)(Z_j, Z_k) = (R(K\wedge Z_i),Z_j\wedge Z_k) = 0.
\end{equation}
We now derive
(\ref{integrability}) from
the structure of the curvature tensor of the Riemannian
symmetric spaces ${\mathbb H}{P}^2, SU(4)/S(U(2)U(2))$
and the corresponding non-compact duals, ${\mathbb H}{H}^2$ and
$SU(2,2)/S(U(2)U(2))$ (we refer to
\cite{salamon, gauduchon}
for a general description of the curvature operator, $R$, of a
Riemannian symmetric space).
We first consider the simplest case
of $N={\mathbb H}{P}^2 = Sp(3)/(Sp(1)Sp(2))$ (or its non-compact
dual). The eigenspaces of $R$ are then
the simple
factors ${\bf sp}(1)$ and ${\bf sp}(2)$
of the isotropy Lie sub-algebra ${\bf h} = {\bf sp}(1) \oplus {\bf sp}(2)$,
and
the
orthogonal complement ${\bf h}^{\perp}$ of ${\bf h}$ in the space
${\rm Skew}({\bf m})$ of
the skew-symmetric endomorphisms of ${\bf m}= {\bf sp}(3)/{\bf h}$
(note that $R$ acts trivially on ${\bf h}^{\perp}$); the
decomposition ${\rm Skew}({\bf m}) = {\bf sp}(1) \oplus {\bf sp}(2)
\oplus {\bf h}^{\perp}$ into eigenspaces of
$R$ then fits with the splitting (\ref{quaternion-split});
$\Lambda^+N$ is thus identified to
${\bf sp}(1)$, and $\Lambda^{1,1}N$ to ${\bf sp}(2)$, whereas
$\Lambda^{\perp}N$
corresponds to the kernel of $R$, the space ${\bf h}^{\perp}$. This
shows that the curvature operator
acts on the first
two
factors in (\ref{quaternion-split}) by multiplication with a
non-zero constant (a certain multiple of the scalar curvature), and
acts trivially on the
third factor (therefore, $R$ has thus
three distinct eigenvalues, $\lambda,\mu$ and $0$); this observation also
shows that any
Killing field on ${\mathbb H}P^2$ is necessarily quaternionic.
As already observed, the almost complex structure $J$
(defined on $H$)
commutes with the $I_l$'s, so that $I_l(Z_k)$ is again a
(1,0)-vector of $(H,J)$; we thus get
$$\Pi^+(Z_j\wedge Z_k) = \sum_{l} (Z_j, I_l(Z_k))\omegaega_l= 0,$$
which means that $Z_j\wedge Z_k$ is an element of
$\Lambda_x^{1,1}M \oplus \Lambda_x^{\perp}N$. It then follows that
\begin{eqnarray}\nonumber
(R(K\wedge Z_i), Z_j\wedge Z_k) &=& (R(Z_j\wedge Z_k), K\wedge Z_i) \\ \nonumber
&=&
\mu(\Pi^{1,1}(Z_j\wedge Z_k), K\wedge Z_i).
\end{eqnarray}
But $\Pi^{1,1}(Z_j\wedge Z_k)$ is again a (2,0)-vector of $(M,J)$
(see formula (\ref{pi11})),
so that
$(\Pi^{1,1}(Z_j\wedge Z_k), K\wedge Z_i)=0$; this implies
(\ref{integrability}).
The same argument holds for the non-compact dual space ${\mathbb H}H^2$.
The case of $N=SU(4)/S(U(2)U(2))$ (or its non-compact dual)
is similar, but $N$ is now a {\it Hermitian symmetric} space, whose
canonical Hermitian structure $I$ comutes with any $I_i \in \Lambda_x^+ N$.
The corresponding
K{\"a}hler form, $\Omega_I$, then belongs to the space $\Lambda^{1,1}N$ and gives rise
to a further splitting
$$\Lambda^{1,1}N = {\mathbb R}\cdot \Omega_I \oplus \Lambda^{1,1}_0 N, $$
where $\Lambda^{1,1}_0 N$ is the orthogonal complement of $\Omega_I$.
Correspondingly, the eigenspaces of the curvature $R$ are the bundles
$\Lambda^+ N$, ${\mathbb R}\cdot \Omega_I$, $\Lambda^{1,1}_0 N$, and $\Lambda^{\perp} N$.
Note that $R$ acts trivially
on $\Lambda^{\perp}N$, whereas $\Omega_I$ is an eigenform
of $R$ corresponding to the simple eigenvalue; in particular,
$K$ must preserve $I$ and $\Omegaega_I$, so that $\Psi$ is of type $(1,1)$
with respect to $I$; in other words, the almost complex structure $I$
commutes with $J$,
when acting on $H$. It follows that $Z_i\wedge Z_j$ belongs to
$\Lambda^{1,1}_0 N \oplus \Lambda^{\perp}N$,
and we conclude as in the case of ${\mathbb H}P^2$.
\end{proof}
\noindent
{\bf Remark 5.} (i) By (\ref{important}) and Lemma \ref{integrab},
we see that $\frac{1}{|K|^2} \Psi^*$ is a harmonic 2-form
on $(M,g^*)$; it is actually the K{\"a}hler form of a self-dual
K{\"a}hler metric in the conformal class of $g^*$
(see \cite[Prop. 2]{AG}). In particular, if $(M,g^*)$ is not a
real space form, then $\Psi^*$ has no zero on $M$. By construction,
$\frac{2}{|K|^2}\Psi^*$ is the curvature form
of the submersion $\pi: L_K \longmapsto M$. It follows that
$L_K$ is a Sasakian manifold fibered over a K{\"a}hler self-dual
--- equivalently, a Bochner-flat ---
four-manifold. It is well known that the corresponding CR-structure
of $L_K$
has vanishing
fourth-order Chern-Moser curvature; therefore $L_{K}$ is uniformized
over $S^5$ with respect to ${\rm Aut}_{CR}(S^5)=PU(3,1)$,
cf. \cite{webster}.
(ii) As observed in \cite[p. 20]{G-L}, the
quaternionic reduction procedure can be applied
to the quaternionic hyperbolic space to obtain
{\it smooth}, {\it complete} (non locally symmetric)
Einstein self-dual metrics
of negative scalar curvature, which are necessarily
Hermitian by Lemma \ref{integrab};
see also \cite{Br} for another construction of complete
Einstein self-dual Hermitian metrics. In view of our first remark,
these examples seem to contradict
some results in \cite{kamishima}.
\end{document} |
\begin{document}
\begin{abstract}
We use an alternative definition of topological complexity to show that the topological complexity of the mapping telescope of a sequence $X_1\stackrel{f_1}{\longrightarrow}X_2\stackrel{f_2}{\longrightarrow}X_3\stackrel{f_3}{\longrightarrow}\ldots$ is bounded above by $2\max\{\mathord{\mathrm{TC}}(X_i);\;i=1,2,\ldots\}$.
\end{abstract}
\title{Topological complexity of the telescope}
\section{Introduction}
The notion of topological complexity was first introduced by Farber in \cite{Farber:TC}:
\begin{definition}\label{FarberDef}
{\em Topological complexity} $\mathord{\mathrm{TC}}(X)$ of a space $X$ is the least integer $n$ for which there exist an open cover $\{U_1, U_2,\ldots, U_n\}$ of $X\times X$ and sections $s_i\colon U_i\rightarrow X^I$ of the fibration $\pi\colon X^I\rightarrow X\times X$, $\alpha\mapsto (\alpha(0),\alpha(1))$. If no such integer exists we write $\mathord{\mathrm{TC}}(X)=\infty$.
\end{definition}
In \cite{IS}, Iwase and Sakai proved that (for nice spaces $X$) topological complexity is a special case of what James and Morris \cite{JamesMorris} call {\em fibrewise pointed LS category}. A {\em fibrewise pointed space} over a {\em base} $B$ is a topological space $E$, supplied with a {\em projection} $p\colon E\rightarrow B$ and a {\em section} $s\colon B\rightarrow E$. Fibrewise pointed spaces over a base $B$ form a category and the notions of fibrewise pointed maps and fibrewise pointed homotopies are defined as one would expect. More details can be found in \cite{James} and \cite{JamesMorris}.
We consider the product $X\times X$ as a fibrewise pointed space over the base $X$ with the projection to the first component and the diagonal section $\Delta\colon X\rightarrow X\times X$. According to Theorem 1.7 of \cite{IS}, we do not have to work with the fibrewise pointed homotopies but can instead use the less restrictive notion of (unpointed) fibrewise homotopies. A fibrewise homotopy in this case is any homotopy $H\colon X\times X\times I\rightarrow X\times X$ that fixes the first coordinate. So, $H(x,y,t)=(x,h(x,y,t))$ for some homotopy $h\colon X\times X\times I\rightarrow X$. For obvious reasons we call them {\em vertical homotopies}.
We can therefore consider the following theorem as an alternative definition of topological complexity:
\begin{theorem}\label{IwaseDef}
{\em Topological complexity} $\mathord{\mathrm{TC}}(X)$ of a space $X$ is the least integer $n$ for which there exists an open cover $\{U_1, U_2,\ldots, U_n\}$ of $X\times X$ such that each $U_i$ is vertically compressible to the diagonal $\Delta(X)$. If no such integer exists we write $\mathord{\mathrm{TC}}(X)=\infty$.
\end{theorem}
Note that we do not require the homotopies to be stationary on the section $\Delta(X)$, nor do we require the sets $U_i$ to contain the section.
Our result is analogous to the statement concerning LS category proven by Ganea in \cite{Ganea}. He gave an example to show that the LS category of the telescope is not necessarily equal to the LS categories of its parts. As we will see, this is also true for topological complexity. In \cite{Hardie}, Hardie improved Ganea's bound by 1 and Ganea's example shows that Hardie's bound is sharp.
\section{Topological complexity of the telescope}
We approach the problem indirectly by first estimating the topological complexity of an increasing union. The increasing union is much easier to handle and we can explicity construct a cover with the required properties. We then use homotopy invariance of topological complexity to apply the result to mapping telescopes.
\begin{theorem}\label{maintheorem}
Let $X=\bigcup_{i=1}^{\infty}X_i$ be the increasing union of closed subspaces with the property that for each $i$ there exists an open set $Y_i\subset X$ such that $X_i\subset Y_i\subset\mathord{\mathrm{cl}}(Y_i)\subset \mathord{\mathrm{int}}{(X_{i+1})}$. If $\mathord{\mathrm{TC}}(X_i)\leq n$ for all $i$, then $\mathord{\mathrm{TC}}(X)\leq 2n$.
\end{theorem}
\begin{proof}
Since $X_i\subset X_{i+1}$ for all $i$, we have $X_i\times X_i\subset X_{i+1}\times X_{i+1}$ for all $i$ and the product $X\times X=\bigcup_{i=1}^{\infty}X_i\times X_i$ is an increasing union of its subspaces. Let $\{U_j^{(i)}\}_{j=1}^n$ be an open cover of $X_i\times X_i$ with sets $U_j^{(i)}$ vertically compressible to the diagonal $\Delta(X_i)\subset\Delta(X)$.
Define $L_i=\mathord{\mathrm{int}}(X_i\times X_i)-\mathord{\mathrm{cl}}(Y_{i-2}\times Y_{i-2})$ for $i>2$, $L_2=\mathord{\mathrm{int}}(X_2\times X_2)$, $L_1=\mathord{\mathrm{int}}(X_1\times X_1)$. Here, $\mathord{\mathrm{int}}(A)$ and $\mathord{\mathrm{cl}}(A)$ denote the interior and the closure of $A$ as a subset of $X\times X$. Let $V_j^{(i)}=U_j^{(i)}\cap L_i$ and consider the sets
$$W_1 = \bigcup_{i=1}^\infty V_1^{(2i-1)}, W_2 = \bigcup_{i=1}^\infty V_1^{(2i)},\ldots, W_{2n-1} = \bigcup_{i=1}^\infty V_n^{(2i-1)}, W_{2n} = \bigcup_{i=1}^\infty V_n^{(2i)}.$$
Figure \ref{figure} illustrates the construction of the first three sets from $W_1$.
\begin{figure}
\caption{The shaded areas represent the sets $V_1^{(1)}
\label{figure}
\end{figure}
We observe the following:
\begin{itemize}
\item Every $(x,y)\in X$ belongs to $L_i$ for some $i$ and is therefore contained in $V_j^{(i)}$ for some $j$. So, $\{W_k\}_{k=1}^{2n}$ covers $X\times X$.
\item Each $V_j^{(i)}$ can be compressed to $\Delta(X_i)\subset\Delta(X)$ by the restriction of the vertical homotopy defined on $U_j^{(i)}$.
For all positive integers $l$ and $m$ we have $L_{l}\cap L_{m}=\emptyset$ as long as $|l-m|\geq 2$, so $V_j^{(l)}\cap V_j^{(m)}=\emptyset$ for $|l-m|\geq 2$.
The vertical homotopies we defined on $V_j^{(i)}$ can therefore be combined to define a (continuous) homotopy that vertically compresses $W_k$ to $\Delta(X)$.
\item The sets $L_i$ are open in $X\times X$, so $V_j^{(i)}=U_j^{(i)}\cap L_i$ are open in $L_i$ and therefore in $X\times X$. Each $W_k$ is defined as a union of open sets, so all $W_k$ are open.
\end{itemize}
From this we infer that $\{W_k\}_{k=1}^{2n}$ is indeed an open cover of $X\times X$ with each $W_k$ vertically compressible to $\Delta(X)$. The conclusion now follows from Theorem \ref{IwaseDef}.
\end{proof}
\begin{remark}
The proof of Theorem \ref{maintheorem} can be reused with only minor alterations to notation to prove a slightly more general statement. For a fibrewise pointed space $p\colon E\rightarrow B$ with section $s$ denote by $\mathord{\mathrm{cat}}_B^*(E)$ the {\em fibrewise unpointed category} as in Definition 1.6 of \cite{IS}. Assume that $E=\bigcup_{i=1}^{\infty}E_i$ is an increasing union of closed subspaces with the property that $s(p(E_i))\subset E_i$ and that there exist open sets $Y_i\subset E$ such that $E_i\subset Y_i\subset\mathord{\mathrm{cl}}{(Y_i)}\subset\mathord{\mathrm{int}}{(E_{i+1})}$. Let $B_i=p(E_i)$ and denote by $p_i\colon E_i\rightarrow B_i$ the restriction of $p$ to $E_i$ with the section $s_i$ being the restriction of section $s$ to $B_i$. If $\mathord{\mathrm{cat}}_{B_i}^*(E_i)\leq n$, then $\mathord{\mathrm{cat}}_{B}^*(E)\leq 2n$.
\end{remark}
We now represent a mapping telescope as an increasing union of subspaces and obtain the following result:
\begin{coro}\label{maincoro}
Let $X=\bigcup_{i=1}^{\infty}X_i\times[i-1,i]$ be the mapping telescope of a sequence of maps $$X_1\stackrel{f_1}{\longrightarrow}X_2\stackrel{f_2}{\longrightarrow}X_3\stackrel{f_3}{\longrightarrow}\ldots$$
and let $\mathord{\mathrm{TC}}(X_i)\leq n$ for all $i$. Then $\mathord{\mathrm{TC}}(X)\leq 2n$.
\end{coro}
\begin{proof}
Define $X'_n=\bigcup_{i=1}^{n}X_i\times[i-1,i]$ to be the union of the first $n$ mapping cylinders in the telescope $X=\bigcup_{i=1}^{\infty}X_i\times[i-1,i]$. Then $X$ is the increasing union $X=\bigcup_{i=1}^{\infty}X'_i$ and we can take
$$Y_i=\left(\bigcup_{i=1}^{n}X_i\times[i-1,i]\right)\cup X_{i+1}\times\left[i,i+1/2\right).$$
Since $X'_i$ are homotopy equivalent to $X_i$ for all $i$, we have $\mathord{\mathrm{TC}}(X'_i)=\mathord{\mathrm{TC}}(X_i)\leq n$ for all $i$. The conclusion now follows from Theorem~\ref{maintheorem}.
\end{proof}
Finally, here is an equivalent formulation of Corollary \ref{maincoro}:
\begin{coro}
Let $X=\bigcup_{i=1}^{\infty}X_i\times[i-1,i]$ be the mapping telescope of a sequence of maps $$X_1\stackrel{f_1}{\longrightarrow}X_2\stackrel{f_2}{\longrightarrow}X_3\stackrel{f_3}{\longrightarrow}\ldots.$$
Then $\mathord{\mathrm{TC}}(X)\leq 2\max\{\mathord{\mathrm{TC}}(X_i);\;i=1,2,\ldots\}$.
\end{coro}
\begin{proof}
If $\mathord{\mathrm{TC}}(X_i)$ are not bounded above, then $\max\{\mathord{\mathrm{TC}}(X_i);\;i=1,2,\ldots\}=\infty$ and the statement is trivially true. If $\max\{\mathord{\mathrm{TC}}(X_i);\;i=1,2,\ldots\}=M<\infty$, then $\mathord{\mathrm{TC}}(X_i)\leq M$ for all $i$ and Corollary~\ref{maincoro} implies that $\mathord{\mathrm{TC}}(X)\leq 2M$.
\end{proof}
\begin{example}
The mapping telescope of the sequence
$$S^1\stackrel{\cdot 2}{\longrightarrow}S^1\stackrel{\cdot 2}{\longrightarrow}S^1\stackrel{\cdot 2}{\longrightarrow}\ldots$$
is $X=K(\mathbb{Z}[\frac{1}{2}],1)$. We have $\mathord{\mathrm{TC}}(S^1)=2$ and Corollary \ref{maincoro} implies that $\mathord{\mathrm{TC}}(X)\leq 4$. The cohomology of $X$ is nontrivial only in dimension 2, and there we have $H^2(X;\mathbb{Z})=\hat{\mathbb{Z}}_2/\mathbb{Z}$, where $\hat{\mathbb{Z}}_2$ denotes the group of $2$-adic integers (detailed calculations can be found in \cite{Hatcher}, Section 3F, in particular Example 3F.9). Elements of finite order in $\hat{\mathbb{Z}}_2/\mathbb{Z}$ are represented by rational numbers. Since $\hat{\mathbb{Z}}_2/\mathbb{Z}$ is uncountable, there exists an element $u\in H^2(X;\mathbb{Z})$ of infinite order and we obtain a non-trivial product of length $2$:
$$(1\otimes u-u\otimes 1)^2=-2u\otimes u\in H^2(X;\mathbb{Z})\otimes H^2(X;\mathbb{Z}).$$
Combining Theorem 7 of \cite{Farber:TC} and Theorem 4 of \cite{Schwarz} we get a lower bound in terms of zero-divisors: $\mathord{\mathrm{TC}}(X)\geq 3$.
So, $3\leq\mathord{\mathrm{TC}}(X)\leq 4$.
Notice how in this example our upper bound is better than the standard upper bounds in terms of dimension and LS category (see \cite{Farber:TC}, Theorem 4 and Theorem 5), although it is not low enough to determine $TC(X)$.
\end{example}
This example shows that the topological complexity of the telescope $X$ can be greater than the topological complexity of its parts $X_i$. The question remains of whether our bound can be improved by 1.
\end{document} |
\begin{document}
\title{The minimum rank of a sign pattern matrix with a $1$-separation}
\begin{abstract}
A sign pattern matrix is a matrix whose entries are from the set $\{+,-,0\}$. If $A$ is an $m\times n$ sign pattern matrix, the qualitative class of $A$, denoted $Q(A)$, is the set of all real $m\times n$ matrices $B=[b_{i,j}]$ with $b_{i,j}$ positive (respectively, negative, zero) if $a_{i,j}$ is $+$ (respectively, $-$, $0$). The minimum rank of a sign pattern matrix $A$, denoted $\mbox{mr}(A)$, is the minimum of the ranks of the real matrices in $Q(A)$. Determination of the minimum rank of a sign pattern matrix is a longstanding open problem.
For the case that the sign pattern matrix has a $1$-separation, we present a formula to compute the minimum rank of a sign pattern matrix using the minimum ranks of certain generalized sign pattern matrices associated with the $1$-separation.
\end{abstract}
\section*{Introduction}
A \emph{sign pattern matrix} (or \emph{sign pattern}) is a matrix whose entries are from the set $\{+,-,0\}$. If $B=[b_{i,j}]$ is a real matrix, then $\mbox{sgn}(B)$ is the sign pattern matrix $A=[a_{i,j}]$ with $a_{i,j}=+$ (respectively, $-$, $0$) if $b_{i,j}$ is positive (respectively, negative, zero). If $A$ is a sign pattern matrix, the \emph{sign pattern class} of $A$, denoted $Q(A)$, is the set of all real matrices $B=[b_{i,j}]$ with $\mbox{sgn}(B)=A$.
The \emph{minimum rank} of a sign pattern matrix $A$, denoted $\mbox{mr}(A)$, is the minimum of the ranks of matrices in $Q(A)$; see \cite{Hall07}. Recently, Li et al. \cite{LivdHolst2013} obtained a characterization of sign pattern matrices $A$ with $\mbox{mr}(A)\leq 2$.
In this paper, we present a formula to compute the minimum rank of a sign pattern matrix with a $1$-separation using the minimum ranks of certain generalized sign pattern matrices associated with the $1$-separation.
The notion of sign pattern matrix can be extended to generalized sign pattern matrices by allowing certain entries to be $\#$; see \cite{Hall07}. For a generalized sign pattern matrix $A$, the generalized sign pattern class of $A$, denoted $Q(A)$, is defined by allowing entries of a matrix $B=[b_{i,j}]\in Q(A)$ to be any real number if the corresponding entries of $A$ are $\#$. The minimum rank $\mbox{mr}(A)$ of a generalized sign pattern matrix $A$ is defined in the same way as for a sign pattern matrix: $\mbox{mr}(A)$ is the minimum of the ranks of matrices in $Q(A)$.
If $A=[a_{i,j}]$ and $C=[c_{i,j}]$ are generalized sign pattern matrices of the same size, we write $A\leq C$ if for each entry of $A$, $a_{i,j} = c_{i,j}$ or $c_{i,j} = \#$. It is clear that if $A\leq C$, then $Q(A)\subseteq Q(C)$. For a generalized sign pattern $C$, let $\mathcal{C}$ be the set of all sign pattern matrices $A$ such that $A\leq C$. Then, clearly, $Q(C) = \cup_{A\in \mathcal{C}} Q(A)$. Hence the minimum rank of a generalized sign pattern matrix $C$ equals $\min_{A\in \mathcal{C}} \mbox{mr}(A)$.
We define subtraction of two elements from $\{+,-,0\}$ as follows:
\begin{enumerate}
\item $(+)-(0)=+,(0)-(-)=+, (+)-(-)=+$,
\item $(-)-(+)=-,(0)-(+)=-,(-)-(0)=-$,
\item $(0)-(0)=0$,
\item $(+)-(+)=\#,(-)-(-)=\#$.
\end{enumerate}
The idea behind the definition of, for example, $(-)-(+)=-$ is that subtracting a positive number from a negative number gives a negative number.
Let
\begin{equation*}
M = \begin{bmatrix}
A_{1,1} & A_{1,2} & 0\\
A_{2,1} & a_{2,2} & A_{2,3}\\
0 & A_{3,2} & A_{3,3}
\end{bmatrix}
\end{equation*}
be a sign pattern matrix, where $A_{1,2}$ has only one column and $A_{2,1}$ only one row. We also say that the sign pattern matrix $M$ has a \emph{$1$-separation}.
For $p\in \{+,-,0\}$, let
\begin{equation*}
R_p = \begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & p
\end{bmatrix}
\text{ and }
S_p = \begin{bmatrix}
a_{2,2}-p & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix}.
\end{equation*}
(Indeed $S_p$ might be a generalized sign pattern matrix.)
In this paper, we prove that the following formula holds:
\begin{equation}\label{mainformula}
\begin{split}
\mbox{mr}(M) = \min \{ & \mbox{mr}(A_{1,1})+ \mbox{mr}(A_{3,3})+2,\\
& \mbox{mr}([A_{1,1}\;A_{1,2}])+\mbox{mr}([A_{3,2}\;A_{3,3}])+1,\\
& \mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix})+\mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1,\\
& \mbox{mr}(R_+)+\mbox{mr}(S_+),\\
& \mbox{mr}(R_0)+\mbox{mr}(S_0),\\
& \mbox{mr}(R_-)+\mbox{mr}(S_-)
\}
\end{split}
\end{equation}
In the next section, we show that each of the terms in the minimum is at least $\mbox{mr}(M)$. In Section~\ref{sec:minrank1sep}, we show that at least one of the terms in the minimum attains $\mbox{mr}(M)$.
Formula~(\ref{mainformula}) is analogous to the formula for the minimum rank of $1$-sums of graphs. This formula was given by Hsieh \cite{Hsieh2001}, and, independently, by Barioli, Fallat, and Hogben \cite{BarFalHog2004a}. In case the graph is permitted to have loops, a formula was given by Mikkelson~\cite{Mikkelson:2008aa}.
The reader can see Fallat and Hogben \cite{FalHog2007} for a survey on the minimum ranks of graphs.
\section{Inequalities}
Let $0\leq k\leq m,n,r,s$ and let
\begin{equation*}
A=\begin{bmatrix}
A_{1,1}&A_{1,2}\\
A_{2,1}&A_{2,2}
\end{bmatrix} \text{ and } B=\begin{bmatrix}
B_{1,1}&B_{1,2}\\
B_{2,1}&B_{2,2}
\end{bmatrix}
\end{equation*}
be $m\times n$ and $r\times s$ matrices with real entries, respectively, where $A_{2,2}$ and $B_{1,1}$ are $k\times k$. In \cite{FalJoh1999a}, the \emph{$k-$subdirect sum} of $A$ and $B$, denoted by $A\oplus_k B$, was introduced; this is the matrix
\begin{equation*}A \oplus_{k} B=
\begin{bmatrix}
A_{1,1}&A_{1,2}&0\\
A_{2,1}&A_{2,2}+B_{1,1} & B_{1,2}\\
0 & B_{2,1} & B_{2,2}
\end{bmatrix}
\end{equation*}
\begin{lemma}\cite{MR3010007}\label{lem:sum}
Let
\begin{equation*}
C = \begin{bmatrix}
C_{1,1} & C_{1,2}\\
C_{2,1} & C_{2,2}
\end{bmatrix}
\quad\text{and}\quad
D = \begin{bmatrix}
D_{1,1} & D_{1,2}\\
D_{2,1} & D_{2,2}
\end{bmatrix},
\end{equation*}
where $C_{2,2}$ and $D_{1,1}$ are $k\times k$ matrices.
Then $\rank(C\oplus_k D)\leq \rank(C\oplus D)$.
\end{lemma}
In the next theorem, we show that each term in the minimum of Formula~(\ref{mainformula}) is at least $\mbox{mr}(M)$.
\begin{thm}
Let
\begin{equation*}
M=\begin{bmatrix}
A_{1,1}&A_{1,2}&0\\
A_{2,1}& a_{2,2} & A_{2,3}\\
0 & A_{3,2} & A_{3,3}\\
\end{bmatrix}
\end{equation*}
be a sign pattern matrix, where $A_{1,1}$ is $m_1\times n_1$, $A_{1,2}$ is $m_1\times 1$, $A_{2,1}$ is $1\times n_1$, $a_{2,2}$ is $1\times 1$, $A_{2,3}$ is $1\times n_2$, $A_{3, 2}$ is $m_2\times 1$ and $A_{3,3}$ is $m_2\times n_2$. Then each of the following inequalities hold:
\begin{enumerate}[(i)]
\item $\mbox{mr}(A_{1,1})+\mbox{mr}(A_{3,3})+2\geq \mbox{mr}(M)$,
\item $\mbox{mr}([A_{1,1}\;A_{1,2}])+\mbox{mr}([A_{3,2}\;A_{3,3}])+1\geq \mbox{mr}(M)$,
\item $\mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}\\
\end{bmatrix})+\mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}\\
\end{bmatrix})+1\geq \mbox{mr}(M)$, and
\item
for each $p\in \{+,-,0\}$,
\begin{equation*}
\mbox{mr}(\begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & p
\end{bmatrix}
)+\mbox{mr}(
\begin{bmatrix}
a_{2,2}-p & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix}
)\geq \mbox{mr}(M).
\end{equation*}
\end{enumerate}
\end{thm}
\begin{proof}
To see that $\mbox{mr}(A_{1,1})+\mbox{mr}(A_{3,3})+2\geq \mbox{mr}(M)$, let $C_{1,1}\in Q(A_{1,1})$ and $C_{3,3} \in Q(A_{3,3})$ such that $\rank(C_{1,1}) = \mbox{mr}(A_{1,1})$ and $\rank(C_{3,3})=\mbox{mr}(A_{3,3})$.
Let $C_{1,2}\in Q(A_{1,2})$, $C_{2,1}\in Q(A_{2,1})$, $C_{2,3} \in Q(A_{2,3})$, $C_{3,2}\in Q(A_{3,2})$, and $\mbox{sgn}(c_{2,2}) = a_{2,2}$. Then
\begin{equation*}
\begin{split}
\mbox{mr}(A_{1,1}) + \mbox{mr}(A_{3,3})+2 & = \rank(\begin{bmatrix}
C_{1,1} & 0\\
0 & C_{3,3}
\end{bmatrix}) + 2\\
&\geq \rank(\begin{bmatrix}
C_{1,1} & C_{1,2} & 0\\
C_{2,1} & c_{2,2} & C_{2,3}\\
0 & C_{3,2} & C_{3,3}
\end{bmatrix})\\
&\geq \mbox{mr}(M),
\end{split}
\end{equation*}
To see that $\mbox{mr}([A_{1,1}\;A_{1,2}])+\mbox{mr}([A_{3,2}\;A_{3,3}])+1\geq \mbox{mr}(M)$, let $[C_{1,1}\;C_{1,2}]\in Q([A_{1,1}\;A_{1,2}])$ and $[C_{3,2}\;C_{3,3}]\in Q([A_{3,2}\;B_{3,3}])$ be such that $\rank([C_{1,1}\;C_{1,2}])=\mbox{mr}([A_{1,1}\;A_{1,2}])$ and $\rank([C_{3,2}\;C_{3,3}])=\mbox{mr}([A_{3,2}\;A_{3,3}])$. Clearly,
\begin{equation*}
\rank([C_{1,1}\;C_{1,2}]) + \rank([C_{3,2}\;C_{3,3}])\geq \rank(\begin{bmatrix}
C_{1,1} & C_{1,2} & 0\\
0 & C_{3,2} & C_{3,3}
\end{bmatrix}).
\end{equation*}
Since
\begin{equation*}
\rank(\begin{bmatrix}
C_{1,1} & C_{1,2} & 0\\
0 & C_{3,2} & C_{3,3}
\end{bmatrix}) + 1\geq \rank(
\begin{bmatrix}
C_{1,1} & C_{1,2} & 0\\
C_{2,1} & c_{2,2} & C_{2,3}\\
0 & C_{3,2} & C_{3,3}
\end{bmatrix}),
\end{equation*}
we obtain $\mbox{mr}([A_{1,1}\;A_{1,2}])+\mbox{mr}([A_{3,2}\;A_{3,3}])+1\geq \mbox{mr}(M)$.
The proof that
$\mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}\\
\end{bmatrix})+\mbox{mr}(\begin{bmatrix}
B_{1,2}\\
B_{2,2}\\
\end{bmatrix})+1\geq \mbox{mr}(M)$ is similar to the proof of the previous case.
Let $p\in \{+,-,0\}$. To shorten notation, let
\begin{equation*}
R_p = \begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & p
\end{bmatrix}
\text{ and }
S_p = \begin{bmatrix}
a_{2,2}-p & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix}.
\end{equation*}
To see that $\mbox{mr}(R_p) + \mbox{mr}(S_p) \geq \mbox{mr}(M)$,
let
\begin{equation*}
C = \begin{bmatrix}
C_{1,1} & C_{1,2}\\
C_{2,1} & c
\end{bmatrix}\in Q(R_p)
\text{ and }
D = \begin{bmatrix}
d & C_{2,3}\\
C_{3,2} & C_{3,3}
\end{bmatrix}\in Q(S_p)
\end{equation*}
be such that $\rank(C) = \mbox{mr}(R_p)$ and $\rank(D)=\mbox{mr}(S_p)$.
We now do a case-checking.
Suppose first that $a_{2,2}-p=0$. Then $p=0$ and $a_{2,2}=0$.
Hence $c=0$ and $d=0$. Then $C\oplus_1 D \in Q(M)$, and, by Lemma~\ref{lem:sum}, $\mbox{mr}(M)\leq \rank(C\oplus_1 D)\leq \rank(C)+\rank(D) = \mbox{mr}(R_p) + \mbox{mr}(S_p)$.
Suppose next that $a_{2,2}-p=+$. Then one of the following holds:
\begin{enumerate}
\item $p=-$ and $a_{2,2}=0$,
\item $p=0$ and $a_{2,2}=+$, and
\item $p=-$ and $a_{2,2}=+$.
\end{enumerate}
Suppose $p=-$ and $a_{2,2}=0$. By scaling $D$ by a positive scalar, we may assume that $d=-c$. Then $C\oplus_1 D\in Q(M)$, and, by Lemma~\ref{lem:sum}, $\mbox{mr}(M) \leq \rank(C\oplus_1 D)\leq \rank(C)+\rank(D) = \mbox{mr}(R_p) + \mbox{mr}(S_p)$. Suppose $p=0$ and $a_{2,2}=+$. Then $C\oplus_1 D\in Q(M)$, and, by Lemma~\ref{lem:sum}, $\mbox{mr}(M) \leq \rank(C\oplus_1 D)\leq \rank(C)+\rank(D) = \mbox{mr}(R_p) + \mbox{mr}(S_p)$. Suppose $p=-$ and $a_{2,2}=+$. By scaling $D$ by a positive scalar, we may assume that $c+d > 0$. Then $C\oplus_1 D\in Q(M)$, and, by Lemma~\ref{lem:sum}, $\mbox{mr}(M)\leq \rank(C\oplus_1 D)\leq \rank(C)+\rank(D) = \mbox{mr}(R_p) + \mbox{mr}(S_p)$.
The case where $a_{2,2}-p=-$ is similar.
Suppose finally that $a_{2,2}-p=\#$. Then one of the following holds:
\begin{enumerate}
\item $p=+$ and $a_{2,2}=+$, and
\item $p=-$ and $a_{2,2}=-$.
\end{enumerate}
Suppose $p=+$ and $a_{2,2}=+$. Then $C\oplus_1 D\in Q(M)$, and, by Lemma~\ref{lem:sum}, $\mbox{mr}(M) \leq \rank(C\oplus_1 D)\leq \rank(C)+\rank(D) = \mbox{mr}(R_p) + \mbox{mr}(S_p)$. The case where $p=-$ and $a_{2,2}=-$ is similar.
\end{proof}
\section{Minimum Rank of Sign Pattern with 1-Separation}\label{sec:minrank1sep}
In this section we finish the proof that Formula~(\ref{mainformula}) is correct. First we prove some lemmas.
\begin{lemma}\label{lem:vertexadd}
For any $m\times n$ real matrix $B$ with $m, n\geq 1$, and any nonzero real numbers $a$ and $c$,
\begin{equation*}
\rank(\begin{bmatrix}
0 & a & 0\\
c & b_{1,1} & B_{1,2}\\
0 & B_{2,1} & B_{2,2}
\end{bmatrix}) = \rank(B_{2,2}) + 2.
\end{equation*}
\end{lemma}
\begin{proof}
Let
\begin{equation*}
P = \begin{bmatrix}
\frac{1}{a} & 0 & 0\\
-\frac{b_{1,1}}{2a} & 1 & 0\\
-\frac{B_{2,1}}{a} & 0 & I_{m-2}
\end{bmatrix}\quad\text{and}\quad
Q = \begin{bmatrix}
\frac{1}{c} & -\frac{b_{1,1}}{2c} & -\frac{B_{1,2}}{c}\\
0 & 1 & 0\\
0 & 0 & I_{n-2}
\end{bmatrix}.
\end{equation*}
Then
\begin{equation*}
P \begin{bmatrix}
0 & a & 0\\
c & b_{1,1} & B_{1,2}\\
0 & B_{2,1} & B_{2,2}
\end{bmatrix} Q =
\begin{bmatrix}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & B_{2,2}
\end{bmatrix}.
\end{equation*}
From this the lemma easily follows.
\end{proof}
\begin{lemma}\label{lem:adjoin}
Let $A = \begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & A_{2,2}
\end{bmatrix}$ be a real matrix, where $A_{1,1}$ is $m_1\times n_1$, $A_{1,2}$ is $m_1\times n_2$, $A_{2,1}$ is $m_2\times n_1$, and $A_{2,2}$ is $m_2\times n_2$.
If $x\in \ker(A_{2,2}^T)$ and $y\in \ker(A_{2,2})$, then
\begin{equation*}
\rank \begin{bmatrix}
0 & x^T A_{2,1} & 0\\
A_{1,2} y & A_{1,1} & A_{1,2}\\
0 & A_{2,1} & A_{2,2}
\end{bmatrix} = \rank A.
\end{equation*}
\end{lemma}
\begin{proof}
Let
\begin{equation*}
P = \begin{bmatrix}
0 & x^T\\
I_{m_1} & 0\\
0 & I_{m_2}
\end{bmatrix}\quad\text{and}\quad
Q= \begin{bmatrix}
0 & I_{n_1} & 0\\
y & 0 & I_{n_2}
\end{bmatrix}.
\end{equation*}
Then
\begin{equation*}
P A Q= \begin{bmatrix}
0 & x^T A_{2,1} & 0\\
A_{1,2} y & A_{1,1} & A_{1,2}\\
0 & A_{2,1} & A_{2,2}
\end{bmatrix}.
\end{equation*}
Hence, $\rank \begin{bmatrix}
0 & x^T A_{2,1} & 0\\
A_{1,2} y & A_{1,1} & A_{1,2}\\
0 & A_{2,1} & A_{2,2}
\end{bmatrix} \leq \rank A$. The other inequality is clear.
\end{proof}
\begin{lemma}\label{lem:decomp}
Let $A = \begin{bmatrix}
A_{1,1} & A_{1,2} & 0\\
A_{2,1} & a_{2,2} & A_{2,3}\\
0 & A_{3,2} & A_{3,3}
\end{bmatrix}$ be an $m\times n$ real matrix, where $A_{1,1}$ is $m_1\times n_1$ and $A_{3,3}$ is $m_2\times n_2$, (and so $m=m_1+m_2+1$ and $n=n_1+n_2+1$). Then at least one of the following holds:
\begin{enumerate}[(i)]
\item There exist vectors $v \in \mathbb{R}^{m_1}$ and $z\in \mathbb{R}^{n_1}$ such that
\begin{equation*}
\rank(\begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & v^T A_{1,1} z
\end{bmatrix})+\rank(
\begin{bmatrix}
a_{2,2} - v^T A_{1,1} z & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix}) = \rank(A).
\end{equation*}
\item\label{item:sumcase2} $\rank(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix}) + \rank(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1 = \rank(A)$.
\item\label{item:sumcase3} $\rank([A_{1,1}\;A_{1,2}]) + \rank([A_{3,2}\;A_{3,3}]) +1 = \rank(A)$.
\item $\rank(A_{1,1}) + \rank(A_{3,3}) + 2=\rank(A)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Suppose first that $[A_{2,1}\;A_{2,3}]x = 0$ for all $x\in \ker(A_{1,1}\oplus A_{3,3})$ and that $y^T \begin{bmatrix}
A_{1,2}\\
A_{3,2}
\end{bmatrix} = 0$ for all $y\in \ker((A_{1,1}\oplus A_{3,3})^T)$.
Then there exist a vector $v\in \mathbb{R}^{m_1}$ such that $v^T A_{1,1} = A_{2,1}$ and a vector $z\in \mathbb{R}^{n_1}$ such that $A_{1,1} z = A_{1,2}$. Let
\begin{equation*}
P = \begin{bmatrix}
I_{m_1} & 0 & 0\\
v^T & 0 & 0\\
-v^T & 1 & 0\\
0 & 0 & I_{m_2}
\end{bmatrix}\quad\text{and}\quad
Q=\begin{bmatrix}
I_{n_1} & z & -z & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & I_{n_2}
\end{bmatrix}.
\end{equation*}
A calculation shows that
\begin{equation*}
P A Q = \begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & v^T A_{1,1} z
\end{bmatrix}\oplus
\begin{bmatrix}
a_{2,2} - v^T A_{1,1} z & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix}.
\end{equation*}
Hence
\begin{equation*}
\rank(A) \geq \rank(\begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & v^T A_{1,1} z
\end{bmatrix}) + \rank(
\begin{bmatrix}
a_{2,2} - v^T A_{1,1} z & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix})
\end{equation*}
By Lemma~\ref{lem:sum}, the opposite inequality also holds.
Suppose next that $[A_{2,1}\;A_{2,3}]x = 0$ for all $x\in \ker(A_{1,1}\oplus A_{3,3})$ and that there exists a $y\in \ker((A_{1,1}\oplus A_{3,3})^T)$ such that $y^T \begin{bmatrix}
A_{1,2}\\
A_{3,2}
\end{bmatrix} = e \not= 0$.
By Lemma~\ref{lem:adjoin},
\begin{equation*}
\rank(\begin{bmatrix}
0 & 0 & e & 0\\
0 & A_{1,1} & A_{1,2} & 0\\
0 & A_{2,1} & a_{2,2} & A_{2,3}\\
0 & 0 & A_{3,2} & A_{3,3}
\end{bmatrix}) = \rank(A).
\end{equation*}
Hence
\begin{equation*}
1 + \rank(
\begin{bmatrix}
A_{1,1} & 0\\
A_{2,1} & A_{2,3}\\
0 & A_{3,3}
\end{bmatrix}) = \rank(A).
\end{equation*}
Since
\begin{equation*}
\mbox{nullity }(\begin{bmatrix}
A_{1,1} & 0\\
A_{2,1} & A_{2,3}\\
0 & A_{3,3}
\end{bmatrix}) =
\mbox{nullity }(
\begin{bmatrix}
A_{1,1} & 0\\
0 & A_{3,3}
\end{bmatrix}),
\end{equation*}
we obtain
\begin{equation*}
\rank(\begin{bmatrix}
A_{1,1} & 0\\
A_{2,1} & A_{2,3}\\
0 & A_{3,3}
\end{bmatrix}) =
\rank(\begin{bmatrix}
A_{1,1} & 0\\
0 & A_{3,3}
\end{bmatrix}).
\end{equation*}
Hence $\rank(A) = \rank(A_{1,1}) + \rank(A_{3,3})+1$.
From $[A_{2,1}\;A_{2,3}]x = 0$ for all $x\in \ker(A_{1,1}\oplus A_{3,3})$, it follows that $\rank(\begin{bmatrix}
A_{1,1}\\
A_{1,2}
\end{bmatrix})
= \rank(A_{1,1})$ and $\rank(
\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix}) = \rank(A_{3,3})$. Thus $\rank(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix}) + \rank(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1 = \rank(A)$.
The case that there exists an $x\in \ker(A_{1,1}\oplus A_{3,3})$ such that $[A_{2,1}\;A_{2,3}]x$ is nonzero and $y^T \begin{bmatrix}
A_{1,2}\\
A_{3,2}
\end{bmatrix} = 0$ for all $y\in \ker((A_{1,1}\oplus A_{3,3})^T)$ yields $\rank([A_{1,1}\;A_{1,2}]) + \rank([A_{3,2}\;A_{3,3}]) +1 = \rank(A)$.
Hence, we are left with the case that there exist an $x\in \ker(A_{1,1}\oplus A_{3,3})$ such that $f = [A_{2,1}\;A_{2,3}]x$ is nonzero and there exists a $y\in \ker((A_{1,1}\oplus A_{3,3})^T)$ such that $e = y^T \begin{bmatrix}
A_{1,2}\\
A_{3,2}
\end{bmatrix}$ is nonzero.
Then, by Lemma~\ref{lem:adjoin},
\begin{equation*}
\rank(\begin{bmatrix}
0 & 0 & e & 0\\
0 & A_{1,1} & A_{1,2} & 0\\
f & A_{2,1} & a_{2,2} & A_{2,3}\\
0 & 0 & A_{3,2} & A_{3,3}
\end{bmatrix}) = \rank(A).
\end{equation*}
By Lemma~\ref{lem:vertexadd},
\begin{equation*}
\rank(\begin{bmatrix}
A_{1,1} & 0\\
0 & A_{3,3}
\end{bmatrix}) + 2 = \rank(A).
\end{equation*}
Thus $\rank(A_{1,1}) + \rank(A_{3,3}) + 2 = \rank(A)$.
\end{proof}
\begin{remark}
The proof shows that if Case~(\ref{item:sumcase2}) happens, then $A_{2,1}$ belongs to the row space of $A_{1,1}$, and $A_{2,3}$ belongs to the row space of $A_{3,3}$. Similarly, if Case~(\ref{item:sumcase3}) happens, then $A_{1,2}$ belongs to the column space of $A_{1,1}$, and $A_{2,3}$ belongs to the column space of $A_{3,3}$.
\end{remark}
\begin{thm}
Let
\begin{equation*}
M = \begin{bmatrix}
A_{1,1} & A_{1,2}& 0\\
A_{2,1} & a_{2,2} & A_{2,3}\\
0 & A_{3,2} & A_{3,3}
\end{bmatrix},
\end{equation*}
where $A_{1,2}$ is $n_1\times 1$, $A_{2,1}$ is $m_1\times 1$, and, for $p\in \{+,-,0\}$, let
\begin{equation*}
R_p = \begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & p
\end{bmatrix}
\text{ and }
S = \begin{bmatrix}
a_{2,2}-p & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix}.
\end{equation*}
Then
\begin{equation*}
\begin{split}
\mbox{mr}(M) = \min \{ & \mbox{mr}(A_{1,1})+ \mbox{mr}(A_{3,3})+2,\\
& \mbox{mr}([A_{1,1}\;A_{1,2}])+\mbox{mr}([A_{3,2}\;A_{3,3}])+1,\\
& \mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix})+\mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1,\\
& \mbox{mr}(R_+)+\mbox{mr}(S_+),\\
& \mbox{mr}(R_0)+\mbox{mr}(S_0),\\
& \mbox{mr}(R_-)+\mbox{mr}(S_-)
\}
\end{split}
\end{equation*}
\end{thm}
\begin{proof}
By the previous section,
\begin{equation}\label{eq:formula2}
\begin{split}
\mbox{mr}(M) \leq \min \{ & \mbox{mr}(A_{1,1})+ \mbox{mr}(A_{3,3})+2,\\
& \mbox{mr}([A_{1,1}\;A_{1,2}])+\mbox{mr}([A_{3,2}\;A_{3,3}])+1,\\
& \mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix})+\mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1,\\
& \mbox{mr}(R_+)+\mbox{mr}(S_+),\\
& \mbox{mr}(R_0)+\mbox{mr}(S_0),\\
& \mbox{mr}(R_-)+\mbox{mr}(S_-)
\}
\end{split}
\end{equation}
We now show that at least one of the terms in the minimum on the right-hand side of $(\ref{eq:formula2})$ equals $\mbox{mr}(M)$.
Let
\begin{equation*}
C = \begin{bmatrix}
C_{1,1} & C_{1,2} & 0\\
C_{2,1} & c_{2,2} & C_{2,3}\\
0 & C_{3,2} & C_{3,3}
\end{bmatrix}\in Q(M)
\end{equation*}
be such that $\rank(C) = \mbox{mr}(M)$.
Then, by Lemma~\ref{lem:decomp}, at least one of the following holds:
\begin{enumerate}[(i)]
\item\label{item1} There exist vectors $v\in\ensuremath{\mathbb{R}}^{n_1}$ and $z\in\ensuremath{\mathbb{R}}^{m_1}$ such that
\begin{equation*}
\rank(\begin{bmatrix}
C_{1,1} & C_{1,2}\\
C_{2,1} & v^T C_{1,1} z
\end{bmatrix})+\rank(
\begin{bmatrix}
c_{2,2} - v^T C_{1,1} z & C_{2,3}\\
C_{3,2} & C_{3,3}
\end{bmatrix}) = \rank(C).
\end{equation*}
\item\label{item2} $\rank(\begin{bmatrix}
C_{1,1}\\
C_{2,1}
\end{bmatrix}) + \rank(\begin{bmatrix}
C_{2,3}\\
C_{3,3}
\end{bmatrix})+1 = \rank(C)$.
\item\label{item3} $\rank([C_{1,1}\;C_{1,2}]) + \rank([C_{3,2}\;C_{3,3}]) +1 = \rank(C)$.
\item\label{item4} $\rank(C_{1,1}) + \rank(C_{3,3}) + 2=\rank(C)$.
\end{enumerate}
Suppose first that $(\ref{item2})$ holds. Then
\begin{equation*}
\begin{split}
\mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix}) + \mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1 & \leq \rank(\begin{bmatrix}
C_{1,1}\\
C_{2,1}
\end{bmatrix}) + \rank(\begin{bmatrix}
C_{2,3}\\
C_{3,3}
\end{bmatrix})+1\\
& = \rank(C) = \mbox{mr}(M).
\end{split}
\end{equation*}
Case $(\ref{item3})$ is similar to $(\ref{item2})$.
Suppose next that $(\ref{item4})$ holds. Then
\begin{equation*}
\begin{split}
\mbox{mr}(A_{1,1}) + \mbox{mr}(A_{3,3}) + 2 & \leq \rank(C_{1,1}) + \rank(C_{3,3}) + 2 \\
& = \rank(C) = \mbox{mr}(M).
\end{split}
\end{equation*}
Suppose finally that $(\ref{item1})$ holds. If $v^T C_{1,1} z > 0$, then
\begin{equation*}
\begin{bmatrix}
C_{1,1} & C_{1,2}\\
C_{2,1} & v^T C_{1,1} z
\end{bmatrix}\in Q(R_+)\quad\text{and}\quad
\begin{bmatrix}
c_{2,2} - v^T C_{1,1} z & C_{2,3}\\
C_{3,2} & C_{3,3}
\end{bmatrix}\in Q(S_+).
\end{equation*}
Hence
\begin{equation*}
\begin{split}
\mbox{mr}(M) = \rank(C) &= \rank(\begin{bmatrix}
C_{1,1} & C_{1,2}\\
C_{2,1} & v^T C_{1,1} z
\end{bmatrix})+\rank(
\begin{bmatrix}
c_{2,2} - v^T C_{1,1} z & C_{2,3}\\
C_{3,2} & C_{3,3}
\end{bmatrix}) \\
&\geq \mbox{mr}(R_+) + \mbox{mr}(S_+).
\end{split}
\end{equation*}
The cases where $v^T C_{1,1} z = 0$ and $v^T C_{1,1} z < 0$ are similar.
\end{proof}
\section{Examples}
We exhibit several examples of sign pattern matrices illustrating that each term in Formula~(\ref{mainformula}) is needed.
To see that the term $\mbox{mr}(A_{1,1})+\mbox{mr}(A_{3,3})+2$ is needed in Formula~(\ref{mainformula}), let
\begin{equation*}
M = \begin{bmatrix}
A_{1,1} & A_{1,2}& 0\\
A_{2,1} & a_{2,2} & A_{2,3}\\
0 & A_{3,2} & A_{3,3}
\end{bmatrix}=\begin{bmatrix}
0 & + & 0\\
+ & 0 & +\\
0 & + & 0
\end{bmatrix},
\end{equation*}
and, for $p\in \{+,-,0\}$, let
\begin{equation*}
R_p = \begin{bmatrix}
0 & +\\
+ & p
\end{bmatrix}
\text{ and }
S_p = \begin{bmatrix}
-p & +\\
+ & 0
\end{bmatrix}.
\end{equation*}
Observe that $\mbox{mr}(M)=2$. Note that $\mbox{mr}(A_{1,1})+ \mbox{mr}(A_{3,3})+2=0+0+2=2$, while $\mbox{mr}([A_{1,1}\;A_{1,2}])+\mbox{mr}([A_{3,2}\;A_{3,3}])+1=1+1+1=3$, $\mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix})+\mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1=1+1+1=3$, $\mbox{mr}(R_+)+\mbox{mr}(S_+)=2+2=4$, $\mbox{mr}(R_0)+\mbox{mr}(S_0)=2+2=4$, and $\mbox{mr}(R_-)+\mbox{mr}(S_-)=2+2=4$.
To see that the term
$\mbox{mr}(\begin{bmatrix}
A_{1,1} & A_{1,2}
\end{bmatrix}) +
\mbox{mr}(\begin{bmatrix}
A_{3,2}& A_{3,3}
\end{bmatrix})+1$
is needed in Formula~(\ref{mainformula}), let
\begin{equation*}
M = \begin{bmatrix}
A_{1,1} & A_{1,2} & 0\\
A_{2,1} & a_{2,2} & A_{2,3}\\
0 & A_{3,2} & A_{3,3}
\end{bmatrix} =
\begin{bmatrix}
+ & + & 0 & 0 & 0\\
+ & + & 0 & 0 & 0\\
0 & + & + & + & 0\\
0 & 0 & + & 0 & +\\
\end{bmatrix},
\end{equation*}
and let
\begin{equation*}
R_p = \begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & p
\end{bmatrix}=
\begin{bmatrix}
+ & + & 0\\
+ & + & 0\\
0 & + & p
\end{bmatrix}
\end{equation*}
and
\begin{equation*}
S_p = \begin{bmatrix}
a_{2,2}-p & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix}=
\begin{bmatrix}
(+)-p & + & 0\\
+ & 0 & +
\end{bmatrix}.
\end{equation*}
Observe that $\mbox{mr}(M) = 3$. Note that $\mbox{mr}(A_{1,1}) + \mbox{mr}(A_{3,3}) + 2 = 1+1+2 = 4$, $\mbox{mr}([A_{1,1}\;A_{1,2}]) + \mbox{mr}([A_{3,2}\;A_{3,3}]) + 1 = 1 + 1 + 1 = 3, \mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix}) +
\mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1 = 2 + 2 + 1 = 5$, $\mbox{mr}(R_+) + \mbox{mr}(S_+) = 2 + 2 = 4$, $\mbox{mr}(R_0) + \mbox{mr}(S_0) = 2 + 2 = 4$, $\mbox{mr}(R_-) + \mbox{mr}(S_-) = 2+2=4$.
Taking the transpose of $M$ in the previous example shows that the term
$\mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix}) +
\mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1$
is needed in Formula~(\ref{mainformula}).
To see that the term $\mbox{mr}(R_+)+\mbox{mr}(S_+)$ is needed in Formula~(\ref{mainformula}), let
\begin{equation*}
M = \begin{bmatrix}
A_{1,1} & A_{1,2}& 0\\
A_{2,1} & a_{2,2} & A_{2,3}\\
0 & A_{3,2} & A_{3,3}
\end{bmatrix}=\begin{bmatrix}
+ & + & 0\\
+ & - & -\\
0 & + & +
\end{bmatrix},
\end{equation*}
and let
\begin{equation*}
R_p = \begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & p
\end{bmatrix}=
\begin{bmatrix}
+ & +\\
+ & p
\end{bmatrix}
\end{equation*}
and
\begin{equation*}
S_p = \begin{bmatrix}
a_{2,2}-p & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix}=
\begin{bmatrix}
(-)-p & -\\
+ & +
\end{bmatrix}.
\end{equation*}
Observe that $\mbox{mr}(M)=2$. Note that $\mbox{mr}(A_{1,1})+ \mbox{mr}(A_{3,3})+2=1+1+2=4, \mbox{mr}([A_{1,1}\;A_{1,2}])+\mbox{mr}([A_{3,2}\;A_{3,3}])+1=1+1+1=3,\mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix})+\mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1=1+1+1=3, \mbox{mr}(R_+)+\mbox{mr}(S_+)=1+1=2, \mbox{mr}(R_0)+\mbox{mr}(S_0)=2+1=3,\mbox{mr}(R_-)+\mbox{mr}(S_-)=2+1=3$.
Taking $-M$ in the previous example shows that the term
$\mbox{mr}(R_-)+\mbox{mr}(S_-)$ is needed in Formula~(\ref{mainformula}).
To see that the term $\mbox{mr}(R_0)+\mbox{mr}(S_0)$ is needed in Formula~(\ref{mainformula}), let
\begin{equation*}
M = \begin{bmatrix}
A_{1,1} & A_{1,2}& 0\\
A_{2,1} & a_{2,2} & A_{2,3}\\
0 & A_{3,2} & A_{3,3}
\end{bmatrix}=\begin{bmatrix}
+ & 0 & 0\\
+ & - & +\\
0 & + & -
\end{bmatrix},
\end{equation*}
and let
\begin{equation*}
R_p = \begin{bmatrix}
A_{1,1} & A_{1,2}\\
A_{2,1} & p
\end{bmatrix}=
\begin{bmatrix}
+ & +\\
+ & p
\end{bmatrix}
\end{equation*}
and
\begin{equation*}
S_p = \begin{bmatrix}
a_{2,2}-p & A_{2,3}\\
A_{3,2} & A_{3,3}
\end{bmatrix}=
\begin{bmatrix}
(-)-p & -\\
+ & +
\end{bmatrix}.
\end{equation*}
Observe that $\mbox{mr}(M)=2$. Note that $\mbox{mr}(A_{1,1})+ \mbox{mr}(A_{3,3})+2=1+1+2=4, \mbox{mr}([A_{1,1}\;A_{1,2}])+\mbox{mr}([A_{3,2}\;A_{3,3}])+1=1+1+1=3,\mbox{mr}(\begin{bmatrix}
A_{1,1}\\
A_{2,1}
\end{bmatrix})+\mbox{mr}(\begin{bmatrix}
A_{2,3}\\
A_{3,3}
\end{bmatrix})+1=1+1+1=3, \mbox{mr}(R_+)+\mbox{mr}(S_+)=2+1=3, \mbox{mr}(R_0)+\mbox{mr}(S_0)=1+1=2,\mbox{mr}(R_-)+\mbox{mr}(S_-)=2+1=3$.
\newcommand{\noopsort}[1]{}
\end{document} |
\betagin{document}
\title[Unbounded Toeplitz-like operators II: the spectrum]{A Toeplitz-like operator with rational symbol having poles on the unit circle II: the spectrum}
\author[G.J. Groenewald]{G.J. Groenewald}
\address{G.J. Groenewald, Department of Mathematics, Unit for BMI, North-West
University,
Potchefstroom, 2531 South Africa}
\email{[email protected]}
\author[S. ter Horst]{S. ter Horst}
\address{S. ter Horst, Department of Mathematics, Unit for BMI, North-West
University, Potchefstroom, 2531 South Africa}
\email{[email protected]}
\author[J. Jaftha]{J. Jaftha}
\address{J. Jaftha, Numeracy Centre, University of Cape Town, Rondebosch 7701; Cape Town; South Africa}
\email{[email protected]}
\author[A.C.M. Ran]{A.C.M. Ran}
\address{A.C.M. Ran, Department of Mathematics, Faculty of Science, VU Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands and Unit for BMI, North-West~University, Potchefstroom, South Africa}
\email{[email protected]}
\thanks{This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 90670 and 93406).\\
Part of the research was done during a sabbatical of the third author, in which time several
research visits to VU Amsterdam and North-West University were made. Support from University
of Cape Town and the Department of Mathematics, VU Amsterdam is gratefully acknowledged.}
\subjclass{Primary 47B35, 47A53; Secondary 47A68}
\keywords{unbounded Toeplitz operator, spectrum, essential spectrum}
\betagin{abstract}
This paper is a continuation of our study of a class of Toeplitz-like operators with a rational symbol which has a pole on the unit circle. A description of the spectrum and its various parts, i.e., point, residual and continuous spectrum, is given, as well as a description of the essential spectrum. In this case, the essential spectrum need not be connected in ${\mathbb C}$. Various examples illustrate the results.
\end{abstract}
\subjclass[2010]{Primary 47B35, 47A53; Secondary 47A68}
\keywords{unbounded Toeplitz operator, spectrum, essential spectrum}
\maketitle
\section{Introduction}
This paper is a continuation of our earlier paper \cite{GtHJR1} where Toeplitz-like operators with rational symbols which may have poles on the unit circle where introduced. While the aim of \cite{GtHJR1} was to determine the Fredholm properties of such Toeplitz-like operators, in the current paper we will focus on properties of the spectrum. For this purpose we further analyse this class of Toeplitz-like operators, specifically in the case where the operators are not Fredholm.
We start by recalling the definition of our Toeplitz-like operators.
Let ${\mathrm{Rat}}$ denote the space of rational complex functions. Write ${\mathrm{Rat}}(\mathbb{T})$ and ${\mathrm{Rat}}_0(\mathbb{T})$ for the subspaces of ${\mathrm{Rat}}$ consisting of the rational functions in ${\mathrm{Rat}}$ with all poles on ${\mathbb T}$ and the strictly proper rational functions in ${\mathrm{Rat}}$ with all poles on the unit circle ${\mathbb T}$, respectively. For $\om \in {\mathrm{Rat}}$, possibly having poles on ${\mathbb T}$, we define a Toeplitz-like operator $T_\omega (H^p \rightarrow H^p)$, for $1 < p<\infty$, as follows:
\betagin{equation}\label{Toeplitz}
{\mathrm{Dom}}(T_\omega)\!=\! \left\{ g\in H^p \! \mid \! \omega g = f + \rho \mbox{ with } f\!\in\! L^p\!\!,\, \rho \!\in\!\textup{Rat}_0(\mathbb{T})\right\},\
T_\omega g = \mathbb{P}f.
\end{equation}
Here ${\mathbb P}$ is the Riesz projection of $L^p$ onto $H^p$.
In \cite{GtHJR1} it was established that this operator is a densely defined, closed operator which is Fredholm if and only if $\om$ has no zeroes on ${\mathbb T}$. In case the symbol $\om$ of $T_\om$ is in ${\mathrm{Rat}}(\mathbb{T})$ with no zeroes on ${\mathbb T}$, i.e., $T_\om$ Fredholm, explicit formulas for the domain, kernel, range and a complement of the range were also obtained in \cite{GtHJR1}. Here we extend these results to the case that $\om$ is allowed to have zeroes on ${\mathbb T}$, cf., Theorem {\mathrm{Re}}f{T:Rat(T)} below. By a reduction to the case of symbols in ${\mathrm{Rat}}(\mathbb{T})$, we then obtain for general symbols in ${\mathrm{Rat}}$, in Proposition {\mathrm{Re}}f{P:injectdenserange} below, necessary and sufficient conditions for $T_\om$ to be injective or have dense range, respectively.
\paragraph{\bf Main results}
Using the fact that $\la I_{H^p}-T_{\om}=T_{\la-\om}$, our extended analysis of the operator $T_{\om}$ enables us to describe the spectrum of $T_\om$, and its various parts. Our first main result is a description of the essential spectrum of $T_\om$, i.e., the set of all $\la\in{\mathbb C}$ for which $\la I_{H^p}-T_{\om}$ is not Fredholm.
\betagin{theorem}\label{T:main1}
Let $\om\in{\mathrm{Rat}}$. Then the essential spectrum $\si_\textup{ess}(T_{\om})$ of $T_{\om}$ is an algebraic curve in ${\mathbb C}$ which is given by
\[
\si_\textup{ess}(T_{\om})=\om({\mathbb T}):=\{\om(e^{i\theta}) \mid 0\leq \theta \leq 2\pi,\, \mbox{$e^{i\theta}$ not a pole of $\om$} \}.
\]
Furthermore, the map $\la\mapsto {\mathrm{Index}} (T_{\la-\om})$ is constant on connected components of ${\mathbb C}\backslash \om({\mathbb T})$ and the intersection of the point spectrum, residual spectrum and resolvent set of $T_\om$ with ${\mathbb C}\backslash \om({\mathbb T})$ coincides with sets of $\la\in{\mathbb C}\backslash \om({\mathbb T})$ with ${\mathrm{Index}} (T_{\la-\om})$ being strictly positive, strictly negative and zero, respectively.
\end{theorem}
Various examples, specifically in Section {\mathrm{Re}}f{S:ExEssSpec}, show that the algebraic curve $\om({\mathbb T})$, and thus the essential spectrum of $T_\om$, need not be connected in ${\mathbb C}$.
Our second main result provides a description of the spectrum of $T_{\om}$ and its various parts. Here and throughout the paper ${\mathcal P}$ stands for the subspace of $H^p$ consisting of all polynomials and ${\mathcal P}_k$ for the subspace of ${\mathcal P}$ consisting of all polynomials of degree at most $k$.
\betagin{theorem}\label{T:main2}
Let $\om\in{\mathrm{Rat}}$, say $\om=s/q$ with $s,q\in{\mathcal P}$ co-prime. Define
\betagin{equation}\label{Kq0-}
\betagin{aligned}
k_q&=\sharp\{\mbox{roots of $q$ inside $\overline{{\mathbb D}}$}\}=\sharp\{\mbox{poles of $\la-\om$ inside $\overline{{\mathbb D}}$}\},\\
k_\la^-&=\sharp\{\mbox{roots of $\la q-s$ inside ${\mathbb D}$}\}=\sharp\{\mbox{zeroes of $\la-\om$ inside ${\mathbb D}$}\},\\
k_\la^0&=\sharp\{\mbox{roots of $\la q-s$ on ${\mathbb T}$}\}=\sharp\{\mbox{zeroes of $\la-\om$ on ${\mathbb T}$}\},
\end{aligned}
\end{equation}
where in all these sets multiplicities of the roots, poles and zeroes are to be taken into account. Then the resolvent set $\rho(T_\om)$, point spectrum $\si_\textup{p}(T_\om)$, residual spectrum $\si_\textup{r}(T_\om)$ and continuous spectrum $\si_\textup{c}(T_\om)$ of $T_\om$ are given by
\betagin{equation}\label{specparts}
\betagin{aligned}
\rho(T_{\om})&=\{\la\in{\mathbb C} \mid k_\la^0=0 \mbox{ and } k_q=k_\la^-\},\\
\si_\textup{p}(T_{\om})=\{\la\in{\mathbb C} &\mid k_q>k_\la^-+k_\la^0\},\quad
\si_\textup{r}(T_{\om})=\{\la\in{\mathbb C} \mid k_q<k_\la^-\},\\
\si_\textup{c}(T_{\om})&=\{\la\in{\mathbb C} \mid k_\la^0>0 \mbox{ and } k_\la^- \leq k_q\leq k_\la^- + k_\la^0\}.
\end{aligned}
\end{equation}
Furthermore, $\si_\textup{ess}(T_\om)=\om({\mathbb T})=\{\la\in{\mathbb C} \mid k_\la^0>0\}$.
\end{theorem}
Again, in subsequent sections various examples are given that illustrate these results. In particular, examples are given where $T_{\om}$ has a bounded resolvent set, even with an empty resolvent set. This is in sharp contrast to the case where $\om$ has no poles on the unit circle ${\mathbb T}$. For in this case the operator is bounded, the resolvent set is a nonempty unbounded set and the spectrum a compact set, and the essential spectrum is connected.
Both Theorems {\mathrm{Re}}f{T:main1} and {\mathrm{Re}}f{T:main2} are proven in Section {\mathrm{Re}}f{S:Spectrum}.
\paragraph{\bf Discussion of the literature}
In the case of a bounded selfadjoint Toeplitz operator on $\ell^2$, Hartman and Wintner in \cite{HW50} showed that the point spectrum is empty when the symbol is real and rational and posed the problem of specifying the spectral properties of such a Toeplitz operator. Gohberg in \cite{G52}, and more explicitly in \cite{G67}, showed that a bounded Toeplitz operator with continuous symbol is Fredholm exactly when the symbol has no zeroes on ${\mathbb T}$, and in this case the index of the operator coincides with the negative of the winding number of the symbol with respect to zero. This implies immediately that the essential spectrum of a Toeplitz operator with continuous symbol is the image of the unit circle.
Hartman and Wintner in \cite{HW54} followed up their earlier question by showing that in the case where the symbol, $\varphi$, is a bounded real valued function on ${\mathbb T}$, the spectrum of the Toeplitz operator on $H^2$ is contained in the interval bounded by the essential lower and upper bounds of $\varphi$ on ${\mathbb T}$ as well as that the point spectrum is empty whenever $\varphi$ is not a constant. Halmos, after posing in \cite{H63} the question whether the spectrum of a Toeplitz operator is connected, with Brown in \cite{BH64} showed that the spectrum cannot consist of only two points. Widom, in \cite{W64}, established that bounded Toeplitz operators on $H^2$ have connected spectrum, and later extended the result for general $H^p$, with $1 \leq p \leq \infty$. That the essential (Fredholm) spectrum of a bounded Toeplitz operator in $H^2$ is connected was shown by Douglas in \cite{D98}. For the case of bounded Toeplitz operators in $H^p$ it is
posed as an open question in B\"ottcher and Silbermann in \cite[Page 70]{BS06} whether the essential (Fredholm) spectrum of a Toeplitz operator in $H^p$ is necessarily connected.
Clark, in \cite{C67}, established conditions on the argument of the symbol $\varphi$ in the case $\varphi\in L^q, q \geq 2$ that would give the kernel index of the Toeplitz operator with symbol
$\varphi$ on $L^p$, where $\frac{1}{p} + \frac{1}{q} = 1$, to be $m\in{\mathbb N}$.
Janas, in \cite{J91},
discussed unbounded Toeplitz operators on the Bargmann-Siegel space and
showed that $\sigma_\tu{ess}(T_\varphi) \subset \cap_{R>0} \textrm{ closure } \{\varphi (z): \vert z\vert\geq R\}$.
\paragraph{\bf Overview} The paper is organized as follows. Besides the current introduction, the paper consists of five sections. In Section {\mathrm{Re}}f{S:Review} we extend a few results concerning the operator $T_\om$ from \cite{GtHJR1} to the case where $T_\om$ need not be Fredholm. These results are used in Section {\mathrm{Re}}f{S:Spectrum} to compute the spectrum of $T_{\om}$ and various of its subparts, and by doing so we prove the main results, Theorems {\mathrm{Re}}f{T:main1} and {\mathrm{Re}}f{T:main2}. The remaining three sections contain examples that illustrate our main results and show in addition that the resolvent set can be bounded, even empty, and that the essential spectrum can be disconnected in ${\mathbb C}$.
\paragraph{\bf Figures}
We conclude this introduction with a remark on the figures in this paper illustrating the spectrum and essential spectrum for several examples. The color coding in these figures is as follows: the white region is the resolvent set, the black curve is the essential spectrum, and the colors in the other regions codify the Fredholm index, where red indicates index $2$, blue indicates index $1$, cyan indicates index $-1$, magenta indicates index $-2$.
\section{Review and new results concerning $T_\omega$}\label{S:Review}
In this section we recall some results concerning the operator $T_\om$ defined in \eqref{Toeplitz} that were obtained in \cite{GtHJR1} and will be used in the present paper to determine spectral properties of $T_\om$. A few new features are added as well, specifically relating to the case where $T_\om$ is not Fredholm.
The first result provides necessary and sufficient conditions for $T_{\om}$ to be Fredholm, and gives a formula for the index of $T_\om$ in case $T_\om$ is Fredholm.
\betagin{theorem}[Theorems 1.1 and 5.4 in \cite{GtHJR1}]\label{T:recall1}
Let $\om\in {\mathrm{Rat}}$. Then $T_\om$ is Fredholm if and only if $\om$ has no zeroes on ${\mathbb T}$. In case $T_\om$ is Fredholm, the Fredholm index of $T_\om$ is given by
\[
{\mathrm{Index}} (T_\om) = \sharp \left\{\betagin{array}{l}\!\!\!
\textrm{poles of } \om \textrm{ in }\overline{{\mathbb D}} \textrm{ multi.}\!\!\! \\
\!\!\!\textrm{taken into account}\!\!\!
\end{array}\right\} -
\sharp \left\{\betagin{array}{l}\!\!\! \textrm{zeroes of } \om\textrm{ in }{\mathbb D} \textrm{ multi.}\!\!\! \\
\!\!\!\textrm{taken into account}\!\!\!
\end{array}\right\},
\]
and $T_\om$ is either injective or surjective. In particular, $T_\om$ is injective, invertible or surjective if and only if ${\mathrm{Index}}(T_\om)\leq 0$, ${\mathrm{Index}}(T_\om)=0$ or ${\mathrm{Index}}(T_\om)\geq 0$, respectively.
\end{theorem}
Special attention is given in \cite{GtHJR1} to the case where $\om$ is in ${\mathrm{Rat}}({\mathbb T})$, since in that case the kernel, domain and range can be computed explicitly; for the domain and range this was done under the assumption that $T_\om$ is Fredholm. In the following result we collect various statements from Proposition 4.5 and Theorems 1.2 and 4.7 in \cite{GtHJR1} and extend to or improve some of the claims regarding the case that $T_\om$ is not Fredholm.
\betagin{theorem}\label{T:Rat(T)}
Let $\om\in {\mathrm{Rat}}({\mathbb T})$, say $\om=s/q$ with $s,q\in{\mathcal P}$ co-prime. Factor $s=s_-s_0s_+$ with $s_-$, $s_0$ and $s_+$ having roots only inside, on, or outside ${\mathbb T}$. Then
\betagin{equation}\label{DomRanId}
\betagin{aligned}
&\qquad {\mathrm{Ker}} (T_\omega) = \left\{r_0/s_+ \mid \deg(r_0) < \deg(q) - \deg(s_-s_0) \right\};\\
&{\mathrm{Dom}}(T_\om)=qH^p+{\mathcal P}_{\deg(q)-1}; \quad
{\mathrm{Ran}}(T_\om)=s H^p+\wtil{\mathcal P},
\end{aligned}
\end{equation}
where $\wtil{\mathcal P}$ is the subspace of ${\mathcal P}$ given by
\betagin{equation}\label{tilP}
\wtil{\mathcal P} = \{ r\in{\mathcal P} \mid r q = r_1 s + r_2 \mbox{ for } r_1,r_2\in\mathcal{P}_{\deg(q)-1}\}\subset {\mathcal P}_{\deg(s)-1}.
\end{equation}
Furthermore, $H^p=\overline{{\mathrm{Ran}}(T_\om)} + \wtil{{\mathcal Q}}$ forms a direct sum decomposition of $H^p$, where
\betagin{equation}\label{tilQ}
\wtil{\mathcal Q}={\mathcal P}_{k-1}\quad \mbox{with}\quad k=\max\{\deg(s_-)-\deg(q) , 0\},
\end{equation}
following the convention ${\mathcal P}_{-1}:=\{0\}$.
\end{theorem}
The following result will be useful in the proof of Theorem {\mathrm{Re}}f{T:Rat(T)}.
\betagin{lemma}\label{L:closure}
Factor $s\in{\mathcal P}$ as $s=s_-s_0s_+$ with $s_-$, $s_0$ and $s_+$ having roots only inside, on, or outside ${\mathbb T}$. Then $sH^p =s_-s_0 H^p$ and $\overline{s H^p}= s_- H^p$.
\end{lemma}
\betagin{proof}[\bf Proof]
Since $s_+$ has no roots inside $\overline{{\mathbb D}}$, we have $s_+ H^p=H^p$. Furthermore, $s_0$ is an $H^\infty$ outer function (see, e.g., \cite{N}, Example 4.2.5)
so that $\overline{s_0 H^p}=H^p$.
Since $s_-$ has all it's roots inside ${\mathbb D}$, $T_{s_-}:H^p\to H^p$ is an injective operator with closed range. Consequently, we have
\[
\overline{s H^p}
=\overline{s_- s_0 s_+ H^p}
=\overline{s_- s_0 H^p}
=s_- \overline{ s_0 H^p}=s_- H^p,
\]
as claimed.
\end{proof}
\betagin{proof}[\bf Proof of Theorem {\mathrm{Re}}f{T:Rat(T)}]
In case $T_\om$ is Fredholm, i.e., $s_0$ constant, all statements follow from Theorem 1.2 in \cite{GtHJR1}. Without the Fredholm condition, the formula for ${\mathrm{Ker}}(T_\om)$ follows from \cite[Lemma 4.1]{GtHJR1} and for ${\mathrm{Dom}}(T_\om)$ and ${\mathrm{Ran}}(T_\om)$ Proposition 4.5 of \cite{GtHJR1} provides
\betagin{equation}\label{DomRanIncl}
\betagin{aligned}
qH^p+{\mathcal P}_{\deg(q)-1}&\subset {\mathrm{Dom}}(T_\om);\\
T_\om (qH^p+{\mathcal P}_{\deg(q)-1})&=s H^p+\wtil{\mathcal P}\subset {\mathrm{Ran}}(T_\om).
\end{aligned}
\end{equation}
Thus in order to prove \eqref{DomRanId}, it remains to show that ${\mathrm{Dom}}(T_{\om})\subset q H^p +{\mathcal P}_{\deg(q)-1}$.
Assume $g\in {\mathrm{Dom}}(T_{\om})$. Thus there exist $h\in H^p$ and $r\in {\mathcal P}_{\deg(q)-1}$ so that $s g= q h +r$. Since $s$ and $q$ are co-prime, there exist $a,b\in{\mathcal P}$ such that $s a+ q b\equiv 1$. Next write $ar=q r_1+r_2$ for $r_1,r_2\in{\mathcal P}$ with $\deg(r_2)<\deg (q)$. Thus $sg=q h +r=q h+ q br + s ar=q(h+br+sr_1)+ s r_2$. Hence $g=q(h+br+sr_1)/s +r_2$. We are done if we can show that $\wtil{h}:=(h+br+sr_1)/s$ is in $H^p$.
The case where $g$ is rational is significantly easier, but still gives an idea of the complications that arise, so we include a proof. Hence assume $g\in{\mathrm{Rat}}\cap H^p$. Then $h=(s g-r)/q$ is also in ${\mathrm{Rat}} \cap H^p$, and $\wtil{h}$ is also rational. It follows that $q(h+br+sr_1)/s=q \wtil{h} =g-r_2\in {\mathrm{Rat}} \cap H^p$ and thus cannot have poles in $\overline{{\mathbb D}}$. Since $q$ and $s$ are co-prime and $h$ cannot have poles inside $\overline{{\mathbb D}}$, it follows that $\wtil{h}=(h+br+sr_1)/s$ cannot have poles in $\overline{{\mathbb D}}$. Thus $\wtil{h}$ is a rational function with no poles in $\overline{{\mathbb D}}$, which implies $\wtil{h}\in H^p$.
Now we prove the claim for the general case. Assume $q\wtil{h} +r_2=g\in H^p$, but $\wtil{h}=(h+br+sr_1)/s\not \in H^p$, i.e., $\wtil{h}$ is not analytic on ${\mathbb D}$ or $\int_{{\mathbb T}} |\wtil{h}(z)|^p\textup{d}z=\infty$. Set $\widehat{h}=h+br+sr_1\in H^p$, so that $\wtil{h}=\what{h}/s$. We first show $\wtil{h}$ must be analytic on ${\mathbb D}$. Since $\wtil{h}=\what{h}/s$ and $\what{h}\in H^p$, $\wtil{h}$ is analytic on ${\mathbb D}$ except possibly at the roots of $s$. However, if $\wtil{h}$ would not be analytic at a root $z_0\in{\mathbb D}$ of $s$, then also $g= q \wtil{h}+r_2$ should not be analytic at $z_0$, since $q$ is bounded away from 0 on a neighborhood of $z_0$, using that $s$ and $q$ are co-prime. Thus $\wtil{h}$ is analytic on ${\mathbb D}$. It follows that $\int_{{\mathbb T}} |\wtil{h}(z)|^p\textup{d}z=\infty$.
Since $s$ and $q$ are co-prime, we can divide ${\mathbb T}$ as ${\mathbb T}_1\cup {\mathbb T}_2$ with ${\mathbb T}_1\cap {\mathbb T}_2=\emptyset$ and each of ${\mathbb T}_1$ and ${\mathbb T}_2$ being nonempty unions of intervals, with ${\mathbb T}_1$ containing all roots of $s$ on ${\mathbb T}$ as interior points and ${\mathbb T}_2$ containing all roots of $q$ on ${\mathbb T}$ as interior points. Then there exist $N_1,N_2>0$ such that $|q(z)|> N_1$ on ${\mathbb T}_1$ and $|s(z)|>N_2$ on ${\mathbb T}_2$.
Note that
\betagin{align*}
\int_{{\mathbb T}_2}|\wtil{h}(z)|^p \textup{d}z
&=\int_{{\mathbb T}_2}|\what{h}(z)/s(z)|^p \textup{d}z
\leq N_2^{-p} \int_{{\mathbb T}_2}|\what{h}(z)|^p \textup{d}z
\leq N_2^{-p} \|\what{h}\|^p_{H^p}<\infty.
\end{align*}
Since $\int_{{\mathbb T}}|\wtil{h}(z)|^p \textup{d}z=\infty$ and $\int_{{\mathbb T}_2}|\wtil{h}(z)|^p \textup{d}z<\infty$, it follows that $\int_{{\mathbb T}_1}|\wtil{h}(z)|^p \textup{d}z=\infty$. However, since $|q(z)|> N_1$ on ${\mathbb T}_1$, this implies that
\betagin{align*}
\|g-r_2\|^p_{H^p}&=\int_{{\mathbb T}}|g(z)-r_2(z)|^p \textup{d}z
=\int_{{\mathbb T}}|q(z)\wtil{h}(z)|^p \textup{d}z
\geq \int_{{\mathbb T}_1}|q(z) \wtil{h}(z)|^p \textup{d}z\\
&\geq N_1^p \int_{{\mathbb T}_1}|\wtil{h}(z)|^p \textup{d}z =\infty,
\end{align*}
in contradiction with the assumption that $g\in H^p$. Thus we can conclude that $\wtil{h}\in H^p$ so that $g=q \wtil{h}+r_2$ is in $q H^p+{\mathcal P}_{\deg(q)-1}$.
It remains to show that $H^p=\overline{{\mathrm{Ran}}(T_\om)}+\wtil{{\mathcal Q}}$ is a direct sum decomposition of $H^p$. Again, for the case that $T_\om$ is Fredholm this follows from \cite[Theorem 1.2]{GtHJR1}. By the preceding part of the proof we know, even in the non-Fredholm case, that ${\mathrm{Ran}}(T_\om)=s H^p+\wtil{{\mathcal P}}$. Since $\wtil{{\mathcal P}}$ is finite dimensional, and thus closed, we have
\[
\overline{{\mathrm{Ran}}(T_\om)}=\overline{s H^p}+\wtil{{\mathcal P}}= s_- H^p + \wtil{{\mathcal P}},
\]
using Lemma {\mathrm{Re}}f{L:closure} in the last identity. We claim that
\[
\overline{{\mathrm{Ran}}(T_\om)}=s_- H^p + \wtil{{\mathcal P}}=s_- H^p + \wtil{{\mathcal P}}_-,
\]
where $\wtil{{\mathcal P}}_-$ is defined by
\[
\wtil{{\mathcal P}}_{-}:=\{r\in{\mathcal P} \mid qr=r_1 s_- +r_2\mbox{ for }r_1,r_2\in{\mathcal P}_{\deg(q)-1}\}\subset {\mathcal P}_{\deg(s_-)-1}.
\]
Once the above identity for $\overline{{\mathrm{Ran}}(T_\om)}$ is established, the fact that $\wtil{{\mathcal Q}}$ is a complement of $\overline{{\mathrm{Ran}}(T_\om)}$ follows directly by applying Lemma 4.8 of \cite{GtHJR1} to $s=s_-$.
We first show that $\overline{{\mathrm{Ran}}(T_\om)}=s_- H^p +\wtil{{\mathcal P}}$ is contained in $s_- H^p+\wtil{{\mathcal P}}_-$. Let $g=s_- h+r$ with $h\in H^p$ and $r\in\wtil{{\mathcal P}}$, say $qr=r_1s +r_2$ with $r_1,r_2\in{\mathcal P}_{\deg(q)-1}$. Write $r_1 s_0s_+=\wtil{r}_1 q + \wtil{r}_2$ with $\deg(\wtil{r}_2)<\deg(q)$. Then
\[
qr= r_1 s_- s_0s_+ +r_2=q\wtil{r}_1 s_-+\wtil{r}_2s_- +r_2,\mbox{ so that }
q(r-\wtil{r}_1s_-)= \wtil{r}_2s_- + r_2,
\]
with $r_2,\wtil{r_2}\in{\mathcal P}_{\deg(q)-1}$. Thus $r-\wtil{r}_1s_-\in\wtil{{\mathcal P}}_-$. Therefore, we have
\[
g=s_-(h+\wtil{r}_1)+(r-\wtil{r}_1s_-)\in s_- H^p +\wtil{{\mathcal P}}_-,
\]
proving that $\overline{{\mathrm{Ran}}(T_\om)}\subset s_- H^p+\wtil{{\mathcal P}}_-$.
For the reverse inclusion, assume $g=s_-h+r\in s_- H^p+\wtil{{\mathcal P}}_-$. Say $qr=r_1 s_- + r_2$ with $r_1,r_2\in{\mathcal P}_{\deg(q)-1}$. Since $s_0s_+$ and $q$ are co-prime and $\deg(r_1)<\deg(q)$ there exit polynomials $\wtil{r}_1$ and $\wtil{r}_2$ with $\deg(\wtil{r}_1)<\deg(q)$ and $\deg(\wtil{r}_2)<\deg(s_0s_+)$ that satisfy the B\'ezout equation $\wtil{r}_1 s_0 s_+ + \wtil{r}_2 q=r_1$. Then
\[
\wtil{r}_1 s+ r_2 = \wtil{r}_1 s_0s_+s_- + r_2= (r_1-\wtil{r}_2 q)s_- + r_2= r_1 s_- + r_2 -q \wtil{r}_2 s_- = q(r-\wtil{r}_2 s_-).
\]
Hence $r-\wtil{r}_2 s_-$ is in $\wtil{{\mathcal P}}$, so that $g= s_- h+ r= s_-(h+\wtil{r}_2)+ (r-\wtil{r}_2 s_-)\in s_- H^p +\wtil{{\mathcal P}}$. This proves the reverse inclusion, and hence completes the proof of Theorem {\mathrm{Re}}f{T:Rat(T)}.
\end{proof}
The following result makes precise when $T_\om$ is injective and when $T_\om$ has dense range, even in the case where $T_\om$ is not Fredholm.
\betagin{proposition}\label{P:injectdenserange}
Let $\om\in {\mathrm{Rat}}$. Then $T_\om$ is injective if and only if
\[
\sharp \left\{\betagin{array}{l}\!\!\!
\textrm{poles of } \om \textrm{ in }\overline{{\mathbb D}} \textrm{ multi.}\!\!\! \\
\!\!\!\textrm{taken into account}\!\!\!
\end{array}\right\} \leq
\sharp \left\{\betagin{array}{l}\!\!\! \textrm{zeroes of } \om\textrm{ in }\overline{{\mathbb D}} \textrm{ multi.}\!\!\! \\
\!\!\!\textrm{taken into account}\!\!\!
\end{array}\right\}.
\]
Moreover, $T_\om$ has dense range if and only if
\[
\sharp \left\{\betagin{array}{l}\!\!\!
\textrm{poles of } \om \textrm{ in }\overline{{\mathbb D}} \textrm{ multi.}\!\!\! \\
\!\!\!\textrm{taken into account}\!\!\!
\end{array}\right\} \geq
\sharp \left\{\betagin{array}{l}\!\!\! \textrm{zeroes of } \om\textrm{ in }{\mathbb D} \textrm{ multi.}\!\!\! \\
\!\!\!\textrm{taken into account}\!\!\!
\end{array}\right\}.
\]
In particular, $T_\om$ is injective or has dense range.
\end{proposition}
\betagin{proof}[\bf Proof]
First assume $\om\in {\mathrm{Rat}}({\mathbb T})$. By Corollary 4.2 in \cite{GtHJR1}, $T_\om$ is injective if and only if the number of zeroes of $\om$ inside $\overline{{\mathbb D}}$ is greater than or equal to the number of poles of $\om$, in both cases with multiplicity taken into account. By Theorem {\mathrm{Re}}f{T:Rat(T)}, $T_\om$ has dense range precisely when $\wtil{{\mathcal Q}}$ in \eqref{tilQ} is trivial. The latter happens if and only if the number of poles of $\om$ is greater than or equal to the number of zeroes of $\om$ inside ${\mathbb D}$, again taking multiplicities into account. Since in this case all poles of $\om$ are in ${\mathbb T}$, our claim follows for $\om\in {\mathrm{Rat}}({\mathbb T})$.
Now we turn to the general case, i.e., we assume $\om\in{\mathrm{Rat}}$. In the remainder of the proof, whenever we speak of numbers of zeroes or poles, this always means that the respective multiplicities are to be taken into account. Recall from \cite[Lemma 5.1]{GtHJR1} that we can factor $\om(z)= \om_-(z)z^\kappappa \om_0(z) \om_+(z)$ with $\om_-,\om_0,\om_+\in{\mathrm{Rat}}$, $\om_-$ having no poles or zeroes outside ${\mathbb D}$, $\om_+$ having no poles or zeroes inside $\overline{{\mathbb D}}$ and $\om_0$ having poles and zeroes only on ${\mathbb T}$, and $\kappappa$ the difference between the number of zeroes of $\om$ in ${\mathbb D}$ and the number of poles of $\om$ in ${\mathbb D}$. Moreover, we have $T_\om=T_{\om_-}T_{z^\kappappa \om_0} T_{\om_+}$ and $T_{\om_-}$ and $T_{\om_+}$ are boundedly invertible on $H^p$. Thus $T_\om$ is injective or has closed range if and only it $T_{z^\kappappa\om_0}$ is injective or has closed range, respectively.
Assume $\kappappa \geq 0$. Then $z^\kappappa \om_0\in {\mathrm{Rat}}({\mathbb T})$ and the results for the case that the symbol is in ${\mathrm{Rat}}({\mathbb T})$ apply. Since the zeroes and poles of $\om_0$ coincide with the zeroes and poles of $\om$ on ${\mathbb T}$, it follows that the number of poles of $z^\kappappa \om_0$ is equal to the number of poles of $\om$ on ${\mathbb T}$ while the number of zeroes of $z^\kappappa \om_0$ is equal to $\kappappa$ plus the number of zeroes of $\om$ on ${\mathbb T}$ which is equal to the number of zeroes of $\om$ in $\overline{{\mathbb D}}$ minus the number of poles of $\om$ in ${\mathbb D}$. It thus follows that $T_{z^\kappappa \om_0}$ is injective, and equivalently $T_\om$ is injective, if and only if the number of zeroes of $\om$ in $\overline{{\mathbb D}}$ is greater than or equal to the number of poles of $\om$ in $\overline{{\mathbb D}}$, as claimed.
Next, we consider the case where $\kappappa<0$. In that case $T_{z^\kappappa \om_0}=T_{z^\kappappa}T_{\om_0}$, by Lemma 5.3 of \cite{GtHJR1}. We prove the statements regarding injectivity and $T_\om$ having closed range separately.
First we prove the injectivity claim for the case where $\kappappa<0$. Write $\om_0 =s_0/q_0$ with $s_0,q_0\in{\mathcal P}$ co-prime. Note that all the roots of $s_0$ and $q_0$ are on ${\mathbb T}$. We need to show that $T_{z^\kappappa \om_0}$ is injective if and only if $\deg(s_0) \geq \deg(q_0)-\kappappa$ (recall, $\kappappa$ is negative).
Assume $\deg(s_0)+\kappappa \geq \deg(q_0)$. Then $\deg(s_0) > \deg(q_0)$, since $\kappappa<0$, and thus $T_{\om_0}$ is injective. We have ${\mathrm{Ker}}(T_{z^{\kappappa}})={\mathcal P}_{|\kappappa|-1}$. So it remains to show ${\mathcal P}_{|\kappappa|-1} \cap {\mathrm{Ran}} (T_{\om_0})=\{0\}$. Assume $r\in{\mathcal P}_{|\kappappa|-1}$ is also in ${\mathrm{Ran}} (T_{\om_0})$. So, by Lemma 2.3 in \cite{GtHJR1}, there exist $g\in H^p$ and $r'\in{\mathcal P}_{\deg(q_0)-1}$ so that $s_0 g=q_0 r+ r'$, i.e., $g=(q_0 r+ r')/s_0$. This shows that $g$ is in ${\mathrm{Rat}}({\mathbb T})\cap H^p$, which can only happen in case $g$ is a polynomial. Thus, in the fraction $(q_0 r+ r')/s_0$, all roots of $s_0$ must cancel against roots of $q_0 r+ r'$. However, since $\deg(s_0)+\kappappa \geq \deg(q_0)$, with $\kappappa<0$, $\deg(r)<\deg |\kappappa|-1$ and $\deg(r')<\deg(q_0)$, we have $\deg(q_0 r + r')<\deg(s_0)$ and it is impossible that all roots of $s_0$ cancel against roots of $q_0 r + r'$, leading to a contradiction. This shows ${\mathcal P}_{|\kappappa|-1} \cap {\mathrm{Ran}} (T_{\om_0})=\{0\}$, which implies $T_{z^{\kappappa\om_0}}$ is injective. Hence also $T_\om$ is injective.
Conversely, assume $\deg(s_0)+\kappappa < \deg(q_0)$, i.e., $\deg(s_0)< \deg(q_0)+|\kappappa|=:b$, since $\kappappa<0$. Then
\[
s_0\in {\mathcal P}_{b-1}=q_0 {\mathcal P}_{|\kappappa|-1} +{\mathcal P}_{\deg(q_0)-1}.
\]
This shows there exist $r\in{\mathcal P}_{|\kappappa|-1}$ and $r'\in{\mathcal P}_{\deg(q_0)-1}$ so that $s_0= q_0 r+ r'$. In other words, the constant function $g\equiv 1\in H^p$ is in ${\mathrm{Dom}} (T_{\om_0})$ and $T_{\om_0}g=r\in {\mathcal P}_{|\kappappa|-1}={\mathrm{Ker}} (T_{z^\kappappa})$, so that $g\in {\mathrm{Ker}} (T_{z^\kappappa \om_0})$. This implies $T_\om$ is not injective.
Finally, we turn to the proof of the dense range claim for the case $\kappappa<0$. Since $\kappappa<0$ by assumption, $\om$ has more poles in $\overline{{\mathbb D}}$ (and even in ${\mathbb D}$) than zeroes in ${\mathbb D}$. Thus to prove the dense range claim in this case, it suffices to show that $\kappappa<0$ implies that $T_{z^\kappappa\om_0}$ has dense range. We have $T_{z^\kappappa \om_0}=T_{z^\kappappa}T_{\om_0}$ and $T_{z^\kappappa}$ is surjective. Also, $\om_0\in {\mathrm{Rat}}({\mathbb T})$ has no zeroes inside ${\mathbb D}$. So the proposition applies to $\om_0$, as shown in the first paragraph of the proof, and it follows that $T_{\om_0}$ has dense range. But then also $T_{z^\kappappa \om_0}=T_{z^\kappappa}T_{\om_0}$ has dense range, and our claim follows.
\end{proof}
\section{The spectrum of $T_\omega$}\label{S:Spectrum}
In this section we determine the spectrum and various subparts of the spectrum of $T_\om$ for the general case, $\om\in{\mathrm{Rat}}$, as well as some refinements for the case where $\om\in{\mathrm{Rat}}({\mathbb T})$ is proper. In particular, we prove our main results, Theorems {\mathrm{Re}}f{T:main1} and {\mathrm{Re}}f{T:main2}.
Note that for $\om\in{\mathrm{Rat}}$ and $\la\in{\mathbb C}$ we have $\la I-T_\om=T_{\la-\om}$. Thus we can relate questions on the spectrum of $T_\om$ to question on injectivity, surjectivity, closed rangeness, etc.\ for Toeplitz-like operators with an additional complex parameter. By this observation, the spectrum of $T_\om$, and its various subparts, can be determined using the results of Section {\mathrm{Re}}f{S:Review}.
\betagin{proof}[\bf Proof of Theorem {\mathrm{Re}}f{T:main1}]
Since $\la I-T_\om=T_{\la-\om}$ and $T_{\la-\om}$ is Fredholm if and only if $\la-\om$ has no zeroes on ${\mathbb T}$, by Theorem {\mathrm{Re}}f{T:recall1}, it follows that $\la$ is in the essential spectrum if and only if $\la=\om(e^{i\theta})$ for some $0\leq \theta\leq 2\pi$. This shows that $\si_\textup{ess}(T_\om)$ is equal to $\om({\mathbb T})$.
To see that $\om({\mathbb T})$ is an algebraic curve, let $\omega=s/q$ with $s,q\in{\mathcal P}$ co-prime. Then $\la=u+iv=\om(z)$ for $z=x+iy$ with $x^2+y^2=1$ if and only if $\la q(z)-s(z)=0$. Denote
$q(z)=q_1(x,y)+iq_2(x,y)$ and $s(z)=s_1(x,y)+is_2(x,y)$, where $z=x+iy$ and the functions
$q_1, q_2, s_1, s_2$ are real polynomials in two variables. Then $\la=u+iv$ is on the curve $\om({\mathbb T})$ if and only if
\betagin{align*}
q_1(x,y)u-q_2(x,y)v&=s_1(x,y),\\
q_2(x,y)u+q_1(x,y)v&=s_2(x,y),\\
x^2+y^2&=1.
\end{align*}
Solving for $u$ and $v$, this is equivalent to
\betagin{align*}
(q_1(x,y)^2+q_2(x,y)^2)u-(q_1(x,y)s_1(x,y)+q_2(x,y)s_2(x,y))&=0,\\
(q_1(x,y)^2+q_2(x,y)^2)v-(q_1(x,y)s_2(x,y)-q_2(x,y)s_1(x,y))&=0,\\
x^2+y^2&=1.
\end{align*}
This describes an algebraic curve in the plane.
For $\lambda$ in the complement of the curve $\om({\mathbb T})$ the operator $\lambda I -T_\om=T_{\la-\om}$ is
Fredholm, and according to Theorem {\mathrm{Re}}f{T:recall1} the index is given by
$$
{\mathrm{Index}} (\lambda-T_\om)=
\sharp\{\textrm{ poles of } \om \textrm{ in } \overline{{\mathbb D}}\}-
\sharp\{\textrm{zeroes of } \om-\lambda \textrm{ inside }{\mathbb D}\},
$$
taking the multiplicities of the poles and zeroes into account. Indeed, $\lambda - \om = \frac{\lambda q - s}{q}$ and since $q$ and $s$ are co-prime, $\lambda q - s$ and $q$ are also co-prime. Thus Theorem {\mathrm{Re}}f{T:recall1} indeed applies to $T_{\la-\om}$. Furthermore, $\lambda - \om$ has the same poles as $\om$, i.e., the roots of $q$. Likewise, the zeroes of $\la-\om$ coincide with the roots of the polynomial $\lambda q - s$. Since the roots of this polynomial depend continuously on the parameter $\lambda$ the number of them is constant on connected components of the complement of the curve $\omega({\mathbb T})$.
That the index is constant on connected components of the complement of the essential spectrum in fact holds for any unbounded densely defined operator (see \cite[Theorem VII.5.2]{S71}; see also \cite[Proposition XI.4.9]{C90} for the bounded case; for a much more refined analysis of this point see \cite{FK}).
Finally, the relation between the index of $T_{\la-\om}$ and $\la$ being in the resolvent set, point spectrum or residual spectrum follows directly by applying the last part of Theorem {\mathrm{Re}}f{T:recall1} to $T_{\la-\om}$.
\end{proof}
Next we prove Theorem {\mathrm{Re}}f{T:main2} using some of the new results on $T_\om$ derived in Section {\mathrm{Re}}f{S:Review}.
\betagin{proof}[\bf Proof of Theorem {\mathrm{Re}}f{T:main2}]
That the two formulas for the numbers $k_q$, $k_\la^-$ and $k_\la^0$ coincides follows from the analysis in the proof of Theorem {\mathrm{Re}}f{T:main1}, using the co-primeness of $\la q -s$ and $q$. By Theorem {\mathrm{Re}}f{T:recall1}, $T_{\la-\om}$ is Fredholm if and only if $k_\la^0=0$, proving the formula for $\si_\textup{ess}(T_\om)$. The formula for the resolvent set follows directly from the fact that the resolvent set is contained in the complement of $\si_\textup{ess}(T_\om)$, i.e., $k_\la^0=0$, and that it there coincides with the set of $\la$'s for which the index of $T_{\la-\om}$ is zero, together with the formula for ${\mathrm{Index}}(T_{\la-\om})$ obtained in Theorem {\mathrm{Re}}f{T:recall1}.
The formulas for the point spectrum and residual spectrum follow by applying the criteria for injectivity and closed rangeness of Proposition {\mathrm{Re}}f{P:injectdenserange} to $T_{\la-\om}$ together with the fact that $T_{\la-\om}$ must be either injective or have dense range.
For the formula for the continuous spectrum, note that $\si_\textup{c}(T_\om)$ must be contained in the essential spectrum, i.e., $k_\la^0>0$. The condition $k_\la^- \leq k_q\leq k_\la^- + k_\la^0$ excludes precisely that $\la$ is in the point or residual spectrum.
\end{proof}
For the case where $\om\in{\mathrm{Rat}}({\mathbb T})$ is proper we can be a bit more precise.
\betagin{theorem}\label{T:spectrum2}
Let $\om \in\textup{Rat}(\mathbb{T})$ be proper, say $\om=s/q$ with $s,q\in{\mathcal P}$ co-prime. Thus ${\mathrm{deg}}( s) \leq {\mathrm{deg}}( q)$ and all roots of $q$ are on ${\mathbb T}$. Let $a$ be the leading coefficient of $q$ and $b$ the coefficient of $s$ corresponding to the monomial $z^{\deg(q)}$, hence $b=0$ if and only if $\om$ is strictly proper. Then $\si_\textup{r}(T_\om)=\emptyset$, and the point spectrum is given by
\[
\si_\textup{p}(T_\om)=\om({\mathbb C}\backslash \overline{{\mathbb D}}) \cup \{b/a\}.
\]
Here $\om({\mathbb C}\backslash \overline{{\mathbb D}})=\{\om (z) \mid z\in {\mathbb C}\backslash \overline{{\mathbb D}}\}$.
In particular, if $\om$ is strictly proper, then $0=b/a$ is in $\si_\textup{p}(T_\om)$.
Finally,
\[
\si_\textup{c}(T_\om)=\{\lambda\in\mathbb{C} \mid k_\la^0 >0 \mbox{ and all roots of } \lambda q-s \mbox{ are in }\overline{{\mathbb D}} \}.
\]
\end{theorem}
\betagin{proof}[\bf Proof]
Let $\om = s/q\in{\mathrm{Rat}}({\mathbb T})$ be proper with $s,q\in{\mathcal P}$ co-prime. Then $k_q=\deg(q)$. Since ${\mathrm{deg}}(s) \leq \deg(q)$, for any $\la\in{\mathbb C}$ we have
\[
k_\la^-+k_\la^0\leq \deg(\la q-s)\leq \deg(q)=k_q.
\]
It now follows directly from \eqref{specparts} that $\si_\textup{r}(T_\om)=\emptyset$ and $\si_\textup{c}(T_\om)=\{\lambda \in\mathbb{C}\mid k_\lambda^0 >0, k_\lambda^-+k_\lambda^0=\deg(q)\}$. To determine the point spectrum, again using \eqref{specparts}, one has to determine when strict inequality occurs. We have $\deg(\la q-s)<\deg(q)$ precisely when the leading coefficient of $\la q$ is cancelled in $\la q-s$ or if $\la=0$ and $\deg(s)<\deg(q)$. Both cases correspond to $\la=b/a$. For the other possibility of having strict inequality, $k_\la^-+k_\la^0<\deg(\la q-s)$, note that this happens precisely when $\la q-s$ has a root outside $\overline{{\mathbb D}}$, or equivalently $\la=\om(z)$ for a $z\not\in \overline{{\mathbb D}}$.
\end{proof}
\section{The spectrum may be unbounded, the resolvent set empty}
\label{S:Examples1}
In this section we present some first examples, showing that the spectrum can be unbounded
and the resolvent set may be empty.
\betagin{example}\label{E:spectrum2}
Let $\om(z) = \frac{z - \alphapha}{z - 1}$ for some $1\neq \alphapha\in{\mathbb C}$, say $\alphapha=a+ib$, with $a$ and $b$ real. Let $L\subset{\mathbb C}$ be the line given by
\betagin{equation}\label{Line}
L=\{z=x+iy\in{\mathbb C} \mid 2by = (a^2 + b^2 - 1) + (2 - 2a)x \}
\end{equation}
Then we have
\betagin{align*}
\rho(T_\om)=\om({\mathbb D}),\quad & \sigma_\textup{ess} (T_\om)=\om({\mathbb T})=L =\si_\tu{c}(T_\om), \\
\si_\tu{p}(T_\om)&=\om({\mathbb C}\backslash\overline{{\mathbb D}}),\quad \si_\tu{r}(T_{\om})=\emptyset.
\end{align*}
Moreover, the point spectrum of $T_\om$ is the open half plane determined by $L$ that contains $1$ and the resolvent set of $T_\om$ is the other open half plane determined by $L$.
\betagin{figure}
\betagin{center}
\includegraphics[height=4cm]{figure_one}
\\
\caption
{Spectrum of $T_\om$ where $\om(z)=\frac{z-\alphapha}{z-1}$, with $\alphapha=-\frac{i}{2}$.}
\end{center}
\end{figure}
To see that these claims are true note that for $\lambda\not= 1$
\[
\lambda - \om(z) = \frac{z(\lambda - 1) + \alphapha - \lambda}{z - 1} = \frac{1}{\lambda - 1}\frac{z + \frac{\alphapha - \lambda}{\lambda - 1}}{z -1},
\]
while for $\lambda = 1$ we have $\lambda - \om(z) = \frac{\alphapha - \lambda}{z - 1}$. Thus $\lambda = 1\in\sigma_\textup{p}(T_\om)$ for every $1\neq \alphapha\in{\mathbb C}$ as in that case $k_q=1> 0=k_\la^-+k_\la^0$. For $\la\neq 1$, $\la-\om$ has a zero at $\frac{\alpha-\alpha}{\la-1}$ of multiplicity one. For $\lambda = x + iy$ we have $\vert \alphapha - \lambda \vert = \vert \lambda - 1 \vert $ if and only if $ (a - x)^2 + (b - y)^2 = (x - 1)^2 + y^2$, which in turn is equivalent to $2by = (a^2 + b^2 - 1) + (2 - 2a)x$. Hence the zero of $\la-\om$ is on ${\mathbb T}$ precisely when $\la$ is on the line $L$. This shows $\si_\tu{ess}=L$. One easily verifies that the point spectrum and resolvent set correspond to the two half planes indicated above and that these coincide with the images of $\om$ under ${\mathbb C}\backslash\overline{\mathbb D}$ and ${\mathbb D}$, respectively. Since $\la-\om$ can have at most one zero, it is clear from Theorem {\mathrm{Re}}f{T:main2} that $\si_\tu{r}(T_\om)=\emptyset$, so that $\si_\tu{c}(T_\om)=L=\si_\tu{ess}(T_\om)$, as claimed.
$\Box$
\end{example}
\betagin{example}\label{E:spectrum4a}
Let $\om(z) = \frac{1}{(z-1)^k}$ for some positive integer $k>1$. Then
\[
\si_\tu{p}(T_\om)=\si(T_\om)={\mathbb C},\quad \si_{r}(T_\om)=\si_\tu{c}(T_\om)=\rho(T_\om)=\emptyset,
\]
and the essential spectrum is given by
\[
\si_\tu{ess}(T_\om)=\om({\mathbb T})=\{(it-\half)^k \mid t\in{\mathbb R} \}.
\]
For $k=2$ the situation is as in Figure 2; one can check that the curve $\om({\mathbb T})$ is the parabola ${\mathrm{Re}}(z)=\frac{1}{4}-{\mathrm{Im}}(z)^2$. (Recall that different colors indicate different Fredholm index, as explained at the end of the introduction.)
\betagin{figure}
\betagin{center}
\includegraphics[height=4cm]{figure_two}
\caption{Spectrum of $T_\om$ where $\om(z)=\frac{1}{(z-1)^2}$}
\end{center}
\end{figure}
To prove the statements, we start with the observation that for $\vert z\vert = 1$, $\frac{1}{z-1}$ is of the form $it-\frac{1}{2} , t\in{\mathbb R}$. Thus for $z\in{\mathbb T}$ with $\frac{1}{z-1}=it-\frac{1}{2}$ we have
\[
\om(z) = \frac{1}{(z-1)^k} = (z-1)^{-k} = (it -\half)^k.
\]
This proves the formula for $\si_\tu{ess}(T_\om)$. For $\la=re^{i\theta}\neq 0$ we have
\[
\la-\om(z)=\frac{\la(z-1)^{k}-1}{(z-1)^k}.
\]
Thus $\la-\om(z)=0$ if and only if $(z-1)^k=\la^{-1}$, i.e., $z=1+r^{-1/k}e^{i(\theta +2\pi l)/k}$ for $l=0,\ldots,k-1$. Thus the zeroes of $\la-\om$ are $k$ equally spaced points on the circle with center 1 and radius $r^{-1/k}$. Clearly, since $k>1$, not all zeroes can be inside $\overline{{\mathbb D}}$, so $k_q> k_\la^{0}+k_\la^{-}$, and thus $\la\in\si_\tu{p}(T_\om)$. It follows directly from Theorem {\mathrm{Re}}f{T:main2} that $0\in\si_\tu{p}(T_\om)$. Thus $\si_\tu{p}(T_\om)={\mathbb C}$, as claimed. The curve $\om({\mathbb T})$ divides the plane into several regions on which the index is a positive constant integer, but the index may change between different regions.
$\Box$
\end{example}
\section{The essential spectrum need not be connected}\label{S:ExEssSpec}
For a continuous function $\omega$ on the unit circle it is obviously the case that the
curve $\om({\mathbb T})$ is a connected and bounded curve in the complex plane, and hence the
essential spectrum of $T_\omega$ is connected in this case. It was proved by Widom \cite{W64}
that also for $\omega$ piecewise continuous the essential spectrum of $T_\omega$ is connected,
and it is the image of a curve related to $\om({\mathbb T})$ (roughly speaking, filling the jumps with
line segments). Douglas \cite{D98} proved that even for $\omega\in L^\infty$ the essential
spectrum of $T_\omega$ as an operator on $H^2$ is connected.
In \cite{BS06} the question is raised whether or not
the essential spectrum of $T_\omega$ as an operator on $H^p$ is always connected when
$\om \in L^\infty$.
Returning to our case, where $\omega$ is a rational function possibly with poles on the unit circle,
clearly when $\omega$ does have poles on the unit
circle it is not a-priori necessary that $\si_\tu{ess}(T_\om)=\om({\mathbb T})$ is connected. We shall present examples that show that indeed the essential spectrum need not be connected, in contrast with the case where $\omega\in L^\infty$.
Consider $\om=s/q\in{\mathrm{Rat}}({\mathbb T})$ with $s,q\in{\mathcal P}$ with real coefficients. In that case $\overline{\om(z)}=\om(\overline{z})$, so that the essential spectrum is symmetric with respect to the real axis. In particular, if $\om({\mathbb T})\cap {\mathbb R}=\emptyset$, then the essential spectrum is disconnected. The converse direction need not be true, since the essential spectrum can consist of several disconnected parts on the real axis, as the following example shows.
\betagin{example}\label{E:disconR}
Consider $\om(z)=\frac{z}{z^2+1}$. Then
\[
\si_\tu{ess}(T_\om)=\om({\mathbb T})=(-\infty,-1] \cup [1,\infty)=\si_\tu{c}(T_\om),\quad
\si_\tu{p}(T_\om)={\mathbb C}\backslash \om({\mathbb T}),
\]
and thus $\si_\tu{r}(T_\om)=\rho(T_\om)=\emptyset$. Further, for $\la\not\in\om({\mathbb T})$ the Fredholm index is 1.
Indeed, note that for $z=e^{i\theta}\in{\mathbb T}$ we have
\[
\om(z)=\frac{1}{z+z^{-1}}=\frac{1}{2\,{\mathrm{Re}}(z)}=\frac{1}{2\cos(\theta)}\in{\mathbb R}.
\]
Letting $\theta$ run from $0$ to $2\pi$, one finds that $\om({\mathbb T})$ is equal to the union of $(-\infty,-1]$ and $[1,\infty)$, as claimed. Since $\om$ is strictly proper, $\si_\tu{r}(T_\om)=\emptyset$ by Theorem {\mathrm{Re}}f{T:spectrum2}. Applying Theorem {\mathrm{Re}}f{T:recall1} to $T_\om$ we obtain that $T_\om$ is Fredholm with index 1. Hence $T_\om$ is not injective, so that $0\in\si_\tu{p}(T_\om)$. However, since ${\mathbb C}\backslash \om({\mathbb T})$ is connected, it follows from Theorem {\mathrm{Re}}f{T:main1} that the index of $T_{\la-\om}$ is equal to 1 on ${\mathbb C}\backslash \om({\mathbb T})$, so that ${\mathbb C}\backslash \om({\mathbb T})\subset\si_\tu{p}(T_\om)$. However, for $\la$ on $\om({\mathbb T})$ the function $\la-\om$ has two zeroes on ${\mathbb T}$ as well as two poles on ${\mathbb T}$. It follows that $\om({\mathbb T})=\si_\tu{c}(T_\om)$, which shows all the above formulas for the spectral parts hold.
\end{example}
As a second example we specify $q$ to be $z^2-1$ and determine a condition on $s$ that guarantees $\si_\tu{ess}(T_\om)=\om({\mathbb T})$ in not connected.
\betagin{example}
Consider $\om(z)=\frac{s(z)}{z^2-1}$ with $s\in{\mathcal P}$ a polynomial with real coefficients. Then for $z\in{\mathbb T}$ we have
\[
\om(z)=\frac{\overline{z}s(z)}{z-\overline{z}}
=\frac{\overline{z}s(z)}{-2i\,{\mathrm{Im}}(z)}
=\frac{i\overline{z}s(z)}{2\,{\mathrm{Im}}(z)},\quad \mbox{so that}\quad
{\mathrm{Im}}(\om(z))=\frac{{\mathrm{Re}}(\overline{z}s(z))}{2\,{\mathrm{Im}}(z)}.
\]
Hence ${\mathrm{Im}}(\om(z))=0$ if and only if ${\mathrm{Re}}(\overline{z}s(z))=0$. Say $s(z)=\sum_{j=0}^k a_j z^j$. Then for $z\in{\mathbb T}$ we have
\betagin{align*}
{\mathrm{Re}}(\overline{z}s(z)) & = \sum_{j=0}^k a_j {\mathrm{Re}}(z^{j-1}).
\end{align*}
Since $|{\mathrm{Re}}(z^j)|\leq 1$, we obtain that $|{\mathrm{Re}}(\overline{z}s(z))|>0$ for all $z\in{\mathbb T}$ in case $2|a_1|>\sum_{j=0}^k|a_j|$. Hence in that case $\om({\mathbb T})\cap {\mathbb R}=\emptyset$ and we find that the essential spectrum is disconnected in ${\mathbb C}$.
We consider two concrete examples, where this criteria is satisfied.
Firstly, take $\omega(z)=\frac{z^3+3z+1}{z^2-1}$. Then
$$
\omega(e^{i\theta})= \frac{1}{2}(2\cos\theta -1) -\frac{i}{2}\frac{2(\cos\theta +1/4)^2+7/4}{\sin\theta},
$$
which is the curve given in Figure 3, that also shows the spectrum and resolvent as well as the essential spectrum.
\betagin{figure}
\includegraphics[height=4cm]{figure_three}
\caption{Spectrum of $T_\omega$, where $\om(z)=\frac{z^3+3z+1}{z^2-1}$}
\end{figure}
Secondly, take $\omega(z)=\frac{z^4+3z+1}{z^2-1}$. Figure 4 shows the spectrum and resolvent and the essential spectrum. Observe that this is also a case where the resolvent is a bounded set.
\betagin{figure}
\includegraphics[height=4cm]{figure_four}
\caption{Spectrum of $T_\omega$, where $\om(z)=\frac{z^4+3z+1}{z^2-1}$}
\end{figure}
\end{example}
\section{A parametric example}\label{S:Examples2}
In this section we take $\om_k(z) = \frac{z^k + \alphapha}{(z - 1)^2}$ for $\alphapha\in{\mathbb C}, \alpha\neq -1$ and for various integers $k\geq 1$. Note that the case $k=0$ was dealt with in Example {\mathrm{Re}}f{E:spectrum4a} (after scaling with the factor $1+\alpha$). The zeroes of $\la-\om$ are equal to the roots of
\[
p_{\lambda,\alphapha,k}(z)=\lambda q(z)- s(z) = \lambda (z-1)^2 - (z^k + \alphapha).
\]
Thus, $\la$ is in the resolvent set $\rho(T_{\om_k})$ whenever $p_{\lambda,\alphapha,k}$ has at least two roots in ${\mathbb D}$ and no roots on ${\mathbb T}$. Note that Theorem {\mathrm{Re}}f{T:spectrum2} applies in case $k=1,2$. We discuss the first of these two cases in detail, and then conclude with some figures that contain possible configurations of other cases.
\betagin{example}\label{E:spectrum4b}
Let $\om(z)=\om_1(z) = \frac{z+\alphapha}{(z-1)^2} $ for $\alphapha\not = -1$. Then
\betagin{equation}\label{EssSpecPara}
\si_\tu{ess}(T_\om)=\om({\mathbb T})=\{(it-\half) + (1+\alphapha)(it-\half)^2 \mid t\in{\mathbb R}\}.
\end{equation}
Define the circle
\[
{\mathbb T}(-\half,\half)=\{z\in{\mathbb C} \mid |z+\half|=\half\},
\]
and write ${\mathbb D}(-\half,\half)$ for the open disc formed by the interior of ${\mathbb T}(-\half,\half)$ and ${\mathbb D}^c(-\half,\half)$ for the open exterior of ${\mathbb T}(-\half,\half)$.
For $\alpha\notin {\mathbb T}(-\half,\half)$ the curve $\om({\mathbb T})$ is equal to the parabola in ${\mathbb C}$ given by
\betagin{align*}
\om({\mathbb T}) &=\left\{ -(\alpha+1)(x(y)+i y) \mid y\in{\mathbb R} \right\},\quad \mbox{ where } \\
x(y) &= \frac{|\alpha+1|^4}{(|\alpha|^2+{\mathrm{Re}}(\alpha))^2}y^2+
\frac{({\mathrm{Re}}(\alpha)+1)|\alpha+1|^2{\mathrm{Im}}(\alpha)}{(|\alpha|^2+{\mathrm{Re}}(\alpha))^2}y+
\frac{|\alpha|^2(1-|\alpha|^2)}{(|\alpha|^2+{\mathrm{Re}}(\alpha))^2},
\end{align*}
while for $\alpha\in {\mathbb T}(-\half,\half)$ the curve $\om({\mathbb T})$ becomes the half line given by
\[
\om({\mathbb T})=\left\{-(\alpha+1)r - \frac{(\alpha+1)(1+2\overline{\alpha})}{4(1-|\alpha|^2)} \mid r\geq 0 \right\}.
\]
As $\om$ is strictly proper, we have $\si_\tu{r}(T_\om)=\emptyset$. For the remaining parts of the spectrum we consider three cases.
\betagin{itemize}
\item[(i)]
For $\alpha\in {\mathbb D}(-\half,\half)$ the points $-\half$ and $0$ are separated by the parabola $\om({\mathbb T})$ and the connected component of ${\mathbb C}\backslash \om({\mathbb T})$ that contains $-\half$ is equal to $\rho(T_{\om})$, while the connected component that contains 0 is equal to $\si_\tu{p}(T_\om)$. Finally, $\si_\tu{ess}(T_\om)=\om({\mathbb T})=\si_\tu{c}(T_\om)$.
\item[(ii)]
For $\alpha\in {\mathbb T}(-\half,\half)$ we have
\[
\rho(T_\om)=\emptyset,\quad \si_\tu{c}(T_\om)=\om({\mathbb T})=\si_\tu{ess}(T_\om),\quad \si_\tu{p}(T_\om)={\mathbb C}\backslash \om({\mathbb T}),
\]
and for each $\la\in \om({\mathbb T})$, $\la-\om$ has two zeroes on ${\mathbb T}$.
\item[(iii)] For $\alpha\in {\mathbb D}^c(-\half,\half)$ we have $\si_\tu{p}(T_\om)={\mathbb C}$, and hence $\rho(T_\om)=\si_\tu{c}(T_\om)=\emptyset$.
\end{itemize}
The proof of these statements will be separated into three steps.
\paragraph{\it Step 1.}
We first determine the formula of $\om({\mathbb T})$ and show this is a parabola. Note that
\[
\om(z) = \frac{z+\alphapha}{(z-1)^2} = \frac{z-1}{(z-1)^2} + \frac{1+\alphapha}{(z-1)^2} = \frac{1}{z-1} + (\alphapha+1)\frac{1}{(z-1)^2}.
\]
Let $|z|=1$. Then $\frac{1}{z-1}$ is of the form $it-\half$ with $t\in{\mathbb R}$. So $\om({\mathbb T})$ is the curve
\betagin{equation*}
\om({\mathbb T})=\{(it-\half) + (\alphapha+1)(it-\half)^2 \mid t\in{\mathbb R}\}.
\end{equation*}
Thus \eqref{EssSpecPara} holds. Now observe that
\betagin{align*}
& (it-\half) + (\alphapha+1)(it-\half)^2=\\
&\qquad = -t^2(\alphapha+1) + t(i - (\alphapha+1)i) + (-\half + \mbox{$\frac{1}{4}$}(\alphapha+1))\\
&\qquad = -t^2(\alphapha+1) + (-\alphapha i)t + (-\mbox{$\frac{1}{4}$} + \mbox{$\frac{1}{4}$}\alphapha)\\
&\qquad = \displaystyle -(\alphapha+1)\left(t^2 + t\frac{\alphapha i}{\alphapha+1} - \frac{1}{4}\left(\frac{\alphapha - 1}{\alphapha + 1}\right )\right ).
\end{align*}
The prefactor $-(1+\alphapha)$ acts as a rotation combined with a real scalar multiplication, so $\om({\mathbb T})$ is also given by
\betagin{equation}\label{omTeq}
\om({\mathbb T})=-(\alpha+1)\left\{t^2 + t\left(\frac{\alphapha i}{\alphapha+1}\right ) - \frac{1}{4}\left (\frac{\alphapha-1 }{\alphapha+1}\right ) \mid t\in{\mathbb R}\right\}.
\end{equation}
Thus if the above curve is a parabola, so is $\om({\mathbb T})$. Write
\betagin{align*}
x(t) &= {\mathrm{Re}} \left(t^2 + t\frac{\alphapha i}{1+\alphapha} - \frac{1}{4}\left(\frac{\alphapha - 1}{\alphapha + 1}\right )\right ),\\
y(t) &= {\mathrm{Im}} \left(t^2 + t\frac{\alphapha i}{1+\alphapha} - \frac{1}{4}\left(\frac{\alphapha - 1}{\alphapha + 1}\right )\right ).
\end{align*}
Since
\[
\frac{\alpha i}{\alpha+1}=\frac{-{\mathrm{Im}}(\alpha)+i(|\alpha|^2+{\mathrm{Re}}(\alpha))}{|\alpha+1|^2}
\quad\mbox{and}\quad
\frac{\alpha-1}{\alpha+1}=\frac{(|\alpha|^2-1)+2i{\mathrm{Im}}(\alpha)}{|\alpha+1|^2}
\]
we obtain that
\[
x(t) = t^2-\frac{{\mathrm{Im}}(\alpha)}{|\alpha+1|^2}t-\frac{|\alpha|^2-1}{4|\alpha+1|^2},\quad
y(t) = \frac{|\alpha|^2+{\mathrm{Re}}(\alpha)}{|\alpha+1|^2}t-\frac{{\mathrm{Im}}(\alpha)}{2|\alpha+1|^2}.
\]
Note that $|\alpha+\half|^2=|\alpha|^2+{\mathrm{Re}}(\alpha)+\frac{1}{4}$. Therefore, we have $|\alpha|^2+{\mathrm{Re}}(\alpha)=0$ if and only if $|\alpha+\half|=\half$. Thus $|\alpha|^2+{\mathrm{Re}}(\alpha)=0$ holds if and only if $\alpha$ is on the circle ${\mathbb T}(-\half,\half)$.
In case $\alpha\notin {\mathbb T}(-\half,\half)$, i.e., $|\alpha|^2+{\mathrm{Re}}(\alpha)\neq 0$, we can express $t$ in terms of $y$, and feed this into the formula for $x$. One can then compute that
\[
x=\frac{|\alpha+1|^4}{(|\alpha|^2+{\mathrm{Re}}(\alpha))^2}y^2+
\frac{({\mathrm{Re}}(\alpha)+1)|\alpha+1|^2{\mathrm{Im}}(\alpha)}{(|\alpha|^2+{\mathrm{Re}}(\alpha))^2}y+
\frac{|\alpha|^2(1-|\alpha|^2)}{(|\alpha|^2+{\mathrm{Re}}(\alpha))^2}.
\]
Inserting this formula into \eqref{omTeq}, we obtain the formula for $\om({\mathbb T})$ for the case where $\alpha\notin {\mathbb T}(-\half,\half)$.
In case $\alpha\in {\mathbb T}(-\half,\half)$, i.e., $|\alpha|^2+{\mathrm{Re}}(\alpha)= 0$, we have
\[
|\alpha+1|^2=1-|\alpha|^2=1+{\mathrm{Re}}(\alpha), \quad {\mathrm{Im}}(\alpha)^2=|\alpha|^2(1-|\alpha|^2)
\]
and using these identities one can compute that
\[
y(t)=\frac{-2{\mathrm{Im}}(\alpha)}{4(1-|\alpha|^2)}\quad\mbox{and}\quad
x(t)=\left(t-\frac{{\mathrm{Im}}(\alpha)}{2(1-|\alpha|^2)}\right)^2+\frac{1+2{\mathrm{Re}}(\alpha)}{4(1-|\alpha|^2)}.
\]
Thus $\{x(t)+iy(t) \mid t\in{\mathbb R}\}$ determines a half line in ${\mathbb C}$, parallel to the real axis and starting in $\frac{1+2\overline{\alpha}}{4(1-|\alpha|^2)}$ and moving in positive direction. It follows that $\om({\mathbb T})$ is the half line
\[
\om({\mathbb T})=\left\{-(\alpha+1)r - \frac{(\alpha+1)(1+2\overline{\alpha})}{4(1-|\alpha|^2)} \mid r\geq 0 \right\},
\]
as claimed.
\paragraph{\it Step 2.} Next we determine the various parts of the spectrum in ${\mathbb C}\backslash \om({\mathbb T})$. Since $\om$ is strictly proper, Theorem {\mathrm{Re}}f{T:spectrum2} applies, and we know $\si_\tu{r}(T_\om)=\emptyset$ and $\si_\tu{p}=\om({\mathbb C}\backslash \overline{{\mathbb D}})\cup \{0\}$.
For $k=1$, the polynomial $p_{\la,\alpha}(z)=p_{\la,\alpha,1}(z)=\la z^2 -(1+2\la)z+\la-\alpha$ has roots
\[
\frac{-(1+2\la)\pm \sqrt{1+4\la(1+\alpha)}}{2\la}.
\]
We consider three cases, depending on whether $\alpha$ is inside, on or outside the circle ${\mathbb T}(-\half,\half)$.
Assume $\alpha\in{\mathbb D}(-\half,\half)$. Then $\om({\mathbb T})$ is a parabola in ${\mathbb C}$. For $\la=-\half$ we find that $\la-\om$ has zeroes $\pm i\sqrt{1+2\alpha}$, which are both inside ${\mathbb D}$, because of our assumption. Thus $-\half\in\rho(T_\om)$, so that $\rho(T_\om)\neq \emptyset$. Therefore the connected component of ${\mathbb C}\backslash \om({\mathbb T})$ that contains $-\half$ is contained in $\rho(T_\om)$, which must also contain $\om({\mathbb D})$. Note that $0\in\om({\mathbb T})$ if and only if $|\alpha|=1$. However, there is no intersection of the disc $\alpha\in{\mathbb D}(-\half,\half)$ and the unit circle ${\mathbb T}$. Thus 0 is in $\si_\tu{p}(T_\om)$, but not on $\om({\mathbb T})$. Hence $0$ is contained in the connected component of ${\mathbb C}\backslash \om({\mathbb T})$ that does not contain $-\half$. This implies that the connected component containing $0$ is included in $\si_\tu{p}(T_\om)$. This proves our claims for the case $\alpha\in{\mathbb D}(-\half,\half)$.
Now assume $\alpha\in{\mathbb T}(-\half,\half)$. Then $\om({\mathbb T})$ is a half line, and thus ${\mathbb C}\backslash \om({\mathbb T})$ consists of one connected component. Note that the intersection of the disc determined by $|\alpha+\half|<\half$ and the unit circle consists of $-1$ only. But $\alpha\neq -1$, so it again follows that $0\notin\om({\mathbb T})$. Therefore the ${\mathbb C}\backslash \om({\mathbb T})=\si_\tu{p}(T_\om)$. Moreover, the reasoning in the previous case shows that $\la=-\half$ is in $\si_\tu{c}(T_{\om})$ since both zeroes of $-\half-\om$ are on ${\mathbb T}$.
Finally, consider that case where $\alpha$ is in the exterior of ${\mathbb T}(-\half,\half)$, i.e., $|\alpha+\half|>\half$. In this case, $|\alpha|=1$ is possible, so that $0\in\si_\tu{p}(T_\om)$ could be on $\om({\mathbb T})$. We show that $\alpha=\om(0)\in\om({\mathbb D})$ is in $\si_\tu{p}(T_\om)$. If $\alpha=0$, this is clearly the case. So assume $\alpha\neq0$. The zeroes of $\alpha-\om$ are then equal to $0$ and $\frac{1+2\alpha}{\alpha}$. Note that $|\frac{1+2\alpha}{\alpha}|> 1$ if and only if $|1+2\alpha|^2-|\alpha|^2>0$. Moreover, we have
\[
|1+2\alpha|^2-|\alpha|^2=3|\alpha|^2+4{\mathrm{Re}}(\alpha)+1=3|\alpha+\mbox{$\frac{2}{3}$}|^2-\mbox{$\frac{1}{3}$}.
\]
Thus, the second zero of $\alpha-\om$ is outside $\overline{{\mathbb D}}$ if and only if $|\alpha+\frac{2}{3}|^2>\frac{1}{9}$. Since the disc indicated by $|\alpha+\frac{2}{3}|\leq\frac{1}{3}$ is contained in the interior of ${\mathbb T}(-\half,\half)$, it follows that for $\alpha$ satisfying $|\alpha+\half|>\half$ one zero of $\alpha-\om$ is outside $\overline{{\mathbb D}}$, and thus $\om(0)=\alpha\in \si_\tu{p}(T_\om)$. Note that
\[
{\mathbb C}=\om({\mathbb C})=\om({\mathbb D})\cup \om({\mathbb T}) \cup \om({\mathbb C}\backslash \overline{{\mathbb D}}),
\]
and that $\om({\mathbb D})$ and $\om({\mathbb C}\backslash \overline{{\mathbb D}})$ are connected components, both contained in $\si_\tu{p}(T_\om)$. This shows that ${\mathbb C}\backslash \om({\mathbb T})$ is contained in $\si_\tu{p}(T_\om)$.
\paragraph{\it Step 3.} In the final part we prove the claim regarding the essential spectrum $\si_\tu{ess}(T_\om)=\om({\mathbb T})$. Let $\la\in\om({\mathbb T})$ and write $z_1$ and $z_2$ for the zeroes of $\la-\om$. One of the zeroes must be on ${\mathbb T}$, say $|z_1|=1$. Then $\la\in\si_\tu{p}({\mathbb T})$ if and only if $|z_1z_2|=|z_2|>1$. From the form of $p_{\la,\alpha}$ determined above we obtain that
\[
\la z^2-(1+2\la)z+\la-\alpha=\la(z-z_1)(z-z_2).
\]
Determining the constant term on the right hand sides shows that $\la z_1z_2=\la-\alpha$. Thus
\[
|z_2|=|z_1z_2|=\frac{|\la-\alpha|}{|\la|}.
\]
This shows that $\la\in\si_\tu{p}(T_\om)$ if and only if $|\la-\alpha| > |\la|$, i.e., $\lambda$ is in the half plane containing zero determined by the line through $\half\alphapha$ perpendicular to the line segment from zero to $\alphapha$.
Consider the line given by $|\la-\alpha| = |\la|$ and the parabola $\om({\mathbb T})$, which is a half line in case $\alpha\in{\mathbb T}(-\half,\half)$. We show that $\om({\mathbb T})$ and the line intersect only for $\alpha\in{\mathbb T}(-\half,\half)$, and that in the latter case $\om({\mathbb T})$ is contained in the line. Hence for each value of $\alpha\neq -1$, the essential spectrum consists of either point spectrum or of continuous spectrum, and for $\alpha\in{\mathbb T}(-\half,\half)$ both zeroes of $\la-\om$ are on ${\mathbb T}$, so that $\om({\mathbb T})$ is contained in $\si_\tu{c}(T_\om)$.
As observed in \eqref{EssSpecPara}, the parabola $\om({\mathbb T})$ is given by the parametrization $(it-\half)^2(\alphapha+1)+(it-\half)$ with $t\in{\mathbb R}$, while the line is given by the parametrization
$\half\alphapha +si\alphapha$ with $s\in{\mathbb R}$. Fix a $t\in{\mathbb R}$ and assume the point on $\om({\mathbb T})$ parameterized by $t$ intersects with the line, i.e., assume there exists a $s\in{\mathbb R}$ such that:
$$
(it-\half)^2(\alphapha+1)+(it-\half)=\half\alphapha +si\alphapha,
$$
Thus
$$
(-t^2-it+\mbox{$\frac{1}{4}$})(\alphapha+1)+(it-\half)=\half\alphapha +si\alphapha,
$$
and rewrite this as
$$
i(-t(\alphapha+1)+t-\alphapha s)+((-t^2+\mbox{$\frac{1}{4}$})(\alphapha+1)-\half -\half \alphapha)=0,
$$
which yields
$$
-\alphapha i(t+s)+(\alphapha+1)(-t^2-\mbox{$\frac{1}{4}$})=0.
$$
Since $t^2+\mbox{$\frac{1}{4}$}>0$, this certainly cannot happen in case $\alpha=0$. So assume $\alpha\neq 0$.
Multiply both sides by $-\overline{\alphapha}$ to arrive at
$$
|\alphapha|^2i (t+s)+(|\alphapha|^2+\overline{\alphapha})(t^2+\mbox{$\frac{1}{4}$})=0.
$$
Separate the real and imaginary part to arrive at
\[
(|\alphapha|^2+{\mathrm{Re}}(\alphapha))(t^2+\mbox{$\frac{1}{4}$})+ i(|\alpha|^2(t+s)-(t^2+\mbox{$\frac{1}{4}$}){\mathrm{Im}}(\alpha))=0.
\]
Thus
\[
(|\alphapha|^2+{\mathrm{Re}}(\alphapha))(t^2+\mbox{$\frac{1}{4}$})=0
\quad\mbox{and}\quad
|\alpha|^2(t+s)=(t^2+\mbox{$\frac{1}{4}$}){\mathrm{Im}}(\alpha).
\]
Since $t^2+\mbox{$\frac{1}{4}$} >0$, the first identity yields $|\alphapha|^2+{\mathrm{Re}}(\alphapha)=0$, which happens precisely when $\alpha\in{\mathbb T}(-\half,\half)$. Thus there cannot be an intersection when $\alpha\notin{\mathbb T}(-\half,\half)$. On the other hand, for $\alpha\in{\mathbb T}(-\half,\half)$ the first identity always holds, while there always exists an $s\in{\mathbb R}$ that satisfies the second equation. Thus, in that case, for any $t\in{\mathbb R}$, the point on $\om({\mathbb T})$ parameterized by $t$ intersects the line, and thus $\om({\mathbb T})$ must be contained in the line.
We conclude by showing that $\om({\mathbb T})\subset \si_{\tu{p}}(T_\om)$ when $|\alpha +\half|>\half$ and that $\om({\mathbb T})\subset \si_{\tu{c}}(T_\om)$ when $|\alpha +\half|<\half$. Recall that the two cases correspond to $|\alpha|^2+{\mathrm{Re}}(\alpha)>0$ and $|\alpha|^2+{\mathrm{Re}}(\alpha)<0$, respectively. To show that this is the case, we take the point on the parabola parameterized by $t=0$, i.e., take $\la=\frac{1}{4}(\alpha+1)-\half=\frac{1}{4}(\alpha-1)$. Then $\la-\alpha=-\frac{1}{4}(3\alpha+1)$. So
\[
|\la-\alpha|^2=\mbox{$\frac{1}{16}$}(9|\alpha|^2+6{\mathrm{Re}}(\alpha)+1) \quad\mbox{and}\quad
|\la|^2=\mbox{$\frac{1}{16}$}(|\alpha^2|-2{\mathrm{Re}}(\alpha)+1).
\]
It follows that $|\la-\alpha|>|\la|$ if and only if
\[
\mbox{$\frac{1}{16}$}(9|\alpha|^2+6{\mathrm{Re}}(\alpha)+1)> |\la|^2=\mbox{$\frac{1}{16}$}(|\alpha^2|-2{\mathrm{Re}}(\alpha)+1),
\]
or equivalently,
\[
8(|\alpha|^2+{\mathrm{Re}}(\alpha))>0.
\]
This proves out claim for the case $|\la+\half|>\half$. The other claim follows by reversing the directions in the above inequalities.
Figure 5 presents some illustrations of the possible situations.
\betagin{figure}
\includegraphics[width=12cm]{figure_five}
\caption{Spectrum of $T_\omega$, where $\om(z)=\frac{z+\alphapha}{(z-1)^2}$ for some values of
$\alphapha$, with $\alphapha = 1$, and $\alphapha=0$ (top row left and right), $\alphapha=1/2$ and $\alphapha=-2$ (middle row left and right), $\alphapha =-\frac{1}{2}+\frac{1}{4}i$ and $\alphapha=-2+i$
(bottom row).}
\end{figure}
$\Box$
\end{example}
The case $k=2$ can be dealt with using the same techniques, and very similar results are obtained in that case.
The next examples deal with other cases of $\om_k$, now with $k>2$.
\betagin{example}\label{E:spectrum4d}
Let $\om = \frac{z^3 + \alphapha}{(z-1)^2}$. Then
{\small
\[
\om(z) = \frac{z^3 + \alphapha}{(z-1)^2} =
(z-1) + 3 + \frac{3}{z-1} + \frac{1+\alphapha}{(z-1)^2}.
\]
}
For $z\in{\mathbb T}$, $\frac{1}{z-1}$ has the form $-\frac{1}{2} + ti, t\in{\mathbb R}$ and so $\om({\mathbb T})$ has the form
\[
\om({\mathbb T})=\left\{
\frac{1}{-\frac{1}{2} + ti} + 3 + 3(-\frac{1}{2} + ti) + (1+\alphapha)\left (- \frac{1}{2} + ti\right)^2,\mid t\in{\mathbb R}\right\}.
\]
Also $\lambda - \om(z) = \frac{\lambda(z-1)^2 - z^3 - \alphapha}{(z-1)^2}$ and so for invertibility we need the polynomial $p_{\lambda,\alphapha}(z) = \lambda(z-1)^2 - z^3 - \alphapha$ to have exactly two roots in ${\mathbb D}$. Since this is a polynomial of degree $3$ the number of roots inside ${\mathbb D}$ can be zero, one, two or three, and the index of $\lambda-T_\om$ correspondingly can be two, one, zero or minus one. Examples are given in Figure 6.
\betagin{figure}
\includegraphics[width=12cm]{figure_seven}
\caption{Spectrum of $T_\om$ where $\om(z)=\frac{z^3+\alphapha}{(z-1)^2}$ for several values of $\alphapha$, with $\alphapha$ being (left to right and top to bottom) respectively, $-2, -1.05, -0.95, 0.3, 0.7, 1, 1.3, 2$.
}
\end{figure}
\end{example}
\betagin{example}\label{E:spectrum5d}
To get some idea of possible other configurations we present some examples with other values of $k$.
For $\om (z)= \frac{z^4 }{(z-1)^2}$ (so $k=4$ and $\alphapha =0$) the essential spectrum of $T_\om$ is the curve in Figure 7, the white region is the resolvent set, and color coding for the Fredholm index is as earlier in the paper. For $\om (z)= \frac{z^6 + 1.7}{(z-1)^2}$ (so $k=6$ and $\alphapha =1.7$) see Figure 8, and as a final example Figure 9 presents the essential spectrum and spectrum for
$\om(z)=\frac{z^7+1.1}{(z-1)^2}$ and $\om(z)=\frac{z^7+0.8}{(z-1)^2}$. In the latter figure
color coding is as follows: the Fredholm index is $-3$ in the yellow region, $-4$ in the green region and $-5$ in the black region.
\betagin{figure}
\includegraphics[height=4cm]{figure_eight}
\caption{The spectrum of $T_\om$, with $k=4$ and $\alphapha=0$.}
\end{figure}
\betagin{figure}
\includegraphics[height=4cm]{figure_nine}
\caption{The spectrum of $T_\om$ with $k=6$ and $\alphapha=1.7$.}
\end{figure}
\betagin{figure}
\includegraphics[height=4cm]{figure_ten}
\includegraphics[height=4cm]{figure_eleven}
\caption{The spectrum of $T_\om$ for $k=7$ and $\alphapha=1.1$ (left) and $k=7$, $\alphapha=0.8$ (right)}
\end{figure}
\end{example}
\paragraph{\bf Acknowledgement}
The present work is based on research supported in part by the National Research Foundation of South Africa. Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard.
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\end{document} |
\begin{document}
\pacs{
03.67.Lx,
03.65.Fd
03.65.Ud
}
\title{Minimal Universal Two-Qubit {\tt CNOT}-based Circuits}
\author{
\begin{tabular}{ccccc}
Vivek V. Shende$^1$ & \;\;\;\; & Igor L. Markov$^2$ & \;\;\;\;
& Stephen S. Bullock$^3$ \\
\footnotesize \tt [email protected] & & \footnotesize \tt [email protected] &
& \footnotesize \tt [email protected] \\
\end{tabular}
}
\affiliation{\small
$^1$ The University of Michigan, Department of Mathematics\\
$^2$ The University of Michigan, Department of Electrical Engineering
and Computer Science \\
$^3$ National Institute of Standards and Technology,
I.T.L.-M.C.S.D. }
\date{\today}
\begin{abstract}
We give quantum circuits that simulate an arbitrary two-qubit unitary
operator up to global phase. For several quantum gate libraries
we prove that gate counts are optimal in worst and average cases.
Our lower and upper bounds compare favorably to previously published results.
Temporary storage is not used because it tends to be expensive in physical
implementations. For each gate library, best gate counts can be achieved by a
single universal circuit.
To compute gate parameters in universal circuits,
we only use closed-form algebraic expressions,
and in particular do not rely on matrix exponentials.
Our algorithm has been coded in C++.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:intro}
Recent empirical work on quantum communication, cryptography and computation
\cite{NielsenC:00}
resulted in a number of experimental systems that can implement two-qubit
circuits. Thus, decomposing arbitrary two-qubit operators into
fewer gates from a universal library may simplify such physical
implementations. While the universality of various gate libraries
has been established in the past \cite{DiVincenzo:95,BarencoEtAl:95},
the minimization of gate counts has only been studied recently.
Universal quantum circuits with six, four and three {\tt CNOT} gates
have been found that can simulate an arbitrary two-qubit operator
up to phase \cite{ZhangEtAl:03,BullockM:03,VidalDawson:03, VatanWilliams:03}.
It has also been shown that if the {\tt CNOT} gate is the only
two-qubit gate available, then three {\tt CNOT} gates
are required \cite{VidalDawson:03, VatanWilliams:03,
OptCUgates:03}. Many of these results rely on the
Makhlin invariants \cite{Makhlin:00} or the related
{\em magic basis} and {\em canonical decomposition}
\cite{BennettEtAl:96,HillWooters:97,LewensteinEtAl:01, KhanejaBG:01a}.
Similar invariants have been investigated previously \cite{Rains:97,
Grassl:98} and more recently in \cite{BullockBrennen:03}.
Our work improves or broadens each of the above circuit constructions
and lower bounds, as summarized in Table \ref{tab:gatecounts}. We
rely on the Makhlin invariants \cite{Makhlin:00}, and simplify them
for mathematical and computational convenience --- our version
facilitates circuit synthesis algorithms.
We have coded the computation of
specific gate parameters in several hundred lines of C++,
and note that it involves only closed-form algebraic expressions
in the matrix elements of the original operator (no matrix logarithms
or exponents) . We articulate the degrees of freedom in our algorithm,
and our program produces multiple circuits for the same operator.
This may be useful with particular implementation technologies where
certain gate sequences are more likely to experience errors.
Additionally, this paper contributes a lower bound
for the number of {\tt CNOT} gates required to simulate an
arbitrary $n$-qubit
operator, which is tighter than the generic bound for arbitrary
two-qubit operators \cite{BarencoEtAl:95, Knill:95}.
The two lines in Table \ref{tab:gatecounts} give gate counts
for circuits consisting of elementary and basic gates, respectively.
Both types were introduced in \cite{BarencoEtAl:95},
but basic gates better reflect gate costs in some physical implementations
where all one-qubit gates are equally accessible. Yet, when working
with ion traps, $R_z$ gates are significantly easier to implement than
$R_x$ and $R_y$ gates \cite{WinelandEtAl:98}. Our work uncovers another
asymmetry,
which is of theoretical nature and does not depend on
the implementation technology --- a subtle complication arises
when only {\tt CNOT}, $R_x$ and $R_z$ gates are available.
Our work shows that basic-gate circuits can be simplified by
temporarily decomposing basic gates into elementary gates, so as to apply
convenient circuit identities summarized in Table \ref{tab:ident}. Indeed,
all lower bounds in Table \ref{tab:gatecounts} and the $n$-qubit
{\tt CNOT} bound above rely on these circuit identities.
Additionally, temporary decompositions into elementary gates
may help optimizing pulse sequences in physical implementations.
\vbox{
The remainder of this paper is structured as follows.
Section \ref{sec:background} discusses gate libraries and
circuit topologies. Section \ref{sec:lowerbounds} derives the lower bounds of
Table \ref{tab:gatecounts}. Section \ref{sec:invariants}
classifies two-qubit operators up to local unitaries.
Section \ref{sec:param} develops some technical lemmata, and
Section \ref{sec:18} constructs small circuits that match upper bounds in
Table \ref{tab:gatecounts}. Subtle complications caused by the lack
of the $R_y$ gate are discussed in the Appendix and Section
\ref{sec:conclusions}.
}
\begin{table}[t]
\begin{center}
\begin{tabular}{|l||r|r|r|r|}
\hline
Gate libraries&\multicolumn{4}{c|}{\ \ \ \ {\bf Lower} and {\bf Upper} Bounds}\\
& {\tt CNOT} & overall & {\tt CNOT} & overall \\
\hline \hline
\{{\tt CNOT}, any 2 or 3 of \{$R_x$, $R_y$, $R_z$\}\} & 3 & 18 & 3 & 18 \\
\{{\tt CNOT}, arbitrary 1-qubit gates \} & 3 & 9 & 3 & 10 \\ \hline
\end{tabular}
\end{center}
\caption{
\label{tab:gatecounts} Constructive upper bounds on gate
counts for generic circuits using several gate libraries. Each
bound given for controlled-not ({\tt CNOT}) gates is compatible
with the respective overall bound. These bounds are tighter than
those from \cite{BullockM:03,ZhangEtAl:03} in all relevant cases.
}
\end{table}
\section{Gate libraries and circuit topologies}
\label{sec:background}
We recall that the Bloch sphere isomorphism \cite{NielsenC:00}
identifies a unit vector $\vec{n} = (n_x, n_y, n_z)$ with
$\sigma_n= n_x \sigma_x + n_y \sigma_y + n_z \sigma_z$. Under this
identification, rotation by the angle $\theta$ around the vector
$\vec{n}$ corresponds to the special unitary operator $R_n(\theta)
= e^{-i\sigma_n\theta/2}$. It is from this identification that the
decomposition of an arbitrary one-qubit gate $U =\mbox{e}^{i\Phi}
R_z(\theta) R_y(\phi) R_z(\psi)$ arises \cite{NielsenC:00}. Of
course, the choice of $y,z$ is arbitrary; one may take any pair of
orthogonal vectors in place of $\vec{y}, \vec{z}$.
\begin{lemma}
\label{lem:rotations} Let $\vec{n},\vec{m} \in \mathbb{R}^3$,
$\vec{n} \perp \vec{m}$, and $U \in SU(2)$. Then one can find
$\theta, \phi$, and $\psi$ such that $U = R_n(\theta) R_m(\phi)
R_n(\psi)$.
\end{lemma}
In the case of $\vec{n} \perp \vec{m}$, we have $\sigma_n
R_m(\theta) \sigma_n = R_m(-\theta)$ and $R_n(\pi/2) R_m(\phi)
R_n(-\pi/2) = R_p (\phi)$ for $\vec{p} = \vec{m} \times \vec{n}$.
For convenience, we set $S_n = R_n(\pi/2)$; then $S_z$ is the
usual $S$ gate, up to phase. In the sequel, we always take $m,n$
out of $x,y,z$.
We denote by $C^a_b$ the controlled-not ({\tt CNOT}) gate with
control on the $a$-th qubit and target on the $b$-th. We recall
that $R_z$ gates commute past {\tt CNOT}s on the control line and
$R_x$ gates commute past {\tt CNOT}s on the target. Finally, for
mathematical convenience, we multiply the {\tt CNOT} gate by a
global phase $\xi$ such that $\xi^4 = -1$; to represent it as an
element of $SU(4)$.
In this work we distinguish two types of gate libraries for
quantum operators that are universal in the exact sense (compare
to approximate synthesis and the Solovay-Kitaev theorem).
The {\em basic-gate} library
\cite{BarencoEtAl:95} contains the {\tt CNOT}, and all one-qubit
gates. {\em Elementary-gate} libraries also {\tt CNOT} gate and
one-qubit gates, but we additionally require that they contain
only finitely many one-parameter subgroups of $SU(2)$. We call
these {\em elementary-gate} libraries, and Lemma
\ref{lem:rotations} indicates that if such a library includes two
one-parameter subgroups of $SU(2)$ (rotations about around
orthogonal axes) then the library is universal. In the literature,
it is common to make assertions like: $\dim[SU(2^n)] = 4^n - 1$.
Thus if a given gate library contains only gates from
one-parameter families and fully-specified gates such as {\tt
CNOT}, at least $4^n - 1$ one-parameter gates are necessary
\cite{BarencoEtAl:95}, \cite[Theorem 3.4]{Knill:95}. Such
dimension-counting arguments lower-bound the number of
$R_x,R_y,R_z$ gates required in the worst case
\cite{BarencoEtAl:95}.
To formalize dimension-counting arguments, we introduce the
concept of {\em circuit topologies} --- underspecified circuits
that may have {\em placeholders} instead of some gates, only with
the gate type specified. Before studying a circuit topology, we
must fix a gate library and thus restrict the types of
fully-specified (constant) gates and placeholders. We say that a
fully-specified circuit $\mathcal{C}$ conforms to a circuit
topology $\mathcal{T}$ if $\mathcal{C}$ can be obtained from
$\mathcal{T}$ by specifying values for the variable gates. All
$k$-qubit gates are to be in $SU(2^k)$, i.e., normalized. For an
$n$-qubit circuit topology $\mathcal{T}$, we define
$Q(\mathcal{T}) \subset SU(2^n)$ to be the set of all operators
that can be simulated, up to global phase, by circuits conforming
to $\mathcal{T}$. We say that $\mathcal{T}$ is universal iff
$Q(\mathcal{T}) = SU(2^n)$. In this work, constant gates are {\tt
CNOT}s, and placeholders represent either all one-qubit gates or a
given one-parameter subgroup of $SU(2)$. We label one-qubit gate
placeholders by $a,b,c,\ldots$, and one-parameter placeholders by
$R_{\ast}$ with subscripts $x$, $y$ or $z$.
We also allow for explicit relations between placeholders. For
example, circuits conforming to the one-qubit circuit topology $a
b a^\dag$ must contain three one-qubit gates and the first and
last must be inverse to each other.
Circuit identities such as $R_n(\theta) R_n(\phi) = R_n(\theta +
\phi)$ can be performed at the level of circuit topologies. This
identity indicates that two $R_n$ gates may always be combined
into one $R_n$ gate, hence anywhere we find two consecutive $R_n$
placeholders in a circuit topology $\mathcal{T}$, we may replace
them with a single one without shrinking $Q(\mathcal{T})$. Of
course, $Q(\mathcal{T})$ does not grow, either, since $R_n(\psi) =
R_n(0) R_n(\psi)$. We may similarly conglomerate arbitrary
one-qubit gate placeholders, pass $R_z$ ($R_x$) placeholders
through the control (target) of {\tt CNOT} gates, decompose
arbitrary one-qubit gate placeholders into $R_n R_m R_n$
placeholders for $n \perp m$, etc.
We now formalize the intuition that the dimension of $SU(2^n)$ should
match the number of one parameter gates.
\begin{lemma}
\label{lem:dimcount} Fix a gate library consisting of constant
gates and finitely many one-parameter subgroups. Then almost all
$n$-qubit operators cannot be simulated by a circuit with fewer
than $4^n - 1$ gates from the one-parameter subgroups.
\end{lemma}
\begin{proof}
Fix a circuit topology $\mathcal{T}$ with fewer than $\ell <
4^n-1$ one-parameter placeholders. Observe that matrix
multiplication and tensor product are infinitely differentiable
mappings and let $f:\mathbb{R}^{\ell}\rightarrow SU(2^n)$ be the
smooth function that evaluates the operator simulated by
$\mathcal{T}$ for specific values of parameters in placeholders.
Accounting for global phase, $Q(\mathcal{T}) = \bigcup_{\xi^{2^n}
= 1} \mbox{Image}(\xi f)$. Sard's theorem
\cite[p.39]{GuilleminPollack:74} demands that $\mbox{Image}(\xi
f)$ be a measure-zero subset of $SU(2^n)$ for dimension reasons,
and a finite union of measure-zero sets is measure-zero.
For a given library, there are only countably many circuit
topologies. Each captures a measure-zero set of operators, and
their union is also a measure-zero set.
\end{proof}
\section{Lower bounds}
\label{sec:lowerbounds}
Lemma \ref{lem:dimcount} implies that for any given elementary gate
library, one can find $n$-qubit operators requiring at least $4^n-1$
one-qubit gates. We use this fact to obtain
a lower bound for the number of {\tt
CNOT} gates required.
\begin{proposition}
\label{prop:cnotcount} Fix any gate library containing only the
{\tt CNOT} and one-qubit gates. Then almost all $n$-qubit
operators cannot be simulated by a circuit with fewer than $\lceil
\frac{1}{4}(4^n - 3n -1)\rceil$ {\tt CNOT} gates.
\end{proposition}
\begin{proof} Enlarging the gate library cannot increase
the minimum number of {\tt CNOT}s in a universal circuit. Thus we
may assume the library is the basic-gate library. We show that any
$n$-qubit circuit topology $\mathcal{T}$ with $k$ {\tt CNOT} gates
can always be replaced with an $n$-qubit circuit topology
$\mathcal{T}'$ with gates from the \{$R_z$, $R_x$, {\tt CNOT}\}
gate library such that $Q(\mathcal{T}) = Q(\mathcal{T}')$ and
$\mathcal{T}'$ has $k$ {\tt CNOT}s and at most $3n + 4k$
one-parameter gates. The proposition follows from $3n + 4k \ge 4^n
- 1$.
We begin by conglomerating neighboring one-qubit gates; this
leaves at most $n + 2k$ one-qubit gates in the circuit. Now
observe that the following three circuit topologies parametrise
the same sets of operators:
\[C_1^2 (a \otimes b) = C_1^2 (R_x R_z R_x \otimes R_z R_x R_z) =
(R_x \otimes R_z) C_1^2 (R_z R_x \otimes R_x R_z)\]
We use this
identity iteratively, starting at the left of the circuit
topology. This ensures that each {\tt CNOT} has exactly four
one-parameter gates to its left.
(Note that we apply gates in circuits left to right,
but read formulae for the same circuits right to left.)
The $n$ one-qubit gates at the far right of the circuit
can be decomposed into three one-parameter gates apiece.
\end{proof}
\begin{corollary}
\label{cor:bounds} Fix an elementary-gate library. Then almost all
two-qubit operators
cannot be simulated without at least
three {\tt CNOT} gates and fifteen one-qubit gates.
\end{corollary}
For elementary-gate libraries containing two out of the three
subgroups $R_x, R_y, R_z$, we give explicit universal two-qubit
circuit topologies matching this bound in Section \ref{sec:18}.
\begin{proposition}
{\label{lem:bounds:basic} Using the basic-gate library, almost all
two-qubit operators require at least three {\tt CNOT} gates, and
at least basic nine gates total.}
\end{proposition}
\begin{proof}
Proposition \ref{prop:cnotcount} implies that at least three {\tt
CNOT} gates are necessary in general; at least five one-qubit
placeholders are required for dimension reasons. The resulting
overall lower bound of eight basic gates can be improved further
by observing that given any placement of five one-qubit gates
around three {\tt CNOT}s, one can find two one-qubit gates on the
same wire, separated only by a {\tt CNOT}. Using the $R_z R_x R_z$
or $R_x R_z R_x$ decomposition as necessary, the 5 one-qubit gates
can be replaced by fifteen one-parameter gates in such a way that
the closest parameterized gates arising from the adjacent
one-qubit gates can be combined. Thus, if five one-qubit
placeholders and three {\tt CNOT}s suffice, then so do fourteen
one-parameter placeholders and three {\tt CNOT}s, which
contradicts dimension-based lower bounds.
\end{proof}
\section{Invariants of two-qubit operators}
\label{sec:invariants}
To study two-qubit operators that differ only by pre- or
post-composing with one-qubit operators, we use the
terminology of {\em cosets}, common in abstract algebra \cite{Artin:91}.
Let $G$ be the group of operators
that can be simulated entirely by one-qubit operations. That is,
$G = SU(2)^{\otimes n} = \{a_1 \otimes a_2 \otimes \ldots \otimes
a_n:a_i \in SU(2)\}$. Then two operators $u, v$ are said to be in
the same left coset of $SU(4)$ modulo $G$ (written: $uG=vG$) iff
$u$ differs from $v$ only by pre-composing with one-qubit
operators; that is, if $u = vg$ for some $g \in G$. Similarly, we
say that $u$ and $v$ are in the same right coset ($Gu = Gv$) if
they differ only by post-composition ($u=hv$ for some $h\in G$),
and we say that $u$ and $v$ are in the same double coset ($u =
GvG$) if they differ by possibly both pre- and post-composition
($u = hvg$ for some $g,h \in G$). In the literature, the double
cosets are often referred to as {\em local equivalence classes}
\cite{ZhangEtAl:03}.
Polynomial invariants classifying the double cosets have been
proposed by Makhlin \cite{Makhlin:00}.
In what follows, we present equivalent invariants which generalize
to $n$-qubits and are more straightforward to compute.
Moreover, the proofs given here detail
an explicit constructive procedure to find $a, b, c, d$ such that
$(a \otimes b) u (c \otimes d) = v$, once it has been determined
by computing invariants that $u, v$ are in the same double coset.
\begin{definition}
\label{def:f}
We define $\gamma_n$ on $2^n \times 2^n$ matrices by the formula
$u \mapsto u \sigma_y^{\otimes n} u^T \sigma_y^{\otimes n}$.
When $n$ is arbitrary or clear from context, we write $\gamma$ for $\gamma_n$.
\end{definition}
\begin{proposition}
\label{prop:f}
$\gamma$ has the following properties:\\
1. $\gamma(I) = I$
2. $\gamma(ab) = a \gamma(b) \gamma(a^T)^T a^{-1}$
3. $\gamma(a \otimes b) = \gamma(a) \otimes \gamma(b)$
4. $g \in M_{2 \times 2}^{\otimes n} \implies \gamma(g) = \det(g) \cdot I$
5. $\gamma$ is constant on the left cosets $u \cdot SU(2)^{\otimes n}$
6. $\chi[\gamma]$ is constant on double cosets $SU(2)^{\otimes n}
\cdot u \cdot SU(2)^{\otimes n}$
\end{proposition}
\begin{proof} (1), (2), and (3) are immediate from the definition.
(4) can be checked explicitly for $n=1$, and then the general case
follows from (3). For (5), note first that $g \in SU(2)^{\otimes
n} \implies \gamma(g) = I$ by (4). Then expressing $\gamma(ag)$ and
$\gamma(a\cdot I)$ using (1) and (2), we see they are equal. For (6),
we use (2), (4), and (5) to see that $g,h \in SU(2)^{\otimes n}
\implies \gamma(gah) = g^{-1} \gamma(ah) g = g^{-1} \gamma(a) g$ thus
$\chi[\gamma(gah)] = \chi[\gamma(a)]$. Incidentally, (6) is closely
related to \cite[Thm I.3]{BullockBrennen:03}.
\end{proof}
While $\gamma$ is constant on left cosets and $\chi[\gamma]$ on double
cosets, these invariants do not in general suffice to classify
cosets. Roughly, a parameter space for double cosets would need
dimension $\dim(SU(2^n)) - 2 \dim (SU(2)^{\otimes n}) = 4^n - 6n
-1$, whereas the space of possible $\chi[\gamma]$ has dimension $2^n
- 1$ (because the $2^n$ roots of $\chi(\gamma)$ must all have unit
length and have unit product). The first dimension is much larger
except for $n=1,2$. In the case $n=1$, there is only one left
coset (and only one double coset), so our invariants trivially
suffice. For $n=2$, these numbers come out exactly equal, and
$\gamma$ and $\chi[\gamma]$ serve to classify respectively the
left cosets and double cosets.
\begin{proposition} \label{prop:invariants}
For $u, v \in SU(4)$, $G = SU(2) \otimes SU(2)$: \\
1. $u \in G \iff \gamma(u) = I$
2. $uG = vG \iff \gamma(u) = \gamma(v)$
3. $GuG = GvG \iff \chi[\gamma(u)] = \chi[\gamma(v)]$
\end{proposition}
\begin{proof}
Recall that $E \in U(4)$ can be found such that $E~SO(4)~E^\dag =
G$; such matrices are characterized by the property that
$EE^T = -\sigma_y \otimes \sigma_y$. This and related issues
have been exhaustively dealt with in several papers
\cite{BennettEtAl:96,HillWooters:97,LewensteinEtAl:01,
KhanejaBG:01a,BullockBrennen:03},
where it is shown that $E$ can be chosen as:
{
\[
\frac{1}{\sqrt{2}} \left(\footnotesize
\begin{array}{cccc}
1 & i & 0 & 0 \\
0 & 0 & i & 1 \\
0 & 0 & i & -1 \\
1 & -i & 0 & 0 \\
\end{array}
\right)
\]
}
Observe that the properties $\gamma(u) = I, \gamma(u)=\gamma(v),
\chi[\gamma(u)] = \chi[\gamma(v)]$ are not changed by replacing
$\gamma$ with $E^\dag \gamma E$. Then using the fact
$- \sigma_y \otimes \sigma_y = EE^T = (EE^T)^\dag$ compute:
\[E^\dag \gamma(g) E = E^\dag g E E^T g^T E^{t\dag} E^\dag E
= (E^\dag g E)(E^\dag g E)^T\]
Therefore it suffices to prove the
proposition after making the following substitutions: $g \mapsto u=E^\dag
g E$, $G \mapsto SO(4)$, $\gamma(g) \mapsto uu^T$.
Now (1) is immediate and (2) follows from
$uu^T = vv^T \iff v^\dag u = (v^\dag u)^{t\dag} \iff v^\dag u \in SO(4)$
To prove (3), note that for $P$ symmetric unitary, $P^{-1} =
\overline{P}$, hence $[P+\overline{P}, P-\overline{P}]=0$. It
follows that the real and imaginary parts of $P$ share an
orthonormal basis of eigenvectors. As they are moreover real
symmetric matrices, we know from the spectral theorem that their
eigenvectors can be taken to be real.
Thus one can find an $a \in SO(4)$ such
that $auu^T a^\dag$ is diagonal. By re-ordering (and negating)
the columns of $a$, we can re-order the diagonal elements of
$auu^T a^\dag$ as desired. Thus if $\chi[uu^T]=\chi[vv^T]$,
we can find $a, b \in SO(4)$ such that
$auu^T a^T = b vv^T b^T$ by diagonalizing both; then
$(v^\dag b^T a u)(v^\dag b^T a u)^T = I$. Let $c = v^\dag b^T a u \in
SO(4)$. We have $ a^T b v c = u$, as desired.
\end{proof}
The proof above gives an algorithm for computing $a,b,c,d$ for
given two-qubit $u$ and $v$ so that $(a \otimes b) u (c \otimes d)
= v$. Also, $u$ may be chosen as a relative-phasing of Bell
states.
\section{Technical Lemmata}
\label{sec:param}
We present two parameterizations of the space of double cosets
described in Section \ref{sec:invariants}. These will be used in
the constructions of universal two-qubit circuit topologies to
follow.
We will use the following general technique to compute $\gamma(u)$.
First, determine a circuit, $C$, simulating the operator $u$.
Given $C$, it is straightforward to obtain a circuit simulating
$\sigma_y^{\otimes 2} u^T \sigma_y^{\otimes 2}$: reverse the order
of gates in $C$, and replace a given gate $g$ by
$\sigma_y^{\otimes 2} g^T \sigma_y^{\otimes 2}$. As will be shown
below, if $g$ is a one-qubit gate, then $\sigma_y^{\otimes 2} g^T
\sigma_y^{\otimes 2} = g^\dag$. For the {\tt CNOT}, we note that
$\sigma_y^{\otimes 2} C_1^2 \sigma_y^{\otimes 2} = C_1^2 (\sigma_x
\otimes \sigma_z)$ and similarly $\sigma_y^{\otimes 2} C_2^1
\sigma_y^{\otimes 2} = C_2^1 (\sigma_z \otimes \sigma_x)$. Now,
combine the circuits for $u$ and $\sigma_y^{\otimes 2} u^T
\sigma_y^{\otimes 2}$ to obtain a circuit simulating $\gamma(u)$.
\begin{proposition} \label{prop:parameters:yz}
For any $u \in SU(4)$, one can find $\alpha, \beta, \delta$
such that $\chi [\gamma(u)] = \chi[\gamma(C_1^2 (I \otimes R_y(\alpha))
C_2^1 (R_z(\delta) \otimes R_y(\beta)) C_1^2)]$.
\end{proposition}
\begin{proof}
Let $v = C_1^2 (I \otimes R_y(\alpha)) C_2^1 (R_z(\delta) \otimes
R_y(\beta)) C_1^2$. As $v$ is given explicitly by a circuit, we
use the technique described above to determine the following
circuit for $\gamma(v)$.
\noindent
\begin{center}
\begin{picture}(26,4)
\put(0,0){\botCNOT}
\put(2,0){\boxGate{$\sigma_z$}}
\put(2,2){\boxGate{$\sigma_x$}}
\put(4,0){\boxGate{$R_y'^\dag$}}
\put(4,2){\hWire}
\put(6,0){\topCNOT}
\put(8,0){\boxGate{$\sigma_x$}}
\put(8,2){\boxGate{$\sigma_z$}}
\put(10,2){\boxGate{$R_z^\dag$}}
\put(10,0){\boxGate{$R_y^\dag$}}
\put(12,0){\botCNOT}
\put(14,0){\boxGate{$\sigma_z$}}
\put(14,2){\boxGate{$\sigma_x$}}
\put(16,0){\botCNOT}
\put(18,0){\boxGate{$R_y$}}
\put(18,2){\boxGate{$R_z$}}
\put(20,0){\topCNOT}
\put(22,0){\boxGate{$R_y'$}}
\put(22,2){\hWire}
\put(24,0){\botCNOT}
\end{picture}
\end{center}
Here, $R_y'= R_y(\alpha)$, $R_y = R_y(\beta)$, and $R_z =
R_z(\delta)$. We now use the circuit identities in Figure
\ref{fig:ident} and $\sigma_i R_j(\theta) = R_j(-\theta)
\sigma_i$ to push all the $\sigma_i$ gates to the left of the
circuit, where they cancel up to an irrelevant global phase of
$-1$. All gates in the wake of their passing become inverted, and
we obtain the following circuit.
\noindent
\begin{center}
\begin{picture}(20,4)
\put(0,0){\botCNOT}
\put(2,0){\boxGate{$R_y'$}}
\put(2,2){\hWire}
\put(4,0){\topCNOT}
\put(6,2){\boxGate{$R_z$}}
\put(6,0){\boxGate{$R_y$}}
\put(8,0){\botCNOT}
\put(10,0){\botCNOT}
\put(12,0){\boxGate{$R_y$}}
\put(12,2){\boxGate{$R_z$}}
\put(14,0){\topCNOT}
\put(16,0){\boxGate{$R_y'$}}
\put(16,2){\hWire}
\put(18,0){\botCNOT}
\end{picture}
\end{center}
For invertible matrices, $\chi(AB) = \chi(A^{-1}(AB)A) =
\chi(BA)$. In view of the fact that we are ultimately interested
only in $\chi[\gamma(V)]$ we may move gates from the left of the
circuit to the right. Thusly conglomerating $R_y'$ gates and
canceling paired {\tt CNOT} gates, we obtain:
\noindent
\begin{center}
\begin{picture}(8,4)
\put(0,0){\boxGate{$R_y'^2$}}
\put(0,2){\hWire}
\put(2,0){\topCNOT}
\put(4,2){\boxGate{$R_z^2$}}
\put(4,0){\boxGate{$R_y^2$}}
\put(6,0){\topCNOT}
\end{picture}
\end{center}
\noindent We have shown $\chi[\gamma(v)] = \chi[C_2^1 (R_z(\delta)
\otimes R_y(\beta)) C_2^1 (I \otimes R_y(\alpha))]$. Again, since
$\chi[B] = \chi[A^{-1} B A]$, we conjugate by $I \otimes S_x$.
This fixes the {\tt CNOT} gate and replace $R_y$ gates with $R_z$:
\[\chi[\gamma(v)] = \chi[C_2^1(R_z(\delta) \otimes R_z(\beta)) C_2^1 (I
\otimes R_z(\alpha))]\]
Finally, we ensure that the entries of the diagonal matrix
$C_2^1(R_z(\delta) \otimes R_z(\beta)) C_2^1 (I \otimes
R_z(\alpha))$ match the spectrum of $\gamma(U)$ by specifying $\alpha
= \frac{x+y}{2}$, $\beta = \frac{x+z}{2}$, and $\delta =
\frac{y+z}{2}$ for $e^{ix}, e^{iy}, e^{iz}$ any three eigenvalues
of $\gamma(U)$.
\end{proof}
\noindent
\begin{figure}
\caption{\label{fig:ident}
\label{fig:ident}
\end{figure}
\begin{proposition} \label{prop:parameters:xz}
For any $u \in SU(4)$, one can find $\theta, \phi, \psi$ such that
$\chi [\gamma(u C_2^1 (I \otimes R_z(\psi)) C_2^1)] = \chi[\gamma(C_2^1
(R_x(\theta) \otimes R_z(\phi)) C_2^1)]$.
\end{proposition}
\begin{proof}
We set $\Delta = C_2^1 (I \otimes R_z(\psi)) C_2^1$ and compute
$\mbox{tr}[\gamma(u\Delta)]$. By Proposition \ref{prop:f}, this is
$\mbox{tr}[\gamma(u^T)^T \gamma(\Delta)]$. Explicit computation as in
the previous proposition gives $ \gamma(\Delta) = \Delta^2$, and one
obtains $\mbox{tr}[\gamma(u\Delta)] = (t_1 + t_4)e^{-i \psi} + (t_2 +
t_3)e^{i \psi}$, where $t_1, t_2, t_3, t_4$ are the diagonal
entries of $\gamma(u^T)^T$. We may ensure that this number is real by
requiring $\tan(\psi) = \frac{\mbox{Im}(t_1 + t_2 + t_3 +
t_4)}{\mbox{Re}(t_1 + t_2 - t_3 - t_4)}$.
Now consider $m \in SU(N)$, $\chi[m] = \sum a_i X^i = \prod(X -
r_i)$, where the $r_i$ form the spectrum of $m$. Since $m \in
SU(N)$, we must have $\prod r_i = 1 = \prod \overline{r_i}$.
Therefore, $\chi[m] = \chi[m] \prod \overline{r_i} =
\prod(\overline{r_i} X - 1)$. Expanding the equality $\prod(X -
r_i) = \prod(\overline{r_i} X - 1)$ gives $\overline{a_i} =
a_{N-i}$. In particular, for $N = 4$, $a_2 \in \mathbb{R}$, and
$\mbox{tr}(m) = a_3 = \overline{a_1}$. Since $a_4 = a_0 = 1$,
$\chi[m]$ has all real coefficients iff $\mbox{tr}[m] \in \mathbb{R}$. In
this case, the roots of $\chi[m]$ must come in conjugate pairs:
$\chi(m) = (X - e^{i r})(X - e^{-i r})(X - e^{is})(X - e^{-is})$.
On the other hand, for $w = C_2^1 (R_x(\frac{r+s}{2})\otimes
R_z(\frac{r-s}{2})) C_2^1$, one can verify that $\chi[\gamma(w)]$
takes this form.
Taking $m = \gamma(U C_2^1 (I \otimes R_z(\psi)) C_2^1)$, with
$\psi$ as determined above, we obtain $\theta = \frac{r+s}{2}$,
$\phi = \frac{r-s}{2}$.
\end{proof}
\section{Minimal two-qubit circuits}
\label{sec:18}
We now construct universal two-qubit circuit topologies that match
the upper bounds of Table \ref{tab:gatecounts}. We consider three
different gate libraries: each contains the {\tt CNOT}, and two
out of the three one-parameter gates \{$R_x$, $R_y$, $R_z$\}. We
will refer to these as the CXY, CYZ, and CXZ gate libraries.
In view of Lemma \ref{lem:rotations}, one might think that there
is no significant distinction between these cases. Indeed,
conjugation by the Hadamard gate transforms will allow us to move
easily between the CXY and CYZ gate libraries. However, we will
see that the CXZ gate library is fundamentally different from the
other two. Roughly, the reason is that $R_x$ and $R_z$ can be
respectively moved past the target and control of the {\tt CNOT}
gate, while no such identity holds for the $R_y$ gate. While the
CXY and CYZ libraries each only contain one of \{$R_x$, $R_z$\},
the CXZ gate library contains both, and consequently has different
characteristics. Nonetheless, gate counts will be the same in all
cases. We begin with the CYZ case, which has been previously
considered in \cite{BullockM:03}.
\begin{theorem}
\label{thm:18:cyz} Fifteen \{$R_y$, $R_z$\} gates and three {\tt
CNOT}s suffice to simulate an arbitrary two-qubit operator.
\end{theorem}
\begin{proof}
Choose $\alpha, \beta, \delta$ as in Proposition
\ref{prop:parameters:yz}. Then by Proposition
\ref{prop:invariants}, one can find $a,b,c,d \in SU(2)$ such that
\[U = (a \otimes b)C_1^2 (I \otimes R_y(\alpha)) C_2^1
(R_z(\delta) \otimes R_y(\beta)) C_1^2 (c \otimes d)\] Thus, the
circuit topology depicted in Figure \ref{fig:18} is universal.
\end{proof}
\begin{figure}
\caption{\label{fig:18}
\label{fig:18}
\end{figure}
\begin{theorem}
\label{thm:18:cxy} Fifteen \{$R_x$, $R_y$\} gates and three {\tt
CNOT}s suffice to simulate an arbitrary two-qubit operator.
\end{theorem}
\begin{proof}
Conjugation by $H^{\otimes n}$ fixes $SU(2^n)$ and $R_y$. It also
flips {\tt CNOT} gates ($H^{\otimes 2} C_1^2 H^{\otimes 2} =
C_2^1$) and swaps $R_x$ with $R_z$.
\end{proof}
Unfortunately, no such trick transforms CYZ into CXZ. Any such
transformation would yield a universal two-qubit circuit topology
in the CXZ library in which only three one-parameter gates occur
in the middle. We show in the Appendix that no such circuit can be
universal and articulate the implications of this distinction
in Section \ref{sec:conclusions}.
Nonetheless, we demonstrate here a universal two-qubit circuit topology
with gates from the \{$R_x$, $R_z$, {\tt CNOT}\} gate library that contains
$15$ one-qubit gates and $3$ {\tt CNOT} gates.
\begin{figure}
\caption{\label{fig:18:cxz}
\label{fig:18:cxz}
\end{figure}
\begin{theorem}
\label{thm:18:cxz}Fifteen \{$R_x$, $R_z$\} gates and three {\tt
CNOT}s suffice to simulate an arbitrary two-qubit operator.
\end{theorem}
\begin{proof}
Let $U'$ be the desired operator; set $U = U' C_2^1 $. Choose
$\theta, \phi, \psi$ for $U'$ as in Proposition
\ref{prop:parameters:xz}. By Proposition \ref{prop:invariants},
one can find $a,b,c,d \in SU(2)$ such that
\[U (I \otimes R_z(\psi)) C_2^1
= (a \otimes b)C_2^1 (R_z(\theta) \otimes R_x(\phi)) C_2^1 (c
\otimes d)\] Solving for $U$ gives the overall circuit topology in
Figure \ref{fig:18:cxz}.
\end{proof}
Unlike the circuit of \ref{thm:18:cyz}, the circuit in Figure
\ref{fig:18:cxz} can be adapted to both other gate libraries. We
can replace $c$ by $S_z ( S_z^\dag c)$ and $a$ by $(a S_z)
S_z^\dag$, then use the $S_z, S_z^\dag$ gates to change the $R_x$
gate into an $R_z$. A similar trick using $R_x$ can change the
bottom $R_z$ gates into $R_y$; this yields a circuit in the CYZ
gate library. As in Theorem \ref{thm:18:cxy}, conjugating by $H
\otimes H$ yields a circuit in the CXY gate library.
Given an arbitrary two-qubit operator, individual gates in
universal circuits can be computed by interpreting proofs of
Propositions \ref{prop:parameters:xz}, \ref{prop:parameters:yz},
and \ref{prop:invariants}, Theorems \ref{thm:18:cyz},
\ref{thm:18:cxy} and \ref{thm:18:cxz} as algorithms. By
re-ordering eigenvalues in the proof of Proposition
\ref{prop:invariants}, one may typically produce several different
circuits. Similar degrees of freedom are discussed in
\cite{BullockM:03}.
To complete Table \ref{tab:gatecounts}, count {\em basic} gates in
Figure \ref{fig:18} or \ref{fig:18:cxz}.
\section{Conclusions}
\label{sec:conclusions}
Two-qubit circuit synthesis is relevant to on-going physics experiments
and can be used in peephole optimization of larger circuits, where
small sub-circuits are identified and simplified one at a time.
This is particularly relevant to quantum communication, where protocols
often transmit one qubit at a time and use encoding/decoding circuits
on three qubits.
We constructively synthesize small circuits for arbitrary two-qubit operators
with respect to several gate libraries. Most of our lower and upper bounds
on worst-case gate counts are tight, and rely on circuit identities
summarized in Table \ref{tab:ident}.
We also prove that $n$-qubit circuits
require $\lceil \frac{1}{4}(4^n - 3n -1)\rceil$
{\tt CNOT} gates in the worst case.
While our techniques do not guarantee
optimal circuits for non-worst-case operators,
they perform well in practice: one run of our
algorithm produced the circuit shown in Figure \ref{fig:qft} for
the two-qubit Quantum Fourier Transform. We show elsewhere
that this circuit has minimal basic-gate count.
\begin{figure}
\caption{\label{fig:qft}
\label{fig:qft}
\end{figure}
A somewhat surprising result of our work is the apparent asymmetry
between $R_x$, $R_y$ and $R_z$ gates. While one would expect any
circuit topology for {\tt CNOT}, $R_z$ and $R_y$ to carry over to
other elementary-gate libraries, we prove a negative result for
the library {\tt CNOT}, $R_z$ and $R_x$. Namely, using $R_y$ gates
appears essential for the minimal universal circuit topology shown
in Figure \ref{fig:18}, which exhibits the maximal possible number
of one-qubit gates that are not between any two {\tt CNOT} gates.
The asymmetry between elementary one-qubit gates directly impacts
peephole optimization of $n$-qubit circuits, where decompositions
like that in Figure \ref{fig:18} are preferrable over that in
Figure \ref{fig:18:cxz}. For example, consider a three-qubit
circuit consisting of two two-qubit blocks on lines (i) one and two,
(ii) two and three. If both blocks are decomposed as in Figure
\ref{fig:18}, then the $b$ gate from the first block and the
$c$ gate from the second block merge into one gate on line two.
However, no such reduction would happen if the decomposition
from Figure \ref{fig:18:cxz} is used.
{
{\bf Acknowledgments and disclaimers.} This work is funded by the
DARPA QuIST program and an NSF grant.
The views and conclusions contained herein are those of the authors
and should not be interpreted as neces\-sarily representing official
policies or endorsements of employers and funding agencies. }
\begin{table}
\begin{center}
\begin{tabular}{|l|l|} \hline
Circuit identities & Descriptions \\
\hline
\hline
$C^k_j C^k_j = 1$ &
{\tt CNOT}-gate cancellation\\
$\omega^{j,k}\omega^{j,k} = 1$ &
{\tt SWAP}-gate cancellation\\
$C_j^k C_k^j = \omega^{j,k} C_j^k$ &
{\tt CNOT}-gate elimination \\
\hline
\hline
$C^j_k R^j_x(\theta) = R^j_x(\theta) C^j_k$,
$C^j_k S^j_x = S^j_x C^j_k$ &
moving $R_x$, $S_x$ via {\tt CNOT} target \\
$C^j_k R^k_z(\theta) = R^k_z(\theta) C^j_k$,
$C^j_k S^k_z = S^k_z C^j_k$ &
moving $R_z$, $S_z$ via {\tt CNOT} control \\
\hline
$\sigma_x^k C_j^k = C_j^k \sigma_x^j \sigma_x^k$ &
moving $\sigma_x$ via {\tt CNOT} control \\
$C_j^k\sigma_z^j = \sigma_z^j \sigma_z^k C_j^k$ &
moving $\sigma_z$ via {\tt CNOT} target \\
\hline
$C_j^k \omega^{j,k} = \omega^{j,k} C_k^j$ &
moving {\tt CNOT} via {\tt SWAP} \\
$V^j \omega^{j,k} = \omega^{j,k} V^k$ &
moving a 1-qubit gate via {\tt SWAP} \\
\hline
\hline
$R_n(\theta) R_n(\phi) = R_n(\theta+\phi)$ &
merging $R_n$ gates. \\
$\vec{n} \perp \vec{m} \implies S_n R_m(\theta)=
R_{n \times m} (\theta) S_n$
& changing axis of rotation \\
\hline
\end{tabular}
\end{center}
\caption{\label{tab:ident} Circuit identities used in out work. Here
$V^j$ represents an arbitrary one-qubit operator acting on wire $j$.}
\end{table}
\section*{Appendix}
We now illustrate the counterintuitive difference between
(i) the CXZ library, and (ii) libraries CYZ and CXY. Namely,
universal circuit topologies with certain properties exist
only for the CYZ and CXY libraries.
The proof of Proposition \ref{thm:18:cyz} contains
a universal generic circuit with three {\tt CNOT} gates and 15 $R_y$ or $R_z$
gates with the property that all but three of the one-qubit gates
appear either before the first or after the last {\tt CNOT} gate.
This is minimal.
\begin{proposition} \label{prop:mid3}
Fix an elementary-gate library. There exist unitary operators $U
\in SU(4)$ that cannot be simulated by any two-qubit circuit in
which all but two of the one-qubit gates appear either before the
first or after the last {\tt CNOT} gate.
\end{proposition}
\begin{proof}
There are four places where the one-parameter gates can appear: at
the left or right of the first or second line. If more than three gates
appear in one such place, conglomerate them into a single one-qubit
gate, and decompose the result into three one-parameter
gates via Lemma \ref{lem:rotations}. By this method, any two-qubit
circuit can be transformed into an equivalent circuit with
at most 12 one-parameter gates on its sides. By Corollary
\ref{cor:bounds}, there exist operators that cannot be simulated
without 15 one-parameter gates; the remaining three must go in the
middle of the circuit.
\end{proof}
We have seen that for the CYZ and the CXY gate libraries, this
lower bound is tight. We will show that this is not the case for
the CXZ gate library. Before beginning the proof, we make several
observations about the CXZ gate library.
Note that conjugating a circuit identity by $H \otimes H$
exchanges $R_x$ and $R_z$ gates, and flips {\tt CNOT}s. Two other
ways to produce new identities from old are: swapping wires, and
inverting the circuit -- reversing the order of gates \& replacing
each with its inverse. For example, one may obtain one of the
commutativity rules below from the other by conjugating by $H
\otimes H$ and then swapping wires.
\begin{center}
\begin{picture}(24,4)
\put(0,0){\topCNOT}
\put(2,0){\boxGate{$R_x$}}
\put(2,2){\hWire}
\put(5,2){$\equiv$}
\put(6,0){\boxGate{$R_x$}}
\put(6,2){\hWire}
\put(8,0){\topCNOT}
\put(14,0){\topCNOT}
\put(16,2){\boxGate{$R_z$}}
\put(16,0){\hWire}
\put(19,2){$\equiv$}
\put(20,0){\hWire}
\put(20,2){\boxGate{$R_z$}}
\put(22,0){\topCNOT}
\end{picture}
\end{center}
When one {\tt CNOT} gate occurs immediately after another in a
circuit, we say that they are {\em adjacent}. When such
pairs of {\tt CNOT}s share control lines, they cancel out, and
otherwise may still lead to reductions as discussed
below. We will be interested in circuits which do not allow
such simplifications. To this end, recall that $R_x$ gates
commute past the target of a {\tt CNOT}, and $R_z$ gates commute
past the control. Moreover, we have the following circuit
identity: $C_2^1 (R_x(\alpha) \otimes R_z(\beta)) C_2^1 = C_1^2
(R_z(\beta) \otimes R_x(\alpha)) C_1^2$. We say that a given
collection of one-qubit gates {\em effectively separates} a chain
of {\tt CNOT}s iff there is no way of applying the aforementioned
transformation rules to force two {\tt CNOT} gates to be adjacent.
For example, there is no way to effectively separate two {\tt
CNOT}s of opposite orientation by a single $R_x$ or $R_z$
gate. This is illustrated below.
\begin{center}
\begin{picture}(14,4)
\put(0,0){\topCNOT}
\put(2,0){\hWire}
\put(2,2){\boxGate{$R_x$}}
\put(4,0){\botCNOT}
\put(7,2){$\equiv$}
\put(8,0){\topCNOT}
\put(10,0){\botCNOT}
\put(12,2){\boxGate{$R_x$}}
\put(12,0){\hWire}
\end{picture}
\end{center}
On the other hand, two {\tt CNOT} gates of the same orientation
can be effectively separated by a single $R_x$ or $R_z$ gate, as
shown below. Up to swapping wires, these are the only ways to
effectively separate two {\tt CNOT}s with a single $R_x$ or $R_z$.
\begin{center}
\begin{picture}(16,4)
\put(0,0){\botCNOT}
\put(2,2){\hWire}
\put(2,0){\boxGate{$R_x$}}
\put(4,0){\botCNOT}
\put(10,0){\botCNOT}
\put(12,2){\boxGate{$R_z$}}
\put(12,0){\hWire}
\put(14,0){\botCNOT}
\end{picture}
\end{center}
\begin{proposition} \label{prop:4eff}
At least four gates from \{$R_x$, $R_z$\} are necessary to
effectively separate four or more {\tt CNOT} gates.
\end{proposition}
\begin{proof}
Clearly it suffices to check this in the case of exactly four {\tt
CNOT}s. If three $R_x$, $R_z$ gates sufficed, then one would have
to go between each pair of {\tt CNOT} gates. Suppose all the {\tt
CNOT} gates have the same orientation, say with control on the
bottom wire. Then the first pair must look like one of the pairs
above. In either case, we may use the identity $C_2^1 (R_x(\alpha)
\otimes R_z(\beta)) C_2^1 = C_1^2 (R_z(\beta) \otimes R_x(\alpha))
C_1^2$ to flip these {\tt CNOT} gates, thus ensuring that there is
a consecutive pair of {\tt CNOT} gates with opposite orientations.
As remarked above, there is no way to effectively separate these
using the single one-qubit gate allotted them.
\end{proof}
Denote by $\omega^{ij}$ the {\tt SWAP} gate which exchanges the
$i$-th and $j$-th qubits. It can be simulated using {\tt CNOT}s as
$C_i^j C_j^i C_i^j = \omega^{ij} = C_j^i C_i^j C_j^i$. {\tt SWAP}
gates can be pushed through an elementary-gate circuit without
introducing new gates. So, consider a two-qubit circuit in which
adjacent {\tt CNOT} gates appear. If they have the same
orientation (eg. $C_1^2 C_1^2$ or $C_2^1 C_2^1$), then they cancel
out and can be removed from the circuit. Otherwise, use the
identity $C_1^2 C_2^1 = C_2^1 \omega^{12}$ or $C_2^1 C_1^2 = C_1^2
\omega^{12}$ and push the {\tt SWAP} to the end of the circuit. We
apply this technique at the level of circuit topologies and
observe that since $Q(\mathcal{T}\omega^{12})$ is measure-zero (or
universal) iff $Q(\mathcal{T})$ is. By the above discussion, we
can always reduce to an effectively separated circuit before
checking these properties.
\begin{proposition} \label{prop:mid4}
Almost all unitary operators $U \in SU(4)$ cannot be simulated
by any two-qubit circuit with CXZ gates
in which all but three of the $R_x, R_z$ gates
appear either before the first or after the last {\tt CNOT}.
\end{proposition}
\begin{proof}
We show that any circuit topology of the form above can only
simulate a measure-zero subset of $SU(4)$; the result then
follows from the fact that a countable union of measure-zero
sets is measure-zero.
The assumption amounts to the fact that only three gates are available
to effectively separate the {\tt CNOT} gates.
By Proposition \ref{prop:4eff} and the discussion immediately
following it, we need only consider circuit topologies with no
more than three {\tt CNOT}s. On the other hand, we know from
Proposition \ref{lem:bounds:basic} that any two-qubit circuit topology with
fewer than three {\tt CNOT} gates can simulate only a measure-zero
subset of $SU(4)$. Thus it suffices to consider circuit
topologies with exactly three {\tt CNOT} gates. Moreover, we
can require that they be effectively separated, since
otherwise we could reduce to a two-{\tt CNOT} circuit.
Three {\tt CNOT}s partition a minimal two-qubit circuit
in four regions. We are particularly interested in
the two regions limited by {\tt CNOTs} on both sides
because single-qubit gates in those regions must effectively separate
the {\tt CNOT}s. To this end, we consider
two pairs of {\tt CNOT}s (the central {\tt CNOT} is in both
pairs), and distinguish these three cases: (1) both pairs of
{\tt CNOT}s consist of gates
of the same orientation, (2) both consist of gates of opposite
orientations, or (3) one pair has gates of the opposite
orientations and the other pair has gates of the same
orientation. In the second case, the {\tt CNOT} gates cannot
be effectively separated, since each pair of gates with
opposite orientations requires two one-parameter gates to be
effectively separated, and only three $R_x$, $R_z$ gates
are available. In the third case, two {\tt CNOT}s with
opposite orientations must be separated by two one-parameter
gates,
leaving only one $R_x$ or $R_z$ to separate the pair with the same
orientation. Thus, the pair with the same orientation may be flipped,
reducing to Case 1, as shown below.
\begin{center}
\begin{picture}(26,4)
\put(0,0){\topCNOT}
\put(2,0){\hWire}
\put(2,2){\boxGate{$R_x$}}
\put(4,0){\topCNOT}
\put(6,2){\hWire}
\put(8,2){\hWire}
\put(6,0){\boxGate{$R_z$}}
\put(8,0){\boxGate{$R_x$}}
\put(10,0){\botCNOT}
\put(13,2){$\equiv$}
\put(14,0){\botCNOT}
\put(16,2){\hWire}
\put(16,0){\boxGate{$R_x$}}
\put(18,0){\botCNOT}
\put(20,2){\hWire}
\put(22,2){\hWire}
\put(22,0){\boxGate{$R_x$}}
\put(20,0){\boxGate{$R_z$}}
\put(24,0){\botCNOT}
\end{picture}
\end{center}
Finally, consider the case in which all three {\tt CNOT} gates
have the same orientation. Each pair of consecutive {\tt
CNOT}s must have at least one $R_x$ or $R_z$ between them, to
be effectively separated. Thus one of the pairs has a single $R_x$ or $R_z$
between its members, and the other has two one-qubit gates.
We refer to these as the 1-pair and the 2-pair, respectively.
Suppose that the one-qubit gates separating the 2-pair of
{\tt CNOT}s occur on different lines. If either one-qubit can commute
past the {\tt CNOT}s of the 2-pair, then it can move to the edge of the circuit;
in this case Proposition \ref{prop:mid3} implies that the circuit
topology we are looking at can only simulate a measure-zero subset of
$SU(4)$ (one can show that two $R_x$, $R_z$ gates cannot effectively
separate three {\tt CNOT}s.) Otherwise, we use the identity
$C_2^1 (R_x(\alpha) \otimes R_z(\beta)) C_2^1 = C_1^2 (R_z(\beta)
\otimes R_x(\alpha)) C_1^2$ to flip the 2-pair,
and thus 1-pair now have opposite orientations. As there is
only one one-qubit gate between them, this pair
is not effectively separated. For example:
\begin{center}
\begin{picture}(22,4)
\put(0,0){\topCNOT}
\put(2,0){\hWire}
\put(2,2){\boxGate{$R_x$}}
\put(4,0){\topCNOT}
\put(6,0){\boxGate{$R_z$}}
\put(6,2){\boxGate{$R_x$}}
\put(8,0){\topCNOT}
\put(11,2){$\equiv$}
\put(12,0){\topCNOT}
\put(14,0){\botCNOT}
\put(16,0){\hWire}
\put(16,2){\boxGate{$R_x$}}
\put(18,0){\boxGate{$R_x$}}
\put(18,2){\boxGate{$R_z$}}
\put(20,0){\botCNOT}
\end{picture}
\end{center}
We are left with the possibility that all the {\tt CNOT} gates have
the same orientation and that the 2-pair's one-qubit gates appear on the
same line. Both $R_z$, $R_x$ must occur, or else we could combine
them and apply Proposition \ref{prop:mid3} to show
that such a circuit topology can only simulate a
measure-zero subset of $SU(2^n)$. Now, if $R_x R_z$ appears
between two {\tt CNOT} gates of the same orientation, then
either the $R_x$ or the $R_z$ can commute past one of them. If
the outermost gate can commute, Proposition \ref{prop:mid3}
again implies that the circuit topology simulates only a
measure-zero subset of $SU(2^n)$. Thus the inner gate can
commute with the 1-pair. We have now interchanged the roles of
the 1-pair and the 2-pair, thus by the previous paragraph,
the gate which originally separated the 1-pair must be on the same line
as the commuting gate. It follows that all gates are on the same line.
Up to conjugating by $H \otimes H$, swapping wires, and
inverting the circuit, this leaves exactly one possibility.
\begin{center}
\begin{picture}(12,4)
\put(0,0){\botCNOT}
\put(2,2){\hWire}
\put(2,0){\boxGate{$R_x$}}
\put(4,0){\botCNOT}
\put(6,2){\hWire}
\put(8,2){\hWire}
\put(6,0){\boxGate{$R_z$}}
\put(8,0){\boxGate{$R_x$}}
\put(10,0){\botCNOT}
\end{picture}
\end{center}
Finally, we add the four one-qubit gates on the sides, decompose
each into $R_x R_z R_x$ via Lemma \ref{lem:rotations}, and observe
that an $R_x$ gate can commute across the top and be absorbed on
the other side. This leaves 14
one-parameter gates, and by Lemma \ref{lem:dimcount}, such a
circuit topology simulates only a measure-zero subset of $SU(4)$.
\end{proof}
\end{document} |
\begin{document}
\title{Phase-dependent fluctuations of resonance fluorescence near the
coherent population trapping condition}
\author{O. de los Santos-S\'anchez}
\email{[email protected]}
\affiliation{Escuela de Ingenier\'{\i}a y Ciencias, Instituto Tecnol\'ogico y de Estudios
Superiores de Monterrey, \\ Avenida San Carlos 100, Campus Santa Fe, Ciudad
de M\'exico, 01389, M\'exico}
\author{H. M. Castro-Beltr\'an}
\email{[email protected]}
\affiliation{Centro de Investigaci\'on en Ingenier\'ia y Ciencias Aplicadas,
Instituto de Investigaci\'on en Ciencias B\'asicas y Aplicadas, Universidad
Aut\'onoma del Estado de Morelos, Avenida Universidad 1001, 62209
Cuernavaca, Morelos, M\'exico}
\date{\today}
\begin{abstract}
We study phase-dependent fluctuations of the resonance fluorescence of a single
$\Lambda$-type three-level atom in the regime near coherent population trapping,
i.e., alongside the two-photon detuning condition. To this end, we employ the method
of conditional homodyne detection (CHD) which considers squeezing in the weak
driving regime, and extends to non-Gaussian fluctuations for saturating and strong
fields. In this framework, and using estimated parameter settings of the resonance
fluorescence of a single trapped $^{138} \mathrm{Ba}^{+}$ ion, the light scattered
from the probe transitions is found to manifest a non-classical character and
conspicuous asymmetric third-order fluctuations in the amplitude-intensity
correlation of CHD.
\end{abstract}
\keywords{resonance fluorescence, coherent population trapping, squeezing,
non-Gaussian fluctuations.}
\maketitle
\section{Introduction}
Quantum interference effects in the interaction between matter and light,
epitomized by coherent population trapping (CPT) and electromagnetically
induced transparency (EIT), have extensively been studied, both theoretically
and experimentally, over the past decades \cite{Arimondo,FlIM05}. The most
common level structure to enable these effects is the three-level system in the
$\Lambda$ configuration ($\Lambda$-3LA). Although early research in this
regard was primarily focused on ensembles of atomic constituents,
state-of-the-art experimental developments in atomic spectroscopy have made
it possible to realize EIT with a single atom in free space \cite{SHG+10}.
Indeed, these achievements have paved the way for exploring new avenues of
spectroscopic analyses, besides their potential applications in the thriving field
of quantum information, demonstrating, for instance, the viability of
single-atom-based optical logic gates and quantum memories \cite{HPL+09}.
Both CPT and EIT are based on the cancellation of absorption when two lasers
are detuned equally on adjacent transitions, thus stopping further fluorescence.
Near this two-photon detuning condition, large quantum fluctuations are thus
expected. Phase-sensitive fluctuations of the electromagnetic field, usually
characterized by the phenomenon of squeezing, are of particular interest.
Squeezing is the shrinking of a field's quadrature fluctuations at the expense of
increasing those of its conjugate, and is signaled by negative spectra or variance
below the shot noise level. For the resonance fluorescence of a single two-level
atom, squeezing was first predicted almost forty years ago \cite{WaZo81,CoWZ84},
but it was only very recently that squeezing of a two-level quantum dot was
observed \cite{SHJ+15}. This achievement required overcoming the large
collection losses of resonance fluorescence and the quantum detection losses
of the standard balanced homodyne detection (BHD) technique. These issues
were addressed, respectively, by the higher photon collection geometry allowed
by the quantum dot, and by using a method called homodyne correlation
measurement (HCM) \cite{Vogel91,Vogel95,KVM+17}.
The HCM method realizes an intensity-intensity correlation of the light of a
previously selected quadrature; by measuring for several phases of a weak local
oscillator, the method gives access to the variance (squeezing) \cite{SHJ+15} and
a third-order moment of the field. The latter signals the evolution of the field after
a photon was detected, as was demonstrated for the resonance fluorescence of a
$\Lambda$-3LA \cite{GRS+09}, in a driving regime not weak enough to obtain
squeezing, and far from EIT. The third-order moment is a reachable step above
squeezing in the quest for high-order non-classicality \cite{ScVo05,ScVo06}.
Conditional homodyne detection (CHD) is another measurement scheme
capable of detecting phase-dependent fluctuations with high efficiency owing
to its conditional character \cite{CCFO00,FOCC00,CFO+04}. It consists of BHD
on the cue of photons recorded in a separate photodetector, giving direct access
to the third-order moment of the field. Squeezing is measured if the source is
weakly excited (in fact, the first motivation for the scheme) since in this case the
third-order fluctuations of the field are small. However, these fluctuations,
non-negligible for stronger excitation, are no less interesting: CHD goes beyond
squeezing \cite{hmcb10,CaGH15} and reveals the non-Gaussian character of a
source.
One manifestation of non-Gaussian fluctuations is the asymmetry of the field's
amplitude-intensity correlation whenever two or more transitions compete \cite{DeCC02}; it is not observed in the resonance fluorescence of a two- or
three-level atom driven by a single laser \cite{hmcb10,CaGH15,CaRG16}. While
this asymmetry was readily observed for cavity QED systems both numerically \cite{CCFO00, DeCC02} and experimentally \cite{FOCC00}, it has been the
resonance fluorescence of several 3LA systems that have provided clear
theoretical access to the understanding of the asymmetry
\cite{MaCa08,GCRH17,XGJM15,XuMo15,GaJM13,WaFO16}.
More recent accounts of asymmetric correlations are found in plasmonics
\cite{Santos19} and collective cavity QED \cite{Zhao+20}.
In the experiment outlined in \cite{GRS+09}, squeezing, far from the two-photon detuning, was explored in the weak field regime. In keeping with the same spirit, quantum fluctuations of the light scattered by a coherently driven V-type 3LA have thoroughly been analyzed \cite{GCRH17}. In this work, near the two-photon detuning, we investigate, within the framework of CHD, the adjoining effect of CPT on the phase-dependent quantum fluctuations of
the emitted light of the probe transition of a $\Lambda$-3LA by amplitude-intensity
correlations. We follow closely the experimental
conditions of observation of EIT in single $^{138} \mathrm{Ba}^+$ resonance
fluorescence of Ref.~\cite{SHG+10}, where saturation is present, and we find the
fluctuations to be predominantly non-Gaussian.
Our work is structured by introducing the atom-laser model in Section 2, discussing
the role of coherent population trapping on the state populations and on the
emission spectrum; section 3 is devoted to the theory of conditional homodyne
detection and the analysis of quadrature fluctuations via the associated
amplitude-intensity correlation. We study, in section 4, the quadrature fluctuations
in the spectral domain, including squeezing and variance. Finally, in section 5, we
present our conclusions and an appendix shows additional calculations.
\begin{figure}
\caption{\label{fig:3LA}
\label{fig:3LA}
\end{figure}
\section{Model} \label{sec:theory}
\subsection{Atom-Laser Interaction}
Our system, pictorially represented in Fig.~\ref{fig:3LA}, consists of a
$\Lambda$-type three-level atom ($\Lambda$-3LA) with a single excited state
$|e \rangle$ coupled by a monochromatic laser
with Rabi frequency $\Omega_a$ to the ground state $|a \rangle$ and decay rate $\gamma_a$, and to a
long-lived state $|b \rangle$ by a monochromatic laser with Rabi frequency
$\Omega_b$ and decay rate $\gamma_b$. Decay from $|b \rangle $ to
$|a \rangle$ is dipole-forbidden. Henceforth, the fields driving the
$|a \rangle \to |e\rangle$ and $|b\rangle \to |e \rangle$ transitions will be referred
to as the probe and control fields. We define the atomic operators as
$\hat{\sigma}_{jk} = |j \rangle \langle k|$.
Under the above considerations, the system's evolution, in free space, and in
the frame rotating at the laser frequencies, $\nu_{a}$ and $\nu_{b}$, is
governed by the master equation
$\dot{\tilde{\rho}}(t)=-i [\hat{H}, \tilde{\rho}]
+\sum_{j} \frac{\gamma_j}{2} \mathcal{L}_{\hat{\sigma}_{je}}[\tilde{\rho}]$,
in which
\begin{equation}
\hat{H} = \sum_{j=a,b} - \Delta_j \hat{\sigma}_{jj}
+ \frac{\Omega_j}{2} ( \hat{\sigma}_{ej} +\hat{\sigma}_{je})
\end{equation}
is the atom-laser Hamiltonian, and $\Delta_{j} = \omega_{ej}-\nu_{j} $ labels
the individual atom-laser detunings. Dissipation is accounted for by the action
of the Lindblad generator
$\mathcal{L}_{\hat{O}}[\tilde{\rho}] = 2\hat{O} \tilde{\rho} \hat{O}^{\dagger}
- \hat{O}^{\dagger} \hat{O} \tilde{\rho} -\tilde{\rho} \hat{O}^{\dagger} \hat{O} $,
with $\hat{O}=\hat{\sigma}_{je}$. With the help of the relationship
$\hat{\sigma}_{jk} \hat{\sigma}_{lm} = \hat{\sigma}_{jm} \delta_{kl}$, the master equation can be explicitly recast as
\begin{equation} \label{masterEq}
\dot{\tilde{\rho}}(t) = -i [\hat{H}, \tilde{\rho}]
+\sum_{j=a,b} \gamma_j \tilde{\rho}_{ee} \hat{\sigma}_{jj}
- \frac{\gamma_j}{2} \left( \hat{\sigma}_{ee} \tilde{\rho}
+\tilde{\rho} \hat{\sigma}_{ee} \right).
\end{equation}
With the relation $\langle \hat{\sigma}_{jk} \rangle = \tilde{\rho}_{kj}$,
Eq.~(\ref{masterEq}) allows us to arrive at the following set of linear
equations for populations:
{\setlength\arraycolsep{2pt}
\begin{eqnarray}
\langle \dot{\hat{\sigma}}_{aa} \rangle
&=& - i \frac{\Omega_a}{2} (\langle \hat{\sigma}_{ae} \rangle
- \langle \hat{\sigma}_{ea} \rangle)
+\gamma_a \langle \hat{\sigma}_{ee} \rangle, \label{eq:pop1} \\
\langle \dot{\hat{\sigma}}_{bb} \rangle
&=& - i \frac{\Omega_b}{2} (\langle \hat{\sigma}_{be} \rangle
- \langle \hat{\sigma}_{eb} \rangle)
+\gamma_b \langle \hat{\sigma}_{ee} \rangle, \label{eq:pop2} \\
\langle \dot{\hat{\sigma}}_{ee} \rangle
&=& i \frac{\Omega_a}{2} (\langle \hat{\sigma}_{ae} \rangle
- \langle \hat{\sigma}_{ea} \rangle)
+ i \frac{\Omega_b}{2} (\langle \hat{\sigma}_{be} \rangle
- \langle \hat{\sigma}_{eb} \rangle) \nonumber \\
& & - \gamma \langle \hat{\sigma}_{ee} \rangle, \label{eq:pop3}
\end{eqnarray}}
with $\gamma= \gamma_a +\gamma_b$, and coherences:
{\setlength\arraycolsep{2pt}
\begin{eqnarray}
\langle \dot{\hat{\sigma}}_{ab} \rangle &=&
i\frac{ \Omega_a}{2} \langle \hat{\sigma}_{eb} \rangle
- i \frac{\Omega_b}{2} \langle \hat{\sigma}_{ae} \rangle
- i (\Delta_a -\Delta_b) \langle \hat{\sigma}_{ab} \rangle, \label{eq:coh1} \\
\langle \dot{\hat{\sigma}}_{ae} \rangle &=&
i \frac{\Omega_a}{2} (\langle \hat{\sigma}_{ee} \rangle
- \langle \hat{\sigma}_{aa} \rangle)
-i\frac{\Omega_b}{2} \langle \hat{\sigma}_{ab} \rangle \nonumber \\
&& - \left( \frac{\gamma}{2}
+ i \Delta_a \right) \langle \hat{\sigma}_{ae} \rangle, \label{eq:coh2} \\
\langle \dot{\hat{\sigma}}_{be} \rangle &=&
- \frac{i\Omega_a}{2} \langle \hat{\sigma}_{ba} \rangle
+ \frac{i\Omega_b}{2} (\langle \hat{\sigma}_{ee} \rangle
- \langle \hat{\sigma}_{bb} \rangle) \nonumber \\
&& - \left( \frac{\gamma}{2}
+i \Delta_b \right) \langle \hat{\sigma}_{be} \rangle, \label{eq:coh3} \\
\langle \dot{\hat{\sigma}}_{jk} \rangle
&=& \langle \dot{\hat{\sigma}}_{kj} \rangle^{\ast}. \label{eq:coh4}
\end{eqnarray}}
The solution to these equations (a set of nine Bloch equations) is to be
obtained numerically and their structure will facilitate the assessment of the
sought correlation functions via the quantum regression formula, combined
with the employment of matrix methods. For later use, we define the values
of the atomic operators in the steady state as
\begin{equation}
\langle \hat{\sigma}_{jk} (t \to \infty) \rangle
= \langle \hat{\sigma}_{jk} \rangle_{ss} = \alpha_{jk} .
\end{equation}
\begin{figure}
\caption{Upper panel: Occupation probability of the excited state,
$\langle \sigma_{ee}
\label{fig:populations}
\end{figure}
Besides, in order for our findings to be possibly put to the test in a given
realization, we shall consider the decays $\gamma_a =14.7$ MHz and
$\gamma_b =5.4$ MHz, observed in $^{138} \mathrm{Ba}^+$ ions \cite{GRS+09,SHG+10,DNG+15}. Although a more accurate description of
Barium resonance fluorescence would entail considering its multilevel structure,
being composed of eight energy levels, it suffices for our purposes to deal with
the simplified three-level system as a proxy for specifying the relevant allowed
dipole transitions that take part in the dynamics. Parenthetically, the isolation of
a single three-level configuration can be implemented through a proper optical
pumping arrangement.
\subsection{Role of Coherent Population Trapping}
The $\Lambda$-type three-level atom is an archetypal system that readily fulfills
the necessary conditions for coherent population trapping (CPT) to take place
\cite{Arimondo,FlIM05}. In such a scenario, the system is known to evolve towards
the trapping state $|u \rangle = (\Omega_b |a \rangle - \Omega_a |b \rangle )/
\sqrt{ \Omega_a^2 +\Omega_b^2 } $ that turns out to be decoupled from the
lasers, thereby dropping the long-term excited-state population $\alpha_{ee}$
to nearly zero. The manifestation of
this effect is exemplified in the upper panel of Fig.~\ref{fig:populations}, where the
steady state population of the excited state is shown as a function of both the
detuning and Rabi frequency of the probe laser ($e \to a$ transition); the
values of the parameters associated with the control field are, henceforth, taken
to be fixed and the same as those reported in \cite{SHG+10}, namely,
$\Omega_{b}/\gamma_{a} \approx 2.15$ and
$\Delta_{b}/\gamma_{a} \approx 2.38$. In accord with the well-established
prescription to determine the frequency region around which the atom is
essentially transparent to the incoming probe field, the so-called Raman
resonance condition, the probe detuning must be such that
$\Delta_{a} \approx \Delta_{b}$ is satisfied; the role of the probe intensity
$\Omega_{a}$ is that of slightly modifying the width of such a transparent
frequency window. Its location is also depicted in the lower panel of
Fig.~\ref{fig:populations} showing the cross sectional profile of the upper figure
(black line) at $\Omega_{a}/\gamma_{a}\approx 1.12$ where, in turn, we can
observe the complete depopulation of the excited state at
$\Delta_{a}/\gamma_{a}\approx 2.38$ (dashed vertical line); the populations of
the $|a\rangle$ and $|b\rangle$ states are also added as a supplementary view
of their behavior as functions of the probe detuning.
The foregoing was not the actual condition under which the experiments
\cite{SHG+10} were performed, but instead the detuning was chosen so as to fit
the value of the corresponding saturation parameter,
$\Omega_{j}^{2}/(\gamma_{j}^{2}+\Delta_{j}^{2})$, and taken to be $\sim 0.1$ for
the $a \to e$ transition. This choice gives rise to a detuning of about
$\Delta_{a}/\gamma_{a} \approx 3.4$, the location of which being also indicated
in the figure (dotted-dashed vertical line); the saturation parameter associated
with the control field was set to $0.8$. So, for this particular choice of probe and
control detunings that drive the $a \to e$ transition out of the Raman resonance
condition, the complete depopulation of the $|e\rangle$ state can be avoided or
delayed, a working situation that will permit us to study the non-classical properties
of the scattered light we seek to assess. It is worth commenting that if, instead, the
$a \to e$ transition were driven more strongly than the $b \to e$ one, such that
$\Omega_a > \Omega_b$, for general detunings, the population would end up in
the $|b\rangle$ state, with $\Omega_b \ll \gamma_a$. So then, the strong transition
would be turned off due to lack of recycling population to $|e\rangle$. \\
\begin{figure}
\caption{Incoherent spectrum, $S_{inc}
\label{spectrum_inc}
\end{figure}
So, having established the present configuration of laser intensities and
frequencies, we find it pertinent, at this stage, to depict the stationary power
spectrum of the re-emitted light obtained by use of the Wiener-Khintchine
formula
\begin{equation}
S(\omega) = \frac{1}{\pi \alpha_{ee}} \textrm{Re} \int_{0}^{\infty}d\tau
e^{-i\omega \tau} \langle \hat{\sigma}_{ea}(0) \hat{\sigma}_{ae} (\tau) \rangle_{ss},
\end{equation}
i.e., the Fourier transform of the autocorrelation function of the dipole field,
$\langle \hat{\sigma}_{ea}(0) \hat{\sigma}_{ae} (\tau) \rangle_{ss}$, where $ss$
indicates that the process is stationary; the
prefactor $(\pi \alpha_{ee})^{-1}$ normalizes the integral over all frequencies.
For convenience, the spectrum is separated into its coherent and incoherent
parts, namely, $S(\omega)=S_{coh}(\omega)+S_{inc}(\omega)$, as a result of considering the dynamics of the atomic variables to be split into their mean and fluctuations, viz.
$\hat{\sigma}_{jk} (t) = \alpha_{jk} +\Delta \hat{\sigma}_{jk}(t)$, with
$\langle \Delta \hat{\sigma}_{jk}(t) \rangle = 0$. In doing so, we get
\begin{equation}
S_{coh}(\omega) = \frac{|\alpha_{ea}|^{2}}{\pi \alpha_{ee}}
\textrm{Re} \int_{0}^{\infty} d \tau e^{-i\omega \tau}
= \frac{|\alpha_{ea}|^{2}}{\pi \alpha_{ee}}\delta (\omega),
\end{equation}
where $\alpha_{ea} = \langle \hat{\sigma}_{ea} \rangle_{ss}$, and
\begin{equation}
S_{inc} (\omega) = \frac{1}{\pi \alpha_{ee}} \textrm{Re} \int_{0}^{\infty}
d\tau e^{-i\omega \tau} \langle
\Delta \hat{\sigma}_{ea}(0) \Delta \hat{\sigma}_{ae}(\tau) \rangle_{ss},
\end{equation}
the former being the coherent constituent of the spectrum owing to elastic
scattering, and the latter the incoherent part of the spectrum that is brought about
by atomic fluctuations. The main features of the $\Lambda$-type three-level atom
spectrum have already been studied from the weak to the strong field limit, both
theoretically and experimentally \cite{SSA+96}. Figure~\ref{spectrum_inc} shows
a three dimensional view of the incoherent part of the spectrum associated with
the $e \to a$ transition, the one of interest to us, as a function of the probe
intensity (upper panel) and the detuning (lower panel); details of the steps
involved in the calculations herein via the matrix analysis are included in
Appendix \ref{sec:appendix}. The general spectral profile can be understood in
terms of dressed-state configuration that follows from properly diagonalizing
the atom-field Hamiltonian \cite{Cohen92}, emphasizing the fact that, above
saturation, the spectrum displays the appearance of Rabi sidebands as the
intensity field increases, as one can see in the upper panel of the figure.
By setting the Rabi frequency at, say, $\Omega_{a}/\gamma_{a}=1.12$, such sidebands become sufficiently conspicuous and the dependency of their profile
upon the detuning is shown in the lower panel.
\section{Amplitude-Intensity Correlation}
In this section we present the theory of conditional homodyne detection (CHD)
in order to assess and discuss the time-asymmetry, the non-Gaussianity and the
non-classicality of the light scattered from the atomic system under study. In the
next section we move to the frequency domain. The CHD setup is sketched in Fig.~\ref{fig:chd}.
\begin{figure}
\caption{Scheme of conditional homodyne detection (CHD). It features a balance
homodyne detection setup to assess the quadrature of the field on the condition
of photon detection via detector $\textrm{D}
\label{fig:chd}
\end{figure}
In one arm of the setup a quadrature of the source light, $E_{\phi} \propto
\hat{\sigma}_{\phi} =(\hat{\sigma}_{ea} e^{-i\phi} +\hat{\sigma}_{ae} e^{i\phi})/2$,
is analyzed in balanced homodyne detection (BHD), where $\phi$ is the phase
of the local oscillator (LO). This signal has a delay $\tau$ with respect to the
measurement of the source's intensity in another arm, proportional to the
excited-state population,
$I \propto \langle \hat{\sigma}_{ea} \hat{\sigma}_{ae}\rangle
= \langle \hat{\sigma}_{ee} \rangle $. Thus, the outcome is an
amplitude-intensity correlation that reads
\begin{eqnarray} \label{eq:haicdef}
h_{\phi}(\tau) = \frac{\langle: \hat{\sigma}_{ea}(0) \hat{\sigma}_{ae}(0)
\hat{\sigma}_{\phi}(\tau): \rangle_{ss} }{ \alpha_{ee} \alpha_{\phi} } \,,
\end{eqnarray}
where the dots $::$ indicate normal and time operator ordering, and
$\alpha_{\phi} = (\alpha_{ea} e^{-i\phi} +\alpha_{ae} e^{i\phi})/2$ is the
stationary value of the quadrature amplitude.
\subsection{Time-asymmetry and Non-Gaussianity}
Resonance fluorescence is a highly non-linear process, preventing its
description in terms of quasi-probability distributions, i.e., it does not admit
a Fokker-Planck type of equation. The non-linearity leads to non-Gaussian
fluctuations, thus giving rise to non-vanishing odd-order moments. Autocorrelation functions such as that for the spectrum,
$\langle \sigma_{ea}(0) \sigma_{ae} (\tau) \rangle$; squeezing,
$\langle \Delta \sigma_{\phi}(0) \Delta \sigma_{\phi} (\tau) \rangle$;
and photon-photon correlation, $\langle \sigma_{ea}(0) \sigma_{ea} (\tau)
\sigma_{ae} (\tau) \sigma_{ae} (0) \rangle$, are of even-order and, as such,
time-symmetric \cite{DeCC02}. These functions do not address the
non-Gaussianity of the field's fluctuations. \\
In amplitude-intensity correlations, Eq.(\ref{eq:haicdef}), on the other hand,
such a symmetry is not guaranteed: being different observables, the outcome
will be dependent on the time order of measurements. For instance, the
quadrature is measured (preselected) for $\tau \geq 0$ and the intensity for
$\tau \leq 0$ (quadrature is post-selected), a process in which time-asymmetry
is expected to be revealed. Moreover, this correlation would allow us to explore non-classical features of light beyond squeezing and antibunching. \\
Applying the time and normal operator orderings in Eq.(\ref{eq:haicdef}) we
arrive at the following expressions for positive and negative time intervals,
\begin{eqnarray}
h_{\phi}(\tau \geq 0)
&=& \frac{\langle \hat{\sigma}_{ea}(0) \hat{\sigma}_{\phi}(\tau)
\hat{\sigma}_{ae}(0) \rangle_{ss} }{ \alpha_{ee} \alpha_{\phi} } \,, \\
h_{\phi}(\tau \leq 0)
&=& \frac{\mathrm{Re} [e^{-i\phi} \langle \hat{\sigma}_{ea}(0)
\hat{\sigma}_{ee}(\tau_-) \rangle_{ss} ]}{ \alpha_{ee} \alpha_{\phi} } \,.
\end{eqnarray}
\begin{figure}
\caption{Amplitude-intensity correlations of light from the $e \to a $ transition,
as a function of the scaled time $\gamma_{a}
\label{fig:htau1}
\end{figure}
The asymmetry in time revealed by CHD, as shown in Fig.~\ref{fig:htau1},
is an indicative of non-Gaussian noise. The correlation (\ref{eq:haicdef})
contains a product of three dipole operators or, more generally, three field
amplitude operators. This means that $h_{\phi}(\tau)$ provides access up
to third order fluctuations; since these are non-Gaussian, this third-order
correlation does not vanish. To better distinguish the asymmetry and the
size of these fluctuations, we proceed, as we did with the spectrum, to split
the dipole dynamics into its mean plus fluctuations,
$\hat{\sigma}_{jk} = \alpha_{\phi} +\Delta \hat{\sigma}_{jk}$ \cite{hmcb10},
\begin{eqnarray}
h_{\phi}(\tau) &=& 1+ h_{\phi}^{(2)}(\tau) +h_{\phi}^{(3)}(\tau),
\label{eq:h_split}
\end{eqnarray}
where
\begin{equation}
h_{\phi}^{(2)}(\tau) = \frac{\langle: [\alpha_{ea} \Delta \hat{\sigma}_{ae}(0)
+\alpha_{ae} \Delta \hat{\sigma}_{ea}(0)] \Delta \hat{\sigma}_{\phi}(\tau) :\rangle_{ss}}
{ \alpha_{ee} \alpha_{\phi} },
\end{equation}
and
\begin{equation}
h_{\phi}^{(3)}(\tau) = \frac{\langle: \Delta \hat{\sigma}_{ea}(0)
\Delta \hat{\sigma}_{ae}(0) \Delta \hat{\sigma}_{\phi}(\tau) :\rangle_{ss}}
{ \alpha_{ee} \alpha_{\phi} },
\end{equation}
are the components of, respectively, second- and third-order in the dipole
fluctuations of $h_{\phi}(\tau)$, where $\Delta \hat{\sigma}_{\phi}
=(\Delta \hat{\sigma}_{ea} e^{-i\phi} +\Delta \hat{\sigma}_{ae} e^{i\phi})/2$ is the
quadrature fluctuation operator. Fluctuations are said to be Gaussian if
$h_{\phi}^{(3)}(\tau) \to 0$, which can occur when the transition is weakly driven.
For positive time intervals between photon and quadrature detections, we get
\begin{eqnarray}
h_{\phi}^{(2)}(\tau \geq 0) &=& \frac{ 2\mathrm{Re} [\alpha_{ae}
\langle \Delta \hat{\sigma}_{ea}(0) \Delta \hat{\sigma}_{\phi}(\tau) \rangle_{st}]}
{ \alpha_{ee} \alpha_{\phi} }, \label{eq:htaup2} \\
h_{\phi}^{(3)}(\tau \geq 0) &=& \frac{\langle \Delta \hat{\sigma}_{ea}(0)
\Delta \hat{\sigma}_{\phi}(\tau) \Delta \hat{\sigma}_{ae}(0) \rangle_{st} }
{ \alpha_{ee} \alpha_{\phi} }. \label{eq:htaup3}
\end{eqnarray}
\begin{figure}
\caption{Splitting of the intensity-field correlations shown in Fig.~\ref{fig:htau1}
\label{fig:htau2}
\end{figure}
We show in Fig.~\ref{fig:htau2} the foregoing second- and third-order
correlations. For both quadratures, the third-order constituent (blue lines)
represents the main contribution, almost that of the total $h_{\phi} (\tau)$
(black lines). This is understandable from the fact that we are above the
saturation threshold \cite{GCRH17}, a regime where non-Gaussian
fluctuations become significant. In this regime the dipole $\alpha_{ea}$,
indicative of the coherence induced by the laser, is small; most of the total
emission is incoherent. This observation can be quantitatively revealed from
the fact that $h_{\phi}(0) =0$ (just as it occurs for photon correlations in
resonance fluorescence), which leads to the relation \cite{GCRH17}
\begin{equation}
h_{\phi}^{(3)}(0) = -\left[ 1+h_{\phi}^{(2)}(0) \right]
= \frac{2(|\alpha_{ea}|^2 -\alpha_{ee})}{\alpha_{ee}}.
\end{equation}
For strong fields, $|\alpha_{ea}| \ll \alpha_{ee}$, $h_{\phi}^{(3)}(0)$
reaches its extremal value -2, thereby making the dipole factor in
Eq.(\ref{eq:htaup2}) small compared to the third-order term.
Even though the splitting itself cannot be directly realizable from the
experimental viewpoint via the measurement scheme, it provides us with
valuable theoretical information to be able to discern the actual contribution to
the system's fluctuations.
For $\tau < 0$, we want to stress the fact that the outcome of CHD correlation
should be taken with special care: it is to be interpreted as the measurement of
the intensity \textit{after} the detection of the amplitude. Thus, as previously
underlined, the asymmetry results from the different fluctuations of the light's
amplitude and intensity. Time and normal operator ordering leads to
\begin{eqnarray}
h_{\phi} (\tau \leq 0) = 1+ \frac{ \mathrm{Re} [ e^{-i\phi}
\langle \Delta \hat{\sigma}_{ea}(0) \Delta \hat{\sigma}_{ee}(|\tau|) \rangle_{st} ]}
{\alpha_{ee} \alpha_{\phi} } \,,
\end{eqnarray}
i. e., the correlation is only of second order in the dipole fluctuations, albeit
with $\Delta \hat{\sigma}_{ee}$ instead of the quadrature amplitude
$\Delta \hat{\sigma}_{\phi}$ fluctuation operator.
\subsection{Non-classicality}
The initial motivation for CHD was to detect squeezing from weak sources,
such as cavity QED \cite{CCFO00,FOCC00}. Resonance fluorescence is also
a producer of weakly squeezed light \cite{WaZo81,CoWZ84}. In order to
produce light in a squeezed state, a non-classical property of light, these
sources must be weakly driven, so that the third-order fluctuations discussed
above are small. As we will see later, the remaining second-order signal is
related to the spectrum of squeezing. CHD, hence, gives non-classical
criteria in the time domain as resulting of violation of the classical inequalities
\cite{CCFO00,FOCC00}
\begin{eqnarray}
0 \le h_{\phi}(\tau)-1 & \le & 1, \\
|h_{\phi}^{(2)}(\tau)| \le |h_{\phi}^{(2)}(0)| & \le & 1,
\end{eqnarray}
where the second relation is derived for Gaussian fluctuations. More recently, it was
found that light in a coherent state obeys \cite{GCRH17}
\begin{eqnarray}
-1 \le h_{\phi}(\tau) & \le & 1;
\end{eqnarray}
light outside these bounds violates Poissonian statistics. According to these
criteria, we see in Figs.~\ref{fig:htau1} and \ref{fig:htau2} that both the in-phase
($\phi=0$, continuous line) and out-of-phase ($\phi=\pi/2$, dashed line)
quadratures of the field display a non-classical character, violating one or more inequalities. The fact that $h_{\phi}(0) =0$ already shows a non-classical
feature, akin to antibunching in the intensity fluctuations. Also, moderately
strong fields easily drive $h_{\phi}(\tau)$ out of the classical bounds.
We see, then, that CHD clearly reveals non-classicality of quadratures in the
time domain. Let us now proceed to scrutinize the spectral profile of
amplitude-intensity correlations in the frequency domain.
\section{Quadrature Spectra}
\begin{figure}
\caption{Spectra, Eqs.~(\ref{eq:S-chd}
\label{fig:fullspect}
\end{figure}
Since in CHD the signal is time-asymmetric, carrying different information for
positive and negative intervals, the spectra of quadratures measured from
the amplitude-intensity correlation should be calculated separately
\cite{GCRH17}:
\begin{eqnarray}
S_{\phi}^{(\tau \ge 0)}(\omega) = 4 \gamma_{a} \alpha_{ee} \int_{0}^{\infty}
d\tau \cos{\omega \tau} \left[ h_{\phi}(\tau \ge 0) -1 \right]
\label{eq:S-chd} \,, \\
S_{\phi}^{(\tau \le 0)} = 4\gamma_{a} \alpha_{ee} \int_{-\infty}^0 d\tau
\cos(\omega \tau)[h_{\phi}(\tau \le 0) -1] \,, \label{eq:Sn-chd}
\end{eqnarray}
for positive and negative time intervals, respectively. The prefactor
$\gamma_{a} \alpha_{ee}$ is the photon emission rate in the probe transition.
In Fig.~\ref{fig:fullspect} we show the spectra calculated from
Eqs.~(\ref{eq:S-chd}) and (\ref{eq:Sn-chd}) for both quadratures and the same
parameter values of Fig.~\ref{fig:htau1}. From the CHD viewpoint, negative
values of the spectrum are signature of non-classical scattered light beyond
squeezing, which is confirmed for both quadratures, with the $\pi/2$ quadrature
exhibiting a more pronounced non-classical behavior than the other.
Fig.~\ref{fig:squeezing-2} also shows the overall spectral profile as a function of
the probe laser's Rabi frequency for the $\pi/2$ quadrature only, and for positive
(upper panel) and negative (lower panel) intervals. This more complete
landscape allows us to verify non-classicality of light revealed by clear-cut
negative valleys even for excitation above saturation.
\begin{figure}
\caption{Fourier cosine transform of $h_{\pi/2}
\label{fig:squeezing-2}
\end{figure}
Following the splitting of $h_{\phi}(\tau \ge 0)$, Eq.~(\ref{eq:h_split}), the
spectra of second- and third-order dipole fluctuations are
\begin{equation}
S_{\phi}^{(N)}(\omega) = 4\gamma_{a} \alpha_{ee} \int_{0}^{\infty} d\tau
\cos{\omega \tau} \ h_{\phi}^{(N)}(\tau), \label{eq:Sk}
\end{equation}
for $N=2,3$, so that $S_{\phi}^{(\tau \ge 0)}(\omega)=S_{\phi}^{(2)}(\omega)
+S_{\phi}^{(3)}(\omega)$. These are shown in Fig.~\ref{fig:splitting} for both
quadratures, corresponding to the CHD signals of Fig.~\ref{fig:htau2}. We
find that the second-order spectra are mostly positive, while the third-order
spectrum is negative for $\phi =0$, there are negative
bands for $\phi =\pi/2$. In Fig.~\ref{fig:squeezing_del} the dependence of $S_{\pi/2}^{(N)}$
on the detuning of the probe laser is shown. A quite similar spectral
landscape (not shown) was found in the second-order correlation for
$\tau \le 0$, Eq.~(\ref{eq:Sn-chd}). The lineshapes are very complicated, but
the dispersive features at the sides reveal the non-Gaussianity of the field \cite{CaRG16,GCRH17}.
\begin{figure}
\caption{Fourier cosine transform of $h_{\phi}
\label{fig:splitting}
\end{figure}
\begin{figure}
\caption{ Fourier transform of $h_{\pi/2}
\label{fig:squeezing_del}
\end{figure}
The above spectra clearly deviate from the more conventional measure of
non-classical phase-dependent fluctuations, squeezing, due to the non-linearity
induced by the strong lasers. Understood operationally as the reduction of
quantum fluctuations below the shot noise limit, squeezing can be obtained in
the spectral domain as the Fourier transform of symmetric photocurrent
fluctuations in homodyne detection \cite{Carmichael87}. For our source,
{\setlength\arraycolsep{2pt}
\begin{eqnarray} \label{eq:specsqueez}
S_{\phi}(\omega) &=&
8\gamma_{a} \eta \int_{0}^{\infty} d\tau \cos{\omega \tau} \,
\langle : \Delta \hat{\sigma}_{\phi}(0) \Delta \hat{\sigma}_{\phi}(\tau) :
\rangle_{ss}, \nonumber \\
&=& 8\gamma_{a} \eta \int_{0}^{\infty} d\tau \cos{\omega \tau} \nonumber \\
&& \times \mathrm{Re} \left[ e^{-i\phi}
\langle \Delta \hat{\sigma}_{ea}(0)
\Delta \hat{\sigma}_{\phi}(\tau) \rangle_{ss} \right],
\end{eqnarray}}
where $\eta$ is a combined collection and detection efficiency, and the dots
$::$ state that the operators must follow time and normal orderings. It was
shown in \cite{CCFO00,FOCC00} that, in the weak-field limit, when third-order
fluctuations can be neglected, the second-order spectrum from CHD, from
Eqs.~(\ref{eq:Sk}) and (\ref{eq:htaup2}), is indeed the spectrum of squeezing,
but unafected by detector losses, i.e.,
$S_{\phi}^{(2)}(\omega) = S_{\phi}(\omega)/\eta$, owing to the conditional
character of CHD.
Figure~\ref{fig:specbhd} displays a 3D plot of the spectrum of squeezing, given
by Eq.~(\ref{eq:specsqueez}) with $\eta =1$, as a function of the probe laser
intensity $\Omega_{a}/\gamma_{a}$ (upper panel) and detuning
$\Delta_{a}/\gamma_{a}$ (lower panel), for the $\phi=\pi/2$ quadrature. The
figure shows up indicatives of squeezing (negative values on the spectral
content) for a moderate detuning, at $\Delta_{a}/\gamma_{a}=2.38$, around
which CPT takes place, even for laser intensities above saturation. A slightly
higher degree of squeezing comes about within certain regions of the
spectrum by fixing the laser intensity, at $\Omega_{a}/\gamma_{a}=0.1$, say,
and varying the detuning (see lower panel).
\begin{figure}
\caption{Spectrum of squeezing of the $\phi=\pi/2$ quadrature as a function
of the probe field intensity (upper panel, for $\Delta_{a}
\label{fig:specbhd}
\end{figure}
An alternative picture of squeezing is the variance
\begin{eqnarray} \label{eq:variance}
V_{\phi} &=& \langle : (\Delta \sigma_{\phi})^2 : \rangle_{st}
=\mathrm{Re} \left[ e^{-i\phi}
\langle \Delta \sigma_{eg} \Delta \sigma_{\phi} \rangle_{ss} \right],
\end{eqnarray}
related to the integrated spectrum as $\int_{-\infty}^{\infty}
S_{\phi}(\omega) d \omega =4\pi \gamma_a \eta V_{\phi}$. This quantity is
depicted in Fig.~\ref{fig:variance} for $\phi=\pi/2$ as a function of the scaled
detuning and laser intensity, revealing a very small degree of squeezing in the quadrature reflected within a restricted region of negative variance. On the
other hand, the region within which CPT takes hold, around
$\Delta_{a}/\gamma_{a}\approx 2.38$, is found to reduce fluctuations,
approximately, to the extent of a coherent state. It was also verified that the
in-phase quadrature (not shown) did not feature squeezed fluctuations in any
parameter regime of the aforesaid transition.
\begin{figure}
\caption{Variance for $\phi=\pi/2$, as a function of the scaled Rabi
frequency $\Omega_{a}
\label{fig:variance}
\end{figure}
\section{Conclusions}
Using the framework of conditional homodyne detection, we have analyzed the nearby
effect of coherent population trapping on the phase-dependent quantum
fluctuations, in both time and frequency domains, of the light fluoresced in the
probe transition from a coherently driven $\Lambda$-type three-level atom.
Given the feasibility of implementing the outlined optical system, a single
$^{138} \mathrm{Ba}^+$ ion \cite{GRS+09,SHG+10}, our findings are expected
to bolster further experimental investigations to be benchmarked against
CHD-based theoretical predictions.
It is worth underlying that the CHD framework proves to be a versatile tool to
discern the contribution of phase-dependent fluctuations of different orders,
concluding that the light scattered under the aforesaid conditions is essentially
non-Gaussian; i.e., the correlation of third-order in the dipole fluctuation operators
prevails. Non-Gaussianity, notably, manifests in two main ways. On the one hand,
the amplitude-intensity correlation is, in general, time-asymmetric, indicating that
amplitude and intensity of the radiated field have different noise properties.
On the other, the non-linearity imposed by a saturating excitation regime leads to
fluctuations away from the ideal weak-field squeezing regime.
The role of CPT in CHD is explored with particular focus on the spectra of
quadratures. In this regard, as a function of the probe detuning, the spectral
content confirms once again the prevailing contribution of third-order fluctuations
to the outcome of the measurements, for both quadratures. This fact is also
reinforced by examining the variance (the integrated spectrum) of fluorescence.
\section{Acknowledgments}
The authors thank Dr. Ir\'an Ramos-Prieto for useful coversations and help
with the figures.
\appendix
\section{Correlations and Spectra} \label{sec:appendix}
Here, we succinctly describe the evaluation of the expectation values of two-time
correlations and spectra used throughout this work.
From the equations of motion of the atomic operators, Eqs.~(\ref{eq:pop1}) to (\ref{eq:coh4}), which can be put into the concise form
$\dot{\mathbf{s}}(t) = \mathbf{M} \mathbf{s} (t)$, with
$\mathbf{s} = \left\{ \sigma_{ee}, \sigma_{ae}, \sigma_{be}, \sigma_{ea},
\sigma_{aa}, \sigma_{ba}, \sigma_{eb}, \sigma_{ab}, \sigma_{bb} \right\}^{T}$
and $\mathbf{M}$ the parameter matrix (to be specified), together with the use
of the quantum regression formula \cite{Carm99}, we seek the general solution
to the equation
\begin{equation}
\partial_{\tau} \mathbf{g}(\tau) = \mathbf{M} \mathbf{g}(\tau),
\end{equation}
where $\mathbf{g}(\tau) = \langle \Delta \sigma_{ea}(0) \Delta \mathbf{s}(\tau)
\Delta A_-(0) \rangle_{ss}$ is the corresponding vector of correlation functions.
For the second-order correlations, $\Delta A_- =\mathbf{1}$; for the
third-order ones, $\Delta A_- =\Delta \sigma_{ae}$. Its solution can be written in
the form $\mathbf{g}(\tau) = e^{ \mathbf{M} \tau } \mathbf{g}(0)$, where the
initial condition $\mathbf{g}(0)=\mathbf{g}_{ss}$, given in terms of the steady
state solution of populations and coherences, is solved numerically.
The incoherent spectrum requires, for instance, handling the time dependence
of the correlation $\{ \mathbf{g} (\tau) \}_{m} = \langle \Delta \sigma_{ea}(0)
\Delta \sigma_{ae}(\tau) \rangle_{ss}$, where the subindex $m$-th denotes the
element of the vector to be taken. The present matrix analysis saves the work
of solving the correlation explicitly followed by time integration, namely, for
$\Delta A_{-}=1$,
\begin{eqnarray}
S_{inc}(\omega)
&=& \frac{1}{ \pi \alpha_{ee} } \mathrm{Re} \left \{ \int_0^{\infty}
d\tau e^{-(i \omega \mathbf{1} -\mathbf{M}) \tau}
\langle \Delta \sigma_{ea} \Delta \mathbf{s} \rangle_{ss} \right \}_m
\nonumber \\
&=& \frac{1}{ \pi \alpha_{ee} } \mathrm{Re} \left \{
-(i \omega \mathbf{1} -\mathbf{M})^{-1}
e^{-(i \omega \mathbf{1} -\mathbf{M}) \tau} \left. \right|_0^{\infty}
\right. \nonumber \\
&& \left. \times
\langle \Delta \sigma_{ea} \Delta \mathbf{s} \rangle_{ss} \right \}_m
\nonumber \\
&=& \frac{1}{ \pi \alpha_{ee} } \mathrm{Re} \left \{
(i \omega \mathbf{1} -\mathbf{M})^{-1}
\langle \Delta \sigma_{ea} \Delta \mathbf{s} \rangle_{ss} \right \}_m ,
\nonumber
\end{eqnarray}
where $\mathbf{1}$ is the $n \times n$ identity matrix. The spectra
corresponding to the CHD correlations are calculated in the same manner,
thus giving us the sought results
\begin{widetext}
{\setlength\arraycolsep{2pt}
\begin{eqnarray}
S_{\phi}^{(2)}(\omega) &=& \frac{2\gamma_{a}}{\alpha_{\phi}}
\textrm{Re} \left \{ \alpha_{ae}e^{-i\phi}\left[ (i\omega \mathbf{1}-\mathbf{M})^{-1}
-(i\omega \mathbf{1}+\mathbf{M})^{-1})\mathbf{g}(0) \right]_{m} \right \}
\nonumber \\
&& + \frac{2\gamma_{a}}{\alpha_{\phi}} \textrm{Re} \left \{ \alpha_{ae} e^{i\phi}
\left[ (i\omega \mathbf{1}-\mathbf{M})^{-1}
-(i\omega \mathbf{1}+\mathbf{M})^{-1}) \mathbf{g}(0) \right]_{n} \right \},
\nonumber \\
S_{\phi}^{(3)}(\omega) &=& \frac{2\gamma_{a}}{\alpha_{\phi}}
\textrm{Re} \{ e^{-i\phi} \left[ (i\omega \mathbf{1}-\mathbf{M})^{-1}
-(i\omega \mathbf{1}+\mathbf{M})^{-1}) \mathbf{g}(0) \right]_{p} \}, \nonumber
\end{eqnarray}}
for the second- and third-order fluctuations, respectively, and
\begin{equation*}
S_{j,\phi}^{(\tau \le 0)}(\omega) = \frac{2\gamma_{j}}{\alpha_{\phi}}
\textrm{Re} \{ e^{-i \phi} [((i\omega \mathbf{1}-\mathbf{M})^{-1}
-(i\omega \mathbf{1}+\mathbf{M})^{-1})\mathbf{g}(0)]_{q} \}
\end{equation*}
for fluctuations associated with negative time intervals. The elements of the
vectors, denoted by subindexes $m,n,p$ and $q$, have to be chosen
appropriately to match the corresponding correlation it seeks to assess.
For the sake of completeness, the initial conditions of the correlations ($\tau=0$)
are encapsulated by using the fluctuation operator approach as
\begin{eqnarray*}
\langle \Delta \sigma_{ij} \Delta \sigma_{kl} \rangle_{ss}
&=& \alpha_{il} \delta_{jk} -\alpha_{ij} \alpha_{kl}, \\
\langle \Delta \sigma_{ig} \Delta \sigma_{jk} \Delta \sigma_{gi} \rangle_{ss}
&=& 2 |\alpha_{ig}|^2 \alpha_{jk}
+\alpha_{ii} (\delta_{gj} \delta_{kg} -\alpha_{jk})
-\alpha_{ik} \alpha_{gi} \delta_{gj}
-\alpha_{ig} \alpha_{ji} \delta_{kg} \,,
\end{eqnarray*}
for the second- and third-order fluctuations, respectively. More explicitly, for the
$e \to a$ transition, they become
\begin{eqnarray*}
\langle \Delta \sigma_{ea} \Delta \mathbf{s} \rangle_{ss}
&=& \left\{-\alpha_{ea} \alpha_{ee}, \alpha_{ee} -\alpha_{ea} \alpha_{ae},
-\alpha_{ea} \alpha_{be},
-\alpha_{ea}^2, \alpha_{ea}(1- \alpha_{aa}), -\alpha_{ea} \alpha_{ba}
-\alpha_{ea} \alpha_{eb}, \alpha_{eb}-\alpha_{ea}\alpha_{ab},
-\alpha_{ea} \alpha_{bb} \right\}^{T},
\end{eqnarray*}
and
\begin{eqnarray}
\left( \begin{array}{c}
\langle \Delta \sigma_{ea} \Delta \sigma_{ee} \Delta \sigma_{ae} \rangle_{ss} \\
\langle \Delta \sigma_{ea} \Delta \sigma_{ae} \Delta \sigma_{ae} \rangle_{ss} \\
\langle \Delta \sigma_{ea} \Delta \sigma_{be} \Delta \sigma_{ae} \rangle_{ss} \\
\langle \Delta \sigma_{ea} \Delta \sigma_{ea} \Delta \sigma_{ae} \rangle_{ss} \\
\langle \Delta \sigma_{ea} \Delta \sigma_{aa} \Delta \sigma_{ae} \rangle_{ss} \\
\langle \Delta \sigma_{ea} \Delta \sigma_{ba} \Delta \sigma_{ae} \rangle_{ss} \\
\langle \Delta \sigma_{ea} \Delta \sigma_{eb} \Delta \sigma_{ae} \rangle_{ss} \\
\langle \Delta \sigma_{ea} \Delta \sigma_{ab} \Delta \sigma_{ae} \rangle_{ss} \\
\langle \Delta \sigma_{ea} \Delta \sigma_{bb} \Delta \sigma_{ae} \rangle_{ss}
\end{array} \right)
&=& \left( \begin{array}{c}
\alpha_{ee} (2 |\alpha_{ea}|^2 -\alpha_{ee} ) \\
-2 \alpha_{ae} (\alpha_{ee}-|\alpha_{ea}|^{2}) \\
\alpha_{be} (2 |\alpha_{ea}|^2 -\alpha_{ee} ) \\
2\alpha_{ea} ( |\alpha_{ea}|^2 -\alpha_{ee} ) \\
( 2|\alpha_{ea}|^2 -\alpha_{ee} ) (\alpha_{aa} -1) \\
\alpha_{ba} ( 2|\alpha_{ea}|^2 -\alpha_{ee} ) -\alpha_{ea} \alpha_{be} \\
\alpha_{eb} ( 2|\alpha_{ea}|^2 -\alpha_{ee} ) \\
\alpha_{ab} ( 2|\alpha_{ea}|^2 -\alpha_{ee} ) -\alpha_{eb} \alpha_{ae} \\
\alpha_{bb} (2 |\alpha_{ea}|^2 -\alpha_{ee} )
\end{array} \right). \nonumber
\end{eqnarray}
\end{widetext}
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\end{document} |
\begin{document}
\markboth{Schlesinger M.I., Vodolazskiy E.V., Yakovenko V.M.}{Similarity of closed polygonal curves in Frechet metric}
\catchline
\author{ Schlesinger M.I.}
\address{
[email protected]
}
\author{Vodolazskiy E.V.}
\address{
[email protected]
}
\author{Yakovenko V.M.}
\address{
[email protected]
}
\address{
International Research and Training Centre \\
of Information Technologies and Systems \\
National Academy of Science of Ukraine \\
Cybernetica Centre, \\
prospect Academica Glushkova, 40, \\
03680, Kiev-680, GSP, Ukraine.
}
\title{Similarity of closed polygonal curves in Frechet metric}
\pub{Received (received date)}{Revised (revised date)}
{Communicated by (Name)}
\begin{abstract}
The article analyzes similarity of closed polygonal curves in Frechet metric,
which is stronger than the well-known Hausdorff metric and therefore is more appropriate in some applications.
An algorithm that determines whether the Frechet distance between two closed polygonal curves with $m$ and $n$ vertices is less than a given number~$\varepsilon$ is described.
The described algorithm takes $O(m n)$ time whereas the previously known algorithms take $O(m n\log (m n))$ time.
\keywords{computational geometry, Frechet distance, computational complexity.}
\end{abstract}
\section{Introduction}
Frechet metric is a tool for
cyclic process analysis and image processing \cite{ICDAR.2007.121} as well as the well-known Hausdorff metric \cite{Chen02}\cite{rockafellar2011variational}
\cite{ismm2011}.
The Frechet metric is stronger than the Hausdorff metric and therefore is more appropriate in some applications \cite{frechet}.
The Frechet metric for closed polygonal curves has been studied in a paper \cite{frechet} by Alt and Godau.
They propose an algorithm that determines whether the distance between two closed polygonal curves with $m$ and $n$ vertices is greater than a given number~$\varepsilon$.
The complexity of the algorithm is $O(m n \log (m n))$ on a random access machine that performs arithmetical operations
and computes square roots in constant time.
Our paper develops the ideas of the original paper \cite{frechet} and solves the same problem in $O(m n)$ time.
Sections \ref{Definitions} and \ref{Diagrams} describe the concepts that are common for both papers.
The end of Section \ref{Diagrams} shows the difference between the known and the proposed approaches.
Sections \ref{Achievability}-\ref{ResultSection} explicate the proposed approach.
\section{\label{Definitions} Problem definition.}
Let $\mathbb{R}^k$ be a $k$-dimensional linear space with a metric $\text{ }d:\mathbb{R}^k\times \mathbb{R}^k \rightarrow \mathbb{R}$,
where $d(x,y)=\sqrt{(x-y)^2}$ is the distance between points $x, y \in \mathbb{R}^k$.
\begin{definition}
A closed polygonal curve $X$ with $m$ vertices is a sequence $\bar{x}=(x_0, x_1, \cdots, x_m=x_0)$, $x_i \in \mathbb{R}^k$, and a function
$f_X: \{ t \in \mathbb{R}|0 \le t \le m \} \rightarrow \mathbb{R}^k$ such that
$f_X(i+\alpha)=(1-\alpha) x_i + \alpha x_{i+1}$
for $i \in \{0, 1, \dots , m-1\}$ and $0 \le \alpha \le 1$.
\end{definition}
\begin{definition}
A cyclic shift of an interval $\{ t|0 \le t \le m \}$ by a value $\tau$, $0 \le \tau \le m $, is a function
$s:\{ t|0 \le t \le m \} \rightarrow \{ t|0 \le t \le m \}$ that depends on a parameter $\tau$ such that
$s(t;\tau) =t+\tau$ for $t+\tau \le m$ and $s(t;\tau) =t+\tau - m$ for $t+\tau > m$.
\end{definition}
For any number $m$ denote $W_m$ a set of all monotonically non-decreasing continuous functions
$w:\{ t|0 \le t \le 1 \}\rightarrow\{ t|0 \le t \le m \}$ such that $w(0)=0,w(1)=m$.
\begin{definition}
A function $\varphi:\{t|0 \le t \le 1\}\rightarrow \mathbb{R}^k$ is called a monotonic traverse of a closed $m$-gonal curve $X$
if there is a function $w \in W_m$ and a number $\tau$, $0 \le \tau \le m$, such that $\varphi(t)=f_X(s(w(t);\tau))$ for all $t$, $0 \le t \le 1$.
\end{definition}
For given closed polygonal curves $X$ and $Y$ denote $\Phi_X$ and $\Phi_Y$ sets of their monotonic traverses.
\begin{definition}
Frechet distance between closed polygonal curves $X$ and $Y$ is a number
\begin{equation} \nonumber
\delta(X,Y) = \min_{\varphi_X \in \Phi_X} \min_ {\varphi_Y \in \Phi_Y } \max_{0\le t \le 1}d(\varphi_X(t) , \varphi_Y(t)).
\end{equation}
\end{definition}
The article solves the problem of developing an algorithm that determines whether $\delta(X,Y) \le \varepsilon$
for given closed polygonal curves $X$ and $Y$ and a number~$\varepsilon$.
\section{\label{Diagrams}The free space diagram.}
The problem's analysis is largely based on its representation in a form of a free space diagram introduced in a paper \cite{frechet}.
For two numbers $m$ and $n$ let us define a rectangle $\widetilde{D} = \{(u,v) \in \mathbb{R}^2|0 \le u \le m, 0 \le v \le n \}$,
$u$ and $v$ being horizontal and vertical coordinates of a point $(u,v)$.
For two closed polygonal curves $X$ and $Y$ with $m$ and $n$ vertices and a number $\varepsilon$ a subset
$\widetilde{D}_\varepsilon= \{(u,v) \in \widetilde{D}| d(f_X(u),f_Y(v)) \le \varepsilon\}$ is defined.
Let us also define a rectangle
$D = \widetilde{D} \cup \{(u+m,v)|(u,v) \in \widetilde{D}\}$ with its subset
$D_\varepsilon = \widetilde{D}_\varepsilon \cup \{(u+m,v)|(u,v) \in \widetilde{D}_\varepsilon\}$ called a free space.
Denote $T$, $B$, $L$ and $R$
the top, bottom, left and right sides of the rectangle $D$.
Two closed polygonal curves $X$ and $Y$ are shown on Figure \ref{Fig0} and the corresponding rectangles
$\widetilde{D}$ and $D$ are shown on Figure \ref{Fig1}, the light region of $D$ being the free space $D_\varepsilon$.
\begin{figure}
\caption{Two closed polygonal curves}
\label{Fig0}
\end{figure}
\begin{figure}
\caption{The free space diagram}
\label{Fig1}
\end{figure}
\begin{definition}
A monotonic path is a connected subset $\gamma \subset D_\varepsilon$ such that $(u~-~u')(v~-~v') \ge 0$ for any two points
$(u,v) \in \gamma$, $(u',v') \in \gamma$.
\end{definition}
\begin{definition}
Two points $(u,v) \in D$ and $(u',v') \in D$ are mutually reachable if and only if a monotonic path $\gamma$ exists such that $(u,v) \in \gamma$, $(u',v') \in \gamma$.
\end{definition}
Mutual reachability defines a subset of $D_\varepsilon \times D_\varepsilon$ and is a symmetric reflexive binary relation, not necessarily transitive.
The expression "a point $(u,v) \in D$ is reachable from $(u',v') \in D$" is further used in a sense that $(u,v)$ and $(u',v')$ are mutually reachable.
\begin{definition}
A point $(u,v) \in D_\varepsilon$ is reachable from the bottom if it is reachable from at least one point from $B$;
a point $(u,v) \in D_\varepsilon$ is reachable from the top if it is reachable from at least one point of $T$.
\end{definition}
Reachability from top and bottom are unary relations. Each of these two relations define a subset of $D_\varepsilon$.
Denote $g_\downarrow \subset D_\varepsilon$ a set of points reachable from the bottom and
$g^\uparrow \subset D_\varepsilon$ a set of points reachable from the top.
\begin{figure}
\caption{The set of points reachable from the bottom}
\label{Fig2}
\end{figure}
\begin{figure}
\caption{The set of points reachable from the top}
\label{Fig3}
\end{figure}
Let us define four functions\\
\phantom{}\quad $l_\downarrow:g_\downarrow \rightarrow \{u|0 \le u \le 2m \} $, \quad $r_\downarrow:g_\downarrow \rightarrow \{u|0 \le u \le 2m \} $, \\
\phantom{}\quad $l^\uparrow:g^\uparrow \rightarrow \{u|0 \le u \le 2m \}, \quad $ $r^\uparrow:g^\uparrow \rightarrow \{u|0 \le u \le 2m \} $
\\
such that $l_\downarrow(u,v)$ (or $r_\downarrow(u,v)$) is the minimum (or maximum) value $u^*$ such that $(u,v)$ is reachable from $(u^*,0) \in B$.
Similarly, $l^\uparrow(u,v)$ (or $r^\uparrow(u,v)$) is the minimum (or maximum) value $u^*$, such that $(u,v)$ is reachable from $(u^*,n) \in T$.
The subset $g_\downarrow \subset D_\varepsilon$ is shown by a light area on Figure \ref{Fig2}.
The same Figure \ref{Fig2} shows the values $l_\downarrow(u,v)$ and $r_\downarrow(u,v)$) for some point $(u,v) \in g_\downarrow$.
Figure \ref{Fig3} shows the subset $g^\uparrow \subset D_\varepsilon$ and values $l^\uparrow(u,v)$ and $r^\uparrow(u,v)$.
The following two lemmas are proved in the paper \cite{frechet} by Alt and Godau (lemma 9 and 10 in \cite{frechet}).
\begin{lemma} \label{AltFirst}
The distance between closed polygonal curves $X$ and $Y$ is not greater than $\varepsilon$ if and only if there exists a number
$u$, $0 \le u \le m$, such that
the points $(u + m, n)$ and $(u,0)$ are mutually reachable.
\end{lemma}
\begin{lemma} \label{AltSecond}
Two points $(u^{top},n) \in T$ and $(u_{bot},0) \in B$ are mutually reachable if and only if
$(u^{top},n) \in g_\downarrow $, $(u_{bot},0) \in g^\uparrow$ and $l^\uparrow(u_{bot},0) \le u^{top} \le r^\uparrow(u_{bot},0)$.
\end{lemma}
According to these two lemmas testing the inequality $\delta(X,Y) \le \varepsilon$ is reduced in the paper \cite{frechet} to determining if there exists a value $u$ that fulfills
\begin{equation} \label{AltDecision}
(u,0) \in g^\uparrow, \quad (u+m,n) \in g_\downarrow, \quad l^\uparrow(u,0) \le u+m \le r^\uparrow(u,0).
\end{equation}
Our algorithm is also based on Lemma \ref{AltFirst}.
However, instead of Lemma \ref{AltSecond} we rely on a similar Lemma \ref{OurSecond}.
We prove Lemma \ref{OurSecond} in the same manner as Lemma \ref{AltSecond} has been proved in \cite{frechet}.
\begin{lemma} \label{OurSecond}
Two points $(u^{top},n) \in T$ and $(u_{bot},0) \in B$ are mutually reachable if and only if\\
\phantom{} \quad\quad $(u^{top},n) \in g_\downarrow $, \quad $(u_{bot},0) \in g^\uparrow$, \quad $u^{top} \le r^\uparrow(u_{bot},0)$,
\quad $u_{bot} \le r_\downarrow(u^{top},n)$.
\end{lemma}
\begin{proof}
Obviously, if $(u^{top},n) \in T$ and $(u_{bot},0) \in B$ are mutually reachable then
$(u^{top},n) \in g_\downarrow $, $(u_{bot},0) \in g^\uparrow$ and $u^{top} \le r^\uparrow(u_{bot},0)$, $u_{bot} \le r_\downarrow(u^{top},n)$.
The reverse implication is also valid, which is illustrated by Figure \ref{Fig5}.\\
\begin{figure}
\caption{There is a path between $(u_{bot}
\label{Fig5}
\end{figure}
It follows from condition $(u^{top},n) \in g_\downarrow $ that a monotonic path $\gamma^{top} \subset D_\varepsilon$
connecting $(u^{top},n)$ and $(r_\downarrow(u^{top},n),0)$ exists.
Similarly, it follows from condition $(u_{bot},0) \in g^\uparrow$
that a monotonic path $\gamma_{bot} \subset D_\varepsilon$ connecting $(u_{bot},0)$ and $(r^\uparrow(u_{bot},0),n)$ exists.
Both paths are connected subsets and therefore they intersect in at least one point $(u_0, v_0)$ according to conditions $u^{top} \le r^\uparrow(u_{bot},0)$, $u_{bot} \le r_\downarrow(u^{top},n)$.
Let us build a path $\gamma$ that consists of a part of the path $\gamma_{bot}$ from $(u_{bot},0)$ to $(u_0, v_0)$
and a part of the path $\gamma^{top}$ from $(u_0, v_0)$ to $(u^{top},n)$.
The path $\gamma$ is monotonic, it is contained inside $D_\varepsilon$ and it connects $(u_{bot},0)$ and $(u^{top},n)$.
\end{proof}
According to Lemmas \ref{AltFirst} and \ref{OurSecond} testing condition $\delta(X,Y) \le \varepsilon$ is reduced to finding such $u$ that fulfills
\begin{equation} \label{OurDecision}
(u,0) \in g^\uparrow, \quad (u+m,n) \in g_\downarrow, \quad u+m \le r^\uparrow(u,0),
\quad u \le r_\downarrow(u+m,n).
\end{equation}
The algorithm presented in the paper by Alt and Godau \cite{frechet} builds a data structure sufficient to test the consistency of conditions (\ref{AltDecision}) using a divide and conquer approach in $O(mn \log(mn))$ time.
Our algorithm is based on condition (\ref{OurDecision}) and therefore uses other data structures that represent functions $r^\uparrow$
and $r_\downarrow$. It is shown that these data structures can be built in two passes in $O(mn)$ time.
Therefore, the subsets $g^\uparrow$, $g_\downarrow$ and functions $r^\uparrow$, $r_\downarrow$ become the main focus of the further considerations.
Section \ref{Achievability} shows that in order to test condition (\ref{OurDecision})
it is sufficient to have a finite set of parameters of subsets $g^\uparrow$, $g_\downarrow$ and functions $r^\uparrow$, $r_\downarrow$,
which take $O(mn)$ amount of space.
Section \ref{SimilarityNew} shows that it takes $O(mn)$ time to test condition (\ref{OurDecision}) based on these data.
Finally, Section \ref{InputData} shows that computing these data takes $O(mn)$ time.
\section{\label{Achievability}Formal properties of reachability.}
\begin{lemma} \label{MyMonotony}
For any two points $(u_1, v_1) \in g_\downarrow$ and $(u_2, v_2) \in g_\downarrow$ such that $u_1 \le u_2$ and $v_1 \ge v_2$
the inequality $r_\downarrow(u_1, v_1) \le r_\downarrow(u_2, v_2)$ holds.\\
For any two points $(u_1, v_1) \in g^\uparrow$ and $(u_2, v_2) \in g^\uparrow$ such that $u_1 \le u_2$ and $v_1 \ge v_2$
the inequality $r^\uparrow(u_1, v_1) \le r^\uparrow(u_2, v_2)$ holds.
\end{lemma}
\begin{proof}
Let $\gamma_1$ and $\gamma_2$ be two monotonic paths that connect $(u_1, v_1)$ with $(r_\downarrow(u_1, v_1), 0)$ and $(u_2, v_2)$ with $(r_\downarrow(u_2, v_2),0)$ respectively.
Let us assume that $r_\downarrow(u_1, v_1) > r_\downarrow(u_2, v_2)$.
As it is shown on Figure~\ref{Fig6} the paths $\gamma_1$ and $\gamma_2$ intersect at some point $(u_0,v_0)$.
Therefore, a monotonic path that connects $(u_2,v_2)$ with $(r_\downarrow(u_1, v_1), 0)$ exists.
This path consists of a part of the path $\gamma_1$ from $(r_\downarrow(u_1, v_1), 0)$ to $(u_0,v_0)$ and a part of the path $\gamma_2$ from $(u_0,v_0)$ to $(u_2, v_2)$.
This means that the point $(u_2,v_2)$ is reachable from a point that is located to the right of the point $(r_\downarrow(u_2, v_2),0)$.
This contradicts with the definition of function $r_\downarrow$.
Therefore, the assumption $r_\downarrow(u_1, v_1) > r_\downarrow(u_2, v_2)$ is proved to be wrong. The first statement of the theorem is proved.
The proof of the second statement is similar.
\begin{figure}
\caption{Monotonicity of $r_{\downarrow}
\label{Fig6}
\end{figure}
\end{proof}
The property of functions $r_\downarrow$ and $r^\uparrow$ stated in Lemma \ref{MyMonotony} will be referred to as monotonicity of these functions.
For each pair $(i,j)$, $1 \le i \le 2m$, $1 \le j \le n$, denote $D(i,j)$ a square cell
$$D(i,j) = \{ (u,v)| i-1 \le u \le i, j-1 \le v \le j \}$$
and denote $T(i,j)$, $B(i,j)$, $L(i,j)$ and $R(i,j)$ the top, bottom, left and right sides of this square cell, respectively.
We extend these definitions so that $T(i,0) = B(i,1)$ and $R(0,j) = L(1,j)$.
Denote $D_\varepsilon(i,j) = D_\varepsilon \cap D(i,j)$. It is significant that $D_\varepsilon(i,j)$ is convex for each pair $(i,j)$ \cite{frechet}.
It is not difficult to prove that for each pair $(i,j)$
the intersections $g_\downarrow \cap T(i,j)$, $g_\downarrow \cap R(i,j)$, $g_\uparrow \cap T(i,j)$ and $g_\uparrow \cap R(i,j)$ are also convex,
that is they are intervals on the sides of the cell $D(i,j)$.
\begin{lemma} \label{Constancy}
For any pair $(i,j)$ the following two statements are valid:\\
if $g_\downarrow \cap R(i,j) \ne \emptyset $ then the function $r_\downarrow$ is constant on $g_\downarrow \cap R(i,j)$;\\
if $g^\uparrow \cap T(i,j) \ne \emptyset $ then the function $r^\uparrow$ is constant on $g^\uparrow \cap T(i,j)$. \phantom{}
\end{lemma}
\begin{proof}
Let us prove the first statement of the lemma.
Let $(i,v_1)$ and $(i,v_2)$, $v_1 < v_2$, be two points from $g_\downarrow \cap R(i,j)$.
Any point from $B$ that is reachable from $(i,v_1)$ is also reachable from $(i,v_2)$.
Therefore, $r_\downarrow(i,v_1) \le r_\downarrow(i,v_2) $.
According to Lemma \ref{MyMonotony} on monotonicity of the function $r_\downarrow$ the inequality $r_\downarrow(i,v_1) \ge r_\downarrow(i,v_2) $ is valid as well.
Therefore, $r_\downarrow(i,v_1) = r_\downarrow(i,v_2)$.
The proof of the second statement is similar.
\end{proof}
Restrictions of $r_\downarrow$ to $g_\downarrow \cap T(i,j)$ and $r^\uparrow$ to $g^\uparrow \cap R(i,j)$ are more complex.
However, they can be presented in a certain standard form.
Let us first take a look at the function $r_\downarrow$ and then extend the result to $r^\uparrow$.
Let $int \subset g_\downarrow \cap T(i,j)$ be a connected subset called an interval. Let $I$ be a set of pairwise disjoint intervals
such that $g_\downarrow \cap T(i,j) = \bigcup\limits_{int \in I}int$.
\begin{definition}
The restriction of $r_\downarrow$ to $g_\downarrow \cap T(i,j)$ can be expressed in a standard form with a set $I$ of intervals if for each interval $int \in I$ one of the following is valid:\\
-- either $r_\downarrow (u,i) = u$ for all $u \in int$;\\
-- or $r_\downarrow (u_1,i) = r_\downarrow (u_2,i)$ for all $u_1 \in int$, $u_2 \in int$.
\end{definition}
The set $I$ of intervals is ordered so that any nonempty subset has the leftmost and the rightmost intervals (possibly equal).
The rightmost interval from $I$ is closed. All the other intervals are left-closed (contain the left endpoint) and right-open.
\begin{lemma}\label{QuantityOfInt}
The restriction of $r_\downarrow$ to $g_\downarrow \cap T(i,j)$ can be expressed in a standard form on a set of no more than $(2j+1)$ intervals.
\end{lemma}
\begin{proof}
Any point $(u,0) \in D_\varepsilon$ is reachable from itself and belongs to $B$. Therefore, $r_\downarrow(u,0)=u$ for all $(u,0) \in g_\downarrow \cap T(i,0)$ and
the restriction of $r_\downarrow$ to $g_\downarrow \cap T(i,0)$ can be expressed in a standard form with a single interval.
Let us choose an arbitrary $j^*$, $0<j^* \le n$, for the following considerations.
We will assume that the restriction of $r_\downarrow$ to $g_\downarrow \cap T(i,j^*-1)$ can be expressed in a standard form on a set $I(i,j^*-1)$
of no more than $(2(j^*-1)+1)$ intervals.
Based on this assumption we will prove that the restriction of $r_\downarrow$ to $g_\downarrow \cap T(i,j^*)$ can be expressed
in a standard form on a set $I(i,j^*)$ of no more than $(2j^*+1)$ intervals.
The proof is partially illustrated by Figure \ref{Fig8}.
\begin{figure}
\caption{The left, middle and right parts of $g_\downarrow \cap T(i,j^*)$}
\label{Fig8}
\end{figure}
Denote $c$ and $d$
the horizontal coordinate of the left-most and the right-most points of $g_\downarrow \cap T(i,j^*-1)$ and express $g_\downarrow \cap T(i,j^*)$
as the union of three (possibly empty) parts: left, middle and right (see Figure \ref{Fig8}).
Let us examine the function $r_\downarrow$ on each of these three parts.
The left part consists of points $(u,j^*) \in g_\downarrow \cap T(i,j^*)$, $u < c$.
Let $(u^*,j^*)$ be one of these points. No point from $(u,j^*-1) \in g_\downarrow \cap T(i,j^*-1)$ is reachable from $(u^*,j^*)$.
Nevertheless, since the point $(u^*,j^*) \in g_\downarrow \cap T(i,j^*)$ exists, $g_\downarrow \cap R(i-1,j^*) \ne \emptyset$ and
since $D_\varepsilon(i,j^*)$ is convex, the point $(u^*,j^*)$ is reachable from any point $(u,v) \in g_\downarrow \cap R(i-1,j^*)$.
According to Lemma \ref{Constancy} the function $r_\downarrow$ is constant on the set $g_\downarrow \cap R(i-1,j^*)$.
Therefore, the value of the function $r_\downarrow$ on the left part is also constant and is the same as the value of $r_\downarrow$ on $g_\downarrow \cap R(i-1,j^*)$.
Therefore, as long as the left part exists, the set $I(i,j^*)$ contains an interval that is not present in $I(i,j^*-1)$. This interval is just the left part.
The middle part consists of points $(u,j^*) \in g_\downarrow \cap T(i,j^*)$, $c \le u \le d$. Let $(u^*,j^*)$ be one of these points.
Since $D_\varepsilon(i,j^*)$ is convex, the point $(u^*,j^*)$ is reachable from all points $(i,v) \in g_\downarrow \cap R(i-1,j^*)$
and from all points $(u,j^*-1) \in g_\downarrow \cap T(i,j^*-1)$, $u \le u^*$.
Due to monotonicity of the function $r_\downarrow$, it takes its maximum value $r_\downarrow(u^*,j^*-1)$ at the rightmost point $(u^*,j^*-1)$,
which is also reachable from $(u^*,j^*)$.
Therefore, $r_\downarrow(u^*,j^*)=r_\downarrow(u^*,j^*-1)$ and this is also valid for any point $(u,j^*)$ on the middle part.
Since the function $r_\downarrow$ on $g_\downarrow \cap T(i,j^*-1)$ can be expressed in a standard form on a set $I(i,j^*-1)$,
it can also be expressed on the middle part of the set $g_\downarrow \cap T(i,j^*)$ in a standard form on a set $I(i,j^*)$.
The set $I(i,j^*)$ is obtained from $I(i,j^*-1)$ by excluding some intervals and changing the endpoints of some other intervals.
Thus the number of intervals of the middle part does not increase.
The right part consists of points $(u,j^*) \in g_\downarrow \cap T(i,j^*)$, $u \ge d$.
Let $(u^*,j^*)$ be one of these points.
The point $(u^*,j^*)$ is reachable from all points $(i,v) \in g_\downarrow \cap R(i-1,j^*)$ and all points $(u,j^*) \in g_\downarrow \cap T(i,j^*-1)$.
The function $r_\downarrow$ takes its maximum value $r_\downarrow(d,j^*-1)$ at the rightmost point, which is reachable from $(u^*,j^*)$.
Therefore, the value $r_\downarrow(u^*,j^*)$ is constant on the right part and equals to $r_\downarrow(d,j^*-1)$.
Therefore, as long as the right part is not empty, the set $I(i,j^*)$ contains an interval that is not present in $I(i,j^*-1)$. This interval coincides with the right part.
One can see that the number of intervals in $I(i,j)$ can change compared to $I(i,j-\nolinebreak 1)$.
However, the number of intervals can not increase by more than $2$ intervals.
These are the intervals that coincide with the left and the right parts of $g_\downarrow \cap T(i,j^*)$.
\end{proof}
Similarly, one can prove the next lemma that we leave without proof.
\begin{lemma}
The restriction of $r^\uparrow$ to $g^\uparrow \cap R(i,j)$ can be expressed in a standard form on a set of no more than $(2i+1)$ intervals.
\end{lemma}
According to Lemma \ref{Constancy} the set $g^\uparrow \cap B$ and the restriction of $r^\uparrow$ to this set can be expressed with subsets $g^\uparrow \cap T(i,0)$
and numbers $r_i^\uparrow$ for $i \in \{1,2, \dots , m \}$.
The number $r_i^\uparrow$ is the value of $r^\uparrow$ on a subset $g^\uparrow \cap T(i,0)$.
The total amount of these data is of order $m$.
According to Lemma \ref{QuantityOfInt} the set $g_\downarrow \cap T$ and the restriction of $r_\downarrow$ to this set can be expressed with
the sets $I(i,n)$ of intervals $int$ and with numbers $r_\downarrow^{int}$, where $int \in I(i,n)$, $i \in \{m+1,m+2, \dots , 2m \}$.
Numbers $r_\downarrow^{int}$ define the function $r_\downarrow$ on the interval $int$ in the following way.
If $r_\downarrow^{int}$ is less than the right endpoint of the interval $int$ then $r_\downarrow(u,n)=r_\downarrow^{int}$ for all $(u,n) \in int$.
Otherwise, $r_\downarrow(u,n)=u$ for all $(u,n) \in int$.
The total amount of these data is of order $m \times n$.
Denote
$$C(X, Y) = \Big(\big\langle g^\uparrow \cap T(i,0), r_i^\uparrow, I(i+m,n), r_\downarrow^{int} \big\rangle \Big| i \in \{1,2, \dots , m \}, int \in I(i+m,n) \Big)$$
the data that are sufficient to test conditions (\ref{OurDecision}) for an $m$-gonal curve $X$ and an $n$-gonal curve $Y$.
Section \ref{SimilarityNew} shows that it takes $O(m n)$ time to test condition (\ref{OurDecision}) based on these data.
Section \ref{InputData} shows how to obtain the data $C(X, Y)$ in $O(m n)$ time.
\section{Testing the similarity of closed polygonal curves}\label{SimilarityNew}
\begin{lemma}\label{TestingOfMainInequality}
If for closed polygonal curves $X$ and $Y$ with $m$ and $n$ vertices the data $C(X, Y)$ are known then testing $\delta(X,Y) \le \varepsilon$ can be done in $O(m n)$ time.
\end{lemma}
\begin{proof} According to Lemmas \ref{AltFirst} and \ref{OurSecond}, condition $\delta(X,Y) \le \varepsilon$ is equivalent to the existence of a number $u$, $0 \le u \le m$,
that fulfills conditions
\begin{equation} \label{OurDecisionCopy}
(u,0) \in g^\uparrow, \quad (u+m,n) \in g_\downarrow, \quad u+m \le r^\uparrow(u,0),
\quad u \le r_\downarrow(u+m,n).
\end{equation}
Since the data $C(X,Y)$ are known, testing condition (\ref{OurDecisionCopy}) is reduced to testing if there exists a triple
$i \in \{ 1,2, \dots , m \}$, $int \in I(i+m,n)$, $u \in \{t|0 \le t \le m\}$ that fulfills conditions
\begin{equation} \label{OurDecisionConcreteNew }
(u,0) \in g^\uparrow \cap T(i,0), \quad (u+m,n) \in int, \quad u+m \le r_i^\uparrow,
\quad u \le r_\downarrow(u+m,n).
\end{equation}
Let us replace the condition $u \le r_\downarrow(u+m,n)$ in (\ref{OurDecisionConcreteNew }) with a condition $u \le r_\downarrow^{int}$
and express (\ref{OurDecisionConcreteNew }) in a form
\begin{equation} \label{OurDecisionAuxiliary}
(u,0) \in g^\uparrow \cap T(i,0), \quad (u+m,n) \in int, \quad u+m \le r_i^\uparrow,
\quad u \le r_\downarrow^{int}.
\end{equation}
Conditions (\ref{OurDecisionConcreteNew }) and (\ref{OurDecisionAuxiliary}) are equivalent.
Indeed, if the function $r_\downarrow$ is constant on $int$ then $r_\downarrow(u+m,n) = r_\downarrow^{int}$ for all $(u,n) \in int$.
If $r_\downarrow(u,n)=u$ on $int$ then condition $u \le r_\downarrow(u+m,n)$ becomes an inequality $u \le u+m$ that is valid for all $u$.
In this case, the value $r_\downarrow^{int}$ equals to a horizontal coordinate of interval's $int$ right endpoint.
Therefore, the inequality $u \le r_\downarrow^{int}$ is also valid for any point $(u,n) \in int$.
Denote $a^{int}$ and $b^{int}$ the left and right endpoints of the interval $int$.
Denote $c_{i}$ and $d_{i}$ the horizontal coordinate of the leftmost and the rightmost points of $g^\uparrow \cap T(i,0)$ respectively.
Using this notation, condition $(u,0) \in g^\uparrow \cap T(i,0)$ in (\ref{OurDecisionAuxiliary}) becomes $c_{i} \le u \le d_{i}$.
Condition $(u+m,n) \in int$ becomes $a^{int}-m \le u \le b^{int}-m$ when $int$ is the rightmost interval in $I(i+m,n)$ and becomes $a^{int}-m \le u < b^{int}-m$ otherwise.
Therefore, for some triples the constraint
\begin{equation} \label{OurDecisionWeekConcrete2 }
c_{i} \le u \le d_{i}, \quad a^{int}-m \le u \le b^{int}-m, \quad u \le r_{i}^\uparrow - m,
\quad u \le r_\downarrow^{int}
\end{equation}
is weaker than (\ref{OurDecisionAuxiliary}). Nevertheless, if a triple
$(i, int, u)$ that fulfills (\ref{OurDecisionWeekConcrete2 }) exists then the triple that fulfills (\ref{OurDecisionAuxiliary}) also exists.
Let $(i^*, int^*, u^*)$ be a triple that fulfills (\ref{OurDecisionWeekConcrete2 }) and does not fulfill (\ref{OurDecisionAuxiliary}).
In other words, $a^{int^*}-m \le u^* \le b^{int^*}-m$ and $ (u+m,n) \notin int^*$.
It is only possible when $u^* = b^{int^*}-m$ and $int^*$ is not the rightmost interval from $I(i^*+m,n)$.
It follows that an interval $int^+ \in I(i^*+m,n)$ exists such that $a^{int^+} =~b^{int^*}$.
The triple $(i^*, int^+, u^*)$ fulfills conditions (\ref{OurDecisionAuxiliary}).
Therefore, the consistency of conditions (\ref{OurDecisionWeekConcrete2 }) is equivalent to the consistency of conditions (\ref{OurDecisionAuxiliary}), although constraints (\ref{OurDecisionWeekConcrete2 }) are weaker than (\ref{OurDecisionAuxiliary}).
In turn, consistency of (\ref{OurDecisionWeekConcrete2 }) is equivalent to the existence of a pair
$i \in \{ 1,2, \dots , m \}$, $int \in I(i+m,n)$, that fulfills the inequality
$$\max \{a_i, \quad a^{int}-m \} \le
\min \{ d_i,\quad b^{int}-m,\quad r_i^\uparrow - m,\quad r_\downarrow^{int} \}.$$
Testing this inequality for any pair $(i,int)$ can be performed in constant time.
According to Lemma \ref{QuantityOfInt} the number of intervals tested for each $i$ does not exceed $(2n+1)$. Consequently, the number of tested pairs $(i,int)$ is $O(m n)$.
\end{proof}
\section{Obtaining the data $C(X, Y)$}\label{InputData}
\subsection{The general scheme}
Let the intervals $D_\varepsilon \cap T(i,j)$, $0 \le i \le 2m$, $1 \le j \le n$, and $D_\varepsilon \cap R(i,j) $, $1 \le i \le 2m$, $0 \le j \le n$,
be built for closed polygonal curves $X$ and $Y$.
Based on these data it is necessary to build $C(X,Y)$, which is used to test curve similarity as described in Section \ref{SimilarityNew}.
The data $C(X,Y)$ consist of sets $g_\downarrow \cap T(i,n)$, $m+1 \le i \le 2m$, restrictions of $r_\downarrow$ to these sets,
sets $g^\uparrow \cap T(i,0)$, $ 1 \le i \le m $, and values of function $r^\uparrow$ on these sets.
These sets and functions are built in two independent stages that we call forward and backward passes.
Both passes consist of $2mn$ steps, pair $(i,j)$ being the number of the step.
A forward pass performs a step number $(i,j)$ after steps $(i-1,j)$ and $(i,j-1)$.
On a step number $(i,j)$ the subsets $g_\downarrow \cap R(i,j) $, $g_\downarrow \cap T(i,j)$
and restrictions of $r_\downarrow $ to these subsets are built.
These data are obtained based on $g_\downarrow \cap R(i-1,j) $,
$g_\downarrow \cap T(i,j-1) $ and restrictions of $r_\downarrow $ to these sets, which had been built on previous steps.
The result of all $2mn$ steps of a forward pass are the sets $g_\downarrow \cap T(i,n)$, $m+1 \le i \le 2m $,
and restrictions of $r_\downarrow$ to these sets.
The proof of Lemma \ref{QuantityOfInt} practically describes the idea of an algorithm for the forward pass.
Nevertheless, Subsection \ref{Forward} describes it in more detail.
The result of the forward pass is only one part of the data needed to test (\ref{OurDecisionConcreteNew }).
Another part is the result of the backward pass.
During the backward pass a step number $(i,j)$ is performed after steps number $(i+1,j)$ and $(i,j+1)$.
A step number $(i,j)$ builds subsets $g^\uparrow \cap L(i,j) $, $g^\uparrow \cap B(i,j)$
and restrictions $r^\uparrow $ to these subsets based on subsets $g^\uparrow \cap L(i+1,j) $, $g^\uparrow \cap B(i,j+1) $ and restrictions of $r^\uparrow $ to these subsets.
The result of the backward pass are subsets $g^\uparrow \cap T(i,0)$ and values of $r^\uparrow$ on these subsets.
The backward pass uses the same approach as the forward pass and therefore is not presented in detail by this paper.
\subsection{The forward pass}\label{Forward}
We define a data structure to store any function $f$ that can be expressed in a standard form on a set $I$ of intervals $int$.
The function $f$ on each of these intervals $int \in I$ is defined by a triple $(beg, val, end )$, numbers $beg$ and $end$ being the endpoints of the interval $int$.
If $val < end$ then $f(x)=val$ for all $x \in int$, otherwise if $val = end$ then $f(x)=x$ for all $x \in int$.
The triples $(beg, val, end )$ are stored in a double-ended queue (deque) $Q$ with the following operations performing in constant time:\\
-- testing whether the deque is empty;\\
-- reading and removing either the leftmost or the rightmost triple;\\
-- inserting a triple either to the left or to the right end of the deque.\\
For any given number $x$ we define additional operations of cutting the deque to the left of $x$ and cutting it to the right of $x$.
When the deque is being cut to the left of $x$ all such triples $(beg, val, end)$ that $end < x$ are removed from the left end of the deque.
Then if the triple $(beg,val,end)$ on the left end is such that $beg \le x < end$, it gets replaced with a triple $(x,val,end)$.
When the deque is being cut to the right of $x$ all such triples $(beg, val, end)$ that $beg > x$ are removed from the right.
Then the triple $(beg,val,end)$ on the right end with $beg \le x \le end$ is replaced either by a triple $(beg,val,x)$ if $val < end$, or by a triple $(beg,x,x)$ if $val = end$.
Since the triples in the deque are sorted, the time spent on cutting the deque is proportional to the number of triples removed from the deque.
The input data for the forward pass are \\
\phantom{} \quad $D_\varepsilon \cap T(i,j)$, $1 \le i \le 2m$, $0 \le j \le n$,\quad and \quad $D_\varepsilon \cap R(i,j)$, $0 \le i \le 2m$, $1 \le j \le n$.\\
The algorithm works with $2m$ deques $Q(i)$, $i \in \{1, 2, \dots , 2m \}$.
Each $(i,j)$-th step starts with a deque $Q(i)$ that represents the set $g_\downarrow \cap T(i,j-1)$ and
the restriction of $r_\downarrow$ to this set
and updates it to represent the set $g_\downarrow \cap T(i,j)$ and
the restriction of $r_\downarrow$ to this set.
Thus, after the algorithm finishes the deque $Q(i)$ represents the set $g_\downarrow \cap T(i,n)$ and the restriction of $r_\downarrow$ to this set.
In order to perform this update on each step $(i,j)$ the auxiliary data in the form of the subset $g_\downarrow \cap R(i,j-1)$
and the value of $r_\downarrow$ on this subset is needed.
On a step number $(i, j)$ the subset $g_\downarrow \cap R(i,j)$ and the value of $r_\downarrow$ on this subset is calculated
and the deque $Q(i)$ is updated.
The algorithm consists of an initialization and $2mn$ steps.
The initialization sets the initial states of the deques and the initial values of the auxiliary data according to the following rules.
If $(0,0) \in D_\varepsilon$ then $g_\downarrow \cap R(0,1)=D_\varepsilon \cap R(0,1)$ and the function $r_\downarrow $ takes value $0$ on this set.
Otherwise $g_\downarrow \cap R(0,1) = \emptyset$.
If $(0,j-1) \in g_\downarrow$ then $g_\downarrow \cap R(0,j)=D_\varepsilon \cap R(0,j)$ and the function
$r_\downarrow $ takes value $0$ on this set.
Otherwise $g_\downarrow \cap R(0,j) = \emptyset$.
The set $g_\downarrow \cap T(i,0)$, $1 \le i \le 2m$, equals $D_\varepsilon \cap T(i,0)$.
If $D_\varepsilon \cap T(i,0) \ne \emptyset $ then the function $r_\downarrow $ takes value $r_\downarrow (u,0) = u$ on this set.
In this case the deque $Q(i)$ has a single triple $(beg, val, end)$, number $beg$ being the horizontal coordinate of the leftmost point of $D_\varepsilon \cap T(i,0)$
and $val=end$ being the coordinate the rightmost point.
If $D_\varepsilon \cap T(i,0) = \emptyset $ then the deque $Q(i)$ is initially empty.
It takes $O(m+n)$ time to perform the initialization.
At the beginning of a step number $(i,j)$ the following data are known.
The set $g_\downarrow \cap R(i-1,j)$ and the value of function $r_\downarrow$ on this set that we denote $r_\downarrow^*$.
The subset $g_\downarrow \cap T(i,j-1)$ and the restriction of $r_\downarrow$ to this subset that are represented by a deque $Q(i)$.
These data are illustrated on Figure~\ref{Fig9}.
The picture also shows the interval $D_\varepsilon \cap T(i,j)$, which is given as input to the forward pass,
numbers $a$ and $b$ being the left and right endpoints of this interval, $c$ and $d$ being the endpoints of the interval $g_\downarrow \cap T(i,j-1)$.
On each step $(i,j)$ the algorithm computes the set $g_\downarrow \cap R(i,j)$ and the value of $r_\downarrow$ on this set and updates the deque $Q(i)$ to
represent the subset $g_\downarrow \cap T(i,j)$ and the restriction of $r_\downarrow$ to this subset.
\begin{figure}
\caption{The data processed on $(i, j)$-th step}
\label{Fig9}
\end{figure}
Computing the set $g_\downarrow \cap R(i,j)$ and the value of $r_\downarrow$ on this set can be done by a rather simple rule in constant time,
which is not described here.
Updating the deque $Q(i)$ is a bit more complex but still consists of several rather straightforward rules.
\\
1. If $D_\varepsilon \cap T(i,j) = \emptyset$ then remove all triples from $Q(i)$.\\
2. If $g_\downarrow \cap R(i-1,j)= \emptyset$ and $g_\downarrow \cap T(i,j-1) = \emptyset$ then the deque $Q(i)$ is empty and it should remain empty.\\
3. If $D_\varepsilon \cap T(i,j) \ne \emptyset$, $g_\downarrow \cap R(i-1,j)\ne \emptyset$ and
$g_\downarrow \cap T(i,j-1) = \emptyset$ then the deque $Q(i)$ was empty at the beginning of the step $(i ,j)$.
Since $D_\varepsilon \cap D(i,j)$ is convex, any point of $D_\varepsilon \cap T(i,j)$ is reachable from any point of $g_\downarrow \cap R(i-1,j)$.
Therefore, a single triple $(a,r_\downarrow^* , b)$ is inserted into the empty deque $Q(i)$.
If $D_\varepsilon \cap T(i,j) \ne \emptyset$ and $g_\downarrow \cap T(i,j-1) \ne \emptyset$ then the update of $Q(i)$
depends on the relative position of intervals $D_\varepsilon \cap T(i,j)$ and $g_\downarrow \cap T(i,j-1)$.
The different cases of relative positions are shown on Figure~\ref{Fig11}.\\
4. Case a): exclude all triples from $Q(i)$; if $g_\downarrow \cap R(i-1,j)\ne \emptyset$ then insert the triple $(a,r_\downarrow^* , b)$
is inserted into the empty deque $Q(i)$.\\
5. Case b): cut the deque to the right of $b$; if $g_\downarrow \cap R(i-1,j)\ne \emptyset$ then insert the triple $(a,r_\downarrow^* , c)$ to the left end of $Q(i)$.\\
6. Case c): if $g_\downarrow \cap R(i-1,j)\ne \emptyset$ then insert $(a,r_\downarrow^* , c)$ to the left end of $Q(i)$;
insert $(d, r_\downarrow(d,0), b)$ to the right end of the deque $Q(i)$. \\
7. Case d): cut the deque to the left of $a$; cut the deque to the right of $b$. \\
8. Case e): cut the deque to the left of $a$; insert $(d, r_\downarrow(d,0), b)$ to the right end of the deque $Q(i)$. \\
9. Case f): remove all triples from $Q(i)$; insert a single triple $(a, r_\downarrow(d,0), b)$ to $Q(i)$.
\begin{figure}
\caption{Relative position of intervals $D_\varepsilon \cap T(i,j)$ and $g_\downarrow \cap T(i,j-1)$}
\label{Fig11}
\end{figure}
\begin{lemma}\label{ComplexityForwardAndBackward}
It takes $O(m n)$ time to complete both the forward and the backward passes of the algorithm.
\end{lemma}
\begin{proof}
We prove this lemma for the forward pass. The proof for the backward pass is similar.\\
No more than two triples are inserted into the deques on each step. Therefore, no more than $4mn$ triples are inserted during all steps $(i,j)$, $1 \le i \le 2m$, $1 \le j \le n$.
The number of triples removed from the deques does not exceed the number of triples inserted into the deques, and therefore is not greater than $4mn$.
Every time some triple is read from the deques it is also removed. Therefore, the number of times the triples are read is also not greater than $4mn$.
Therefore, the forward pass includes the initialization, which takes $O(m+n)$ time, and the work with deques, which takes $O(m n)$ time.
In addition, the forward pass also computes the set $g_\downarrow \cap R(i,j)$ and the value of $r_\downarrow$ on this set on each step.
It takes constant time on each step to do this. Therefore, the total amount of time spent on these computations is $O(m n)$.
\end{proof}
\section{The result}\label{ResultSection}
\begin{theorem}
Let $X$ and $Y$ be closed polygonal curves with $m$ and $n$ vertices and $\delta(X,Y)$ be the Frechet distance between them.
Testing the inequality $\delta(X,Y) \le \varepsilon$ takes $O(m n)$ time.
\end{theorem}
\begin{proof}
Testing the inequality $\delta(X,Y) \le \varepsilon$ is reduced to the following computations.
\\
For the given polygonal curves $X$ and $Y$ the sets $D_\varepsilon \cap T(i,j)$, $1 \le i \le 2m$, $0 \le j \le n$, and $D_\varepsilon \cap R(i,j)$, $0 \le i \le 2m$, $1 \le j \le n$, are computed.
This is equivalent to solving $2mn$ quadratic equations.
\\
Then based on the sets $D_\varepsilon \cap T(i,j)$ and $D_\varepsilon \cap R(i,j)$ the subsets $g_\downarrow \cap T(i,n)$, $m+1 \le i \le 2m$,
and restrictions of $r_\downarrow$ to these subsets are computed along with the subsets $g^\uparrow \cap T(i,0)$, $ 1 \le i \le m $, and the values of $r^\uparrow$ on these subsets.
According to Lemma \ref{ComplexityForwardAndBackward} it takes $O(m n)$ time to build these.
\\
Finally, testing the inequality $\delta(X,Y) \le \varepsilon$ based on these data can be done in $O(m n)$ time according to Lemma \ref{TestingOfMainInequality}.
\end{proof}
\end{document} |
\begin{document}
\title{Existence of renormalized solutions to elliptic equation \ in Musielak-Orlicz space} \sloppy
{\tau}hispagestyle{empty}
\partialrindent 1em
\begin{abstract}
We prove existence of renormalized solutions to general nonlinear elliptic equation in~Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \begin{equation*}
-\mathrm{div} A(x,\nabla u)= f\in L^1(\Omega),
\end{equation*}
on a Lipschitz bounded domain in ${\mathbb{R}}n$. The growth of the monotone vector field $A$ is controlled by a generalized nonhomogeneous and anisotropic $N$-function $M $. The approach does not require any particular type of growth condition of $M$ or its conjugate $M^*$ (neither ${\Delta}elta_2$, nor $\nabla_2$). The condition we impose is log-H\"older continuity of $M$, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.
\end{abstract}
{\small {\bf Key words and phrases:} elliptic problems, existence of solutions, Musielak-Orlicz spaces, renormalized solutions}
{\small{\bf Mathematics Subject Classification (2010)}: 35J60, 35D30. }
\section{Introduction}
Our aim is to find a way of proving the existence of renormalized solutions to a strongly nonlinear elliptic equation with $L^1$-data under minimal restrictions on the growth of the leading part of the operator. We investigate operators $A$, which are monotone, but not necessarily strictly. The~modular function $M$, which controls the growth of the operator, is not assumed to be isotropic, i.e. $M=M(x,\xi)$ not $M=M(x,|\xi|)$. In turn, we can expect different behaviour of $M(x,\cdot)$ in various directions. We {\tau}extbf{do not} require $M\in{\Delta}elta_2$, nor $M^*\in{\Delta}elta_2$, nor~any particular growth of $M$, such as $M(x,\xi)\geq c|\xi|^{1+\nu}$ for $\xi>\xi_0$. The price we pay for relaxing the conditions on the growth is requirement of~log-H\"older-type regularity of the modular function (cf. condition (M)).
We study the problem
\begin{equation}\label{intro:ell}\left\{\begin{array}{cl}
-\mathrm{div} A(x,\nabla u)= f &\qquad \mathrm{ in}\qquad \Omega,\\
u(x)=0 &\qquad \mathrm{ on}\qquad \partialrtial\Omega,
\end{array}{\mathbb{R}}ight.
\end{equation}
where $\Omega$ is a bounded Lipschitz domain in $ {\mathbb{R}}n$, $N>1$, $f:\Omega{\tau}o{\mathbb{R}}$, $f\in L^1(\Omega)$.
We consider $A$~belonging to an Orlicz class with respect to the second variable. Namely, we assume that function $A:\Omega{\tau}imes{\mathbb{R}}n{\tau}o{\mathbb{R}}n$ satisfies the following conditions.
\begin{itemize}
\item[(A1)] $A$ is a Carath\'eodory's function;
\item[(A2)] There exists an $N$-function $M:\Omega{\tau}imes{\mathbb{R}}n{\tau}o{\mathbb{R}}$ and a constant $c_A\in(0,1]$ such that for all $\xi\in{\mathbb{R}}n$ we have
\[A(x,\xi)\xi\geq c_A\left(M(x,\xi)+M^*(x,A(x,\xi)){\mathbb{R}}ight),\]
where $M^*$ is conjugate to $M$ (see Definition~{\mathbb{R}}ef{def:conj});
\item[(A3)] For all $\xi,\eta\in{\mathbb{R}}n$ and $x\in\Omega$ we have
\[(A(x,\xi) - A(x, \eta)) \cdot (\xi-\eta)\geq 0.\]
\end{itemize}
Existence of solutions to~\eqref{intro:ell} is considered in
\[ V^M_0 =\{u\in W_0^{1,1}(\Omega):\ \nabla u\in L_M(\Omega;{\mathbb{R}}n)\}.\]
The space $L_M$ (Definition~{\mathbb{R}}ef{def:MOsp}) is equipped with the modular function $M$ being an $N$-function (Definition~{\mathbb{R}}ef{def:Nf}) controlling the growth of $A$, cf.~(A2).
Unlike other studies, instead of growth conditions we assume regularity of~$M$.
\begin{itemize}
\item[(M)] Suppose for every measurable set $G\subset\Omega$ and every $z\in{\mathbb{R}}n$ we have
\begin{equation}
\label{ass:M:int}\int_G M(x,z)dx<\infty.
\end{equation} Let us consider a family of $N$-dimensional cubes covering the set $\Omega$. Namely, a family $\{{Q_j^\delta}\}_{j=1}^{N_{\delta}elta}$ consists of closed cubes of edge $2{\delta}elta$, such that $\mathrm{int}{Q_j^\delta}\cap\mathrm{int} Q^{\delta}elta_i=\emptyset$ for $i \neq j$ and $\Omega\subset\bigcup_{j=1}^{N_{\delta}elta}{Q_j^\delta}$. Moreover, for each
cube ${Q_j^\delta}$ we define the cube ${\tau}Qd$ centered at the same point and with parallel corresponding edges of length $4{\delta}elta$. Assume that there exist constants $a,b,c,{\delta}elta_0 >0$, such that for all ${\delta}elta<{\delta}elta_0$, $x\in{Q_j^\delta}$ and all $\xi\in{\mathbb{R}}n$ we have
\begin{equation}
\label{M2}
\frac{M(x,\xi )}{{\cal M}ss}
\leq c \left(1+ |\xi|^{-\frac{a}{\log(b{\delta}elta )}} {\mathbb{R}}ight),
\end{equation}
where \begin{equation}
\label{Mjd}{\cal M}jd(\xi ):= \inf_{x\in \widetildeidetilde{Q}_j^{\delta}elta\cap\Omega}M(x,\xi ),
\end{equation}
while ${\cal M}ss=({({\cal M}jd(\xi))}^*)^* $ is the greatest convex minorant of ${\cal M}jd(\xi )$ (coinciding with the second conjugate cf. Definition~{\mathbb{R}}ef{def:conj}).
\end{itemize}
In further parts of the paper we describe the cases, when the above condition is not necessary. Let us only point out that to get (M) in the isotropic case, i.e. when we consider $M(x,\xi)=M(x,|\xi|)$, it suffices to assume log-H\"older-type condition with respect to $x$ \eqref{M2'}, cf.~Lemma~{\mathbb{R}}ef{lem:Mass}.
We apply the truncation techniques. Let truncation $T_k(f)(x)$ be defined as follows\begin{equation}T_k(f)(x)=\left\{\begin{array}{ll}f & |f|\leq k,\\
k\frac{f}{|f|}& |f|\geq k.
\end{array}{\mathbb{R}}ight. \label{Tk}
\end{equation}
We call a function $u$ a renormalized solution to~\eqref{intro:ell}, when it satisfies the following conditions.\begin{itemize}
\item[(R1)] $u:\Omega{\tau}o{\mathbb{R}}$ is measurable and for each $k>0$ \[T_k(u)\in V_0^M \cap L^\infty (\Omega),\qquad A(x,\nabla T_k(u))\in L_{M^*}(\Omega;{\mathbb{R}}n).\]
\item[(R2)] For every $h\in C^1_c({\mathbb{R}})$ and all $\varphi\in V_0^M\cap L^\infty (\Omega)$ we have
\[\int_\Omega A(x,\nabla u)\cdot\nabla(h(u)\varphi)dx=\int_\Omega fh(u)\varphi\,dx.\]
\item[(R3)] $\int_{\{l<|u|<l+1\}}A(x,\nabla u)\cdot\nabla u\, dx{\tau}o 0$ as $l{\tau}o\infty$.
\end{itemize}
Our main result reads as follows.
\begin{theo}\label{theo:main} Suppose $f\in L^1(\Omega)$, an $N$-function $M$ satisfies assumption (M) and function $A$ satisfies assumptions (A1)-(A3). Then there exists at least one renormalized weak solution to the problem \begin{equation*}
\left\{\begin{array}{cl}
-\mathrm{div} A(x,\nabla u)= f &\qquad \mathrm{ in}\qquad \Omega,\\
u(x)=0 &\qquad \mathrm{ on}\qquad \partialrtial\Omega,
\end{array}{\mathbb{R}}ight.
\end{equation*} Namely, there exists $u \in V_0^M $ which satisfies (R1)-(R3). Moreover, $A(\cdot,\nabla u)\in L_{M^∗}(\Omega)$.
\end{theo}
\begin{ex} We give below pairs of functions $M$ and $A$ satisfying conditions (M) and (A1)-(A3), respectively.\begin{itemize}
\item Consider $M(x,\xi)=|\xi|^{p(x)}$ with log-H\"older $p:\Omega{\tau}o[p^-,p^+]$, where $p^-=\inf_{x\in\Omega}p(x)>1$ and $p^+=\sup_{x\in\Omega}p(x)<\infty,$ then $V_0^M= W_0^{1,p(\cdot)}(\Omega)$ and we admitt $A(x,\xi)=|\xi|^{p(x)-2}\xi$ ($p(\cdot)$-Laplacian case) as well as \[A(x,\xi)={{\cal A }_{l+1}}pha(x)|\xi|^{p(x)-2}\xi\quad{\tau}ext{ with }\quad 0<<{{\cal A }_{l+1}}pha(x)\in L^\infty(\Omega)\cap C(\Omega);\]
\item $M(x,\xi)=\sum_{i=1}^N|\xi_i|^{p_i(x)}$, where $\xi=(\xi_1,{\delta}ots,\xi_N)\in{\mathbb{R}}n$, log-H\"older functions $p_i:\Omega{\tau}o[p_i^-,p_i^+]$, $i=1,{\delta}ots,N$, where $p_i^-=\inf_{x\in\Omega}p_i(x)>1$ and $p_i^+=\sup_{x\in\Omega}p_i(x)<\infty,$ then $V_0^M= W_0^{1,{\varepsilon}c{p}(\cdot)}(\Omega)$ and we admitt \[A(x,\xi)=\sum_{i=1}^N{{\cal A }_{l+1}}pha_i(x)|\xi|^{p_i(x)-2}\xi\quad{\tau}ext{ with }\quad 0<<{{\cal A }_{l+1}}pha_i(x)\in L^\infty(\Omega)\cap C(\Omega).\]
\end{itemize}
\end{ex}
\begin{rem}[cf.~\cite{martin}] {\mathbb{R}}m When the modular function has a special form we can simplify our assumptions. In the case of $M(x,\xi)=M(x,|\xi|)$, via Lemma~{\mathbb{R}}ef{lem:Mass}, we replace condition (M) in the above theorem by log-H\"older continuity of M, cf.~\eqref{M2'}. If $M$ has a form
\[M(x,\xi)=\sum_{i=1}^jk_i(x)M_i(\xi)+M_0(x,|\xi|),\quad j\in{\mathbb{N}},\]
instead of~whole (M) we assume only that $M_0$ is log-H\"older continuous~\eqref{M2'}, all $M_i$ for $i=1,{\delta}ots,j$ are $N$-functions and all $k_i$ are nonnegative and satisfy $\frac{k_i(x)}{k_i(y)}\leq C_i^{\log \frac{1}{|x-y|}}$ with $C_i>0$ for $i=1,{\delta}ots,j$.
\end{rem}
\begin{rem} {\mathbb{R}}m
Note that according to (A2) and the Fenchel-Young inequality we have
\[c_A\left(M(x,\xi)+M^*(x,A(x,\xi)){\mathbb{R}}ight)\leq A(x,\xi)\xi\leq M(x,\xi)+M^*(x,A(x,\xi)) \]
satisfied with a certain $N$-function $M:\Omega{\tau}imes{\mathbb{R}}n{\tau}o{\mathbb{R}}$. This observation results in $A(x,0)=0$. However, the framework admitts considering in (A2)
\[A(x,\xi)\xi\geq c_A\left(M(x,\xi)+M^*(x,A(x,\xi)){\mathbb{R}}ight)-k(x),\qquad 0\leq k(x)\in L^1(\Omega),\]
despite it does not imply $A(x,0)=0$.
\end{rem}
The Musielak-Orlicz spaces equipped with the modular function satisfying ${\Delta}elta_2$-condition (cf.~Definition~{\mathbb{R}}ef{def:D2}) have strong properties, however there is a vast range of $N$-functions not satisfying it, e.g.
\begin{itemize}
\item[i)] $M(x,\xi)=a(x)\left( \exp(|\xi|)-1+|\xi|{\mathbb{R}}ight)$;
\item[ii)] $M(x,\xi)= |\xi_1|^{p_1(x)}\left(1+|\log|\xi||{\mathbb{R}}ight)+\exp(|\xi_2|^{p_2(x)})-1$, when $(\xi_1,\xi_2)\in{\mathbb{R}}^2$ and $p_i:\Omega{\tau}o[1,\infty]$. It is a model example to imagine what we mean by anisotropic modular function.
\end{itemize}
Nonetheless, our assumption that $M,M^*$ are $N$-functions (Definition~{\mathbb{R}}ef{def:Nf}) in the variable exponent setting restrict us to the case of $1<p_-\leq p(x)\leq p^+<\infty$.
\subsubsection*{State of art}
Existence to problems like~\eqref{intro:ell} is very well understood, when $A$ is independent of the spacial variable and has a polynomial growth. In~particular, there is vast literature for analysis of the case involving the $p$-Laplace operator $A(x,\xi)=|\xi|^{p-2}\xi$ and problems stated in the Lebesgue space setting (the modular function is then $M(x,\xi)=|\xi|^p$). Let us note that the variable exponent Lebesgue spaces (for $M(x, \xi ) = | \xi |^{p(x)}$ with $1 < p_{{\mathbb{R}}m min} \leq p(x) \leq p^{{\mathbb{R}}m max} < \infty $) are still reflexive. Despite the~methods of~analysis of problems in this setting are more advanced, they are in the same spirit.
Studies on renormalized solutions comes from DiPerna and Lions~\cite{diperna-lions} investigations on~the~Boltzmann equation. In the elliptic setting the foundations of the branch were laid by Boccardo et.~al.~\cite{boc-g-d-m}, Dall'Aglio~\cite{dall} and Murat~\cite{murat}, providing results for operators with polynomial growth. Their generalisations to the variable exponent setting can be find in~\cite{andreianov,benboubker,wit-zim}.
Investigations of nonlinear elliptic boundary value problems in~non-reflexive Orlicz-Sobolev-type setting was initiated by Donaldson~\cite{Donaldson} and continued by Gossez~\cite{Gossez2,Gossez3,Gossez}. For a~summary of~the~results we refer to~\cite{Mustonen} by Mustonen and Tienari. The generalization to the case of~vector Orlicz spaces with possibly anisotropic modular function, but independent of spacial variables was investigated in~\cite{Gparabolic}.
The existence theory for problems in this setting arising from fluids mechanics is developed from various points of view~\cite{gwiazda-non-newt,gwiazda-tmna,gwiazda2,Aneta}. For the recent existence results for elliptic problems we refer to~\cite{renel2,renel1,Benb,renel3,Dong,fan12,le-ex,gwiazda-ren-ell,gwiazda-ren-cor,hhk,le-ex,liuzhao15}. In~\cite{fan12,hhk,liuzhao15} isotropic, separable and reflexive Musielak-Orlicz spaces are employed,~\cite{Benb} concerns anisotropic variable exponent spaces, \cite{Dong}~studies separable, but not reflexive Musielak-Orlicz spaces, while~\cite{le-ex} anisotropic, but separable and reflexive Orlicz spaces.
Renormalized solutions to~elliptic problems in~Orlicz spaces are explored in~\cite{renel2,renel1,renel3}, while in Musielak-Orlicz spaces in~\cite{gwiazda-ren-ell,gwiazda-ren-cor}.
\subsubsection*{Approximation in Musielak-Orlicz spaces}
The highly challenging part of analysis in the general Musielak-Orlicz spaces is giving a relevant structural condition implying approximation properties of the space. However, we are equipped not only with the weak-* and strong topology of the gradients, but also with the intermediate one, namely - the modular topology.
In the mentioned existence results even in the case, when the growth conditions imposed on~the~modular function were given by a~general $N$-function, besides the growth condition on $M^*$, also ${\Delta}elta_2$-condition on $M$ was assummed (which entails separability of~$L_{M^*}$, see~\cite{Aneta}). It results further in density of smooth functions in $L_M$ with respect to the weak-$*$ topology. In the case of~classical Orlicz spaces, the crucial density result was provided by Gossez~\cite{Gossez}. The improvement of this result for the vector Orlicz spaces was given in~\cite{Gparabolic}, while for the $x$--dependent log-H\"{o}lder continuous modular functions in~\cite{BenkiraneDouieb}, developed in~\cite{GMWK,ASGcoll} and further in~\cite{ASGpara} in the case of log-H\"{o}lder continuous modular functions dependent on $x$, as well as on $t$.
Let us discuss our assumption (M). First we shall stress that it is applied only in the proof of approximation result (Theorem~2.2). When we deal with the space equipped with the approximation properties, we can simply skip (M). Namely, this is the case e.g. of the following modular functions:
\begin{itemize}
\item $M(x,|\xi|)=|\xi|^p+a(x)|\xi|^q$, where $1\leq p<q$ and function $a$ is nonnegative a.e. in $\Omega$ and $a\in L^\infty(\Omega)$, covering the celebrated double-phase case~\cite{min-double-reg};
\item $M(x,\xi)=M_1(\xi)+a(x)M_2(\xi)$, where $M_1,M_2$ satisfy conditions ${\Delta}elta_2$ and $\nabla_2$, moreover a~function $a$ is nonnegative a.e. in $\Omega$ and $a\in L^\infty(\Omega)$.
\end{itemize}
In the both above cases modular approximation sequence obtained in the spirit of Theorem~{\mathbb{R}}ef{theo:approx} can be replaced by existence of a strongly converging affine combination of the weakly converging sequence (ensured in any reflexive Banach space via Mazur's Lemma).
In the variable exponent case typical assumption resulting in approximation properties of the space is log-H\"older continuity of the exponent. In the isotropic case (when $M(x,\xi)=M(x,|\xi|)$)
Lemma~{\mathbb{R}}ef{lem:Mass} shows that to get (M), it suffices to impose on $M$ continuity condition of log-H\"older-type with respect to $x$, namely for each $\xi\in{\mathbb{R}}n$ and $x,y,$ such that $|x-y|<\frac{1}{2}$ we have\begin{equation}
\label{M2'} \frac{M(x,\xi)}{M(y,\xi)}\leq\max\left\{ |\xi|^{-\frac{a_1}{\log|x-y|}}, b_1^{-\frac{a_1}{\log|x-y|}}{\mathbb{R}}ight\},\ {\tau}ext{with some}\ a_1>0,\,b_1\geq 1.
\end{equation} Note that condition~\eqref{M2'}
for $M(x,\xi)=|\xi|^{p(x)}$ relates to the log-H\"older continuity condition for the variable exponent $p$, namely there exists $a>0$, such that for $x,y$ close enough and each $\xi\in{\mathbb{R}}n$
\[|p(x)-p(y)|\leq \frac{a}{\log\left(\frac{1}{|x-y|}{\mathbb{R}}ight)}.\]
Indeed,
\[ \frac{M(x,\xi)}{M(y,\xi)}= \frac{|\xi|^{p(x)}}{|\xi|^{p(y)}}=|\xi|^{p(x)-p(y)}\leq |\xi|^\frac{a}{\log\left(\frac{1}{|x-y|}{\mathbb{R}}ight)}=|\xi|^{-\frac{a}{\log {|x-y|} }}.\]
There are several types of understanding generalisation of log-H\"older continuity to the case of general $x$-dependent isotropic modular functions (when $M(x,\xi)=M(x,|\xi|)$). The important issue is the interplay between types of continuity with respect to each of the variables separately. Besides our condition~\eqref{M2'} (sufficient for (M) via Lemma~{\mathbb{R}}ef{lem:Mass}), we refer to the approaches of~\cite{hhk,hht} and~\cite{mmos:ap,mmos2013}, where the authors deal with the modular function of the form $M(x,\xi)=|\xi|\phi(x,|\xi|)$. We proceed without their doubling assumptions (${\Delta}elta_2$). Since we are restricted to bounded domains, condition $\phi(x,1)\sim 1$ follows from our definition of $N$-function (Definition~{\mathbb{R}}ef{def:Nf} ). As for the types of continuity, in~\cite{mmos:ap,mmos2013} the authors restrict themselves to the case when $\phi(x,|\xi|)\le c \phi(y,|\xi|)$ when $|\xi|\in [1,|x-y|^{-n}].$ This condition implies~\eqref{M2'} and consequently~(M). Meanwhile in~\cite{hhk,hht}, the proposed condition yields $\phi(x, b|\xi|)\le \phi(y,|\xi|)$ when $\phi(y,|\xi|)\in [1, |x-y|^{-n}],$ which does not imply~\eqref{M2'} directly. However, we shall mention that all three conditions are of the same spirit and balance types of continuity with respect to each of the variables separately.
\subsubsection*{Our approach}
The challenges resulting from the lack of the growth conditions are significant and require precise handling with general $x$-dependent and anisotropic $N$-functions. The space we deal with is, in~general, neither separable, nor reflexive. Resigning from imposing ${\Delta}elta_2$-condition on the conjugate of the modular function $M$ complicates understanding of the dual pairing. As a further consequence of relaxing growth condition, we cannot use classical results, such as the Sobolev embeddings or~the~Rellich-Kondrachov compact embeddings. We extend the main goal of~\cite{GMWK}, where the authors deal with bounded data. Lack of~precise control on the growth of~the~ leading part of the operator, together with the low integrability
of~the~right-hand side results in noticeable difficulties in studies on convergence.
Besides the refined version of approximation result of~\cite{GMWK} (Theorem~{\mathbb{R}}ef{theo:approx}), we prove general modular Poincar\'{e}-type inequality (Theorem~{\mathbb{R}}ef{theo:Poincare}). The main goal, i.e. the existence of renormalized solutions to general nonlinear elliptic equation, is given in Theorem~{\mathbb{R}}ef{theo:main}. Our methods leading to~this result are based on the scheme of~\cite{gwiazda-ren-ell,gwiazda-ren-cor}, i.e. we employ truncation arguments, the Minty-Browder monotonicity trick and the Young measures. However, unlike in the latter papers we put regularity restrictions on the modular function instead of the growth conditions.
\section{Preliminaries}
In this section we give only the general preliminaries concerning the setting. All necessary definitions and technical tools, as well as an introduction to the setting and general theorems are given in Appendix.
\subsubsection*{Classes of functions}
\begin{defi}\label{def:MOsp} Let $M$ be an $N$-function (cf.~Definition~{\mathbb{R}}ef{def:Nf}).\\ We deal with the three Orlicz-Musielak classes of functions.\begin{itemize}
\item[i)]${\cal L}_M(\Omega;{\mathbb{R}}n)$ - the generalised Orlicz-Musielak class is the set of all measurable functions $\xi:\Omega{\tau}o{\mathbb{R}}n$ such that
\[\int_\Omega M(x,\xi(x))\,dx<\infty.\]
\item[ii)]${L}_M(\Omega;{\mathbb{R}}n)$ - the generalised Orlicz-Musielak space is the smallest linear space containing ${\cal L}_M(\Omega;{\mathbb{R}}n)$, equipped with the Luxemburg norm
\[||\xi||_{L_M}=\inf\left\{\lambda>0:\int_\Omega M\left(x,\frac{\xi(x)}{\lambda}{\mathbb{R}}ight)\,dx\leq 1{\mathbb{R}}ight\}.\]
\item[iii)] ${E}_M(\Omega;{\mathbb{R}}n)$ - the closure in $L_M$-norm of the set of bounded functions.
\end{itemize}
\end{defi}
Then
\[{E}_M(\Omega;{\mathbb{R}}n)\subset {\cal L}_M(\Omega;{\mathbb{R}}n)\subset { L}_M(\Omega;{\mathbb{R}}n),\]
the space ${E}_M(\Omega;{\mathbb{R}}n)$ is separable and $({E}_M(\Omega;{\mathbb{R}}n))^*=L_{M^*}(\Omega;{\mathbb{R}}n)$, see~\cite{gwiazda-non-newt,Aneta}.
Under the so-called ${\Delta}elta_2$-condition (Definition~{\mathbb{R}}ef{def:D2}) we would be equipped with stronger tools. Indeed, if $M\in{\Delta}elta_2$, then
\[{E}_M(\Omega;{\mathbb{R}}n)= {\cal L}_M(\Omega;{\mathbb{R}}n)= {L}_M(\Omega;{\mathbb{R}}n)\]
and $L_M(\Omega;{\mathbb{R}}n)$ is separable. When both $M,M^*\in{\Delta}elta_2$, then $L_M(\Omega;{\mathbb{R}}n)$ is separable and reflexive, see~\cite{GMWK,gwiazda-non-newt}. We face the problem without this structure.
\begin{rem} Definition~{\mathbb{R}}ef{def:Nf} (see points 3 and 4) implies
$\lim_{|\xi|{\tau}o \infty}\inf_{x\in\Omega}\frac{M^*(x,\xi)}{|\xi|}=\infty$ and
$\inf_{x\in\Omega}M^*(x,\xi)>0$ for any $\xi\neq 0$. Then, consequently, Lemma~{\mathbb{R}}ef{lem:M*<M} ensures
\begin{equation}
\label{LinfinLM}L^\infty(\Omega;{\mathbb{R}}n) \subset L_M(\Omega;{\mathbb{R}}n).\end{equation}
\end{rem}
\subsubsection*{Comments on assumptions on $A$ }
The following consideration explains how condition (A2) settles growth and coercivity condition on the leading part of the operator.
In the standard $L^p$-setting it is enough to note that (A2) implies directly \[A(x,\xi)\xi\geq c_A |\xi|^p\] and $|A(x,\xi)|\cdot|\xi|\geq \widetildeidetilde{c}_A |A(x,\xi)|^{p'},$ leading further to the condition \[\widetildeidetilde{c}_A |\xi|^{p-1} \geq |A(x,\xi)|.\]
In the nonstandard growth setting, considering the first counterpart of the above condition, i.e.
\begin{equation}
\label{Adown}A(x,\xi)\xi\geq c_A M(x,\xi),
\end{equation} we get the minimal growth. As for the bound from above, we define an increasing function $P:{\mathbb{R}}\cup\{0\}{\tau}o{\mathbb{R}}\cup\{0\}$ by the following formula
\[P(s):=\sup_{\xi:\ |\xi|=s}\left(\inf_{x\in\Omega, } M^*(x, \xi ) {\mathbb{R}}ight)^*.\]
Notice that for every $x\in\Omega$ and $\xi\in{\mathbb{R}}n$ such that $|\xi|=s$ it holds $ P(s) \geq M(x,\xi)$. Moreover, we have an upper bound for the growth of the operator
\begin{equation}
\label{Aup}
|A(x,\xi)| \leq 2 (P^*)^{-1}\left(\frac{1}{c_A}P\left(\frac{2}{c_A}|\xi |{\mathbb{R}}ight){\mathbb{R}}ight) .
\end{equation}
Indeed, to prove
\[c_A P^*\left(\frac{1}{2}|A(x,\xi)|{\mathbb{R}}ight) \leq P\left(\frac{2}{c_A}|\xi | {\mathbb{R}}ight)\] it suffices to notice that
Fechel-Young inequality~\eqref{inq:F-Y} yields
\[A(x,\xi)\xi\leq P\left(\frac{2}{c_A}|\xi|{\mathbb{R}}ight)+P^*\left(\frac{c_A}{2}|A(x,\xi)|{\mathbb{R}}ight)\leq P\left(\frac{2}{c_A}|\xi|{\mathbb{R}}ight)+c_A P^*\left(\frac{1}{2}|A(x,\xi)|{\mathbb{R}}ight),\]
whereas on the other hand
\[A(x,\xi)\xi\geq {c_A} M^* (x,A(x,\xi))\geq c_A P^*\left(|A(x,\xi)|{\mathbb{R}}ight)\geq 2c_A P^*\left(\frac{1}{2}|A(x,\xi)|{\mathbb{R}}ight).\]
Conditions of this form are considered in classical Orlicz setting, when $M(x,\xi)=M(|\xi|)$ by e.g.~\cite{Gossez,Mustonen}. Note that then we can take $P(s)=M(s)$. Since (A2) implies~\eqref{Adown} and~\eqref{Aup}, we assume particular growth and coercivity of the leading part of the operator corresponding to the modular function of the space, where the solutions are defined. Nonetheless, conditions~\eqref{Adown} and~\eqref{Aup} are not sufficient in our approach. Note that they do not ensure that the operator and the solution are in the proper dual spaces. Let us stress further that the consequences of (A2) are expressed by $N$-functions of general type of growth.
\subsubsection*{Main tools}
The existence of solutions to the truncated problem follows directly from~\cite[Theorem~1.5]{GMWK}.
\begin{theo}[Existence with bounded data, cf.~\cite{GMWK}]\label{theo:boundex} Suppose $g\in L^\infty(\Omega)$, an $N$-function $M$ satisfies assumption (M) and function $A$ satisfies assumptions (A1)-(A3). Then there exists a weak solution to the problem \[
\left\{\begin{split}
-\mathrm{div} A(x,\nabla u)= g &\qquad \mathrm{ in}\qquad \Omega,\\
u(x)=0 &\qquad \mathrm{ on}\qquad \partialrtial\Omega,
\end{split}{\mathbb{R}}ight.
\] Namely, there exists $u \in W_0^{1,1} (\Omega)$ such that $\nabla u \in L_M (\Omega)$ satisfies
\[\int_\Omega A(x, \nabla u) \cdot \nabla \varphi dx =
\int_\Omega g\varphi\,dx,\]
for all $\varphi \in C_0^\infty(\Omega)$. Moreover, $A(\cdot,\nabla u)\in L_{M^∗}(\Omega)$.
\end{theo}
In fact, \cite[Theorem~1.5]{GMWK} is proven under the assumption that there exists $F:\Omega{\tau}o{\mathbb{R}}n$, such that $g= \mathrm{div} F$ and $F\in E_{M^*}(\Omega)$. Nevertheless, each bounded $g$ is of this form. Existence of such $F$ is clear, while the fact that $F\in E_{M^*}(\Omega)$ is a consequence of properties of the Bogovski operator, see e.g.~[\cite{NavierStokes}, Lemma II.2.1.1].
The following refined approximation result of~\cite[Theorem~2.7]{GMWK} being an~improvement of the case from~\cite{BenkiraneDouieb} is proven in Appendix.
\begin{theo}[Approximation theorem]\label{theo:approx}
Let $\Omega$ be a Lipschitz domain and an $N$-function $M$~satisfy condition (M). Then for any $\varphi$ such that $\varphi\in V_0^M\cap L^\infty(\Omega)$ there exists a sequence $\{\varphi_{\delta}elta\}_{{\delta}elta>0}\in C_0^\infty(\Omega)$ converging modularly to $\varphi$, i.e. such that $\nabla\varphi_{\delta}elta\xrightarrow[]{M}\nabla \varphi$.
\end{theo}
The vital tool in our study is the following modular Poincar\'{e}-type inequality. The proof is also included in Appendix.
\begin{theo}[Modular Poincar\'e inequality]\label{theo:Poincare}
Let $m:{\mathbb{R}}p{\tau}o{\mathbb{R}}p$ be an arbitrary function satisfying ${\Delta}elta_2$-condition and $\Omega\subset{\mathbb{R}}n$ be a bounded domain,
then there exist $c=c(\Omega,N,m)>0$ such that for every $g\in W^{1,1}(\Omega)$, such that $\int_\Omega m(|\nabla g|)dx<\infty$, we have
\[\int_\Omega m(|g|)dx\leq c
\int_\Omega m(|\nabla g|)dx.\]
\end{theo}
\section{The main proof}
\begin{proof}[Proof of Theorem~{\mathbb{R}}ef{theo:main}] The proof is divided into several steps.
{\tau}extbf{Step 1. Truncated problem.} Existence to a truncated problem
\begin{equation}
\label{prob:trunc}\left\{\begin{split}
-\mathrm{div} A(x,\nabla u_s)= T_s(f) &\qquad \mathrm{ in}\qquad \Omega,\\
u_s(x)=0 &\qquad \mathrm{ on}\qquad \partialrtial\Omega,
\end{split}{\mathbb{R}}ight.
\end{equation}
for $s>0$ is a direct consequence of Theorem~{\mathbb{R}}ef{theo:boundex} with $g=T_s(f)$ (truncation $T_s$ comes from~\eqref{Tk}).
{\tau}extbf{Step 2. A priori estimates.} In order to obtain uniform integrability of sequences $\{A(x,\nabla T_k(u_s))\}_{s>0}$ and $\{\nabla T_k(u_s)\}_{s>0}$ we need to obtain the following a priori estimates.
For $u_s$ being a weak solution to~\eqref{prob:trunc}, $s>0$ and $f\in L^1(\Omega)$, we have the following estimates for any $k>0$
\begin{eqnarray}
\int_\Omega M(x, \nabla T_k(u_s))dx&\leq& c k \|f\|_{L^1(\Omega)},\label{Mapriori}\\
\int_\Omega M^*(x, A(x,\nabla T_k(u_s)))dx&\leq& c k \|f\|_{L^1(\Omega)},\label{M*apriori}
\end{eqnarray}
where the constant $c$ depends only on the growth condition~(A2).
Indeed, considering $(T_k(u_s))_{\delta}elta$ -- a sequence approximating $T_k(u_s)$ as in Theorem~{\mathbb{R}}ef{theo:approx}, we get
\begin{multline*}
\int_\Omega A(x, \nabla T_k(u_s))\nabla T_k(u_s)dx=\lim_{{\delta}elta{\tau}o 0}\int_\Omega A(x, \nabla T_k(u_s))\nabla T_k(u_s)dx= \\
=\lim_{{\delta}elta{\tau}o 0}\int_\Omega T_s(f) ( T_k(u_s))_{\delta}elta dx=\int_\Omega T_s(f) T_k(u_s)dx.
\end{multline*}
We observe that due to Assumption (A2) we have
\begin{multline*}\int_\Omega c_A\left(M(x,\nabla T_k(u_s))+M^*(x,A(x, \nabla T_k(u_s))){\mathbb{R}}ight)dx\leq\\\leq \int_\Omega A(x,\nabla T_k(u_s))\nabla T_k(u_s)dx=\int_\Omega T_s(f) T_k(u_s)dx\leq k \|f\|_{L^1(\Omega)}.\end{multline*}
Estimates \eqref{Mapriori} and \eqref{M*apriori} are direct consequences of the above one. Then, according to Lemma~{\mathbb{R}}ef{lem:unif}, we reach the goal of this step.
{\tau}extbf{Step 3. Controlled radiation.} The proof of this step is a modification of~\cite[Lemma~5.1, Corollary~5.2]{gwiazda-ren-ell}. We consider the $N$-function ${\delta}m:{\mathbb{R}}_+\cup\{0\}{\tau}o{\mathbb{R}}$ defined as follows. Let
\begin{equation}
\label{dm}
m_*(r)=\left(\inf_{x\in\Omega,\ |\xi|=r} M(x,\xi){\mathbb{R}}ight)^{**}.
\end{equation}
Then, let ${\delta}m$ be a solution to the differential equation
\[{\delta}m'(s)=\left\{\begin{array}{ll}
m'_*(s)&{\tau}ext{for }s:\ m'_*(s)\leq {{\cal A }_{l+1}}pha\frac{m_*(s)}{s},\\
{{\cal A }_{l+1}}pha\frac{m_*(s)}{s}&{\tau}ext{for }s:\ m'_*(s)> {{\cal A }_{l+1}}pha\frac{m_*(s)}{s},
\end{array}{\mathbb{R}}ight.\]
with the initial condition ${\delta}m(0)=0=m_*(0)$ and a certain ${{\cal A }_{l+1}}pha>1$. Note that ${\delta}m'(s)\leq m'_*(s)$ for every $s$, so ${\delta}m(s)\leq m_*(s)$. Due to Lemma~{\mathbb{R}}ef{rem:2ndconj} also $ m_*(s)\leq \inf_{x\in\Omega,\ |\xi|=s} M(x,\xi)$ for every $s$. Thus
\[{\delta}m(s)\leq \inf_{x\in\Omega,\ |\xi|=s} M(x,\xi)\qquad\forall_{s\in{\mathbb{R}}_+\cup\{0\}}.\]
Moreover, by~\cite[Chapter~II.2.3, Theorem~3, point~1. (ii)]{rao-ren} ${\delta}m$ satisfies ${\Delta}elta_2$-condition (cf.~\eqref{D2} without dependence on $x$).
\begin{prop}
Suppose $u_s$ is a weak solution to~\eqref{prob:trunc}, $s>0$ and $f\in L^1(\Omega)$. Then there exist $c>0$ and $\gamma:{\mathbb{R}}_+{\tau}o{\mathbb{R}}_+$, such that for every $l>0$
\begin{equation}
\label{a<gamma}
\int_{\{l<|u_s|<l+1\}}A(x,\nabla u_s)\nabla u_s dx\leq \gamma\left(\frac{l}{{\delta}m(l)}{\mathbb{R}}ight),
\end{equation}
and $\gamma$ is independent of $l,s$ and $\lim_{r{\tau}o 0}\gamma(r)=0$.
\end{prop}
\begin{proof}
Note that for ${\delta}m$ given by~\eqref{dm} we have\[|\{|u_s|\geq l\}|=|\{|T_l(u_s)|= l\}|=|\{|T_l(u_s)|\geq l\}|=|\{{\delta}m(|T_l(u_s)|)\geq {\delta}m(l)\}|.\]
Moreover, for $l>0$ we have
\begin{equation*}
\begin{split}
|\{|u_s|\geq l\}|&\leq \int_{\Omega} \frac{{\delta}m(|T_l(u_s)|)}{{\delta}m(l)} dx\leq \frac{c(N,\Omega)}{{\delta}m(l)} \int_\Omega{\delta}m( |\nabla T_l(u_s)|)dx\leq \\
&\leq \frac{c(N,\Omega)}{{\delta}m(l)} \int_\Omega M(x, \nabla T_l(u_s) )dx \leq
\frac{C(M,N,\Omega)}{{\delta}m(l)} \cdot l\|f\|_{L^1(\Omega)}\leq\\&\leq C(f,M,N,\Omega)\frac{l}{{\delta}m(l)}
.\end{split}\end{equation*}
In the above estimates we apply (respectively) the Chebyshev inequality, the Poincar\'{e} inequality (Theorem~{\mathbb{R}}ef{theo:Poincare}), a priori estimate~\eqref{Mapriori} and the facts that $f\in L^1(\Omega)$ and that ${\delta}m$ is an $N$-function (cf.~Definition~{\mathbb{R}}ef{def:Nf}). Thus, there exists $\gamma:{\mathbb{R}}_+{\tau}o{\mathbb{R}}_+$ independent of $l,s$, for which $\lim_{r{\tau}o \infty}\gamma(r)=0$. Moreover, $\int_E|f|\, dx\leq \gamma(|E|).$ In particular,
\begin{equation}
\label{intfgammaul}
\int_{\{|u_s|\geq l\}} |f| dx \leq \gamma\left(\frac{l}{{\delta}m(l)}{\mathbb{R}}ight).
\end{equation}
As for the second assertion let us define $\psi_l:{\mathbb{R}}{\tau}o{\mathbb{R}}$ by
\begin{equation}
\psi_l(r):=
\min\{(l+1-|r|)^+,1\}
\label{psil}\end{equation}
and consider $(\psi_l(u_s))_{\delta}elta$ -- a sequence approximating $\psi_l(u_s)$ as in Theorem~{\mathbb{R}}ef{theo:approx}. Using $(1-\psi_l(u_s))_{\delta}elta$ as a test function in~\eqref{prob:trunc} we get
\begin{multline*}\int_\Omega A(x, \nabla u_s)\nabla (1-\psi_l(u_s))dx=\lim_{{\delta}elta{\tau}o 0} \int_\Omega A(x, \nabla u_s)\nabla (1-\psi_l(u_s))dx=\\
=\lim_{{\delta}elta{\tau}o 0} \int_\Omega T_s(f)(1-\psi_l(u_s))_{\delta}elta dx=
\int_\Omega T_s(f)(1-\psi_l(u_s))dx.\end{multline*}
We notice that the meaning of truncations and the form of $\psi_l$, together with~\eqref{intfgammaul} implies
\[\begin{split}&\int_{\{l<|u_s|<l+1\}} A(x, \nabla u_s)\nabla u_s\,dx=\\
&=\int_{\{l<|u_s|<l+1\}} A(x, \nabla T_{l+1}(u_s))\nabla T_{l+1}(u_s)dx=\int_\Omega A(x, \nabla u_s)\nabla (1-\psi_l(u_s))dx=\\
&=\int_\Omega T_s(f)(1-\psi_l(u_s))dx\leq \int_{\{|u_s|\geq l\}}|f| dx\leq \gamma\left(\frac{l}{{\delta}m(l)}{\mathbb{R}}ight),\end{split}\]
which was the aim.
\end{proof}
{\tau}extbf{Step 4. Convergence of truncations}
\begin{prop}
Suppose an $N$-function $M$ satisfies assumption (M) and function $A$ satisfies assumptions (A1)-(A3). For $s>0$ and $f\in L^1(\Omega)$ let $u_s$ be a weak solution to~\eqref{prob:trunc}. Then there exists a measurable function $u:\Omega{\tau}o{\mathbb{R}}$, such that $T_k(u)\in V_0^M$, being a limit of some subsequence of $\{u_s\}_s$ in the following sense
\begin{eqnarray}
&u_s{\tau}o u &a.e.\ {\tau}ext{in}\ \Omega,\label{conv:usae}\\
&|\{|u|>l\}|\leq \gamma\left(\frac{l}{{\delta}m(l)}{\mathbb{R}}ight),& l\in{\mathbb{N}},\label{conv:umeas}
\end{eqnarray}
and for each $k\in{\mathbb{N}}$ and $s{\tau}o\infty$
\begin{eqnarray}
&T_k(u_s)\xrightarrow{} T_k(u)\quad {\tau}ext{strongly\ in}\ L^p(\Omega)\ {\tau}ext{for}\ p\in[1,\infty),&\label{conv:TuLp}\\
&\nabla T_k(u_s)\xrightharpoonup{} \nabla T_k(u)\quad {\tau}ext{weakly\ in}\ L^1(\Omega),&\label{conv:nTuwL}\\
&\nabla T_k(u_s)\xrightharpoonup{*} \nabla T_k(u)\quad {\tau}ext{weakly}-*\ {\tau}ext{in}\ L_M(\Omega;{\mathbb{R}}n),&\label{conv:nTuwLM}\\
&A(x,\nabla T_k(u_s))\xrightharpoonup{*} A(x,\nabla T_k(u))\quad {\tau}ext{weakly}-*\ {\tau}ext{in}\ L_{M^*}(\Omega;{\mathbb{R}}n).&\label{conv:ATuwLMs}
\end{eqnarray}
\end{prop}
\begin{proof}The proven a priori estimate~\eqref{Mapriori} \begin{eqnarray*}
\int_\Omega M(x, \nabla T_k(u_s))dx&\leq& c k \|f\|_{L^1(\Omega)}
\end{eqnarray*}
implies that for each $k$ the sequence $(T_k(u_s))_{s=1}^{\infty}$ is bounded in $W^{1,1}_0(\Omega)$. Hence, there exists a function $u$ such that
\begin{eqnarray*}
T_k(u_s)&\xrightarrow[s{\tau}o\infty]{}& T_k(u)\ {\tau}ext{strongly in } L^1(\Omega),\\
\nabla T_k(u_s)&\xrightharpoonup[s{\tau}o\infty]{ }& \nabla T_k(u)\ {\tau}ext{weakly in } L^1(\Omega;{\mathbb{R}}n),\\
\nabla T_k(u_s)&\xrightharpoonup[s{\tau}o\infty]{*}& \nabla T_k(u)\ {\tau}ext{weakly-$*$ in } L_M(\Omega;{\mathbb{R}}n),
\end{eqnarray*}
in particular implying~\eqref{conv:nTuwL} and~\eqref{conv:nTuwLM}. Furthermore, the Lebesgue Monotone Convergence Theorem implies
\[
u_s \xrightarrow[s{\tau}o\infty]{} u \quad {\tau}ext{strongly in } L^1(\Omega),\]
and up to a subsequence we have~\eqref{conv:usae}, i.e.
\[
u_s \xrightarrow[s{\tau}o\infty]{} u \quad {\tau}ext{a.e. in } \Omega.\]
Since $\Omega$ is bounded, for fixed $k\in{\mathbb{N}}$ convergence in~\eqref{conv:TuLp} results from uniform integrability in~$L^p(\Omega)$ of bounded functions $T_k(u_s)$ combined with the Vitali Convergence Theorem (Theorem~{\mathbb{R}}ef{theo:VitConv}). Meanwhile, the Dominated Convergence Theorem (due to~\eqref{intfgammaul}) gives~\eqref{conv:umeas}.
On the other hand, if for every $k$ we denote
\[{{\cal A }_{s,k}}=A(x,\nabla T_{k}(u_s(x))),\]
then it follows from~\eqref{M*apriori} that there exists ${\cal{A}}_k\in L_{M^*}(\Omega;{\mathbb{R}}n)$ such that
\begin{equation}
\label{a-conv-ca}{{\cal A }_{s,k}}\xrightharpoonup{*} {\cal{A}}_k \quad {\tau}ext{weakly}-*\ {\tau}ext{in}\ L_{M^*}(\Omega;{\mathbb{R}}n).
\end{equation}
Our aim is now to show that in~\eqref{a-conv-ca}
\begin{equation}
\label{lim=ca}
{\cal{A}}_k(x)=A(x,\nabla T_k(u )).
\end{equation}
We take approximating sequence of smooth functions $\nabla (T_k(u))_{{\delta}elta}\xrightarrow[{\delta}elta{\tau}o 0]{M} \nabla T_k(u)$ (cf. Theorem~{\mathbb{R}}ef{theo:approx}) and show that\begin{equation}
\label{limsup2}
\lim_{l{\tau}o\infty}
\lim_{{\delta}elta{\tau}o 0}\limsup_{s{\tau}o\infty}
\int_\Omega {{\cal A }_{s,l+1}} \psi_l(u_s)\nabla \left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx= 0.
\end{equation}
Testing~\eqref{prob:trunc} by $\varphi=\psi_l(u_s)(T_k(u_s)-(T_k(u))_{\delta}elta),$ where $\psi_l$ is given by~\eqref{psil}, we get
\begin{equation}
\label{cf.C5}
\int_\Omega A(x,\nabla u_s)\nabla \left[ \psi_l(u_s)(T_k(u_s)-(T_k(u))_{\delta}elta){\mathbb{R}}ight]dx=\int_\Omega T_s(f)\psi_l(u_s)(T_k(u_s)-(T_k(u))_{\delta}elta)dx.
\end{equation}
We observe that the right-hand side of~\eqref{cf.C5} tends to zero, i.e.
\[\lim_{l{\tau}o\infty}\lim_{{\delta}elta{\tau}o 0}\lim_{s{\tau}o\infty} \int_\Omega T_s(f)\psi_l(u_s)(T_k(u_s)-(T_k(u))_{\delta}elta)dx=0.\]
Indeed, the convergence a.e. is ensured by~\eqref{conv:usae} and to apply the Lebesgue Dominated Convergence Theorem we note
\[\begin{split}&\lim_{{\delta}elta{\tau}o 0}\lim_{s{\tau}o\infty}\left|\int_\Omega T_s(f)\psi_l(u_s)(T_k(u_s)-(T_k(u))_{\delta}elta)dx{\mathbb{R}}ight|\leq\\
& \leq \lim_{{\delta}elta{\tau}o 0}\lim_{s{\tau}o\infty}\int_\Omega |T_s(f)|\psi_l(u_s)\cdot|T_k(u_s)-T_k(u)|dx+\lim_{{\delta}elta{\tau}o 0}\lim_{s{\tau}o\infty}\int_\Omega |T_s(f)|\psi_l(u_s)\cdot|T_k(u)-(T_k(u))_{\delta}elta|dx\leq \\
&\leq \lim_{{\delta}elta{\tau}o 0}\lim_{s{\tau}o\infty}\int_\Omega |f|\cdot 2 k\,dx+\lim_{{\delta}elta{\tau}o 0}\lim_{s{\tau}o\infty}\int_\Omega |f|\cdot|T_k(u)-(T_k(u))_{\delta}elta|dx=\\
&=2k\|f\|_{L^1(\Omega)}+\lim_{{\delta}elta{\tau}o 0} \int_\Omega |f|\cdot|T_k(u)-(T_k(u))_{\delta}elta|dx.\end{split}
\] The last expression is convergent due to Lemma~{\mathbb{R}}ef{lem:TM1}.
Let us now concentrate on the left-hand side of~\eqref{cf.C5}:
\begin{equation*}
\begin{split}
&\int_\Omega A(x,\nabla u_s)\nabla \left[ \psi_l(u_s)(T_k(u_s)-(T_k(u))_{\delta}elta){\mathbb{R}}ight]dx=\\
=&\int_\Omega A(x,\nabla u_s)\nabla \psi_l(u_s)\left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx+\int_\Omega A(x,\nabla u_s) \psi_l(u_s)\nabla \left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx=\\
=&I_1+I_2,\end{split}
\end{equation*}
where due to~\eqref{a<gamma} we have
\[\begin{split}
\lim_{l{\tau}o\infty}\left(\lim_{{\delta}elta{\tau}o 0} \limsup_{s{\tau}o\infty} |I_1|{\mathbb{R}}ight)&\leq\lim_{l{\tau}o\infty}\left(\lim_{{\delta}elta{\tau}o 0} \limsup_{s{\tau}o\infty}\int_{\{l<|u_s|<l+1\}} A(x,\nabla u_s)\nabla u_s|T_k(u_s)-T_k(u) |dx{\mathbb{R}}ight)+\\&+\lim_{l{\tau}o\infty}\left(\lim_{{\delta}elta{\tau}o 0} \limsup_{s{\tau}o\infty}\int_{\{l<|u_s|<l+1\}} A(x,\nabla u_s)\nabla u_s|T_k(u)-(T_k(u))_{\delta}elta|dx{\mathbb{R}}ight)=\\
&= II_1+II_2.\end{split}\]
Moreover,
\[\begin{split}II_1&\leq \lim_{l{\tau}o\infty}\left(2k\limsup_{s{\tau}o\infty}\int_{\{l<|u_s|<l+1\}} A(x,\nabla u_s)\nabla u_sdx{\mathbb{R}}ight)\leq \lim_{l{\tau}o\infty}\left[2k\gamma\left(\frac{l}{{\delta}m(l)}{\mathbb{R}}ight){\mathbb{R}}ight]=0,\end{split}\]
meanwhile the convergence of $II_2$ results from Lemma~{\mathbb{R}}ef{lem:TM1}.
Then passing to the limit in~\eqref{cf.C5} we obtain\begin{equation}
\label{limsup}
\lim_{l{\tau}o\infty}\lim_{{\delta}elta{\tau}o 0}\limsup_{s{\tau}o\infty} I_2
=\lim_{l{\tau}o\infty}\lim_{{\delta}elta{\tau}o 0}\limsup_{s{\tau}o\infty}
\int_\Omega A(x,\nabla u_s) \psi_l(u_s)\nabla \left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx= 0.
\end{equation}
Then~\eqref{limsup} is equivalent to~\eqref{limsup2}.
Before we apply monotonicity trick, we need to show that
\begin{equation}
\label{limtrunc}
\lim_{l{\tau}o\infty}\lim_{{\delta}elta{\tau}o 0}\limsup_{s{\tau}o\infty}
\int_\Omega {{\cal A }_{s,k}} \psi_l(u_s)\nabla \left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx= 0.
\end{equation}
Taking into account~\eqref{limsup2}, the equality~\eqref{limtrunc} will be proven when the following expression is shown to tend to $0$ (still $k\leq l$)\begin{equation}
\begin{split}
&III=\int_\Omega ({{\cal A }_{s,k}} -{{\cal A }_{s,l+1}}) \psi_l(u_s)\nabla \left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx=\\
=&\int_{\Omega} ({{\cal A }_{s,l+1}} -A(x,0)) \mathds{1}_{\{k<|u_s| \}}\psi_l(u_s) \nabla (T_k(u))_{\delta}elta dx=\\
=&\int_{\Omega} {{\cal A }_{s,l+1}} \mathds{1}_{\{k<|u_s| \}} \psi_l(u_s) \nabla (T_k(u))_{\delta}elta dx.
\end{split}
\end{equation}
We prove that
\begin{equation}\label{asl<al}
\begin{split}
\lim_{{\delta}elta{\tau}o 0}\limsup_{s{\tau}o\infty} |III|&\leq \lim_{{\delta}elta{\tau}o 0}\limsup_{s{\tau}o\infty}\int_{\Omega} |{{\cal A }_{s,l+1}}| \mathds{1}_{\{k<|u_s| \}} \psi_l(u_s) |\nabla (T_k(u))_{\delta}elta|\, dx\leq \\
& \leq \lim_{{\delta}elta{\tau}o 0} \int_{\Omega} |{\cal A}_{l+1}| \mathds{1}_{\{k<|u| \}} \psi_l(u) |\nabla (T_k(u))_{\delta}elta|\, dx.
\end{split}
\end{equation}
For this we will use Lemma~{\mathbb{R}}ef{lem:TM1} with
\[w^s=|{{\cal A }_{s,l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|\xrightharpoonup[s{\tau}o\infty]{L^1(\Omega)}|{\cal A}_{l+1}|\cdot|\nabla (T_k(u))_{\delta}elta|=w.\]
The convergence $w^s\xrightharpoonup{} w$ is a consequence of~\eqref{a-conv-ca}. Let
$v^s=\mathds{1}_{\{k<|u_s| \}}\psi_l(u_s)$ and $v^s_{\varepsilon}\in C(\Omega)\cap L^\infty (\Omega)$ with ${\varepsilon}\geq 0$ be given by
\[v^s_{\varepsilon}=\left\{\begin{array}{ll}
1 & k<|u_s|<l,\\
{\tau}ext{affine} & k-{\varepsilon}\leq |u_s|\leq k,\ l\leq |u_s|\leq l+1\\
0 & |u_s|<k-{\varepsilon},\ |u_s|>l+1.\\
\end{array}{\mathbb{R}}ight.\]
Notice that for $s{\tau}o\infty$ and every ${\varepsilon}>0$, due to continuity of $v_{\varepsilon}^s$, we have
\[v^s_{\varepsilon}\xrightarrow{a.e.}v_{\varepsilon}:=\left\{\begin{array}{ll}
1 & k<|u|<l,\\
{\tau}ext{affine} & k-{\varepsilon}\leq |u|\leq k,\ l\leq |u|\leq l+1\\
0 & |u|<k-{\varepsilon},\ |u|>l+1.\\
\end{array}{\mathbb{R}}ight.\]
Furthermore, for every $s$ we have
\begin{equation}
\label{g-ep-s}
\int_\Omega |{{\cal A }_{s,l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|v^s\,dx\leq\int_\Omega |{{\cal A }_{s,l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|v^s_{\varepsilon}\,dx.
\end{equation}
Since $v_{\varepsilon}\in L^\infty (\Omega)$, Lemma~{\mathbb{R}}ef{lem:TM1} yields $\int_\Omega w^s v^s_{\varepsilon}\, dx{\tau}o \int_\Omega w v_{\varepsilon}\, dx,$ that is
\[\lim_{s{\tau}o\infty}\int_\Omega |{{\cal A }_{s,l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|v^s_{\varepsilon}\,dx=\int_\Omega |{{\cal A }_{l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|v_{\varepsilon}\,dx.\]
The Lebesgue Monotone Convergence Theorem implies
\begin{equation}
\label{g-ep-ve}\lim_{{\varepsilon}{\tau}o 0} \int_\Omega |{{\cal A }_{l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|v_{\varepsilon}\,dx=\int_\Omega |{{\cal A }_{l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|v_0\,dx=\int_\Omega |{{\cal A }_{l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|\mathds{1}_{\{k<|u| \}}\psi_l(u)\,dx.
\end{equation}
Thus~\eqref{g-ep-s} together with~\eqref{g-ep-ve} give
\[\begin{split}
\limsup_{s{\tau}o \infty} \int_\Omega |{{\cal A }_{s,l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|g^s\,dx\leq\int_\Omega |{{\cal A }_{l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|\mathds{1}_{\{k<|u| \}}\psi_l(u)\,dx\end{split}\]
and we get~\eqref{asl<al}.
Our aim now is to prove
\begin{equation}
\label{al-delta}\lim_{{\delta}elta{\tau}o 0}\int_\Omega |{{\cal A }_{l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|\mathds{1}_{\{k<|u| \}}\psi_l(u)\,dx=0
\end{equation}
Recall that $\nabla (T_k (u))_{\delta}elta\xrightarrow{M}\nabla T_k (u)$. Therefore by Definition~{\mathbb{R}}ef{def:convmod} ii), the sequence $\{M(x,\nabla(T_k (u))_{\delta}elta/\lambda)\}_{\delta}elta$ is uniformly bounded in $L^1(\Omega;{\mathbb{R}}n)$ for some $\lambda$ and consequently, by~Lemma~{\mathbb{R}}ef{lem:unif} $\{\nabla(T_k (u))_{\delta}elta\}_{\delta}elta$ is uniformly integrable. Hence the Vitali Convergence Theorem (Theorem~{\mathbb{R}}ef{theo:VitConv}) gives
\[\lim_{{\delta}elta{\tau}o 0}\int_\Omega |{{\cal A }_{l+1}}|\cdot|\nabla (T_k(u))_{\delta}elta|\mathds{1}_{\{k<|u| \}}\psi_l(u)\,dx=\int_\Omega |{{\cal A }_{l+1}}|\cdot|\nabla T_k(u) |\mathds{1}_{\{k<|u| \}}\psi_l(u)\,dx,\]
which is equal to zero, because $T_k(u) |\mathds{1}_{\{k<|u| \}}=0$. Thus~\eqref{al-delta} and~\eqref{limtrunc} hold.
We observe that we can remove $\psi_l(u_s)$ from~\eqref{limtrunc}. Indeed, notice that for $l\geq k$ due to~Lemma~{\mathbb{R}}ef{lem:M*<M} we have
\begin{equation*}
\begin{split}
&\int_\Omega {{\cal A }_{s,k}} \psi_l(u_s)\nabla \left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx=\\
&=\int_\Omega {{\cal A }_{s,k}} \nabla \left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx-\int_{\{|u_s|>l\}} A(x,0) (\psi_l(u_s)-1)\nabla (T_k(u))_{\delta}elta dx=\\
&=\int_\Omega {{\cal A }_{s,k}} \nabla \left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx.\end{split}
\end{equation*}
Therefore,~\eqref{limtrunc} is equivalent to
\begin{equation}\label{lim<<}
\lim_{{\delta}elta{\tau}o 0}\limsup_{s{\tau}o\infty}\int_\Omega {{\cal A }_{s,k}}\nabla \left[T_k(u_s)-(T_k(u))_{\delta}elta{\mathbb{R}}ight]dx= 0.
\end{equation}
Now we apply the Minty-Browder monotonicity trick. Since~\eqref{a-conv-ca}, then for each ${\delta}elta$
\begin{equation}
\label{lim<A}\lim_{s{\tau}o\infty}
\int_\Omega {{\cal A }_{s,k}} \nabla (T_k(u))_{\delta}elta dx= \int_\Omega {\cal A}_k\cdot \nabla (T_k(u))_{\delta}elta dx.
\end{equation}
Then~\eqref{lim<<} together with~\eqref{lim<A} imply
\begin{equation}
\label{lim<A'} \limsup_{s{\tau}o\infty}
\int_\Omega {{\cal A }_{s,k}} \nabla T_k(u_s) dx=\lim_{{\delta}elta{\tau}o 0} \int_\Omega {\cal A}_k\cdot \nabla (T_k(u))_{\delta}elta dx=\int_\Omega {\cal A}_k\cdot \nabla T_k(u) dx,
\end{equation}
where the last equality is obtained analogically as~\eqref{al-delta}.
Monotonicity of~$A$ results in
\[\int_\Omega {{\cal A }_{s,k}} \nabla T_k(u_s) dx\geq \int_\Omega {{\cal A }_{s,k}} \eta\ dx+\int_\Omega A(x,\eta) (\nabla T_k(u_s)-\eta)\ dx\]
for any $\eta\in {\mathbb{R}}n$. Taking upper limit with ${s{\tau}o\infty}$ above (due to~\eqref{lim<A'},~\eqref{a-conv-ca}, and~\eqref{conv:nTuwLM}) we get
\begin{equation*}
\int_\Omega {\cal A}_k\cdot \nabla T_k(u) dx\geq \int_\Omega {\cal A}_k\cdot \eta\ dx+\int_\Omega A(x,\eta) (\nabla T_k(u)-\eta)\ dx.\end{equation*}
Note that it is equivalent to
\begin{equation}
\label{Ak-mon} \int_\Omega( {\cal A}_k- A(x,\eta) )( \nabla T_k(u)-\eta) dx\geq 0.
\end{equation}
Let us define\begin{equation}
\label{omm}
\Omega_K=\{x\in\Omega:\ |\nabla T_k(u)|\leq K\}.
\end{equation}
Then, in~\eqref{Ak-mon} we choose
\[\eta=\nabla T_k(u)\mathds{1}_{\Omega_i}+hz\mathds{1}_{\Omega_j},\]
where $0<j<i$, $h\in{\mathbb{R}}p$ and $z\in L^\infty(\Omega;{\mathbb{R}}n)$, to get
\begin{equation*}
\int_\Omega( {\cal A}_k- A(x,\nabla T_k(u)\mathds{1}_{\Omega_i}+hz\mathds{1}_{\Omega_j}) )( \nabla T_k(u)-\nabla T_k(u)\mathds{1}_{\Omega_i}-hz\mathds{1}_{\Omega_j}) dx\geq 0.
\end{equation*}
Notice that it is equivalent to
\begin{equation}
\label{Ak-po-mon} \int_{\Omega\setminus\Omega_i} {\cal A}_k\nabla T_k(u)dx- \int_{\Omega\setminus\Omega_i}A(x,0)\nabla T_k(u)dx +h\int_{ \Omega_j} (A(x,\nabla T_k(u)+hz)-{\cal A}_k)z dx\geq 0.
\end{equation}
The first and the second expression above tend to zero when $i{\tau}o\infty.$ Indeed, since ${\cal A}_k,A(x,0)\in L_{M^*}(\Omega;{\mathbb{R}}n)$ and $\nabla T_k(u)\in L_M(\Omega;{\mathbb{R}}n)$, the H\"older inequality~\eqref{inq:Holder} gives boudedness of integrands in $L^1(\Omega)$. Then we take into account shrinking domain of integration to get the desired convergence to $0$. In~particular, we can erase these expressions in~\eqref{Ak-po-mon} and divide the remaining expression by $h>0$, to obtain
\begin{equation*} \int_{ \Omega_j} (A(x,\nabla T_k(u)+hz)-{\cal A}_k)z dx\geq 0.
\end{equation*}
Note that
\[A(x,\nabla T_k(u)+hz)\xrightarrow[h{\tau}o 0]{}A(x,\nabla T_k(u))\quad{\tau}ext{a.e. in}\quad \Omega_j.\]
Moreover, as $A(x,\nabla T_k(u)+hz)$ is bounded on $\Omega_j$, Lemma~{\mathbb{R}}ef{lem:M*<M} results in
\[\int_{ \Omega_j} M^*\left(x,A(x,\nabla T_k(u)+hz){\mathbb{R}}ight)dx\leq \frac{2}{c_A} \sup_{h\in(0,1)} \int_{ \Omega_j} M\left(x,\frac{2}{c_A}A(x,\nabla T_k(u)+hz){\mathbb{R}}ight)dx.\]
The right-hand side is bounded, because $(\nabla T_k(u)+hz)_h$ is uniformly bounded in $L^\infty(\Omega_j;{\mathbb{R}}n)\subset L_M(\Omega;{\mathbb{R}}n)$ (cf.~\eqref{omm} and~\eqref{LinfinLM}). Hence, Lemma~{\mathbb{R}}ef{lem:unif} gives uniform integrability of $\left(A(x,\nabla T_k(u)+hz){\mathbb{R}}ight)_h$. When we notice that $|\Omega_j|<\infty$, we can apply the Vitali Convergence Theorem (Theorem~{\mathbb{R}}ef{theo:VitConv}) to get
\[A(x,\nabla T_k(u)+hz)\xrightarrow[h{\tau}o 0]{}A(x,\nabla T_k(u))\quad{\tau}ext{in}\quad L^1(\Omega_j;{\mathbb{R}}n).\]
Thus
\begin{equation*} \int_{ \Omega_j} (A(x,\nabla T_k(u)+hz)-{\cal A}_k)z dx\xrightarrow[h{\tau}o 0]{} \int_{ \Omega_j} (A(x,\nabla T_k(u))-{\cal A}_k)z dx.
\end{equation*}
Consequently,
\begin{equation*} \int_{ \Omega_j} (A(x,\nabla T_k(u))-{\cal A}_k)z dx\geq 0,
\end{equation*}
for any $z\in L^\infty(\Omega;{\mathbb{R}}n)$. Let us take
\[z=\left\{\begin{array}{ll}-\frac{A(x,\nabla T_k(u))-{\cal A}_k}{|A(x,\nabla T_k(u))-{\cal A}_k|}&\ {\tau}ext{if}\quad A(x,\nabla T_k(u))-{\cal A}_k\neq 0,\\
0&\ {\tau}ext{if}\quad A(x,\nabla T_k(u))-{\cal A}_k\neq 0.
\end{array}{\mathbb{R}}ight.\]
We obtain
\begin{equation*} \int_{ \Omega_j} |A(x,\nabla T_k(u))-{\cal A}_k| dx\leq 0,
\end{equation*}
hence \[A(x,\nabla T_k(u))={\cal A}_k\qquad {\tau}ext{a.e.}\quad{\tau}ext{in}\quad \Omega_j.\]
Since $j$ is arbitrary, we have the equality a.e. in $\Omega$ and~\eqref{lim=ca} is satisfied.
\end{proof}
{\tau}extbf{Step 5. Renormalized solutions.}
We aim at proving that $u$ is a renormalized solution (see Introduction). At first we observe that $u$ satisfies (R1) and concentrate on (R2).
Since $T_k(u)\in V_0^M\cap L^\infty(\Omega)$, Theorem~{\mathbb{R}}ef{theo:approx} ensures that there exists a sequence $\{u_r\}_r\subset C_0^\infty(\Omega)$ indexed with $r{\tau}o \infty$, such that
\begin{eqnarray*}
&u_r\xrightarrow{} u\quad {\tau}ext{a.e.\ in}\ \Omega,&\\
&\nabla T_k(u_r)\xrightharpoonup{*} \nabla T_k(u)\quad {\tau}ext{weakly}-*\ {\tau}ext{in}\ L_M(\Omega;{\mathbb{R}}n),&\\
&\nabla h(u_r)\xrightharpoonup{*} \nabla h(u)\quad {\tau}ext{weakly}\ {\tau}ext{in}\ L_{M}(\Omega),&
\end{eqnarray*}
where $h\in C_c^1({\mathbb{R}})$ is arbitrary.
We test~\eqref{prob:trunc} by $\psi_l(u_s)h(u_r)\phi$ with $\phi\in W^{1,\infty}_0(\Omega)$ and get
\[L_{s,r,l}=\int_\Omega A(x,\nabla u_s) \nabla [\psi_l(u_s)h(u_r)\phi]dx= \int_\Omega T_s(f) \psi_l(u_s)h(u_r)\phi\,dx=R_{s,r,l}.\]
We notice at first that due to the Lebesgue Dominated Convergence Theorem it holds that
\[\lim_{l{\tau}o\infty}\lim_{r{\tau}o \infty}\limsup_{s{\tau}o\infty} R_{s,r,l}=\int_\Omega f h(u)\phi dx. \]
Meanwhile on the left-hand side\[L_{s,r,l}=\int_\Omega A(x,\nabla u_s) \nabla \psi_l(u_s)h(u_r)\phi dx+\int_\Omega A(x,\nabla u_s) \psi_l(u_s)\nabla [h(u_r)\phi]dx=L^1_{s,r,l}+L^2_{s,r,l},\]
where
\[\lim_{l{\tau}o\infty}\lim_{r{\tau}o \infty}\limsup_{s{\tau}o\infty}|L^1_{s,r,l}|\leq \|h\|_{L^\infty(\Omega)}\|\phi\|_{L^\infty(\Omega)}\lim_{l{\tau}o\infty}\lim_{r{\tau}o \infty} \left(\sup_{s}\int_{\{l<|u_s|<l+1\}} A_{s,l}(x) \nabla T_{l+1} (u_s) dx{\mathbb{R}}ight)=0\]
due to~\eqref{a<gamma}. As for $L^2_{s,r,l}$ we notice that when $s{\tau}o\infty$, up to a subsequence,
\begin{equation*}{{\cal A }_{s,l+1}}\xrightharpoonup{ } A(x,\nabla T_{l+1}(u))\quad {\tau}ext{weakly} \ {\tau}ext{in}\ L^{1}(\Omega).
\end{equation*}
Indeed, a priori estimate~\eqref{M*apriori} combined with Lemma~{\mathbb{R}}ef{lem:unif} give uniform integrability. Then, taking into account weak-* convergence~\eqref{conv:ATuwLMs}, the Dunford-Pettis Theorem (Theorem~{\mathbb{R}}ef{theo:dunf-pet}) ensures weak $L^1$-convergence up to a subsequence.
Moreover, note that\begin{eqnarray*}
&|\psi_l(u_s)|\leq 1,&\\
&\nabla (h(u_r)\phi)\in L^\infty(\Omega;{\mathbb{R}}n).&
\end{eqnarray*} and for $s{\tau}o\infty$
\[\psi_l(u_s)\xrightarrow{} \psi_l(u)\quad {\tau}ext{a.e.\ in}\ \Omega.\]
The sequence $\{ A(x,\nabla u_s) \psi_l(u_s)\nabla [h(u_r)\phi] \}_s$ is uniformly integrable.
Due to the consequence of Chacon's Biting Lemma, Theorem~{\mathbb{R}}ef{theo:bitinglemma}, we notice that \[\limsup_{r{\tau}o\infty}\limsup_{s{\tau}o\infty}\int_\Omega {{\cal A }_{s,l+1}}\nabla h(u_r)\psi_l(u_s)dx=\int_\Omega A(x,\nabla T_{l+1}(u )) \psi_l(u )\nabla [h(u )\phi]dx.\]
Since $\mathrm{supp}\, h(u)\subset[-m,m]$ for some $m\in{\mathbb{N}}$ and we can consider only $l>m+1$. Then
\[\lim_{l{\tau}o\infty}\limsup_{r{\tau}o\infty}\limsup_{s{\tau}o\infty}L^2_{s,r,l}
=\lim_{l{\tau}o\infty}\int_\Omega A(x,\nabla T_{l+1}(u )) \psi_l(u )\nabla [h(u )\phi]dx=\int_\Omega A(x,\nabla u ) \nabla [h(u )\phi]dx.\]
and our solution $u$ satisfies condition (R2).
Let us consider radiation control condition (R3), i.e.
\[\int_{\{l<|u|<l+1\}}A(x,\nabla u)\cdot\nabla u\, dx=\int_{\{l<|u|<l+1\}}A(x,\nabla T_{l+1}( u))\cdot\nabla T_{l+1}(u )\, dx\xrightarrow[l{\tau}o\infty]{} 0 .\]
We follow the ideas of~\cite{gwiazda-ren-para} involving the Chacon Biting Lemma and the Young measure approach to show that for $s{\tau}o\infty$ it holds that
\begin{equation}
\label{117} {{\cal A }_{s,l+1}}\cdot\nabla T_{l+1}(u_s)\xrightharpoonup{} A(x,\nabla T_{l+1}(u) )\cdot \nabla T_{l+1}(u)\qquad {\tau}ext{weakly in } L^1(\Omega).
\end{equation}
First we observe that the sequence $\{[{{\cal A }_{s,l+1}}-A(x,\nabla T_{l+1}(u ))]\cdot [ \nabla T_{l+1}(u_s)-\nabla T_{l+1}(u )]\}_s$ is uniformly bounded in $L^1(\Omega)$. Indeed,
\[\begin{split}&\int_\Omega[{{\cal A }_{s,l+1}}-A(x,\nabla T_{l+1}(u ))]\cdot [ \nabla T_{l+1}(u_s)-\nabla T_{l+1}(u )]dx\leq\\
&\qquad\leq \int_\Omega {{\cal A }_{s,l+1}} \nabla T_{l+1}(u_s) dx
+\int_\Omega {{\cal A }_{s,l+1}} \nabla T_{l+1}(u ) dx+\\
&\qquad+\int_\Omega A(x,\nabla T_{l+1}(u )) \nabla T_{l+1}(u_s) dx+\int_\Omega A(x,\nabla T_{l+1}(u )) \nabla T_{l+1}(u ) dx=IV_1+IV_2+IV_3+IV_4,
\end{split}\]
where $IV_1$ is uniformly bounded due to~\eqref{a<gamma} and $IV_4$ is independent of $s$. As for $IV_2$ we note
\[\begin{split}
\limsup_{s{\tau}o\infty}IV_2&\leq\lim_{{\delta}elta{\tau}o 0}\limsup_{s{\tau}o\infty}\int_\Omega {{\cal A }_{s,l+1}} (\nabla T_{l+1}(u)- \nabla( T_{l+1}(u))_{\delta}elta) dx +\lim_{{\delta}elta{\tau}o 0}\limsup_{s{\tau}o\infty}\int_\Omega {{\cal A }_{s,l+1}} \nabla (T_{l+1}(u) )_{\delta}elta dx\leq\\
&= 0+\lim_{{\delta}elta{\tau}o 0} \int_\Omega {{\cal A }_{l+1}} \nabla (T_{l+1}(u) )_{\delta}elta dx=\int_\Omega {{\cal A }_{l+1}} \nabla T_{l+1}(u) dx,\end{split}\]
where we applied~\eqref{lim<<},~\eqref{conv:ATuwLMs}, and then~\eqref{lim<A'}. Moreover, in the case of $IV_3$ the Fenchel-Young inequality and~\eqref{Mapriori} gives boundedness.
Then monotonicity of $A(x,\cdot)$ and Theorem~{\mathbb{R}}ef{theo:bitinglemma} give, up to~a~subsequence, convergence
\begin{equation}
\label{110}
\begin{split}0&\leq [{{\cal A }_{s,l+1}}-A(x,\nabla T_{l+1}(u ))]\cdot [\nabla T_{l+1}(u_s)-\nabla T_{l+1}(u )]\\
&\xrightarrow{b}\int_{\mathbb{R}}n [A(x,\lambda)-A(x,\nabla T_{l+1}(u ))]\cdot [\lambda-\nabla T_{l+1}(u )]d\nu_{x}(\lambda),\end{split}
\end{equation}
where $\nu_{x}$ denotes the Young measure generated by the sequence $\{\nabla T_{l+1}(u_s)\}_s$.
Since $\nabla T_{l+1}(u_s)\xrightharpoonup{}\nabla T_{l+1}(u )$ in $L^1(\Omega)$, we have $\int_{\mathbb{R}}n\lambda \, d\nu_x(\lambda)=\nabla T_{l+1}(u)$ for a.e. $x\in\Omega$. Then
\[\int_{\mathbb{R}}n {{\cal A }_{s,l+1}} \cdot [\lambda-\nabla T_{l+1}(u )]d\nu_{x}(\lambda)=0\]
and the limit in~\eqref{110} is equal for a.e. $x\in\Omega$ to
\begin{equation}
\label{111} \int_{\mathbb{R}}n [A(x,\lambda)-A(x,\nabla T_{l+1}(u ))]\cdot [\lambda-\nabla T_{l+1}(u )]d\nu_{x}(\lambda)=
\int_{\mathbb{R}}n A(x,\lambda) \cdot \lambda\, d\nu_{x}(\lambda)-\int_{\mathbb{R}}n A(x,\lambda) \cdot \nabla T_{l+1}(u )d\nu_{x}(\lambda).
\end{equation}
Uniform boundedness of the sequence $\{ {{\cal A }_{s,l+1}}\nabla T_{l+1}(u_s) \}_s$ in $L^1(\Omega)$ (cf.~\eqref{a<gamma}) enables us to apply once again Theorem~{\mathbb{R}}ef{theo:bitinglemma} to obtain
\begin{equation*}
{{\cal A }_{s,l+1}}\nabla T_{l+1}(u_s)\xrightarrow{b} \int_{\mathbb{R}}n A(x,\lambda) \cdot \lambda\,d\nu_{x}(\lambda).
\end{equation*}
Moreover, assumption (A2) implies ${{\cal A }_{s,l+1}}\nabla T_{l+1}(u_s)\geq 0$. Therefore, due to~\eqref{111} and~\eqref{110}, we have
\[\limsup_{s{\tau}o\infty} A(x,\nabla T_{l+1}(u_s) ) \nabla T_{l+1}(u_s)\geq \int_{\mathbb{R}}n A(x,\lambda) \cdot \lambda\,d\nu_{x}(\lambda).\]
Taking into account that in~\eqref{lim<A'} we can put ${\cal A}_k=A(x,\nabla T_{l+1}(u) )=\int_{\mathbb{R}}n A(x,\lambda) \,d\nu_{x}(\lambda)$, the above expression implies\[\nabla T_{l+1}(u) \int_{\mathbb{R}}n A(x,\lambda) \,d\nu_{x}(\lambda)\geq \int_{\mathbb{R}}n A(x,\lambda) \cdot \lambda\,d\nu_{x}(\lambda).\]
When we apply it, together with~\eqref{111}, the limit in~\eqref{110} is non-positive. Hence,
\begin{equation*} [{{\cal A }_{s,l+1}}-A(x,\nabla T_{l+1}(u ))]\cdot [\nabla T_{l+1}(u_s)-\nabla T_{l+1}(u )]\xrightarrow{b} 0.
\end{equation*}
Observe further that $A(x,\nabla T_{l+1}(u ))\in L_{M^*}(\Omega;{\mathbb{R}}n)$ and we can choose ascending family of sets $E^{l+1}_j$, such that $| E^{l+1}_j|{\tau}o 0$ for $j{\tau}o \infty$ and $A(x,\nabla T_{l+1}(u ))\in L^\infty(\Omega\setminus E^{l+1}_j).$ Then, since $\nabla T_{l+1}(u_s)\xrightharpoonup{}\nabla T_{l+1}(u )$, we get\begin{equation*} A(x,\nabla T_{l+1}(u )) \cdot [\nabla T_{l+1}(u_s)-\nabla T_{l+1}(u )]\xrightarrow{b} 0
\end{equation*}
and similarly we conclude
\begin{equation*} {{\cal A }_{s,l+1}}\cdot\nabla T_{l+1}(u )\xrightarrow{b} A(x,\nabla T_{l+1}(u))\cdot\nabla T_{l+1}(u ).
\end{equation*}
Summing it up we get
\begin{equation*}
{{\cal A }_{s,l+1}}\cdot\nabla T_{l+1}(u_s )\xrightarrow{b} A(x,\nabla T_{l+1}(u))\cdot\nabla T_{l+1}(u ).
\end{equation*}
Recall that Theorem~{\mathbb{R}}ef{theo:bitinglemma} together with~\eqref{lim<A'} and~\eqref{conv:ATuwLMs} results in~\eqref{117}.
We turn back to prove (R3). Note that $\nabla u_s=0$ a.e. in $\{x\in\Omega:|u_s|\in\{l,l+1\}\}$. Then~\eqref{a<gamma} implies
\begin{equation*}
\lim_{l{\tau}o\infty}\sup_{s>0}\int_{\{l-1<|u_s|<l+2\}}A(x,\nabla u_s)\cdot\nabla u_s\,dx=0.
\end{equation*}
For $g_l:{\mathbb{R}}{\tau}o{\mathbb{R}}$ defined by
\[g_l(r)=\left\{\begin{array}{ll}1&{\tau}ext{if }\ l\leq |r|\leq l+1,\\
0&{\tau}ext{if }\ |r|<l-1{\tau}ext{ or } |r|> l+2,\\
{\tau}ext{is affine} &{\tau}ext{otherwise},
\end{array}{\mathbb{R}}ight.\]
we have
\begin{equation}
\label{128}
\int_{\{l-1<|u|<l+2\}}A(x,\nabla u)\cdot\nabla u\,dx\leq \int_{\Omega}g_l(u)A(x,\nabla T_{l+2}( u))\cdot\nabla T_{l+2}( u)\,dx.
\end{equation}
Let us remind that we know that $u_s{\tau}o u$ a.e. in $\Omega$ (cf.~\eqref{conv:usae}) and $|\{x:|u_s|>l\}|\leq\gamma\left(l/{\delta}m(l){\mathbb{R}}ight)$ (cf.~\eqref{conv:umeas}). Moreover, we have weak convergence~\eqref{117}, $A(x,\nabla T_{l+2}( u_s))\cdot\nabla T_{l+2}( u_s)>0$ and function $g_l$ is continuous and bounded. Thus, we infer that we can estimate the limit of the right-hand side of~\eqref{128} in the following way
\[\begin{split}
0&\leq \lim_{l{\tau}o\infty} \int_{\{l-1<|u|<l+2\}}A(x,\nabla u)\cdot\nabla u\,dx\leq \lim_{l{\tau}o\infty}\int_{\Omega}g_l(u)A(x,\nabla T_{l+2}( u))\cdot\nabla T_{l+2}( u)\,dx=\\
&= \lim_{l{\tau}o\infty} \lim_{s{\tau}o\infty} \int_{\Omega}g_l(u)A(x,\nabla T_{l+2}(u_s))\cdot\nabla T_{l+2}(u_s)\,dx\leq\\
&\leq \lim_{l{\tau}o\infty} \lim_{s{\tau}o\infty}\int_{\{l-1<|u|<l+2\}}A(x,\nabla T_{l+2}(u_s))\cdot\nabla T_{l+2}(u_s)\,dx=0,
\end{split}\]
where the last equality comes from~\eqref{a<gamma}.
Hence, our solution $u$ satisfies condition (R3) and is a renormalized solution.
\end{proof}
\section*{Appendix A}
\begin{defi}[$N$-function]\label{def:Nf} Suppose $\Omega\subset{\mathbb{R}}n$ is an open bounded set. A~function $M:\Omega{\tau}imes{\mathbb{R}}n{\tau}o{\mathbb{R}}$ is called an $N$-function if it satisfies the
following conditions:
\begin{enumerate}
\item $ M$ is a Carath\'eodory function (i.e. measurable with respect to $x$ and continuous with respect to the last variable), such that $M(x,\xi) = 0$ if and only if $\xi = 0$; and $M(x,\xi) = M(x, -\xi)$ a.e. in $\Omega$,
\item $M(x,\xi)$ is a convex function with respect to $\xi$,
\item $\lim_{|\xi|{\tau}o 0}\mathrm{ess\,sup}_{x\in\Omega}\frac{M(x,\xi)}{|\xi|}=0$,
\item $\lim_{|\xi|{\tau}o \infty}\mathrm{ess\,inf}_{x\in\Omega}\frac{M(x,\xi)}{|\xi|}=\infty$.
\end{enumerate}
\end{defi}
\begin{defi}[Complementary function] \label{def:conj}
The complementary~function $M^*$ to a function $M:\Omega{\tau}imes{\mathbb{R}}n{\tau}o{\mathbb{R}}$ is defined by
\[M^*(x,\eta)=\sup_{\xi\in{\mathbb{R}}n}(\xi\cdot\eta-M(x,\xi)),\qquad \eta\in{\mathbb{R}}n,\ x\in\Omega.\]
\end{defi}
\begin{rem}\label{rem:f*<g*}
If $f(x,\xi)\leq g(x,\xi)$, then $g^*(x,\xi)\leq f^*(x,\xi)$.
\end{rem}
\begin{rem} If $M$ is an $N$-function and $M^*$ its complementary, we have\begin{itemize}
\item the Fenchel-Young inequality \begin{equation}
\label{inq:F-Y}|\xi\cdot\eta|\leq M(x,\xi)+M^*(x,\eta)\qquad \mathrm{for\ all\ }\xi,\eta\in{\mathbb{R}}n\mathrm{\ and\ }x\in\Omega.
\end{equation}
\item the generalised H\"older's inequality \begin{equation}
\label{inq:Holder}
\left|\int_{\Omega} \xi\cdot\eta\,dx{\mathbb{R}}ight|\leq 2\|\xi\|_{L_M }\|\eta\|_{L_{M^*} }\quad \mathrm{for\ all\ }\xi\in L_M(\Omega;{\mathbb{R}}n),\eta\in L_{M^*}(\Omega;{\mathbb{R}}n).
\end{equation}
\end{itemize}
\end{rem}
\begin{lem}\label{lem:M*<M} Suppose $M$ and $A$ are such that (A2) is satisfied, then
\begin{equation*}
\int_\Omega M^*(x,A(x,\eta))dx\leq \frac{2}{c_A}\int_\Omega M\left(x, \frac{2}{c_A}\eta {\mathbb{R}}ight)dx\quad{\tau}ext{for}\quad \eta\in L^\infty(\Omega;{\mathbb{R}}n).\end{equation*}
\end{lem}
\begin{proof}
Since $M^*$ is convex, $M^*(x,0)=0$ and $c_A\in (0,1]$, we notice that
\[M^*\left(x,\frac{c_A}{2}A\left(x,\eta{\mathbb{R}}ight){\mathbb{R}}ight)\leq \frac{c_A}{2}M^*(x,A(x,\eta)).\]
Taking this into account together with~(A2) and~\eqref{inq:F-Y} we have
\[\begin{split}
c_A\left(M(x,\eta)+M^*(x,A(x,\eta)){\mathbb{R}}ight)\leq \frac{c_A}{2} A(x,\eta)\cdot \frac{2}{c_A}\eta &\leq M\left(x,\frac{2}{c_A}\eta{\mathbb{R}}ight)+M^*\left(x,\frac{c_A}{2}A(x,\eta){\mathbb{R}}ight)\leq\\ &\leq M\left(x,\frac{2}{c_A}\eta{\mathbb{R}}ight)+\frac{c_A}{2}M^*\left(x,A(x,\eta){\mathbb{R}}ight).\end{split}\]
We can ignore $M(x,\eta)>0$ on the left-hand side above, rearrange the remaining terms and integrate both sides over $\Omega$ (cf.~\eqref{LinfinLM}) to get the claim.
\end{proof}
\begin{rem}\label{rem:2ndconj} For any function $f:{\mathbb{R}}^M{\tau}o{\mathbb{R}}$ the second conjugate function $f^{**}$ is convex and $f^{**}(x)\leq f(x)$. In fact, $f^{**}$ is a convex envelope of $f$, namely it is the biggest convex function smaller or equal to~$f$.
\end{rem}
\begin{lem}\label{lem:Mass} Suppose a cube ${Q_j^\delta}$ is an arbitrary one defined in (M) with ${\delta}elta_0=1/(8\sqrt{N})$ and function $M:{\mathbb{R}}n{\tau}imes[0,\infty){\mathbb{R}}ightarrow[0,\infty)$ is log-H\"older continuous, that is there exist constants $a_1>0$ and $b_1\geq 1$, such that for all $x,y\in\Omega$ with $|x-y|\leq \frac{1}{2}$ and all $\xi\in{\mathbb{R}}n$ we have~\eqref{M2'}. Let us consider function $ {\cal M}jd $ given by~\eqref{Mjd} and its greatest convex minorant $({\cal M}jd)^{**}$. Then there exist constants $a,c>0$, such that~\eqref{M2} is satisfied.
\end{lem}
\begin{proof}[Proof. cf.~\cite{martin}]First, we fix an arbitrary $y\in Q^{\delta}elta_j$ and note that
\begin{equation}\label{Quotient}
\frac{M(y,\xi)}{(M^{\delta}elta_j)^{**}(\xi)}=\frac{M(y,\xi)}{M^{\delta}elta_j(\xi)}\frac{M^{\delta}elta_j(\xi)}{(M^{\delta}elta_j)^{**}(\xi)}.
\end{equation}
We estimate separately both quotients on the right hand side of the latter equality. By continuity of $M$ we find $\bar{y}\in \widetildeidetilde{Q}^{\delta}elta_j$ such that $M^{\delta}elta_j(\xi)=M(\bar{y},\xi)$. Then using condition~\eqref{M2'} and the fact that $|y-\bar{y}|\leq 3{\delta}elta\sqrt{d}<\frac{1}{2}$ we get
\begin{equation}\label{FirQuoEst}
\frac{M(y,\xi)}{M(\bar y,\xi)}\leq \max\{\xi^{-\frac{a_1}{\log|y-\bar y|}}, b_1^{-\frac{a_1}{\log|y-\bar y|}}\}\leq \max\{\xi^{-\frac{a_1}{\log(3{\delta}elta\sqrt{N})}}, b_1^{-\frac{a_1}{\log(3{\delta}elta\sqrt{N})}}\}.
\end{equation}
In order to estimate the second quotient in \eqref{Quotient} we observe first that if $\xi\in[0,\infty)$ is such that $M^{\delta}elta_j(\xi)=(M^{\delta}elta_j)^{**}(\xi)$ then the statement is obvious. Therefore we assume that $M^{\delta}elta_j(\xi_0)>(M^{\delta}elta_j)^{**}(\xi_0)$ at some $\xi_0$. Due to continuity of $M^{\delta}elta_j$ and $(M^{\delta}elta_j)^{**}$ there is a neighborhood $U$ of $\xi_0$ such that $M^{\delta}elta_j>(M^{\delta}elta_j)^{**}$ on $U$. Consequently, $(M^{\delta}elta_j)^{**}$ is affine on $U$. Moreover, Definition~{\mathbb{R}}ef{def:Nf} implies that $m_1\leq M^{\delta}elta_j\leq m_2$, where $m_1$ and $m_2$ are convex. Therefore there are $\xi_1,\xi_2$ such that $U\subset(\xi_1,\xi_2)$, $M^{\delta}elta_j>(M^{\delta}elta_j)^{**}$ on $(\xi_1,\xi_2)$, $(M^{\delta}elta_j)^{**}(\xi_i)=M^{\delta}elta_j(\xi_i)$, $i=1,2$ and $(M^{\delta}elta_j)^{**}$ is an affine function on $[\xi_1,\xi_2]$, i.e.
\begin{equation}\label{ConvexificationIsAffine}
(M^{\delta}elta_j)^{**}(t\xi_1+(1-t)\xi_2)=tM^{\delta}elta_j(\xi_1)+(1-t)M^{\delta}elta_j(\xi_2),\qquad{\tau}ext{for}\quad t\in[0,1].
\end{equation}
We note that we consider $\xi_1>0$, because it follows that $0=M^{\delta}elta_j(0)=(M^{\delta}elta_j)^{**}(0)$. Now, thanks to the continuity of $M$ we find $y_i\in\widetildeidetilde{Q}^{\delta}elta_j$ such that $M^{\delta}elta_j(\xi_i)=M(y_i,\xi_i)$, $i=1,2$. Consequently, it follows from \eqref{ConvexificationIsAffine} that
\begin{equation}\label{ConvexificationIsAffineII}
(M^{\delta}elta_j)^{**}(t\xi_1+(1-t)\xi_2)=tM(y_1,\xi_1)+(1-t)M(y_2,\xi_2).
\end{equation}
Denoting ${\tau}ilde\xi=t\xi_1+(1-t)\xi_2$ we get
\begin{equation}\label{QuoBiConEst}
\frac{M^{\delta}elta_j\left({\tau}ilde\xi{\mathbb{R}}ight)}{(M^{\delta}elta_j)^{**}\left({\tau}ilde\xi{\mathbb{R}}ight)}\leq \frac{M\left(y_2,{\tau}ilde\xi{\mathbb{R}}ight)}{tM(y_1,\xi_1)+(1-t)M(y_2,\xi_2)}\leq\frac{tM(y_2,\xi_1)+(1-t)M(y_2,\xi_2)}{tM(y_1,\xi_1)+(1-t)M(y_2,\xi_2)}.
\end{equation}
Next, we observe that the definition of $M^{\delta}elta_j$ implies $M(y_1,\xi_1)=M^{\delta}elta_j(\xi_1)\leq M(y_2,\xi_1)$. We can assume without loss of generality that
\begin{equation}\label{MXiOneIneq}
M(y_1,\xi_1)< M(y_2,\xi_1)
\end{equation}
because for $M(y_1,\xi_1)= M(y_2,\xi_1)$ inequality \eqref{QuoBiConEst} implies $M^{\delta}elta_j\leq(M^{\delta}elta_j)^{**}$ on $[\xi_1,\xi_2]$. Since we have always $M^{\delta}elta_j\geq(M^{\delta}elta_j)^{**}$ we arrive at $M^{\delta}elta_j=(M^{\delta}elta_j)^{**}$ on $[\xi_1,\xi_2]$.
Let us consider a function $h:[0,1]{\mathbb{R}}ightarrow{\mathbb{R}}$ defined by
\begin{equation*}
h(t)=\frac{tM(y_2,\xi_1)+(1-t)M(y_2,\xi_2)}{tM(y_1,\xi_1)+(1-t)M(y_2,\xi_2)}.
\end{equation*}
Then we compute
\begin{equation*}
h'(t)=\frac{(M(y_2,\xi_1)-M(y_1,\xi_1))M(y_2,\xi_2)}{(t(M(y_1,\xi_1)-M(y_2,\xi_2))+M(y_2,\xi_2))^2}.
\end{equation*}
Obviously, we have $h'>0$ on $(0,1)$ due to \eqref{MXiOneIneq}. Therefore the maximum of $h$ is attained at $t=1$, which implies
\begin{equation}
\frac{M^{\delta}elta_j\left({\tau}ilde\xi{\mathbb{R}}ight)}{(M^{\delta}elta_j)^{**}\left({\tau}ilde\xi{\mathbb{R}}ight)}\leq\frac{M(y_2,\xi_1)}{M(y_1,\xi_1)}.
\end{equation}
Next, we apply condition~\eqref{M2'} and $\xi_1\leq{\tau}ilde\xi$ to infer
\begin{equation}\label{SecQuoEst}
\frac{M^{\delta}elta_j\left({\tau}ilde\xi{\mathbb{R}}ight)}{(M^{\delta}elta_j)^{**}\left({\tau}ilde\xi{\mathbb{R}}ight)}\leq \max\{\xi_1^{\frac{-a_1}{\log|y_2-y_1|}}, b_1^{\frac{-a_1}{\log|y_2-y_1|}}\}\leq \max\{\xi^{\frac{-a_1}{\log|y_2-y_1|}}, b_1^{\frac{-a_1}{\log|y_2-y_1|}}\}\leq\max\{\xi^{\frac{-a_1}{\log(4{\delta}elta\sqrt{N})}}, b_1^{\frac{-a_1}{\log(4{\delta}elta\sqrt{N})}}\}
\end{equation}
since $y_1,y_2\in{\tau}ilde{Q}^{\delta}elta_j$ implies $|y_1-y_2|\leq 4{\delta}elta\sqrt{N}<\frac{1}{2}$. Combining \eqref{Quotient} with \eqref{FirQuoEst} and \eqref{SecQuoEst} yields
\begin{equation*}
\begin{split} \frac{M(y,\xi)}{(M^{\delta}elta_j)^{**}(\xi)}\leq \max\{\xi^{\frac{-a_1}{\log(3{\delta}elta\sqrt{N})}}, b_1^{\frac{-a_1}{\log(3{\delta}elta\sqrt{N})}}\}\cdot \max\{\xi^{\frac{-a_1}{\log(4{\delta}elta\sqrt{N})}}, b_1^{\frac{-a_1}{\log(4{\delta}elta\sqrt{N})}}\}\leq \max\{\xi^{\frac{-2a_1}{\log(4{\delta}elta\sqrt{N})}}, b_1^{\frac{-2a_1}{\log(4{\delta}elta\sqrt{N})}}\}\\
\leq \xi^{\frac{-2a_1}{\log(4{\delta}elta\sqrt{N})}}+ b_1^{\frac{-2a_1}{\log(4{\delta}elta\sqrt{N})}}\leq c \left(1+ |\xi|^{-\frac{a}{\log(b{\delta}elta )}} {\mathbb{R}}ight),\end{split}
\end{equation*}
which is the desired conclusion.
\end{proof}
\begin{defi}[${\Delta}elta_2$-condition]\label{def:D2}
We say that an $N$-function $M:\Omega{\tau}imes{\mathbb{R}}n{\tau}o{\mathbb{R}}$ satisfies ${\Delta}elta_2$ condition if for a.e. $x\in\Omega$, there exists a constant $c>0$ and nonnegative integrable function $h:\Omega{\tau}o{\mathbb{R}}$ such that
\begin{equation}
\label{D2} M(x,2\xi)\leq cM(x,\xi)+h(x).
\end{equation}
\end{defi}
\section*{Appendix B}
We have two equivalent definitions of modular convergence.
\begin{defi}[Modular convergence]\label{def:convmod}
We say that a sequence $\{\xi_i\}_{i=1}^\infty$ converges modularly to $\xi$ in~$L_M(\Omega;{\mathbb{R}}n)$ (and denote it by $\xi_i\xrightarrow[i{\tau}o\infty]{M}\xi$), if
\begin{itemize}
\item[i)] there exists $\lambda>0$ such that
\begin{equation*}
\int_{\Omega}M\left(x,\frac{\xi_i-\xi}{\lambda}{\mathbb{R}}ight)dx{\tau}o 0,
\end{equation*}
equivalently
\item[ii)] there exists $\lambda>0$ such that
\begin{equation*}
\left\{M\left(x,\frac{\xi_i}{\lambda}{\mathbb{R}}ight){\mathbb{R}}ight\}_i \ {\tau}ext{is uniformly integrable in } L^1(\Omega)\quad {\tau}ext{and}\quad \xi_i\xrightarrow[]{i{\tau}o\infty}\xi \ {\tau}ext{in measure};
\end{equation*}
\end{itemize}
\end{defi}
\begin{defi}[Biting convergence]\label{def:convbiting}
Let $f_n,f\in L^1(\Omega)$ for every $n\in{\mathbb{N}}$. We say that a sequence $\{f_n\}_{n=1}^\infty$ converges in the sense of biting to $f$ in~$L^1(\Omega)$ (and denote it by $f_n\xrightarrow[]{b}f$), if there exists a sequence of measurable $E_k$ -- subsets of $\Omega$, such that $\lim_{k{\tau}o\infty} |E_k|=0$, such that for every $k$ we have $f_n{\tau}o f$ in $L^1(\Omega\setminus E_k)$.
\end{defi}
\begin{defi}[Uniform integrability] We call a sequence $\{f_n\}_{n=1}^\infty$ of measurable functions $f_n:\Omega{\tau}o {\mathbb{R}}n$
uniformly integrable if
\[\lim_{R{\tau}o\infty}\left(\sup_{n\in\mathbb{N}}\int_{\{x:|f_n(x)|\geq R\}}|f_n(x)|dx{\mathbb{R}}ight)=0,\]
equivalently (cf.~\cite{pgasgaw-stokes}) if
\begin{equation}
\label{uni-int-con2}
\forall_{{\varepsilon}>0}\quad\exists_{{\delta}elta>0}\qquad \sup_{n\in\mathbb{N}}\int_\Omega \left(|f_n(x)|-\frac{1}{\sqrt{{\delta}elta}}{\mathbb{R}}ight)_+ dx\leq{\varepsilon},
\end{equation}
where we denote the positive part of function $f$ by $(f(x))_+:=\max\{f(x),0\}$.
\end{defi}
We use the following results.
\begin{lem}[Modular-uniform integrability,~\cite{gwiazda2}]\label{lem:unif}
Let $M$ be an $N$-function and $\{f_n\}_{n=1}^\infty$ be a sequence of measurable functions such that $f_n:\Omega{\tau}o {\mathbb{R}}n$ and $\sup_{n\in{\mathbb{N}}}\int_\Omega M(x,f_n(x))dx<\infty$. Then the sequence $\{f_n\}_{n=1}^\infty$ is uniformly integrable.
\end{lem}
\begin{lem}[Density of simple functions, \cite{Musielak}]\label{lem:dens}
Suppose~\eqref{ass:M:int}. Then the set of simple functions integrable on $\Omega$ is dense in $L_M(\Omega)$ with respect to the modular topology.
\end{lem}
The above result can be obtained by the method of the proof of~\cite[Theorem~7.6]{Musielak}.
We need the following consequence of the Chacon Biting Lemma, \cite[Lemma~6.9]{pedr}.
\begin{theo}\label{theo:bitinglemma}Let $f_n\in L^1(\Omega)$ for every $n\in{\mathbb{N}}$, $f_n(x)\geq 0$ for every $n\in{\mathbb{N}}$ and a.e. $x$ in $\Omega$. Moreover, suppose $f_n\xrightarrow[]{b}f$ (cf.~Definition~{\mathbb{R}}ef{def:convbiting}) and $\limsup_{n{\tau}o\infty}\int_\Omega f_n dx\leq \int_\Omega f dx.$ Then $f_n\xrightharpoonup{}f$ in $L^1(\Omega)$ for $n{\tau}o\infty$.
\end{theo}
\begin{theo}[The Vitali Convergence Theorem]\label{theo:VitConv} Let $(X,\mu)$ be a positive measure space, $\mu(X)<\infty $, and $1\leq p<\infty$. If $\{f_{n}\}$ is uniformly integrable in $L^p_\mu$, $f_{n}(x){\tau}o f(x)$ in measure and $|f(x)|<\infty $ a.e. in $X$, then $f\in {L}^p_\mu(X)$
and $f_{n}(x){\tau}o f(x)$ in ${L}^p_\mu(X)$.
\end{theo}
\begin{theo}[The Dunford-Pettis Theorem]\label{theo:dunf-pet}
A sequence $\{f_n\}_n$ is uniformly integrable in $L^1(\Omega)$ if and only if it is relatively compact in the weak topology.
\end{theo}
\begin{lem} \label{lem:TM1}
Suppose $w_n\xrightharpoonup[n{\tau}o\infty]{}w$ in $L^1(\Omega)$, $v_n,v\in L^\infty(\Omega)$, and $v_n\xrightarrow[n{\tau}o\infty]{a.e.}v$. Then \[\int_\Omega w_n v_n\,dx \xrightarrow[n{\tau}o\infty]{}\int_\Omega w v\,dx.\]
\end{lem}
\section*{Appendix C}
\begin{proof}[Proof of Theorem~{\mathbb{R}}ef{theo:approx}]The proof is divided into four steps. We start with the case of star-shaped domain and then, in the fourth step, we turn to any Lipschitz domain.
{\tau}extbf{Step 1.} Let us assume, that $\Omega$ is a star-shape domain with respect to the ball $B(0, r)$ (i.e. with respect to any point of this ball). For $0 < {\delta}elta < r/4$, we set $\kappa_{\delta}elta=1-\frac{2{\delta}elta}{r}$. It holds that
\[\kappa_{\delta}elta \Omega + {\delta}elta B(0, 1) \subset \Omega.\]
For a measurable function $\xi:{\mathbb{R}}n{\tau}o{\mathbb{R}}n$ with $\mathrm{supp}\,\xi\subset\Omega$, we define
\begin{equation}
\label{xid}\xi_{\delta}elta(x) = \int_\Omega {\varrho}_{\delta}elta( x-y)\xi (\kappa_{\delta}elta y)dy=
\int_{B(0,{\delta}elta)} {\varrho}_{\delta}elta(y)\xi (\kappa_{\delta}elta (x-y))dy,
\end{equation}
where $ {\varrho}_{\delta}elta(x)={\varrho}(x/{\delta}elta)/{\delta}elta^N$ is a standard regularizing kernel on ${\mathbb{R}}n$ (i.e. ${\varrho}\in C^\infty({\mathbb{R}}n)$,
$\mathrm{supp}\,{\varrho}\subset\subset B(0, 1)$ and ${\int_{\Omega}} {\varrho}(x)dx = 1$, ${\varrho}(x) = {\varrho}(-x)\geq 0$). Let us notice that $\xi_{\delta}elta\in C_0^\infty({\mathbb{R}}n;{\mathbb{R}}n)$.
{\tau}extbf{Step 2.} We show that the family of operators $(\xi_{\delta}elta)_{\delta}elta$ is uniformly bounded from $L_M(\Omega;{\mathbb{R}}n)$ to $L_M(\Omega;{\mathbb{R}}n)$. Without loss of generality we assume
\begin{equation}
\label{xi<1}\|\xi\|_{L^1(\Omega;{\mathbb{R}}n)}\leq 1.
\end{equation}
We have to show that\begin{equation}
\label{unifMxid}{\int_{\Omega}} M(x,\xi_{\delta}elta(x))dx\leq C{\int_{\Omega}} M(x,\xi (x))dx
\end{equation}
for every suffciently small ${\delta}elta$.
We consider $M_j^{\delta}elta(\xi)$ given by~\eqref{Mjd} and ${\cal M}ss$, see~Remark~{\mathbb{R}}ef{rem:2ndconj}. Since $M(x,\xi_{\delta}elta(x))=0$ whenever $\xi_{\delta}elta(x)=0$, we have
\begin{equation}
\label{M:div-mult}\begin{split}
{\int_{\Omega}} M(x,\xi_{\delta}elta(x))dx=\sum_{j=1}^{N_{\delta}elta} {\int_{Q_j^\delta\cap\Omega} } M(x,\xi_{\delta}elta(x))dx=\\=\sum_{j=1}^{N_{\delta}elta} {\int_{Q_j^\delta\cap\Omega} }n \frac{M(x,\xi_{\delta}elta(x))}{{\cal M}sdx}{{\cal M}sdx}dx.\end{split}
\end{equation}
Our aim is to show now the following uniform bound
\begin{equation}
\label{M/M<c}\frac{M(x,\xi_{\delta}elta(x))}{{\cal M}sdx}\leq c
\end{equation}
for sufficiently small ${\delta}elta>0$, $x\in{Q_j^\delta}\cap\Omega$ with $c$ independent of ${\delta}elta,x$ and $j$. Let us fix an arbitrary cube and take $x\in {Q_j^\delta}$. For sufficiently small ${\delta}elta$ (i.e. ${\delta}elta< {\delta}elta_1:=\min\{ {r}/{4},{\delta}elta_0\}$), due to~\eqref{M2}, we obtain \begin{equation}
\label{M/M<xi}\frac{M(x,\xi_{\delta}elta(x))}{{\cal M}sdx}
\leq c \left(1+ |\xi_{\delta}elta(x)|^{-\frac{a}{\log(b{\delta}elta )}} {\mathbb{R}}ight).
\end{equation}
To estimate the right--hand side of~\eqref{M/M<xi} we consider~\eqref{xid}. Denote \[K=\sup_{B(0,1)}|{\varrho}(x)|.\]
Note that for any $x,y\in\Omega$ and each ${\delta}elta>0$ we have
\[{\varrho}_{\delta}elta(x-y)\leq {K}/{{\delta}elta^N}.\]
Therefore, taking into account~\eqref{xi<1} we get
\begin{equation}
\label{xidest}\begin{split}|\xi_{\delta}elta(x)|& = \left| \int_\Omega {\varrho}_{\delta}elta( x-y)\xi (\kappa_{\delta}elta y)dy{\mathbb{R}}ight|\\&\leq \frac{K}{{\delta}elta^N } \int_{\Omega} |\xi (\kappa_{\delta}elta y)|dy \leq \frac{K}{{\delta}elta^N\kappa_{\delta}elta}\|\xi\|_{L^1(\Omega;{\mathbb{R}}n)}\leq \frac{2K}{{\delta}elta^N}.\end{split}
\end{equation}
Note that $(2 K)^{-a/\log (b{\delta}elta )}\leq (4 K)^{-a/\log (b{\delta}elta_0 )}$ and \[\left| {{\delta}elta^N} {\mathbb{R}}ight|^{ \frac{a}{\log (b{\delta}elta )}}=\exp \frac{aN \log {\delta}elta}{\log (b{\delta}elta )},\]
which is bounded for ${\delta}elta\in [0,{\delta}elta_0]$. We combine this with~\eqref{M/M<xi} and~\eqref{xidest} to get\begin{equation}
\label{M/M<bezxi}\frac{M(x,\xi_{\delta}elta(x))}{(M_j^{\delta}elta(\xi_{\delta}elta(x)))^{**}} \leq c \left(1+ \left|4\frac{K}{{\delta}elta^N }{\mathbb{R}}ight|^{-\frac{a}{\log(b{\delta}elta )}} {\mathbb{R}}ight)\leq c.
\end{equation}
Thus, we have obtained~\eqref{M/M<c}. Now, starting from~\eqref{M:div-mult}, noting~\eqref{M/M<c} and the fact that ${\cal M}ss$=0 if and only if $\xi=0$, we observe \[
\begin{split}
{\int_{\Omega}} M(x,\xi_{\delta}elta(x))dx &=\sum_{j=1}^{N_{\delta}elta} {\int_{Q_j^\delta\cap\Omega} }n \frac{M(x,\xi_{\delta}elta(x))}{{\cal M}sdx}{{\cal M}sdx}dx\leq \\
&\leq c\sum_{j=1}^{N_{\delta}elta} {\int_{Q_j^\delta\cap\Omega} }n {{\cal M}sdx}dx\leq\\
&\leq c\sum_{j=1}^{N_{\delta}elta} {\int_{Q_j^\delta\cap\Omega} } \ {{{\cal M}sd}\left( \int_{B(0,{\delta}elta)} {\varrho}_{\delta}elta(y)\xi (\kappa_{\delta}elta (x-y))dy{\mathbb{R}}ight)}\mathds{1}_{{Q_j^\delta}\cap\Omega}(x) dx\leq\\
&\leq c\sum_{j=1}^{N_{\delta}elta} \int_{\mathbb{R}}n \ {{{\cal M}sd}\left( \int_{B(0,{\delta}elta)} {\varrho}_{\delta}elta(y)\xi (\kappa_{\delta}elta (x-y))\mathds{1}_{{Q_j^\delta}\cap\Omega}(x)dy{\mathbb{R}}ight)} dx\leq\\
&\leq c\sum_{j=1}^{N_{\delta}elta} \int_{\mathbb{R}}n {{{\cal M}sd}\left( \int_{B(0,{\delta}elta)} {\varrho}_{\delta}elta(y)\xi (\kappa_{\delta}elta (x-y))\mathds{1}_{{\tau}Qd\cap\Omega}(x-y)dy{\mathbb{R}}ight)} dx.\end{split}
\]
Note that by applying the Jensen inequality the right-hand side above can be estimated by the following quantity
\[ \begin{split}
& \quad \ c\sum_{j=1}^{N_{\delta}elta} \int_{\mathbb{R}}n \int_{{\mathbb{R}}n} {\varrho}_{\delta}elta(y) {{{\cal M}sd}\left( \xi (\kappa_{\delta}elta (x-y))\mathds{1}_{{\tau}Qd\cap\Omega}(x-y) {\mathbb{R}}ight)} dy\,dx\leq\\
& \leq c \| {\varrho}_{\delta}elta\|_{L^1({B(0,{\delta}elta);{\mathbb{R}}n})}\sum_{j=1}^{N_{\delta}elta}\int_{\mathbb{R}}n {{{\cal M}sd}\left( \xi (\kappa_{\delta}elta z)\mathds{1}_{{\tau}Qd\cap\Omega}(z) {\mathbb{R}}ight)} dz\leq\\
&\leq c \sum_{j=1}^{N_{\delta}elta} \int_{{\tau}Qd\cap\Omega} {{{\cal M}sd}\left( \xi (\kappa_{\delta}elta z) {\mathbb{R}}ight)} dz.\end{split}
\]
We applied inequality for convolution, boundedness of ${\varrho}_{\delta}elta$, once again the fact that ${\cal M}ss$=0 if~and only if~$\xi=0$. Then, by the definition of $M_j^{\delta}elta(\xi)$, i.e.~\eqref{Mjd} and properties of ${\cal M}ss$, see~Remark~{\mathbb{R}}ef{rem:2ndconj}, we realize that
\[ \begin{split} c \sum_{j=1}^{N_{\delta}elta} \int_{{\tau}Qd\cap\Omega} {{{\cal M}sd}\left( \xi (\kappa_{\delta}elta z) {\mathbb{R}}ight)} dz&\leq c'\sum_{j=1}^{N_{\delta}elta} \int_{{\kappa_{\delta}elta}{\tau}Qd} {M\left(x, \xi (x) {\mathbb{R}}ight)} dx\leq c'\sum_{j=1}^{N_{\delta}elta} \int_{{2}{\tau}Qd} {M\left(x, \xi (x) {\mathbb{R}}ight)} dx\leq \\&\leq C\int_\Omega {M\left(x, \xi (x) {\mathbb{R}}ight)} dx.\end{split}
\]
The last inequality above stands for computation of~a~sum taking into account the measure of~repeating parts of cubes.
We get~\eqref{unifMxid} by summing up the estimates of this step.
{\bf Step 3.} Fix arbitrary $\varphi\in V_0^M$ and recall definition of the cadidate for approximating family~\eqref{xid}. We are going to show that (still in the case of star-shape domains) it holds that
$$\int_\Omega M\left(x, \frac{(\nabla \varphi)_{\delta}elta- \nabla \varphi}{\lambda}{\mathbb{R}}ight) dx \xrightarrow[]{{\delta}elta{\tau}o 0} 0. $$
Fix $\sigma$ to be specified later and recall $C$ from~\eqref{unifMxid}. By Lemma~{\mathbb{R}}ef{lem:dens} and continuity of $M$ we can choose family of measurable sets $\{ E_j \}_{j=0}^n$ such that $\bigcup_{j=0}^n E_j = \Omega$ and a simple vector valued function
\[E^n(x)=\sum_{j=0}^n \mathds{1}_{E_j}(x) \va_{j}(x),\]
such that
\begin{equation}\label{IE:aw15}
\int_\Omega M\left( x, \frac{ E^n - \nabla \varphi }{\frac{1}{3}\lambda} {\mathbb{R}}ight) dx <
\frac{\sigma}{C}.
\end{equation} Then by \eqref{unifMxid} we have
\begin{equation}\label{IE:aw16}
\int_\Omega M\left( x,
\frac{ (\nabla \varphi -E^{n} )_{\delta}ep }{ \frac{1}{3} \lambda } {\mathbb{R}}ight) \,dx=\int_\Omega M\left( x,
\frac{ (\nabla \varphi)_{\delta}ep -(E^{n})_{\delta}ep }{ \frac{1}{3} \lambda } {\mathbb{R}}ight) \,dx <
\sigma.
\end{equation} Convexity of $M(x, \cdot)$ implies
\begin{equation*}
\begin{split}
&\int_\Omega
M \left( x, \frac{ (\nabla \varphi)_{{\delta}ep} - \nabla \varphi }{ \lambda }{\mathbb{R}}ight) \,dx =\\
& = \int_\Omega
M \left( x, \frac{ (\nabla \varphi)_{{\delta}ep} -(E^n)_{{\delta}ep}
+ (E^n)_{{\delta}ep} -E^n +E^n - \nabla \varphi}{ \lambda }{\mathbb{R}}ight) \,dx\\
& \leq
\frac{1}{3} \int_\Omega M\left( x,
\frac{ (\nabla \varphi)_{{\delta}ep} - (E^n)_{\delta}ep }{ \frac{1}{3} \lambda } {\mathbb{R}}ight) \,dx
+ \frac{1}{3} \int_\Omega M\left( x,
\frac{ (E^n)_{{\delta}ep} -E^n }{
\frac{1}{3} \lambda } {\mathbb{R}}ight) \,dx \\
& + \frac{1}{3} \int_\Omega M\left( x, \frac{ E^n - \nabla \varphi }{\frac{1}{3}\lambda} {\mathbb{R}}ight) \,dx .
\end{split}
\end{equation*}
Since we have already estimated the first and the last expression on the right-hand side above, let us concentrate on the second one.
The Jensen inequality and then the Fubini theorem lead to
\begin{equation}\label{IE:aw17}
\begin{split}
& \int_\Omega M\left( x,
\frac{ (E^n)_{{\delta}ep} -E^n }{
\frac{1}{3} \lambda } {\mathbb{R}}ight) \,dx\\
&=\int_\Omega M\left( x,
\frac{ \sum_{j=0}^n (\mathds{1}_{E_j}(x) \va_j(x))_{{\delta}ep} -\sum_{j=0}^n \mathds{1}_{E_j}(x) \va_j(x) }{
\frac{1}{3} \lambda } {\mathbb{R}}ight) \,dx\\
& = \int_\Omega M \left( x, \frac{3}{ \lambda} \int_{B(0,{\delta}elta)} \varrho_{\delta}elta(y) \sum_{j=0}^n [ \mathds{1}_{E_j}(\kappa_{\delta}elta(x - y)) \va_j(\kappa_{\delta}elta(x - y)) - \mathds{1}_{E_j}(x) \va_j (x) ]\,dy {\mathbb{R}}ight) \,dx
\\ &
\leq
\int_{B(0,{\delta}elta)} \varrho_{\delta}elta(y) \left( \int_\Omega M \left( x, \frac{3}{\lambda} \sum_{j=0}^n [ \mathds{1}_{E_j}(\kappa_{\delta}elta(x - y)) \va_j(\kappa_{\delta}elta(x - y)) - \mathds{1}_{E_j}(x) \va_j (x) ] {\mathbb{R}}ight) \,dx {\mathbb{R}}ight) \,dy.
\end{split}
\end{equation}
Using the continuity of the shift operator in $L^1$ we observe that poinwisely
\[\frac{3}{\lambda} \sum_{j=0}^n [ \mathds{1}_{E_j}(\kappa_{\delta}elta(x - y)) \va_j (\kappa_{\delta}elta(x - y))- \mathds{1}_{E_j}(x) \va_j (x) ] \xrightarrow[]{{\delta}ep{\tau}o 0} 0.\]
Moreover, note that
$$
M \left( x, \frac{3}{\lambda} \sum_{j=0}^n [ \mathds{1}_{E_j}(\kappa_{\delta}elta(x - y)) \va_j (\kappa_{\delta}elta(x - y))- \mathds{1}_{E_j}(x) \va_j (x) ] {\mathbb{R}}ight) \leq \sup_{ |{\varepsilon}c\eta|=1}M \left( x, \frac{3}{ \lambda} \sum_{j=0}^n| \va_j| {\varepsilon}c\eta {\mathbb{R}}ight) <\infty
$$
and
the Lebesgue Dominated Convergence Theorem provides the right-hand side of \eqref{IE:aw17} converges to zero as ${\delta}elta{\tau}o 0$.
To sum up, regarding to arbitrariness of $\sigma >0$ in~\eqref{IE:aw15} and~\eqref{IE:aw16}, and to the convergence of the second term we get the claim.
{\bf Step 4.} If $\Omega$ is a bounded Lipschitz domain in~${\mathbb{R}}n$, then there exists a finite family of open sets
$\{\Omega_i\}_{i\in I}$ and a finite family of balls $\{ B^i\}_{i\in I}$ such that
$$\Omega=\bigcup\limits_{i\in I}\Omega_i$$
and every set $\Omega_i$ is star-shaped with respect to ball $B^i$ of radius $r_i$ (see e.g. \cite{Novotny}).
Let us introduce the partition of unity ${\tau}heta_i$ with
for $x\in\Omega$.
Then one can decompose function $\varphi$ in the following way
$$\varphi(x) = \sum_{i\in I} ({\tau}heta_i \varphi )(x).$$
Let us notice that if $\nabla \varphi \in L_M(\Omega;{\mathbb{R}}n)$ and $\varphi \in L^\infty(\Omega)$, then
$\nabla ({\tau}heta_i \varphi) = (\varphi \nabla {\tau}heta_i + {\tau}heta_i \nabla \varphi) \in L_M(\Omega;{\mathbb{R}}n)$.
Therefore we can apply the previous arguments to every function ${\tau}heta_i \varphi$ of a support on a star-shaped domain $\Omega_i\subset\Omega$.
\end{proof}
\begin{proof}[Proof of Theorem~{\mathbb{R}}ef{theo:Poincare}] The proof consist of three steps starting with the case of smooth and compactly supported functions on small cube, then turning to the Orlicz class and concluding the~claim on arbitrary bounded set.
{\bf Step 1.} We start the proof for $u\in C_0^\infty(\Omega)$ with $\mathrm{supp} u\subset\subset [-\frac{1}{4},\frac{1}{4}]^N$. Let $u$ be extended by $0$ outside $\Omega$ and ${\omega_{N}}=(1,{\delta}ots,1)\in{\mathbb{R}}n.$ Note that
\[u(x)=\int_{-\frac{1}{2}}^0\sum_{j=1}^N \partialrtial_j u(x+s{\omega_{N}})ds=\int_0^{\frac{1}{2}}\sum_{j=1}^N \partialrtial_j u(x+s{\omega_{N}})ds\]
and so
\[2u(x)=\int_{-\frac{1}{2}}^{\frac{1}{2}}\sum_{j=1}^N \partialrtial_j u(x+s{\omega_{N}})ds.\]
Then we realize that for the constant $c=\sqrt{N}/2$ we have
\[ u(x)\leq \int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{1}{2}\sum_{j=1}^N |\partialrtial_j u(x+s{\omega_{N}})|ds\leq \int_{-\frac{1}{2}}^{\frac{1}{2}}c \|\nabla u(x+s{\omega_{N}})\|ds.\]
Applying $m$, which is increasing, to both sides above and the Jensen inequality (note that our interval with the Lebesgue measure is a probability space) we get
\[m(|u(x)|)\leq m\left(\int_{-\frac{1}{2}}^{\frac{1}{2}}c \|\nabla u(x+s{\omega_{N}})\|ds{\mathbb{R}}ight)\leq \int_{-\frac{1}{2}}^{\frac{1}{2}} m\left(c\|\nabla u(x+s{\omega_{N}})\|{\mathbb{R}}ight)ds.\]
Integrating over $\Omega$ and changing the order of integration we obtain
\[\begin{split}
\int_\Omega m(|u(x)|)dx&\leq \int_\Omega \int_{-\frac{1}{2}}^{\frac{1}{2}} m\left(c\|\nabla u(x+s{\omega_{N}})\|{\mathbb{R}}ight)dsdx= \int_{-\frac{1}{2}}^{\frac{1}{2}} \int_\Omega m\left(c\|\nabla u(x+s{\omega_{N}})\|{\mathbb{R}}ight) dx ds\leq\\
&\leq \| 1\|_{L^1\left(-d,d{\mathbb{R}}ight)}\sup_{s\in \left(-d,d{\mathbb{R}}ight)} \int_\Omega m\left(c\|\nabla u(x+s{\omega_{N}})\|{\mathbb{R}}ight) dx = \int_\Omega m\left(c\|\nabla u(x)\|{\mathbb{R}}ight) dx.\end{split}\]
Since $m\in{\Delta}elta_2$, we apply~\eqref{D2} (with constant $c_{m,{\Delta}elta_2}$ and no $x$-dependence) $k$ times with the~smallest $k$, such that $c(\Omega,N)<2^k$. Then, due to monotonicity of $m$, we get
\[\int_{\Omega_1} m(c(\Omega,N)|\nabla {u}|)dx\leq (c_{m,{\Delta}elta_2})^k\int_{\Omega_1} m(|\nabla {u}|)dx.\]
{\bf Step 2.} Let us consider now an open set $\widetildeidetilde{\Omega}$, such that $\overline{\Omega}\subset\widetildeidetilde{\Omega}\subset [-\frac{1}{4},\frac{1}{4}]^N$. Step~1. provides that for $u\in C_0^\infty(\widetildeidetilde{\Omega})$ we have \begin{equation}
\label{inq:m-poi} \|m(|u|)dx\|_{L^1({\mathbb{R}}n)}\leq C\|m(|\nabla u|)dx\|_{L^1({\mathbb{R}}n)}.
\end{equation}
Now, we aim at showing that for each $u\in V_0^m$ the inequality also holds. Of course, each such $u$ can be regularised by convolution with a standard mollifier $\varrho_\frac{1}{n}$
\[u_n(x):=\varrho_\frac{1}{n} * u(x),\]
where $\frac{1}{n}<\frac{1}{2}{{\mathbb{R}}m dist}(\partialrtial \widetildeidetilde{\Omega},\Omega)$. Such $u_n$ is smooth and compactly supported in $\widetildeidetilde{\Omega}$, so we have~\eqref{inq:m-poi} for~ $u_n$. Passing to the limit with $n{\tau}o \infty$ gives $u_n{\tau}o u$ and $\nabla u_n{\tau}o\nabla u$ a.e. in~${\mathbb{R}}n$. Then continuity of $m$ gives
\[ m(|u_n|){\tau}o m(|u|) \quad{\tau}ext{and}\quad m(|\nabla u_n|){\tau}o m(|\nabla u|)\quad {\tau}ext{a.e. in}\ {\mathbb{R}}n.\]
To get the strong convergence in $L^1(\Omega)$ of the sequence, we are going to apply the Vitali Convergence Theorem (Theorem~{\mathbb{R}}ef{theo:VitConv}). It suffices to show uniform integrability of the sequence via condition~\eqref{uni-int-con2}.
Function $u\in W^{1,1}(\Omega)$, so $\nabla u_n=\varrho_\frac{1}{n}*\nabla u$.
The Jensen inequality implies
\[\int_{\widetildeidetilde{\Omega}} m(|\nabla u_n|)dx\leq\int_{\widetildeidetilde{\Omega}} m(|\nabla u|)dx.\]
Observe that $t\mapsto |m(t)-1/\sqrt{{\delta}elta}|_+$ is a convex function and the Jensen inequality implies
\[\int_{\widetildeidetilde{\Omega}}\left(m(|\nabla u_n|)-\frac{1}{\sqrt{{\delta}elta}}{\mathbb{R}}ight)_+dx\leq\int_{\widetildeidetilde{\Omega}}\left(m(|\nabla u|)-\frac{1}{\sqrt{{\delta}elta}}{\mathbb{R}}ight)_+dx.\] Moreover, $m(|\nabla u|)\in L^1(\widetildeidetilde{\Omega})$, so for every ${\varepsilon}>0$ there exists ${\delta}elta>0$, such the right-hand side above is smaller than ${\varepsilon}$, i.e. condition~\eqref{uni-int-con2} is satisfied and we get uniform integrability of $\{m(|\nabla u_n|)\}_n$. From~\eqref{inq:m-poi} we notice that
$m(| u|)\in L^1(\widetildeidetilde{\Omega})$ and due to the same arguments the sequence $\{m(| u_n|)\}_n$ is uniformly integrable.
{\bf Step 3.} Suppose that $\Omega$ is arbitrary bounded set containing $0$. It is contained in the cube of~the~edge $D={{\mathbb{R}}m diam} \Omega$. Then $\widetildeidetilde{u}(x)=u\left(4Dx{\mathbb{R}}ight)$ has ${{\mathbb{R}}m supp}\,\widetildeidetilde u\subset \Omega_1\subset \left[-\frac{1}{4},\frac{1}{4}{\mathbb{R}}ight]^N.$ We have
\[\int_\Omega m(|u|)dx=(4D)^N\int_{\Omega_1} m(|\widetildeidetilde{u}|)dx\leq (4D)^NC\int_{\Omega_1} m(|\nabla\widetildeidetilde{u}|)dx=C\int_{\Omega} m(4D|\nabla {u}|)dx.\]
Moreover, we estimate the right-hand side as in Step~1 in order to put a constant outside the~integral and the claim follows for such $\Omega$. To obtain it on an arbitrary domain we need only to observe that the Lebesgue measure is translation-invariant.
\end{proof}
\end{document} |
\begin{equation}gin{document}
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\sloppy
\tilde{t}le{Quantum Data-Fitting}
\author{ Nathan Wiebe$^1$, Daniel Braun$^{2,3}$, and Seth Lloyd$^4$
}
\affiliation{$^1$ Institute for Quantum Computing, Waterloo, {\cal O}n, Canada}
\affiliation{$^2$ Universit\'e de Toulouse, UPS, Laboratoire
de Physique Th\'eorique (IRSAMC), F-31062 Toulouse, France}
\affiliation{$^3$ CNRS, LPT (IRSAMC), F-31062 Toulouse, France}\
\affiliation{$^4$ MIT - Research Laboratory for Electronics and
Department of Mechanical Engineering, Cambridge, MA 02139, USA}
\begin{equation}gin{abstract}
We provide a new quantum algorithm that efficiently determines the quality
of a least-squares fit over an
exponentially large data set by building upon an
algorithm for solving systems of linear
equations efficiently (Harrow et al.,
Phys.~Rev.~Lett.~{\bf 103}, 150502 (2009)). In many
cases, our algorithm can also efficiently find a concise function that
approximates the data to be fitted
and bound the approximation error. In cases where the input data is a pure quantum state,
the algorithm can be used to provide an efficient parametric estimation of the quantum state and therefore can be
applied as an alternative to full quantum state tomography given a fault tolerant quantum computer.
\end{abstract}
\pacs{03.67.-a, 03.67.Ac, 42.50.Dv }
\title{Quantum Data-Fitting}
Invented as early as 1794 by Carl Friedrich Gauss, fitting data to
theoretical models has become over the centuries one of the most important
tools in all
of quantitative science \cite{Bretscher95}. Typically, a theoretical model
depends on a number
of parameters, and leads to functional relations between data that will depend
on those parameters. Fitting a large amount of experimental data to the
functional
relations allows one to obtain reliable estimates of the parameters. If the
amount of data becomes very large, fitting can become very costly. Examples
include inversion problems of X-ray or
neutron scattering data for structure analysis, or high-energy physics with
giga-bytes of data
produced per second at the LHC. Typically, structure analysis starts from a
first guess of the structure, and then iteratively tries to improve the
fit to the experimental data by testing variations of the structure. It is
therefore often desirable to test many {\em different
models}, and compare the best possible fits they provide before committing
to one for which one extracts then the parameters from the fit. Obtaining a
good fit with a relatively small number of parameters compared to the amount
of data can be considered a form of data compression. Indeed, also for
numerically calculated data, such as many-body wave-functions in molecular
engineering, efficient fitting of the wave-functions to simpler models would
be highly desirable.
With the rise of quantum information theory, one might
wonder if a quantum algorithm can be found that solves these problems
efficiently. The discovery that exploiting
quantum mechanical effects might lead to enhanced computational power
compared to
classical information processing has triggered large-scale research
aimed at finding quantum algorithms which are more efficient
than the best classical counterparts
\cite{Shor94,Simon94,Grover97,Dam01,Aharonov06,Childs02}. Although
fault--tolerant quantum computation remains out of reach at present,
quantum simulation is already now on the verge of providing answers to
questions concerning the states of complex systems that are beyond classical
computability \cite{BMS+11,SBM+11}.
Recently, a quantum algorithm (called HHL in the following) was
introduced that efficiently
solves a linear equation, ${\bf F}{\bf x}={\bf b}$, with given vector ${\bf b}$ of dimension
$N$
and sparse Hermitian matrix ${\bf F}$ \cite{Harrow09.2}. ``Efficient
solution'' means
that the expectation value ${\bf r}a{{\bf x}}{\bf M}\ket{{\bf x}}$ of an arbitrary
poly-size Hermitian operator ${\bf M}$ can be found in roughly ${
{\cal O}}(s^4\kappa^2\log(N)/\epsilon)$ steps~\cite{HHLcorrection}, where
$\kappa$ is the condition
number of ${\bf F}$, i.e.~the ratio between the largest and smallest
eigenvalue of ${\bf F}$, $s$ denotes the sparsenes (i.e.~the maximum number of
non-zero matrix
elements of
${\bf F}$ in any given row or column), and $\epsilon$ is the maximum allowed distance
between the $\ket{x}$ found by the computer and the exact solution.
In contrast, they show that it is unlikely that classical computers can efficiently
solve similar problems because it would imply that quantum computers
are no more powerful than classical computers.
While it has remained unclear so far whether expectation values of the form
${\bf r}a{\bf x}{\bf M}\ket{\bf x}$ provide answers to computationally important
questions, we provide here an adaption of the algorithm to the problem of
data fitting that allows one to
efficiently obtain
the quality of a fit without having to learn the fit-parameters. {\cal O}ur
algorithm is particularly useful for fitting data {\em
efficiently computed} by a quantum computer or quantum simulator,
especially if an evolution
can be efficiently simulated but no known method exists to efficiently
learn the resultant state.
For example, our algorithm could be used to efficiently find
a concise matrix--product state approximation to
a groundstate yielded by a quantum many--body simulator and assess the
approximation error.
More complicated states can be used in the fit if the quantum computer
can efficiently prepare them. Fitting quantum states to a set of
known functions is an interesting alternative to performing
full quantum-state tomography \cite{H97}.
\emph{Least-squares fitting}--
The goal in least--squares fitting is to find a simple continuous function that well approximates a discrete set of $N$ points $\{x_i,y_i\}$. The function is constrained to be linear in the fit parameters
$\mbox{\boldmath{$\lambda$}}\in\mathbb{C}^M$, but it can be non-linear in
${\bf x}$. For simplicity we consider
$x\in \mathbb{C}$, but the generalization to higher dimensional $x$ is
straight-forward.
{\cal O}ur fit function is then of the form $$f(x,\mbox{\boldmath{$\lambda$}}):=\sum_{j=1}^M
f_j(x)\lambda_j$$ where $\lambda_j$ is a component of $\mbox{\boldmath{$\lambda$}}$ and
$f(x,\mbox{\boldmath{$\lambda$}}):\mathbb{C}^{M+1}\mapsto\mathbb{C}$.
The optimal fit parameters can be found by minimizing
\begin{equation}gin{equation} \label{E}
E=\sum_{i=1}^N|f(x_i,\mbox{\boldmath{$\lambda$}})-y_i|^2=|{\bf F}\mbox{\boldmath{$\lambda$}}-{\bf y}|^2
\end{equation}
over all $\mbox{\boldmath{$\lambda$}}$, where we have defined the $N\times M$ matrix ${\bf F}$ through
${\bf F}_{ij}=f_j(x_i)$, ${\bf F}^t$ is its transpose, and ${\bf y}$ denotes the column
vector $(y_1,\ldots,y_N)^t$. Also, following HHL, we assume without loss of generality that $\frac{1}{\kappa^2}\le\|{{\bf F}}^\dagger{\bf F}\|\le 1$ and $\frac{1}{\kappa^2}\le\|{{\bf F}}{\bf F}^\dagger\|\le 1$~\cite{Harrow09.2}.
Throughout this Letter we use
$\|\cdot\|$ to denote the spectral norm.
Given that ${\bf F}^\dagger{\bf F}$ is invertible, the fit parameters that give
the least square error are
found by applying
the Moore--Penrose pseudoinverse \cite{BiG+74} of ${\bf F}$, ${\bf F}^+$, to ${\bf y}$:
\begin{equation}gin{equation}
\mbox{\boldmath{$\lambda$}}={\bf F}^+{\bf y}=({\bf F}^\dagger {\bf F})^{-1}{\bf F}^\dagger {\bf y}.\label{eq:pseudoinverse}
\end{equation}
A proof that~\eqref{eq:pseudoinverse} gives an optimal $\mbox{\boldmath{$\lambda$}}$ for a least--square fit
is given in the appendix.
The algorithm consists of three subroutines: a quantum algorithm for performing the pseudo--inverse,
an algorithm for estimating the fit quality and an algorithm for learning
the fit-parameters $\mbox{\boldmath{$\lambda$}}$.\\
\emph{1. Fitting Algorithm}---
{\cal O}ur algorithm uses a quantum computer and oracles that
output quantum states that encode the matrix elements of ${\bf F}$ to approximately prepare ${\bf F}^{+}{\bf y}$. The matrix multiplications,
and inversions, are implemented using an improved version of the HHL
algorithm~\cite{Harrow09.2} that utilizes recent developments in
quantum simulation algorithms.
{\flushleft\emph{Input}: A quantum state
$\ket{{\bf y}}=\sum_{p=M+1}^{M+N} {\bf y}_p \ket{p}/|{\bf y}|$
that
stores the data ${\bf y}$, an upper bound (denoted $\kappa$)
for the square roots of the condition numbers of ${\bf F}{\bf F}^\dagger$ and ${\bf F}^\dagger{\bf F}$,
the
sparseness of ${\bf F}$ (denoted $s$) and an error tolerance
$\epsilon$.}
{\flushleft\emph{Output}: A quantum state $\ket{\mbox{\boldmath{$\lambda$}}}$ that is approximately proportional to the
optimal fit parameters
$\mbox{\boldmath{$\lambda$}}/|\mbox{\boldmath{$\lambda$}}|$ up to error $\epsilon$ as measured by the Euclidean--norm.}
{\flushleft\emph{Computational Model}: We have a universal quantum computer equipped
with oracles that, when queried about a non--zero matrix element in a given row, yield a quantum state
that encodes a requested bit of a binary encoding
the column number or value of a nonzero matrix element of ${\bf F}$ in a manner
similar to those in~\cite{WBHS11}. We also assume
a quantum blackbox is provided
that yields copies of the input state $\ket{{\bf y}}$ on demand.}
{\flushleft \emph{Query Complexity}: The number of oracle queries used is }
\begin{equation}gin{equation}
\tilde {\cal O}\left(\log(N)(s^3\kappa^6)/\epsilon{\rm i}ght),\label{eq:alg1cost}
\end{equation}
where $\tilde {\cal O}$ notation implies an upper bound on the scaling of a function, suppressing all sub-polynomial functions.
Alternatively, the simulation method
of~\cite{Chi09,BC12} can be used to achieve a query complexity of
\begin{equation}gin{equation}
\tilde {\cal O}\left( \log(N)(s\kappa^6)/\epsilon^2{\rm i}ght).
\end{equation}
\emph{Analysis of Algorithm}---
The operators ${\bf F}$ and ${\bf F}^{\dagger}$ are implemented using an isometry superoperator ${\bf I}$ to represent them as Hermitian operators on $\mathbb{C}^{N+M}$.
The isometry has the following action on a matrix ${\bf X}$:
\begin{equation}gin{equation}
{\bf I}:{\bf X}\mapsto\left(\begin{equation}gin{array}{cc} 0 &{\bf X}\\ {\bf X}^\dagger & 0 \end{array} {\rm i}ght).\label{iso}
\end{equation}
These choices are convenient because ${\bf I}({\bf F}^{\dagger})\ket{{\bf y}}$ contains ${\bf F}^{\dagger} {\bf y}/|{\bf y}|$ in its first $M$ entries.
We also assume for simplicity that $|{\bf I}({\bf F}^\dagger)\ket{{\bf y}}|=1$. This can easily be relaxed by dividing ${\bf I}({\bf F}^\dagger)\ket{{\bf y}}$ by $|{\bf F}^\dagger{\bf y}|$.
\emph{Preparing ${\bf I}({\bf F}^{\dagger})\ket{{\bf y}}$}---
The next step is to prepare the state ${\bf I}({\bf F}^{\dagger})\ket{{\bf y}}$. This is not straightforward because ${\bf I}({\bf F}^{\dagger})$ is a Hermitian, rather than unitary, operator.
We implement the Hermitian operator using the same phase estimation trick that HHL use to enact the inverse of a Hermitian operator, but instead of
dividing by the eigenvalues of each eigenstate we multiply each eigenstate by its eigenvalue. We describe the relevant steps below. For more details, see~\cite{Harrow09.2}.
The algorithm first prepares an ancilla state for a large integer $T$
that is of order $N$
\begin{equation}gin{equation}
\ket{\Psi_0}=\sqrt{\frac{2}{T}}\sum_{\tau=0}^{T-1} \sin \left(\frac{\pi(\tau+1/2)}{T} {\rm i}ght)\ket{\tau}\otimes\ket{{\bf y}}.
\end{equation}
It then maps $\ket{\Psi_0}$ to,
\begin{equation}gin{align}
\sqrt{\frac{2}{T}}\sum_{\tau=0}^{T-1} \sin \left(\frac{\pi(\tau+1/2)}{T}{\rm i}ght)\ket{\tau} \otimes e^{-i{\bf I}({\bf F}^{\dagger})\tau t_0/T}\ket{{\bf y}},\label{eq:phasest1}
\end{align}
for $t_0\in {\cal O}(\kappa/\epsilon)$. We know from work on quantum simulation
that $\exp(-i{\bf I}({\bf F}^{\dagger})\tau t_0/T)$ can be implemented within error ${\cal O}(\epsilon)$
in the 2-norm using $\tilde {\cal O}(\log(N)s^3t_0/T)$ quantum operations, if
${\bf F}$ has sparseness $s$~\cite{CK11}. Alternatively, the method
of~\cite{Chi09,BC12} gives
query complexity $\tilde {\cal O}(\log(N)s \tau t_0/(\epsilon T))$.
If we write $\ket{{\bf y}}=\sum_{j=1}^N \begin{equation}ta_j \ket{\mu_j}$, where $\ket{\mu_j}$ are the eigenvectors of
${\bf I}({\bf F}^\dagger)$ with eigenvalue $E_j$ we obtain
\begin{equation}gin{align}
\sqrt{\frac{2}{T}}\sum_{\tau=0}^{T-1} \sin \left(\frac{\pi(\tau+1/2)}{T}{\rm i}ght)e^{-iE_j\tau t_0/T}\ket{\tau} \otimes \begin{equation}ta_j\ket{\mu_j},\label{eq:phasest1}
\end{align}
The quantum Fourier transform is then applied to the first register and, after labeling the Fourier coefficients $\alphapha_{k|j}$,
the state becomes
\begin{equation}gin{equation}
\sum_{j=1}^N \sum_{k=0}^{T-1} \alphapha_{k|j} \begin{equation}ta_j \ket{k}\ket{\mu_j},\label{eq:fouriertrans}
\end{equation}
HHL show that the Fourier
coefficients are small unless the eigenvalue $E_j\approx \tilde E_k:=2\pi
k/t_0$, and $t_0\in {\cal O}(\kappa/\epsilon)$ is needed to ensure that the error from approximating the eigenvalue is at most $\epsilon$.
It can be seen using the analysis in~\cite{Harrow09.2} that after
re-labeling $\ket{k}$ as $\ket{\tilde E_k}$, and taking $T\in {\cal O}(N)$, \eqref{eq:fouriertrans} is
exponentially close to
$\sum_{j=1}^N \begin{equation}ta_j \ket{\tilde E_j}\ket{\mu_j}$.
The final step is to introduce an ancilla system and perform a controlled
unitary on it that rotates the ancilla state from $\ket{0}$ to
$\sqrt{1-C^2\tilde E_j^2}\ket{0}+C\tilde E_j\ket{1}$, where $C\in {\cal O}(\max_j |E_j|)^{-1}$ because the state would not be properly normalized if $C$ were larger. The probability of
measuring the ancilla to be $1$ is ${\cal O}(1/\kappa^2)$ since $CE_j$ is at least ${\cal O}(1/\kappa)$. ${\cal O}(\kappa^2)$ repetitions are therefore needed to guarantee success with high probability, and amplitude amplification can be used to reduce the number of repetitions to ${\cal O}(\kappa)$~\cite{Harrow09.2}. HHL show that either ${\cal O}(1/\kappa^2)$ or ${\cal O}(1/\kappa)$ attempts are also needed to successfully perform ${\bf I}({\bf F})^{-1}$ depending on whether amplitude amplification is used.
The cost of implementing ${\bf I}({\bf F}^\dagger)$ is the product of
the cost of simulating ${\bf I}({\bf F}^\dagger)$ for time $\kappa/\epsilon$ and the number of repetitions required to obtain a successful result, which scales as $O(\kappa)$.
The
improved simulation
method of Childs and Kothari~\cite{CK11} allows the simulation to be performed in time $\tilde{\cal O}(\log(N)s^3\kappa/\epsilon)$, where $s$ is the sparseness of ${\bf F}$; therefore, ${\bf I}({\bf F}^\dagger)\ket{{\bf y}}$ can be prepared using $\tilde {\cal O}( \log(N) s^3 \kappa^2/\epsilon)$
oracle calls.
The cost of performing the inversion using the simulation method of~\cite{Chi09,BC12} is found by substituting $s{\rm i}ghtarrow s^{1/3}/\epsilon$ into this or any of our subsequent results.
\emph{Inverting ${\bf F}^{\dagger}{\bf F}$}---
We then finish the algorithm by applying $({\bf F}^{\dagger}{\bf F})^{-1}$ using the method
of HHL~\cite{Harrow09.2}. Note that the existence of $({\bf F}^{\dagger}{\bf F})^{-1}$ is
implied by a well-defined fitting-problem, in the sense that a zero eigenvalue
of ${\bf F}^{\dagger}{\bf F}$ would result in a degenerate direction of the quadratic form
({\rm e}f{E}).
The operator ${\bf F}^{\dagger} {\bf F}\in \mathbb{C}^{M\times M}$ is Hermitian and hence amenable to the linear systems algorithm.
We do, however, need to extend the domain of the operator to make it
compatible with $\ket{{\bf y}}$ which is in a Hilbert space of dimension $N+M$.
We introduce ${\bf A}$ to denote the corresponding operator,
\begin{equation}gin{equation}
{\bf A}:= \left(\begin{equation}gin{array}{cc} {\bf F}^{\dagger}{\bf F} &0\\ 0 & {\bf F}{\bf F}^\dagger \end{array} {\rm i}ght)={\bf I}({\bf F})^2.
\end{equation}
If we define $\ket{\mbox{\boldmath{$\lambda$}}}\in \mathbb{C}^{N+M}$ to be a state of the form $\ket{\mbox{\boldmath{$\lambda$}}}=\sum_{j=1}^{M} \lambda_j\ket{j}$ up to a normalizing constant, then
${\bf F}^{\dagger} {\bf F} \mbox{\boldmath{$\lambda$}}$ is proportional to ${\bf A} \ket{\mbox{\boldmath{$\lambda$}}}$ up to a normalizing
constant. This means that we can find a vector that is proportional
to the least-squares fit parameters by inversion via
\begin{equation}gin{equation}
\ket{\mbox{\boldmath{$\lambda$}}}= {\bf A}^{-1}{\bf I}({\bf F}^{\dagger})\ket{{\bf y}}.\label{lamend}
\end{equation}
This can be further simplified by noting that
\begin{equation}gin{equation}
{\bf A}^{-1}={\bf I}({\bf F})^{-2}.
\end{equation}
Amplitude amplification does not decrease the number of attempts needed to implement ${\bf A}^{-1}$ in~\eqref{lamend} because the algorithm require reflections about ${\bf I}({\bf F}^\dagger)\ket{{\bf y}}$, which requires ${\cal O}(\kappa)$ repetitions to prepare.
Since amplitude amplification provides no benefit for implementing ${\bf A}^{-1}$, ${\cal O}(\kappa^5)$ repetitions are needed to implement ${\bf A}^{-1}{\bf I}({\bf F}^\dagger)$. This is a consequence of the fact that the probability of successfully performing each ${\bf I}({\bf F})^{-1}$ is ${\cal O}(1/\kappa^2)$ and the probability of performing ${\bf I}({\bf F}^\dagger)$ is ${\cal O}(1/\kappa)$ (if amplitude amplification is used). The cost of performing the simulations involved in each attempt is $\tilde{\cal O}(\log(N) s^3\kappa/\epsilon)$ and hence the required number of oracle calls scales as
\begin{equation}gin{equation}
\tilde{{\cal O}}\left(\log(N)(s^3\kappa^6/\epsilon) {\rm i}ght).~\label{eq:totalcost}
\end{equation}
Although the algorithm yields $\ket{\mbox{\boldmath{$\lambda$}}}$ efficiently, it may be exponentially expensive to learn
$\ket{\mbox{\boldmath{$\lambda$}}}$ via tomography; however, we show below that a quantum computer can assess the quality of the fit efficiently.
\emph{2. Estimating Fit Quality}---
We will now show that we can efficiently estimate
the fit quality $E$
even if $M$ is exponentially large and without having to determine the
fit-parameters. For this problem, note that due to the isometry ({\rm e}f{iso})
${E}=|\ket{{\bf y}}-
{\bf I}({\bf F})\ket{\mbox{\boldmath{$\lambda$}}}|^2$. We assume the prior computational model. We are
also provided a desired error tolerance, $\epsilon$,
and wish to determine the quality of the fit within error $\delta$.
{\flushleft\emph{Input}: A constant $\delta>0$ and all inputs required by algorithm 1.}
{\flushleft\emph{Output}: An estimate of $\gamfb|{\bf r}a{{\bf y}}{\bf I}({\bf F})\ket{\mbox{\boldmath{$\lambda$}}}|^2$ accurate within error $\delta$.}
{\flushleft\emph{Query Complexity}:
\begin{equation}gin{equation}
\tilde {\cal O}\left(\log(N)\frac{ s^3 \kappa^4}{\epsilon\delta^2}{\rm i}ght).
\end{equation}
}
\emph{Algorithm}--- We begin by preparing the state $\ket{{\bf y}}\otimes
\ket{{\bf y}}$ using the provided state preparation blackbox. We then use the prior
algorithm to construct
the state
\begin{equation}gin{equation}
{\bf I}({\bf F}){\bf A}^{-1}{\bf I}({\bf F}^{\dagger})\ket{{\bf y}}\otimes \ket{{\bf y}}={\bf I}({\bf F})^{-1}{\bf I}({\bf F}^{\dagger})\ket{{\bf y}}\otimes \ket{{\bf y}},\label{eq:deltalearna}
\end{equation}
within error ${\cal O}(\epsilon)$. The cost of implementing ${\bf I}({\bf F})^{-1}{\bf I}({\bf F}^{\dagger})$ (with high probability) within error $\epsilon$ is
\begin{equation}gin{equation}
\tilde {\cal O}\left(\log(N)\frac{ s^3\kappa^4}{\epsilon}{\rm i}ght).\label{eq:deltalearn}
\end{equation}
The swap test~\cite{BCWdW01} is then used to determine the accuracy of
the fit. The swap test is a method that can be used to distinguish $\ket{{\bf y}}$ and $\gamf{\bf I}({\bf F})\ket{\mbox{\boldmath{$\lambda$}}}$ by performing a swap operation on the
two quantum states controlled by a qubit in the state $(\ket{0}+\ket{1})/\sqrt{2}$. The Hadamard operation is then applied to the
control qubit and the control qubit is then measured in the computational basis. The test concludes that the states are different if the outcome
is ``1''. The probability of observing an outcome of ``1'' is $(1-\gamfb|{\bf r}a{{\bf y}}{\bf I}({\bf F})\ket{\mbox{\boldmath{$\lambda$}}}|^2)/2$ for our problem.
The overlap between the two quantum states can be learned by statistically sampling the outcomes from many instances of the swap
test. The value of $\gamfb|{\bf r}a{{\bf y}}{\bf I}({\bf F})\ket{\mbox{\boldmath{$\lambda$}}}|^2$ can be approximated using the sample mean of this distribution. It follows from estimates of the standard deviation of the mean that ${\cal O}(1/\delta^2)$ samples are required to estimate the mean within error
${\cal O}(\delta)$. The cost of algorithm 2 is then found by multiplying~\eqref{eq:deltalearn} by $1/\delta^2$.
The quantity $E$ can be estimated from the output of algorithm 2 by $E\le 2(1-\gamf|{\bf r}a{{\bf y}}{\bf I}({\bf F})\ket{\mbox{\boldmath{$\lambda$}}}|)$. Taylor series analysis shows
that the error in the upper bound for $E$ is also ${\cal O}(\delta)$.
There are several important limitations to this technique. First, if ${\bf F}$ is not sparse (meaning $s\in {\cal O}({\rm poly}(N))$) then the algorithm may not be efficient
because the quantum simulation step used in the algorithm may not be efficient.
As noted in previous results~\cite{Chi09,BC12,WBHS11}, we can generalize our results to systems where ${\bf F}$ is non-sparse if there exists a set of efficient unitary transformations
$U_j$ such that ${\bf I}({\bf F})=\sum_j U_j H_j U_j^\dagger$ where each $H_j$ is sparse and Hermitian. Also, in many important cases (such as fitting to experimental data) it may not be posible to prepare the initial
state $\ket{{\bf y}}$ efficiently. For this reason, our algorithm is better suited for approximating
the output of quantum devices than the classical outputs of
experiments. Finally, algorithm 2 only provides an efficient
estimate of the fit quality and does not provide $\mbox{\boldmath{$\lambda$}}$;
however, we can use it to determine whether
a quantum state has a concise representation within a family of states. If algorithm 2 can be used to find such a representation, then the parameters
$\ket{\mbox{\boldmath{$\lambda$}}}$ can be learned via state tomography. We discuss this
approach below.
{\emph{3. Learning $\mbox{\boldmath{$\lambda$}}$}-- This method can also be used to find a concise fit function that approximates ${\bf y}$. Specifically, we use statistical sampling
and quantum state tomography to find a concise representation for the quantum state using $M'$ parameters. The resulting
algorithm is efficient if $M'\in {\cal O}({\rm polylog}(N))$.}
{\flushleft\emph{Input}: As algorithm 2, but in addition with an integer $M'\in {\cal O}({\rm polylog}(M))$ that gives the
maximum number of fit functions allowed in the fit.}
{\flushleft\emph{Output}: A classical bit string approximating $\ket{\mbox{\boldmath{$\lambda$}}}$ to precision $\epsilon$, a list of the $M'$ fit functions that comprise $\ket{\mbox{\boldmath{$\lambda$}}}$ and $|{\bf r}a{{\bf y}}{\bf I}({\bf F})\ket{\mbox{\boldmath{$\lambda$}}}|^2$ calculated to precision $\delta$.}
{\flushleft\emph{Computational Model}: As algorithm 1, but the oracles can be controlled to either fit the state to all $M$ fit functions or any
subset consisting of $M'$ fit functions.}
{\flushleft \emph{Query Complexity}: $$\tilde {\cal O}\left( \log(N)s^3\left(\frac{\kappa^4}{\epsilon\delta^2}+\frac{M'^2\kappa^6}{\epsilon^3}{\rm i}ght){\rm i}ght).$$}
\emph{Algorithm}---
The first step of the algorithm is to prepare the state $\ket{\mbox{\boldmath{$\lambda$}}}$ using algorithm 1. The state is then measured ${\cal O}(M')$ times and a histogram
of the measurement outcomes is constructed. Since the probability of measuring each of these outcomes is proportional to their relevance to the fit, we are
likely to find the $M'$ of the most likely outcomes by sampling the state ${\cal O}(M')$ times.
After choosing the $M'$ most significant fit functions, we remove all other fit functions from the fit and prepare the state $\ket{\mbox{\boldmath{$\lambda$}}}$ using the reduced set of
fit functions. Compressed sensing~\cite{GYF+10,SKM+11,SML+10} is then used to reconstruct $\ket{\mbox{\boldmath{$\lambda$}}}$ within ${\cal O}(\epsilon)$ error. The idea of compressed sensing is that a low--rank density matrix can be uniquely determined (with high probability) by a small number of randomly chosen measurements. A convex optimization routine is then used to reconstruct the density matrix from the expectation values found for each of the measurements.
Compressed sensing requires ${\cal O}(M' \log(M')^2)$ measurement
settings to reconstruct pure states, and observation 1 of~\cite{GYF+10} implies that ${\cal O}(M'/\epsilon^2)$ measurements are needed for each setting to ensure that the reconstruction error is ${\cal O}(\epsilon)$; therefore, ${\cal O}(M'^2 \log(M')^2/\epsilon^2)$ measurements are needed to approximate the state within error ${\cal O}(\epsilon)$. The total cost of learning $\ket{\mbox{\boldmath{$\lambda$}}}$
is the number of measurements needed for tomography multiplied by the cost of preparing the state and thus scales as
\begin{equation}gin{equation}
\tilde {\cal O}\left(\log(N)\frac{ s^3M'^2\kappa^6}{\epsilon^3}{\rm i}ght),\label{eq:delta4}
\end{equation}
which subsumes the cost of measuring $\ket{\mbox{\boldmath{$\lambda$}}}$ to find the most significant $M'$ fit functions.
Finally, we measure the quality of the fit using algorithm 2. The total cost of estimating $\ket{\mbox{\boldmath{$\lambda$}}}$ and the fit quality is thus the sum of~\eqref{eq:delta4} and~\eqref{eq:deltalearn}, as claimed.
\emph{Remark}: The quality of the resulting fit that is yielded by this algorithm depends strongly on the set of fit functions that are used. If the fit functions are
chosen well, fewer than $M'$ fit functions are used to estimate $\ket{{\bf y}}$ with high fidelity. Conversely, ${\cal O}(N)$ fit functions may be needed to achieve the desired error tolerance if the fit functions are chosen
poorly.
Fortunately, the efficiency of algorithm 2 allows the
user to search many sets of possible fit functions for a concise and accurate model within a large set of potential models.
{\em Acknowledgements: } DB would like to thank the Joint Quantum Institute
(NIST and University of Maryland) and the Institute for Quantum Computing
(University of Waterloo), for
hospitality, and Arram Harrow and Avinatan Hassidim for useful
correspondence. NW would like to thank Andrew Childs and Dominic Berry for useful
feedback and acknowledges support from USAR{\cal O}/DT{\cal O}. SL is supported by NSF, DARPA, ENI and ISI.
\appendix
\section{Moore--Penrose Pseudoinverse}
Here we review an elementary proof~\cite{Pen56} of why applying Moore--Penrose pseudoinverse to the complex--valued vector ${\bf y}$ yields
parameters that minimize the least--squares fit of the initial state. To begin, we need to prove some properties of the pseudoinverse.
First,
\begin{equation}gin{equation}
({\bf F}{\bf F}^+)^\dagger = {\bf F} {\bf F}^+.\label{eq:suppprop1}
\end{equation}
The proof of this property is
\begin{equation}gin{equation}
({\bf F}{\bf F}^+)^\dagger = ({\bf F} ({\bf F}^\dagger{\bf F})^{-1}{\bf F}^\dagger)^{\dagger}=({\bf F} (({\bf F}^\dagger{\bf F})^{-1})^\dagger{\bf F}^\dagger).
\end{equation}
The result of~\eqref{eq:suppprop1} then follows by noting that ${\bf F}^\dagger{\bf F}$ is self--adjoint.
Next, we need the property that
\begin{equation}gin{equation}
{\bf F}{\bf F}^+{\bf F}={\bf F}.\label{eq:supprop2}
\end{equation}
This property follows directly from substituting in the definition of ${\bf F}^+$ into the expression.
The final property we need is
\begin{equation}gin{equation}
{\bf F}^\dagger({\bf F}{\bf F}^+{\bf y}-{\bf y})=0.\label{eq:suppprop3}
\end{equation}
Using property~\eqref{eq:suppprop1} we find that
\begin{equation}gin{equation}
{\bf F}^\dagger({\bf F}{\bf F}^+{\bf y}-{\bf y})=({\bf F}{\bf F}^+{\bf F})^\dagger{\bf y} - {\bf F}^\dagger{\bf y}.
\end{equation}
Property~\eqref{eq:supprop2} then implies that
\begin{equation}gin{equation}
({\bf F}{\bf F}^+{\bf F})^\dagger{\bf y} - {\bf F}^\dagger{\bf y}={\bf F}^\dagger{\bf y} - {\bf F}^\dagger{\bf y}=0.
\end{equation}
For simplicity, we will express ${\bf z}={\bf F}^+{\bf y}$ and then find
\begin{equation}gin{equation}
\|{\bf F} \mbox{\boldmath{$\lambda$}} - {\bf y}\|^2 = \|{\bf F}{\bf z} - {\bf y} +({\bf F}\mbox{\boldmath{$\lambda$}}-{\bf F}{\bf z})\|^2.
\end{equation}
Expanding this relation yields,
\begin{equation}gin{widetext}
\begin{equation}gin{equation}
\|{\bf F} \mbox{\boldmath{$\lambda$}} - {\bf y}\|^2=\|{\bf F}{\bf z}-{\bf y}\|^2 + \|{\bf F}(\mbox{\boldmath{$\lambda$}}-{\bf z})\|^2+({\bf F}{\bf z}-{\bf y})^\dagger{\bf F}(\mbox{\boldmath{$\lambda$}}-{\bf z})+(\mbox{\boldmath{$\lambda$}}-{\bf z})^\dagger{\bf F}^\dagger({\bf F}{\bf z}-{\bf y}).
\end{equation}
\end{widetext}
Property~\eqref{eq:suppprop3} then implies that ${\bf F}^\dagger({\bf F}{\bf z}-{\bf y})=0$ and hence
\begin{equation}gin{align}
\|{\bf F} \mbox{\boldmath{$\lambda$}} - {\bf y}\|^2&=\|{\bf F}{\bf z}-{\bf y}\|^2 + \|{\bf F}(\mbox{\boldmath{$\lambda$}}-{\bf z})\|^2\nonumber\\
&\ge \|{\bf F}{\bf F}^+{\bf y}-{\bf y}\|,
\end{align}
which holds with equality if $\mbox{\boldmath{$\lambda$}}={\bf z}={\bf F}^+{\bf y}$. Therefore, applying the Moore--Penrose peudoinverse to ${\bf y}$ provides
fit parameters that minimize the least--square error.
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\begin{document}
\title{
The Carath\'{e}odory-Cartan-Kaup-Wu theorem on an
infinite-dimensional Hilbert space}
\author{Joseph A. Cima, Ian Graham,\footnote{Author
supported in part by the Natural Sciences and Engineering Research
Council of Canada Grant \# A9221.} \\ Kang-Tae
Kim,\footnote{Author supported in part by the grant
KRF-2002-070-C00005 of The Korea Research Foundation.} and Steven
G. Krantz\footnote{Author supported in part by NSF Grant
DMS-9988854.}}
\maketitle
\begin{quote} \small
{\small \bf Abstract:} This paper treats a holomorphic
self-mapping $f: \Omega \rightarrow \Omega$ of a bounded domain
$\Omega$ in a separable Hilbert space ${\cal H}$ with a fixed
point $p$. In case the domain is convex, we prove an
infinite-dimensional version of the Cartan-Carath\'eodory-Kaup-Wu
Theorem. This is basically a rigidity result in the vein of the
uniqueness part of the classical Schwarz lemma. The main
technique, inspired by an old idea of H. Cartan, is iteration of
the mapping $f$ and its derivative. A normality result for
holomorphic mappings in the compact-weak-open topology, due to Kim
and Krantz, is used.
\end{quote}
\begin{quote}
{\bf AMS Subject Classification}:
Primary 32H02, 46G20
\end{quote}
\setcounter{section}{-1}
\section{Introduction}
Perhaps the most important part of the classical Schwarz lemma is
the uniqueness statement: If $f: D \rightarrow D$ is a holomorphic
function from the unit disc $D$ to itself, $f(0) = 0$, and
$|f'(0)| = 1$, then $f$ is a rotation. This rigidity statement has
had considerable effect in the subject of complex differential
geometry. It is the wellspring of many holomorphically invariant
metrics, and has had a notable influence in the subject of mapping
theory.
The analogous result in higher (finite) dimensions has been
explored by Carath\'{e}odory, Cartan, Kaup, and Wu. See [KRA] and
[WU] for a careful discussion of the matter. The theorem is that,
if a holomorphic mapping $f:\Omega \to \Omega$ of a bounded domain
$\Omega$ in ${\mathbb C}^n$ satisfies $f(p)=p$ for some $p \in \Omega$ and
$|\det (Df(p))|=1$, then $f$ is a biholomorphic mapping. We remark
here that the condition on the Jacobian determinant is, in this
case, equivalent to saying that every eigenvalue of the Jacobian
matrix has modulus 1.
In recent years, the study of holomorphic functions and mappings
on a Hilbert or Banach space has received much attention. Work of
Lempert (see for instance [LEM]) has served as a catalyst to this
activity. The more recent work of Kim/Krantz (see [KIK1], [KIK2])
is the more direct genesis of the present paper. In this rather
general setting, many of the familiar tools of finite-dimensional
analysis are no longer available. The geometry is much more
difficult. Yet there is interest in developing these ideas because
of potential applications to mathematical physics and partial
differential equations.
This paper develops a version of the
Carath\'{e}odory-Cartan-Kaup-Wu theorem on a separable Hilbert
space ${\cal H}$. We begin by formulating and proving a result on
a convex domain $\Omega \subseteq {\cal H}$. This is Theorem 1.1.
Afterward, in Section 3, we present a more refined result
(Proposition 3.1) on the ball in a separable Hilbert space.
The authors thank the Banff International Research Station (BIRS)
for hosting us during the work on this problem, and for providing
a stimulating working environment. We thank John McCarthy, Peter
Rosenthal and Warren Wogen for advice and information about
analysis on Hilbert spaces.
\section{Statement of the main results}
Let ${\mathcal H}$ be a separable, complex Hilbert space, and let $B$
denote its open unit ball. Let $S^1$ represent the unit circle in
the complex plane ${\mathbb C}$. For a bounded linear operator $T$ on
${\mathcal H}$, denote by $\sigma(T)$ its spectrum, i.e.
$$
\sigma(T) = \set{ \lambda \in {\mathbb C} \mid T - \lambda I
\hbox{ is not
invertible} } \, .
$$
A bounded linear operator $T: {\mathcal H} \to {\mathcal H}$ is said to be {\it
triangularizable} if there exists a basis $\set{e_1,e_2,\ldots}$
such that $T({\mathbb C} e_1 + \ldots + {\mathbb C} e_N) \subset {\mathbb C} e_1 + \ldots
+ {\mathbb C} e_N$ for every positive integer $N$. In this case, we shall
sometimes say that $T$ is upper-triangular with respect to the
basis $\set{e_1, e_2, \ldots}$. Note that the basis that renders
an operator upper-triangular can always be taken to be
orthonormal, just because the Gram-Schmidt process preserves the
flags $E_N = {\mathbb C} e_1 + \cdots + {\mathbb C} e_N$.
Now we formulate the principal result of this paper:
\begin{theorem} \label{Main} \sl
Let $\Omega \subseteq {\cal H}$ be a bounded convex domain.
Fix a
point $p \in \Omega$. Let $f:\Omega \rightarrow \Omega$ be
a
holomorphic mapping such that
\begin{enumerate}
\item[{\bf (a)}] $f(p) = p$ \, ; \item[{\bf (b)}] the
differential $df_p$ is triangularizable; \item[{\bf (c)}]
$\sigma(df_p) \subseteq S^1$ \, .
\end{enumerate}
Then $f$ is a biholomorphic mapping.
\end{theorem}
To put the theorem in perspective, notice that in dimension $n =
1$ the hypothesis specializes to just $|f'(p)|=1$. In several (but
finitely many) variables, the hypothesis is equivalent to {\bf
(i)} non-degeneracy of the Jacobian matrix at $p$ and {\bf (ii)}
all eigenvalues of the Jacobian having modulus 1. In the
finite-dimensional case, one can always triangularize. In infinite
dimensions there are geometric conditions (involving the Fredholm
index) for triangularization.
Even if $T$ is a unitary operator on a Hilbert space ${\cal H}$
(so that certainly $\sigma(T) \subseteq S^1$), it does not
necessarily follow that $T$ is triangularizable. For example, take
$T$ to be the forward shift on $\ell^2({\mathbb Z})$. See [HER] for more
on the triangular operators.
Concerning the assumption {\bf (b)} in Theorem {\rm Re}\,f{Main}, it
seems appropriate to present an example of a bounded convex domain
with an automorphism whose derivative at a fixed point is
upper-triangular. Consider the map $A:{\mathbb C}^2 \to {\mathbb C}^2$ defined by
$A(z,w) = (z+bw, -w)$ for an arbitrary choice for $b \in
\mathbb{{\mathbb C}} \setminus \set{0}$. Then $A$ is an involution (i.e.
$A \circ A = I$) with eigenvalues $\pm 1$; also $A$ is an
upper-triangular linear map. Take any bounded domain $D$ in
${\mathbb C}^2$ containing the origin. Let
$$
V = D \cup A(D).
$$
Then let $\Omega$ be the convex hull of $V$ in ${\mathbb C}^2$. It is
obviously a bounded convex domain containing the origin. Then $A$
is an automorphism of $\Omega$ satisfying $A(0)=0$. Since $dA_0 =
A$, the differential $dA_0$ is upper-triangular. Furthermore, if
one would sacrifice the convexity of the domain, it is possible
also to obtain a non-linear example. Let $G(z,w) = (z, w+z^2)$ for
instance, let $f = G \circ A \circ G^{-1}$, and let $W =
G(\Omega)$. Then $W$ is a bounded domain in ${\mathbb C}^2$, and $f$ is an
automorphism of $W$ with $f(0)=0$. We see that $df_0$ is clearly
upper-triangularizable. It is simple to modify this example to
give an example in an infinite dimensional Hilbert space; one
simply needs to consider a map that is equal to $A$ on a
two-dimensional subspace and the identity on the orthogonal
complement of this subspace.
The referee has raised the question of how large is the class of
holomorphic mappings $f:\Omega\rightarrow \Omega$ satisfying the
conditions {\bf (a)}, {\bf (b)} and {\bf (c)} of Theorem 1.1,
within the class of such mappings satisfying only {\bf (a)} and
{\bf (c)}. We believe that rigidity phenomena for automorphisms of
domains in infinite dimensions would imply that the answer depends
very much on the domain. A related question is how large is the
class of triangularizable operators on a separable Hilbert space
$H$ within the class of all bounded linear mappings of $H$. In
this connection we mention the Weyl-von Neumann-Berg theorem [DAV,
p.\ 59], which asserts that every normal operator is a small
compact perturbation of a diagonalizable (in particular
triangularizable) operator. We thank the referee and the editors
for their remarks.
\section{Proof of Theorem~{\rm Re}\,f{Main}}
The proof has several steps, some of which are of
independent interest.
\subsection{Basic Facts on the Differential}
Let $\Omega_1$ and $\Omega_2$ be domains in the separable Hilbert
space ${\mathcal H}$ and let $f:\Omega_1\to\Omega_2$ be a holomorphic
mapping. The differential of $f$ at a point $p\in\Omega_1$ is
the bounded linear operator on ${\mathcal H}$ defined by
$$
df_p (v) = \lim_{{\mathbb C}\ni h \to 0} \frac{f(p+hv) -
f(p)}{h}.
$$
Write $B(p;r)=\{z\in {\mathcal H} \mid \|z-p\| < r\}$.
Suppose that $\Omega_2$ is bounded, say $\Omega_2\subseteq B(0;M)$.
In this situation we have
\begin{lemma} \sl
If $p\in\Omega_1$ and $\hbox{dist}(p,\partial \Omega_1) = d >0$, then
$$\|df_p\|\le \frac{M}{d}\, .$$
\label{diffone}
\end{lemma}
\noindent\it Proof. \rm
Choose $\rho > 0$ so that $\overline B(p;\rho) \subset \Omega_1$.
Let $v$ be a unit vector in ${\mathcal H}$. The integral representation
\begin{equation} \label{Cauchyformula}
df_p(v)=\frac{1}{2\pi
i}\int_{|\zeta|=\rho}\frac{f(p+\zeta
v)}{\zeta^2} d\zeta
\end{equation}
leads immediately to the Cauchy estimates
$$
\|df_p(v)\|\le \frac{M}{\rho},
$$
and we may let $\rho$ tend to $d$.
$\Box\;$
\\
\begin{lemma} \sl
If $\|df_z\|\le A$ for every $z\in B(p;r)$, then for any
pair $z,w\in B(p;r)$ one has
$$
\|f(z)-f(w)\| < 2rA \, .
$$
\label{difftwo}\end{lemma}
\noindent\it Proof. \rm
Write
$$
f(z)-f(w)=\int_0^1 \frac{d}{dt}[f((1-t)w+tz)] dt
$$
and use the chain rule.
$\Box\:$
\\
Now, in the situation of Theorem {\rm Re}\,f{Main}, $f$ is a holomorphic
self-map of a bounded convex domain in ${\mathcal H}$ and $p$ is a fixed
point. Let us write $T=df_p$.
We are assuming that $T$ is triangularizable. Hence it is
possible to choose a basis $e_1, e_2, \ldots$ such that
$$
T({\mathbb C} e_1 + \cdots + {\mathbb C} e_N) \subset {\mathbb C} e_1 + \cdots + {\mathbb C}
e_N
$$
for every positive integer $N$. For convenience set
$$
E_N = {\mathbb C} e_1 + \cdots + {\mathbb C} e_N
$$
for every positive integer $N$. We shall call such $E_N$ a
\it
flag. \rm The union of these flags yields a vector space
that is
dense in ${\mathcal H}$.
\\
{From} here on we assume that the operator $T$ is
upper-triangular
with respect to a {\sl fixed} orthonormal basis $e_1,e_2,
\ldots$.
\\
Notice that $\sigma(T)$ is contained in the unit circle. Since $T$
is upper-triangular, the diagonal entries are contained in its
spectrum $\sigma(T)$, and hence are of modulus 1.
\\
We also note the following consequence of Lemma {\rm Re}\,f{diffone}:
\begin{corollary} \sl
There exists a constant $C$ such that
$$
\|T^m\| \le C,
$$
for every positive integer $m$. \label{corol}
\end{corollary}
\subsection{Iteration of $T$}
In this section $T$ is an upper triangular matrix whose spectrum
$\sigma(T)$ is contained in $S^1$ and whose positive powers are
uniformly bounded in norm. Denote by $\lambda_j$ the $(j,j)$-th
diagonal entry of $T$. Then $|\lambda_j|=1$, $j\in{\mathbb N}$.
\begin{lemma} \sl
There exists a sequence of natural numbers $\{m(k)\}_k$
such that for each fixed $j\in{\mathbb N}$, $\lambda_j^{m(k)}\to 1$
as $k\to\infty$.
\label{T1}
\end{lemma}
\noindent \it Proof. \rm
The sequence of powers $\{\lambda_1^k\}$ is bounded, so there
is a convergent subsequence, say $\lambda_1^{m(1,k)}\to \alpha_1$ as
$k\to\infty$, where $|\alpha_1|=1$. Similarly the sequence of powers
$\{\lambda_2^{m(1,k)}\}$ is bounded, so there is a subsequence
$\{m(2,k)\}$ of $\{m(1,k)\}$ such that
$\lambda_2^{m(2,k)}\to\alpha_2$ as $k\to\infty$, where $|\alpha_2|=1$.
Continuing in this way and using a diagonal sequence argument, we
obtain a subsequence $\{m(k,k)\}$ of the natural numbers and
complex numbers $\alpha_j$, $j\in{\mathbb N}$, of modulus one such that
$\lambda_j^{m(k,k)}\to\alpha_j$ as $k\to\infty$ for each $j\in{\mathbb N}$.
Now the sequence
$$m(k)=m(k+1,k+1)-m(k,k), \quad k\in{\mathbb N}$$
is easily seen to have the property that
$$
\lambda_j^{m(k)}\to\frac{\alpha_j}{\alpha_j} = 1
$$
as $k\to\infty$, for each $j\in{\mathbb N}$.
$\Box\;$
\\
If $A$ is an infinite square matrix we denote the $(i,j)$-th entry
by $A_{i,j}$, $1\le i,j\le\infty$. Also, we order the positions above
the diagonal in such a matrix first by column and then by row, i.e.
the ordering is $(1,2)$, $(1,3)$, $(2,3)$, $(1,4)\ldots$. If $N$
is a positive integer, we denote the $N\times N$ truncation of such
a matrix by $A_N$. The $N\times N$ identity matrix will be denoted
by $I_N$.
\begin{lemma} \sl
There is a sequence of natural numbers $\{\mu(k)\}_k$
such that, for each $N\in{\mathbb N}$,
$$
(T_N)^{\mu(k)}\to I_N, \quad k\to\infty \, .
$$
\label{T2}
\end{lemma}
\noindent \it Proof. \rm
Write $T = S+V$, where $S$ is a diagonal matrix with diagonal
entries of modulus 1 and $V$ has zeros on and below the main
diagonal. Denote the diagonal entries of $S$ by $\lambda_j,
j=1,2,\ldots.$
By Lemma {\rm Re}\,f{T1}, there exists a sequence of natural
numbers $\{m(k)\}$ such that $\lambda_j^{m(k)}\to 1$ as
$k\to\infty$, for each fixed $j$.
The entries $\{(T^{m(k)})_{1,2}\}_k$ are bounded, so there is
a subsequence $\{\mu(k;1,2)\}$ of $\{m(k)\}$ such that
$\{(T^{\mu(k;1,2)})_{1,2}\}_k$ converges. By similar reasoning, a further
subsequence $\{\mu(k;1,3)\}$ has the property that
$\{(T^{\mu(k;1,3)})_{1,3}\}_k$ converges, and a further subsequence
of that one, denoted by $\{\mu(k;2,3\}$, has the property that
$\{(T^{\mu(k;2,3)})_{2,3}\}_k$ converges. Continuing in this way and
extracting a diagonal subsequence yields a subsequence $\{\mu(k)\}$
of $\{m(k)\}$ such that $\{(T^{\mu(k)}))_{i,j}\}_k$ converges for all
$(i,j),\quad 1\le i < j < \infty$.
Thus there is an infinite square matrix $W$, whose entries on and below
the main diagonal are zero, such that for each $N\in\mathbb{N}$
we have $(T_N)^{\mu(k)}\to I_N + W_N$.
(For fixed $N$ the convergence
may be taken to be in norm, but the norm convergence is not uniform in $N$.)
Now choose the smallest value of $N \ge 2$ with the
property that $W_{N}\ne 0$. Then at least one of the entries
in the last column of $W_N$ is nonzero, and all entries in the other
columns of $W_N$ are zero. If $\ell$ is a positive integer, then
$$((T_N)^{\mu(k)})^\ell = (T_N)^{\mu(k)\ell}
\to (I_N + W_N)^\ell, \quad k\to\infty,$$
and the entries in the last column of the matrix on the right
are given by $\ell$ times the corresponding entries in $(I_N + W_N)$
(except for a 1 in the $(N,N)$-position).
But this is a contradiction to the power boundedness of $T$ for
sufficiently large $\ell$ (see Corollary {\rm Re}\,f{corol}).
We conclude that no such $N$ exists, i.e. $W=0$.
Therefore the sequence $\{\mu(k)\}$ has the property that
$(T_N)^{\mu(k)}\to I_N$ as $k\to\infty$ for each fixed $N$.
$\Box\;$
\subsection{Iteration of $f$}
Next, consider the iteration given by $f^1 = f$, $f^m =
f\circ
f^{m-1}$ for each integer $m > 1$.
We need the following two fundamental results. There is in fact a Banach
space version of the theorem of Kim and Krantz [KIK2]; we indicate
a proof here for the Hilbert space case.
\begin{theorem}[Kim/Krantz \hbox{ \bf [KIK2]}] \sl
Let $\Omega_1, \Omega_2$ be domains in a separable Hilbert space
${\mathcal H}$, and let $\Omega_2$ be bounded. Then every sequence
$\set{h_n : \Omega_1\to \Omega_2 \mid n=1,2,\ldots}$ of
holomorphic mappings admits a subsequence $\{h_{n(k)}\}_k$ that
converges to a holomorphic mapping $\widehat h$ from $\Omega_1$
into the closed convex hull of $\Omega_2$, in the
compact-weak-open topology (i.e., the compact-open topology in
which the strong topology is used on the domain space and the weak
topology is used on the target space). \label{KK}
\end{theorem}
\noindent \it Proof. \rm
Let $\langle \, \cdot \, , \, \cdot \, \rangle$ denote the Hilbert
space inner product (linear in the first variable,
conjugate-linear in the second). One needs to show that for every
sequence $\set{h_n: \Omega_1 \to \Omega_2 \mid n=1,2, \ldots}$ of
holomorphic mappings, there is a subsequence $\{h_{n(k)}\}_k$ and
a holomorphic mapping $\widehat h : \Omega_1 \to {\mathcal H}$ such that,
for all $g$ in the unit ball of ${\mathcal H}$, $\langle
h_{n(k)},g\rangle \to \langle \widehat h,g \rangle$, uniformly on
compact subsets, as $k\to\infty$. If $g$ is such that $\Re \langle
w,g\rangle < 1$ for all $w\in\Omega_2$, then it is clear that $\Re
\langle \widehat h,g\rangle \le 1$, i.e. the image of $\widehat h$
must be contained in the closed convex hull of $\Omega_2$.
Let $\{z_n\}_{n\in{\mathbb N}}$ be a dense sequence in $\Omega_1$.
We are going to do the usual diagonal sequence construction.
The sequence $\{h_n\}$ has a subsequence $\{h_{n(1,k)}\}_k$
such that $h_{n(1,k)}(z_1)$ converges weakly to an element
$\widehat h(z_1)$ as $k\to\infty$. This just uses the boundedness
of $\Omega_2$.
Now choose a subsequence of $\{n(1,k)\}_k$, denoted by
$\{n(2,k)\}_k$, so that $h_{n(2,k)}(z_2)$ converges weakly to
an element $\widehat h(z_2)$ as $k\to\infty$. Continue and then
choose the diagonal sequence $\{h_{n(k)}\}_k=\{h_{n(k,k)}\}_k$
generated by this process.
Since $\Omega_2$ is bounded, as we did earlier in the proof of
Lemma 2.1, we let $M$ be a positive constant such that $\Omega_2
\subset B(0; M)$.
Assume that $K$ is a compact subset of $\Omega_1$ and let
$\epsilon > 0$. Denote by $D= \hbox{dist }(K,\partial\Omega_1)$
and let $\delta =\min\{\frac{D}{3}, \frac{D\epsilon}{18M}\}$.
Cover $K$ with a finite number of balls $B(x_1;\delta),\ldots,
B(x_m; \delta)$ such that $B(x_\ell; \delta) \cap K \neq
\emptyset$ for any $\ell = 1,2,\ldots,m$, where each $x_\ell$
belongs to the dense sequence $\{z_n\}_{n\in{\mathbb N}}$. Note that for
$z\in B(x_\ell;\delta)$ one has $\hbox{dist}(z,\partial
\Omega_1)
> D/3$. By Lemma {\rm Re}\,f{diffone} we have $\|dh_n|_z \| <
\frac{3M}{D}$ for such $z$. This holds for every $n$. By Lemma
{\rm Re}\,f{difftwo} and the choice of $\delta$, we have
\begin{equation} \label{equicontinuity}
\|h_n(z)-h_n(w)\| < \frac{6M\delta}{D} \le \frac{\epsilon}{3}
\end{equation}
for all $z,w\in B(x_\ell,\delta)$.
For any $g$ in the unit ball of $H$, choose $J=J(g)$ so that,
for $j,k > J$, we have
$$
|\langle h_{n(j)}(x_\ell)-h_{n(k)}(x_\ell),g\rangle| < \frac{\epsilon}{3}
$$
for $1\le \ell\le m$.
Now for any choice of $z\in K$ we have the existence of some $\ell$
with $z\in B(x_\ell,\delta)$.
Then for $j,k > J$ we have
\begin{eqnarray*}
|\langle h_{n(j)}(z)-h_{n(k)}(z),g\rangle|\, & \le & \,
|\langle h_{n(j)}(z)-h_{n(j)}(x_\ell),g\rangle| \\
&& \quad \, + \, |\langle h_{n(j)}(x_\ell)-h_{n(k)}(x_\ell),g\rangle| \\
&& \quad \, + \, |\langle h_{n(k)}(x_\ell)-h_{n(k)}(z), g \rangle| \, .
\end{eqnarray*}
Each of the terms on the right hand side is less than
$\epsilon/3$.
For the first and third terms this follows from ({\rm Re}\,f{equicontinuity}),
and for the second term it follows from the choice of $J$.
This shows that the sequence $h_{n(k)}(z)$ converges weakly to
the ``assignment'' $\widehat h(z)$ uniformly on compacta. It remains
only to show that the assignment is an analytic mapping on $\Omega_1$.
But given a fixed $z_0\in\Omega_1$ and a unit vector $v\in{\mathcal H}$,
we can find $a>0$ such that the closed disc
$$
S=\{z_0+\zeta v \mid |\zeta|\le a\}
$$
is contained in $\Omega_1$, and hence the analytic functions
$\langle h_{n(k)}(z_0+\zeta v),g\rangle$ converge uniformly to
$\langle \widehat h(z_0+\zeta v),g\rangle$. The mapping
$\zeta\mapsto \widehat h(z_0+\zeta v)$ from the disc of radius $a$
in ${\mathbb C}$ into ${\mathcal H}$ is therefore holomorphic. This says
precisely that $\widehat h$ is Gateaux holomorphic [HIP], [MUJ].
Since a bounded Gateaux holomorphic mapping is holomorphic, we are
done.
$\Box\;$
\begin{remark} \sl
By the Cauchy estimates, it follows from Theorem {\rm Re}\,f{KK} that
the
derivative $dh_{n(k)}|_z (v)$ converges to $d\widehat h_z
(v)$
weakly, uniformly on compact subsets of $\Omega_1 \times
{\mathcal H}$
(i.e., $z \in \Omega_1$ and $v \in {\cal H}$).
\end{remark}
To see this,
let $L$ be a compact subset of $\Omega_1\times{\mathcal H}$.
Then for all $(z,v)\in L$,
we have $\hbox{dist}(z,\partial\Omega_1) \ge a > 0$ and $\|v\| \le b$,
for some positive constants $a$ and $b$.
It is elementary to see that there exists $r > 0$ such that
$$
K=\{z+\zeta v \mid (z,v)\in L, |\zeta|\le r \}
$$
is a compact subset of $\Omega_1$.
Now the relation ({\rm Re}\,f{Cauchyformula}) gives
$$
dh_{n(k)}|_z(v) - dh_z(v) = \frac{1}{2\pi i}\int_{|\zeta|=r}
\frac{h_{n(k)}(z+\zeta v)-h(z+\zeta v)}{\zeta^2} d\zeta \quad ,
$$
for all $(z,v)\in L$. Hence for any linear functional $\tau$
on ${\mathcal H}$,
$$
\tau\circ dh_{n(k)}|_z(v) - \tau\circ dh_z(v)
$$
$$
=\frac{1}{2\pi i}\int_{|\zeta|=r} \frac{\tau\circ h_{n(k)}(z+\zeta
v) -\tau\circ h(z+\zeta v)}{\zeta^2} d\zeta \, .
$$
The assertion follows easily from this in view of the compactness
of $K$.
\\
\begin{theorem}[H. Cartan] \sl
Let $\Omega$ be a bounded domain in ${\mathcal H}$ and let $p \in
\Omega$. Let $f:\Omega \to \Omega$ be a holomorphic mapping with
$f(p)=p$ and $df_p = I$. Then $f$ coincides with the identity
mapping of $\Omega$. \label{Cartan}
\end{theorem}
See [FRV], [KRA] for the proof of Cartan's theorem.
\\
Now apply Theorem {\rm Re}\,f{KK} to the sequence $\{f^{\mu(k)}\}$, where
$f$ is the mapping in Theorem 1.1 and $\{\mu(k)\}$ is the sequence
constructed in Lemma {\rm Re}\,f{T2}. We obtain a subsequence
$\{f^{\nu(k)}\}$ that converges to some $\widehat{f}$ in the
compact-weak-open-topology. The sequence of derivatives at $p$ is
$\{T^{\nu(k)}\}$. The discussion of the preceding section shows
that
$$
\lim_{k\to\infty} (T_N)^{\nu(k)} = I_N
$$
for every positive integer $N$.
Thus
$$
d{\widehat f}_p|_{E_N} = I_{E_N} \hbox{ for every } N =
1,2,\ldots
\, .
$$
Note that $\displaystyle{\bigcup_N E_N}$ is dense in ${\cal
H}$.
Since $d\widehat{f}_p$ is bounded and $d\widehat{f}_p \bigr
|_{E_N} = I_N$ for all $N$, it follows that $d\widehat{f}_p
= I$.
Finally, the fact that $\widehat{f}(p) = p$ together with the
convexity of $\Omega$ implies that $\widehat f(\Omega)\subseteq
\Omega$. Now Cartan's theorem (Theorem {\rm Re}\,f{Cartan}) implies that
$\widehat{f} \equiv \hbox{id}$.
\subsection{Proof of Theorem {\rm Re}\,f{Main}}
It is time to complete the proof of Theorem {\rm Re}\,f{Main}. From
the
last part of the preceding subsection, we have
\begin{equation} \label{tada-1}
\lim_{k \to \infty} f (f^{\nu(k)-1} (z)) = z = \lim_{k \to \infty}
f^{\nu(k)-1} (f(z))
\end{equation}
in the compact-weak-open topology on $\Omega$.
\\
By Theorem {\rm Re}\,f{KK}, choose a subsequence $\set{f_\ell}$ of
$\set{f^{\nu(k)-1}}$ that converges to a holomorphic mapping
$\widehat h:\Omega \to \Omega$ in the compact-weak-open topology.
(Recall that $\Omega$ is convex and $\widehat h(p)=p$).
The second identity in ({\rm Re}\,f{tada-1}) implies that
\begin{equation}
\widehat h \circ f = \hbox{\rm id}. \label{tada-2}
\end{equation}
We see therefore that the holomorphic mapping $\widehat h$ is a
left inverse of $f$, and that $f \circ \widehat{h}$ is a
holomorphic mapping from $\Omega$ into itself.
\\
On the other hand, one cannot immediately deduce from the first
identity in ({\rm Re}\,f{tada-1}) that $f \circ \widehat h = \hbox{\rm
id}$, because it is only known at this point that $f_\ell$
converges to $\widehat h$ weakly. (With respect to the weak
topology on the source-domain and the strong topology on the
target-domain, holomorphic mappings need not be continuous.) So it
is necessary to show that $f \circ \widehat h = \hbox{\rm id}$.
\\
Now ({\rm Re}\,f{tada-2}) implies that
$$
d{\widehat h}_p \circ df_p = I. \label{tada-3}
$$
Since $df_p$ is invertible, we see that $d{\widehat h}_p$ is
also
the right inverse of $df_p$. This implies that
$$
d(f \circ \widehat h)_p = df_p \circ d\widehat{h}_p = I.
$$
Applying Cartan's Theorem (Theorem {\rm Re}\,f{Cartan}) again to
$f\circ
\widehat h:\Omega \to \Omega$, we see that
$$
f \circ \widehat h = \hbox{\rm id}.
$$
Therefore $f$ is a biholomorphic mapping of $\Omega$ onto itself.
This completes the proof.
$\Box\;$
\section{Closing Remarks}
For holomorphic functions (i.e. ${\mathbb C}$-valued functions) on a
domain in a separable Banach space, there is a normality result
for the compact-open topology, similar to the finite-dimensional
case [HUY], [KIK2], [MUJ]. For holomorphic mappings there is no
such result unless one makes further restrictions [HUY]. However,
it is possible to obtain interesting theorems about holomorphic
mappings using normality with respect to the compact-weak-open
topology.
The assumption that the differential be triangularizable at the
fixed point is not necessary for the conclusion of Theorem
{\rm Re}\,f{Main} to be valid. In the case of the unit ball $B$ of the
Hilbert space ${\mathcal H}$, a unitary map conjugated by a M\"obius
transformation is a holomorphic automorphism, but in general the
differential at the fixed point is not triangularizable. This
phenomenon reflects the difficulty in the case of infinite
dimensional holomorphy caused by the excessive size of the
isotropy group in such cases as the ball (in finite dimensions,
large isotropy group characterizes the ball---see [GRK]). However,
the ball case has a reasonable formulation as follows.
\begin{proposition} \sl
Let $f:B \to B$ be a holomorphic mapping, let $p \in B$, and let
$M_p$ be a M\"obius transformation of the ball sending $p$ to the
origin. If $f$ satisfies:
\begin{itemize}
\item[\rm (1)] $f(p)=p$,
\item[\rm (2)] $U \circ d[M_p]|_p \circ df_p \circ d[M_p^{-1}]|_0$
is triangularizable, for some unitary transform $U$, and
\item[\rm (3)] $\sigma(U \circ d[M_p]|_p \circ df_p \circ
d[M_p^{-1}]|_0)$ lies in the unit circle in ${\mathbb C}$,
\end{itemize}
then $f$ is an automorphism of $B$.
\end{proposition}
The arguments given in this article surely yield a proof of
Proposition 3.1. However, one can simplify the argument, thanks to
the fact that {\it the domain in consideration is the unit ball}.
Indeed, from Schwarz's lemma, upper triangularity and the spectrum
condition, one obtains that the operator $V = U \circ d[M_p]|_p
\circ df_p \circ d[M_p^{-1}]|_0$ has norm 1. It is known [DIN]
that such a linear transformation $V$ is in fact unitary. Thus
Cartan's Theorem applied to $V^{-1} \circ U \circ M_p \circ f
\circ M_p^{-1}$ implies that this map is the identity map. Hence
$f$ is a holomorphic automorphism of $B$. The authors would like
to thank Warren Wogen for pointing out this line of reasoning.
The convexity assumption on $\Omega$ was necessary due to the use
of weak convergence in several places. Whether one can remove
this additional assumption should be an interesting problem to
explore in future work. It would also be of interest to know
whether there is a result of Schwarz-Pick type in our
infinite-dimensional context.
\\
\noindent {\Large \bf References} \vspace*{.15in}
\begin{enumerate}
\item[{\bf [BOM]}] S. Bochner and W. T. Martin, {\it
Functions of
Several Complex Variables}, Princeton University Press,
Princeton,
1936.
\item[{\bf [CON]}] J. B. Conway, {\it A Course in Operator
Theory}, American Mathematical Society, Providence, RI,
2000.
\item[{\bf [DAV]}] K. R. Davidson, {\it $C^*$-algebras by
example}, Fields Institute Monographs, American Mathematical
Society, Providence, RI, 1996.
\item[{\bf [DIN]}] S. Dineen, {\it The Schwarz Lemma}, The
Clarendon Press, Oxford University Press, Oxford, 1989.
\item[{\bf [DUS]}] N. Dunford and J. T. Schwartz, {\it
Linear
Operators}, Interscience, New York, 1988.
\item[{\bf [FRV]}] T. Franzoni and E. Vesentini, {\it
Holomorphic
Maps and Invariant Distances}, North-Holland, Amsterdam,
1980.
\item[{\bf [GRK]}] R. E. Greene and S. G. Krantz,
Characterization of complex manifolds by the isotropy subgroups of
their automorphism groups, {\it Indiana Univ. Math. J.} 34(1985),
865-879.
\item[{\bf [HER]}] D.A. Herrero, Triangular operators, {\it Bull.
London Math. Soc.}, 23 (1991), 513-554.
\item[{\bf [HIP]}] E. Hille and R. S. Phillips, {\it Functional
Analysis and Semigroups}, Amer. Math. Soc. Coll. Publ. 31,
Providence, R. I., 1957.
\item[{\bf [HUY]}] C.-G. Hu and T.-H. Yue, Normal families
of holomorphic mappings, {\it J. Math. Anal. Appl.} 171(1992),
436-447.
\item[{\bf [KIK1]}] K.-T. Kim and S. G. Krantz,
Characerization of
the Hilbert ball by its automorphism group, {\it Trans.\
Amer.\
Math.\ Soc.} 354(2002), 2797--2818.
\item[{\bf [KIK2]}] K.-.T. Kim and S. G. Krantz, Normal
families
of holomorphic functions and mappings on a Banach space,
{\it
Expo.\ Math.} 21(2003), 193--218.
\item[{\bf [KRA]}] S. G. Krantz, {\it Function Theory of
Several
Complex Variables}, American Mathematical Society-Chelsea,
Providence, RI, 2001.
\item[{\bf [LEM]}] L. Lempert, The Dolbeault complex in
infinite
dimensions, {\it J. Amer.\ Math.\ Soc.} 11(1998), 485--520.
\item[{\bf [MUJ]}] J. Mujica, {\it Complex Analysis in Banach
Spaces}, North-Holland, Amsterdam and New York, 1986.
\item[{\bf [NAR]}] R. Narasimhan, {\it Several Complex
Variables},
University of Chicago Press, Chicago, 1971.
\item[{\bf [WU]}] H. H. Wu, Normal families of holomorphic
mappings, {\it Acta Math.} 119(1967), 193--233.
\end{enumerate}
\vspace*{.2in}
\noindent Department of Mathematics \\
University of North Carolina \\
Chapel Hill, North Carolina 27514 \ \ USA \\
{\tt [email protected]}
\\
\noindent Department of Mathematics \\
University of Toronto \\
Toronto, CANADA \\
M5S 3G3 \\
{\tt [email protected]}
\\
\noindent Department of Mathematics \\
Pohang University of Science and Technology \\
Pohang 790-784 KOREA \\
{\tt [email protected]}
\\
\noindent Department of Mathematics \\
Campus Box 1146 \\
Washington University in St. Louis \\
St.\ Louis, Missouri 63130 \ \ USA \\
{\tt [email protected]}
\end{document} |
\begin{equation}gin{document}
\begin{equation}gin{center}
\lambdarge{ \bf Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation}
\end{center}
\begin{equation}gin{center}
{Tao Luo, Zhouping Xin, Huihui Zeng}
\end{center}
\begin{equation}gin{abstract} This paper concerns the well-posedness theory of the motion of physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in \cite{10',7,16'} by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary. \end{abstract}
\thetableofcontents
\section{Introduction}
Due to its great physical importance and mathematical challenges, the motion of physical vacuum in compressible fluids has received much attention recently (cf. \cite{29,39,23, 24, 25}), and significant progress has been made in particularly on the Euler equations (cf. \cite{7, 10, 10', 16, 16', 15}). Physical vacuum problems arise in many physical situations naturally, for example, in the study of the evolution and structure of gaseous stars (cf. \cite{6',cox}) for which vacuum boundaries are natural boundaries. This paper is concerned with the evolving boundary of stars (the interface of fluids and vacuum states) in a compressible self-gravitating gas flow, which is modeled by
\begin{equation}\lambdabel{2.1} \begin{equation}gin{split}
& \partial_t \rho + {\rm div}(\rho {\bf u}) = 0 & {\rm in}& \ \ \Omega(t), \\
& \partial_t (\rho {\bf u}) + {\rm div}(\rho {\bf u}\otimes {\bf u})+\nablabla_{\bf x} p(\rho) = -\kappappa \rho \nablabla_{\bf x} \Psi & {\rm in}& \ \ \Omega(t),\\
&\rho>0 &{\rm in } & \ \ \Omega(t),\\
& \rho=0 & {\rm on}& \ \ \Gamma(t):=\partial \Omega(t),\\
& \mathcal{V}(\Gamma(t))={\bf u}\cdot {\bf n}, & &\\
&(\rho,{\bf u})=(\rho_0, {\bf u}_0) & {\rm on} & \ \ \Omega:= \Omega(0).
\end{split} \end{equation}
Here $({\bf x},t)\in \mathbb{R}^3\times [0,\infty)$, $\rho $, ${\bf u} $, $p$ and $\Psi$ denote, respectively, the space and time variable, density, velocity, pressure and gravitational potential given by
\begin{equation}\lambdabel{potential}\Psi({\bf x}, t)=-G\int_{\Omega(t)} \frac{\rho({\bf y}, t)}{|{\bf x}-{\bf y}|}d{\bf y} \ \ {\rm satisfying} \ \ \Delta \Psi=4\pi G \rho \ \ {\rm in} \ \ \Omega(t)
\end{equation}
with the gravitational constant $G$ taken to be unity; $\Omega(t)\subset \mathbb{R}^3$, $\Gamma(t)$, $\mathcal{V}(\Gamma(t))$ and ${\bf n}$ represent, respectively, the changing volume occupied by a fluid at time $t$, moving interface of fluids and vacuum states, normal velocity of $\Gamma(t)$ and exterior unit normal vector to $\Gamma(t)$; {$\kappappa=1$ or $0$ corresponds to the Euler equations with or without self-gravitation}. We consider a polytropic star: the equation of state is then given by
\begin{equation}\lambdabel{polytropic} p(\rho)=A \rho^{\gammamma} \ \ {\rm for} \ \ \gammamma>1 \end{equation}
with the adiabatic constant $A >0$ set to be unity. Let $c=\sqrt {p'(\rho)}$ be the sound speed, the following condition:
\begin{equation}\lambdabel{pv} -\infty<\nablabla_{\bf n}(c^2)<0 \ \ {\rm on} \ \ \Gamma(t), \end{equation}
defines a physical boundary (cf. \cite{7,10',16',23,24,25}). Equations $\eqref{2.1}_{1,2}$ describe the balance laws of mass and momentum, respectively; conditions $\eqref{2.1}_{3,4}$ state that $\Gamma(t)$ is the interface to be investigated; $\eqref{2.1}_5$ indicates that the interface moves with the normal component of the fluid velocity; and $\eqref{2.1}_6$ is the initial conditions for the density, velocity and domain.
The physical vacuum appears in the stationary solutions of system \eqref{2.1} naturally. Since for a stationary solution, one has
$$\nablabla_{\bf x} p(\rho)=-\rho \nablabla_{\bf x} \Psi,$$
which yields that in many
physical situations,
$$\nablabla_{\bf N}(c^2)=-\frac {(\gamma-1)}{\gammamma}\nablabla_{\bf N} \Psi \in (-\infty, 0)$$
on the interface,
where ${\bf N}$ is the exterior unit normal to the interface pointing from fluids to vacuum. The physical vacuum makes the study of free boundary problems of compressible fluids challenging and very interesting, because standard methods of symmetric hyperbolic systems (cf. \cite{17}) do not apply directly. Recently, important progress has been made in the local-in-time well-posedness theory for the one- and three-dimensional compressible
Euler equations (cf. \cite{16, 10, 7, 10', 16'}). But
for the three-dimensional compressible Euler-Poisson equations, the gravitational potential $\Psi$ defined by \eqref{potential} is non-local and depends on the unknown domain $\Omega(t)$. This will cause certain difficulties in the analysis.
Moreover, the self-gravitation leads to very rich and interesting physical phenomena for compressible fluids with vacuum (cf. \cite{rein, {17'}, luosmoller1, luosmoller2, liebyau, 6'}).
First, we address the uniqueness of classical solutions for the above free boundary problem. The uniqueness problem of free boundary problems for the equations of compressible fluids
is subtle. This is particularly so in the presence of vacuum states. For the physical vacuum free boundary problem of the 3-dimensional compressible Euler equations, a uniqueness theorem is proved in \cite{10'} in functional spaces which are smoother of one more degree than the spaces in which the existence theorems are established. This functional space in \cite{10} involves the weighted Sobolev norms of solutions. In the present paper, we prove a general uniqueness theorem of classical solutions for $1<\gammamma\le 2$ (the most physically relevant regime) only requiring the derivatives appearing in the equations are continuous (indeed, we can only require that the solutions are in $W^{1, \infty}$ in the whole domain and
$C^1$ in a neighborhood of the boundary). {The strategy is to extend the solutions of \eqref{2.1} to those of Cauchy problems, for which the physical vacuum \eqref{pv} is crucial.}
Due to vacuum, the uniqueness of the extended solutions to the Cauchy problem is nontrivial because the standard symmetrization method of hyperbolic system does not apply in the presence of physical vacuum. We use the relative entropy argument (cf. \cite{dafermos}) and potential estimates (cf. \cite{AB}). The advantage of the relative entropy argument is making the full use of the nonlinear structure of the equations and requiring less regularity as realized by R. DiPerna (cf. \cite{diperna}). The proof of the uniqueness theorem is valid for both the compressible Euler-Poisson equations and the compressible Euler equations without self-gravitation. The above approach works for the case when $1<\gammamma\le 2$. For the general case of $\gammamma>1$, {we study the vacuum dynamics of free boundary problems of the compressible Euler equations without self-gravitation for spherically symmetric motions, and prove the uniqueness theorem in the class of $C^1\cap W^{1, \infty}(\{{\bf x}\in\mathbb{R}^3: 0<|{\bf x}|\le R(t)\})$ without requiring that the solutions are differentiable at the center of symmetry. Here the ball $B_{R(t)}$ is the moving domain. It should be noted that we do not require the vacuum boundary being physical in this case.}
We now turn to the existence problem. For a gaseous star, it is important to consider spherically symmetric motions since the stable equilibrium configurations are spherically symmetric which minimize the energy among all possible configurations (cf. \cite{liebyau}). As aforementioned, there have been some existence theories available for the free boundary problems of the three-dimensional compressible Euler equations with physical vacuum (cf. \cite{10', 16'}). However, for spherically symmetric motions, if the compatibility condition of the first derivatives of solutions being zero at the center of the symmetry is not imposed for the initial data, the initial data are $C^1$ only in the region excluding the origin as 3-spatial dimensional functions, but may not be differentiable at the origin. In this case, the general existence theories in the three spatial-dimensions in \cite{10', 16'} do not apply. Moreover, without imposing this compatibility condition at the center of symmetry, the coordinate singularity is very strong and the equation becomes very degenerate near the center of the symmetry. Indeed, the initial density, $\rho_0$,
appears as the coefficients in equation \eqref{e1-2'} in the Lagrangian coordinates. This gives tremendous difficulties when we use the elliptic estimates to estimate the derivatives in the direction normal to the boundary. In those estimates, whether the first-order derivatives of the initial density is zero at the origin or not makes a
distinct difference since we differentiate the system in the direction normal to the boundary in the elliptic estimates. We will choose deliberately a cut-off function whose effective width depending on the initial density to capture more singular behavior of the solutions near the origin for the case of the non-zero first derivatives of the initial density.
The spherically symmetric solution we construct in this paper without imposing the above mentioned compatibility condition at origin is $C^1$ smooth only in the region excluding the origin, but $W^{1, \infty}$ in the region including the
origin as the functions of 3-spatial dimensional functions. Therefore, the solution constructed in this paper is
different from those in \cite{16',10'} and exhibits some specially interesting features. For instance, in the currently available well-posedness theory for the free boundary problems of the three-dimensional compressible Euler equations with physical vacuum, it requires by \cite{16'} or \cite{10'} a weighted norm involving $ \nablabla_{\bf x}^{2\left[ {1}/({\gammamma-1})\right]+9} {\bf u} |_{t=0}$ or $ \partial_t^{7+\lceil (2-\gamma)/(\gamma-1) \rceil}\nablabla_{\bf x} {\bf u} |_{t=0} $ to be finite. However, for the three-dimensional spherically symmetric motion without imposing the compatibility condition of the first order derivatives of solutions being zero at the center of the symmetry, we find in the present work a new well-posedness theory with the initial data less regular than those required in \cite{16',10'}.
As mentioned above, one of interesting features and challenges in the exploration of spherically symmetric motions is to deal with the difficulty caused by the coordinates singularity at the origin (the center of the symmetry), besides the one caused by physical vacuum on the boundary. This is particularly so without imposing the compatibility condition of the first order derivatives of the solution being zero at the center of symmetry. Indeed, in the well-posedness theory for spherically symmetric motions of viscous gaseous stars modeled by the compressible Navier-Stokes-Poisson equations with vacuum boundary was shown in \cite{15}, a higher-order energy functional was constructed which consists of two parts, called the Eulerian energy near the origin expressed in Eulerian coordinates and the Lagrangian energy described in Lagrangian coordinates away from the origin. This indicates the subtlety of the behavior of solutions near the origin and vacuum boundary. In this paper, we find a uniform way to construct a higher-order energy functional only in Lagrangian coordinates by choosing suitable weights and cutoff functions which work for both the origin and the physical vacuum boundary of which the construction is elaborative. It is noted such a strategy works also for the compressible Navier-Stokes-Poisson equations .
It should be noted here that the detailed proofs of the existence theorems in \cite{16',10'} are given for a initial flat domain of the form $\mathbb{T}^2 \times (0, 1)$, where $\mathbb{T}^2$ is a two-dimensional period box in $x_1$ and $x_2$. Initially, the reference vacuum boundary is the top boundary
$\Gamma(0) =\{x_3 = 1\}$ while the bottom boundary $\{x_3 = 0\}$ is fixed. The moving vacuum boundary is given by $\Gamma(t) = \eta(t)(\Gamma(0))$ with the flow map $\eta(t)$. In principle, it would be feasible to extend flat domains to general non-flat ones, for example, utilizing local coordinate charts and changes of coordinates to straighten out the boundary for each chart. However, it seems quite complicated and technically involved. In this article, we give a direct proof for non-flat initial domains (balls) of the existence theorem for the free boundary problem with physical vacuum. {It should be noted that the general approach we use here is motivated by \cite{16'}, in particular on the choice of the weights near the vacuum boundary.}
Before closing this introduction, we would like to review some prior results on the free boundary problems besides the ones aforementioned. There has been a recent explosion of interests in the analysis of inviscid flows, one may refer to \cite{zhenlei,24,25,22,20,26, 29} for compressible motions and to \cite{1,9,19,21,34,40} for incompressible motions. Among these works, it should be mentioned that in \cite{24} a smooth existence theory ({for the sound speed $c$,} $c^\alpha$ is smooth across the interface with $0 < \alpha \le 1$) was developed for the one-dimensional Euler equations with damping, based on the adaptation of the theory of symmetric hyperbolic systems which is not applicable to physical vacuum boundary problems for which only $c^2$, the square of sound speed in stead of $c^{\alpha}$ ( $0 < \alpha \le 1$) , is required to be smooth across the interface); in \cite{zhenlei} the well-posedness of the physical vacuum free boundary problem is investigated for the one-dimensional Euler-Poisson equations,
using the methods motivated by those in \cite{10} for the one-dimensional Euler equations; {existence and uniqueness for the three-dimensional compressible Euler equations modeling a liquid rather than a gas were established in \cite{22} by using Lagrangian variables combined with Nash-Moser iteration to construct solutions. For a compressible liquid, the density is assumed to be a strictly positive constant on the moving boundary. As such, the compressible liquid provides a uniformly hyperbolic, but characteristic, system. An alternative proof for the existence of a compressible liquid was given in \cite{35}, employing a solution strategy based on symmetric hyperbolic systems combined with Nash- Moser iteration.} As for viscous flows, there have been many results on the free-boundary Navier-Stokes equations which cause quite different difficulties in analyses from that for inviscid flows, so we do not discuss the works in that regime here.
The rest of this paper is organized as follows. In the next section, we present and prove the uniqueness of classical solutions to the three-dimensional physical vacuum problem \eqref{2.1} when $1<\gammamma\le 2$. The rest is devoted to the study of spherically symmetric motions. In Section 3, we formulate the three-dimensional spherically symmetric problem and state the main existence result.
Sections 4-8 are devoted to the case of $\gammamma=2$. In Section 4, we describe a degenerate parabolic approximation to the original degenerate hyperbolic system. The uniform estimates for the higher-order energy functional are given in Sections 5-7: some preliminaries are presented in Section 5, the energy estimates in the tangential directions are given in Section 6, and the elliptic estimates in the normal direction for interior and boundary regions are presented respectively in Section 7. With those
estimates, the existence can be shown in Section 8. In sections 9 and 10, we will outline, but with enough details, the existence theory for the cases of $1<\gammamma<2$ and $\gammamma>2$, respectively. Section 11 is devoted to the uniqueness theorem of classical solutions for the vacuum free-boundary problem of the compressible Euler equations without the self-gravitation in the spherical symmetry setting for all the values of $\gammamma>1$, without assuming that the vacuum boundary is physical in the sense of \eqref{pv}.
\section{Uniqueness for three-dimensional Euler-Poisson equations with physical vacuum when $1<\gammamma\le 2$. }
For the three-dimensional free-boundary problem \eqref{2.1} with a physical vacuum, we prove the following quite general uniqueness theorem for $1<\gammamma\le 2$ in a natural functional space. It should be remarked that the uniqueness theorems proved in \cite{10,10'} are in the functional spaces which are one more derivative smoother than the spaces in which the existence theorems are established. Before stating the uniqueness theorem, we give a definition of classical solutions to problem \eqref{2.1}.
\begin{equation}gin{defi}
A triple $(\rho, {\bf u}, \Omega(t))$ is called a classical solution to the physical vacuum free boundary
problem \eqref{2.1} on $[0, T]$ for $T>0$ if the following conditions hold:
{\rm 1)} $\Omega(t)=\cup_{k=1}^{m} \Omega^k(t)\subseteq \mathbb{R}^3$ is an open bounded set and $\partial\Omega(0)\in C^2$, where $\Omega^k(t)$ $(k=1,\cdots, m)$ are the connected component of $\Omega(t)$ satisfying
\begin{equation}\lambdabel{omega}
\overline{\Omega^j (t) } \ \cap \ \overline{\Omega^k(t)}=\emptyset, \ \ 1\le j\ne k \le m, \ \ t\in [0,T];
\end{equation}
{\rm 2)} $(\rho, {\bf u})\in C^1 (\begin{equation}tar{ D})$ satisfies system \eqref{2.1} and the physical vacuum condition:
\begin{equation}\lambdabel{physical}
-\infty<\nablabla_{\bf n}\left(\rho^{\gammamma-1}\right)<0 \ \ {\rm on} \ \ \Gamma(t)=\partial\Omega(t), \end{equation}
where ${\bf n}$ is the spatial unit outer norm to $\Gamma(t)$ and
$$D=\{({\bf x}, t): \ \ {\bf x}\in \Omega(t),\ \ t\in [0, T]\}, \ \ \begin{equation}tar{D}=D\cup \partial D.$$
\end{defi}
{Due to the regularities of the solution ${\bf u}\in C^1 (\begin{equation}tar{ D})$ and $\partial\Omega(0)\in C^2$ in the definition above, we can see easily that
\begin{equation}\lambdabel{oac2}
\bigcup_{0\le t\le T} \Gamma(t)=\bigcup_{0\le t\le T} \partial \Omega(t)=:\widetilde{\partial D} \in C^2.
\end{equation}
Indeed, the interface $\Gamma(t)$ is moving with the fluids given by $ \mathcal{V}(\Gamma(t))={\bf u}\cdot {\bf n}$ on $\partial \Omega(t)$, where $\mathcal{V}(\Gamma(t))$ is the normal velocity of $\Gamma(t)$; which is equivalent
to saying that $\widetilde{\partial D}$ is foliated by the integral curves of the vector fields $\partial_t +{\bf u}\cdot \nablabla_x$.
}
The uniqueness theorem is as follows:
\begin{equation}gin{thm}\lambdabel{uniqueness1} {\rm (uniqueness for the 3-d problem)} Suppose $1<\gammamma\le 2$. Let $(\rho_1, {\bf u}_1, \Omega_1(t))$ and $(\rho_2, {\bf u}_2, \Omega_2(t))$ be two classical solutions to
problem \eqref{2.1} on $[0, T]$ for $T>0$ in the sense of Definition 2.1, then for $t\in[0,T]$,
\begin{equation}\lambdabel{uniq1}\begin{equation}gin{split}
\Omega_1(t)=\Omega_2(t) \ \ {\rm and} \ \
(\rho_1, {\bf u}_1)({\bf x}, t)=(\rho_2, {\bf u}_2)({\bf x}, t), \ \ {\bf x}\in \Omega_1(t)=\Omega_2(t),
\end{split}\end{equation}
provided that \eqref{uniq1} holds for $t=0$.
\end{thm}
{\begin{equation}gin{rmk}\lambdabel{rmk2.2} It follows easily from the proof that the uniqueness result stated in Theorem \ref{uniqueness1} holds true for the solutions to \eqref{2.1} as stated in definition 2.1 but with the regularity condition $\left(\rho, {\bf u} \right)\in C^1(\begin{equation}tar D)$ replaced by a less regular one:
\begin{equation}\lambdabel{regular'}
\ (\rho, {\bf u})\in W^{1, \infty} (\begin{equation}tar{ D}) {~\rm and~} \ (\rho, {\bf u})\in C^1( {D}_{\delta} \cup \partial {D}_{\delta} ),
\end{equation}
where $D_{\delta}\subset D$ is a neighborhood of $\widetilde{\partial D}$.
\end{rmk}}
{\bf Proof of Theorem \ref{uniqueness1}}. The proof is divided into two steps. In step 1, we extend the solutions of \eqref{2.1} to those of Cauchy problems. After that, we use the relative entropy argument and potential estimates to prove the uniqueness .
{\bf Step 1} (extension). Suppose that the triple $(\rho, {\bf u}, \Omega(t))$ is a classical solution to
problem \eqref{2.1} on $[0, T]$ in the sense of Definition 2.1.
We will first extend the solution $(\rho, {\bf u})$ from the domains $\Omega(t)$ to the whole domain $\mathbb{R}^3$ for
$t\in [0, T]$ such that the extended functions $(\widetilde{\rho}, \widetilde{{\bf u}})$ satisfy
\begin{equation}\lambdabel{extend'}(\widetilde{\rho}, \widetilde{{\bf u}})({\bf x}, t)\in W^{1, \infty}(\mathbb{R}^3\times [0, T]), \end{equation}
and solve the Euler-Poisson equations.
{\bf Step 1.1}. The extension of $\rho$ is clearly given by
\begin{equation}\lambdabel{vacuumextend}
\widetilde{\rho}({\bf x}, t)= {\rho}({\bf x}, t) \ \ {\rm in} \ \ D \ \ {\rm and} \ \ \widetilde{\rho}({\bf x}, t)\equiv 0 \ \ {\rm in} \ \ \mathbb{R}^3\times[0,T]\setminus D .\end{equation}
The extension of the vector field ${\bf u}$ is more complicated. In what follows, we extend it from $\Omega(t)$ to a neighborhood of $\Omega(t)$, and then to the rest region.
It follows from the condition \eqref{omega} that there exists a small positive constant $\epsilon$ such that
\begin{equation}e\lambdabel{x1}
\Omega_{\epsilon}^j(t)\cap \Omega_{\epsilon}^k(t)=\emptyset, \ \ 1\le j\ne k \le m, \ \ t\in[0,T] ,\end{equation}e
where
\begin{equation}e\lambdabel{4211}\Omega_{\epsilon}^j(t) = \Omega^j(t) \cup \{\begin{equation}tar {\bf x}+s {\bf n} (\begin{equation}tar {\bf x}, t): \ \ \begin{equation}tar {\bf x}\in \partial \Omega^j(t), \ \ 0\le s\le \epsilon\}, \ \ 1\le j \le m.\end{equation}e
Moreover, $\epsilon>0$ is chosen so small that the exponential map:
\begin{equation}\lambdabel{mapin}\begin{equation}gin{split}
\partial \Omega^j(t)\times [0, \epsilon] &\to \mathbb{R}^3 :
\ \ (\begin{equation}tar{\bf x}, s )& \mapsto \begin{equation}tar {\bf x}+s {\bf n} (\begin{equation}tar {\bf x}, t)
\end{split}\end{equation}
is injective for $1\le j\le m$ (that is, $\epsilon$ is less than the injectivity radius of $\partial \Omega^j(t)$). It should be noted that the number $\epsilon>0$ can be chosen uniformly for $t\in [0, T]$, because $ \widetilde{\partial D}\in C^2$ (see \eqref{oac2} for details).
Indeed, denote the second fundamental form of $\partial \Omega(t)$ by $\theta (\begin{equation}tar {\bf x}, t)$, then $\|\theta (\begin{equation}tar {\bf x}, t)\|_{C({\widetilde{\partial D}})}\le K_T$ for some positive constant $K_T$ which
may depends on $T$. Therefore, the injectivity radius of $\partial\Omega(t)$ has a positive lower bound for $t\in [0, T]$ (cf. \cite{CL00}).
Let $\eta\in C^{\infty}([0, \ \epsilon])$ be a cut-off function satisfying
$$0\le \eta \le 1, \ \ \eta (s)=1 \ \ {\rm for} \ \ 0\le s\le \frac{\epsilon}{3}, \ \ \eta (s)=0 \ \ {\rm for} \ \ \frac{2\epsilon}{3}\le s\le{\epsilon}.$$
For any ${\bf x} \in \Omega_{\epsilon}^j(t) \setminus \Omega^j(t)$, define the extension of ${\bf u}$ as
\begin{equation}\lambdabel{extensionforu}\begin{equation}gin{split}
\widetilde{{\bf u}}\left({\bf x},t\right)
= \widetilde{{\bf u}}(\begin{equation}tar{\bf x}+s {\bf n} (\begin{equation}tar {\bf x}, t), t)
=&\eta(s)\left[{\bf u}(\begin{equation}tar {\bf x}, t)+s\nablabla_{\bf x} {\bf u}(\begin{equation}tar{\bf x}, t)\cdot {\bf n}(\begin{equation}tar{\bf x}, t)\right] \\
=&\eta(s)\left[{\bf u}(\begin{equation}tar {\bf x}, t)+\nablabla_{\bf x} {\bf u}(\begin{equation}tar{\bf x}, t)\cdot
({\bf x}-\begin{equation}tar{\bf x})\right], \ \ 0\le s \le \epsilon.
\end{split}\end{equation}
So, we have extended the vector field ${\bf u}$ from $\Omega(t)$ to $\cup_{j=1}^m \Omega_\epsilon(t)=: \Omega_\epsilon (t)$, a neighborhood of $\Omega(t)$.
For the rest region, we simply define
\begin{equation}\lambdabel{uu}
\widetilde{{\bf u}}({\bf x}, t)= {{\bf u}}({\bf x}, t) \ \ {\rm in }\ \ D, \ \
\widetilde{{\bf u}}({\bf x}, t)={\bf 0} \ \ {\rm for} \ \ {\bf x}\in\mathbb{R}^3 \setminus \ \Omega_\epsilon(t) \ \ {\rm and} \ \ t\in [0,T].
\end{equation}
{\bf Step 1.2}. Next, we verify that the extended functions $(\widetilde{\rho}, \widetilde{{\bf u}})({\bf x}, t)$
defined on $\mathbb{R}^3\times [0, T]$ satisfy \eqref{extend'}. The key is the differentiability across the boundary $\widetilde{\partial D}:=\cup_{0\le t\le T} \partial \Omega(t)$.
Before doing so, some notations are needed. For any point $(\begin{equation}tar {\bf x}, \begin{equation}tar t)\in \widetilde{\partial D}$, let $(\thetau_0, \thetau_1, \thetau_2)$ be a basis of the space-time tangent space of $\widetilde{\partial D}$ at $(\begin{equation}tar {\bf x}, \begin{equation}tar t)$ and ${\bf N}={\bf n}(\begin{equation}tar {\bf x}, \begin{equation}tar t)$ be the spatial unit outer normal to $\partial \Omega(\begin{equation}tar{t})$ at $\begin{equation}tar{\bf x}$. Then $(\thetau_0, \thetau_1, \thetau_2, {\bf N})$ forms a basis of $\mathbb{R}^4$. So
$\nablabla_{\thetau_j}$ ($j=0, 1, 2$) and $\nablabla_{{\bf N}}$ determine all the derivatives $\partial_t$ and $\nablabla_{\bf x}$ at the point $(\begin{equation}tar{\bf x}, \begin{equation}tar t)$.
For $t\in [0, T]$, denote the interior and exterior sides of $\widetilde{\partial D}$ (or $\partial \Omega(t)$) by $\widetilde{\partial D}-$ (or $\partial \Omega(t)-$) and $\widetilde{\partial D}+$ (or $\partial \Omega(t)+$), respectively.
For $\widetilde{\rho}$, it follows from
$$\widetilde{\rho}\in C^1(\begin{equation}tar D) \ \ {\rm and} \ \ \widetilde{ \rho}=0 \ \ {\rm on} \ \ \mathbb{R}^3\times [0, T]\setminus D$$
that $\nablabla_{\thetau_i} \widetilde{\rho}=0$ on both $\widetilde{\partial D}-$ and $\widetilde{\partial D}+$ for $i=0,1,2$; which implies that the tangential derivatives of $\widetilde{\rho}$ is continuous
across $\widetilde{\partial D}$. For the spatial normal derivative, it follows from the physical vacuum
condition:
$$-\infty<\nablabla_{\bf N}(\widetilde{\rho}^{\gammamma-1})<0 \ \ {\rm on } \ \ \partial \Omega(t)- , $$
that
$$\nablabla_{\bf N}(\widetilde{\rho})=0 \ \ {\rm if} \ \ 1<\gamma<2 \ \ {\rm and} \ \ -\infty<\nablabla_{\bf N}(\widetilde{\rho})<0
\ \ {\rm if} \ \ \gamma=2 \ \ {\rm on} \ \ {\partial \Omega(t)}-;$$
because of $\widetilde{\rho}=0$ on $\widetilde{\partial D}$ and the fact
$$\nablabla_{\bf N}(\widetilde{\rho})=\frac{1}{\gammamma-1}\widetilde{\rho}^{2-\gammamma}
\nablabla_{\bf N}(\widetilde{\rho}^{\gammamma-1}) $$
As on $\widetilde{\partial D}+$, it is easy to see that both the tangential and normal derivatives of $\widetilde{\rho}$ are zero due to $\widetilde{\rho}=0$ in $\mathbb{R}^3\times[0,T]\setminus D$. Thus, we have the following regularity of $\widetilde{\rho}$:
\begin{equation}gin{equation}\lambdabel{extentionrho1} \begin{equation}gin{cases}
\widetilde{\rho}\in C^1\left(\mathbb{R}^3\times [0, T]\right)\cap W^{1, \infty}\left(\mathbb{R}^3\times[0, T]\right), \qquad {\rm if~} \ 1<\gammamma<2, \\
\widetilde{\rho}\in C^1\left(\overline {D}\right)\cap C^1\left(\overline {\mathbb{R}^3\times[0,T] \setminus D}\right)\cap W^{1, \infty}\left(\mathbb{R}^3\times[0, T]\right), \qquad {\rm if~} \ \gammamma=2.
\end{cases}\end{equation}
For $\widetilde{{\bf u}}$, it follows from $\widetilde{{\bf u}}\in C^1(\begin{equation}tar {D})$ and \eqref{extensionforu} that
${\widetilde{\bf u}}$ is continuous across the interface $\widetilde{\partial D}$ which implies that the tangential derivatives of $\widetilde{{\bf u}}$ are continuous across $\widetilde{\partial D}$; and that $\nablabla_{{\bf N}}\widetilde{{\bf u}}$ is continuous across $\widetilde{\partial D}$. Therefore, it holds that
\begin{equation}\lambdabel{2.12} \widetilde{{\bf u}}\in C^1(\mathbb{R}^3\times [0, T]) \cap W^{1, \infty}(\mathbb{R}^3\times[0, T]). \end{equation}
{\bf Step 1.3} We now verify that $(\widetilde{\rho}, \widetilde{{\bf u}})({\bf x}, t)$ solves the isentropic Euler-Poisson equations point-wisely. Note that
$$\widetilde{\rho} (\cdot, t) \in C^1\left(\overline {\Omega(t)}\right)\cap C(\mathbb{R}^3) \ \ {\rm and} \ \ \widetilde{\rho}\equiv 0 \ \ {\rm in} \ \ \mathbb{R}^3\setminus \Omega(t), \ \ t\in [0, T], $$
then we have, by the potential theory (cf. \cite{AB}), that for each $t\in [0, T]$,
\begin{equation}gin{equation}\lambdabel{psies}
\widetilde{\psi}(x, t)=-\int_{\Omega(t)}\frac{\widetilde{\rho}(y, t)}{|x-y|}dy=-\int_{\mathbb{R}^3}\frac{\widetilde{\rho}(y, t)}{|x-y|}dy\in C^1(\mathbb{R}^3)\cap W^{1, \infty}(\mathbb{R}^3). \end{equation}
In view of \eqref{extentionrho1}, \eqref{2.12} and \eqref{psies}, we see that the extended functions $(\widetilde{\rho}, \widetilde{{\bf u}})$ solves the Euler-Poisson equations in $\mathbb{R}^3\times[0, T]\setminus \overline {D}$,
since $\widetilde{\rho}\equiv 0$ in this region.
As in $D$, by Definition 2.1, $(\widetilde{\rho}, \widetilde{{\bf u}})$ of course solves the Euler-Poisson equations.
The remaining task is to verify this on $\widetilde{\partial D}$. Since
the vector field $\partial_t+\widetilde{{\bf u}}\cdot \nablabla_{{\bf x}} $ is tangential to
$\widetilde{\partial D}$ and $\widetilde{\rho}=0$ on $\widetilde{\partial D}$, then
$$\left(\partial_t+\widetilde{{\bf u}}\cdot \nablabla_{{\bf x}} \right) \widetilde{\rho} =0 \ \ {\rm on} \ \
\widetilde{\partial D}+ \ \ {\rm and} \ \ \widetilde{\partial D}-. $$
It follows from \eqref{2.12} and $\widetilde{\rho}=0$ on $\widetilde{\partial D}$ that
\begin{equation}e\lambdabel{xu}\widetilde{\rho} {\rm div}\widetilde{u}=0 \ \ {\rm on} \ \
\widetilde{\partial D}+ \ \ {\rm and} \ \ \widetilde{\partial D}-. \end{equation}e
Therefore, the equation of conservation of mass is verified. Similarly, we have
\begin{equation}gin{equation}\lambdabel{2.15}
\widetilde{\rho}(\partial_t+\widetilde{{\bf u}}\cdot \nablabla_{{\bf x}} ) \widetilde{{\bf u}}=0 \ \ {\rm on~} \ \ {\widetilde{\partial D}}.
\end{equation}
Moreover, for any tangent vector $\thetau$ to $\widetilde{\partial D}$, we have
\begin{equation}\lambdabel{tangentiala}
\nablabla_{\thetau} P(\widetilde{\rho})\equiv 0 \ \ {\rm on~} \ \widetilde{\partial D}, \end{equation}
because of $\widetilde{\rho}\equiv 0$ on $\widetilde{\partial D}$. For any spatial normal ${\bf N}$ to $\partial\Omega(t)$, it holds that
$$\nablabla_{\bf N} P(\widetilde{\rho})=\nablabla_{\bf N} (\widetilde{\rho}^{\gammamma})=\frac{\gammamma}{\gammamma-1}\widetilde{\rho} \nablabla_{\bf N} (\widetilde{\rho}^{\gammamma-1})$$
and
$$-\infty<\nablabla_{\bf N} (\widetilde{\rho}^{\gammamma-1})<0 \ \ {\rm on} \ \ \widetilde{\partial D}- , \ \ \widetilde{\rho}=0 \ \ {\rm on} \ \ \widetilde{\partial D}.$$
Thus, we have
$$\nablabla_{\bf N} P(\widetilde{\rho})=0 \ \ {\rm on} \ \ \widetilde{\partial D}-.$$
This, together
with \eqref{tangentiala}, verifies that
\begin{equation}\lambdabel{tango}
\nablabla_{\bf x} P(\widetilde{\rho})\equiv 0 \ \ {\rm on~} \ \ \widetilde{\partial D}-. \end{equation}
Since
$$\widetilde{\rho}\equiv 0 \ \ {\rm in} \ \ \mathbb{R}^3\times [0, T]\setminus D,$$
then
$$\nablabla_{\bf N} P(\widetilde{\rho})=0 \ \ {\rm on} \ \ \widetilde{\partial D}+,$$
which together with \eqref{tangentiala} implies that
\begin{equation}\lambdabel{tango1}
\nablabla_{\bf x} P(\widetilde{\rho})\equiv 0 \ {\rm on~} \widetilde{\partial D}+. \end{equation}
Therefore, it follows from \eqref{2.15}, \eqref{tango} and \eqref{tango1} that the left-hand side of the equation of the balance law of the momentum is zero on $\widetilde{\partial D}$. On the other hand,
in view of \eqref{psies} and the fact that $\widetilde{\rho}\equiv 0$ on $\widetilde{\partial D}$, the right-hand side is also zero on $\widetilde{\partial D}$.
{\bf Step 2} (uniqueness).
Now, let $(\rho_1, {\bf u}_1, \Omega_1(t))$ and $(\rho_2, {\bf u}_2, \Omega_2(t))$ be two classical solutions of
problem \eqref{2.1} on $[0, T]$ for $T>0$ in the sense of Definition 2.1. We extend those solutions as above by replacing $(\rho, {\bf u}, \Omega(t))$ by $(\rho_i, {\bf u}_i, \Omega_i(t))$ ($i=1, 2$), and denote these extended functions still by $(\rho_i, {\bf u}_i)$ ($i=1, 2$). It is easy to see that, for $i=1, 2,$
\begin{equation}\lambdabel{gep1}\begin{equation}gin{split}
& \partial_t \rho_i + {\rm div}(\rho_i {\bf u}_i) = 0 & {\rm in}& \ \ \mathbb{R}^3\times (0, T], \\
& \partial_t (\rho_i {\bf u}_i) + {\rm div} (\rho_i {\bf u}_i\otimes {\bf u}_i)+\nablabla_{\bf x} p(\rho_i) = -\kappappa\rho_i \nablabla_{\bf x} \Psi_i & {\rm in}& \ \ \mathbb{R}^3\times (0, T],\\
&\rho_i>0 &{\rm in } & \ \ \Omega_i(t),\\
& \rho_i=0 & {\rm in}& \ \ \mathbb{R}^3\setminus \Omega_i(t),\\
\end{split} \end{equation}
where
\begin{equation}\lambdabel{potentiali}
\Psi_i ({\bf x}, t)=-\int_{\mathbb{R}^3}\frac{\rho_i({\bf y}, t)}{|{\bf x}-{\bf y}|}d{\bf y}, \quad
{\bf x}\in \mathbb{R}^3, \ \ t\in [0, T],\end{equation}
$\kappappa=0$ or $1$,
and \eqref{extend'}, \eqref{extentionrho1}, \eqref{2.12}, \eqref{psies} hold for $(\widetilde{\rho}, \widetilde{{\bf u}}, \Omega(t), \widetilde{\psi})$=$(\rho_i, {\bf u}_i, \Omega_i(t), \psi_i )$, $i=1, 2$. In what follows, we define the relative entropy-entropy flux pairs and derive some potential estimates.
{\bf Step 2.1}. For $i=1,2$, set
$${\bf u}_i=\left(u_i^1, u_i^2, u_i^3\right)^{\rm T}, \ \ {\bf m}_i=\left(m_i^1, m_i^2, m_i^3\right)^{\rm T} \ \ {\rm and } \ \ {\bf U}_i=\left(U_i^0, U_i^1, U_i^2, U_i^3\right)^{\rm T}, $$
where
$${ m}_i^j =\rho_i { u}_i^j , \ \ { U}_i^0 =\rho_i ,\ \ { U}_i^j ={ m}_i^j, \ \ j=1,2,3.$$
Here and thereafter $(\cdot)^{\rm T}$ denotes the transpose. Equations $\eqref{gep1}_{1,2}$ can be written as
\begin{equation}\lambdabel{U}
\partial_t {\bf U}_i + \sum_{j=1}^3 \partial_{{\bf x}_j} {\bf F}_j({\bf U}_i) = {\bf R}_i, \ \ i=1,2,
\end{equation}
where ${\bf R}_i=\kappappa(0, -\rho_i (\nablabla_{\bf x} \Psi_i)^{\rm T})^{\rm T}$ and the flux functions ${\bf F}_j=({ F}_j^0,{ F}_j^1,{ F}_j^2, { F}_j^3)^{\rm T}$ are given by
\begin{equation}e
{\bf F}_1({\bf U}_i ) =\left( { m}_i^1, \ \frac{{ m}_i^1 { m}_i^1}{\rho_i} + p(\rho_i) ,\
\frac{{ m}_i^1 { m}_i^2}{\rho_i},\ \frac{{ m}_i^1 { m}_i^3}{\rho_i}\right)^{\mathrm T},
\end{equation}e
\begin{equation}e
{\bf F}_2({\bf U}_i ) =\left( { m}_i^2, \ \frac{{ m}_i^1 { m}_i^2}{\rho_i} ,\
\frac{{ m}_i^2 { m}_i^2}{\rho_i}+ p(\rho_i) ,\ \frac{{ m}_i^2 { m}_i^3}{\rho_i}\right)^{\mathrm T},
\end{equation}e
\begin{equation}e
{\bf F}_3({\bf U}_i ) =\left( { m}_i^3, \ \frac{{ m}_i^1 { m}_i^3}{\rho_i} ,\
\frac{{ m}_i^2 { m}_i^3}{\rho_i},\ \frac{{ m}_i^3 { m}_i^3}{\rho_i}+ p(\rho_i)\right)^{\mathrm T}.
\end{equation}e
Denote the entropy $\eta$ and entropy flux function ${\bf q}=({ q}^1, { q}^2,{ q}^3)^{\rm T}$ by
\begin{equation}\lambdabel{eeflux} \begin{equation}gin{split}
\eta( {\bf U}_i) =\frac{|{\bf m}_i|^2}{2\rho_i}+\frac{1}{\gammamma-1}\rho_i^\gammamma \ \ {\rm and} \ \ {\bf q} ({\bf U}_i) =\left(\frac{|{\bf m}_i|^2}{2\rho_i}+\frac{\gammamma}{\gammamma-1}\rho_i^\gammamma\right)\frac{{\bf m}_i}{\rho_i}, \ \ i=1,2.
\end{split} \end{equation}
(For ${\bf x}\in \mathbb{R}^3\setminus \Omega_i(t)$ and $ t\in [0, T]$ where $\rho_i=0$, we set
$ \left({\bf m}_i/ \rho_i\right)({\bf x}, t)={\bf u}_i({\bf x}, t) $, $i=1,2$.) Then, we have
\begin{equation}e
D \eta ({\bf U}_i) D {\bf F}_j({\bf U}_i ) = D { q}^j ({\bf U}_i), \ \ j=1,2,3, \ \ i=1, 2,
\end{equation}e
where
\begin{equation}e
D \eta ({\bf U}_i)= \left(\frac{\partial \eta( {\bf U}_i)}{\partial { U}_i^0},\frac{\partial \eta( {\bf U}_i)}{\partial { U}_i^1},\frac{\partial \eta( {\bf U}_i)}{\partial { U}_i^2},\frac{\partial \eta( {\bf U}_i)}{\partial { U}_i^3}\right),
\end{equation}e
\begin{equation}e
D { q}^j ({\bf U}_i) = \left(\frac{\partial { q}^j ( {\bf U}_i)}{\partial { U}_i^0},\frac{\partial { q}^j ( {\bf U}_i)}{\partial { U}_i^1},\frac{\partial { q}^j ( {\bf U}_i)}{\partial { U}_i^2},\frac{\partial { q}^j ( {\bf U}_i)}{\partial { U}_i^3}\right),
\end{equation}e
and $ D {\bf F}_j({\bf U}_i ) $ represents the Jacobian matrix whose $(k,l)$ element is ${\partial { F}_j^k( {\bf U}_i)}/{\partial { U}_i^l}$.
Easily, one can derive the equation for the entropy $\eta$ when ${\bf U}_i\in W^{1, \infty}$:
\begin{equation}\lambdabel{etaeta}
\partial_t \eta({\bf U}_i) + \sum_{j=1}^3 \partial_{{\bf x}_j} { q}^j({\bf U}_i) +\kappappa {\bf m}_i \cdot \nablabla_{\bf x} \Psi_i =0, \ \ i=1,2.
\end{equation}
We can therefore define the relative entropy-entropy flux pairs by
\begin{equation}e\lambdabel{re}\begin{equation}gin{split}
&\eta^*({\bf U}_1, {\bf U}_2)=\eta({\bf U}_2)-\eta({\bf U}_1)-D \eta({\bf U}_1) ({\bf U}_2-{\bf U}_1), \\
& { q}^{* j}({\bf U}_1, {\bf U}_2)={ q}^j({\bf U}_2)-{ q}^j({\bf U}_1)-D \eta({\bf U}_1) ({\bf F}_j({\bf U}_2)-{\bf F}_j({\bf U}_1)), \ \ j=1, 2, 3. \end{split}\end{equation}e
where $\eta$ and ${\bf q}$ are defined by \eqref{eeflux}. It follows from \eqref{gep1}, \eqref{U} and \eqref{etaeta} that
\begin{equation}\lambdabel{etastar}\begin{equation}gin{split}
&\partial_t \eta^*+\sum_{j=1}^3 \partial_{{\bf x}_j}{ q}^{*j} \\
= & \left[D\eta({\bf U}_2)- D \eta({\bf U}_1)\right]{\bf R}_2
-D^2 \eta({\bf U}_1) \left( {\bf R}_1, {\bf U}_2- {\bf U}_1\right)\\
&- \sum_{j=1}^3 D^2 \eta({\bf U}_1) \left( \partial_{{\bf x}_j}{\bf U}_1 , {\bf F}_j({\bf U}_2)- {\bf F}_j({\bf U}_1) - D {\bf F}_j({\bf U}_1) ({\bf U}_2-{\bf U}_1) \right)\\
= & \kappappa \rho_2 ({\bf u}_1 -{\bf u}_2) \cdot \nablabla_{{\bf x}}(\Psi_2-\Psi_1)\\
& - \sum_{j=1}^3D^2 \eta({\bf U}_1) \left( \partial_{{\bf x}_j}{\bf U}_1 , {\bf F}_j({\bf U}_2)- {\bf F}_j({\bf U}_1) - D {\bf F}_j({\bf U}_1) ({\bf U}_2-{\bf U}_1) \right),
\end{split}\end{equation}
where
\begin{equation}gin{equation*}\lambdabel{}
D^2 \eta({\bf U}_1) =\begin{equation}gin{pmatrix}
&{|{\bf m}_1|^2}/(\rho_1)^3 + \gamma (\rho_1)^{\gamma-2} & -{m_1^1}/(\rho_1)^2 & -{m_1^2}/(\rho_1)^2 & -{m_1^3}/(\rho_1)^2\\
&-{m_1^1}/(\rho_1)^2 & {1}/{\rho_1} & 0 & 0\\
&-{m_1^2}/(\rho_1)^2 & 0 & {1}/{\rho_1} &
0\\
&-{m_1^3}/(\rho_1)^2 & 0 & 0 & {1}/{\rho_1}
\end{pmatrix}.\end{equation*}
{\bf Step 2.2}. Next, we will estimate the terms on the right-hand side of \eqref{etastar}. Note that
\begin{equation}\lambdabel{eta}\begin{equation}gin{split}
\eta^*=&\frac{1}{\gamma-1}\left[\rho_2^\gamma - \rho_1^\gamma -\gamma \rho_1^{\gamma-1}(\rho_2-
\rho_1)\right]+\frac{1}{2}\rho_2|{\bf u}_2-{\bf u}_1|^2 \end{split}\end{equation}
and
\begin{equation}gin{equation*}\lambdabel{122}\begin{equation}gin{split}
& \sum_{j=1}^3D^2 \eta({\bf U}_1) \left( \partial_{{\bf x}_j}{\bf U}_1 , {\bf F}_j({\bf U}_2)- {\bf F}_j({\bf U}_1) - D {\bf F}_j({\bf U}_1) ({\bf U}_2-{\bf U}_1) \right)\\
= &[p(\rho_2)-p(\rho_1)-p'(\rho_1)(\rho_2-\rho_1)]\sum_{j=1}^3
\partial_{{\bf x}_j} u_1^j\notag
+\frac{1}{2}\sum_{i, j=1}^3\rho_2(u_2^i-u_1^i)(u_2^j-u_1
^j)(\partial_{{\bf x}_j} u_1^i+\partial_{{\bf x}_i} u_1^j).
\end{split}\end{equation*}
Then, we have
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left|\sum_{j=1}^3D^2 \eta({\bf U}_1) \left( \partial_{{\bf x}_j}{\bf U}_1 , {\bf F}_j({\bf U}_2)- {\bf F}_j({\bf U}_1) - D {\bf F}_j({\bf U}_1) ({\bf U}_2-{\bf U}_1) \right)\right|
\le C \left\| \nablabla_{\bf x} {\bf u}_1 (\cdot, t) \right\|_{L^\infty} \eta^*,
\end{split}\end{equation}e
for some constant $C>0$.
Therefore, we can integrate \eqref{etastar} to get
\begin{equation}\begin{equation}gin{split}\lambdabel{test5}
\int_{\mathbb{R}^3} \eta^*({\bf x}, t)d{\bf x}
\le & \int_{\mathbb{R}^3} \eta^*({\bf x}, 0)d{\bf x} +\kappappa \int_0^t\int_{\mathbb{R}^3}\rho_2 ({\bf u}_1 -{\bf u}_2)\cdot\nablabla_{{\bf x}}(\Psi_2-\Psi_1)d{\bf x} d\thetau\\
&+C\sup_{0\le \thetau \le t}||\nablabla_{\bf x} {\bf u}_1(\cdot, \thetau)||_{L^\infty}\int_0^t\int_{\mathbb{R}^3}\eta^*({\bf x}, \thetau)d{\bf x}d\thetau.
\end{split}\end{equation}
To bound the second term on the right-hand side of \eqref{test5}, we need a lemma presented in \cite{AB}: suppose $h\in L^{\infty}(\mathbb{R}^3)$ is a function having a compact support, then
$$
\left\|\nablabla_{{\bf x}} \int_{\mathbb{R}^3}\frac{ {h}({\bf y})}{|{\bf x}-{\bf y}|}d{\bf y}\right\|_{L^2(\mathbb{R}^3)}^2\le C \left(\int_{\mathbb{R}^3}|h({\bf x})|^{4/3}d{\bf x}\right)\left(\int_{\mathbb{R}^3}|h({\bf x})|d{\bf x}\right)^{2/3}<\infty, $$
where $C$ is a universal constant. By applying this fact and noting \eqref{potentiali}, we obtain
\begin{equation}\begin{equation}gin{split}\lambdabel{x10}
& \int_{\mathbb{R}^3}\left|\nablabla_{{\bf x}}(\Psi_2-\Psi_1)({\bf x}, \thetau)\right|^2 d{\bf x} \\
\le & C \left(\int_{\mathbb{R}^3} |\rho_2-\rho_1|^{4/3}({\bf x},\thetau) d{\bf x}\right)\left(\int_{\mathbb{R}^3}|\rho_2-\rho_1| ({\bf x},\thetau)d{\bf x}\right)^{2/3} \\
\le &C \left(\int_{S(\thetau)} |\rho_2-\rho_1|^{4/3}({\bf x},\thetau) d{\bf x}\right)\left(\int_{S(\thetau)}|\rho_2-\rho_1| ({\bf x},\thetau)d{\bf x}\right)^{2/3}
\end{split}\end{equation}
for any $\thetau\in [0, T]$, where
$$S(\thetau):= \{{\bf x}: |\rho_1-\rho_2|({\bf x}, \thetau)>0\}, \ \ \thetau\in [0, T].$$
By virtue of H${\rm \ddot{o}}$lder's inequality, one gets
$$\int_{S(\thetau)}|\rho_2-\rho_1|^{4/3}({\bf x}, \thetau)d{\bf x}\le \left(\int_{S(\thetau)}|\rho_2-\rho_1|^2({\bf x}, \thetau)d{\bf x}\right)^{2/3}\left({\rm Vol} S(\thetau)\right)^{1/3}$$
and
$$\left(\int_{S(\thetau)}|\rho_2-\rho_1|({\bf x}, \thetau)d{\bf x}\right)^{2/3}\le \left(\int_{S(\thetau)}|\rho_2-\rho_1|^2({\bf x}, \thetau)d{\bf x}\right)^{1/3}\left({\rm Vol} S(\thetau)\right)^{1/3}.$$
We thus achieve, using \eqref{x10}, that
$$
\int_{\mathbb{R}^3}\left|\nablabla_{{\bf x}}(\Psi_2-\Psi_1)({\bf x}, \thetau)\right|^2 d{\bf x}
\le C \left(\int_{S(\thetau)}|\rho_2-\rho_1|^2({\bf x}, \thetau)d{\bf x}\right) \left({\rm Vol} S(\thetau)\right)^{2/3}.
$$
Note from \eqref{eta} that for $1<\gammamma\le 2$,
\begin{equation}\lambdabel{hheta}\begin{equation}gin{split}
\eta^*({\bf x}, \thetau)
\ge & C(\gamma)\left(\left||\rho_2(\cdot, \thetau)\right||_{L^\infty}+\left||\rho_1(\cdot, \thetau)\right||_{L^\infty}\right)^{\gammamma-2}(\rho_2-\rho_1)^2 +\frac{1}{2}\rho_2|{\bf u}_2-{\bf u}_1|^2\ge 0. \end{split}\end{equation}
Then, it yields that
\begin{equation}e\begin{equation}gin{split}\lambdabel{}
& \int_{\mathbb{R}^3}\left|\nablabla_{{\bf x}}(\Psi_2-\Psi_1)({\bf x}, \thetau)\right|^2 d{\bf x}\\
\le & C \left(\left||\rho_2(\cdot, \thetau)\right||_{L^\infty}+\left||\rho_1(\cdot, \thetau)\right||_{L^\infty}\right)^{2-\gamma} \left({\rm Vol} S(\thetau)\right)^{2/3}\int_{\mathbb{R}^3} \eta^*({\bf x} ,\thetau) d{\bf x}.
\end{split}\end{equation}e
Using this and the Cauchy inequality, we have
\begin{equation}\lambdabel{finaa}\begin{equation}gin{split}
& \left|\int_{\mathbb{R}^3}\rho_2 ({\bf u}_1 -{\bf u}_2)\cdot\nablabla_{{\bf x}}(\Psi_2-\Psi_1)d{\bf x}\right|\\
\le & \int_{\mathbb{R}^3}\rho_2 \left|{\bf u}_1 -{\bf u}_2\right|^2 d{\bf x} + \int_{\mathbb{R}^3}\rho_2 \left|\nablabla_{{\bf x}}(\Psi_2-\Psi_1)({\bf x}, \thetau)\right|^2 d{\bf x}\\
\le & C (1+ { Z}(\thetau))\int_{\mathbb{R}^3} \eta^*({\bf x} ,\thetau) d{\bf x},
\end{split}\end{equation}
where
$$Z(\thetau)=\left||\rho_2(\cdot, \thetau)\right||_{L^\infty} \left(\left||\rho_2(\cdot, \thetau)\right||_{L^\infty}+\left||\rho_1(\cdot, \thetau)\right||_{L^\infty}\right)^{2-\gamma} \left({\rm Vol} S(\thetau)\right)^{2/3}.$$
Now, it follows from \eqref{test5} and \eqref{finaa} that for $t\in[0,T]$,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\int_{\mathbb{R}^3} \eta^*({\bf x}, t)d{\bf x}
\le &
C\sup_{0\le \thetau \le T}\left(||\nablabla_{\bf x} {\bf u}_1(\cdot, \thetau)||_{L^\infty}+ Z(\thetau)\right)\int_0^t\int_{\mathbb{R}^3}\eta^*({\bf x}, \thetau)d{\bf x}d\thetau,
\end{split}\end{equation}e
when
$$\Omega_1(0)=\Omega_2(0) \ \ {\rm and} \ \ (\rho_1, {\bf u}_1)(x,0)=(\rho_2, {\bf u}_2)(x,0).$$
So, one concludes from \eqref{extend'}, \eqref{hheta} and Grownwall's inequality that
$$
\int_{\mathbb{R}^3} \eta^*({\bf x}, t)d{\bf x}=0, \ \ ({\bf x},t)\in\mathbb{R}^3 \times [0,T],
$$
and
$$
\rho_1({\bf x}, t)= \rho_2 ({\bf x}, t), \ \ ({\bf x},t)\in\mathbb{R}^3 \times [0,T].
$$
In particular,
$${\rm Spt} \rho_1(\cdot, t)= {\rm Spt}\rho_2(\cdot, t), \ \ t\in[0,T],$$
where
$${\rm Spt} \rho_i(\cdot, t)=\{{\bf x}\in \mathbb{R}^3:\ \rho_i({\bf x}, t)>0\}.$$
This implies that
$$\Omega_1(t)=\Omega_2(t), \ \ t\in [0,T].$$
In view of \eqref{hheta} and $\eqref{gep1}_{3,4}$, we then see that
$${\bf u}_1({\bf x}, t)={\bf u}_2({\bf x}, t), \ \ ({\bf x},t)\in \Omega_1(t) \times [0,T]. $$
This finishes the proof of Theorem \ref{uniqueness1}.
\section{Formulation and main existence results for spherically symmetric motions}
Starting from this section, we will focus on spherically symmetric motions.
For a three-dimensional spherically symmetric motion, that is,
\begin{equation}gin{equation}\lambdabel{3.1}\rho({\bf x}, t) = \rho(r, t), \ \ {\bf u}({\bf x}, t) = u(r, t) {\bf x} /r , \ \ {\rm where} \ \ u\in \mathbb{R} \ \ {\rm and} \ \ r=|{\bf x}|,\end{equation}
system \eqref{2.1} can be written as follows: for $0\le t\le T$,
\begin{equation}gin{equation}\lambdabel{103}
\begin{equation}gin{split}
&\partial_t (r^2\rho)+\partial_r(r^2\rho u)=0 & {\rm in } & \ \ \left(0, \ R(t)\right) , \\
&\rho(\partial_t u+u\partial_r u)+\partial_r p+ {4\pi\rho}r^{-2}\int_0^r\rho(s,t) s^2ds=0 & {\rm in } & \ \ \left(0, \ R(t)\right), \\
& \rho>0 & {\rm in } & \ \ \left[0, \ R(t)\right)\\
& \rho(R(t),t)=0, \ \ u(0,t)=0, & & \\
& \dot R(t)=u(R(t), t) \ \ {\rm with} \ \ R(0)=1 , & &\\
& (\rho, u)= (\rho_0, u_0) & {\rm on } & \ \ I:= (0, 1).
\end{split}
\end{equation}
Here $\eqref{103}_{3,4}$ state that $r=R(t)$ is the free boundary and the center of the symmetry does not move; $\eqref{103}_{5}$ describes that the free boundary issues from $r=1$ and moves with the fluid velocity; the initial conditions are prescribed in $\eqref{103}_{6}$. The initial domain is taken to be a unit ball $\{0\le r\le 1\}$. And the initial density of interest is supposed to satisfy
\begin{equation}\lambdabel{156}
\rho_0(r)>0 \ \ {\rm for} \ \ 0\le r<1 \ \ {\rm and} \ \ \rho_0(1)=0 ;\end{equation}
and the physical vacuum condition:
\begin{equation}\lambdabel{142}
-\infty< \partial_r \left(\rho_0^{\gamma-1}\right) <0 \ \ {\rm at} \ \ r=1.
\end{equation}
To fix the boundary, we transform system \eqref{103} into Lagrangian variables.
Without abusing notations and for convenience, we use $x$ ($0\le x\le 1$) as the initial reference variable, and define the Lagrangian variable $r(x, t)$ by
\begin{equation}\lambdabel{rxt}
\partial_t r(x, t)= u(r(x, t), t) \ \ {\rm for} \ \ t>0 \ \ {\rm and} \ \ r(x, 0)=x.
\end{equation}
Thus $\eqref{103}_1$ implies that
$$\int_0^{r(x, t)}\rho(s, t)s^2ds=\int_0^x \rho_0(y) y^2dy.$$
Define the Lagrangian density and velocity by
$$f(x, t)=\rho(r(x, t), t) \ \ {\rm and} \ \ v(x, t)=u(r(x,t), t).$$
Then the Lagrangian version of system \eqref{103} can be written on the reference domain $I$ as
\begin{equation}\lambdabel{419} \begin{equation}gin{split}
&\partial_t(r^2f) +r^2f(\partial_x v)/(\partial_x r)=0 & {\rm in}& \ \ I\times (0, T],\\
& f \partial_t v+ {\partial_x (f^{\gammamma})}/(\partial_x r)+ {4\pi f} {r^{-2}}\int_0^x \rho_0(y) y^2dy=0 \ \ &{\rm in}& \ \ I\times (0, T] \\
& f(1, t)=0, \ \ v(0, t)=0 & {\rm on}& \ \ (0,T],\\
& (f, v) =(\rho_0, u_0) & {\rm on}& \ \ I \times \{t=0\}.
\end{split}
\end{equation}
It follows from solving $\eqref{419}_1$ that
$$
f(x, t)=\left(\frac{x}{r}\right)^2\frac{\rho_0(x)}{\partial_x r(x, t)}.$$
So that system \eqref{419} can be rewritten as
\begin{equation}\lambdabel{419'}\begin{equation}gin{split}
& \rho_0 \left( \frac{x}{r}\right)^2\partial_t v + \partial_x \left[ \left(\frac{x^2}{r^2}\frac{\rho_0}{\partial_x r}\right)^\gamma \right] +4\pi \rho_0 \frac{x^2}{r^4} \int_0^x \rho_0 y^2 dy =0 & {\rm in} & \ \ I \times (0,T],\\
& v(0, t)=0 & {\rm on} & \ \ \{x=0\}\times (0,T],\\
& v(x, 0)= u_0(x) & {\rm on} & \ \ I \times \{t=0\},
\end{split}
\end{equation}
where the initial density $\rho_0$ satisfying \eqref{156} and \eqref{142} has been viewed as a parameter.
With the notations
$$ \sigma (x) :=\rho_0^{\gamma-1} x \ \ {\rm and} \ \ \phi(x) := 4\pi x^{-3}\int_0^x \rho_0 y^2 dy,$$
and the fact $r, \rho_0>0$ in $I \times (0,T]$, equation $\eqref{419'}_1$ can be rewritten as
\begin{equation}\lambdabel{e1-2'}\begin{equation}gin{split}
x \sigma \partial_t v + \partial_x \left[ \sigma^2 \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{\partial_x r}\right)^\gamma \right] - 2 \frac{\sigma^2}{x}\left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{\partial_x r} \right)^{\gamma -1}+ \phi \sigma x^2 \left(\frac{x}{r}\right)^2 \\
+ \frac{ 2-\gamma }{\gamma-1}
\sigma x \partial_x \left(\frac{\sigma}{x}\right) \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{\partial_x r}\right)^\gamma =0, \ \ {\rm in} \ \ I \times (0,T].
\end{split}\end{equation}
As $\gamma=2$, equation \eqref{e1-2'} becomes relatively simple. However, it should be noted that the essential parts for $\gamma=2$ and $\gamma \neq 2$ are the same
(see equations \eqref{eie} and \eqref{gi1} later), so that the analysis for $\gamma=2$ is applicable for general $\gamma$. Therefore, we first present the main results for $\gamma=2$ in the rest of this section, following the proof of the results we will then discuss the case for general $\gamma$ in Sections 9 and 10.
For $\gamma=2$, we will consider a higher-order energy functional. To this end, we choose a cut-off function $\zeta$ satisfying
$$
\zeta=1 \ \ {\rm on} \ \ [0,\delta], \ \ \zeta=0 \ \ {\rm on} \ \ [2\delta, 1], \ \ |\zeta'|\le s_0/\delta,
$$
for some constant $s_0$, where $\delta=\delta(\rho_0) $ is a small positive constant depending only on the initial density $\rho_0$ to be determined in Section 7.1.1.
The higher-order energy functional is defined to be
\begin{equation}\lambdabel{norm}\begin{equation}gin{split}
{E}(v, t):= & \left\| \sigma \partial_t^{4} v(\cdot,t) \right\|_{1}^2 +\left\|\partial_t^{4 } v (\cdot,t)\right\|_{0}
+ \sum_{j=1}^2 \left\{ \left\| \sigma \partial_t^{4-2j } v (\cdot,t)\right\|_{j+1}^2
+\left\| \partial_t^{4-2j } v (\cdot,t)\right\|_{j }^2 \right.\\
& \left. +
\left\| \frac{ \partial_t^{4-2j } v }{x} (\cdot,t)\right\|_{j-1}^2
+\left\| \sigma^{3/2} \partial_t^{5-2j } \partial_x^{ j +1 } v (\cdot,t)\right\|_{0}^2
+\left\| \sigma^{1/2} \partial_t^{5-2j } \partial_x^{ j } v (\cdot,t)\right\|_{0}^2 \right. \\
&\left.
+ \left\| \partial_t^{5-2j } v (\cdot,t)\right\|_{j-1/2}^2 +\left\| \frac{\partial_t^{5-2j } v }{x}(\cdot,t)\right\|_{j -1}^2 \right \}\\
& +\sum_{j=1}^2 \left\{ \left\|\zeta \sigma \partial_t^{5-2j } v (\cdot,t)\right\|_{j+1}^2
+\left\|\zeta \partial_t^{5-2j } v (\cdot,t)\right\|_{j }^2 \right\} .
\end{split}\end{equation}
Here and thereafter, we use $\| \cdot\|_s$ to denote the norm of the standard Sobolev space $\|\cdot\|_{H^s(I)}$ for $s\ge 0$; and define the polynomial function $M_0$ by
\begin{equation}\lambdabel{559} M_0= P(E(v,0)),\end{equation}
where $P$ denotes a generic polynomial function of its argument. Now, we are ready to state the main result.
\begin{equation}gin{thm}\lambdabel{existence1} {\rm (existence for $\gammamma=2$)} Given initial data
$(\rho_0, u_0)$ such that $M_0 < \infty$, conditions \eqref{156} and \eqref{142} hold and $\rho_0\in C^3([0,1])$, there exists a solution $v(x,t)$ to problem \eqref{419'} on $[0, T]$ for
$T >0$ taken sufficiently small such that
\begin{equation}\lambdabel{612}\sup_{0\le t\le T}E(v, t)\le 2M_0.\end{equation}
\end{thm}
{This section will be closed by several comments in order. First, the time derivatives of $v(x, t)$ at time $t=0$ involved in the definition of $M_0$ can be given in terms of the corresponding spatial derivatives of the initial data $\rho_0$ and $u_0$ due to the compatibility conditions of equation $\eqref{419'}_1$. Second, the solution to the spherically symmetric problem \eqref{103} in Eulerian coordinates can be obtained from the solution constructed in Theorem \ref{existence1}, since the Lagrangian variable $r\in H^2$ and $\partial_x r$ has a positive lower-bound. Finally, we can transform the solution of problem \eqref{103} back to solve the three-dimensional problem \eqref{2.1} in $W^{1, \infty}(D_T)$, where
$$D_T=\{({\bf x}, t): \ {\bf x} \in \Omega(t), \ \ t\in [0, T]\} \ \ {\rm and} \ \ \Omega(t)=\{{\bf x}\in \mathbb{R}^3: \ |{\bf x}| < R(t)\}.$$
In fact, one can obtain a function
$\rho({\bf x}, t)$ and a vector field ${\bf u}({\bf x}, t)$ via \eqref{3.1} for $({\bf x}, t)\in D_T$ since $u({\bf 0}, t)=0$, and verify that $(\rho, {\bf u})\in C^1(D_T^0)\cap W^{1, \infty}(D_T) $ and \eqref{2.1} holds in $D_T^0$, where
$$\ D_T^0=D_T\setminus \{{\bf 0}\}\times [0, T]. $$
However, $(\rho, {\bf u})$ may not be in $C^1(D_T)$ if the compatibility condition of the first derivative being zero at the origin is not required.
}
\section{Parabolic approximations}
Let $\gamma=2$ from this section to Section 8. Equation $\eqref{419'}_1$ reads
\begin{equation}\lambdabel{e1-3}\begin{equation}gin{split}
x \sigma \partial_t v + \left[ x^2 \sigma^2 /(r^2 r'^2) \right]' -2 x^2 \sigma^2 /(r^3 r') + x^4 \phi \sigma /r^2 =0, \ \ {\rm in} \ \ I\times (0, T],
\end{split}\end{equation}
where and in what follows, the notation $'$ denotes the $\partial_x$.
For $\mu>0$, we use the following degenerate parabolic problem to approximate \eqref{419'}:
\begin{equation}\lambdabel{pe1-3}\ \begin{equation}gin{split}
&x \sigma \partial_t v + \left[\sigma^2 \frac{x^2 } {r^2 r'^2} \right]' -2 \frac{\sigma^2}{x} \frac{x^3}{r^3 r'} + x^2 \phi \sigma \frac{x^2} {r^2} =\frac{2\mu}{x}\left[(x\sigma)^2\left(\frac{v}{x}\right)'\right]' &{\rm in} &\ \ I\times (0, T],\\
& v(0, t)=0 & {\rm on} &\ \ (0, T],\\
&v(x, 0)= u_0(x) & {\rm on} & \ \ I.
\end{split} \end{equation}
As in \cite{10,10'}, one can show easily the existence and uniqueness of the solution $v_{\mu}$ to the above degenerate parabolic problem in a time interval $[0, T_{\mu}]$ with sufficient smoothness for which our later arguments are legitimate by smoothing the initial data and using the fixed point argument. Next, we will give the uniform estimates independent of $\mu$ to obtain the compactness of the sequence $\{v_{\mu}\}$ and a common time interval $[0, T]$ in which the problem \eqref{pe1-3} is solvable for any $\mu>0$, that is,
\begin{equation}gin{lem}\lambdabel{mainlem}
For any fixed $\mu>0$, let $v_{\mu}$ be the smooth solution of \eqref{pe1-3} in $[0, T_{\mu}]$. Then there exist constants $C>0$ and $T\in (0, T_{\mu}]$ independent of $\mu$ such that for the higher-order energy functional
\begin{equation}
E(t):= E(v_{\mu}, t)
\end{equation}
defined in \eqref{norm} satisfies the inequality
\begin{equation}\lambdabel{mainestimate}
\sup_{t\in [0, T]}E(t)\le M_0+CT P\left(\sup_{t\in [0, T]}E(t)\right),
\end{equation}
where $P(\cdot)$ denotes a generic polynomial function of its argument, and $M_0$ is defined in \eqref{559}.
\end{lem}
We will establish the energy estimates in the tangential directions of the boundaries and the elliptic
estimates in the normal direction to prove this lemma. In what follows, for the sake of notational convenience, we omit $\mu$ in $v_{\mu}$, i.e., we denote $v_{\mu}$ by $v$ without ambiguity. Before performing the detailed estimate, we list some preliminaries which will be often used later.
\section{Some preliminaries}
In this section, we will present some embedding estimates for weighted Sobolev spaces, and derive some bounds which
follows directly from the definition of the high order energy functional \eqref{norm} and the a priori assumption.
{\bf Embedding of weighted Sobolev spaces}. Set
\begin{equation}\lambdabel{distant}
d(x)=dist(x, \partial I)=\min\{x, 1-x\} \ \ {\rm for} \ \ x\in I.
\end{equation}
For any $a>0$ and nonnegative integer $b$, the weighted Sobolev space $H^{a, b}(I)$ is given by
$$ H^{a, b}(I) := \left\{d^{a/2}F\in L^2(I): \ \ \int d^a|D^k F|^2dx<\infty, \ \ 0\le k\le b\right\}$$
with the norm
$$ \|F\|^2_{H^{a, b}} := \sum_{k=0}^b \int d^a|D^k F|^2dx.$$
Here and thereafter, we use $\int dx := \int_I dx$ to denote the spatial integral over the interval $I$.
Then for $b\ge {a}/{2}$, it holds the following embedding (cf. \cite{18'}):
$$ H^{a, b}(I)\hookrightarrow H^{b- {a}/{2}}(I)$$
with the estimate
\begin{equation}\lambdabel{wsv} \|F\|_{b- {a}/{2}} \le C \|F\|_{H^{a, b}} .\end{equation}
In particular, we have
\begin{equation}\lambdabel{weightedsobolev}
\|F\|^2_{1- a/2}\le C\int d(x)^a \left(|F(x)|^2+|DF(x)|^2\right)dx, \ \ a=1 \ \ {\rm or} \ \ 2.
\end{equation}
{\bf Some consequences of \eqref{norm}}. It follows from conditions \eqref{156} and \eqref{142} that $\sigma(x)$ is equivalent to the distance function $d(x)$ defined in \eqref{distant}.
Hence, the definition of the energy norm \eqref{norm} and the embedding \eqref{wsv} yield that
\begin{equation}\lambdabel{odd}\begin{equation}gin{split}
\left\| \left(\sigma \partial_t^3 v', \ \sigma \partial_t v'' \right ) (\cdot,t)\right\|_{1/2}^2 + \left\| \left(\sigma \partial_t^3 v\right)(\cdot,t)\right\|^2_{3/2} + \left\| \left(\sigma \partial_t v\right)(\cdot,t) \right \|_{5/2}^2 \le C {E(t)},
\end{split}\end{equation}
Therefore, it holds that for any $p\in(1,\infty)$,
\begin{equation}\lambdabel{egn}\begin{equation}gin{split}
\left\| \left(x^{-1}v, \ v', \ \sigma v'',\ x^{-1} \partial_t v, \ \sigma\partial_t v',\ \ \partial_t^2 v, \ \sigma\partial_t^2 v', \sigma \partial_t^3 v, \sigma \partial_t^4v
\right)(\cdot,t)\right\|_{L^\infty}
\\
+ \left\| \left( \partial_t v', \ \sigma \partial_t v'', \ \partial_t^3 v, \ \sigma \partial_t^3 v'
\right)(\cdot,t)\right\|_{L^p}\le C\sqrt{E(t)},
\end{split}\end{equation}
where one has used the fact that in one space dimension, $\|\cdot\|_{L^\infty}\le C\|\cdot\|_1$ and $\|\cdot\|_{L^p}\le C\|\cdot\|_{1/2}$ ($1<p<\infty$).
Besides,
another type of estimates are also needed.
Noting from \eqref{norm}, \eqref{odd}, and the simple fact that for any norm,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\|\partial_t^j v(\cdot, t)\right\| = & \left\|\partial_t^j v(\cdot, 0) + \int_0^t \partial_t^{j+1} v(\cdot,s)ds\right\| \\
\le & \left\|\partial_t^j v(\cdot, 0)\right\| + \int_0^t\left\| \partial_t^{j+1} v(\cdot,s)\right\| ds \\
\le & \left\|\partial_t^j v(\cdot, 0)\right\| + t \sup_{s\in [0,t]}\left\| \partial_t^{j+1} v(\cdot,s)\right\|, \ \ j=0,1,2,3;
\end{split}\end{equation}e
one can get
\begin{equation}\lambdabel{tnorm}\begin{equation}gin{split}
&\sum_{j=0}^3 \left\{\left\|\left(\sigma \partial_t^j v \right)(\cdot,t)\right\|_{(5-j)/2}^2+\left\| \partial_t^j v(\cdot,t)\right\|_{(3-j)/2}^2\right\}
\\
&\ \ \ \ \ + \left\|\left(\frac{\partial_t^2 v}{x}\right)(\cdot,t)\right\|_0^2 +\left\|\left(\frac{\partial_t v}{x}\right)(\cdot,t)\right\|_0^2+\left\|\left(\frac{ v}{x}\right)(\cdot,t)\right\|_1^2
\le M_0 + CtP\left(\sup_{[0,t]}E\right);
\end{split}\end{equation}
which implies in the same way as in the derivation of \eqref{egn} that for $p\in(1,\infty)$,
\begin{equation}\lambdabel{tegn}\begin{equation}gin{split}
&\left\| \left( \ x^{-1} v, \ \sigma v', \ \partial_t v, \ \sigma\partial_t v',\ \sigma \partial_t^2 v,
\ \sigma \partial_t^3 v
\right)(\cdot,t)\right\|_{L^\infty}^2 \\
& \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \
+ \left\| \left( v', \ \sigma v'', \ \partial_t^2 v, \ \sigma \partial_t^2 v'
\right)(\cdot,t)\right\|_{L^p}^2 \le M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}
{It should be noted that this paper concerns the local existence, so we always assume the time variable $t\le 1$.}
{\bf The a priori assumptions}. Let $\mathcal{M}>0$ be a large constant (for instance, $\mathcal{M}=2{M}_0 +1$). Suppose that for $T\in (0,\mathcal{M}/2] $,
$$
\|v(\cdot,t)\|_2\le \mathcal{M}, \ \ t\in[0,T].
$$
Then it holds that for $(x,t)\in(0,1)\times[0,T]$,
\begin{equation}\lambdabel{xr}
\frac{1}{2}\le \frac{r(x,t)}{x}\le \frac{3}{2}, \ \ \frac{1}{2}\le {r'(x,t)} \le \frac{3}{2}.
\end{equation}
This can be achieved by noticing that $r(x,0)=x$ and
for any $(x,t)\in(0,1)\times(0,T]$,
\begin{equation}e\begin{equation}gin{split}
\left|\frac{r}{x}-1 \right| =\left| \int_0^t \frac{v(x,s)}{x}ds \right|= \left|\int_0^t \int_0^1 v'(\theta x,s)d\theta ds \right| \le t \sup_{s\in[0,t]}\|v'(s)\|_1\le \mathcal{M}T \le \frac{1}{2},
\end{split}\end{equation}e
\begin{equation}e
\left|r'-1\right|=\left|\int_0^t v'( x,s)ds \right| \le t \sup_{s\in[0,t]}\|v'(s)\|_1\le\mathcal{ M} T \le \frac{1}{2}. \end{equation}e
In the proof of Lemma \ref{mainlem}, the time $t>0$ is taken sufficiently small so that the bounds \eqref{xr} are always true.
\section{Energy estimates}
The purpose of this section is to derive a bound for
$$\sup_{[0,t]} \left(\left\| \sqrt{x \sigma } \partial_t^{5} v \right\|_0^2 + \left\| \sigma \partial_t^{4} v\right\|_1^2 + \left \| \partial_t^{4} v\right\|_0^2\right).$$
It should be noted that the estimate $ \| \partial_t^{4} v\|_0$ is needed because the solution,we seek, satisfies $v(\cdot, t)\in C^1(I)$. By the Sobolev
embedding, one needs to estimate $\|v(\cdot, t)\|_2$. Due to the degeneracy of the equation, one time derivative of the solution is equivalent to
the half of the spatial derivative.
We first derive a general equation for time derivatives. Taking the $(k+1)$-th time derivative of equation $\eqref{pe1-3}_1$ gives
\begin{equation}\lambdabel{e1-11}\begin{equation}gin{split}
&x \sigma \partial_t^{k+2} v -2\left\{ \sigma^2 \left[ \frac{x^3} {r^3 r'^2} \frac{\partial_t^{k } v } {x} + \frac{x^2 } {r^2 r'^3}\partial_t^{k } v'\right]\right\}' + 2 \frac{\sigma^2}{x}\left[ 3 \frac{x^4} {r^4 r' } \frac{\partial_t^{k } v } {x} + \frac{x^3\partial_t^{k } v'}{r^3 r'^2} \right] \\
= & \frac{2\mu}{x}\left[(x\sigma)^2\left(\frac{\partial^{k+1}_t v}{x}\right)'\right]'+ 2\left\{ \sigma^2\left[ I_{11} + I_{12}\right]\right\}'- 2 \frac{\sigma^2}{x}\left[ 3 I_{21} + I_{22} \right]
- \phi \sigma x^2 \partial_t^{k+1} \left(\frac{x^2}{r^2}\right) ,
\end{split}\end{equation}
where
\begin{equation}\lambdabel{e1-6}\begin{equation}gin{split}
&I_{11}= \partial_t^{k } \left( \frac{x^3 } {r^3 r'^2} \frac{v}{x}\right)- \frac{x^3} {r^3 r'^2} \frac{\partial_t^{k } v } {x} = \sum\limits_{\alpha=0}^{k-1} C_\alpha^{k-1} \partial_t^{k -\alpha}\left( \frac{x^3}{r^3r'^2}\right)\left(\frac{\partial_t^{\alpha} v}{x}\right) ,\\
&I_{12}= \partial_t^{k } \left( \frac{x^2 v' } {r^2 r'^3}\right)- \frac{x^2 } {r^2 r'^3}\partial_t^{k } v'
=\sum\limits_{\alpha=0}^{k-1} C_\alpha^{k-1} \ \partial_t^{k -\alpha}\left( \frac{x^2}{r^2r'^3}\right)(\partial_t^{\alpha} v') ,\\
&I_{21}= \partial_t^{k } \left( \frac{x^4 } {r^4 r' } \frac{v}{x}\right)- \frac{x^4} {r^4 r' } \frac{\partial_t^{k } v } {x} = \sum\limits_{\alpha=0}^{k-1} C_\alpha^{k-1} \partial_t^{k -\alpha}\left( \frac{x^4}{r^4r' }\right)\left(\frac{\partial_t^{\alpha} v}{x}\right) ,\\
&I_{22}= \partial_t^{k }\left( \frac{x^3v'}{r^3r'^2}\right)-\frac{x^3\partial_t^{k } v'}{r^3 r'^2} =\sum\limits_{\alpha=0}^{k-1} C_\alpha^{k-1} \partial_t^{k -\alpha}\left( \frac{x^3}{r^3r'^2}\right)(\partial_t^{\alpha} v').
\end{split}\end{equation}
Here and thereafter, $C_{\alpha}^{k-1}=(k-1)!/[(k-1-\alpha)!\alpha!]$.
Multiplying \eqref{e1-11} with $k=4 $ by $\partial_t^{5} v$ and integrating the resulting equation with respect to space and time yield, by virtue of integration by parts, that
\begin{equation}\lambdabel{energy}\begin{equation}gin{split}
&\left.\int \left\{ \frac{x \sigma }{2} \left(\partial_t^{5} v \right)^2 + \frac{ x^2 }{r^2 r'} \left[ {\frac{1}{r'^2}(\sigma \partial_t^{4} v')^2+ 3\frac{ x^2}{r^2} \left(\frac{\sigma}{x}\partial_t^{4} v\right)^2 +2\frac{ x}{rr'}\left(\frac{\sigma}{x}\partial_t^{4} v\right)(\sigma\partial_t^{4} v')} \right] \right\} dx \right|_0^t\\
& +2\mu\int_0^t\int \left[x\sigma \left(\frac{\partial_t^5 v}{x}\right)'\right]^2dxds\\
=& \int_0^t\int \left\{ \partial_t \left(\frac{x^2 }{r^2r'^3} \right)(\sigma\partial_t^{4} v')^2 +3\partial_t \left(\frac{x^4 }{r^4r'}\right) \left(\frac{\sigma}{x}\partial_t^{4} v \right)^2 +2\partial_t\left(\frac{x^3 }{r^3r'^2}\right) \left(\frac{\sigma}{x}\partial_t^{4} v\right)(\sigma\partial_t^{4} v') \right\}\\
&\times dxds - 2 \int_0^t\int \left[ \sigma^2(I_{11}+I_{12})(\partial_t^{5} v') + (\sigma/x) \sigma(3I_{21}+I_{22}) (\partial_t^{5} v) \right]dxds\\
& - \int_0^t \int \phi { x^2 \sigma} \partial_t^{5}\left(x^2/r^2\right) (\partial_t^{5} v) dxds\\
=: & J_1 -2 J_2 - J_3 .
\end{split}\end{equation}
In order to estimate the terms on the right-hand side of \eqref{energy}, we notice that for all nonnegative integers $m$ and $n$,
\begin{equation}\lambdabel{14}\begin{equation}gin{split}
\left|\partial_t^{k+1}\left(\frac{x^m}{r^mr'^n}\right)\right| \le C \mathfrak{J}_k,\ \
k=0,\cdots,4,
\end{split}\end{equation}
which follows from simple calculations and the a priori bounds \eqref{xr}. Here
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\mathfrak{J}_{0}= |{x^{-1}} { v}| + | v'|,
\ \
\mathfrak{J}_{1}=|{x^{-1}} {\partial_t v}| + |\partial_t v'| + \mathfrak{J}_{0}^2,
\ \
\mathfrak{J}_2=|{x^{-1}} {\partial_t^{2}v} | + | \partial_t^{2}v' | +\mathfrak{J}_{1}\mathfrak{J}_{0},
\\
&\mathfrak{J}_{3}=|{x^{-1}} {\partial_t^{3}v} | + | \partial_t^{3}v' | + \mathfrak{J}_{2}\mathfrak{J}_{0}+\mathfrak{J}_{1}^2,
\ \
\mathfrak{J}_{4}= |{x^{-1}} {\partial_t^{4 }v} | + | \partial_t^{4}v' | + \mathfrak{J}_{3}\mathfrak{J}_{0}+\mathfrak{J}_{2}\mathfrak{J}_{1}.
\end{split}\end{equation}e
It follows from \eqref{norm}, \eqref{egn}, the H$\ddot{o}$lder inequality and $\|(\sigma,\rho_0)\|_{L^\infty}\le C$ that
\begin{equation}\lambdabel{jk} \begin{equation}gin{split}
&\left\|\mathfrak{J}_0\right\|_{L^\infty}\le\left\| {x^{-1}} { v} \right\|_{L^\infty} + \left\| v'\right\|_{L^\infty}\le C E^{1/2},\\
&\left\|\mathfrak{J}_{1}\right\|_{L^p}\le \left\| {x^{-1}} {\partial_t v} \right\|_{L^\infty} + \left\|\partial_t v'\right\|_{L^p} + \left\| \mathfrak{J}_{0} \right\|_{L^\infty}^2 \le CP(E^{1/2}),\\
& \left\|\mathfrak{J}_{2}\right\|_{0}\le \left\| {x^{-1}} {\partial_t^2 v} \right\|_0 + \left\|\partial_t^2 v'\right\|_0 + \left\| \mathfrak{J}_{1} \right\|_0 \left\| \mathfrak{J}_{0} \right\|_{L^\infty} \le CP(E^{1/2}),
\\
& \left\|\sigma \mathfrak{J}_{2}\right\|_{L^p}\le C \left\| {\partial_t^2 v} \right\|_{L^\infty} + \left\|\sigma \partial_t^2 v'\right\|_{L^\infty} + C \left\| \mathfrak{J}_{1} \right\|_{L^p} \left\| \mathfrak{J}_{0} \right\|_{L^\infty} \le CP(E^{1/2}),
\\
&\left\|\sigma\mathfrak{J}_{3}\right\|_{L^p}\le C \left\| {\partial_t^3 v} \right\|_{L^p} + \left\|\sigma \partial_t^3 v'\right\|_{L^p} + \left\| \sigma\mathfrak{J}_{2} \right\|_{L^p} \left\| \mathfrak{J}_{0} \right\|_{L^\infty} + C \left\| \mathfrak{J}_{1}\right\|_{L^{2p}}^2 \le CP(E^{1/2}),\\
&\left\|\sigma\mathfrak{J}_{4}\right\|_{0}\le C \left\| {\partial_t^4 v} \right\|_{0} + \left\|\sigma \partial_t^4 v'\right\|_{0} + \left\| \sigma\mathfrak{J}_{3} \right\|_0 \left\| \mathfrak{J}_{0} \right\|_{L^\infty} + \left\| \sigma\mathfrak{J}_{2} \right\|_{L^4} \left\| \mathfrak{J}_{1} \right\|_{L^4} \le CP(E^{1/2}),
\end{split}\end{equation}
for any $p\in(1,\infty)$. Here and thereafter $P(\cdot)$ denotes a generic polynomial function. In particular, we have for $m\ge 1$ and $k=0,\cdots,4$,
\begin{equation}\lambdabel{ik}\begin{equation}gin{split}
\left|\partial_t^{k+1}\left(\frac{x^m}{r^m }\right)\right| \le C \mathcal{I}_k \ \ {\rm satisfying} \ \ \left\|x \mathcal{{I}}_{4}\right\|_0 \le CP(E^{1/2}),
\end{split}\end{equation}
where $\mathcal{I}_k$ equals $ {\mathfrak{J}}_k$ modular the terms involving spatial derivatives such as $\partial_t^i v'$ $(i=1,2,3,4)$. Similarly, one can use \eqref{tnorm} and \eqref{tegn} to show that for $p\in(1,\infty)$,
\begin{equation}\lambdabel{tjk} \begin{equation}gin{split}
&\left\|\mathfrak{J}_0(t)\right\|_{L^p}^2 +
\left\|\mathfrak{J}_{1}(t)\right\|_{0}^2+ \left\|\sigma \mathfrak{J}_{2}(t)\right\|_{L^p}^2+ \left\|\sigma\mathfrak{J}_{3}(t)\right\|_{0}^2\le M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}
\begin{equation}\lambdabel{pjk} \begin{equation}gin{split}
&\left\|\mathcal{I}_0(t)\right\|_{L^\infty}^2 +
\left\|\mathcal{I}_{1}(t)\right\|_{0}^2+ \left\|x \mathcal{I}_{1}(t)\right\|_{L^\infty}^2+ \left\| \mathcal{I}_{2}(t)\right\|_{0}^2 + \left\|x \mathcal{I}_{3}(t)\right\|_{0}^2\le M_0 + CtP\left(\sup_{[0,t]}E\right) .
\end{split}\end{equation}
Next, we estimate the terms on the right-hand side of \eqref{energy}. For $J_1$, it follows from \eqref{14}, $\eqref{jk}_1$ and the Cauchy inequality that
\begin{equation}\lambdabel{J1}\begin{equation}gin{split}
J_1
\le & C
\int_0^t \left\{\left\|\mathfrak{J}_{0}\right\|_{L^\infty} \int \left[(\sigma \partial_t^{4} v')^2
+ \left(\frac{\sigma}{x}\partial_t^{4} v \right)^2 \right] dx \right\}ds
\le C t \left( \sup_{[0,t]} E ^{3/2}\right).
\end{split}\end{equation}
For $J_2$, an integration by parts leads to
\begin{equation}\lambdabel{J2}\begin{equation}gin{split}
J_2 = &
\left. \int \left[ \sigma (I_{11}+I_{12})(\sigma\partial_t^{4} v') + \sigma (3I_{21}+I_{22}) \left(\frac{\sigma}{x}\partial_t^{4 } v\right) \right]dx \right|_0^t \\
& -\int_0^t \int \left[ \sigma (\partial_t I_{11}+\partial_t I_{12})(\sigma\partial_t^{4 } v') + \sigma (3\partial_t I_{21}+\partial_t I_{22} ) \left(\frac{\sigma}{x}\partial_t^{4 } v\right) \right]dxds\\
=: & J_{21} - J_{22}.
\end{split}\end{equation}
For $J_{22}$, noting from \eqref{e1-6} and \eqref{14} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left| \partial_t I_{11}\right|
= \sum\limits_{\alpha=0}^{4} C_{ \alpha } \left| \partial_t^{5-\alpha}\left( \frac{x^3}{r^3r'^2}\right)\left(\frac{\partial_t^{\alpha} v}{x}\right)\right|
\le C \sum\limits_{\alpha=0}^{4} \left| \mathfrak{J}_{4-\alpha} \left(\frac{\partial_t^{\alpha} v}{x}\right)\right| ;
\end{split}\end{equation}e
we can then obtain, using \eqref{egn}, \eqref{jk} and the H${\rm \ddot{o}}$lder inequality, that
\begin{equation}\lambdabel{j21}\begin{equation}gin{split}
\left\|\sigma \partial_t I_{11}\right\|_0
\le & C \left\| \mathfrak{J}_{0}\right\|_{L^\infty} \left\|\partial_t^4 v \right\|_0 + C\left\| \mathfrak{J}_{1}\right\|_{L^4} \left\|\partial_t^{3} v \right\|_{L^4} + C\left\| \mathfrak{J}_{2}\right\|_{0} \left\|\partial_t^{2} v \right\|_{L^\infty} \\
&+ C\left\| \sigma \mathfrak{J}_{3}\right\|_{0} \left\|x^{-1}\partial_t v \right\|_{L^\infty}
+C\left\|\sigma\mathfrak{J}_{4}\right\|_{0} \left\|x^{-1} v \right\|_{L^\infty}\\
\le & C P(E^{1/2}) E^{1/2}.
\end{split}\end{equation}
Similarly, one can show that
\begin{equation}\lambdabel{j22}\begin{equation}gin{split}
\left\|\sigma \partial_t I_{21}\right\|_0 \le C P(E^{1/2}) E^{1/2}.
\end{split}\end{equation}
It follows from \eqref{14}, \eqref{egn}, \eqref{jk} and the H${\rm \ddot{o}}$lder inequality that
\begin{equation}\lambdabel{j23}\begin{equation}gin{split}
&\left\|\sigma \partial_t I_{12}\right\|_0 +\left\|\sigma \partial_t I_{22}\right\|_0 \\
\le & C \left\| \mathfrak{J}_{0}\right\|_{L^\infty} \left\|\sigma\partial_t^{4} v' \right\|_0 + C\left\| \mathfrak{J}_{1}\right\|_{L^4} \left\|\sigma\partial_t^{3} v' \right\|_{L^4} + C\left\| \mathfrak{J}_{2}\right\|_{0} \left\|\sigma\partial_t^{2} v' \right\|_{L^\infty} \\
&+ C\left\| \sigma \mathfrak{J}_{3}\right\|_{L^4} \left\|\partial_t v' \right\|_{L^4}
+C\left\|\sigma\mathfrak{J}_{4}\right\|_{0} \left\| v' \right\|_{L^\infty}\\
\le & C P(E^{1/2}) E^{1/2}.
\end{split}\end{equation}
Therefore, it follows from \eqref{J2}-\eqref{j23} and the H$\ddot{o}$lder inequality that
\begin{equation}\lambdabel{J22}\begin{equation}gin{split}
\left|J_{22}\right | \le & C\int_0^t \left[\left(\left\|\sigma \partial_t I_{11}\right\|_0 + \left\|\sigma \partial_t I_{12}\right\|_0 \right) \left\|\sigma \partial_t^{4} v' \right\|_0 \right.\\
&+ \left.\left(\left\|\sigma \partial_t I_{21}\right\|_0 + \left\|\sigma \partial_t I_{22}\right\|_0 \right) \left\| (\sigma/x) \partial_t^{4} v \right\|_0 \right] ds \le C t P \left(\sup\limits_{ [0,t]}E\right).
\end{split}\end{equation}
The term $J_{21}$ can be estimated as
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
|J_{21}| \le & M_0 + \epsilon \left(\left\|\sigma \partial_t^{4} v'(t)\right\|_0^2 + \left\| (\sigma/x) \partial_t^{4 } v(t)\right\|_0^2 \right) \\
&+ C(\epsilon) \left\| \sigma (|I_{11}|+|I_{12}|+|I_{21}|+|I_{22}|)(t)\right\|_0^2 \\
\le & M_0 + \epsilon \left(\left\|\sigma \partial_t^{4} v'(t)\right\|_0^2 + \left\|(\sigma/x) \partial_t^{4 } v(t)\right\|_0^2 \right) \\
&+ C(\epsilon) \sum_{\alpha=0}^3 \left( \left\| \frac{\sigma}{x} \mathfrak{J}_{3-\alpha} (t)\partial_t^\alpha v (t)\right\|_0^2 + \left\| \sigma \mathfrak{J}_{3-\alpha}(t) \partial_t^\alpha v' (t)\right\|_0^2 \right),
\end{split}\end{equation}e
where $\epsilon$ is a small positive constant to be determined later. Here we have used \eqref{wsv}, \eqref{14}, the Holder inequality and the Cauchy inequality.
By virtue of \eqref{norm}, \eqref{egn}, \eqref{jk} and \eqref{tjk}, we obtain
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\left\|\mathfrak{J}_0 (t) \partial_t^3 v (t) \right\|_0+ \left\| \mathfrak{J}_0 (t) \sigma \partial_t^3 v' (t) \right\|_0 \\
= & \left\| \mathfrak{J}_0 (t) \left(\partial_t^3 v (0)+\int_0^t \partial_t^4 v (s)ds \right) \right\|_0+ \left\|\mathfrak{J}_0 (t) \left(\sigma \partial_t^3 v '(0)+\int_0^t \sigma \partial_t^4 v'(s)ds \right) \right\|_0\\
\le & \left\| \mathfrak{J}_0 (t) \right\|_{L^4} \left(\left\| \partial_t^3 v(0) \right\|_ {L^4} +\left\|\sigma \partial_t^3 v'(0) \right\|_{L^4} \right) \\
&+ \int_0^t \left( \left\| \partial_t^4 v (s) \right\|_0+ \left\|\sigma \partial_t^4 v' (s) \right\|_0\right) ds \left\| \mathfrak{J}_0 (t)\right\|_{L^\infty} \\
\le & M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}e
Similarly, one can show that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\sum_{\alpha=0}^2 \left( \left\| \frac{\sigma}{x} \mathfrak{J}_{3-\alpha} (t)\partial_t^\alpha v (t) \right\|_0 + \left\| \sigma \mathfrak{J}_{3-\alpha}(t) \partial_t^\alpha v' (t) \right\|_0 \right)\\
\le & \left\| \mathfrak{J}_1 \right\|_{0} \left(\left\| \partial_t^2 v(0) \right\|_ {L^\infty} +\left\|\sigma \partial_t^2 v'(0) \right\|_{L^\infty} \right) + \int_0^t \left( \left\| \partial_t^3 v (s) \right\|_{L^4}+ \left\|\sigma \partial_t^3 v' (s) \right\|_{L^4}\right) ds \left\| \mathfrak{J}_1 \right\|_{L^4}\\
&+\left\| \sigma \mathfrak{J}_2 \right\|_{L^4} \left(\left\|\frac{ \partial_t v(0)} {x} \right\|_ {L^4} +\left\| \partial_t v'(0) \right\|_{L^4} \right) + \int_0^t \left( \left\| \frac{\partial_t^2 v (s) }{x} \right\|_0+ \left\|\partial_t^2 v' (s) \right\|_0\right) ds \left\| \sigma \mathfrak{J}_2 \right\|_{L^\infty}\\
&+\left\| \sigma \mathfrak{J}_3 \right\|_{0} \left(\left\|\frac{ v(0)} {x} \right\|_ {L^\infty} +\left\| v'(0) \right\|_{L^\infty} \right) + \int_0^t \left( \left\|\frac{ \partial_t v (s) }{x} \right\|_{L^\infty}+ \left\| \partial_t v' (s) \right\|_{L^4}\right) ds \left\|\sigma \mathfrak{J}_3 \right\|_{L^4}\\
\le & M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}e
Therefore, we have arrived at
\begin{equation}\lambdabel{J21}\begin{equation}gin{split}
|J_{21}| \le C(\epsilon)\left[ M_0 + CtP\left(\sup_{[0,t]}E\right)\right] + \epsilon \left(\left\|\sigma \partial_t^{4} v'(t)\right\|_0^2 + \left\|(\sigma/x) \partial_t^{4 } v(t)\right\|_0^2 \right) .
\end{split}\end{equation}
It remains to bound $J_3$. Note from \eqref{ik} that
\begin{equation}\lambdabel{J3}\begin{equation}gin{split}
\left|J_3\right| \le & \int_0^t \left\| \phi\right\|_{L^\infty}\left\| x \partial_t^{5}\left(x^2/ r^2\right) (s) \right\|_{0} \left\| x\sigma \partial_t^{5} v(s) \right\|_0 ds \\
\le & C \left\|\rho_0\right\|_{L^\infty} \left(\sup_{s\in[0,t]} \left\| x\sigma \partial_t^5 v (s)\right\|_0 \right) \int_0^t \left\| x \partial_t^5\left(x^2/r^2\right)(s) \right\|_{0} ds \\
\le & C (\epsilon) \left(\int_0^t \left\| x \partial_t^5\left(x^2/r^2\right)(s) \right\|_{0} ds \right)^2+ \epsilon \left(\sup_{[0,t]} \left\| x\sigma \partial_t^5 v \right\|_0 \right)^2 \\
\le & C(\epsilon) t P\left(\sup_{[0,t]} E\right) + \epsilon \sup_{[0,t]} \left\| x\sigma \partial_t^{5} v \right\|_0^2 ,
\end{split}\end{equation}
where $\epsilon>0$ is a small constant to be determined later.
In view of \eqref{energy}, \eqref{J1}, \eqref{J2}, \eqref{J22}, \eqref{J21} and \eqref{J3}, we see that
\begin{equation}e\begin{equation}gin{split}
&\left.\int \left\{ \frac{x \sigma }{2} \left(\partial_t^{5} v \right)^2 + \frac{ x^2 }{r^2 r'} \left[ {\frac{1}{r'^2}(\sigma \partial_t^{4} v')^2+ 3\frac{ x^2}{r^2} \left(\frac{\sigma}{x}\partial_t^{4} v\right)^2 +2\frac{ x}{rr'}\left(\frac{\sigma}{x}\partial_t^{4} v\right)(\sigma\partial_t^{4} v')} \right] \right\} dx \right|_0^t\\
& +2\mu\int_0^t\int \left[x\sigma \left(\frac{\partial_t^5 v}{x}\right)'\right]^2dxds\\
\le & C(\epsilon) \left[ M_0+ C t P\left(\sup_{[0,t]} E\right) \right]+ \epsilon \left(\left\|\sigma \partial_t^{4} v'\right\|_0^2 + \left\| (\sigma/x)\partial_t^{4 } v\right\|_0^2 + \sup_{[0,t]} \left\| x\sigma \partial_t^{5} v \right\|_0^2 \right).
\end{split}\end{equation}e
Since $\left\| \left(\sqrt{x \sigma } \partial_t^{5} v \right)(\cdot, 0)\right\|_0^2$ can be bounded by $M_0$ due to \eqref{e1-11} with $k=3$, and
\begin{equation}e\begin{equation}gin{split}
&\frac{ x^2 }{r^2 r'} \left[ {\frac{1}{r'^2}(\sigma \partial_t^{4} v')^2+ 3\frac{ x^2}{r^2} \left(\frac{\sigma}{x}\partial_t^{4} v\right)^2 +2\frac{ x}{rr'}\left(\frac{\sigma}{x}\partial_t^{4} v\right)(\sigma\partial_t^{4} v')} \right] \\
=&\frac{ x^2 }{r^2 r'} \left[ \frac{1}{2r'^2}(\sigma \partial_t^{4} v')^2+ \frac{ x^2}{r^2} \left(\frac{\sigma}{x}\partial_t^{4} v\right)^2 +\left(\frac{1}{\sqrt{2}r' }(\sigma \partial_t^{4} v')+\sqrt{2}\frac{ x }{r } \left(\frac{\sigma}{x}\partial_t^{4} v\right) \right)^2 \right] \\
\ge & \frac{ x^2 }{r^2 r'} \left[ \frac{1}{2r'^2}(\sigma \partial_t^{4} v')^2+ \frac{ x^2}{r^2} \left(\frac{\sigma}{x}\partial_t^{4} v\right)^2 \right]\\
\ge & C \left[(\sigma \partial_t^{4} v')^2+ (\sigma x^{-1} \partial_t^{4} v)^2\right],
\end{split}\end{equation}e
where the a priori lower bounds for $1/r'$ and $x/r$ were used; then we have
\begin{equation}e\begin{equation}gin{split}
&\left\| \sqrt{x \sigma } \partial_t^{5} v (t) \right\|_0^2 + \left\| \sigma \partial_t^{4} v'(t)\right\|_0^2 + \left\| \sigma x^{-1} \partial_t^{4} v(t)\right\|_0^2 + \mu\int_0^t\int \left[x\sigma \left(\frac{\partial_t^5 v}{x}\right)'\right]^2dxds\\
\le & C(\epsilon) \left[ M_0+ C t P\left(\sup_{[0,t]} E\right) \right]+ C \epsilon \left(\left\|\sigma \partial_t^{4} v'(t)\right\|_0^2 + \left\|\frac{\sigma}{x}\partial_t^{4 } v(t)\right\|_0^2 + \sup_{[0,t]} \left\| x\sigma \partial_t^{5} v \right\|_0^2 \right),
\end{split}\end{equation}e
which implies, by choosing $\epsilon$ suitably small, that
\begin{equation}\begin{equation}gin{split}
&\sup_{[0,t]} \left(\left\| \sqrt{x \sigma } \partial_t^{5} v \right\|_0^2 + \left\| \sigma \partial_t^{4} v'\right\|_0^2 + \left \| (\sigma/x) \partial_t^{4} v\right\|_0^2\right)+\mu \int_0^t \left\| x\sigma \left(\frac{\partial_t^5 v}{x}\right)'(s) \right\|^2 ds \\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
The weighted Sobolev embedding \eqref{weightedsobolev} implies
\begin{equation}e\begin{equation}gin{split}
\left\| \partial_t^{4} v\right\|_0^2 \le & C \left(\left\| \sigma \partial_t^{4} v\right\|_0^2 + \left\| \sigma \partial_t^{4} v' \right\|_0^2\right) = C \left(\left\| x(\sigma /x) \partial_t^{4} v\right\|_0^2 + \left\| \sigma \partial_t^{4} v' \right\|_0^2\right)\notag\\
\le & C \left(\left\| (\sigma /x) \partial_t^{4} v \right\|_0^2 + \left\| \sigma \partial_t^{4} v' \right\|_0^2\right),
\end{split}\end{equation}e
and we then obtain that
\begin{equation}e\begin{equation}gin{split}
&\sup_{[0,t]} \left(\left\| \sqrt{x \sigma } \partial_t^{5} v \right\|_0^2 + \left\| \sigma \partial_t^{4} v'\right\|_0^2 + \left \| \partial_t^{4} v\right\|_0^2\right)+\mu \int_0^t \left\| x\sigma \left(\frac{\partial_t^5 v}{x}\right)'(s) \right\|^2 ds \\
\le& M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
or equivalently
\begin{equation}\lambdabel{et4}\begin{equation}gin{split}
&\sup_{[0,t]} \left(\left\| \sqrt{x \sigma } \partial_t^{5} v \right\|_0^2 + \left\| \sigma \partial_t^{4} v\right\|_1^2 + \left \| \partial_t^{4} v\right\|_0^2\right)+\mu \int_0^t \left\| x\sigma \left(\frac{\partial_t^5 v}{x}\right)'(s) \right\|^2 ds \\
\le& M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
\section{Elliptic estimates}
In order to estimate the derivatives in the normal direction (the spatial derivatives in Lagrangian coordinates) which can not be obtained by energy estimates as in the last section,
we employ the equation to perform the elliptic estimates. Since the degeneracy of the equation near the origin $x=0$ and the boundary $x=1$ is of different orders, for example, in equation \eqref{e1-3}, the coefficient of $\partial_t v$ is of the order $x^2$ as $x\to 0$, and of the order $(1-x)$ as $x\to 1$, we separate the interior estimates and the estimates near the boundary by choosing suitable cut-off functions. To this end, we first identify the leading terms and lower order terms of the equation. Notice that
\begin{equation}\lambdabel{}\begin{equation}gin{split}
- \left\{ \sigma^2 \left[ \frac{x^3} {r^3 r'^2} \frac{\partial_t^{k } v } {x} + \frac{x^2 } {r^2 r'^3}\partial_t^{k } v'\right]\right\}' + \frac{\sigma^2}{x}\left[ 3 \frac{x^4} {r^4 r' } \frac{\partial_t^{k } v } {x} + \frac{x^3}{r^3 r'^2} \partial_t^{k } v'\right]=-\sigma(\mathfrak{{H}}_0+\mathfrak{H}_1+\mathfrak{H}_2),
\end{split}\end{equation}
where
\begin{equation}\lambdabel{h12}\begin{equation}gin{split}
\mathfrak{H}_0=&\sigma\partial_t^k v''+\sigma\left(\frac{\partial_t^k v}{x}\right)'+\left[2\sigma'-\frac{\sigma}{x}\right]\partial_t^k v'+\left[2\sigma'-3\frac{\sigma}{x}\right]\frac{\partial_t^k v}{x}
= H_0 +4 \left(\frac{\sigma}{x}\right)' \partial_t^k v,\\
\mathfrak{H}_1 = & \left\{
2\sigma' \left( \frac{x^3 } {r^3 r'^2}-1\right)
- \frac{3\sigma }{x } \left(\frac{ x^4 } {r^4 r' }-1\right) \right\}\frac{\partial_t^{k } v}{x} \\
& +\left\{
2\sigma' \left( \frac{x^2 } {r^2 r'^3}-1\right)
- \frac{ \sigma }{x } \left(\frac{ x^3 } {r^3 r'^2 }-1\right) \right\} {\partial_t^{k } v'} , \\
\mathfrak{H}_2= & \sigma \left[ \left(\frac{x^3} {r^3 r'^2} \right)'\frac{\partial_t^{k } v } {x} + \left( \frac{x^2 } {r^2 r'^3}\right)'\partial_t^{k } v' + \left( \frac{x^3 } {r^3 r'^2}-1\right)\left(\frac{\partial_t^{k } v}{x}\right)'+ \left( \frac{x^2 } {r^2 r'^3}-1\right) {\partial_t^{k } v''}\right]
\end{split}\end{equation}
and
\begin{equation}\lambdabel{H0}
H_0=\sigma \partial_t^{k } v''+2\sigma' \partial_t^{k } v' -2\sigma'
{ \partial_t^{k } v }/{x}=\frac{1}{x\sigma}\left[(x\sigma)^2\left(\frac{\partial_t^k v}{x}\right)'\right]'.
\end{equation}
We can then rewrite \eqref{e1-11} as
\begin{equation}\lambdabel{eie}\begin{equation}gin{split}
H_0+\mu \partial_t H_0 = &
\frac{1}{2}x \partial_t^{k+2} v - 4 \left(\frac{\sigma}{x}\right)' \partial_t^k v - \mathfrak{H}_1- \mathfrak{H}_2 -\frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]' \\
& + \left(\frac{\sigma}{x}\right) (3I_{21}+I_{22})
+ \frac{1}{2}\phi { x^2 } \partial_t^{k+1}\left(\frac{x^2}{r^2}\right) =: \mathcal{G},
\end{split}\end{equation}
where $I_{11}$, $I_{12}$, $I_{21}$ and $I_{22}$ are given by \eqref{e1-6}.
In order to obtain estimates independent of the regularization parameter $\mu$, we will also need the following lemma, whose proof can be found in \cite{10}:
\begin{equation}gin{lem}\lambdabel{lem1} Let $\mu>0$ and $g\in L^{\infty}(0, T; H^s(I))$ be given, and let $f\in H^1(0, T; H^s(I))$ be such that
$$ f+\mu f_t=g, \qquad {\rm in~} (0, T)\times I.$$
Then
\begin{equation}\lambdabel{lemma}
\|f\|_{L^{\infty}(0, T; H^s(I))}\le C \max \left\{\|f(0)\|_s, \|g\|_{L^{\infty}(0, T; H^s(I))}\right\}.\end{equation}
\end{lem}
As an immediate consequence of \eqref{eie} and \eqref{lemma}, we see that for any smooth function $\begin{equation}ta(x)$,
\begin{equation}\lambdabel{ik3''}
\sup_{[0, t]}\|\begin{equation}ta H_0\|_0\le C\left( \|\begin{equation}ta H_0(0)\|_0+\sup_{[0, t]} \|\begin{equation}ta \mathcal{G}\|_0\right),\end{equation}
\begin{equation}\lambdabel{ik3'''}
\sup_{[0, t]}\|\begin{equation}ta H_0'\|_0\le C\left( \|\begin{equation}ta H_0'(0)\|_0+\sup_{[0, t]} \|\begin{equation}ta \mathcal{G}'\|_0\right).\end{equation}
{Clearly, the weighted norm of $\partial_t^k v''$ (or $\partial_t^k v'''$) can be derived from the corresponding weighted norm of $\partial_t^{k+2} v$ (or $\partial_t^{k+2} v'$).
Based on the energy estimate \eqref{et4}, we can then obtain the estimates of $\partial_t^3 v''$ and $\partial_t^2 v''$ associated with weights. Furthermore, with the estimates of spatial derivatives of $\partial_t^3 v$ and $\partial_t^2 v$, one can get the weighted estimates of higher-order spatial derivatives of $\partial_t v$ and $v$.}
\subsection{Elliptic estimates -- Interior Estimates}
For the elliptic estimates, since the degeneracy of the equation near the origin $x=0$ and the boundary $x=1$ is of different orders, we will first choose a suitable cut-off function to separate the interior and boundary estimates. {The key is to match the interior and boundary norms in the intermediate region.}
\subsubsection{Interior cut-off functions}
The interior cut-off function $\zeta(x)$ is chosen to satisfy
\begin{equation}\lambdabel{delta}
\zeta=1 \ \ {\rm on} \ \ [0,\delta], \ \ \zeta=0 \ \ {\rm on} \ \ [2\delta, 1], \ \ |\zeta'|\le s_0/\delta,
\end{equation}
for some constant $s_0$, where $\delta$ is a constant to be chosen so that the estimates \eqref{v''} and \eqref{vt'''} below hold for all $k=0,1,2,3$. The choice of $\delta$ will depend on the initial
density $\rho_0$. Since
\begin{equation}e
\sigma'(x)=\rho_0(x)-x\rho_0'(x), \ \ \sigma'(0)=\rho_0(0)>0,
\end{equation}e
there exists a constant $\delta_0$ (depending only on $\rho_0(x)$) such that for all $ x\in[0,\delta_0]$,
\begin{equation}
m_0\le \rho_0(x)\le 3m_0, \ \ m_0\le \sigma'(x)\le 3m_0, \ \ {\rm where} \ \ m_0=\rho_0(0)/2;
\end{equation}
and then
\begin{equation}
m_0x\le \sigma(x)\le 3m_0 x, \ \ x\in [0,\delta_0].
\end{equation}
Set $m_1=\max_{0\le x\le \delta_0} \left\{ \left| \rho_0'(x)\right| , \left| \rho_0''(x)\right| \right\}$. Then for all $ x\in[0,\delta_0]$,
\begin{equation}
\ \ \left|\sigma(x)/x-\sigma'(x)\right| =\left| x\rho_0'(x)\right| \le m_1 x, \ \ \left| \sigma''(x)\right| \le 3m_1.
\end{equation}
{\bf Analysis for $H_0$}. To this end, we rewrite $H_0$ as
\begin{equation}\lambdabel{tl1}
H_0=\sigma f''+2\sigma' f' -2\sigma'
\frac{ f }{x}, \ \ {\rm where} \ \ f=\partial_t^{k } v.
\end{equation}
Multiplying $H_0$ by the cut-off function $\zeta$ with $\delta\in[0,\delta_0/2]$, one may get
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \zeta {H}_{0 }\right\|_0^2 =&
\left\| \zeta \sigma f'' \right\|_0^2 + 4 \left\| \zeta \sigma' f' \right\|_0^2 + 4\left\| \zeta \sigma' \left(
\frac{ f }{x} \right) \right\|_0^2
+4 \int \zeta \sigma f'' \zeta \sigma' f' dx
\\
&-4 \int \zeta \sigma f'' \zeta \sigma' \left(
\frac{ f }{x} \right) dx - 8 \int \zeta \sigma' f' \zeta \sigma' \left(
\frac{ f }{x} \right) dx.
\end{split}\end{equation}e
Observing that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
2 \int \zeta \sigma f'' \zeta \sigma' f' dx =& - \left\| \zeta \sigma' f' \right\|_0^2 - \int \left( \zeta^2 \sigma' \right)' \sigma \left| f' \right|^2 dx\\
\ge & - \left\| \zeta \sigma' f' \right\|_0^2 - C(m_0,s_0) \int_\delta^{2\delta} \left| f' \right|^2 dx -C(m_0,m_1) \delta \left\| \zeta f' \right\|_0^2
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
- \int \zeta \sigma f'' \zeta \sigma' \left(
\frac{ f }{x} \right) dx
= & \int \left(\zeta^2 \sigma'\right)' \sigma \left(
\frac{ f }{x} \right) f' dx + \int \zeta^2 \sigma' \left(\sigma' -\frac{\sigma}{x}\right) \left(
\frac{ f }{x} \right) f' dx \\
&+ \left\| \zeta \sigma' f' \right\|_0^2 + \int \zeta^2 \sigma' \left( \frac{ \sigma }{x} -\sigma'\right) \left| f' \right|^2 dx\\
\ge & \left\| \zeta \sigma' f' \right\|_0^2 -C(m_0,s_0) \int_\delta^{2\delta} \left(\left| f' \right|^2 + \left|
\frac{ f }{x} \right|^2
\right) dx
\\&-C(m_0, m_1) \delta \left[ \left\| \zeta \left(
\frac{ f }{x} \right) \right\|_0^2 + \left\| \zeta f' \right\|_0^2\right],
\end{split}\end{equation}e
we have, using the fact $\sigma'(x)\ge m_0$ on $[0,2\delta]$, that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \zeta {H}_{0 }\right\|_0^2 \ge &
\left\| \zeta \sigma f'' \right\|_0^2 + \frac{2}{3} \left\| \zeta \sigma' f' \right\|_0^2 + \left\| \zeta \sigma' \left(
\frac{ f }{x} \right) \right\|_0^2\\
& +\left( \frac{16}{3} \left\| \zeta \sigma' f' \right\|_0^2 + 3\left\| \zeta \sigma' \left(
\frac{ f }{x} \right) \right\|_0^2 - 8 \int \zeta \sigma' f' \zeta \sigma' \left(
\frac{ f }{x} \right) dx\right) \\
&-C(m_0,s_0) \int_\delta^{2\delta} \left( \left| f' \right|^2 + \left|
\frac{ f }{x} \right|^2
\right) dx -C(m_0, m_1) \delta \left[ \left\| \zeta \left(
\frac{ f }{x} \right) \right\|_0^2 + \left\| \zeta f' \right\|_0^2\right]\\
\ge &
\left\| \zeta \sigma f'' \right\|_0^2 + \frac{2}{3} m_0^2 \left\| \zeta f' \right\|_0^2 + m_0^2 \left\| \zeta \left(
\frac{ f }{x} \right) \right\|_0^2-C(m_0,s_0) \int_\delta^{2\delta} \left( \left| f' \right|^2 + \left|
\frac{ f }{x} \right|^2
\right) dx\\
& -C(m_0, m_1) \delta \left[ \left\| \zeta \left(
\frac{ f }{x} \right) \right\|_0^2 + \left\| \zeta f' \right\|_0^2\right].
\end{split}\end{equation}e
Therefore, there exists a positive constant $\delta_1=\delta_1(m_0,m_1)$ such that if $\delta \le \min\{\delta_0/2, \delta_1\}$,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \zeta {H}_{0 }\right\|_0^2 \ge &
\left\| \zeta \sigma f'' \right\|_0^2 + \frac{1}{3} m_0^2 \left\| \zeta f' \right\|_0^2 + \frac{1}{2} m_0^2 \left\| \zeta \left(
\frac{ f }{x} \right) \right\|_0^2-C(m_0,s_0) \int_\delta^{2\delta} \left( \left| f' \right|^2 + \left|
\frac{ f }{x} \right|^2
\right) dx;
\end{split}\end{equation}e
or equivalently
\begin{equation}\lambdabel{v''}\begin{equation}gin{split}
&\left\| \zeta \sigma {\partial_t^{k } v''} \right\|_0^2+ \left\| \zeta {\partial_t^{k } v'} \right\|_0^2
+ \left\| \zeta \left( \frac{\partial_t^{k } v}{x} \right) \right\|_0^2\\
\le & C(m_0) \left\|\zeta H_{ 0 } \right\|_0^2 + C(m_0,s_0) \int_\delta^{2\delta} \left[ ({\partial_t^{k } v'})^2 + \left( \frac{\partial_t^{k } v}{x} \right)^2 \right]dx.
\end{split}\end{equation}
{\bf Analysis for $H_0'$}. To estimate $H_0'$, one needs also to compute the 1st spatial derivative of $H_0$. Clearly,
\begin{equation}\lambdabel{i10}\begin{equation}gin{split}
H_0'+ 2\sigma''\left(\frac{f}{x}-f'\right)= \sigma f''' + 3 \sigma' f'' -2\sigma' \left(
\frac{ f }{x} \right)'
=: \widetilde{H}_{0 } , \ \ {\rm where} \ \ f=\partial_t^{k } v.
\end{split}\end{equation}
For any function $ \mathfrak{f}=\mathfrak{f}(x,t)$, it holds that
\begin{equation}\lambdabel{formula}
\partial_x^j \mathfrak{f} =\partial_x^j \left(x \frac{\mathfrak{f}}{x} \right) = x\partial_x^j \left(\frac{\mathfrak{f}}{x} \right) +j\partial_x^{j-1}\left(\frac{\mathfrak{f}}{x} \right) , \ \ \ \ j=1,2,3;
\end{equation}
so $\widetilde{H}_{0}$ can be rewritten as
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\widetilde{H}_{ 0 } = \sigma x g'' +3 \left(\sigma x\right)' g'
+ 4\sigma' g , \ \ {\rm where}
\ \ g=\left(\frac{f}{x} \right)'=\left(\frac{\partial_t^{k } v}{x} \right)'.
\end{split}\end{equation}e
Thus,
\begin{equation}\lambdabel{i1}\begin{equation}gin{split}
{ \widetilde{ H}_{ 0 }-3 \left(\sigma'x-\sigma\right) g' }
= & \sigma x g'' +6 \sigma g'
+ 4 \sigma' g. \ \
\end{split}\end{equation}
Multiplying this equality by the cut-off function $\zeta$ with $\delta\in[0,\delta_0]$ and taking the $L^2$-norm of the product yield
\begin{equation}\lambdabel{hi1}\begin{equation}gin{split}
\left\|\zeta \widetilde{H}_{ 0 } +3\zeta \left(\sigma'x-\sigma\right) g' \right\|_0^2
=
\left\| \zeta \sigma x g'' \right\|_0^2+36 \left\| \zeta \sigma g'\right\|_0^2
+ 16 \left\| \zeta \sigma' g \right\|_0^2 \\
+ 12\int \zeta \sigma x g'' \zeta \sigma g' dx
+ 8 \int \zeta \sigma x g'' \zeta \sigma' g dx + 48 \int \zeta \sigma g' \zeta \sigma' g dx.
\end{split}\end{equation}
The last three terms on the right-hand side of \eqref{hi1} can be bounded as follows:
\begin{equation}e\lambdabel{hi2}\begin{equation}gin{split}
-2 \int \zeta \sigma x g'' \zeta \sigma g' dx = \int \left( \zeta^2 \sigma^2 x \right)'\left|g'\right|^2 dx =& 3 \left\| \zeta \sigma g'\right\|_0^2 + 2 \int \zeta \zeta ' x \left|\sigma g'\right|^2 dx\\
&+ 2 \int \zeta^2 \sigma (\sigma'x -\sigma ) \left| g'\right|^2 dx,
\end{split}\end{equation}e
\begin{equation}e\lambdabel{hi3}\begin{equation}gin{split}
& \int \zeta \sigma x g'' \zeta \sigma' g dx = \int \zeta^2 \sigma \left(\sigma' x -\sigma \right) g'' g dx + \int \zeta^2 \sigma^2 g'' g dx \\
=& \int \zeta^2 \sigma \left(\sigma' x -\sigma \right) g g'' dx - 2 \int \zeta \zeta' \sigma^2 g g' dx -2 \int \zeta^2 \sigma \sigma' g g' dx - \left\| \zeta \sigma g'\right\|_0^2
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{hi4}\begin{equation}gin{split}
-2 \int \zeta^2 \sigma \sigma' g g' dx = \left\| \zeta \sigma' g \right\|_0^2 + 2 \int \zeta \zeta' \sigma \sigma' g^2 dx
+ \int \zeta^2 \sigma \sigma'' g^2 dx.
\end{split}\end{equation}e
It then follows from \eqref{hi1} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \zeta \sigma x g'' \right\|_0^2+10 \left\| \zeta \sigma g'\right\|_0^2 \\
= & \left\|\zeta \widetilde{H}_{ 0 } +3\zeta \left(\sigma'x-\sigma\right) g' \right\|_0^2
+ 12\left[ \int \zeta \zeta ' x \left|\sigma g'\right|^2 dx + \int \zeta^2 \sigma (\sigma'x -\sigma ) \left| g'\right|^2 dx \right] \\
& - 8 \left[ \int \zeta^2 \sigma \left(\sigma' x -\sigma \right) g g'' dx - 2 \int \zeta \zeta' \sigma^2 g g' dx \right]
+ 24 \left[2 \int \zeta \zeta' \sigma \sigma' g^2 dx +\int \zeta^2 \sigma \sigma'' g^2 dx\right]\\
\le & 2\left\|\zeta \widetilde{H}_{ 0 } \right\|_0^2 + C(m_0,m_1) \delta \left[ \left\| \zeta \sigma x g'' \right\|_0^2 + \left\| {\zeta} \sigma g'\right\|_0^2 + \left\| {\zeta} g \right\|_0^2 \right]+ C(m_0,s_0) \int_\delta^{2\delta} \left[(\sigma g')^2 + g^2 \right] dx.
\end{split}\end{equation}e
Therefore, there exists a constant $\delta_2=\delta_2(m_0,m_1)$ such that for $\delta\le \min\{\delta_0/2, \delta_2\}$, it holds that
\begin{equation}\lambdabel{hha}\begin{equation}gin{split}
& \frac{1}{2}\left\| \zeta \sigma x g'' \right\|_0^2+ 5\left\| \zeta \sigma g'\right\|_0^2 \\
\le & 2\left\|\zeta \widetilde{H}_{ 0 } \right\|_0^2 +C(m_0,m_1) \delta \left\| {\zeta} g \right\|_0^2 +C(m_0,s_0) \int_\delta^{2\delta} \left[(\sigma g')^2 + g^2 \right] dx.
\end{split}\end{equation}
To handle the term $\left\| \zeta g \right\|_0^2$, we need an additional estimate which follows from \eqref{i1}, that is
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \zeta \sigma' g \right\|_0^2 \le
& C(m_0) \left[ \left\|\zeta \widetilde{H}_{ 0 } \right\|_0^2 +\left\| \zeta \sigma x g'' \right\|_0^2+ \left\| \zeta \sigma g'\right\|_0^2 \right] \\
\le & C(m_0) \left\|\zeta \widetilde{H}_{ 0 } \right\|_0^2 +C(m_0,m_1) \delta \left\| {\zeta} g \right\|_0^2 +C(m_0,s_0) \int_\delta^{2\delta} \left[(\sigma g')^2 + g^2 \right] dx ,
\end{split}\end{equation}e
where we have used \eqref{hha}. Hence, it holds that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \zeta \sigma x g'' \right\|_0^2+ \left\| \zeta \sigma g'\right\|_0^2
+ \left\| \zeta \sigma' g \right\|_0^2 \\
\le & C(m_0) \left\|\zeta \widetilde{H}_{ 0 } \right\|_0^2 +C(m_0,m_1) \delta \left\| {\zeta} g \right\|_0^2 +C(m_0,s_0) \int_\delta^{2\delta} \left[(\sigma g')^2 + g^2 \right] dx .
\end{split}\end{equation}e
Thus, there exists a constant $\delta_3=\delta_3(m_0,m_1)$ such that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \zeta \sigma x g'' \right\|_0^2+ \left\| \zeta \sigma g'\right\|_0^2
+ \frac{1}{2}\left\| \zeta g \right\|_0^2
\le C(m_0) \left\|\zeta \widetilde{H}_{ 0 } \right\|_0^2 +C(m_0,s_0) \int_\delta^{2\delta} \left[(\sigma g')^2 + g^2 \right] dx ,
\end{split}\end{equation}e
provided $\delta\le \min\{\sigma_0/2, \delta_2,\delta_3\}$; where we have used the fact $\sigma'(x)\ge m_0$ on $[0,\delta_0]$. It then follows from \eqref{formula} and \eqref{i10} that
\begin{equation}\lambdabel{vt'''}\begin{equation}gin{split}
&\left\| \zeta \sigma {\partial_t^{k } v} ''' \right\|_0^2+ \left\| \zeta {\partial_t^{k } v} ''\right\|_0^2
+ \left\| \zeta \left( \frac{\partial_t^{k } v}{x} \right)' \right\|_0^2 \\
\le & C(m_0) \left\|\zeta \widetilde{H}_{ 0 } \right\|_0^2 + C( m_0,s_0) \int_\delta^{2\delta} \left[ \left| {\partial_t^{k } v} '' \right|^2+ \left|\left( \frac{\partial_t^{k } v}{x} \right)'\right|^2 \right] dx \\
\le & C(m_0) \left\|\zeta {H}_{ 0 }' \right\|_0^2 + C(m_0,m_1) \left(
\left\| \zeta {\partial_t^{k } v} '\right\|_0^2
+ \left\| \zeta \frac{\partial_t^{k } v}{x} \right\|_0^2\right)
\\
& +C( m_0,s_0) \int_\delta^{2\delta} \left[ \left| {\partial_t^{k } v} '' \right|^2+ \left|\left( \frac{\partial_t^{k } v}{x} \right)'\right|^2 \right] dx.
\end{split}\end{equation}
{\bf A Choice of $\delta$}. Choose
\begin{equation}\lambdabel{da}\delta=\min\{\delta_0/2,\delta_1,\delta_2,\delta_3\},\end{equation}
then the estimates \eqref{v''} and \eqref{vt'''} hold for all $k=0,1,2,3$.
\subsubsection{Interior estimates for $\partial_t^3 v$ and $\partial_t^2 v$}
Consider equation \eqref{eie} with $k=3$, that is
\begin{equation}\lambdabel{ik3}\begin{equation}gin{split}
H_0+\mu \partial_t H_0 = &
\frac{1}{2}x \partial_t^{5} v - 4 \left(\frac{\sigma}{x}\right)' \partial_t^3 v - \mathfrak{H}_1- \mathfrak{H}_2 -\frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]' \\
& + \left(\frac{\sigma}{x}\right) (3I_{21}+I_{22}) + \frac{1}{2} \phi { x^2 } \partial_t^{4}\left(\frac{x^2}{r^2}\right).\\
\end{split}\end{equation}
In order to bound $\left\|\zeta H_0\right\|$ by applying \eqref{ik3''} with $\begin{equation}ta=\zeta$ given by \eqref{delta}, we need to estimate the $L^2$-norm of the right-hand side of \eqref{ik3} term by term. For this purpose, we first derive some estimates which will be used later. In addition to \eqref{egn}, \eqref{tnorm} and \eqref{tegn}, we have some interior bounds:
\begin{equation}\lambdabel{3egn}\begin{equation}gin{split}
& \left\| \left( \zeta \partial_t v', \ \zeta \sigma \partial_t v'', \ \zeta \partial_t^3 v, \ \zeta \sigma \partial_t^3 v'
\right)(\cdot,t)\right\|_{L^\infty}\le C\sqrt{E(t)},\\
& \left\|\left(\zeta {\partial_t^2 v} \right)(\cdot,t)\right\|_1^2+ \left\|\left(\zeta \sigma {\partial_t^2 v}, \zeta v \right)(\cdot,t)\right\|_2^2 + \left\|\left(\zeta \sigma { v} \right)(\cdot,t)\right\|_3^2
\le M_0 + CtP\left(\sup_{[0,t]}E\right),\\
&\left\| \left( \zeta v', \ \zeta \sigma v'', \ \zeta \partial_t^2 v, \ \zeta \sigma \partial_t^2 v'
\right)(\cdot,t)\right\|_{L^\infty}^2 \le M_0 + CtP\left(\sup_{[0,t]}E\right);
\end{split}\end{equation}
which implies
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\|\zeta \mathfrak{J}_{1}\right\|_{L^\infty}\le \left\| {x^{-1}} {\partial_t v} \right\|_{L^\infty} + \left\|\zeta \partial_t v'\right\|_{L^\infty} + \left\| \mathfrak{J}_{0} \right\|_{L^\infty}^2 \le CP(E^{1/2}),\\
&\left\|\zeta \mathfrak{J}_0(t)\right\|_{L^\infty}^2 \le 2 \left(\left\| {x^{-1}} { v} \right\|_{L^\infty}^2 + \left\| \zeta v' \right\|_{L^\infty}^2 \right)\le M_0 + CtP\left(\sup_{[0,t]}E\right),\\
& \left\|\zeta \mathfrak{J}_{2}(t)\right\|_{0}^2
\le
C\left(\left\| {x^{-1}} {\partial_t^2 v} \right\|_0^2 + \left\|\zeta \partial_t^2 v' \right\|_0^2 + \left\| \mathfrak{J}_{1} \right\|_0 ^2\left\| \zeta \mathfrak{J}_{0} \right\|_{L^\infty}^2\right)
\le M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}e
This, together with \eqref{jk} and \eqref{tjk}, yields that for $p\in(1,\infty)$,
\begin{equation}\lambdabel{3jk} \begin{equation}gin{split}
&\left\|\mathfrak{J}_0\right\|_{L^\infty} +
\left\|\mathfrak{J}_{1}\right\|_{L^p} + \left\|\zeta \mathfrak{J}_{1}\right\|_{L^\infty}+ \left\| \mathfrak{J}_{2}\right\|_{0} \le CP(E^{1/2}) , \ \ \\
&\left\|\mathfrak{J}_0(t)\right\|_{L^p}^2+
\left\|\zeta \mathfrak{J}_{0}(t)\right\|_{L^\infty}^2 +
\left\|\mathfrak{J}_{1}(t)\right\|_{0}^2+ \left\|\zeta \mathfrak{J}_{2}(t)\right\|_{0}^2 \le M_0 + CtP\left(\sup_{[0,t]}E\right) .
\end{split}\end{equation}
In a similar way as the derivation of \eqref{14}, we have that for nonnegative integers $m$ and $n$,
\begin{equation}\lambdabel{314}\begin{equation}gin{split}
\left|\partial_t^{k+1}\left(\frac{x^m}{r^mr'^n}\right)'\right| \le C \mathfrak{L}_k,\ \
k=0,1,2;
\end{split}\end{equation}
where
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \mathfrak{L}_{0} = \left|v''\right| + \left| \left(\frac{ v}{x} \right)'\right|+ \mathcal{R}_0 \mathfrak{J}_{0}, \ \ {\rm with} \ \ \mathcal{R}_0 =\left|\left(\frac{r}{x}\right)'\right|+\left|r''\right| ,
\\
& \mathfrak{L}_{1} =\left| \partial_t v''\right|+ \left| \left(\frac{\partial_t v}{x} \right)'\right| + \mathcal{R}_0 \mathfrak{J}_{1}
+ \mathfrak{L}_{0} \mathfrak{J}_{0}, \\
& \mathfrak{L}_{2} = \left|\partial_t^2 v'' \right|+ \left| \left(\frac{\partial_t^2 v}{x} \right)'\right|
+ \mathcal{R}_0 \mathfrak{J}_{2} + \mathfrak{L}_{0} \mathfrak{J}_{1}+\mathfrak{L}_{1} \mathfrak{J}_{0}.
\end{split}\end{equation}e
It can be checked (see the Appendix) that the following estimates hold:
\begin{equation}\lambdabel{3lk} \begin{equation}gin{split}
&\left\| \mathcal{R}_0 \right\|_{0} +
\left\| \sigma \mathcal{R}_0\right\|_{L^\infty } \le C t \sup_{[0,t]} \sqrt{E} ,
\\
&\left\| \mathfrak{L}_0\right\|_{0}^2+ \left\|\sigma \mathfrak{L}_0\right\|_{L^\infty }^2 + \left\| \sigma \mathfrak{L}_{1}\right\|_{L^p}^2+
\left\|\zeta \sigma \mathfrak{L}_{1}\right\|_{L^\infty}^2+ \left\| \sigma \mathfrak{L}_{2}\right\|_{0}^2 \le C P\left(E(t)\right) + C t P\left(\sup_{[0,t]} E\right),
\\
&\left\| \zeta \mathfrak{L}_0\right\|_{0}^2+ \left\| \sigma \mathfrak{L}_0\right\|_{L^p}^2 + \left\|\zeta \sigma \mathfrak{L}_0\right\|_{L^\infty}^2 +
\left\|\sigma \mathfrak{L}_{1}\right\|_{0}^2+ \left\|\zeta \sigma \mathfrak{L}_{2}\right\|_{0}^2 \le M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}
with $\|\cdot\|$ denoting $\| \cdot(t)\|$.
Next, we will bound $\|\zeta H_0 \|$ by the terms on the right-hand side of \eqref{ik3}. It follows from \eqref{tnorm}, \eqref{et4} and the lower bound of $\rho_0(x)$ in the interior region that
\begin{equation}\lambdabel{t31}\begin{equation}gin{split}
\left\|
\zeta\left(\frac{1}{2}x \partial_t^{5} v - 4 \left(\frac{\sigma}{x}\right)' \partial_t^3 v\right)(t)\right\|_0^2 \le C \left\| \zeta \sqrt{x \frac{\sigma}{\rho_0}} \partial_t^{5} v(t)\right\|_0^2 +C \left\| \partial_t^3 v (t)\right\|_0^2 \\
\le C \left\| \sqrt{x {\sigma} } \partial_t^{5} v(t)\right\|_0^2 +C \left\| \partial_t^3 v (t)\right\|_0^2 \le
M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For $\mathfrak{H}_1$, noting from \eqref{egn} that
\begin{equation}\lambdabel{r0}\begin{equation}gin{split}
\left\| \frac{ x }{r(x,t) }-1 \right\|_{L^\infty} +
\left\| \frac{1 }{r'(x,t)}-1\right\|_{L^\infty}
\le \left\| \frac{ x }{r } \left(1-\frac{r}{x}\right) \right\|_{L^\infty} +
\left\| \frac{1 }{r'}\left(1-r'\right)\right\|_{L^\infty} \\
\le C \int_0^t \left(\left\| \frac{v}{x}\right\|_{L^\infty}+\left\|v'\right\|_{L^\infty}\right)ds \le C t \left(\sup_{[0,t]} \sqrt{E}\right) ,
\end{split}\end{equation}
we have
\begin{equation}\lambdabel{t32}\begin{equation}gin{split}
\left\| \zeta \mathfrak{ H}_1 (t) \right\|_0^2 \le C \left\{ \left\| \frac{ x^4 } {r^4 r' }-1 \right\|_{L^\infty}^2 +
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2 +\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2
\right\} \\
\times \left\{ \left\| \frac{\partial_t^{3 } v}{x} \right\|_0^2 + \left\| \zeta \partial_t^{3 } v' \right\|_0^2 \right\}
\le C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For $\mathfrak{H}_2$, it follows from $\eqref{norm}$, \eqref{r0} and $\eqref{3lk}_1$ that
\begin{equation}\lambdabel{t33}\begin{equation}gin{split}
\left\| \zeta \mathfrak{H}_2 (t)\right\|_0^2 \le &\left\{
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2 +\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2
\right\} \left\{ \left\| \zeta \sigma \left(\frac{\partial_t^{3 } v}{x} \right)'\right\|_0^2 + \left\| \zeta \sigma\partial_t^{3 } v'' \right\|_0^2 \right\} \\
&+ C\left\| \sigma \mathcal{R}_0 \right\|_{L^\infty}^2 \left(\left\| {\partial_t^{3} v }/{x}\right\|_{0}^2 + \left\| \zeta \partial_t^{3} v' \right\|_{0}^2 \right)
\le C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}
since
$$ \left\| \zeta \sigma \left(\frac{\partial_t^{3 } v}{x} \right)'\right\|_0
\le C \left\| \zeta x \left(\frac{\partial_t^{3 } v}{x} \right)'\right\|_0
= C \left\| \zeta \partial_t^{3 } v'- \zeta \left(\frac{\partial_t^{3 } v}{x} \right)\right\|_0.$$
Next, we will handle the terms involving $I_{11}$ and $I_{12}$ as follows,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \zeta \frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]' \right\|_0^2
\le C \left\|\zeta (I_{11}+I_{12}) \right\|_0^2 + C\left\|\zeta \sigma ( I_{11}+ I_{12})' \right\|_0^2
\\
\le & C \sum_{\alpha=0}^2 \left\|\zeta \mathfrak{J}_{2-\alpha} \left( \left| {\partial_t^\alpha v} /x \right|+|\partial_t^\alpha v'| \right)\right\|_0^2 +C \sum_{\alpha=0}^2 \left\|\zeta \sigma \mathfrak{L}_{2-\alpha} \left( |\partial_t^\alpha v/x| + | \partial_t^\alpha v'|
\right) \right\|_0^2 \\
&+ C \sum_{\alpha=0}^2 \left\|\zeta \mathfrak{J}_{2-\alpha} \left( \left| \sigma \left( {\partial_t^\alpha v} /x \right)' \right| + |\sigma \partial_t^\alpha v''|
\right)\right\|_0^2 \\
\le & C \sum_{\alpha=0}^2 \left\{ \left\|\zeta \mathfrak{J}_{2-\alpha} \left( \left| {\partial_t^\alpha v} /x \right|+|\partial_t^\alpha v'| + |\sigma \partial_t^\alpha v''|\right)\right\|_0^2 + \left\|\zeta \sigma \mathfrak{L}_{2-\alpha} \left( |\partial_t^\alpha v/x| + | \partial_t^\alpha v'|
\right) \right\|_0^2 \right\}.
\end{split}\end{equation}e
Here we have used \eqref{14} and \eqref{314}. It follows from \eqref{norm}, \eqref{egn}, \eqref{tnorm}, $\eqref{3egn}_1$ and $\eqref{3jk}$ that
\begin{equation}\lambdabel{abc}\begin{equation}gin{split}
&\sum_{\alpha=0}^2 \left\|\zeta \mathfrak{J}_{2-\alpha}(t)\left( \left| \partial_t^\alpha v' (t) \right|+\left| x^{-1} {\partial_t^\alpha v} (t) \right| \right) \right\|_0
\\
\le & \left\| \zeta \mathfrak{J}_0 \right\|_{L^\infty} \left(\left\| \frac{\partial_t^2 v(0) }{x} \right\|_ {0} +\left\| \partial_t^2 v'(0) \right\|_{0} \right) + \int_0^t \left( \left\| \frac{\partial_t^3 v}{x} \right\|_{0}+ \left\|\zeta \partial_t^3 v' \right\|_{0}\right) ds \left\| \mathfrak{J}_0 \right\|_{L^\infty}\\
&+\left\| \mathfrak{J}_1 \right\|_{0} \left(\left\|\frac{ \partial_t v(0)} {x} \right\|_ {L^\infty} +\left\| \zeta \partial_t v'(0) \right\|_{L^\infty} \right) + \int_0^t \left( \left\| \frac{\partial_t^2 v }{x} \right\|_0+ \left\|\partial_t^2 v' \right\|_0\right) ds \left\| \zeta \mathfrak{J}_1 \right\|_{L^\infty}\\
&+\left\| \zeta \mathfrak{J}_2 \right\|_0 \left(\left\| \frac{ v(0) }{x} \right\|_ {L^\infty} +\left\| v'(0) \right\|_{L^\infty}\right) + \int_0^t \left( \left\| \frac{\partial_t v}{x} \right\|_{L^\infty}+ \left\|\zeta \partial_t v' \right\|_{L^\infty}\right) ds \left\| \mathfrak{J}_2 \right\|_{0}\\
\le & M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}
Similarly,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sum_{\alpha=0}^2 \left\|\zeta \mathfrak{J}_{2-\alpha} (t) \left( \sigma {\partial_t^\alpha v''}\right)(t) \right\|_0
\le M_0 + CtP\left(\sup_{[0,t]}E\right);
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sum_{\alpha=0}^2 \left\|\zeta \sigma \mathfrak{L}_{2-\alpha}(t) \left( |\partial_t^\alpha v/x| + | \partial_t^\alpha v'|
\right)(t) \right\|_0^2
\le & M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}e
Here we have used $\eqref{3lk}_{2,3}$ to derive the last inequality. Therefore, it holds that
\begin{equation}\lambdabel{t34}\begin{equation}gin{split}
\left\| \zeta \frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]'(t) \right\|_0^2 \le M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}
In view of \eqref{abc}, we obtain
\begin{equation}\lambdabel{t35}\begin{equation}gin{split}
\left\|\zeta \frac{\sigma}{x} (3I_{21}+I_{22})(t)\right\|_0 ^2 \le &C \sum_{\alpha=0}^2 \left\|\zeta \mathfrak{J}_{2-\alpha}(t)\left( \left| \partial_t^\alpha v' \right|+\left| \frac{ {\partial_t^\alpha v}}{x} \right| \right)(t) \right\|_0\\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Noting from \eqref{ik} and \eqref{pjk} that
\begin{equation}e\lambdabel{phi3}\begin{equation}gin{split}
\left\| \phi { x^2 } \partial_t^{4}\left(\frac{x^2}{r^2}\right)(t)
\right\|_0^2
\le & C \left\| x {\mathcal{I}}_3(t)
\right\|_0^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
one then derives from \eqref{ik3''}, \eqref{ik3}, \eqref{t31}, \eqref{t32}, \eqref{t33}, \eqref{t34}-\eqref{t35} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0, t]}\left\|\zeta H_0\right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
In view of \eqref{v''} and \eqref{tnorm}, we can therefore obtain, for any $s\in [0, t]$,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \zeta \sigma {\partial_t^3 v''} (s) \right\|_0^2+ \left\| \zeta {\partial_t^3 v'} (s) \right\|_0^2
+ \left\| \zeta \frac{\partial_t^{3} v (s)}{x} \right\|_0^2 \\
\le & C(m_0) \left\|\zeta H_{ 0 } (s) \right\|_0^2 + C(m_0,s_0) \int_\delta^{2\delta} \left[ ({\partial_t^{3} v'(s)})^2 + \left( \frac{\partial_t^{3 } v(s)}{x} \right)^2 \right] dx\\
\le & C \sup_{[0, t]}\left\|\zeta H_0\right\|_0^2 + C \int_\delta^{2\delta} \left[ (\sigma {\partial_t^{3} v'(s)})^2
+ \left( {\partial_t^{3 } v(s)} \right)^2 \right] dx\\
\le &M_0+ C t P\left(\sup_{[0,t]} E\right) +M_0+ C s P\left(\sup_{[0,s]} E\right)\\
\le &M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
where we used the fact that $\sigma(x)\ge m_0 \delta $ on $[\delta,2\delta]$. This, together with \eqref{tnorm}, implies that
\begin{equation}\lambdabel{hk3}\begin{equation}gin{split}
\sup_{[0, t]}\left(\left\| \zeta \sigma {\partial_t^3 v } \right\|_2^2+ \left\| \zeta {\partial_t^3 v } \right\|_1^2
+ \left\| \zeta \left( \frac{\partial_t^{3} v}{x} \right) \right\|_0^2\right)
\le &M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
It follows from \eqref{hk3} and \eqref{norm} that
\begin{equation}\lambdabel{iry2}\begin{equation}gin{split}
\sup_{[0, t]}\left( \left\| \zeta \sigma {\partial_t^2 v } \right\|_2^2+ \left\| \zeta {\partial_t^2 v } \right\|_1^2
+ \left\| \zeta \left( \frac{\partial_t^{2} v}{x} \right) \right\|_0^2\right)
\le &M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
\subsubsection{Interior estimates for $\partial_t v$ and $ v$.}
Consider \eqref{eie} with $k=1$. The basic idea is to apply \eqref{ik3'''} with $\begin{equation}ta=\zeta$. As before, we first list some useful estimates here and then deal with
$\left\|\zeta H_0'\right\|_0$ later.
Note that for all nonnegative integers $m$ and $n$,
\begin{equation}\lambdabel{114}\begin{equation}gin{split}
\left|\partial_t \left(\frac{x^m}{r^mr'^n}\right)''\right| \le C \mathcal{Q},
\end{split}\end{equation}
where
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\mathcal{Q}= \left|\left(\frac{v}{x}\right)''\right|+ \left|v'''\right| + \mathcal{R}_0 \mathfrak{L}_0 + \left(\mathcal{R}_1 + \mathcal{R}_0^2 \right)\mathfrak{J}_0 \ \ {\rm with} \ \ \mathcal{R}_1= \left|r'''\right|+ \left|\left(\frac{r}{x}\right)''\right|.
\end{split}\end{equation}e
It follows from \eqref{norm}, \eqref{tnorm}, \eqref{formula} and $\eqref{3egn}_2$ that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\left\| \sigma \left(\frac{v}{x}\right)''(t)\right\|_0+ \left\|\sigma v'''(t)\right\|_{0}
\le C \left\|v''- 2\left(\frac{v}{x}\right)'\right\|_0+ \left\| \sigma v'''\right\|_0 \le C \sqrt{E(t)},
\\
&\left\|\zeta \sigma \left(\frac{v}{x}\right)''(t)\right\|_0^2+ \left\|\zeta \sigma v'''(t)\right\|_0^2
\le 2C \left\|\zeta v''(t)- 2\zeta \left(\frac{v}{x}\right)'(t)\right\|_0^2+ 2\left\|\zeta \sigma v'''(t)\right\|_0^2\\
&\qquad \qquad\qquad\qquad \qquad\qquad \quad\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
We then have, by $\eqref{3jk}_1$ and $\eqref{3lk}_{1,2}$, that
\begin{equation}\lambdabel{rrr}\begin{equation}gin{split}
\left\|\sigma \mathcal{R}_1(t)\right\|_{0}
\le & \int_0^t \left( \left\|\sigma \left(\frac{v}{x}\right)''\right\|_0+ \left\|\sigma v'''\right\|_{0} \right)ds
\le C tP\left(\sup_{[0,t]} \sqrt{E}\right),
\end{split}\end{equation}
\begin{equation}\lambdabel{eq}\begin{equation}gin{split}
\left\| \sigma \mathcal{Q} (t)\right\|_{0}
\le & C\left\|\sigma\left(\frac{v}{x}\right)''(t)\right\|_0+ \left\|\sigma v'''(t)\right\|_{0}
+ \left\|\sigma \mathcal{R}_0 (t) \right\|_{L^\infty }\left\| \mathfrak{L}_0(t)\right\|_0
\\
&+ \left( \left\| \sigma \mathcal{R}_1(t)\right\|_{0} + \left\|\sigma \mathcal{R}_0 (t) \right\|_{L^\infty }\left\| \mathcal{R}_0 (t)\right\|_0\right)\left\|\mathfrak{J}_0 (t) \right\|_{L^\infty }\\
\le & C \sqrt{E(t)}+ C tP\left(\sup_{[0,t]} \sqrt{E}\right),
\end{split}\end{equation}
\begin{equation}\lambdabel{tq}\begin{equation}gin{split}
\left\| \zeta \sigma \mathcal{Q}(t) \right\|_{0}^2
\le & C \left\|\zeta \sigma \left(\frac{v}{x}\right)''(t)\right\|_0^2 + C \left\|\zeta \sigma v'''(t)\right\|_{0}^2
+ C \left\|\sigma \mathcal{R}_0 (t) \right\|_{L^\infty }^2\left\| \mathfrak{L}_0(t)\right\|_0^2
\\
&+ C \left( \left\| \sigma \mathcal{R}_1(t)\right\|_{0} + \left\|\sigma \mathcal{R}_0(t) \right\|_{L^\infty }\left\| \mathcal{R}_0 (t)\right\|_0\right)^2\left\|\mathfrak{J}_0 (t) \right\|_{L^\infty }^2\\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Now, we are ready to deal with $\left\|\zeta H_0'\right\|_0$. For $\mathfrak{H}_1$, it follows from \eqref{norm}, \eqref{r0}, \eqref{egn}, $\eqref{3egn}_1$ and $\eqref{3lk}_1$ that
\begin{equation}\lambdabel{l11}\begin{equation}gin{split}
\left\| \zeta \mathfrak{H}_1'(t) \right\|_0^2 \le & C \left\{ \left\| \frac{ x^4 } {r^4 r' }-1 \right\|_{L^\infty}^2 +
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2
+\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2\right\}\\
&\times \left(\left\| \frac{\partial_t v}{x} \right\|_1^2 + \left\| \partial_t v' \right\|_0^2+\left\| \zeta \partial_t v'' \right\|_0^2\right) \\
&+ C \left\|\mathcal{R}_0\right\|_0^2 \left\{\left\| \zeta \partial_t v' \right\|_{L^\infty}^2 + \left\| {\partial_t v}/ {x} \right\|_{L^\infty}^2\right\}
\le C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For $\mathfrak{H}_2$, it follows from \eqref{norm}, \eqref{egn}, $\eqref{3lk}_1$, \eqref{3egn} and \eqref{rrr} that
\begin{equation}\lambdabel{l12}\begin{equation}gin{split}
\left\| \zeta \mathfrak{H}_2'(t) \right\|_0^2 \le &
C \left\|\mathcal{R}_0\right\|_0^2 \left\{\left\| \partial_t v/x \right\|_{L^\infty}^2+ \left\| \zeta \partial_t v' \right\|_{L^\infty}^2 + \left\| \zeta \sigma ( \partial_t v/x)' \right\|_{L^\infty}^2 + \left\| \zeta \sigma \partial_t v'' \right\|_{L^\infty}^2 \right\} \\
& +C \left(\left\|\mathcal{R}_0\right\|_0^2 \left\|\sigma\mathcal{R}_0\right\|_{L^\infty}^2
+ \left\|\sigma \mathcal{R}_1\right\|_0^2 \right) \left\{\left\| \partial_t v/x \right\|_{L^\infty}^2+ \left\|\zeta \partial_t v' \right\|_{L^\infty}^2 \right\} \\
&+C\left\{
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2 +\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2
\right\} \left\{ \left\| \zeta \sigma \left(\frac{\partial_t v}{x} \right)''\right\|_0^2
+ \left\| \zeta \sigma\partial_t v''' \right\|_0^2 \right.\\
&\left. +\left\| \left(\frac{\partial_t v}{x} \right)'\right\|_0^2 + \left\| \zeta \partial_t v'' \right\|_0^2 \right\}
\le C t P\left(\sup_{[0,t]} E\right) ,
\end{split}\end{equation}
since
$$\left\| \zeta \sigma ( \partial_t v/x)' \right\|_{L^\infty}\le C\left\| \zeta x ( \partial_t v/x)' \right\|_{L^\infty} \le C\left\| \zeta \left(\partial_t v' - \partial_t v /x\right) \right\|_{L^\infty},$$
$$\left\| \zeta \sigma \left(\frac{\partial_t v}{x} \right)''\right\|_0 \le C\left\| \zeta x \left(\frac{\partial_t v}{x} \right)''\right\|_0
\le C \left\| \zeta \partial_t v '' - 2\zeta (\partial_t v /x)'\right\|_0.$$
For the term involving $I_{11}$ and $I_{12}$, we have from \eqref{14}, \eqref{314} and \eqref{114} that
\begin{equation}e\lambdabel{ehk}\begin{equation}gin{split}
& \left\| \zeta \left\{\frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]' \right\}' \right\|_0\\
\le& C \left\|\zeta (I_{11}+I_{12}) \right\|_0 + C \left\|\zeta (I_{11}+I_{12})' \right\|_0 +\left\|\zeta \sigma (I_{11}+I_{12})'' \right\|_0 \\
\le & C \left\|\zeta \sigma \mathcal{Q} \left(\left|v/x\right| + \left| v'\right|\right) \right\|_0 + C\left\|\zeta \mathfrak{L}_0 \left(|v/x|+\left|v'\right| + |\sigma(v/x)'|+\left|\sigma v''\right|\right) \right\|_0 \\
& + C\left\| \zeta\mathfrak{J}_0 \left(|v/x|+|v'|+|(v/x)'|+\left|v''\right| + |\sigma(v/x)''| + \left|\sigma v'''\right|\right) \right\|_0 .
\end{split}\end{equation}e
Note that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\left\|\zeta \sigma \mathcal{Q} (t)\left(\left|v/x\right| + \left| v'\right|\right)(t)\right\|_0^2 \\ \le & 2 \left\|\zeta \sigma \mathcal{Q} (t)\right\|_0^2 \left\|v(t)/x\right\|_{L^\infty}^2 + 2\left(\left\| \zeta \sigma\mathcal{Q}(t) \right\|_0 \left\| v'(0)\right\|_{L^\infty}
+ \left\| \sigma \mathcal{Q} \right\|_{0} \int_0^t \left\| \zeta \partial_t v' \right\|_{L^\infty} ds\right)^2 \\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
where we have used \eqref{egn}, \eqref{tegn}, $\eqref{3egn}_1$, \eqref{eq} and \eqref{tq};
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\|\zeta \mathfrak{L}_0 (t)\left(|v/x|+\left|v'\right| + |\sigma(v/x)'|+\left|\sigma v''\right|\right)(t)\right\|_0^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
due to $\eqref{3lk}_{2,3}$; and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \zeta\mathfrak{J}_0(t) \left(|v/x|+|v'|+|(v/x)'|+\left|v''\right| + |\sigma(v/x)''| + \left|\sigma v'''\right|\right)(t)\right\|_0^2 \le
M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
since
$$|\sigma(v/x)''|\le C |x(v/x)''|=C |v''-2(v/x)'|$$
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\|\zeta \mathfrak{J}_0(t) \left(|v''|+\left|\sigma v'''\right| \right)(t)\right\|_0^2
\\ \le & \left[\left\| \zeta \mathfrak{J}_0 (t) \right\|_{L^\infty} \left(\left\| v''(0)\right\|_{0} +\left\|\sigma v'''(0)\right\|_{0}\right)
+ \left\| \mathfrak{J}_0(t) \right\|_{L^\infty} \int_0^t \left( \left\| \zeta \partial_t v'' \right\|_{0} +\left\| \zeta \sigma \partial_t v''' \right\|_{0}\right)ds \right]^2\\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
Then, we have arrived at
\begin{equation}\lambdabel{l13}\begin{equation}gin{split}
& \left\| \zeta \left\{\frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]' \right\}' (t)\right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
In a similar but easier way as for \eqref{l13}, one can show
\begin{equation}\lambdabel{l14}\begin{equation}gin{split}
\left\|\zeta \left[(\sigma/x)(3I_{21}+I_{22})\right]'(t)\right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Finally, the last term in $\zeta \mathcal{G}'$ can be bounded as
\begin{equation}\lambdabel{l15}\begin{equation}gin{split}
\left\| \zeta \left[ \phi { x^2 } \partial_t^{2}\left(\frac{x^2}{r^2}\right)\right]'(t)
\right\|_0^2
\le & C \left\| \zeta x \partial_t^2 \left(\frac{x^2}{r^2}\right)
\right\|_0^2+ C\left\| \zeta x^2 \partial_t^2 \left(\frac{x^2}{r^2}\right)'
\right\|_0^2\\
\le &C \left\| \zeta x \mathfrak{J}_1
\right\|_0^2+ C\left\| \zeta x^2 \mathfrak{L}_1
\right\|_0^2 \\
\le & C\left\| \zeta \mathfrak{J}_1
\right\|_0^2+ C\left\| \zeta (x^2/\sigma) \sigma \mathfrak{L}_1
\right\|_0^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}
due to $\eqref{3jk}_2$, $\eqref{3lk}_3$ and the lower bound of $\rho_0$ in the interior region.
It follows from \eqref{ik3'''}, \eqref{eie}, \eqref{hk3}, \eqref{l11}, \eqref{l12}, \eqref{l13}-\eqref{l15} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]} \left\|\zeta H_0'\right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right) + C \sup_{[0,t]}\left\|\zeta \partial_t^3 v\right\|_1^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
In view of \eqref{vt'''} and \eqref{tnorm}, we can then obtain
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\sup_{[0,t]}\left(\left\| \zeta \sigma {\partial_t v} ''' \right\|_0^2+ \left\| \zeta {\partial_t v} ''\right\|_0^2
+ \left\| \zeta \left( {\partial_t v}/{x} \right)' \right\|_0^2\right) \\
\le & C \sup_{[0,t]}\left[\left\|\zeta H_0' \right\|_0^2 + \left\|\zeta {\partial_t v}/{x}\right\|_0^2
+\left\|\zeta \partial_t v'\right\|_0^2
\right] + C(\delta) \sup_{[0,t]}\left[\left\|\sigma \partial_t v''\right\|_0^2 + \left\| \partial_t v'\right\|_0^2 +\left\| \partial_t v\right\|_0^2 \right]\\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
where we used the fact that $\sigma(x)\ge m_0 \delta $ on $[\delta,2\delta]$. This, together with \eqref{tnorm} and $\eqref{3egn}_2$ produces that
\begin{equation}\lambdabel{hk1}\begin{equation}gin{split}
\sup_{[0,t]}\left(\left\| \zeta \sigma {\partial_t v} \right\|_3^2+ \left\| \zeta {\partial_t v} \right\|_2^2
+ \left\| \zeta \left( \frac{\partial_t v}{x} \right) \right\|_1^2\right)
\le & M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Then we can derive from \eqref{hk1} and \eqref{norm} that
\begin{equation}\lambdabel{iry0}\begin{equation}gin{split}
\sup_{[0,t]}\left( \left\| \zeta \sigma v \right\|_3^2+ \left\| \zeta v \right\|_2^2
+ \left\| \zeta \left( \frac{ v}{x} \right) \right\|_1^2 \right)
\le & M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
\subsection{Elliptic estimates -- boundary estimates}
For the boundary estimates, we introduce a cut-off function $\chi(x)$ satisfying
\begin{equation}\lambdabel{chi}
\chi=1 \ \ {\rm on} \ \ [\delta,1], \ \ \chi=0 \ \ {\rm on} \ \ [0, \delta/2], \ \ |\chi'|\le s_0/\delta,
\end{equation}
for some constant $s_0$, where $\delta$ is given by \eqref{da}. Let
\begin{equation}\lambdabel{h-B}\begin{equation}gin{split}
B =& \sigma \partial_t^{k } v''+2\sigma' \partial_t^{k } v'=H_0+2\sigma'\partial^k_t v/x.
\end{split}\end{equation}
Since for any function $h=h(x,t)$ and integer $i\ge 2$, it holds that
\begin{equation}\lambdabel{hzz}\begin{equation}gin{split}
&\left\| \chi \sigma h'\right\|_0^2 + \left\| \chi \sigma' h \right\|_0^2
\le \left\| \chi \left(\sigma h' + i \sigma' h \right) \right\|_0^2 + C \left\| \sigma^{1/2} h \right\|_0^2 ,
\\
& \left\| \chi \sigma^{3/2} h'\right\|_0^2 + \left\| \chi \sigma^{1/2} \sigma' h \right\|_0^2 \le
4 \left\| \chi \sigma^{1/2} \left(\sigma h' + i \sigma' h \right) \right\|_0^2 + C \left\| \sigma h \right\|_0^2.
\end{split}\end{equation}
We can see that
\begin{equation}\lambdabel{bht}\begin{equation}gin{split}
&\left\| \chi \sigma^{3/2} \partial_t^3 v''\right\|_0^2 + \left\| \chi \sigma^{1/2} \sigma' \partial_t^3 v' \right\|_0^2 \le
4 \left\| \chi \sigma^{1/2} B \right\|_0^2 + C \left\| \sigma \partial_t^3 v' \right\|_0^2 , \ \ k=3;
\\
&\left\| \chi \sigma \partial_t^2 v'' \right\|_0^2 + \left\| \chi \sigma' \partial_t^2 v' \right\|_0^2
\le \left\| \chi B \right\|_0^2 + C \left\| \sigma^{1/2} \partial_t^2 v' \right\|_0^2 , \ \ k=2;
\\
&\left\| \chi \sigma^{3/2} \partial_t v'''\right\|_0^2 + \left\| \chi \sigma^{1/2} \sigma' \partial_t v'' \right\|_0^2 \le
4 \left\| \chi \sigma^{1/2} \left( B' - 2\sigma'' \partial_t v'\right) \right\|_0^2 + C \left\| \sigma \partial_t v'' \right\|_0^2 , \ \ k=1;
\\
&\left\| \chi \sigma v''' \right\|_0^2 + \left\| \chi \sigma' v'' \right\|_0^2
\le \left\| \chi \left( B' - 2\sigma'' v'\right) \right\|_0^2 + C \left\| \sigma^{1/2} v'' \right\|_0^2 , \ \ k=0.
\end{split}\end{equation}
Thus, we need to deal with $\|\sigma^{1/2}\chi B \|_0$ when $k=3$, $\|\chi B\|_0$ for $k=2$, $\|\sigma^{1/2}\chi B' \|_0$ when $k=1$ and $\|\chi B'\|_0$ for $k=0$. The proof of \eqref{hzz} is left to the appendix.
\subsubsection{Boundary estimates for $\partial_t^2 v$}
To estimate $\|\chi B\|_0$ with $k=2$, we consider equation \eqref{eie} with $k=2$. To this end,
we will first list some useful facts. Similar to \eqref{tnorm}, one can obtain also
\begin{equation}\lambdabel{ho2}\begin{equation}gin{split}
\left\| \left( \sigma^{1/2}\partial_t^2 v', \ \sigma^{3/2} \partial_t^2 v'', \ \sigma^{1/2} v'', \ \sigma^{3/2} v'''
\right)(\cdot,t)\right\|_{0}^2
\le M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}
Setting $\|\cdot\|=\|\cdot(t)\|$, we can summarize from $\eqref{jk}$, \eqref{tjk}, $\eqref{3lk}$, \eqref{rrr} and \eqref{r0} that
\begin{equation}\lambdabel{lb1}\begin{equation}gin{split}
&\left\| x/r-1 \right\|_{L^\infty} +
\left\| 1/r'-1\right\|_{L^\infty} +\left\| \mathcal{R}_0 \right\|_{0} +
\left\| \sigma \mathcal{R}_0\right\|_{L^\infty } + \left\|\sigma \mathcal{R}_1\right\|_0 \le C t P\left( \sup_{[0,t]} \sqrt{E}\right) ,
\\
&\left\| \mathfrak{J}_0\right\|_{L^\infty}^2+ \left\| \mathfrak{J}_1\right\|_{L^4}^2+ \left\| \sigma \mathfrak{L}_0\right\|_{L^\infty}^2+ \left\| \sigma \mathfrak{L}_1\right\|_{L^4}^2 \le C P\left(E(t)\right) + C t P\left(\sup_{[0,t]} E\right),
\\
&\left\| \mathfrak{J}_0\right\|_{L^4}^2+ \left\| \mathfrak{J}_1\right\|_{0}^2 +\left\| \sigma \mathfrak{L}_0\right\|_{L^4}^2+ \left\| \sigma \mathfrak{L}_1\right\|_{0}^2 \le M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}
Next, we will deal with the terms on the right-hand side of \eqref{eie}. It follows from \eqref{tnorm} and \eqref{et4} that
\begin{equation}\lambdabel{bt21}\begin{equation}gin{split}
&\left\|\chi \left(\frac{1}{2}x \partial_t^{4} v - 4 \left(\frac{\sigma}{x}\right)' \partial_t^2 v \right)(t)\right\|_0^2\\
\le & C \left\| \partial_t^{4} v (t) \right\|_0^2 + C(\delta)\left\|
{ \partial_t^{2} v } (t)\right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For $\mathfrak{H}_1$ and $\mathfrak{H}_2$, by virtue of $\eqref{lb1}_1$, \eqref{norm}, and Hardy's inequality, one has
\begin{equation}\lambdabel{bt22}\begin{equation}gin{split}
\left\| \chi \mathfrak{ H}_1 (t)\right\|_0^2 \le & C \left\{ \left\| \frac{ x^4 } {r^4 r' }-1 \right\|_{L^\infty}^2 +
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2 +\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2
\right\} \left\| {\partial_t^{2} v} \right\|_1^2 \\
\le & C t P\left(\sup_{[0,t]} E\right) ,
\end{split}\end{equation}
\begin{equation}\lambdabel{bt23}\begin{equation}gin{split}
\left\| \chi \mathfrak{H}_2 (t)\right\|_0^2 \le &C\left\| \sigma \mathcal{R}_0 \right\|_{L^\infty}^2 \left\| {\partial_t^{2} v } \right\|_{1}^2
+\left\{
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2 +\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2
\right\} \\
&\times \left( \left\| \sigma {\partial_t^{2 } v} \right\|_2^2 +\left\| {\partial_t^{2 } v} \right\|_1^2 \right) \\
\le & C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For the term involving $I_{11}$ and $I_{12}$, we derive from \eqref{14} and \eqref{314} that
\begin{equation}\lambdabel{bt24}\begin{equation}gin{split}
\left\| \chi \frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]'(t) \right\|_0^2 \
\le C \sum_{\alpha=0,1} \left\{ \left\|\chi \mathfrak{J}_{1-\alpha}(t) \left( \left| {\partial_t^\alpha v} \right|+|\partial_t^\alpha v'| + |\sigma \partial_t^\alpha v''|\right)(t)\right\|_0^2 \right.\\
\left. + \left\|\chi \sigma \mathfrak{L}_{1-\alpha} (t)\left( |\partial_t^\alpha v | + | \partial_t^\alpha v'|
\right)(t) \right\|_0^2 \right\}
\le M_0 + CtP\left(\sup_{[0,t]}E\right).
\end{split}\end{equation}
Indeed, it follows from \eqref{egn}, \eqref{tegn} and \eqref{lb1} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \sum_{\alpha=0,1} \left( \left\| \mathfrak{J}_{1-\alpha}(t) {\partial_t^\alpha v} (t) \right\|_0^2 + \left\| \sigma \mathfrak{L}_{1-\alpha} (t) \partial_t^\alpha v (t) \right\|_0^2 \right)\\
\le & C \left(\left\| \mathfrak{J}_{1 }(t) \right\|_0^2 + \left\| \sigma \mathfrak{L}_{1 }(t) \right\|_0^2 \right) \left\| v (t)\right\|_{L^\infty}^2 + C \left(\left\| \mathfrak{J}_{0 } (t)\right\|_0^2 + \left\| \sigma \mathfrak{L}_{0}(t) \right\|_0^2 \right) \left\| \partial_t v (t)\right\|_{L^\infty}^2 \\
\le & M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\sum_{\alpha=0,1} \left\| \mathfrak{J}_{1-\alpha}(t)\left( \left| \partial_t^\alpha v' \right|+\left| \sigma {\partial_t^\alpha v''} \right| \right)(t) \right\|_0
\\
\le & \left\| \mathfrak{J}_0(t) \right\|_{L^4} \left(\left\| {\partial_t v'(0) } \right\|_ {L^4} +\left\| \sigma \partial_t v''(0) \right\|_{L^4} \right) + \int_0^t \left( \left\| {\partial_t^2 v'} \right\|_{0}+ \left\|\sigma \partial_t^2 v'' \right\|_{0}\right) ds \left\| \mathfrak{J}_0 (t)\right\|_{L^\infty}\\
&+\left\| \mathfrak{J}_1 (t)\right\|_0 \left(\left\| { v'(0) } \right\|_ {L^\infty} +\left\| \sigma v''(0) \right\|_{L^\infty}\right) + \int_0^t \left( \left\| {\partial_t v'} \right\|_{L^4}+ \left\|\sigma \partial_t v' \right\|_{L^4}\right) ds \left\| \mathfrak{J}_1 (t) \right\|_{L^4}\\
\le & M_0 + CtP\left(\sup_{[0,t]}E\right)
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sum_{\alpha=0,1} \left\| \sigma \mathfrak{L}_{1-\alpha} (t) \partial_t^\alpha v'(t)
\right\|_0^2 \le
M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}e
so \eqref{bt24} follows. Similarly, one can also obtain
\begin{equation}\lambdabel{bt25}\begin{equation}gin{split}
\left\|\chi (\sigma/x ) (3I_{21}+I_{22})(t)\right\|_0 ^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Finally, one has
\begin{equation}\lambdabel{bt26}\begin{equation}gin{split}
\left\| \chi \phi { x^2 } \partial_t^{3}\left(\frac{x^2}{r^2}\right)(t)
\right\|_0^2
\le C \left\| \mathcal{I}_2(t)
\right\|_0^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Here \eqref{ik} and \eqref{pjk} have been used. Applying \eqref{ik3''} with $k=2$ and $\begin{equation}ta=\chi$, with the help of \eqref{eie}, \eqref{bt21}-\eqref{bt26} , we obtain
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left\|\chi H_0 \right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
In view of \eqref{h-B} and \eqref{tnorm}, one can thus get
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left\|\chi B \right\|_0^2 \le 2\sup_{[0,t]}\left\|\chi H_0 \right\|_0^2
+ C(\delta)\sup_{[0,t]}\left\|\partial_t^2 v\right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
It follows from this, $\eqref{bht}_2$ and \eqref{ho2} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left( \left\| \chi \sigma \partial_t^2 v'' \right\|_0^2 + \left\| \chi \sigma' \partial_t^2 v' \right\|_0^2 \right)
\le \sup_{[0,t]}\left( \left\| \chi B \right\|_0^2 + C \left\| \sigma^{1/2} \partial_t^2 v' \right\|_0^2 \right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
This, together with \eqref{tnorm}, yields that
\begin{equation}\lambdabel{bde2}\begin{equation}gin{split}
\sup_{[0,t]} \left(\left\| \chi \sigma \partial_t^2 v \right\|_2^2 + \left\| \chi \partial_t^2 v \right\|_1^2 \right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}
due to the estimate:
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]} \left\| \chi \partial_t^2 v ' \right\|_0^2 \le \sup_{[0,t]}\left( C \left\| \chi \sigma \partial_t^2 v' \right\|_0^2 + C \left\| \chi \sigma' \partial_t^2 v' \right\|_0^2\right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
\subsubsection{Boundary estimates for $ v$}
Consider now \eqref{eie} with $k=0$. Our goal is to bound $\left\|\chi B'\right\|_0$. It follows from \eqref{tnorm} and \eqref{bde2} that
\begin{equation}\lambdabel{bt01}\begin{equation}gin{split}
\left\|\chi \left( \frac{1}{2}x \partial_t^{2} v - 4 \left(\frac{\sigma}{x}\right)' v \right)'(t)\right\|_0^2 \le C \left\| \chi \partial_t^{2} v (t)\right\|_1^2 + C \left\|
{ v } (t)\right\|_1^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For $\mathfrak{H}_1$ and $\mathfrak{H}_2$, it follows from \eqref{norm}, \eqref{egn} and $\eqref{lb1}_1$ that
\begin{equation}\lambdabel{bt02}\begin{equation}gin{split}
\left\| \chi \mathfrak{H}_1' (t)\right\|_0^2 \le & C \left\{ \left\| \frac{ x^4 } {r^4 r' }-1 \right\|_{L^\infty}^2 +
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2
+\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2\right\} \left\| { v} \right\|_2^2 \\
&+ C \left\|\mathcal{R}_0\right\|_0^2 \left\{\left\| v' \right\|_{L^\infty}^2 + \left\| v \right\|_{L^\infty}^2\right\}
\le C t P\left(\sup_{[0,t]} E\right)
\end{split}\end{equation}
and
\begin{equation}\lambdabel{bt03}\begin{equation}gin{split}
\left\| \chi \mathfrak{H}_2' (t)\right\|_0^2 \le &
C \left\|\mathcal{R}_0\right\|_0^2 \left\{\left\| v \right\|_{L^\infty}^2+ \left\| v' \right\|_{L^\infty}^2 + \left\| \sigma v'' \right\|_{L^\infty}^2 \right\} \\
& +C \left(\left\|\mathcal{R}_0\right\|_0^2 \left\|\sigma\mathcal{R}_0\right\|_{L^\infty}^2
+ \left\|\sigma \mathcal{R}_1\right\|_0^2 \right) \left\{\left\| v \right\|_{L^\infty}^2+ \left\| v' \right\|_{L^\infty}^2 \right\} \\
&+C\left\{
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2 +\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2
\right\} \left\{ \left\| \sigma v''' \right\|_0^2 + \left\| v \right\|_2^2 \right\} \\
\le & C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Using \eqref{tnorm}, one has
\begin{equation}\lambdabel{bt04}\begin{equation}gin{split}
\left\| \chi \left[ \phi { x^2 } \partial_t \left(\frac{x^2}{r^2}\right)\right]'(t)
\right\|_0^2 \le & C(\delta) \left\| v(t)\right\|_1^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
It yields from \eqref{ik3'''}, \eqref{eie}, \eqref{bt01}-\eqref{bt04} that
\begin{equation}e\lambdabel{hh}\begin{equation}gin{split}
\sup_{[0,t]}\left\|\chi H'_0 \right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
In view of \eqref{h-B} and \eqref{tnorm}, one gets
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left\|\chi B' \right\|_0^2 \le 2\sup_{[0,t]}\left\|\chi H_0 ' \right\|_0^2
+ C(\delta)\sup_{[0,t]}\left\| v\right\|_1^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
We can then obtain, using $\eqref{bht}_4$, \eqref{tnorm} and \eqref{ho2}, that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left( \left\| \chi \sigma v''' \right\|_0^2 + \left\| \chi \sigma' v'' \right\|_0^2 \right)
\le \sup_{[0,t]}\left( \left\| \chi \left( B' - 2\sigma'' v'\right) \right\|_0^2 + C \left\| \sigma^{1/2} v'' \right\|_0^2 \right)\\
\le C \sup_{[0,t]} \left(\left\| \chi B' \right\|_0^2 + \left\| v' \right\|_0^2 + \left\| \sigma^{1/2} v'' \right\|_0^2 \right)\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
This, together with \eqref{tnorm}, yields
\begin{equation}\lambdabel{bde0}\begin{equation}gin{split}
\sup_{[0,t]}\left(\left\| \chi \sigma v \right\|_3^2 + \left\| \chi v \right\|_2^2 \right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}
since
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]} \left\| \chi v '' \right\|_0^2 \le \sup_{[0,t]} \left(C\left\| \chi \sigma v'' \right\|_0^2 + C \left\| \chi \sigma' v'' \right\|_0^2\right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
\subsubsection{Boundary estimates for $\partial_t^3 v$}
Consider equation \eqref{eie} with $k=3$.
As before, we list here some estimates which will be used later. First, it follows from \eqref{iry2}, \eqref{bde2}, \eqref{iry0} and \eqref{bde0} that
\begin{equation}\lambdabel{f0}\begin{equation}gin{split}
\sup_{[0,t]}\left(\left\| \sigma v \right\|_3^2 + \left\| v \right\|_2^2 +\left\| \sigma \partial_t^2 v \right\|_2^2 + \left\| \partial_t^2 v \right\|_1^2 \right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Moreover, we have the following estimates for $\partial_t v$ and $\partial_t^3 v$:
\begin{equation}\lambdabel{ho3}\begin{equation}gin{split}
& \left\| \left( \sigma^{1/2} \partial_t v', \ \sigma^{1/2} \partial_t^3 v
\right)(\cdot,t)\right\|_{L^\infty}^2 + \left\| \left( \sigma^{3/2} \partial_t v'', \ \sigma^{3/2} \partial_t^3 v'
\right)(\cdot,t)\right\|_{L^\infty}^2 \le C{E(t)};
\end{split}\end{equation}
and those for $\mathfrak{J}$ and $\mathfrak{L}$:
\begin{equation}\lambdabel{lb0}\begin{equation}gin{split}
&\left\| \sigma^{1/2} \left( \mathfrak{J}_1 , \sigma \mathfrak{L}_1\right)(\cdot,t)\right\|_{L^\infty}^2 \le CP\left(E(t) \right),\\
& \left\| \left(\mathfrak{J}_0, \sigma \mathfrak{L}_0\right)(\cdot,t)\right\|_{L^\infty}^2 + \left\| \left( \mathfrak{J}_1, \sigma \mathfrak{L}_1\right) (\cdot,t)\right\|_{0}^2 + \left\| \left( \mathfrak{J}_2, \sigma \mathfrak{L}_2\right) (\cdot,t)\right\|_{0}^2 \le
M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
The proofs of \eqref{ho3} and \eqref{lb0} will be given in the appendix.
We are now ready to do the estimates. First, \eqref{tnorm} and \eqref{et4} imply that
\begin{equation}\lambdabel{bt31}\begin{equation}gin{split}
\left\|
\chi \sigma^{1/2}\left(\frac{1}{2}x \partial_t^{5} v - 4 \left(\frac{\sigma}{x}\right)' \partial_t^3 v\right)(t)\right\|_0^2 \le & C(\delta) \left( \left\| (x\sigma)^{1/2} \partial_t^{5} v (t)\right\|_0^2 + \left\|
\partial_t^{3} v (t) \right\|_0^2 \right)\\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For $\mathfrak{H}_1$ and $\mathfrak{H}_2$, it follows from \eqref{norm} and $\eqref{lb1}_1$ that
\begin{equation}\lambdabel{bt32}\begin{equation}gin{split}
\left\| \chi \sigma^{1/2} \mathfrak{ H}_1 (t) \right\|_0^2 \le C \left\{ \left\| \frac{ x^4 } {r^4 r' }-1 \right\|_{L^\infty}^2 +
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2 +\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2
\right\} \\
\times \left\{ \left\| \sigma^{1/2} {\partial_t^{3 } v} \right\|_0^2 + \left\| \sigma^{1/2} \partial_t^{3 } v' \right\|_0^2 \right\}
\le C t P\left(\sup_{[0,t]} E\right);
\end{split}\end{equation}
\begin{equation}\lambdabel{bt33}\begin{equation}gin{split}
\left\| \chi \sigma^{1/2} \mathfrak{H}_2 (t)\right\|_0^2 \le &C\left\| \sigma \mathcal{R}_0 \right\|_{L^\infty}^2 \left(\left\| \sigma^{1/2} {\partial_t^{3} v } \right\|_{0}^2 + \left\| \sigma^{1/2} \partial_t^{3} v' \right\|_{0}^2 \right) \\
&+C\left\{
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2 +\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2
\right\} \left\| \sigma^{3/2} \left(\partial_t^3 v, \partial_t^3 v', \partial_t^{3 } v'' \right)\right\|_0^2 \\
\le & C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For the term involving $I_{11}$ and $I_{12}$, one can derive from \eqref{14} and \eqref{314} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \chi \sigma^{1/2} \frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]' \right\|_0^2 \\
\le & C \sum_{\alpha=0}^2 \left\{ \left\|\chi \sigma^{1/2} \mathfrak{J}_{2-\alpha} \left( \left| {\partial_t^\alpha v} \right|+|\partial_t^\alpha v'| + |\sigma \partial_t^\alpha v''|\right)\right\|_0^2 + \left\|\chi \sigma^{3/2} \mathfrak{L}_{2-\alpha} \left( |\partial_t^\alpha v | + | \partial_t^\alpha v'|
\right) \right\|_0^2 \right\}.
\end{split}\end{equation}e
Note that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\sum_{\alpha=0 }^2 \left\|\sigma^{1/2} \mathfrak{J}_{2-\alpha}(t)\left( \left| \partial_t^\alpha v' \right|+\left| \sigma {\partial_t^\alpha v''} \right| \right)(t) \right\|_0
\\
\le & \left\| \mathfrak{J}_0 \right\|_{L^\infty} \left(\left\| {\partial_t^2 v' } \right\|_ {0} +\left\| \sigma \partial_t^2 v'' \right\|_{0} \right) +\left\| \mathfrak{J}_1 \right\|_{0} \left(\left\| {\sigma^{1/2} \partial_t v'(0) } \right\|_ {L^\infty} +\left\| \sigma^{3/2} \partial_t v''(0) \right\|_{L^\infty} \right)
\\
&+ \int_0^t \left( \left\| {\partial_t^2 v'} \right\|_{0}+ \left\|\sigma \partial_t^2 v'' \right\|_{0}\right) ds \left\| \sigma^{1/2} \mathfrak{J}_1 \right\|_{L^\infty} +\left\| \mathfrak{J}_2 \right\|_{0} \left(\left\| { v' } \right\|_ {L^\infty} +\left\| \sigma v'' \right\|_{L^\infty} \right) \\
\le & M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}e
where we have used \eqref{norm}, \eqref{f0}-\eqref{lb0} and $\|\cdot\|_{L^\infty}\le C \|\cdot\|_1$. Similarly, one has
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sum_{\alpha=0 }^2 \left\| \sigma^{3/2} \mathfrak{L}_{2-\alpha} (t) \partial_t^\alpha v'(t)
\right\|_0^2 \le & \sum_{\alpha=0 }^2 \left\| \sigma^{1/2} \left( \sigma \mathfrak{L}_{2-\alpha}\right) \partial_t^\alpha v' \right\|_0^2
\le M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \sum_{\alpha=0}^2 \left( \left\| \sigma^{1/2} \mathfrak{J}_{2-\alpha} (t) {\partial_t^\alpha v} (t)\right\|_0^2 + \left\| \sigma^{3/2} \mathfrak{L}_{2-\alpha} (t) \partial_t^\alpha v (t)\right\|_0^2 \right)\\
\le & C \left(\left\| \mathfrak{J}_{2 } \right\|_0^2 + \left\| \sigma \mathfrak{L}_{2 } \right\|_0^2 \right) \left\| v \right\|_{L^\infty}^2 + C \left(\left\| \mathfrak{J}_{1 } \right\|_0^2 + \left\| \sigma \mathfrak{L}_{1} \right\|_0^2 \right) \left\| \partial_t v \right\|_{L^\infty}^2 \\
&+ C \left(\left\| \mathfrak{J}_{0 } \right\|_{L^\infty}^2 + \left\| \sigma \mathfrak{L}_{ 0 } \right\|_{L^\infty}^2 \right) \left\| \partial_t^2 v \right\|_{0}^2\\
\le & M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}e
where we have used \eqref{tnorm}, \eqref{tegn}, \eqref{f0}, $\eqref{lb0}_2$ and $\|\cdot\|_{L^\infty}\le C\|\cdot\|_1$.
Hence, it holds that
\begin{equation}\lambdabel{bt34}\begin{equation}gin{split}
\left\| \chi \sigma^{1/2} \frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]' (t) \right\|_0^2 \le M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}
Similarly, one can also obtain easily that
\begin{equation}\lambdabel{bt35}\begin{equation}gin{split}
\left\|\chi \sigma^{1/2}(\sigma/x) (3I_{21}+I_{22})(t)\right\|_0 ^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Finally, one has
\begin{equation}\lambdabel{bt36}\begin{equation}gin{split}
\left\| \chi \sigma^{1/2} \phi { x^2 } \partial_t^{4}\left(\frac{x^2}{r^2}\right)(t)
\right\|_0^2
\le C \left\| x \mathcal{I}_3(t)
\right\|_0^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
Here \eqref{ik} and \eqref{pjk} were used. Now, it follows from \eqref{eie}, \eqref{bt31}-\eqref{bt36}, by applying \eqref{ik3''} with $\begin{equation}ta=\chi\sigma^{1/2}$, that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left\|\chi \sigma^{1/2} H_0 \right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
Thanks to \eqref{h-B} and \eqref{tnorm}, one can then get
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left\|\chi \sigma^{1/2} B \right\|_0^2 \le 2\sup_{[0,t]}\left\|\chi\sigma^{1/2} H_0 \right\|_0^2
+ C(\delta)\sup_{[0,t]}\left\|\partial_t^3 v\right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
It then follows from $\eqref{bht}_1$ and \eqref{tnorm} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left( \left\| \chi \sigma^{3/2} \partial_t^3 v''\right\|_0^2 + \left\| \chi \sigma^{1/2} \sigma' \partial_t^3 v' \right\|_0^2 \right)\le & \sup_{[0,t]}\left(
4 \left\| \chi \sigma^{1/2} B \right\|_0^2 + C \left\| \sigma \partial_t^3 v' \right\|_0^2\right)\\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
This, together with \eqref{tnorm} and the Sobolev embedding \eqref{weightedsobolev}, yields
\begin{equation}\lambdabel{bde3}\begin{equation}gin{split}
\sup_{[0,t]}\left( \left\| \chi \sigma^{3/2} \partial_t^3 v''\right\|_0^2 + \left\| \chi \sigma^{1/2} \partial_t^3 v' \right\|_0^2 + \left\| \chi \partial_t^3 v \right\|_{1/2}^2 \right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}
because of
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]} \left\| \chi \sigma^{1/2} \partial_t^3 v ' \right\|_0^2 \le & \sup_{[0,t]}\left(C \left\| \chi \sigma^{3/2} \partial_t^3 v' \right\|_0^2 + C \left\| \chi \sigma^{1/2} \sigma' \partial_t^3 v ' \right\|_0^2\right)\\
\le & \sup_{[0,t]}\left(C \left\| \chi \sigma \partial_t^3 v' \right\|_0^2 + C \left\| \chi \sigma^{1/2} \sigma' \partial_t^3 v ' \right\|_0^2\right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right)
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left\| \chi \partial_t^3 v \right\|_{1/2}^2 \le &\sup_{[0,t]}\left( C \left\| \sigma^{1/2} \partial_t^3 v \right\|_0^2 + C \left\| \chi \sigma^{1/2} \partial_t^3 v ' \right\|_0^2
\right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
\subsubsection{Boundary estimates for $ \partial_t v$}
Consider equation \eqref{eie} with $k=1$. Our goal is to bound $\left\|\chi \sigma^{1/2} B'\right\|_0$. It follows from \eqref{tnorm} and \eqref{bde3} that
\begin{equation}\lambdabel{bt11}\begin{equation}gin{split}
&\left\|\chi \sigma^{1/2}\left( \frac{1}{2}x \partial_t^{3} v - 4 \left(\frac{\sigma}{x}\right)' \partial_t v \right)'(t)\right\|_0^2 \\
\le & C \left\| \partial_t^{3} v (t) \right\|_0^2 + \left\| \chi \sigma^{1/2} \partial_t^{3} v' (t)\right\|_0^2+ C(\delta)\left\|
\partial_t v (t)\right\|_1^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For $\mathfrak{H}_1$ and $\mathfrak{H}_2$, it follows from \eqref{norm}, \eqref{egn}, \eqref{tegn}, $\eqref{lb1}_1$ and \eqref{ho3} that
\begin{equation}\lambdabel{bt12}\begin{equation}gin{split}
\left\| \chi \sigma^{1/2} \mathfrak{H}_1' (t)\right\|_0^2 \le & C \left\{ \left\| \frac{ x^4 } {r^4 r' }-1 \right\|_{L^\infty}^2 +
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2
+\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2\right\}\\
&\times \left(\left\| {\partial_t v} \right\|_1^2 + \left\| \partial_t v' \right\|_0^2+\left\| \sigma^{1/2} \partial_t v'' \right\|_0^2\right) \\
&+ C \left\|\mathcal{R}_0\right\|_0^2 \left\{\left\| \sigma^{1/2} \partial_t v' \right\|_{L^\infty}^2 + \left\| {\partial_t v} \right\|_{L^\infty}^2\right\}
\le C t P\left(\sup_{[0,t]} E\right)
\end{split}\end{equation}
and
\begin{equation}\lambdabel{bt13}\begin{equation}gin{split}
&\left\|\chi \sigma^{1/2} \mathfrak{H}_2' (t)\right\|_0^2 \\
\le &
C \left\|\mathcal{R}_0\right\|_0^2 \left\{\left\| \partial_t v \right\|_{L^\infty}^2+ \left\| \sigma^{1/2} \partial_t v' \right\|_{L^\infty}^2 + \left\| \sigma^{3/2} \partial_t v'' \right\|_{L^\infty}^2 \right\} \\
&+C \left(\left\|\mathcal{R}_0\right\|_0^2 \left\|\sigma\mathcal{R}_0\right\|_{L^\infty}^2
+ \left\|\sigma \mathcal{R}_1\right\|_0^2 \right) \left\{\left\| \partial_t v \right\|_{L^\infty}^2 + \left\| \sigma^{1/2} \partial_t v' \right\|_{L^\infty}^2 \right\}
\\
& +C\left\{
\left\| \frac{x^3 } {r^3 r'^2}-1\right\|_{L^\infty}^2 +\left\|\frac{x^2 } {r^2 r'^3}-1\right\|_{L^\infty}^2
\right\} \left\{ \left\| \sigma^{1/2} \partial_t v'' \right\|_0^2 + \left\| \sigma^{3/2}\partial_t v''' \right\|_0^2
+\left\| {\partial_t v} \right\|_1^2 \right\} \\
\le & C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
For the term involving $I_{11}$ and $I_{12}$, it follows from \eqref{14}, \eqref{314} and \eqref{114} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \chi \sigma^{1/2} \left\{\frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]' \right\}' \right\|_0 \\
\le & C \left\| \chi \sigma^{3/2} \mathcal{Q} \left(\left|v/x\right| + \left| v'\right|\right)\right\|_0 + C\left\| \chi \sigma^{1/2} \mathfrak{L}_0 \left(|v/x|+\left|v'\right| + |\sigma(v/x)'|+\left|\sigma v''\right|\right)\right\|_0 \\
& + C\left\|\chi \sigma^{1/2}\mathfrak{J}_0 \left(|v/x|+|v'|+|(v/x)'|+\left|v''\right| + |\sigma(v/x)''| + \left|\sigma v'''\right|\right)\right\|_0 \\
\le & C \left\| \chi \sigma^{3/2} \mathcal{Q} \left(\left|v\right| + \left| v'\right|\right)\right\|_0 + C\left\| \chi \sigma^{1/2} \mathfrak{L}_0 \left(|v|+\left|v'\right| +\left|\sigma v''\right|\right)\right\|_0 \\
& + C\left\|\chi \sigma^{1/2}\mathfrak{J}_0 \left(|v|+|v'| + \left|v''\right| + \left|\sigma v'''\right|\right)\right\|_0 .
\end{split}\end{equation}e
Note that one can derive from \eqref{f0} and $\eqref{lb1}_{1,2}$ that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\|\chi \mathfrak{L}_{0}(t) \right\|_{0}^2 \le & C\left(\left\| v(t)\right\|_{2} + \left\| \mathcal{R}_0(t)\right\|_{0} \left\|\mathfrak{J}_{0}(t)\right\|_{L^\infty} \right)^2
\le M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \chi \sigma^{} \mathcal{Q} (t)\right\|_{0}^2
\le & C \left\|v(t)\right\|_2^2 + C \left\|\sigma v'''(t)\right\|_{0}^2
+ C \left\| \mathcal{R}_0 (t) \right\|_0^2\left\| \sigma \mathfrak{L}_0(t)\right\|_{L^\infty }^2
+ C \left( \left\| \sigma \mathcal{R}_1(t)\right\|_{0} \right. \\
&\left.+ \left\| \sigma\mathcal{R}_0 (t) \right\|_{L^\infty}\left\| \mathcal{R}_0 (t) \right\|_0\right)^2\left\|\mathfrak{J}_0 (t) \right\|_{L^\infty }^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right);
\end{split}\end{equation}e
which implies, due to \eqref{f0} and $\eqref{lb0}_2$, that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\left\|\chi \sigma \mathcal{Q}(t) \right\|_0^2 + \left\|\chi \mathfrak{L}_0(t)\right\|_0^2 + \left\| \chi \mathfrak{J}_0(t)\right\|_{L^\infty}^2 + \|\sigma v(t)\|_3^2+ \|v(t)\|_2^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
So, we obtain
\begin{equation}\lambdabel{bt14}\begin{equation}gin{split}
& \left\| \chi \sigma^{1/2} \left\{\frac{1}{\sigma} \left[ \sigma^2 (I_{11}+I_{12}) \right]' \right\}' (t)\right\|_0^2 \\
\le & C \left\|\chi \sigma \mathcal{Q}\right\|_0^2 \|v\|_2^2 + C\left\|\chi \mathfrak{L}_0 \right\|_0 \left(\|v\|_2^2 +\|\sigma v\|_3^2 \right) + C\left\|\chi \mathfrak{J}_0\right\|_{L^\infty}^2 \left(\|v\|_2^2 +\|\sigma v\|_3^2 \right) \\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}
where we have used the fact that $\|\cdot\|_{L^\infty}\le C \|\cdot\|_1$. Similarly, one can show that
\begin{equation}\lambdabel{bt15}\begin{equation}gin{split}
\left\|\chi \sigma^{1/2} \left[ (\sigma/x) (3I_{21}+I_{22})\right]'(t)\right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
It follows from \eqref{tnorm}, $\eqref{3lk}_1$ and \eqref{ho3} that
\begin{equation}\lambdabel{bt16}\begin{equation}gin{split}
& \left\| \chi \sigma^{1/2} \left[ \phi { x^2 } \partial_t^2 \left(\frac{x^2}{r^2}\right)\right]'(t)
\right\|_0^2 \\
\le & C \left(\left\| \partial_t v\right\|_1 + \left\| v \right\|_{L^\infty} \|v \|_0 + \|v\|_{L^\infty} \|v'\|_0
\right)^2 + C \|\mathcal{R}_0\|_0^2\left( \|\partial_t v\|_{L^\infty} + \|v\|_{L^\infty}^2 \right)^2 \\
\le & C \left(\left\| \partial_t v\right\|_1 + \|v \|_2^2 \right)^2 + C \|\mathcal{R}_0\|_0^2\left( \|\partial_t v\|_1 + \|v\|_1^2 \right)^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
It yields from \eqref{eie}, \eqref{ik3'''} and \eqref{bt11}-\eqref{bt16} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left\|\sigma^{1/2}\chi H_0' \right\|_0^2 \le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
which implies
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}\left\|\sigma^{1/2}\chi B' \right\|_0^2 \le \sup_{[0,t]}\left(2\left\|\sigma^{1/2}\chi H_0' \right\|_0^2 + C(\delta)\left\| \partial_t v \right\|_1^2 \right) \le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
due to \eqref{h-B} and \eqref{tnorm}.
We can then obtain, using $\eqref{bht}_3$ and \eqref{tnorm}, that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\sup_{[0,t]} \left(\left\| \chi \sigma^{3/2} \partial_t v'''\right\|_0^2 + \left\| \chi \sigma^{1/2} \sigma' \partial_t v'' \right\|_0^2 \right)\\
\le&\sup_{[0,t]} \left(
4 \left\| \chi \sigma^{1/2} \left( B' - 2\sigma'' \partial_t v'\right) \right\|_0^2 + C \left\| \sigma \partial_t v'' \right\|_0^2 \right)\\
\le & C\sup_{[0,t]} \left(\left\| \chi \sigma^{1/2} B' \right\|_0^2 + \left\| \partial_t v' \right\|_0^2 + \left\| \sigma \partial_t v'' \right\|_0^2 \right)\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
This, together with \eqref{tnorm} and the Sobolev embedding \eqref{wsv}, yields
\begin{equation}\lambdabel{bde1}\begin{equation}gin{split}
\sup_{[0,t]}\left( \left\| \chi \sigma^{3/2} \partial_t v'''\right\|_0^2 + \left\| \chi \sigma^{1/2} \partial_t v'' \right\|_0^2 + \left\| \chi \partial_t v \right\|_{3/2}^2\right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}
\section{Existence for the case $\gammamma=2$}
Summing over inequalities \eqref{et4}, \eqref{hk3}, \eqref{iry2}, \eqref{hk1}, \eqref{iry0}, \eqref{bde2}, \eqref{bde0}, \eqref{bde3} and \eqref{bde1}, we find that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{[0,t]}E
\le M_0+ C t P\left(\sup_{ [0,t]} E \right), \ \ t\in [0,T];
\end{split}\end{equation}e
which implies that for small $T$,
\begin{equation}\lambdabel{final}\begin{equation}gin{split}
\sup_{t\in[0,T]}E(t)
\le 2 M_0.
\end{split}\end{equation}
With this $\mu$-independent estimate, one can use the standard compactness argument \cite{10} to show the existence of the solutions to the problem \eqref{419'} for some time $T$.
\section{Case $1<\gamma< 2$}
In this section, we use similar arguments to those used to deal with the case for $\gamma=2$ to handle the case for general $\gamma$. {It should be noted that the value of $\gammamma$ determines the rate of degeneracy near the vacuum boundary, since $\rho_0$ appears as the coefficient in front of $\partial_t v$ in (3.7) and the physical vacuum condition indicates that $\rho_0(x)\sim (1-x)^{\frac{1}{\gammamma-1}}$ as
$x\to 1$. Thus the smaller value of $\gammamma$ is, the more degenerate equation (3.7) is near the vacuum boundary. Although the rate of degeneracy near the origin is the same no matter what $\gamma$ is,
we need higher order derivatives in the energy functional to control the $H^2$-norm of $v$ (and thus the $C^1$-norm of $v$) for smaller $\gammamma$, since we have to match the norms in the intermediate region.}
{We first define the higher-order energy functional for $1<\gamma<2$.} Set
$$\nu:=(2-\gamma)/(2\gamma-2)>0, \ \ l:= 3 + 2 \lceil 1/2 +\nu \rceil,$$
where $\lceil\cdot\rceil$ is the ceiling function defined for any real number $q\ge 0$ as
$$\lceil q\rceil :=\min\{m: \ \ m\ge q, \ m \ {\rm is~an~integer}\}.$$
Define
\begin{equation}\lambdabel{ggnorm}\begin{equation}gin{split}
\widetilde{{E}}(v, t):= & \left\| \sigma (\sigma/x)^\nu \partial_t^l v'(\cdot,t) \right\|_0^2 + \left\| (\sigma/x)^{1+\nu} \partial_t^l v (\cdot,t)\right\|_0^2 \\
& + \sum_{j=1}^{ \frac{l+1}{2} } \left\{ \left\| \sigma^{3/2+\nu} \partial_t^{l-2j +1 } \partial_x^{ j +1 } v (\cdot,t)\right\|_{0}^2
+ \sum_{i=0}^j \left\| \sigma^{1/2+\nu} \partial_t^{l-2j +1 } \partial_x^{ i } v (\cdot,t)\right\|_{0}^2 \right\} \\
& + \sum_{j=1}^{ \frac{l-1}{2} } \left\{ \left\| \sigma^{2+\nu} \partial_t^{l-2j } \partial_x^{ j +2 } v (\cdot,t)\right\|_{0}^2
+\sum_{i=-1}^j \left\| \sigma^{1+\nu} \partial_t^{l-2j } \partial_x^{ i +1 } v (\cdot,t)\right\|_{0}^2 \right\}\\ &+\sum_{j=1}^{ \frac{l+1}{2}} \left\{ \left\|\zeta \sigma \partial_t^{l-2j +1 } v (\cdot,t)\right\|_{j+1}^2
+\left\|\zeta \partial_t^{l-2j +1 } v (\cdot,t)\right\|_{j }^2
+\left\|\zeta \frac {\partial_t^{l-2j +1 } v}{x} (\cdot,t)\right\|_{j-1}^2 \right\}\\
&+ \sum_{j=1}^{\frac{l-1}{2}} \left\{ \left\|\zeta \sigma \partial_t^{l-2j } v (\cdot,t)\right\|_{j+2}^2
+\left\|\zeta \partial_t^{l-2j } v (\cdot,t)\right\|_{j +1 }^2
+\left\|\zeta \frac{ \partial_t^{l-2j } v }{x} (\cdot,t)\right\|_{j}^2 \right\}
,
\end{split}\end{equation}
where, as before,
\begin{equation}e
\zeta=1 \ \ {\rm on} \ \ [0,\delta_\gamma], \ \ \zeta=0 \ \ {\rm on} \ \ [2\delta_\gamma, 1], \ \ |\zeta'|\le s_0/\delta_\gamma.
\end{equation}e
Here $\delta_\gamma$ is a given constant depending on $\rho_0$ and $\gamma$ which will be determined in \eqref{daga} later.
It follows from the Hardy type embedding for the weighted Sobolev spaces \eqref{wsv} that
\begin{equation}\lambdabel{}\begin{equation}gin{split}
\|v\|_2^2 \le \|v \|_{\frac{l+1}{2}-\left(\frac{1}{2}+\nu\right)}^2 \le C \sum_{i=0}^ {\frac{l+1}{2}} \left\| \sigma^{1/2+\nu} \partial_x^{ i } v \right\|_{0}^2 \le C\widetilde{{E}},
\end{split}\end{equation}
which indicates that the high-order energy functional $\widetilde{{E}}$ is suitable for the study of the physical vacuum problem \eqref{419'} when $\gamma\in(1,2)$. {In fact, the norm chosen in \eqref{ggnorm}
is in the same spirit of but slightly different from that in \eqref{norm} for $\gamma=2$. Since the energy estimate gives
the bound of
$$\left\| \sqrt{x \sigma } (\sigma/x)^\nu \partial_t^{l+1} v (t)\right\|_0 + \left\| \sigma (\sigma/x)^\nu \partial_t^l v' (t)\right\|_0 + \left\| (\sigma/x)^{1+\nu} \partial_t^l v (t)\right\|_0 ,$$
from which we can derive the bound of $\|\partial_t^l v\|_0$ for $\gamma=2$. But for $\gamma\in(1,2)$, we cannot improve the spatial regularity as that for $\gamma=2$ due to $\nu>0$ (or equivalently, the higher degeneracy of the equation). So, the norm chosen for $\partial_t^{l-2i}v$ ($i=1,2,\cdots$) is based on $\|\partial_t^l v\|_0$ for $\gamma=2$ and on $\left\| \sigma (\sigma/x)^\nu \partial_t^l v' (t)\right\|_0$ for $\gamma\in(1,2)$. This is the difference between \eqref{norm} and \eqref{ggnorm}.}
For $\mu>0$, we use the following parabolic approximation to $\eqref{419'}_1$:
\begin{equation}\lambdabel{93}\begin{equation}gin{split}
x \sigma \partial_t v + \left[ \sigma^2 \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^\gamma \right]' - 2 \frac{\sigma^2}{x}\left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'} \right)^{\gamma-1} + \phi \sigma x^2 \left(\frac{x^2}{r^2}\right) \\
+ \frac{ 2-\gamma }{\gamma-1}
\sigma x \left(\frac{\sigma}{x}\right)' \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^\gamma = \frac{\gamma\mu}{x} \left[(x\sigma)^2 \left(\frac{\sigma}{x}\right)^{2\nu}\left(\frac{v}{x}\right)'\right]',
\end{split}\end{equation}
{which is the general form of $\eqref{pe1-3}_1$ for $\gamma=2$. This approximation matches the energy estimates and elliptic estimates in the sense that one can derive the uniform estimates with respect to $\mu$.
The existence and uniqueness of the solution to the approximate parabolic problem with the same initial and boundary data as in \eqref{419'} can be checked easily as before. To reduce the length of this paper, we will only derive the a priori estimates that guarantees the existence of the solution to problem \eqref{419'}.}
\subsection{Energy estimates}
{As for $\gamma=2$, taking the $(k+1)-$th time derivative of \eqref{93} yields}
\begin{equation}\lambdabel{gg1}\begin{equation}gin{split}
&x \sigma \partial_t^{k+2} v -\left\{ \sigma^2 \left[(2\gamma-2) \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^\gamma \frac{\partial_t^k v}{x} +\gamma \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^{\gamma+1} \partial_t^k v'\right]\right\}' \\
& + 2 \frac{\sigma^2}{x}\left[ (2\gamma-1) \left( \frac{x}{r}\right)^{2\gamma } \left(\frac{1}{r'}\right)^{\gamma-1} \frac{\partial_t^k v}{x} +(\gamma-1) \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^{\gamma} \partial_t^k v' \right]\\
& - 2\nu
\sigma x \left(\frac{\sigma}{x}\right)' \left[ (2\gamma-2) \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^\gamma \frac{\partial_t^k v}{x} +\gamma \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^{\gamma+1} \partial_t^k v'\right] \\
= & \left\{ \sigma^2\left[ (2\gamma-2) W_{11} +\gamma W_{12}\right]\right\}'- 2 \frac{\sigma^2}{x}\left[ (2\gamma-1) W_{21} +(\gamma-1) W_{22} \right]
\\&+2\nu
\sigma x \left(\frac{\sigma}{x}\right)'\left[ (2\gamma-2) W_{11} +\gamma W_{12}\right]
- \phi \sigma x^2 \partial_t^{k+1}\left(\frac{x^2}{r^2}\right),
\end{split}\end{equation}
where
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&W_{11}= \partial_t^{k } \left( \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^\gamma \frac{v}{x}\right)- \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^\gamma \frac{\partial_t^{k } v } {x} ,\\
&W_{12}= \partial_t^{k } \left( \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^{\gamma+1} v'\right)- \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^{\gamma+1}\partial_t^{k } v' ,\\
&W_{21}= \partial_t^{k } \left( \left( \frac{x}{r}\right)^{2\gamma } \left(\frac{1}{r'}\right)^{\gamma-1} \frac{v}{x}\right)- \left( \frac{x}{r}\right)^{2\gamma } \left(\frac{1}{r'}\right)^{\gamma-1} \frac{\partial_t^{k } v } {x} ,\\
&W_{22}= \partial_t^{k }\left( \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^{\gamma} v'\right)- \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^{\gamma}\partial_t^{k } v' .
\end{split}\end{equation}e
{Comparing it with \eqref{e1-11} for $\gamma=2$, we have to deal with an additional term, the last term on the left-hand side of \eqref{gg1}, which does not appear in \eqref{e1-11}. To do so, we introduce a weight $(\sigma/x)^{2\nu}$ (or equivalently, $\rho_0^{2-\gamma}$), which is $1$ for $\gamma=2$.} Multiply \eqref{gg1} with $k=l$ by $(\sigma/x)^{2\nu} \partial_t^{l+1} v$ and integrate the resulting equation with respect to time and space to get
\begin{equation}\lambdabel{geg}\begin{equation}gin{split}
&\left\| \sqrt{x \sigma } (\sigma/x)^\nu \partial_t^{l+1} v (t)\right\|_0^2 + \left\| \sigma (\sigma/x)^\nu \partial_t^l v' (t)\right\|_0^2 + \left\| (\sigma/x)^{1+\nu} \partial_t^l v (t)\right\|_0^2 \\
\le & \widetilde{M}_0+ C t P\left(\sup_{[0,t]} \widetilde{E}\right),
\end{split}\end{equation}
provided that $t$ is small. Here $\widetilde{M}_0=P(\widetilde{E}(0,v))$ is determined by the initial density $\rho_0$.
{It should be noted that \eqref{geg} is the energy estimate parallelling to \eqref{et4} for $\gamma=2$.}
{Based on this energy estimate, we can derive the higher-order spatial derivative of $\partial_t^{l-1}v$ and $\partial_t^{l-2}v$ associated with weights, respectively. Inductively, the weighted spatial derivative of $\partial_t^{l-{2i}+1}v$ and $\partial_t^{l-2i}v$ ($i=2,3,\cdots$) can then be achieved. Next, we use elliptic estimates to
obtain the other norms in the higher-order energy functional. This is done by the interior and boundary estimates}.
\subsection{Elliptic estimates -- interior part}
{To obtain the interior estimates, the key is to choose a suitable cut-off function to separate the whole region into interior and boundary regions such that the energy norms can be matched in the intermediate regions. For this purpose,} note that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\frac{1}{\sigma}\left\{ \sigma^2 \left[(2\gamma-2) \frac{\partial_t^k v}{x} +\gamma \partial_t^k v'\right]\right\}'
- 2 \frac{\sigma }{x}\left[ (2\gamma-1) \frac{\partial_t^k v}{x} +(\gamma-1) \partial_t^k v' \right] + 2 \gamma \nu x \left(\frac{\sigma}{x}\right)' \partial_t^k v ' \\
& = \gamma H_0 + (6\gamma-4 ) (\sigma/x)' \partial_t^k v + 2 \gamma \nu x \left(\frac{\sigma}{x}\right)' \partial_t^k v ' \\
& = \gamma \left[ H_0 + 2\nu x(\sigma/x)' \partial_t^k v' - 2\nu (\sigma/x)' \partial_t^k v \right] + (6\gamma-4 + 2\nu\gamma) (\sigma/x)' \partial_t^k v\\
&= \gamma ({x\sigma})^{-1} \left[(x\sigma)^2 \left( {\sigma}/{x}\right)^{2\nu}\left( {\partial_t^k v}/{x}\right)'\right]' + (6\gamma-4 + 2\nu\gamma) (\sigma/x)' \partial_t^k v,
\end{split}\end{equation}e
where $H_0$ is defined in \eqref{H0}. Then equation \eqref{gg1} reads
\begin{equation}\lambdabel{gi1}\begin{equation}gin{split}
&\gamma \left[ H_0 + 2\nu x \left(\frac{\sigma}{x}\right)' \partial_t^k v' - 2\nu \left(\frac{\sigma}{x}\right)' \partial_t^k v \right] \\
= &x \partial_t^{k+2} v - (6\gamma-4+2\nu \gamma) \left(\frac{\sigma}{x}\right)'\partial_t^k v- \frac{1}{\sigma} \left\{ \sigma^2\left[ (2\gamma-2) W_{11} +\gamma W_{12}\right]\right\}'\\
&+ 2 \frac{\sigma }{x}\left[ (2\gamma-1) W_{21} +(\gamma-1) W_{22} \right]
-2\nu
x \left(\frac{\sigma}{x}\right)'\left[ (2\gamma-2) W_{11} +\gamma W_{12}\right]
+ \phi x^4 \partial_t^{k+1} \left(\frac{1}{r^2}\right)
\\
&- \frac{1}{\sigma}\left\{ \sigma^2 \left[(2\gamma-2) \left[ \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^\gamma -1\right] \frac{\partial_t^k v}{x} +\gamma \left[\left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^{\gamma+1} -1\right] \partial_t^k v'\right]\right\}' \\
& + 2 \frac{\sigma }{x}\left[ (2\gamma-1) \left[ \left( \frac{x}{r}\right)^{2\gamma } \left(\frac{1}{r'}\right)^{\gamma-1}-1\right] \frac{\partial_t^k v}{x} +(\gamma-1) \left[ \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^{\gamma} -1\right] \partial_t^k v' \right]\\
& - 2\nu
x \left(\frac{\sigma}{x}\right)' \left[ (2\gamma-2) \left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'}\right)^\gamma \frac{\partial_t^k v}{x} +\gamma \left[\left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^{\gamma+1}-1\right] \partial_t^k v'\right].\\
\end{split}\end{equation}
In the interior region, one can see easily that the main part of the left-hand side of \eqref{gi1} is $H_0$. So, we analyze $H_0$ to determine the length of the interior region, $\delta_\gamma$. Taking the $i$-th ($i\ge 2$) spatial derivative of $H_0$ ($i=0,1$ has been treated in the case of $\gamma=2$) leads to
\begin{equation}\lambdabel{g54}\begin{equation}gin{split}
H_0^{(i)}-{{H}}_{0 i} = \sigma f^{(i+2)} + (i+2) \sigma' f^{(i+1)} -2\sigma' \left(
\frac{ f }{x} \right)^{(i)}
=: {\widetilde{H}}_{0 i} , \ \ {\rm where} \ \ f=\partial_t^k v
\end{split}\end{equation}
and
\begin{equation}e
{{H}}_{0 i} = \sum_{\alpha=2}^{i} C_\alpha^i \sigma^{(\alpha)} f^{(i+2-\alpha)} + 2\sum_{\alpha=1}^{i} C_\alpha^i \sigma^{(\alpha+1)} f^{(i+1-\alpha)} - 2\sum_{\alpha=1}^{i} C_\alpha^i \sigma^{(\alpha+1)} \left(
\frac{ f }{x} \right)^{(i -\alpha)}
\end{equation}e
is the lower-order term. Here $\mathfrak{g}^{(i)}$ denotes $\partial_x^i \mathfrak{g}(x,t)$ for any function $\mathfrak{g}(x,t)$.
Note that
\begin{equation}e
f^{(j)}=\left(x \frac{f}{x} \right)^{(j)}= x \left(\frac{f}{x} \right)^{(j)} +j\left(\frac{f}{x} \right)^{(j-1)}, \ \ \ \ j=1,2,\cdots.
\end{equation}e
Then ${\widetilde{H}}_{0 i}$ $(i=2, 3, \cdots)$ can be rewritten as
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
{\widetilde{H}}_{0 i}
= & \sigma x g'' +(i+2) \left(\sigma x\right)' g'
+ i(i+3) \sigma' g , \ \ {\rm where}
\ \ g=\left(\frac{f}{x} \right)^{(i )}=\left(\frac{\partial_t^{k } v}{x} \right)^{(i )};
\end{split}\end{equation}e
or equivalently,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\widetilde{H}_{0i} -(i+2) \left(\sigma'x-\sigma\right) g'
= & \sigma x g'' +2 (i+2) \sigma g'
+ i(i+3) \sigma' g .
\end{split}\end{equation}e
Therefore, we obtain that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
&\left\|\zeta {\widetilde{H}}_{0 i}-(i+2)\zeta \left(\sigma'x-\sigma\right) g' \right\|_0^2 \\
=
& \left\| \zeta \sigma x g'' \right\|_0^2+4 (i+2)^2 \left\| \zeta \sigma g'\right\|_0^2
+ i^2(i+3)^2 \left\| \zeta \sigma' g \right\|_0^2 + 4(i+2) \int \zeta \sigma x g'' \zeta \sigma g' dx \\
&+ 2i(i+3) \int \zeta \sigma x g'' \zeta \sigma' g dx + 4 i(i+2) (i+3) \int \zeta \sigma g' \zeta \sigma' g dx,
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \zeta \sigma x g'' \right\|_0^2+2\left[ (i+1)^2+1\right] \left\| \zeta \sigma g'\right\|_0^2
+ i (i+3)\left( i^2+i-2 \right) \left\| \zeta \sigma' g \right\|_0^2 \\
= &\left\|\zeta \widetilde{H}_{ 0i } -(i+2)\zeta \left(\sigma'x-\sigma\right) g' \right\|_0^2\\
& + 4(i+2)\left[ \int \zeta \zeta ' x \left|\sigma g'\right|^2 dx + \int \zeta^2 \sigma (\sigma'x -\sigma ) \left| g'\right|^2 dx \right] \\
&- 2i(i+3) \left[ \int \zeta^2 \sigma \left(\sigma' x -\sigma \right) g g'' dx - 2 \int \zeta \zeta' \sigma^2 g g' dx \right]\\
& + 2 i(i+2) (i+3) \left[2 \int \zeta \zeta' \sigma \sigma' g^2 dx +\int \zeta^2 \sigma \sigma'' g^2 dx\right]\\
\le & 2\left\|\zeta \widetilde{H}_{ 0i } \right\|_0^2 + C(i,m_0,m_1)\delta \left[ \left\| \zeta \sigma x g'' \right\|_0^2 + \left\| {\zeta} \sigma g'\right\|_0^2 + \left\| {\zeta} g \right\|_0^2 \right]\\
&+ C(i,m_0,s_0) \int_\delta^{2\delta} \left[(\sigma g')^2 + g^2 \right] dx.
\end{split}\end{equation}e
So, there exist constants $\begin{equation}tar{\delta}_{i}=\begin{equation}tar{\delta}_{i}(i,m_0,m_1)$ ($i=2,3,\cdots$) such that for $\delta\le \min\{\delta_0/2, \begin{equation}tar{\delta}_{i}\}$,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \frac{1}{2}\left\| \zeta \sigma x g'' \right\|_0^2+ \left[ (i+1)^2+1\right] \left\| \zeta \sigma g'\right\|_0^2
+ \frac{1}{2} i (i+3)\left( i^2+i-2 \right) m_0^2 \left\| \zeta g \right\|_0^2 \\
\le & 2\left\|\zeta \widetilde{H}_{ 0i } \right\|_0^2 + C(i,m_0,s_0) \int_\delta^{2\delta} \left[(\sigma g')^2 + g^2 \right] dx,
\end{split}\end{equation}e
where one has used the fact $\sigma'(x)\ge m_0$ on $[0,\delta_0]$. Consequently,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \zeta \sigma x g'' \right\|_0^2+ \left\| \zeta \sigma g'\right\|_0^2
+ \left\| \zeta g \right\|_0^2
\le C (i,m_0 ) \left\|\zeta \widetilde{H}_{ 0i } \right\|_0^2 + C(i,m_0,s_0) \int_\delta^{2\delta} \left[(\sigma g')^2 + g^2 \right] dx .
\end{split}\end{equation}e
It then follows from \eqref{g54} that, for each $i\ge 2$,
\begin{equation}\lambdabel{ghaha}\begin{equation}gin{split}
&\left\| \zeta \sigma \left( {\partial_t^{k } v} \right)^{(i+2 )} \right\|_0^2+ \left\| \zeta \left( {\partial_t^{k } v} \right)^{(i+1 )}\right\|_0^2
+ \left\| \zeta \left( \frac{\partial_t^{k } v}{x} \right)^{(i )} \right\|_0^2 \\
\le & C(i,m_0 ) \left[\left\|\zeta {H}_{ 0 }^{(i)} \right\|_0^2 + \left\|\zeta {H}_{ 0i } \right\|_0^2 \right] + C(i,m_0,s_0) \int_\delta^{2\delta} \left[ \left|\left( {\partial_t^{k } v} \right)^{(i+1 )} \right|^2+ \left|\left( \frac{\partial_t^{k } v}{x} \right)^{(i )}\right|^2 \right] dx .
\end{split}\end{equation}
Choose
\begin{equation}\lambdabel{daga}
\delta_\gamma=\min\{\delta_0/2,\delta_1,\delta_2,\delta_3, \begin{equation}tar{\delta}_2, \cdots, \begin{equation}tar{\delta}_{ \frac{l-1}{2} } \}.
\end{equation}
(Thus $\delta_\gamma$ depends on the initial density $\rho_0(x)$ and $\gamma$.)
With this $\delta_\gamma$, we can derive from \eqref{ghaha}, \eqref{gi1} and \eqref{geg} that
\begin{equation}\lambdabel{gie}\begin{equation}gin{split}
& \sum_{j=1}^{ \frac{l+1}{2}} \left\{ \left\|\zeta \sigma \partial_t^{l-2j +1 } v \right\|_{j+1}^2
+\left\|\zeta \partial_t^{l-2j +1 } v \right\|_{j }^2
+\left\|\zeta \frac {\partial_t^{l-2j +1 } v}{x} \right\|_{j-1}^2 \right\}(t) \\
&+
\sum_{j=1}^{\frac{l-1}{2}} \left\{ \left\|\zeta \sigma \partial_t^{l-2j } v \right\|_{j+2}^2
+\left\|\zeta \partial_t^{l-2j } v \right\|_{j +1 }^2
+\left\|\zeta \frac{ \partial_t^{l-2j } v }{x} \right\|_{j}^2 \right\}(t)
\le \widetilde{M}_0+ C t P\left(\sup_{[0,t]} \widetilde{E}\right) .
\end{split}\end{equation}
{This completes the interior estimates. Next, we show the boundary estimates using the same argument as that in Section 7.2.}
\subsection{Elliptic estimates -- boundary part}
As before, we can introduce a cut-off function as
$$
\chi=1 \ \ {\rm on} \ \ [\delta_\gamma,1], \ \ \chi=0 \ \ {\rm on} \ \ [0, \delta_\gamma/2], \ \ |\chi'|\le s_0/\delta_\gamma,
$$
for some constant $s_0$, where $\delta_\gamma$ is given by \eqref{daga}.
Note that in the boundary region, $x\in [\delta_\gamma/2, 1]$, the main part of the left-hand side of \eqref{gi1} is
$$
B_\gamma:=\sigma \partial_t^k v''+ (2+2\nu)\sigma'\partial_t^k v'.
$$
Taking the $i$-th $(i\ge 0)$ spatial derivative of $B_\gamma$ yields
$$
B^{(i)}_\gamma-B_{\gamma i }
= {\sigma \partial_t^k v ^{(i+2)} + (i+2+2\nu) \sigma' \partial_t^k v^{(i+1)}} =: \widetilde{B}_{\gamma i},
$$
where $${B}_{\gamma i}=\sum_{\alpha=2}^{i} C_\alpha^i \sigma^{(\alpha)} \partial_t^k v^{(i+2-\alpha)} + 2(1+\nu)\sum_{\alpha=1}^{i} C_\alpha^i \sigma^{(\alpha+1)} \partial_t^k v^{(i+1-\alpha)}$$
denotes the lower-order term.
Since for any function $h=h(x,t)$ and integer $i\ge 0$, it holds that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \chi \sigma^{3/2+\nu} h'\right\|_0^2 + \left\| \chi \sigma^{1/2+\nu} \sigma' h \right\|_0^2 \le
4 \left\| \chi \sigma^{1/2+\nu} \left(\sigma h' + (i+2+2\nu) \sigma' h \right) \right\|_0^2 + C \left\| \sigma^{1+\nu} h \right\|_0^2,\\
&\left\| \chi \sigma^{2+\nu} h'\right\|_0^2 + \left\| \chi \sigma^{1+\nu} \sigma' h \right\|_0^2
\le 4 \left\| \chi \sigma^{1+\nu} \left(\sigma h' + (i+3+2\nu) \sigma' h \right) \right\|_0^2 + C \left\| \sigma^{3/2+\nu} h \right\|_0^2 ;
\end{split}\end{equation}e
then we have
\begin{equation}\lambdabel{gbe}\begin{equation}gin{split}
& \sum_{j=1}^{ \frac{l+1}{2} } \left\{ \left\|\chi \sigma^{3/2+\nu} \partial_t^{l-2j +1 } \partial_x^{ j +1 } v \right\|_{0}^2
+\left\|\chi \sigma^{1/2+\nu} \partial_t^{l-2j +1 } \partial_x^{ j } v \right\|_{0}^2 \right\}\le \widetilde{M}_0+ C t P\left(\sup_{[0,t]} \widetilde{E}\right) \\
&\sum_{j=1}^{ \frac{l-1}{2} } \left\{ \left\|\chi \sigma^{2+\nu} \partial_t^{l-2j } \partial_x^{ j +2 } v \right\|_{0}^2
+\left\|\chi \sigma^{1+\nu} \partial_t^{l-2j } \partial_x^{ j +1 } v \right\|_{0}^2 \right\}\le \widetilde{M}_0+ C t P\left(\sup_{[0,t]} \widetilde{E}\right).
\end{split}\end{equation}
{This yields the desired is elliptic estimates on the boundary.}
\subsection{Existence for case $1<\gamma<2$}
It follows from \eqref{ggnorm}, \eqref{geg}, \eqref{gie} and \eqref{gbe} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\widetilde{E}(t)
\le \widetilde{M}_0+ C t P\left(\sup_{s\in[0,t]} \widetilde{E}(s)\right), \ \ t\in [0,T];
\end{split}\end{equation}e
which implies that for small $T$,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{t\in[0,T]}\widetilde{E}(t)
\le 2 \widetilde{M}_0.
\end{split}\end{equation}e
{With this a priori estimates, one can then obtain the local existence of smooth solutions in the functional space for which $\sup_{t\in[0,T]}\widetilde{{E}}(v, t)<\infty$
provided that $\widetilde{{E}}(v, 0)<\infty $ ($\widetilde{{E}}(v, 0)$ is determined by the initial data and their spatial derivatives via the equation), by using the parabolic approximation in \eqref{93} in a similar way as before.}
\section{Case $\gamma >2$}
{In this section, we deal with the case when $\gamma>2$, which is easier than
the case when $1<\gamma<2$ because the rate of degeneracy of equation $\eqref{419'}_1$ near vacuum states
is lower and less derivatives are needed to control the $H^2$-norm of $v$.}
Set $$\nu=(2-\gamma)(2\gamma-2)\in (-1/2,0).$$ The higher-order energy norm is chosen as follows:
\begin{equation}\lambdabel{less2}\begin{equation}gin{split}
\widehat{E}(v, t)= & \left\| \sigma (\sigma/x)^\nu \partial_t^4 v' (\cdot,t)\right\|_0^2 + \left\| (\sigma/x)^{1+\nu} \partial_t^4 v (\cdot,t)\right\|_0^2 \\
& + \sum_{j=1}^{ 2 } \left\{ \left\| \sigma^{3/2+\nu} \partial_t^{5-2j } \partial_x^{ j +1 } v (\cdot,t)\right\|_{0}^2
+ \sum_{i=0}^j \left\| \sigma^{1/2+\nu} \partial_t^{5-2j } \partial_x^{ i } v (\cdot,t)\right\|_{0}^2 \right\} \\
& + \sum_{j=1}^{ 2 } \left\{ \left\| \sigma^{2+ \nu} \partial_t^{4-2j } \partial_x^{ j +2 } v (\cdot,t)\right\|_{0}^2
+\sum_{i=-1}^j \left\| \sigma^{ 1+ \nu} \partial_t^{4-2j } \partial_x^{ i +1 } v (\cdot,t) \right\|_{0}^2 \right\}\\
&+\sum_{j=1}^{ 2} \left\{ \left\|\zeta \sigma \partial_t^{5-2j } v (\cdot,t)\right\|_{j+1}^2
+\left\|\zeta \partial_t^{5-2j } v (\cdot,t)\right\|_{j }^2
+\left\|\zeta \frac {\partial_t^{5-2j } v}{x} (\cdot,t)\right\|_{j-1}^2 \right\}\\
&+ \sum_{j=1}^{2} \left\{ \left\|\zeta \sigma \partial_t^{4-2j } v (\cdot,t)\right\|_{j+2}^2
+\left\|\zeta \partial_t^{4-2j } v (\cdot,t)\right\|_{j+1 }^2
+\left\|\zeta \frac{ \partial_t^{4-2j } v }{x} (\cdot,t)\right\|_{j }^2 \right\}.
\end{split}\end{equation}
Here $\zeta$ is defined in \eqref{delta}. It follows from Sobolev embedding \eqref{wsv} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\|v\|_2^2 \le \|v\|_{2-\nu}^2 \le \sum_{i=0}^3 \| \sigma^{1+\nu} \partial_x^i v\|_0 \le C\widehat{E}.
\end{split}\end{equation}e
As before, we can show
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\widehat{E}(t)
\le P(\widehat{E}(0)) + C t P\left(\sup_{s\in[0,t]} \widehat{E}(s)\right), \ \ t\in [0,T];
\end{split}\end{equation}e
which implies that for small $T$,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\sup_{t\in[0,T]}\widehat{E}(t)
\le 2 P(\widehat{E}(0)) .
\end{split}\end{equation}e
{With the above estimates, one can then obtain the local existence of smooth solutions in the functional space $\sup_{t\in[0,T]}\widehat{E}(t)<\infty$.}
\section{Uniqueness of spherically symmetric motions for the three-dimensional compressible Euler equations}
For the free-boundary problem of the compressible Euler equations without self-gravitation, we can prove that the uniqueness theorem is true for all values of $\gammamma>1$ in a natural functional space for the spherically symmetric motion. (Indeed, a similar argument can be extended to the general three-dimensional motion.) In this case, problem \eqref{419'} becomes
\begin{equation}\lambdabel{419''}\begin{equation}gin{split}
& \rho_0 \left( \frac{x}{r}\right)^2\partial_t v + \partial_x \left[ \left(\frac{x^2}{r^2}\frac{\rho_0}{\partial_x r}\right)^\gamma \right] =0 & {\rm in} & \ \ I \times (0,T],\\
& v(0, t)=0 & {\rm on} & \ \ \{x=0\}\times (0,T],\\
& v(x, 0)= u_0(x) & {\rm on} & \ \ I \times \{t=0\},
\end{split}
\end{equation}
where the initial density $\rho_0$ satisfies \eqref{156}. For problem \eqref{419''}, we have the following result:
\begin{equation}gin{thm}\lambdabel{unique2} {\rm(uniqueness for Euler equations)}
Suppose $\gammamma>1$. Let $v_1$ and $v_2$ be two solutions to the problem \eqref{419''} on $[0, T]$ for $T >0$ with
$$r_i(x, t)=x+\int_0^t v_i(x, s)ds, \ \ i=1, 2.$$
If there exist some positive constants $w_1$, $w_2$ and $w_3$ such that
\begin{equation}\lambdabel{a1'}w_1\le r_i'(x, t)\le w_2 \ \ {\rm and} \ \ |v_i'(x, t)|\le w_3, \ \ \ (x, t)\in [0, 1]\times [0, T] , \ \ i=1, 2,\end{equation}
then
\begin{equation}\lambdabel{v1v2}v_1(x, t)=v_2(x, t), \ \ (x, t)\in [0, 1]\times [0, T]\end{equation}
provided that $v_1(x, 0)=v_2(x, 0)$ for $x\in [0,1]$.
\end{thm}
The solution to the spherically symmetric problem of Euler equations in Eulerian coordinates can be obtained from the solution to \eqref{419''}. Denote this solution by
$(\rho, u)(r, t)$ ($0\le r\le R(t)$, $0\le t\le T$). For $({\bf x}, t)\in \mathbb{R}^3\times [0, T]$ with $|{\bf x}|<R(t)$, we set
\begin{equation}gin{equation}\lambdabel{3.1'}\rho({\bf x}, t) = \rho(|{\bf x}|, t), \ \ {\bf u}({\bf x}, t) = u(|{\bf x}|, t) {\bf x} /|{\bf x}|. \end{equation}
Then $(\rho, {\bf u}, R(t))$ is a solution of the following free boundary problem:
\begin{equation}\lambdabel{2.1euler} \left\{ \begin{equation}gin{split}
& \partial_t \rho + {\rm div}(\rho {\bf u}) = 0 , & 0<|{\bf x}|< R(t), \ \ t\in [0, T], \\
&\partial_t (\rho {\bf u}) + {\rm div}(\rho {\bf u}\otimes {\bf u})+\nablabla_{\bf x} (\rho^{\gammamma}) =0 , & 0<|{\bf x}|< R(t), \ \ t\in [0, T],\\
&\rho>0, & 0 \le |{\bf x}|< R(t), \ \ t\in [0, T],\\
& \rho =0, & |{\bf x}|= R(t), \ \ t\in [0, T], \\
&u({\bf 0}, t)=0, & t\in [0, T],\\
& \mathcal{V}(\partial B_{R(t)})={\bf u}|_{\partial B_{R(t)}}\cdot {\bf n}, & t\in [0, T], \\
&(\rho,{\bf u})({\bf x}, 0)=(\rho_0, {\bf u}_0)(|{\bf x}|), \ & |{\bf x}|\le R_0,
\end{split} \right. \end{equation}
where $R_0>0$ is a constant, $B_{R(t)}=\{{\bf x}\in \mathbb{R}^3: |{\bf x}|<R(t)\}$, $\mathcal{V}(\partial B_{R(t)})$ and ${\bf n}$ represent, respectively, the normal velocity of $\partial B_{R(t)}$ and exterior unit normal vector to $\partial B_{R(t)}$.
As a direct consequence of Theorem \ref{unique2}, we have
\begin{equation}gin{coro}\lambdabel{corollary} Let $\gammamma>1$. The solutions $(\rho, {\bf u}, R(t))$ of the form \eqref{3.1'} to the free boundary problem \eqref{2.1euler} are unique provided they satisfy the following regularity conditions:
$$R(t)\in C^1\left([0, T]\right) \ \ {\rm and} \ \
(\rho, {\bf u})\in C^1\cap W^{1, \infty}\left(\{({\bf x}, t)\in \mathbb{R}^3\times [0, T]: 0<|{\bf x}|\le R(t)\}\right).$$
\end{coro}
\noindent{\bf Proof of Theorem \ref{unique2}}. We first present the proof for the case of $\gammamma=2$ for simplicity. When $\gammamma=2$, equation $\eqref{419''}_1$ reduces to
$$
x \sigma \partial_t^2 r + \left[\sigma^2 \frac{x^2 } {r^2 r'^2} \right]' -2 \frac{\sigma^2}{x} \frac{x^3}{r^3 r'} =0 \ \ {\rm in} \ \ I\times (0, T].
$$
Set
$$\theta(x,t)=r_2(x,t)-r_1(x,t),$$
then
\begin{equation}\lambdabel{1}\begin{equation}gin{split}
x \sigma \partial_t^2 \theta - \left[\sigma^2 \left( \frac{x^2 } {r_1^2 {r_1'}^{2}} - \frac{x^2 }{r_2^2 {r_2'}^{2}}\right) \right]' + 2 \frac{\sigma^2}{x} \left( \frac{x^3}{r_1^3 r'_1} -\frac{x^3}{r_2^3 r'_2}\right) =0 \ \ {\rm in} \ \ I\times (0, T].
\end{split}\end{equation}
Multiplying \eqref{1} by $\partial_t \theta$ and integrating the resulting equation with respect to $x$, we have
\begin{equation}e\lambdabel{2}\begin{equation}gin{split}
\frac{1}{2}\frac{d}{dt}\int x \sigma \left(\partial_t \theta \right)^2 dx
= & -\int \sigma^2 \left( \frac{x^2 } {r_1^2 {r_1'}^{2}} - \frac{x^2 }{r_2^2 {r_2'}^{2}}\right) \left(\partial_t \theta'\right) dx
\\
&- 2 \int \frac{\sigma^2}{x} \left( \frac{x^3}{r_1^3 r'_1} -\frac{x^3}{r_2^3 r'_2}\right) \left(\partial_t \theta \right)dx.
\end{split}\end{equation}e
Note that
\begin{equation}e
\frac{x^2 } {r_1^2 {r_1'}^{2}} - \frac{x^2 }{r_2^2 {r_2'}^{2}}= \mathcal{A}_1 \theta' + \mathcal{A}_2 (\theta/x) \ \ {\rm and} \ \
\frac{x^3}{r_1^3 r'_1} -\frac{x^3}{r_2^3 r'_2} = \mathcal{A}_3 \theta' +\mathcal{A}_4 (\theta /x),
\end{equation}e
where
\begin{equation}e\begin{equation}gin{split}
& \mathcal{A}_1 = \left(\frac{x}{r_1}\right)^2 \left(\frac{1}{r_1'}+ \frac{1}{r_2'}\right) \frac{1}{r_1' r_2'}, &
\mathcal{A}_2 = \left(\frac{1}{r_2'}\right)^2 \left(\frac{x}{r_1}+ \frac{x}{r_2}\right) \frac{x}{r_1 }\frac{x}{r_2 },\\
&\mathcal{A}_3 = \left(\frac{x}{r_1}\right)^3 \frac{1}{r_1' r_2'}, &
\mathcal{A}_4= \frac{1}{r_2'} \left[\left(\frac{x}{r_1} \right)^2 + \frac{x}{r_1} \frac{x}{r_2} +\left( \frac{x}{r_2}\right)^2 \right]\frac{x}{r_1 }\frac{x}{r_2 }.
\end{split}\end{equation}e
Since $r_i(0, t)=0$ and $v_i(0, t)=0$ ($i=1,\ 2$) for $t\in [0, T]$, the bounds in \eqref{a1'} give the following bounds:
$$ w_1\le r_i (x, t)/x\le w_2 \ \ {\rm and } \ \ |v_i(x,t)/x|\le w_3,\ \ (x, t)\in [0, 1]\times [0, T], \ \ i=1, 2. $$
Then, using the integration by parts and the Cauchy inequality, we can get that
\begin{equation}\lambdabel{2}\begin{equation}gin{split}
&\frac{1}{2}\frac{d}{dt}\int \left\{ x \sigma \left(\partial_t \theta \right)^2
+ \sigma^2 \left[ \mathcal{A}_1 (\theta')^2 + 2\mathcal{A}_2 (\theta/x) \theta' + 2\mathcal{A}_4 (\theta/x)^2 \right] \right\} dx\\
= & \int \sigma^2 \left[ \frac{1}{2}(\partial_t \mathcal{A}_1) (\theta')^2 + (\partial_t \mathcal{A}_2) (\theta/x) \theta' + (\partial_t \mathcal{A}_4)(\theta/x)^2 \right]dx \\
& + \int \sigma^2 (\mathcal{A}_2-2 \mathcal{A}_3) (\partial_t \theta/x) \theta' dx \\
\le & C(w_1,w_2,w_3) \int \sigma^2 \left[ (\theta')^2 + (\theta/x)^2 \right]dx + 2 w_3 \int \sigma^2 \left|\mathcal{A}_2-2 \mathcal{A}_3\right| \left|\theta'\right| dx ,
\end{split}\end{equation}
where $C(w_1,w_2,w_3)$ is a positive constant depending on $w_1,w_2,w_3$; because
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left|\partial_t \mathcal{A}_1\right| + \left|\partial_t \mathcal{A}_2\right|+\left|\partial_t \mathcal{A}_4\right| \le C(w_1,w_2) \left(|v_1/x|+|v_2/x| + |v_1'| + |v_2'|\right) \le C(w_1,w_2, w_3)
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left|\partial_t \theta/x\right| = \left|(v_2-v_1)/x \right| \le 2 w_3.
\end{split}\end{equation}e
To estimate \eqref{2}, we need the following a priori assumption: there exists a small positive constant $\epsilon_0$ such that
\begin{equation}\lambdabel{aa1} |\theta'(x,t)|+ |(\theta/x)(x,t)|\le \epsilon_0 \ \ {\rm for } \ \ {\rm all} \ \ (x, t)\in (0, 1)\times [0, T].\end{equation}
Thus, a simple calculation yields that
\begin{equation}e\begin{equation}gin{split}
& \mathcal{A}_1 \ge (2- C(w_1,w_2) \epsilon_0) \left(\frac{x}{r_1}\right)^2 \left( \frac{1}{r_1'} \right)^3 \ge \frac{7}{4}\left(\frac{x}{r_1}\right)^2 \left( \frac{1}{r_1'} \right)^3 ,\\
&
\mathcal{A}_2 \le (2+ C(w_1,w_2) \epsilon_0) \left(\frac{x}{r_1}\right)^3 \left( \frac{1}{r_1'} \right)^2 \le \frac{9}{4}\left(\frac{x}{r_1}\right)^3 \left( \frac{1}{r_1'} \right)^2 ,\\
&
\mathcal{A}_4 \ge (3- C(w_1,w_2) \epsilon_0) \left(\frac{x}{r_1}\right)^4 \frac{1}{r_1'} \ge \frac{11}{4} \left(\frac{x}{r_1}\right)^4 \frac{1}{r_1'} ;
\end{split}\end{equation}e
which implies that for $(x, t)\in (0, 1)\times [0, T]$,
\begin{equation}\lambdabel{3}\begin{equation}gin{split}
\mathcal{A}_1 (\theta')^2 + 2\mathcal{A}_2 (\theta/x) \theta' + 2\mathcal{A}_4 (\theta/x)^2 \ge &\frac{1}{4} \left(\frac{x}{r_1}\right)^2 \left( \frac{1}{r_1'} \right)^3 (\theta')^2 + \frac{17}{8} \left(\frac{x}{r_1}\right)^4 \frac{1}{r_1'} \left(\frac{\theta}{x}\right)^2\\
\ge & k_1 (\theta')^2 + k_2 (\theta/x)^2 .
\end{split}\end{equation}
Here $k_1$ and $k_2$ are positive constants depending on $w_1$ and $w_2$. We use the cancelation of the leading terms to estimate of $\int \sigma^2 \left|\mathcal{A}_2-2 \mathcal{A}_3\right| \left|\theta'\right| dx$. Note that
\begin{equation}e\begin{equation}gin{split}
&
\mathcal{A}_2 =2 \left(\frac{x}{r_1}\right)^3 \left( \frac{1}{r_1'} \right)^2+ C(w_1,w_2) \left(\left|\frac{\theta}{x}\right|+|\theta'|\right) , \\
&\mathcal{A}_3 = \left(\frac{x}{r_1}\right)^3 \left( \frac{1}{r_1'} \right)^2+ C(w_1,w_2) |\theta'|.
\end{split}\end{equation}e
It then follows from the Cauchy's inequality that
\begin{equation}\lambdabel{4}\begin{equation}gin{split}
\int \sigma^2 \left|\mathcal{A}_2-2 \mathcal{A}_3\right| \left|\theta'\right| dx \le
C(w_1,w_2) \int \sigma^2 \left[ (\theta')^2 + (\theta/x)^2 \right]dx .
\end{split}\end{equation}
In view of \eqref{2}, \eqref{3} and \eqref{4}, we see that
\begin{equation}e\lambdabel{5}\begin{equation}gin{split}
&\frac{1}{2} \int \left[ x \sigma \left(\partial_t \theta \right)^2
+ k_1 ( \sigma \theta')^2 + k_2 (\sigma \theta/x)^2 \right] dx (t) \\
\le & \frac{1}{2} \int \left\{ x \sigma \left(\partial_t \theta \right)^2
+ \sigma^2 \left[ \mathcal{A}_1 (\theta')^2 + 2\mathcal{A}_2 (\theta/x) \theta' + 2\mathcal{A}_4 (\theta/x)^2 \right] \right\} dx (t=0)\\
& +C(w_1,w_2,w_3) \int_0^t \int \left[ (\sigma \theta')^2 + (\sigma \theta/x)^2 \right]dx \\
\le & C(w_1,w_2,w_3)\int_0^t \int \left[ (\sigma \theta')^2 + (\sigma \theta/x)^2 \right]dx,
\end{split}\end{equation}e
provided that $v_1(x,0)=v_2(x,0)$. So, it gives from Grownwall's inequality that for $t\in[0, T]$,
\begin{equation}e\lambdabel{5}\begin{equation}gin{split}
& \int \left[ x \sigma \left(\partial_t \theta \right)^2
+ k_1 ( \sigma \theta')^2 + k_2 (\sigma \theta/x)^2 \right](x, t) dx \\
\le & \exp\left\{C(w_1,w_2,w_3)T\right\} \int \left[ x \sigma \left(\partial_t \theta \right)^2
+ k_1 ( \sigma \theta')^2 + k_2 (\sigma \theta/x)^2 \right] (x, 0) dx
= 0 ,
\end{split}\end{equation}e
if $v_1(x,0)=v_2(x,0)$; which implies directly that
$$v_2-v_1=\partial_t \theta = \theta' = \theta /x =0, \ \ (x,t)\in (0,1)\times [0,T] ,$$
because of $\sigma(x)>0$ for all $x\in(0,1)$. This verifies the a priori assumption \eqref{aa1} and completes the proof of Theorem \ref{unique2} when $\gamma=2$.
When $\gamma\neq 2$, equation $\eqref{419''}_1$ reduces to
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
x \sigma \partial_t^2 r + \left[ \sigma^2 \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^\gamma \right]' - &2 \frac{\sigma^2}{x}\left( \frac{x}{r}\right)^{2\gamma-1} \left(\frac{1}{r'} \right)^\gamma
\\&+ \frac{ 2-\gamma }{\gamma-1}
\sigma x \left(\frac{\sigma}{x}\right)' \left( \frac{x}{r}\right)^{2\gamma-2} \left(\frac{1}{r'}\right)^\gamma =0,
\end{split}\end{equation}e
which implies that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
x \sigma \partial_t^2 \theta -& \left[ \sigma^2 \left( \frac{x}{r_1}\right)^{2\gamma-2} \left(\frac{1}{r'_1}\right)^\gamma -\sigma^2 \left( \frac{x}{r_2}\right)^{2\gamma-2} \left(\frac{1}{r'_2}\right)^\gamma \right]' \\
&+ 2 \frac{\sigma^2}{x}\left[\left( \frac{x}{r_1}\right)^{2\gamma-1} \left(\frac{1}{r'_1} \right)^\gamma-\left( \frac{x}{r_2}\right)^{2\gamma-1} \left(\frac{1}{r'_2} \right)^\gamma\right]\\
&-\frac{ 2-\gamma }{\gamma-1}
\sigma x \left(\frac{\sigma}{x}\right)' \left[\left( \frac{x}{r_1}\right)^{2\gamma-2} \left(\frac{1}{r'_1}\right)^\gamma-\left( \frac{x}{r_2}\right)^{2\gamma-2} \left(\frac{1}{r'_2}\right)^\gamma\right]=0.
\end{split}\end{equation}e
Set
$$\nu:=(2-\gamma)/(2\gamma-2).$$
Multiply the preceding equation with $(\sigma/x)^{2\nu} \partial_t \theta$ and integrate the product with respect to time and space. Then using the same argument as to the proof of $\gamma=2$, we can show that \eqref{v1v2} is true for $\gamma\neq2$.
This finishes the proof of Theorem \ref{unique2}.
\vskip 1cm
\centerline{\bf Acknowledgements}.
Xin's research was partially supported by the Zheng Ge Ru Foundation, and Hong
Kong RGC Earmarked Research Grants CUHK-4041/11P, CUHK-4048/13P, a Focus Area Grant from
The Chinese University of Hong Kong, and a grant from Croucher Foundation. Zeng's research was partially supported by NSFC grant \#11301293/A010801.
\vskip 1cm
\section*{Appendix}
In this appendix, we verify \eqref{3lk}, \eqref{hzz}, \eqref{ho3} and \eqref{lb0}.
\vskip 0.5cm
\noindent{\bf Verification of \eqref{3lk}}. For $\mathcal{R}_0$, it follows from \eqref{norm} and \eqref{egn} that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \mathcal{R}_0(t)\right\|_0
\le \int_0^t \left( \left\|\left(\frac{v}{x}\right)'\right\|_0+ \left\|v''\right\|_0 \right)ds
\le C t \sup_{[0,t]} \sqrt{E} ,
\end{split}\end{equation}e
\begin{equation}e\lambdabel{rrr}\begin{equation}gin{split}
\left\|\sigma \mathcal{R}_0(t)\right\|_{L^\infty } \le &
\left\|\int_0^t \sigma \left(\frac{v}{x}\right)' ds\right\|_{L^\infty }+ \left\|\int_0^t \sigma v'' ds\right\|_{L^\infty }\\
\le & C\left\|\int_0^t x \left(\frac{v}{x}\right)' ds\right\|_{L^\infty }+ \left\|\int_0^t \sigma v'' ds\right\|_{L^\infty }\\
\le & C\int_0^t \left(\left\|v'- \frac{v}{x} \right\|_{L^\infty}+ C \left\|\sigma v''\right\|_{L^\infty}\right)ds \le Ct \sup_{[0,t]} \sqrt{E}.
\end{split}\end{equation}e
Next, we will show $\eqref{3lk}_2$. It follows from $\eqref{3jk}_1$, \eqref{egn} and $\eqref{3egn}_1$ that for $p\in(1,\infty)$,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \mathfrak{L}_{0} (t) \right\|_{0} \le \left\| v''\right\|_{0} + \left\| (v/x)'\right\|_{0}+ \left\| \mathcal{R}_0\right\|_0
\left\|\mathfrak{J}_{0}\right\|_{L^\infty}
\le C \sqrt{E(t)} + Ct \sup_{[0,t]} {E},
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\|\sigma \mathfrak{L}_{0} (t)\right\|_{L^\infty } \le \left\|\sigma v''\right\|_{L^\infty} + C\left\| v'- \frac{v}{x}\right\|_{L^\infty}+ \left\| \sigma \mathcal{R}_0\right\|_{L^\infty }
\left\|\mathfrak{J}_{0}\right\|_{L^\infty}
\le C \sqrt{E(t)} + Ct \sup_{[0,t]} {E} ,
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma \mathfrak{L}_{1} (t)\right\|_{L^p } \le & \left\| \sigma \partial_t v''\right\|_{L^p} + C \left\| \partial_t v ' - \frac{ \partial_t v}{x} \right\|_{L^p}+ \left\|\sigma \mathcal{R}_0\right\|_{L^\infty }
\left\| \mathfrak{J}_{1}\right\|_{L^p}
\\& + \left\| \sigma \mathfrak{L}_{0}\right\|_{L^\infty } \left\| \mathfrak{J}_{0}\right\|_{L^\infty}
\le C P \left(\sqrt{E(t)}\right) + Ct P \left(\sup_{[0,t]} \sqrt{E}\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\|\zeta \sigma \mathfrak{L}_{1}(t) \right\|_{L^\infty} \le & \left\|\zeta \sigma \partial_t v''\right\|_{L^\infty} + C\left\|\zeta \left(\partial_t v ' - \frac{ \partial_t v}{x} \right)\right\|_{L^\infty}+ \left\|\sigma \mathcal{R}_0\right\|_{L^\infty }
\left\|\zeta \mathfrak{J}_{1}\right\|_{L^\infty} \\
& + \left\| \sigma \mathfrak{L}_{0}\right\|_{L^\infty } \left\| \mathfrak{J}_{0}\right\|_{L^\infty}
\le C P \left(\sqrt{E(t)}\right) + Ct P \left(\sup_{[0,t]} \sqrt{E}\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma \mathfrak{L}_{2} (t) \right\|_{0} \le & \left\| \sigma \partial_t^2 v''\right\|_{0} + C\left\| \partial_t^2 v ' - \frac{ \partial_t^2 v}{x} \right\|_{0}+ \left\| \sigma \mathcal{R}_0\right\|_{L^\infty }
\left\| \mathfrak{J}_{2}\right\|_{0}
+ \left\| \sigma \mathfrak{L}_{0}\right\|_{L^\infty } \left\| \mathfrak{J}_{1}\right\|_{0}\\
& +
\left\| \sigma \mathfrak{L}_{1}\right\|_{0} \left\| \mathfrak{J}_{0}\right\|_{L^\infty}
\le C P \left(\sqrt{E(t)}\right) + Ct P \left(\sup_{[0,t]} \sqrt{E}\right).
\end{split}\end{equation}e
We now turn to the proof of $\eqref{3lk}_3$. It follows from \eqref{tnorm}, \eqref{tegn}, $\eqref{3jk}_2$ and $\eqref{3egn}_{2,3}$ that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \zeta \mathfrak{L}_{0} (t) \right\|_{0}^2 \le \left( \left\| \zeta v''\right\|_{0} + \left\| \zeta (v/x)'\right\|_{0}+ \left\| \mathcal{R}_0\right\|_0
\left\|\mathfrak{J}_{0}\right\|_{L^\infty}\right)^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right) ,
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma \mathfrak{L}_{0} (t) \right\|_{L^p }^2 \le & \left( \left\| \sigma v''\right\|_{L^p} + C\left\| v'- {v}/{x}\right\|_{L^p} + \left\| \sigma \mathcal{R}_0\right\|_{L^\infty }
\left\| \mathfrak{J}_{0}\right\|_{L^p} \right)^2 \\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\|\zeta \sigma \mathfrak{L}_{0} (t)\right\|_{L^\infty }^2 \le & \left( \left\|\zeta \sigma v''\right\|_{L^\infty} + C\left\| \zeta v'- \zeta {v}/{x}\right\|_{L^\infty} +
\left\| \sigma \mathcal{R}_0\right\|_{L^\infty} \left\|\zeta \mathfrak{J}_{0}\right\|_{L^\infty} \right)^2\\
\le & M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma \mathfrak{L}_{1} (t)\right\|_{0}^2 \le & \left( \left\| \sigma \partial_t v''\right\|_{0} + C \left\| \partial_t v ' - { (\partial_t v)}/{x} \right\|_{0} +
\left\| \sigma \mathcal{R}_0\right\|_{L^\infty }\left\| \mathfrak{J}_{1}\right\|_{0}\right. \\
&\left.
+ \left\| \sigma \mathfrak{L}_{0}\right\|_{L^4} \left\| \mathfrak{J}_{0}\right\|_{L^4} \right)^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\|\zeta \sigma \mathfrak{L}_{2} (t)\right\|_{0}^2 \le & \left( \left\| \zeta \sigma \partial_t^2 v''\right\|_{0} + C\left\| \zeta\left( \partial_t^2 v ' - ({ \partial_t^2 v})/{x}\right) \right\|_{0} + \left\| \sigma \mathcal{R}_0\right\|_{L^\infty }
\left\| \zeta \mathfrak{J}_{2}\right\|_{0}
\right.\\
& \left.
+ \left\| \zeta \sigma \mathfrak{L}_{0}\right\|_{L^\infty } \left\| \mathfrak{J}_{1}\right\|_{0} +
\left\| \sigma \mathfrak{L}_{1}\right\|_0 \left\| \zeta \mathfrak{J}_{0}\right\|_{L^\infty}^2\right)^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right).
\end{split}\end{equation}e
\vskip 0.2cm
\noindent {\bf Verification of \eqref{hzz}.} Note that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \chi \left(\sigma h' + i \sigma' h \right) \right\|_0^2 = \left\| \chi \sigma h'\right\|_0^2 + i^2 \left\| \chi \sigma' h \right\|_0^2 +2i \int \chi^2 \sigma \sigma' h h' dx
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& 2i \int \chi^2 \sigma \sigma' h h' dx = - i\int \left(\chi^2 \sigma \sigma'\right)' h^2 dx \\ \ge & - i \left\| \chi \sigma' h \right\|_0^2 - C(i) \left\| \chi \sigma^{1/2} h \right\|_0^2 - C(i,\delta)\int_{\delta/2}^\delta \chi \sigma h^2 dx \\
\ge & - i \left\| \chi \sigma' h \right\|_0^2 - C (i,\delta) \left\| \sigma^{1/2} h \right\|_0^2 .
\end{split}\end{equation}e
Then we have for $i\ge 2$ that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \chi \sigma h'\right\|_0^2 + \left\| \chi \sigma' h \right\|_0^2 \le \left\| \chi \sigma h'\right\|_0^2 + i(i-1) \left\| \chi \sigma' h \right\|_0^2
\le \left\| \chi \left(\sigma h' + i \sigma' h \right) \right\|_0^2 + C \left\| \sigma^{1/2} h \right\|_0^2 .
\end{split}\end{equation}e
This is $\eqref{hzz}_1$. Next, we will show $\eqref{hzz}_2$.
Note that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma^{1/2}\chi \left(\sigma h' + i \sigma' h \right) \right\|_0^2 = \left\| \chi \sigma^{3/2} h'\right\|_0^2 + i^2 \left\| \chi \sigma^{1/2} \sigma' h \right\|_0^2 +2i \int \chi^2 \sigma^2 \sigma' h h' dx
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& 2i \int \chi^2 \sigma^2 \sigma' h h' dx = - i\int \left(\chi^2 \sigma^2 \sigma'\right)' h^2 dx \\ \ge & - 2 i \left\| \chi \sigma^{1/2}\sigma' h \right\|_0^2 - C(i) \left\| \chi \sigma h \right\|_0^2 - C(i,\delta)\int_{\delta/2}^\delta \chi \sigma^2 h^2 dx \\
\ge & - 2 i \left\| \chi \sigma^{1/2}\sigma' h \right\|_0^2 - C (i,\delta) \left\| \sigma h \right\|_0^2 .
\end{split}\end{equation}e
Then, one has for $i\ge 2$
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \chi \sigma^{3/2} h'\right\|_0^2\le & \left\| \chi \sigma^{3/2} h'\right\|_0^2 + i(i-2) \left\| \chi \sigma^{1/2} \sigma' h \right\|_0^2
\le \left\| \chi \sigma^{1/2} \left(\sigma h' + i \sigma' h \right) \right\|_0^2 + C \left\| \sigma h \right\|_0^2 .
\end{split}\end{equation}e
Since the estimate on $\left\| \chi \sigma^{1/2} \sigma' h \right\|_0$ is missed, one has to
use Minkowski's inequality to find it again. That is,
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \chi \sigma^{1/2} \sigma' h \right\|_0^2 \le i^2 \left\| \chi \sigma^{1/2} \sigma' h \right\|_0^2 \le & \left(\left\| \chi \sigma^{1/2} \left(\sigma h' + i \sigma' h \right) \right\|_0 + \left\| \chi \sigma^{3/2} h'\right\|_0\right)^2\\
\le & 2\left(\left\| \chi \sigma^{1/2} \left(\sigma h' + i \sigma' h \right) \right\|_0^2 + \left\| \chi \sigma^{3/2} h'\right\|_0^2\right) \\
\le & 3 \left\| \chi \sigma^{1/2} \left(\sigma h' + i \sigma' h \right) \right\|_0^2 + C \left\| \sigma h \right\|_0^2,
\end{split}\end{equation}e
provided that $i\ge 1$. Therefore, we obtain
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \chi \sigma^{3/2} h'\right\|_0^2 + \left\| \chi \sigma^{1/2} \sigma' h \right\|_0^2 \le
4 \left\| \chi \sigma^{1/2} \left(\sigma h' + i \sigma' h \right) \right\|_0^2 + C \left\| \sigma h \right\|_0^2.
\end{split}\end{equation}e
\vskip 1cm
\noindent {\bf Verification of \eqref{ho3}. }
In view of \eqref{norm}, one has
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \left( \sigma^{3/2} \partial_t v'', \ \sigma^{3/2} \partial_t^3 v'
\right)(\cdot,t)\right\|_{1}^2
\le C E(t),
\end{split}\end{equation}e
which implies
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \left( \sigma^{3/2} \partial_t v'', \ \sigma^{3/2} \partial_t^3 v'
\right)(\cdot,t)\right\|_{L^\infty}^2 \le C{E(t)}.
\end{split}\end{equation}e
Using the embedding $W^{1,4/3}(\mathbb{R}) \subset W^{3/4,2}(\mathbb{R})$, one has
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma^{1/2} \partial_t^3 v(t) \right\|_{L^\infty} \le & C\left\|\sigma^{1/2} \partial_t^3 v \right\|_{3/4}=C\left\|\sigma^{1/2} \partial_t^3 v \right\|_{W^{3/4,2}}\le C \left\|\sigma^{1/2} \partial_t^3 v \right\|_{W^{1,4/3}}\\
\le & C \left\|\sigma^{1/2} \partial_t^3 v \right\|_{L^{4/3}} +C \left\|\left(\sigma^{1/2} \partial_t^3 v \right)'\right\|_{L^{4/3}} \le C \sqrt{E(t)},
\end{split}\end{equation}e
since
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\|\sigma^{1/2} \partial_t^3 v (t)\right\|_{L^{4/3}}\le \left\|\sigma^{1/2} \partial_t^3 v \right\|_{L^{2}}\|1\|_{L^4} \le C \left\| \partial_t^3 v \right\|_{0} \le C \sqrt{E(t)}
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\|\left(\sigma^{1/2} \partial_t^3 v \right)' (t) \right\|_{L^{4/3}}\le & \left\| \sigma^{1/2} \partial_t^3 v ' \right\|_{L^{4/3}} + C\left\| \sigma^{-1/2} \partial_t^3 v \right\|_{L^{4/3}}\\
\le & \left\| \sigma^{1/2} \partial_t^3 v ' \right\|_{0} + C\left\| \sigma^{-1/2} \right\|_{L^{5/3}}\left\| \partial_t^3 v \right\|_{L^{20/3}} \\
\le & \left\| \sigma^{1/2} \partial_t^3 v ' \right\|_{0} + C\left\| \partial_t^3 v \right\|_{1/2} \le C \sqrt{E(t)}.
\end{split}\end{equation}e
Here we have used the H$\ddot{o}$lder inequality and the fact $\|\cdot\|_{L^p} \le C \|\cdot\|_{1/2}$ for any $p\in(1,\infty)$.
Similarly,
$$ \left\| \sigma^{1/2} \partial_t v' \right\|_{L^\infty} \le C \sqrt{E(t)}.
$$
\vskip 1cm
\noindent {\bf Verification of \eqref{lb0}. }
One can obtain $\eqref{lb0}_1$ by using \eqref{egn}, \eqref{ho3} and $\eqref{lb1}_{1,2}$, since
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
& \left\| \sigma^{1/2} \mathfrak{J}_1 (t)\right\|_{L^\infty}\le C \|\partial_t v/x\|_{L^\infty} + \left\| \sigma^{1/2} \partial_t v' \right\|_{L^\infty} + \left\| \mathfrak{J}_0 \right\|_{L^\infty}^2 \le
P\left(\sqrt{E (t)}\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma^{3/2} \mathfrak{L}_{1}(t) \right\|_{L^\infty} \le & C\left(\left\|\sigma^{3/2} \partial_t v''\right\|_{L^\infty} + \left\| \sigma^{1/2} \left( \partial_t v ' - \partial_t v/x\right) \right\|_{L^\infty} +\left\| \partial_t v \right\|_{L^\infty}\right. \\
&\left. + \left\| \sigma \mathcal{R}_0\right\|_{L^\infty } \left\| \sigma^{1/2}\mathfrak{J}_{1}\right\|_{L^\infty} + \left\| \sigma {\mathfrak{L}}_0\right\|_{L^\infty} \left\|\mathfrak{J}_{0}\right\|_{L^\infty} \right) \\
\le & C P\left(\sqrt{E(t)}\right) +C t P\left( \sup_{[0,t]} \sqrt{E}\right).
\end{split}\end{equation}e
For $\eqref{lb0}_2$, it follows from \eqref{f0}, \eqref{tnorm}, \eqref{tjk} and $\|\cdot\|_{L^\infty}\le \|\cdot\|_1$ that
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \mathfrak{J}_0(t)\right\|_{L^\infty}^2\le C \left(\left\| { v} /x \right\|_{L^\infty}^2 + \left\| v'\right\|_{L^\infty}^2\right)\le C \left(\left\| { v} /x \right\|_{1}^2 + \left\| v\right\|_{2}^2\right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \mathfrak{J}_1(t)\right\|_{0}^2
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \mathfrak{J}_2(t)\right\|_{0}^2\le & C \left(\left\| \partial_t^2 { v} /x\right\|_{0}^2 + \left\| \partial_t^2 v'\right\|_{0}^2 +
\left\| \mathfrak{J}_0\right\|_{L^\infty}^2\left\| \mathfrak{J}_1\right\|_{0}^2\right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
and
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma \mathfrak{L}_{0} (t)\right\|_{L^\infty}^2 \le & C\left(\left\|\sigma v''\right\|_{L^\infty}^2 + \left\| v '-v/x \right\|_{L^\infty}^2 + \left\| \sigma \mathcal{R}_0\right\|_{L^\infty}^2 \left\|\mathfrak{J}_{0}\right\|_{L^\infty }^2 \right)\\
\le & C\left(\left\|\sigma v \right\|_{3}^2 + \left\| v \right\|_{2}^2+ \left\| v /x \right\|_{1}^2 + \left\| \sigma \mathcal{R}_0\right\|_{L^\infty}^2 \left\|\mathfrak{J}_{0}\right\|_{L^\infty }^2 \right)
\le M_0+ C t P\left(\sup_{[0,t]} E\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma \mathfrak{L}_{1}(t) \right\|_{0}^2
\le M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}e
\begin{equation}e\lambdabel{}\begin{equation}gin{split}
\left\| \sigma \mathfrak{L}_{2} (t)\right\|_{0}^2 \le & C\left(\left\|\sigma \partial_t^2 v''\right\|_{0} + \left\| \partial_t^2 v' - \partial_t^2 v/x \right\|_{0} + \left\| \sigma \mathcal{R}_0\right\|_{L^\infty } \left\| \mathfrak{J}_{2}\right\|_{0} + \left\| \sigma \mathfrak{L}_0\right\|_{L^\infty} \left\|\mathfrak{J}_{1}\right\|_{0}\right.\\
&\left. +
\left\| \sigma \mathfrak{L}_1\right\|_{0} \left\|\mathfrak{J}_{0}\right\|_{L^\infty } \right)^2
\le M_0 + CtP\left(\sup_{[0,t]}E\right),
\end{split}\end{equation}e
where we have used $\eqref{3lk}_{1,3}$.
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\bibitem{40} Zhang, P., Zhang, Z.: On the free boundary problem of three-dimensional incompressible
Euler equations. Commun. Pure Appl. Math. 61, 877-940 (2008)
\end{thebibliography}
\noindent Tao Luo\\
Dept. of Mathematics and Statistics,\\
Georgetown University,\\
Washington DC, USA.\\
[email protected]\\
\noindent Zhouping Xin\\
Institute of Mathematical Sciences,\\
Chinese University of Hong Kong,\\
Hong Kong, China. \\
[email protected]\\
\noindent Huihui Zeng\\
Mathematical Sciences Center,\\
Tsinghua University, \\
Beijing, China. \\
[email protected]
\end{document} |
\begin{equation*}gin{document}
\begin{equation*}gin{abstract}
We consider the space of elliptic hypergeometric functions of the
$sl_{2}$ type associated with elliptic curves with one marked point. This
space represents conformal blocks in the $sl_2$ WZW model of CFT. The
modular group acts on this space. We give formulas for the matrices of
the action in terms of values at roots of unity of Macdonald polynomials
of the $sl_2$ type.
\end{abstract}
\maketitle
\begin{equation*}gin{center}
{\it
$^\star$ Departement Mathematik, ETH-Zentrum, 8092 Z\"urich, Switzerland,
[email protected]
$^\diamond$ Department of Mathematics, University of North Carolina
at Chapel Hill,
Chapel Hill, NC 27599-3250, USA,
[email protected], [email protected]}
\end{center}
\centerline{February, 2002}
\centerline{\emph{Dedicated to V.~I.~Arnold on his 65th birthday}}
\section{Introduction} In the WZW model of conformal field theory associated
with a simple complex Lie algebra $\mathfrak{g}$,
one defines a holomorphic
vector bundle, the bundle of conformal blocks,
on the moduli space of smooth complex compact curves with
marked points labeled by representations of $\mathfrak{g}$. This vector bundle comes with a projectively flat connection,
see \cite{TUY}. For curves of genus zero and one, this
connection is flat and can be described in explicit classical terms.
Moreover, horizontal sections admit
integral representations
\cite{SV}, \cite{FV1}.
For genus zero curves, the connection
is the Knizhnik--Zamolodchikov (KZ) connection and the
equation for
horizontal sections (KZ equation) is a generalization of the
Gauss hypergeometric equation. The KZ equation reduces to the Gauss
hypergeometric equation in a special case.
The solutions to the KZ equations are given by integrals
of certain differential forms over
$m$-dimensional cycles
where $m$ depends on $\mathfrak g$ and on the representations
labeling marked points. The classical
integral representation of the hypergeometric function
is recovered in the case of four points on the
Riemann sphere when $m=1$.
The horizontal sections (conformal blocks)
are holomorphic functions on the
universal covering of the configuration space of points
on the complex plane and integral representation may be
used to compute the action of covering transformations
on conformal blocks. In this way representations of the
pure braid group are obtained as monodromy representations.
We consider here the case of genus one curves with one
marked point (elliptic curves)
and the Lie algebra $sl_2$. The equation for
horizontal sections is the
Kni\-zhnik--Za\-mo\-lo\-dchi\-kov--Ber\-nard (KZB) heat
equation \eqref{KZB}: it is essentially the heat equation
associated to the Hamilton operator of a particle
in a Weierstrass function potential. Again, solutions are
given by generalizations of hypergeometric integrals,
which are appropriately called elliptic hypergeometric
integrals. Of particular interest is a finite dimensional
subspace of the space of solutions, the space of conformal
blocks of the WZW model. It may be characterized by
symmetry and holomorphy conditions, see Sect.~2. This
subspace is invariant under the (projective)
action of the modular group
$\mathrm{SL}(2,\mathbb{Z})$ of covering transformations
of the upper half plane, viewed as the universal cover of
the moduli space of elliptic curves.
We compute the projective
action of the modular group on this space
and relate the matrix elements of the $S$-transformation
$\tau\mapsto-1/\tau$ to values of $A_1$-Macdonald polynomials. This
implies a
special case of Kirillov's theorem, stating that the
representation of the modular group on certain conformal blocks
on elliptic curves
for $sl_N$ is equivalent to a representation where the
matrix elements of $S$ are given in terms of
$A_{N-1}$-Macdonald polynomials.
One new feature of our computation is that it gives
a formula for the matrix elements of the
projective representation of the
modular group (not just its conjugacy class)
with respect to an explicit basis of
elliptic hypergeometric integral solutions of the KZB equation
\eqref{KZB}.
Moreover, the elliptic hypergeometric integrals give
naturally a recursive procedure, developed in \cite{FS}
in a similar situation,
to compute the matrix elements of $S$ by repeated application
of the Stokes theorem. This recursive
procedure shows the role of the {\em shift operator} \cite{AI}
in this context: it is identical to the recursive construction
of Macdonald polynomials out of Schur functions
by repeated application of the shift operator.
We also discuss the relation of the matrix elements of $S$ with
the traces of intertwining operators of the quantum group
$U_q(sl_2)$ at root of unity.
Finally, let us point out that the KZB connection is unitary
with respect to a hermitian form which can also
be given by integrals of elliptic hypergeometric type.
They are discussed in \cite{FG}.
The authors thank P. Etingof and A. Kirillov, Jr., for useful discussions
and the referee for useful suggestions.
\section{Conformal Blocks on the Torus}
Let $\kappa$ and $p$ be non-negative integers satisfying $\kappa\geq2p+2$. Let
$q=e^{\frac{\pi i}{\kappa}}$. Denote
\begin{equation*}gin{gather*}
[n]_{q}=[n]=\frac{q^n-q^{-n}}{q-q^{-1}},\qquad
[n]!=[1][2]\cdots[n], \qquad
\begin{equation*}gin{bmatrix}n\\j\end{bmatrix}=\frac{[n]!}{[j]![n-j]!},\\
(n,q)_{j}=[n][n+1]\cdots[n+j-1].
\end{gather*}
Let $\tau\in\mathbb{C}$ be such that $\operatorname{Im}\tau>0$.
The KZB-heat equation is the partial differential equation
\begin{equation*}gin{equation}\lambdabel{KZB}
2\pi i\kappa{\partial u\over\partial\tau}(\lambda,\tau)={\partial^2
u\over\partial\lambda^{2}}(\lambda,\tau)+p(p+1)\rho\,'(\lambda,\tau)u(\lambda,\tau).
\end{equation}
Here, the prime denotes the derivative with respect to the first
argument, and $\rho$ is defined in terms of
the first Jacobi theta function \cite{WW},
\begin{equation*}gin{equation*}
\vartheta_{1}(t,\tau)=-\sum_{j\in\mathbb{Z}}
e^{\pi i(j+\frac{1}{2})^2\tau+2\pi i(j+\frac{1}{2})(t+\frac{1}{2})},\qquad
\rho(t,\tau)=\frac{\vartheta_{1}'(t,\tau)}{\vartheta_{1}(t,\tau)}.
\end{equation*}
Holomorphic solutions of the KZB-heat equation with the properties,
\begin{equation*}gin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}\renewcommand{\lambdabelenumi}{(\theenumi)}
\item $u(\lambda+2,\tau)=u(\lambda,\tau)$,
\item $u(\lambda+2\tau,\tau)=e^{-2\pi i\kappa(\lambda+\tau)}\,u(\lambda,\tau)$,
\item $u(-\lambda,\tau)=(-1)^{p+1}\,u(\lambda,\tau)$,
\item $u(\lambda,\tau)=\mathcal{O}((\lambda-m-n\tau)^{p+1})$ as $\lambda\to
m+n\tau$ for any $m,n\in\mathbb{Z}$,
\end{enumerate}
are called conformal blocks associated with the family of elliptic curves
$\mathbb{C}/\mathbb{Z}+\tau\mathbb{Z}$ with the marked point $z=0$ and the
irreducible $sl_{2}$ representation of dimension $2p+1$. It is known
that the space of conformal blocks has dimension $\kappa-2p-1$.
{\bf Remark.}
The relation of this definition of conformal blocks
with the more standard one of \cite{TUY} is the following.
Recall the definition of \cite{TUY} in this case:
integrable irreducible representations $V_{k,m}$
of the affine Kac--Moody Lie algebra $\widehat{sl}_2$
are labeled by two non-negative integers: $k$, the level and
$m\in\{0,\dots,k\}$, the highest weight. To each such pair $(k,m)$
and a curve $E_\tau=\mathbb{C}/\mathbb{Z}+\tau\mathbb{Z}$ one associates
a space of conformal blocks: it
is the space of linear functions on $V_{k,m}$ invariant under a
natural action
the Lie algebra $R(\tau)$ of $sl_2$-valued functions on $E_\tau$
with poles at $0$. As one varies $\tau$ in the upper half plane,
the spaces of conformal blocks form a vector bundle with a flat
connection. The description of this bundle and connection in
terms of theta functions was explained in \cite{FW}:
the family of Lie algebras
$R(\tau)$ is extended to a two parameter family $R(\lambdambda,\tau)$
which also acts on $V_{k,m}$. The $R(\lambdambda,\tau)$-invariant linear
forms on $V_{k,m}$ form
a vector bundle of twisted conformal blocks with a flat connection.
The space of horizontal sections is then isomorphic to the space
of twisted conformal blocks at any $(\lambdambda,\tau)$, in particular
if $\lambdambda=0$, where (untwisted) conformal blocks are recovered.
We have an injective restriction map from the space of horizontal sections
of the bundle of twisted conformal blocks
to the space of holomorphic functions of $\lambdambda,\tau$ with
values in the the dual of the zero weight space $V_m[0]\subset V_{k,m}$ of the
$2m+1$-dimensional irreducible representation $V_m$ of $sl_2$. If
$m$ is odd, $V_m[0]=0$. If $m=2p$ is even, then $V_m[0]=\mathbb{C}$ and the
image of the restriction map
consists of functions of the form $v(\lambdambda,\tau)=
\vartheta_1(\lambdambda,\tau)u(\lambdambda,\tau)$, where $u$ is a solution
of the KZB-heat equation obeying (i)-(iv) with $\kappappa=k+2$.
Conformal blocks may also be defined in other ways which also
lead to the same description in terms of theta functions:
in \cite{FG} this is done for conformal blocks defined
as states of the Chern--Simons--Witten theory and
in \cite{EFK} for the definition
as spherical functions on the Kac--Moody group.
Introduce two transformations
\begin{equation*}gin{equation}\lambdabel{defnsofst}
Tu(\lambda,\tau)=u(\lambda,\tau+1),\qquad
Su(\lambda,\tau)=e^{-\pi i \kappa{\lambda^{2}\over2\tau}}\tau^{-\frac{1}{2}-{p(p+1)\over\kappa}}\,u\left({\lambda\over\tau},-{1\over\tau}\right),
\end{equation}
where we fix $\arg\tau\in(0,\pi)$.
\begin{equation*}gin{thm}\lambdabel{kzbsols}\cite{EK}
If $u(\lambda,\tau)$ is a solution of the KZB-heat equation, then
$Tu(\lambda,\tau)$ and $Su(\lambda,\tau)$
are solutions too. Moreover, the transformations $T$ and $S$ preserve
the properties (i)-(iv).
\end{thm}
The proof of Theorem \ref{kzbsols} is by direct verification.
Restricted to the space of
conformal blocks, the transformations $T$ and $S$ satisfy the relations
\begin{equation*}gin{equation*}
S^{2}=(-1)^{p}\,iq^{-p(p+1)}I,\qquad (ST)^{3}=(-1)^{p}\,iq^{-p(p+1)}I,
\end{equation*}
where $I$ is the identity transformation.
The modular group is the group generated by two elements $T$ and $S$
with relations
\begin{equation*}gin{equation*}
S^{2}=1,\qquad (ST)^{3}=1.
\end{equation*}
The modular group is naturally isomorphic to the quotient group
$\mathrm{SL}(2,\mathbb{Z})/\{\pm I\}$.
The formulas \operatorname{Re}f{defnsofst} define a
projective representation of the modular group in the space of
conformal blocks.
The isomorphism class of the projective representation of the modular
group in the space of conformal blocks is described in \cite{K} using
the Kazhdan-Lusztig-Finkelberg isomorphism between the modular tensor
categories arising from affine Lie algebras
and the modular tensor categories arising from quantum groups at roots of unity.
In this paper, we describe the action of the modular group on the
space of conformal blocks using integral representations of solutions
of the KZB-heat equation.
\section{Integral Representations of Solutions of the KZB-Heat Equation}
Solutions of the KZB-heat equation can be realized as elliptic hypergeometric
integrals depending on parameters.
Introduce special functions
\begin{equation*}gin{equation*}
\sigma_{\lambda}(t,\tau)=
\frac{\vartheta_{1}(\lambda-t,\tau)\vartheta_{1}'(0,\tau)}
{\vartheta_{1}(\lambda,\tau)\vartheta_{1}(t,\tau)},
\qquad
E(t,\tau)=\frac{\vartheta_{1}(t,\tau)}{\vartheta_{1}'(0,\tau)}.
\end{equation*}
They have the properties
\begin{equation*}gin{gather}\lambdabel{tp1}
E(t+1,\tau)=-E(t,\tau),\qquad E(t+\tau,\tau)=-e^{-\pi i\tau-2\pi i t}E(t,\tau),\\
\lambdabel{tp2}
\sigma_{\lambda}(t+1,\tau)=\sigma_{\lambda}(t,\tau),
\qquad
\sigma_{\lambda}(t+\tau,\tau)=e^{2\pi i\lambda}\sigma_{\lambda}(t,\tau).
\end{gather}
The modular transformation properties of the functions $\sigma_{\lambda}$
and $E$ are calculated using the modular transformation properties of
the function $\vartheta_{1}$,
\begin{equation*}gin{gather}\lambdabel{thetatrans}
\vartheta_{1}(t,\tau+1)=e^{\frac{\pi i}{4}}\vartheta_{1}(t,\tau),\qquad
\vartheta_{1}\left(\frac{t}{\tau},-\frac{1}{\tau}\right)=\frac{\sqrt{-i\tau}}{i}e^{\pi i\frac{t^2}{\tau}}\vartheta_{1}(t,\tau),
\end{gather}
where $\sqrt{-i\tau}$ is to be interpreted by the convention $|\arg(-i\tau)|<\pi/2$.
We have
\begin{equation*}gin{gather*}
\sigma_{\lambda}(t,\tau+1)=\sigma_{\lambda}(t,\tau),\qquad
\sigma_{\frac{\lambda}{\tau}}\left(\frac{t}{\tau},-\frac{1}{\tau}\right)=\tau
e^{-\frac{2\pi it\lambda}{\tau}}\sigma_{\lambda}(t,\tau),\\
E(t,\tau+1)=E(t,\tau),\qquad
E\left({t\over\tau},-{1\over\tau}\right)=\tau^{-1}e^{\pi i
{t^{2}\over\tau}}E(t,\tau).
\end{gather*}
Let $\Phi_{p,\kappa}$ be the multi-valued function
\begin{equation*}gin{equation*}
\Phi_{p,\kappa}=\Phi_{\kappa}(t_{1},\dots,t_{p},\tau)=
\prod_{j=1}^{p}E(t_{j},\tau)^{-{2p\over\kappa}}\,
\prod_{1\leq i<j\leq p}E(t_i - t_j ,\tau)^{2\over\kappa}.
\end{equation*}
For $\kappa\geq2$ and $n\in\mathbb{Z}$, let $\theta_{\kappa,n}$ be the
theta function of level $\kappa$
\begin{equation*}gin{equation*}
\theta_{\kappa,n}(t,\tau)=
\sum_{j\in\mathbb{Z}}e^{2\pi i\kappa(j+\frac{n}{2\kappa})^{2}\tau+2\pi
i\kappa(j+\frac{n}{2\kappa})t}.
\end{equation*}
We have
\begin{equation*}gin{gather}\lambdabel{reflp}
\theta_{\kappa,n+2\kappa}(t,\tau)=\theta_{\kappa,n}(t,\tau),\qquad
\theta_{\kappa,n}(-t,\tau)=\theta_{\kappa,-n}(t,\tau),\\\lambdabel{tp3}
\theta_{\kappa,n}\left(t + {2\over\kappa},\tau\right)=q^{2n}\,\theta_{\kappa,n}(t,\tau),\qquad
\theta_{\kappa,n}\left(t + {2\tau\over\kappa},\tau\right)=e^{-2\pi it-{2\pi
i\tau\over\kappa}}\,\theta_{\kappa,n+2}(t,\tau).
\end{gather}
The modular transformation properties of $\theta_{\kappa,n}$ are given by
\begin{equation*}gin{gather*}
\theta_{\kappa,n}(t,\tau+1)=q^{\frac{n^{2}}{2}}\theta_{\kappa,n}(t,\tau),\\
\theta_{\kappa,n}\left({t\over\tau},-{1\over\tau}\right)=\sqrt{-{i\tau\over2\kappa}}e^{\pi
i\kappa{t^{2}\over2\tau}}\,\sum_{m=0}^{2\kappa-1}q^{-mn}\,\theta_{\kappa,m}(t,\tau).
\end{gather*}
Let $\Delta_{k}\subset\mathbb{R}^{k}\subset\mathbb{C}^{k}$ be the simplex
\begin{equation*}gin{equation*}
\Delta_{k}=\{(t_{1},t_{2},\dots ,t_{k})\in\mathbb{R}^{k}\subset\mathbb{C}^{k}\,|\,0\leq t_{k}\leq
t_{k-1}\leq\cdots\leq t_{1}\leq1\}.
\end{equation*}
Let
$\widetilde{\Delta}_{k}$ be the image of ${\Delta_{k}}$ under the map $(t_{1},\dots,t_{k})\mapsto(\tau
t_{1},\dots,\tau t_{k})$.
For $0\leq k\leq p$,
define
\begin{equation*}gin{equation*}
J_{\kappa,n}^{[k]}(\lambda,\tau)=\int\Phi_{\kappa}(t_{1},\dots,t_{p},\tau)\theta_{\kappa,n}\left(\lambda+{2\over\kappa}\sum_{j=1}^{p}t_{j},\tau\right)\,\prod_{j=1}^{p}\sigma_{\lambda}(t_{j},\tau)dt_{j}\,,
\end{equation*}
where the integral is over
\begin{equation*}gin{equation*}
\{(t_{1},t_{2},\dots,t_{p})\in\mathbb{C}^{p}\,|\,(t_{1},\dots,t_{k})\in\Delta_{k},\,(t_{k+1},\dots,t_{p})\in\widetilde{\Delta}_{p-k}\}.
\end{equation*}
The integral is a meromorphic function of the exponents of
$\Phi_{p,\kappa}$. It is well-defined when all of the exponents in the function
$\Phi_{p,\kappa}$ have positive real parts. We are interested in the
case when the exponents in the function $\Phi_{p,\kappa}$ are negative
real numbers. In this case, the integral is understood as an analytic
continuation from the region where the exponents have positive real parts.
The branch of $\Phi_{p,\kappa}$ is determined by fixing the arguments of all
factors of $\Phi_{p,\kappa}$ for the case $\tau=i$ and deforming
continuously for arbitrary values of $\tau$. For $\tau=i$, we fix $\arg E(t_{j},\tau)=0$
for $j=1,\dots,k$, $\arg E(t_{j},\tau)=\frac{\pi}{2}$ for
$j=k+1,\dots,p$, and $\arg E(t_{i}-t_{j},\tau)\in(-\pi,\pi)$ for $1\leq
i<j\leq p$.
Introduce
\begin{equation*}gin{equation*}
u_{n}^{[k]}(\lambda,\tau)=u^{[k]}_{\kappa,n}(\lambda,\tau)=J_{\kappa,n}^{[k]}(\lambda,\tau)+(-1)^{p+1}
J_{\kappa,n}^{[k]}(-\lambda,\tau).
\end{equation*}
\begin{equation*}gin{lemma}\lambdabel{l1}
The integrals $u_{n}^{[k]}$ have the properties
\begin{equation*}gin{equation*}
u_{n}^{[k]}=u_{n+2\kappa}^{[k]},\qquad
u^{[k]}_{n}=-q^{2k(n+p-k)}u^{[k]}_{-n-2(p-k)}.
\end{equation*}
\end{lemma}
\begin{equation*}gin{proof}
The first equation is an immediate consequence of the $2\kappa$
periodicity of the functions $\theta_{\kappa,n}$. To derive the second
equation, we consider the integrals $u_{n}^{[k]}$ after the change of variables
$t_{j}\mapsto-t_{j}+1$ for $1\leq j\leq k$, $t_{j}\mapsto-t_{j}+\tau$
for $k+1\leq j\leq p$. The result follows from the formulas
\operatorname{Re}f{tp1}, \operatorname{Re}f{tp2}, \operatorname{Re}f{reflp}, and \operatorname{Re}f{tp3}.
\end{proof}
\begin{equation*}gin{thm}\cite{FV1}
For $0\leq k\leq p$ and any $n$, the integrals $u_{\kappa,n}^{[k]}(\lambda,\tau)$ are solutions of the KZB-heat
equation having the properties (i)-(iv).
\end{thm}
\begin{equation*}gin{thm} The set
\begin{equation*}gin{equation*}
\{u_{n}^{[p]}(\lambda,\tau)\,\vert\, p+1\leq n\leq \kappa -p-1\}
\end{equation*}
is a basis for the space of conformal blocks.
\end{thm}
\begin{equation*}gin{proof}
We prove that for $n\in\{p+1,\dots,\kappa-p-1\}$, the integrals $u_{n}^{[p]}$
are linearly independent over $\mathbb{C}$.
In Theorem \ref{zerothm}, we show that
for all other values of $n$ in the interval $0\leq n\leq \kappa$, the
integrals $u_{n}^{[p]}$ are identically zero. In the limit as $\tau\to i\infty$, the leading term of
$u_{n}^{[p]}$ is of the form
\begin{equation*}gin{equation*}
\frac{A_{p}e^{\frac{\pi i n^{2}}{2\kappa}\tau}}{(\sin(\pi\lambda))^{p}}
\left(\left(\prod_{j=1}^{p}(q^{2(n+j)}-1)\right)
B_{p}\left(\frac{n+1}{\kappa},-\frac{2p}{\kappa},\frac{1}{\kappa}\right)
\sin(\pi\lambda(n+p))+C_{n}(e^{\pi i\lambda},e^{-\pi i\lambda})\right),
\end{equation*}
where $C_{n}(e^{\pi i\lambda},e^{-\pi i\lambda})$ is a Laurent
polynomial of degree $< n+p$ in $e^{\pi i\lambda}$. Here $A_{p}$ is a non-zero constant depending only on $p$,
and $B_{p}(\alpha,\begin{equation*}ta,\gamma)$ is the Selberg integral,
\begin{equation*}gin{equation*}
B_{p}(\alpha,\begin{equation*}ta,\gamma)=
{1\over p!}
\prod_{j=0}^{p-1}
\frac{\Gamma(1+\gamma+j\gamma)
\Gamma(\alpha+j\gamma)
\Gamma(\begin{equation*}ta+j\gamma)}
{\Gamma(1+\gamma)
\Gamma(\alpha+\begin{equation*}ta+(p+j-1)\gamma)}.
\end{equation*}
It
follows that the $u_{n}^{[p]}$ are linearly independent provided that the
coefficients $\prod_{j=1}^{p}(q^{2(n+j)}-1)$ and
$B_{p}\left((n+1)/\kappa,-2p/\kappa,1/\kappa\right)$ are nonzero functions for each $n$ in the range
$p+1\leq n\leq\kappa-p-1$. It is straightforward to check that the product
$\prod_{j=1}^{p}(q^{2(n+j)}-1)$ is never zero for $n$ in this
interval. The Selberg integral
$B_{p}\left((n+1)/\kappa,-2p/\kappa, 1/\kappa\right)$ is
also nonzero for $n$ in this interval since
$1+1/\kappa\notin\mathbb{Z} _{\leq 0}$, and
$(n-p+j)/\kappa\notin\mathbb{Z} _{\leq0}$ for any $j$ satisfying $0\leq j\leq p-1$.
\end{proof}
We use the modular transformation properties of the functions $E$,
$\sigma_{\lambdambda}$ and $\theta_{\kappa,n}$ to
obtain the following two lemmas.
\begin{equation*}gin{lemma}\lambdabel{ttrans}
For any $n$, the action of the operator $T$ on $u_{n}^{[p]}$ is given
by the formula
\begin{equation*}gin{equation*}
Tu_{n}^{[p]}(\lambda,\tau)=q^{\frac{n^{2}}{2}}u_{n}^{[p]}(\lambda,\tau).\qed
\end{equation*}
\end{lemma}
\begin{equation*}gin{lemma}\lambdabel{strans} For any $n$, we have
\begin{equation*}gin{equation*}
Su_{n}^{[p]}(\lambda,\tau)=\frac{e^{-\frac{\pi i}{4}}}{\sqrt{2\kappa}}\sum_{m=0}^{2\kappa-1}q^{-mn}u^{[0]}_{m}(\lambda,\tau). \qed
\end{equation*}
\end{lemma}
In Lemma \ref{strans}, we have written $Su_{n}^{[p]}$ as a linear
combination of $u_{m}^{[0]}$.
Our goal is to express $Su_{n}^{[p]}$ in terms of the basis $
\{u_{m}^{[p]}(\lambda,\tau)\,\vert\, p+1\leq m\leq \kappa -p-1\}$.
This is accomplished using the Stokes theorem. Repeated applications
of the Stokes theorem give us a recursive procedure for expressing
the integrals $u_{m}^{[0]}$
as linear combinations of the integrals $u_{m}^{[p]}$.
\begin{equation*}gin{lemma}\lambdabel{st}For $0\leq k\leq p-1$ and any $n$, we have
\begin{equation*}gin{equation*}
[p-k](q^{n+p-k}-q^{-n-p+k})u^{[k]}_{ n}=q^{-n-k-1}[k+1]
\left(q^{-2(k+1)}u^{[k+1]}_{n+2}-u^{[k+1]}_{n}\right).
\end{equation*}
\end{lemma}
\begin{equation*}gin{proof}
Recall that the
$u_{n}^{[k]}$ are defined as analytic continuations of integrals where
all of the exponents in the function $\Phi_{p,\kappa}$ have positive real
parts. So the identity in the lemma relates objects which are
understood as analytic continuations.
To prove the identity, we begin by considering simpler objects.
Namely, we consider the case when all of the exponents in
$\Phi_{p,\kappa}$ have positive real parts. In this case, we can apply
the Stokes theorem which gives an identity for integrals with positive
exponents. Then, we analytically continue all terms of the identity
to get the statement of the lemma. More precisely,
let $A\subset\mathbb{C}$ be the parallelogram with vertices at
$0,\,1,\,\tau,\,1+\tau$. Consider the $(p+1)$-dimensional cell
\begin{equation*}gin{equation*}
B_{k}=\{(t_{1},t_{2},\dots,t_{p})\in\mathbb{C}^{p}\,|\,(t_{1},\dots,t_{k})\in\Delta_{k},\,t_{k+1}\in
A,\,(t_{k+2},\dots,t_{p})\in\widetilde{\Delta}_{p-k-1}\}.
\end{equation*}
Applying the Stokes theorem to $B_{k}$ gives
\begin{equation*}gin{multline}\lambdabel{stidentity}
\int_{\delta
B_{k}}\Phi_{\kappa}(t_{1},\dots,t_{p},\tau)\theta_{\kappa,n}\left(\lambda+{2\over\kappa}\sum_{j=1}^{p}t_{j},\tau\right)\prod_{j=1}^{p}\sigma_{\lambda}(t_{j},\tau)dt_{j}\\
+(-1)^{p+1}\int_{\delta
B_{k}}\Phi_{\kappa}(t_{1},\dots,t_{p},\tau)\theta_{\kappa,n}\left(-\lambda+{2\over\kappa}\sum_{j=1}^{p}t_{j},\tau\right)\prod_{j=1}^{p}\sigma_{-\lambda}(t_{j},\tau)dt_{j}=0.
\end{multline}
The boundary of $B_{k}$ consists of $2p+2$
components of dimension $p$. When all of the exponents in
the function $\Phi_{p,\kappa}$ have positive real parts, the restrictions
of the integrand to all but four of the boundary
components are zero. Those four components are
\begin{equation*}gin{align*}
&\gamma^{1}_{k}=\{(t_{1},t_{2},\dots,t_{p})\in\mathbb{C}^{p}\,|\,(t_{1},\dots,t_{k})\in\Delta_{k},\,t_{k+1}\in
[0,1],\,(t_{k+2},\dots,t_{p})\in\widetilde{\Delta}_{p-k-1}\},\\
&\gamma^{2}_{k}=\{(t_{1},t_{2},\dots,t_{p})\in\mathbb{C}^{p}\,|\,(t_{1},\dots,t_{k})\in\Delta_{k},\,t_{k+1}\in
[1,1+\tau],\,(t_{k+2},\dots,t_{p})\in\widetilde{\Delta}_{p-k-1}\},\\
&\gamma^{3}_{k}=\{(t_{1},t_{2},\dots,t_{p})\in\mathbb{C}^{p}\,|\,(t_{1},\dots,t_{k})\in\Delta_{k},\,t_{k+1}\in
[\tau,1+\tau],\,(t_{k+2},\dots,t_{p})\in\widetilde{\Delta}_{p-k-1}\},\\
&\gamma^{4}_{k}=\{(t_{1},t_{2},\dots,t_{p})\in\mathbb{C}^{p}\,|\,(t_{1},\dots,t_{k})\in\Delta_{k},\,t_{k+1}\in
[0,\tau],\,(t_{k+2},\dots,t_{p})\in\widetilde{\Delta}_{p-k-1}\},
\end{align*}
where $[a,b]$ denotes the straight line segment connecting $a$ and
$b$. Thus, the Stokes theorem applied to $B_{k}$ gives us that the sum
of the integrals over $\gamma_{1}^{k}$, $\gamma_{2}^{k}$,
$\gamma_{3}^{k}$, and $\gamma_{4}^{k}$ is equal to zero.
We obtain the result of the lemma as follows. Let $B_{k}$ be defined
as above. Take the analytic continuation of all terms in the identity
\operatorname{Re}f{stidentity}. Let $\gamma^{*}_{k}$ be any component of
$\delta B_{k}\backslash\cup_{i=1}^{4}\gamma^{i}_{k}$. In the case
when the exponents in $\Phi_{p,\kappa}$ had positive real parts, we had
that the integral over $\gamma^{*}_{k}$ was zero. Thus we must have
that the analytic continuation of the integral over $\gamma^{*}_{k}$
is also zero. So we have that the sum of the integrals (understood
as analytic continuations) over $\gamma_{1}^{k}$, $\gamma_{2}^{k}$,
$\gamma_{3}^{k}$, and $\gamma_{4}^{k}$ is equal to zero.
The integrals over the boundary components $\gamma_{1}^{k}$ and
$\gamma_{4}^{k}$ are $c^{k}_{1}u_{n}^{[k+1]}$ and
$c^{k}_{4}u_{n}^{[k]}$, respectively, where $c_{1}^{k},c_{4}^{k}\in\mathbb{C}$. To recognize the integral over
$\gamma_{2}^{k}$ as $c_{2}^{k}u_{n}^{[k]}$, where $c_{2}^{k}\in\mathbb{C}$, we make the change of
variables $t_{k+1}\to t_{k+1}+1$. To recognize the
integral over
$\gamma_{3}^{k}$ as $c_{3}^{k}u_{n+2}^{[k+1]}$, where $c_{3}^{k}\in\mathbb{C}$, we make the change of
variables $t_{k+1}\to t_{k+1}+\tau$.
We calculate the constants
$c_{j}^{k}$ using the formulas \operatorname{Re}f{tp1}, \operatorname{Re}f{tp2}, and \operatorname{Re}f{tp3}.
\end{proof}
\begin{equation*}gin{thm} \lambdabel{zerothm} For $0\leq k\leq p$ and
$n\in\{-p,-p+1,\dots,-p+2k\}\cup\{\kappa-p,\kappa-p+1,\dots,\kappa-p+2k\}$,
we have
\begin{equation*}gin{equation*}
u_{n}^{[k]}=0.
\end{equation*}
\end{thm}
\begin{equation*}gin{proof}
We prove the statement by induction on $k$. Setting $k=0$ in Lemma \ref{l1} gives
us the identity
\begin{equation*}gin{equation*}
u_{n}^{[0]}=-u_{-n-2p}^{[0]}.
\end{equation*}
Hence, using the $2\kappa$ periodicity of the $u_{n}^{[k]}$, for
$n\equiv-p\mod\kappa$, we have
\begin{equation*}gin{equation*}
u_{n}^{[0]}=0.
\end{equation*}
Now assume the theorem is true for some
$k$ in the interval from $0$ to $p-1$. We must show that $u_{n}^{[k+1]}=0$ for
$n\in\{-p,-p+1,\dots,-p+2k+2\}\cup\{\kappa-p,\kappa-p+1,\dots,\kappa-p+2k+2\}$.
We use the reflection identity of Lemma \ref{l1},
\begin{equation*}gin{gather}\notag
u^{[k+1]}_{n}=-q^{2(k+1)(n+p-k-1)}u^{[k+1]}_{-n-2(p-k-1)},
\intertext{and the relation of Lemma \ref{st},}
\lambdabel{st2}
[p-k](q^{n+p-k}-q^{-n-p+k})u^{[k]}_{n}=q^{-n-k-1}[k+1]
\left(q^{-2(k+1)}u^{[k+1]}_{n+2}-u^{[k+1]}_{n}\right).
\end{gather}
By the induction hypothesis, the left hand side of
equation \operatorname{Re}f{st2} is zero for
$n\in\{-p,-p+1,\dots,-p+2k\}\cup\{\kappa-p,\kappa-p+1,\dots,\kappa-p+2k\}$.
Thus, for
$n\in\{-p,-p+1,\dots,-p+2k\}\cup\{\kappa-p,\kappa-p+1,\dots,\kappa-p+2k\}$, we
have the recursion relations
\begin{equation*}gin{equation}\lambdabel{rr}
u_{n}^{[k+1]}=q^{-2(k+1)}u_{n+2}^{[k+1]}.
\end{equation}
We use the identities
\begin{equation*}gin{equation*}
u^{[k+1]}_{-p+k+1}=u^{[k+1]}_{\kappa-p+k+1}=0, \quad u^{[k+1]}_{-p+k}=u^{[k+1]}_{\kappa-p+k}=0,
\end{equation*}
to obtain, recursively on the subscripts, the result. The first identity is an
immediate consequence of the reflection identity, and the second
identity follows from comparing the recursion relations \operatorname{Re}f{rr} for $n=-p+k$ and
$n=\kappa-p+k$, respectively, with the reflection identities
\begin{equation*}gin{equation*}
u^{[k+1]}_{-p+k}=-q^{-2(k+1)}u^{[k+1]}_{-p+k+2}, \qquad u^{[k+1]}_{\kappa-p+k}=-q^{-2(k+1)}u^{[p]}_{\kappa-p+k+2}.
\end{equation*}
\end{proof}
\begin{equation*}gin{proposition}\lambdabel{5c0} For $0\leq k\leq p$ and any $n$,
\begin{equation*}gin{equation*}
Su_{n}^{[p]}(\lambda,\tau)=\frac{e^{-\frac{\pi i}{4}}}{\sqrt{2\kappa}}\left(\sum_{m=-p+2k+1}^{\kappa-p-1}f^{(k)}_{m,n}u_{m}^{[k]}(\lambdambda,\tau)+\sum_{m=\kappa-p+2k+1}^{2\kappa-p-1}f^{(k)}_{m,n}u_{m}^{[k]}(\lambdambda,\tau)\right),
\end{equation*}
where
\begin{equation*}gin{equation*}
f_{m,n}^{(k)}=\frac{q^{-m(n+k)-\frac{k(k+1)}{2}}}{(q^{-1}-q)^{k}}\begin{equation*}gin{bmatrix}p\\k\end{bmatrix}^{-1}
\sum_{j=0}^{k}\begin{equation*}gin{bmatrix}k\\j\end{bmatrix}\frac{q^{2jn}}{(-m-p+k+1,q)_{j}\,(m+p-k+1,q)_{k-j}}.
\end{equation*}
\end{proposition}
\begin{equation*}gin{proof}
The proof is by induction on $k$. The result from Lemma
\ref{strans} (where we have used the periodicity of the $u_{n}^{[k]}$
to shift the interval of summation) combined with the identity
$u_{-p}^{[0]}=u_{\kappa-p}^{[0]}=0$ of Theorem \ref{zerothm} give us
\begin{equation*}gin{equation*}
e^{\frac{\pi i}{4}}\sqrt{2\kappa}\,Su_{n}^{[p]}(\lambda,\tau)=S'u_{n}^{[p]}=\sum_{m=-p+1}^{\kappa-p-1}q^{-mn}u^{[0]}_{m}(\lambda,\tau)+
\sum_{m=\kappa-p+1}^{2\kappa-p-1}q^{-mn}u^{[0]}_{m}(\lambda,\tau).
\end{equation*}
But this is exactly the result for $k=0$ since
\begin{equation*}gin{equation*}
f^{(0)}_{m,n}=q^{-mn}.
\end{equation*}
Assuming the result for some $k$ satisfying $0\leq k\leq p-1$, we have
\begin{equation*}gin{equation}\lambdabel{ih}
S'u_{n}^{[p]}(\lambda,\tau)=\sum_{m=-p+2k+1}^{\kappa-p-1}f^{(k)}_{m,n}u_{m}^{[k]}(\lambdambda,\tau)+\sum_{m=\kappa-p+2k+1}^{2\kappa-p-1}f^{(k)}_{m,n}u_{m}^{[k]}(\lambdambda,\tau).
\end{equation}
Rewriting \operatorname{Re}f{ih} using the identity of Lemma \ref{st},
\begin{equation*}gin{equation*}
u^{[k]}_{m}=\frac{q^{-m-k-1}}{(q^{m+p-k}-q^{-m-p+k})}\frac{[k+1]}{[p-k]}
\left(q^{-2(k+1)}u^{[k+1]}_{m+2}-u^{[k+1]}_{m}\right),
\end{equation*}
which makes sense for all $m$ in the interval of summation since
those values of $m$ never give zero denominators, we obtain
\begin{equation*}gin{multline}\lambdabel{ih2}
S'u_{n}^{[p]}(\lambda,\tau)=
\frac{q^{-k-1}}{(q^{-1}-q)}\frac{[k+1]}{[p-k]}\,\times\\
\biggl(\,
\sum_{m=-p+2k+1}^{\kappa-p-1}
\frac{q^{-m}f^{(k)}_{m,n}}{[m+p-k]}u_{m}^{[k+1]}(\lambdambda,\tau)
-
\sum_{m=-p+2k+3}^{\kappa-p+1}
\frac{q^{-m-2k}f^{(k)}_{m-2,n}}{[m+p-k-2]}u_{m}^{[k+1]}(\lambdambda,\tau)\\
+
\sum_{m=\kappa-p+2k+1}^{2\kappa-p-1}
\frac{q^{-m}f^{(k)}_{m,n}}{[m+p-k]}u_{m}^{[k+1]}(\lambdambda,\tau)
-
\sum_{m=\kappa-p+2k+3}^{2\kappa-p+1}
\frac{q^{-m-2k}f^{(k)}_{m-2,n}}{[m+p-k-2]}u_{m}^{[k+1]}(\lambdambda,\tau)
\biggl)
.
\end{multline}
By Theorem \ref{zerothm}, the integrals $u_{m}^{[k+1]}$ are identically
zero for $m\in\{-p+2k+1,-p+2k+2,\kappa-p,\kappa-p+1,\kappa-p+2k+1,\kappa-p+2k+2,2\kappa-p,2\kappa-p+1\}$.
Thus, we may combine the first two terms on the right hand side of
\operatorname{Re}f{ih2} into one sum over $m$ in the range $-p+2k+3\leq m \leq \kappa-p-1$.
Similarly, the last two terms give us a sum over $m$ in the range $\kappa-p+2k+3
\leq m \leq 2\kappa-p-1$. In each of these sums, the coefficient of
$u_{m}^{[k+1]}$ is given by the expression
\begin{equation*}gin{multline}\lambdabel{coeff}
\frac{q^{-m(n+k+1)-\frac{(k+1)(k+2)}{2}}}{(q^{-1}-q)^{k+1}}
\begin{equation*}gin{bmatrix}p\\k+1\end{bmatrix}^{-1}\biggl(
\sum_{j=0}^{k}\begin{equation*}gin{bmatrix}k\\j\end{bmatrix}
\frac{q^{2jn}}{(-m-p+k+1,q)_{j}\,(m+p-k,q)_{k-j+1}}\\
+
\sum_{j=1}^{k+1}\begin{equation*}gin{bmatrix}k\\j-1\end{bmatrix}
\frac{q^{2jn}}{(-m-p+k+2,q)_{j}\,(m+p-k-1,q)_{k-j+1}}\biggl).
\end{multline}
To complete the proof of the theorem, we must show that the expression \operatorname{Re}f{coeff}
is equal to the expression
\begin{equation*}gin{equation*}
\frac{q^{-m(n+k+1)-\frac{(k+1)(k+2)}{2}}}{(q^{-1}-q)^{k+1}}\begin{equation*}gin{bmatrix}p\\k+1\end{bmatrix}^{-1}
\sum_{j=0}^{k+1}\begin{equation*}gin{bmatrix}k+1\\j\end{bmatrix}\frac{q^{2jn}}{(-m-p+k+2,q)_{j}\,(m+p-k,q)_{k-j+1}}.
\end{equation*}
This is proved by direct calculation.
\end{proof}
\begin{equation*}gin{corollary}\lambdabel{5c1}
For $0\leq k\leq p-1$, $m\in\{-p+2k+1,\dots,\kappa-p-1\}\cup\{\kappa-p+2k+1,\dots,2\kappa-p-1\}$, and
any $n$, we have
\begin{equation*}gin{equation*}
f_{m,n}^{(k+1)}=\frac{q^{-m-k-1}}{q-q^{-1}}\,\frac{[k+1]}{[p-k]}\left(\frac{q^{-2k}f^{(k)}_{ m-2,n}}{[m-2+p-k]}-\frac{f^{(k)}_{ m,n}}{[m+p-k]}\right).
\end{equation*}
\end{corollary}
\begin{equation*}gin{lemma}\lambdabel{l2}
The functions $f^{(k)}_{m,n}$ have the property
\begin{equation*}gin{equation*}
f^{(k)}_{m,n}=q^{-2k(m+p-k)+2pn}f^{(k)}_{-m-2p+2k,-n}.\qed
\end{equation*}
\end{lemma}
We rewrite the sum in Proposition \ref{5c0} using Lemmas \ref{l1} and \ref{l2}.
\begin{equation*}gin{corollary}
For $0\leq k\leq p$ and any $n$,
\begin{equation*}gin{equation*}
Su^{[p]}_{
n}(\lambda,\tau)=\frac{e^{-\frac{\pi i}{4}}}{\sqrt{2\kappa}}\sum_{m=-p+2k+1}^{\kappa-p-1}
\left(f_{m,n}^{(k)}-q^{2pn}f_{m,-n}^{(k)}\right)u_{
m}^{[k]}(\lambda,\tau).
\end{equation*}
\end{corollary}
\section{Macdonald Polynomials and the Shift Operator}
The Macdonald polynomials of type $A_{1}$ are $x$-even polynomials
in terms of $q^{mx}$, where $m\in\mathbb{Z}$. They depend on two
parameters $k$ and $n$, where $k$ and $n$ are non-negative
integers. They are
defined by the conditions:
\begin{equation*}gin{enumerate}
\item
$P_{n}^{(k)}(x)=q^{nx}+q^{-nx}+$ lower order terms, except for $P_{0}^{(k)}(x)=1$,
\item
$\lambdangle P_{m}^{(k)},P_{n}^{(k)}\rangle=0$ for $m\neq n$, where
\begin{equation*}gin{equation*}
\lambdangle f,g\rangle= \,\frac{1}{2}\text{Const Term}\,\left(fg\prod_{j=0}^{k-1}(1-q^{2(j+x)})(1-q^{2(j-x)})\right).
\end{equation*}
\end{enumerate}
\begin{equation*}gin{example} For $n>0$, we have
\begin{equation*}gin{equation*}
P_{n}^{(0)}(x)=q^{nx}+q^{-nx}.
\end{equation*}
\end{example}
The shift operator $D$ is an operator acting on functions $f(x)$ by
\begin{equation*}gin{equation*}
Df(x)=\frac{f(x-1)-f(x+1)}{q^{x}-q^{-x}}.
\end{equation*}
The name is ``shift operator'' because its action on the basic
($q$-difference) hypergeometric functions results in a shift of the
parameters \cite{AW,Ch}.
\begin{equation*}gin{thm}\cite{AI}\lambdabel{so}
For $n\geq 1$ and $k\geq 0$, we have
\begin{equation*}gin{equation*}
DP_{n}^{(k)}(x)=(q^{-n}-q^{n})\,P_{n-1}^{(k+1)}(x).
\end{equation*}
\end{thm}
\begin{equation*}gin{remark}
It follows from Theorem \ref{so} that all of the Macdonald polynomials
can be calculated from the polynomials $P^{(0)}_{n}$ using the shift
operator provided that $q^{-n}-q^{n}\neq0$.
\end{remark}
\begin{equation*}gin{example} For $n>0$, we have
\begin{equation*}gin{equation*}
P^{(1)}_{n-1}(x)=\frac{q^{nx}-q^{-nx}}{q^x-q^{-x}}.
\end{equation*}
\end{example}
\section{Identification of the $f_{m,n}^{(k)}$ with values of the
Macdonald polynomials}
\begin{equation*}gin{thm}\lambdabel{mp=f} For $0\leq k\leq p$, $k+1\leq n \leq \kappa$, and
$m\in\{-p+2k+1,\dots,\kappa-p-1\}$, we have
\begin{equation*}gin{multline*}
f_{m,n}^{(k)}-q^{2pn}f_{m,-n}^{(k)}= q^{pn-km-\frac{k(k+1)}{2}}\begin{equation*}gin{bmatrix}p\\k\end{bmatrix}^{-1}(q^{-m-p+k}-q^{m+p-k})\;\times\\
\left(\prod_{j=1}^{k}(q^{-n+j}-q^{n-j})\right)P^{(k+1)}_{n-k-1}(m+p-k).
\end{multline*}
\end{thm}
\begin{equation*}gin{proof}
We prove the statement by induction on $k$. The case $k=0$ follows directly
from the formulas
\begin{equation*}gin{equation*}
f_{m,n}^{(0)}=q^{-mn},
\qquad
P^{(1)}_{n-1}(m+p)=\frac{q^{n(m+p)}-q^{-n(m+p)}}{q^{m+p}-q^{-m-p}}.
\end{equation*}
We assume that the theorem is true for some $k$ in the range $0\leq k\leq p-1$.
To prove the theorem, we must verify the identity
\begin{equation*}gin{multline}\lambdabel{i1}
f_{m,n}^{(k+1)}-q^{2pn}f_{m,-n}^{(k+1)}=
q^{pn-(k+1)m-\frac{(k+1)(k+2)}{2}}
\begin{equation*}gin{bmatrix}p\\k+1\end{bmatrix}^{-1}
(q^{-m-p+k+1}-q^{m+p-k-1})\,\times\\
\left(\prod_{j=1}^{k+1}(q^{-n+j}-q^{n-j})\right)P^{(k+2)}_{n-k-2}(m+p-k-1)
\end{multline}
for $k+2\leq n\leq \kappa$ and
$m\in\{-p+2k+3,\dots,\kappa-p-1\}$.
By Corollary \ref{5c1}, the left hand side of the equation \operatorname{Re}f{i1} equals
\begin{equation*}gin{equation*}
\frac{q^{-m-k-1}}{(q-q^{-1})}\frac{[k+1]}{[p-k]}\left(q^{-2k}\,\frac{f_{m-2,n}^{(k)}-q^{2pn}f_{m-2,-n}^{(k)}}{[m+p-k-2]}-\frac{f_{m,n}^{(k)}-q^{2pn}f_{m,-n}^{(k)}}{[m+p-k]}\right).
\end{equation*}
By Theorem \ref{so}, the right hand side of the equation \operatorname{Re}f{i1} equals
\begin{equation*}gin{multline*}
q^{pn-(k+1)m-\frac{(k+1)(k+2)}{2}}\begin{equation*}gin{bmatrix}p\\k+1\end{bmatrix}^{-1}
\left(\prod_{j=1}^{k}(q^{-n+j}-q^{n-j})\right)\,\times\\
\left(P^{(k+1)}_{n-k-1}(m+p-k)-P^{(k+1)}_{n-k-1}(m+p-k-2)\right).
\end{multline*}
Thus, the equation \operatorname{Re}f{i1} is equivalent to the equation
\begin{equation*}gin{multline*}
q^{-2k}\,\frac{f_{m-2,n}^{(k)}-q^{2pn}f_{m-2,-n}^{(k)}}{q^{m+p-k-2}-q^{-m-p+k+2}}+\frac{f_{m,n}^{(k)}-q^{2pn}f_{m,-n}^{(k)}}{q^{-m-p+k}-q^{m+p-k}}
=
q^{pn-km-\frac{k(k+1)}{2}}\begin{equation*}gin{bmatrix}p\\k\end{bmatrix}^{-1}\,\times\\
\left(\prod_{j=1}^{k}(q^{-n+j}-q^{n-j})\right)\left(P^{(k+1)}_{n-k-1}(m+p-k)-P^{(k+1)}_{n-k-1}(m+p-k-2)\right),
\end{multline*}
which follows from the induction hypothesis.
\end{proof}
\begin{equation*}gin{corollary}\lambdabel{smf}
For $0\leq k\leq p$ and $k+1\leq n \leq \kappa$,
\begin{equation*}gin{multline*}
Su_{ n}^{[p]}(\lambda,\tau)=\frac{e^{\frac{-\pi i}{4}}}{\sqrt{2\kappa}}\sum_{m=-p+2k+1}^{\kappa-p-1}q^{pn-km-\frac{k(k+1)}{2}}\begin{equation*}gin{bmatrix}p\\k\end{bmatrix}^{-1}(q^{-m-p+k}-q^{m+p-k})\;\times\\
\left(\prod_{j=1}^{k}(q^{-n+j}-q^{n-j})\right)P^{(k+1)}_{n-k-1}(m+p-k)u_{ m}^{[k]}(\lambda,\tau).
\end{multline*}
\end{corollary}
The left hand side of the equation in Corollary \ref{smf} is equal to zero if
$n=k+1,...,p$ or $n=\kappappa-p, \kappappa -p + 1,...,\kappappa - k - 1$.
This gives $2(p-k)$ relations for the $\kappappa - 2k-1$ possibly nonzero functions
$u^{[k]}_m$, $m=k+1,...,p-k-1$.
\begin{equation*}gin{lemma}
The $2(p-k)$ relations between the functions $u^{[k]}_m$,
$m=k+1,...,p-k-1$, given by Corollary \ref{smf} are linearly
independent and thus generate all linear relations between those functions.
\end{lemma}
\begin{equation*}gin{proof}
The coefficients of the above relations form $2(p-k)$ columns of the
$S$-matrix corresponding to $\kappappa'=\kappappa$ and $p'=k$.
That $S$-matrix is a non-degenerate matrix whose square is a root of
unity. Thus any set of its columns is linearly independent.
\end{proof}
\section{Representation of the Modular Group}
Let $T=(t_{m,n})$ and $S=(s_{m,n})$ be
the matrices of the transformations $T$ and $S$, respectively, with
respect to the basis
\begin{equation*}gin{equation*}
\{u_{\kappa,n}^{[p]}(\lambda,\tau)\,\vert\, p+1\leq n\leq \kappa -p-1\}.
\end{equation*}
Here the matrices $T$ and $S$ are defined by
\begin{equation*}gin{gather*}
Tu_{\kappa,n}^{[p]}=\sum_{m=p+1}^{\kappa-p-1}t_{m,n}u_{\kappa,m}^{[p]},\\
Su_{\kappa,n}^{[p]}=\sum_{m=p+1}^{\kappa-p-1}s_{m,n}u_{\kappa,m}^{[p]}.
\end{gather*}
\begin{equation*}gin{thm}
For $p+1\leq m,n\leq \kappa-p-1$, we have
\begin{equation*}gin{gather*}
t_{m,n}=q^{\frac{n^2}{2}}\delta_{mn},\\
s_{m,n}=
\frac{e^{-\frac{\pi i}{4}}}{\sqrt{2\kappa}}q^{p(n-m)-\frac{p(p+1)}{2}}(q^{-m}-q^{m})\left(\prod_{j=1}^{p}(q^{-n+j}-q^{n-j})\right)P^{(p+1)}_{n-p-1}(m),
\end{gather*}
where $\delta_{mn}=1$ for $m=n$ and $0$ otherwise.
\end{thm}
The theorem follows directly from Lemma \ref{ttrans} and Theorem \ref{smf}.
\begin{equation*}gin{example} Let $\kappappa= 2p+2$. In this case, the only element of the S-matrix
is
\begin{equation*}gin{equation*}
s_{p+1,\,p+1} = \frac{(-i)^{p+1}}{\sqrt{p+1}}e^{-\pi i\frac{p+1}{4}}\prod_{j=1}^{p}(q^{j}+q^{-j}).
\end{equation*}
On the other hand, according to \cite{FSV},
\begin{equation*}gin{equation*}
u^{[p]}_{2p+2,p+1} =a_{p}
\vartheta_{1}(\lambda,\tau)^{p+1},
\end{equation*}
where $a_{p}$ is a constant,
and hence,
\begin{equation*}gin{equation*}
Su^{[p]}_{2p+2,p+1} = a_{p}(-i)^{p+1}e^{-\pi i\frac{p+1}{4}}\theta (\lambdambda,\tau)^{p+1}.
\end{equation*}
This, in particular, implies
\begin{equation*}gin{equation*}
\prod_{j=1}^p(q^j+q^{-j})=\sqrt{p+1}.
\end{equation*}
This formula and its relations to the classical Gauss sums can be
found in \cite{Ch}.
\end{example}
The projective representation of the modular
group in the space of conformal blocks described in \cite{K}
is given by the matrices $\tilde{T}=(\tilde{t}_{m,n})$
and $\tilde{S}=(\tilde{s}_{m,n})$, $p+1\leq m,n\leq \kappa-p-1$,
where
\begin{equation*}gin{gather*}
\tilde{t}_{m,n}=e^{-\frac{\pi i}{4}}q^{\frac{n^2}{2}}\delta_{mn},\\
\tilde{s}_{m,n}=
\frac{i}{\sqrt{2\kappa}}q^{-\frac{p(p+1)}{2}}\left(\prod_{j=0}^{p}(q^{-m+j}-q^{m-j})\right)P^{(p+1)}_{n-p-1}(m).
\end{gather*}
\begin{equation*}gin{proposition}
Let $D$ be the diagonal matrix such that
\begin{equation*}gin{equation*}
d_{j}=q^{pj}\,\prod_{l=1}^{p}(q^{-j+l}-q^{j-l}),\qquad p+1\leq j\leq \kappa-p-1.
\end{equation*}
Then
\begin{equation*}gin{equation*}
T=e^{\frac{\pi i}{4}}D^{-1}\tilde{T}D,\qquad S=e^{-\frac{3\pi i}{4}}D^{-1}\tilde{S}D.\qed
\end{equation*}
\end{proposition}
{\bf Remark.} For the standard definition of conformal blocks
$v(\lambdambda,\tau)=\vartheta_1(\lambdambda,\tau)^{-1}u(\lambdambda,\tau)$
(see the remark in Sect.~2), the $S$ and
$T$ transformations are defined as
\[
\hat Tv(\lambdambda,\tau)=v(\lambdambda,\tau+1),\qquad
\hat Sv(\lambdambda,\tau)=e^{-\pi i (\kappappa-2){\lambdambda^{2}\over2\tau}}\tau^{-
\frac{p(p+1)}{\kappappa}}\,v\left({\lambdambda\over\tau},-{1\over\tau}\right).
\]
Using the transformation rules \operatorname{Re}f{thetatrans} of
$\vartheta_1$ we see that
the corresponding matrices $\hat T$, $\hat S$ are related to $T$, $S$
by
\[
\hat T=e^{-\frac{\pi i}{4}}T,\qquad \hat S=e^{\frac{3\pi i}{4}}S.
\]
Thus $\hat T$, $\hat S$ are conjugated to $\tilde T$, $\tilde S$.
\section{Trace Functions for $U_{q}(sl_{2})$}
Let $q=e^{\frac{\pi i}{\kappa}}$. The quantum group $U_{q}(sl_{2})$ has
generators $E,F,q^{ch}$, where $c\in\mathbb{C}$, with relations
\begin{equation*}gin{gather*}
q^{ch}q^{c'h}=q^{(c+c')h},\,\quad
q^{ch}Eq^{-ch}=q^{2c}E,\,\quad q^{ch}Fq^{-ch}=q^{-2c}F,\,\quad
EF-FE=\frac{q^{h}-q^{-h}}{q-q^{-1}},
\intertext{and comultiplication defined by}
\Delta(E)=E\otimes q^{h}+1\otimes
E,\qquad\Delta(F)=F\otimes1+q^{-h}\otimes F,\qquad
\Delta(q^{ch})=q^{ch}\otimes q^{ch}.
\end{gather*}
Identify weights for
$U_{q}(sl_{2})$ with complex numbers as follows. Say that a
vector $v$ in a $U_{q}(sl_{2})$-module has weight $\nu\in\mathbb{C}$ if
$q^{h}v=q^{\nu}v$. Let $M_{\mu}$ be the Verma module over $U_{q}(sl_{2})$ with highest
weight $\mu$, and let $v_{\mu}$ be its highest weight vector. Let $k$
be a non-negative integer such that $\kappa\geq2k+2$. Let $U$ be
the irreducible finite dimensional representation of $U_{q}(sl_{2})$
of weight $2k$. Let $U[0]$ denote the zero weight subspace of $U$.
Let $u\in U[0]$. For generic $\mu$, let
$\varPhi_{\mu}^{u}:M_{\mu}\to M_{\mu}\otimes U$ be the intertwining
operator defined by
\begin{equation*}gin{equation*}
\varPhi_{\mu}^{u}v_{\mu}=v_{\mu}\otimes u +\frac{Fv_{\mu}}{[-\mu]}\otimes
Eu+\cdots+\frac{F^{j}v_{\mu}}{[j]!(-\mu,q)_{j}}\otimes
E^{j}u+\cdots.
\end{equation*}
Introduce an $\operatorname{End}(U[0])$-valued function $\psi^{(k)}(q,\nu,\mu)$ defined by
\begin{equation*}gin{equation*}
\psi^{(k)}(q,\nu,\mu)u=\operatorname{Tr}|_{M_{\mu}}(\varPhi_{\mu}^{u}q^{\nu h}).
\end{equation*}
Since $U[0]$ is one-dimensional, this function is a scalar function.
\begin{equation*}gin{thm}\cite{EV} The function $\psi^{(k)}(q,\nu,\mu)$ is given by the formula
\begin{equation*}gin{equation*}
\psi^{(k)}(q,\nu,\mu)=q^{\nu\mu}\sum_{j=0}^{k}(-1)^{j}q^{\frac{j(j-3)}{2}}(q-q^{-1})^{-j-1}\frac{[k+j]!}{[j]![k-j]!}\frac{q^{-j\mu-(j-1)\nu}}{\prod_{l=0}^{j-1}[\mu-l]\prod_{l=0}^{j}[\nu-l]}.
\end{equation*}
\end{thm}
Introduce renormalized trace functions $\Psi^{(k)}(q,\nu,\mu)$
defined by
\begin{equation*}gin{equation*}
\Psi^{(k)}(q,\nu,\mu)=\prod_{j=1}^{k}\left(\frac{q^{\mu+1-j}-q^{-\mu-1+j}}{q^{\nu+j}-q^{-\nu-j}}\right)\psi^{(k)}(q,\nu,\mu).
\end{equation*}
The function $\Psi^{(k)}$ is a holomorphic function of $\mu$.
\section{Identification of the $f_{m,n}^{(k)}$ with values
of the
trace functions $\Psi^{(k)}$ }
\begin{equation*}gin{thm}\lambdabel{tf}
For $0\leq k\leq p$, $m\in\{-p+2k+1,\dots,\kappa-p-1\}\cup\{\kappa-p+2k+1,\dots,2\kappa-p-1\}$, and
any $n$,
\begin{equation*}gin{equation*}
f_{m,n}^{(k)}=q^{pn-km-k(k+1)}\,(q^{m+p-k}-q^{-m-p+k})\begin{equation*}gin{bmatrix}p\\k\end{bmatrix}^{-1}\Psi^{(k)}(q^{-1},-m-p+k,-n-1).
\end{equation*}
\end{thm}
\begin{equation*}gin{proof}
The statement of the theorem is equivalent to the identity
\begin{equation*}gin{multline}\lambdabel{f=psi}
(q-q^{-1})^{k}\sum_{j=0}^{k}\begin{equation*}gin{bmatrix}k\\j\end{bmatrix}(-m-p+k+j+1,q)_{k-j}(m+p-j+1,q)_{j}q^{2jn}\\
=q^{-\frac{k(k+1)}{2}+kn}\sum_{j=0}^{k}q^{-\frac{j(j-1)}{2}-j(m+p-k+n)}(q-q^{-1})^{j}\frac{[k+j]!}{[j]![k-j]!}\,\times\\
\prod_{l=j+1}^{k}(q^{m+p-k+l}-q^{-m-p+k-l})(q^{n+l}-q^{-n-l}).
\end{multline}
Let $x=q^{n}$. Using the
$q$-binomial theorem, we rewrite the identity \operatorname{Re}f{f=psi} as
\begin{equation*}gin{multline}\lambdabel{f=psi2}
(q-q^{-1})^{k}\sum_{j=0}^{k}\begin{equation*}gin{bmatrix}k\\j\end{bmatrix}(-m-p+k+j+1,q)_{k-j}(m+p-j+1,q)_{j}x^{2j-k}\\
=q^{-k(k+1)}\sum_{j=0}^{k}q^{-j(m+p-k-1)}(q-q^{-1})^{j}\frac{[k+j]!}{[j]![k-j]!}\,\times\\
\prod_{l=j+1}^{k}(q^{m+p-k+l}-q^{-m-p+k-l})\sum_{i=0}^{k-j}(-1)^{k-j-i}q^{i(k+j+1)}\begin{equation*}gin{bmatrix}k-j\\i\end{bmatrix}x^{2i-k}.
\end{multline}
For a fixed value of $j$ in the range $0\leq j\leq k$, the coefficient
of $x^{2j-k}$ on the left hand side of \operatorname{Re}f{f=psi2} is
\begin{equation*}gin{equation*}
(q-q^{-1})^{k}\begin{equation*}gin{bmatrix}k\\j\end{bmatrix}(-m-p+k+j+1,q)_{k-j}(m+p-j+1,q)_{j}.
\end{equation*}
The coefficient
of $x^{2j-k}$ on the right hand side of \operatorname{Re}f{f=psi2} is
\begin{equation*}gin{multline*}
q^{-(k+1)(k-j)}(q-q^{-1})^{k}\begin{equation*}gin{bmatrix}k\\j\end{bmatrix}(m+p-j+1,q)_{j}(-m-p+j,q)_{k-j}
\,\times\\\sum_{i=0}^{k-j}q^{-i(m+p-k-j-1)}\frac{(k+1,q)_{i}(j-k,q)_{i}}{[i]!(m+p-k+1,q)_{i}}.
\end{multline*}
We observe that the sum
\begin{equation*}gin{equation*}
\sum_{i=0}^{k-j}q^{-i(m+p-k-j-1)}\frac{(k+1,q)_{i}(j-k,q)_{i}}{[i]!(m+p-k+1,q)_{i}}
\end{equation*}
is the hypergeometric series
\begin{equation*}gin{equation}\lambdabel{hgs}
_{2}\phi_{1}\biggl(\begin{equation*}gin{array}{c}q^{-2(k+1)},\,q^{-2(j-k)}\\q^{-2(m+p-k+1)}\end{array};\,q^{-2},\,q^{-2(m+p-k-j)}\biggl).
\end{equation}
The series \operatorname{Re}f{hgs} is equal to
\begin{equation*}gin{equation*}
q^{(k-j)(k+1)}\frac{(m+p-2k,q)_{k+1}}{(m+p-k-j,q)_{k+1}}.
\end{equation*}
(For a proof of this formula, see \cite{GR}).
Hence the coefficients of $x^{2j-k}$ on both sides of \operatorname{Re}f{f=psi2} are
equal. This proves the theorem.
\end{proof}
Under the identification in Theorem \ref{tf}, Corollary
\ref{5c1} takes the following form.
\begin{equation*}gin{corollary}\lambdabel{so2} For $0\leq k\leq p-1$, $m\in\{-p+2k+1,\dots,\kappa-p-1\}\cup\{\kappa-p+2k+1,\dots,2\kappa-p-1\}$,
and any $n$,
\begin{equation*}gin{equation*}
q^{-k-1}\Psi^{(k+1)}(q^{-1},m,n+k)=
\frac{\Psi^{(k)}(q^{-1},m-1,n+k)-\Psi^{(k)}(q^{-1},m+1,n+k)}{q^m - q^{-m}}.
\end{equation*}
\end{corollary}
\begin{equation*}gin{remark} The right hand side of the equation in Corollary
\ref{so2} is the shift operator applied to
$\Psi^{(k)}(q^{-1},m,n+k)$. Thus, for $m$ as above, $k\geq0$, and
any $n$, we have
\begin{equation*}gin{equation*}
D\Psi^{(k)}(q^{-1},m,n)=q^{-k-1}\,\Psi^{(k+1)}(q^{-1},m,n).
\end{equation*}
\end{remark}
Together with Theorem \ref{mp=f}, the identification in Theorem \ref{tf}
gives us a formula for the Macdonald polynomials evaluated at roots of unity in terms of
values of the renormalized trace functions.
\begin{equation*}gin{corollary}\lambdabel{es}
For $\kappa\geq 2k+2$, $k+1\leq n\leq \kappa$, and
$m\in\{-p+2k+1,\dots,\kappa-p-1\}$, we have
\begin{equation*}gin{multline*}
\Psi^{(k)}(q^{-1},-m-p+k,n-1)-\Psi^{(k)}(q^{-1},-m-p+k,-n-1)\\=
P^{(k+1)}_{n-k-1}(m+p-k)\prod_{j=1}^{k}(q^{-n+2j}-q^{n}).
\end{multline*}
\end{corollary}
\begin{equation*}gin{remark}
For a generalization of Corollary \ref{es}, see \cite{ES}.
\end{remark}
Using the identification in Theorem \ref{tf}, we have the following expression for the $S$ transformation in terms of
the renormalized trace functions.
\begin{equation*}gin{corollary}
For $0\leq k\leq p$ and any $n$, we have
\begin{equation*}gin{multline*}
Su_{ n}^{[p]}(\lambda,\tau)=\frac{e^{-\frac{\pi i}{4}}}{\sqrt{2\kappa}}\sum_{m=-p+2k+1}^{\kappa-p-1}q^{pn-km-k(k+1)}\begin{equation*}gin{bmatrix}p\\k\end{bmatrix}^{-1}(q^{-m-p+k}-q^{m+p-k})\;\times\\
\left(\Psi^{(k)}(q^{-1},-m-p+k,n-1)-\Psi^{(k)}(q^{-1},-m-p+k,-n-1)\right)u_{ m}^{[k]}(\lambda,\tau)\,.
\end{multline*}
\end{corollary}
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\end{document} |
\begin{document}
\begin{abstract}
We develop a semiclassical second microlocal calculus of pseudodifferential operators associated to linear coisotropic submanifolds $\mathcal{C}\subset T^* \mathbb{T}^n$, where $\mathbb{T}^n = \mathbb{R}^n / \mathbb{Z}^n$. First microlocalization is localization in phase space $T^* \mathbb{T}^n$; second microlocalization is finer localization near a submanifold of $T^* \mathbb{T}^n$.
Our second microlocal operators test distributions on $\mathbb{T}^n$ (e.g., Laplace eigenfunctions) for a coisotropic wavefront set, a second microlocal measure of absence of \emph{coisotropic regularity}. This wavefront set tells us where, in the coisotropic, and in what directions, approaching the coisotropic, a distribution lacks coisotropic regularity.
We prove propagation theorems for coisotropic wavefront that are analogous to H\"{o}rmander's theorem for pseudodifferential operators of real principal type. Furthermore, we study the propagation of coisotropic regularity for quasimodes of semiclassical pseudodifferential operators. We Taylor expand the relevant Hamiltonian vector field, partially in the characteristic variables, at the spherical normal bundle of the coisotropic. Provided the principal symbol is real valued and depends only on the fiber variables in the cotangent bundle, and the subprincipal symbol vanishes, we show that coisotropic wavefront is invariant under the first two terms of this expansion.
\end{abstract}
\title{Semiclassical second microlocalization at linear coisotropic submanifolds in the torus}
\author{Rohan Kadakia}
\date{\today}
\maketitle
2010 \emph{Mathematics Subject Classification}: 35S05, 35A18, 35A21.
\section{Introduction}
A submanifold $\mathcal{C}$ of the symplectic manifold $(T^* X,\omega)$
is said to be \emph{coisotropic} if $(T\mathcal{C})^\omega \subset T\mathcal{C}$; in words, $\mathcal{C}$ is coisotropic if the symplectic orthocomplement to its tangent bundle is a subbundle of the tangent bundle itself.
Most distributions encountered while studying PDE are not $O(h^\infty)$. ($O(h^\infty)$ is the semiclassical analogue of $C^\infty$ smoothness.) Some of these distributions, however, possess a different type of regularity---they are \emph{coisotropic distributions}, associated to a coisotropic submanifold $\mathcal{C}$. We say that $u = u_h\in L^2(X)$ (uniformly as $h\downarrow 0$) is coisotropic if for all $k$ and all semiclassical pseudodifferential operators (PsDO) $A_1,\ldots,A_k\in\Psi_h(X)$ with $\sigma_\mathrm{pr}(A_j)\restriction_\mathcal{C}\equiv 0$, we have
\[
h^{-k} A_1\ldots A_k u\in L^2(X).
\]
Here, $\sigma_\mathrm{pr}(A)$ is the semiclassical principal symbol of $A$. We may, by restricting the microsupports of the characteristic operators $A_j$, refine coisotropic regularity to be local on $\mathcal{C}$.
The central idea in this paper is that of second microlocalization at a linear coisotropic submanifold $\mathcal{C}$ of $T^*\mathbb{T}^n$, where $\mathbb{T}^n = \mathbb{R}^n / \mathbb{Z}^n$. To (first) microlocalize is to localize in symplectic phase space, i.e., to simultaneously localize in position and momentum (up to the uncertainty principle). Second microlocalization is more refined localization near a submanifold. The first step is to \emph{blow up} $\mathcal{C}$, which technically means that we replace $\mathcal{C}$ with its spherical normal bundle $SN(\mathcal{C})$ (cf.\ \cite{Me-1}, \cite[Chapter~5]{Me-3}).
Next, for our second microlocal pseudodifferential calculus $\Psi_{2,h}(\mathcal{C})$, the collection of symbols consists of functions that are smooth on the blown up space. In particular, this includes functions singular on the original (i.e., blown down) space with conormal singularities, resolved in the blowup. Thus, $\Psi_{2,h}(\mathcal{C})$ contains the ordinary pseudodifferential calculus $\Psi_h(\mathbb{T}^n)$ as a subalgebra\footnote{Technically, as we see later, only PsDO with compactly supported symbols may be regarded as elements of $\Psi_{2,h}(\mathcal{C})$.}.
Associated to the calculus $\Psi_{2,h}(\mathcal{C})$ is a wavefront set ${}^2\mathrm{WF}$ in $SN(\mathcal{C})$, whose absence (together with absence of standard semiclassical wavefront $\mathrm{WF}_h$ in $T^*\mathbb{T}^n\backslash\mathcal{C}$) is equivalent to $u$ being a coisotropic distribution. A helpful analogy is that presence of second wavefront set in $SN(\mathcal{C})$ is to failure of coisotropic regularity in $\mathcal{C}$ as presence of homogeneous wavefront in the cotangent bundle is to singular support in the base. ${}^2\mathrm{WF}$ tells us where, in the coisotropic, \emph{and} in what directions, approaching the coisotropic, a distribution lacks coisotropic regularity; for instance, see Example \ref{ex:second wavefront}.
Whether in the homogeneous or semiclassical setting, several instances of second microlocalization exist in the literature: A.\ Vasy and J.\ Wunsch in the special case of Lagrangian submanifolds \cite{VW}; N.\ Anantharaman, C.\ Fermanian, and F.\ Maci\`{a} in \cite{AM-1,AM-2,AFM} to study \emph{defect measures}; and J-M.\ Bony's \cite{Bon} second microlocalization at conic Lagrangians. To study resonances, J.\ Sj\"{o}strand and M.\ Zworski \cite{SZ} construct a second microlocal calculus for hypersurfaces. Additional sources are \cite{BoLe,Del,CFK,KaKa,KaLa,Lau,Leb}.
Every coisotropic submanifold is endowed with a \emph{characteristic foliation}. The leaves of the characteristic foliation of $\mathcal{C}$ are the integral curves of the Hamiltonian vector fields of its defining functions. All of our results pertain specifically to linear coisotropic submanifolds. As coordinates on $(T^* \mathbb{T}^n,\omega)$, we take $(x,\xi)$, where $\omega = d\xi\wedge dx$. Then, a $d$-codimensional linear coisotropic is of the form $\mathcal{C} = \mathbb{T}^n \times \{\mathbf{v}_1\cdot\xi = \ldots = \mathbf{v}_d\cdot\xi = 0\}$ for $\mathbf{v}_j\in\mathbb{R}^n$.
\subsection{Propagation of coisotropic second wavefront set}
First, we show by commutator methods the analogue of H\"{o}rmander's real principal type theorem \cite{Ho-2}:
\begin{theorem} \label{primary propagation simple}
Let $P \in \Psi_{2,h}(\mathcal{C})$ and suppose that $P$ has real valued second principal symbol ${}^2 \sigma_\mathrm{pr}(P)$. Assume also that the distribution $u$ satisfies $Pu = f$. Then ${}^2 \mathrm{WF}(u) \backslash {}^2 \mathrm{WF}(f)$ is invariant under Hamiltonian flow at $SN(\mathcal{C})$.
\end{theorem}
For the next result, we consider an ordinary semiclassical PsDO $P\in\Psi_h(\mathbb{T}^n)$, regarded as an element of the second microlocal calculus. Further, we require that the principal symbol of $P$ depends only on the fiber variables $\xi$, and that its subprincipal symbol vanishes. We calculate the Hamiltonian vector field $H$ for ${}^2 \sigma_\mathrm{pr}(P)$. Then we Taylor expand $H$ at $SN(\mathcal{C})$: if $SN(\mathcal{C}) = \{\rho=0\}$, then $H = H_1 + \rho H_2 + O(\rho^2)$.
\begin{theorem} \label{secondary propagation simple}
Assume $P\in\Psi_h(\mathbb{T}^n)$ has real principal symbol depending only on the fiber variables in the cotangent bundle, that its subprincipal symbol vanishes, and that $Pu = O_{L^2(\mathbb{T}^n)}(h^\infty)$. Then ${}^2 \mathrm{WF}(u)\cap SN(\mathcal{C})$ is invariant under both $H_1$ and $H_2$.
\end{theorem}
\noindent This crucially hinges on $P$ being an ordinary semiclassical PsDO, so having a total symbol that is smooth even on the blown down space $T^* \mathbb{T}^n$.
The author is supported in part by NSF RTG grant 1045119. He would like to thank Jared Wunsch for introducing him to the problems addressed in this paper, and for helpful discussions throughout. The author is grateful also to Dean Baskin and Alejandro Uribe. Much of this work was carried out at Northwestern University.
\section{Preliminaries}
Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold. We will be interested in the case $M = T^* \mathbb{T}^n$. For a submanifold $\mathcal{C}\subset M$, consider the \emph{symplectic orthocomplement} $(T\mathcal{C})^\omega$ of the tangent bundle $T\mathcal{C}$, defined as the union of its fibers:
\[
(T_p \mathcal{C})^\omega := \{ v\in T_p M \ | \ \omega(v,w)=0 \ \mathrm{for \ all} \ w\in T_p \mathcal{C} \}.
\]
\begin{definition}[Coisotropic submanifold]
$\mathcal{C}$ is said to be a \emph{coisotropic} submanifold of $M$ if $(T\mathcal{C})^\omega$ is a subbundle of $T\mathcal{C}$; that is, $\mathcal{C}$ is coisotropic if $(T_p \mathcal{C})^\omega$ is a subspace of $T_p \mathcal{C}$ for each $p\in\mathcal{C}$.
\end{definition}
In particular, this means that $\dim{(\mathcal{C})}\geq n$. If $\dim{(\mathcal{C})} = n$, then $\mathcal{C}$ is a Lagrangian submanifold.
\subsection{Coisotropic regularity}
For $m,k\in\mathbb{R}$, let $\Psi^{m,k}_h(\mathbb{T}^n) = h^{-k} \Psi^m_h(\mathbb{T}^n)$ be the space of semiclassical pseudodifferential operators of differential order $m$. For treatments of the semiclassical pseudodifferential calculus, see \cite{DS,Ma,Zw}. Let
\begin{equation} \label{symbol inequality}
S^m (T^* \mathbb{T}^n) := \{a(x,\xi)\in C^\infty (T^* \mathbb{T}^n) \ | \ \forall\alpha,\beta \ \exists C_{\alpha\beta}>0 \ \mathrm{such \ that} \ |\partial^\alpha_x \partial^\beta_\xi a| \leq C_{\alpha\beta}\left\langle \xi \right\rangle^{m-|\beta|}\}.
\end{equation}
The symbol class $S^m$ is due to J.J.\ Kohn and L.\ Nirenberg \cite{KN}. Then $\Psi^{m,k}_h(\mathbb{T}^n)$ consists \emph{locally} of quantizations of symbols in $h^{-k} C^\infty ([0,1)_h;S^m(T^* \mathbb{T}^n))$.
Let $A\in\Psi^{m,k}_h(\mathbb{T}^n)$. We will generally not be interested in the differential order of $A$, which corresponds to the behavior of its total symbol at infinity in the fibers of the cotangent bundle. Thus, we define the subalgebra $\widetilde{\Psi}^k_h(\mathbb{T}^n) \subset \Psi^{-\infty,k}_h(\mathbb{T}^n)$ consisting locally of quantizations of $h^{-k} C^\infty_c (T^* \mathbb{T}^n \times [0,1)_h)$ (i.e., the total symbols are compactly supported in the fibers) plus quantizations of $h^\infty C^\infty([0,1);S^{-\infty}(T^* \mathbb{T}^n))$ (symbols residual in both semiclassical and differential filtrations). This is the same subalgebra considered in the motivating paper \cite{VW}.
\subsubsection{Characteristic operators}
Let $\mathcal{C}$ be any coisotropic submanifold of $T^* \mathbb{T}^n$. Consider
\[
\mathfrak{M}_\mathcal{C} := \{A\in\Psi_h^0(\mathbb{T}^n) \ | \ \sigma_0(A)\restriction_\mathcal{C} = 0\}.
\]
We call this the module of \emph{characteristic} operators associated to $\mathcal{C}$. An application of Taylor's theorem proves that $\mathfrak{M}_\mathcal{C}$ is finitely generated. It is closed under commutators, as well. To see this, take $A,B\in\mathfrak{M}_\mathcal{C}$.
Recall that $[A,B]$ has principal symbol $\frac{h}{i}\{\sigma_0(A),\sigma_0(B)\}$. Since $\{\sigma_0(A),\sigma_0(B)\} = \mathcal{L}_{H_{\sigma_0(A)}}(\sigma_0(B))$, $H_{\sigma_0(A)}$ is tangent to $\mathcal{C}$, and $\sigma_0(B)$ is constant on $\mathcal{C}$, then $\{\sigma_0(A),\sigma_0(B)\}\restriction_\mathcal{C} = 0$.
As an example, if $\mathcal{C} = \{\xi_1=\ldots=\xi_d=0\}$, the module $\mathfrak{M}_\mathcal{C}$ is generated by the differential operators $h D_{x_j}$, $1\leq j\leq d$.
\begin{remark}
If we write $u_h\in L^2(\mathbb{T}^n)$, we mean that $u_h$ lies in $L^2$ uniformly in $h$ as $h\downarrow 0$. We will suppress $h$ dependence of families of distributions. Likewise, we will write $P$ instead of $P_h$ when referring to the family of operators $P_h$.
\end{remark}
\subsubsection{Definition of coisotropic regularity}
\begin{definition}[Coisotropic regularity]
Let $\mathcal{C}$ be a coisotropic submanifold of $T^* \mathbb{T}^n$. A distribution $u\in L^2(\mathbb{T}^n)$ is said to exhibit \emph{coisotropic regularity} with respect to $\mathcal{C}$ at the point $(x,\xi)\in\mathcal{C}$ if $(x,\xi)$ has a neighborhood $U\subset T^* \mathbb{T}^n$ such that for all $P_j\in\mathfrak{M}_\mathcal{C}$ microsupported in $U$, we have
\begin{equation} \label{iterated regularity condition}
P_1\ldots P_k u\in h^k L^2(\mathbb{T}^n)
\end{equation}
(for all $k$).
Suppose that $u$ has coisotropic regularity everywhere on $\mathcal{C}$. Then we call $u$ a \emph{coisotropic distribution} or simply say that $u$ is coisotropic. Or, if condition \eqref{iterated regularity condition} holds only for $k_0\leq k$ for some $k$, then $u$ has coisotropic regularity of order $k$ at $(x,\xi)$.
\end{definition}
\begin{remark}
(1) We may write the defining condition for coisotropic regularity of $u\in L^2(\mathbb{T}^n)$ equivalently as
\[
(h^{-1} P_1) \ldots (h^{-1} P_k) u \in L^2(\mathbb{T}^n), \ \forall k, \ P_j\in\mathfrak{M}_\mathcal{C}.
\]
Note that since each $P_j$ is a semiclassical PsDO of order 0 in $h$, it is a bounded operator on $L^2$, so necessarily $P_1 \ldots P_k u \in L^2(\mathbb{T}^n)$. By contrast, $h^{-1} P_j\in\Psi^1_h(\mathbb{T}^n)$ will generally \emph{not} be $L^2$ bounded.
(2) We are certainly not the first to define a notion of regularity by means of iterated application of characteristic operators. The original iterated regularity characterization, of conic Lagrangian distributions, is given by L.\ H\"{o}rmander and R.\ Melrose \cite[Section~25.1]{Ho85}.
\end{remark}
\begin{example}
To determine whether some $u\in L^2(\mathbb{T}^3)$ is (globally) coisotropic with respect to $\mathcal{C} = \{\xi_1 = \xi_2 = 0,\xi_3\in\mathbb{R}\}$, amounts to checking for $L^2$-Sobolev regularity \emph{in the directions} $x_1$ and $x_2$. For instance, $e^{i x_3 / h}$ is a coisotropic distribution with respect to $\mathcal{C}$, but $e^{i x_1 / h}$ is nowhere coisotropic at $\mathcal{C}$.
\end{example}
\subsubsection{Coisotropic regularity, extended}
For $s\in\mathbb{R}$, we may consider the set of distributions that are coisotropic of order $k\in\mathbb{Z}_{\geq 0}$ relative to $h^s L^2(\mathbb{T}^n)$:
\begin{equation} \label{eq:distributions}
I^k_{(s)}(\mathcal{C}) := \{u \ | \ h^{-j-s} A_1 \ldots A_j u \in L^2(\mathbb{T}^n) \ \forall A_i\in\mathfrak{M}_\mathcal{C}, 0\leq j\leq k\}.
\end{equation}
However, because $\mathfrak{M}_\mathcal{C}$ is (locally) finitely generated, let $\{B_1,\ldots,B_J\}$ be a generating set. To check whether $u$ lies in $I^k_{(s)}(\mathcal{C})$, it suffices to check $h^{-|\beta|-s} \mathbf{B}^\beta u \in L^2(\mathbb{T}^n)$, where $\mathbf{B}^\beta = B_1^{\beta_1} \cdots B_J^{\beta_J}$ and $0\leq |\beta|\leq k$.
We may extend the definition of $I^k_{(s)}(\mathcal{C})$ to $k\in\mathbb{R}$ by interpolation (to $k\in\mathbb{R}_{\geq 0}$) and duality (to negative $k$). For each $s$, $I^\infty_{(s)}(\mathcal{C})$ is a space of semiclassical coisotropic distributions.
\subsection{Real blowup of submanifolds}
For a thorough treatment of this topic, see R.\ Melrose's notes in \cite{Me-1} and \cite[Chapter~5]{Me-3}. If $M$ is a manifold without boundary and $Y$ is any submanifold of $M$, then $Y$ blown up in $M$ is $[M;Y] = M \backslash Y \cup SN(Y)$. Here, $SN(Y)$ is the spherical (unit) normal bundle of $Y$. $[M;Y]$ is a manifold with boundary, and the boundary of $[M;Y]$ is $SN(Y)$.
Next, suppose $M$ is a manifold with boundary and $Y$ is a submanifold \emph{lying in the boundary of $M$}. Then the blowup of $M$ along $Y$ is $[M;Y] = M \backslash Y \cup SN^+ (Y)$, where $SN^+ (Y)$ is the \emph{inward pointing} part of the spherical normal bundle to $Y$ in $M$. $SN^+ (Y)$ is the \emph{front face} of the blowup, and the \emph{side face} is $\partial M \backslash Y$. There is a smooth \emph{blowdown map} $\beta: [M;Y] \longrightarrow M$ projecting the front face to $Y$; $\beta$ is a diffeomorphism away from the front face. Thus, crucially, there are more smooth functions on the blown up space than on the original manifold. In our case, $M = T^* \mathbb{T}^n \times [0,1)_h$ (which has boundary $T^* \mathbb{T}^n \times \{h=0\}$) and $Y = \mathcal{C} \times \{h=0\}$.
\subsection{Linear coisotropics}
We study linear coisotropics in $T^* \mathbb{T}^n$, defined as follows:
\begin{definition}
A $d$-codimensional \emph{linear coisotropic submanifold} $\mathcal{C}\subset T^* \mathbb{T}^n$ has the form
\[
\mathcal{C} = \mathcal{C}(\mathbf{v}_1,\ldots,\mathbf{v}_d) = \mathbb{T}^n_x \times \{\mathbf{v}_1\cdot\xi = \ldots = \mathbf{v}_d\cdot\xi = 0\},
\]
where $\{\mathbf{v}_1,\ldots,\mathbf{v}_d\}\subset\mathbb{R}^n$ is linearly independent over $\mathbb{R}$.
\end{definition}
A simple linear coisotropic is $\{\xi_1 = \ldots = \xi_d = 0\}$. In fact, locally every coisotropic is of this form \cite[Theorem~21.2.4]{Ho85}.
\subsection{Second microlocal symbols}
The domain of our total symbols is the manifold with corners $S_\mathrm{tot} := [T^* \mathbb{T}^n \times [0,1)_h;\mathcal{C} \times \{h=0\}]$. The corner occurs at the intersection of the front face $SN^+(\mathcal{C}\times\{h=0\})$ and the side face. Second principal symbols (defined in Section \ref{sec:second principal}) live on $S_\mathrm{pr} := [T^* \mathbb{T}^n;\mathcal{C}]$, which is identifiable with the side face of $S_\mathrm{tot}$.
\begin{example}
In Figure \ref{fig:totsymb}, we see the total symbol space for the coisotropic $\mathcal{C} = \mathbb{T}^2 \times \{\xi_1\in\mathbb{R},\xi_2 = 0\}$, with base variables omitted. Notice that the front face is a \emph{half} cylinder. Since we are not interested in negative values of $h$, only (unit) normal vectors pointing into the interior of $T^* \mathbb{T}^2 \times [0,1)_h$ are considered.
\end{example}
\begin{figure}
\caption{$S_\mathrm{tot}
\label{fig:totsymb}
\end{figure}
In $S_\mathrm{pr}$, we replace $\mathcal{C}\subset T^*\mathbb{T}^n$ with its \emph{full} spherical normal bundle. Let $\rho_{\mathrm{ff}}$ and $\rho_{\mathrm{sf}}$ denote boundary defining functions for the front and side faces of $S_\mathrm{tot}$, respectively.
\begin{definition}[Total symbols]
For $m,l\in\mathbb{R}$, let $$S^{m,l}(S_\mathrm{tot}) := \rho_{\mathrm{sf}}^{-m} \rho_{\mathrm{ff}}^{-l} C_{c}^{\infty}(S_\mathrm{tot}).$$
\end{definition}
We define
\[
S^{-\infty,l}(S_\mathrm{tot}) := \bigcap_{m\in\mathbb{R}} S^{m,l}(S_\mathrm{tot}).
\]
\begin{remark}
Here we clarify that we are considering \emph{two different} spherical normal bundles. First, there is the inward-pointing spherical normal bundle $SN^+ (\mathcal{C}\times\{h=0\})$, which is the front boundary face of $S_\mathrm{tot}$. Second, there is the full spherical normal bundle to $\mathcal{C}\subset T^* \mathbb{T}^n$, namely $SN(\mathcal{C})$. The principal symbol space $S_\mathrm{pr}$ is a manifold with boundary, with $\partial S_\mathrm{pr} = SN(\mathcal{C})$. Finally, $SN(\mathcal{C})$ is identifiable with the corner of $S_\mathrm{tot}$.
\end{remark}
Let $d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi := (2\pi h)^{-n} d\xi$. Let ${}^h \mathrm{Op_l}$, ${}^h \mathrm{Op_W}$, and ${}^h \mathrm{Op_r}$ represent semiclassical left, Weyl, and right quantization, respectively: For $a\in S^{m,l}(S_\mathrm{tot})$,
\begin{align*}
{}^h \mathrm{Op_l}(a) &= \int{e^{\frac{i}{h}(x-y)\cdot\xi} \chi(x,y) a(x,\xi;h) \ d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi}; \\
{}^h \mathrm{Op_W}(a) &= \int{e^{\frac{i}{h}(x-y)\cdot\xi} \chi(x,y) a\left(\frac{x+y}{2},\xi;h\right) d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi}; \ \mathrm{and} \\
{}^h \mathrm{Op_r}(a) &= \int{e^{\frac{i}{h}(x-y)\cdot\xi} \chi(x,y) a(y,\xi;h) \ d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi}.
\end{align*}
Also, if $a\in C^\infty(\mathbb{T}^n;S^{m,l}(S_\mathrm{tot}))$, then
\begin{equation} \label{general quantization}
I_h(a) := \int{e^{\frac{i}{h}(x-y)\cdot\xi}\chi(x,y)a(x,y,\xi;h) \ d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi}.
\end{equation}
$\chi$ is any cutoff function supported in a neighborhood of the diagonal, and $\chi\equiv 1$ in a smaller neighborhood of the diagonal. The purpose of $\chi$ is to make sense of the difference $(x-y)$ appearing in the phase. A stationary phase argument shows that the choice of a particular $\chi$ does not matter, up to $O(h^\infty)$.
Our calculus will be denoted $\Psi_{2,h}(\mathcal{C})$. The calculus will consist of, say, left quantizations of elements of $S^{m,l}(S_\mathrm{tot})$ ($m,l\in\mathbb{R}$), as well as \emph{residual operators} to be introduced in Section \ref{section on residual ops}.
\section{\textsc{Composition and Invariance} \label{section on residual ops}}
Let $\mathcal{C} = \mathbb{T}^n \times \{\mathbf{v}_1\cdot\xi = \ldots = \mathbf{v}_d\cdot\xi = 0\}$, with $\mathbf{v}_i\in\mathbb{R}^n$ linearly independent.
\subsection{Residual operators}
We next define the ``residual'' elements which, along with quantizations of symbols, comprise the second microlocal calculus $\Psi_{2,h}(\mathcal{C})$. Residual operators play the same role as smoothing operators in the $C^\infty$ pseudodifferential calculus.
Let $B_j = \mathbf{v}_j \cdot h D_x$, where $D_x = (D_{x_1} \ \cdots \ D_{x_n})^t$. Then $B_1,\ldots,B_d\in\Psi^0_h(\mathbb{T}^n)$ generate the module $\mathfrak{M}_\mathcal{C}$ of operators characteristic on $\mathcal{C}$, and let $\mathbf{B}^\beta := B_1^{\beta_1}\cdots B_d^{\beta_d}$ be a monomial formed from these generators. Also, let $\widetilde{B}_j = \mathbf{v}_j \cdot D_x$ and $\widetilde{\mathbf{B}}^\beta = \widetilde{B}_1^{\beta_1}\cdots\widetilde{B}_d^{\beta_d}$.
\begin{definition}[Residual operator]
For $l\in\mathbb{R}$, the bounded linear operator $R$ is in the residual space $\Re^l$ if: \begin{condition} \label{involutizing condition}
$R$ is \emph{involutizing}: for $u_h\in L^2(\mathbb{T}^n)$ and multi-indices $\beta,\gamma$, $R$ satisfies
\[
h^{-|\beta+\gamma| + l}\left(\mathbf{B}^\beta R \mathbf{B}^\gamma\right) u_h \in L^2(\mathbb{T}^n).
\]
(Since we are composing $\mathbf{B}$ on the right of $R$, if this condition is fulfilled, then the adjoint operator $R^*$ is also involutizing.)
\end{condition}
\end{definition}
Let $\Re := \bigcup_{l\in\mathbb{R}} \Re^l$. Then:
\begin{definition}
The second microlocal calculus associated to $\mathcal{C}$ is
\[
\Psi_{2,h}(\mathcal{C}) := \Re + \bigcup_{m,l\in\mathbb{R}}{{}^h \mathrm{Op_l}(S^{m,l}(S_\mathrm{tot}))}.
\]
\end{definition}
We will show that $\Psi_{2,h}(\mathcal{C})$ is closed under composition. The key result is a reduction theorem, \autoref{composition theorem}. The calculus is also closed under asymptotic summation: if $A_j\in\Psi^{m-j,l}_{2,h}(\mathcal{C})$, then there exists $A\in\Psi^{m,l}_{2,h}(\mathcal{C})$ for which
\[
A - \sum_{j=0}^{N-1} A_j \in \Psi^{m-N,l}_{2,h}(\mathcal{C})
\]
for all $N$.
\subsection{Reduction and Composition}
\begin{theorem}[Right reduction] \label{composition theorem}
Let $a(x,y,\xi;h)\in C^\infty(\mathbb{T}^n;S^{m,l}(S_\mathrm{tot}))$. Then there exists $b\in S^{m,l}(S_\mathrm{tot})$ such that $I_h(a)={}^h \mathrm{Op_r}(b)+R$, and the remainder $R$ belongs to $\Re^l$.
\end{theorem}
Before we prove this theorem, we state and prove a result concerning boundedness.
\begin{prop}[$L^2$ boundedness] \label{boundedness prop}
Let $s\in C^{\infty}_{c}(\mathbb{T}^n \times S_\mathrm{tot})$. Then $I_h(s)\in\Psi^{0,0}_{2,h}(\mathcal{C})$ is bounded on $L^2(\mathbb{T}^n)$.
\end{prop}
This parallels the standard result that $h$-pseudodifferential operators in $\Psi^0_h(\mathbb{T}^n)$ (here, $0$ is the order in the $h$-filtration, not the differential order) are $L^2$ bounded. The following is an essential ingredient of our proof of $L^2$ boundedness.
\begin{lemma}[Calder\'{o}n--Vaillancourt theorem]
Let $a(x,y,\xi;h)\in C_c^\infty(\mathbb{T}^n\times \mathbb{T}^n\times\mathbb{R}^n\times[0,1))$. Suppose that for all multi-indices $\alpha,\beta$, there exists a constant $C_{\alpha\beta} > 0$ (independent of $h$) such that
\begin{equation} \label{C-V estimate}
|\partial^{\alpha}_{x,y} \partial^{\beta}_{\xi} a(x,y,\xi;h)|\leq C_{\alpha\beta}.
\end{equation}
Then
\[
I(a) = (2\pi)^{-n} \int e^{i(x-y)\cdot\xi} \chi(x,y) a(x,y,\xi;h)\ d\xi
\]
is bounded, as an operator on $L^2(\mathbb{T}^n)$. (Note that this is a non-semiclassical quantization.)
\end{lemma}
\begin{proof}
See \cite{CaVa}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{boundedness prop}]
Let $s\in C^\infty_c(\mathbb{T}^n \times S_\mathrm{tot})$ and change variables $\eta = \xi/h$. Then the estimate in \eqref{C-V estimate} becomes
\begin{equation} \label{estimate in C-V}
h^{|\beta|} \left|\left(\partial^{\alpha}_{x,y} \partial^{\beta}_{\xi} s\right)(x,y,\xi;h)\right| = |\partial^{\alpha}_{x,y} \partial^{\beta}_{\eta}[s(x,y,h\eta,h)]| \leq C_{\alpha\beta}.
\end{equation}
Using a partition of unity, we may decompose $s$ into pieces supported on $\mathbb{T}^n$ times the lift to $S_\mathrm{tot}$ of $\{\mathbf{v}_j\cdot\xi \neq 0\} \times [0,1)_h \subset T^* \mathbb{T}^n\times [0,1)_h$ for $j=1,\ldots,d$. By symmetry, it is enough to study the part of $s$ supported on $\mathbb{T}^n$ times the lift of $\{\mathbf{v}_1\cdot\xi \neq 0\} \times [0,1)$.
We may extend the linearly independent set $\{\mathbf{v}_1,\ldots,\mathbf{v}_d\}$ to a basis
\[
\{\mathbf{v}_1,\ldots,\mathbf{v}_d,\mathbf{w}_{d+1},\ldots,\mathbf{w}_n\}
\]
for $\mathbb{R}^n$. Locally, in a neighborhood of the corner of $S_\mathrm{tot}$, we employ the coordinates $x$, $y$, $H = h / {(\mathbf{v}_1\cdot\xi)}$, $\zeta = \mathbf{v}_1\cdot\xi$,
\[
\Xi = (\Xi_2,\ldots,\Xi_d) = \left(\frac{\mathbf{v}_2\cdot\xi}{\mathbf{v}_1\cdot\xi},\ldots,\frac{\mathbf{v}_d\cdot\xi}{\mathbf{v}_1\cdot\xi}\right),
\]
and $\mathbf{W} = (\mathbf{w}_{d+1}\cdot\xi,\ldots,\mathbf{w}_n\cdot\xi)$.
\emph{We lift $h^{|\beta|} \partial^{\alpha}_{x,y} \partial^{\beta}_{\xi}$ to the coordinates just introduced, then show that $s$ satisfies the estimate in \eqref{estimate in C-V} under application of the lifted vector field.} In particular, we will lift $h \partial_{\xi_i}$ for $1\leq i\leq n$. We have
\begin{equation*}
\partial_{\xi_i} = v_1^i \partial_\zeta - \frac{h v_1^i}{\zeta^2} \partial_H + \mathbf{w}^i\cdot\partial_{\mathbf{W}} + \frac{\partial \Xi}{\partial \xi_i}\cdot\partial_\Xi,
\end{equation*}
where $\mathbf{w}^i = \left(w_{d+1}^i,\ldots,w_n^i\right)$. Let $\mathbf{v}^i = (v_2^i,\ldots,v_d^i)$. We calculate that
\[
\frac{\partial \Xi}{\partial \xi_i} = \left(\frac{v_2^i \zeta - v_1^i(\mathbf{v}_2\cdot\xi)}{\zeta^2},\ldots,\frac{v_d^i \zeta - v_1^i(\mathbf{v}_d\cdot\xi)}{\zeta^2}\right).
\]
Therefore,
\begin{equation} \label{hard lift}
h \partial_{\xi_i} = H \left(v_1^i \zeta \partial_\zeta - v_1^i H \partial_H + \mathbf{w}^i \cdot \zeta \partial_{\mathbf{W}} + \mathbf{v}^i \cdot \partial_\Xi - v_1^i \Xi\cdot\partial_\Xi\right) =: H \vec{V}.
\end{equation}
For $1\leq i\leq n$, $|h \partial_{\xi_i} s|$ is thus bounded (independently of $h$), because $s\in C^\infty_c(\mathbb{T}^n \times S_\mathrm{tot})$ is compactly supported in the variables $\zeta$, $H$, $\mathbf{W}$, and $\Xi$.
Therefore, since \eqref{estimate in C-V} is satisfied, we may apply the Calder\'{o}n--Vaillancourt theorem to conclude that $I_h(s):L^2(\mathbb{T}^n)\rightarrow L^2(\mathbb{T}^n)$.
\end{proof}
Let $\mathcal{V}_b$ represent the set of vector fields on $S_\mathrm{tot}$ tangent to both front and side faces. Notice that $\vec{V}\in\mathcal{V}_b$. Now we prove \autoref{composition theorem}.
\begin{proof}
We use the same coordinates as in the previous proof, and let $\rho_\mathrm{sf} = H = h / {(\mathbf{v}_1\cdot\xi)}$, $\rho_\mathrm{ff} = \zeta = \mathbf{v}_1\cdot\xi$.
Let $a(x,y,\xi;h)\in C^\infty(\mathbb{T}^n;S^{m,l}(S_\mathrm{tot}))$. Taylor's formula yields the asymptotic sum
\[
a(x,y,\xi;h)\sim\sum_\alpha{\frac{(x-y)^\alpha}{\alpha!}(\partial^{\alpha}_{x}a)(y,y,\xi;h)}.
\]
We see from \eqref{hard lift} that for all $1\leq j\leq n$, application of $h \partial_{\xi_j}$ to a second microlocal total symbol improves its decay at the side face. Therefore, for any multi-index $\alpha$,
\[
(h\partial_\xi)^\alpha(\partial^{\alpha}_{x}a)(y,y,\xi;h)\in H^{-m+|\alpha|} \zeta^{-l} C^{\infty}_{c}(S_\mathrm{tot}),
\]
so there exists $b\in H^{-m} \zeta^{-l} C^{\infty}_{c}(S_\mathrm{tot})$ for which
\[
b\sim\sum_\alpha \frac{i^{|\alpha|}}{\alpha!} (h\partial_\xi)^\alpha(\partial^\alpha_x a)(y,y,\xi;h).
\]
Fix any $N$. Then
\[
a - \sum_{|\alpha|<N} \frac{(x-y)^\alpha}{\alpha!}(\partial^{\alpha}_{x}a)(y,y,\xi;h) =: a_N\in C^\infty(\mathbb{T}^n;S^{m-N,l}(S_\mathrm{tot})),
\]
and
\[
b - \sum_{|\alpha|<N} \frac{i^{|\alpha|}}{\alpha!} (h\partial_\xi)^\alpha(\partial^\alpha_x a)(y,y,\xi;h) =: b_N\in S^{m-N,l}(S_\mathrm{tot}).
\]
We have
\begin{align*}
I_h(a) &= \sum_{|\alpha|<N} \frac{1}{\alpha!} \int e^{\frac{i}{h}(x-y)\cdot\xi} \chi(x,y) (x-y)^\alpha (\partial^{\alpha}_{x}a)(y,y,\xi;h) d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi + I_h(a_N) \\
&= \sum_{|\alpha|<N} \frac{1}{i^{|\alpha|}\alpha!} \int (h\partial_\xi)^\alpha e^{\frac{i}{h}(x-y)\cdot\xi} \chi(x,y) (\partial^{\alpha}_{x}a)(y,y,\xi;h) d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi + I_h(a_N) \\
&= \sum_{|\alpha|<N} \frac{i^{|\alpha|}}{\alpha!} \int e^{\frac{i}{h}(x-y)\cdot\xi} \chi(x,y) (h\partial_\xi)^\alpha(\partial^{\alpha}_{x}a)(y,y,\xi;h) d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi +I_h(a_N) \\
&= {}^h \mathrm{Op_r}(b - b_N) + I_h(a_N).
\end{align*}
Let $R_N := I_h(a_N) - {}^h \mathrm{Op_r}(b_N)\in \Psi^{m-N,l}_{2,h}(\mathcal{C})$. We will show that the operator $I_h(a_N)$ is residual as defined above, `up to order $N$' (see below). The proof for ${}^h \mathrm{Op_r}(b_N)$ is the same, except $b_N$ has one fewer component.
\textit{Proof of} Condition \ref{involutizing condition}: We are interested in $h^{-|\beta|} \mathbf{B}^\beta I_h(a_N)$. By the Leibniz rule, this equals
\[
\sum_{\mu+\nu=\beta} \frac{\beta!}{\mu!\nu!} \int \left(\frac{1}{H}\right)^{\mu_1}\left(\frac{\Xi_2}{H}\right)^{\mu_2}\cdots\left(\frac{\Xi_d}{H}\right)^{\mu_d} e^{\frac{i}{h}(x-y)\cdot\xi}\left[\widetilde{\mathbf{B}}^\nu a_N\right] d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi.
\]
Recall that $H^{-1} = (\mathbf{v}_1\cdot\xi) / h$ is the reciprocal of the boundary defining function for the side face. Hence, as long as $|\beta| \leq -m + N + l$, the net contribution is a positive power of $\rho_{\mathrm{sf}}$.
$I_h(a_N) \widetilde{\mathbf{B}}^\gamma$ is handled similarly. Note that the amplitude $a_N$ is smooth in $x$ and $y$, so application of $\widetilde{\mathbf{B}}$ does not change the symbol class to which $a_N$ belongs. By Proposition \ref{boundedness prop}, we conclude that $I_h(a_N)$ is involutizing for $|\beta| + m - l \leq N$.
Similarly, $I_h(a-a_N),{}^h \mathrm{Op_r}(b-b_N)\in\Psi^{m-N,l}_{2,h}(\mathcal{C})$ are finitely residual in this sense. Then, since
\[
I_h(a) - {}^h \mathrm{Op_r}(b) = I_h(a-a_N) - {}^h \mathrm{Op_r}(b-b_N) + I_h(a_N) - {}^h \mathrm{Op_r}(b_N),
\]
we may conclude that $I_h(a) - {}^h \mathrm{Op_r}(b)$ is residual.
\color{black}
\end{proof}
We can likewise prove a \emph{left} reduction or Weyl reduction. The point is that we can convert one quantization map into any other, at the (low) cost of one of these residual operators.
\begin{corollary}[Composition Law] \label{composition law}
If $a\in S^{m,l}(S_\mathrm{tot})$, $b\in S^{m',l'}(S_\mathrm{tot})$, then ${}^h \mathrm{Op_l}(a)\hspace{.1cm}\circ\hspace{.1cm}{}^h \mathrm{Op_l}(b) = {}^h \mathrm{Op_l}(c) + R$ for $c\in S^{m+m',l+l'}(S_\mathrm{tot})$ and $R\in\Re^{l+l'}$.
\end{corollary}
\begin{proof} By \autoref{composition theorem}, it is sufficient to consider ${}^h \mathrm{Op_l}(a)\circ {}^h \mathrm{Op_r}(b)$. We have
\begin{align*}
& {}^h \mathrm{Op_l}(a) {}^h \mathrm{Op_r}(b) = \\
&= (2\pi h)^{-2n}\int{\int{\int{e^{\frac{i}{h}(x-w)\cdot\xi}e^{\frac{i}{h}(w-y)\cdot\eta} \chi(x,w)\chi(w,y) a(x,\xi;h)b(y,\eta;h)\hspace{.05cm}dw}\hspace{.05cm}d\eta}\hspace{.05cm}d\xi}.
\end{align*}
For this integral, we use the stationary phase theorem \cite[Theorem~3.17]{Zw}, and we concisely write the amplitude as $s(w,\eta;h)$. By this theorem, for all $N$
\begin{align}
{}^h \mathrm{Op_l}(a) {}^h \mathrm{Op_r}(b) &= (2\pi h)^{-2n}\int{\int{\int{e^{\frac{i}{h}((x-y)\cdot\xi + (w-y)\cdot(\eta-\xi))} s(w,\eta;h)\hspace{.05cm}dw}\hspace{.05cm}d\eta}\hspace{.05cm}d\xi} \nonumber \\
&= (2\pi h)^{-n} \left(\int e^{\frac{i}{h}(x-y)\cdot\xi} \sum^{N-1}_{k=0} \frac{1}{i^k k!} \left[(D_w\cdot h D_\eta)^k s(w,\eta;h)\right]_{w=y,\eta=\xi}\ d\xi + O(h^N)\right) \label{stationary phase}.
\end{align}
Note that $(\partial_{w_j} \chi)(y,y) = 0$ and likewise for higher order derivatives of $\chi$, since these are supported off the diagonal. Thus, writing out the first few terms of the sum,
\begin{align*}
& \sum^{N-1}_{k=0} \frac{1}{i^k k!} \left[(D_w\cdot h D_\eta)^k s(w,\eta;h)\right]_{w=y,\eta=\xi} = \\
&= \chi(x,y)a(x,\xi;h)b(y,\xi;h) + \sum^n_{j=1} (\partial_{w_j} \chi)(x,y)a(x,\xi;h)(h \partial_{\eta_j} b)(y,\xi;h) + O(h^2).
\end{align*}
We know $(\partial_{w_j} \chi)(x,y)$, and all higher derivatives evaluated at $w=y$, is supported off-diagonal. So the contributions of these terms are residual (i.e., the kernel of a second microlocal operator is a coisotropic distribution, associated to $\mathcal{C}\times\mathcal{C}$, away from the diagonal). Note that for all $j$, $(h \partial_{\eta_j} b)(y,\eta;h)$ is a second microlocal symbol, due to \eqref{hard lift}.
By \autoref{composition theorem}, we can then write \eqref{stationary phase} as the left quantization of some $c\in S^{m+m',l+l'}(S_\mathrm{tot})$, modulo a residual remainder.
\end{proof}
From now on, we will not explicitly write $\chi$. By definition, the adjoint of a residual operator in $\Re^l$ is again an element of $\Re^l$. In addition, it is routine to prove:
\begin{prop}
Let $A = {}^h \mathrm{Op_l}(a)$ for $a\in S^{m,l}(S_\mathrm{tot})$. Then $A^* \in\Psi^{m,l}_{2,h}(\mathcal{C})$.
\end{prop}
Moreover, it is reassuring that quantizations of elements of $S^{-\infty,l}(S_\mathrm{tot})$, $l\in\mathbb{R}$, are residual.
\begin{prop} \label{quants are residual}
Suppose $\mathcal{C}$ is a linear coisotropic. If $A = {}^h \mathrm{Op_r}(a)$ for $a\in S^{-\infty,l}(S_\mathrm{tot})$, then $A\in\Re^l$.
\end{prop}
\begin{proof} Locally in $S_\mathrm{tot}$, we may take $\mathbf{v}_1\cdot\xi$ as defining function for the front face and $h / {(\mathbf{v}_1\cdot\xi)}$ for side face defining function.
We show that $A$ satisfies Condition \ref{involutizing condition}. Recall that $\widetilde{B}_j = \mathbf{v}_j \cdot D_x$ and $\widetilde{\mathbf{B}}^\beta = \widetilde{B}_1^{\beta_1}\circ\cdots\circ\widetilde{B}_d^{\beta_d}$; also, let $\mathbf{V} = (\mathbf{v}_1\cdot\xi,\ldots,\mathbf{v}_d\cdot\xi)$. We have
\begin{equation*}
\widetilde{\mathbf{B}}^\beta A = \int \left(\frac{\mathbf{V}}{h}\right)^\beta e^{\frac{i}{h}(x-y)\cdot\xi} a(y,\xi;h) \ d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi = \int \rho_\mathrm{sf}^{-|\beta|} \Xi^{(\beta_2,\ldots,\beta_d)} e^{\frac{i}{h}(x-y)\cdot\xi} a(y,\xi;h) \ d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi.
\end{equation*}
No matter how large $\beta$ is, $\rho_\mathrm{sf}^{-|\beta|} a\in S^{0,l}(S_\mathrm{tot})$. Therefore, $\widetilde{\mathbf{B}}^\beta A$ maps $L^2(\mathbb{T}^n)$ to $h^{-l} L^2(\mathbb{T}^n)$.
\end{proof}
Finally, we show:
\begin{prop} \label{two-sided ideal prop}
The residual algebra is an ideal in $\Psi_{2,h}(\mathcal{C})$.
\end{prop}
\begin{proof}
Let $a\in S^{m,l}(S_\mathrm{tot})$ and $A = {}^h \mathrm{Op}_r(a)$. Let $R\in\Re^{l'}$. We show that $AR\in\Re^{l+l'}$. Taking adjoints then implies $RA\in\Re^{l+l'}$.
Let $\Theta_{x_j} = \mathbf{v}_j\cdot D_x$ and $\Theta_x^\beta = \Theta_{x_1}^{\beta_1}\cdots\Theta_{x_d}^{\beta_d}$. Then, for $u\in L^2(\mathbb{T}^n)$,
\begin{align*}
h^{l+l'} \Theta_x^\beta A R u(x) &= (-1)^{|\beta|} h^{l+l'} \int\int \left(\frac{\mathbf{V}}{h}\right)^\beta \left(\frac{h}{\mathbf{V}} \Theta_y\right)^\beta e^{\frac{i}{h}(x-y)\cdot\xi} a(y,\xi;h) Ru(y)\ dy d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi \\
&= \sum_{\mu+\nu = \beta} \frac{\beta!}{\mu!\nu!} h^l \int\int e^{\frac{i}{h}(x-y)\cdot\xi} \Theta_y^\mu a(y,\xi;h) \left[h^{l'} \Theta_y^\nu Ru(y)\right] dy d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi.
\end{align*}
The bracketed term lies in $L^2$, since $R$ is involutizing. And $h^l\ {}^h \mathrm{Op}_r(\Theta_y^\mu a)\in\Psi^{m-l,0}_{2,h}(\mathcal{C})$ so, if $m\leq l$, it is $L^2$ bounded. Instead, if $m>l$, choose $p\geq m-l$. As before, locally we may take $\rho_\mathrm{sf} = h/{\mathbf{v}_1\cdot\xi}$ as side face defining function. Then for $\mu+\nu = \beta$,
\[
(-1)^p h^l \int\int \left(\frac{h}{\mathbf{v}_1\cdot\xi} \Theta_{y_1}\right)^p e^{\frac{i}{h}(x-y)\cdot\xi} \Theta_y^\mu a(y,\xi;h) \left[h^{l'} \Theta_y^\nu Ru(y)\right] dy d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi
\]
\[
= \sum_{i+j = p} \frac{p!}{i!j!} h^l \int\int \rho_\mathrm{sf}^p e^{\frac{i}{h}(x-y)\cdot\xi} \Theta_{y_1}^i \Theta_y^\mu a(y,\xi;h) \left[h^{l'} \Theta_{y_1}^j \Theta_y^\nu Ru(y)\right] dy d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi.
\]
This time, the amplitude belongs to $S^{m-l-p,0}(S_\mathrm{tot})$, so by Proposition \ref{boundedness prop} the operator is $L^2$ bounded. Thus, $AR$ satisfies Condition \ref{involutizing condition}.
Meanwhile, it is easily seen that for $R_1\in\Re^l$, $R_2\in\Re^{l'}$, we have $R_1 R_2\in\Re^{l+l'}$.
\end{proof}
\section{\textsc{Microsupport, parametrices \& second wavefront}\label{sec:second principal}}
\subsection{Second microsupport}
Let $S_\mathrm{tot}$ and $S_\mathrm{pr}$ be symbol spaces associated to any linear coisotropic $\mathcal{C}$. Recall that $S_\mathrm{pr}$ may be identified with the side face of $S_\mathrm{tot}$.
\begin{definition}
For $a\in S^{m,l}(S_\mathrm{tot})$, define the \emph{essential support} of $a$ by:
\[
S_\mathrm{pr}\backslash\mathrm{ess \ supp}_l(a) := \{p\in S_\mathrm{pr} \ | \ \exists \varphi\in C^\infty_c (S_\mathrm{tot}), \varphi(p)\neq 0, \varphi a\in S^{-\infty,l}(S_\mathrm{tot})\}.
\]
If $A = {}^h \mathrm{Op_r}(a)$ for $a\in S^{m,l}(S_\mathrm{tot})$, then the \emph{second microsupport} ${}^2 \mathrm{WF}_l'(A)$ of $A$ is given by ${}^2 \mathrm{WF}_l'(A) := \mathrm{ess \ supp}_l(a)$. If $A \in \Re^l$, define ${}^2 \mathrm{WF}_l'(A) = \emptyset$.
\end{definition}
Second microsupport obeys the usual laws of microsupports:
\begin{prop} \label{laws of microsupports}
Let $A,B\in\Psi^{m,l}_{2,h}(\mathcal{C})$, $D\in\Psi^{m',l'}_{2,h}(\mathcal{C})$. Then ${}^2 \mathrm{WF}'$ satisfies $${}^2 \mathrm{WF}_l'(A+B) \subset {}^2 \mathrm{WF}_l'(A) \cup {}^2 \mathrm{WF}_l'(B), \ \ {}^2 \mathrm{WF}_{l+l'}'(AD) \subset {}^2 \mathrm{WF}_l'(A) \cap {}^2 \mathrm{WF}_{l'}'(D).$$
\end{prop}
\subsection{Principal symbols of second microlocal operators}
Let $A\in\Psi^{m,l}_{2,h}(\mathcal{C})$. Suppose $A = {}^h \mathrm{Op_r}(a) + R$ for some $a\in S^{m,l}(S_\mathrm{tot})$ and residual operator $R\in\Re^l$.
\begin{definition}
The \emph{principal symbol} of $A$ is
\[
{}^2 \sigma_{m,l}(A) := (h^m a)\restriction_\mathrm{sf} \ \in S^{l-m}(S_\mathrm{pr}).
\]
\end{definition}
But the pair $(a,R)$ is not unique. Suppose we also have $a'\in S^{m,l}(S_\mathrm{tot})$, $R'\in\Re^l$ such that $A = {}^h \mathrm{Op_r}(a') + R'$. Then
\[
{}^h \mathrm{Op_r}(a - a') = R' - R \in \Re^l.
\]
Thus, while \emph{a priori} ${}^h \mathrm{Op_r}(a - a')\in\Psi^{m,l}_{2,h}(\mathcal{C})$, in fact ${}^h \mathrm{Op_r}(a - a')\in\Re^l$. This principal symbol will be well defined if it is independent of the choice of right reduction. This amounts to showing that the difference $a-a'$, which \emph{a priori} belongs to $S^{m,l}(S_\mathrm{tot})$ for whatever value of $m\in\mathbb{R}$, in fact decays at the side face of $S_\mathrm{tot}$. We claim the stronger result that $a-a'\in S^{-\infty,l}(S_\mathrm{tot})$.
\begin{lemma} \label{lem:more modest}
For $b\in S^{m,l}(S_\mathrm{tot})$, suppose that ${}^h \mathrm{Op_r}(b)\in\Re^l$. Then $b\in S^{-\infty,l}(S_\mathrm{tot})$.
\end{lemma}
\begin{proof}
Let $\mathcal{C} = \{\mathbf{v}_1\cdot\xi = \ldots = \mathbf{v}_d\cdot\xi = 0\}$ as usual. We use the same coordinates in $S_\mathrm{tot}$ as in the proof of Proposition \ref{boundedness prop}: $\rho_\mathrm{ff} = \mathbf{v}_1\cdot\xi$, $\Xi$, and $\mathbf{W}$. (Note that $\rho_\mathrm{sf} = h / {(\mathbf{v}_1\cdot\xi)}$, as we must have $\rho_\mathrm{ff}\times\rho_\mathrm{sf} = h$.)
Expand $b$ in powers of $\rho_\mathrm{sf}$, near the corner $\{\rho_\mathrm{sf} = \rho_\mathrm{ff} = 0\}$:
\[
b(x,\rho_\mathrm{sf},\rho_\mathrm{ff},\Xi,\mathbf{W}) \sim \sum_j \rho_\mathrm{sf}^j b_j(x,\rho_\mathrm{ff},\Xi,\mathbf{W});
\]
the coefficients $b_j$ can be taken to be smooth functions on the side face.
Next, suppose $b\neq O(\rho_\mathrm{sf}^\infty)$. Then there must exist a coefficient $b_{j_0}$ which is nontrivial at the corner. By continuity of $b_{j_0}$, we have $b_{j_0}$ nontrivial in a neighborhood $\{\rho_\mathrm{ff} < \epsilon\}$ (also localized in $x$, $\Xi$, and $\mathbf{W}$); i.e., there exists a point $(x^0,\rho_\mathrm{ff}^0 > 0,\Xi^0,\mathbf{W}^0)$ at which $b_{j_0}$ is nonzero. So $\rho_\mathrm{sf}^{j_0} b_{j_0}\neq O(h^\infty)$ at $(x^0,\rho_\mathrm{sf} = 0,\rho_\mathrm{ff}^0,\Xi^0,\mathbf{W}^0)$. Hence ${}^h \mathrm{Op_r}(b)$ cannot lie in $\Re^l$, since $\Re^l$ reduces to $O(h^\infty)$ along the side face. (More generally, $\Psi_{2,h}(\mathcal{C})$ is really just $\widetilde{\Psi}_h(\mathbb{T}^n)$ microlocally away from the coisotropic.)
\end{proof}
Set ${}^2 \mathrm{char}_{m,l}(A) := \{\rho_\mathrm{ff}^{l-m} \ {}^2 \sigma_{m,l}(A) = 0\}$. That is, the second characteristic set is the zero set of a smooth function on the principal symbol space. We abuse notation by omitting the indices and writing ${}^2 \mathrm{char}_{m,l}$ simply as ${}^2 \mathrm{char}$. The complementary notion is
\[
{}^2 \mathrm{ell}(A) = {}^2 \mathrm{ell}_{m,l}(A) := \{\rho_\mathrm{ff}^{l-m} \ {}^2 \sigma_{m,l}(A)\neq 0\}.
\]
Note that these definitions are independent of choice of front face defining function. Next, we state some essential properties of the principal symbol map.
\begin{lemma} \label{short exact sequence}
\[
0\longrightarrow\Psi^{m-1,l}_{2,h}(\mathcal{C})\longrightarrow\Psi^{m,l}_{2,h}(\mathcal{C})\xrightarrow{{}^2 \sigma_{m,l}}S^{l-m}(S_\mathrm{pr})\longrightarrow 0
\]
is a short exact sequence. Furthermore, the principal symbol map is a homomorphism: if $A\in\Psi^{m,l}_{2,h}(\mathcal{C})$, $B\in\Psi^{m',l'}_{2,h}(\mathcal{C})$, then ${}^2 \sigma_{m+m',l+l'}(AB)={}^2 \sigma_{m,l}(A) {}^2 \sigma_{m',l'}(B)$.
\end{lemma}
The proof is straightforward, so we omit it.
\begin{remark}
If $A\in\Psi^{m,l}_{2,h}(\mathcal{C})$, $B\in\Psi^{m',l'}_{2,h}(\mathcal{C})$, then
\[
{}^2 \sigma_{m+m'-1,l+l'}(i[A,B]) = \{{}^2 \sigma_{m,l}(A),{}^2 \sigma_{m',l'}(B)\},
\]
where the Poisson bracket is computed with respect to the symplectic form on $S_\mathrm{pr}$ lifted from the symplectic form on $T^* \mathbb{T}^n$. See also Remark \ref{rmk:scaling}.
\end{remark}
\begin{remark} \label{one calculus in the other}
For each $m\in\mathbb{R}$, $\widetilde{\Psi}^{m}_{h}(\mathbb{T}^n)$ can be identified with a subset of $\Psi^{m,m}_{2,h}(\mathcal{C})$. Locally, each element $A\in\widetilde{\Psi}^{m}_{h}(\mathbb{T}^n)$ is a quantization of a symbol $a\in h^{-m} C^\infty_c (T^* \mathbb{T}^n \times [0,1)_h)$ (or $a$ is residual in both semiclassical and differential filtrations). If $\beta_\mathrm{tot}: S_\mathrm{tot} \rightarrow T^* \mathbb{T}^n \times [0,1)$ is the (smooth) blowdown map, then the pullback $\beta_\mathrm{tot}^* a$ quantizes to the corresponding element $A\in\Psi^{m,m}_{2,h}(\mathcal{C})$. Likewise, we may identify the principal symbol of $A\in\widetilde{\Psi}^{m}_{h}(\mathbb{T}^n)$ with the second principal symbol of $A\in\Psi^{m,m}_{2,h}(\mathcal{C})$.
\end{remark}
\subsection{Global Parametrices}
\begin{lemma} \label{global parametrix}
Suppose that $A\in\Psi^{m,l}_{2,h}(\mathcal{C})$ is globally elliptic (i.e., ${}^2 \sigma_{m,l}(A)$ vanishes nowhere). Then there exist $B,C\in\Psi^{-m,-l}_{2,h}(\mathcal{C})$ for which $AB-\mathrm{Id}\in\Re^0$, $CA-\mathrm{Id}\in\Re^0$; and $B$ and $C$ differ by an element of $\Re^{-l}$.
\end{lemma}
\begin{proof}
An iterative construction similar to the proof of \cite[Theorem~3.4]{Wu13}.
\end{proof}
\subsection{Microlocal Parametrices}
Let $\mathcal{C}=\{\mathbf{v}_1\cdot\xi = \ldots = \mathbf{v}_d\cdot\xi = 0\}$ for $\mathbf{v}_i\in\mathbb{R}^n$. Let $A\in\Psi^{m,l}_{2,h}(\mathcal{C})$. We partition the principal symbol space $S_\mathrm{pr}$, and restrict our focus to the lift $L_1$ of $\{\mathbf{v}_1\cdot\xi\neq 0\}\subset T^* \mathbb{T}^n$ (we have likewise partitioned $S_\mathrm{tot}$). Extend $\{\mathbf{v}_1,\ldots,\mathbf{v}_d\}$ to a basis
\[
\{\mathbf{v}_1,\ldots,\mathbf{v}_d,\mathbf{w}_{d+1},\ldots,\mathbf{w}_n\}
\]
for $\mathbb{R}^n$. Near the corner $SN(\mathcal{C}) = \partial S_\mathrm{pr}$, valid coordinates are $x$, $\zeta = \mathbf{v}_1\cdot\xi$, $$\Xi:=\left(\frac{\mathbf{v}_2\cdot\xi}{\mathbf{v}_1\cdot\xi},\ldots,\frac{\mathbf{v}_d\cdot\xi}{\mathbf{v}_1\cdot\xi}\right), \ \mathrm{and} \ \mathbf{W}:=(\mathbf{w}_{d+1}\cdot\xi,\ldots,\mathbf{w}_n\cdot\xi);$$ $h=0$ on the entirety of $S_\mathrm{pr}$.
\begin{lemma} \label{microlocal parametrix}
Take a point $\overline{p} = \left(\overline{x},\overline{\zeta},\overline{\Xi},\overline{\mathbf{W}}\right)\in {}^2 \mathrm{ell}(A) \cap L_1$. Then there exists $B\in\Psi^{-m,-l}_{2,h}(\mathcal{C})$ such that
\[
\overline{p}\notin{}^2 \mathrm{WF}_0'(AB-\mathrm{Id}) \cup {}^2 \mathrm{WF}_0'(BA-\mathrm{Id}).
\]
\end{lemma}
\begin{proof}
Similar to the construction of microlocal parametrix presented in Lemma 4.3 of \cite{Me-2}, but adapted to the second microlocal setting.
\end{proof}
A neat consequence of microlocal parametrices is the following elliptic regularity result:
\begin{corollary} \label{elliptic regularity}
Suppose $P\in\Psi^{m,l}_{2,h}(\mathcal{C})$ is elliptic on the microsupport of $A\in\Psi^{m',l'}_{2,h}(\mathcal{C})$:
\[
{}^2 \mathrm{WF}_{l'}'(A) \subset {}^2 \mathrm{ell}(P) \subset S_\mathrm{pr}.
\]
Then there exists a second microlocal operator $A_0\in\Psi^{-m+m',-l+l'}_{2,h}(\mathcal{C})$ such that $A = A_0 P + \Re^{l'}$.
\end{corollary}
\subsection{Mapping Properties}
As usual, let $\mathcal{C} = \{\mathbf{v}_1\cdot\xi = \ldots = \mathbf{v}_d\cdot\xi = 0\}$. Next, for $k,s\in\mathbb{R}$, we give an alternative characterization of $I^k_{(s)} (\mathcal{C})$ (these spaces of distributions were defined in \eqref{eq:distributions}):
\begin{lemma} \label{alternative characterization lemma}
\begin{equation} \label{alternative characterization}
I^k_{(s)} (\mathcal{C}) = \{u\in h^s L^2(\mathbb{T}^n) \ | \ \exists \ \mathrm{globally \ elliptic} \ A\in\Psi^{k,0}_{2,h}(\mathcal{C}), \ Au\in h^s L^2(\mathbb{T}^n)\}
\end{equation}
\end{lemma}
\begin{proof}
For simplicity, we may as well assume $s = 0$. We prove the lemma directly for $k\in\mathbb{Z}_{\geq 0}$; interpolation and duality then give the full result.
Assume $u\in I^k_{(0)} (\mathcal{C})$. For $x\in\mathbb{T}^n$, $(x,\xi)\in\mathcal{C}$ if and only if $(\mathbf{v}_1\cdot\xi)^2 + \ldots + (\mathbf{v}_d\cdot\xi)^2 = 0$. So
\[
B := h^{2} ((\mathbf{v}_1\cdot D_x)^2 + \ldots + (\mathbf{v}_d\cdot D_x)^2)
\]
belongs in $\mathfrak{M}_\mathcal{C}$. Hence, $h^{-1} B\in\Psi^1_h (\mathbb{T}^n)$ and $(h^{-1} B)^k u\in L^2(\mathbb{T}^n)$. At the same time, we take $\rho_\mathrm{ff} = |(\mathbf{v}_1\cdot\xi,\ldots,\mathbf{v}_d\cdot\xi)|$ as front face defining function. The symbol of $(h^{-1} B)^k$ is $h^{-k}|(\mathbf{v}_1\cdot\xi,\ldots,\mathbf{v}_d\cdot\xi)|^{2k}$, which lifts to $\rho_\mathrm{sf}^{-k} \rho_\mathrm{ff}^k$, so $(h^{-1} B)^k\in\Psi^{k,-k}_{2,h}(\mathcal{C})\subset\Psi^{k,0}_{2,h}(\mathcal{C})$\footnote{Really we should first `cut off' $B$ so its symbol is compactly supported in the fibers. Then only can it be regarded as a second microlocal operator. Similarly for the $B_j$ in the subsequent paragraph.} (since $k\geq 0$). Then $A = (I + h^{-1} B)^k$
is an elliptic operator satisfying $Au\in L^2(\mathbb{T}^n)$.
Conversely, consider the generators $B_1,\ldots,B_d$, $B_j = \mathbf{v}_j \cdot hD_x$, of $\mathfrak{M}_\mathcal{C}$. Suppose there exists elliptic $A\in\Psi^{k,0}_{2,h}(\mathcal{C})$ such that $Au\in L^2(\mathbb{T}^n)$. Let $0\leq |\beta|\leq k$. Then there exists a parametrix $A'\in\Psi^{-k,0}_{2,h}(\mathcal{C})$ such that
\[
h^{-|\beta|} \mathbf{B}^\beta u = h^{-|\beta|} \mathbf{B}^\beta A' A u + h^{-|\beta|} \mathbf{B}^\beta R u, \ \ R\in\Re^0.
\]
Note that $h^{-|\beta|} \mathbf{B}^\beta\in\Psi^{|\beta|,0}_{2,h}(\mathcal{C})$ (for each $1\leq j\leq d$, $\mathbf{v}_j\cdot\xi$ is a locally valid defining function for the front face). By Proposition \ref{two-sided ideal prop}, $h^{-|\beta|} \mathbf{B}^\beta R\in\Re^0$, so the latter summand in the RHS above lies in $I^\infty_{(0)}(\mathcal{C})\subset L^2(\mathbb{T}^n)$. Since $|\beta| \leq k$, $h^{-|\beta|} \mathbf{B}^\beta A'\in\Psi^{0,0}_{2,h}(\mathcal{C})$, so Proposition \ref{boundedness prop} implies that $h^{-|\beta|} \mathbf{B}^\beta A' (A u)\in L^2(\mathbb{T}^n)$.
\end{proof}
\begin{lemma}[Mapping Property] \label{mapping property lemma}
For $m\in\mathbb{R}$ and $l\geq 0$, $P\in\Psi^{m,l}_{2,h}(\mathcal{C})$ satisfies
\begin{equation} \label{mapping property}
P:I^k_{(s)}(\mathcal{C})\longrightarrow I^{k-m}_{(s-l)}(\mathcal{C})
\end{equation}
for each $k,s\in\mathbb{R}$. In particular, if $R\in\Re^l$, then
\[
R:I^{-\infty}_{(s)}(\mathcal{C})\longrightarrow I^\infty_{(s-l)}(\mathcal{C}).
\]
\end{lemma}
Since $I^0_{(0)}(\mathcal{C}) = L^2(\mathbb{T}^n)$, this property generalizes Proposition \ref{boundedness prop}. Define
\begin{equation} \label{partial Sobolev norm}
\|u_h\|_{I^k_{(s)}} := \left\|\mathcal{F}^{-1}_h \left(1 + |(\mathbf{v}_1\cdot\xi,\ldots,\mathbf{v}_d\cdot\xi)|^2\right)^{k/2} \mathcal{F}_h(h^{-s} u_h)\right\|_{L^2(\mathbb{T}^n)}.
\end{equation}
$\mathcal{F}_h$ is the semiclassical Fourier transform. We see that the $I^k_{(s)}$-norm is a partial Sobolev norm with respect to the characteristic derivatives.
\begin{proof}
Take $u\in I^k_{(s)}(\mathcal{C})$. By Lemma \ref{alternative characterization lemma}, there is an elliptic operator $A\in\Psi^{k,0}_{2,h}(\mathcal{C})$ for which $Au\in h^s L^2(\mathbb{T}^n)$. Choose $P\in\Psi_{2,h}^{m,l}(\mathcal{C})$. We want to prove there exists elliptic $\widetilde{A}\in\Psi^{k-m,0}_{2,h}(\mathcal{C})$ satisfying $\widetilde{A}Pu\in h^{s-l} L^2(\mathbb{T}^n)$. Let $\widetilde{A}$ be any elliptic element of $\Psi^{k-m,0}_{2,h}(\mathcal{C})$. At the same time, since $A$ is elliptic, there exists $B\in\Psi^{-k,0}_{2,h}(\mathcal{C})$ such that $BA - \mathrm{Id} = R\in\Re^0$. Therefore,
\[
\widetilde{A}Pu = \widetilde{A}P(BA - R)u = \widetilde{A}PB(Au) - (\widetilde{A}PR)u.
\]
$\widetilde{A}PR\in\Re^l$ (by Proposition \ref{two-sided ideal prop}) and $\widetilde{A}PB\in\Psi^{0,l}_{2,h}(\mathcal{C})$. Since $u\in h^s L^2(\mathbb{T}^n)$, certainly we have $(\widetilde{A}PR)u\in h^{s-l} L^2(\mathbb{T}^n)$. And since $l\geq 0$, $h^l \widetilde{A}PB\in\Psi^{-l,0}_{2,h}(\mathcal{C})$ maps $h^s L^2$ into itself, by Proposition \ref{boundedness prop}.
\end{proof}
\subsection{Second Wavefront Set}
\begin{definition} \label{def of 2-wavefront}
For any $m,l\in\mathbb{R}$, the $m,l$-graded \emph{second wavefront set} of a distribution $u\in I^{-\infty}_{(l)}(\mathcal{C})$ is
\[
{}^2 \mathrm{WF}^{m,l}(u) = \bigcap\{{}^2 \mathrm{char}(A) \ | \ A\in\Psi^{m,l}_{2,h}(\mathcal{C}),Au\in L^2(\mathbb{T}^n)\}.
\]
\end{definition}
Let $u\in I^{-\infty}_{(l)}(\mathcal{C})$. We will later use the containment property
\begin{equation} \label{eqn:containment}
m \leq m' \ \Longrightarrow \ {}^2 \mathrm{WF}^{m,l}(u) \subset {}^2 \mathrm{WF}^{m',l}(u).
\end{equation}
Also, put ${}^2 \mathrm{WF}^{\infty,l}(u) = \overline{\bigcup_{m\in\mathbb{R}} {}^2 \mathrm{WF}^{m,l}(u)}$, so that
\[
{}^2 \mathrm{WF}^{\infty,l}(u) = \bigcap\{{}^2 \mathrm{char}(A) \ | \ A\in\Psi^{m,l}_{2,h}(\mathcal{C}),Au\in I^\infty_{(0)}(\mathcal{C})\}.
\]
Notice that ${}^2 \mathrm{WF}^{\infty,l}(u) = \emptyset$ if and only if $u\in I^\infty_{(l)}(\mathcal{C})$, i.e., $u$ is a coisotropic distribution.
We see that ${}^2 \mathrm{WF}^{m,l}(u)\subset S_\mathrm{pr}$. Away from the coisotropic, this new wavefront is the same as standard semiclassical wavefront set:
\[
{}^2 \mathrm{WF}^{m,l}(u) \backslash SN(\mathcal{C}) \simeq \mathrm{WF}_h^m(u) \backslash \mathcal{C}.
\]
Just as the semiclassical pseudodifferential calculus $\Psi_h$ does not spread singularities as measured by $\mathrm{WF}_h$, we have:
\begin{lemma} \label{lem:pseudolocality}
Let $A\in\Psi^{m',l'}_{2,h}(\mathcal{C})$. For $u\in I^{-\infty}_{(l)}(\mathcal{C})$, we have
\[
{}^2 \mathrm{WF}^{m-m',l-l'}(Au) \subset {}^2 \mathrm{WF}_{l'}'(A) \cap {}^2 \mathrm{WF}^{m,l}(u).
\]
\end{lemma}
\begin{proof}
Analogous to the proof of microlocality in \cite[Proposition~4.2]{Me-2}.
\end{proof}
As a consequence, there is a partial converse to Lemma \ref{lem:pseudolocality}.
\begin{corollary} \label{wavefront is characteristic}
Let $A\in\Psi^{m',l'}_{2,h}(\mathcal{C})$ and $u\in I^{-\infty}_{(l)}(\mathcal{C})$. Then for any $m\in\mathbb{R}$,
\[
{}^2 \mathrm{WF}^{m,l}(u) \subset {}^2 \mathrm{WF}^{m-m',l-l'}(Au) \cup {}^2 \mathrm{char}(A).
\]
\end{corollary}
\begin{proof}
Similar to the proof of \cite[Proposition~4.3]{Me-2}.
\end{proof}
\begin{example} \label{ex:second wavefront}
Let $k\in\mathbb{Z}$. In $\mathbb{T}^n$, we consider the distribution
\[
u_k = \exp[i(k(x_1 + \ldots + x_{n-1}) + k^2 x_n)].
\]
Letting
\[
h^2_k = \frac{1}{(n-1)k^2 + k^4} \xrightarrow{|k|\rightarrow\infty} 0,
\]
we see that $u_k$ satisfies $(h^2_k \Delta - 1)u_k = 0$.
At the same time, we find that
\begin{align}
\mathrm{WF}_{h_k}(u_k) &= \{\xi_1=\ldots=\xi_{n-1}=0, \xi_n = 1\} \label{fiber} \\
&\subset \{\xi_1=\ldots=\xi_{n-1}=0, \xi_n\in\mathbb{R}\} =: \mathcal{C} \nonumber
\end{align}
Therefore, whatever second wavefront there is will lie in $SN(\mathcal{C})$.
For $\mathcal{C}$ as defined above, $\mathfrak{M}_\mathcal{C}$ is generated by $hD_{x_j}$, $j < n$. Fix $j$. Then we have
\[
h^{-1}(h D_{x_j}) u_k = D_{x_j} u_k = k u_k,
\]
so $D_{x_j} u_k\notin L^2$ uniformly as $|k|\rightarrow\infty$. Thus, $u_k$ is not a coisotropic distribution, so $\emptyset \neq {}^2 \mathrm{WF}^{\infty,0}(u_k)\subset SN(\mathcal{C})$.
Let $\xi' = (\xi_1,\ldots,\xi_{n-1})$ and $\hat{\xi}' = \xi' / |\xi'|$. Due to \eqref{fiber}, ${}^2 \mathrm{WF}^{\infty,0}(u_k) \subset \{|\hat{\xi}'| = 1, \xi_n = 1\}$.
Next, consider the operators $h(D_{x_i} - D_{x_j})\in\Psi^0_h(\mathbb{T}^n)$, $1\leq i,j\leq (n-1)$, formed from the generators of $\mathfrak{M}_\mathcal{C}$. Unlike the generators themselves, $h^{-1}(hD_{x_i} - hD_{x_j}) u_k = 0$. We may regard $D_{x_i} - D_{x_j}$ as a second microlocal operator\footnote{Again, after truncation.}.
Since $(D_{x_i} - D_{x_j}) u_k = 0$,
\[
{}^2 \mathrm{WF}^{\infty,0}(u_k) \subset \left\{{}^2 \sigma(D_{x_i} - D_{x_j}) = \frac{\xi_i}{h} - \frac{\xi_j}{h} = 0\right\}.
\]
So we must have $\hat{\xi}_i = \hat{\xi}_j$ for these $i,j$. This leads to the following claim:
\emph{Claim:} If $k\rightarrow +\infty$, then
\[
{}^2 \mathrm{WF}^{\infty,0}(u_k) = \left\{\hat{\xi}_1 = \ldots = \hat{\xi}_{n-1} = \frac{1}{\sqrt{n-1}}, \xi_n = 1\right\}.
\]
If instead $k\rightarrow -\infty$,
\[
{}^2 \mathrm{WF}^{\infty,0}(u_k) = \left\{\hat{\xi}_1 = \ldots = \hat{\xi}_{n-1} = -\frac{1}{\sqrt{n-1}}, \xi_n = 1\right\}.
\]
\emph{Proof of claim:} In both cases, all of the $\hat{\xi}_j$ ($j < n$) have the same sign. So it suffices to construct a second microlocal operator $A$ that `distinguishes' signs for, say, $\hat{\xi}_1$, and also for which $A u_k\in I^\infty_{(0)}(\mathcal{C})$.
We partition $S_\mathrm{tot}$ into $(n-1)$ symmetric pieces, the blowup of $\{\xi_j\neq 0\}\times\{h\geq 0\}$ for each of $j = 1,\ldots,n-1$. Since WLOG we chose to distinguish $\hat{\xi}_1>0$ from $\hat{\xi}_1<0$, we write a symbol for $A$ which is supported in the lift of $\{\xi_1\neq 0\}\times\{h\geq 0\}$ (and which is zero on the other $(n-2)$ pieces).
Now, suppose $k\rightarrow +\infty$. We write down $a\in C^\infty_c (S_\mathrm{tot})$ so that $A = {}^h \mathrm{Op_r}(a)$ is elliptic for $\hat{\xi}_1<0$ (but characteristic for $\hat{\xi}_1>0$), and so that $A u_k\in I^\infty_{(0)}(\mathcal{C})$. This will exclude the point
\[
\left(x,\hat{\xi}_1=-\frac{1}{\sqrt{n-1}},\ldots,\hat{\xi}_{n-1}=-\frac{1}{\sqrt{n-1}},\xi_n=1\right)
\]
from ${}^2 \mathrm{WF}^{\infty,0}(u_k)$.
Let $\psi\in C^\infty(\mathbb{R})$ be supported in $(0,\infty)$. Let $\varphi_j(x_j)\in C^\infty_c(S^1)$ be a bump function supported near some (arbitrary) $\bar{x}_j$, and let $\varphi(x) = \prod^n_{j=1} \varphi_j(x_j)$. Define
\[
a(x,\xi;h) := \psi\left(-\frac{\xi_1}{h}\right) \varphi(x)\in C^\infty_c(S_\mathrm{tot}).
\]
Since $A = {}^h \mathrm{Op_r}(a)$, we have
\[
A u_k(x) = \frac{1}{(2\pi h)^n} \int_{\mathbb{R}^n} \int_{\mathbb{T}^n} e^{\frac{i}{h}(x-y)\cdot\xi} \psi\left(-\frac{\xi_1}{h}\right) \varphi(y) u_k(y)\ dyd\xi.
\]
Let $\mathcal{F}$ be the \emph{non-semiclassical} Fourier transform; recall this takes smooth, compactly supported functions to Schwartz functions. We change variables by setting $\eta_j = \xi_j / h$, so that
\[
A u_k(x) = \mathcal{F}^{-1}_{\eta\rightarrow x} (\psi(-\eta_1) \mathcal{F}_{y\rightarrow\eta} \varphi u_k)(x).
\]
We compute
\begin{align*}
\mathcal{F}_{y\rightarrow\eta} (\varphi u_k)(\eta) &= \int_{\mathbb{T}^n} e^{-iy\cdot\eta} \varphi(y) e^{i(k(y_1+\ldots+y_{n-1})+k^2 y_n)}\ dy \\
&= \left[\prod^{n-1}_{j=1} \int_{S^1} e^{-iy_j\eta_j} \varphi_j(y_j) e^{iky_j}\ dy_j\right]\left[\int_{S^1} e^{-iy_n\eta_n} \varphi_n(y_n) e^{ik^2 y_n}\ dy_n\right].
\end{align*}
Hence,
\[
A u_k(x) = \mathcal{F}^{-1}_{\eta\rightarrow x} (\psi(-\eta_1) \widehat{\varphi}_1(\eta_1-k)\cdots\widehat{\varphi}_{n-1}(\eta_{n-1}-k)\widehat{\varphi}_n(\eta_n-k^2))(x).
\]
Finally, since $\psi(-\eta_1) = \psi(-\xi_1/h)$ is supported in $\{\eta_1\leq 0\}$, yet $\widehat{\varphi}_1(\eta_1-k)$ is a Schwartz function whose supported is translated to the right as $k\rightarrow +\infty$, we may conclude that $A u_k = O_{L^2}(h_k^\infty)$. This is stronger than $A u_k\in I^\infty_{(0)}(\mathcal{C})$, so certainly
\[
{}^2 \mathrm{WF}^{\infty,0}(u_k) = \mathbb{T}^n \times \left\{\hat{\xi}_1=\ldots=\hat{\xi}_{n-1}=\frac{1}{\sqrt{n-1}}, \xi_n=1\right\}.
\]
\end{example}
\section{\textsc{Results on propagation of second wavefront} \label{propagation results}}
\subsection{Real principal type propagation}
Let $\mathcal{C}\subset T^* \mathbb{T}^n$ be any linear coisotropic.
\begin{lemma} \label{lem:bounding}
For $s,r,\alpha,\beta\in\mathbb{R}$, let $f\in I^\alpha_{(r)} (\mathcal{C})$, $g\in I^\beta_{(s)} (\mathcal{C})$, and $P\in\Re^{r+s}$. Then
\[
|\left\langle Pf,g \right\rangle_{L^2(\mathbb{T}^n)}| \lesssim \|f\|_{I^\alpha_{(r)}} \|g\|_{I^\beta_{(s)}}
\]
\end{lemma}
For the definition of the norm $\|\cdot\|_{I^\alpha_{(r)}}$, refer to \eqref{partial Sobolev norm}.
\begin{proof}
Essentially, we factor $P\in\Re^{r+s}$ as the product of a residual operator of order $r$ and a residual operator of order $s$. For all $N$, we may choose globally elliptic $T\in\Psi^{-N,s}_{2,h}(\mathcal{C})$. Then there exists $T'\in\Psi^{N,-s}_{2,h}(\mathcal{C})$ such that $TT' + Q = \mathrm{Id}, \ Q\in\Re^0$. Thus,
\[
\left\langle Pf,g \right\rangle_{L^2(\mathbb{T}^n)} = \left\langle T'Pf,T^* g \right\rangle_{L^2(\mathbb{T}^n)} + \left\langle QPf,g \right\rangle_{L^2(\mathbb{T}^n)}.
\]
We have
\[
\left\langle QPf,g \right\rangle_{L^2(\mathbb{T}^n)} = \left\langle Pf,Q^* g \right\rangle_{L^2(\mathbb{T}^n)} = \left\langle h^s Pf,h^{-s} Q^* g \right\rangle_{L^2(\mathbb{T}^n)}.
\]
Since $h^s P\in\Re^r$, $h^{-s} Q^*\in\Re^s$, then $h^s Pf, h^{-s} Q^* g\in I^\infty_{(0)}(\mathcal{C})\subset L^2(\mathbb{T}^n)$. Thus,
\[
|\left\langle h^s Pf,h^{-s} Q^* g \right\rangle_{L^2(\mathbb{T}^n)}| \leq \|h^s Pf\|_{L^2(\mathbb{T}^n)} \|h^{-s} Q^* g\|_{L^2(\mathbb{T}^n)} \lesssim \|f\|_{I^\alpha_{(r)}} \|g\|_{I^\beta_{(s)}}.
\]
At the same time, $T'Pf\in I^\infty_{(0)}(\mathcal{C})\subset L^2(\mathbb{T}^n)$, and $T^* g\in L^2(\mathbb{T}^n)$ for all $N \geq -\beta$, so
\[
|\left\langle T'Pf,T^* g \right\rangle_{L^2(\mathbb{T}^n)}| \leq \|T'Pf\|_{L^2(\mathbb{T}^n)} \|T^* g\|_{L^2(\mathbb{T}^n)} \lesssim \|f\|_{I^\alpha_{(r)}} \|g\|_{I^\beta_{(s)}}. \qedhere
\]
\end{proof}
Fix $s\in\mathbb{R}$. Let $P\in\Psi^{m,l}_{2,h}(\mathcal{C})$, and suppose that $p_0 = {}^2 \sigma_{m,l}(P)$ is real valued. Let $H_{p_0}$ be the corresponding Hamiltonian vector field. We first show that if $u\in I^{-\infty}_{(s)}(\mathcal{C})$, and if $u$ satisfies the equation $Pu = f$,
then ${}^2\mathrm{WF}(u) \backslash {}^2\mathrm{WF}(f)$ propagates along null bicharacteristics.
\begin{remark} \label{rmk:scaling}
$S_\mathrm{pr}$ is a manifold with boundary $SN(\mathcal{C})$, so $H_{p_0}$ must be rescaled to be tangent to this boundary. If $\rho_\mathrm{ff}$ is a defining function for $SN(\mathcal{C})$, the rescaled vector field $\rho_\mathrm{ff}^{l-m+1} H_{p_0}$ is tangent to $SN(\mathcal{C})$. In particular, if any point in an orbit of $\rho_\mathrm{ff}^{l-m+1} H_{p_0}$ lies in $SN(\mathcal{C})$, then the entire orbit lies in $SN(\mathcal{C})$.
\end{remark}
\begin{theorem} \label{primary propagation}
For $P\in\Psi^{m,l}_{2,h}(\mathcal{C})$, if $Pu = f$ and $p_0$ is real valued, then for any $k\in\mathbb{R}$
\[
{}^2\mathrm{WF}^{k,s}(u) \backslash {}^2\mathrm{WF}^{k-m+1,s-l}(f)
\]
propagates along the flow of $\rho_\mathrm{ff}^{l-m+1} H_{p_0}$.
\end{theorem}
\begin{proof}
First, note that
\[
{}^2 \mathrm{WF}^{k,s}(u) \backslash {}^2 \mathrm{WF}^{k-m,s-l}(f) \subset {}^2 \mathrm{char}(P),
\]
due to Corollary \ref{wavefront is characteristic}.
Let $\overline{H}_{p_0} = \rho_\mathrm{ff}^{l-m+1} H_{p_0}$. Away from the boundary $SN (\mathcal{C})$, this result is no different from the usual semiclassical real principal type propagation. For this reason, we take a point $q\in SN (\mathcal{C})$ (at which $\overline{H}_{p_0}$ does not vanish, i.e., is not radial).
Since $u\in I^{-\infty}_{(s)}(\mathcal{C})$, there exists $\beta$ for which ${}^2 \mathrm{WF}^{\beta,s}(u) = \emptyset$.
(More precisely, if $u\in I^\gamma_{(s)}(\mathcal{C})$, then ${}^2 \mathrm{WF}^{\beta,s}(u) = \emptyset$ for any $\beta\leq\gamma$.) For some $\alpha > \beta$, assume that ${}^2 \mathrm{WF}^{\alpha - 1/2,s}(u)$ propagates along $\overline{H}_{p_0}$ flow. Moreover, assume absence of ${}^2 \mathrm{WF}^{\alpha,s}(u)$ at one end of a bicharacteristic segment (the end opposite $q$):
\[
\exp{(t_0 \overline{H}_{p_0})} q \notin {}^2 \mathrm{WF}^{\alpha,s}(u)
\]
for some small $t_0 > 0$. This implies, by \eqref{eqn:containment}, that $\exp{(t_0 \overline{H}_{p_0})} q \notin {}^2 \mathrm{WF}^{\alpha - 1/2,s}(u)$. Therefore,
\[
\exp{(t \overline{H}_{p_0})} q \notin {}^2 \mathrm{WF}^{\alpha - 1/2,s}(u), \ t\in [0,t_0].
\]
Finally, assume absence of ${}^2 \mathrm{WF}^{\alpha-m+1,s-l}(f)$ along the whole segment:
\begin{equation} \label{assumption on inhomogeneity}
\exp{(t \overline{H}_{p_0})} q \notin {}^2 \mathrm{WF}^{\alpha-m+1,s-l}(f), \ t\in [0,t_0].
\end{equation}
We will then show that
\[
\exp{(t \overline{H}_{p_0})} q \notin {}^2 \mathrm{WF}^{\alpha,s}(u), \ t\in [0,t_0].
\]
The idea is that $\alpha$ is increased, in increments of a half, until it reaches $k$. (It may be necessary to make minor numerological adjustments so that $\alpha$ equals $\beta$ plus an integral multiple of one-half, and $k$ is $\alpha$ plus an integral multiple of one-half.)
Next, define $\chi_0(s) = 0$ for $s \leq 0$, $\chi_0(s) = e^{-M/s}$ for $s > 0$ ($M>0$ to be specified); so $\chi_0$ is increasing on $(0,\infty)$ (but bounded above by one). Let $\chi \geq 0$ be a smooth function with $\chi(s) = 0$ for $s\leq 0$ which increases to $\chi(s) = 1$ for $s \geq 1$, with $\sqrt{\chi}$ and $\sqrt{\chi'}$ both smooth. Finally, let $\varphi$ be any smooth cutoff supported in $(-1,1)$.
We can choose coordinates $\rho_1,\ldots,\rho_{2n}$ on $S_\mathrm{pr}$ centered at $q$ in which (i) $\overline{H}_{p_0} = \partial_{\rho_1}$ and (ii) $SN (\mathcal{C}) = \{\rho_{2n}=0\}$, so that we may take $\rho_{2n}$ as front face boundary defining function. Set $\rho'=(\rho_2,\ldots,\rho_{2n})$. So, we assumed $\alpha,s$-regularity of $u$ at the point $(\rho_1=t_0,\rho'=0)$, and the other end of the piece of bicharacteristic is $q=(\rho_1=0,\rho'=0)$. Let $\lambda$ be a positive parameter, and define the principal symbol of the commutant as follows:
\begin{align*}
a_\lambda(\rho_1,\rho')&=\rho_{2n}^{-s+l/2+\alpha-(m-1)/2} \varphi^2(\lambda^2 |\rho'|^2) \ \chi_0(\lambda\rho_1 + 1) \ \chi(\lambda(t_0 - \rho_1)-1)\in S^{s-l/2-\alpha+(m-1)/2}(S_\mathrm{pr}).
\end{align*}
Then on the support of $a$, we have $|\rho'| < \lambda^{-1}$ and $\rho_1 \geq -\lambda^{-1}$, $\rho_1 \leq t_0 - \lambda^{-1}$. If the parameter $\lambda > 0$ is taken to be large, then $a$ is supported in a neighborhood of the bicharacteristic segment. By the short exact sequence for second principal symbol, there exists $A\in\Psi^{\alpha-(m-1)/2, s - l/2}_{2,h}(\mathcal{C})$ which has (real valued) principal symbol $a$ and such that ${}^2 \mathrm{WF}_{s - l/2}'(A) = \mathrm{supp}(a)$.
``Commuting'' this $A$ with $P$,
\begin{align*}
{}^2 \sigma_{2\alpha,2s}(-i(A^* AP - P^* A^* A)) &= {}^2 \sigma_{2\alpha,2s}(-i[A^* A,P]) + {}^2 \sigma_{2\alpha,2s}(-i(P - P^*)A^* A) \\
&= \overline{H}_{p_0}\left(a^2\right) - i \ {}^2 \sigma_{m-1,l}(P - P^*) \ {}^2 \sigma_{2\alpha - m + 1,2s - l}(A^* A) \\
&= b^2 - e^2 + g a^2.
\end{align*}
(The $2\alpha+1,2s$-principal symbol of the commutator vanishes.) Also, notice that the $m,l$-principal symbol of $P-P^*$ vanishes, since $p_0$ is real valued. Above, $b^2,e^2\in S^{2s-2\alpha}(S_\mathrm{pr})$ arise when $\overline{H}_{p_0} = \partial_{\rho_1}$ is applied to $\chi_0^2(\lambda\rho_1 + 1)$ and $\chi^2(\lambda(t_0 - \rho_1)-1)$, respectively:
\begin{align*}
b^2 &= 2\lambda \ \rho_\mathrm{ff}^{-2s+l+2\alpha-m+1} \varphi^4\left(\lambda^2 |\rho'|^2\right) \ \chi_0(\lambda\rho_1 + 1) \ \chi_0'(\lambda\rho_1 + 1) \ \chi^2(\lambda(t_0 - \rho_1) - 1) \\
e^2 &= -2\lambda \ \rho_\mathrm{ff}^{-2s+l+2\alpha-m+1} \varphi^4\left(\lambda^2 |\rho'|^2\right) \ \chi_0^2(\lambda\rho_1 + 1) \ \chi(\lambda(t_0 - \rho_1)-1) \ \chi'(\lambda(t_0 - \rho_1)-1)
\end{align*}
Note that since $\chi(\lambda(t_0 - \rho_1)-1)$ is decreasing for $\rho_1 \in [t_0 - 2\lambda^{-1},t_0 - \lambda^{-1}]$ (and otherwise constant), $\chi'(\lambda(t_0 - \rho_1)-1)$ is negative in that interval, which means $e^2$ is actually positive. So for large $\lambda$, $e$ is supported in the complement of ${}^2 \mathrm{WF}^{\alpha,s}(u)$. Our assumption that $\sqrt{\chi}$ and $\sqrt{\chi'}$ are smooth ensures that $e\in S^{s-\alpha}(S_\mathrm{pr})$. Conversely, since $\chi_0(\lambda\rho_1 + 1)$ is increasing for $\rho_1 \in (-\lambda^{-1},\infty)$, $\chi_0'(\lambda\rho_1 + 1)$ is positive in that interval (but $b^2$ is ``turned off'' at $t_0 - \lambda^{-1}$ by the $\chi^2$ term); hence, $b$ is supported along the whole segment.
Thus, again by the short exact sequence,
\begin{equation} \label{remainder}
-i(A^* AP - P^* A^* A) = B^* B + E^* E + A^* GA + R,
\end{equation}
where $R\in\Psi^{2\alpha - 1,2s}_{2,h}(\mathcal{C})$ satisfies ${}^2 \mathrm{WF}_{2s}'(R)\subset {}^2 \mathrm{WF}_{s - l/2}'(A)$, $B\in\Psi^{\alpha,s}_{2,h}(\mathcal{C})$ satisfies ${}^2\sigma_{\alpha,s}(B) = b$ and ${}^2 \mathrm{WF}_s'(B) = \mathrm{supp}(b)$, likewise for $E\in\Psi^{\alpha,s}_{2,h}(\mathcal{C})$; and $G\in\Psi^{m - 1,l}_{2,h}(\mathcal{C})$ has second principal symbol $g$ ($g$ may not be real valued). By construction, ${}^2 \mathrm{WF}_s'(E)$ is contained in $|\rho'| < \lambda^{-1}$, $\rho_1\in[t_0 - 2\lambda^{-1},t_0 - \lambda^{-1}]$, so for large $\lambda$, ${}^2 \mathrm{WF}_s'(E)$ is contained in the complement of ${}^2 \mathrm{WF}^{\alpha,s}(u)$; thus $\|Eu\|_{L^2(\mathbb{T}^n)} < \infty$. Similarly, ${}^2 \mathrm{WF}_{2s}'(R)$ is contained in $|\rho'| < \lambda^{-1}$, $\rho_1\in[-\lambda^{-1},t_0 - \lambda^{-1}]$, so for large $\lambda$, ${}^2 \mathrm{ell}(R) \subset {}^2 \mathrm{WF}_{2s}'(R)$ is disjoint from ${}^2 \mathrm{WF}^{\alpha - 1/2,s}(u)$; hence $|\left\langle Ru,u \right\rangle_{L^2(\mathbb{T}^n)}|$ is bounded as well. On the other hand, ${}^2 \mathrm{WF}_s'(B)$ is contained in $|\rho'| < \lambda^{-1}$, $\rho_1\in[-\lambda^{-1},t_0 - \lambda^{-1}]$,
so \emph{a priori} $\|Bu\|_{L^2(\mathbb{T}^n)}$ is unbounded.
Pairing both sides of equation \eqref{remainder} with the distribution $u$, and using $Pu = f$, we have
\begin{equation} \label{ineq:inequality}
\|Bu\|^2_{L^2(\mathbb{T}^n)} \leq |\left\langle Ru,u \right\rangle_{L^2(\mathbb{T}^n)}| \ + \ |\left\langle Au,G^* Au \right\rangle_{L^2(\mathbb{T}^n)}| \ + 2 \ |\left\langle Au,Af \right\rangle_{L^2(\mathbb{T}^n)}| \ + \|Eu\|^2_{L^2(\mathbb{T}^n)}.
\end{equation}
Here we have used
\[
\left\langle (P^*A^*A - A^*AP)u,u \right\rangle_{L^2(\mathbb{T}^n)} = 2i \ \mathrm{Im}\left\langle Au,APu \right\rangle_{L^2(\mathbb{T}^n)}.
\]
We have already showed that the first and last terms on the right side of \eqref{ineq:inequality} are uniformly bounded as $h\downarrow 0$.
Let $T\in\Psi^{(m-1)/2,l/2}_{2,h}(\mathcal{C})$ be globally elliptic, with parametrix $T'$: $T' T + Q = \mathrm{Id}$ for some $Q\in\Re^0$. We have
\begin{equation} \label{the other term}
|\left\langle Au,G^* Au \right\rangle_{L^2(\mathbb{T}^n)}| \leq \|TAu\|^2_{L^2(\mathbb{T}^n)} + \|(T')^* G^* Au\|^2_{L^2(\mathbb{T}^n)} + |\left\langle QAu,G^* Au \right\rangle_{L^2(\mathbb{T}^n)}|.
\end{equation}
We may control $\left\langle QAu,G^* Au \right\rangle_{L^2(\mathbb{T}^n)}$ by application of Lemma \ref{lem:bounding}. Also, the principal symbols of $TA,(T')^* G^* A\in\Psi^{\alpha,s}_{2,h}(\mathcal{C})$ are multiples of $a$, so:
\textit{Claim:} The first two terms on the RHS of \eqref{the other term} may be absorbed into $\|Bu\|^2_{L^2(\mathbb{T}^n)}$, for sufficiently large $M$. Comparing $a^2$ and $b^2$, it will suffice to show that
\[
\chi_0 (\lambda \rho_1 + 1) \leq 2 \lambda \chi_0'(\lambda \rho_1 + 1) \Longleftrightarrow (\log\chi_0)' \geq \frac{1}{2\lambda}.
\]
This amounts to finding $M$ for which $M\geq (\lambda \rho_1 + 1)^2 / (2\lambda)$. Since $0\leq \rho_1 \leq t_0$, for all $\lambda$, no matter how large, there exists such an $M$.
Also, Cauchy--Schwarz gives
\begin{equation} \label{inhomogeneity}
|\left\langle Au,Af \right\rangle_{L^2(\mathbb{T}^n)}| \leq \|TAu\|^2_{L^2(\mathbb{T}^n)} + \|(T')^* Af\|^2_{L^2(\mathbb{T}^n)} + |\left\langle QAu,Af \right\rangle_{L^2(\mathbb{T}^n)}|.
\end{equation}
$\|TAu\|^2_{L^2(\mathbb{T}^n)}$ can be absorbed into $\|Bu\|^2_{L^2(\mathbb{T}^n)}$, as justified by the above claim.
$\|(T')^* Af\|^2_{L^2(\mathbb{T}^n)}$ is controlled as a result of assumption \eqref{assumption on inhomogeneity}, as $(T')^* A\in\Psi^{\alpha-m+1,s-l}_{2,h}(\mathcal{C})$. We may directly apply Lemma \ref{lem:bounding} to control $\left\langle QAu,Af \right\rangle_{L^2(\mathbb{T}^n)}$.
Thus, $\|Bu\|_{L^2(\mathbb{T}^n)}<\infty$, so ${}^2 \mathrm{WF}^{\alpha,s}(u)$ is absent on the whole bicharacteristic segment.
\end{proof}
\begin{example}
We explicitly write down the vector field $H_{p_0}$ in a simple case. The example is $\mathcal{C} = \{\xi_1=\xi_2=0\}$ in $T^* \mathbb{T}^3$. The coordinates on $S_\mathrm{pr}$ are $x,\theta,\xi_3$, and $\rho_\mathrm{ff} = \sqrt{\xi_1^2 + \xi_2^2}$, where
\[
\cos(\theta) = \frac{\xi_1}{\sqrt{\xi_1^2 + \xi_2^2}}, \ \ \sin(\theta) = \frac{\xi_2}{\sqrt{\xi_1^2 + \xi_2^2}}.
\]
These coordinates\footnote{In codimension 3 and higher, we would not be able to use cosine and sine, but instead projective coordinates} are valid up to the boundary of $S_\mathrm{pr}$, namely $SN(\mathcal{C}) = \{\rho_\mathrm{ff} = 0\}$.
Let $\omega = \sum_{j=1}^3 dx_j\wedge d\xi_j$ denote the standard symplectic form on $T^* \mathbb{T}^3$. The blowdown map is
\[
\beta: S_\mathrm{pr}\longrightarrow T^* \mathbb{T}^3, \ \ \beta(x,\rho_\mathrm{ff},\theta,\xi_3) = (x,\rho_\mathrm{ff}\cos\theta,\rho_\mathrm{ff}\sin\theta,\xi_3).
\]
Then the Hamiltonian vector field of $p_0\in C^\infty(S_\mathrm{pr}) = S^0 (S_\mathrm{pr})$ with respect to $\beta^* \omega$ is
\begin{equation} \label{Hamiltonian vf}
H_{p_0} = \left(\frac{\partial p_0}{\partial\rho_\mathrm{ff}} \cos\theta - \frac{1}{\rho_\mathrm{ff}} \frac{\partial p_0}{\partial\theta} \sin\theta\right) \frac{\partial}{\partial x_1}
+ \left(\frac{\partial p_0}{\partial\rho_\mathrm{ff}} \sin\theta + \frac{1}{\rho_\mathrm{ff}} \frac{\partial p_0}{\partial\theta} \cos\theta\right) \frac{\partial}{\partial x_2}
+ \frac{\partial p_0}{\partial\xi_3} \frac{\partial}{\partial x_3} +
\end{equation}
\[
+ \frac{1}{\rho_\mathrm{ff}} \left(\frac{\partial p_0}{\partial x_1} \sin\theta - \frac{\partial p_0}{\partial x_2} \cos\theta\right) \frac{\partial}{\partial\theta}
- \left(\cos\theta \frac{\partial p_0}{\partial x_1} + \sin\theta \frac{\partial p_0}{\partial x_2}\right) \frac{\partial}{\partial\rho_\mathrm{ff}} - \frac{\partial p_0}{\partial x_3} \frac{\partial}{\partial\xi_3}.
\]
Note that $\rho_\mathrm{ff} H_{p_0}$ is well defined up to the boundary. \emph{However, as explained below, it may not be necessary to multiply by $\rho_\mathrm{ff}$ to ensure tangency to the boundary.}
Moreover, if $p_0(x,\xi_1,\xi_2,\xi_3)$ is smooth (in particular, smooth in $\xi_1$ and $\xi_2$), then since $\xi_1 = \rho_\mathrm{ff}\cos\theta$, $\xi_2 = \rho_\mathrm{ff}\sin\theta$, we have
\begin{equation} \label{cancellation}
\frac{\partial p_0}{\partial\theta} = \rho_\mathrm{ff}\left(\cos\theta\frac{\partial p_0}{\partial\xi_2} - \sin\theta\frac{\partial p_0}{\partial\xi_1}\right).
\end{equation}
Notice that $\rho_\mathrm{ff}$ cancels with $\rho_\mathrm{ff}^{-1}$ in the coefficients of $\partial_{x_1}$ and $\partial_{x_2}$ above.
\end{example}
\subsection{Principal type propagation, version 2}
We start with smooth, real valued principal symbol $p_0(\xi)$ depending only on the fiber variables in $T^* \mathbb{T}^n$.
Then the corresponding Hamiltonian vector field $H_{p_0}$ is \emph{a priori} tangent to the boundary $SN(\mathcal{C})$. This can be seen in \eqref{Hamiltonian vf} by setting $\partial_{x_j} p_0 = 0$, together with \eqref{cancellation}. Thus, even without rescaling, $H_{p_0}$ is well defined up to $SN(\mathcal{C})$.
So we have the following theorem, with the same proof as that of \autoref{primary propagation}:
\begin{theorem} \label{take two}
More generally, let $u\in I^{-\infty}_{(s)}(\mathcal{C})$. For $P\in\Psi^{m,l}_{2,h}(\mathcal{C})$ with real principal symbol $p_0$, suppose $Pu = O_{L^2(\mathbb{T}^n)}(h^\infty)$. If $p_0$ depends only on $\xi$, and is smooth in $\xi$, then for any $k$ ${}^2 \mathrm{WF}^{k,s}(u)$ propagates along the flow lines of $H_{p_0}$.
\end{theorem}
The key step in the proof of \autoref{primary propagation} is the ability to choose coordinates $\rho_j$ satisfying (i) $\overline{H}_{p_0} = \partial_{\rho_1}$ and (ii) $SN(\mathcal{C}) = \{\rho_{2n}=0\}$. This is possible precisely because $\overline{H}_{p_0} = \rho_\mathrm{ff}^{l-m+1} H_\mathrm{p_0}$ is tangent to $SN (\mathcal{C})$. Thus, if we assume $p_0 = p_0(\xi)$ (and choose not to rescale), then $H_\mathrm{p_0}$ is again tangent to $SN(\mathcal{C})$, and this choice of coordinates is again possible.
\begin{remark}
(1) For \emph{any} $p_0\in C^\infty(S_\mathrm{pr})$ (whether or not it is smooth `downstairs', whether or not it is independent of $x$), we may decompose $H_{p_0}$ as follows:
\begin{equation} \label{vf decomposition}
H_{p_0} = \vec{V}_1 + \frac{1}{\rho_\mathrm{ff}} \vec{V}_2,
\end{equation}
where $\vec{V}_1$ and $\vec{V}_2$ are each smooth up to the boundary $SN(\mathcal{C}) = \{\rho_\mathrm{ff} = 0\}$. This decomposition is not canonical, however. To see this, notice that we may further decompose $\vec{V}_2$ as $\vec{V}_2 = \vec{Y}_1 + \rho_\mathrm{ff} \vec{Y}_2$. Thus,
\begin{equation} \label{vf decomposition 2}
H_{p_0} = (\vec{V}_1 + \vec{Y}_2) + \rho_\mathrm{ff}^{-1} \vec{Y}_1,
\end{equation}
where $\vec{V}_1 + \vec{Y}_2$, $\vec{Y}_1$ are each smooth up to $SN(\mathcal{C})$. While the role of $\vec{V}_2$ is therefore not unique, we always have
\[
{\vec{V}_2}|_{SN(\mathcal{C})} = {\vec{Y}_1}|_{SN(\mathcal{C})} = (\rho_\mathrm{ff} H_{p_0})|_{SN(\mathcal{C})}.
\]
(2)
If we rescale: in \autoref{primary propagation}, second wavefront is propagating along $\rho_\mathrm{ff} H_{p_0} = \rho_\mathrm{ff} \vec{V}_1 + \vec{V}_2$.
(If we just wanted to prove propagation away from the boundary of $S_\mathrm{pr}$, there was no need to rescale at all. Away from $SN(\mathcal{C})$, rescaling merely reparametrizes the same flow lines.)
(3) If instead we assume $p_0 = p_0(\xi)$ and smoothness in $\xi$, then as we have showed, $H_{p_0}$ need not be rescaled.
Under these assumptions, in the example,
\[
\vec{Y}_1 = \left(\frac{\partial p_0}{\partial x_1} \sin\theta - \frac{\partial p_0}{\partial x_2} \cos\theta\right) \frac{\partial}{\partial\theta}, \ \ \vec{Y}_2 = \left(\cos\theta\frac{\partial p_0}{\partial\xi_2} - \sin\theta\frac{\partial p_0}{\partial\xi_1}\right)\left(\cos\theta\frac{\partial}{\partial x_2} - \sin\theta\frac{\partial}{\partial x_1}\right).
\]
And as we see, this $\vec{Y}_1$ is zero when $p_0 = p_0(\xi)$, so the problematic term $\vec{Y}_1 / {\rho_\mathrm{ff}}$ in \eqref{vf decomposition 2} drops out. So then $H_{p_0} = \vec{V}_1 + \vec{Y}_2$, and \autoref{take two} gives propagation along this vector field.
\end{remark}
\subsection{Secondary propagation of coisotropic wavefront\label{secondary propagation subsection}}
Let $P\in\widetilde{\Psi}^{0}_{h}(\mathbb{T}^n)$ have real valued principal symbol $p_0$. Recall from Remark \ref{one calculus in the other} that $P$ can be regarded as a second microlocal operator: $P\in\widetilde{\Psi}^{0}_{h}(\mathbb{T}^n)\subset\Psi^{0,0}_{2,h}(\mathcal{C})$.
We will abuse notation and refer to the second principal symbol of $P$ also as $p_0$.
The linear coisotropic $\mathcal{C}$ is given by
\[
\mathbb{T}^n_x \times \{\mathbf{v}_1 \cdot \xi = \ldots = \mathbf{v}_d \cdot \xi = 0\}.
\]
for linearly independent $\{\mathbf{v}_1,\ldots,\mathbf{v}_d\}\subset\mathbb{R}^n$. Let $\widetilde{\xi}' = (\mathbf{v}_1 \cdot \xi,\ldots,\mathbf{v}_d \cdot \xi)$. We extend $\{\mathbf{v}_1,\ldots,\mathbf{v}_d\}$ to a basis
\[
\{\mathbf{v}_1,\ldots,\mathbf{v}_d,\mathbf{w}_{d+1},\ldots,\mathbf{w}_n\}
\]
for $\mathbb{R}^n$, as before. Let $\widetilde{\xi}'' = (\mathbf{w}_{d+1} \cdot \xi,\ldots,\mathbf{w}_n \cdot \xi)$ and $\widetilde{\xi} = (\widetilde{\xi}',\widetilde{\xi}'')$. Locally, we may define $\widetilde{x}'$, $\widetilde{x}''$, and $\widetilde{x}$ analogously. In the coordinates $(\widetilde{x},\widetilde{\xi})$, $\mathcal{C} = \mathbb{T}^n_{\widetilde{x}} \times \{\widetilde{\xi}' = 0\}$.
Further assume that $p_0$ is a function only of $\widetilde{\xi}$, i.e., is independent of $\widetilde{x}$. Hence, $H_{p_0}$ is tangent to $\mathcal{C}$. Set $\rho_\mathrm{ff} := |\widetilde{\xi}'|$, and $\Gamma' := \widetilde{\xi}' / |\widetilde{\xi}'|$. As the notation suggests, $\rho_\mathrm{ff}$ is a front face defining function: $\partial S_\mathrm{pr} = SN(\mathcal{C}) = \{\rho_\mathrm{ff} = 0\}$.
For our next result, we impose a further condition, on the subprincipal symbol of $P$. Locally, we may write $P = {}^h \mathrm{Op_W}(p)$, where the total symbol $p$ decomposes as
\[
p = p_0 + h\mathrm{sub}(P) + O(h^2).
\]
(We specified the Weyl quantization here, so that the subprincipal symbol is the $O(h)$ term of the total symbol.) We assume that $\mathrm{sub}(P) \equiv 0$.
We remarked earlier that $P$ can be regarded as an element of $\Psi^{0,0}_{2,h}(\mathcal{C})$. We can Taylor expand $p_0$ at $\mathcal{C}$, partially in the characteristic variables $\widetilde{\xi'}$, to obtain
\[
{p_0}\restriction_\mathcal{C}\left(\widetilde{\xi}''\right) + \left.\frac{\partial p_0}{\partial\widetilde{\xi}'}\right|_\mathcal{C}\left(\widetilde{\xi}''\right)\cdot\widetilde{\xi}' + \frac{1}{2}\left.\frac{\partial^2 p_0}{{\partial\widetilde{\xi}'}^2}\right|_\mathcal{C}\left(\widetilde{\xi}''\right) \cdot \left(\widetilde{\xi}'\right)^2 + O\left(\left(\widetilde{\xi}'\right)^3\right).
\]
Note that $p_0\restriction_\mathcal{C}\left(\widetilde{\xi}''\right) = p_0\left(0,\widetilde{\xi}''\right)$.
The corresponding Hamiltonian vector field, computed term-by-term, is
\small
\[
H_{p_0} = \frac{\partial {p_0}\restriction_\mathcal{C}}{\partial\widetilde{\xi}''}\cdot\partial_{\widetilde{x}''} + \left[\left(\left.\frac{\partial^2 p_0}{\partial\widetilde{\xi}'\partial\widetilde{\xi}''}\right|_\mathcal{C}\left(\widetilde{\xi}''\right)\right) \partial_{\widetilde{x}''}\right]\cdot\widetilde{\xi}' + \left.\frac{\partial p_0}{\partial\widetilde{\xi}'}\right|_\mathcal{C}\left(\widetilde{\xi}''\right)\cdot\partial_{\widetilde{x}'} + \left[\left(\left.\frac{\partial^2 p_0}{{\partial\widetilde{\xi}'}^2}\right|_\mathcal{C}\left(\widetilde{\xi}''\right)\right) \widetilde{\xi}'\right]\cdot\partial_{\widetilde{x}'} + O\left(\left(\widetilde{\xi}'\right)^2\right).
\]
\normalsize
In the coordinates of $S_\mathrm{pr}$, $H_{p_0}$ is lifted to
\begin{align}
\mathbf{H} &= \left.\frac{\partial p_0}{\partial\widetilde{\xi}''}\right|_{SN(\mathcal{C})}\left(\widetilde{\xi}''\right)\cdot\partial_{\widetilde{x}''} + \left.\frac{\partial p_0}{\partial\widetilde{\xi}'}\right|_{SN(\mathcal{C})}\left(\widetilde{\xi}''\right)\cdot\partial_{\widetilde{x}'} + \label{Hamilton} \\
&+ \rho_\mathrm{ff}\left(\left[\left(\left.\frac{\partial^2 p_0}{\partial\widetilde{\xi}'\partial\widetilde{\xi}''}\right|_{SN(\mathcal{C})}\left(\widetilde{\xi}''\right)\right) \partial_{\widetilde{x}''}\right]\cdot\Gamma' + \left[\left(\left.\frac{\partial^2 p_0}{{\partial\widetilde{\xi}'}^2}\right|_{SN(\mathcal{C})}\left(\widetilde{\xi}''\right)\right) \Gamma'\right]\cdot\partial_{\widetilde{x}'}\right) + \rho_\mathrm{ff}^2 H'. \nonumber
\end{align}
$H'$ is tangent to $SN (\mathcal{C}) = \{\rho_\mathrm{ff} = 0\}$, as it does not contain $\partial_{\widetilde{\xi}}$. Let $H_1$ refer to the sum of the first two terms of $\mathbf{H}$, and $H_2$ to the $O(\rho_\mathrm{ff})$-piece of $\mathbf{H}$ (i.e., $H_2$ is the next order jets to $H_1$). Before stating the theorem, we give an example.
\begin{example} \label{ex:symbol class}
Suppose $a\in C^\infty_c(S_\mathrm{tot})$ for the coisotropic $\{\mathbf{v}_1 \cdot \xi = \mathbf{v}_2 \cdot \xi = 0\} \subset T^*\mathbb{T}^3$. For later use, we explicitly determine the symbol class to which $h^{|\alpha|+|\beta|-1}\left(\partial^\alpha_{\widetilde{\xi}} \partial^\beta_{\widetilde{x}} a\right)$ belongs. We employ the following coordinates in one coordinate patch of $S_\mathrm{tot}$: $\widetilde{x}$, $H = h / {\widetilde{\xi}_1}$, $\widetilde{\xi}_1$, $\Xi_2 = \widetilde{\xi}_2 / \widetilde{\xi}_1$, and $\widetilde{\xi}_3 = \mathbf{w}_3 \cdot \xi$. $H$ is a defining function for the side face, $\widetilde{\xi}_1$ for the front face. We compute $h\partial_{\widetilde{\xi}_1} = H(\widetilde{\xi}_1\partial_{\widetilde{\xi}_1} - H\partial_{\Xi_2} - H\partial_H)$. The lifted vector field in parentheses is tangent to both side and front faces. We also have $h\partial_{\widetilde{\xi}_2} = H\partial_{\Xi_2}$. Therefore,
\[
h^{|\alpha|+|\beta|-1}\left(\partial^\alpha_{\widetilde{\xi}} \partial^\beta_{\widetilde{x}} a\right) \in S^{1-|\alpha|-|\beta|,1-|\beta|-\alpha_3}(S_\mathrm{tot}).
\]
More generally, for $\{\mathbf{v}_1 \cdot \xi = \ldots = \mathbf{v}_d \cdot \xi = 0\}\subset T^*\mathbb{T}^n$,
\[
h^{|\alpha|+|\beta|-1}\left(\partial^\alpha_{\widetilde{\xi}} \partial^\beta_{\widetilde{x}} a\right)\in S^{1-|\alpha|-|\beta|,1-|\beta|-(\alpha_{d+1}+\ldots+\alpha_n)}(S_\mathrm{tot}).
\]
\end{example}
\begin{theorem} \label{secondary propagation}
Let $\mathcal{C}$ be a linear coisotropic submanifold. Assume $P\in\widetilde{\Psi}^0_h(\mathbb{T}^n)$ has real valued principal symbol depending only on the fiber variables in $T^* \mathbb{T}^n$, and subprincipal symbol identically equal to zero. Let $u \in L^2(\mathbb{T}^n)$ satisfy $Pu = O_{L^2(\mathbb{T}^n)}(h^\infty)$. Then for all $l \leq 0$, ${}^2 \mathrm{WF}^{\infty,l}(u) \cap SN(\mathcal{C})$ propagates along the flow of $H_1$; and for all $l \leq -1$, ${}^2 \mathrm{WF}^{\infty,l}(u) \cap SN(\mathcal{C})$ propagates along the flow of both $H_1$ and $H_2$.
\end{theorem}
Since $u \in L^2(\mathbb{T}^n) = I^0_{(0)}(\mathcal{C})$ and $I^0_{(0)}(\mathcal{C})\subset I^0_{(l)}(\mathcal{C})$ for $l\leq 0$, it makes sense to consider the $m,l$-wavefront set of $u$ for any $m\in\mathbb{R}$.
\begin{proof}
The vector field $H_1$ is the same as $(H_{p_0})|_{SN(\mathcal{C})}$ of \autoref{take two} in the case $m = l = 0$. Therefore, we obtain $H_1$ invariance by simply quoting \autoref{take two}.
Since the coordinates $(\widetilde{x},\widetilde{\xi})$ are valid locally in a neighborhood of $\mathcal{C}$, we may extend the vector field $H_1$ to a neighborhood of $SN(\mathcal{C})$. For $\epsilon > 0$, we extend $H_1$ to $N := \{\rho_\mathrm{ff} = |\widetilde{\xi}'| < \epsilon\}$. Let $\widetilde{H}_2$ be defined on $N$ by $\widetilde{H}_2 := \rho_\mathrm{ff}^{-1}(\mathbf{H} - H_1)$. We see that $\widetilde{H}_2$ coincides with $H_2$ on $SN(\mathcal{C})$. Since the principal symbol of $P$ is independent of $\widetilde{x}$, then $\widetilde{\xi}$ is constant along the flows of $H_1$ and $\widetilde{H}_2$, so the $(H_1,\widetilde{H}_2)$ joint flow from $N$ stays in this neighborhood of $SN(\mathcal{C})$ for all times.
Next, there exists $m_0\in\mathbb{R}$ for which ${}^2 \mathrm{WF}^{m_0,l}(u) = \emptyset$. For some $m > m_0$, assume ${}^2 \mathrm{WF}^{m - 1/2,l}(u)$ is invariant under the $\widetilde{H}_2$ flow. Take any $\zeta \in SN(\mathcal{C})$ and suppose that $\zeta \notin {}^2 \mathrm{WF}^{m,l}(u)$. We seek to prove that (the closure of) the $\widetilde{H}_2$ orbit through $\zeta$ is disjoint from ${}^2 \mathrm{WF}^{m,l}(u)$. This argument may then be iterated to show $\widetilde{H}_2$ invariance of ${}^2 \mathrm{WF}^{\infty,l}(u)$. Let $\overline{\mathcal{O}}_{H_1}(\zeta)$ refer to (the closure of) the $H_1$ orbit through $\zeta$; due to $H_1$-invariance, we know that $\overline{\mathcal{O}}_{H_1}(\zeta) \cap {}^2 \mathrm{WF}^{m,l}(u)=\emptyset$. Let $\overline{\mathcal{O}}_{\widetilde{H}_2}(\zeta)$ refer to (the closure of) the $\widetilde{H}_2$ orbit through $\zeta$; since $\zeta \notin {}^2 \mathrm{WF}^{m - 1/2,l}(u)$, we know $\overline{\mathcal{O}}_{\widetilde{H}_2}(\zeta) \cap {}^2 \mathrm{WF}^{m - 1/2,l}(u)=\emptyset$.
Since $\zeta\in\{\rho_\mathrm{ff} = 0\}$, $\rho_\mathrm{ff} = |\widetilde{\xi}'|$, and $\widetilde{\xi}$ is constant on $H_1$ orbits, the entire $H_1$ orbit containing $\zeta$ lies in $SN(\mathcal{C})$. Since $\overline{\mathcal{O}}_{H_1}(\zeta) \cap {}^2 \mathrm{WF}^{m,l}(u)=\emptyset$, there exists a neighborhood of this orbit closure in $S_\mathrm{pr}$ that is disjoint from ${}^2 \mathrm{WF}^{m,l}(u)$. Choose nonnegative $a_0\in C^\infty(S_\mathrm{pr})$ whose support contains $\overline{\mathcal{O}}_{H_1}(\zeta)$, whose support is contained in this neighborhood (so is disjoint from ${}^2 \mathrm{WF}^{m,l}(u)$), whose support is in $N$ (see previous paragraph), and such that $H_1(a_0) = 0$. Note that in particular, $\mathrm{supp}(a_0)$ is disjoint from ${}^2 \mathrm{WF}^{m - 1/2,l}(u)$.
Fix $\delta \in (0,1)$. We consider $\widetilde{H}_2$ flow segments with one endpoint $\exp\left(0 \widetilde{H}_2\right)$ on the $H_1$ orbit through $\zeta$ (so one of these segments has endpoint $\zeta$ itself) and other endpoint $\exp\left(-\delta \widetilde{H}_2\right)$. Informally, we are taking the $H_1$ orbit passing through $\zeta$ and ``smearing'' it along the $\widetilde{H}_2$ flow; see Figure \ref{fig:smear}.
Next, for $p \in N$, put
\[
a_1(p) := -\int^\delta_0 (1 - s) a_0\left(\exp\left(-s \widetilde{H}_2\right) p \right) ds.
\]
Away from $N$, we are certainly off the support of $a_0$, so we may define $a_1$ to be zero on $S_\mathrm{pr} \backslash N$; then $a_1\in C^\infty(S_\mathrm{pr})$. We are most interested in $p\in\overline{\mathcal{O}}_{H_1}(\zeta)\subset SN (\mathcal{C})$. We have
\begin{equation} \label{eqn:symbol support}
\mathrm{supp}(a_1)\supseteq \{\exp(-s \widetilde{H}_2) p \ | \ 0\leq s\leq\delta, \ p\in\overline{\mathcal{O}}_{H_1}(\zeta)\}.
\end{equation}
Since $\mathrm{supp}(a_0) \cap {}^2 \mathrm{WF}^{m - 1/2,l}(u) = \emptyset$ and ${}^2 \mathrm{WF}^{m - 1/2,l}(u)$ is $\widetilde{H}_2$ invariant, we have $\mathrm{supp}(a_1) \cap {}^2 \mathrm{WF}^{m - 1/2,l}(u) = \emptyset$.
\begin{figure}\label{fig:smear}
\end{figure}
Then define $\widetilde{a}_1 := \rho_\mathrm{ff}^{-(2l+1)+2m} a_1 \in S^{2l+1-2m}(S_\mathrm{pr})$. We compute
\begin{align*}
\rho_\mathrm{ff} \widetilde{H}_2(\widetilde{a}_1) &= \rho_\mathrm{ff}^{-2l+2m} \widetilde{H}_2(a_1) = \rho_\mathrm{ff}^{-2l+2m} \int^\delta_0 (1-s) \frac{\partial}{\partial s}\left[a_0\left(\exp\left(-s \widetilde{H}_2\right)\right)\right] ds \\
&= \rho_\mathrm{ff}^{-2l+2m} \int^\delta_0 a_0\left(\exp\left(-s \widetilde{H}_2\right)\right) ds + (1-\delta) \rho_\mathrm{ff}^{-2l+2m} a_0\left(\exp\left(-\delta \widetilde{H}_2\right)\right) - \rho_\mathrm{ff}^{-2l+2m} a_0 \\
&=: \int^\delta_0 b^2_s \ ds + c^2 - d^2.
\end{align*}
We have $[H_1,\widetilde{H}_2]=0$. This, combined with $H_1(a_0)=0$, implies $H_1(a_1)=0$, which in turn implies $H_1(\widetilde{a}_1)=0$.
Choose $A\in\Psi^{2m,2l+1}_{2,h}(\mathcal{C})$ with principal symbol $\widetilde{a}_1$, such that ${}^2 \mathrm{WF}_{2l+1}'(A) = \mathrm{supp}(\widetilde{a}_1)$. Locally, we may write $A := {}^h \mathrm{Op_W}(a)$ for $a\in S^{2m,2l+1}(S_\mathrm{tot})$. Assume further that $a$ is real valued, hence $A$ is formally self-adjoint (and actually self-adjoint for $m\leq 0$, $l\leq -1$). We also have $P = {}^h \mathrm{Op_W}(p)$ locally. Since in particular $P\in\widetilde{\Psi}^0_h(\mathbb{T}^n)$, $p$ is smooth on the blown down space $T^*\mathbb{T}^n \times [0,1)_h$; this is not true of $a$.
We ``commute'' $h^{-1} P\in\Psi^{1,1}_{2,h}(\mathcal{C})$ with $A$ to obtain
\small
\[
i((h^{-1}P)^* A - A(h^{-1}P)) \sim {}^h \mathrm{Op_W}\left(\sum_{\alpha,\beta}\frac{ih^{-1} (-1)^{|\alpha|}}{(2i)^{|\alpha + \beta|}\alpha!\beta!}\left((\partial^\alpha_{\widetilde{x}} \partial^\beta_{\widetilde{\xi}} \bar{p})((h \partial_{\widetilde{\xi}})^\alpha \partial^\beta_{\widetilde{x}} a)-(\partial^\alpha_{\widetilde{x}} (h \partial_{\widetilde{\xi}})^\beta a)(\partial^\alpha_{\widetilde{\xi}} \partial^\beta_{\widetilde{x}} p)\right)\right) + R,
\]
\normalsize
where $R\in\Re^{2l+2}$. This is the Weyl formula for the total symbol of a composition (cf.\ \cite[Section~2.7]{Ma}).
``Slicing'' this sum ($|\alpha + \beta|=0$, $|\alpha + \beta|=1$, $|\alpha + \beta|=2$, etc.) and examining each ``slice'' separately, we determine that
\[
{}^2 \sigma_{2m,2l+2}(i((h^{-1}P)^* A - A(h^{-1}P))) = \mathbf{H}(\widetilde{a}_1) + 2 \widetilde{a}_1 \mathrm{Im}(\mathrm{sub}(P)).
\]
(Note that the $2m+1,2l+2$-principal symbol of the ``commutator'' vanishes.) The first summand $\mathbf{H}(\widetilde{a}_1)$ arises from the ``slice'' $|\alpha + \beta|=1$ by replacing $p$ with its real valued principal part $p_0$
and $a$ with $\widetilde{a}_1$. The latter summand comes from the ``slice'' $\alpha=\beta=0$, replacing $p$ with $h\mathrm{sub}(P)$ and $a$ with $\widetilde{a}_1$. Now, since $H_1(\widetilde{a}_1) = 0$, and since we assumed $\mathrm{sub}(P)\equiv 0$, we conclude that
\[
{}^2 \sigma_\mathrm{2m,2l+2}(i((h^{-1}P)^* A - A(h^{-1}P))) = \rho_\mathrm{ff} \widetilde{H}_2(\widetilde{a}_1).
\]
Thus, the $2m,2l+2$-principal symbol of the ``commutator'' vanishes to first order at $SN(\mathcal{C})$.
Let $B_s$, $C$, $D\in\Psi^{m,l}_{2,h}(\mathcal{C})$ have principal symbols $b_s$, $c$, $d\in S^{l-m}(S_\mathrm{pr})$, respectively, from earlier; and such that ${}^2 \mathrm{WF}_{l}'(B_s) = \mathrm{supp}(b_s)$, and likewise for $C,D$.
Thus, since $\int^\delta_0 B_s^* B_s \ ds + C^* C - D^* D$ and $i((h^{-1}P)^* A - A(h^{-1}P))$ share the same principal symbol, we must have
\begin{equation} \label{remainder 2}
i\left((h^{-1}P)^* A - A(h^{-1}P)\right) = \int^\delta_0 B_s^* B_s \ ds + C^* C - D^* D + L,
\end{equation}
where $L\in\Psi^{2m-1,2l+2}_{2,h}(\mathcal{C})$ satisfies ${}^2 \mathrm{WF}_{2l+2}'(L) \subset {}^2 \mathrm{WF}_{2l+1}'(A)$. Note that $D$ is microsupported along the $H_1$ orbit containing $\zeta$, at which we have absence of ${}^2 \mathrm{WF}^{m,l}(u)$; and that $C$ lives at the opposite ends of the $H_2$ flow segments, so $C^* C$ is the term we wish to control.
Unfortunately, the decay of $L$ at the front face is two orders worse than that of $C^* C$. To overcome this issue, we present the following lemma, to be followed by the end of the proof of the theorem.
\begin{lemma}[Decomposition of $L$]
$L\in\Psi^{2m-1,2l+2}_{2,h}(\mathcal{C})$ in Equation \eqref{remainder 2} may be decomposed as $L = L_1 + L_2$, where $L_1\in\Psi^{2m-1,2l}_{2,h}(\mathcal{C})$ and $L_2\in\Re^{2l+2}$.
\end{lemma}
\begin{proof}
We make full use of the assumption that $\mathrm{sub}(P)\equiv 0$. We expand the total symbol $p$ (of $P = {}^h \mathrm{Op_W}(p)$) in powers of $h$:
\[
p(\widetilde{x},\widetilde{\xi}) = p_0(\widetilde{\xi}) + O\left(h^2\right) = p_0 + p_1,
\]
where $p_1$ signifies everything but the principal part of $P$.
Next, we again make use of the Weyl formula, in conjunction with equation \eqref{remainder 2}. If we replace $p$ by $p_0$ in the asymptotic sum, and replace $a$ by $\widetilde{a}_1$, the term $\alpha=\beta=0$ directly cancels, and the term $|\alpha+\beta| = 1$ is quantized to give ${}^h \mathrm{Op_W}(\{p_0,\widetilde{a}_1\})\in\Psi^{2m,2l}_{2,h}(\mathcal{C})$.
This differs from $\int^\delta_0 B_s^* B_s \ ds + C^* C - D^* D$ by a remainder $\tilde{L}\in\Psi^{2m-1,2l}_{2,h}(\mathcal{C})$, since ${}^h \mathrm{Op_W}(\{p_0,\widetilde{a}_1\})$ and $\int^\delta_0 B_s^* B_s \ ds + C^* C - D^* D$ both have symbol $\mathbf{H}(\widetilde{a}_1) = \rho_\mathrm{ff} \widetilde{H}_2(\widetilde{a}_1)$.
Therefore, $L\in\Psi^{2m-1,2l+2}_{2,h}(\mathcal{C})$ is generated by quantizing all terms involving $p_1$, plus all terms with $|\alpha+\beta| > 1$ with $p_0$ in place of $p$, plus $\tilde{L}$ plus the residual operator $L_2 := R\in\Re^{2l+2}$.
First, we carefully study the terms in the Weyl expansion arising from $p=p_0 + p_1$ from all ``slices'' $|\alpha+\beta| > 1$. To do this, we introduce local coordinates in one coordinate patch of $S_\mathrm{tot}$: $\widetilde{x}$, $\widetilde{\xi}_1$, $H := h / {\widetilde{\xi}_1}$, $\Xi_2 := \widetilde{\xi}_2 / \widetilde{\xi}_1,\ldots,\Xi_d := \widetilde{\xi}_d / \widetilde{\xi}_1$, and $\widetilde{\xi}''$. As usual, $H$ is a defining function for the side face and $\widetilde{\xi}_1$ for the front face. At this point, we use our earlier observation that $p$ is smooth on $T^*\mathbb{T}^n \times [0,1)$ (in particular $p$ is smooth in $\widetilde{\xi}'$, unlike $a$), so for all multi-indices $\gamma$ we have $\partial^\gamma_{\widetilde{\xi}} p = O(1)$ and $\partial^\gamma_{\widetilde{x}} p = O(1)$. ``Slices'' for which $|\alpha+\beta| > 1$ satisfy (1) $|\alpha|\geq 2$ or (2) $|\beta|\geq 1$, $|\alpha|\leq |\beta|$ (these cases are not exclusive). Example \ref{ex:symbol class} gives
\[
h^{|\alpha+\beta| - 1}\left(\partial^\alpha_{\widetilde{x}} \partial^\beta_{\widetilde{\xi}} a\right)\left(\partial^\alpha_{\widetilde{\xi}} \partial^\beta_{\widetilde{x}} p\right)\in S^{2m + 1-|\alpha+\beta|,2l+2-|\alpha|-(\beta_{d+1}+\ldots+\beta_n)}(S_\mathrm{tot}).
\]
If $\gamma\neq 0$, we can improve on $\partial^\gamma_{\widetilde{x}} p = O(1)$. We have:
\begin{align}
p(\widetilde{x},\widetilde{\xi}) &= p_0(\widetilde{\xi}) + O\left(h^2\right) \nonumber \\
&= {p_0}\restriction_\mathcal{C}(\widetilde{\xi}'') + \rho_\mathrm{ff} \left[\frac{\partial p_0}{\partial\widetilde{\xi}'}\right]_\mathcal{C}(\widetilde{\xi}'')\cdot\Gamma' + \frac{\rho_\mathrm{ff}^2}{2} \left[\frac{\partial^2 p_0}{{\partial\widetilde{\xi}'}^2}\right]_\mathcal{C}(\widetilde{\xi}'')\cdot (\Gamma')^2 + O\left(\rho_\mathrm{ff}^3\right) + O\left(\rho_\mathrm{sf}^2 \rho_\mathrm{ff}^2\right), \label{eqn:Taylor expansion}
\end{align}
where $SN^+(\mathcal{C}\times\{0\}) = \{\rho_\mathrm{ff} = 0\}$.
The $O\left(\rho_\mathrm{ff}^3\right)$ term is the remainder in the Taylor expansion of the principal symbol, so it is independent of $\widetilde{x}$. Then we differentiate \eqref{eqn:Taylor expansion} to get $\partial^\gamma_{\widetilde{x}} p = O\left(\rho^2_\mathrm{sf} \rho^2_\mathrm{ff}\right) = O\left(H^2 \widetilde{\xi}_1^2\right)$. This implies, in case (2), that
\[
h^{|\alpha+\beta| - 1}\left(\partial^\alpha_{\widetilde{x}} \partial^\beta_{\widetilde{\xi}} a\right)\left(\partial^\alpha_{\widetilde{\xi}} \partial^\beta_{\widetilde{x}} p\right)\in S^{2m-1-|\alpha+\beta|,2l-|\alpha|-(\beta_{d+1}+\ldots+\beta_n)}(S_\mathrm{tot}).
\]
We are thus able to conclude that all terms of the form $h^{|\alpha+\beta| - 1}\left(\partial^\alpha_{\widetilde{x}} \partial^\beta_{\widetilde{\xi}} a\right)\left(\partial^\alpha_{\widetilde{\xi}} \partial^\beta_{\widetilde{x}} p\right)$ for which $|\alpha+\beta| > 1$ are $O\left(\rho_\mathrm{sf}^{-2m+1} \rho_\mathrm{ff}^{-2l}\right)$.
Finally, we consider the terms arising when $p = p_1 = O\left(\rho_\mathrm{sf}^2 \rho_\mathrm{ff}^2\right)$ and $|\alpha+\beta|\leq 1$. If $\alpha=\beta=0$, it is easily seen that $h^{-1} a p_1\in S^{2m-1,2l}(S_\mathrm{tot})$. Next suppose $\alpha=0$, $|\beta|=1$. We assume the worst: one of the first $d$ components of $\beta$ is equal to one. Then, since $\partial^\beta_{\widetilde{x}} p_1 = O\left(\rho_\mathrm{sf}^2 \rho_\mathrm{ff}^2\right)$ and $h^{-1}(h \partial_{\widetilde{\xi}})^\beta a\in S^{2m,2l+2}(S_\mathrm{tot})$, we have
\[
h^{-1} \left(\partial^\beta_{\widetilde{x}} p_1\right) \left(h\partial_{\widetilde{\xi}}\right)^\beta a\in S^{2m-2,2l}(S_\mathrm{tot}).
\]
The remaining case is $|\alpha|=1$, $\beta=0$. We again assume the worst: $\alpha_1 = 1$. Since $\partial_{\widetilde{\xi}_1} p_1 = O\left(H^2 \widetilde{\xi}_1\right)$ and $\partial_{\widetilde{x}_1} a\in S^{2m,2l+1}(S_\mathrm{tot})$, we have
\[
(\partial_{\widetilde{x}_1} a)(\partial_{\widetilde{\xi}_1} p_1)\in S^{2m-2,2l}(S_\mathrm{tot}),
\]
which is even better than we need.
A similar calculation holds if we interchange $\alpha$ and $\beta$ and take complex conjugates. As a result, we may Borel sum, then quantize, then add on $\tilde{L}$ to obtain the desired operator $L_1\in\Psi^{2m-1,2l}_{2,h}(\mathcal{C})$. This completes the proof of the lemma.
\end{proof}
First, using $A = A^*$, we find that
\[
\left\langle i((h^{-1}P)^* A - A(h^{-1}P))u,u \right\rangle_{L^2(\mathbb{T}^n)} = -2 \ \mathrm{Im}\left\langle Au,h^{-1}Pu \right\rangle_{L^2(\mathbb{T}^n)}.
\]
Then, applying the lemma to Equation \eqref{remainder 2},
\begin{equation} \label{eq:rearranged eqn}
\int^\delta_0 \|B_s u\|^2 ds + \|Cu\|^2 \leq \|Du\|^2 + \ |\left\langle L_1 u,u \right\rangle| \ + \ |\left\langle L_2 u,u \right\rangle| \ + 2 \ |\left\langle Au,h^{-1}Pu \right\rangle|.
\end{equation}
Since $u\in L^2(\mathbb{T}^n)$ (uniformly in $h$), in order to ensure $L_2 u\in L^2(\mathbb{T}^n)$, we need $2l+2\leq 0$, which holds if and only if $l\leq -1$. This is exactly what we assumed. Then by Cauchy--Schwarz,
\[
|\left\langle L_2 u,u \right\rangle_{L^2(\mathbb{T}^n)}| \leq \|L_2 u\|_{L^2(\mathbb{T}^n)} \|u\|_{L^2(\mathbb{T}^n)} < \infty.
\]
Since $D\in\Psi^{m,l}_{2,h}(\mathcal{C})$ is microsupported on the $H_1$ orbit through $\zeta$, $\zeta\notin {}^2 \mathrm{WF}^{m,l}(u)$, and we know $H_1$ invariance, then $Du\in L^2(\mathbb{T}^n)$.
By construction,
\[
{}^2 \mathrm{WF}_{2l+1}'(A) \cap {}^2 \mathrm{WF}^{m-1/2,l}(u) = \emptyset.
\]
Therefore, since
\[
{}^2 \mathrm{WF}_{2l}'(L_1) = {}^2 \mathrm{WF}_{2l+2}'(L) \subset {}^2 \mathrm{WF}_{2l+1}'(A),
\]
$L_1\in\Psi^{2m-1,2l}_{2,h}(\mathcal{C})$ satisfies $|\left\langle L_1 u,u \right\rangle| < \infty$. Finally, the last remaining term is controlled by our assumption that $Pu = O_{L^2(\mathbb{T}^n)}(h^\infty)$.
Since the RHS of Equation \eqref{eq:rearranged eqn} is bounded (as $h\rightarrow 0$), each term on the LHS is bounded. The boundedness of $\|Cu\|^2$ demonstrates absence of ${}^2 \mathrm{WF}^{m,l}(u)$ on the microsupport of $C$. Since $C$ is microsupported near the $\widetilde{H}_2$ flow-segment-ends opposite $\overline{\mathcal{O}}_{H_1}(\zeta)$ and since $\delta$ can be made arbitrarily small, we have proved that (lack of) ${}^2 \mathrm{WF}^{m,l}(u)$ spreads along each piece of $\widetilde{H}_2$ flow.
\end{proof}
\begin{remark}
In fact, to control the last term in \eqref{eq:rearranged eqn}, it suffices to assume that $Pu = O_{L^2(\mathbb{T}^n)}(h^s)$ for $s\geq\max(2m+1,2l+2)$,
since if $Pu = h^s g$ for $s\geq 2m+1$, $s\geq 2l+2$, and $g\in L^2(\mathbb{T}^n)$, then
\[
|\left\langle Au,h^{-1}Pu \right\rangle_{L^2(\mathbb{T}^n)}| = |\left\langle h^{s-1}Au,g \right\rangle_{L^2(\mathbb{T}^n)}| \leq \|h^{s-1}Au\|_{L^2(\mathbb{T}^n)} \|g\|_{L^2(\mathbb{T}^n)} < \infty.
\]
\end{remark}
\noindent Therefore, if we only wish to prove invariance of the \emph{graded} second wavefront ${}^2 \mathrm{WF}^{m,l}(u)$:
\begin{theorem}
Let $\mathcal{C}\subset T^* \mathbb{T}^n$ be a linear coisotropic submanifold. Assume $P\in\widetilde{\Psi}^0_h(\mathbb{T}^n)$ has real valued principal symbol depending only on the fiber variables in $T^* \mathbb{T}^n$, and subprincipal symbol identically equal to zero. Let $u \in L^2(\mathbb{T}^n)$. Then for all $m$ and $l \leq -1$, ${}^2 \mathrm{WF}^{m,l}(u) \cap SN(\mathcal{C})$ propagates along the flow of $H_2$, if $u$ satisfies $Pu = O_{L^2(\mathbb{T}^n)}(h^s)$ for $s\geq\max(2m+1,2l+2)$.
\end{theorem}
The Hamiltonian flow of $P\in\Psi_h(\mathbb{T}^n)$ on $\mathcal{C}\subset T^* \mathbb{T}^n$ is described by quasi-periodic motion with respect to a set of frequencies $\overline{\omega}_1,\ldots,\overline{\omega}_n$. By definition, these frequencies are the derivatives of $\sigma_\mathrm{pr}(P)(\xi)$ with respect each $\xi_j$, restricted to $\mathcal{C}$. Hence, if all frequencies are irrationally related, then coisotropic regularity fills out the coisotropic, in the base variables.
We also consider the complementary case, in which $\overline{\omega}_i / \overline{\omega}_j \in \mathbb{Q}$ for some $i,j$.
Again, coisotropic regularity occurs on whole Hamiltonian orbits, but in this case, these orbits need not be dense. Here coisotropic regularity is invariant under two separate flows, according to \autoref{secondary propagation}.
\subsection{Propagation Examples\label{ex:propag examples}}
We wish to apply our real principal type and secondary propagation theorems to quasimodes of the Laplacian $h^2 \Delta$. However, note that \autoref{secondary propagation} is only valid for $h$-pseudodifferential operators with compactly supported symbols, belonging to the subalgebra $\widetilde{\Psi}^0_h(\mathbb{T}^n) \subset \Psi^0_h(\mathbb{T}^n)$. Suppose $u = u_h\in L^2(\mathbb{T}^n)$ satisfies $Pu = O_{L^2(\mathbb{T}^n)}(h^\infty)$ for $P = h^2 \Delta - 1$. Then
\begin{equation} \label{why this cutoff}
\mathrm{WF}_h(u) \subset \mathrm{char}(P) = \{|\xi| = 1\}.
\end{equation}
Clearly, $P\notin\widetilde{\Psi}^0_h(\mathbb{T}^n)$. Let $\chi$ be a smooth and compactly supported function satisfying $\chi(x)\equiv 1$ in some neighborhood of the point $x=1$. Then $P = \chi(h^2 \Delta)(h^2 \Delta - 1)\in\widetilde{\Psi}^0_h(\mathbb{T}^n)$ and $${}^2 \mathrm{WF}^{\infty,l}(u) \subset {}^2 \mathrm{char}(P).$$ Notice that we abuse notation and refer to the truncated operator also as $P$. We may then use \autoref{secondary propagation} to study propagation of ${}^2 \mathrm{WF}^{\infty,l}(u)$ at $SN(\mathcal{C})$ for any linear coisotropic $\mathcal{C} \subset T^* \mathbb{T}^n$.
\begin{example}
Consider the Laplace operator $P = \left(h^2 \Delta_x - 1\right)/2$, and $\mathcal{C} = \{\xi'=(\xi_1,\xi_2)=(0,0)\} \subset T^*\mathbb{T}^4$. We cut off $P$ as described above. The Hamiltonian vector field determined by the principal symbol of $P$ is $\xi\cdot\partial_x$. Coordinates near $SN (\mathcal{C})$ are $\rho_\mathrm{ff} = |\xi'|$, $\hat{\xi}' = \xi' / |\xi'|$, as well as $x$ and $\xi''$. Then $\mathbf{H} = \xi''\cdot\partial_{x''} + \rho_\mathrm{ff}\left(\hat{\xi}'\cdot\partial_{x'}\right)$. This is consistent with \eqref{Hamilton}. For $Pu = 0$, invariance of ${}^2 \mathrm{WF}(u) \cap SN(\mathcal{C})$ under $H_1 = \xi''\cdot\partial_{x''}$ only gives propagation in directions not tangent to the leaves of the characteristic foliation, whereas invariance under $\widetilde{H}_2 = \hat{\xi}'\cdot\partial_{x'}$ gives propagation along the leaves only.
Elements of ${}^2 \mathrm{WF}(u) \cap SN(\mathcal{C})$ satisfy $|\xi''| = 1$ (since ${}^2 \mathrm{WF}(u)\subset {}^2 \mathrm{char}(P)$), and also satisfy $|\hat{\xi}'| = 1$. For generic values of $\xi''$ and $\hat{\xi}'$ subject to these constraints, $H_1$ and $\widetilde{H}_2$ together flow to all of $\mathbb{T}^4$, but there are exceptional cases.
\end{example}
\begin{example}
For the same operator $P$ and the coisotropic $\mathcal{C} = \{\xi_1 + \xi_3 = \xi_2 + \xi_4 = 0\}$ (so $\mathbf{v}_1 = (1 \ 0 \ 1 \ 0)^t$, $\mathbf{v}_2 = (0 \ 1 \ 0 \ 1)^t$), define
\[
\hat{\xi}_1 = \frac{\xi_1 + \xi_3}{\sqrt{(\xi_1 + \xi_3)^2 + (\xi_2 + \xi_4)^2}}, \ \ \hat{\xi}_2 = \frac{\xi_2 + \xi_4}{\sqrt{(\xi_1 + \xi_3)^2 + (\xi_2 + \xi_4)^2}}
\]
(regarding $\xi_3$ and $\xi_4$ as free variables), and take $\rho_\mathrm{ff} = \sqrt{(\xi_1 + \xi_3)^2 + (\xi_2 + \xi_4)^2}$. Then $\xi\cdot\partial_x$ lifts to
\[
\xi_3(\partial_{x_3} - \partial_{x_1}) + \xi_4(\partial_{x_4} - \partial_{x_2}) + \rho_\mathrm{ff}\left(\hat{\xi}_1 \partial_{x_1} + \hat{\xi}_2 \partial_{x_2}\right),
\]
with $H_1 = \xi_3(\partial_{x_3} - \partial_{x_1}) + \xi_4(\partial_{x_4} - \partial_{x_2})$ and $\widetilde{H}_2 = \hat{\xi}_1 \partial_{x_1} + \hat{\xi}_2 \partial_{x_2}$. Again, this is consistent with \eqref{Hamilton}. (In the formulation of \eqref{Hamilton}, for $d<j\leq n$, $\widetilde{x}_j = \mathbf{w}_j\cdot x$. To match the coordinates above, we would choose $\mathbf{w}_3 = (0 \ 0 \ 1 \ 0)^t$, $\mathbf{w}_4 = (0 \ 0 \ 0 \ 1)^t$.) For $Pu = 0$, note that elements of ${}^2 \mathrm{WF}(u) \cap SN(\mathcal{C})$ satisfy $\xi_3^2 + \xi_4^2 =\frac{1}{2}$.
\end{example}
\footnotesize
\noindent \textsc{Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109} \\
\textit{E-mail address}: {\tt \href{mailto:[email protected]}{[email protected]}}
\end{document} |
\begin{document}
\tildetle{Hessian estimates for the sigma-2 equation in dimension four}
\author{Ravi Shankar and Yu Yuan}
\date{}
\title{Hessian estimates for the sigma-2 equation in dimension four}
\begin{abstract}
We derive a priori interior Hessian estimates and interior regularity for the $\sigma_2$ equation in dimension four. Our method provides respectively a new proof for the corresponding three dimensional results and a Hessian estimate for smooth solutions satisfying a dynamic semi-convexity condition in higher $n\ge 5$ dimensions.
\end{abstract}
\title{Hessian estimates for the sigma-2 equation in dimension four}
\section{Introduction}
\footnotetext[1]{\today}
In this article, we resolve the question of the interior a priori Hessian estimate
and regularity for the $\sigma_{2}$ equation
\begin{equation}
\sigma_{2}\left( D^{2}u\right) =\sum_{1\leq i<j\leq n}\lambda_{i}\lambda
_{j}=1 \label{s2}
\end{equation}
in dimension $n=4,$ where $\lambda_{i}'s$ are the eigenvalues of the
Hessian $D^{2}u.$
\begin{thm}
\label{thm:s2}
Let $u$ be a smooth solution to (\mbb Rf{s2}) in the positive branch
$\bigtriangleup u>0$ on $B_{1}(0)\subset\mathbb{R}^{4}$. Then $u$ has an
implicit Hessian estimate
\[
|D^{2}u(0)|\leq C(\left\Vert u\right\Vert _{C^{1}\left( B_{1}\left(
0\right) \right) })\ \ \
\]
with$\ \left\Vert u\right\Vert _{C^{1}\left( B_{1}\left( 0\right) \right)
}=\left\Vert u\right\Vert _{L^{\infty}\left( B_{1}\left( 0\right) \right)
}+\left\Vert Du\right\Vert _{L^{\infty}\left( B_{1}\left( 0\right) \right)
}.$
\end{thm}
From the gradient estimate for $\sigma_{k}$-equations by Trudinger \cite{T2} and
also Chou-Wang \cite{CW} in the mid 1990s, we can bound $D^{2}u$ in terms of the
solution $u$ in $B_{2}\left( 0\right) $ as
\[
|D^{2}u(0)|\leq C(\left\Vert u\right\Vert _{L^{\infty}\left( B_{2}\left(
0\right) \right) }).
\]
In higher $n\geq5$ dimensions, our method provides a Hessian estimate for
smooth solutions satisfying a semi-convexity type condition with movable lower
bound (\mbb Rf{dscx}), which is unconditionally valid in four dimensions by \eqref{sharp}.
\begin{thm}
\label{thm:n5}
Let $u$ be a smooth solution to (\mbb Rf{s2}) in the positive branch
$\bigtriangleup u>0$ on $B_{1}(0)\subset\mathbb{R}^{n}$ with $n\geq5,$
satisfying a dynamic semi-convex condition
\begin{equation}
\lambda_{\min}\left( D^{2}u\right) \geq-c\left( n\right) \bigtriangleup
u\ \ \ \text{with \ \ }c\left( n\right) =\frac{\sqrt{3n^{2}+1}-n+1}{2n}.
\label{dscx}
\end{equation}
Then $u$ has an implicit Hessian estimate
\[
|D^{2}u(0)|\leq C(n,\left\Vert u\right\Vert _{L^{\infty}\left( B_{1}\left(
0\right) \right) }).
\]
\end{thm}
One application of the above estimates is the interior regularity
(analyticity) of $C^{0}$ viscosity solutions to (\mbb Rf{s2}) in four dimensions,
when the estimates are combined with the solvability of the Dirichlet problem
with $C^{4}$ boundary data by Caffarelli-Nirenberg-Spruck \cite{CNS} and also
Trudinger \cite{T1}. In particular, the solutions of the Dirichlet problem with
$C^{0}$ boundary data to four dimensional (\mbb Rf{s2}) of both positive branch
$\bigtriangleup u>0$ and negative branch $\bigtriangleup u<0$ respectively,
enjoy interior regularity.
Another consequence is a rigidity result for entire solutions to (\mbb Rf{s2}) of
both branches with quadratic growth, namely all such solutions must be
quadratic, provided the smooth solutions in dimension $n\geq5$ also satisfying
the dynamic semi-convex assumption (\mbb Rf{dscx}), or the symmetric one
$\lambda_{\max}\left( D^{2}u\right) \leq -c\left( n\right) \bigtriangleup
u$ in the symmetric negative branch case. Warren's rare saddle entire solution
to (\mbb Rf{s2}) shows certain convexity condition is necessary \cite{W}. Other
earlier related results can be found \cite{BCGJ} \cite{Y1} \cite{CY} \cite{CX} \cite{SY3}.
In two dimensions, an interior Hessian bound for \eqref{s2}, the
Monge-Amp\`{e}re equation $\sigma_{2}=\det D^{2}u=1$ was found via isothermal coordinates, which are readily available under Legendre-Lewy transform, by
Heinz \cite{H} in the 1950s. The dimension three case was done via the
minimal surface structure of equation (\mbb Rf{s2}) and a full strength Jacobi inequality by Warren-Yuan in the late 2000s \cite{WY}. In higher dimensions $n\geq4$ any effective geometric structure of (\mbb Rf{s2}) appears hidden, although the level set of non-uniformly elliptic equation (\mbb Rf{s2}) is convex.
In recent years, Hessian estimates for convex smooth solutions of (\mbb Rf{s2})
have been obtained via a pointwise approach by Guan and Qiu \cite{GQ}. Hessian
estimates for almost convex smooth solutions of (\mbb Rf{s2}) have been derived
by a compactness argument in \cite{MSY}, and for semi-convex smooth solutions
in \cite{SY1} by an integral method. However, we cannot extend these a priori
estimates, including Theorem 1.2, to interior regularity statements for
viscosity solutions of (\mbb Rf{s2}), because the smooth approximations may not
preserve the convexity or semi-convexity constraints. Taking advantage of an
improved regularity property for the equation satisfied by the Legendre-Lewy
transform of almost convex viscosity solutions, interior regularity was
reached in \cite{SY2}.
For higher order $\sigma_{k}\left( D^{2}u\right) =1$ with $k\geq3$
equations, which is the Monge-Amp\`{e}re equation in $k$ dimensions, there are
the famous singular solutions constructed by Pogorelov \cite{P} in the 1970s, and
later generalized in \cite{U1}. Worse singular solutions have been produced in
recent years. Hessian estimates for solutions with certain strict $k$-convexity
constraints to Monge-Amp\`{e}re equations and $\sigma_{k}$ equation ($k\geq2$)
were derived by Pogorelov \cite{P} and Chou-Wang \cite{CW} respectively using the
Pogorelov's pointwise technique. Urbas \cite{U2} \cite{U3} obtained (pointwise) Hessian
estimates in term of certain integrals of the Hessian for $\sigma_{k}$ equations. Recently, Mooney \cite{M} derived the strict
2-convexity of convex viscosity solutions to (\mbb Rf{s2}), consequently, relying on
the solvability \cite{CNS} and a priori estimates \cite{CW}, gave a different proof of
the interior regularity of those convex viscosity solutions.
\setminusallskip
Our proof of Theorem \mbb Rf{thm:s2} synthesizes the ideas of Qiu \cite{Q} with Chaudhuri-Trudinger \cite{CT} and Savin \cite{S}. Qiu showed that in dimension three, where a Jacobi inequality is valid (see Section \mbb Rf{sec:Jac} for definitions of the operators)
$$
F_{ij}\partial_{ij}\ln\,\Deltalta u\ge \varepsilon F_{ij}(\ln\,\Deltalta u)_i(\ln\,\Deltalta u)_j,
$$
a maximum principle argument leads to a doubling, or ``three-sphere'' inequality:
$$
\sup_{B_1(0)}\,\Deltalta u\le C(n,\|u\|_{C^1(B_2(0))})\sup_{B_{1/2}(0)}\,\Deltalta u.
$$
(A lower bound condition on $\sigma_3(D^2u),$ satisfied by convex solutions of (\mbb Rf{s2}) in general dimensions permitted Guan-Qiu to exclude the inner ``sphere" term $B_{1/2}(0)$ in the above inequality for their eventual Hessian estimates earlier in \cite{GQ}.) Iterating this ``three-sphere'' inequality shows that the Hessian is controlled by its maximum on any arbitrarily small ball. To put it another way, any blowup point propagates to a dense subset of $B_1(0)$. To rule out Weierstrass nowhere twice differentiable counterexamples, it suffices to find a single smooth point; Savin's small perturbation theorem \cite{S} guarantees a smooth ball if there is a smooth point. It more than suffices to establish partial regularity, such as Alexandrov's theorem. Chaudhuri and Trudinger \cite{CT} showed that $k$-convex functions have an Alexandrov theorem if $k>n/2$. This gives a new compactness proof of Hessian estimate and regularity for \eqref{s2} in dimension three without minimal surface arguments, and also Hessian estimate for \eqref{s2} in general dimensions with semi-convexity assumption in \cite{SY1}, where a Jacobi inequality and Alexandrov twice differentiability are available.
\setminusallskip
In higher dimensions $n\ge 4$, there are three new difficulties. Although the H\"older estimate for $k$-convex \textit{functions} may not be valid for $k\le n/2$, we can replace it with the interior gradient estimate for 2-convex \textit{solutions} in \cite{T2} \cite{CW}; this gives an Alexandrov theorem. The main hurdle is the Jacobi inequality, which fails for four and higher dimensions without a priori control on the minimum eigenvalue $\lambda_{min}$ of $D^2u$; the Jacobi inequality was discovered in \cite{SY1, SY3} for semi-convex solutions. Instead, we can only establish an ``almost-Jacobi inequality", where $\varepsilon\sim 1+2\lambda_{min}/\,\Deltalta u$ in four dimensions. This choice of $\varepsilon$ degenerates to zero for the extreme configurations $(\lambda_1,\lambda_2,\lambda_{3},\lambda_4)=(a,a,a,-a+O(1/a))$. At first glance, $\varepsilon\to 0$ means Qiu's maximum principle argument fails; the positive term $\varepsilon|\nabla_F b|^2$ can no longer absorb bad terms. On the other hand, for the extreme configurations, the equation becomes conformally uniformly elliptic. The, usually defective, lower order term $\,\Deltalta_F|Du|^2\gtrsim \sigma_1\lambda_{min}^2$, is large enough to take control of the bad terms. The dynamic semi-convexity assumption \eqref{dscx} allows the outlined four dimensional arguments to continue working in higher $n\ge5$ dimensions.
\setminusallskip
Using similar methods, a new proof of regularity for strictly convex solutions to the Monge-Amp\`ere equation is found in \cite{SY4}. Extrinsic curvature estimates for the scalar curvature equation in dimension four are found in \cite{Sh}, extending the dimension three result of Qiu \cite{Q1}. In forthcoming work, investigation will be done on conformal geometry's $\sigma_2$ Schouten tensor equation with negative scalar curvature and the improvement of the $W^{2,6+\delta}$ to $C^{1,1}$ estimate in \cite{D} to a $W^{2,6}$ to $C^{1,1}$ estimate.
\setminusallskip
In still higher dimensions $n\ge 5$, we are not even able to establish that $\ln\,\Deltalta u$ is sub-harmonic, $\varepsilon\ge 0$, without a priori conditions on the Hessian. There is still the problem of regularity for such solutions. Combining the Alexandrov theorem with small perturbation [S, Theorem 1.3] only shows that the singular set is closed with Lebesgue measure zero.
\section{Almost Jacobi inequality}
\label{sec:Jac}
In \cite{SY3}, we established a Jacobi inequality for $b=\ln(\,\Deltalta u+C(n,K))$ under the semi-convexity assumption $\lambda_{min}(D^2u)\ge -K$, namely the quantitative subsolution inequality
$$
\,\Deltalta_Fb:=F_{ij}\partial_{ij}b\ge \varepsilon\,F_{ij}b_ib_j=: \varepsilon|\nabla_Fb|^2,
$$
where $\varepsilon=1/3$, and for the sigma-2 equation $F(D^2u)=\sigma_2(\lambda)=1$, we denote the linearized operator by the positive definite matrix
\eqal{
\label{linearized}
(F_{ij})=\,\Deltalta u\,I-D^2u=\sqrt{2\sigma_2+|D^2u|}\,I-D^2u>0.
}
In dimension three, the above Jacobi inequality holds for $C(3,K)=0$ unconditionally; see \cite[$\text{p. 3207}$]{SY3} and Remark \mbb Rf{rem:3D}. In dimension four, we can establish an inequality for $b=\ln\,\Deltalta u$ without any Hessian conditions. The cost is that $\varepsilon$ depends on the Hessian, and $\varepsilon(D^2u)\to 0$ is allowed. We obtain an ``almost" Jacobi inequality.
\begin{prop}
\label{prop:Jac}
Let $u$ be a smooth solution to $\sigma_2(\lambda)=1$, and $b=\ln\,\Deltalta u$. In dimension $n=4$, we have
\eqal{
\label{aJac}
\,\Deltalta_Fb\ge\varepsilon\,|\nabla_Fb|^2,
}
where
$$
\varepsilon=\frac{2}{9}\left(\frac{1}{2}+\frac{\lambda_{min}}{\,\Deltalta u}\right)> 0.
$$
In higher dimension $n\ge 5$, \eqref{aJac} holds for
$$
\varepsilon=
\frac{\sqrt{3n^2+1}-(n+1)}{3(n-1)}\left(\frac{\sqrt{3n^2+1}-(n-1)}{2n}+\frac{\lambda_{min}}{\,\Deltalta u}\right)
$$
under the condition
$$
\frac{\lambda_{min}}{\,\Deltalta u}\ge -\frac{\sqrt{3n^2+1}-(n-1)}{2n}
$$
Here, $\lambda_{min}$ is the minimum eigenvalue of $D^2u$.
\end{prop}
An important ingredient for Proposition \mbb Rf{prop:Jac} is the following sharp control on the minimum eigenvalue.
\begin{lem}
\label{lem:sharp}
Let $\lambda=(\lambda_1,\dots,\lambda_n)$ solve $\sigma_2(\lambda)=1$ with $\lambda_1+\cdots+\lambda_n>0$ and $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n$. Then the following bound holds for $n>2$ and is sharp:
\eqal{
\label{sharp}
\sigma_1(\lambda)>\frac{n}{n-2}|\lambda_{n}|.
}
\begin{comment}
More generally, let $\lambda'=(\lambda_1,\dots,\lambda_{n-1},0)$, $\tau=(1,\dots,1,0)/\sqrt {n-1}$ and $\lambda'^\bot=\lambda'-(\lambda'\cdot\tau)\tau$ be its traceless part. Then
\eqal{
\label{sharper}
\frac{|\lambda'^\bot|^2}{\sigma_1(\lambda)^2}\le 2\left(\frac{n-2}{n}\right)^2 \left(\frac{\sigma_1(\lambda)}{|\lambda_n|}-\frac{n}{n-2}\right).
}
\end{comment}
\end{lem}
\begin{proof}
The sharpness follows from the configurations
\eqal{
\label{config}
\lambda=\left(a,a,\dots,a,-\frac{(n-2)}{2}a+\frac{1}{(n-1)a}\right).
}
Next, if $\lambda_n\ge 0$, we have
$$
\sigma_1=\lambda_1+\cdots+\lambda_n\ge n\lambda_n.
$$
For $\lambda_n<0$, we write $\lambda'=(\lambda_1,\dots,\lambda_{n-1})$ and observe that $\lambda_n=(1-\sigma_2(\lambda'))/\sigma_1(\lambda')$. We must have $\sigma_2(\lambda')>1$, as $\sigma_1(\lambda')>0$ from (\mbb Rf{linearized}), so we write
$$
\frac{\sigma_1(\lambda)}{-\lambda_n}=-1+\frac{\sigma_1(\lambda')^2}{\sigma_2(\lambda')-1}>-1+\frac{\sigma_1(\lambda')^2}{\sigma_2(\lambda')}.
$$
We write $\sigma_1(\lambda')^2$ in terms of the traceless part $\lambda'^\bot$ of $\lambda'$ and $\sigma_2(\lambda')$:
$$
\sigma_1(\lambda')^2=\frac{n-1}{n-2}(2\sigma_2(\lambda')+|\lambda'^\bot|^2).
$$
It then follows
\begin{align*}
\frac{\sigma_1(\lambda)}{-\lambda_n}&>-1+\frac{2(n-1)}{n-2}= \frac{n}{n-2}.
\end{align*}
\begin{comment}
For $\lambda_n<0$, we write $\lambda'=(\lambda_1,\dots,\lambda_{n-1})$ and observe that $\lambda_n=(1-\sigma_2(\lambda'))/\sigma_1(\lambda')$. We must have $\sigma_2(\lambda')>1$, so we write
$$
\frac{\sigma_1(\lambda)}{-\lambda_n}=-1+\frac{\sigma_1(\lambda')^2}{\sigma_2(\lambda')-1}>-1+\frac{\sigma_1(\lambda')^2}{\sigma_2(\lambda')}.
$$
We eliminate $\sigma_1(\lambda')^2$ in terms of the traceless part $\lambda'^\bot$:
$$
\sigma_1(\lambda')^2=\frac{n-1}{n-2}(2\sigma_2(\lambda')+|\lambda'^\bot|^2).
$$
By \eqref{sharp}, $\sigma_1(\lambda)>\frac{n}{2(n-1)}\sigma_1(\lambda')$, so we obtain
\begin{align*}
\frac{\sigma_1(\lambda)}{-\lambda_n}&>-1+\frac{2(n-1)}{n-2}+\frac{n-1}{n-2}\frac{|\lambda'^\bot|^2}{\sigma_2(\lambda')}\ge \frac{n}{n-2}+2\left(\frac{n-1}{n-2}\right)^2\frac{|\lambda'^\bot|^2}{\sigma_1(\lambda')^2}\\
&> \frac{n}{n-2}+\frac{n^2}{2(n-2)^2}\frac{|\lambda'^\bot|^2}{\sigma_1(\lambda)^2}.
\end{align*}
\end{comment}
\end{proof}
As a consequence, we obtain the following quantitative ellipticity for equation (\mbb Rf{s2}).
\begin{cor}
\label{cor:ellipticity}
Let $\lambda=(\lambda_1,\dots,\lambda_n)$ solve $f(\lambda)=\sigma_2(\lambda)=1$, with $\lambda_1+\cdots+\lambda_n>0$. For $\lambda_1\ge \lambda_2\ge\cdots\ge \lambda_n$ and $f_i=\partial f/\partial\lambda_i$, we have
\eqal{
\label{ellipticity}
\frac{1}{\sigma_1}&\le f_1\le \left(\frac{n-1}{n}\right)\sigma_1,\\
\left(1-\frac{1}{\sqrt 2}\right)\sigma_1&\le f_i\le 2\left(\frac{n-1}{n}\right)\sigma_1,\qquad i\ge 2.
}
\end{cor}
\begin{proof}
The upper bound for $f_1=\sigma_1-\lambda_1$ comes from the easy bound $n\lambda_1\ge\sigma_1$. The sharp upper bound for $f_n$ follows from \eqref{sharp}:
$$
f_i\le f_n=\sigma_1-\lambda_n<\left(1+\frac{n-2}{n}\right)\sigma_1.
$$
The $i=1$ lower bound goes as follows:
$$
f_1=\sigma_1-\lambda_1=\frac{2+|(0,\lambda_2,\dots,\lambda_n)|^2}{\sigma_1+\lambda_1}>\frac{2}{\sigma_1+\lambda_1}>\frac{1}{\sigma_1}.
$$
The $i\ge 2$ lower bounds for $f_i=\sigma_1-\lambda_i$ are true if $\lambda_i\le 0$. For $\lambda_i>0$,
$$
f_i=\sigma_1-\lambda_i>\sigma_1-\sqrt{\frac{\lambda_1^2+\cdots+\lambda_i^2}{i}}>\left(1-i^{-1/2}\right)\sigma_1,
$$
where we used
$$
\sigma_1=\sqrt{2+|\lambda|^2}>\sqrt{\lambda_1^2+\cdots+\lambda_i^2},
$$
in the last inequality.
\end{proof}
\begin{rem}
A sharp form of \eqref{ellipticity} for the $i\ge 2$ lower bounds and rougher upper bounds was first shown in [LT, (16)]. A rougher form of the lower bounds in \eqref{ellipticity}, enough for our proof of doubling Proposition \mbb Rf{prop:doub}, also follows from \cite[Lemma 3.1]{CW}, \cite[Lemma 2.1]{CY}, and \cite[(2.4)]{SY1}.
\begin{comment}
Let us now derive \eqref{2ndLower}. Recalling from Lemma \mbb Rf{lem:sharp} the definitions $\lambda':=(\lambda_1,\dots,\lambda_{n-1},0)$, $\tau=(1,\dots,1,0)/\sqrt{n-1}$, and $\lambda'^\bot=\lambda'-(\lambda'\cdot\tau)\tau$, we write
\begin{align*}
\lambda_{n-1}&>\frac{\sigma_1(\lambda')}{n-1}-\frac{1}{\sqrt{n-1}}|\lambda'^\bot|\ge \frac{\sigma_1(\lambda)-\lambda_n}{n-1}-\frac{\sqrt{2}(n-2)}{n\sqrt{n-1}}\sigma_1(\lambda)\left(\frac{\sigma_1}{|\lambda_n|}-\frac{n}{n-2}\right)^{1/2}\\
&>\frac{\sigma_1(\lambda)}{n}\left[\frac{2}{n-1}-\frac{\sqrt 2(n-2)}{\sqrt{n-1}}\left(\frac{\sigma_1}{|\lambda_n|}-\frac{n}{n-2}\right)^{1/2}\right].
\end{align*}
\setminusallskip
\end{comment}
\end{rem}
\begin{proof}[Proof of Proposition \mbb Rf{prop:Jac}]
Step 1. Expression of the Jacobi inequality. After a rotation at $x=p$, we assume that $D^2u(p)$ is diagonal. Then $(F_{ij})=\text{diag}(f_i)$, where $f(\lambda)=\sigma_2(\lambda)$. The following calculation was performed in \cite[$\text{p. 4}$]{SY3} for $b=\ln(\,\Deltalta u+J)$ for some constant $J$. We repeat it below with $J=0$, for completeness. We start with the following formulas at $x=p$:
\begin{align}
\label{gradb}
&|\nabla_Fb|^2=\sum_{i=1}^nf_i\frac{(\,\Deltalta u_i)^2}{(\,\Deltalta u)^2},\\
\label{Deltab}
&\,\Deltalta_Fb=\sum_{i=1}^nf_i\left[\frac{\partial_{ii}\,\Deltalta u}{\,\Deltalta u}-\frac{(\partial_i\,\Deltalta u)^2}{(\,\Deltalta u)^2}\right]
\end{align}
Next, we replace the fourth order derivatives $\partial_{ii}\,\Deltalta u=\sum_{k=1}^n\partial_{ii}u_{kk}$ in \eqref{Deltab} by third derivatives. By differentiating \eqref{s2}, we have
\eqal{
\label{Ds2}
\,\Deltalta_FDu=(F_{ij}u_{ijk})_{k=1}^n=0.
}
Differentiating \eqref{Ds2} and using \eqref{linearized}, we obtain at $x=p$,
\begin{align*}
\sum_{i=1}^nf_i\partial_{ii}\,\Deltalta u&=\sum_{k=1}^n\,\Deltalta_Fu_{kk}=\sum_{i,j,k=1}^nF_{ij}\partial_{ij}u_{kk}=-\sum_{i,j,k=1}^n\partial_kF_{ij}\partial_{ij}u_k\\
&=\sum_{i,j,k=1}^n-(\,\Deltalta u_k\delta_{ij}-u_{kij})u_{kij}=\sum_{i,j,k=1}^nu_{ijk}^2-\sum_{k=1}^n(\,\Deltalta u_k)^2.
\end{align*}
Substituting this identity into \eqref{Deltab} and regrouping terms of the forms $u_{ \clubsuit\heartsuit\spadesuit}^2$, $u_{\clubsuit\clubsuit\heartsuit}^2,$ $u_{\heartsuit\heartsuit\heartsuit}^2$, and $(\,\Deltalta u_\clubsuit)^2$, we obtain
$$
\,\Deltalta_Fb=\frac{1}{\sigma_1}\left\{6\sum_{i<j<k}u_{ijk}^2+\left[3\sum_{i\neq j}u_{jji}^2+\sum_iu_{iii}^2-\sum_i\left((1+\frac{f_i}{\sigma_1}\right)
(\,\Deltalta u_i)^2\right]\right\}
$$
Accounting for \eqref{gradb}, we obtain the following quadratic:
$$
(\,\Deltalta_Fb-\varepsilon|\nabla_Fb|^2)\sigma_1\ge 3\sum_{i\neq j}u_{jji}^2+\sum_{i}u_{iii}^2-\sum_i(1+\delta f_i/\sigma_1)(\,\Deltalta u_i)^2,
$$
where $\delta:=1+\varepsilon$ here. As in \cite{SY3}, we fix $i$ and denote $t_i=(u_{11i},\dots,u_{nni})$ and $e_i$ the $i$-th basis vector of $\mbb R^n$. Then we recall equation (2.9) from \cite{SY3} for the $i$-th term above:
$$
Q:=3|t|-2\langle e_i,t\rangle^2-(1+\delta f_i/\sigma_1)\langle(1,\dots,1),t\rangle^2.
$$
The objective is to show that $Q\ge 0$. The idea in \cite{SY3} was to reduce the quadratic form to a two dimensional subspace. In that paper, $Q\ge 0$ was shown under a semi-convexity assumption of the Hessian. Here, we show how to remove this assumption in dimension four. For completeness, we repeat that reduction below.
\setminusallskip
Step 2. Anisotropic projection. Equation \eqref{Ds2} at $x=p$ shows that $\langle Df,t_i\rangle=0$, so $Q$ is zero along a subspace. We can thus replace the vectors $e_i$ and $(1,\dots,1)$ in $Q$ with their projections:
$$
Q=3|t|^2-2\langle E,t\rangle^2-(1+\delta f_i/\sigma_1)\langle L,t\rangle ^2,
$$
where
$$
E=e_i-\frac{\langle e_i,Df\rangle}{|Df|^2} Df,\qquad L=(1,\dots,1)-\frac{\langle (1,\dots,1),Df\rangle}{|Df|^2} Df.
$$
Their rotational invariants can be calculated as in [SY3, equation (2.10)]:
\eqal{
\label{invariants}
|E|^2=1-\frac{f_i^2}{|Df|^2},\qquad |L|^2=1-\frac{2(n-1)}{|Df|^2},\qquad E\cdot L=1-\frac{(n-1)\sigma_1 f_i}{|Df|^2}.
}
The quadratic is mostly isotropic: if $t$ is orthogonal to both $E$ and $L$, then $Q=3|t|^2\ge 0$, so it suffices to assume that $t$ lies in the $\{E,L\}$ subspace. The matrix associated to the quadratic form is
$$
Q=3I-2E\otimes E-\eta L\otimes L,
$$
where $\eta=1+\delta f_i/\sigma_1=1+(1+\varepsilon)f_i/\sigma_1$. Since $Q$ is a quadratic form, its matrix is symmetric and has real eigenvalues. In the non-orthogonal basis $\{E,L\}$, the eigenvector equation is
$$
\begin{pmatrix}
3-2|E|^2&-2E\cdot L\\
-\eta L\cdot E&3-\eta|L|^2
\end{pmatrix}
\begin{pmatrix}
\alphapha\\
\beta
\end{pmatrix}
=
\xi
\begin{pmatrix}
\alphapha\\
\beta
\end{pmatrix}.
$$
The real eigenvalues of this matrix have the explicit form
$$
\xi=\frac{1}{2}\left(tr\pm\sqrt{tr^2-4det}\right),
$$
where the trace and determinant are given by
$$
tr=6-2|E|^2-\eta|L|^2,\qquad det=9-6|E|^2-3\eta|L|^2+2\eta\left[|E|^2|L|^2-(E\cdot L)^2\right].
$$
It thus suffices to show that $tr\ge 0$ and $\det\ge 0$.
Step 3. Non-negativity of the trace of the quadratic form. In [SY3], the trace was shown positive; indeed, by \eqref{invariants},
\begin{align*}
tr&=6-2\left(1-\frac{f_i^2}{|Df|^2}\right)-\left(1+\delta\frac{f_i}{\sigma_1}\right)\left(1-\frac{2(n-1)}{|Df|^2}\right)\\
&>3-\delta\frac{f_i}{\sigma_1}=\frac{(3-\delta)\sigma_1+\delta\lambda_i}{\sigma_1}\\
&\ge 3-\delta\left(1+\frac{n-2}{n}\right)\ge 0,
\end{align*}
for any
\eqal{
\label{trace}
\delta\le \frac{3n}{2(n-1)},
}
using the bound \eqref{sharp} in the case that $\lambda_i<0$.
Step 4. Non-negativity of the determinant of the quadratic form. Our new contribution here is to analyze the determinant in general. Again by \eqref{invariants}, the determinant is
\begin{align*}
det&=\frac{6f_i^2}{|Df|^2}-\frac{3\delta f_i}{\sigma_1}+3\left(1+\frac{\delta f_i}{\sigma_1}\right)\boxed{\frac{2(n-1)}{|Df|^2}}\\
&+2\left(1+\frac{\delta f_i}{\sigma_1}\right)\left[\frac{2(n-1)\sigma_1 f_i}{|Df|^2}-\frac{n f_i^2}{|Df|^2}-\boxed{\frac{2(n-1)}{|Df|^2}}\,\,\right]\\
&>-\frac{3\delta f_i}{\sigma_1}+4\left(1+\frac{\delta f_i}{\sigma_1}\right)\frac{(n-1)\sigma_1 f_i}{|Df|^2}+\left[6-2n\left(1+\frac{\delta f_i}{\sigma_1}\right)\right]\frac{f_i^2}{|Df|^2}.
\end{align*}
Since $f_i=\sigma_1-\lambda_i$ and $\sigma_1^2=2+|\lambda|^2$, we get $|Df|^2=(n-1)\sigma_1^2-2$, so we obtain an inequality in terms of $y:=f_i/\sigma_1$:
\eqal{
\label{det}
det\cdot\frac{|Df|^2}{\sigma_1f_i}&>\boxed{\frac{6\delta}{\sigma_1^2}}-3(n-1)\delta+4(n-1)\left(1+\delta \frac{f_i}{\sigma_1}\right)+\left[6-2n\left(1+\delta\frac{f_i}{\sigma_1}\right)\right]\frac{f_i}{\sigma_1}\\
&>
\left(n-1\right)\left(4-3\delta\right)+\Big[6-2n+4\left(n-1\right)\delta\Big] y-2n\delta y^{2}\\
&=:q_\delta(y).
}
\begin{rem}
\label{rem:3D}
In three dimensions, the almost Jacobi inequality \eqref{aJac} becomes a
full strength one $\bigtriangleup_{F}b\geq\frac{1}{3}\left\vert \nabla
_{F}b\right\vert ^{2},\ $because in \eqref{det}, $q_{4/3}\left( y\right)
=\frac{8}{3}\frac{f_{i}}{\sigma_{1}}\left( 1+3\lambda_{i}/\sigma_{1}\right)
>0\ $ by \eqref{sharp}. This was observed in \cite[$\text{p. 3207}$]{SY3}.
\end{rem}
We write $q_\delta(y)=q_1(y)+\varepsilon\,r(y)$. The remainder:
$$
r(y)=-3(n-1)+4(n-1)y-2ny^2=-3(n-1)+2ny\left(\frac{2(n-1)}{n}-y\right)>-3(n-1),
$$
where we used $0<y=f_i/\sigma_1 \le f_n/\sigma_1 < 2(n-1)/n$; see \eqref{ellipticity} in Corollary \mbb Rf{cor:ellipticity}. To estimate $q_1(y)$, let us solve $0=q_1(y)=n-1+2(n+1)y-2ny^2$:
\eqal{
y_n^\pm:=\frac{n+1\pm\sqrt{1+3n^2}}{2n},\qquad y^+_n\stackrel{n=4}{=}\,\frac{3}{2}.
}
Then $q_1(y)/(y_n^+-y)=2n(y-y_n^-)$. This linear function is minimized at the endpoint $y=0$, so if $y_n^+-y\ge 0$, we conclude
$$
q_\delta(y)\ge -2ny_n^-(y_n^+-y)-3(n-1)\varepsilon\ge -2ny_n^-\left(y_n^+-\frac{f_n}{\sigma_1}\right)-3(n-1)\varepsilon=0,
$$
provided
\eqal{
\varepsilon&:= -\frac{2ny_n^-}{3(n-1)}\left(y_n^+-\frac{f_n}{\sigma_1}\right)\\
&=\frac{\sqrt{3n^2+1}-(n+1)}{3(n-1)}\left(\frac{\sqrt{3n^2+1}-(n-1)}{2n}+\frac{\lambda_n}{\sigma_1}\right)\\
&\stackrel{n=4}{=}\frac{2}{9}\left(\frac{1}{2}+\frac{\lambda_n}{\sigma_1}\right).
}
The condition $y_n^+-y=y_n^+-\frac{f_i}{\sigma_1}\ge 0$ for all $i$ is equivalent to dynamic semi-convexity,
$$
\frac{\lambda_n}{\sigma_1}\ge -\frac{\sqrt{3n^2+1}-(n-1)}{2n}.
$$
If $n=4$, all solutions satisfy this unconditionally, using \eqref{sharp}.
Let us now check that the trace condition \eqref{trace} is also satisfied. It suffices to have $\varepsilon<1/2$. Writing $\varepsilon=c(n)(c_n+\lambda_n/\sigma_1)$, it can be shown that $c(n)$ is an increasing function of $n$ bounded by $ (\sqrt{3}-1)/3<1/4$, and $c_n$ is a decreasing function bounded by $(\sqrt{13}-1)/4<2/3$. Combined with $\lambda_n/\sigma_1\le 1/n\le 1/2$ (see Lemma \mbb Rf{lem:sharp}), we find that $\varepsilon<7/24$ for $n\ge 2$.
This completes the proof of Proposition \mbb Rf{prop:Jac} in dimension $n=4$ and higher dimension $n\ge 5$.
\end{proof}
\section{The doubling inequality}
We now use the almost-Jacobi inequality in Proposition \mbb Rf{prop:Jac} to show an a priori doubling inequality for the Hessian.
\begin{prop}
\label{prop:doub}
Let $u$ be a smooth solution of sigma-2 equation \eqref{s2} on $B_4(0)\subset \mbb R^n$. If $n=4$, then the following inequality is valid:
$$
\sup_{B_{2}(0)}\,\Deltalta u\le C(n)\exp\left(C(n)\|u\|_{C^1(B_3(0))}^2\right)\sup_{B_{1}(0)}\,\Deltalta u.
$$
If $n\ge 5$, the inequality is true, if we suppose also that on $B_3(0)$, there is a semi-convexity type condition
\eqal{
\label{lower}
\frac{\lambda_{min}(D^2u)}{\,\Deltalta u}\ge -c_n,\qquad c_n:=\frac{\sqrt{3n^2+1}-n+1}{2n}.
}
\end{prop}
\begin{proof}
The following test function on $B_3(0)$ is taken from \cite[Theorem 4]{GQ} and \cite[Lemma 4]{Q}:
\eqal{
\label{Pdef}
P_{\alphapha\beta\gamma}:=2\ln\rho(x)+\alphapha (x\cdot Du-u)+\beta|Du|^2/2+\ln \max(\bar b,\gamma^{-1}).
}
Here, $\rho(x)=3^2-|x|^2$, and $\bar b=b-\max_{B_1(0)}b$ for $b=\ln\,\Deltalta u$. We also define $\Gamma:=4+\|u\|_{L^\infty(B_3(0))}+\|Du\|_{L^\infty(B_3(0))}$ to gauge the lower order terms, and denote by $C=C(n)$ a dimensional constant which changes line by line and will be fixed in the end. Small dimensional positive $\gamma$, and smaller positive constants $\alphapha, \beta$ depending on $\gamma$ and $\Gamma$, will be chosen later. We also assume summation over repeated indices for simplicity of notation, where it is impossible in Section \mbb Rf{sec:Jac}.
\setminusallskip
Suppose the maximum of $P_{\alphapha\beta\gamma}$ occurs at $x^*\in B_3(0)$. If $\bar b(x^*)\le \gamma^{-1}$, then we conclude that for $C$ large enough,
\eqal{
\label{Pmax1}
\max_{B_2(0)}P_{\alphapha\beta\gamma}\le C+3\alphapha\Gamma+\frac{1}{2}\beta\Gamma^2+\ln\gamma^{-1}.
}
So we suppose that $\bar b(x^*)>\gamma^{-1}$. If $|x^*|\le 1$, then again we obtain \eqref{Pmax1}, so we also assume that $1<|x^*|<3$.
\setminusallskip
After a rotation about $x=0$, we assume that $D^2u(x^*)$ is diagonal, $u_{ii}=\lambda_i$, with $\lambda_1\ge \lambda_2\ge\cdots\ge \lambda_n$. At the maximum point $x^*$, we have $DP_{\alphapha\beta\gamma}=0$,
\eqal{
\label{max1}
-\frac{\bar b_i}{\bar b}&=2\frac{\rho_i}{\rho}+\alphapha x_ku_{ik}+\beta u_ku_{ik}\\
&=2\frac{\rho_i}{\rho}+\alphapha x_i\lambda_i+\beta u_i\lambda_i,
}
and for $0\ge D^2P_{\alpha\beta\gamma}= (\partial_{ij}P_{\alpha\beta\gamma})$, we get
\eqal{
0\ge\Big(&-\frac{4\delta_{ij}}{\rho}-2\frac{\rho_i\rho_j}{\rho^2}+\alphapha ( x_ku_{ijk}+ u_{ij})+\beta(u_ku_{ijk}+u_{ik}u_{jk})+\frac{\bar b_{ij}}{\bar b}-\frac{\bar b_i\bar b_j}{\bar b^2}\Big)
}
Contracting with $F_{ij}=\partial \sigma_2/\partial u_{ij}$ and using
$$
F_{ij}u_{ijk}=0,\qquad F_{ij}u_{ij}=2\sigma_2=2,\qquad F_{ij}\delta_{ij}=(n-1)\sigma_1,
$$
as well as diagonality at $x^*$, $(F_{ij})=(f_{i}\delta_{ij})$ for $f(\lambda)=\sigma_2(\lambda)$, we obtain at maximum point $x^*$,
$$
0\ge F_{ij}\partial_{ij}P_{\alphapha\beta\gamma}> -4(n-1)\frac{\sigma_1}{\rho}-2\frac{f_i\rho_i^2}{\rho^2}+\beta f_i\lambda_i^2+\frac{f_i\bar b_{ii}}{\bar b}-\frac{f_i\bar b_{i}^2}{\bar b^2}.
$$
Under the assumption that $n=3,4$, or instead that $n\ge 5$ with Hessian constraint \eqref{lower}, almost-Jacobi inequality Proposition \mbb Rf{prop:Jac} is valid, and we get for larger $C$,
\eqal{
\label{maxP}
0\ge-C\frac{\sigma_1}{\rho}-2\frac{f_i\rho_i^2}{\rho^2}+\beta f_i\lambda_i^2+\left(c_n+\frac{\lambda_n}{\,\Deltalta u}\right)\frac{f_i\bar b_{i}^2}{\bar b}-\frac{f_i\bar b_{i}^2}{\bar b^2}.
}
If the nonnegative coefficient of $f_i\bar b_i^2/\bar b$ is positive, we can proceed as in Qiu's proof.
In the alternative case, we must use the $\beta$ term.
We start with the latter case. Note that from \eqref{sharp} in Lemma 1, condition \eqref{lower} $\lambda_n /\,\Deltalta u > -1/2 = - c_n$ is automatically satisfied for $n=4$, and $\lambda_n /\,\Deltalta u > -1/3 > -c_n/2$ for $n=3$.
\textbf{CASE} $-c_n\le \lambda_n/\,\Deltalta u\le -c_n/2$: It follows from \eqref{ellipticity} that $f_n\lambda_n^2\ge c(n)\sigma_1^3$. For larger $C$,
$$
0\ge -C\frac{\sigma_1}{\rho^2}+\beta \sigma_1^3-Cf_i\frac{\bar b_i^2}{\bar b^2}.
$$
Using \eqref{max1} and ellipticity \eqref{ellipticity}, we obtain
$$
\beta \sigma_1^3\le C\frac{\sigma_1}{\rho^2}+C(\alphapha^2+\beta^2\Gamma^2)\sigma_1^3.
$$
If the small parameters satisfy
\eqal{
\label{small1}
\alphapha^2\le \beta/(3C),\qquad \beta\le 1/(3C\Gamma^2),
}
we obtain $\rho^2\sigma_1^2\le C/\beta$. Since $\sigma_1=\sqrt{2+|\lambda|^2}>\sqrt 2$, we have $\sigma_1^2>2\ln\sigma_1$, and we conclude from \eqref{Pdef} and \eqref{small1} that
\eqal{
\label{Pmax2}
P_{\alphapha\beta\gamma}\le C+\ln\beta^{-1}.
}
We next show that Qiu's argument goes through, in the case that ``almost" Jacobi becomes a regular Jacobi.
\textbf{CASE $\lambda_n/\,\Deltalta u\ge -c_n/2$.} It follows that, after enlarging $C$, \eqref{maxP} can be reduced to
$$
0\ge -C\frac{\sigma_1}{\rho}-C\frac{f_i\rho_i^2}{\rho^2}+\beta f_i\lambda_i^2+(\bar b-C)f_{i}\frac{\bar b_i^2}{\bar b^2}.
$$
Using $\bar b(x^*)\ge \frac{1}{2}\bar b(x^*)+\frac{1}{2}\gamma^{-1}$, we assume that $\gamma$ satisfies
\eqal{
\label{small1a}
\frac{1}{2}\gamma^{-1}\ge C,
}
so after enlarging $C$ again, we can further reduce it to
\eqal{
\label{maxP1}
0\ge -C\frac{\sigma_1}{\rho}-C\frac{f_i\rho_i^2}{\rho^2}+\beta f_i\lambda_i^2+\bar bf_{i}\frac{\bar b_i^2}{\bar b^2}.
}
\setminusallskip
\textbf{SUBCASE} $1<|x^*|<3$ and $x_1^2 > 1/n$:
If the small parameters satisfy the condition
\eqal{
\label{small2}
\beta\le \alphapha/(2n\Gamma),\qquad
}
we then obtain from \eqref{max1},
$$
\frac{\bar b_1^2}{\bar b^2}\ge \frac{1}{2}(\alphapha/n-\beta\Gamma)^2\lambda_1^2-\frac{C}{\rho^2}\ge\frac{1}{8n^2}\alphapha^2\lambda_1^2-\frac{C}{\rho^2}.
$$
We assume that this gives a lower bound, or that $C/\rho^2\le \alphapha^2\lambda_1^2/(16n^2)$:
\eqal{
\label{B1}
\frac{\bar b_1^2}{\bar b^2}\ge \frac{1}{16n^2}\alphapha^2\lambda_1^2.
}
For if not, we get $\rho^2\lambda_1^2\le C/\alphapha^2$. Since $\lambda_{1}\ge\sigma_1/n$, we can get $\rho^2\ln\sigma_1\le C/\alphapha^2$. Using \eqref{Pdef} and \eqref{small1}, we would obtain
\eqal{
\label{Pmax3}
P_{\alphapha\beta\gamma}\le C+2\ln\alphapha^{-1}.
}
It follows then, from \eqref{B1} and \eqref{ellipticity}, that \eqref{maxP1} can be simplified to
$$
0\ge -C\frac{\sigma_1}{\rho^2}+\bar b\,f_1(\alphapha^2 \lambda_1^2).
$$
From \eqref{ellipticity}, there holds $f_1\lambda_1^2\ge \sigma_1/n^2$, so we conclude $\rho^2\bar b\le C/\alphapha^2$. By \eqref{Pdef} and \eqref{small1}, we conclude a similar bound \eqref{Pmax3}:
$$
P_{\alphapha\beta\gamma}\le C+2\ln\alphapha^{-1}.
$$
\setminusallskip
\textbf{SUBCASE} $1<|x^*|<3$ and $x_k^2>1/n$ for some $k\ge 2$: Let us first note that $\sigma_1/\rho\le C f_k\rho_k^2/\rho^2$, by \eqref{ellipticity}. We apply $\bar b>\gamma^{-1}$ to \eqref{maxP1}:
$$
0\ge -C\frac{f_i\rho_i^2}{\rho^2}+\beta f_i\lambda_i^2+\gamma^{-1}f_{i}\frac{\bar b_i^2}{\bar b^2}.
$$
Using the $DP=0$ equation \eqref{max1} and enlarging $C$, we obtain
\eqal{
\label{maxP2}
0&\ge -C\frac{f_i\rho_i^2}{\rho^2}+\beta f_i\lambda_i^2+\gamma^{-1}f_i\frac{\rho_i^2}{\rho^2}-C\gamma^{-1}\alphapha^2f_ix_i^2\lambda_i^2-C\gamma^{-1}\Gamma^2\beta^2 f_i\lambda_i^2\\
&\ge \frac{f_i\rho_i^2}{\rho^2}(\gamma^{-1}-C)+\Gamma^{-2}f_i\lambda_i^2\Big((\Gamma^2\beta)-C\gamma^{-1}(\Gamma\alphapha)^2-C\gamma^{-1}(\Gamma^2\beta)^2\Big).
}
The first term is handled if $\gamma^{-1}$ is large enough:
$$
\gamma^{-1}\ge 2C.
$$
We choose $\alphapha,\beta$ as follows:
\eqal{
\label{small3}
\alphapha=\gamma^4/\Gamma,\qquad \beta=\gamma^{6}/\Gamma^2.
}
Let us check that the previous $\alphapha,\beta$ conditions \eqref{small1} and \eqref{small2} are satisfied for any $\gamma^{-1}\ge 2C$, if $C$ is large enough:
$$
\frac{\alphapha^2}{\beta}=\gamma^{2}\le \frac{1}{4C^2}<\frac{1}{3C},\qquad \frac{\Gamma\beta}{\alphapha}=\gamma^2\le \frac{1}{4C^2}<\frac{1}{2n}.
$$
Finally, the coefficient of $\Gamma^{-2}f_i\lambda_i^2$ in \eqref{maxP2} is
$$
\gamma^6-C\gamma^7-C\gamma^{11}=\gamma^6(1-C\gamma-C\gamma^5)\ge \gamma^6\left(1-\frac{1}{2}-\frac{\gamma^4}{2}\right)>0.
$$
Overall, we obtain a contradiction to \eqref{maxP2}.
We conclude that for all large $\gamma^{-1}\ge 2C$ and $\alphapha,\beta$ satisfying \eqref{small3}, the maximum of $P_{\alphapha\beta\gamma}$ obeys the largest of the $P$ bounds \eqref{Pmax1}, \eqref{Pmax2}, and \eqref{Pmax3}:
$$
\max_{B_2(0)}P_{\alphapha\beta\gamma}\le C+\ln\max(\gamma^{-1},\beta^{-1},\alphapha^{-2})=C+\ln(\Gamma^2\gamma^{-8}).
$$
We now choose large $\gamma^{-1}=2C=C(n)$. By \eqref{Pdef}, we obtain the doubling estimate
$$
\frac{\max_{B_2(0)}\sigma_1}{\max_{B_1(0)}\sigma_1}\le \exp\exp\left(C+\ln\Gamma^2\right)=\exp (C\Gamma^2).
$$
\end{proof}
We now modify the doubling inequality to account for ``moving centers". We may control the global maximum by the maximum on any small ball.
\begin{cor}
\label{cor:doub}
Let $u$ be a smooth solution of sigma-2 equation \eqref{s2} on $B_4(0)\subset\mbb R^n$. If $n=4$, or if lower bound \eqref{lower} holds for $n\ge 5$, then the following inequality is true for any $y\in B_{1/3}(0)$ and $0<r<4/3$:
\eqal{
\label{doubley}
\sup_{B_{2}(0)}\,\Deltalta u\le C(n,r,\|u\|_{C^1(B_3(0))})\sup_{B_{r}(y)}\,\Deltalta u.
}
\end{cor}
\begin{proof}
We first note that
$$
B_1(0)\subset B_{4/3}(y)\subset B_{5/3}(y)\subset B_2(0),
$$
for any $|y|<1/3$. By Proposition \mbb Rf{prop:doub}, we find an inequality independent of the center:
\eqal{
\label{doublex}
\sup_{B_{5/3}(y)}\,\Deltalta u\le C(n)\exp\Big(C(n)\|u\|^2_{C^1(B_3(0))}\Big)\sup_{B_{4/3}(y)}\,\Deltalta u.
}
We iterate this inequality about $y$ using the rescalings
$$
u_{k+1}(\bar x)=\left(\frac{5}{4}\right)^2u_k\left(\frac{4}{5}(\bar x-y)+y\right),\qquad u_0=u,\qquad k=0,1,2,\dots
$$
It follows that each $u_k$ satisfies \eqref{doublex}. Denoting
$$
C_k=C(n)\exp \left[C(n)\|u_k\|^2_{C^1(B_3(0))}\right]\le C(n)\exp\left[ \left(\frac{5}{4}\right)^{2k}C(n)\|u\|_{C^1(B_3(0))}\right],
$$
we obtain for $k=1,2,\dots$,
$$
\sup_{B_{5/3}(y)}\,\Deltalta u\le C_0C_1\cdots C_{k}\sup_{B_{r_{k+1}}(y)}\,\Deltalta u\le C(k,n,\|u\|_{C^1(B_3(0))})\sup_{B_{r_{k+1}}(y)}\,\Deltalta u,\qquad r_k=\frac{5}{3}\left(\frac{4}{5}\right)^{k}
$$
Letting $r_{k+1}\le r< r_k$ for some $k$, we combine this inequality with Proposition \mbb Rf{prop:doub} again, to arrive at \eqref{doubley}.
\end{proof}
\begin{rem}
In the uniformly elliptic case, or $a^{ij}b_{ij}\ge a^{ij}b_ib_j$ for $\lambda I\le (a^{ij})\le \Lambda I$, it follows from Trudinger \cite[$\text{p. 70}$]{T3} that a local Alexandrov maximum principle argument gives an integral doubling inequality:
$$
\sup_{B_1(0)}b\le C\left(n,r,\frac{\Lambda}{\lambda}\right)\left(1+\|b\|_{L^n(B_r(0)}\right).
$$
In the $\sigma_2$ case, we can find an integral doubling inequality by modifying Qiu's argument, but the non-uniform ellipticity adds a nonlinear weight to the integral:
$$
\sup_{B_1(0)}\ln\,\Deltalta u\le C(n,r)\Gamma^2\left(1+\|(\,\Deltalta u)^{2/n}\ln\,\Deltalta u\|_{L^n(B_{r}(0)}\right).
$$
This \textit{nonlinear} doubling inequality can be employed to reach Theorems \mbb Rf{thm:s2} and \mbb Rf{thm:n5},
as in Section \mbb Rf{sec:proof}, Step 3.
\end{rem}
\section{Alexandrov regularity for viscosity solutions}
We modify the approach of Evans-Gariepy \cite{EG} and Chaudhuri-Trudinger \cite{CT} to show the following Alexandrov regularity. In \cite[Theorem 1, section 6.4]{EG}, the Alexandrov theorem is seen to arise from combining a gradient estimate with a ``$W^{2,1}$ estimate" for convex functions. The latter can be heuristically understood from the a priori divergence structure calculation
$$
\int_{B_1(0)}|D^2u|\,dx \leq \int_{B_1(0)} \,\Deltalta u\,dx\le C(n)\|u\|_{L^\infty(B_2(0))}.
$$
However, for $k$-convex functions, there is no gradient estimate, in general, and only H\"older and $W^{1,n+}$ estimates for $k>n/2$. We are not able to use Chaudhuri and Trudinger's result in dimension $n=4$. Yet, 2-convex solutions of $\sigma_2=1$ have an even stronger interior Lipschitz estimate, by Trudinger \cite{T2}, and also Chou-Wang \cite{CW}, with a similar ``$W^{2,1}$ estimate" from $\,\Deltalta u=\sqrt{2+|D^2u|^2}$, so the method of \cite{EG} and \cite{CT} can be applied verbatim. We record the modifications below, for completeness.
\begin{prop}
\label{prop:Alex}
Let $u$ be a viscosity solution of sigma-2 equation \eqref{s2} on $B_4(0)$ with $\,\Deltalta u>0$. Then $u$ is twice differentiable almost everywhere in $B_4(0)$, or for almost every $x\in B_4(0)$, there is a quadratic polynomial $Q$ such that
$$
\sup_{y\in B_{r}(x)}|u(y)-Q(y)|=o(r^2).
$$
\end{prop}
We begin the proof of this proposition by first recalling the weighted norm Lipschitz estimate \cite[Corollary 3.4, $\text{p. 587}$]{TW} for smooth solutions of $\sigma_2=1,\,\Deltalta u>0$ on a smooth, strongly convex
domain $\Omegaega\subset\mbb R^n$:
\eqal{
\label{gradp}
\sup_{x,y\in\Omegaega:x\neq y}d_{x,y}^{n+1}\frac{|u(x)-u(y)|}{|x-y|}\le C(n)\int_\Omega |u|dx,
}
where $d_{x,y}=\min(d_x,d_y)$, and $d_x=\text{dist}(x,\partial\Omegaega)$. By solving the Dirichlet problem \cite{CNS} with smooth approximating boundary data, this pointwise estimate holds for viscosity solution $u$, if $\Omegaega\subset\subset B_4(0)$, i.e. $u$ is locally Lipschitz. By Rademacher's theorem, $u$ is differentiable almost everywhere, with $Du\in L^\infty_{loc}$ equal the weak (distribution) gradient. By Lebesgue differentiation, for almost every $x\in B_4(0)$,
\eqal{
\label{gradi}
&\lim_{r\to 0}\Xint-_{B_r(x)}|Du(y)-Du(x)|dy=0.
}
For second order derivatives, we next recall the definition \cite[$\text{p. 306}$]{CT} that a continuous 2-convex function satisfies both $\,\Deltalta u>0$ and $\sigma_2>0$ in the viscosity sense. Since viscosity solution $u$ to $\sigma_2=1$ and $\,\Deltalta u>0$ is 2-convex, we deduce from \cite[Theorem 2.4]{CT} that the weak Hessian $\partial^2u$, interpreted as a vector-valued distribution, gives a vector-valued Radon measure $[D^2u]=[\mu^{ij}]$:
$$
\int u\,\varphi_{ij}\, dx=\int \varphi \,d\mu^{ij},\qquad\varphi\in C^\infty_0(B_4(0)).
$$
Let us outline another proof. Noting that $\sum_{j\neq i}D_{jj}u\ge \bigtriangleup u-\lambda_{\max}\geq0$ for 2-convex smooth function $u,$ where the last inequality follows from \eqref{linearized} with $2\sigma_{2}\geq0,$ via smooth approximation in $C^{0}/L^{\infty}$ norm, we see that $\mu^{i}$ and also $\mu^{e}$ for any unit vector on $\mathbb{R}^{n}$ in \cite[(2.7)]{CT}, are non-negative Borel measures, in turn, bounded on compact sets, that is, Radon measures for 2-convex continuous $u.$ Readily $\mu^{I_{n}}$ for 2-convex continuous $u$ in \cite[(2.6)]{CT} is also a Radon measure. Consequently, for 2-convex continuous $u,$ $D_{ii}u=\mu^{I_{n}}-\mu^{i}$ and also $D_{ee}u=\mu^{I_{n}}-\mu^{e}$ in \cite[(2.8)]{CT} are Radon
measures. This leads to another way in showing that the Hessian measures $D_{ij}u=\mu^{ij}=\left(\mu^{e_{+}e_{+}}-\mu^{e_{-}e_{-}}\right) /2$ with $e_{+}=\left( \partial_{i}+\partial_{j}\right) /\sqrt{2}$ and $e_{-}=\left(\partial_{i}-\partial_{j}\right) /\sqrt{2}$ in \cite[(2.9)]{CT}, are Radon measures for all $1\leq i,j\leq n$ and 2-convex continuous $u.$
\setminusallskip
By Lebesgue decomposition, we write $[D^2u]=D^2u\,dx+[D^2u]_s$, where $D^2u\in L^1_{loc}$ denotes the absolutely continuous part with respect to $dx$, and $[D^2u]_s$ is the singular part. In particular, for $dx$-almost every $x$ in $B_4(0)$,
\begin{align}
\label{Hess1}
&\lim_{r\to 0}\Xint-_{B_r(x)}|D^2u(y)-D^2u(x)|dy=0,\\
\label{Hess2}
&\lim_{r\to 0}\frac{1}{r^n}\|[D^2u]_s\|(B_r(x))=0.
\end{align}
Here, we denote by $\|[D^2u]_s\|$ the total variation measure of $[D^2u]_s$. In fact, these conditions plus \eqref{gradi} are precisely conditions (a)-(c) in \cite[Theorem 1, section 6.4]{EG}. We state their conclusion as a lemma, and include their proof of this fact in the Appendix.
\begin{lem}
\label{lem:o(r^2)}
Let $u\in C(B_4(0))$ have a weak gradient $Du\in L^1_{loc}$ which satisfies \eqref{gradi} for a.e. $x$, and a weak Hessian $\partial ^2u$ which induces a Radon measure $[D^2u]=D^2u\,dx+[D^2u]_s$ obeying conditions \eqref{Hess1} and \eqref{Hess2} for a.e. $x$. Then for a.e. $x\in B_4(0)$, it follows that
\eqal{
\Xint-_{B_r(x)}\Big |u(y)-u(x)-(y-x)\cdot Du(x)-\frac{1}{2}(y-x)D^2u(x)(y-x)\Big|dy=o(r^2).
}
\end{lem}
Choose $x$ for which conditions \eqref{gradi}, \eqref{Hess1}, and \eqref{Hess2} are valid. Let $h(y)=u(y)-u(x)-(y-x)\cdot Du(0)-(y-x)\cdot D^2u(0)\cdot (y-x)/2$. Using
\eqal{
\label{L^1}
\Xint-_{B_r(x)}|h(y)|dy=o(r^2),
}
we will upgrade this to the desired $\|h\|_{L^\infty(B_{r/2}(x))}=o(r^2)$. The crucial ingredient is a pointwise estimate: for $0<2r<4-|x|$,
\eqal{
\label{point}
\sup_{y,z\in B_r(x),y\neq z}\frac{|h(y)-h(z)|}{|y-z|}\le \frac{C(n)}{r}\Xint-_{B_{2r}(x)}|h(y)|dy+Cr,
}
where $C=C(n)|D^2u(x)|$. This was shown as \cite[Lemma 3.1]{CT} for $k$-convex functions with $k>n/2$ using the H\"older estimate \cite[Theorem 2.7]{TW}, and \cite[Claim \#1, $\text{p. 244}$]{EG} for convex functions using a gradient estimate, respectively.
\begin{proof}[Proof of \eqref{point}]
To establish \eqref{point}, we first let $g(y)=u(y)-u(x)-(y-x)\cdot Du(x)$; then $\sigma_2(D^2g(y))=1$ with $\,\Deltalta g(y)>0$, so gradient estimate \eqref{gradp} yields
\begin{align}
\label{pointLem}
\nonumber
r^{n+1}\sup_{y,z\in B_r(x),y\neq z}&\frac{|g(y)-g(z)|}{|y-z|}\\
\nonumber
&=\text{dist}(\boxed{\partial B_r(x)},\partial B_{2r}(x))^{n+1}\sup_{y,z\in \boxed{B_r(x)},y\neq z}\frac{|g(y)-g(z)|}{|y-z|}\\
\nonumber
&\le \sup_{y,z\in B_{2r}(x),y\neq z}d_{y,z}^{n+1}\frac{|g(y)-g(z)|}{|y-z|}\\
\nonumber
&\stackrel{\eqref{gradp}}{\le}C(n)\int_{B_{2r}(x)}|g(y)|dy\\
&\le C(n)\int_{B_{2r}(x)}|h(y)|dy+C(n)|D^2u(x)|\,r^{n+2},
\end{align}
where $d_{y,z}:=\min(2r-|y-x|,2r-|z-x|)$. Next, we polarize
$$
(y-x)\cdot D^2u(x)\cdot(y-x)-(z-x)\cdot D^2u(x)\cdot(z-x)=(y-x+z-x)\cdot D^2u(x)\cdot(y-z),
$$
which gives
$$
r^{n+1}\sup_{y,z\in B_r(x),y\neq z}\frac{|h(y)-h(z)|}{|y-z|}\le r^{n+1}\sup_{y,z\in B_r(x),y\neq z}\frac{|g(y)-g(z)|}{|y-z|}+C(n)r^{n+2}|D^2u(x)|.
$$
This inequality and \eqref{pointLem} lead to \eqref{point}.
\end{proof}
The rest of the proof follows \cite[Claim \#2, $\text{p. 244}$]{EG} or \cite[Proof of Theorem 1.1, $\text{p. 311}$]{CT} verbatim. We summarize the conclusion as a lemma and include its proof in the appendix.
\begin{lem}
\label{lem:pto(r^2)}
Let $h(y)\in C(B_4(0))$ and $x\in B_4(0)$ satisfy integral \eqref{L^1} and pointwise \eqref{point} bounds for $0<2r<4-|x|$. Then $\sup_{B_{r/2}(x)}|h(y)|=o(r^2)$.
\end{lem}
This completes the proof of Proposition \mbb Rf{prop:Alex}.
\begin{rem} In fact, Proposition \mbb Rf{prop:Alex} holds true for (continuous)
viscosity solutions to $\sigma_{k}\left( D^{2}u\right) =1$ for $2\leq k\leq
n/2$ in $n$ dimensions, because the needed conditions \eqref{gradp}-\eqref{Hess2} in the proof
are all available. The twice differentiability a.e. for all $k$-convex functions
and $k>n/2,$ without satisfying any equation in $n$ dimensions, is the content
of the theorems by Alexandrov \cite[$\text{p. 242}$]{EG} and Chaudhuri-Trudinger \cite{CT}.
\end{rem}
\section{Proof of Theorems \mbb Rf{thm:s2} and \mbb Rf{thm:n5}}
\label{sec:proof}
Step 1. After scaling $4^{2}u(x/4)$, we claim that the Hessian $D^2u(0)$ is controlled by $\|u\|_{C^1(B_4(0))}$. Otherwise, there exists a sequence of smooth solutions $u_k$ of \eqref{s2} on $B_4(0)$ with bound $\|u_k\|_{C^1(B_{3}(0))}\le A$, but $|D^2u_k(0)|\to\infty$, in either dimension $n=4$, or in higher dimension $n\ge 5$ with dynamic semi-convexity \eqref{lower}. By Arzela-Ascoli, a subsequence, still denoted by $u_k$, uniformly converges on $B_3(0)$. By the closedness of viscosity solutions (cf.\cite{CC}), the subsequence $u_k$ converges uniformly to a continuous viscosity solution, abusing notation, still denoted by $u$, of \eqref{s2} on $B_3(0)$; we included the non-uniformly elliptic convergence proof in the appendix, Lemma \mbb Rf{lem:conv}. By Alexandrov Proposition \mbb Rf{prop:Alex}, we deduce that $u$ is second order differentiable almost everywhere on $B_3(0)$. We fix such a point $x=y$ inside $B_{1/3}(0)$, and let $Q(x)$ be such that $u-Q=o(|x-y|^2)$.
Step 2. We apply Savin's small perturbation theorem \cite{S} to $v_k=u_k-Q$. Given small $0<r<4/3$, we rescale near $y$:
$$
\bar v_k(\bar x)=\frac{1}{r^2}v_k(r\bar x+y).
$$
Then
\begin{align*}
\|\bar v_k\|_{L^\infty(B_1(0))}&\le \frac{\|u_k(r\bar x+y)-u(r \bar x+y) \|_{L^\infty(B_1(0))}} {r^2} + \frac{\|u(r\bar x+y)-Q(r \bar x+y) \|_{L^\infty(B_1(0))}}{r^2} \\
&\le \frac{\|u_k(r\bar x+y)-u(r \bar x+y) \|_{L^\infty(B_1(0))}} {r^2} + \sigma(r)
\end{align*}
for some modulus $\sigma(r)=o(r^2)/r^2$. And also $\bar v_k$ solves the elliptic PDE in $B_1(0)$
$$
G(D^2\bar w)=\,\Deltalta \bar w+\,\Deltalta Q-\sqrt{2+|D^2\bar w+D^2Q|^2}=0.
$$
Note that $\sigma_2(D^2Q)=1$ with $\,\Deltalta Q>0$, so $G(0)=0$ with $G(M)$ smooth. Moreover, $|D^2G|\le C(n)$, and $G(M)$ is uniformly elliptic for $|M|\le 1$, with elliptic constants depending on $n,Q$.
Now we fix $r=r(n,Q,\sigma)=:\rho$ small enough such that $\sigma(\rho)<c_1/2,$ where $c_1$ is the small constant in \cite[Theorem 1.3]{S}. As $u_k$ uniformly converges to $v$, we have $\|\bar v_k\|_{L^\infty(B_1(0))} \le c_1$ for all large enough $k$.
It follows from \cite[Theorem 1.3]{S} that
$$
\|u_k-Q\|_{C^{2,\alphapha}(B_{\rho/2}(y))}\le C(n,Q,\sigma),
$$
with $\alphapha=\alphapha(n,Q,\sigma)\in(0,1)$. This implies $\,\Deltalta u_k\le C(n,Q,\sigma)$ on $B_{\rho/2}(y)$, uniform in $k$.
Step 3. Finally we apply doubling inequality \eqref{doubley} in Corollary \mbb Rf{cor:doub} to $u_k$ with $r=\rho/2$:
$$
\sup_{B_{2}(0)}\,\Deltalta u_k\le C(n,\rho/2,\|u_k\|_{C^1(B_3(0))})C(n,Q,\sigma)\le C(n,Q,\sigma, A).
$$
We deduce a contradiction to the ``otherwise blowup assumption" at $x=0$.
\begin{rem}
In fact, a similar proof directly establishes interior regularity for viscosity solution $u$ of \eqref{s2} in four dimensions, and then the Hessian estimate, instead of first obtaining the Hessian estimate, then the interior regularity as indicated in the introduction. By rescaling $\bar u(\bar x)=u(r\bar x+x_0)/r^2$ at various centers, it suffices to show smoothness in $B_1(0)$, if $u\in C(B_5(0))$. By Alexandrov Proposition \mbb Rf{prop:Alex}, we let $x=y$ be a second order differentiable point of $u$ in $B_{1/3}(0)$, with quadratic approximation $Q(x)$ and error $\sigma$ at $y$. By Savin's small perturbation theorem \cite[Theorem 1.3]{S}, we find a ball $B_\rho(y)$ with $\rho=\rho(n,Q,\sigma)$ on which $u$ is smooth, with estimates depending on $n,Q,\sigma$. Using \cite{CNS}, we find smooth approximations $u_k\to u$ uniformly on $B_{4}(0)$, with $|Du_k(x)|\le C(\|u\|_{L^\infty(B_4(0))})$ in $B_3(0)$ by the gradient estimate in \cite{T2} and also \cite{CW}. By the small perturbation theorem \cite[Theorem 1.3]{S}, it follows that $u_k\to u$ in $C^{2,\alphapha}$ on $B_{\rho/2}(y)$. Applying doubling \eqref{doubley} to $u_k$ with $r=\rho/2$, we find that $\,\Deltalta u_k\le C(n,Q,\sigma,\|u\|_{L^\infty(B_4(0))})$ on $B_{2}(0)$. By Evans-Krylov, $u_k\to u$ in $C^{2,\alphapha}(B_1(0))$. It follows that $u$ is smooth on $B_1(0)$.
\setminusallskip
From interior regularity, a compactness proof for a Hessian estimate would then follow by an application of the small perturbation theorem. Suppose $u_k\to u$ uniformly but $|D^2u_k(0)|\to\infty$. We observe that the limit $u$ is interior smooth. Applying Savin's small perturbation theorem to $u_k-u$, which solves a fully nonlinear elliptic PDE with smooth coefficients, implies a uniform bound on $D^2u_k(0)$ for large $k$, a contradiction.
\end{rem}
\begin{rem}
By combining Alexandrov Proposition \mbb Rf{prop:Alex} with [S, Theorem 1.3] as above, we find that general viscosity solutions of $\sigma_2=1$ on $B_1(0)\subset\mbb R^n$ with $\,\Deltalta u>0$ have partial regularity: the singular set is closed with Lebesgue measure zero. The same partial regularity also holds for (k-convex) viscosity solutions of equation $\sigma_k=1$, because Alexandrov Proposition \mbb Rf{prop:Alex} is valid for such solutions as noted in Remark 4.1.
\end{rem}
\section{Appendix}
\begin{proof}[Proof of Lemma \mbb Rf{lem:o(r^2)}]
Choose $x\in B_4(0)$ for which conditions \eqref{gradi}, \eqref{Hess1}, and \eqref{Hess2} are valid. Given $r>0$ small enough for $B_{2r}(x)\subset B_4(0)$, we just assume $x=0$. Letting $\eta_\varepsilon(y)=\varepsilon^{-n}\eta(y/\varepsilon)$ be the standard mollifier, we set $u^\varepsilon(y)=\eta_\varepsilon\ast u(y)$ for $|y|<r$. Letting $Q^\varepsilon(y)=u^\varepsilon(0)+y\cdot Du^\varepsilon(0)+y\cdot D^2u(0)\cdot y/2$, we use Taylor's theorem for the linear part:
$$
u^\varepsilon(y)-Q^\varepsilon(y)=\int_0^1(1-t)y\cdot [D^2u^\varepsilon(ty)-D^2u(0)]\cdot y\,dt.
$$
Letting $\varphi\in C^2_c(B_r(0))$ with $|\varphi(y)|\le 1$, we average over $B_r=B_r(0)$:
\eqal{
\label{avg}
\Xint-_{B_r}\varphi(y)(u^\varepsilon(y)-Q^\varepsilon(y))dy&=\int_0^1(1-t)\left(\Xint-_{B_r}\varphi(y)y\cdot[D^2u^\varepsilon(ty)-D^2u(0)]\cdot y\,dy\right)dt\\
&=\int_0^1\frac{1-t}{t^2}\left(\Xint-_{B_{rt}}\varphi(t^{-1}z)z\cdot[D^2u^\varepsilon(z)-D^2u(0)]\cdot z\,dz\right)dt.
}
The first term converges to the Radon measure representation of the Hessian:
\begin{align*}
g^\varepsilon(t)&:=\int_{B_{rt}}\varphi(t^{-1}z)z\cdot D^2u^\varepsilon(z)\cdot z\,dz\\
&\to\int_{B_{rt}}u(z)\partial_{ij}(z^iz^j\varphi(t^{-1}z)dz\qquad\text{as }\varepsilon\to0\\
&=\int_{B_{rt}}\varphi(t^{-1}z)z^iz^jd\mu^{ij}\\
&=\int_{B_{rt}}\varphi(t^{-1}z)z\cdot D^2u(z)\cdot z\,dz+\int_{B_{rt}}\varphi(t^{-1}z)z^iz^jd\mu^{ij}_s.
\end{align*}
It also has a bound which is uniform in $\varepsilon$:
\begin{align*}
\frac{g^\varepsilon(t)}{r^nt^{n+2}}&\le\frac{r^2}{(rt)^n}\int_{B_{rt}}|D^2u^\varepsilon(z)|dz\\
&=\frac{r^2}{(rt)^n}\int_{B_{rt}}\left|\int_{\mbb R^n}D^2\eta_\varepsilon(z-\zeta)u(\zeta)\right|dz\\
&=\frac{r^2}{(rt)^n}\int_{B_{rt}}\left|\int_{\mbb R^n}\eta_\varepsilon(z-\zeta)d[D^2u](\zeta)\right|dz\\
&\le \frac{Cr^2}{\varepsilon^n(rt)^n}\int_{B_{rt+\varepsilon}}|B_{rt}(0)\cap B_\varepsilon(\zeta)|\,d\|D^2u\|(\zeta)\\
&\le \frac{Cr^2}{\varepsilon^n(rt)^n}\min(rt,\varepsilon)^n\|D^2u\|(B_{rt+\varepsilon})\\
&\le Cr^2\frac{\|D^2u\|(B_{rt+\varepsilon})}{(rt+\varepsilon)^n}\\
&\le Cr^2.
\end{align*}
In the last inequality, we used \eqref{Hess1} and \eqref{Hess2}, and denoted by $\|D^2u\|$ the total variation measure of $[D^2u]$. Note also, by \eqref{gradi},
\begin{align*}
|Du^\varepsilon(0)-Du(0)|&\le \int_{B_\varepsilon}\eta_\varepsilon(z)|Du(z)-Du(0)|dz\\
&\le C\Xint-_{B_\varepsilon}|Du(z)-Du(0)|dz\\
&=o(1)_\varepsilon.
\end{align*}
By the dominated convergence theorem, we send $\varepsilon\to 0$ in \eqref{avg}:
\begin{align*}
\Xint-_{B_r}\varphi(y)(u(y)-Q(y))dy&\le Cr^2\int_0^1\Xint-_{B_{rt}}|D^2u(z)-D^2u(0)|dzdt+Cr^2\int_0^1\frac{\|[D^2u]_s\|(B_{rt})}{(rt)^n}dt\\
&=o(r^2),
\end{align*}
using \eqref{Hess1} and \eqref{Hess2}. Taking the supremum over all such $|\varphi(y)|\le 1$, we conclude $\Xint-_{B_r}|h(y)|dy=o(r^2)$. This completes the proof.
\end{proof}
\begin{proof}[Proof of Lemma \mbb Rf{lem:pto(r^2)}]
Given $x\in B_4(0)$ such that \eqref{L^1} and \eqref{point} are true, we let $0<2r<4-|x|$ and $0<\varepsilon<1/2$. Then by \eqref{L^1},
\begin{align*}
\left|\{z\in B_r(x):|h(z)|\ge \varepsilon r^2\}\right|&\le \frac{1}{\varepsilon r^2}\int_{B_r(x)}|h(z)|dz\\
&=\varepsilon^{-1}o(r^n)\\
&< \varepsilon|B_r(x)|,
\end{align*}
provided $r<r_0(\varepsilon,n,h)$. Then for each $y\in B_{r/2}(x)$, there exists $z\in B_r(x)$ such that
$$
|h(z)|\le \varepsilon r^2\qquad\text{ and }\qquad |y-z|\le \varepsilon r.
$$
By \eqref{point} and \eqref{L^1}, we obtain for such $y$,
\begin{align*}
|h(y)|&\le |h(z)|+\frac{|h(y)-h(z)|}{|y-z|}\varepsilon r\\
&\le \varepsilon r^2+C(n)\varepsilon\Xint-_{B_{2r}(x)}|h(\zeta)|d\zeta+C(n,h)\varepsilon r^2\\
&\le C(n,h)\varepsilon r^2.
\end{align*}
We conclude $\sup_{B_{r/2}(x)}|h(y)|=o(r^2)$.
\end{proof}
The following is standard, but for lack of reference, we include a proof.
\begin{lem}
\label{lem:conv}
If $u_k\to u$ is a uniformly convergent sequence of viscosity solutions on $B_1(0)$ of a fully nonlinear elliptic equation $F(D^2u,Du,u,x)=0$ continuous in all variables, then $u$ is a viscosity solution of $F$ on $B_1(0)$.
\end{lem}
\begin{proof}
We show it is a subsolution. Suppose for some $x_0\in B_1(0)$, $0<r<\text{dist}(x_0,\partial B_1(0))$, and smooth $Q$ that $Q\ge u$ on $B_r(x_0)$ with equality at $x_0$. Set
$$
Q_\varepsilon=Q+\varepsilon |x-x_0|^2-\varepsilon^4.
$$
We observe that
$$
u_k(x_0)-Q_\varepsilon(x_0)\ge u(x_0)-Q(x_0)+\varepsilon^4-o(1)_k>0
$$
for $k=k(\varepsilon)$ large enough. In the ring $B_{\varepsilon}(x_0)\setminus B_{\varepsilon/2}(x_0)$, we have
$$
u_k(x)-Q_\varepsilon(x)<u(x)-Q(x)-\varepsilon^3/4+\varepsilon^4+o(1)_k<0
$$
for $\varepsilon=\varepsilon(r)$ small enough, and $k=k(\varepsilon)$ large enough. This means the maximum of $u_k-Q_\varepsilon$ occurs at some in $x_\varepsilon\in B_{\varepsilon/2}(x_0)$. Since $u_k$ is a subsolution, we get
$$
0\le F(D^2Q_\varepsilon(x_\varepsilon),DQ_\varepsilon(x_\varepsilon),Q_\varepsilon(x_\varepsilon),x_\varepsilon)\to F(D^2Q(x_0),DQ(x_0),Q(x_0),x_0),
$$
as $\varepsilon\to 0$. This completes the proof.
\end{proof}
\noindent
\textbf{Acknowledgments.} Y.Y. is partially supported by an NSF grant.
\setminusallskip
\noindent
DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NJ 08544-1000
\textit{Email address:} [email protected]
\setminusallskip
\noindent
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WASHINGTON, BOX 354350, SEATTLE, WA 98195
\textit{Email address:} [email protected]
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Inclusion Criteria for Subclasses of Functions and Gronwall's Inequality \tnoteref{t1}}
\tnotetext[t1]{The work presented here was supported in part by a grant from Universiti Sains Malaysia. This work was completed during the
last two authors' visit to Universiti Sains Malaysia.}
\author[rma]{Rosihan M. Ali\corref{cor1}}
\address[rma]{School of Mathematical Sciences,
Universiti Sains Malaysia, 11800 USM, Penang, Malaysia}
\ead{[email protected]}
\author[rma]{Mahnaz M. Nargesi}
\ead{[email protected]}
\author[rma,vr]{V. Ravichandran\fnref{fn3}}
\address[vr]{Department of Mathematics, University of Delhi,
Delhi 110 007, India} \ead{[email protected]}
\author[swami]{A. Swaminathan}
\address[swami]{Department of Mathematics,
Indian Institute of Technology Roorkee, Roorkee 247 667, India}
\ead{[email protected]}
\cortext[cor1]{Corresponding author}
\begin{abstract}
A normalized analytic function $f$ is shown to be univalent in the open unit disk $\mathbb{D}$ if its second coefficient is sufficiently small
and relates to its Schwarzian derivative through a certain inequality. New criteria for
analytic functions to be in certain subclasses of
functions are established in terms of the Schwarzian derivatives
and the second coefficients. These include obtaining a sufficient condition for functions to be strongly $\alpha$-Bazilevi\v c of order $\beta$.
\end{abstract}
\begin{keyword}
Univalent functions, Bazilevi\v c functions, Gronwall's inequality, Schwarzian derivative, second coefficient.
\MSC[2010] 30C45
\end{keyword}
\end{frontmatter}
\section{Introduction} Let $\mathcal{A}$ be the set of all
normalized analytic functions $f$ of the form $f(z) = z + \sum_{k=2}^{\infty}a_k z^k $ defined in the open unit disk $\mathbb{D}:=\{z\in
\mathbb{C}: |z|<1\}$ and denote by $\mathcal{S}$ the subclass of $\mathcal{A}$ consisting of univalent functions. A function $f\in \mathcal{A}$
is starlike if it maps $\mathbb{D}$ onto a starlike domain with respect to the origin, and $f$ is convex if $f(\mathbb{D})$ is a convex domain.
Analytically, these are respectively equivalent to the conditions $\RE(zf'(z)/f(z))>0$ and $1+\RE(zf''(z)/f'(z))>0$ in $\mathbb{D}$. Denote by
$\mathcal{ST}$ and $\mathcal{CV}$ the classes of starlike and convex functions respectively. More generally, for $0\leq \alpha <1$, a function
$f\in \mathcal{A}$ is starlike of order $\alpha$ if $\RE(zf'(z)/f(z))>\alpha$, and is convex of order $\alpha$ if
$1+\RE(zf''(z)/f'(z))>\alpha$. We denote these classes by $\mathcal{ST}(\alpha)$ and $\mathcal{CV}(\alpha)$ respectively. For $0 <\alpha \leq
1$, let $\mathcal{SST}(\alpha)$ be the subclass of $\mathcal{A}$ consisting of functions $f$ satisfying the inequality
\begin{align*}
\left|\arg
\dfrac{zf'(z)}{f(z)}\right| \leq \frac{\alpha\pi}{ 2}.
\end{align*}Functions in $\mathcal{SST}(\alpha)$ are called strongly starlike functions of order $\alpha$.\\
The Schwarzian derivative $S(f,z)$ of a locally univalent analytic function $f$ is
defined by
\[
S(f,z):= \left(\frac{f''(z)}{f'(z)}\right)' - \frac{1}{2}
\left(\frac{f''(z)}{f'(z)}\right)^2.
\]
The Schwarzian derivative is invariant under M\"obius
transformations. Also, the Schwarzian derivative of an analytic
function $f$ is identically zero if and only if it is a M\"obius
transformation.
Nehari showed that the univalence of an analytic function in ${\mathbb{D}}$ can be guaranteed if its Schwarzian derivative is dominated by a
suitable positive function \cite[Theorem I, p.\ 700]{N2}. In \cite{N1}, by considering two particular positive functions, a bound on the
Schwarzian derivative was obtained that would ensure univalence of an analytic function in $\mathcal{A}$. In fact, the following theorem was
proved.
\begin{theorem}\cite[Theorem II, p.\ 549]{N1}
If $f\in {\mathcal{A}}$ satisfies
\[
|S(f,z)|\leq \dfrac{\pi^2}{2} \quad (z\in {\mathbb{D}}),\]then
$f\in \mathcal{S}$. The result is sharp for the function $f$ given
by $f(z)=(\exp(i\pi z)-1)/i\pi$.
\end{theorem}
The problems of finding similar bounds on the Schwarzian derivatives that would imply univalence, starlikeness or convexity of functions were
investigated by a number of authors including Gabriel \cite{gaber}, Friedland and Nehari \cite{F}, and Ozaki and Nunokawa \cite{ Oza}.
Corresponding results related to meromorphic functions were dealt with in \cite{gaber, Haimo, N1, P}. For instance, Kim and Sugawa
\cite{sugawa} found sufficient conditions in terms of the Schwarzian derivative for locally univalent meromorphic functions in the unit disk to
possess specific geometric properties such as starlikeness and convexity. The method of proof in \cite{sugawa} was based on comparison theorems
in the theory of ordinary differential equations with real coefficients.
Chiang \cite{chiang} investigated strong-starlikeness of order
$\alpha$ and convexity of functions $f$ by requiring the Schwarzian
derivative $S(f ,z)$ and the second coefficient $a_2$ of $f$ to
satisfy certain inequalities. The following results were proved:
\begin{theorem} \cite[Theorem 1, pp.\ 108-109]{chiang} \label{ch:th1}
Let $f\in \mathcal{A}$, $0<\alpha\leq 1$ and
$|a_2|=\eta<\sin(\alpha\pi/2)$. Suppose
\begin{equation}\label{ch:eq1}
\sup_{z\in\mathbb{D}}| S(f,z)| =2\delta(\eta),
\end{equation}
where $\delta(\eta)$ satisfies the inequality
\[\sin^{-1}\left(\frac{1}{2}\delta e^{\delta/2}\right) +
\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2}\right)
\leq \frac{\alpha\pi}{2}. \] Then $f\in \mathcal{SST}(\alpha)$.
Further, $|\arg(f(z)/z)| \leq \alpha\pi/2$.
\end{theorem}
\begin{theorem}\cite[Theorem 2, p.\ 109]{chiang}\label{thm:suff-cv-chiang}
Let $f\in \mathcal{A}$, and $|a_2|=\eta<1/3$. Suppose \eqref{ch:eq1}
holds where $\delta(\eta)$ satisfies the inequality
\[
6\eta+5(1+\eta)\delta e^{\delta/2} < 2.
\]
Then
\[ f\in\mathcal{CV}\left( \frac{2-6\eta-5(1+\eta)\delta e^{\delta/2} }{
2-2\eta-(1+\eta)\delta e^{\delta/2} } \right).
\]
In particular, if $a_2=0$ and $2\delta\leq 0.6712$, then
$f\in\mathcal{CV}$.
\end{theorem}
Chiang's proofs in \cite{chiang} rely on Gronwall's inequality (see Lemma \ref{gronineq} below). In this paper, Gronwall's inequality is used
to obtain sufficient conditions for analytic functions to be univalent. Also, certain inequalities related to the Schwarzian derivative and the
second coefficient will be formulated that would ensure analytic functions to possess certain specific geometric properties. The sufficient
conditions of convexity obtained in \cite{chiang} will be seen to be a special case of our result, and similar conditions for starlikeness
will also be obtained.
\section{Consequences of Gronwall's Inequality } Gronwall's
inequality and certain relationships between the Schwarzian derivative of $f$ and the solution of the linear second-order differential equation
$y''+A(z)y=0$ with $A(z):=S(f;z)/2$ will be revisited in this section. We first state \textit{Gronwall's inequality}, which is needed in our
investigation.
\begin{lemma}\cite[p.\ 19]{gron}\label{gronineq}
Suppose $A$ and $g$ are non-negative continuous real functions for
$t\geq 0$. Let $k> 0$ be a constant. Then the inequality
\[ g(t) \leq k+\int_0^t g(s)A(s)ds \]
implies
\[ g(t) \leq k \exp\left( \int_0^t A(s)ds \right) \quad ( t>0). \]
\end{lemma}
For the linear second-order differential equation $y''+A(z)y=0$
where $A(z):=\frac{1}{2} S(f;z)$ is an analytic function, suppose
that $u$ and $v$ are two linearly independent solutions with
initial conditions $u(0)=v'(0)=0$ and $u'(0)=v(0)=1$. Such solutions
always exist and thus the function $f$ can be represented by
\begin{equation} \label{f}
f(z)= \frac{u(z)}{c u(z)+v(z)}, \quad (c:=-a_2).
\end{equation}
It is evident that
\begin{equation} \label{fd}
f'(z)= \frac{1}{(c u(z)+v(z))^2}.
\end{equation}
Estimates on bounds for various expressions related to $u$ and $v$
were found in \cite{chiang}. Indeed, using the integral
representation of the fundamental solutions
\begin{equation}\label{*}
\begin{array}{ll}
u(z) & = z + \int_0^z (\eta-z) A(\eta)u(\eta)d\eta ,\\[10pt]
v(z) & = 1 + \int_0^z (\eta-z) A(\eta)v(\eta)d\eta,
\end{array}
\end{equation}
and applying Gronwall's inequality, Chiang obtained the following inequalities \cite{chiang}
which we list for easy reference:
\begin{align}\label{eqn-mod-u}
&|u(z)| < e^{\delta/2}, \\
\label{eqn-mod-u-z}
&\left|\frac{u(z)}{z}-1\right| <\frac{1}{2}\delta e^{\delta/2}, \\
\label{eqn-mod-cu+v}
&|cu(z)+v(z)| < (1+\eta)e^{\delta/2}, \\
\label{eqn-mod-cu+v-1}
&|cu(z)+v(z)-1| < \eta + \frac{1}{2}(1+\eta)\delta e^{\delta/2}.
\end{align}
For instance, by
taking the path of integration $\eta(t)=te^{i\theta}$,
$t\in[0,r]$, $z=re^{i\theta}$, Gronwall's inequality
shows that, whenever $|A(z)|<\delta$ and $0<r<1$,
\begin{align*}
|u(z)| & \leq 1+\int_0^r(r-t)|A(te^{i\theta})|\,|u(te^{i\theta})|dt \\
& \leq \exp(\int_o^r (r-t)|A(te^{i\theta})|dt) \leq \exp(\delta/2).
\end{align*} This proves inequality \eqref{eqn-mod-u}.
Note that there was a typographical error in \cite[ Inequality (8), p.\ 112]{chiang}, and that inequality \eqref{eqn-mod-u-z} is the right
form.
\section{Inclusion Criteria for Subclasses of Analytic Functions}
The first result leads to sufficient conditions for univalence.
\begin{theorem}{\label{Thm-f'}}Let $0<\alpha\leq1$, $0\leq\beta<1$,
$f\in \mathcal{A}$ and $|a_2|=\eta $, where
$\alpha$, $\beta$ and $\eta$ satisfy
\begin{equation}\label{Q14a} \sin^{-1}\big(\beta(1+\eta)^2\big)
+2 \sin^{-1}\eta<\frac{\alpha \pi}{2}.\end{equation}
Suppose \eqref{ch:eq1} holds
where $\delta (\eta )$ satisfies the inequality
\begin{equation}\label{Q14}
\sin^{-1}\big(\beta(1+\eta)^2 e^{\delta}\big)+2
\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta
e^{\delta/2}\right)\leq \frac{\alpha\pi}{2}
\end{equation}
Then $|\arg(f'(z)-\beta) | \leq \alpha \pi/2$.
\end{theorem}
\begin{proof}Using a limiting argument as $\delta\rightarrow 0$, the condition \eqref{Q14a} shows that there is a real
number $\delta(\eta)\geq0$ satisfying inequality \eqref{Q14}. The
representation of $f'$ in terms of the linearly independent
solutions of the differential equation $y''+A(z)y=0$ with $ A(z):=
S(f;z)/2$ as given by
equation \eqref{fd} yields
\begin{align}
f'(z)-\beta =\frac{1-\beta(c\ u(z)+v(z))^2}{(c\
u(z)+v(z))^2}\label{th1e0}.
\end{align}
In view of the fact that for $w\in \mathbb{C}$, \[|w-1|\leq r\Leftrightarrow|\arg w|\leq \sin^{-1} r, \]
inequality $(\ref{eqn-mod-cu+v})$ implies
\begin{align}\label{th1e2}
|\arg[1-\beta(c\ u(z)+v(z))^2]|\leq \sin^{-1}\big(\beta(1+\eta)^2
e^{\delta}\big).
\end{align} Similarly, inequality \eqref{eqn-mod-cu+v-1} shows
\begin{equation} \label{th1e3}
|\arg[c\ u(z)+v(z)]| \leq \sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2}\right).
\end{equation}
Hence, it follows from \eqref{th1e0}, \eqref{th1e2} and \eqref{th1e3} that
\begin{align*}
|\arg(f'(z)-\beta)|&\leq|\arg[1-\beta(c\ u(z)+v(z))^2]|+2|\arg[c\ u(z)+v(z)]|\\
&\leq\sin^{-1}(\beta(1+\eta)^2 e^{\delta})+2
\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2}\right)\\
&\leq \frac{\alpha\pi}{2},
\end{align*}
where the last inequality follows from \eqref{Q14}. This completes the
proof.
\end{proof}
By taking $\beta=0$ in Theorem \ref{Thm-f'}, the following univalence criterion is obtained.
\begin{corollary}\label{Thm-suff-univ}
Let $f\in \mathcal{A}$, and $|a_2|=\eta < \sin (\alpha\pi /4)$,
$0<\alpha\leq1$. Suppose $(\ref{ch:eq1})$ holds where $\delta (\eta
)$ satisfies the inequality
\begin{equation*}
\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2} \leq \sin \left(\frac{\alpha\pi}{4}\right).
\end{equation*}
Then $|\arg f'(z)|\leq \alpha\pi /2$, and in particular
$f\in\mathcal{S}$.
\end{corollary}
\begin{example}\label{example-g} Consider the univalent function $g$ given by
\[g(z)=\frac{z}{1+cz},\ \quad |c|\leq 1, \quad z\in {\mathbb{D}}.\]
Since the Schwarzian derivative of an analytic function is zero if
and only if it is a M\"obius transformation, it is evident that $S(g,z)=0$. Therefore
the condition \eqref{ch:eq1} is satisfied with $\delta=0$.
It is enough to take $\eta=|c|$ and to assume that
$\eta $, $\alpha $ and $\beta$ satisfy the inequality \eqref{Q14a}.
Now
\begin{align*}|\arg (g'(z)-\beta)|&=\left|\arg \frac{1}{(1+cz)^2}-\beta\right|
\leq|\arg(1-\beta (1+cz)^2)|+2|\arg (1+cz)|\\
&\leq\sin^{-1}(\beta(1+|c|)^2)+2 \sin^{-1}|c|.\end{align*} In view of the latter inequality, it is necessary to assume inequality \eqref{Q14a}
for $g$ to satisfy $|\arg(g'(z)-\beta) | \leq \alpha \pi/2$.
\end{example}
Let $0\leq\rho< 1$, $0\leq\lambda<1$, and $\alpha$ be a positive integer. A function $f\in\mathcal{A}$ is called an $\alpha$-Bazilevi\v c
function of order $\rho$ and type $\lambda$, written $f\in \mathcal{B}(\alpha,\rho,\lambda)$, if \[ \RE \left(
\frac{zf'(z)}{f(z)^{1-\alpha}g(z)^\alpha}\right)>\rho\quad (z\in\mathbb{D})\] for some function $g\in \mathcal{ST}(\lambda)$. The following
subclass of $\alpha$-Bazilevi\v c functions is of interest. A function $f\in \mathcal{A}$ is called strongly $\alpha$-Bazilevi\v c of order
$\beta$ if
\begin{align*}\label{eqn-Bazil}
\left|\arg\left(\left(\frac{z}{f(z)}\right)^{1-\alpha}f'(z)\right)\right|
<\frac{\beta\pi}{2},\quad (\alpha>0;\ 0<\beta\leq 1),\end{align*}
(see Gao \cite{Gao}). For the class of strongly $\alpha$-Bazilevi\v
c functions of order $\beta$, the following sufficient condition is
obtained.
\begin{theorem}\label{Thm: suff-bazil} Let $\alpha>0$, $0<\beta\leq
1$, $f\in \mathcal{A}$ and $|a_2|=\eta $, where $\eta$, $\alpha$
and $\beta$ satisfy \[
\eta<\sin\left(\frac{\beta\pi}{2(1+\alpha)}\right).\] Suppose
\eqref{ch:eq1} holds where $\delta (\eta )$ satisfies the inequality
\begin{equation}\label{6}
|1-\alpha|\sin^{-1}\left(\frac{1}{2}\delta e^{\delta/2}\right)
+(1+\alpha )\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2}\right)
\leq\frac{\beta\pi}{2}.
\end{equation}
Then $f$ is strongly $\alpha$-Bazilevi\v c of order $\beta$.
\end{theorem}
\begin{proof}The condition $\eta<\sin(\beta\pi/2(1+\alpha))$ ensures that there is a real
number $\delta(\eta)$ satisfying \eqref{6}. Using \eqref{f} and \eqref{fd} lead to
\begin{align*}
\left|\arg\left(\left(\frac{z}{f(z)}\right)^{1-\alpha}f'(z)\right)\right|
& =\left| \arg\left( \left(\frac{u(z)}{z}\right)^{\alpha-1} \left(cu(z)+v(z)\right)^{-(\alpha+1)} \right)\right|\\
&\leq |1-\alpha|\left|\arg\left(\frac{u(z)}{z}\right)\right|
+|\alpha+1|\,\left|\arg(cu(z)+v(z))\right|.
\end{align*}
It now follows from \eqref{eqn-mod-u-z}, \eqref{th1e3} and
\eqref{6} that
\begin{align*}
\left|\arg\left(\left(\frac{z}{f(z)}\right)^{1-\alpha}f'(z)\right)\right|& \leq |1-\alpha|\sin^{-1}\left(\frac{1}{2}\delta e^{\delta/2}\right)
+(1+\alpha )\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2}\right)\\
&\leq\frac{\beta\pi}{2}. \qedhere
\end{align*}
\end{proof}
For $\alpha\geq0$, consider the class $R(\alpha)$ defined by
\[\mathcal{R}(\alpha)=\{f\in \mathcal{A}: \RE \left(f'(z)+\alpha zf''(z)\right)>0, \ \alpha\geq0\}.\]
For this class, the following sufficient condition is obtained.
\begin{theorem}\label{Thm-libera(f)}Let $\alpha \geq 0$, $f\in
\mathcal{A}$ and $|a_2|=\eta $, where $\eta$ and $\alpha$ satisfy
\begin{equation}\label{4}
2\sin^{-1}\eta+ \sin^{-1}\left(\frac{2\eta\alpha}{1-\eta}\right)<\frac{\pi}{2}.\end{equation}
Suppose \eqref{ch:eq1} holds
where $\delta (\eta )$ satisfies the inequality
\begin{equation}\label{5}
2 \sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta
e^{\delta/2}\right)+\sin^{-1}\left(\frac{4\alpha\big(\eta+(1+\eta)\delta
e^{\delta/2}\big)}{2-2\eta-(1+\eta)\delta
e^{\delta/2}}\right)\leq\frac{\pi}{2}.
\end{equation}
Then $ f\in\mathcal{R}(\alpha) $.
\end{theorem}
\begin{proof} Again it is easily seen from a limiting argument that the condition \eqref{4} guarantees the existence of
a real number $\delta(\eta)\geq0$ satisfying the
inequality \eqref{5}. It is sufficient to show that
\begin{align*}
\left|\arg\left(f'(z)\left(1+\alpha\frac{zf''(z)}{f'(z)}\right)\right)\right|<\frac{\pi}{2}.
\end{align*}
The equation \eqref{fd} yields
\begin{align}\label{ksp1a}
\frac{zf''(z)}{f'(z)}&=-2z \frac{c u'(z)+v'(z)}{cu(z)+v(z)}.
\end{align}
A simple calculation from \eqref{*} shows that
\[ cu'(z)+v'(z)= c-\int_0^zA(\eta)[cu(\eta)+v(\eta)]d\eta, \]
and an application of \eqref{eqn-mod-cu+v} leads to
\begin{align}\label{ksp3}
|cu'(z)+v'(z)|\leq \eta+(1+\eta)\delta e^{\delta/2}.
\end{align}
Use of \eqref{eqn-mod-cu+v-1} results in
\begin{align}\label{ksp2}
|cu(z)+v(z)| \geq 1-|cu(z)+v(z)-1| \geq 1- \eta-\frac{1}{2}(1+\eta)\delta e^{\delta/2}.
\end{align}
The lower bound in \eqref{ksp2} is non-negative from the assumption made in \eqref{5}. From \eqref{ksp1a}, \eqref{ksp3} and \eqref{ksp2} , it
is evident that
\begin{align}\notag
\left|\left(1+\alpha\frac{zf''(z)}{f'(z)}\right)-1\right|& =\left|2z\alpha\frac{c u'(z)+v'(z)}{c
u(z)+v(z)}\right| \\
& \leq \frac{2\alpha\big(\eta+(1+\eta)\delta
e^{\delta/2}\big)}{1-\eta-\frac{1}{2}(1+\eta)\delta
e^{\delta/2}}.\notag
\end{align}
Hence,
\begin{equation}\label{Q2}
\left|\arg \left(1+\alpha \frac{zf''(z)}{f'(z)}\right)\right|
\leq\sin^{-1}\left(\frac{4\alpha\big(\eta+(1+\eta)\delta
e^{\delta/2}\big)}{2-2\eta-(1+\eta)\delta e^{\delta/2}}\right).
\end{equation}
From \eqref{th1e3} it follows that
\begin{align}\label{ksp1}
|\arg f'(z)|=2|\arg(cu(z)+v(z))|\leq 2
\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2}\right).
\end{align} Using \eqref{fd} and \eqref{th1e3},
the inequality \eqref{ksp1} together with \eqref{Q2} and \eqref{5}
imply that
\begin{align*}
\left|\arg \left( f'(z)\left(1+\alpha\frac{zf''(z)}{f'(z)}\right)\right) \right|
&\leq |\arg f'(z)|+\left|\arg\left(1+\alpha \frac{zf''(z)}{f'(z)}\right)\right|\\
&\leq 2 \sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2}\right)
+\sin^{-1}\left(\frac{4\alpha\big(\eta+(1+\eta)\delta e^{\delta/2}\big)}{2-2\eta-(1+\eta)\delta e^{\delta/2}}\right)\\
&\leq\frac{\pi}{2}.\qedhere
\end{align*}
\end{proof}
\begin{theorem}\label{thm-suff-nonlinear-combination}
Let $f\in \mathcal{A}$, $|a_2|=\eta \leq1/3$, and $\beta$, $\alpha$ be real numbers satisfying
\begin{equation}\label{9}|\alpha|\sin^{-1}\eta+
|\beta|\sin^{-1}\left(\frac{2\eta}{1-\eta}\right)<\frac{\pi}{2}.\end{equation}
Suppose \eqref{ch:eq1} holds
where $\delta (\eta )$ satisfies the inequality
\begin{equation}\label{8}\begin{split}
|\alpha|\sin^{-1}\left(\frac{1}{2}\delta e^{\delta/2}\right)
+|\alpha|\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2}\right) \\
{}+|\beta|\sin^{-1}\left(\frac{4\big(\eta+(1+\eta)\delta
e^{\delta/2}\big)}{2-2\eta-(1+\eta)\delta
e^{\delta/2}}\right)\leq\frac{\pi}{2}.
\end{split}
\end{equation}
Then
\begin{align}\label{eqn-nonlinear-st-cv}
{\rm \RE\,}
\left(\left(\dfrac{zf'(z)}{f(z)}\right)^{\alpha}\left(1+\dfrac{zf''(z)}{f'(z)}\right)^{\beta}\right)>0.
\end{align}
\end{theorem}
\begin{proof} The inequality \eqref{9} assures the existence of
$\delta$ satisfying \eqref{8}. From \eqref{f} and
\eqref{fd} it follows that
\begin{align}\label{eq-zf'/f}
\dfrac{z f'(z)}{f(z)}=\dfrac{z}{u(z)}\dfrac{1}{cu(z)+v(z)}, \quad z\in {\mathbb{D}}.
\end{align}
By \eqref{eqn-mod-u-z} and \eqref{th1e3},
\begin{equation}\label{3}\left|\arg\left(\frac{zf'(z)}{f(z)}\right)\right|\leq\sin^{-1}\left(\frac{1}{2}\delta
e^{\delta/2}\right)+\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta
e^{\delta/2}\right).\end{equation} Using \eqref{Q2} with $\alpha=1$,
\eqref{3} and \eqref{8} lead to
\begin{align*}
\left|\arg\left(\left(\frac{zf'(z)}{f(z)}\right)^{\alpha}
\left(1+\frac{zf''(z)}{f'(z)}\right)^{\beta}\right)\right|
&\leq|\alpha|\left|\arg\left(\frac{zf'(z)}{f(z)}\right)\right|
+|\beta|\left|\arg\left(1+\frac{zf''(z)}{f'(z)}\right)\right|\\
&\leq|\alpha|\sin^{-1}\left(\frac{1}{2}\delta
e^{\delta/2}\right)+|\alpha|\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta
e^{\delta/2}\right)\\
& \quad{}+|\beta|\sin^{-1}\frac{4\big(\eta+(1+\eta)\delta
e^{\delta/2}\big)}{2-2\eta-(1+\eta)\delta e^{\delta/2}}\\
&\leq\frac{\pi}{2}.\end{align*} This shows that
\eqref{eqn-nonlinear-st-cv} holds.
\end{proof}
\begin{remark}
Theorem \ref{thm-suff-nonlinear-combination} yields
the following interesting special cases.
\begin{enumerate}[(i)]
\item If $\alpha=0$, $\beta=1$, a sufficient condition for
convexity is obtained. This case reduces to a result in
\cite[Theorem 2, p.\ 109]{chiang}.
\item For $\alpha=1$, $\beta=0$, a sufficient condition for
starlikeness is obtained.
\item For $\alpha=-1$ and $\beta=1$, then the class of functions
satisfying \eqref{eqn-nonlinear-st-cv} reduces to the class of
functions
\[ \mathcal{G}:= \left\{f\in \mathcal{A}\Bigg| \ {\rm \RE\,}
\left(\dfrac{1+\frac{zf''(z)}{f'(z)}}{\frac{zf'(z)}{f(z)}}\right)>0\right\}.\]
This class $\mathcal{G}$ was considered by Silverman \cite{Sil}
and Tuneski \cite{Tuneski}.
\end{enumerate}
\end{remark}
\begin{theorem}
Let $\beta\geq0$, $f\in \mathcal{A}$ and $|a_2|=\eta $, where $\eta$
satisfies
\begin{align}\label{lasteqn} \sin^{-1}\left(\eta\right)
+\sin^{-1}\left(\frac{2\beta\eta}{1-\eta }\right)<\frac{\pi}{2}.
\end{align}
Suppose \eqref{ch:eq1} holds
where $\delta (\eta )$ satisfies the inequality
\begin{align*}
\sin^{-1}\left(\frac{1}{2}\delta e^{\delta/2}\right)
+\sin^{-1}\left(\eta+\frac{1}{2}(1+\eta)\delta e^{\delta/2}\right)
+\sin^{-1}\left(\frac{4\beta\big(\eta+(1+\eta)\delta
e^{\delta/2}\big)}{2-2\eta-(1+\eta)\delta
e^{\delta/2}}\right)\leq\frac{\pi}{2}.
\end{align*}
Then
\begin{align}\label{kspcon}
\RE\left( \dfrac{zf'(z)}{f(z)}+ \beta \dfrac{z^2f''(z)}{f(z)}\right)>0.
\end{align}
\end{theorem}
The proof is similar to the proof of Theorem
$\ref{thm-suff-nonlinear-combination}$, and is therefore omitted.
The inequality \eqref{lasteqn} is equivalent to the condition \[
\eta\left( 1+\sqrt{(1-\eta)^2-4\beta^2\eta^2}+2\beta\sqrt{1-\eta^2}
\right) <1.
\]
For $\beta=1$, the above equation simplifies to
\begin{align*}
\eta^8-4 \eta^7+12 \eta^6-12 \eta^5+6 \eta^4+20 \eta^3-4 \eta^2-4
\eta+1 = 0;
\end{align*} the value of the root $\eta$ is approximately 0.321336.
Functions satisfying inequality \eqref{kspcon}
were investigated by Ramesha \emph{et al.}\ \cite{KSP}.\\
Consider the class $P(\gamma)$, $0\leq\gamma\leq 1$, given by
\[
P(\gamma):= \left\{f\in {\mathcal{A}}:
\left|\arg\left((1-\gamma)\dfrac{f(z)}{z}+\gamma f'(z)\right)\right|<\dfrac{\pi}{2}, \quad z \in {\mathbb{D}}\right\}.
\]
The same approach applying Gronwall's inequality leads to the
following result about the class $P(\gamma)$.
\begin{theorem}\label{thm-p-gamma} Let $0\leq \gamma <1$, $f\in {\mathcal{A}}$ and $|a_2|=\eta$, where $\eta$ and $\gamma$ satisfy
\begin{equation}\label{7}
\sin^{-1}\left(\dfrac{\gamma}{1-\gamma}\dfrac{1}{\eta-1}\right)+\sin^{-1} \eta<\dfrac{\pi}{2}.
\end{equation}
Suppose $(\ref{ch:eq1})$ holds
where $\delta(\eta)$ satisfies the inequality
\begin{equation}\begin{split}\label{eq-p-gamma}
&\sin^{-1}\left(\dfrac{1}{2}\delta e^{\delta/2}\right)
+\sin^{-1}\left(\eta+\dfrac{1}{2}(1+\eta)\delta e^{\delta/2}\right)\\
&+ \sin^{-1}\left(\dfrac{2\gamma}{1-\gamma}\dfrac{1}{2-2\eta-(1+\eta)\delta
e^{\delta/2}}\dfrac{1}{1-2e^{\delta/2}}\right)\leq\dfrac{\pi}{2}.
\end{split}
\end{equation}
Then $f\in P(\gamma)$.
\end{theorem}
\begin{proof} Condition \eqref{7} assures the existence of a real
number $\delta(\eta)\geq0$ satisfying the inequality \eqref{eq-p-gamma}. A simple calculation from \eqref{*} and Lemma \ref{gronineq} shows
that
\begin{align*}
|u(z)-1|& \leq |z-1|+\left|\int_0^z (\zeta-z)A(\zeta)u(\zeta)d\zeta \right| \\
& \leq 2 \displaystyle e^{\delta/2}.
\end{align*}
The above inequality gives
\begin{align}\label{eq-z/u}
\left|\dfrac{z}{u(z)}\right| \leq \dfrac{1}{|u(z)|}\leq \dfrac{1}{1-|u(z)-1|}\leq \dfrac{1}{1-2e^{\delta/2}}.
\end{align}
Therefore, for some $0<\beta\leq\gamma/(1-\gamma)$, \eqref{eq-zf'/f}, \eqref{eq-z/u} and \eqref{ksp2} lead to
\begin{align*}
\left|1+\beta \dfrac{zf'(z)}{f(z)}-1\right|
& = \beta \left|\dfrac{z}{u(z)}\right|\, \dfrac{1}{\left|cu(z)+v(z)\right|}\\
& \leq \dfrac{\beta}{1-2e^{\delta/2}}\dfrac{1}{1-\eta-\frac{1}{2}(1+\eta)\delta e^{\delta/2}}\\
&= \dfrac{2\beta}{1-2e^{\delta/2}}\dfrac{1}{2-2\eta-(1+\eta)\delta e^{\delta/2}}.
\end{align*}
Hence
\begin{equation}\label{1}
\left| \arg\left(1+\beta \dfrac{zf'(z)}{f(z)}\right)\right|
\leq\sin^{-1}\left(\dfrac{2\beta}{1-2e^{\delta/2}}\dfrac{1}{2-2\eta-(1+\eta)\delta
e^{\delta/2}}\right).
\end{equation}
Also, \eqref{eqn-mod-u-z} and \eqref{th1e3} yield
\begin{align}\label{2}
\left|\arg \dfrac{f(z)}{z}\right|&=\left|\arg \dfrac{u(z)}{z(cu(z)+v(z))}\right|\notag \\
& \leq
\left|\arg \dfrac{u(z)}{z}\right|+ \left|\arg (cu(z)+v(z))\right|\notag\\
&\leq \sin^{-1}\left(\dfrac{1}{2}\delta e^{\delta/2}\right)+
\sin^{-1} \left(\eta+\dfrac{1}{2}(1+\eta)\delta e^{\delta/2}\right).
\end{align}
Replacing $\beta$ by $\gamma/(1-\gamma)$ in inequality \eqref{1}, and using \eqref{2} and \eqref{eq-p-gamma} yield
\begin{align*}
\left|\arg\left((1-\gamma)\dfrac{f(z)}{z}+\gamma f'(z)\right)\right|
&\leq \left|\arg\dfrac{f(z)}{z}\right|
+ \left|\arg\left(1+\dfrac{\gamma}{1-\gamma} \dfrac{zf'(z)}{f(z)}\right)\right| \\
& \leq \sin^{-1}\left(\dfrac{1}{2}\delta e^{\delta/2}\right)
+\sin^{-1}\left(\eta+\dfrac{1}{2}(1+\eta)\delta e^{\delta/2}\right)\\
&\quad{}
+\sin^{-1}\left(\dfrac{2\gamma}{1-\gamma}\dfrac{1}{1-2e^{\delta/2}}
\dfrac{1}{2-2\eta-(1+\eta)\delta e^{\delta/2}}\right)\\
&\leq\dfrac{\pi}{2},
\end{align*}
and hence $f\in P(\gamma)$.
\end{proof}
\end{document} |
\begin{document}
\maketitle
\begin{abstract}
The concept of quasi-isometric embedding maps between $*$-algebras is introduced. We have obtained some basic results related to this notion and similar to quasi-isometric embedding maps on metric spaces, under some conditions, we give a necessary and sufficient condition
on a $*$-homomorphism to be a quasi-isometric embedding between $*$-algebras.
\end{abstract}
\section{Introduction}
Throughout this paper by a $*$-algebra we mean a Banach algebra with the involution $*$.
Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces. A map $f:X\longrightarrow Y$ is called quasi-isometric embedding if there exist $\alpha\geq1$ and $\beta\geq0$ such that
\begin{equation}\label{eqm}
\frac{1}{\alpha}d_X(x,y)-\beta\leq d_Y(f(x),f(y))\leq\alpha d_X(x,y)+\beta,
\end{equation}
for every $x,y\in X$. In the above inequalities if $\beta=0$ then the quasi-isometric embedding $f$ is called bi-Lipschitz map where for some results we refer to \cite{be, ca, hu, le, va}. The concept of quasi-isometric embedding is a very useful tool for investigating Cayley and hyperbolic graphs, for more details and applications, we refer to \cite{bo, gr, ham, sc}. Authors in \cite{cc} considered the notion of quasi-isometric on finitely generated algebras, where they have obtained many interesting geometric properties. For more details about geometric properties and other works related to these properties we refer to \cite[Introduction]{cc} and the references there in.
In this paper, we consider $*$-algebras and quasi-isometric embedding maps on these algebras where our definition is different from the quasi-isometric embedding maps on metric spaces and algebras that was defined before.
\section{Quasi-isometric Embedding}
We start off with the following definitions:
\begin{definition}\label{d2.1}
Let $A$ and $B$ two $*$-algebras. We say the map $\varphi:A\longrightarrow B$ is a quasi-isometric embedding of $A$ into $B$ if the following two conditions hold:
\begin{itemize}
\item[(i)] for every finite subset $F\subset A$ there is a finite subset $F'\subset B$ such that, if, $a_1^*a_2\in F$, then $\varphi(a_1)^*\varphi(a_2)\in F'$, for every $a_1,a_2\in A$.
\item[(ii)] for every finite subset $F'\subset B$ there is a finite subset $F\subset A$ such that, if, $\varphi(a_1)^*\varphi(a_2)\in F'$, then $a_1^*a_2\in F$, for every $a_1,a_2\in A$.
\end{itemize}
\end{definition}
\begin{definition}\label{d2.2}
Let $A$ and $B$ two $*$-algebras. A quasi-isometry from $A$ into $B$ is a quasi-isometric embedding $\varphi:A\longrightarrow B$ for which there is a finite dimensional subspaces $K\subset B$ such that $\varphi(A)+K=B$ i.e. for every $b\in B$ there exists $k\in K$ and $a\in A$ such that $\varphi(a)+k=b$. If there is a quasi-isometry from $A$ into $B$, then we say that $A$ is quasi-isometric to $B$.
\end{definition}
We continue with the following example that says there is an example of quasi-isometric embedding between $*$-algebras that is not isometry in the classical case.
\begin{ex}\label{ex2.3}
Let $A$,$B$ two $*$-algebras and $\varphi:A\longrightarrow B$ be an injective $*$-homomorphism. We claim that $\varphi$ is a quasi-isometric embedding of $A$ into $B$. We must show that $\varphi$ satisfies in the conditions of Definition \ref{d2.1}. Assume that $F\subset A$ is a finite subset and set $\varphi(F)=F'$. Clearly, $F'\subset B$ is finite. Then for every $a_1,a_2\in A$ such that $a_1^*a_2\in F$, we have
\begin{equation}\label{eq1ex2.3}
\varphi(a_1)^*\varphi(a_2)= \varphi(a_1^*a_2)\in F'.
\end{equation}
This implies that the condition \emph{(}i\emph{)} holds. For \emph{(}ii\emph{)}, suppose that $F'$ is a finite subset of $B$ and set $F=\varphi^{-1}(F')$. Then $F$ is a finite subset of $A$, because of that $\varphi$ is injective. For every $a_1,a_2\in A$ satisfying in $\varphi(a_1)^*\varphi(a_2)\in F'$, we have $\varphi(a_1^*a_2)\in F'$. This implies that $a_1^*a_2\in F$.
\end{ex}
Let $A$ be a $*$-algebra and $F_1, F_2\subset A$ be arbitrary sets, by $\bigoplus_{f\in F_2}(F_1+f)$ we means disjoint union of $F_1+f$'s. In the following, we give necessary and sufficient conditions on defined map in the Example \ref{ex2.3} that becomes a quasi-isometry.
\begin{proposition}\label{p2.4}
Let $A$,$B$ be two $*$-algebras and $\varphi:A\longrightarrow B$ be an injective $*$-homomorphism with closed range. Then $\varphi$ is a quasi-isometry if and only if ${\varphi(A)}$ is a subspace of $B$ with finite codimension.
\end{proposition}
\begin{proof}
Assume that $\varphi$ is a quasi-isometry. Hence, there is a finite dimensional subspace $K$ of $B$ such that $\varphi(A)+K=B$. Then
\begin{equation}\label{eq1p2.4}
\dim \frac{B}{{\varphi(A)}}\leq \dim K<\infty.
\end{equation}
Conversely, suppose that ${\varphi(A)}$ has finite codimension in $B$. Let $K$ be the set of representatives for the right cosets of ${\varphi(A)}$ in $B$. Thus, $K$ is a finite dimensional subspace of $B$ and
\begin{equation}\label{eq1p2.4}
B=\bigoplus_{k\in K}\left(\varphi(A) +k\right)=\varphi(A)+K.
\end{equation}
This implies that $\varphi$ is a quasi-isometry.
\end{proof}
Amenability of Banach algebras was introduced by Johnson in \cite{j1} and its relation between homological properties of Banach algebras was introduced by Helemskii in \cite{he}. Let $A$ and $B$ be two Banach algebras and $\varphi$ be a continuous dense range homomorphism, if, $A$ is amenable, then $B$ is too \cite[Proposition 5.3]{j1}. A Banach algebra $A$ is called has property (\textbf{G}) if there exists an amenable locally compact group $G$ and a continuous homomorphism $\varphi:L^1(G)\longrightarrow A$ with dense range, this concept is introduced by Kepert \cite{ke} that he used the above mentioned result in \cite{j1}. The following result is a special case of the inverse of \cite[Proposition 5.3]{j1}.
\begin{proposition}\label{p2.4}
Let $A$,$B$ be two $*$-algebras and $\varphi:A\longrightarrow B$ be an injective quasi-isometry.
If $B$ is amenable, then $A$ is amenable.
\end{proposition}
\begin{proof}
Assume that $B$ is amenable and $K$ be a finite dimensional subspace of $B$ such that $\varphi(A)+K=B$. This implies that $\varphi(A)$ is amenable. Since $A\cong\varphi(A)$, $A$ is amenable.
\end{proof}
Let $A$,$B$ be two $*$-algebras and $\varphi:A\longrightarrow B$ be a $*$-map i.e. $\varphi(a^*)=\varphi(a)^*$ for all $a\in A$. Then $\varphi$ is called unitary preserving $*$-map if for every unitary element $a\in A$, $\varphi(a)$ is a unitary element in $B$. Assume that $e_A$ and $e_B$ are the unit elements of $A$ and $B$, respectively. We denote the set of all unitary elements of a $*$-algebra $A$ by $U(A)$ and by $|U(A)|$ we mean the cardinal number of $U(A)$. If $\varphi$ is a quasi-isometric embedding, then we could not say that $\varphi(e_A)=e_B$, in general. By the following, we are seeking a quasi-isometric embedding that has this property.
\begin{lemma}\label{l2.5}
Let $A$ and $B$ be two unital $*$-algebras with units $e_A$ and $e_B$, respectively. If there is a quasi-isometric embedding from $A$ into $B$ that is a unitary preserving $*$-map, then there is a quasi-isometric embedding $\phi$ from $A$ into $B$ such that $\phi(e_A)=e_B$. Moreover, if the exited map is a quasi-isometry, then $\phi$ is a quasi-isometry.
\end{lemma}
\begin{proof}
Suppose that $\varphi:A\longrightarrow B$ is a quasi-isometric embedding that satisfies in the stated conditions. Define $\phi:A\longrightarrow B$ by $\phi(a)=\varphi(e_A)^*\varphi(a)$ for all $a\in A$. Then
\begin{equation}\label{eq1l2.5}
\phi(a_1)^*\phi(a_2)=\varphi(a_1)^*\varphi(e_A)\varphi(e_A)^*\varphi(a_2)=\varphi(a_1)^*\varphi(a_2),
\end{equation}
for every $a_1,a_2\in A$. Thus, $\phi$ is a quasi-isometric embedding. Also, according to definition of $\phi$, we have $\phi(e_A)=e_B$.
Now, let $\varphi$ be a a quasi-isometry. Thus, there is a finite dimensional subspace $K$ of $B$ such that $\varphi(A)+K=B$. Set $K'=\varphi(e_A)^*K$. Clearly, $K'$ is a finite dimensional subspace of $B$. Then
\begin{equation}\label{eq2l2.5}
\phi(A)+K'=\varphi(e_A)^*\varphi(A)+\varphi(e_A)^*K=\varphi(a)^*B=B.
\end{equation}
This shows that $\phi$ is a quasi-isometry.
\end{proof}
By the following result we show that the composition of the quasi-isometric embedding maps is a quasi-isometric embedding map.
\begin{lemma}\label{l2.6}
Let $A, B$ and $C$ be three $*$-algebras. Assume that $\varphi:A\longrightarrow B$ and $\psi:B\longrightarrow C$ are quasi-isometric embedding maps, then $\psi\circ\varphi:A\longrightarrow C$ is a quasi-isometric embedding map.
\end{lemma}
\begin{proof}
It is straightforward to check that $\psi\circ\varphi:A\longrightarrow C$ is a quasi-isometric embedding.
\end{proof}
Let $A$ and $B$ be two $*$-algebras, let $\varphi:A\longrightarrow B$ be a map and $b\in B$. Define the sets
\begin{itemize}
\item[] $S_A=\{a^*a':\ \|a-a'\|=1,\ \text{for every}\ a, a'\in A\}$,
\item[] $S_B=\{\varphi(a)^*\varphi(a'):\ \|\varphi(a)-\varphi(a')\|=1,\ \text{for every}\ a, a'\in A\}$,
\end{itemize}
and
\begin{itemize}
\item[] $C_\varphi(b)=\{\varphi(a):\ \varphi(a)=b\ \text{for every}\ a\in A\}.$
\end{itemize}
For a finite subset $F$ of $A$ by $\text{Diam}(F)$ we mean the diameter of $F$ i.e. $$\text{Diam}(F)=\max_{a,a'\in F}\|a-a'\|.$$
Now, we are ready to prove the one of the main results of this paper as follows:
\begin{theorem}\label{mt}
Let $\varphi$ be a quasi-isometric embedding between $*$-algebras $A$ and $B$, then there exist $\alpha\geq1$ and $\beta\geq0$ such that
\begin{equation}\label{eq1mt}
\frac{1}{\alpha}\|a-a'\|-\beta\leq \|\varphi(a)-\varphi(a')\|\leq\alpha \|a-a'\|+\beta,
\end{equation}
for every $a,a'\in A$.
\end{theorem}
\begin{proof}
Let $F\subset A$ be a finite subset such that $a_1^*a_2\in F$ whenever $\varphi(a_1)^*\varphi(a_2)\in S_B$ for every $a_1,a_2\in A$. Similarly, assume that $F'\subset B$ is a finite subset such that $\varphi(a_1)^*\varphi(a_2)\in F'$ whenever $a_1^*a_2\in S_A$ for every $a_1,a_2\in A$. Also, let $F_1\subset A$ be a finite subset and $b\in B$ such that $a_1^*a_2\in F_1$ whenever $\varphi(a_1),\varphi(a_2)\in C_\varphi(b)$ for every $a_1,a_2\in A$. Now; set
\begin{equation}\label{eq2mt}
\alpha=\max\left\{\text{Diam}(F), \text{Diam}(F')\right\}\ \ \text{and}\ \ \beta=\frac{1}{\alpha}\text{Diam}(F_1).
\end{equation}
Clearly, if $a=a'$ then \eqref{eq1mt} holds. Suppose that $\|a-a'\|=1$, then $a^*a'\in S_A$ and consequently, $\varphi(a)^*\varphi(a')\in F'$. Since, $\varphi$ is a quasi-isometric embedding, we have $\|\varphi(a)-\varphi(a')\|\leq\alpha$. Now, assume that $1\leq\|a-a'\|=\gamma\leq n$, where $\gamma\in\mathbb{R}^+$ and assume that $a_0,a_1,\ldots,a_n\in A$ such that $a_0=a$, $a_n=a'$ and $\|a_i-a_{i+1}\|=1$ for $0\leq i\leq n-1$. Then
\begin{eqnarray}\label{eq3mt}
\nonumber
\|\varphi(a)-\varphi(a')\|&\leq&\sum_{i=0}^{n-1}\|\varphi(a_i)-\varphi(a_{i+1})\| \\
\nonumber
&\leq& n\alpha \\
&=& \alpha\|a-a'\|.
\end{eqnarray}
Hence,
\begin{equation}\label{eq3mt}
\|\varphi(a)-\varphi(a')\|\leq \alpha\|a-a'\|,
\end{equation}
for all $a,a'\in A$. Let $\|\varphi(a)-\varphi(a')\|=1$, then $\varphi(a)^*\varphi(a')\in S_B$ and consequently, $a^*a'\in F$. Since, $\varphi$ is a quasi-isometric embedding, we have $\|a-a'\|\leq\alpha$. Similar to the above statements, if assume that $\|\varphi(a)-\varphi(a')\|=n\geq1$ one can show that
\begin{equation}\label{eq4mt}
\|a-a'\| \leq \alpha\|\varphi(a)-\varphi(a')\|,
\end{equation}
for all $a,a'\in A$. If $\varphi(a_1),\varphi(a_2)\in C_\varphi(b)$, then by \eqref{eq2mt}, we have $a^*a'\in F_1$. Thus
\begin{equation}\label{eq5mt}
\frac{1}{\alpha} \|a-a'\| \leq \beta.
\end{equation}
This implies that
\begin{equation}\label{eq6mt}
\frac{1}{\alpha} \|a-a'\| -\beta \leq 0= \|\varphi(a)-\varphi(a')\|.
\end{equation}
Thus, the inequalities \eqref{eq4mt} and \eqref{eq6mt} imply the inequality \eqref{eq1mt}.
\end{proof}
As a special case of the converse of the above obtained result, we have the following:
\begin{theorem}\label{mt2}
Let $\varphi$ be a $*$-homomorphism between $*$-algebras $A$ and $B$ that satisfies in \eqref{eq1mt}. Then $\varphi$ is a quasi-isometric embedding.
\end{theorem}
\begin{proof}
Let $F\subset A$ be a finite subset. Pick $a\in F$ and define
\begin{equation}\label{eq1mt2}
F'=\{\varphi(a'):\ a'\in F\ \text{and}\ \|\varphi(a)-\varphi(a')\|<\delta\},
\end{equation}
where $\delta=\alpha\max_{a'\in F}\|a-a'\|+\beta$. Clearly, $F'\subset B$ is a finite subset. For every $a_1, a_2\in A$ such that $a_1^*a_2\in F$, the relation \eqref{eq1mt} implies that
\begin{eqnarray}\label{eq2mt2}
\nonumber
\|\varphi(a)-\varphi(a_1)^*\varphi(a_2)\|&=& \|\varphi(a)-\varphi(a_1^*a_2)\|\leq\alpha\|a-a_1^*a_2\|+\beta\\
&<&\delta.
\end{eqnarray}
Hence, $\varphi(a_1)^*\varphi(a_2)\in F'$. Thus, the condition (i) of Definition \ref{d2.1} holds. For the condition (ii), suppose that $F'$ is a finite subset of $B$. Let $f\in F'$ and assume that $a$ is an element of $A$ such that $\varphi^{-1}(f)=a$ (inverse map). Define
\begin{equation}\label{eq3mt2}
F=\{a':\ \text{there is a}\ f'\in F'\ \text{such that}\ \varphi^{-1}(f')=a'\ \text{and}\ \|a-a'\|<\delta\},
\end{equation}
where $\delta=\alpha\max_{f'\in F}\|f-f'\|+\alpha\beta$. Note that in $F$, we just choose one of elements that $\varphi^{-1}(f')$ contains. Clearly, $F'\subset B$ is a finite subset. For every $a_1, a_2\in A$ such that $\varphi(a_1)^*\varphi(a_2)\in F'$, then \eqref{eq1mt} implies that
\begin{eqnarray}\label{eq4mt2}
\nonumber
\|a-a_1^*a_2\|&\leq&\alpha\|\varphi(a)-\varphi(a_1^*a_2)\|+\alpha\beta\\
\nonumber
& = &\alpha\|\varphi(a)-\varphi(a_1)^*\varphi(a_2)\|+\alpha\beta\\
&<&\delta.
\end{eqnarray}
This means that $a_1^*a_2\in F$ and this completes the proof.
\end{proof}
Theorems \ref{mt} and \ref{mt2} follow the following result:
\begin{corollary}\label{mc}
Let $\varphi$ be a $*$-homomorphism between $*$-algebras $A$ and $B$. Then $\varphi$ is a quasi-isometric embedding if and only if there exist $\alpha\geq1$ and $\beta\geq0$ such that
\begin{equation}\label{eq1mc}
\frac{1}{\alpha}\|a-a'\|-\beta\leq \|\varphi(a)-\varphi(a')\|\leq\alpha \|a-a'\|+\beta,
\end{equation}
for every $a,a'\in A$.
\end{corollary}
\begin{ex}
Let $A$ be a $*$-algebra. Consider $A\times A$ with the coordinate-wise product. It becomes a $*$-algebra with respect to the norm $\|(a,b)\|=\|a\|+\|b\|$ and the involution $(a,b)^*=(a^*,b^*)$ for every $a,b\in A$. Define $\varphi:A\longrightarrow A\times A$ by $\varphi(a)=(a,a)$ for every $a\in A$. Clearly, $\varphi$ is a $*$-homomorphism and
$$\|\varphi(a)-\varphi(a')\|=\|(a-a',a-a')\|=2\|a-a'\|,$$
for every $a,a'\in A$. Obviously, $\varphi$ satisfies in \eqref{eq1mc} for $\alpha\geq 2$ and $\beta\geq0$. Thus, it is a quasi-isometric embedding.
\end{ex}
Let $A$ and $B$ two $C^*$-algebras. Suppose that $\varphi:A\longrightarrow B$ is a $*$-homomorphism, then $\varphi$ is norm-decreasing \cite[Corollary 3.2.4]{da}. This fact follows the following result.
\begin{corollary}\label{mc}
Every $*$-homomorphism between $C^*$-algebras is a quasi-isometric embedding.
\end{corollary}
\end{document} |
\begin{document}
\title{Showcase of Blue Sky Catastrophes}
\author{Leonid Shilnikov, Andrey Shilnikov and Dmitry Turaev}
\institute{Leonid Shilnikov \at Institute for Applied Mathematics \& Cybernetics,
10 Ulyanov Street, Nizhny Novgorod, 603005, Russia\\
Andrey Shilnikov \at Neuroscience Institute, and
Department of Mathematics and Statistics,
Georgia State University, Atlanta, 100 Piedmont Ave SE, Atlanta, GA, 30303, USA, \email{[email protected]}, \\
Dmitry Turaev \at Department of Mathematics,
Imperial College, London, SW7 2AZ, UK, \email{[email protected]}}
\maketitle
\abstract{Let a system of differential equations possess a saddle-node periodic orbit such that
every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global)
stable manifold. We study several bifurcation cases where the splitting of such a homoclinic connection
causes the Blue Sky Catastrophe, including the onset of complex dynamics.
The birth of an invariant torus or a Klein bottle is also described.}
\section{Introduction}
In the pioneering works by A.A.~Andronov and E.A.~Leontovich \cite{AL1,AL2}
all main bifurcations of stable periodic orbits of dynamical systems in a plane had been studied: the emergence
of a limit cycle from a weak focus,
the saddle-node bifurcation through a merger of a stable limit cycle with an unstable one and their consecutive
annihilation, the birth of a limit cycle from a separatrix loop to a saddle, as well as from a separatrix loop
to a saddle-node equilibrium. Later, in the 50-60s these bifurcations were generalized for the multi-dimensional case,
along with two additional bifurcations: period doubling and the birth of a two-dimensional torus. Apart from that,
in \cite{lp1,lp2} L.~Shilnikov had studied the main bifurcations of saddle periodic orbits out of homoclinic loops
to a saddle and discovered a novel bifurcation of homoclinic loops to a saddle-saddle\footnote{an equilibrium state, alternatively called a Shilnikov saddle-node,
due to a merger of two saddles of different topological types}.
Nevertheless, an open problem still remained: could there be other types of codimension-one bifurcations
of periodic orbits? Clearly, the emphasis was put on bifurcations of {\em stable} periodic orbits, as only
they generate robust self-sustained periodic oscillations, the original paradigm of
nonlinear dynamics. One can pose the problem as follows:\\ {\em
In a one-parameter family $X_{\mu}$ of systems of differential equations,
can both the period and the length of a structurally stable periodic orbit ${\cal L}_\mu$
tend to infinity as the parameter $\mu$ approaches some bifurcation value, say $\mu_0=0$?} \\
Here, structural stability means that none of the multipliers of the periodic orbit ${\cal L}_\mu$ crosses the unit circle, i.e. ${\cal L}_\mu$
does not bifurcate at $\mu\neq\mu_0$. Of particular interest is the case where ${\cal L}_\mu$ is stable, i.e.
all the multipliers are strictly inside the unit circle.
A similar formulation was given by J.~Palis and Ch.~Pugh \cite{PP} (notable Problem~37), however the structural stability
requirement was missing there. Exemplary bifurcations of a periodic orbit whose period becomes arbitrarily large
while the length remains finite as the bifurcation moment is approached are
a homoclinic bifurcation of a saddle with a negative saddle value and that
of a saddle-node \cite{lp0,book2}. These were well-known at the time, so in \cite{PP} an additional condition
was imposed, in order to ensure that the sought bifurcation is really of a new type: the periodic orbit ${\cal L}_\mu$ must stay
away from any equilibrium states (this would immediately imply that the length of the orbit grows to infinity
in proportion to the period). As R.~Abraham put it, the periodic orbit must ``disappear in the blue-sky'' \cite{Ab}.
In fact, a positive answer to ``Problem 37'' could be found in an earlier paper \cite{F}. In explicit form, a solution
was proposed by V.~Medvedev \cite{Me}. He constructed examples of flows on a torus and a Klein bottle with
stable limit cycles whose lengths and periods tend to infinity as $\mu\to\mu_0$, while at $\mu=\mu_0$
both the periodic orbits disappear and new, structurally unstable saddle-node periodic orbits appear
(at least two of them, if the flow is on a torus). The third example of \cite{Me} was a flow on a 3-dimensional torus
whose all orbits are periodic and degenerate, and for the limit system the torus is foliated by two-dimensional invariant tori.
Medvedev's examples are not of codimension-1: this is obvious for the torus case that requires at least two saddle-nodes, i.e.
$X_{\mu_0}$ is of codimension 2 at least. In case of the Klein bottle one may show \cite{book2,AfS,TSh3,Li,Il}
that for a generic perturbation of the Medvedev family the periodic orbits existing at $\mu\neq\mu_0$
will not remain stable for all $\mu$ as they undergo an infinite sequence of
forward and backward period-doubling bifurcations (this is a typical behavior of fixed points of a non-orientable
diffeomorphism of a circle).
A blue-sky catastrophe of codimension 1 was found only in 1995 by L.~Shilnikov and D.~Turaev \cite{TSh3,TSh1,TSh2,ShT}.
The solution was based on the study of bifurcations of a saddle-node periodic orbit whose entire unstable manifold
is homoclinic to it. The study of this bifurcation was initiated by V.~Afraimovich and L.~Shilnikov \cite{AfS,AfS1,AfS2,AfS3}
for the case where the unstable manifold of the saddle-node is a torus or a Klein bottle (see Fig.~\ref{fig1}).
As soon as the saddle-node disappears, the Klein bottle may persist, or it may break down to cause chaotic
dynamics in the system \cite{AfS4,NPT,TSh,Sync}. In these works, most of attention was paid to the torus case,
as its breakdown provides a geometrical model of the quasiperiodicity-toward-chaos transition encountered
universally in Nonlinear Dynamics, including the onset of turbulence \cite{Sh00}.
\begin{figure}
\caption{Two cases of the unstable manifold $W^u_L$ homoclinic to the saddle-node periodic orbit $L$:
a 2D torus (A) or a Klein bottle (B).}
\label{fig1}
\end{figure}
In the hunt for the blue sky catastrophe, other distinct configurations of the unstable manifold of the saddle-node were suggested
in \cite{TSh1}. In particular, it was shown that in the phase space of dimension 3 and higher
the homoclinic trajectories may spiral back onto the saddle-node orbit in the way shown in Fig.~\ref{fig2}.
If we have a one-parameter family $X_\mu$ of systems of differential equations
with a saddle-node periodic orbit at $\mu=\mu_0$ which possesses this special kind of the homoclinic unstable
manifold and satisfy certain additional conditions, then as the saddle-node disappears the inheriting attractor
consists of a single stable periodic orbit ${\cal L}_\mu$ which
undergoes no bifurcation as $\mu\to\mu_0$ while its length tends to infinity. Its topological limit, $M_0$, is
the entire unstable manifold of the saddle-node periodic orbit.
\begin{figure}
\caption{Original construction of the blue sky catastrophe from \cite{TSh1}
\label{fig2}
\end{figure}
The conditions found in \cite{TSh1} for the behavior of the homoclinic orbits ensuring the blue-sky catastrophe are open,
i.e. a small perturbation of the one-parameter family $X_\mu$ does not destroy the construction. This implies
that such a blue-sky catastrophe occurs any time a family of systems of differential equations crosses
the corresponding codimension-1 surface in the Banach space of smooth dynamical systems. This surface constitutes
a stability boundary for periodic orbits. This boundary is drastically new comparable to those
known since the 30-60s and has no analogues in planar systems. There are reasons to conjecture that this
type of the blue-sky catastrophe closes the list of main stability boundaries for periodic orbits (i.e. any new stability
boundary will be of codimension higher than 1).
In addition, another version of blue-sky catastrophe leading to the birth of a uniformly-hyperbolic strange attractor
(the Smale-Williams solenoid \cite{Sm,W}) was also discovered in \cite{TSh1,TSh2}. This codimension-1 bifurcation
of a saddle-node which corresponds yet to a different configuration of the homoclinic unstable manifold of the periodic orbit
(the full classification is presented in \cite{book2}). Here, the structurally stable attractor existing all the way
up to $\mu=\mu_0$ does not bifurcate so that the length of each and every (saddle) periodic orbit in it tends to
infinity as $\mu\to\mu_0$.
Initially we believed that the corresponding configuration of the unstable manifold would be too exotic for
the blue-sky catastrophe to occur naturally in a plausible system. In contrast, soon after, a first explicit
example of the codimension-1 blue-sky catastrophe
was proposed by N.~Gavrilov and A.~Shilnikov \cite{GSh}, in the form of a family of 3D systems of differential equations
with polynomial right-hand sides. A real breakthrough came in when the blue-sky catastrophe has turned out to be a typical
phenomenon for slow-fast systems. Namely, in \cite{book2,mmj} we described a number of very general scenarios leading
to the blue-sky catastrophe in such
systems with at least two fast variables; for systems with one fast variable the blue-sky catastrophe was found in \cite{GKR}.
In this way, the blue-sky catastrophe has found numerous applications in mathematical neuroscience, namely, it explains a smooth
and reversible transition between tonic spiking and bursting in exact Hodgkin-Huxley type models of
interneurons \cite{leech1,leech2} and in mathematical models of square-wave bursters \cite{hr}.
The great variability of the burst duration near the blue-sky catastrophe
was shown to be the key mechanism ensuring the diversity of rhythmic patterns generated by small neuron complexes
that control invertebrate locomotion \cite{DG1,DG2,DG3}.
In fact, the term ``blue sky catastrophe" should be naturally treated in a broader way. Namely, under this term we allow
to embrace a whole class of dynamical phenomena that all are due to the existence of a stable (or, more generally, structurally stable)
periodic orbit, ${\cal L}_\mu$, depending continuously on the parameter $\mu$ so that both, the length and the period of ${\cal L}_\mu$ tend
to infinity as the bifurcation parameter value is reached. As for the topological limit, $M_0$,
of the orbit ${\cal L}_\mu$ is concerned, it may possess a rather degenerate structure that does not prohibit $M_0$ from
having equilibrium states included. As such, the periodic regime ${\cal L}_\mu$ could emerge as a composite construction
made transiently of several quasi-stationary states: nearly constant, periodic, quasiperiodic, and even chaotic fragments.
As one of the motivations (which we do not pursue here) one may think on slow-fast model where the fast
3D dynamics is driven by a periodic motion in a slow subsystem.
\section{Results}
In this paper we focus on an infinitely degenerate case where $M_0$
is comprised of a saddle periodic orbit with a continuum of homoclinic trajectories.
Namely, we consider a one-parameter family of sufficiently smooth systems of differential equations
$X_\mu$ defined in $R^{n+1}$, $n\geq 2$, for which we need to make a number of assumptions as follows.\\
\noindent {\bf (A)} There exists a saddle periodic orbit $L$ (we assume the period equals $2\pi$) with the
multipliers\footnote{the eigenvalues of the linearization of the Poincare map} $\rho_1,\dots,\rho_n$. Let the multipliers satisfy
\begin{equation}\label{rh1}
\max_{i=2,\dots,n-1} |\rho_i|<|\rho_1|<\;1\;<|\rho_n|.
\end{equation}
Once this property is fulfilled at $\mu=0$, it implies that the saddle periodic orbit $L=L_\mu$ exists
for all small $\mu$ and smoothly depends on $\mu$. Condition (\ref{rh1}) also holds for all small $\mu$.
This condition implies that the
stable manifold $W^s_\mu$ is $n$-dimensional\footnote{the intersection of $W^s_\mu$ with any cross-section to $L_\mu$ is $(n-1)$-dimensional}
and the unstable manifold $W^u_\mu$ is two-dimensional.
If the unstable multiplier $\rho_n$ is positive (i.e. $\rho_n>1$), then
the orbit $L_\mu$ divides $W^u_\mu$ into two halves, $W^+_\mu$ and $W^-_\mu$, so
$W^u_\mu=L_\mu\cup W^+_\mu\cup W^-_\mu$. If $\rho_n$ is negative ($\rho_n<-1$), then
$W^u_\mu$ is a M\"obius strip, so $L_\mu$ does not divide $W^u_\mu$; in this case we denote
$W^+_\mu=W^u_\mu\backslash L_\mu$.
Concerning the stable manifold, condition (\ref{rh1}) implies that in $W^s_\mu$ there
exists (at $n\geq 3$) an $(n-1)$-dimensional strong-stable invariant manifold $W^{ss}_\mu$ whose
tangent at the points of $L_\mu$ contains the eigen-directions
corresponding to the multipliers $\rho_2,\dots,\rho_{n-1}$, and the orbits in $W^s_\mu\backslash W^{ss}_\mu$
tend to $L_\mu$ along the direction which correspond to the leading multiplier $\rho_1$.\\
\noindent {\bf (B)} At $\mu=0$ we have $W^+_0\subset W^s_0\backslash W^{ss}_0$,
i.e. we assume that {\em all} orbits from $W^+_0$ are
homoclinic to $L$. Moreover, as $t\to +\infty$, they tend to $L$ along the leading direction.\\
\noindent {\bf (C)} We assume that the flow near $L$ contracts three-dimensional volumes, i.e.
\begin{equation}\label{contr}
|\rho_1\rho_n| <1.
\end{equation}
This condition is crucial, as the objects that we obtain by bifurcations of the homoclinic surface $W^+_0\cup L$ are meant to
be attractors. Note that this condition is similar to the negativity of the saddle value condition from the theory of
homoclinic loops to a saddle equilibrium \cite{AL1,AL2,lp0}, see (\ref{sadl}).\\
\noindent {\bf (D)} We assume that one can introduce linearizing coordinates near $L$. Namely, a small neighborhood $U$ of $L$
is a solid torus homeomorphic to $S^1\times R^n$, i.e. we can coordinatize it by an angular variable $\theta$
and by normal coordinates $u\in R^n$. Our assumption is that these coordinates are chosen so that
the system in the small neighborhood of $L$ takes the form
\begin{equation}\label{lfr}
\dot u=C(\theta,\mu) u, \qquad \dot \theta=1,
\end{equation}
where $C$ is $2\pi$-periodic in $\theta$. The smooth linearization is not always possible, and our results
can be obtained without this assumption. We, however, will avoid discussing the general case here, in order
to make the construction more transparent.
It is well-known that by a $4\pi$-periodic transformation of the coordinates $u$ system (\ref{lfr}) can be brought to
the time-independent form. Namely, we may write the system as follows
\begin{equation}\label{lcfr}
\begin{array}{l}
\dot x=-\lambda(\mu) x, \qquad \dot y=B(\mu) y,\\
\dot z=\gamma(\mu) z,\\
\dot \theta=1,\end{array}
\end{equation}
where $x\in R^1$, $y\in R^{n-2}$, $z\in R^1$, and $\lambda=-\frac{1}{2\pi}\ln|\rho_1|>0$, $\gamma=\frac{1}{2\pi}\ln|\rho_n|>0$
and, if $n\geq 2$, $B(\mu)$ is an $(n-2)\times(n-2)$-matrix such that
\begin{equation}\label{nev}
\|e^{Bt}\|=o(e^{-\lambda t}) \qquad (t\to+\infty).
\end{equation}
Note also that condition {\bf (C)} implies
\begin{equation}\label{sadl}
\gamma-\lambda<0.
\end{equation}
By (\ref{lcfr}), the periodic orbit $L(\mu)$ is given by $x=0$, $y=0$, $z=0$, its local stable manifold is given
by $z=0$, and the leading direction in the stable manifold is given by $y=0$; the local unstable manifold
is given by $\{x=0,y=0\}$.
Recall that the $4\pi$-periodic transformation we used to bring system (\ref{lfr}) to the autonomous form (\ref{lcfr})
is, in fact, $2\pi$-periodic or $2\pi$-antiperiodic. Namely, the points $(\theta,x,z,y)$ and $(\theta+2\pi,\sigma(x,z,y))$
are equal (they represent the same point in the solid torus $U$), where $\sigma$
is an involution which changes signs
of some of the coordinates $x,z,y_1,\dots,y_{n-2}$. More precisely, $\sigma$ changes the orientation of each of the directions
which correspond to the real negative multipliers $\rho$. In particular, if all the multipliers $\rho$ are positive, then $\sigma$
is the identity, i.e. our coordinates are $2\pi$-periodic in this case.\\
\begin{figure}
\caption{Poincar\'e map $T_1$ takes a cross-section $S_1$ transverse to the unstable manifold $W^u$
to a cross-section $S_0$ transverse to the stable manifold $W^s$.}
\label{fig3}
\end{figure}
\noindent {\bf (E)} Consider two cross-sections $S_0:\{x=d,\quad \|y\|\leq \varepsilon_1,\quad |z|\leq \varepsilon_1\}$ and
$S_1:\{z=d,\quad \|y\|\leq\varepsilon_2,\quad |x|\leq\varepsilon_2\}$ for some small positive $d$ and $\varepsilon_{1,2}$.
Denote the coordinates on $S_0$ as $(y_0,z_0,\theta_0)$ and the coordinates on $S_1$ as $(x_1,y_1,\theta_1)$.
The set $S_0$ is divided by the stable manifold $W^s$ into two regions, $S_0^+:\{z_0>0\}$ and $S_0^-:\{z_0<0\}$.
Since $W^+_0\subset W^s_0$ by assumption 2, it follows that the orbits starting at $S_1$
define a smooth map $T_1:S_1\to S_0$ (see Fig.~\ref{fig3}) for all small $\mu$:
\begin{equation}\label{glom}
\begin{array}{l}
z_0 =f(x_1,y_1,\theta_1,\mu)\\
y_0 =g(x_1,y_1,\theta_1,\mu)\\
\theta_0 =m\theta_1 + h(\theta_1,\mu)+\tilde h(x_1,y_1,\theta_1,\mu),
\end{array}
\end{equation}
where $f,g,h,\tilde h$ are smooth functions $4\pi$-periodic in $\theta_1$, and the function $\tilde h$ vanishes at $(x_1=0,y_1=0)$.
Condition $W^+_0\subset W^s_0$ reads as
$$f(0,0,\theta_1,0)\equiv 0.$$
We assume that
\begin{equation}\label{qqfff}
f(0,0,\theta_1,\mu)=\mu\alpha(\theta_1,\mu),
\end{equation}
where
\begin{equation}\label{alpt}
\alpha(\theta_1,\mu)>0
\end{equation}
for all $\theta_1$, i.e. {\em all the homoclinics are split simultaneously and in the same direction}, and
the intersection $W^+_\mu\cap S_0$ moves inside $S_0^+$ with a non-zero velocity as $\mu$ grows across zero.
The coefficient $m$ in the last equation of (\ref{glom}) is an integer. In order to see this, recall that
two points $(\theta,x,z,y)$ and $(\hat\theta,\hat x,\hat z,\hat y)$ in $U$ are the same if and only if
$\hat\theta=\theta+2\pi k, (\hat x,\hat z,\hat y)=\sigma^k (x,z,y)$ for an integer $k$. Thus, if we increase $\theta_1$ to $4\pi$
in the right-hand side of (\ref{glom}), then the corresponding value of $\theta_0$ in the left-hand side
may change only to an integer multiple of $2\pi$, i.e. $m$ must be an integer or a half-integer. Let us show that
the half-integer $m$ are forbidden by our assumption (\ref{alpt}). Indeed, if the multiplier $\rho_n$ is positive, then
the involution $\sigma$ keeps the corresponding variable $z$ constant. Thus, $(z=d,\theta=\theta_1, x=0, y=0)$ and
$(z=d,\theta=\theta_1+2\pi, x=0, y=0)$ correspond, in this case, to the same point on $W^+_\mu\cap S_1$, hence their image
by (\ref{glom}) must give the same point on $S_0$, i.e. the corresponding values of $\theta_0$ must differ on an integer multiple
of $2\pi$, which means that $m$ must be an integer. If $\rho_n<0$, then $\sigma$ changes the sign of $z$, i.e. if two values of
$\theta_0$ which correspond to the same point on $S_0$ differ on $2\pi k$, the corresponding values of $z$ differ to a factor of
$(-1)^k$. Now, since the increase of $\theta_1$ to $4\pi$ leads to the increase of $\theta_0$ to $4\pi m$ in (\ref{glom}),
we find that $f(0,0,4\pi,\mu)=(-1)^{2m}f(0,0,0,\mu)$ in the case $\rho_n<0$. This implies
that if $m$ is a half-integer, then $f(0,0,\theta)$ must have zeros at any $\mu$ and (\ref{alpt}) cannot be satisfied.
The number $m$ determines the shape of $W^+\cap S_0$. Namely, the equation of the curve $W^+_0\cap S_0$ is
$$\theta_0 =m\theta_1 + h_1(\theta_1,0),\qquad y_0 =g(0,0,\theta_1,0), \qquad z_0=0,$$
so $|m|$ defines the homotopic type of this curve in $S_0\cap W^s_0$, and the sign of $m$ is responsible for the orientation.
In the case $n=2$, i.e. when the system is defined in $R^3$, the only possible case is $m=1$. At $n=3$ (the system in $R^4$)
the curve $W^+_0\cap S_0$ lies in the two-dimensional intersection of $W^s$ with $S_0$. This is either an annulus (if $\rho_1>0$),
or a M\"obius strip (if $\rho_1<0$). Since the smooth curve $W^+_0\cap S_0$ cannot have self-intersections, it follows that
the only possible cases are $m=0,\pm1$ when $W^s\cap S_0$ is a two-dimensional annulus and $m=0,\pm1,\pm2$ when
$W^+_0\cap S_0$ is a M\"obius strip. At large $n$ (the system in $R^5$ and higher) all integer values of $m$ are possible.\\~\\
Now we can formulate the main results of the paper.\\
\noindent{\bf Theorem.}
Let conditions {\bf (A-E)} hold. Consider a sufficiently small neighborhood $V$ of the homoclinic surface $\Gamma=W^+_0\cap L$.\\
\begin{enumerate}
\item If $m=0$ and, for all $\theta$,
\begin{equation}\label{bsky}
|h'(\theta,0)-\frac{\alpha'(\theta,0)}{\gamma \alpha(\theta,0)}|<1,
\end{equation}
then a single stable periodic orbit ${\cal L}_\mu$ is born as $\Gamma$ splits. The orbit ${\cal L}_\mu$ exists at all small $\mu>0$; its
period and length tend to infinity as $\mu\to+0$.
All orbits which stay in $V$ for all positive times and
which do not lie in the stable manifold of the saddle orbit $L_\mu$ tend to ${\cal L}_\mu$.\\
\item If $|m|=1$ and, for all $\theta$,
\begin{equation}\label{tor}
1+m \left[h'(\theta,0)-\frac{\alpha'(\theta,0)}{\gamma \alpha(\theta,0)}\right]>0,
\end{equation}
then a stable two-dimensional invariant torus (at $m=1$) or a Klein bottle (at $m=-1$) is born as $\Gamma$ splits. It exists
at all small $\mu>0$ and attracts all the orbits which stay in $V$ and which do not lie in the stable manifold of $L_\mu$.\\
\item If $|m|\geq 2$ and, for all $\theta$,
\begin{equation}\label{hypat}
|m+h'(\theta,0)-\frac{\alpha'(\theta,0)}{\gamma \alpha(\theta,0)}|>1,
\end{equation}
then, for all small $\mu>0$, the system has a hyperbolic attractor (a Smale-Williams solenoid) which is an $\omega$-limit set
for all orbits which stay in $V$ and
which do not lie in the stable manifold of $L_\mu$. The flow on the attractor is topologically conjugate to
suspension over the inverse spectrum limit of a degree-$m$ expanding map of a circle. At $\mu=0$, the attractor
degenerates into the homoclinic surface $\Gamma$.\\
\end{enumerate}
\noindent{\em Proof.} Solution of (\ref{lcfr}) with the initial conditions $(x_0=d,y_0,z_0,\theta_0)\in S_0$ gives
$$\begin{array}{l}
x(t)=e^{-\lambda t} d, \qquad y(t)=e^{B t} y_0,\\
z(t)=e^{\gamma t} z_0,\\
\theta(t)=\theta_0+t.\end{array}
$$
The flight time to $S_1$ is found from the condition
$$d = e^{\gamma t} z_0,$$
which gives $\displaystyle t=-\frac{1}{\gamma}\ln\frac{z_0}{d}$. Thus the orbits in $U$ define the map $T_0: S_0^+\to S_1$:
$$\begin{array}{l}
x_1=d^{1-\nu} z_0^\nu, \qquad y_1=Q(z_0) y_0,\\
\theta_1=\theta_0-\frac{1}{\gamma}\ln\frac{z_0}{d}\end{array}
$$
where $\nu=\lambda/\gamma>1$ and $\|Q(z_0)\|=o(z_0^\nu)$ (see (\ref{nev}),(\ref{sadl})). By (\ref{glom}), we may write the
map $T=T_0T_1$ on $S_1$ as follows (we drop the index ``$1$''):
$$\begin{array}{l}
\bar x=d^{1-\nu} (\mu\alpha(\theta,\mu)+O(x,y))^\nu, \qquad \bar y=Q(\mu\alpha+O(x,y)) g(x,y,\theta,\mu),\\
\bar\theta=m\theta+h(\theta,\mu)-\frac{1}{\gamma}\ln(\frac{\mu}{d}\alpha(\theta,\mu)+O(x,y))+O(x,y).\end{array}
$$
For every orbit which stays in $V$, its consecutive intersections with the cross-section $S_1$ constitute an orbit of
the diffeomorphism $T$. Since $\nu>1$, the map $T$ is contracting in $x$ and $y$, and it is easy to see
that all the orbits eventually enter a neighborhood of $(x,y)=0$ of size $O(\mu^\nu)$.
We therefore rescale the coordinates $x$ and $y$ as follows:
$$x=d^{1-\nu}\mu^\nu X,\qquad y=\mu^\nu Y.$$
The map $T$ takes the form
\begin{equation}\label{mapt}
\begin{array}{l}
\bar X= \alpha(\theta,0)^\nu +o(1), \qquad \bar Y=o(1),\\
\bar\theta=\omega(\mu)+m\theta +h(\theta,0)-\frac{1}{\gamma}\ln\alpha(\theta,0)+o(1),
\end{array}
\end{equation}
where $o(1)$ stands for terms which tend to zero as $\mu\to+0$, along with their first derivatives,
and $\omega(\mu)=\frac{1}{\gamma}\ln(\mu/d)\to\infty$ as $\mu\to+0$. Recall that $\alpha>0$ for all $\theta$
and that $\alpha$ and $h$ are periodic in $\theta$.
It is immediately seen from (\ref{mapt}) that all orbits
eventually enter an invariant solid torus $\{|x-\alpha(\theta,0)^\nu|< K_\mu,\;\|y\|<K_\mu\}$
for appropriately chosen $K_\mu$, $K_\mu\to 0$ as $\mu\to +0$ (see Fig.~\ref{fig4}). Thus, there is an attractor in $V$ for
all small
positive $\mu$, and it merges into $\Gamma$ as $\mu\to+0$. Our theorem claims that the structure of the attractor depends
on the value of $m$, so we now consider different cases separately.
\begin{figure}
\caption{Case $m=0$: the image of the solid torus is contractible to a point; case $m = 1$: contraction transverse to the longitude;
case $m = 2$: the solid-torus is squeezed,
doubly stretched and twisted within the original and so on, producing the solenoid in the limit.}
\label{fig4}
\end{figure}
If $m=0$ and (\ref{bsky}) holds, then map (\ref{mapt}) is, obviously, contracting at small $\mu$,
hence it has a single stable fixed point. This fixed point corresponds to the sought periodic orbit
$A_\mu$. Its period tends to infinity as $\mu\to+0$: the orbit intersects both the
cross-sections $S_0$ and $S_1$, and the flight time from $S_0$ to $S_1$ is of order $\frac{1}{\gamma}|\ln\mu|$.
The length of the orbit also tends to infinity, since the phase velocity never vanishes in $V$.
In the case $m=\pm 1$ we prove the theorem by referring to the ``annulus principle'' of \cite{AfS3}. Namely, consider a map
$$\bar r=p(r,\theta),\qquad \bar\theta=q(r,\theta)$$
of a solid torus into itself (here $\theta$ is the angular variable and $r$ is the vector of normal variables).
Let the map $r\mapsto p(r,\theta)$ be a contraction for every fixed $\theta$, i.e.
$$\left\|\frac{\partial p}{\partial r}\right\|_\circ<1$$
(where by $\|\cdot\|_\circ$ we denote the supremum of the norm over the solid torus under consideration)
and let the map $\theta\mapsto q(r,\theta)$ be a diffeomorphism of a circle for every fixed $r$. Then
it is well-known \cite{AfS3,book2} that if
$$1-\left\|\left(\frac{\partial q}{\partial \theta}\right)^{-1}\right\|_\circ \cdot
\left\|\frac{\partial p}{\partial r}\right\|_\circ >
2\sqrt{\left\|\left(\frac{\partial q}{\partial \theta}\right)^{-1}\right\|_\circ \cdot
\left\|\frac{\partial q}{\partial r}\right\|_\circ
\left\|\frac{\partial p}{\partial \theta}\left(\frac{\partial q}{\partial \theta}\right)^{-1}\right\|_\circ},$$
then the map has a stable, smooth, closed invariant curve $r=r^*(\theta)$ which attracts all orbits from the solid torus.
These conditions are clearly satisfied by map (\ref{mapt}) at $|m|=1$ if (\ref{tor}) is true (here $r=(X,Y)$,
$p=(\alpha(\theta,0)^\nu +o(1), o(1))$, $q=\omega(\mu)+m\theta +h(\theta,0)-\frac{1}{\gamma}\ln\alpha(\theta,0)+o(1)$).
Thus, the map $T$ has a a closed invariant curve in this case. The restriction of $T$ to the invariant curve preserves
orientation if $m=1$, while at $m=-1$ it is orientation-reversing. Therefore, this invariant curve on the cross-section
corresponds to an invariant torus of the flow at $m=1$ or to a Klein bottle at $m=-1$.
It remains to prove the theorem for the case $|m|\geq 2$. The proof is based on the following result.\\
\noindent{\bf Lemma.} Consider a diffeomorphism $T:(r,\theta)\mapsto (\bar r,\bar\theta)$ of a solid torus, where
\begin{equation}\label{maptr}
\bar r=p(r,\theta),\qquad \bar\theta=m\theta+s(r,\theta)=q(r,\theta),
\end{equation}
where $s$ and $p$ are periodic functions of $\theta$
Let $|m|\geq 2$, and
\begin{equation}\label{frc}
\left\|\frac{\partial p}{\partial r}\right\|_\circ <1,
\end{equation}
\begin{equation}\label{cndir}
\left(1-\left\|\frac{\partial p}{\partial r}\right\|_\circ\right)
\left(1-\left\|\left(\frac{\partial q}{\partial \theta}\right)^{-1}\right\|_\circ\right)>
\left\|\frac{\partial p}{\partial \theta}\right\|_\circ\; \left\|\left(\frac{\partial q}{\partial \theta}\right)^{-1}
\frac{\partial q}{\partial r}\right\|_\circ.
\end{equation}
Then the map has a uniformly-hyperbolic attractor, a Smale-Williams solenoid,
on which it is topologically conjugate to the inverse spectrum limit
of $\bar \theta=m\theta$, a degree-$m$ expanding map of the circle.\\
\noindent{\em Proof.} It follows from (\ref{frc}),(\ref{cndir}) that
$\|(\frac{\partial q}{\partial \theta})^{-1}\|$ is uniformly bounded. Therefore, $\theta$
is a uniquely defined smooth function of $(\bar\theta, r)$, so we may rewrite (\ref{maptr})
in the ``cross-form''
\begin{equation}\label{crmps}
\bar r=p^\times(r,\bar\theta),\qquad \theta=q^\times(r,\bar\theta),
\end{equation}
where $p^\times$ and $q^\times$ are smooth functions. It is easy to see that conditions (\ref{frc}),
(\ref{cndir}) imply
\begin{equation}\label{frc0}
\left\|\frac{\partial p^\times}{\partial r}\right\|_\circ <1,\qquad
\left\|\frac{\partial q^\times}{\partial \bar\theta}\right\|_\circ <1
\end{equation}
\begin{equation}\label{cncrs}
\left(1-\left\|\frac{\partial p^\times}{\partial r}\right\|_\circ\right)
\left(1-\left\|\frac{\partial q^\times}{\partial \theta}\right\|_\circ\right)\geq
\left\|\frac{\partial p^\times}{\partial \bar\theta}\right\|_\circ\; \left\|\frac{\partial q^\times}{\partial r}\right\|_\circ.
\end{equation}
These inequalities imply the uniform hyperbolicity of the map $T$ (note that (\ref{cndir}) coincides with
the hyperbolicity condition for the Poincare map for the Lorenz attractor from \cite{ABS}).
Indeed, it is enough to show that there exists
$L>0$ such that the derivative $T'$ of $T$ takes every cone $\|\Delta r\|\leq L\|\Delta \theta\|$ inside
$\|\Delta \bar r\|\leq L\|\Delta \bar \theta\|$ and is uniformly expanding in $\theta$ in this cone,
and that the inverse of $T'$ takes
every cone $\|\Delta \bar\theta\|\leq L^{-1}\|\Delta \bar r\|$ inside
$\|\Delta \theta\|\leq L^{-1}\|\Delta r\|$ and is uniformly expanding in $r$ in this cone.
Let us check these properties. When $\|\Delta r\|\leq L\|\Delta \theta\|$, we find from (\ref{crmps}) that
\begin{equation}
\|\Delta\theta\|\leq \frac{\left\|\frac{\partial q^\times}{\partial \bar\theta}\right\|_\circ}
{1-L\left\|\frac{\partial q^\times}{\partial r}\right\|_\circ} \|\Delta\bar\theta\|
\end{equation}
and
\begin{equation}
\|\Delta\bar r\|\leq \left\{\frac{L \left\|\frac{\partial p^\times}{\partial r}\right\|_\circ
\left\|\frac{\partial q^\times}{\partial \bar\theta}\right\|_\circ}
{1-L\left\|\frac{\partial q^\times}{\partial r}\right\|_\circ} +
\left\|\frac{\partial p^\times}{\partial \bar\theta}\right\|_\circ\right\}
\|\Delta\bar\theta\|.
\end{equation}
Similarly, if
$\|\Delta \bar\theta\|\leq L^{-1}\|\Delta \bar r\|$, we find from (\ref{crmps}) that
\begin{equation}
\|\Delta\bar r\|\leq \frac{\left\|\frac{\partial p^\times}{\partial r}\right\|_\circ}
{1-L^{-1}\left\|\frac{\partial p^\times}{\partial \bar\theta}\right\|_\circ} \|\Delta r\|
\end{equation}
and
\begin{equation}
\|\Delta\theta\|\leq \left\{\frac{L^{-1} \left\|\frac{\partial q^\times}{\partial \bar\theta}\right\|_\circ
\left\|\frac{\partial p^\times}{\partial r}\right\|_\circ}
{1-L^{-1}\left\|\frac{\partial p^\times}{\partial \bar\theta}\right\|_\circ} +
\left\|\frac{\partial q^\times}{\partial r}\right\|_\circ\right\}
\|\Delta r\|.
\end{equation}
Thus, we will prove hyperbolicity if we show that there exists $L$ such that
$$\left\|\frac{\partial q^\times}{\partial \bar\theta}\right\|_\circ < 1-
L\left\|\frac{\partial q^\times}{\partial r}\right\|_\circ$$
and
$$\left\|\frac{\partial p^\times}{\partial r}\right\|_\circ < 1-
L^{-1}\left\|\frac{\partial p^\times}{\partial \bar\theta}\right\|_\circ.$$
These conditions are solved by any $L$ such that
$$\frac{\left\|\frac{\partial p^\times}{\partial \bar\theta}\right\|_\circ}
{1-\left\|\frac{\partial p^\times}{\partial r}\right\|_\circ}<L<
\frac{1-\left\|\frac{\partial q^\times}{\partial \bar\theta}\right\|_\circ}
{\left\|\frac{\partial q^\times}{\partial r}\right\|_\circ}.$$
It remains to note that such $L$ exist indeed when (\ref{frc0}),(\ref{cncrs}) are satisfied.
We have proved that the attractor $A$ of the map $T$ is uniformly hyperbolic. Such attractors are structurally stable,
so $T|_A$ is topologically conjugate to the restriction to the attractor of any diffeomorphism which
can be obtained by a continuous deformation of the map $T$ without violation of conditions
(\ref{frc}) and (\ref{cndir}). An obvious example of such a diffeomorphism is given by the map
\begin{equation}\label{epd}
\bar r=p(\delta r,\theta),\qquad \bar\theta=q(\delta r,\theta)
\end{equation}
for any $0<\delta\leq 1$. Fix small $\delta>0$ and consider a family of maps
$$\bar r=p(\delta r,\theta),\qquad \bar\theta=q(\varepsilon r,\theta),$$
where $\varepsilon$ runs from $\delta$ to zero. When $\delta$ is sufficiently small, every map in this family is a diffeomorphism
(otherwise we would get that the curve $\{\bar r=p(0,\theta), \bar\theta= q(0,\theta)\}$ would have points of self-intersection,
which is impossible since this curve is the image of the circle $r=0$ by the diffeomorphism $T$), and each satisfies
inequalities (\ref{frc}),(\ref{cndir}). This family is a continuous deformation of map (\ref{epd}) to the map
\begin{equation}\label{skd}
\bar r=p(\delta r,\theta),\qquad \bar\theta=q(0,\theta)=m\theta+s(0,\theta).
\end{equation}
Thus, we find that $T|_A$ is topologically conjugate to the restriction of diffeomorphism (\ref{skd}) to its attractor.
It remains to note that map (\ref{skd}) is a skew-product map of the solid torus, which contracts along the fibers $\theta=const$
and, in the base, it is an expanding degree-$m$ map of a circle. By definition, the attractor of such map is the sought
Smale-Williams solenoid \cite{Sm,W}. This completes the proof of the lemma.
Now, in order to finish the proof of the theorem, just note that map (\ref{mapt}) satisfies the conditions of the Lemma
when (\ref{hypat}) is fulfilled.
\section*{Acknowledgment}
This work was supported by RFFI Grant No.~08-01-00083 and the Grant 11.G34.31.0039 of the Government of the Russian Federation
for state support of ``Scientific research conducted under supervision of leading scientists in
Russian educational institutions of higher professional education" (to L.S); NSF grant DMS-1009591,
MESRF ``Attracting leading scientists to Russian universities" project 14.740.11.0919 (to A.S) and
the Royal Society Grant "Homoclinic bifurcations" (to L.S. and D.T.)
\end{document} |
\begin{document}
\title{Quantifying and Estimating the Predictive Accuracy for Censored Time-to-Event
Data with Competing Risks}
\author[1,2]{Cai Wu}
\author[2]{Liang Li*}
\authormark{ C. WU AND L. LI}
\address[1]{\orgdiv{Department of Biostatistics}, \orgname{The University of Texas Health Science Center at Houston}, \orgaddress{\state{TX}, \country{USA}}}
\address[2]{\orgdiv{Department of Biostatistics}, \orgname{The University of Texas MD Anderson Cancer Center}, \orgaddress{\state{TX}, \country{USA}}}
\corres{*Liang Li, Department of Biostatistics, The University of Texas MD Anderson Cancer Center, Houston, TX 77054. \email{[email protected]}}
\abstract{This paper focuses on quantifying and estimating the predictive accuracy
of prognostic models for time-to-event outcomes with competing events.
We consider the time-dependent discrimination and calibration metrics,
including the receiver operating characteristics curve and the
Brier score, in the context of competing risks. To address censoring,
we propose a unified nonparametric estimation framework for both discrimination
and calibration measures, by weighting the censored subjects with the conditional probability of the event
of interest given the observed data.
We demonstrate through simulations
that the proposed estimator is unbiased, efficient and robust against
model misspecification in comparison to other methods published in
the literature. In addition, the proposed method can be extended to
time-dependent predictive accuracy metrics constructed from a general
class of loss functions. We apply the methodology to a data set from the African American Study of Kidney Disease and Hypertension to
evaluate the predictive accuracy of a prognostic risk score in predicting
end-stage renal disease (ESRD), accounting for the competing risk of pre-ESRD death.}
\keywords{Brier Score; Competing Risks; Diagnostic Medicine; Predictive
Accuracy; Prognostic Model; Time-dependent ROC}
\maketitle
\section{Introduction}
\renewcommand\[{\begin{equation}}
\renewcommand\]{\end{equation}}
In modern evidence-based medicine, decisions on a diagnosis or personalized
treatment plan are often guided by risk scores generated from prognostic
models \cite{baskin2007recipient,hernandez2009novel,lorent2016mortality}.
Such prognostic risk scores can be either a single risk factor, such as a biomarker, or a risk probability calculated from multiple
risk factors. For a risk score to be utilized in clinical practice,
its predictive accuracy is often assessed through two types of metrics:
(1) the discrimination metric, which measures how well the risk score
can distinguish subjects with and without the disease condition,
and (2) the calibration metric, which measures how well the predicted
risk matches the observed risk in the target population. Motivated
by the prediction of end-stage renal disease (ESRD) among a cohort of patients with chronic kidney disease, the goal
of this paper is to propose a framework to estimate the predictive
accuracy of a risk score from a prognostic model, accounting
for right censoring and competing events.
For a continuous time-to-event outcome, the presence and absence
of a disease condition at any time point $\tau$ can be viewed as a binary outcome. To study the relationship between a continuous risk
score and this binary outcome at any prespecified time point $\tau$, the
time-dependent receiver operating characteristics (ROC) curve is widely used for assessing discrimination, i.e., the separation of subjects with and without a given disease at time $\tau$ by the risk score \cite{heagerty2000time}. For example, the risk score is the $\tau$-year (e.g., $\tau=5$) survival probability
calculated based on the characteristics of a cancer patient at initial
diagnosis, and the disease presence or absence is defined by whether
the patient died of cancer within $\tau$ years after the initial
diagnosis. For such a risk score, the area under the ROC curve (AUC)
presents the probability that a subject with the disease at time $\tau$ has a higher predicted risk score than a subject without the disease. A challenge of estimating such time-dependent ROC curve is that the disease
status at $\tau$ is unknown among subjects who are censored prior to $\tau$. A number of methods
have been developed
to address this issue, including the nearest neighboring estimator (NNE)
\cite{heagerty2000time} and inverse probability censoring weighting
(IPCW) \cite{blanche2013review,chiang2010non,uno2007evaluating}.
In addition to the metrics for discrimination, metrics for calibration \cite{graf1999assessment} quantify the absolute
deviance of the risk score from the observed outcome, known as the prediction error. Time-dependent prediction error
metrics for survival outcomes have been proposed \cite{graf1999assessment,gerds2006consistent,korn1990measures,schemper2000predictive}.
The prediction error can be constructed through a class of
loss functions that link the risk score and the binary disease outcome
at $\tau$ \cite{graf1999assessment}. Among those, the quadratic loss, known as the Brier score
\cite{brier1950verification}, is a popular choice \cite{blanche2015quantifying,cortese2013comparing,parast2012landmark}. Censoring remains a challenge when estimating the Brier score, and an IPCW method
was proposed to deal with it \cite{graf1999assessment,gerds2006consistent}.
Competing risks are common in clinical research that involves time-to-event data. For example, in a cardiovascular study, one may be interested in the time to the first myocardial infarction after cardiovascular surgery, but patients may die before experiencing the event of interest. Limited statistical methodology is available to estimate the predictive accuracy metrics in the context of competing risks. To estimate the time-dependent
ROC, Saha \& Heagerty \cite{saha2010time} extended the NNE method \cite{heagerty2000time}
to the competing risk context. Zheng et al. \cite{zheng2012evaluating} further
extended the method of Saha \& Heagerty \cite{saha2010time} to covariate-adjusted time-dependent
ROC. Blanche et al. \cite{blanche2013estimating} studied the use of IPCW in estimating the time-dependent ROC with competing risk data.
For the estimation of the Brier score with competing risk data, the available published methods are based on the IPCW \cite{blanche2015quantifying,liu2016robust,schoop2011quantifying}, with the censoring distribution estimated either by the Kaplan-Meier (KM) method without conditioning on the risk score \cite{graf1999assessment} or by the Cox proportional hazards model conditional on the risk score \cite{gerds2006consistent}.
This paper focuses on the time-dependent discrimination and calibration
estimation in the context of competing risk outcomes. We propose a novel
nonparametric kernel-weighted estimation framework for both time-dependent discrimination and calibration measures. The proposed method first estimates the conditional probability of experiencing an event of interest at $\tau$
given the observed data of the subjects. This is done through nonparametric
kernel regression for the cumulative incidence function. Then the
time-dependent predictive accuracy metrics, such as sensitivity, specificity,
and Brier score, are estimated by weighting each subject with their
own conditional probabilities.
The proposed method has some attractive properties. First, it is fully
nonparametric, without any distributional or modeling assumptions.
This is desirable for estimating predictive accuracy metrics since
it reduces the bias from the estimation procedure itself. Second, the
proposed method, unlike other nonparametric methods such as NNE
\cite{heagerty2000time}, is insensitive to the bandwidth choice. This
is shown in this paper with both numerical and methodological
justifications. Third, the method automatically accommodates correlation between
the censoring time and the risk score. Furthermore, the proposed method can be invariant
to monotone transformation of the risk score when the tuning parameter
is specified by the span, the proportion of subjects included in the kernel estimation.
Also, the estimated sensitivity, specificity, and ROC curve are monotone in
the cut-off point $c$.
Our simulation shows that the proposed method has competitive performance
in terms of bias and the mean squared error (MSE) when compared with other published
methods. Section 2 presents the notations and definitions for the time-dependent
ROC and time-dependent prediction error. Section 3 describes the proposed
estimators for the predictive accuracy metrics. Then the finite sample
performance is evaluated by simulations in Section 4. In Section
5, we illustrate the method with data from the African American Study of Kidney
Disease and Hypertension (AASK) in evaluating the prediction
of ESRD. Section 6 concludes the paper by discussing the findings and
providing some perspective.
\section{Predictive Accuracy for Time-to-Event Data with Competing Risks}
\subsection{Notation}
Let $T$ denote the event time, $C$ the censoring time, $\delta$
the event type, and $\Delta=1(T\le C)$ the censoring indicator, where
$1(\cdot)$ is the indicator function. We observe independent and
identically distributed (i.i.d.) samples of $\{(\tilde{T}_{i},U_{i},\tilde{\delta}_{i}),i=1,2,\ldots n\}$ in a validation data set,
where $\tilde{T}_{i}=\textrm{min}(T_{i},C_{i})$ is the observed
time to the event or censoring, whichever comes first. The observed status
$\tilde{\delta}_{i}=\Delta_{i}\delta_{i}$, which equals zero for censored
subjects and equals one of the $K$ possible causes, $\delta_{i}\in\{1,2,\ldots K\}$,
for uncensored subjects. Without loss of generality, we present our
methodology with $K=2$ to match the data application in Section
5. The methodology still applies with other choices of $K\ (K>2)$. For clarity,
suppose that we are interested in assessing the predictive accuracy
of event type $\delta=1$. Let $U_{i}$ denote the risk
score for subject $i$, with higher values of $U_{i}$ indicating higher
risk of the event. For example, $U_{i}$
can be the predicted cumulative incidence probability from a competing
risk regression model that we want to evaluate, i.e., $U_{i}=\pi_{1}(\tau\vert\boldsymbol{Z}_{i})=P(T_{i}\le\tau,\delta_{i}=1\vert\boldsymbol{Z}_{i})$, where $\boldsymbol{Z}$ denotes the predictor and $\tau$ is the predictive horizon. The predictive model is often developed from a training data set that is different from the validation data set.
This paper focuses on estimating the predictive accuracy metrics in a validation data set. We do not study how the model for the risk score $U$ is estimated or whether the model is correctly estimated. We assume that this model has already been developed, needs to be evaluated, and the risk score $U$ has the interpretation of being the subject-specific predicted cumulative incidence probability at horizon $\tau$.
\subsection{Definitions of the time-dependent ROC curve and AUC\label{subsec:tdROC_comprsk}}
In the presence of competing events, the definition of cases is straightforward.
The cases at time $\tau$ for event type $k$ are defined as subjects
who undergo event $\delta=k$ before time $\tau$, i.e., $Case_{k}=\{i:T_{i}\le\tau,\delta_{i}=k\}$.
At a given threshold $c$, the cause-specific sensitivity at time
$\tau$ is defined as
\begin{equation}
Se(c,\tau)=P(U>c\vert T\le\tau,\delta=k).\label{eq:def_sen}
\end{equation}
This is the definition of $cumulative/dynamic$ sensitivity
\cite{heagerty2000time}. When $U$ is higher than the threshold value
$c$, the patient is predicted to experience event $k$ within the time
window $(0,\tau]$.
We consider two definitions of controls that lead
to two different definitions of time-dependent specificity. Saha \& Heagerty \cite{saha2010time} originally
defined the control group at time $\tau$ as the event-free subjects,
i.e., $\{i:T_{i}>\tau\}$. According to this definition, subjects who
experienced competing events other than $k$ are neither cases nor
controls. Therefore, Zheng et al. \cite{zheng2012evaluating} introduced an alternative
definition of the control group $\{i:T_{i}>t\}\cup\{i:T_{i}\le t,\delta_{i}\ne k\}$,
which includes both event-free subjects and subjects who experience other competing
events. We study the estimation under both definitions:
\begin{lyxlist}{00.00.0000}
\item [{\textbf{Definition}}] \textbf{A. }Case $k$: $T\le\tau,\delta=k$;
Control$_{A}$: $(T>\tau)\cup(T\le\tau\cap\delta\ne k).$
\end{lyxlist}
\begin{lyxlist}{00.00.0000}
\item [{\textbf{Definition}}] \textbf{B.} Case $k$: $T\le\tau,\delta=k$;
Control$_{B}$: $T>t$ .
\end{lyxlist}
The specificity at time $\tau$ with respect to the two types of definitions
is
\begin{align}
Sp_{A}(c,\tau) & =P(U\le c\vert\{T>\tau\}\cup\{T\le\tau,\delta\ne k\})\nonumber \\
Sp_{B}(c,\tau) & =P(U\le c\vert T>\tau).\label{eq:def_sp}
\end{align}
Two different time-dependent ROC curves can be obtained by plotting
$Se(c,\tau)$ versus either $1-Sp_{A}(c,\tau)$ or $1-Sp_{B}(c,\tau)$,
i.e., $ROC_{A}(x,\tau)=Se(Sp_{A}^{-1}(1-x,\tau),\tau)$ and $ROC_{B}(x,\tau)=Se(Sp_{B}^{-1}(1-x,\tau),\tau)$
for $x\in[0,1]$. The corresponding $AUC$s can be defined as $AUC(\tau)=\int_{0}^{1}ROC(x,\tau)dx$ or
as the proportion of concordance pairs among the population \cite{blanche2013estimating}:
\begin{align}
AUC_{A}(\tau) & =P(U_{i}>U_{j}\vert T_{i}\le\tau,\delta_{i}=k,\{T_{j}>\tau\}\cup\{T_{j}\le\tau,\delta_{j}\ne k\})\nonumber \\
AUC_{B}(\tau) & =P(U_{i}>U_{j}\vert T_{i}\le\tau,\delta_{i}=k,T_{j}>\tau),\label{eq:AUC def}
\end{align}
where $i$ and $j$ indicate two independent subjects under comparison.
The subjects who experienced the competing events before $\tau$ contribute
to $AUC_{A}(\tau)$ but not $AUC_{B}(\tau)$. The justification for
both definitions is related to the clinical interpretation \cite{zheng2012evaluating}.
\subsection{Definitions of the time-dependent prediction error }
The time-dependent prediction error in the competing risk framework
is defined as the distance between the event-specific status $1(T\le\tau,\delta=k)$
and the subject-specific predicted cumulative incidence function at horizon $\tau$, $\pi_{k}(\tau\vert \boldsymbol{Z})=P(T\le\tau,\delta=k\vert\boldsymbol{Z})$.
Suppose we are interested in evaluating the prediction for event type
1, three types of prediction error measurements can be defined as
follows \cite{van2011dynamic}:
\[
AbsErr(\tau)=E\Big|1\{T\le\tau,\delta=1\}-\pi_{1}(\tau\vert\mathbf{Z})\Big|
\]
\[
Brier(\tau)=E\Big[1\{T\le\tau,\delta=1\}-\pi_{1}(\tau\vert\mathbf{Z})\Big]^{2}
\]
\[
KL(\tau)=-E\Big[1\{T\le\tau,\delta=1\}\cdot\textrm{ln}\pi_{1}(\tau\vert\mathbf{Z})+1\{(T>\tau)\cup(T\le\tau,\delta\ne1)\}\cdot\textrm{ln(}1-\pi_{1}(\tau\vert\mathbf{Z}))\Big].
\]
Among the three measures, $AbsErr(\tau)$ is not \textquotedblleft proper\textquotedblright{}
in the sense that it is not minimized by the predicted cumulative incidence function (CIF) from the true model \cite{graf1999assessment}.
$Brier(\tau)$ is not only \textquotedblleft proper\textquotedblright,
but has the attractive property that it can be decomposed into a term related to the
bias of the predictive survival probability and a term related to the variance of disease status \cite{schoop2011quantifying}.
The Kullback-Leibler score, $KL(\tau)$, has a close connection to the
likelihood ratio test and the Akaike information criteria (AIC), but its
disadvantage is that $KL(\tau)$ goes to infinity when $\pi_{1}(\tau\vert\mathbf{Z})=0$
and $\{T\le\tau,\delta=1\}$, or when $\pi_{1}(\tau\vert\mathbf{Z})=1$
and $\{T>\tau$ or $T\le\tau,\delta\ne1\}$ \cite{van2011dynamic}. The Brier score is more widely used than the other two, and we will
focus on the Brier score for the rest of this paper, even though our methodology also applies to the other two metrics.
\section{The Proposed Nonparametric Weighting Estimators}
Without censoring, sensitivity and specificity can be estimated
empirically as the fraction of true positives and true negatives.
However, when subjects are censored before $\tau$, the true disease
status at $\tau$ is unknown. The empirical fractions can no longer
be used and proper adjustment for censoring is needed. In the context
of right-censored data without competing events, Li et al. \cite{li2016simple} proposed to weigh each
subject by their respective conditional probability of having the
disease at $\tau$ given all the observed data for that subject. The
conditional probability equals 0 if a subject survives beyond $\tau$
without the disease or 1 if the subject acquires the disease prior to $\tau$.
If a subject is censored prior to $\tau$, the conditional probability
is estimated through a nonparametric kernel regression. In this paper,
we extend that approach to the context of competing risk data. The weight
is defined as the conditional probability of being a case prior to
time $\tau$ given the observed time to the event, event status and prognostic
risk score:
\begin{align}
W_{1i} & =P(T_{i}\le\tau,\delta_{i}=1\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i})\nonumber \\
& =\Big\{1(\tilde{\delta}_{i}=0)\cdot\frac{F_{1}(\tau\vert U_{i})-F_{1}(\tilde{T}_{i}\vert U_{i})}{S(\tilde{T}_{i}\vert U_{i})}+1(\tilde{\delta}_{i}=1)\Big\}\cdot1(\tilde{T}_{i}\le\tau),\label{eq:wt_cmprsk}
\end{align}
where $F_{1}(t\vert U_{i})=P(T_{i}\le t,\delta_{i}=1\vert U_{i})$ is the conditional cumulative incidence
function for event 1, and $S(t\vert U_{i})=P(T_{i}>t \vert U_{i})$ is the conditional overall survival probability.
According to equation (\ref{eq:wt_cmprsk}), we have $W_{1i}=1$ for
subjects with observed event 1 before $\tau$: $\{i:\tilde{T}_{i}\le\tau,\tilde{\delta}_{i}=1\}$;
$W_{1i}=0$ for subjects without any events before $\tau$ or with
competing events before $\tau$: $\Big\{ i:\{\tilde{T}_{i}>\tau\}\cup\{\tilde{T}_{i}\le\tau,\tilde{\delta}_{i}\notin\{0,1\}\}\Big\}$
; and $W_{1i}=\dfrac{F_{1}(\tau\vert U_{i})-F_{1}(\tilde{T}_{i}\vert U_{i})}{S(\tilde{T}_{i}\vert U_{i})}$
for subjects censored before $\tau$: $\{i:\tilde{T}_{i}\le\tau,\tilde{\delta}_{i}=0\}$.
This weighting approach uses the observed status for uncensored subjects and only
imputes the unknown status for censored subjects with a probability.
A heuristic justification is that the case group includes not
only those who are known to have experienced event 1 but also fractions
of those whose status is unknown due to censoring. Similar justification applies to the controls. This differs from
the IPCW method \cite{blanche2013estimating,schoop2011quantifying}, which uses only uncensored subjects and reweights them to account for
censoring. The IPCW weight is defined as $W_{i}^{IPCW}(\tau)=\dfrac{1(T_{i}\le\tau,\tilde{\delta}_{i}\ne0)}{G_{n}(\tilde{T_{i}}\vert \cdot)}+\dfrac{1(T_{i}>\tau)}{G_{n}(\tau\vert \cdot)}$. It is
the inverse of the probability of being censored, where $G(t\vert \cdot)$
is the censoring distribution that can be estimated by the Kaplan-Meier
estimator or conditionally given covariates.
Estimation of the proposed weight (\ref{eq:wt_cmprsk}) includes estimation
of two quantities: the conditional CIF $F_{1}(\cdot\vert U_{i})$
and the conditional overall survival probability $S(\cdot\vert U_{i})$. We propose
to use a nonparametric kernel-weighted Kaplan-Meier estimator \cite{li2016simple}:
\begin{equation}
\widehat{S}_{T}(t\vert U_{i})=\prod_{\zeta\in\Omega,\zeta\le t}\Big\{1-\frac{\sum_{j}K_{h}(U_{j},U_{i})\cdot1(\tilde{T}_{j}=\zeta,\tilde{\delta}_{j}\ne0)}{\sum_{j}K_{h}(U_{j},U_{i})\cdot1(\tilde{T}_{j}\ge\zeta)}\Big\},\label{eq:kernel_surv}
\end{equation}
and the kernel-weighted CIF \cite{kalbfleisch2011statistical}:
\begin{equation}
\widehat{F}_{1}(t\vert U_{i})=\sum_{\zeta\in\Omega,\zeta\le t}\frac{\sum_{j}K_{h}(U_{j},U_{i})1(\tilde{T_{j}}=\zeta,\tilde{\delta_{j}}=1)}{\sum_{j}K_{h}(U_{j},U_{i})1(\tilde{T}_{j}\ge\zeta)}\cdot\widehat{S}_{T}(\zeta-\vert U_{i}).\label{eq:kernel_cif}
\end{equation}
$\Omega$ is the set of distinct $\tilde{T}_{i}$'s for $\tilde{\delta_{j}}\ne0$
; and $K_{h}(x,x_{0})=\frac{1}{h}K(\frac{x-x_{0}}{h})$ is the kernel
weight with kernel function $K(\cdot)$ and bandwidth $h$. Alternatively,
we can specify a $span$ instead of a fixed bandwidth. A span is the
proportion of subjects around the neighborhood involved in the kernel
estimation with a uniform kernel function. In implementation, the CIF in (\ref{eq:kernel_cif}) can be estimated as a Kaplan-Meier type product-limit estimator, with the hazard function being replaced by the sub-distribution hazard. The at-risk set in the sub-distribution hazard is obtained by reweighting the individuals who had competing events. This process can be achieved by reformatting the competing risk data into a counting
process with \texttt{crprep()} function from the \texttt{mstate} package, and using \texttt{survfit()} in the \texttt{survival}
package by specifying a time-dependent $weight$ in \texttt{R} \cite{geskus2011cause}.
\subsection{The proposed weighting estimators for the time-dependent ROC curve and
AUC }
The estimated weight $\widehat{W}_{1i}$ can be obtained by replacing the CIF and survival functions in (\ref{eq:wt_cmprsk}) with their estimators given by (\ref{eq:kernel_cif}) and (\ref{eq:kernel_surv}). The
$Se(c,\tau)$, $Sp_{A}(c,\tau)$ and $Sp_{B}(c,\tau)$ can be estimated by
\begin{align}
\widehat{Se}(c,\tau) & =\frac{\sum_{i=1}^{n}\widehat{W}_{1i}\cdot1(U_{i}>c)}{\sum_{i=1}^{n}\widehat{W}_{1i}}\nonumber \\
\widehat{Sp}_{A}(c,\tau) & =\frac{\sum_{i=1}^{n}(1-\widehat{W}_{1i})\cdot1(U_{i}\le c)}{\sum_{i=1}^{n}(1-\widehat{W}_{1i})}\label{eq:sen_est}\\
\widehat{Sp}_{B}(c,\tau) & =\frac{\sum_{i=1}^{n}(1-\sum_{k=1}^{K}\widehat{W}_{ki})\cdot1(U_{i}\le c)}{\sum_{i=1}^{n}(1-\sum_{k=1}^{K}\widehat{W}_{ki}).}\nonumber
\end{align}
The estimator of sensitivity can be justified theoretically
as
\begin{align*}
Se(c,\tau) & =P(U>c\vert T\le\tau,\delta=1)\\
& =\frac{E\big(1\{U>c\}\times1\{T\le\tau,\delta=1\}\big)}{E\big(1\{T\le\tau,\delta=1\}\big)}\\
& =\frac{E\Big\{1\{U>c\}\times E\big(1\{T\le\tau,\delta=1\}\vert\tilde{T},\tilde{\delta},U\big)\Big\}}{E\Big\{ E\big(1\{T\le\tau,\delta=1\}\vert\tilde{T},\tilde{\delta},U\big)\Big\}}\\
& =\textrm{lim}_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}1(U_{i}>c)\cdot P(T_{i}\le\tau,\delta_{i}=1\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i})}{\sum_{i=1}^{n}P(T_{i}\le\tau,\delta_{i}=1\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i}).}
\end{align*}
The justification for the specificity estimator is similar. The time-dependent
ROC curve is an increasing function obtained by plotting the time-dependent
sensitivity and 1-specificity over a range of threshold $c$'s. By definition,
the AUC can be calculated by trapezoidal integration: $\int_{0}^{1}\widehat{ROC}_{A}(x,\tau)dx=\int_{0}^{1}\widehat{Se}(\widehat{Sp}_{A}^{-1}(1-x,\tau),\tau)dx$
and $\int_{0}^{1}\widehat{ROC}_{B}(x,\tau)dx=\int_{0}^{1}\widehat{Se}(\widehat{Sp}_{B}^{-1}(1-x,\tau),\tau)dx$.
Alternatively, it can be estimated by the empirical estimator of the proportion
of concordance pairs, with the proposed weight estimator $\widehat{W}_{1i}$:
\begin{align}
\widehat{AUC}_{A}(\tau) & =\frac{\sum_{i}\sum_{j}\widehat{W}_{1i}(1-\widehat{W}_{1i})\cdot1(U_{i}>U_{j})}{\sum_{i}\sum_{j}\widehat{W}_{1i}(1-\widehat{W}_{1i})}\nonumber \\
\widehat{AUC}_{B}(\tau) & =\frac{\sum_{i}\sum_{j}\widehat{W}_{1i}(1-\sum_{k=1}^{K}\widehat{W}_{ki})\cdot1(U_{i}>U_{j})}{\sum_{i}\sum_{j}\widehat{W}_{1i}(1-\sum_{k=1}^{K}\widehat{W}_{ki}).}\label{eq:AUC_est}
\end{align}
In practice, we can add $0.5\times1(U_{i}=U_{j})$ to the group of
$1(U_{i}>U_{j})$ to account for ties between the $U$'s. The theoretical
justification for the AUC estimators above is as follows.
\begin{align}
AUC_{A}(\tau) & =P(U_{i}>U_{j}\vert T_{i}\le\tau,\delta_{i}=1,\{T_{j}>\tau\}\cup\{T_{j}\le\tau,\delta_{j}\ne1\})\nonumber \\
& =\frac{E\big(1(T_{i}\le\tau,\delta_{i}=1)\times1(\{T_{j}>\tau\}\cup\{T_{j}\le\tau,\delta_{j}\ne1\})\times1(U_{i}>U_{j})\big)}{E\big(1(T_{i}\le\tau,\delta_{i}=1)\times1(\{T_{j}>\tau\}\cup\{T_{j}\le\tau,\delta_{j}\ne1\})\big)}\nonumber \\
& =\frac{E\Big\{1(U_{i}>U_{j})\cdot E\big(1(T_{i}\le\tau,\delta_{i}=1)\cdot1(\{T_{j}>\tau\}\cup\{T_{j}\le\tau,\delta_{j}\ne1\})\vert\tilde{T}_{i},\tilde{\delta_{i}},U_{i},\tilde{T_{j}},\tilde{\delta_{j}},U_{j}\big)\Big\}}{E\Big\{ E\big(1(T_{i}\le\tau,\delta_{i}=1)\cdot1(\{T_{j}>\tau\}\cup\{T_{j}\le\tau,\delta_{j}\ne1\})\vert\tilde{T}_{i},\tilde{\delta_{i}},U_{i},\tilde{T_{j}},\tilde{\delta_{j}},U_{j}\big)\Big\}}\nonumber \\
& =\textrm{lim}_{n\rightarrow\infty}\frac{\sum_{i}\sum_{j}1(U_{i}>U_{j})\cdot P(T_{i}\le\tau,\delta_{i}=1\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i})\cdot\big(1-P(T_{i}\le\tau,\delta_{i}=1\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i})\big)}{\sum_{i}\sum_{j}P(T_{i}\le\tau,\delta_{i}=1\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i})\cdot\big(1-P(T_{i}\le\tau,\delta_{i}=1\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i})\big)}\nonumber \\
& =\textrm{lim}_{n\rightarrow\infty}\frac{\sum_{i}\sum_{j}1(U_{i}>U_{j})\times W_{1i}\times\big(1-W_{1i}\big)}{\sum_{i}\sum_{j}W_{1i}\times\big(1-W_{1i}\big)}\label{eq:AUC_justify-1}
\end{align}
A similar justification for $AUC_{B}(\tau)$ is obtained by replacing $\big(1-W_{1i}\big)$ in the
formula (\ref{eq:AUC_justify-1}) with $(1-\sum_{k=1}^{K}W_{ki})$ for the control definition B.
In our numerical studies, the estimator in (\ref{eq:AUC_est}) is
almost identical (up to four digits after the decimal) to the AUC estimator obtained by trapezoidal integration.
The confidence intervals for sensitivity, specificity and AUC can be
estimated numerically by bootstrapping.
\subsection{The Proposed Weighting Estimators for the Brier Score}
By definition, the Brier score is the expected quadratic loss function between the
true disease status $1(T_{i}\le\tau,\delta_{i}=1)$ and the risk score for
event 1, $U_{i}=\pi_{1}(\tau\vert\mathbf{Z}_{i})$, calculated from a prognostic
model to be evaluated. We propose the following estimator for the Brier score, weighting observations according to their probability of having the event of interest:
\[
\widehat{Brier}(\tau)=\dfrac{1}{n}\sum_{i=1}^{n}\Big(\widehat{W}_{1i}\cdot(1-U_{i})^{2}+(1-\widehat{W}_{1i})\cdot(0-U_{i})^{2}\Big).
\]
The justification for consistency of the above estimator is
\begin{align*}
Brier(\tau) & =E\Big\{1(T_{i}\le\tau,\delta_{i}=1)-U_{i}\Big\}^{2}\\
& =E\Big\{ E\Big(\Big[1(T_{i}\le\tau,\delta_{i}=1)-U_{i}\Big]{}^{2}\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i}\Big)\Big\}\\
& =E\Big\{ P(T_{i}\le\tau,\delta_{i}=1\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i})\cdot(1-U_{i})^{2}+(1-P(T_{i}\le\tau,\delta_{i}=1\vert\tilde{T}_{i},\tilde{\delta}_{i},U_{i}))\cdot(0-U_{i})^{2}\Big\}\\
& =\textrm{lim}_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\Big(W_{1i}\cdot(1-U_{i})^{2}+(1-W_{1i})\cdot(0-U_{i})^{2}\Big).
\end{align*}
Similarly, the $AbsErr(\tau)$ and $KL(\tau)$ can be estimated with
the proposed conditional probability weight:
\[
\widehat{KL}(\tau)=-\frac{1}{n}\sum_{i=1}^{n}\Big(\widehat{W}_{1i}\text{\ensuremath{\cdot}log}U_{i}+(1-\widehat{W}_{1i})\text{\ensuremath{\cdot}log}(1-U_{i})\Big)
\]
and
\[
\widehat{AbsErr}(\tau)=\frac{1}{n}\sum_{i=1}^{n}\Big(\widehat{W}_{1i}\cdot(1-U_{i})+(1-\widehat{W}_{1i})\cdot U_{i}\Big).
\]
\\
\\
To summarize, the proposed method is a nonparametric
method for estimating the time-dependent predictive accuracy for competing
risk data. It extends the methodology in Li et al. \cite{li2016simple} for a single right-censored time-to-event outcome to competing risk outcomes and to time-dependent calibration metrics. The proposed methodology has a connection to some existing methods. In the context of semi-competing risks with interval censoring, Jacqmin-Gadda et al. \cite{jacqmin2016receiver} proposed an imputation estimator that weights the data with a similar conditional probability of observing an
event in the presence of interval censoring. But their estimator of the conditional probability
is calculated from a parametric illness-death model using the survival and
marker. Schemper \& Henderson \cite{schemper2000predictive} also proposed an imputation method with a Cox model-based estimator for $AbsErr(\tau)$. But this method was shown to be biased when the
prognostic model was misspecified, and an alternative IPCW estimator was proposed in that situation \cite{schmid2011robust}. In contrast, our method is nonparametric, without modeling assumptions, and is applicable to both time-dependent discrimination and calibration metrics. We demonstrated the robustness of the nonparametric method to the selection of tuning parameters in Section 4.3.
\section{Simulation}
In this section, we present simulation studies to evaluate the performance
of the proposed method in estimating both the time-dependent ROC and time-dependent
Brier score in the context of competing risk data. The performance of the proposed
method is compared with those of NNE \cite{saha2010time,zheng2012evaluating}
and IPCW \cite{blanche2013estimating,schoop2011quantifying} methods
from the published literature.
\subsection{Simulation design}
We generate two independent baseline covariates $\boldsymbol{Z}_{i}=(Z_{i1},Z_{i2})$,
where $Z_{i1}$ is a biomarker variable of standard normal distribution,
and $Z_{i2}$ is a baseline characteristic (e.g., gender) of Bernoulli distribution with probability 0.5. The event times are generated according to a Fine-Gray model by using the procedure described in Fine \& Gray \cite{fine1999proportional} with a baseline
sub-distribution hazard (SDH) function and additive covariate effects
on the log SDH. The baseline SDH of event 1 follows a mixture
of Weibull distribution with scale $\lambda_{1}$ and shape $\alpha_{1}$, and a point mass with probability $1-p$ at $\infty$. The log SDH
ratios for covariates $Z_{i1}$ and $Z_{i2}$ are denoted by $\boldsymbol{\beta}=(\beta_{1},\beta_{2})'$ for event 1 and $\boldsymbol{\gamma}=(\gamma_{1},\gamma_{2})'$ for event 2. In our simulations, we set $\boldsymbol{\beta}= (-0.6,0.5)'$,
and $\boldsymbol{\gamma}= (-0.1,-0.2)'$. The event indicator is generated from a Bernoulli distribution with the probability of event 1 being $P_{1}=F_{1}(\infty\vert \boldsymbol{Z})=1-(1-p)^{\textrm{exp}(\boldsymbol{Z \beta}))}$.
The values of $p$ are set to be $(0.22,0.42,0.61)$ to achieve 30\%,
50\% and 70\% of event 1 given the covariate effects. Unless otherwise specified, the random censoring times are generated from a mixture
of uniform distributions on the intervals of $(0,3]\cup(3,6]\cup(6,9]\cup(9,12]\cup(12,15]\cup(15,18]$.
We adjust the probability of falling into each interval to control
the censoring rate. Each simulated data set consists of i.i.d. samples
of $\{(\tilde{T}_{i},U_{i},\tilde{\delta}_{i}),i=1,2,\ldots n\}$:
the observed event time $\tilde{T_{i}}$ is the true event time or censoring
time, whichever comes first; the prognostic score $U_{i}$ is the probability
of experiencing event 1 prior to $\tau$; and the event indicator $\tilde{\delta}_{i}$
takes values of 0, 1, or 2. We use the simulated data sets as validation
data sets to evaluate the predictive accuracy of prognostic score
$U_{i}$ at horizon $\tau$.
We organize the simulation scenarios into a $3\times2\times2$ factorial
design. We consider three proportions for event type 1 (70\%, 50\%
and 30\%), two levels of censoring rates (medium: 25\%-30\% and high:
45\%-50\%) and two sample sizes ($300$ and $600$). The predictive
accuracy is estimated at a time horizon $\tau$, which is approximately
at the 65\% quantile of the observed event time distribution for each
scenario. We compute the true values of $AUC(\tau)$ and $Brier(\tau)$
by a Monte Carlo method using 20,000 independent data sets without
censoring. The prognostic score $U_{i}$ is computed from the true
CIF at $\tau$: $F_{1}(\tau;\boldsymbol{Z})=P(T\le\tau,\delta=1\vert\boldsymbol{Z})=1-\{1-p(1-e^{-\lambda_{1}\tau^{\alpha_{1}}})\}^{\textrm{exp}(\boldsymbol{Z \beta})}$.
In each setting, $500$ Monte Carlo repetitions are performed and
the results are aggregated to compute the bias percentage (bias\%) and MSE in estimating $AUC(\tau)$ and $Brier(\tau)$.
The results are presented in Sections 4.2, 4.3 and 4.4. In Section
4.2, we compare the finite sample performance of the proposed method
with those of some existing methods. For the estimation of the time-dependent
ROC, we compare the proposed estimator with those of the
NNE \cite{saha2010time,zheng2012evaluating} and IPCW methods \cite{blanche2013estimating}.
The NNE method is available in the \texttt{R} package \texttt{CompRisksROC} \cite{saha2010time} for
Definition B of Section \ref{subsec:tdROC_comprsk}, and package
\texttt{SurvCompetingRisk} \cite{zheng2012evaluating} for Definition A. The IPCW method is available
in the R package \texttt{timeROC} \cite{blanche2013estimating}.
For the estimation of the Brier score, the proposed estimator is compared
with that of the IPCW method \cite{schoop2011quantifying}. Since the proposed
method is nonparametric with a tuning parameter (bandwidth or span),
we study the sensitivity of the results to the tuning parameter selection in Section 4.3 and compare the performance with that of another
nonparametric method (NNE) that also uses a bandwidth. In Section 4.4,
we take a closer examination of the relative performance of the proposed
method and IPCW when the censoring time is correlated with the risk score. We consider two
versions of IPCW methods that have been reported in the literature. The first one is the IPCW.KM method \cite{blanche2013estimating,graf1999assessment}, where the censoring distribution in the weight function is estimated by the Kaplan-Meier estimator without conditioning on the risk score:
\begin{equation}
\widehat{W}_{i}^{IPCW.KM}(\tau)=\frac{1(\tilde{T_{i}}\le\tau,\tilde{\delta_{i}}\ne0)}{\widehat{G}(\tilde{T_{i}})}+\frac{1(\tilde{T_{i}}>\tau)}{\widehat{G}(\tau)}.\label{eq:IPCW.KM}
\end{equation}
The second one is the IPCW.Cox method \cite{gerds2006consistent,schoop2011quantifying}, where the censoring distribution in the weight function is estimated from a Cox proportional hazard model, conditioning on the risk score
\begin{equation}
\widehat{W}_{i}^{IPCW.Cox}(\tau)=\frac{1(\tilde{T_{i}}\le\tau,\tilde{\delta_{i}}\ne0)}{\widehat{G}(\tilde{T_{i}}\vert U)}+\frac{1(\tilde{T_{i}}>\tau)}{\widehat{G}(\tau\vert U)}.\label{eq:IPCW.Cox}
\end{equation}
The sensitivity, specificity and Brier score based on the IPCW weight
$\widehat{W}^{IPCW}(t)=1/\widehat{G}(t\vert \cdot)$ from the equations above are estimated as
\begin{align*}
\widehat{Se}^{IPCW}(c,\tau) & =\dfrac{\sum_{i=1}^{n}1(U_{i}>c)\times1(\tilde{T}_{i}\le\tau,\tilde{\delta}_{i}=1)\times\widehat{W}_{i}^{IPCW}(\tilde{T}_{i})}{\sum_{i=1}^{n}1(\tilde{T}_{i}\le\tau,\tilde{\delta}_{i}=1)\times\widehat{W}_{i}^{IPCW}(\tilde{T}_{i})}\\\\
\widehat{Sp}_{A}^{IPCW}(c,\tau) & = \dfrac{\sum_{i=1}^{n}1(U_{i}\le c)\times1(\tilde{T}_{i}>\tau)\times\widehat{W}_{i}^{IPCW}(\tau)}{\sum_{i=1}^{n}1(\tilde{T}_{i}>\tau)\times\widehat{W}_{i}^{IPCW}(\tau)}\\\\
\widehat{Sp}_{B}^{IPCW}(c,\tau) & = \dfrac{\sum_{i=1}^{n}1(U_{i}\le c)\times\Big(1(\tilde{T}_{i}>\tau)\cdot\widehat{W}_{i}^{IPCW}(\tau)+1(\tilde{T}_{i}\le\tau,\tilde{\delta}_{i}\notin\{0,1\}\cdot\widehat{W}_{i}^{IPCW}(\tilde{T}_{i})\Big)}{\sum_{i=1}^{n}\Big\{1(\tilde{T}_{i}>\tau)\cdot\widehat{W}_{i}^{IPCW}(\tau)+1(\tilde{T}_{i}\le\tau,\tilde{\delta}_{i}\notin\{0,1\}\cdot\widehat{W}_{i}^{IPCW}(\tilde{T}_{i})\Big\}}\\\\
\widehat{Brier}^{IPCW}(\tau) & =\frac{1}{n}\sum_{i=1}^{n}\Big(1(\tilde{T}_{i}\le\tau,\tilde{\delta}_{i}=1)-\pi(\tau\vert \boldsymbol{Z})\Big)^{2}\times\widehat{W_{i}}^{IPCW}.
\end{align*}
\subsection{Simulation results on the finite sample performance of the proposed method.}
Table 1 shows the performances of the proposed method, IPCW
and NNE for estimating $\widehat{AUC}_{A}(\tau)$ and $\widehat{AUC}_{B}(\tau)$
under 12 simulation scenarios. For IPCW,
we use the estimator with the weight calculated by (\ref{eq:IPCW.KM}).
In general, the proposed method has smaller bias than the IPCW, and
the magnitude of the bias is negligible ($<1\% $ in most settings).
The NNE method has notably larger bias, especially for $\widehat{AUC}(\tau)$.
The MSE for the proposed method is also the smallest among the three methods
studied. Table 2 shows the performance of the proposed estimators
and IPCW estimators for estimating the Brier score. The bias percentages
of the proposed estimator are less than 1.5\% in all settings and are
in general smaller than those from the IPCW method. The MSEs of the
proposed estimators are also similar to or smaller than those from the
IPCW method. The NNE method was proposed in the literature only for estimating the AUC and hence was not included in the simulation about the Brier score. We conclude
that the proposed method performs similarly or better than the IPCW
method, and both methods are substantially better than the NNE method.
\subsection{Simulation results on the sensitivity to tuning parameter selection.}
One advantage of the proposed method is that it is nonparametric,
which prevents the predictive accuracy from being affected
by the modeling assumptions involved in calculating the predictive accuracy metrics themselves. However, it
does involve a tuning parameter, which is the bandwidth or span that
is used in the kernel weight calculation. Therefore, it is important
to study whether this estimator is sensitive to the tuning parameter
selection. Since the NNE method also uses the tuning parameter, and
to our knowledge no previous work has studied its sensitivity
to the tuning parameter selection, we include that method in the comparison.
Table 3 presents the performance of the proposed and NNE
methods in estimating the AUC under different spans. This table
only includes the results with 70\% of event
1; the results under other scenarios lead to the same general conclusion
and are hence omitted for brevity. When the $span$ varies from 0.05
to 0.5, the proposed method is quite stable and the bias remains under
1.5\% in all scenarios. Slightly larger biases are observed under
two scenarios: small sample size ($n=300$) with small span $(span=0.05$),
and large sample size $(n=600$) with unrealistically large span ($span=0.5$).
When both the sample size and span are small, there is not enough data
for estimation; and when both the sample size and span are
large, bias may be introduced. In contrast, the NNE estimator
is very sensitive to the span and can result in a large bias when the
span is not chosen properly. We speculate that this led to the relatively
large bias shown in Table 1. A similar performance is observed in Table 4
when the Brier score is estimated. A heuristic explanation of the robustness
of the proposed method to the tuning parameter selection is as follows. First,
the tuning parameter only affects subjects who are censored prior
to time $\tau$ because their disease status at $\tau$ is unknown.
This is a smaller proportion than the overall censoring proportion
of the data. Second, the probability weight $W_{1i}=\dfrac{F_{1}(\tau\vert U_{i})-F_{1}(\tilde{T}_{i}\vert U_{i})}{S(\tilde{T}_{i}\vert U_{i})}$
is defined as the ratio of two conditional probabilities for subjects
censored before $\tau$.
The numerator of $W_{1i}$ can be expressed as the cause-specific
survival probability between $\tilde{T}_{i}$ and $\tau$: $S_{1}(\tilde{T}_{i}\vert U_{i})-S_{1}(\tau\vert U_{i})=Pr(\tilde{T}_{i}<T\le\tau,\delta_{i}=1\vert U_{i})$;
and the denumerator is the overall survival probability beyond $\tilde{T}_{i}$. The asymptotic bias of two conditional survival probabilities as
a function of bandwidth are in the same direction \cite{bordes2011uniform}.
Therefore, the bias of their ratio can be canceled out to some extent,
particularly when $\tilde{T}_{i}$ and $\tau$ are close.
\subsection{Simulation results for the performance of the proposed method under dependent
censoring. }
In this section, we compare the proposed method and IPCW under a
dependent censoring scenario where the event time $T$ and censoring
time $C$ are marginally dependent but are conditionally independent
given the risk score $U$. In practice, the censoring time is often correlated with baseline covariates. Since $U$ is a function of these covariates,
$C$ and $U$ may also be correlated. Literature on the time-dependent
ROC and time-dependent Brier score describes estimation under dependent censoring of this kind using the
IPCW approach, where a Cox model is used to estimate the censoring distribution, conditioning on the risk score \cite{gerds2006consistent,schoop2011quantifying}. In contrast,
our proposed method does not model the censoring distribution, which is a nuisance for scientific purposes. We directly estimate the conditional survival and CIF nonparametrically. In this simulation, we consider two settings. In setting $(a)$, we generate censoring
time $C_{i}$ from a Weibull$(\lambda_{c},\alpha_{c})$ distribution
with the mean $\mu_{C}=\frac{\Gamma(1+1/\alpha_{c})}{\lambda_{c}^{1/\alpha_{c}}}=a*1\{(\zeta >0.4)\cup(\zeta <-0.6)\}+b*1\{-0.6\le \zeta \le0.4\}$, where $\zeta=\boldsymbol{Z \beta}$ is a monotone transformation of $U$.
Different values of $(a,b)$ and $\alpha_{c}$ are chosen to achieve
a medium or high censoring rate. The dependency between the censoring
distribution and $U$ is not monotone and cannot be correctly estimated
by a proportional hazard model. We use setting (a) to study the robustness of the methods to model misspecification. In setting $(b)$, we generate
the censoring time from a Cox model on $\zeta$, so that the censoring time is correctly modeled by the IPCW. For both
settings, we compare the performance of the proposed method and IPCW methods with both weight estimators (\ref{eq:IPCW.KM}) and (\ref{eq:IPCW.Cox}).
Tables 5 and 7 compare the performance of the proposed method with that of the IPCW
in estimating $AUC(\tau)$. All bias percentages for the proposed
method are under 1.5\% and 1\% for settings $(a)$ and $(b)$, respectively.
In contrast, the IPCW.KM method, which ignores the dependent censoring,
produces results with a large bias under both mechanisms. Compared to
IPCW.KM, the IPCW.Cox estimator in setting $(a)$ alleviates the bias
by accounting for the dependence but still has larger bias and MSE
than the proposed method, especially when the type 1 event rate is
low (e.g., 30\%). When the censoring times are generated from the
Cox model in setting $(b)$, the bias from the IPCW.Cox method is controlled
under 1.5\% but is still slightly larger than that from the proposed method
in general. This indicates that the proposed method is more robust
than the IPCW methods under different dependence structures of $C$ and
$U$.
Tables 6 and 8 present similar comparisons between the proposed method
and IPCW in estimating the Brier score. The overall performance
is similar to that of $\widehat{AUC}(\tau)$. However, we notice that when the IPCW.Cox method is used under a misspecified censoring mechanism in setting $(a)$, it produces a larger bias in the estimation of the Brier score than the AUC. In contrast, the performance of IPCW.Cox under
setting $(b)$ is similar in both estimands, with the biases well
controlled under 1.5\%. The results indicate that estimation of $\widehat{Brier}(\tau)$
appears to be more sensitive to misspecification than that of $\widehat{AUC}(\tau)$.
We speculate that this is because $AUC(\tau)$ is based on the rankings
of the data, whereas $Brier(\tau)$ measures the actual deviation
from the true status in quantity and therefore is more sensitive to the
misspecification of the estimation procedure.
The results above suggest that our nonparametric method does not suffer from
bias caused by model dependence. The rationale for developing a nonparametric
estimation method is that the estimator of a predictive accuracy metric
should be an objective reflection of the model under evaluation, without
introducing another source of bias due to the modeling assumption of
the estimation method. In this spirit, one can extend
the IPCW method by using a nonparametric estimator
for the conditional distribution of the censoring time given the risk score. But from a clinical perspective, this conditional distribution is less intuitive than directly
modeling the conditional survival distribution, which offers additional
insight into the relationship between the risk score and disease development. In addition, the relationship between the risk score and the survival time is expected to be monotone by the definition of the ROC, but this is not necessarily the case for the relationship between the risk score and the censoring time. The nonparametric smoothing literature suggests that the nonparametric regression result is less sensitive to the tuning parameters when the relationship between the outcome and covariate is monotone \cite{meyer2008inference}.
In summary, the simulation results from Table 1 to Table 8 demonstrate
that the proposed method has similar or better performance than other
published methods. While the NNE method only estimates the time-dependent
ROC, the proposed method works with the time-dependent ROC, time-dependent
Brier score and other predictive accuracy metrics, with notably smaller
bias and MSE. Unlike the NNE, the proposed method is
robust to tuning parameter selection, which makes it easy to use in
practice. As a nonparametric method, the proposed method outperforms
the IPCW under dependent censoring, particularly in light of the
possibility that IPCW may use a misspecified model for the censoring
distribution.
\section{Application}
We illustrate the proposed method with a data set from AASK, a randomized
clinical trial for 1,094 patients with chronic kidney disease, whose
baseline estimated glomerular filtration rates (eGFRs) were between $20-65\ \textrm{mL/min/1.73m}^{2}$
\cite{wright2002effect}. The patients were followed for 6.5 years
during the trial period. Among them, 179 developed ESRD and 85 died before developing ESRD. We evaluate the predictive
accuracy of a prognostic risk score developed from a proportional
sub-distributional hazard model with five baseline covariates: the
eGFR, urine protein creatinine
ratio, age, gender, the randomized blood pressure group (low and medium) and the randomized anti-hypertensive therapy (ramipril, metoprolol, amlodipine). The prognostic
score is the predicted CIF for ESRD at prespecified horizons.
Figure 1 compares the time-dependent ROC curves estimated from the
proposed method (red), IPCW.KM (black), IPCW.Cox (blue) and NNE (green)
at three predictive horizons: 3, 4 and 5 years from baseline. The span used in the proposed and NNE methods is 0.05, which includes 5\% of the neighborhood data. The two rows in the panel present the estimated ROC curves based on the two definitions (Section 2.2).
Definition A discriminates patients with ESRD within $\tau$ years from ESRD-free patients, which include patients who are event-free and who die by year $\tau$.
Definition B discriminates patients with ESRD within $\tau$ years from those who are event-free at year $\tau$.
The ROC curves from the two IPCW methods, IPCW.KM and IPCW.Cox, are
almost identical. The curves by IPCW and the proposed method are also very
close, and the differences between the $\widehat{AUC}(\tau)$ are within
5\%. The estimated $\widehat{AUC}^{A}(\tau)$ and $\widehat{AUC}^{B}(\tau)$ are also very close within the different estimation methods except
for NNE. This indicates that the sub-distribution hazard
model we used can discriminate well between ESRD patients and ESRD-free or event-free patients. A possible explanation is that the patients who died in the study period are a relatively small population and may have died from causes unrelated to kidney disease. Therefore, adding these patients to the control group may not substantially change the discrimination
of the risk score, which primarily consists of risk factors for ESRD. There
is some discussion of how to use different definitions of controls in the ROC estimation \cite{zheng2012evaluating}; the choice is related to the clinical context and here we provide estimation methods
for both.
In Figure 2, we show further results of our study of the proposed and NNE methods with varying spans of
0.05, 0.1, and 0.3. The proposed
method produces stable $\widehat{AUC}(\tau)$ around 0.88 while the NNE
method is very sensitive to the span specification. This result is consistent
with the simulation results in Table 4. Such robustness to the tuning parameter
selection is a very attractive feature for our nonparametric estimator.
The Brier scores over all the predictive horizons are plotted
in Figure 3, along with the percentages of ESRD and censoring at each predictive horizon. The prediction error increases with the predictive horizon. This result implies that the predictive accuracy decreases as the predictive horizon moves away from the time of prediction. Overall the estimated Brier scores are small, between 0 and 0.11. Prior to year 3.5, when there is little censoring,
the three estimation methods produce almost identical results.
When the percentage of censoring increases beyond 3.5 years, the results from the three
methods begin to diverge but the absolute differences among them remain
small.
\section{Discussion }
In this paper, we propose an analytical framework for estimating
time-dependent predictive accuracy metrics with competing risk data
that are subject to right censoring. The method is illustrated with
the time-dependent ROC and time-dependent Brier score. The
proposed framework first computes a nonparametric estimator of the
conditional probability of the true event status given the observed
data and then uses it to weigh the data in an empirical calculation
of the time-dependent metrics. This is a unified
approach to estimating
the time-dependent ROC, time-dependent Brier score, and time-dependent
metrics constructed from other loss functions. The proposed method requires
no parametric assumptions about the marginal, conditional or joint
distribution of the risk score and time to the event of interest. It can be applied to evaluate the discrimination for a
single biomarker or a risk score constructed from a prognostic model with multiple biomarkers, and to evaluate the
calibration of the prognostic model.
The method is applicable when the censoring time and the risk score are correlated. It is also insensitive to the tuning parameter specification. Such robustness to the tuning
parameter specification has not been studied in nonparametric estimations of time-dependent predictive accuracy metrics \cite{heagerty2000time,saha2010time,zheng2012evaluating}
and no guidelines are yet available for practical users. When compared with competing methods in simulations, our proposed method demonstrates better
overall performance and robustness to tuning parameters, particularly when the censoring is correlated with the risk score. The R code that implements the proposed methodology is available upon request and will be added to the \texttt{tdROC} package in \texttt{R}.
One limitation with the proposed method is that, like many other nonparametric
methods, it works better with larger sample sizes. When the sample size is very
small, there may not be enough subjects with events for calculating $\widehat{F}_{1}(t\vert U_{i})$ and $\widehat{S}_{T}(t\vert U_{i})$
within some local neighborhoods defined by the kernel. In such case, the bandwidth may need to be increased for those neighborhoods.
\section{Acknowledgment }
The authors declare no potential conflicts of interest with respect
to the research, authorship and publication of this article. This
research was supported by the U.S. National Institutes of Health (grants
5P30CA016672 and 5U01DK103225).
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\begin{table}[ph]
\begin{centering}
\caption{Simulation results of $\widehat{AUC}_{A}(\tau)$ and $\widehat{AUC}_{B}(\tau)$
for the proposed method, IPCW, and NNE under different event 1 rate (70\%,
50\% and 30\%), censoring rate (Medium: 25-30\%, High: 45-50\%).}
\par\end{centering}
\centering{}
\begin{tabular}{ccccccccccc}
\hline
\multirow{2}{*}{Event 1} & \multirow{2}{*}{Censoring} & \multirow{2}{*}{True} & \multirow{2}{*}{n} & \multicolumn{3}{c}{Bias\% $\widehat{AUC}_{A}(\tau)$} & & \multicolumn{3}{c}{MSE$\times10^{-3}$$\widehat{AUC}_{A}(\tau)$}\tabularnewline
\cline{5-7} \cline{9-11}
& & & & {\small{}proposed} & {\small{}IPCW} & {\small{}NNE} & & {\small{}proposed} & {\small{}IPCW} & {\small{}NNE}\tabularnewline
\hline
\multirow{4}{*}{70\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.698} & 300 & 0.358 & 0.720 & -3.024 & & 1.281 & 1.299 & 1.600\tabularnewline
& & & 600 & 0.789 & 1.129 & -1.577 & & 0.591 & 0.638 & 0.646\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.693} & 300 & 0.487 & 0.950 & -2.986 & & 1.605 & 1.620 & 1.853\tabularnewline
& & & 600 & 0.700 & 1.158 & -1.714 & & 0.797 & 0.846 & 0.869\tabularnewline
\hline
\multirow{4}{*}{50\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.691} & 300 & 0.519 & 0.877 & -2.869 & & 1.567 & 1.593 & 1.826\tabularnewline
& & & 600 & 0.447 & 0.800 & -2.046 & & 0.774 & 0.789 & 0.931\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.685} & 300 & 0.106 & 0.651 & -3.430 & & 1.850 & 1.886 & 2.255\tabularnewline
& & & 600 & 0.827 & 1.311 & -1.718 & & 0.974 & 1.054 & 1.030\tabularnewline
\hline
\multirow{4}{*}{30\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.685} & 300 & 0.544 & 0.924 & -2.907 & & 2.020 & 2.050 & 2.290\tabularnewline
& & & 600 & 0.885 & 1.272 & -1.662 & & 0.985 & 1.020 & 1.042\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.683} & 300 & 0.580 & 0.981 & -2.949 & & 2.305 & 2.436 & 2.560\tabularnewline
& & & 600 & 0.246 & 0.759 & -2.359 & & 1.173 & 1.253 & 1.375\tabularnewline
\hline
& & & & & & & & & & \tabularnewline
& & & & \multicolumn{3}{c}{Bias\% $\widehat{AUC}_{B}(\tau)$} & & \multicolumn{3}{c}{MSE$\times10^{-3}$ $\widehat{AUC}_{B}(\tau)$}\tabularnewline
\cline{5-7} \cline{9-11}
& & & & proposed & IPCW & NNE & & proposed & IPCW & NNE\tabularnewline
\hline
\multirow{4}{*}{70\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.661} & 300 & 0.500 & 0.887 & -1.027 & & 1.656 & 1.721 & 1.710\tabularnewline
& & & 600 & 0.985 & 1.347 & -0.796 & & 0.733 & 0.797 & 0.757\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.663} & 300 & 0.585 & 1.054 & -1.270 & & 1.848 & 1.890 & 1.975\tabularnewline
& & & 600 & 0.693 & 1.164 & -1.336 & & 0.904 & 0.982 & 1.017\tabularnewline
\hline
\multirow{4}{*}{50\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.652} & 300 & 0.711 & 1.065 & -1.354 & & 2.080 & 2.154 & 2.207\tabularnewline
& & & 600 & 0.650 & 1.020 & -1.709 & & 0.934 & 0.956 & 1.063\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.653} & 300 & 0.403 & 0.983 & -1.820 & & 2.093 & 2.208 & 2.355\tabularnewline
& & & 600 & 1.051 & 1.519 & -1.664 & & 1.136 & 1.264 & 1.203\tabularnewline
\hline
\multirow{4}{*}{30\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.672} & 300 & 0.914 & 1.330 & -1.375 & & 2.404 & 2.487 & 2.492\tabularnewline
& & & 600 & 1.122 & 1.508 & -2.020 & & 1.162 & 1.229 & 1.325\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.672} & 300 & 0.728 & 1.081 & -1.853 & & 2.632 & 2.828 & 2.726\tabularnewline
& & & 600 & 0.343 & 0.841 & -2.981 & & 1.294 & 1.418 & 1.619\tabularnewline
\hline
\end{tabular}
\end{table}
\begin{table}[ph]
\centering{}\caption{Simulation results of $\widehat{Brier}(\tau)$ for the proposed method and IPCW
under different event 1 rate (70\%, 50\% and 30\%), censoring rate
(Medium: 25-30\%, High: 45-50\%).}
\begin{tabular}{ccccccccc}
\hline
\multirow{2}{*}{Event 1} & \multirow{2}{*}{Censoring} & \multirow{2}{*}{True} & \multirow{2}{*}{n} & \multicolumn{2}{c}{Bias\% BS} & & \multicolumn{2}{c}{MSE$\times10^{-3}$ BS}\tabularnewline
\cline{5-6} \cline{8-9}
& & & & proposed & IPCW.KM & & proposed & IPCW.KM\tabularnewline
\hline
\multirow{4}{*}{70\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.195} & 300 & -0.282 & -0.702 & & 0.143 & 0.146\tabularnewline
& & & 600 & -0.759 & -1.105 & & 0.066 & 0.070\tabularnewline
\cline{2-9}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.182} & 300 & -0.125 & -0.530 & & 0.179 & 0.176\tabularnewline
& & & 600 & 0.178 & -0.223 & & 0.096 & 0.097\tabularnewline
\hline
\multirow{4}{*}{50\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.165} & 300 & -1.337 & -1.575 & & 0.187 & 0.187\tabularnewline
& & & 600 & -1.478 & -1.698 & & 0.102 & 0.105\tabularnewline
\cline{2-9}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.149} & 300 & -0.697 & -1.052 & & 0.227 & 0.230\tabularnewline
& & & 600 & -1.137 & -1.401 & & 0.116 & 0.121\tabularnewline
\hline
\multirow{4}{*}{30\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.140} & 300 & -1.069 & -1.085 & & 0.213 & 0.211\tabularnewline
& & & 600 & -0.906 & -0.977 & & 0.100 & 0.101\tabularnewline
\cline{2-9}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.122} & 300 & 0.079 & -0.006 & & 0.235 & 0.237\tabularnewline
& & & 600 & -0.103 & -0.201 & & 0.112 & 0.112\tabularnewline
\hline
\end{tabular}
\end{table}
\begin{table}[ph]
\caption{Simulation results of $\widehat{AUC}_{A}(\tau)$ and $\widehat{AUC}_{B}(\tau)$
for the proposed method and NNE on the robustness of span specification.
Setting: 70\% event 1 rate, censoring rate (Medium: 25-30\%, High:
45-50\%).}
\centering{}
\begin{tabular}{ccccccccc}
\hline
\multirow{2}{*}{Censoring} & \multirow{2}{*}{True} & \multirow{2}{*}{n} & \multirow{2}{*}{Span} & \multicolumn{2}{c}{Bias\% $\widehat{AUC}_{A}(\tau)$} & & \multicolumn{2}{c}{MSE$\times10^{-3}$ $\widehat{AUC}_{A}(\tau)$}\tabularnewline
\cline{5-6} \cline{8-9}
& & & & proposed & NNE & & proposed & NNE\tabularnewline
\hline
\multirow{8}{*}{Medium} & \multirow{8}{*}{0.698} & \multirow{4}{*}{300} & 0.05 & 1.089 & -2.775 & & 1.164 & 1.343\tabularnewline
& & & 0.1 & 0.798 & -6.821 & & 1.342 & 3.141\tabularnewline
& & & 0.3 & -0.202 & -21.348 & & 1.131 & 22.313\tabularnewline
& & & 0.5 & -0.523 & -25.287 & & 1.009 & 31.188\tabularnewline
\cline{3-9}
& & \multirow{4}{*}{600} & 0.05 & 0.625 & -2.518 & & 0.697 & 0.912\tabularnewline
& & & 0.1 & 0.543 & -6.790 & & 0.606 & 2.649\tabularnewline
& & & 0.3 & -0.278 & -21.502 & & 0.604 & 22.588\tabularnewline
& & & 0.5 & -0.530 & -25.392 & & 0.551 & 31.441\tabularnewline
\hline
\multirow{8}{*}{High} & \multirow{8}{*}{0.693} & \multirow{4}{*}{300} & 0.05 & 1.128 & -3.017 & & 1.543 & 1.689\tabularnewline
& & & 0.1 & 0.550 & -7.950 & & 1.517 & 3.944\tabularnewline
& & & 0.3 & -0.355 & -21.845 & & 1.478 & 23.002\tabularnewline
& & & 0.5 & -1.368 & -25.278 & & 1.415 & 30.690\tabularnewline
\cline{3-9}
& & \multirow{4}{*}{600} & 0.05 & 0.754 & -2.796 & & 0.862 & 1.099\tabularnewline
& & & 0.1 & 0.756 & -7.537 & & 0.703 & 3.143\tabularnewline
& & & 0.3 & -0.355 & -21.991 & & 0.781 & 23.262\tabularnewline
& & & 0.5 & -1.371 & -25.383 & & 0.723 & 30.938\tabularnewline
\hline
& & & & & & & & \tabularnewline
& & & & \multicolumn{2}{c}{Bias\% $\widehat{AUC}_{B}(\tau)$} & & \multicolumn{2}{c}{MSE$\times10^{-3}$$\widehat{AUC}_{B}(\tau)$}\tabularnewline
\cline{5-6} \cline{8-9}
& & & & proposed & NNE & & proposed & NNE\tabularnewline
\hline
\multirow{8}{*}{Medium} & \multirow{8}{*}{0.661} & \multirow{4}{*}{300} & 0.05 & 1.222 & 0.705 & & 1.407 & 1.440\tabularnewline
& & & 0.1 & 0.790 & -0.718 & & 1.641 & 1.717\tabularnewline
& & & 0.3 & -0.161 & -4.624 & & 1.358 & 2.079\tabularnewline
& & & 0.5 & -0.247 & -6.180 & & 1.197 & 2.528\tabularnewline
\cline{3-9}
& & \multirow{4}{*}{600} & 0.05 & 0.810 & 0.144 & & 0.819 & 0.855\tabularnewline
& & & 0.1 & 0.620 & -0.956 & & 0.736 & 0.790\tabularnewline
& & & 0.3 & -0.073 & -5.270 & & 0.754 & 1.814\tabularnewline
& & & 0.5 & -0.202 & -6.651 & & 0.655 & 2.368\tabularnewline
\hline
\multirow{8}{*}{High} & \multirow{8}{*}{0.663} & \multirow{4}{*}{300} & 0.05 & 1.158 & 0.534 & & 1.914 & 1.998\tabularnewline
& & & 0.1 & 0.527 & -1.291 & & 1.711 & 1.836\tabularnewline
& & & 0.3 & -0.267 & -5.147 & & 1.671 & 2.528\tabularnewline
& & & 0.5 & -1.199 & -7.061 & & 1.550 & 3.260\tabularnewline
\cline{3-9}
& & \multirow{4}{*}{600} & 0.05 & 0.796 & -0.092 & & 0.953 & 0.947\tabularnewline
& & & 0.1 & 0.786 & -1.296 & & 0.816 & 0.956\tabularnewline
& & & 0.3 & -0.286 & -6.143 & & 0.904 & 2.336\tabularnewline
& & & 0.5 & -1.231 & -7.893 & & 0.800 & 3.271\tabularnewline
\hline
\end{tabular}
\end{table}
\begin{table}[ph]
\begin{centering}
\caption{Simulation results of $\widehat{Brier}(\tau)$ for the proposed method on
the robustness of span specification. Setting: 70\% event 1 rate,
censoring rate (Medium: 25-30\%, High: 45-50\%).}
\par\end{centering}
\centering{}
\begin{tabular}{cccccc}
\hline
Censoring & True & n & Span & Bias\% $\widehat{Brier}(\tau)$ & MSE$\times10^{-3}$ $\widehat{Brier}(\tau)$\tabularnewline
\hline
\multirow{8}{*}{Medium} & \multirow{8}{*}{0.195} & \multirow{4}{*}{300} & 0.05 & -1.159 & 0.143\tabularnewline
& & & 0.1 & -0.470 & 0.162\tabularnewline
& & & 0.3 & 0.065 & 0.121\tabularnewline
& & & 0.5 & 0.614 & 0.140\tabularnewline
\cline{3-6}
& & \multirow{4}{*}{600} & 0.05 & -0.831 & 0.078\tabularnewline
& & & 0.1 & -0.466 & 0.069\tabularnewline
& & & 0.3 & 0.320 & 0.073\tabularnewline
& & & 0.5 & 0.439 & 0.071\tabularnewline
\hline
\multirow{8}{*}{High} & \multirow{8}{*}{0.182} & \multirow{4}{*}{300} & 0.05 & -1.004 & 0.182\tabularnewline
& & & 0.1 & -0.232 & 0.182\tabularnewline
& & & 0.3 & 0.360 & 0.194\tabularnewline
& & & 0.5 & 1.086 & 0.180\tabularnewline
\cline{3-6}
& & \multirow{4}{*}{600} & 0.05 & -0.439 & 0.099\tabularnewline
& & & 0.1 & -0.069 & 0.079\tabularnewline
& & & 0.3 & 0.425 & 0.096\tabularnewline
& & & 0.5 & 1.086 & 0.095\tabularnewline
\hline
\end{tabular}
\end{table}
\begin{sidewaystable}
\centering{}\caption{Simulation results of $\widehat{AUC}_{A}(\tau)$ and $\widehat{AUC}_{B}(\tau)$
for the proposed method and IPCW methods under dependent censoring setting
$(a)$. Setting: event 1 rate (70\%, 50\% and 30\%), censoring rate
(Medium: 25-30\%, High: 45-50\%).}
\begin{tabular}{ccccccccccc}
\hline
\multirow{2}{*}{Event 1} & \multirow{2}{*}{Censoring} & \multirow{2}{*}{True} & \multirow{2}{*}{n} & \multicolumn{3}{c}{Bias\% $\widehat{AUC}_{A}(\tau)$} & & \multicolumn{3}{c}{MSE$\times10^{-3}$ $\widehat{AUC}_{A}(\tau)$}\tabularnewline
\cline{5-7} \cline{9-11}
& & & & proposed & IPCW.KM. & IPCW.Cox & & proposed & IPCW.KM & IPCW.Cox\tabularnewline
\hline
\multirow{4}{*}{70\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.698} & 300 & 0.980 & -5.587 & 1.600 & & 1.359 & 3.042 & 1.308\tabularnewline
& & & 600 & 0.510 & -5.810 & 1.193 & & 0.588 & 2.300 & 0.586\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.692} & 300 & 0.838 & -9.515 & 0.975 & & 1.718 & 6.109 & 1.463\tabularnewline
& & & 600 & 0.927 & -9.175 & 1.098 & & 0.864 & 4.910 & 0.740\tabularnewline
\hline
\multirow{4}{*}{50\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.702} & 300 & -0.673 & 3.351 & 0.322 & & 1.413 & 1.884 & 2.014\tabularnewline
& & & 600 & -0.720 & 3.264 & 0.490 & & 0.739 & 1.193 & 1.126\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.687} & 300 & 1.312 & -6.183 & 1.073 & & 2.090 & 3.676 & 1.746\tabularnewline
& & & 600 & 1.332 & -6.192 & 0.995 & & 1.076 & 2.811 & 0.892\tabularnewline
\hline
\multirow{4}{*}{30\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.685} & 300 & 0.906 & -4.493 & 1.821 & & 1.897 & 2.758 & 1.763\tabularnewline
& & & 600 & 0.745 & -4.512 & 1.748 & & 0.854 & 1.791 & 0.863\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.686} & 300 & -0.602 & 4.627 & 1.951 & & 2.352 & 3.287 & 5.925\tabularnewline
& & & 600 & -0.217 & 4.915 & 2.057 & & 1.237 & 2.310 & 6.800\tabularnewline
\hline
& & & & & & & & & & \tabularnewline
& & & & \multicolumn{3}{c}{Bias\% $\widehat{AUC}_{B}(\tau)$} & & \multicolumn{3}{c}{MSE$\times10^{-3}$ $\widehat{AUC}_{B}(\tau)$}\tabularnewline
\cline{5-7} \cline{9-11}
& & & & proposed & IPCW.KM. & IPCW.Cox & & proposed & IPCW.KM & IPCW.Cox\tabularnewline
\hline
\multirow{4}{*}{70\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.661} & 300 & 1.041 & -8.929 & 1.443 & & 1.684 & 5.483 & 1.609\tabularnewline
& & & 600 & 0.561 & -9.103 & 0.996 & & 0.752 & 4.533 & 0.717\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.665} & 300 & 0.831 & -13.040 & 0.350 & & 1.992 & 9.687 & 1.705\tabularnewline
& & & 600 & 0.820 & -12.736 & 0.409 & & 1.002 & 8.257 & 0.835\tabularnewline
\hline
\multirow{4}{*}{50\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.674} & 300 & -0.784 & 6.233 & 0.596 & & 1.809 & 3.367 & 2.972\tabularnewline
& & & 600 & -0.536 & 6.322 & 0.876 & & 0.874 & 2.601 & 1.622\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.663} & 300 & 1.373 & -9.341 & 0.484 & & 2.368 & 6.113 & 1.981\tabularnewline
& & & 600 & 1.559 & -9.275 & 0.472 & & 1.222 & 5.000 & 0.994\tabularnewline
\hline
\multirow{4}{*}{30\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.670} & 300 & 1.096 & -8.818 & 2.223 & & 2.356 & 5.948 & 2.141\tabularnewline
& & & 600 & 0.948 & -8.836 & 2.098 & & 1.011 & 4.679 & 1.028\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.676} & 300 & -0.266 & 7.490 & 2.504 & & 2.648 & 5.034 & 7.830\tabularnewline
& & & 600 & -0.113 & 7.651 & 2.043 & & 1.329 & 3.928 & 8.537\tabularnewline
\hline
\end{tabular}
\end{sidewaystable}
\begin{sidewaystable}
\caption{Simulation results of $\widehat{Brier}(\tau)$ for the proposed method and
IPCW methods under dependent censoring setting $(a)$. Setting: event
1 rate (70\%, 50\% and 30\%), censoring rate (Medium: 25-30\%, High:
45-50\%).}
\centering{}
\begin{tabular}{ccccccccccc}
\hline
\multirow{2}{*}{Event 1} & \multirow{2}{*}{Censoring} & \multirow{2}{*}{True} & \multirow{2}{*}{n} & \multicolumn{3}{c}{Bias\% BS} & & \multicolumn{3}{c}{MSE$\times10^{-3}$ BS}\tabularnewline
\cline{5-7} \cline{9-11}
& & & & proposed & IPCW.KM & IPCW.Cox & & proposed & IPCW.KM & IPCW.Cox\tabularnewline
\hline
\multirow{4}{*}{70\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.195} & 300 & -0.849 & 6.737 & 1.533 & & 0.160 & 0.318 & 0.152\tabularnewline
& & & 600 & -0.625 & 6.823 & 1.729 & & 0.082 & 0.249 & 0.082\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.175} & 300 & -0.576 & 11.201 & 3.742 & & 0.272 & 0.570 & 0.242\tabularnewline
& & & 600 & -0.494 & 11.240 & 3.776 & & 0.128 & 0.478 & 0.137\tabularnewline
\hline
\multirow{4}{*}{50\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.187} & 300 & -0.072 & -5.733 & -1.541 & & 0.155 & 0.289 & 0.209\tabularnewline
& & & 600 & -0.098 & -5.775 & -1.709 & & 0.080 & 0.203 & 0.108\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.153} & 300 & -0.768 & 8.387 & 2.426 & & 0.286 & 0.390 & 0.244\tabularnewline
& & & 600 & -1.277 & 8.219 & 2.280 & & 0.126 & 0.258 & 0.110\tabularnewline
\hline
\multirow{4}{*}{30\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.142} & 300 & -0.584 & 5.406 & 1.359 & & 0.222 & 0.258 & 0.206\tabularnewline
& & & 600 & -0.892 & 4.900 & 0.881 & & 0.097 & 0.134 & 0.089\tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.130} & 300 & -0.081 & -8.491 & -2.294 & & 0.236 & 0.343 & 0.252\tabularnewline
& & & 600 & -0.555 & -8.617 & -2.244 & & 0.109 & 0.233 & 0.127\tabularnewline
\hline
\end{tabular}
\end{sidewaystable}
\begin{sidewaystable}
\centering{}
\caption{Simulation results of $\widehat{AUC}_{A}(\tau)$ and $\widehat{AUC}_{B}(\tau)$
for the proposed method and IPCW methods under dependent censoring setting
$(b)$. Setting: event 1 rate (70\%, 50\% and 30\%), censoring rate
(Medium: 25-30\%, High: 45-50\%).}
\begin{tabular}{ccccccccccc}
\hline
\multirow{2}{*}{Event 1} & \multirow{2}{*}{Censoring} & \multirow{2}{*}{True} & \multirow{2}{*}{n} & \multicolumn{3}{c}{Bias\% $\widehat{AUC}_{A}(\tau)$} & & \multicolumn{3}{c}{MSE$\times10^{-3}$ $\widehat{AUC}_{A}(\tau)$}\tabularnewline
\cline{5-7} \cline{9-11}
& & & & proposed & IPCW.KM. & IPCW.Cox & & proposed & IPCW.KM & IPCW.Cox\tabularnewline
\hline
\multirow{4}{*}{70\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.702} & 300 & 0.267 & 2.709 & 0.668 & & 1.277 & 1.576 & 1.335 \tabularnewline
& & & 600 & -0.069 & 2.426 & 0.412 & & 0.562 & 0.829 & 0.596 \tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.693} & 300 & 0.251 & 4.718 & 1.216 & & 1.542 & 2.463 & 1.945 \tabularnewline
& & & 600 & 0.336 & 4.888 & 1.401 & & 0.690 & 1.839 & 1.029 \tabularnewline
\hline
\multirow{4}{*}{50\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.691} & 300 & 0.180 & 2.689 & 1.035 & & 1.578 & 1.861 & 1.811 \tabularnewline
& & & 600 & 0.094 & 2.474 & 0.899 & & 0.703 & 0.985 & 0.843 \tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.693} & 300 & -0.347 & 4.802 & 1.190 & & 1.782 & 2.708 & 2.734 \tabularnewline
& & & 600 & 0.008 & 4.890 & 1.313 & & 0.857 & 1.970 & 2.133 \tabularnewline
\hline
\multirow{4}{*}{30\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.685} & 300 & 0.613 & 2.621 & 1.148 & & 1.669 & 1.963 & 1.800 \tabularnewline
& & & 600 & 0.453 & 2.471 & 1.125 & & 0.843 & 1.125 & 0.915 \tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.687} & 300 & -0.444 & 3.525 & 0.742 & & 2.029 & 2.590 & 2.558 \tabularnewline
& & & 600 & 0.123 & 4.125 & 1.282 & & 0.991 & 1.772 & 1.355 \tabularnewline
\hline
& & & & & & & & & & \tabularnewline
& & & & \multicolumn{3}{c}{Bias\% $\widehat{AUC}_{B}(\tau)$} & & \multicolumn{3}{c}{MSE$\times10^{-3}$ $\widehat{AUC}_{B}(\tau)$}\tabularnewline
\cline{5-7} \cline{9-11}
& & & & proposed & IPCW.KM. & IPCW.Cox & & proposed & IPCW.KM & IPCW.Cox\tabularnewline
\hline
\multirow{4}{*}{70\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.662} & 300 & 0.181 & 3.931 & 0.603 & & 1.610 & 2.231 & 1.733 \tabularnewline
& & & 600 & -0.183 & 3.630 & 0.322 & & 0.743 & 1.286 & 0.802 \tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.663} & 300 & 0.193 & 6.244 & 1.211 & & 1.812 & 3.330 & 2.358 \tabularnewline
& & & 600 & 0.265 & 6.467 & 1.353 & & 0.814 & 2.651 & 1.251 \tabularnewline
\hline
\multirow{4}{*}{50\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.652} & 300 & 0.537 & 4.874 & 1.383 & & 1.986 & 2.862 & 2.403 \tabularnewline
& & & 600 & 0.353 & 4.542 & 1.154 & & 0.898 & 1.738 & 1.156 \tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.671} & 300 & -0.277 & 7.155 & 1.311 & & 2.115 & 4.140 & 3.395 \tabularnewline
& & & 600 & 0.214 & 7.323 & 1.401 & & 1.016 & 3.346 & 2.802 \tabularnewline
\hline
\multirow{4}{*}{30\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.671} & 300 & 0.997 & 4.794 & 1.467 & & 2.205 & 3.153 & 2.512 \tabularnewline
& & & 600 & 0.617 & 4.454 & 1.236 & & 1.115 & 1.975 & 1.231 \tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.676} & 300 & -0.227 & 5.541 & 0.913 & & 2.264 & 3.572 & 3.054 \tabularnewline
& & & 600 & 0.412 & 6.265 & 1.494 & & 1.136 & 2.876 & 1.668 \tabularnewline
\hline
\end{tabular}
\end{sidewaystable}
\begin{sidewaystable}
\centering{}
\caption{Simulation results of $\widehat{Brier}(\tau)$ for the proposed method and
IPCW methods under dependent censoring setting $(b)$. Setting: event
1 rate (70\%, 50\% and 30\%), censoring rate (Medium: 25-30\%, High:
45-50\%).}
\begin{tabular}{ccccccccccc}
\hline
\multirow{2}{*}{Event 1} & \multirow{2}{*}{Censoring} & \multirow{2}{*}{True} & \multirow{2}{*}{n} & \multicolumn{3}{c}{Bias\% BS} & & \multicolumn{3}{c}{MSE$\times10^{-3}$ BS}\tabularnewline
\cline{5-7} \cline{9-11}
& & & & proposed & IPCW.KM & IPCW.Cox & & proposed & IPCW.KM & IPCW.Cox\tabularnewline
\hline
\multirow{4}{*}{70\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.199} & 300 & -0.434 & -4.019 & -0.935 & & 0.133 & 0.207 & 0.146 \tabularnewline
& & & 600 & -0.311 & -3.907 & -0.747 & & 0.064 & 0.128 & 0.069 \tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.182} & 300 & -0.387 & -6.949 & -1.165 & & 0.166 & 0.339 & 0.191 \tabularnewline
& & & 600 & 0.107 & -6.543 & -0.530 & & 0.083 & 0.238 & 0.108 \tabularnewline
\hline
\multirow{4}{*}{50\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.165} & 300 & -1.326 & -5.477 & -1.638 & & 0.174 & 0.262 & 0.188 \tabularnewline
& & & 600 & -0.979 & -5.048 & -1.266 & & 0.092 & 0.166 & 0.099 \tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.171} & 300 & -0.615 & -8.321 & -1.468 & & 0.190 & 0.402 & 0.202 \tabularnewline
& & & 600 & -0.923 & -8.610 & -1.463 & & 0.092 & 0.315 & 0.121 \tabularnewline
\hline
\multirow{4}{*}{30\%} & \multirow{2}{*}{Medium} & \multirow{2}{*}{0.150} & 300 & -0.566 & -3.731 & -0.777 & & 0.189 & 0.228 & 0.196 \tabularnewline
& & & 600 & -0.966 & -4.094 & -1.149 & & 0.090 & 0.127 & 0.089 \tabularnewline
\cline{2-11}
& \multirow{2}{*}{High} & \multirow{2}{*}{0.132} & 300 & -0.840 & -6.320 & -0.954 & & 0.213 & 0.286 & 0.223 \tabularnewline
& & & 600 & -0.675 & -6.127 & -0.818 & & 0.103 & 0.168 & 0.102 \tabularnewline
\hline
\end{tabular}
\end{sidewaystable}
\begin{figure}
\caption{$\widehat{ROC}
\end{figure}
\begin{figure}
\caption{$\widehat{ROC}
\end{figure}
\begin{figure}
\caption{$\widehat{Brier}
\end{figure}
\pagebreak
\end{document} |
\begin{document}
\begin{frontmatter}
\title{\bf Edge Detection Methods Based on Modified Differential Phase Congruency of Monogenic Signal}
\author{Yan Yang}
\address{School of Mathematics (Zhuhai), Sun Yat-Sen University, \\
Zhuhai, 519080, China.\\ Email: [email protected]}
\author{Kit Ian Kou}
\address{Department of Mathematics, University of Macau, Macao (Via Hong
Kong).\\ Email: [email protected]}
\author{Cuiming~Zou}
\address{Department of Mathematics, University of Macau, Macao (Via Hong
Kong).\\ Email: [email protected]}
\begin{abstract}
Monogenic signal is regarded as a generalization of analytic
signal from one dimensional to higher dimensional space, which has been received considerable attention in the literature. It is defined by an original signal with its isotropic Hilbert transform (the combination of Riesz transform). Like the analytic signal, monogenic signal can be written in the polar form. Then it provides the signal features representation, such as the local attenuation and the local phase vector.
The aim of the paper is twofold: first, to analyze the relationship between the local phase vector and the local attenuation in the higher dimensional spaces. Secondly, a study on image edge detection using modified differential phase congruency is presented. Comparisons with competing methods on real-world images consistently show the superiority of the proposed methods.
{\bf e}nd{abstract}
\begin{keyword}
Hilbert transform; Phase space, Poisson operator
{\bf AMS Mathematical Subject Classification:} 44A15; 70G10; 35105
{\bf e}nd{keyword}
{\bf e}nd{frontmatter}
\section{Introduction}\label{S1}
In the scale-space literature, there are a lot of
papers discussing Gaussian scale-space as the only linear
scale-space \cite{BWBD, I, L}. The Gaussian scale space is obtained as the solution of the heat
equation. In \cite{FS2}, M. Felsberg
and G. Sommer proposed a new linear scale-space which is generated by
the Poisson kernel, it is the so-called Poisson scale-space in
two-dimensional (2D) spaces.
The Poisson scale-space is obtained by Poisson filtering (the
convolution of the original signal and the Poisson kernel). The
harmonic conjugate (the convolution of the original signal and the
conjugate Poisson kernel) yields the corresponding figure flow at
all scales. The Poisson scale-space and its corresponding figure
flow form the monogenic scale-space \cite{FS2}. In mathematics, monogenic
scale-space is the Hardy space in the upper half complex plane. The
boundary value of a monogenic function in the upper half space is the
monogenic signal. The monogenic scale signal gives deeper insight to low level image processing.
Monogenic signal is regarded as a generalization of analytic
signal from one dimensional space to higher dimensional case, which is first studied by M.
Felsberg and G. Sommer in 2001 \cite{FS1}.
It is defined by an original signal with its Riesz transform in higher dimensions.
Under certain assumptions, monogenic function can be representation in the polar form
and then it provides the signal features, such as the local attenuation and the
local phase vector \cite{FS1, YQS, YDQ, F}. In \cite{YQS}, we first defined the scalar-valued phase derivative
(local frequency) of a multivariate signal in higher dimensions.
Then we studied the applications in signal processing \cite{YDQ, YK, LZa, LZb, LZc}.
Monogenic signals at any scale $s>0$ form monogenic scale-space.
The representation of monogenic scale-space is just a monogenic function in the upper half-space.
Therefore, considering the monogenic scale space with scale $s$ instead of monogenic signals, provides us more analysis tools.
In the monogenic scale-space,
the important features in image processing, such as local phase-vector,
and local attenuation (the log of local amplitude) involving through scale $s$ are given in \cite{FS2}.
The relationship between the local attenuation
and the local phase in the intrinsically 1D cases are derived in \cite{FS2}. However, the problem is open if the signal is
not intrinsically 1D signal.
The contributions of this paper are summarized as follows.
\begin{itemize}
{\bf i}tem[1.] We give the solution of the problem: if the higher dimensional signal is not intrinsically 1D signal, we derive the relationship between the local phase-vector and and local attenuation.
{\bf i}tem[2.] We proposed the local attenuation (LA) method for edge detection operator. We establish the theoretical and experiment results on the newly methods.
{\bf i}tem[3.] We proposed the modified differential phase congruency (MDPC) method for the edge detection operator. We establish the theoretical and experiment results on the newly methods.
{\bf i}tem[4.] We show that in higher dimensional space, the instantaneous frequency in higher dimensional spaces defined by
is equal to the minus of the scale derivative of the
local attenuation.
{\bf i}tem[5.] We show that the zero points of the differential phase congruency is {\bf not} equal to the extrema of the local attenuation. The nonzero extra term $$-{\rm
Vec} \left[ \left(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|} \right)\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|} \right]\sin^2
\theta+(\sin \theta \cos \theta - \theta){\partial \over \partial s} \left({\underline{n}derline{v} \over |\underline{n}derline{x}|} \right)$$ appears in high dimensional cases.
{\bf e}nd{itemize}
The rest of this paper is organized as follows. In order to make it self-contained, Section \ref{S2} gives a brief introduction to some general definitions and basic properties of Hardy space, analytic signal, Clifford algebra, monogenic signal and monogenic scale space. In Section \ref{S3} we derive the relationship between the local phase-vector and and local attenuation. Various edge detection methods are provided in Section \ref{S4}. Finally, experiment results are drawn in Section \ref{S5}.
\section{Preliminaries}\label{S2}
In the present section, we begin by reviewing some definitions and basic properties of analytic signal and Hardy space \cite{Co, G1981, H1996}.
\subsection{Analytic Signal and Hardy Space}
\begin{Def}[Analytic Signal] For a square integrable real-valued function $f$, the complex-valued signal $f_A$ whose imaginary part is the Hilbert transform of its real part is called the {{\bf i}t analytic signal}. That is, $$f_A(x) :=f(x)+{\bf i} \mathcal{H}(f)(x),$$
where $\mathcal{H}(f)(x)$ is the Hilbert transform (HT) of $f$ defined by
\begin{eqnarray*}
\mathcal{H}[f](x) := \frac{1}{\pi}{\rm p.v.}{\bf i}nt_{-{\bf i}nfty}^{+{\bf i}nfty}\frac{f(s)}{x-s}ds=\frac{1}{\pi} \lim\limits_{\varepsilon\to 0^+} {\bf i}nt_{\varepsilon \leq |x-s|} \frac{f(s)}{x-s}ds,\label{liu11}
{\bf e}nd{eqnarray*} provided this integral exists as a principal value (${\rm p.v.}$ means the Cauchy principle value).
{\bf e}nd{Def}
Due to its definition, the real $u$ and imaginary parts $v$ of analytic signal $f_A=u+{\bf i} v$ form the {{\bf i}t Hilbert transform pairs}
\begin{eqnarray}\label{HTP}
\mathcal{H}[u]=v.
{\bf e}nd{eqnarray}
To proceed the properties of analytic signal, we introduce the notion of Hardy space \cite{G1981, H1996}, we will notice that the class of analytic signals is the class of boundary values of Hardy space functions.
\begin{Def}[Hardy Space]
The {{\bf i}t Hardy space} $H^2({\bf C}^+)$ is the class of analytic functions $f$ on the upper half complex plane ${\bf C}^+:=\{ x+{\bf i} y \,|\, x {\bf i}n {\bf R}, y>0\}$ which satisfies the growth condition
$$ \left( {\bf i}nt_{-{\bf i}nfty}^{{\bf i}nfty} |f(x+{\bf i} y)|^2 dx \right)^{1/2} < {\bf i}nfty,$$ for all $y>0$.
{\bf e}nd{Def}
Important properties of Hardy functions are given by Titchmarsh's Theorem \cite{N1972}.
\begin{Th}[Titchmarsh's Theorem] Let $g:=u+{\bf i} v {\bf i}n H^2({\bf C}^+)$. Then the following two assertions are equivalent:
\begin{itemize}
{\bf i}tem[1.] The Hardy function $g$ has no negative-frequency components. That is, $$g(z)={1 \over 2 \pi} {\bf i}nt_0^{{\bf i}nfty} G(\omega) e^{{\bf i} \omega z} d\omega,$$ where $G(\omega):= {\bf i}nt_{{\bf R}} g(x) e^{-{\bf i} \omega x} dx$ is the Fourier transform of $g$.
{\bf i}tem[2.] The real and imaginary parts verify the formulas: $$u(x+{\bf i} y)= u \ast P_y(x)={\bf i}nt_{{\bf R}} P_y(x-t) u(t) dt,$$ and $$v(x+{\bf i} y)=v \ast Q_y(x)={\bf i}nt_{{\bf R}} Q_y(x-t)v(t) dt,$$ for all $y>0$, where $P_y(x)={1 \over \pi} {y \over x^2+y^2}$ and $Q_y(x)={1 \over \pi} {x \over x^2+y^2}$ are the Poisson and conjugate Poisson kernel in ${\bf C}^+$.
{\bf e}nd{itemize}
{\bf e}nd{Th}
In this way, an analytic signal $f_A =f+{\bf i} \mathcal{H}[f]$ represents the boundary values of Hardy function $u+{\bf i} v$ in the upper half plane ${\bf C}^{+}$ \cite{G1981}. That is, $$f(x)=\lim_{y \rightarrow 0} u(x+{\bf i} y)$$ and $$\mathcal{H}[f](x) =\lim_{y \rightarrow 0} v(x+{\bf i} y).$$
Starting from this concept we are going to study the higher dimensional generalization on Clifford algebra.
\subsection{Clifford Algebra}
For all what follows we will work in ${{\bf i}t Cl}_{0, m}$, the real {{\bf i}t Clifford algebra}. Most of the basic knowledge and notations in relation to
Clifford algebra are referred to \cite{BDS, DSS}. Let ${\bf e}_1, ..., {\bf e}_m $ be {{\bf i}t basic elements} satisfying
${\bf e}_i{\bf e}_j+{\bf e}_j{\bf e}_i=-2\delta_{ij}$, where
$\delta_{ij}=1$ if $i=j,$ and $\delta_{ij}=0$ otherwise, $i, j=1, 2,
\cdots, m.$ The Clifford algebra ${{\bf i}t Cl}_{0, m}$ is the associative algebra over
the real field ${\bf R}$. A general element in ${{\bf i}t Cl}_{0, m}$, therefore, is of the form $x=\sum_S x_S {\bf e}_S,
x_s{\bf i}n {{\bf R}}$, where ${\bf e}_S={\bf e}_{i_1}{\bf
e}_{i_2}\cdots {\bf e}_{i_l},$ and $S$ runs over all the ordered
subsets of $\{1,2,\cdots,m\},$ namely
$S=\{1 \leq i_1 <i_2< \cdots < i_l \leq m \}, \quad 1\leq l \leq
m.$
Let \[\ {\bf R}^m =\{\underline{n}derline{x} \; |\; \underline{n}derline{x}=x_1 {\bf e}_1 + \cdots + x_m {\bf e}_m, x_j
{\bf i}n {\bf R}, j=1, 2, \cdots, m \}\] be identical with the usual Euclidean
space and an element in ${\bf R}^m$ is called a {{\bf i}t vector}. Moreover, let $$
{\bf R}_1^m =\{x \; |\; x=x_0+\underline{n}derline{x}, x_0 {\bf i}n {\bf R}, \underline{n}derline{x} {\bf i}n {\bf R}^m \}$$ be the {{\bf i}t para-vector} space and an element in ${\bf R}_1^m $ is called a {{\bf i}t para-vector}.
The multiplication of two para-vectors
$x_0+\underline{n}derline{x}=\sum_{j=0}^{m}x_j {\bf e}_j$ and $y_0+\underline{n}derline{y}=\sum_{j=0}^{m}y_j {\bf e}_j$
is given by
$(x_0+\underline{n}derline{x})(y_0+\underline{n}derline{y})=(x_0y_0+\underline{n}derline{x}\cdot \underline{n}derline{y}) +(x_0 \underline{n}derline{y}+y_0\underline{n}derline{x})+(\underline{n}derline{x}\wedge\underline{n}derline{y})$
with
$
\underline{n}derline{x}\cdot \underline{n}derline{y}=-<\underline{n}derline{x}, \underline{n}derline{y}>=-\sum_{j=1}^{m}x_jy_j
$
and
$\underline{n}derline{x}\wedge\underline{n}derline{y}=\sum_{i<j}{\bf e}_{ij}(x_iy_j-x_jy_i).$
Clearly, we have
\begin{eqnarray}\label{addeq1}
\underline{n}derline{x}\cdot \underline{n}derline{y}=\frac{1}{2}(\underline{n}derline{x} \underline{n}derline{y}+\underline{n}derline{y}\underline{n}derline{x})
{\bf e}nd{eqnarray}
and
$$\underline{n}derline{x}\wedge\underline{n}derline{y}=\frac{1}{2}(\underline{n}derline{x}\underline{n}derline{y}-\underline{n}derline{y}\underline{n}derline{x}).$$
There are three parts of $(x_0+\underline{n}derline{x})(y_0+\underline{n}derline{y})$. We denote them as follows
\begin{itemize}
{\bf i}tem the {{\bf i}t scalar part}: $x_0y_0+\underline{n}derline{x}\cdot \underline{n}derline{y}={\rm{Sc}}[(x_0+\underline{n}derline{x})(y_0+\underline{n}derline{y})]$,
{\bf i}tem the {{\bf i}t vector part }: $x_0 \underline{n}derline{y}+y_0\underline{n}derline{x}={\rm{Vec}}[(x_0+\underline{n}derline{x})(y_0+\underline{n}derline{y})]$,
{\bf i}tem the {{\bf i}t bi-vector part }: $\underline{n}derline{x}\wedge\underline{n}derline{y}={\rm{Bi}}[(x_0+\underline{n}derline{x})(y_0+\underline{n}derline{y})]$.
{\bf e}nd{itemize}
In particular, we have $\underline{n}derline{x}^2=-<\underline{n}derline{x}, \underline{n}derline{x}>=-|\underline{n}derline{x}|^2=-\sum_{j=1}^{m}x_j^2, \mbox{ for }\underline{n}derline{x}{\bf i}n {\bf R}^m$.
The conjugation and reversion of ${\bf e}_S $ are defined by
$\overline{\bf e}_{S}=\overline{\bf e}_{il}\cdots\overline{\bf
e}_{i1}$ and ${\overline{\bf e}_j=-{\bf e}_j}$, respectively. Therefore, the Clifford conjugate
of a para-vector $x_0+\underline{n}derline{x}$ is
$\overline{x_0+\underline{n}derline{x}}=x_0-\underline{n}derline{x}$. It is
easy to verify that $0 \not= x_0+\underline{n}derline{x} {\bf i}n {\bf R}_1^m$ implies
$$(x_0+\underline{n}derline{x})^{-1} := \frac{\overline{x_0+\underline{n}derline{x}}}{|x_0+\underline{n}derline{x}|^2}.$$
The open ball with center $0$ and radius $r$ in ${\bf R}_1^m $
is denoted by $B(0, r)$ and the unit sphere in ${\bf R}_1^m $ is
denoted by $S^m$.
The natural inner product between $x$ and $y$ in ${{\bf i}t Cl}_{0, m}$ is defined by $<x, y> :=\sum_Sx_S\overline{y_S},$ where $x :=\sum_Sx_S{\bf e}_S, x_S {\bf i}n {\bf R}$
and $y :=\sum_Sy_S{\bf e}_S, y_S {\bf i}n {\bf R}$. The norm associated with
this inner product is defined by $|x|=<x, x>^{1\over 2}=(\sum_S|x_S|^2)^{{1\over 2}}.$
Let $\Omega$ be an open subset of ${\bf R}_1^m$ with a piecewise smooth boundary. We say that function $f$ defined on $\Omega$ such that $f(x_0+\underline{n}derline{x})=\sum_S f_S
(x_0+\underline{n}derline{x}) {\bf e}_S$ is a Clifford-valued function or, briefly, a ${{\bf i}t Cl}_{0,m}$-valued function, where $f_S$ are real-valued functions defined in $\Omega$.
A possibility to generalize complex analytic is offered by following the Riemann approach, which is introduced by means of the {\bf e}mph{generalized Cauchy-Riemann
operator} $\frac{\partial}{\partial {x_0}}+\underline{n}derline{D}$, where
$\underline{n}derline D={\partial\over \partial x_1}{\bf e}_1+\cdots
+{\partial\over
\partial x_m}{\bf e}_m$ is the {\bf e}mph{Dirac operator}. Nullsolutions to this operator provide us with the class of the so-called {{\bf i}t monogenic functions}.
\begin{Def} (Monogenic Function)
An ${{\bf i}t Cl}_{0, m}$-valued function $f$ is called left (resp. right) monogenic in $\Omega$ if $\left( \frac{\partial}{\partial {x_0}}+\underline{n}derline{D}\right) f=0$ (resp. $f \left(\frac{\partial}{\partial {x_0}}+\underline{n}derline{D} \right) =0$) in $\Omega$.
{\bf e}nd{Def}
In the following, let
$$ E(x_0+\underline{n}derline{x})={\overline{x_0+\underline{n}derline{x}} \over |x_0+\underline{n}derline{x}|^{m+1}}$$ be the {{\bf i}t Cauchy kernel} defined in ${\bf R}_1^{m}\setminus
\{0\}$. It is easy to verify that $E(x_0+\underline{n}derline{x})$ is a monogenic function in ${\bf R}_1^{m}\setminus
\{0\}$ \cite{BDS, DSS}.
\begin{Rem}
\begin{itemize}
{\bf i}tem
For a ${{\bf i}t Cl}_{0, m}$-valued function defined on an open subset of ${\bf R}^m$, we apply the {{\bf i}t Dirac operator} $\underline{n}derline{D}$ for the monogenic function.
{\bf i}tem
Throughout the paper, and unless otherwise stated, we only use left ${{\bf i}t Cl}_{0, m}$-valued monogenic functions that, for simplicity, we call monoginic. Nevertheless, all results accomplished to left ${{\bf i}t Cl}_{0, m}$-valued monogenic functions can be easily adapted to right ${{\bf i}t Cl}_{0, m}$-valued monogenic functions.
{\bf e}nd{itemize}
{\bf e}nd{Rem}
We further introduce the right linear Hilbert space of integrable and square integrable ${{\bf i}t Cl}_{0, m}$-valued functions in $\Omega \subset {\bf R}^m$ that we denote by $L^1{(\Omega, {{\bf i}t Cl}_{0, m})}$ and $L^2{(\Omega, {{\bf i}t Cl}_{0, m})}$, respectively.
If $f {\bf i}n L^1{({\bf R}^m, {{\bf i}t Cl}_{0, m})}$, the {{\bf i}t Fourier
transform} of $f$ is defined by
\begin{eqnarray}\label{FT}\hat{f}(\underline{n}derline{x}i)={\bf i}nt_{{\bf R}^m}e^{-{\bf i}<\underline{n}derline{x},\underline{n}derline{x}i>}f(\underline{n}derline{x})d\underline{n}derline{x},{\bf e}nd{eqnarray}
if in addition, $\hat{f} {\bf i}n L^1{({\bf R}^m, {{\bf i}t Cl}_{0, m})}$, then function $f$ can be recovered by the {{\bf i}t inverse Fourier transform}
$$f(\underline{n}derline{x})=\frac{1}{(2\pi)^m}{\bf i}nt_{{\bf R}^m}e^{{\bf i}<\underline{n}derline{x},\underline{n}derline{x}i>}\hat{f}(\underline{n}derline{x}i)d\underline{n}derline{x}i.$$
The well-known Plancherel Theorem for Fourier transform of $f$ and $g {\bf i}n L^2({\bf R}^m, {{\bf i}t Cl}_{0, m})$ holds
\begin{eqnarray*}\label{PT}
{\bf i}nt_{{\bf R}^m} f(\underline{n}derline{x})g(\underline{n}derline{x}) d\underline{n}derline{x}=\frac{1}{(2\pi)^m}{\bf i}nt_{{\bf R}^m} \hat{f}(\underline{n}derline{x}i)\overline{\hat{g}(\underline{n}derline{x}i)} d\underline{n}derline{x}i.{\bf e}nd{eqnarray*}
In a recent paper \cite{FS1}, the authors defined the notion of the monogenic signal. It is regarded as an extension of the notion of the analytic signal to multidimensional signals.
\subsection{Monogenic Signal and Monogenic Scale Space}
The monogenic signal was defined by an
original signal and its "isotropic Hilbert transform" in the higher dimensional
spaces (a combination of the Riesz transforms).
\begin{Def}[Monogenic Signal]\label{MS} For $f {\bf i}n L^2({\bf R}^m, {{\bf i}t Cl}_{0,m})$, the monogenic signal $f_M {\bf i}n L^2({\bf R}^m, {{\bf i}t Cl}_{0,m})$ is defined by
$$f_{M}(\underline{n}derline{x}):=f(\underline{n}derline{x})+H[f](\underline{n}derline{x}),$$ where $H[f]$ is the {{\bf i}t isotropic Hilbert transform} of $f$
defined by
\begin{eqnarray*}
H[f](\underline{n}derline{x})&:=& p.v.
\frac{1}{\omega_{m}}{\bf i}nt_{{\bf R}^m}\frac{\overline{\underline{n}derline{x}-\underline{t}}}{|\underline{n}derline{x}-\underline{t}|^{m+1}}f(\underline{t})d\underline{t}\\
&=&\lim_{{\bf e}psilon\rightarrow
0^{+}}\frac{1}{\omega_{m}}{\bf i}nt_{|\underline{n}derline{x}-\underline{t}|>{\bf e}psilon}\frac{\overline{\underline{n}derline{x}-\underline{t}}}{|\underline{n}derline{x}-\underline{t}|^{m+1}}f(\underline{t})d\underline{t}\\
&=&-\sum_{j=1}^{m}R_j(f)(\underline{n}derline{x}){\bf e}_j.
{\bf e}nd{eqnarray*}
Furthermore, $$R_j(f)(\underline{n}derline{x}):=\lim_{{\bf e}psilon\rightarrow
0^{+}}\frac{1}{\omega_{m}}{\bf i}nt_{|\underline{n}derline{x}-\underline{t}|>{\bf e}psilon}\frac{x_j-t_j}{|\underline{n}derline{x}-\underline{t}|^{m+1}}f(\underline{t})d\underline{t},$$
is the jth-Reisz transform of $f$ and $\omega_m=\frac{2\pi^{\frac{m+1}{2}}}{\Gamma(\frac{m+1}{2})}$ is the area of the unit sphere $S^m$ in ${\bf R}^m_1$.
{\bf e}nd{Def}
\begin{Rem}
If $f(\underline{n}derline{x})$ is real-valued, then by Definition \ref{MS}, $H[f](\underline{n}derline{x})$ is
vector-valued.
{\bf e}nd{Rem}
Let us now generalize the notion of Hardy space to multidimensional space.
\begin{Def}[Monogenic Scale Space] \label{MSS}
The monogenic scale space $M^2({\bf R}_1^{m,+})$ is the class of monogenic functions $f^+(\underline{n}derline{x}, s)$ defined on half space $$
{\bf R}_1^{m,+} =\{x \; |\; x=(\underline{n}derline{x}, s), \underline{n}derline{x} {\bf i}n {\bf R}^m, s >0 \},$$ which satisfies the growth condition
$$ \left( {\bf i}nt_{{\bf R}^m} |f^+(\underline{n}derline{x}, s)|^2 d \underline{n}derline{x} \right)^{1/2} < {\bf i}nfty,$$
for all scale $s>0$.
{\bf e}nd{Def}
Like in the complex case, a monogenic signal is the boundary value of the monogenic scale function in the half space ${\bf R}_1^{m,+}$ \cite{LMcQ}. Some basic properties of the Monogenic scale space $M^2({\bf R}_1^{m,+})$ in the half space are summarized as follows. For the proof of Theorem \ref{MT} we refer the reader to \cite{LMcQ} and \cite{KQ}.
\begin{Th}\label{MT}
Suppose $f^+(\underline{n}derline{x}, s) := u(\underline{n}derline{x},s) + v(\underline{n}derline{x},s) {\bf i}n M^2({\bf R}_1^{m,+})$. Then the following two assertions are equivalent:
\begin{itemize}
{\bf i}tem[1.] The inverse Fourier transform of $f^+$ vanishes for $s<0$. That is, the ${{\bf i}t Cl}_{0,m}$-valued function $f^+(\underline{n}derline{x}, s)$ has the form
\begin{eqnarray*}\label{eq1}
f^{+}(\underline{n}derline{x}, s)=\frac{1}{(2\pi)^m}{\bf i}nt_{{\bf R}^m}e^{+}(s+\underline{n}derline{x},
\underline{t})\hat{f}(\underline{t})d\underline{t}
{\bf e}nd{eqnarray*}
in the half space $s >0$, where $$e^{+}(s+\underline{n}derline{x}, \underline{t})=e^{-s|\underline{t}|}e^{{\bf i}<\underline{n}derline{x},
\underline{t}>}(1+{\bf i}\frac{\underline{t}}{|\underline{t}|}),$$ is monogenic in ${{\bf R}}_1^m$ and $\hat{f}$ is the Fourier transform of $f$ given by (\ref{FT}).
{\bf i}tem[2.] The functions $u$ and $v$ are constructed by the Poisson and the conjugate Poisson integrals, respectively.
That is, \begin{eqnarray}\label{PK} u(\underline{n}derline{x}, s)=u*P_s(\underline{n}derline{x})=\frac{1}{\omega_{m}}{\bf i}nt_{{\bf R}^m}\frac{s}{|s+(\underline{n}derline{x}-\underline{t})|^{m+1}} u(\underline{t})d\underline{t}{\bf e}nd{eqnarray}
and \begin{eqnarray}\label{CPK}
\underline{n}derline{v}(\underline{n}derline{x}, s)=v*Q_s(\underline{n}derline{x})=\frac{1}{\omega_{m}}{\bf i}nt_{{\bf R}^m}
\frac{\overline{\underline{n}derline{x}-\underline{t}}}{|s+(\underline{n}derline{x}-\underline{t})|^{m+1}}v(\underline{t})d\underline{t},{\bf e}nd{eqnarray} where
$P_s(\underline{n}derline{x}) :=\frac{1}{\omega_{m}}\frac{s}{|s+\underline{n}derline{x}|^{m+1}}$ and
$Q_s(\underline{n}derline{x}) :=\frac{1}{\omega_{m}}\frac{\overline{\underline{n}derline{x}}}{|s+\underline{n}derline{x}|^{m+1}}$ are the Poisson and
the conjugate Poisson kernel in ${\bf R}_1^{m,+}$, respectively.
{\bf e}nd{itemize}
{\bf e}nd{Th}
\section{Local Attenuation and Local Phase Vector}\label{S3}
Note that it is possible to write the monogenic scale function $f {\bf i}n M^2({\bf R}^{m,+}_1)$ in polar coordinate. Let us review the local feature \cite{YQS} as follows.
\begin{Def}[Local Features Representation I]
Suppose $f :=u+\underline{n}derline{v} {\bf i}n M^2({\bf R}^{m,+}_1)$ has the polar form \begin{eqnarray}\label{polar}
f(\underline{n}derline{x}, s) =A(f)(\underline{n}derline{x}, s)e^{\frac{\underline{n}derline{v}(\underline{n}derline{x}, s)}{|\underline{n}derline{v}(\underline{n}derline{x},
s)|}\theta(\underline{n}derline{x}, s)},{\bf e}nd{eqnarray} then
\begin{eqnarray}\label{LA}
A(f)(\underline{n}derline{x}, s) : =|f(\underline{n}derline{x}, s)| =\sqrt{u(\underline{n}derline{x}, s)^2+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}{\bf e}nd{eqnarray} is
is called the {{\bf i}t local amplitude}.
\begin{eqnarray}\label{PA}
\theta(\underline{n}derline{x}, s) :=\arctan \left(\frac{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}{u(\underline{n}derline{x}, s)}\right){\bf e}nd{eqnarray}
is called the {{\bf i}t phase angle} that is between $0$ and $\pi$.
\begin{eqnarray}\label{lpv}
\underline{n}derline{r}(\underline{n}derline{x}, s) :=\frac{\underline{n}derline{v}(\underline{n}derline{x},
s)}{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}\theta(\underline{n}derline{x}, s),
{\bf e}nd{eqnarray}
is called the {{\bf i}t local phase vector}. ${\rm{Sc}}\left[(\underline{n}derline{D}\theta(\underline{n}derline{x},
s))\frac{\underline{n}derline{v}(\underline{n}derline{x}, s)}{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}\right]$ is called the {{\bf i}t directional phase derivative} and $e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}$ is called the {{\bf i}t phase
direction}.
The {\bf e}mph{phase derivative} or {{\bf i}t instantaneous frequency} is defined
by
\begin{eqnarray}\label{pd}
{\rm{Sc}}\left[\left({\underline{n}derline{D}f(\underline{n}derline{x}, s)}\right)\left({f(\underline{n}derline{x},
s)}\right)^{-1}\right].
{\bf e}nd{eqnarray}
{\bf e}nd{Def}
Building on the ideas of \cite{FS2}, we can have the alternative form.
\begin{Def}[Local Features Representation II]
For nontrivial function $f:=u+\underline{n}derline{v} {\bf i}n M^2({\bf R}^{m,+}_1)$, the local amplitude is nonzero. We can rewrite (\ref{polar}) as
\begin{eqnarray}\label{eq22}
f(\underline{n}derline{x}, s) =e^{a(\underline{n}derline{x}, s)+\underline{n}derline{r}(\underline{n}derline{x}, s)},
{\bf e}nd{eqnarray} where
\begin{eqnarray}\label{atten}
a(\underline{n}derline{x}, s) :=\ln A(f)(\underline{n}derline{x}, s)=\frac{1}{2}\ln(u(\underline{n}derline{x},
s)^2+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2)
{\bf e}nd{eqnarray} is called the {{\bf i}t local attenuation}.
{\bf e}nd{Def}
\begin{Rem} In one-dimensional case,
$\frac{\underline{n}derline{v}(x, s)}{|\underline{n}derline{v}(x, s)|}={\bf i},$ therefore the local phase vector
$\underline{n}derline{r}(x, s)={\bf i} \theta(x, s)$.
{\bf e}nd{Rem}
Suppose $f(x,s):=u(x,s)+{\bf i} v(x,s) {\bf i}n H^2({{\bf C}^+})$ has the form (\ref{eq22}). That is, $f(x,s)=e^{a(x,s)+{\bf i} \theta(x,s)}$ has no zeros and isolated singularities in the half plane ${\bf C}^+$, then the local attenuation $a(x,
s)=\frac{1}{2}\ln(u^2+v^2)$ and the local phase
$\theta(x,s)=\arctan \left( \frac{v}{u}\right)$ are related by the Cauchy-Riemann system. The reason is that the composition of analytic function is analytic. If $f(x, s)=u(x, s)+ {\bf i} v(x, s)$
is analytic and has no zeros and isolated singularities in the half plane ${\bf C}^+$, then $a(x, s)+{\bf i} \theta(x, s)$ is also analytic in ${\bf C}^+$.
Using the Cauchy-Riemann system for $a(x, s)+{\bf i} \theta(x, s)$, we
have
\begin{eqnarray*}\label{eq3}
\frac{\partial a}{\partial s}+\frac{\partial \theta}{\partial x}=0,
{\bf e}nd{eqnarray*}
\begin{eqnarray*}\label{eq4}
\frac{\partial a}{\partial x}-\frac{\partial \theta}{\partial s}=0
{\bf e}nd{eqnarray*}
From the above system, we notice that:
\begin{itemize}
{\bf i}tem The instantaneous frequency $\frac{\partial \theta}{\partial x}$ can be
obtained by the minus of the scale derivative of the local
attenuation ${\partial a \over \partial s}$.
{\bf i}tem The zero points of the scale derivative of the local
phase ${\partial \theta \over \partial s}$ is given by the extrema of the local attenuation.{\bf e}nd{itemize}
Building on the ideas of 1D, the authors \cite{FS2} considered the {{\bf i}t intrinsically 1D monogenic
signals}.
\begin{Def} If $f(x, s):=u(x,s)+{\bf i} v(x,s) {\bf i}n H^2({\bf C}^{+})$ has no zeros and isolated singularities in the half plane ${\bf C}^+$, then the {{\bf i}t intrinsically 1D monogenic signal} is defined by
\begin{eqnarray}\label{1DMS}
f(<\underline{n}derline{x}, \underline{n}>, s)
&=&u(<\underline{n}derline{x}, \underline{n}>, s)+\overline{\underline{n}}v(<\underline{n}derline{x}, \underline{n}>, s)\nonumber\\
&=&u(<\underline{n}derline{x}, \underline{n}>, s)+\underline{n}derline{v}(<\underline{n}derline{x}, \underline{n}>,
s\nonumber\\&=&e^{a(<\underline{n}derline{x}, \underline{n}>, s)+\underline{n}derline{r}(<\underline{n}derline{x}, \underline{n}>, s)},
{\bf e}nd{eqnarray}
where $\underline{n}derline{x}, \underline{n} {\bf i}n {\bf R}^{m}$ and $\underline{n}$ is a fixed unit vector. The local attenuation is given by $a(<\underline{n}derline{x}, \underline{n}>, s)=\frac{1}{2}\ln(u^2+v^2)$ and the local phase vector is given by
$\underline{n}derline{r}(<\underline{n}derline{x}, \underline{n}>, s)=\overline{n}\arctan \left(\frac{v}{u}\right)$ for intrinsically 1D signal.
{\bf e}nd{Def}
Felsberg et al. \cite{FS2} proved that for the intrinsically 1D signals, the local
attenuation $a(<\underline{n}derline{x}, \underline{n}>, s)$ and the local phase-vector $\underline{n}derline{r}(<\underline{n}derline{x},
\underline{n}>, s)$ are related by the Hilbert transform pairs (\ref{HTP}). Moreover, by the analyticity, using the generalized Cauchy-Riemann operator ${\partial \over \partial s}+\underline{n}derline{D}$ on $a(<\underline{n}derline{x}, \underline{n}>,
s)+\underline{n}derline{r}(<\underline{n}derline{x}, \underline{n}>, s)$, we have
\begin{eqnarray}\label{1Da}
\frac{\partial a}{\partial s}+\underline{n}derline{D}(\underline{n}derline{r})=0,
{\bf e}nd{eqnarray}
\begin{eqnarray}\label{1Db}
\underline{n}derline{D}a+\frac{\partial \underline{n}derline{r}}{\partial s}=0.
{\bf e}nd{eqnarray}
In \cite{FS2}, the {{\bf i}t local frequency } of the
intrinsically 1D signal and the
{{\bf i}t differential phase congruency} are defined by $\underline{n}derline{D}(\underline{n}derline{r})$ and $\frac{\partial
\underline{n}derline{r}}{\partial s}$, receptively. From system (\ref{1Da}) and (\ref{1Db}), we notice that:\begin{itemize}
{\bf i}tem The local frequency
in an intrinsically 1D signal $\underline{n}derline{D}(\underline{n}derline{r})$ can also be obtained by the minus of the
scale derivative of the local attenuation ${\partial a \over \partial s}$. {\bf i}tem The zero points of the differential
phase congruency ${\partial r \over \partial s}$ is given by the extrema of the local attenuation.{\bf e}nd{itemize}
\begin{Rem}
In the recent paper \cite{YQS}, the instantaneous frequency of $f :=u+\underline{n}derline{v}=e^{a+\underline{n}derline{r}}$ is given by (\ref{pd})
\begin{eqnarray}\label{phd}
&&{\rm{Sc}}\left[({\underline{n}derline{D}f(\underline{n}derline{x}, s)})({f(\underline{n}derline{x},
s)})^{-1}\right]\nonumber\\
&=&{\rm{Sc}}\left[\left(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\right)\sin\theta(\underline{n}derline{x},
s)\cos\theta(\underline{n}derline{x},
s)\right]+{\rm{Sc}}\left[\left(\underline{n}derline{D}\theta(\underline{n}derline{x},
s)\right)\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\right].\label{1}
{\bf e}nd{eqnarray}
In particular, if $\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}$ is a constant, the first term in (\ref{1}) vanishes,
then the instantaneous frequency is ${\rm{Sc}}\left[\left(\underline{n}derline{D}\theta(\underline{n}derline{x},
s)\right)\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\right]$. It coincides with the local frequency defined in \cite{FS2}. That is, when
$\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}=\overline{\underline{n}}$ is
a constant, the local frequency $\underline{n}derline{D}(\underline{n}derline{r})$ is given by
$(\underline{n}derline{D}\theta(<\underline{n}derline{x}, \underline{n}>, s)) \overline{\underline{n}}$.
{\bf e}nd{Rem}
\begin{Rem} \begin{itemize}
{\bf i}tem In Clifford analysis \cite{DSS}, we notice that if $f(x,
s)=u(x, s)+{\bf i} v(x, s) {\bf i}n H^2({\bf C}^+)$, then for fixed unit
vector $\underline{n} {\bf i}n {\bf R}^{m}$, the function
$$f(<\underline{n}derline{x}, \underline{n}>,
s)=u(<\underline{n}derline{x}, \underline{n}>, s)+\overline{\underline{n}}v(<\underline{n}derline{x}, \underline{n}>, s),$$ is monogenic in
${\bf C}^+$. It is called {{\bf i}t monogenic plane wave}.
{\bf i}tem Clearly, if $f(x, s)=e^{a(x,s)+i \theta (x,s)} {\bf i}n H^2({\bf C}^+)$ has no zeros and isolated singularities in ${\bf C}^+$, then
$a(x, s)+{\bf i} \theta(x, s)$ is also analytic in ${\bf C}^+$. Consequently, the function $a(<\underline{n}derline{x}, \underline{n}>, s)+\overline{\underline{n}}\theta(<\underline{n}derline{x}, \underline{n}>, s)=a(<\underline{n}derline{x}, \underline{n}>, s)+\underline{n}derline{r} (<\underline{n}derline{x}, \underline{n}>, s)$
is monogenic in ${\bf R}^{m,+}_1$.
{\bf e}nd{itemize}
{\bf e}nd{Rem}
\begin{Prob}\label{P1} What is the situation in the higher dimension if the signal is not intrinsically 1D signal?{\bf e}nd{Prob}
The solution was not considered in \cite{FS2} and \cite{FDF}.
While in higher dimension, the situation is more complicated. The theory does not hold in general.
In fact, if $f(\underline{n}derline{x}, s)=u(\underline{n}derline{x}, s)+\underline{n}derline{v}(\underline{n}derline{x}, s)=e^{a(\underline{n}derline{x}, s)+\underline{n}derline{r}(\underline{n}derline{x}, s)}$ is monogenic
in the half space ${\bf R}_1^{m,+}$, $a(\underline{n}derline{x}, s)+\underline{n}derline{r}(\underline{n}derline{x}, s)$ is not monogenic in
general. Therefore, the local attenuation $a$ and the local phase vector $r$ are
not related by the generalized Cauchy-Riemann system in higher dimensions. Let us now look at an example to illustrate the topic discussed above.
\begin{Exa} Let $f(\underline{n}derline{x},
s)=\frac{s}{|s+\underline{n}derline{x}|^{m+1}}+\frac{\overline{\underline{n}derline{x}}}{|s+\underline{n}derline{x}|^{m+1}}=E(s+\underline{n}derline{x})$
be the Cauchy kernel in ${\bf R}_1^m \setminus \{0\}$, which is monogenic in ${\bf
R}_1^{m}\setminus \{0\}$. Then, by straightforward computations, we have
$$a(\underline{n}derline{x}, s)+\underline{n}derline{r}(\underline{n}derline{x},
s)=-\frac{m}{2}\ln(s^2+|\underline{n}derline{x}|^2)+\frac{\overline{\underline{n}derline{x}}}{|\underline{n}derline{x}|}\arctan \left(\frac{|\underline{n}derline{x}|}{s}\right).$$
Then we apply the generalized Cauchy-Riemann operator ${\partial \over \partial s}+\underline{n}derline{D}$ on it, we have
\begin{eqnarray*}
&&\left(\frac{\partial}{\partial s}+\underline{n}derline{D}\right)\left[-\frac{m}{2}\ln(s^2+|\underline{n}derline{x}|^2)+\frac{\overline{\underline{n}derline{x}}}{|\underline{n}derline{x}|}
\arctan \left(\frac{|\underline{n}derline{x}|}{s}\right)\right]\\
&=&\frac{(1-m)(s+\underline{n}derline{x})}{s^2+|\underline{n}derline{x}|^2}+\frac{m-1}{|\underline{n}derline{x}|}\arctan \left(\frac{|\underline{n}derline{x}|}{s}\right)\neq 0.
{\bf e}nd{eqnarray*}
Therefore, $a(\underline{n}derline{x}, s)+\underline{n}derline{r}(\underline{n}derline{x}, s)$ is not monogenic.
{\bf e}nd{Exa}
Let us now describe the solution for Problem \ref{P1}, Theorem \ref{th1} gives the relationship
between the local phase vector $r$ and the local attenuation $a$ in higher
dimensional spaces.
\begin{Th}\label{th1}
Let $f(\underline{n}derline{x}, s)=u(\underline{n}derline{x}, s)+\underline{n}derline{v}(\underline{n}derline{x}, s)=e^{a(\underline{n}derline{x},
s)+\underline{n}derline{r}(\underline{n}derline{x}, s)} {\bf i}n M^2({\bf R}_1^{m,+})$, where $a(\underline{n}derline{x}, s)$ and $\underline{n}derline{r}(\underline{n}derline{x}, s)$ are the local
attenuation and the local
phase-vector defined by (\ref{atten}) and (\ref{lpv}), respectively. If $f$ has no zeros and isolated singularities in the half space ${\bf R}_1^{m,+}$. Then we have
\begin{eqnarray}\label{eq7}
\frac{\partial a}{\partial s}+{\rm
Sc}[(\underline{n}derline{D}e^{\underline{n}derline{r}})e^{-\underline{n}derline{r}}]=0,
{\bf e}nd{eqnarray}
\begin{eqnarray}\label{eq8}
\frac{\partial \underline{n}derline{r}}{\partial s}+\underline{n}derline{D}a -{\rm
Vec} \left[ \left(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|} \right)\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|} \right]\sin^2
\theta+(\sin \theta \cos \theta - \theta){\partial \over \partial s} \left({\underline{n}derline{v} \over |\underline{n}derline{x}|} \right)=0.
{\bf e}nd{eqnarray}
{\bf e}nd{Th}
In particular, if $\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}$ is independent of $s$, that is ${\partial \over \partial s} \left({\underline{n}derline{v} \over |\underline{n}derline{x}|} \right)=0$, then we
have the following corollary.
\begin{Cor}\label{cor1}
Let $f(\underline{n}derline{x}, s)=u(\underline{n}derline{x}, s)+\underline{n}derline{v}(\underline{n}derline{x}, s)=e^{a(\underline{n}derline{x},
s)+\underline{n}derline{r}(\underline{n}derline{x}, s)}{\bf i}n M^2({\bf R}_1^{m,+})$, where $a$ and $\underline{n}derline{r}$ are the local
attenuation and local phase-vector defined by (\ref{atten}) and (\ref{lpv}), respectively.
If $f$ has no zeros and isolated singularities in the half space ${\bf R}_1^{m,+}$
and the local orientation $\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}$ does not change
through scale $s$, then we have
\begin{eqnarray}\label{eq88}
\frac{\partial \underline{n}derline{r}}{\partial s}+\underline{n}derline{D}a -{\rm
Vec}[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}]\sin^2
\theta=0.
{\bf e}nd{eqnarray}
{\bf e}nd{Cor}
Combining (\ref{eq7}), (\ref{eq8}) and (\ref{phd}), we conclude that
\begin{Th}\label{th2} [Instantaneous Frequency]
\begin{itemize}
{\bf i}tem
The instantaneous frequency in higher dimensional spaces defined by
(\ref{pd}) is equal to the minus of the scale derivative of the
local attenuation ${\partial a \over \partial s}$.
{\bf i}tem The zero points of the differential phase congruency ${\partial r \over \partial s}$ is {\bf not} equal to the extrema of the local attenuation.
{\bf e}nd{itemize}
{\bf e}nd{Th}
\begin{Rem} By Theorem \ref{th2}, we notice that, like in one dimensional case,
the phase derivative in higher dimensions
can also be given by the minus of the scale derivative of the local
attenuation. However, the zero points of
the phase congruency is {\bf not} equal to the extrema of the local
attenuation in high dimensional case. The nonzero extra term $$-{\rm
Vec} \left[ \left(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|} \right)\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|} \right]\sin^2
\theta+(\sin \theta \cos \theta - \theta){\partial \over \partial s} \left({\underline{n}derline{v} \over |\underline{n}derline{x}|} \right)$$ appears in high dimensional cases.
{\bf e}nd{Rem}
We have divided the proof of Theorem \ref{th1} into a series of lemmas.
\begin{Lem}\label{lem2}
Let $f(\underline{n}derline{x}, s)=u(\underline{n}derline{x}, s)+\underline{n}derline{v}(\underline{n}derline{x}, s)=e^{a(\underline{n}derline{x},
s)+\underline{n}derline{r}(\underline{n}derline{x}, s)} {\bf i}n M^2({\bf R}_1^{m,+})$, where $a(\underline{n}derline{x}, s)$ and $\underline{n}derline{r}(\underline{n}derline{x}, s)$ are the local
attenuation and the local
phase-vector defined by (\ref{atten}) and (\ref{lpv}), respectively. If $f$ has no zeros and isolated singularities in the half space ${\bf R}_1^{m,+}$. Then we have
\begin{eqnarray}\label{eq10}
{\rm{Sc}}\left[(\frac{\partial}{\partial s}e^{\underline{n}derline{r}(\underline{n}derline{x},
s)})e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}\right]=0.
{\bf e}nd{eqnarray}
{\bf e}nd{Lem}
\noindent {\bf Proof: } By the generalized Euler formula
$e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}=e^{\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\theta}=\cos\theta+\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|} \sin
\theta$, we have
\begin{eqnarray}\label{qq}
&& \left(\frac{\partial}{\partial s}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}\right) e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}\nonumber\\
&=&\frac{\partial}{\partial s} \left(\cos \theta+\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\sin\theta\right) \left(\cos\theta-\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\sin\theta\right) \nonumber\\
&=&\left(-\sin\theta \frac{\partial \theta}{\partial
s}+\frac{\partial \frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial
s}\sin\theta+\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\cos\theta \frac{\partial
\theta}{\partial s}\right)
\left(\cos\theta-\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\sin\theta\right)\nonumber\\
&=&\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|} \frac{\partial\theta}{\partial
s}+\sin\theta\cos\theta \frac{\partial \frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial
s} -\sin^2\theta\frac{\partial \frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial
s}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}.
{\bf e}nd{eqnarray}
Clearly, the scalar part of $\left(\frac{\partial}{\partial s}
e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}\right) e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}$ is decided by the third
part of equation (\ref{qq}). Let us now prove the following
$$
{\rm{Sc}}\left[(\frac{\partial \frac{\underline{n}derline{v}(\underline{n}derline{x},
s)}{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}}{\partial s})\frac{\underline{n}derline{v}(\underline{n}derline{x},
s)}{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}\right] =0.
$$
Denote $\underline{n}derline{I}(\underline{n}derline{x}, s):=\frac{\underline{n}derline{v}(\underline{n}derline{x}, s)}{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}$, we have $\underline{n}derline{I}^2(\underline{n}derline{x}, s)=-1$. Then $\frac{\partial [{\underline{n}derline{I}^2(\underline{n}derline{x}, s)}]}{\partial s}=0$.
By equation (\ref{addeq1}), we have
\begin{eqnarray*}\label{addeq3}
\frac{\partial [{\underline{n}derline{I}^2(\underline{n}derline{x}, s)}]}{\partial s}&=&\frac{\partial [{\underline{n}derline{I}(\underline{n}derline{x}, s)}]}{\partial s}\underline{n}derline{I}(\underline{n}derline{x}, s)+\underline{n}derline{I}(\underline{n}derline{x}, s)\frac{\partial [{\underline{n}derline{I}(\underline{n}derline{x}, s)}]}{\partial s}\nonumber\\
&=&2{\rm {Sc}}[\frac{\partial {\underline{n}derline{I}(\underline{n}derline{x}, s)}}{\partial s}\underline{n}derline{I}(\underline{n}derline{x}, s)]
=0.
{\bf e}nd{eqnarray*}
Thus, we obtain the desired result.
\begin{Lem}\label{lem3}
Let $f(\underline{n}derline{x}, s)=u(\underline{n}derline{x}, s)+\underline{n}derline{v}(\underline{n}derline{x}, s)=e^{a(\underline{n}derline{x},
s)+\underline{n}derline{r}(\underline{n}derline{x}, s)} {\bf i}n M^2({\bf R}_1^{m,+})$, where $a(\underline{n}derline{x}, s)$ and $\underline{n}derline{r}(\underline{n}derline{x}, s)$ are the local
attenuation and the local
phase-vector defined by (\ref{atten}) and (\ref{lpv}), respectively. If $f$ has no zeros and isolated singularities in the half space ${\bf R}_1^{m,+}$. Then we have
\begin{eqnarray}\label{eq23}
{\rm{Vec}}\left[(\frac{\partial}{\partial s}e^{\underline{n}derline{r}(\underline{n}derline{x},
s)})e^{-\underline{n}derline{r}(\underline{n}derline{x},
s)}\right]=(\sin\theta\cos\theta-\theta)\frac{\partial\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial
s}+\frac{\partial \underline{n}derline{r}}{\partial s}.
{\bf e}nd{eqnarray}
\begin{eqnarray}\label{eq24}
{\rm{Vec}}\left[(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x},
s)})e^{-\underline{n}derline{r}(\underline{n}derline{x},
s)}\right]=-\sin^2\theta{\rm{Vec}}\left[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\right].
{\bf e}nd{eqnarray}
{\bf e}nd{Lem}
\noindent {\bf Proof: } From (\ref{qq}), we know that the vector part of
$(\frac{\partial}{\partial s}e^{\underline{n}derline{r}(\underline{n}derline{x},
s)})e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}$ is decided by $\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}
\frac{\partial\theta}{\partial s}+\sin\theta\cos\theta
\frac{\partial \frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial s}$. Since $\underline{n}derline{r}=\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\theta$, we have
$$\frac{\partial \underline{n}derline{r}}{\partial s}=\frac{\partial\theta}{\partial
s}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}+\theta\frac{\partial \frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial
s}.$$ Therefore, we obtain equation (\ref{eq23}).
To prove equation (\ref{eq24}), by direct calculation, we have
\begin{eqnarray}\label{qqq}
&& \left(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}\right) e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}\nonumber\\
&=&\underline{n}derline{D} \left(\cos\theta+\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\sin\theta\right) \left(\cos\theta-\frac{\underline{n}derline{r}}{|\underline{n}derline{r}|}\sin\theta\right) \nonumber\\
&=&\left[-\sin\theta (\underline{n}derline{D}\theta)+
(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\sin\theta+\cos\theta
(\underline{n}derline{D}\theta)\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\right]
\left[\cos\theta-\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\sin\theta\right]\nonumber\\
&=&\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}(\underline{n}derline{D}\theta)+\sin\theta\cos\theta
(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})
-\sin^2\theta(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}.
{\bf e}nd{eqnarray}
The fist part and the second part of equation (\ref{qqq}) are scalar and bi-vector, respectively. Therefore the vector part of $(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x},
s)})e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}$ is decided by the third part of
equation (\ref{qqq}). Thus we obtain (\ref{eq24}).
We can now prove Theorem \ref{th1}.
\noindent {\bf Proof of Theorem \ref{th1}: } Since $f(\underline{n}derline{x},
s)=e^{a(\underline{n}derline{x}, s)+\underline{n}derline{r}(\underline{n}derline{x}, s)} {\bf i}n M^2({\bf R}_1^{m,+})$, we have
$$\left(\frac{\partial}{\partial s}+\underline{n}derline{D}\right) e^{a(\underline{n}derline{x},
s)+\underline{n}derline{r}(\underline{n}derline{x}, s)}=0.$$ By straightforward computation, we have
$$e^{a(\underline{n}derline{x}, s)}\frac{\partial a(\underline{n}derline{x}, s)}{\partial s}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}+e^{a(\underline{n}derline{x}, s)}\frac{\partial e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}}{\partial s}
+e^{a(\underline{n}derline{x}, s)}[\underline{n}derline{D}a(\underline{n}derline{x}, s)]e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}+e^{a(\underline{n}derline{x},
s)}(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)})=0.$$ That is
\begin{eqnarray}\label{eq13}
\frac{\partial a(\underline{n}derline{x}, s)}{\partial s}+\frac{\partial e^{\underline{n}derline{r}(\underline{n}derline{x},
s)}}{\partial s}e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)} +\underline{n}derline{D}a(\underline{n}derline{x},
s)+(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)})e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}=0.
{\bf e}nd{eqnarray}
Therefore, the scalar part of (\ref{eq13}) is zero. By combining Lemma \ref{lem2}, we have
\begin{eqnarray}\label{eq14}
&&{\rm Sc}\left[\frac{\partial a(\underline{n}derline{x}, s)}{\partial s}+\frac{\partial
e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}}{\partial s}e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)} +\underline{n}derline{D}a(\underline{n}derline{x},
s)+(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)})e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}\right]\nonumber\\
&=&\frac{\partial a(\underline{n}derline{x}, s)}{\partial s}+{\rm Sc}[\frac{\partial
e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}}{\partial s}e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}]+{\rm
Sc}[(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)})e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}]\\
&=&\frac{\partial a(\underline{n}derline{x}, s)}{\partial s}+{\rm
Sc}[(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)})e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}]
=0\nonumber.
{\bf e}nd{eqnarray}
Therefore, we get the desired result (\ref{eq7}).
The vector part of Eq. (\ref{eq13}) is also zero. By using Lemma \ref{lem3}, we obtain
\begin{eqnarray}\label{eq15}
&&{\rm Vec}\left[\frac{\partial a(\underline{n}derline{x}, s)}{\partial
s}+\frac{\partial e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}}{\partial s}e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}
+\underline{n}derline{D}a(\underline{n}derline{x}, s)+(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)})e^{-\underline{n}derline{r}(\underline{n}derline{x},
s)}\right]\nonumber\\
&=&{\rm Vec}\left[\frac{\partial e^{\underline{n}derline{r}(\underline{n}derline{x}, s)}}{\partial
s}e^{-\underline{n}derline{r}(\underline{n}derline{x}, s)}\right] +\underline{n}derline{D}a(\underline{n}derline{x}, s)+{\rm
Vec}\left[(\underline{n}derline{D}e^{\underline{n}derline{r}(\underline{n}derline{x}, s)})e^{-\underline{n}derline{r}(\underline{n}derline{x},
s)}\right]\\
&=&\frac{\partial \underline{n}derline{r}}{\partial
s}+\underline{n}derline{D}a+(\sin\theta\cos\theta-\theta)\frac{\partial\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial
s}-{\rm{Vec}}\left[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\right]\sin^2\theta
=0\nonumber.
{\bf e}nd{eqnarray}
This completes the proof.
If $f {\bf i}n M^2({\bf R}_1^{m,+})$ has the {{\bf i}t axial form}
$$f(\underline{n}derline{x}, s)=u(\rho, s)+\frac{\overline{\underline{n}derline{x}}}{|\underline{n}derline{x}|}v(\rho, s),
\mbox{ }\rho=|\underline{n}derline{x}|.
$$ Then, in this case, the local orientation $\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}=\frac{\overline{\underline{n}derline{x}}}{|\underline{n}derline{x}|}$ does not change
through the scale $s$. By polar coordinate, $f(\underline{n}derline{x}, s)=e^{a(\rho,
s)+\frac{\overline{\underline{n}derline{x}}}{|\underline{n}derline{x}|}\theta(\rho, s)}$, using Theorem
\ref{th1}, we have the following corollary.
\begin{Cor}
Let $f(\underline{n}derline{x}, s)=u(\rho, s)+\frac{\overline{\underline{n}derline{x}}}{|\underline{n}derline{x}|}v(\rho,
s)=e^{a(\rho, s)+\frac{\overline{\underline{n}derline{x}}}{|\underline{n}derline{x}|}\theta(\rho, s)} {\bf i}n M^2({\bf R}_1^{m,+})$. Then we have
\begin{eqnarray}\label{eq11}
-\frac{\partial a}{\partial s}=\frac{\partial \theta}{\partial
\rho}+\frac{m-1}{\rho}\sin\theta\cos\theta,
{\bf e}nd{eqnarray}
\begin{eqnarray}\label{eq12}
\frac{\partial \theta}{\partial s}=\frac{\partial a}{\partial \rho}
+\frac{m-1}{\rho}\sin^2\theta.
{\bf e}nd{eqnarray}
{\bf e}nd{Cor}
It is easy to see that when $m=1$, the above system ((\ref{eq11}) and (\ref{eq12})) is just the Cauchy-Riemann system in the one dimensional case.
\section{Edge Detection Methods}\label{S4}
Edge detection by means of quadrature filters has two ways:
either by detecting local maxima of the local amplitude or by detecting
points of stationary phase in scale-space. In this section, we begin by reviewing the differential phase congruency method \cite{FS2}.
\subsection{Differential Phase Congruency Methods}
\begin{Method}[DPC] For intrinsically 1D monogenic signal $f {\bf i}n H^2({\bf C}^+)$ given by (\ref{1DMS}), if $f$ has no zero and isolated singularities in the half plane ${\bf C}^+$. Then the differential phase congruency (DPC) has the following formula
\begin{eqnarray}\label{ma}
\frac{\partial \underline{n}derline{r}_{in1}(\underline{n}derline{x}, s)}{\partial s}=\frac{u(\underline{n}derline{x},
s)\frac{\partial \underline{n}derline{v}(\underline{n}derline{x}, s)}{\partial
s}-\underline{n}derline{v}(\underline{n}derline{x}, s)\frac{\partial u(\underline{n}derline{x}, s)}{\partial s}
}{u(\underline{n}derline{x}, s)^2+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}=0,
{\bf e}nd{eqnarray} where $\underline{n}derline{r}_{in1}(\underline{n}derline{x}, s):=\underline{n}derline{r}(<\underline{n}derline{x}, \underline{n}>, s)$.
{\bf e}nd{Method}
By (\ref{1Db}), we notice that formula (\ref{ma}) can also be obtained by the $-\underline{n}derline{D}a$. However, the zero points of the differential phase congruency is {\bf not} given by the extrema of the local attenuation in higher dimension.
\subsection{Proposed Methods}
Let's introduce the local attenuation (LA) method for monogenic signals.
\begin{Method}[LA]
For $f {\bf i}n M^2({\bf R}_1^{m,+})$ has no zeros and isolated singularities in the half space ${\bf R}^{m,+}_1$, the local attenuation has the formula
\begin{eqnarray}\label{LAM}
\underline{n}derline{D}a(\underline{n}derline{x}, s)=\frac{u(\underline{n}derline{x}, s)\underline{n}derline{D}[u(\underline{n}derline{x},
s)]+|\underline{n}derline{v}(\underline{n}derline{x}, s)|\underline{n}derline{D}[|\underline{n}derline{v}(\underline{n}derline{x},
s)|]}{u^2(\underline{n}derline{x}, s)+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}.
{\bf e}nd{eqnarray}
{\bf e}nd{Method}
Applying Dirac operator $\underline{n}derline{D}$ on the local attenuation $a$, by direct computation on (\ref{LA}), formula (\ref{LAM}) follows. Using (\ref{1Db}), we know that for intrinsically 1D signals,
the zero points of the differential phase congruency is given by the extrema of the local attenuation. Notice that formula (\ref{LAM}) is equivalent to (\ref{ma}) for the intrinsically 1D monogenic signal.
Our second method is the modified differential phase congruency (MDPC) method. To proceed, we need the following technical lemma.
\begin{Lem}\label{newth2}
\begin{eqnarray}\label{addeq5}
\frac{\partial \underline{n}derline{r}(\underline{n}derline{x}, s)}{\partial
s}=\left(\theta-\sin\theta\cos\theta\right)\frac{\partial
\frac{\underline{n}derline{v}(\underline{n}derline{x}, s)}{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}}{\partial s}+
\frac{u(\underline{n}derline{x}, s)\frac{\partial \underline{n}derline{v}(\underline{n}derline{x}, s)}{\partial
s}-\underline{n}derline{v}(\underline{n}derline{x}, s)\frac{\partial u(\underline{n}derline{x}, s)}{\partial s}
}{u^2(\underline{n}derline{x}, s)+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}.
{\bf e}nd{eqnarray}
{\bf e}nd{Lem}
\noindent{\bf Proof: } By Eq. (\ref{lpv}), we have
\begin{eqnarray}\label{eq99}
\frac{\partial \underline{n}derline{r}(\underline{n}derline{x}, s)}{\partial s}&=&\frac{\partial}
{\partial s}(\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\theta)=\frac{\partial
\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}} {\partial
s}\theta+\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\frac{\partial} {\partial s}\theta.
{\bf e}nd{eqnarray}
By straightforward computation, we have
\begin{eqnarray}\label{eq999}
\frac{\partial} {\partial s}\theta=\frac{\partial} {\partial
s}(\arctan \left(\frac{|\underline{n}derline{v}|}{u}\right))=\frac{\frac{\partial |\underline{n}derline{v}|}{\partial
s}u-|\underline{n}derline{v}|\frac{\partial u}{\partial s}}{u^2+|\underline{n}derline{v}|^2}.
{\bf e}nd{eqnarray}
Then,
\begin{eqnarray}\label{eq100}
\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\frac{\partial} {\partial
s}(\arctan \left(\frac{|\underline{n}derline{v}|}{u}\right))=\frac{\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\frac{\partial
|\underline{n}derline{v}|}{\partial s}u-\underline{n}derline{v}\frac{\partial u}{\partial s}}{u^2+|\underline{n}derline{v}|^2}.
{\bf e}nd{eqnarray}
Using the equation
\begin{eqnarray*}
\frac{\partial \underline{n}derline{v}}{\partial s}=\frac{\partial}{\partial
s}(\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}|\underline{n}derline{v}|)
=|\underline{n}derline{v}|\frac{\partial \frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial
s}+\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\frac{\partial |\underline{n}derline{v}|}{\partial s},
{\bf e}nd{eqnarray*}
we obtain
\begin{eqnarray}\label{eq101}
\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\frac{\partial |\underline{n}derline{v}|}{\partial s}=\frac{\partial
\underline{n}derline{v}}{\partial s}-|\underline{n}derline{v}|\frac{\partial \frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial s}.
{\bf e}nd{eqnarray}
Applying Eq. (\ref{eq101}) to Eq. (\ref{eq100}), we have
\begin{eqnarray}\label{eq102}
\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\frac{\partial} {\partial
s}(\arctan \left(\frac{|\underline{n}derline{v}|}{u}\right))=\frac{u\frac{\partial \underline{n}derline{v}}{\partial
s}-\underline{n}derline{v}\frac{\partial u}{\partial
s}}{u^2+|\underline{n}derline{v}|^2}-\frac{u|\underline{n}derline{v}|}{u^2+|\underline{n}derline{v}|^2}\frac{\partial
\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial s}.
{\bf e}nd{eqnarray}
Combining Eq. (\ref{eq99}) and Eq. (\ref{eq102}), we obtain Eq. (\ref{addeq5}).
\begin{Rem} Note that equation (\ref{ma}) is a
special case of (\ref{addeq5}). The reason is in the
intrinsically 1D neighborhood, the local orientation
$\frac{\underline{n}derline{v}(\underline{n}derline{x}, s)}{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}=\overline{\underline{n}}$ is a
constant. So $\frac{\partial \frac{\underline{n}derline{v}(\underline{n}derline{x},
s)}{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}}{\partial s}=0$. In fact, formula (\ref{ma}) always holds if the local orientation is independent of
$s$.
{\bf e}nd{Rem}
Let us now define the points of modified differential phase congruency.
\begin{Def} Let $\underline{n}derline{r}(\underline{n}derline{x}, s)$ be the local phase vector, given by (\ref{lpv}),
of function $f {\bf i}n M^2({\bf R}_1^{m,+}$. Points where
$$\frac{\partial \underline{n}derline{r}(\underline{n}derline{x}, s)}{\partial s}-{\rm
Vec}[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}]\sin^2
\theta+(\sin \theta \cos \theta - \theta)\frac{\partial
\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial s}=0$$ are called
points of modified differential phase congruency (MDPC).
{\bf e}nd{Def}
\begin{Rem} From Theorem \ref{th1} we know that in
any higher dimensional cases, edge detection by means of local
amplitude maxima is equivalent to edge detection by modified
differential phase congruency.
{\bf e}nd{Rem}
Using Eq. (\ref{addeq5}), we can now proposed our second method, the so-called modified differential phase congruency (MDPC) method.
\begin{Method}[MDPC]
For $f {\bf i}n M^2({\bf R}_1^{m,+})$ has no zeros and isolated singularities in the half space ${\bf R}^{m,+}_1$, the MDPC has the formula
\begin{eqnarray}\label{eq9999}
&&\frac{\partial \underline{n}derline{r}(\underline{n}derline{x}, s)}{\partial s}-{\rm
Vec}[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}]\sin^2
\theta+\left(\sin \theta \cos \theta - \theta\right)\frac{\partial
\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial s}\nonumber\\
&=&\frac{u(\underline{n}derline{x}, s)\frac{\partial \underline{n}derline{v}(\underline{n}derline{x}, s)}{\partial
s}-\underline{n}derline{v}(\underline{n}derline{x}, s)\frac{\partial u(\underline{n}derline{x}, s)}{\partial s}
}{u^2(\underline{n}derline{x}, s)+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}-{\rm
Vec}\left[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\right]\sin^2
\theta.
{\bf e}nd{eqnarray}
{\bf e}nd{Method}
Finally, we introduce a mixed method by combining local attenuation and modified differential phase congruency
(LA+MDPC) for edge detection.
\begin{Method}[LA+MDPC]
For $f {\bf i}n M^2({\bf R}_1^{m,+})$ has no zeros and isolated singularities in the half space ${\bf R}^{m,+}_1$, the MDPC has the formula
\begin{eqnarray}\label{eq.mix}
&&\frac{\partial \underline{n}derline{r}}{\partial s}-\underline{n}derline{D}a -{\rm
Vec}[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}]\sin^2
\theta+(\sin \theta \cos \theta - \theta)\frac{\partial
\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial s}\nonumber\\
&=&\frac{u(\underline{n}derline{x}, s)\frac{\partial \underline{n}derline{v}(\underline{n}derline{x}, s)}{\partial
s}-\underline{n}derline{v}(\underline{n}derline{x}, s)\frac{\partial u(\underline{n}derline{x}, s)}{\partial s}
}{u^2(\underline{n}derline{x}, s)+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}-\underline{n}derline{D}a -{\rm
Vec}[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}]\sin^2
\theta
{\bf e}nd{eqnarray}
{\bf e}nd{Method}
\section{Experiments}\label{S5}
In this section, we begin by showing the details of our proposed methods. Two classical edge detection methods, such as Canny and Sobel edge detectors, will be compared with our algorithms. The Canny edge detector will begin by applying Gaussian filter to the test images. Then Canny edge detector computes the gradients on the images.
For the Sobel edge detector, we only apply its gradients to the original test images.
For the DPC and our proposed methods, we first apply the Poisson filter to the test images, then we compute apply their formulas to the images.
By comparing with the classical methods, phased based methods may show more detail on image.
\begin{figure}[!]
\centering
{\bf i}ncludegraphics[height=3.2cm]{3example.png}
\caption{Original images}
\label{fig.Original}
{\bf e}nd{figure}
\subsection{Algorithms}\label{S5.1}
Let us now give the details of the phase based algorithms. They are divided by the following steps.
\begin{description}
{\bf i}tem[Step 1.]
Input image $f(\underline{n}derline{x})$. For simplicity, the color image is converted to the gray image.
{\bf i}tem[Step 2.]
Poisson filtering: $u(\underline{n}derline{x}, s)=f*P_s(\underline{n}derline{x})$ and and $\underline{n}derline{v}(\underline{n}derline{x}, s)=f*Q_s(\underline{n}derline{x})$ for a fixed scale $s>0$.
We will discuss how to choose $s$ in Section \ref{5.1}. We consider $s$ in $0.1$, $0.5$, $1.0$ and $5.0$. Moreover, we choose $s=0.5$ for all test images to compare with different methods.
{\bf i}tem[Step 3.]
Compute the local attenuation $a(\underline{n}derline{x}, s)=\frac{1}{2}\ln(u(\underline{n}derline{x},s)^2+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2)$ and
local phase vector $\underline{n}derline{r}(\underline{n}derline{x}, s)=\frac{\underline{n}derline{v}(\underline{n}derline{x},s)}{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}\theta(\underline{n}derline{x}, s)$, where
the phase angle is given by $\theta(\underline{n}derline{x}, s)=\arctan \left(\frac{|\underline{n}derline{v}(\underline{n}derline{x}, s)|}{u(\underline{n}derline{x}, s)}\right)$.
{\bf i}tem[Step 4.]
Compute gradients by different methods to get the gradient maps.
\begin{itemize}
{\bf i}tem
The differential phase congruency (DPC) method: compute $\frac{\partial \underline{n}derline{r}_{in1}(\underline{n}derline{x}, s)}{\partial s}$ by the formula
$$
\frac{u(\underline{n}derline{x},s)\frac{\partial \underline{n}derline{v}(\underline{n}derline{x}, s)}{\partial
s}-\underline{n}derline{v}(\underline{n}derline{x}, s)\frac{\partial u(\underline{n}derline{x}, s)}{\partial s}
}{u(\underline{n}derline{x}, s)^2+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}.
$$
{\bf i}tem
The local amplitude (LA) method: compute $\underline{n}derline{D}a(\underline{n}derline{x}, s)$, where $\underline{n}derline{D}$ is the sum for the derivatives of image in vertical and horizontal directions.
By theoretical analysis, $\underline{n}derline{D}a(\underline{n}derline{x}, s)$ can be computed by
$$\frac{u(\underline{n}derline{x}, s)\underline{n}derline{D}[u(\underline{n}derline{x},
s)]+|\underline{n}derline{v}(\underline{n}derline{x}, s)|\underline{n}derline{D}[|\underline{n}derline{v}(\underline{n}derline{x},
s)|]}{u^2(\underline{n}derline{x}, s)+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}.$$
{\bf i}tem
The modified differential phase congruency (MDPC) method: compute
$\frac{\partial \underline{n}derline{r}(\underline{n}derline{x}, s)}{\partial s}-{\rm
Vec}[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}]\sin^2
\theta+\left(\sin \theta \cos \theta - \theta\right)\frac{\partial
\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}}{\partial s}$,
which equals to
$$\frac{u(\underline{n}derline{x}, s)\frac{\partial \underline{n}derline{v}(\underline{n}derline{x}, s)}{\partial
s}-\underline{n}derline{v}(\underline{n}derline{x}, s)\frac{\partial u(\underline{n}derline{x}, s)}{\partial s}
}{u^2(\underline{n}derline{x}, s)+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}-{\rm
Vec}\left[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\right]\sin^2
\theta.$$
{\bf i}tem
The mixed method by using local attenuation and modified differential phase congruency (LA+MDCP): compute
$$\frac{u(\underline{n}derline{x}, s)\frac{\partial \underline{n}derline{v}(\underline{n}derline{x}, s)}{\partial s }-\underline{n}derline{v}(\underline{n}derline{x}, s)\frac{\partial u(\underline{n}derline{x}, s)}{\partial s}}{u^2(\underline{n}derline{x}, s)+|\underline{n}derline{v}(\underline{n}derline{x}, s)|^2}
-\underline{n}derline{D}a -{\rm Vec}[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}]\sin^2\theta.$$
{\bf e}nd{itemize}
{\bf i}tem[Step 5.]
Applying Non-maximum suppress to these gradient maps, which is the same as for the Canny edge detector.
After non-maximum suppression, the edge will become thinner \cite{PK99, PK03}.
For a fair comparison, all the six methods aforementioned
utilize the non-maximum suppression method with the same parameters.
Concretely, we choose the radius $r=1.5$ and the lower and upper threshold values are $1.0$ and $3.5$, respectively.
{\bf e}nd{description}
\begin{figure}[!]
\centering
{\bf i}ncludegraphics[height=20cm]{3results.png}
\caption{ Results for Canny, Sobel, DPC, LA, MDPC and LA+MDPC from top to the bottom.}
\label{fig.3results}
{\bf e}nd{figure}
\subsection{Experiment Results}\label{S5.2}
We will use three different images (Fig. \ref{fig.Original}) for the comparison of different edge detectors.
Fig. \ref{fig.results} shows the edge detection results of various methods with the fixed scale $s=0.5$. From top to down of Fig. \ref{fig.3results}, there are six rows. Each row shows one comparison method. They are
Canny, Sobel, DPC, LA, MDPC and LA+MDPC methods, respectively. From the experiment results, we can draw the following conclusions.
\begin{itemize}
{\bf i}tem First, the mixed method yields decent edge detection results with fewer mistakes, outperforming other algorithms in some cases.
{\bf i}tem
Second, the comparison between the results of LA, DPC, MDPC and the mixed methods also
suggest that both local attenuation and local phase are important in edge detection.
{\bf i}tem
Our proposed method MDPC can achieve very good performances in dealing with the details. For the pepper in Fig. \ref{fig.3results}, we found that our method and canny's results are similar, we can find the edge of pepper in the results. However, for the shadows of the house and liver in Fig. \ref{fig.3results}, where the human eye is relatively subtle. Fortulately, the DPC and MDPC methods have found the details in the shadow. However,
Canny, Sobel and LA methods cannot give the information about the shadows.
Canny uses the Gaussian filter which will make part of these shadows as noise and removed them.
While Sobel directly generate the horizontal and vertical differences, because the shadows
and the surrounding area is not much difference, which may not find the shadow of the image.
By applying the phase based method, these details can be clearly found in our experiment results.
This shows that our method can detect the whole smooth region and local small change region.
Applications can be useful in places where it is difficult for the human eye to find the details.
{\bf e}nd{itemize}
\begin{Rem} For intrinsically 1D signals, we know
that edge detection by means of local amplitude maxima is equivalent
to edge detection by phase congruency. While, in intrinsically 2D
signals, we know that it dose not hold. In \cite{FS2}, the authors
said: $\lq\lq$ We cannot give an exhaustive answer to this question.
In this paper, since the behavior of phase and attenuation in
intrinsically 2D neighborhoods is still work in progress." From
Eq. (\ref{eq9999}), we know that difference between the DPC and MDPC methods
is ${\rm
Vec}\left[(\underline{n}derline{D}\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|})\frac{\underline{n}derline{v}}{|\underline{n}derline{v}|}\right]\sin^2
\theta$. By experiment, we know that the effect is not obvious.
{\bf e}nd{Rem}
\subsection{Effect of Scale}\label{S5.3}
\begin{figure}[!]
\centering
{\bf i}ncludegraphics[height=13.5cm]{4S.png}
\caption{ Results for $s=0.1,~0.5,~1.0,~5.0$ from top to bottom.
The first column show the differential phase congruency (DPC) method,
the second column is the modified differential phase congruency (MDPC) method ,
and the third column is the proposed mixed (LA+MDPC) method . }
\label{fig.results}
{\bf e}nd{figure}
Monogenic signals at any scale $s>0$ form the monogenic scale-space $M^2({\bf R}^{m,+}_1)$.
The representation of monogenic scale-space is just a monogenic function in the upper half space ${\bf R}^{m,+}_1$.
Therefore, considering the monogenic scale space instead of monogenic signal, it has a scale parameter $s>0$ to choose which provides us more analysis tools for different purposes.
We found that when $s$ tends to $0$, the Poisson integral tends to Hilbert transform.
Moreover, in the paper \cite{P}, Hilbert transform has been proved to be useful for image edge extraction.
In the choice of scale, we compare $s$ from $0.1$ to $5$, as can be seen in Fig. \ref{fig.3results},
when $s$ is smaller, the edge is more fine.
But too much detail is not good at all, so in the comparative experiment in Fig. \ref{fig.results},
we chose the case of 0.5 for $s$ to compare with the various methods.
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\begin{document}
\title{``Graph Paper'' Trace Characterizations of Functions of Finite Energy}
\author{Robert S. Strichartz}
\thanks{Research supported in part by the National Science Foundation, grant DMS-1162045.}
\address{Mathematics Department, Malott Hall, Cornell University, Ithaca, NY 14853}
\email{[email protected]}
\date{}
\begin{abstract}
We characterize functions of finite energy in the plane in terms of their
traces on the lines that make up ``graph paper'' with squares of side length
$m^n$ for all $n$, and certain $\frac 1 2$-order Sobolev norms on the graph
paper lines. We also obtain analogous results for functions of finite energy
on two classical fractals: the Sierpinski gasket and the Sierpinski carpet.
\end{abstract}
\maketitle
\section{Introduction}\label{sec:!}
Functions of finite energy play an important role in analysis and probability.
On Euclidean space or a domain in Euclidean space, these are just the
functions whose gradient in the distribution sense belongs to $L^2$, with
the energy given by
\begin{equation}\label{eq:1.1}
\int \left| \nabla F\right|^2\, dx
\end{equation}
As such they make up a homogeneous Sobolev space that we will denote here as
$H^1$. The more usual inhomogeneous Sobolev space is smaller, requiring that
$F\in L^2$ as well \cite{9,10}. There are many ways to generalize the notion
of finite energy to other contexts. For example, as the functions in the
domain of a Dirichlet form \cite{4}. In this paper we will only consider
functions of finite energy in regions in the plane, and on two classical
fractals, the Sierpinski gasket \cite{8,12} and the Sierpinski carpet
\cite{1,2}.
It is well-known that functions of finite energy in the plane (or in higher
dimensions) do not have to be continuous, so the value $F(x,y)$ at a point is
not meaningful. Nevertheless, the trace on a line, say $TF(x) = F(x,0)$, is
well-defined, and belongs to a certain $\frac 1 2$-order homogeneous Sobolev
space that we will denote here by $H^{1/2}(\mathbb{R})$, defined by the finiteness of
\begin{equation}\label{eq:1.2}
\int_{--\infty}^\infty \int_{-\infty}^\infty \frac{\left|f(x)-f(y)\right|^2}{|x-y|^2}\, dxdy\text{,}
\end{equation}
with a corresponding norm estimate. Of course it is the norm estimate that is
important since it implies the existence of the trace by routine arguments.
The result is sharp, meaning that there is an extension operator from
$H^{1/2}(\mathbb{R})$ to $H^1(\mathbb{R}^2)$. There are in fact two rather natural
$\frac 1 2$-order Sobolev spaces on $\mathbb{R}$. The other one, which we denote by
$\tilde H^{1/2}(\mathbb{R})$ is larger, and only requires the finiteness of an integral like
(\ref{eq:1.2}) where the integration is restricted to the region
$|x-y|\leqslant 1$. We will show that the trace of a function $F$ of finite
energy in the strip $\{(x,y) : 0<y<1\}$ only belongs to $\tilde H(\mathbb{R})$. In
particular this implies that there does not exist a Sobolev extension theorem
from $H^1$ of the strip to $H^1(\mathbb{R}^2)$, even though such a result for
inhomogeneous Sobolev spaces is well-known and essentially trivial.
The trace of a function of finite energy on a single line does not, of course,
determine the function. What about the trace of an infinite collection of
lines that together form a dense subset of the plane? A simple example is the
set of lines of ``graph paper,'' where we take the graph paper squares to have
side length $m^n$, where $m$ is an integer ($m\geqslant 2$) and $n$ varies
over $\mathbb{Z}$, so the graph papers $\mathsf{GP}_{m^n}$ are nested. The main results of
this paper are first a trace theorem that characterizes the traces of
$H^1(\mathbb{R}^2)$ functions on $\mathsf{GP}_{m^n}$ in terms of a Sobolev space
$H^{1/2}(\mathsf{GP}_{m^n})$ with a given norm, and then the characterization of
$H^1(\mathbb{R}^2)$ in terms of a uniform bound on the norms of the traces on
$\mathsf{GP}_{m^n}$ as $n\to -\infty$.
The trace theorem is discussed in section \ref{sec:3} in the context of
Sobolev spaces $H^{1/2}$ on \emph{metric graphs} (graphs whose edges have
specified length, \cite{3}), as discussed in section \ref{sec:2}. Because the
functions in these spaces need not be continuous, the key issue is to
understand a kind of ``gluing'' condition at the vertices of the graph. It
turns out that this condition was given in \cite{11}. For the
convenience of the reader we give all the proofs in section \ref{sec:2},
although many of the results are already known, because they are usually
treated in the context of inhomogeneous Sobolev spaces. In section \ref{sec:4}
we discuss the trace characterizations of $H^1(\mathbb{R}^2)$. In section \ref{sec:5}
we discuss the analogous results on the two fractals. It turns out that the
trace theorems are already known \cite{5,6,7}, and the Sobolev spaces are
$H^\beta$ for values satisfying $\frac 1 2<\beta<1$. The spaces of functions
of finite energy on these fractals consist of continuous functions, as do the
trace spaces, so there is no difficulty defining the traces, and the
``gluing'' condition at vertices is simply continuity. Thus the fractal analog
of the trace characterization is perhaps simpler than the theorem in the
plane. We also characterize the traces on Julia sets of functions of finite
energy in the unbounded component of the complement of the Julia set. We
believe strongly that there is a great benefit to thinking about problems in
both the smooth and the fractal contexts, and looking for interactions in the
ideas that emerge. We hope this paper gives some support to this point of
view.
\section{Metric Graphs}\label{sec:2}
A \emph{metric graph} $G=(V,E,L_e)$ consists of a graph $(V,E)$ with vertices
$V$ and edges $E$, and a function that assigns a length $L_e$ in $(0,\infty]$
to each edge $e\in E$.
\begin{definition}\label{dfn:2.1}
For a metric graph $G=(V,E,L_e)$, define the homogeneous Sobolev norm
\begin{equation}\label{eq:2.1}
\begin{aligned}
\left\|f\right\|_{H^{1/2}(G)}^2
&= \sum_{e\in E} \int_0^{L_e} \int_0^{L_e} \frac{\left|f\left(e(x)\right) - f\left(e(y)\right)\right|^2}{|x-y|^2}\, dxdy \\
&+ \sum_{e\sim e'} \int_0^L \frac{\left|f\left(e(x)\right) - f\left(e'(x)\right)\right|^2}{x}\, dx
\end{aligned}
\end{equation}
(in the second sum $L=\min(L_e,L_{e'})$, and the parameterizations of $e$ and
$e'$ are chosen so that $e(0)$ and $e'(0)$ correspond to the intersection
point). We define the Sobolev space $H^{1/2}(G)$ to be the equivalence classes
(modulo constants) of locally $L^2$ functions for which the norm is finite. It
is easy to see that $H^{1/2}(G)$ is a Hilbert space.
\end{definition}
\begin{example}\label{ex:1}
Let $G=\mathbb{R}$, so $G$ has no vertices and a single edge of infinite length. We
need to modify (\ref{eq:2.1}) in this case to read
\begin{equation}\label{eq:2.2}
\left\|f\right\|_{H^{1/2}(\mathbb{R})}^2 = \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{\left|f(x)-f(y)\right|^2}{|x-y|^2}\, dxdy\text{.}
\end{equation}
For this example we also want to consider the smaller norm
\begin{equation}\label{eq:2.3}
\left\|f\right\|_{\tilde H^{1/2}(\mathbb{R})}^2 = \iint_{|x-y|\leqslant 1} \frac{\left|f(x)-f(y)\right|^2}{|x-y|^2} \, dxdy
\end{equation}
and corresponding larger Sobolev space $\tilde H^{1/2}(\mathbb{R})$.
\end{example}
We note that the space $H^{1/2}(\mathbb{R})$ is M\"obius invariant, meaning that
$f\in H^{1/2}(\mathbb{R})$ if and only if $f\circ M\in H^{1/2}(\mathbb{R})$ with equal
norms, for $M(x)=\frac{ax+b}{cx+d}$ with $\smat a b c d\in \SL(2,\mathbb{R})$.
Indeed it suffices to verify this for translations $M(x)=x+b$, dilations
$M(x) = a x$ and the inversion $M(x)=\frac 1 x$, where it follows by a
simple change of variable in the integral defining the norm. We note that the
same statement is false for $\tilde H^{1/2}(\mathbb{R})$.
We may easily characterize these norms and spaces in terms of the Fourier
transform $\hat f$. The finiteness of the norm easily implies that $f$ is a
tempered distribution so $\hat f$ is well defined as a tempered distribution,
and the equivalence of the functions that differ by a constant means $\hat f$
is only defined up to the addition of an arbitrary multiple of the delta
function. Note that there is no ``canonical'' choice of $f$ and $\hat f$
within each equivalence class.
\begin{theorem}\label{thm:2.2a}
a) $f\in H^{1/2}(\mathbb{R})$ if and only if $\hat f$ may be identified with a
function that is locally in $L^2$ in the complement of the origin with
\begin{equation}\label{eq:2.4}
\int_{-\infty}^\infty |\hat f(\xi)|^2 |\xi|\, d\xi < \infty\text{,}
\end{equation}
and (\ref{eq:2.4}) is in fact a constant multiple of (\ref{eq:2.2})
b) $f\in \tilde H^{1/2}(\mathbb{R})$ if and only if $\hat f$ may be identified with a
function that is locally in $L^2$ in the compliment of the origin, with
\begin{equation}\label{eq:2.5}
\int_{|\xi|\geqslant 1} |\hat f(\xi)|^2|\xi|\, d\xi + \int_{|\xi|\leqslant 1} |\hat f(\xi)|^2|\xi|^2\, d\xi < \infty\text{,}
\end{equation}
and (\ref{eq:2.5}) is bounded above and below by a multiple of (\ref{eq:2.3}).
\end{theorem}
\begin{proof}
a) is of course well-known, and follows from the formal computation
\begin{align*}
\|f\|_{H^{1/2}(\mathbb{R})}^2
&= \int_{-\infty}^\infty \int_{-\infty}^\infty |f(x+t)-f(x)^2\, dx\frac{dt}{t^2} \\
&= \int_{-\infty}^\infty \int_{-\infty}^\infty |\hat f(\xi)|^2 |e^{2\pi i \xi t}-1|^2\, d\xi\frac{dt}{t^2} \\
&= c \int_{-\infty}^\infty |\hat f(\xi)|^2|\xi|\, d\xi
\end{align*}
for $c=\int_{-\infty}^\infty \frac{|e^{2\pi i t}-1|^2}{t^2}\, dt$.
To prove b) we similarly compute
\begin{align*}
\|f\|_{\tilde H^{1/2}(\mathbb{R})}^2
&= \int_{-1}^1 \int_{-\infty}^\infty |f(x+t)-f(x)^2\, dx\frac{dt}{t^2} \\
&= \int_{-\infty}^\infty |\hat f(\xi)|^2 \left(\int_{-1}^1 |e^{2\pi i \xi t}-1|^2\, \frac{dt}{t^2}\right)\, d\xi
\end{align*}
Now
\[
\int_{-1}^1 |e^{2\pi i \xi t} - 1|^2\, \frac{dt}{t^2} = |\xi| \int_{-|\xi|}^{|\xi|} |e^{2\pi i t}-1|^2\, \frac{dt}{t^2}\text{,}
\]
and for $|\xi|\geqslant 1$ the last integral is bounded above and below by a
constant. On the other hand, for $|\xi|\leqslant 1$, the integrand is bounded
above and below by a constant, so the integral is bounded above and below by
the length of the interval. This shows the equivalence of (\ref{eq:2.3}) and
(\ref{eq:2.5}).
The formal computation easily implies that any $f\in \tilde H^{1/2}(\mathbb{R})$ has a
Fourier transform satisfying (\ref{eq:2.5}). To complete the proof we need to
show that any locally $L^2$ function $g(\xi)$ with
\begin{equation}\label{eq:2.6}
\int_{|\xi|\geqslant 1} |g(\xi)|^2|\xi|\, d\xi + \int_{|\xi|\leqslant 1} |g(\xi)|^2 |\xi|^2\, d\xi < \infty
\end{equation}
is in fact the Fourier transform of a function in $\tilde H^{1/2}(\mathbb{R})$. Since the
only problem is near the origin, we may assume that $g$ is supported in
$[-1,1]$. Let $h(\xi) = \xi g(\xi)$. Note that $h\in L^2$ by (\ref{eq:2.6}).
We define a distribution $\tilde g$ associated to $g$ as follows. Note that
$\langle \tilde g,\varphi\rangle = \int h(\xi)\varphi(\xi)\,\frac{d\xi}{\xi}$
is well-defined for any $\varphi\in \mathcal{S}$ with $\varphi(0) = 0$. Choose
$\psi\in \mathcal{S}$ with $\psi(0) = 1$. Then
$\varphi(\xi) = \left(\varphi(\xi)-\varphi(0)\psi(\xi)\right)+\varphi(0)\psi(\xi)$,
with the first summand vanishing at the origin. We will choose to have
$\langle \tilde g,\psi\rangle = 0$, so our definition of $\tilde g$ is
\begin{equation}\label{eq:2.7}
\langle \tilde g,\varphi\rangle = \int h(\xi) \left(\varphi(\xi)-\varphi(0)\psi(\xi)\right)\,\frac{d\xi}{\xi}\text{.}
\end{equation}
It follows that $h(\xi)=\xi \tilde g(\xi)$ in the distribution sense. The
inverse Fourier transform of $\tilde g$ is the function $f$. Note that $f$ has
a derivative in $L^2$ so it is continuous, and the formal computation shows
that $f\in \tilde H^{1/2}(\mathbb{R})$.
\end{proof}
A trivial consequence of the theorem is that the space $\tilde H^{1/2}(\mathbb{R})$ is
strictly larger than $H^{1/2}(\mathbb{R})$. On the other hand,
$L^2(\mathbb{R})\cap \tilde H^{1/2}(\mathbb{R}) = L^2(\mathbb{R})\cap H^{1/2}(\mathbb{R})$.
\begin{example}\label{ex:2}
Let $G$ be the graph with one vertex and two edges of infinite length meeting
at the vertex. We may realize $G$ as the real line with edges $(-\infty,0]$
and $[0,\infty)$, and we write it as $\mathbb{R}^-\cup \mathbb{R}^+$. We see that
(\ref{eq:2.1}) explicitly is
\begin{equation}\label{eq:2.8}
\begin{aligned}
\|f\|_{H^{1/2}(\mathbb{R}^-\cup \mathbb{R}^+)}^2
&= \int_{-\infty}^0 \int_{-\infty}^0 \frac{|f(x)-f(y)|}{|x-y|}\, dxdy \\
&+ \int_0^\infty \int_0^\infty \frac{|f(x)-f(y)|^2}{|x-y|^2}\, dxdy\\
&+ \int_0^\infty \frac{|f(x)-f(-x)|^2}{x}\, dx
\end{aligned}
\end{equation}
\end{example}
\begin{theorem}\label{thm:2.2b}
The spaces $H^{1/2}(\mathbb{R}^-\cup \mathbb{R}^+)$ and $H^{1/2}(\mathbb{R})$ are identical
with equivalent norms.
\end{theorem}
\begin{proof}
This result is essentially contained in \cite{11}, section III.3. Let $x,y$
stand for variables that are always positive. Since
$\int_0^\infty \frac{dy}{(x+y)^2} = \frac 1 x$ we have
\[
\int_0^\infty \frac{|f(x)-f(-x)|}{x}\, dx = \int_0^\infty \int_0^\infty \frac{|f(x)-f(-x)|^2}{(x+y)^2}\, dydx\text{.}
\]
Writing $f(x)-f(-x) = \left(f(x)-f(y)\right)+\left(f(y)-f(-x)\right)$,
we have by the triangle inequality
\begin{align*}
\left(\int_0^\infty \frac{|f(x)-f(-x)|^2}{x}\, dx\right)^{1/2}
&\leqslant \left(\int_0^\infty \int_0^\infty \frac{|f(x)-f(y)|^2}{(x+y)^2}\, dydx\right)^{1/2} \\
&+ \left(\int_0^\infty \int_0^\infty \frac{|f(y)-f(-x)|^2}{|x+y|^2}\, dydx\right)^{1/2} \\
&\leqslant 2\|f\|_{H^{1/2}(\mathbb{R})}
\end{align*}
since $\frac{1}{(x+y)^2}\leqslant \frac{1}{(x-y)^2}$. This yields the bound of
(\ref{eq:2.8}) by a multiple of $\|f\|_{H^{1/2}(\mathbb{R})}$. A similar argument
gives
\begin{align*}
\left(\int_0^\infty\int_0^\infty\frac{|f(y)-f(-x)|^2}{|x+y|^2}\ dydx\right)^{1/2}
&\leqslant \left(\int_0^\infty \int_0^\infty \frac{f(x)-f(y)|^2}{|x+y|^2}\, dydx\right)^{1/2} \\
&+ \left(\int_0^\infty \frac{|f(x)-f(-x)|^2}{x}\, dx\right)^{1/2}
\end{align*}
for the bound in the other direction.
\end{proof}
\begin{example}\label{ex:3}
Let $G$ be the graph $\mathbb{Z}$; in other words the vertices are the integers and
the edges are $[k,k+1]$ for $k\in\mathbb{Z}$ of length $1$. Then (\ref{eq:2.1}) is
explicitly
\begin{equation}\label{eq:2.9}
\|f\|_{H^{1/2}(\mathbb{Z})}^2 = \sum_{k\in\mathbb{Z}} \int_k^{k+1}\int_k^{k+1} \frac{|f(x)-f(y)|^2}{|x-y|^2} \, dxdy + \sum_{k\in\mathbb{Z}} \int_0^1 \frac{|f(k+t)-f(k-t)|}{t}\, dt\text{.}
\end{equation}
\end{example}
\begin{theorem}\label{thm:2.3}
The spaces $H^{1/2}(\mathbb{Z})$ and $\tilde H^{1/2}(\mathbb{R})$ are identical with equivalent
norms.
\end{theorem}
\begin{proof}
The first term on the right side of (\ref{eq:2.9}) is clearly bounded by
$\|f\|_{\tilde H^{1/2}(\mathbb{R})}^2$. For the second term we note that an argument as in
the proof of Theorem \ref{thm:2.2b} gives the estimate
\[
\int_0^1 \frac{|f(k+t)-f(k-t)|^2}{t}\, dt\leqslant c\int_{k-1}^{k+1}\int_{k-1}^{k+1} \frac{|f(x)-f(y)|^2}{|x-y|^2}\, dxdy\text{,}
\]
and summing over $k\in\mathbb{Z}$ we obtain
\[
\sum_{k\in\mathbb{Z}} \int_0^1 \frac{|f(k+t)-f(k-t)|^2}{t}\, dt \leqslant c \iint_{|x-y|\leqslant 2} \frac{|f(x)-f(y)|^2}{|x-y|^2}\, dxdy\text{.}
\]
A straightforward estimate controls the integral over
$1\leqslant |x-y|\leqslant 2$ by a multiple of the integral over
$|x-y|\leqslant 1$, so we have
\[
\|f\|_{H^{1/2}(\mathbb{Z})}^2\leqslant c \|f\|_{\tilde H^{1/2}(\mathbb{R})}^2\text{.}
\]
For the reverse estimate we use an argument in the proof of Theorem
\ref{thm:2.2b} to obtain
\begin{align*}
\int_{k-1}^k \int_k^{k+1} \frac{|f(x)-f(y)|^2}{|x-y|^2}\, dxdy
&\leqslant \int_{k-1}^k\int_{k-1}^k \frac{|f(x)-f(y)|^2}{|x-y|^2}\, dxdy \\
&+ \int_k^{k+1}\int_k^{k+1} \frac{|f(x)-f(y)|^2}{|x-y|^2}\, dxdy \\
&+ \int_0^t \frac{|f(k+t)-f(k-t)|^2}{t}\, dt
\end{align*}
and then sum over $k\in\mathbb{Z}$.
\end{proof}
\begin{example}\label{ex:4}
Let $G$ be the square graph $\mathsf{SQ}_\delta$ with side length $\delta$. So
$\mathsf{SQ}_\delta$ has $4$ vertices that we will identify with the points $(0,0)$,
$(\delta,0)$, $(\delta,\delta)$, $(0,\delta)$ in the plane and $4$ edges of
length $\delta$. Then
\begin{equation}\label{eq:2.10}
\begin{aligned}\
\|f\|_{H^{1/2}(\mathsf{SQ}_\delta)}^2
&= \int_0^\delta\int_0^\delta \frac{|f(x,0)-f(y,0)|^2}{|x-y|^2}\, dxdy + \int_0^\delta\int_0^\delta \frac{|f(\delta,x)-f(\delta,y)|^2}{|x-y|^2}\, dxdy \\
&+ \int_0^\delta\int_0^\delta\frac{|f(x,\delta)-f(y,\delta)|^2}{|x-y|^2}\, dxdy + \int_0^\delta\int_0^\delta\frac{|f(0,x)-f(0,y)|^2}{|x-y|^2}\, dxdy \\
&+ \int_0^\delta |f(x,0)-f(0,x)|^2\, \frac{dx}{x} + \int_0^\delta |f(x,0)-f(\delta,\delta-x)|^2\, \frac{dx}{x} \\
&+ \int_0^\delta |f(x,\delta)-f(\delta,x)|^2\, \frac{dx}{x} + \int_0^\delta |f(0,x)-f(\delta-x,\delta)|^2\, \frac{dx}{x}\text{.}
\end{aligned}
\end{equation}
Although the $H^{1/2}(\mathsf{SQ}_\delta)$ norm does not involve comparisons between
values on opposite edges, it is not difficult to show bounds
\begin{equation}\label{eq:2.11}
\begin{aligned}
\int_0^1 |f(x,0)-f(x,\delta)|^2\, dx &\leqslant c\delta \|f\|_{H^{1/2}(\mathsf{SQ}_\delta)}^2 \\
\int_0^1 |f(0,y)-f(\delta,y)|^2\, dy &\leqslant c\delta \|f\|_{H^{1/2}(\mathsf{SQ}_\delta)}^2\text{.}
\end{aligned}
\end{equation}
\end{example}
\begin{example}\label{ex:5}
Let $G$ be the graph paper graph $\mathsf{GP}_\delta$ with vertices at
$\{(j\delta,k\delta)\}$, $j,k\in\mathbb{Z}$ and horizontal and vertical edges of
length $\delta$ joining $(j\delta,k\delta)$ with $((j+1)\delta,k\delta)$ and
$(j\delta,k\delta)$ with $(j\delta,(k+1)\delta)$. The norm is given by
\begin{equation}\label{eq:2.12}
\begin{aligned}
\|f\|_{H^{1/2}(\mathsf{GP}_\delta)}^2 &= \sum_j \sum_k \int_0^\delta\int_0^\delta \frac{|f(j\delta+x,k\delta)-f(j\delta+y,k\delta)|^2}{|x-y|^2}\, dxdy \\
&+ \sum_j\sum_k\int_0^\delta\int_0^\delta \frac{|f(j\delta,k\delta+x)-f(j\delta,k\delta+y)|^2}{|x-y|^2}\, dxdy \\
&+ \sum_j\sum_k\int_{-\delta}^\delta |f(j\delta+x,k\delta)-f(j\delta,k\delta+x)|^2\, \frac{dx}{|x|} \\
&+ \sum_j\sum_k \int_0^\delta |f(j\delta+x,k\delta)-f(j\delta-x,k\delta)|^2\, \frac{dx}{x} \\
&+ \sum_j\sum_k \int_0^\delta |f(j\delta,k\delta+x)-f(j\delta,k\delta-x)|^2\, \frac{dx}{x}\text{.}
\end{aligned}
\end{equation}
Of course we could get an equivalent norm by deleting the last two sums in
(\ref{eq:2.12}), as they are controlled by the third sum. We may regard
$\mathsf{GP}_\delta$ as a countable union of square graphs $\mathsf{SQ}_\delta$,and it is
easily seen that $f\in H^{1/2}(\mathsf{GP}_\delta)$ if and only if the restriction of
$f$ to each of the square graphs is in $H^{1/2}(\mathsf{SQ}_\delta)$ with the sum of
the squares of the norms $\|f\|_{H^{1/2}(\mathsf{SQ}_\delta)}^2$ finite, and this
gives an equivalent norm.
\end{example}
\section{Traces of functions of finite energy}\label{sec:3}
Consider the homogeneous Sobolev space $H^1(\mathbb{R}^2)$ of functions with finite
energy
\begin{equation}\label{eq:3.1}
\|F\|_{H^1(\mathbb{R})}^2 = \int_{\mathbb{R}^2} |\nabla F(x,y)|^2\, dxdy\text{.}
\end{equation}
These form a Hilbert space modulo constants. Functions of finite energy do not
have to be continuous, as the example $F(x,y)=\log|\log(x^2+y^2)|$ (multiplied
by an appropriate cutoff function) shows. However, it is well-known that these
functions have well-defined traces on straight lines that are in
$H^{1/2}(\mathbb{R})$, and $H^{1/2}(\mathbb{R})$ is the exact space of traces. Since the
usual treatment of traces involves inhomogeneous Sobolev spaces we give the
proof for the convenience of the reader. We omit the routine step of actually
defining the traces and just prove the norm estimates.
\begin{theorem}\label{thm:3.1}
The trace map $T:H^1(\mathbb{R})\to H^{1/2}(\mathbb{R})$ given formally by $TF(x)=F(x,0)$ is
continuous,
\begin{equation}\label{eq:3.2}
\|T F\|_{H^{1/2}(\mathbb{R})} \leqslant c \|F\|_{H^1(\mathbb{R}^2)}\text{.}
\end{equation}
Moreover there exists a continuous extension map
$E:H^{1/2}(\mathbb{R}) \to H^1(\mathbb{R}^2)$ with $TEf=f$ and
\begin{equation}\label{eq:3.3}
\|E f\|_{H^1(\mathbb{R}^2)} \leqslant c \|f\|_{H^{1/2}(\mathbb{R})}
\end{equation}
\end{theorem}
\begin{proof}
We work on the Fourier transform side, where
\begin{align}\label{eq:3.4}
\|F\|_{H^1(\mathbb{R}^2)}^2
&= \int_{\mathbb{R}^2} (\xi^2+\eta^2)|\hat F(\xi,\eta)|^2\, d\xi d\eta \qquad \text{and } \\ \label{eq:3.5}
(T f)^\wedge(\xi) &= \int_{-\infty}^\infty \hat F(\xi,\eta)\, d\eta\text{.}
\end{align}
By Theorem \ref{thm:2.2a} we have
\[
\|T f\|_{H^{1/2}(\mathbb{R})}^2 = \int_{-\infty}^\infty \left| \int_{-\infty}^\infty \hat F(\xi,\eta)\, d\eta\right|^2\, |\xi|\, d\xi\text{.}
\]
By Cauchy-Schwarz we have
\begin{align*}
\left|\int_{-\infty}^\infty \hat F(\xi,\eta)\ d\eta\right|^2
&\leqslant \left(\int_{-\infty}^\infty (\xi^2+\eta^2) |\hat F(\xi,\eta)|^2\, d\eta\right)\left(\int_{-\infty}^\infty \frac{1}{\xi^2+\eta^2}\, d\eta\right) \\
&= \frac{\pi}{|\xi|} \left(\int_{-\infty}^\infty (\xi^2+\eta^2) |\hat F(\xi,\eta)|^2\, d\eta\right) \qquad\text{so} \\
\|T f\|_{H^{1/2}(\mathbb{R})}^2
&\leqslant \pi \int_{-\infty}^\infty \int_{-\infty}^\infty (\xi^2+\eta^2) |\hat F(\xi,\eta)|^2\, d\eta \\
&= \pi \|F\|_{H^1(\mathbb{R}^2)}^2
\end{align*}
proving (\ref{eq:3.2}).
Conversely, given $f\in H^{1/2}(\mathbb{R})$ define $E f=F$ by the Poisson integral
\begin{equation}\label{eq:3.6}
F(x,y) = \frac{|y|}{\pi} \int \frac{f(x-t)}{t^2+y^2}\, dt
\end{equation}
so that $T F=f$. Then
\begin{equation}\label{eq:3.7}
\hat F(\xi,\eta) = \frac{1}{\pi} \frac{\hat f(\xi) |\xi|}{\eta^2+|\xi|^2} \text{.}
\end{equation}
By (\ref{eq:3.4}) we have
\begin{align*}
\|F\|_{H^1(\mathbb{R}^2)}^2
&= \frac{1}{\pi^2} \int_{\mathbb{R}^2} \frac{|\hat f(\xi)|^2 |\xi|^2}{\eta^2+\xi^2}\, d\xi d\eta \\
&= \frac{1}{\pi} \int_{-\infty}^\infty |\hat f(\xi)|^2 |\xi|\, d\xi
\end{align*}
so we obtain (\ref{eq:3.3}) by Theorem \ref{thm:2.2a}.
\end{proof}
Note that we define the extension $E f$ to be harmonic in each half-plane
$y>0$ and $y<0$. Since harmonic functions minimize energy, our extension
achieves the minimum $H^1(\mathbb{R}^2)$ norm.
There is a virtually identical trace theorem for functions of finite energy
in the half-plane, say $y>0$ denoted $\mathbb{R}_+^2$. To see this we only have to
observe that an even reflection
\begin{equation}\label{eq:3.8}
R F(x,y) = F(x,-y) \qquad \text{for $y<0$}
\end{equation}
maps $H^1(\mathbb{R}_+^2)$ continuously to $H^1(\mathbb{R}^2)$.
\begin{theorem}\label{thm:3.2}
The trace map $T:H^1(\mathbb{R}_+^2)\to H^{1/2}(\mathbb{R})$ given formally by
$T F(x)=F(x,0)$ is well-defined and bounded, and there exists a bounded
extension map $E:H^{1/2}(\mathbb{R}) \to H^1(\mathbb{R}_+^2)$ with $T E f=f$, and the
analogues of (\ref{eq:3.2}) and (\ref{eq:3.3}) hold.
\end{theorem}
If we combine this with the well-known observation that energy is conformally
invariant in the plane (not true in other dimensions, however), we obtain a
powerful tool for obtaining trace theorems for other domains: find a conformal
map between the domain and the half-space $\mathbb{R}_+^2$, and transfer the
$H^{1/2}(\mathbb{R})$ norm from the boundary of $\mathbb{R}_+^2$ to the boundary of the
domain, assuming the conformal map extends continuously to the boundary.
A simple example is the strip $S=\{(x,y) : 0<y<\pi\}$. In complex notation
$\varphi(z) = \log z$ is the conformal map from $\mathbb{R}_+^2$ to $S$, with
$\psi(z) = e^z$ its inverse. So $F\in H^1(S)$ if and only if
$F\circ\varphi\in H^1(\mathbb{R}_+^2)$ with equal norms. Then
$f(t)=F(\varphi(t))\in H^{1/2}(\mathbb{R})$. Using Theorem \ref{thm:2.2a} this means
\begin{equation}\label{eq:3.9}
\begin{aligned}
\int_0^\infty \int_0^\infty \frac{|F(\log t)-F(\log s)|^2}{|t-s|^2}\, dtds
&+ \int_0^\infty\int_0^\infty \frac{|F(\log t+i\pi)-F(\log s+i\pi)|^2}{|t-s|^2}\, dtds \\
&+ \int_0^\infty |F(\log t)-F(\log t+i\pi)|^2\, \frac{dt}{t} \\
&\leqslant c \|F\|_{H^1(S)}^2 \text{.}
\end{aligned}
\end{equation}
The change of variable $x=\log t$, $y=\log s$ transforms the let hand side of
(\ref{eq:3.9}) into
\begin{equation}\label{eq:3.10}
\begin{aligned}
\int_{-\infty}^\infty \int_{-\infty}^\infty |F(x)-F(y)|^2 \frac{e^x e^y}{|e^x-e^y|^2}\, dxdy
&+ \int_{-\infty}^\infty\int_{-\infty}^\infty |F(x+y)-F(x+i\pi)|^2 \frac{e^x e^y}{|e^x-e^y|}\, dxdy \\
&+ \int_{-\infty}^\infty |F(x)-F(x+i\pi)|^2\, dx\text{.}
\end{aligned}
\end{equation}
To simplify the notation we split the trace of $F$ on the boundary of $S$ into
two pieces, $T_0 F(x) = F(x)$ and $T_1 F(x) = F(x+i\pi)$, so that $T_0 F$ and
$T_1 F$ are functions on $\mathbb{R}$.
\begin{theorem}\label{thm:3.3}
If $F\in H^1(S)$ then $T_0 F$ and $T_1 F$ are in $\tilde H^{1/2}(\mathbb{R})$ and
$T_0 F-T_1 F\in L^2(\mathbb{R})$, with
\begin{equation}\label{eq:3.11}
\|T_0 F\|_{\tilde H^{1/2}(\mathbb{R})}^2 + \|T_1 F\|_{\tilde H^{1/2}(\mathbb{R})}^2 + \|T_0 F - T_1 F\|_2^2 \leqslant c \|F\|_{H^1(S)}\text{.}
\end{equation}
Conversely, given $f_0$ and $f_1$ in $\tilde H^{1/2}(\mathbb{R})$ with
$f_0-f_1\in L^2(\mathbb{R})$, there exists $F=E(f_0,f_1)$ with $T_0 F=f_0$,
$T_1 F = f_1$, $F\in H^1(S)$ with the reverse estimate of (\ref{eq:3.11})
holding.
\end{theorem}
\begin{proof}
In view of (\ref{eq:3.10}) it suffices to show that
\begin{equation}\label{eq:3.12}
\int_{-\infty}^\infty\int_{-\infty}^\infty |F(x)-F(y)|^2 \frac{e^x e^y}{|e^xe^y|}{|e^x-e^y|^2}\, dxdy
\end{equation}
is bounded above and below by a constant multiple of
\begin{equation}\label{eq:3.13}
\iint_{|x-y|\leqslant 1} \frac{|F(x)-F(y)|^2}{|x-y|^2}\, dxdy = \|F\|_{\tilde H^{1/2}(\mathbb{R})}\text{.}
\end{equation}
Note that we may rewrite (\ref{eq:3.12}) as
\begin{equation}\label{eq:3.14}
\frac 1 4 \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{|F(x)-F(y)|^2}{\left|\sinh\left(\frac{x-y}{2}\right)\right|^2}\, dxdy\text{.}
\end{equation}
It is clear that (\ref{eq:3.14}) is bounded below by a multiple of
(\ref{eq:3.13}), and for the upper bound we need
$\iint_{|x-y|\geqslant 1} \frac{|f(x)-f(y)|^2}{\left|\sinh\left((x-y)/2\right)\right|^2}$
bounded above by a multiple of (\ref{eq:3.13}), but this is a routine
exercise because of the exponential decay of
$\left|\sinh\left(\frac{x-y}{2}\right)\right|^{-2}$.
\end{proof}
It might seem perplexing that the trace space on each of the lines is larger
than $H^{1/2}(\mathbb{R})$, since in particular this implies that there are functions
in $H^1(S)$ that do not extend to $H^1(\mathbb{R}^2)$. However, it is easy to give an
example of such a function: just take $F(x,y) = g(x)$ where $g(0) = 0$ for
$x\leqslant 0$ and $g(x) = 1$ for $x\geqslant 1$ and $g$ is smooth in
$[0,1]$. Then $\nabla F$ has compact support in $S$ so $F\in H^{1/2}(S)$, but
$g\notin H^{1/2}(\mathbb{R})$.
Another simple example is the first quadrant
$Q=\{(x,y) : x>0\text{ and }y>0\}$. Then $\varphi(z) = \mathsf{SQ}rt z$ is the
conformal map of $\mathbb{R}_+^2$ to $Q$, with inverse $\psi(z)=z^2$. Again it is
convenient to split the trace into two parts mapping to functions on $\mathbb{R}_+$,
namely $T_0F(x) = F(x,0)$ and $T_1 F(x) = F(0,x)$. Since $F\in H^1(Q)$ if and
only if $F\circ\varphi\in H^1(\mathbb{R}_+^2)$, again by Theorem \ref{thm:2.2a} we
have the expression
\begin{equation}\label{eq:3.15}
\begin{aligned}
\int_0^\infty\int_0^\infty\frac{|T_0 F(\mathsf{SQ}rt t)-T_0 F(\mathsf{SQ}rt s)|^2}{|t-s|^2}\, dsdt
&+ \int_0^\infty\int_0^\infty \frac{|T_1 F(\mathsf{SQ}rt t)-T_1 F(\mathsf{SQ}rt s)|^2}{|t-s|^2}\, dsdt \\
&+ \int_0^\infty |T_0 F(\mathsf{SQ}rt t)-T_1 F(\mathsf{SQ}rt t)|^2\, \frac{dt}{t}
\end{aligned}
\end{equation}
for the trace norm. With the substitutions $t=x^2$, $s=y^2$ this becomes
\begin{equation}\label{eq:3.16}
\begin{aligned}
4 \int_0^\infty\int_0^\infty \frac{|T_0 F(x)-T_0 F(y)|^2}{|x-y|^2}\, \frac{xy}{|x+y|^2}\, dxdy
&+ 4\int_0^\infty\int_0^\infty\frac{|T_1 F(x)-T_1 F(y)|^2}{|x-y|^2}\, \frac{xy}{|x+y|^2}\, dxdy \\
&+ 2\int_0^\infty |T_1 F(x)-T_1 F(x)|^2\, \frac{dx}{x}\text{.}
\end{aligned}
\end{equation}
It is easy to see that if $f_0,f_1\in H^{1/2}(\mathbb{R}_+)$ and
\begin{equation}\label{eq:3.17}
\int_0^\infty |f_0(x)-f_1(x)|^2\, \frac{dx}{x} < \infty
\end{equation}
then there exists $F\in H^1(Q)$ with $T_0 F=f_0$ and $T_1 F=f_1$, because
$\frac{xy}{|x+y|^2}$ is bounded. In other words, the function
\[
f(x) = \begin{cases}
f_0(x) & \text{if $x>0$} \\
f_1(x) & \text{if $x<0$}
\end{cases}
\]
is in $H^{1/2}(\mathbb{R})$, and $\|F\|_{H^1(Q)} \leqslant c \|f\|_{H^{1/2}(\mathbb{R})}$.
It is possible to show the converse statement as well, but this involves some
technicalities since $\frac{xy}{|x+y|^2}$ is not bounded below. It is easier
to observe that $F\in H^1(Q)$ may be extended by even reflection across the
axes to a function in $H^1(\mathbb{R}^2)$, so the even reflections of $T_0 F$ and
$T_1 F$ must be in $H^{1/2}(\mathbb{R}^2)$, so $T_0 F$ and $T_1 F$ must be in
$H^{1/2}(\mathbb{R}_+)$, and we already have (\ref{eq:3.17}) for $f_0=T_0 F$,
$f_1=T_1 F$. A direct proof of (\ref{eq:3.17}) is possible but involves
technicalities.
Another simple example is the unit disk $D$, with
$\varphi(z) = \frac{1-z}{1+z}$ the conformal mapping of $\mathbb{R}_+^2$ to $D$.
The trace space of $H^1(D)$ is $H^{1/2}(C)$ for $C$ the unit circle with norm
\begin{equation}\label{eq:3.18}
\|f\|_{H^{1/2}(C)}^2 = \int_0^{2\pi}\int_0^{2\pi} \frac{|f(e^{i\theta})-f(e^{i\theta'})|^2}{4\left|\sin \frac 1 2(\theta-\theta')\right|^2}\, d\theta d\theta' \text{.}
\end{equation}
Of course $2\left|\sin \frac 1 2(\theta-\theta')\right|$ is exactly the
chordal distance $|e^{i\theta}-e^{i\theta'}|$. It is interesting to observe
that exactly the same trace space arises from the exterior of the circle
$\{|z|>1\}$, as $z\mapsto 1/\bar z$ is an anticonformal map of $D$ to this
exterior domain that agrees with $\varphi(z)$ on the circle. Similarly, for a
circle $C_r$ of radius $r$, the analog of (\ref{eq:3.18}) is
\begin{equation}\label{eq:3.19}
\|f\|_{H^{1/2}(C_r)}^2 = \int_0^{2\pi}\int_0^{2\pi} \frac{|f(re^{i\theta}-f(re^{i\theta'})|^2}{4\left|r\sin \frac 1 2(\theta-\theta')\right|^2}\, rd\theta r d\theta'\text{.}
\end{equation}
Of course it is not necessary to use a conformal map $\varphi$. A Lipschitz
map or even a quasiconformal map changes the $H^1$ norm by a bounded amount.
So for $\mathsf{SQ}_\delta^\circ$, the interior of the square $\mathsf{SQ}_\delta$, the trace
space of $H^1(\mathsf{SQ}_\delta^\circ)$ is $H^{1/2}(\mathsf{SQ}_\delta)$ with norm given by
(\ref{eq:2.10}), since one can ``square the circle'' with a Lipschitz map.
Next we consider traces on infinite collections of lines. First consider the
horizontal line collection $\mathsf{HLC}=\{(x,n\pi) : x\in \mathbb{R},n\in\mathbb{Z}\}$. For a
function $F$ in $H^1(\mathbb{R}^2)$ define the traces $T_n F(x)=F(x,\pi n)$.
\begin{theorem}\label{thm:3.4}
A set of functions $\{f_n\}$ on $\mathbb{R}$ are the traces $f_n=T_n F$ for
$F\in H^1(\mathbb{R}^2)$ if and only if $f_n\in \tilde H^{1/2}(\mathbb{R})$ and
$f_n-f_{n+1}\in L^2(\mathbb{R})$ with
\begin{equation}\label{eq:3.20}
\sum_n \|f_n\|_{\tilde H^{1/2}(\mathbb{R})}^2 + \sum_n \|f_n-f_{n+1}\|_{L^2(\mathbb{R})}^2 < \infty\text{,}
\end{equation}
and the corresponding norm equivalence holds.
\end{theorem}
\begin{proof}
Basically we just have to apply Theorem \ref{thm:3.3} to each of the strips
$\{n\pi < y < (n+1)\pi\}$ and sum (\ref{eq:3.11}) over all the strips. To do
this we just have to observe that a function belongs to $H^1(\mathbb{R}^2)$ if and
only if its restriction to each strip is in $H^1$ of that strip, the traces
agree on neighboring strips, and the sum of the energies is finite.
\end{proof}
There is something a bit unsettling about this result. We know that
$f_n=T_n F$ actually belongs to the smaller space $H^{1/2}(\mathbb{R})$ for
$F\in H^1(\mathbb{R}^2)$, yet this space plays no role in the characterization
(\ref{eq:3.20}). It is an indirect consequence of the theorem that if
$\{f_n\}$ is a family of functions satisfying (\ref{eq:3.20}), then each $f_n$
is indeed in $H^{1/2}(\mathbb{R})$. It should be possible to prove this directly, but
again this seems rather technical. Note that we only get a uniform bound for
$\|f_n\|_{H^{1/2}(\mathbb{R})}^2$. The following example shows that we can't do too
much better than this (most likely $\|f_n\|_{H^{1/2}(\mathbb{R})}^2 = o(1)$).
Consider the function $F(x,y)=(1+x^2+y^2)^{-\alpha}$ for $\alpha>0$. A direct
computation shows that
$|\nabla F(x,y)|\leqslant 2\alpha (1+x^2+x^y)^{-\alpha-\frac 1 2}$, so
$F\in H^1(\mathbb{R}^2)$. Now
\[
T_n F(x) = (1+\pi^2 n^2 + x^2)^{-\alpha}
= (1+\pi^2 n^2)^{-\alpha} g\left(\frac{x}{\mathsf{SQ}rt{1+\pi^2 n^2}}\right)
\]
for $g(x) = (1+x^2)^{-\alpha}$. It is easy to see that $g\in H^{1/2}(\mathbb{R})$,
so by dilation invariance of the $H^{1/2}(\mathbb{R})$ norm we see that
$\|T_n F\|_{H^{1/2}(\mathbb{R})}^2 = c(1+\pi^2 n^2)^{-2\alpha}$ so
$\sum \|T_n F\|_{H^{1/2}(\mathbb{R})}^2 = \infty$ for $\alpha\leqslant \frac 1 4$.
Next we consider the trace on the graph paper graph $\mathsf{GP}_\delta$.
\begin{theorem}\label{thm:3.5}
The trace space of $H^1(\mathbb{R}^2)$ on $\mathsf{GP}_\delta$ is exactly $H^1(\mathsf{GP}_\delta)$
with norm given by (\ref{eq:2.12}).
\end{theorem}
\begin{proof}
We simply use the trace theorem of $H^1$ on each $\delta$-square that makes
up $\mathsf{GP}_\delta$ and add.
\end{proof}
In place of square graph paper we could consider triangular graph paper
$\mathsf{TG}p_\delta$ consisting of the tiling of the plane by equilateral triangles
of side length $\delta$. Then the analog of Theorem \ref{thm:3.5} holds with
essentially the same proof.
\section{The graph paper trace characterization}\label{sec:4}
In this section we fix an integer $m\geqslant 2$, and consider the sequence of
graph paper graphs $\mathsf{GP}_{m^n}$, thought of as the unions of the edges, or
equivalently the countable union of horizontal and vertical lines in the
plane with $m^n$ separation. These are nested subsets of the plane,
$\mathsf{GP}_{m^n} \subset \mathsf{GP}_{m^{n'}}$ if $n'<n$ and we are interested in the limit
as $n\to -\infty$, so the graph paper gets increasingly finer.
We let $T_n$ denote the trace map from functions defined on $\mathbb{R}^2$ to
$\mathsf{GP}_{m^n}$. By the nesting property we may also consider $T_n$ to be defined
on functions on $\mathsf{GP}_{m^{n'}}$ with $n'<n$. Our goal is to characterize
functions in $H^1(\mathbb{R}^2)$ by their traces $T_n F$.
\begin{theorem}\label{thm:4.1}
a) Let $F\in H^1(\mathbb{R}^2)$. Then $T_n F\in H^{1/2}(\mathsf{GP}_{m^n})$ for all $n$ with
uniformly bounded norms, and
\begin{equation}\label{eq:4.1}
\sup_{n\in\mathbb{Z}} \|T_n F\|_{H^{1/2}(\mathsf{GP}_{m^n})}^2 \leqslant c \|F\|_{H^1(\mathbb{R})}^2
\end{equation}
b) Let $f_n\in H^{1/2}(\mathsf{GP}_{m^n})$ be a sequence of functions with uniformly
bounded norms satisfying the consistency condition $T_n f_{n'} = f_n$ if
$n'<n$. Then there exists $F\in H^1(\mathbb{R}^2)$ such that $T_n F=f_n$ and
\begin{equation}\label{eq:4.2}
\|F\|_{H^1(\mathbb{R}^2)}^2 \leqslant c \sup_{n\in\mathbb{Z}} \|f_n\|_{H^{1/2}(\mathsf{GP}_{m^n})}^2
\end{equation}
\end{theorem}
\begin{proof}
Part a) is an immediate consequence of Theorem \ref{thm:3.5}. To prove b) we
define $F_n$ to be the harmonic extension of $f_n$ into each of the graph
paper spaces. Since these harmonic extensions minimize energy, we have
$F_n\in H^1(\mathbb{R}^2)$ and
\[
\|F_n\|_{H^1(\mathbb{R}^2)} \leqslant c \|f_n\|_{H^{1/2}(\mathsf{GP}_{m^n})}\text{,}
\]
again by Theorem \ref{thm:3.5}. Thus there exists a subsequence
$n_j\to -\infty$ such that $F_{n_j}$ converges in the weak topology of
$H^1(\mathbb{R}^2)$ to a function $F$ satisfying (\ref{eq:4.2}). It remains to show
that the weak convergence respects traces, so that $T_n F_{n_j} = f_n$ for all
$n_j$ implies $T_n F = f_n$.
But the equality of traces on $\mathsf{GP}_{m^n}$ is the same as equality of traces on
each of the lines that make up $\mathsf{GP}_{m^n}$; and since all lines are
essentially equivalent, it suffices to show that $F_{n_j}(x,0)$ converges
weakly in $H^{1/2}(\mathbb{R})$ to $F(x,0)$. This is most easily seen on the Fourier
transform side, where both $H^1(\mathbb{R}^2)$ and $H^{1/2}(\mathbb{R})$ are just weighted
$L^2$ spaces.
The weak convergence $F_{n_j}\to F$ in $H^1(\mathbb{R}^2)$ says
\begin{equation}\label{eq:4.3}
\iint \hat F_{n_j}(\xi,\eta) G(\xi,\eta)(\xi^2+\eta^2)\, d\xi d\eta \to \iint \hat F(\xi,\eta) G(\xi,\eta)(\xi^2+\eta^2)\, d\xi d\eta
\end{equation}
for every $G\in L^2\left((\xi^2+\eta^2)d\xi d\eta\right)$. The weak
convergence $F_{n_j}(x,0)\to F(x,0)$ requires that we show
\begin{equation}\label{eq:4.4}
\int\left(\int \hat F_{n_j}(\xi,\eta)\, d\eta\right) H(\xi)|\xi|\, d\xi \to \int\left(\int \hat F(\xi,\eta)\, d\eta\right) H(\xi) |\xi|\, d\xi
\end{equation}
for every $H\in L^2\left(|\xi|\, d\xi\right)$. So given $H$, choose
\begin{equation}\label{eq:4.5}
G(\xi,\eta) = \frac{|\xi| H(\xi)}{\xi^2+\eta^2} \text{.}
\end{equation}
Since
\begin{align*}
\iint |G(\xi,\eta)|^2(\xi^2+\eta^2)\, d\xi d\eta
&= \int\left(\int \frac{|\xi|^2}{\xi^2+\eta^2}\, d\eta\right)|H(\xi)|^2\, d\xi \\
&= \pi \int |H(\xi)|^2 |\xi|\, d\xi
\end{align*}
we may use the choice of $G$ in (\ref{eq:4.2}). But then (\ref{eq:4.3}) and
(\ref{eq:4.4}) are identical.
\end{proof}
This result localizes in several ways. For example, if $F\in H^1(\mathbb{R}^2)$ and
we wish to estimate the amount of energy that is contained in an open set
$\Omega$, that is
\begin{equation}\label{eq:4.6}
\int_\Omega |\nabla F|^2\, dxdy\text{,}
\end{equation}
we just have to take the sum of the terms in (\ref{eq:2.12}) that correspond
to edges contained in $\Omega$. Denote this sum by
$\|T_n F\|_{H^{1/2}(\Omega\cap \mathsf{GP}_{m^n})}^2$. Then (\ref{eq:4.6}) is bounded
above and below by a constant times
\begin{equation}\label{eq:4.7}
\sup_{n\in\mathbb{Z}} \|T_n F\|_{H^{1/2}(\Omega\cap\mathsf{GP}_{m^n})} \text{.}
\end{equation}
We obtain the same norm equivalence if we only assume $F\in H^1(\Omega)$,
meaning (\ref{eq:4.6}) is finite. (Note that this does not say anything about
the trace of $F$ on the boundary of $\Omega$.) Also, we may start by assuming
that $F\in H_\text{loc}^1(\mathbb{R}^2)$, meaning that (\ref{eq:4.6}) is finite
whenever $\Omega$ is bounded, and obtain the norm equivalence of
(\ref{eq:4.6}) and (\ref{eq:4.7}).
The same result will also hold if we replace $\mathsf{GP}_{m^n}$ by the triangular
$\mathsf{TG}p_{m^n}$.
It is clear that we may replace the $\sup$ in (\ref{eq:4.2}) and
(\ref{eq:4.7}) by the $\limsup$ as $n\to-\infty$. It is not clear that a limit
has to exist, however, since we only have estimates above and below, rather
than identity, for our norms.
We can also characterize functions of finite energy by their traces on pencils
of parallel lines of equal separation; in other words, the horizontal lines in
$\mathsf{GP}_{m^n}$. Denote this by $\mathsf{PP}_{m^n}$. We will use Theorem \ref{thm:3.4},
but the norms defined by (\ref{eq:3.20}) are not dilation invariant. That
means we want to define $\tilde H^{1/2}(\mathsf{PP}_{m^n})$ by the finiteness of
\begin{equation}\label{eq:4.8}
\begin{aligned}
\sum_{k\in\mathbb{Z}} \iint_{|x-y|\leqslant m^n} &\frac{|f(x,k m^n) - f(y,k m^n)|^2}{|x-y|^2}\, dxdy \\
&+ \sum_{k\in\mathbb{Z}} m^{-n} \int_{-\infty}^\infty |f(x,(k+1) m^n)-f(x,k m^n)|^2\, dx\text{,}
\end{aligned}
\end{equation}
and we define this to be $\|f\|_{\tilde H^{1/2}(\mathsf{PP}_{m^n})}^2$. Then the analog of
Theorem \ref{thm:4.1} holds with $T_n F$ equal to the trace on $\mathsf{PP}_{m^n}$ and
$H^{1/2}(\mathsf{GP}_{m^n})$ replaced by $\tilde H^{1/2}(\mathsf{PP}_{m^n})$. The proof is
essentially the same, using the scaled version of Theorem \ref{thm:3.4} with
(\ref{eq:4.8}) in place of (\ref{eq:3.20}).
\section{Fractals}\label{sec:5}
The Sierpinski gasket ($\mathsf{SG}$) is the self-similar fractal defined by the
identity
\begin{equation}\label{eq:5.1}
\mathsf{SG} = \bigcup_{i=0}^2 \Phi_i(\mathsf{SG})
\end{equation}
where $\Phi_i$ are the homothety maps of the plane
$\Phi_i(x) = \frac 1 2 x + \frac 1 2 q_i$ and $\{q_0,q_1,q_2\}$ are the
vertices of an equilateral triangle with side length $1$. $\mathsf{SG}$ is the unique
nonempty compact subset of the plane satisfying (\ref{eq:5.1}). The mappings
$\{\Phi_i\}$ comprise what is called an \emph{iterated function system}, and
the iterates of the mappings are denoted
$\Phi_w = \Phi_{w_1}\circ\cdots\circ\Phi_{w_m}$ where $w=(w_1,\dotsc,w_m)$ is
a word of length $|w|=m$ and each $w_j=0$, $1$, or $2$. Then by iterating
(\ref{eq:5.1}) we obtain
\begin{equation}\label{eq:5.2}
\mathsf{SG}=\bigcup_{|w|=m} \Phi_w(\mathsf{SG})
\end{equation}
expressing $\mathsf{SG}$ as a union of $3^m$ miniature gaskets (called
\emph{$m$-cells}) that are similar to $\mathsf{SG}$ with similarity ratio $2^{-m}$.
Note that $\mathsf{SG}$ has the \emph{post-critically finite} (PCF) property that
distinct $m$-cells can intersect only at the vertices $\Phi_w q_i$. For this
reason we refer to $\{q_i\}$ as the \emph{boundary} of $\mathsf{SG}$, and
$\{\Phi_w q_i\}$ as the boundary of the $m$-cell $\Phi_m(\mathsf{SG})$, although these
are not boundaries in the topological sense.
We may approximate $\mathsf{SG}$ by the metric graphs $\mathsf{SG}_m=\mathsf{SG}\cap \mathsf{TG}_{2^{-m}}$. So
the vertices are $\{\Phi_w q_i\}$, for $|w|=m$ and $i=0,1,2$, the edges are
$\{\Phi_w e_{i j}\}$ for $|w|=m$ and $e_{i j}$ is the edge of the original
triangle joining $q_i$ and $q_j$, and $\Phi_w e_{ij}$ has length $2^{-m}$. Let
\begin{equation}\label{eq:5.3}
E_m(f) = \sum_{i\ne j} \sum_{|w|=m} |f(\Phi_w q_i) - f(\Phi_w q_j)|^2
\end{equation}
denote the unrenormalized graph energy on $\mathsf{SG}_m$. Kigami (see \cite{8,12})
defines an energy on $\mathsf{SG}$ by
\begin{equation}\label{eq:5.4}
\mathcal{E}(f) = \lim_{m\to \infty} \left(\frac 5 3\right)^m E_m (f)\text{.}
\end{equation}
The renormalization factor $(5/3)^m$ may be explained as follows: the
sequence $(5/3)^m E_m(f)$ is always nondecreasing, and there exists a
$3$-dimensional space of harmonic functions for which it is constant. We can
then define $\dom \mathcal{E}$, the space of functions of finite energy, as those
functions for which (\ref{eq:5.4}) is finite. This is a space of continuous
functions on $\mathsf{SG}$ that forms an infinite dimensional Hilbert space (after
modding out by the constants) with norm $\mathcal{E}(f)^{1/2}$. This energy satisfies
the self-similar identity
\begin{equation}\label{eq:5.5}
\mathcal{E}(f) = \sum_{i=0}^2 \left(\frac 5 3\right) \mathcal{E}(f\circ \Phi_i)
\end{equation}
and satisfies the axioms for a local regular Dirichlet form (\cite{4}). Up to
a constant multiple it is the only Dirichlet form with those properties. It is
also symmetric with respect to the $D_3$ symmetry group of the triangle. This
energy forms the basic building block for a whole theory of analysis on $\mathsf{SG}$,
including a theory of Laplacians. We will not be using this wider theory here,
but direct the curious reader to \cite{8,12} for details.
Since the functions in $\dom\mathcal{E}$ are continuous, there is no problem defining
traces $T_m$ on $\mathsf{SG}_m$. The problem of characterizing the trace space
$T_n(\mathsf{SG})$ on the boundary of the triangle has been solved by Jonsson
\cite{6,7} (see \cite{5} for a different proof) in terms of Sobolev spaces of
order $\beta$, with $\beta=\frac 1 2+\frac{\log 5/3}{\log 4}$. Note that
$\frac 1 2<\beta<1$. For any metric graph $G$ we define $H^\beta(G)$ (for any
$\beta$ in the above range) to be the space of continuous functions such that
\begin{equation}\label{eq:5.6}
\|F\|_{H^\beta(G)}^2 = \sum_{e\in E}\int_0^{L_e}\int_0^{L_e} \frac{|F(e(x))-F(e(y))|^2}{|x-y|^{1+2\beta}}\, dxdy
\end{equation}
is finite. Note that in contrast to (\ref{eq:2.1}), there is no term comparing
values on intersecting edges, since the continuity condition takes care of the
comparison (this idea is also used in \cite{11}). We then have the following
result analogous to Theorem \ref{thm:3.1}.
\begin{proposition}[\cite{5,6,7}]\label{prop:5.1}
The trace map $T_0$ is continuous from $\dom\mathcal{E}$ to $H^\beta(\mathsf{SG}_0)$ with
$\beta=\frac 1 2+\frac{\log 5/3}{\log 4}$ with
\begin{equation}\label{eq:5.7}
\|T_0 F\|_{H^\beta(\mathsf{SG}_0}^2 \leqslant c \mathcal{E}(F)\text{.}
\end{equation}
Moreover, there exists a continuous linear extension map
$E_0:H^\beta(\mathsf{SG}_0) \to \dom\mathcal{E}$ with $T_0 E_0 f = f$ and
\begin{equation}\label{eq:5.8}
\mathcal{E}(E_0 f) \leqslant c \|f\|_{H^\beta(\mathsf{SG}_0)}^2\text{.}
\end{equation}
\end{proposition}
We note that \cite{5,6,7} use a slightly different, but equivalent norm for
$H^\beta(\mathsf{SG}_0)$.
Next we need to obtain the analogous statement for the trace map $T_m$ to
$\mathsf{SG}_m$. We note that energy is additive for continuous functions, and in view
of the self-similarity (\ref{eq:5.5}) iterated,
\begin{equation}\label{eq:5.9}
\mathcal{E}(F)=\sum_{|w|=m} \left(\frac 5 3\right)^m \mathcal{E}(F\circ \Phi_w)\text{,}
\end{equation}
and if we apply (\ref{eq:5.8}) to $F\circ\Phi_w$ we have
\begin{equation}\label{eq:5.10}
\sum_{|w|=m}\left(\frac 5 3\right)^m \|T_0 F\circ \Phi_w\|_{H^\beta(\mathsf{SG}_0)}^2 \leqslant c \sum_{|w|=m} \left(\frac 5 3\right)^m \mathcal{E}(F\circ \Phi_w) = c \mathcal{E}(F)
\end{equation}
by (\ref{eq:5.9}). Now we observe that $\mathsf{SG}_m=\bigcup_{|w|=m} \Phi_w(\mathsf{SG}_0)$,
and this is a disjoint union of edges, since each edge is just a side of a
triangle $\Phi_w(\mathsf{SG}_0)$ for some $w$ with $|w|=m$.
So consider one of these edges, $\Phi_w(e_{i j})$. It is parameterized by $x$
in the interval $[0,2^{-m}]$, and the contribution (\ref{eq:5.6}) is
\begin{equation}\label{eq:5.11}
\begin{aligned}
\int_0^{2^{-m}}\int_0^{2^{-m}} &\frac{|F(e(x))-F(e(y))|^2}{|x-y|^{1+2\beta}}\, dxdy \\
&= \frac{4^m}{2^{1+2\beta}} \int_0^1\int_0^1 \frac{|F(\Phi_w(e_{i j}(x))) - F(\Phi_w(e_{i j}(y)))|^2}{|x-y|^{1+2\beta}}\, dxdy
\end{aligned}
\end{equation}
after a change of variables. Summing all the contributions over all the edges
in $\mathsf{SG}_m$ yields
\begin{equation}\label{eq:5.12}
\|T_m F\|_{H^\beta(\mathsf{SG}_m)}^2 = \sum_{|w|=m} \frac{4^m}{2^{(1+2\beta)m}} \|T_0 F\circ \Phi_w\|_{H^\beta(\mathsf{SG}_0)}^2
\end{equation}
by (\ref{eq:5.11}). But the choice of $\beta$ makes
$\frac{4}{2^{1+2\beta}} = \frac 5 3$, so (\ref{eq:5.12}) combined with
(\ref{eq:5.10}) yields
\begin{equation}\label{eq:5.13}
\|T_m F\|_{H^\beta(\mathsf{SG}_m)}^2 \leqslant c \mathcal{E}(F)\text{.}
\end{equation}
This is the exact analog of (\ref{eq:5.7}).
\begin{theorem}\label{thm:5.2}
The trace map $T_m$ is continuous from $\dom \mathcal{E}$ to $H^\beta(\mathsf{SG}_m)$ for
$\beta$ as in Proposition \ref{prop:5.1} and the estimate (\ref{eq:5.13})
holds. Moreover, there exists a continuous linear extension map
$E_m:H^\beta(\mathsf{SG}_m)\to \dom\mathcal{E}$ with $T_m E_m f = f$ and
\begin{equation}\label{eq:5.14}
\mathcal{E}(E_m f) \leqslant c \|f\|_{H^\beta(\mathsf{SG}_m)}^2 \text{.}
\end{equation}
\end{theorem}
\begin{proof}
We have already established (\ref{eq:5.13}). To define the extension map $E_m$
we set
\begin{equation}\label{eq:5.15}
E_m(f) = \Phi_w^{-1} E_0(f\circ \Phi_w) \qquad \text{on $\Phi_w(\mathsf{SG})$.}
\end{equation}
Note that $E_m(f)$ is continuous, because at the boundary points of the
$m$-cells that make up $\mathsf{SG}_m$ we have $E_m(f) = f$. The same reasoning that
obtains (\ref{eq:5.13}) from (\ref{eq:5.7}) also leads from (\ref{eq:5.8}) to
(\ref{eq:5.14}).
\end{proof}
Next we have the analog of Theorem \ref{thm:4.1}.
\begin{theorem}\label{thm:5.3}
a) Let $F\in \dom\mathcal{E}$. Then $T_m F\in H^\beta(\mathsf{SG}_m)$ for all $m$ with
uniformly bounded norms, and
\begin{equation}\label{eq:5.16}
\sup_m \|T_m F\|_{H^\beta(\mathsf{SG}_m)}^2 \leqslant c \mathcal{E}(F)
\end{equation}
b) Let $f_m\in H^\beta(\mathsf{SG}_m)$ be a sequence of functions with
uniformly bounded norms satisfying the consistency condition
$T_m f_{m'} = f_m$ if $m\leqslant m'$. Then there exists $F\in\dom\mathcal{E}$ such
that $T_m f=f_m$ and
\begin{equation}\label{eq:5.17}
\mathcal{E}(F)\leqslant c\sup_m \|f_m\|_{H^\beta(\mathsf{SG}_m)}^2 \text{.}
\end{equation}
\end{theorem}
\begin{proof}
(\ref{eq:5.16}) is an immediate ff consequence of (\ref{eq:5.13}). To prove
b) construct a sequence of functions $F_m$ by taking the harmonic (energy
minimizing) extension of $f_m$ from $\mathsf{SG}_m$ to $\mathsf{SG}$. Then by (\ref{eq:5.14}),
the sequence $\{F_m\}$ is uniformly bounded in $\dom\mathcal{E}$. A quantitative
version of the continuity of functions in $\dom\mathcal{E}$ implies that the sequence
$\{F_m\}$ is also uniformly equicontinuous. Thus by passing to a subsequence
twice we can find a subsequence $\{F_{m_j}\}$ that converges both weakly in
the Hilbert space $\dom\mathcal{E}$ and uniformly to a function $F$ in $\dom\mathcal{E}$ with
the estimate (\ref{eq:5.17}) holding. Because the convergence is pointwise and
the consistency condition holds we have $T_{m_j} F = F_{m_j} = f_{m_j}$ on
$\mathsf{SG}_{m_j}$, so $T_m F = f_m$.
\end{proof}
The second example of a fractal we consider is the Sierpinski carpet
($\mathsf{SC}$), again defined by a self-similar identity
\begin{equation}\label{eq:5.18}
\mathsf{SC} = \bigcup_{i=1}^8 \Phi_i(\mathsf{SC})
\end{equation}
where now $\Phi_i$ are the homothety maps of the plane with contraction ratio
$1/3$ mapping the unit square into $8$ of the $9$ subsquares of side length
$1/3$ (all except the central subsquare). This self-similar fractal is not
PCF, so the method of Kigami cannot be used to construct an energy.
Nevertheless, two approaches due to Barlow and Bass and Kusuoka and Zhou
\cite{1} were given in the late 1980's, and recently in \cite{2} it was shown
that up to a constant multiple there is a unique self-similar energy, so both
approaches yield the same energy. Once again, all functions in $\dom\mathcal{E}$ are
continuous. The self-similar identity for the energy here is
\begin{equation}\label{eq:5.19}
\mathcal{E}(F) = \sum_{i=1}^8 r \mathcal{E}(F\circ \Phi_i)\text{,}
\end{equation}
where $r$ is a constant whose exact value has not been determined ($r$ is
slightly larger than $1.25$).
Again we may approximate $\mathsf{SC}$ by a sequence of metric graphs, $\{\mathsf{SC}_m\}$,
with $\mathsf{SC}_m = \mathsf{SC}\cap \mathsf{GP}_{3^{-m}}$. Thus, the edges of $\mathsf{SC}_m$ have length
$3^{-m}$ and are of the form $\Phi_w(e_i)$ with $|w|=m$, where
$e_1,e_2,e_3,e_4$ are the boundary edges of the unit square. Again let $T_m$
denote the trace map onto $\mathsf{SC}_m$. The trace space for $T_0$ has been
identified by Hino and Kumagai \cite{5} as the Sobolev space $H^\beta(\mathsf{SC}_0)$
with $\beta=\frac 1 2+\frac{\log r}{\log 9}$. Note that again
$\frac 1 2<\beta<1$.
\begin{proposition}[\cite{5}]\label{prop:5.4}
The trace map $T_0$ is continuous from $\dom\mathcal{E}$ to $H^\beta(\mathsf{SC}_0)$ for
$\beta=\frac 1 2+\frac{\log r}{\log 9}$ with
\begin{equation}\label{eq:5.20}
\|T_0 F\|_{H^\beta(\mathsf{SC}_0)}^2 \leqslant c \mathcal{E}(F)\text{.}
\end{equation}
Moreover, there exists a continuous linear extension map
$E_0:H^\beta(\mathsf{SC}_0)\to \dom\mathcal{E}$ with $T_0 E_0 f = f$ and
\begin{equation}\label{eq:5.21}
\mathcal{E}(E_0 f) \leqslant c \|f\|_{H^\beta(\mathsf{SC}_0)}^2 \text{..}
\end{equation}
\end{proposition}
We now claim that the analogs of Theorem \ref{thm:5.2} and \ref{thm:5.3} hold
for $\mathsf{SC}$ in place of $\mathsf{SG}$, with essentially the same proof. The only detail
that needs to be checked is the dilation argument. In this case the
contribution to (\ref{eq:5.6}) from the edge $e=F_w(e_1)$ is
\begin{equation}\label{eq:5.22}
\begin{aligned}
\int_0^{3^{-m}}\int_0^{3^{-m}} &\frac{|F(e_i(x))-F(e_i(y))|^2}{|x-y|^{1+2\beta}}\, dxdy \\
&= \frac{9^m}{3^{1+2\beta}} \int_0^1\int_0^1 \frac{|F(\Phi_w(e_i(x)))-F(\Phi_w(e_i(y)))|^2}{|x-y|^{1+2\beta}}\, dxdy
\end{aligned}
\end{equation}
after a change of variable, as the analog of (\ref{eq:5.11}). We note that
$\frac{9}{3^{1+2\beta}}=r$ in this case, so summing (\ref{eq:5.22}) yields the
analog of (\ref{eq:5.13}) as a subsequence of (\ref{eq:5.20}). The rest of the
arguments are the same.
For our final fractal example we consider the classical Julia sets of complex
polynomials. Fix a polynomial $P(z)$ (of degree at least two) and let
$\mathcal{J}$ denote its Julia set. We assume $\mathcal{J}$ is connected. In many cases (see
\cite{13}) it is possible to parameterize $\mathcal{J}$ by the unit circle as follows.
Let $\Omega$ denote the unbounded component of the complement of $\mathcal{J}$ in
$\mathbb{C}$, so $\Omega\cup\{\infty\}$ is simply connected, and let $\varphi$ be a
conformal map from $\{z:|z|>1\}$ to $\Omega$. In many cases $\varphi$ extends
continuously to the boundary circle, and this maps $C$ onto $\mathcal{J}$ (usually
not one-to-one). Although there is usually no useful formula for $\varphi$, in
many cases it is possible to describe explicitly the points on $C$ that are
identified under $\varphi$. There have been a number of papers that utilize
this parametrization to construct an energy on $\mathcal{J}$ \cite{14,15,16,17}.
Here we deal with a different question: how to characterize the traces on
$\mathcal{J}$ of functions of finite energy on $\Omega$. The answer is almost
immediate using the methods of section \ref{sec:3}. We know that
$F\in H^1(\Omega)$ if and only if $F\circ\varphi\in H^1(|z|>1)$, and the space
of traces of $F\circ\varphi$ on $C$ is exactly $H^{1/2}(C)$. Thus the space of
traces of $F$ on $\mathcal{J}$, that we should denote $H^{1/2}(\mathcal{J})$, is characterized
by the finiteness of
\begin{equation}\label{eq:5.23}
\|F\|_{H^{1/2}(\mathcal{J})}^2 = \int_0^{2\pi}\int_0^{2\pi} \frac{|F(\varphi(e^{i\theta}))-F(\varphi(e^{i\theta'}))|^2}{4\sin^2\frac 1 2(\theta-\theta')}\, d\theta d\theta'\text{.}
\end{equation}
One could perhaps hope for a more direct characterization in terms of an
integral involving $|F(z)-F(z')|^2$ as $z$ and $z'$ vary over $\mathcal{J}$. This
would involve choosing a measure on $\mathcal{J}$ (there are more than one natural
choices) and finding the appropriate denominator in terms of a distance from
$z$ to $z'$ on $\mathcal{J}$. Good luck!
\end{document} |
\begin{equation}gin{document}
\title{A dual de Finetti theorem}
\author{Graeme \surname{Mitchison}}
\email[]{[email protected]} \affiliation{Centre for
Quantum Computation, DAMTP,
University of Cambridge,
Cambridge CB3 0WA, UK}
\begin{equation}gin{abstract}
The quantum de Finetti theorem says that, given a symmetric state, the
state obtained by tracing out some of its subsystems approximates a
convex sum of power states. The more subsystems are traced out, the
better this approximation becomes. Schur-Weyl duality suggests that
there ought to be a dual result that applies to a unitarily invariant
state rather than a symmetric state. Instead of tracing out a number
of subsystems, one traces out part of every subsystem. The theorem
then asserts that the resulting state approximates the fully mixed
state, and the larger the dimension of the traced-out part of each
subsystem, the better this approximation becomes. This paper gives a
number of propositions together with their dual versions, to show how
far the duality holds.
\end{abstract}
\pacs{03.67.-a, 02.20.Qs}
\date{\today}
\maketitle
\pagestyle{plain}
\section{Introduction}
Suppose we have a state space $H=(\mathbb{C}^d)^{\otimes n}$ consisting
of $n$ identical subsystems. The quantum de Finetti theorem
~\cite{KoeRen05,Ren05} tells us that, given a symmetric state on $H$,
the state obtained by tracing out $n-k$ of the subsystems can be
approximated by a convex sum of powers, i.e. by a convex sum of states
of the form $\sigma^{\otimes k}$; the smaller $k/n$, the better the
approximation. This is a useful result, because such power states are
often rather easy to analyse.
Now, the symmetric group $S_n$ and the unitary group ${\cal U}(d)$
both act on the space $(\mathbb{C}^d)^{\otimes n}$, the former by
permuting the factors and the latter by applying any $g \in {\cal
U}(d)$ to each factor, so the action is given by $g^{\otimes
n}$. These actions commute, and this leads to a type of duality,
called Schur-Weyl duality \cite{GooWal98}. Given any result that
holds for the symmetric group, one can hope to find a dual result for
the unitary group.
Here I show that there is a dual to the de Finetti theorem, obtained
by swapping the roles of $S_n$ and ${\cal U}(d)$. The situation is
summed up in Table \ref{dual1}. Instead of symmetric states, we
consider unitarily-invariant states. And instead of tracing out a
number of subsystems, we trace out part of each subsystem; more
precisely, we replace each individual subsystem $\mathbb{C}^d$ by
$\mathbb{C}^p \otimes \mathbb{C}^q$, and we trace out the $\mathbb{C}^q$
part from all the subsystems in $(\mathbb{C}^p \otimes
\mathbb{C}^q)^{\otimes n}$. The theorem then states that, when $q$ is
large relative to $n$, the resulting traced-out state approximates the
fully mixed state. This is different in character from the standard de
Finetti theorem, in that all information about the original state is
lost. However, this fact in itself may lead to some interesting
applications.
As far as possible, the results are laid out as pairs of propositions
that are duals of each other. Some of these pairs are exact analogues;
in other cases, one of the pair is less meaningful or even
trivial. This gives some insight into the nature of the duality.
\begin{equation}gin{table}
\caption{
\label{dual1}}
\begin{equation}gin{center}
\begin{equation}gin{tabular}{|c|c|}
\hline
{\bf Standard de Finetti theorem.} & {\bf Dual theorem.}\\
\hline
Symmetric state $\rho$. & Unitarily-invariant state $\rho$.\\
\hline
State space is $(\mathbb{C}^d)^{\otimes n}$. & State space is $(\mathbb{C}^p \otimes \mathbb{C}^q)^{\otimes n}$.\\
\hline
Trace out $n-k$ subsystems. & Trace out $\mathbb{C}^q$ from each subsystem.\\
\hline
$\tr_{n-k} \rho \approx \mbox{convex sum of powers.}$ & $\tr_{\mathbb{C}^q} \rho \approx \mbox{fully mixed state.}$\\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Duality for symmetric Werner states.}
We will refer to unitarily-invariant states as {\em Werner} states
\cite{Werner89}. Rather than considering general Werner states, we
begin by looking at a special class, the symmetric Werner states,
i.e. states that are invariant under both the unitary and symmetric
groups. The de Finetti theorem and its dual can then be applied to the
same state, so the pattern becomes particular clear, as shown in Table
\ref{dual2}.
The Schur-Weyl decomposition \cite{FultonHarris91} of
$H=(\mathbb{C}^d)^{\otimes n}$ is given by:
\begin{equation} \label{SchurWeyl}
(\mathbb{C}^d)^{\otimes n} \cong
\bigoplus_{\lambda\in \Par{n}{d}} U_\lambda \otimes V_\lambda,
\end{equation}
where $U_\lambda$ is the irrep (irreducible representation) of
$\cU(d)$ with highest weight $\lambda_1 \ge \lambda_2 \ge \ldots \ge
\lambda_d$, and $V_\lambda$ is the irrep of $S_n$ defined by the same
partition $\lambda$. Here $\Par{n}{d}$ denotes the ordered partitions
of $n$ with at most $d$ rows. We will also refer to a $\lambda \in
\Par{n}{d}$ as a (Young) diagram.
Let $P_\lambda$ denote the projector onto the subspace $U_\lambda
\otimes V_\lambda$ in the Schur-Weyl decomposition. Write
$f_\lambda=\dim(V_\lambda)$, and $e^d_\lambda=\dim(U_\lambda)$, where
$d$ is the dimension of the unitary group $\cU(d)$. Then the
normalised projector $\rho_\lambda=P_\lambda/(e^d_\lambda f_\lambda)$
is a symmetric Werner state, and in fact any symmetric Werner state
$\rho$ can be written as a weighted sum of such projectors, $\sum_\mu
a_\mu \rho_\mu$, with $\sum_\mu a_\mu=1$ \cite{ChrEtal06}. Let
$\tr_{n-k} \rho_\lambda$ denote the state obtained by tracing out
$n-k$ of the $n$ subsystems from the state $\rho_\lambda$. Lemma III.4
in \cite{ChrEtal06} can be restated as follows:
\begin{equation}gin{proposition}[Trace formula] \label{original-trace}
Let $\lambda\in\Par{n}{d}$. Then
\[
\tr_{n-k} \rho_\lambda =\frac{1}{f_\lambda} \sum_\mu \rho_\mu f_\mu
\left(\sum_\nu c_{\mu \nu}^\lambda f_\nu\right),
\]
where the sums extends over all $\mu \in \Par{k}{d}$ and $\nu \in
\Par{n-k}{d}$, and $c_{\mu \nu}^\lambda$ is the Littlewood-Richardson
coefficient, i.e. the coefficient in the Clebsch-Gordan series for
$\cU(d)$: $U_\mu \otimes U_\nu=\sum_\lambda c^\lambda_{\mu\nu}
U_\lambda$.
\end{proposition}
From now on, we assume each individual subsystem $\mathbb{C}^d$ is
bipartite, so it can be written as $\mathbb{C}^p \otimes
\mathbb{C}^q$. Let $\tr_{\mathbb{C}^q}$ denote the result of tracing out
$\mathbb{C}^q$ from each subsystem in the total state space $(\mathbb{C}^p
\otimes \mathbb{C}^q)^{\otimes n}$. The dual of the preceding
Proposition is:
\begin{equation}gin{proposition}[Dual trace formula] \label{dual-trace}
Let $\lambda\in\Par{n}{pq}$. Then
\[
\tr_{\mathbb{C}^q} \rho_\lambda =\frac{1}{e^{pq}_\lambda} \sum_\mu \rho_\mu e^p_\mu \left(\sum_\nu g_{\lambda \mu \nu} e^q_\nu
\right),
\]
where the sums extend over all diagrams $\mu \in \Par{n}{p}$ and $\nu
\in \Par{n}{q}$, and $g_{\lambda \mu \nu}$ is the Kronecker
coefficient, i.e. the coefficient in the Clebsch-Gordan series for
$S_n$: $V_\mu \otimes V_\nu=\sum_\lambda g_{\lambda\mu\nu}
V_\lambda$.
\end{proposition}
\begin{equation}gin{proof}
We can restrict the action of the group $\cU(pq)$ on $\mathbb{C}^p
\otimes \mathbb{C}^q$ to the subgroup $\cU(p) \times \cU(q)$. This
gives an expansion in tensor products of irreps \cite{ChrMit06}:
\[
U_\lambda = \sum_{\mu \nu} g_{\lambda \mu \nu} U_\mu \otimes U_\nu,
\]
where $\mu \in \Par{n}{p}$ and $\nu \in \Par{n}{q}$. If
$P_{U_\lambda}$ denotes the projector onto $U_\lambda$, we can rewrite
this as
\begin{equation} \label{content-projection}
P_{U_\lambda} = \sum_{\mu \nu} \sum_{i=1}^{g_{\lambda \mu \nu}} P^i_{U_\mu} \otimes P^i_{U_\nu}.
\end{equation}
Taking the trace over $\mathbb{C}^q$ gives
\begin{equation} \label{content-trace}
\tr_{\mathbb{C}^q} P_{U_\lambda} = \sum_{\mu \nu} \sum_{i=1}^{g_{\lambda \mu \nu}} P^i_{U_\mu} e^q_\nu.
\end{equation}
Now define the symmetric average, $\symtwirl$, by
\begin{equation} \label{stwirl}
\symtwirl(\tau)=\frac{1}{n!}\sum_{\pi \in S_n}\pi \tau \pi^{-1},
\end{equation}
for any operator $\tau$. Applying $\symtwirl$ to both sides of
(\ref{content-trace}), Schur's lemma implies
\[
\tr_{\mathbb{C}^q}
\frac{P_\lambda}{f_\lambda} = \sum_{\mu}
\frac{P_\mu}{f_\mu}\left(\sum_\nu g_{\lambda \mu \nu} e^q_\nu \right).
\]
Substituting $\rho_\lambda=P_\lambda/(e^{pq}_\lambda f_\lambda)$,
$\rho_\mu=P_\mu/(e^p_\mu f_\mu)$ gives the result we seek.
\end{proof}
This shows, incidentally, why the dual operation to tracing out over
$n-k$ subsystems is to trace out over part of each subsystem: the
analogue of the subgroup $S_k \times S_{n-k} \subset S_n$ is the
subgroup $\cU(p) \times \cU(q) \subset \cU(pq)$.
\begin{equation}gin{table}
\caption{
\label{dual2}}
\begin{equation}gin{center}
\begin{equation}gin{tabular}{|c|c|}
\hline
\multicolumn{2}{|c|}{{\bf Duality Dictionary for symmetric Werner states.}}\\
\hline
$f_\lambda \ \ (\dim V_\lambda).$ & $e^d_\lambda \ \ (\dim U_\lambda).$ \\
\hline
Littlewood-Richardson coefficient $c^\lambda_{\mu \nu}$. & Kronecker coefficient $g_{\lambda \mu \nu}$.\\
\hline
Unitary group character (Schur function) $s_\lambda$. & Symmetric group character $\chi^\lambda$.\\
\hline
Shifted Schur function $s^*_\mu(\lambda)$. & Character polynomial $\chi^{\lambda \mu}(q)$ (Definition \ref{character-polynomial-definition}).\\
\hline
Twirled power state. & Symmetrised cycle operator.\\
\hline
\end{tabular}
\end{center}
\end{table}
Theorem 8.1 in \cite{OkoOls96} allows one to evaluate the bracketted
inner sum in Proposition \ref{original-trace}. We restate this result
as follows:
\begin{equation}gin{proposition}[Inner sum formula] \label{fsum}
\[
\sum_\nu c_{\mu \nu}^\lambda f_\nu = \frac{f_\lambda
s_\mu^*(\lambda)}{n \downharpoonright k},
\]
where $s_\mu^*(\lambda)$ is the shifted Schur function defined in
\cite{OkoOls96} and $n \downharpoonright k =n(n-1) \ldots (n-k+1)$.
\end{proposition}
Likewise, one can evaluate the bracketed inner sum in Proposition
\ref{dual-trace}. First we introduce a symmetric-group analogue of the
shifted Schur function:
\begin{equation}gin{definition} \label{character-polynomial-definition}
Suppose $\lambda$ and $\mu$ are arbitrary diagrams with $n$ boxes. The
{\em character polynomial} $\chi^{\lambda \mu}(q)$ is the polynomial
in $q$ defined by
\[
\chi^{\lambda \mu}(q)=\sum_{\pi \in S_n}
q^{c(\pi)} \chi^\lambda(\pi) \chi^\mu(\pi),
\]
where $\chi^\mu(\pi)$ is the character of the symmetric group evaluated
at the permutation $\pi$ and $c(\pi)$ is the number of cycles in $\pi$.
\end{definition}
The character polynomial can sometimes be more conveniently calculated by
summing over cycle types $\alpha$ rather than permutations, giving
\[
\chi^{\lambda \mu}(q)=\sum_{\alpha \in \Par{n}{n}} h_\alpha q^{c(\alpha)} \chi^\lambda(\alpha)
\chi^\mu(\alpha),
\]
where $h_\alpha$ is the number of elements in the conjugacy class
$\alpha$ \cite{digest}, and $c(\alpha)$ is the number of rows in the
diagram $\alpha$ representing the cycle type.
\begin{equation}gin{proposition}[Dual inner sum formula] \label{gsum}
\[
\sum_\nu g_{\lambda \mu \nu} e^q_\nu= \frac{\chi^{\lambda \mu}(q)}{n!}.
\]
\end{proposition}
\begin{equation}gin{proof}
First observe that
\begin{equation} \label{e-formula}
e^q_\nu=\frac{1}{n!} \sum_{\pi \in S_n}
q^{c(\pi)} \chi^\nu(\pi).
\end{equation}
This follows from the fact \cite{digest} that the projector $P_\nu$ on
$(\mathbb{C}^q)^{\otimes n}$ is defined by
\[
P_\nu=\frac{f_\nu}{n!}\sum_\pi \chi^\nu(\pi) \pi,
\]
and it vanishes on all components of the Schur-Weyl decomposition
(\ref{SchurWeyl}) except $U_\nu \otimes V_\nu$, where it has trace
$e^q_\nu f_\nu$. On the other hand, the trace of $\pi$ acting on
$(\mathbb{C}^q)^{\otimes n}$ is given by $q^{c(\pi)}$ since the basis
elements $e_{i_1} \otimes \ldots \otimes e_{i_n}$ of
$(\mathbb{C}^q)^{\otimes n}$ that are fixed by $\pi$, i.e. that
contribute to $\tr \pi$, are those that assign the same $e_i$ to all
the elements of each cycle of $\pi$, and there are $q$ ways of picking
an $e_i$ and $c(\pi)$ cycles. Thus $P_\nu$ has trace $\frac{f_\nu}{n!}
\sum_\pi q^{c(\pi)}\chi^\nu(\pi)$, and equating these two expressions
for the trace gives (\ref{e-formula}).
Now the Kronecker coefficient can be defined by
\[
g_{\lambda \mu \nu}=\frac{1}{n!}\sum_\pi \chi^\lambda(\pi)\chi^\mu(\pi)\chi^\nu(\pi).
\]
Combining this with (\ref{e-formula}), we have
\[
\sum_\nu g_{\lambda \mu \nu} e^q_\nu= \frac{1}{n!} \sum_{\pi, \pi^\prime}
q^{c(\pi^\prime)} \chi^\lambda(\pi) \chi^\mu(\pi)\left(\frac{1}{n!} \sum_\nu
\chi^\nu(\pi) \chi^\nu(\pi^\prime) \right).
\]
The orthogonality relations for characters imply that the expression
in brackets is zero if $\pi$ and $\pi^\prime$ are in different
conjugacy classes, and is otherwise the inverse of $h_{[\pi]}$, the
number of elements in the conjugacy class of $\pi$. As $c(\pi^\prime)$
only depends on the conjugacy class of $\pi^\prime$, the result
follows.
\end{proof}
Propositions \ref{fsum} and \ref{original-trace} can be
used to prove the de Finetti theorem for symmetric Werner states
\cite{ChrEtal06}:
\begin{equation}gin{theorem}[de Finetti theorem] \label{deFinetti}
Let $\rho_\lambda$ be the normalised projector onto the Young subspace
of $(\mathbb{C}^d)^{\otimes n}$ with diagram $\lambda$. Then
\[
||\tr_{n-k} \rho_\lambda-\tau|| \leq\frac{3}{4}\cdot
\frac{k(k-1)}{\lambda_\ell}+O(\frac{k^4}{\lambda_\ell^2})\ ,
\]
where $\tau$ is a convex sum of power states and $\lambda_\ell$ is the
smallest non-zero component of $\lambda$.
\end{theorem}
\begin{equation}gin{theorem}[Dual de Finetti theorem] \label{dual-deFinetti}
\[
||\tr_{C^q}\rho_\lambda - \frac{\cI}{p^n}|| \le 2-2\left(\frac{q-n+1}{q}\right)^n = \frac{2n(n-1)}{q} + O(n^4/q^2),
\]
where $\cI$ is the identity on $(\mathbb{C}^p)^{\otimes n}$.
\end{theorem}
We leave the proof till section \ref{general}, where the theorem is
proved for all Werner states, not just symmetric ones.
\begin{equation}gin{example}
The simplest example is the symmetric subspace for $n=2$. Using
Propostions \ref{dual-trace} and \ref{gsum}, we find
\[
\tr_{\mathbb{C}^q} \rho_{(2)}=\frac{(p+1)(q+1)}{2(pq+1)} \rho_{(2)}+\frac{(p-1)(q-1) }{2(pq+1)}\rho_{(1^2)}.
\]
Also
\[
\frac{\cI}{p^2}=\frac{(p+1)}{2p}\rho_{(2)}+\frac{(p-1)}{2p} \rho_{(1^2)},
\]
from which one gets
\[
|| \tr_{\mathbb{C}^q} \rho_{(2)}-\frac{\cI}{p^2}||=\frac{p^2-1}{p^2q+p}.
\]
Note that the bound tends to zero with $q \to \infty$ but its
behaviour does not depend sensitively upon $p$; in particular, there
is no requirement for $p$ to be small relative to $q$ (see
Discussion).
\end{example}
\section{Twirled power states and their duals}
Theorem \ref{deFinetti} in \cite{ChrEtal06} actually makes the
stronger claim that the approximating state $\tau$ is the twirl of a
power state $\sigma^k$. We describe this now and also its dual
version, where the analogue of the power state is a permutation
matrix. However, the rewards of the dual approach diminish rapidly,
and one does not get a stronger version of Theorem
\ref{dual-deFinetti} as will become clear at the end of this section.
Let us define the twirl of an arbitrary state $\tau$ on
$(\mathbb{C}^d)^{\otimes k}$ as follows:
\[
\twirl(\tau)=\int U^{\otimes k} \tau (U^\dagger)^{\otimes k} dU,
\]
where $dU$ is the Harr measure. Suppose $r=(r_1, \ldots, r_d)$ is the
spectrum of a state $\sigma$ on $\mathbb{C}^d$. Then the {\em twirled
power state} $\tau(r)=\twirl(\sigma^{\otimes k})$ depends only on $r$
and not on the particular state $\sigma$ chosen. Lemma III.1 from
\cite{ChrEtal06} expresses $\tau(r)$ in terms of basic Werner states.
\begin{equation}gin{proposition}[Twirl sum]\label{original-twirl}
Given a spectrum $r=(r_1, \ldots, r_d)$,
\[
\tau(r)=\sum_\mu f_\mu s_\mu(r) \rho_\mu,
\]
where $s_\mu(r)$ is the Schur function.
\end{proposition}
To define the dual version of a twirled power state, let $\pi$
be a permutation, and let $b_1, \ldots b_d$ be a basis in
$\mathbb{C}^d$. Define the permutation matrix $\tau_\pi$ by
$\tau_\pi=\pi \cI$, i.e.
\[
\tau_\pi=\sum_{0 \le i_1 \ldots i_n \le d} \ket{(b_{i_{\pi(1)}}, \ldots, b_{i_{\pi(n)}})}\bra{(b_{i_1}, \ldots, b_{i_n})}.
\]
Let $\lambda$ be a Young diagram with $n$ boxes representing a
permutation cycle type. Pick any permutation $\pi$ with cycle type
$\lambda$. The {\em symmetrised cycle operator} $\sigma(\lambda)$ is
defined to be $\symtwirl(\tau_\pi/d^n)$, where $\symtwirl$ is defined by
(\ref{stwirl}). This does not depend on the choice of a permutation
$\pi$ having the cycle type $\lambda$. We can regard $\symtwirl$ as
the ``dual twirl'', with the symmetric group replacing the unitary
group. Thus we have:
\begin{equation}gin{proposition}[Dual twirl sum] \label{dual-twirl}
Given a cycle type $\lambda$,
\begin{equation} \label{dual-twirlsum}
\sigma(\lambda)=\frac{1}{d^n}\sum_\mu e^d_\mu \chi^\mu(\lambda) \rho_\mu.
\end{equation}
\end{proposition}
\begin{equation}gin{proof}
By construction, $\sigma(\lambda)$ is symmetric; it is also unitarily
invariant, since $U\tau_\pi U^\dagger=U\pi I U^\dagger=\pi U I
U^\dagger=\pi I=\tau_\pi$. Thus $\sigma(\lambda)$ can be expressed as
a sum $\sum_\mu c_\mu \rho_\mu$, where the coefficients are given by
\[
c_\mu=\tr[P_\mu\sigma(\lambda)]=\tr[P_\mu\symtwirl(\tau_\pi/d^n)]=\tr[\symtwirl(P_\mu)\tau_\pi]/p^n=\tr[P_\mu \tau_\pi]/p^n,
\]
$\pi$ being a permutation of cycle type $\lambda$. But $\tr[P_\mu
\tau_\pi]$ is the character of the representation $\pi \to P_\mu
\tau_\pi P_\mu$, and as this is equivalent to $e^d_\mu$ copies of the
irrep $V_\lambda$, we have $c_\lambda=e^d_\mu\chi^\mu(\lambda)/p^n$.
\end{proof}
Note that $\sigma(\lambda)$ is in general not a state, since its
eigenvalues, the coefficients in (\ref{dual-twirlsum}), can be
negative. For instance, with $d=3$,
$\sigma((2,1))=\frac{10}{27}\rho_{(3)}-\frac{8}{27}\rho_{(1^3)}$.
Returning to the standard de Finetti theorem, Propositions
\ref{original-trace} and \ref{fsum} tell us that, for
$\lambda\in\Par{n}{d}$,
\[
\tr_{n-k} \rho_\lambda =\sum_\mu \rho_\mu f_\mu
\frac{s_\mu^*(\lambda)}{n \downharpoonright k},
\]
The shifted Schur function \cite{OkoOls96}, $s_\mu^*(\lambda)$, which
appears on the right-hand side of this equation, is a polynomial in
the $\lambda_i$, and its highest degree terms are the ordinary Schur
function $s_\mu(\lambda)$. It follows that
\begin{equation} \label{approx}
\frac{s_\mu^*(\lambda)}{n
\downharpoonright k} \to s_\mu(\bar \lambda) \mbox{ as } n \to \infty,
\end{equation}
where $\bar \lambda=(\lambda_1/\sum \lambda_i, \ldots , \lambda_d/\sum
\lambda_i)$. Putting this together with Proposition
\ref{original-twirl}, we can restate Theorem \ref{deFinetti}, showing
that the approximating state can be taken to be the twirled power
state $\tau(\bar \lambda)$.
\begin{equation}gin{proposition}[Twirl limit for de Finetti theorem] \label{to-twirl}
\begin{equation}
||\tr_{n-k} \rho_\lambda - \tau(\bar \lambda)||\leq\frac{3}{4}\cdot
\frac{k(k-1)}{\lambda_\ell}+O(\frac{k^4}{\lambda_\ell^2})\ .
\end{equation}
\end{proposition}
Dually, Propositions \ref{dual-trace} and \ref{gsum} tell us that
\begin{equation} \label{full-expansion}
\tr_{\mathbb{C}^q} \rho_\lambda =\frac{1}{e^{pq}_\lambda} \sum_\mu \rho_\mu e^p_\mu \frac{\chi^{\lambda \mu}(q)}{n!},
\end{equation}
We can imitate the approximation of the shifted Schur function by the
ordinary Schur function, and take the highest degree term in
$\chi^{\lambda \mu}(q)$, which is $q^n\chi^\lambda(1^n)\chi^\mu(1^n)$.
Using equation \ref{dual-twirlsum} and the fact that
$\chi^\lambda(1^n)=f_\lambda$, we get
\[
\tr_{\mathbb{C}^q} \rho_\lambda \to \frac{(pq)^n f_\lambda}{e^{pq}_\lambda n!} \sigma(1^n) \mbox{ as } q \to \infty.
\]
We shall see later that the rather complicated coefficient of
$\sigma(1^n)$ tends to 1 for large $q$ (see inequality
\ref{finalbound}). This enables us to write
\begin{equation}gin{proposition}[Twirl limit for dual de Finetti theorem]
\begin{equation}
||\tr_{C^q}\rho_\lambda - \sigma(1^n)|| \le \frac{2n(n-1)}{q} + O(n^4/q^2).
\end{equation}
\end{proposition}
Unlike Proposition \ref{to-twirl}, however, this adds nothing to the
preceding result (Theorem \ref{dual-deFinetti}), since
$\sigma(1^n)=\cI/p^n$. A more interesting result is obtained from
equation (\ref{full-expansion}) without taking the limit of
large $q$:
\begin{equation} \label{cyclesum}
\tr_{\mathbb{C}^q} \rho_\lambda=\frac{1}{n! e^{pq}_\lambda} \sum_\pi
q^{c(\pi)} \chi^\lambda(\pi) \sigma(\pi),
\end{equation}
This shows how symmetrised cycle operators other than $\sigma(1^n)$
contribute to the trace.
\section{The quantum marginal problem and Horn's conjecture}
We have now compared most of the ingredients of the de Finetti theorem and
their dual versions. In this section we complete this process by
comparing the shifted Schur functions that appear in Proposition
\ref{fsum} with the character polynomials that appear in Proposition
\ref{gsum}. We do this by relating each of them to a mathematical
problem of some historical interest. For the shifted Schur functions
this is Horn's conjecture \cite{Knutson00}, whereas for the character
polynomials it is the quantum marginal problem \cite{Kly04}. We begin
with the latter.
Let $\rho_A=\tr_B (\rho_{AB})$ and $\rho_B=\tr_A (\rho_{AB})$ be the
two marginal states of a bipartite state $\rho_{AB}$. Let
$\Sigma^{p,q}$ denote the set of triples of spectra
$\{\spec{\rho_{AB}}$, $\spec{\rho_A}$, $\spec{\rho_B}\}$ for all
operators $\rho_{AB}$ on $\mathbb{C}^p \otimes \mathbb{C}^q$. It was shown
in \cite{ChrMit06}, \cite{Kly04}, \cite{CHM06} that $\Sigma^{p,q}$ can
be defined in terms of the Kronecker coefficients. Given a diagram
$\lambda$, define $\bar \lambda=(\lambda_1/\sum \lambda_i, \ldots
\lambda_d/\sum \lambda_i)$, and let $K$ be the set of all triples
$(\bar \lambda, \bar \mu, \bar \nu)$ with $\lambda \in \Par{n}{pq}$,
$\mu \in \Par{n}{p}$, $\nu \in \Par{n}{q}$, for some $n$, satisfying
$g_{\lambda \mu \nu} >0$. Then $\Sigma^{p,q}$ is $\bar K$, the closure
of $K$.
One can also focus on a single marginal, and ask which {\em pairs},
$\{\spec{\rho_{AB}}$, $\spec{\rho_A}\}$ of spectra can occur
\cite{DH04}. From the characterisation of $\Sigma^{p,q}$, it follows
that this set, $\Gamma^{p,q}$ say, is the closure of the set of pairs
$(\bar \lambda, \bar \mu)$ where $\lambda \in \Par{n}{pq}$, $\mu \in
\Par{n}{p}$, and there is some $\nu \in \Par{n}{q}$ satisfying
$g_{\lambda \mu \nu} >0$. For a given $\lambda$, the $\mu$'s
satisfying this condition correspond to the $\rho_\mu$'s that have
non-zero coefficients in the expansion of $\tr_{\mathbb{C}^q}
\rho_\lambda$ given by Proposition \ref{dual-trace}. This, together
with Proposition \ref{gsum}, implies
\begin{equation}gin{proposition}[Character polynomial condition for the marginal problem]\label{gamma}
Suppose $\lambda \in \Par{n}{pq}$, $\mu \in \Par{n}{p}$, and
$\chi^{\lambda \mu}(q) > 0$. Then $(\bar \lambda, \bar \mu) \in
\Gamma^{p,q}$.
\end{proposition}
The converse does not follow from the characterisation of
$\Sigma^{p,q}$ by Kronecker coefficients. If $\lambda \in \Par{n}{pq}$
and $\mu \in \Par{n}{p}$, and $(\bar \lambda, \bar \mu) \in
\Gamma^{p,q}$ then we know there is a state $\rho_{AB}$ with
$\spec{\rho_{AB}}=\bar \lambda$ and $\spec{\rho_A}=\bar \mu$, but it
does not follow that $\spec{\rho_B}$ has the form $\bar \nu$ for some
$\nu \in \Par{n}{q}$, or even that $\spec{\rho_B}$ is rational. Even
if it were true that $\spec{\rho_B}=\bar \nu$ with $\nu \in
\Par{n}{q}$, we could only conclude \cite{Kly04,CHM06} that
$g_{m\lambda \ m\mu \ m\nu}>0$ for some integer $m>0$ and hence that
$\chi^{m\lambda \ m\mu}(q) > 0$ for some $m>0$.
\begin{equation}gin{proposition} \label{zeros}
For any $\lambda \in \Par{n}{pq}$, $\mu \in \Par{n}{p}$, there is an
integer $q_+$ in the range $1 \le q_+ \le n$ such that
$\chi^{\lambda \mu}(q)>0$ for $q \ge q_+$ and $\chi^{\lambda
\mu}(q)=0$ for $0 \le q < q_+$. If $\lambda \ne \mu$, $q_+ \ge 2$.
\end{proposition}
\begin{equation}gin{proof}
Clearly $\chi^{\lambda \mu}(q)=0$ for $q=0$, and as
$\chi^{\lambda \mu}(q)$ is a polynomial of degree $n$ and can
therefore have at most $n$ distinct roots, there must be some integer
$q$ in the range $1 \le q \le n$ for which $\chi^{\lambda
\mu}(q)=0$. Let $q_+$ be the least such $q$. Then by Proposition
\ref{gsum}, $\sum_\nu g_{\lambda \mu \nu} e^{q_+}_\nu >0$, and thus
$g_{\lambda \mu \nu}>0$ and $e^{q_+}_\nu >0$ for some $\nu$. Thus
$e^q_\nu>0$ for all $q \ge q_+$, and $\chi^{\lambda \mu}(q)=\sum_\nu
g_{\lambda \mu \nu} e^q_\nu >0$ for all $q \ge q_+$. If $\lambda \ne
\mu$, $\chi^{\lambda \mu}(1)=0$ by the orthogonality relations for
characters, so $q_+ \ge 2$.
\end{proof}
This result is also a consequence of a theorem of Berele and
Imbo \cite{BerImbo01}, which says that $g_{\lambda \mu \nu}>0$ for some
$\nu$ with $c(\nu) \le \max \{c(\lambda),c(\mu)\}$. This implies
the stronger result that $q_+ \le \max \{c(\lambda),c(\mu)\}$.
\begin{equation}gin{corollary} \label{nbound}
For any $\lambda \in \Par{n}{pq}$, $\mu \in \Par{n}{p}$, there is an
integer $q_+$ in the range $1 \le q_+ \le n$ such that $(\bar \lambda,
\bar \mu) \in \Gamma^{p,q}$ for $q \ge q_+$.
\end{corollary}
\begin{equation}gin{example} \label{same}
Take $\lambda =\mu$. Since every term in $\chi^{\lambda \lambda}(1)$ is
non-negative, and the term with $\alpha=(1^n)$ is $f^2_\lambda/n!>0$,
we have $\chi^{\lambda \lambda}(1)>0$ and hence $(\bar \lambda, \bar
\lambda) \in \Gamma^{p,1}$. It is easy to see why this is true: take
$\rho_{AB}=\rho_A \otimes \proj{0}_B$, and
$\spec{\rho_{AB}}=\spec{\rho_A}$.
\end{example}
\begin{equation}gin{example} \label{extreme}
Take $\lambda=(1^n)$, $\mu=(n)$. Then
$\chi^\lambda(\pi)=(-1)^{n+c(\pi)}$, by the Murnaghan-Nakayama rule
\cite{FultonHarris91}, and $\chi^\mu(\pi)=1$ for all $\pi$. It follows
that
\[
\chi^{\lambda \mu}(q)=(-1)^n\sum_\pi (-q)^{c(\pi)}=q(q-1)
\ldots (q-n+1).
\]
Thus $\chi^{\lambda \mu}(q)=0$ for $q=1, \ldots, n-1$. Hence $(\bar
\lambda, \bar \mu) \in \Gamma^{p,n}$. For $q \ge n$, a state with the
appropriate spectra for $\rho_{AB}$ and $\rho_A$ is
$\rho_{AB}=\frac{1}{n}\proj{0}_A \otimes \sum_{i=1}^n \proj{i}_B$.
Since $\chi^{\lambda \mu}(q)=\chi^{\mu \lambda}(q)$, if $\lambda=(n)$,
$\mu=(1^n)$ then $(\bar \lambda, \bar \mu) \in \Gamma_n$. A state with
the appropriate spectra is $\rho_{AB}=\proj{\psi_{AB}}$, where
$\psi_{AB}=\frac{1}{\sqrt{n}}\ket{11+ \dots +nn}_{AB}$. (Note that
this form of $\mu$ implies $p \ge n$.)
\end{example}
We can extend Proposition \ref{zeros} as follows
\begin{equation}gin{proposition}
For any $\lambda \in \Par{n}{pq}$, $\mu \in \Par{n}{p}$, there is a
positive integer $q_+$ and a negative integer $q_-$ such that
$\chi^{\lambda \mu}(q) \ne 0$ for $q \ge q_+$ and $q \le q_-$, and
$\chi^{\lambda \mu}(q)=0$ for $q_- < q < q_+$.
\end{proposition}
\begin{equation}gin{proof}
Let $\lambda^\prime$ denote the diagram conjugate to $\lambda$,
obtained by interchanging rows and columns. Then
$\chi^{\lambda^\prime}(\pi)=(-)^{n+c(\pi)}\chi^{\lambda}(\pi)$, so
$\chi^{\lambda^\prime \mu}(q)=(-1)^n\chi^{\lambda \mu}(-q)$. It
follows that the negative range of integral roots has the same
properties as the positive range, and the result follows from
Proposition \ref{zeros}.
\end{proof}
\begin{equation}gin{example}
Table \ref{polynomials} gives some examples of $\chi^{\lambda \mu}(q)$
for $n=5$, illustrating the fact that the integral roots form a
sequence without a gap. Note that $\chi^{\lambda^\prime
\mu^\prime}(q)=\chi^{\lambda \mu}(q)$, since
$\chi^{\lambda^\prime}(\pi)=(-)^{n+c(\pi)}\chi^{\lambda}(\pi)$. To
illustrate the property $\chi^{\lambda^\prime
\mu}(q)=(-1)^n\chi^{\lambda \mu}(-q)$, for each $(\lambda,\mu)$,
either $(\lambda^\prime,\mu)$ or $(\lambda,\mu^\prime)$ is also given.
For each $\lambda$, $\mu$, states with $q=q_+$ and the appropriate
spectra are described in Example \ref{same} for the cases where
$\lambda=\mu$, and in Example \ref{extreme} for the case $(5),
(1^5)$. States for the other cases are easy to construct; eg for
$(4,1), (2,1^3)$, where $q_+=3$, we can take $p=4$ and
\begin{equation}astar
\rho_{AB}&=&\frac{1}{5}\proj{11}_{AB}+\frac{4}{5}\proj{\psi_{AB}},\\
\mbox{ where } \ket{\psi}_{AB}&=&\frac{1}{2}\ket{22+33}_{AB}+\frac{1}{\sqrt{2}}\ket{41}_{AB}.\\
\end{equation}astar
\end{example}
\begin{equation}gin{table}
\caption{Some examples of the polynomials $\chi^{\lambda \mu}(q)$ for $n=5$.
\label{polynomials}}
\begin{equation}gin{center}
\begin{equation}gin{tabular}{|c|c|c|}
\hline
$\lambda$, $\mu$ & $\chi^{\lambda \mu}(q)$ & integral roots\\
\hline
\hline
$(5), (5)$ ; $(1^5), (1^5)$ & $q^5+10q^4+35q^3+50q^2+24q$ & $-4,-3,-2,-1,0$\\
\hline
$(5), (4,1)$ ; $(1^5), (2,1^3)$ & $4q^5+20q^4+20q^3-20q^2-24q$ & $-3,-2,-1,0,1$ \\
\hline
$(4,1), (4,1)$ ;$(2,1^3), (2,1^3)$& $16q^5+40q^4+20q^3+20q^2+24q$ & $-2,-1,0$\\
\hline
$(4,1), (2,1^3)$& $16q^5-40q^4+20q^3-20q^2+24q$ & $0,1,2$\\
\hline
$(5), (2,1^3)$; $(1^5), (4,1)$ & $4q^5-20q^4+20q^3+20q^2-24q$ & $-1,0,1,2,3$\\
\hline
$(5), (1^5)$ & $q^5-10q^4+35q^3-50q^2+24q$ & $0,1,2,3,4$\\
\hline
\end{tabular}
\end{center}
\end{table}
Turning now to shifted Schur functions, Horn's conjecture -- now a
theorem \cite{Klyachko98} -- states that, given $\lambda, \mu,
\nu \in \Par{n}{d}$, $c^\lambda_{\mu \nu}>0$ if and only if there is
triple of $n \times n$ Hermitian matrices $A$, $B$ and $C$ with
eigenvalues $\lambda$, $\mu$ and $\nu$, respectively, such that
$A+B=C$. Thus, if we know that $\sum_\nu c_{\mu \nu}^\lambda f_\nu
>0$, we can infer that
\begin{equation}gin{proposition}[Shifted Schur function condition for Horn's conjecture] \label{horn}
Suppose $\lambda \in \Par{n}{d}$, $\mu \in \Par{k}{d}$. Then
$s^*_\mu(\lambda) > 0$ implies that there are $n \times n$ Hermitian
matrices $A$, $B$ and $C$ such that $A+B=C$ and the eigenvalues of $C$
are $\lambda_i$, and those of $A$ are $\mu_i$.
\end{proposition}
\begin{equation}gin{proof}
By Proposition \ref{fsum}, $s^*_\mu(\lambda) > 0$ implies there is
some $\nu \in \Par{n-k}{d}$ such that $c^\lambda_{\mu \nu} >0$, and by
the Horn-Klyachko theorem there are Hermitian matrices $A$, $B$, $C$
with eigenvalues $\lambda$, $\mu$, $\nu$, respectively, satisfying
$A+B=C$.
\end{proof}
Unlike Proposition \ref{gamma}, there is a simple criterion for the
conditions of Proposition \ref{horn} to hold, since $s^*_\mu(\lambda)
> 0$ if and only if $\mu \subset \lambda$. This follows immediately
from the fact that $f_\lambda s^*_\mu(\lambda)/(n \downharpoonright
k)=\sum_\nu c^\lambda_{\mu \nu} f_\nu= \dim \lambda/\mu$, where $\dim
\lambda/\mu$ is the number of standard numberings of the skew diagram
$\lambda/\mu$. This is a positive integer when $\mu \subset \lambda$
and zero otherwise. Indeed, when $\mu \subset \lambda$ it is clear
that the matrix $B$ with $\lambda_i-\mu_i$ down the diagonal satisfies
$A+B=C$, where $A$ and $C$ are diagonal with spectra $\mu$ and
$\lambda$, respectively. Thus this ``two matrix'' version of Horn's
conjecture of the single marginal problem is essentially trivial,
unlike its dual counterpart, the single marginal problem. However,
note that the single marginal problem is also trivial, by Proposition
\ref{zeros}, in the sense that the condition $\chi^{\lambda \mu}(q)>0$ is
always satisfied, unless one also specifies the dimension $q$ of the
traced-out subsystem.
\section{General Werner states.} \label{general}
We now drop the assumption that the state is symmetrical, and consider
a general Werner state. First we characterise such states.
\begin{equation}gin{proposition} [Werner state characterisation]\label{Werner-form}
Any Werner state $\rho$ can be written
\[
\rho= \sum_\lambda \sum_i r_\lambda^i P^i_{U_\lambda},
\]
where $r_\lambda^i$ are positive constants, and $P^i_{U_\lambda}$ are
projectors onto unitary irreps $U^i_\lambda$.
\end{proposition}
\begin{equation}gin{proof}
If $\rho=\sum \gamma_i \proj{a_i}$ is the eigenvalue decomposition of
$\rho$, unitary invariance implies
\[
\rho=\sum \gamma_i \twirl(\proj{a_i}).
\]
From the Schur-Weyl
decomposition, we can write
\[
\ket{a_i}=\sum_{\lambda} \gamma_{i,\lambda} \ket{a_{i,\lambda}},
\]
and Schur's lemma then tells us that
\[
\twirl(\proj{a_i})=\sum_{\lambda} |\gamma_{i,\lambda}|^2 \twirl(\proj{a_{i,\lambda}}).
\]
Let $U^i_\lambda$ be the subspace generated by the set
$\{U\ket{a_{i,\lambda}}\ |\ U \in {\cal U}(d)\}$. This is a unitary
irrep, and $\twirl(\proj{a_{i,\lambda}})$ is an intertwining operator from
$U^i_\lambda$ to itself, and hence by Schur's lemma is proportional to
the projector $P^i_{U_\lambda}$.
\end{proof}
\begin{equation}gin{corollary} \label{Werner-dimension}
The number of (real) degrees of freedom of the set of Werner states
on $(\mathbb{C}^d)^{\otimes n}$ is $d_W=\sum f_\lambda^2 -1$, where the
sum is over $\lambda$ in $\Par{n}{d}$.
\end{corollary}
\begin{equation}gin{proof}
Another way of stating the result of the Proposition is that the
$\lambda$-isotypic part of any Werner state is isomorphic to $\rho
\otimes P_{U_\lambda}$, where $\rho$ is any density matrix on
$V_\lambda$. But $\rho$ has $f_\lambda$ real terms down the diagonal,
with one constraint due to the sum of eigenvalues being 1, and there
are $f_\lambda(f_\lambda-1)$ real components in the non-diagonal terms
above the diagonal, and those below the diagonal are the conjugates of
those above.
\end{proof}
We are now ready to prove the main theorem:
\begin{equation}gin{theorem}[General dual de Finetti theorem] \label{general-dual}
If $\rho$ is a Werner state on $(\mathbb{C}^p \otimes
\mathbb{C}^q)^{\otimes n}$ and $q \ge n$, then
\[
||\tr_{\mathbb{C}^q} \rho - \frac{\cI}{p^n}|| \le 2-2\left(\frac{q-n+1}{q}\right)^n = \frac{2n(n-1)}{q} + O(n^4/q^2).
\]
\end{theorem}
\begin{equation}gin{proof}
By Proposition \ref{Werner-form}, it suffices to consider a state $\rho$ that
is a normalised projector onto a unitary irrep, i.e. a state of the
form
\begin{equation} \label{rho}
\rho=P_{U_\lambda}/e^{pq}_\lambda.
\end{equation}
Let $\{ a_i\}$ and $\{b_j\}$ be bases for $\mathbb{C}^p$ and
$\mathbb{C}^q$, respectively. We can define the Cartan subgroup of
$\cU(pq)$ as the set of matrices diagonal in the product basis $\{ a_i
\otimes b_j\}$. Let $\cF$ denote the set of lexicographically ordered
$n$-tuples of elements of this basis, which we write as $((i_1j_1)
\dots (i_nj_n))$; these define the weights of $U_\lambda$.
Let $\cD$ be the subset of $\cF$ where the $j$ indices are distinct;
this set is non-empty because we are assuming $q \ge n$. The
corresponding weight vectors are linear combinations of terms whose
indices are permutations of those that occur in the weight, i.e.
$((i_{\pi(1)}j_{\pi(1)}) \dots (i_{\pi(n)}j_{\pi(n)}))$ for some
permutation $\pi \in S_n$.
Let $U^\cD_\lambda$ be the subspace of $U_\lambda$ consisting of all
the weight spaces for elements of $\cD$. A permutation of the product
basis $\{ a_i \otimes b_j\}$, which can be regarded as an element of
$S_{pq}$, induces a unitary map on $U^\cD_\lambda$, and hence
$P_{U_\lambda}^\cD$, the projector on $U^\cD_\lambda$, is invariant
under such permutations. This implies that terms of the form
\[
\left(\proj{a_{i_{\pi(1)}}} \otimes \proj{b_{j_{\pi(1)}}}\right) \otimes \dots \otimes \left(\proj{a_{i_{\pi(n)}}} \otimes \proj{b_{j_{\pi(n)}}}\right)
\]
in $P_{U_\lambda}^\cD$ all have the same coefficients, since any two
such terms with different permuations $\pi$ in $S_n$ can be mapped
into each other by an appropriate basis permutation in $S_{pq}$. Thus
$tr_{\mathbb{C}^q} P_{U_\lambda}^\cD$, i.e. the result of tracing out
the $\ket{b_j}s$ from $P_{U_\lambda}^\cD$, is a sum of terms
\[
\proj{a_{i_{\pi(1)}}} \otimes \dots \otimes \proj{a_{i_{\pi(n)}}} ,
\]
for all $\pi \in S_n$, all terms having equal coefficients. Therefore
$tr_{\mathbb{C}^q} P_{U_\lambda}^\cD$ is proportional to the identity
$\cI$ on $(\mathbb{C}^p)^{\otimes n}$.
Now $U^\cD_\lambda$ is the union of weight spaces, all of which are
isomorphic and have dimension given by the Kostka number
$K_{\lambda,(1^n)}$, which is $f_\lambda$ (see
\cite[p. 56-57]{FultonHarris91}). As there are $p^n$ sets of possible
$i$-indices in $\cD$ and ${q \choose n}$ sets of distinct $j$-indices,
$U^\cD_\lambda$ has dimension $f_\lambda {q \choose n}p^n$. Thus,
\[
tr_{\mathbb{C}^q} P_{U_\lambda}^\cD= f_\lambda {q \choose n}\cI.
\]
From this and eq. (\ref{rho}),
\[
tr_{\mathbb{C}^q}\rho=\frac{tr_{\mathbb{C}^q} P_{U_\lambda}}{e^{pq}_\lambda}=\frac{tr_{\mathbb{C}^q} P_{U_\lambda}^\cD}{e^{pq}_\lambda} + A=\frac{f_\lambda {q \choose n}\cI}{e^{pq}_\lambda}+A,
\]
where A is a positive operator comes from tracing out the
remaining weight subspaces in $P_{U_\lambda}-P_{U_\lambda}^D$. Thus,
from the triangle inequality
\[
||tr_{\mathbb{C}^q}\rho - \frac{\cI}{p^n}|| \le \left(1- \frac{f_\lambda {q \choose n}p^n}{e^{pq}_\lambda}\right)+||A|| = 2 \left(1- \frac{f_\lambda {q \choose n}p^n}{e^{pq}_\lambda}\right).
\]
The remainder of the proof consists in finding a lower bound for
$f_\lambda {q \choose n}/e^{pq}_\lambda$. To do this, we use the Weyl
dimension formula for $e^{pq}_\lambda$ and the hooklength formula for
$f_\lambda$ \cite{FultonHarris91} to write
\[
\frac{f_\lambda}{e^{pq}_\lambda}=\frac{n!(pq-1)!(pq-2)! \ldots
1!}{(pq+\lambda_1-1)!(pq+\lambda_2-2)! \ldots \lambda_d!}.
\]
This ratio decreases when a box in the diagram $\lambda$ is moved
upwards, so it achieves its minimum for the diagram $(n)$, giving
\[
\frac{f_\lambda}{e^{pq}_\lambda}
\ge \frac{n!(pq-1)!}{(pq+n-1)!}.
\]
Combining this with the inequality
\[
\frac{p(q-i+1)}{(pq+n-i)} \ge \frac{q-n+1}{q},
\]
which holds for $1 \le i \le n$, one concludes
\begin{equation} \label{finalbound}
\frac{f_\lambda}{e^{pq}_\lambda} {q \choose n}p^n = \frac{q(q-1) \ldots (q-n+1) p^n}{(pq+n-1)(pq+n-2) \ldots pq}
\ge \left(\frac{q-n+1}{q}\right)^n.
\end{equation}
\end{proof}
\begin{equation}gin{example} \label{tableau}
\begin{equation}gin{figure}
\centerline{\epsfig{file=young.eps,width=0.2\textwidth}}
\caption{The two tableaux for $(2,1)$; see Example \ref{tableau}.
\label{young}}
\end{figure}
The simplest diagram $\lambda$ where $V_\lambda$ is non-trivial is
$(2,1)$. Here $f_\lambda=2$, corresponding to the fact that there are
two standard tableaux (numberings of $\lambda$ that increase downwards
and to the right), shown in Figure \ref{young} as $T_1$ and $T_2$. Let
$U_{\lambda,1}$ denote the unitary representation obtained by applying
the Young projector \cite{FultonHarris91} for the tableau $T_1$. The
normalised projector $\rho=P_{U_{\lambda,1}}/e^{pq}_\lambda$ onto this
representation is an example of a Werner state that is not
symmetric. We explicitly calculate an approximation to the trace
$tr_{\mathbb{C}^q} \rho$ of this state.
As in the above proof, let $\{ a_i\}$ and $\{b_i\}$ be bases for
$\mathbb{C}^p$ and $\mathbb{C}^q$. Given $(i_1,i_2,i_3)$ and
distinct $(j_1,j_2,j_3)$, let us write
\[
\ket{u_{xyz}}=(\ket{a_{i_x}} \otimes \ket{b_{j_x}}) \otimes (\ket{a_{i_y}} \otimes \ket{b_{j_y}}) \otimes (\ket{a_{i_z}} \otimes \ket{b_{j_z}}),
\]
where $x,y,z$ is some permutation of $1,2,3$. Applying the Young
projector to the $\ket{u_{xyz}}$ for all possible permutations of
$1,2,3$ gives the set of vectors
\begin{equation}astar
\ket{\psi_1}&=&\left(\ket{u_{123}}+\ket{u_{213}}-\ket{u_{321}}-\ket{u_{312}}\right)/2,\\
\ket{\psi_2}&=&\left(\ket{u_{132}}+\ket{u_{312}}-\ket{u_{231}}-\ket{u_{213}}\right)/2,\\
\ket{\psi_3}&=&\left(\ket{u_{321}}+\ket{u_{231}}-\ket{u_{123}}-\ket{u_{132}}\right)/2,\\
\end{equation}astar
These are linearly dependent, since
$\ket{\psi_1}+\ket{\psi_2}+\ket{\psi_3}=0$, and make the same angle
with each other, since $\braket{\psi_i}{\psi_j}=-1/2$ for all $i \ne
j$. Thus the projector onto the 2D subspace they span is
\[
\frac{2}{3}\left(\proj{\psi_1}+\proj{\psi_2}+\proj{\psi_3}\right).
\]
Summing this expression over all $(i_1,i_2,i_3)$ and distinct
$(j_1,j_2,j_3)$ gives the projector $P_{U^D_{\lambda,1}}$.
Observe that $P_{U^D_{\lambda,1}}$ is not symmetric; for instance
$\ket{u_{123}}\bra{u_{321}}$ occurs with coefficient $-1/3$, whereas
$\ket{u_{213}}\bra{u_{231}}$ has coefficient $1/6$. However,
$\proj{u_{xyz}}$ has the same coefficient, $1/3$, for all permutations
$x,y,z$ of $1,2,3$, and it is only these terms that contribute to the
trace $tr_{\mathbb{C}^q} P_{U^D_{\lambda,1}}$. Summing over
distinct indices $(j_1,j_2,j_3)$ we therefore find
\[
tr_{\mathbb{C}^q} P_{U^D_{\lambda,1}}=3!{q \choose 3}\left(\frac{1}{3}\right) \cI.
\]
The factor $3!$ here arises because, for distinct $(i_1,i_2,i_3)$,
there are $3!$ ways of combining them with a set of distinct
$(j_1,j_2,j_3)$; eg $(i_1j_1,i_2j_2,i_3j_3)$,
$(i_1j_2,i_2j_1,i_3j_3)$, etc. When $(i_1,i_2,i_3)$ are not distinct,
there are fewer ways of combining them with $(j_1,j_2,j_3)$,
but on tracing out we regain the lost factor.
Since $e^d_\lambda=d(d-1)(d+1)/3$ for $\lambda=(2,1)$, we can write
\[
tr_{\mathbb{C}^q}
\frac{P_{U^D_{\lambda,1}}}{e^{pq}_\lambda}=\left[\frac{(q-1)(q-2)}{(q-1/p)(q+1/p)}\right]\frac{\cI}{p^3}.
\]
Let $\alpha$ denote the term in square brackets. Then
\[
tr_{\mathbb{C}^q} \rho = \alpha \frac{\cI}{p^3} + A,
\]
and we see that $\alpha \to 1$ for $q \to \infty$.
\end{example}
To conclude this section, we look at the dual to Proposition
\ref{Werner-form} and its corollary.
\begin{equation}gin{proposition} [Symmetric state characterisation]\label{Symmetric-form}
Any symmetric state $\rho$ can be written
\[
\rho= \sum_\lambda \sum_i r_\lambda^i P^i_{V_\lambda},
\]
where $r_\lambda^i$ are positive constants, and $P^i_{V_\lambda}$ are
projectors onto irreps $V^i_\lambda$ of the symmetric group.
\end{proposition}
\begin{equation}gin{corollary} \label{symmetric-dimension}
The number of degrees of freedom of the set of symmetric states on
$(\mathbb{C}^d)^{\otimes n}$ is $d_S=\sum
(e^d_\lambda)^2 -1$, where the sum is over $\lambda$ in $\Par{n}{d}$.
\end{corollary}
One might wonder if the standard deFinetti theorem could be proved by
methods like those used for Theorem \ref{general-dual}. It seems that
this is not possible, as the symmetric group representations have no
analogue of the weight spaces that are essential for this proof.
\section{Discussion}
The de Finetti theorem and its dual seem very different in
character. In the case of a symmetric Werner state $\rho_\lambda$, the
standard de Finetti theorem tells us that $\tr_{n-k} \rho_\lambda$,
the residual state when $n-k$ subsystems are traced out, can be
approximated by the twirled power state $\twirl(\sigma^{\otimes k})$,
where $\sigma$ has spectrum $\bar \lambda$ (see Proposition
\ref{to-twirl}). If one carries out a measurement on
$\twirl(\sigma^{\otimes k})$ of the projections onto the subspaces
$U_\mu \otimes V_\mu$ in the Schur-Weyl decomposition of
$(\mathbb{C}^d)^{\otimes k}$, the measured $\mu$, normalised to $\bar
\mu$, approximates $\bar \lambda$ \cite{KeyWer01PRA,Alicki88}. One
will only get an accurate estimate if $k \gg d$; when this condition
is satisfied, most of the information about the initial state is
encoded in the traced-out state. By contrast, when part of each
subsystem of a unitary-invariant state is traced out, the resulting
state approximates a fully mixed state, which conveys no information
about the initial state.
One might wonder whether this difference between the standard and dual
de Finetti theorems is related to the number of parameters, $d_S$ and
$d_W$, needed to specify symmetric and Werner states, respectively. Is
there a large reduction in $d_W$ in tracing out $\mathbb{C}^q$ from each
subsystem? If so, the loss of information about the initial state
would be explained. However, this is not the case. In fact, for $p >
n$, $d_W$ is given by $\sum f_\lambda^2 -1$ over $\lambda \in
\Par{n}{d}$ (Corollary \ref{Werner-dimension}), and is the same for
the whole system, where $d=pq$, and for the traced-out system where
$d=p$. There is actually more of a reduction in the number of
parameters with the standard de Finetti theorem, since $d_S$ is given
by $\sum_{\lambda \in \Par{n}{d}} (e^d_\lambda)^2-1$ (Corollary
\ref{symmetric-dimension}), which does increase, though only
polynomially, with $n$.
For the approximation to the fully mixed state to be
close, the dimension $q$ of the traced-out part of each subsystem must
be large relative to $n(n-1)$, where $n$ is the number of
subsystems. Note that one does not require that $p/q$ is small, where
$p$ is the dimension of the remaining part of each subsystem after
tracing-out. The situation is therefore not directly analogous to the
standard de Finetti theorem, where a good approximation requires that
$(n-k)/n$, the ratio of the number of subsystems traced out to the
total number of subsystems, be close to 1.
When $n=1$, the bound in the dual de Finetti theorem is zero, which
tells us that no tracing-out is needed; this just conveys the familiar
fact that averaging the action of ${\cal U}(d)$ on a state on
$\mathbb{C}^d$ gives the fully mixed state. One can ask which finite
subsets $S$ of ${\cal U}(d)$ have the property that the average
$\sum_S U \rho U^{\dagger}/|S|$ gives a good approximation to the
fully mixed state for any $\rho$, and it is known \cite{HLSW04} that
there are such sets with $|S| \approx d\log d$. The same question can
be posed for $n>1$, though now we expect to have to trace out part of
each subsystem to get an approximation to the completely mixed state.
The dual de Finetti theorem has a certain resemblance to a theorem
proved in \cite{PopescuShortWinter05}. This asserts that if $H_E$, the
state space of the environment, is traced-out from a random state
$\rho$ on the product of the system and environment $H_S \otimes H_E$,
then $\tr_E \rho$ is approximately a fully mixed state, the
approximation improving as $\dim H_E/\dim H_S$ increases. (Actually
the theorem holds more generally, for a state defined on an arbitrary
subspace of $H_S \otimes H_E$.) This suggests that obtaining the fully
mixed state after tracing out should be a property that holds for
``almost all states'', and not just for those with the special
structure of Werner states. One might therefore hope to be able to
extend the dual de Finetti theorem to a larger class of states (though
mathematics abounds with propositions known to be almost always true,
yet where specific instances are rather hard to find).
A natural application is to quantum secret-sharing: the theorem tells
us that this can be achieved by splitting up the subsystems of a Werner
state and giving them to two or more parties. With two parties, for
instance, each can have half of each subsystem, though the dimension
of each subsystem has to be large relative to $n$ for this to
work. Note that the procedure relies on the fact that $p/q$ does not
have to be small; we need to be able to regard both $\mathbb{C}^{\otimes
q}$ and $\mathbb{C}^{\otimes p}$ as the traced-out part (and similarly
for more than two parties).
Finally, one can ask whether the de Finetti theorem and its dual are
facets of some more all-embracing version of the theorem.
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\expandafter\ifx\csname bibnamefont\endcsname\relax
\def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
\def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
\def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
\def\url#1{\texttt{#1}}\fi
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\end{document} |
\begin{document}
\title{Online Matrix Factorization via Broyden Updates}
\author{\"Omer Deniz Aky{\i}ld{\i}z
\thanks{The author is with Bogazici University. Email: [email protected]}}
\markboth{}
{Shell \MakeLowercase{\textit{et al.}}: Bare Demo of IEEEtran.cls for Journals}
\maketitle
\begin{abstract}
In this paper, we propose an online algorithm to compute matrix factorizations. Proposed algorithm updates the dictionary matrix and associated coefficients using a single observation at each time. The algorithm performs low-rank updates to dictionary matrix. We derive the algorithm by defining a simple objective function to minimize whenever an observation is arrived. We extend the algorithm further for handling missing data. We also provide a mini-batch extension which enables to compute the matrix factorization on big datasets. We demonstrate the efficiency of our algorithm on a real dataset and give comparisons with well-known algorithms such as stochastic gradient matrix factorization and nonnegative matrix factorization (NMF).
\end{abstract}
\begin{IEEEkeywords}
Matrix factorizations, Online algorithms.
\end{IEEEkeywords}
\IEEEpeerreviewmaketitle
{\sigma}ection{Introduction}
\IEEEPARstart{P}{roblem} of online factorization of data matrices is of special interest in many domains of signal processing and machine learning. The interest comes from either applications with streaming data or from domains where data matrices are too wide to use batch algorithms. Analysis of such datasets is needed in many popular application domains in signal processing where batch matrix factorizations is successfully applied \cite{LeeSeungNMF}. Some of these applications include processing and restoration of images \cite{cemgil09-nmf}, source separation or denoising of musical \cite{FevotteISNMF,smaragdis2003nmftranscription} and speech signals \cite{wilson2008speech}, and predicting user behaviour from the user ratings (collaborative filtering) \cite{korenMFrecommender}. Nowadays, since most applications in these domains require handling streams or large databases, there is a need for online factorization algorithms which updates factors only using a subset of observations.
Formally, matrix factorization is the problem of factorizing a data matrix $Y\in {\mathbb R}^{m\times n}$ into \cite{LeeSeungNMF},
\begin{align}\label{NMFproblem}
Y \approx C X
\end{align}
where $C \in {\mathbb R}^{m\times r}$ and $X \in {\mathbb R}^{r\times n}$. Intuitively, $r$ is the approximation rank which is typically selected by hand. These methods can be interpreted as dictionary learning where columns of $C$ defines the elements of the dictionary, and columns of $X$ can be thought as associated coefficients. \textit{Online} matrix factorization problem consists of updating $C$ and associated columns of $X$ by only observing a subset of columns of $Y$ which is the problem we are interested in this work.
In recent years, many algorithms were proposed to tackle online factorization problem. In \cite{YildirimSMCNMF}, authors propose an algorithm which couples the expectation-maximization with sequential Monte Carlo methods for a specific Poisson nonnegative matrix factorization (NMF) model to develop an online factorization algorithm. The model makes Markovian assumptions on the columns of $X$, and it is similar to the classical probabilistic interpretation of NMF \cite{cedric-eusipco2009}, but a dynamic one. They demonstrate the algorithm on synthetic datasets. In \cite{onlineisnmf}, authors propose an online algorithm to solve the Itakura-Saito NMF problem where only one column of data matrix is used in each update. They also provide a mini-batch extension to apply it in a more efficient manner and demonstrate audio applications. In \cite{mairal2010online}, authors propose several algorithms for online matrix factorization using sparsity priors. In \cite{BucakGunselNMF}, authors propose an incremental nonnegative matrix factorisation algorithm based on an incremental approximation of the overall cost function with video processing applications. In \cite{sismanisSGD}, authors implement a stochastic gradient algorithm for matrix factorization which can be used in many different settings.
In this paper, we propose an online algorithm to compute matrix factorizations, namely the online matrix factorization via Broyden updates (OMF-B). We do not impose nonnegativity conditions although they can be imposed in several ways. At each time, we assume only observing a column of the data matrix (or a mini-batch), and perform low-rank updates to dictionary matrix $C$. We do not assume any structure between columns of $X$, and hence $Y$, but it is possible to extend our algorithm to include a temporal structure. OMF-B is very straightforward to implement and has a single parameter to tune aside from the approximation rank.
The rest of the paper is organised as follows. In Section \ref{SecObjFunc}, we introduce our cost function and the motivation behind it explicitly. In Section \ref{SecOnlineAlg}, we derive our algorithm and give update rules for factors. In Section \ref{SecModifications}, we provide two modifications to implement mini-batch extension and update rules for handling missing data. In Section \ref{SecExperiment}, we compare our algorithm with stochastic gradient matrix factorization and NMF on a real dataset. Section \ref{SecConc} concludes this paper.
{\sigma}ection{The Objective Function}\label{SecObjFunc}
We would like to solve the approximation problem \eqref{NMFproblem} by using only columns of $Y$ at each iteration. For notational convenience, we denote the $k$'th column of $Y$ with $y_{k} \in {\mathbb R}^m$. In the same manner, we denote the $k$'th column of $X$ as $x_{k} \in {\mathbb R}^r$ where $r$ is the approximation rank. This notation is especially suitable for matrix factorisations when columns of the data matrix represent different instances (e.g. images). We set $[n] = \{1,\ldots,n\}$ for $n\in {\mathbb N}$.
We assume that we observe \textit{random} columns of $Y$. To develop an appropriate notation, we use $y_{k_t}$ to denote the data vector observed at time $t$ where $k_t$ is sampled from $[n]$ uniformly random. The use of this notation implies that, at time $t$, we can sample any of the columns of $Y$ denoted by $y_{k_t}$. This randomization is not required and one can use sequential observations as well by putting simply $k_t = t$. We denote the estimates of the dictionary matrix $C$ at time $t$ as $C_t$. As stated before, we would like to update dictionary matrix $C$ and a column of the $X$ matrix $x_{k_t}$ after observing a single column $y_{k_t}$ of the dataset $Y$. For this purpose, we make the following crucial observations:
\begin{itemize}
\item[(i)] We need to ensure $y_{k_t} \approx C_t x_{k_t}$ at time $t$ for $k_t\in [n]$,
\item[(ii)] We need to penalize $C_t$ estimates in such a way that it should be ``common to all observations", rather than being overfitted to each observation.
\end{itemize}
As a result we need to design a cost function that satisfies conditions (i) and (ii) simultaneously. Therefore, for fixed $t$, we define the following objective function which consists of two terms. Suppose we are given $y_{k_t}$ for $k_t\in [n]$ and $C_{t-1}$, then we solve the following optimization problem for each $t$,
\begin{align}\label{Cost}
(x_{k_t}^*, C_t^*) = \operatornamewithlimits{argmin}_{x_{k_t},C_t} \big\| y_{k_t} - C_t x_{k_t} \big\|_2^2 + \lambda \big\| C_t - C_{t-1} \big\|_{F}^2
\end{align}
where $\lambda \in {\mathbb R}$ is a parameter which simply chooses how much emphasis should be put on specific terms in the cost function. Note that, Eq.~\eqref{Cost} has an analytical solution both in $x_{k_t}$ and $C_t$ separately. The first term ensures the condition (i), that is, $y_{k_t} \approx C_t x_{k_t}$. The second term ensures the condition (ii) which keeps the estimate of dictionary matrix $C$ ``common" to all observations. Intuitively, the second term penalizes the change of entries of $C_t$ matrices. In other words, we want to restrict $C_t$ in such a way that it is still close to $C_{t-1}$ after observing $y_{k_t}$ but also the error of the approximation $y_{k_t} \approx C_t x_{k_t}$ is small enough. One can use a weighted Frobenius norm to define a correlated prior structure on $C_t$ \cite{hennig2013quasi}, but this is left as a future work.
{\sigma}ection{Online Factorization Algorithm}\label{SecOnlineAlg}
For each $t$, we solve \eqref{Cost} by fixing $x_{k_t}$ and $C_t$. In other words, we will perform an alternating optimisation scheme at each step. In the following subsections, we derive the update rules explicitly.
{\sigma}ubsection{Derivation of the update rule for $x_{k_t}$}
To derive an update for $x_{k_t}$, $C_t$ is assumed to be fixed. To solve for $x_{k_t}$, let $G_{k_t}$ denote the cost function such that,
\begin{align*}
G_{k_t} =\big\| y_{k_t} - C_t x_{k_t} \big\|_F^2 + \lambda \big\| C_t - C_{t-1} \big\|_F^2,
\end{align*}
and set $\nabla_{x_{k_t}} G_{k_t} = 0$. We are only interested in the first term. As a result, solving for $x_{k_{t}}$ becomes a least squares problem, the solution is the following pseudoinverse operation [12],
\begin{align}\label{updateX}
x_{k_t} = (C_t^\top C_t)^{-1} C_t^\top y_{k_t},
\end{align}
for fixed $C_t$.
{\sigma}ubsection{Derivation of the update rule for $C_t$}
If we assume $x_{k_t}$ is fixed, the update with respect to $C_t$ can be derived by setting $\nabla_{C_t} G_{k_t} = 0$. We leave the derivation to the appendix and give the update as,
\begin{align}\label{beforeShermanMorrison}
C_t &= (\lambda C_{t-1} + y_{k_t} x_{k_t}^\top) (\lambda I + x_{k_t} x_{k_t}^\top)^{-1},
\end{align}
and by using Sherman-Morrison formula \cite{matrixcookbook} for the term $(\lambda I + x_{k_t} x_{k_t}^\top)^{-1}$, Eq.~\eqref{beforeShermanMorrison} can be written more explicitly as,
\begin{align}\label{updateC}
C_t &= C_{t-1} + \frac{(y_{k_t} - C_{t-1} x_{k_t}) x_{k_t}^\top}{\lambda + x_{k_t}^\top x_{k_t}},
\end{align}
which is same as the Broyden's rule of quasi-Newton methods as $\lambda \to 0$ \cite{hennig2013quasi}. We need to do some subiterations between updates \eqref{updateX} and \eqref{updateC} for each $t$. As it turns out, empirically, even $2$ inner iterations are enough to obtain a reliable overall approximation error.
\begin{algorithm}[t]
\begin{algorithmic}[1]
\caption{OMF-B}\label{OMFB}
\State Initialise $C_0$ randomly and set $t = 1$.
\Repeat
\State Pick $k_t \in [n]$ at random.
\State Read $y_{k_t} \in {\mathbb R}^m$
\For{$\text{Iter} = 1:2$}
\begin{align*}
x_{k_t} &= (C_t^\top C_t)^{-1} C_t^\top y_{k_t} \\
C_t &= C_{t-1} + \frac{(y_{k_t} - C_{t-1}x_{k_t})x_{k_t}^\top}{\lambda + x_{k_t}^\top x_{k_t}}
\end{align*}
\EndFor
\State $t \leftarrow t+1$
\Until{convergence}
\end{algorithmic}
\end{algorithm}
{\sigma}ection{Some Modifications}\label{SecModifications}
In this section, we provide two modifications of the Algorithm~\ref{OMFB}. The first modification is an extension to a mini-batch setting and requires no further derivation. The second modification provides the rules for handling missing data.
{\sigma}ubsection{Mini-Batch Setting}
In this subsection, we describe an extension of the Algorithm~\ref{OMFB} to the mini-batch setting. If $n$ is too large (e.g. hundreds of millions), it is crucial to use subsets of the datasets. We use a similar notation, where instead of $k_t$, now we use an index set $v_t {\sigma}ubset [n]$. We denote a mini-batch dataset at time $t$ with $y_{v_t}$. Hence $y_{v_t} \in {\mathbb R}^{m \times |v_t|}$ where $|v_t|$ is the cardinality of the index set $v_t$. In the same manner, $x_{v_t} \in {\mathbb R}^{|v_t|\times n}$ denotes the corresponding columns of the $X$. We can not use the update rule \eqref{updateC} immediately by replacing $y_{k_t}$ with $y_{v_t}$ (and $x_{k_t}$ with $x_{v_t}$) because now we can not use the Sherman-Morrison formula for \eqref{beforeShermanMorrison}. Instead we have to use Woodbury matrix identity \cite{matrixcookbook}. However, we just give the general version \eqref{beforeShermanMorrison} and leave the use of this identity as a choice of implementation. Under these conditions, the following updates can be used for mini-batch OMF-B algorithm. Update for $x_{v_t}$ reads as,
\begin{align}\label{updateMBX}
x_{v_t} = (C_t^\top C_t)^{-1} C_t^\top y_{v_t}
\end{align}
and update rule for $C_t$ can be given as,
\begin{align}\label{updateMBC}
C_t &= (\lambda C_{t-1} + y_{v_t} x_{v_t}^\top) (\lambda I + x_{v_t} x_{v_t}^\top)^{-1}
\end{align}
which is no longer same as the Broyden's rule for mini-batch observations.
{\sigma}ubsection{Handling Missing Data}
\begin{algorithm}[t]
\begin{algorithmic}[1]
\caption{OMF-B with Missing Data}\label{MissingOMFB}
\State Initialise $C_0$ randomly and set $t = 1$.
\Repeat
\State Pick $k_t \in [n]$ at random.
\State Read $y_{k_t} \in {\mathbb R}^m$
\For{$\text{Iter} = 1:2$}
\begin{align*}
x_{k_t} =& \left((M_{C_t} \odot C_t)^\top (M_{C_t} \odot C_t)\right)^{-1} \times \\
&(M_{C_t} \odot C_t)^\top (m_{k_t}\odot y_{k_t}) \\
C_t &= C_{t-1} + \frac{(m_{k_t} \odot (y_{k_t} - C_{t-1}x_{k_t})) x_{k_t}^\top}{\lambda + x_{k_t}^\top x_{k_t}}
\end{align*}
\EndFor
\State $t \leftarrow t+1$
\Until{convergence}
\end{algorithmic}
\end{algorithm}
In this subsection, we give the update rules which can handle the missing data. We only give the updates for single data vector observations because deriving the mini-batch update for missing data is not obvious and also become computationally demanding as $|v_t|$ increases. So we only consider the case $|v_t| = 1$ i.e. we assume observing only a single-column at a time.
We define a mask $M \in \{0,1\}^{m\times n}$, and we denote the data matrix with missing entries with $M \odot Y$ where $\odot$ denotes the Hadamard product. We need another mask to update related entries of the estimate of the dictionary matrix $C_t$, which is denoted as $M_{C_t}$ and naturally, $M_{C_t} \in \{0,1\}^{m\times r}$. Suppose we have an observation $y_{k_t}$ at time $t$ and some entries of the observation are missing. We denote the mask vector for this observation as $m_{k_t}$ which is $k_t$'th column of $M$. We construct $M_{C_t}$ for each $t$ in the following way:
\begin{align*}
M_{C_t} = \underbrace{[m_{k_t},\ldots, m_{k_t}]}_{r \textnormal{ times}}.
\end{align*}
The use of $M_{C_t}$ stems from the following fact. We would like to solve the following least squares problem for $x_{k_t}$ (for fixed $C_t$),
\begin{align}\label{LSmissing}
\min_{x_{k_t}} \big\| m_{k_t} \odot \left(y_{k_t} - C_t x_{k_t}\right) \big\|_2^2.
\end{align}
One can easily verify that,
\begin{align*}
m_{k_t} \odot \left(C_t x_{k_t}\right) = \left(M_{C_t} \odot C_t \right) x_{k_t}.
\end{align*}
Then \eqref{LSmissing} can equivalently be written as,
\begin{align*}
\min_{x_{k_t}} \big\| \left(m_{k_t} \odot y_{k_t}\right) - \left( M_{C_t} \odot C_t\right) x_{k_t} \big\|_2^2.
\end{align*}
As a result the update rule for $x_{k_t}$ becomes the following pseudoinverse operation,
\begin{align*}
x_{k_t} =& ((M_{C_t} \odot C_t)^\top (M_{C_t} \odot C_t))^{-1} \times \\
&(M_{C_t} \odot C_t)^\top (m_{k_t}\odot y_{k_t}),
\end{align*}
and the update rule for $C_t$ (for fixed $x_{k_t}$) can trivially be given as,
\begin{align*}
C_t &= C_{t-1} + \frac{(m_{k_t} \odot (y_{k_t} - C_{t-1}x_{k_t})) x_{k_t}^\top}{\lambda + x_{k_t}^\top x_{k_t}}.
\end{align*}
We denote the results on dataset with missing entries in Experiment~\ref{ExperimentMissing}.
{\sigma}ection{Experimental Results}\label{SecExperiment}
In this section, we demonstrate two experiments on the Olivetti faces dataset\footnote{Available at: \url{http://www.cs.nyu.edu/~roweis/data.html}} consists of $400$ faces with size of $64\times 64$ grayscale pixels. We first compare our algorithm with stochastic gradient descent matrix factorization in the sense of error vs. runtimes. In the second experiment, we randomly throw away the \%25 of each face in the dataset, and try to fill-in the missing data. We also compare our results with NMF \cite{LeeSeungNMF}.
{\sigma}ubsection{Comparison with stochastic gradient MF}
In this section, we compare our algorithm with the stochastic gradient descent matrix factorization (SGMF) algorithm \cite{sismanisSGD}. Notice that one can write the classical matrix factorization cost as,
\begin{align*}
\big\| Y - W H\big\|_F^2 = {\sigma}um_{k=1}^n \big\| y_k - W h_k \big\|_2^2
\end{align*}
so it is possible to apply alternating stochastic gradient algorithm \cite{sismanisSGD}. We derive and implement the following updates for SGMF,
\begin{align*}
W_t &= W_{t-1} - \gamma^W_t \nabla_{W} \big\| y_{k_t} - W h_{k_t}\big\|_2^2 \Big|_{W = W_{t-1}} \\
h^{t}_{k_t} &= h^{t-1}_{k_{t}} - \gamma^{h}_t \nabla_{h} \big\| y_k - W_t h \big\|_2^2 \Big|_{h = h^{t-1}_{k_{t}}}
\end{align*}
for uniformly random $k_t \in [n]$ for each $t$. The following conditions hold for convergence: ${\sigma}um_{t=1}^\infty \gamma^W_t = \infty$ and ${\sigma}um_{t=1}^\infty \left(\gamma^W_t\right)^2 < \infty$ and same conditions hold for $\gamma_t^h$. In practice we scale the step-sizes like $\alpha/t^{\beta}$ where $0<\alpha<\infty$ and $0.5 < \beta < 1$. These are other parameters we have to choose for both $W$ and $h$. It is straightforward to extend this algorithm to mini-batches \cite{sismanisSGD}. We merely replace $k_t$ with $v_t$.
\begin{figure}
\caption{Comparison with SGMF on Olivetti faces dataset. (a) This plot shows that although SGMF is faster than our algorithm (since we employ two iterations for each mini-batch), and SGMF processes the dataset in a much less wall-clock time, we achieve a lower error in the same wall-clock time. (b) This plot shows that our algorithm uses samples in a more efficient manner. We obtain lower errors for the same processed amount of data.}
\label{SGMF}
\end{figure}
In this experiment, we set identical conditions for both algorithms, where we use the Olivetti faces dataset, set $r = 30$, and use mini-batch size $10$ for both algorithms. We have carefully tuned and investigated step-size of the SGMF to obtain the best performance. We used scalar step sizes for the matrix $W$ and we set a step-size for each mini-batch-index, i.e. we use a matrix step-size for updating $h_{v_t}$. We set $\lambda = 10$. At the end, both algorithms passed $30$ times over the whole dataset taking mini-batch samples at each time. We measure the error by taking Frobenius norm of the difference between real data and the approximation.
The results are given in Fig.~\ref{SGMF}. We compared error vs. runtimes and observed that SGMF is faster than our algorithm in the sense that it completes all passes much faster than OMF-B as can be seen from Fig.~\ref{OMFB}(a). However our algorithm uses data much more efficiently and achieves much lower error rate \textit{at the same runtime} by using much fewer data points than SGMF. In the long run, our algorithm achieves a lower error rate within a reasonable runtime. Additionally, our algorithm has a single parameter to tune to obtain different error rates. In contrast, we had to carefully tune the SGMF step-sizes and even decay rates of step-sizes. Compared to SGMF, our algorithm is much easier to implement and use in applications.
{\sigma}ubsection{Handling missing data on Olivetti dataset}\label{ExperimentMissing} In this experiment, we show results on the Olivetti faces dataset with missing values where \%25 of the dataset is missing (we randomly throw away \%25 of the faces). Although this dataset is small enough to use a standard batch matrix factorisation technique such as NMF, we demonstrate that our algorithm competes with NMF in the sense of Signal-to-Noise Ratio (SNR). We compare our algorithm with NMF in terms of number of passes over data vs. SNR. We choose $\lambda = 2$, and set inner iterations as $2$. Our algorithm achieves approximately same SNR values with NMF (1000 batch passes over data) with only 30 online passes over dataset. This shows that our algorithm needs much less low-cost passes over dataset to obtain comparable results with NMF. Numbers and visual results are given in Fig.~\ref{figFaces}.
\begin{figure}
\caption{A demonstration on Olivetti faces dataset consists of 400 faces of size $64\times 64$ with \%25 missing data. Some example faces with missing data are on the left. Comparison of results of OMF-B (middle) with 30 online passes over dataset and NMF with 1000 batch iterations (right). Signal-to-noise ratios (SNR) are: OMF-B: 11.57, NMF: 12.13 where initial SNR is 0.75.}
\label{figFaces}
\end{figure}
{\sigma}ection{Conclusions and Future Work}\label{SecConc}
We proposed an online and easy-to-implement algorithm to compute matrix factorizations, and demonstrated results on the Olivetti faces dataset. We showed that our algorithm competes with the state-of-the-art algorithms in different contexts. Although we demonstrated our algorithm in a general setup by taking random subsets of the data, it can be used in a sequential manner as well, and it is well suited to streaming data applications. In the future work, we plan to develop probabilistic extensions of our algorithm using recent probabilistic interpretations of quasi-Newton algorithms, see e.g. \cite{hennig2013quasi} and \cite{hennig2015probabilistic}. The powerful aspect of our algorithm is that it can also be used with many different priors on columns of $X$ such as the one proposed in \cite{piecewiseNMF}. As a future work, we think to elaborate more complicated problem formulations for different applications.
{\sigma}ection*{Acknowledgements}
The author is grateful to Philipp Hennig for very helpful discussions. He is also thankful to Taylan Cemgil and S. Ilker Birbil for discussions, and to Burcu Tepekule for her careful proofreading. This work is supported by the TUBITAK under the grant number 113M492 (PAVERA).
{\sigma}ection*{Appendix}
We derive $\nabla_{C_t} G_{k_t}$ as the following. First we will find $\nabla_{C_t} \big\|y_{k_t} - C_t x_{k_t}\big\|_2^2$ which is the derivative of the first term. Notice that
\begin{align*}
\big\|y_{k_t} - C_t x_{k_t} \big\|_2^2 = {\operatorname{Tr}}\left(y_{k_t}^\top y_{k_t} - 2 y_{k_t}^\top C_t x_{k_t} + x_{k_t}^\top C_t^\top C_t x_{k_t}\right)
\end{align*}
First of all the first term is not important for us, since it does not include $C_t$. Using standard formulas for derivatives of traces \cite{matrixcookbook}, we arrive,
\begin{align}\label{derivativeOfFirst}
\nabla_{C_t} \big\|y_{k_t}-C_t x_{k_t} \big\|_2^2 = -2 y_{k_t} x_{k_t}^\top + 2 C_t x_{k_t} x_{k_t}^\top
\end{align}
The second term of the cost function can be written as,
\begin{align*}
\lambda \big\|C_t - C_{t-1} \big\|_F^2 = \lambda {\operatorname{Tr}}\left((C_t - C_{t-1})^\top (C_t - C_{t-1})\right)
\end{align*}
If we take the derivative with respect to $C_t$ using properties of traces \cite{matrixcookbook},
\begin{align}\label{derivativeOfSecond}
\nabla_{C_t} \lambda\big\|C_t - C_{t-1} \big\|_F^2 = 2\lambda C_t - 2\lambda C_{t-1}
\end{align}
By summing \eqref{derivativeOfFirst} and \eqref{derivativeOfSecond}, setting them equal to zero, and leaving $C_t$ alone, one can show \eqref{beforeShermanMorrison} easily. Using Sherman-Morrison formula, one can obtain the update rule given in the Eq.~\eqref{updateC}.
\ifCLASSOPTIONcaptionsoff
\fi
\end{document} |
\begin{document}
\title[the viscous Saint-Venant system
for shallow waters]{Well-posedness of classical solutions to the vacuum free boundary
problem of the viscous Saint-Venant system
for shallow waters}
\author{Hai-Liang Li}
\author{Yuexun Wang}
\author{Zhouping Xin}
\address{ School of Mathematics and CIT, Capital Normal University, Beijing 100048, P. R. China.}
\email{[email protected]}
\address{
School of Mathematics and Statistics,
Lanzhou University, 730000 Lanzhou, China.}
\email{[email protected]}
\address{The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong}
\email{[email protected]}
\thanks{}
\begin{abstract} We establish the local-in-time well-posedness of classical solutions to the vacuum free boundary
problem of the viscous Saint-Venant system
for shallow waters derived rigorously from incompressible
Navier-Stokes system with a moving free surface by Gerbeau-Perthame \cite{MR1821555}. Our solutions (the height and velocity) are smooth (the solutions satisfy the equations point-wisely) all the way to the moving boundary, although the height degenerates as a singularity of the distance to the vacuum boundary. The proof is built on some new higher-order weighted energy functional and weighted estimates associated to the degeneracy near the moving vacuum boundary.
\end{abstract}
\maketitle
\section{Introduction }
The one-dimensional compressible isentropic Navier-Stokes equations with the density-dependent
viscosity coefficient are given by
\begin{equation}\label{eq:intro-1}
\left\{
\begin{aligned}
&\rho_t+(\rho u)_x=0,\\
&(\rho u)_t+(\rho u^2)_x+p_x
=(\mu(\rho)u_x)_x,
\end{aligned}
\right.
\end{equation}
where $(x,t)\in\mathbb{R}\times\mathbb{R}_+ $, and $\rho(x,t)\geq 0,u(x,t)$ and $p=\rho^\gamma\ (\gamma>1)$ stand for the density, velocity, and pressure, respectively. And $\mu(\rho)=\rho^\alpha\ (\alpha\geq 0)$ is the viscosity coefficient.
There is a vast body of literature on the long time existence and asymptotic behavior of solutions to the system \eqref{eq:intro-1} in the case that the viscosity $\mu(\rho)$ is constant, i.e., \(\alpha=0\). When the initial density is strictly away from vacuum (\(\inf_{x\in\mathbb{R}} \rho_0(x) > 0\)),
the global existence of strong solutions was addressed for sufficiently smooth data by Kazhikhov et
al. \cite{MR0468593}, and for discontinuous initial data by Serre \cite{MR870700} and Hoff \cite{MR896014}, respectively.
The crucial point to establish such global existence of strong solutions lies in the fact that if the initial density is positive, then the density is positive for any later-on time as well. This fact is also proved to be true
for weak solutions by Hoff and Smoller \cite{MR1814847}, namely weak solutions
do not contain vacuum states in finite time as long as there is no vacuum initially.
When the initial density contains vacuum, the problem becomes subtle.
In fact, the appearance of vacuum indeed leads to some singular behaviors of solutions, such as the failure of continuous dependence of weak solutions on
initial data \cite{MR1117422} and the finite time blow-up of smooth solutions \cite{MR3360663, MR1488513}, and even non-existence of classical solutions with
finite energy \cite{MR3925527}.
Thus, when the solutions may contain vacuum states, it seems natural to investigate
the compressible Navier-Stokes equations with density-dependent viscosity.
Indeed, in the derivation of the
compressible Navier-Stokes equations from the Boltzmann equation by the Chapman-Enskog expansions,
as pointed out and investigated by Liu-Xin-Yang \cite{MR1485360}, the viscosity shall depend on the temperature and thus correspondingly depend on the density for isentropic flows.
Moreover, Gerbeau-Perthame \cite{MR1821555} derived rigorously a viscous Saint-Venant system for the shallow waters
which is expressed exactly to \eqref{eq:intro-1} with \(\alpha=1\) and \(\gamma=2\), from the incompressible Navier-Stokes equation with a moving free surface. Such viscous compressible models with density-dependent viscosity coefficients and its variants also appear in geophysical flows \cite{MR1978317, MR1989675, MR2257849} (see also P.-L. Lions's book \cite{MR1637634}).
There are also extensive studies on the compressible Navier-Stokes equations with density-dependent viscosity.
When the initial density was assumed to be connected to vacuum with
discontinuities,
the local well-posedness of weak solutions to this problem was first established by Liu-Xin-Yang \cite{MR1485360}, and
the global existence of weak solutions for \(0<\alpha<1\) was considered by many authors,
see \cite{MR2254008} and the references therein. The above analysis relies heavily
on the fact that the density
of the approximate solutions has a uniform positive lower bound in the non-vacuum region.
When the density connects to
vacuum continuously, the density has no positive lower bound and thus the viscosity
coefficient vanishes at vacuum. This degeneracy in the viscosity coefficient gives rise to
some new difficulties. Despite of this, there is still much progress, for instance, one may refer to \cite{MR1929151} when \(\alpha>1/2\) for the local existence result, and \cite{MR1990849} for the global existence results of
weak solutions when \(0<\alpha<1/2\), in the free boundary
setting.
For \(\alpha>1/2\), some phenomena of vacuum vanishing and blow-up of solutions were found by Li-Li-Xin \cite{MR2410901}, more precisely, the authors proved that for any global entropy weak solution, the vacuum state must vanish within finite time, and the velocity blows up in finite time as the vacuum states vanish.
For the study on the asymptotic stability of rarefaction waves to this problem, one may refer to \cite{MR3116645} and the references therein.
Since P.-L. Lions' breakthrough work \cite{MR1226126, MR1637634},
there have also been much important progress for the multi-dimensional isentropic Navier-Stokes equations with the constant coefficients or density-dependent viscosity coefficients, see \cite{MR1779621, MR2914266, MR3862947, MR1978317, MR1867887, MR2877344,
LX2015, MR1810944, MR0564670, MR3573976, MR4011697, MR4195559} and the references therein.
The vacuum free boundary problem
of \eqref{eq:intro-1} had attracted a vast of attractions in the past two decades.
In the case that the viscosity is constant, Luo-Xin-Yang \cite{MR1766564} studied the global regularity and behavior of the weak solutions near the interface
when the initial density connects to vacuum states in a very smooth manner. Zeng \cite{MR3303172} showed that the global existence of smooth solutions for which the smoothness extends all the way to
the boundary. In the case that the viscosity is density-dependent, the global existence of weak solutions was studied by many authors, see \cite{MR1990849} without external force, and \cite{MR2771263, MR2259332, MR2068308} with external force and the references therein.
By taking the effect of external force into account, Ou-Zeng \cite{MR3397339} obtained the global well-posedness of strong solutions and the global regularity uniformly up to the vacuum boundary.
Although there have been much important progress as aforementioned, it is still not clear whether the above solutions are smooth or not even locally in time when the viscosity coefficient vanishes at vacuum. In the present paper, we study the local well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system
for shallow waters derived rigorously from the incompressible
Navier-Stokes system with a moving free surface by Gerbeau-Perthame \cite{MR1821555}, which corresponds to \eqref{eq:intro-1} with \(\alpha=1\) and \(\gamma=2\), i.e.,
\begin{equation}\label{eq:intro-2}
\begin{cases}
\rho_t+(\rho
u)_x=0 &\quad \text{in}\ I(t),\\
(\rho u)_t+(\rho u^2+\rho^2)_x=(\rho u_x)_x &\quad \text{in}\ I(t),\\
\rho>0 &\quad \text{in}\ I(t),\\
\rho=0 &\quad \text{on}\ \Gamma(t),\\
\mathcal{V}(\Gamma(t))=u,\\
(\rho,u)=(\rho_0,u_0) &\quad \text{on}\ I(0),\\
I(0)=I=\{x: 0<x<1\}.
\end{cases}
\end{equation}
To solve the system \eqref{eq:intro-2}, we need to solve the four pairs \((\rho, u, I(t), \Gamma(t))\) (in fact it suffices to solve the triple \((\rho, u, \Gamma(t))\)).
Here \(\rho\) denotes the \emph{height} of the fluid (we use this terminology
from its original meaning), and \(u\) denotes the Eulerian velocity, respectively.
The open, bounded interval \(I(t)\) denotes the changing domain occupied by the
fluid, \(\Gamma(t)=\colon \partial I(t)\) denotes the moving vacuum boundary, and \(\mathcal{V}(\Gamma(t))\) denotes the velocity of \(\Gamma(t)\), respectively.
Equation \(\eqref{eq:intro-2}_1\) stands for the conservation of mass, and Equation \(\eqref{eq:intro-2}_2\) describes the conservation of
momentum, the condition \(\eqref{eq:intro-2}_3\) means that there is no vacuum inside of fluid,
the conditions \(\eqref{eq:intro-2}_4\) tell the dynamical boundary conditions to be investigated, \(\eqref{eq:intro-2}_5\) states that the vacuum boundary is moving with the fluid velocity, and \(\eqref{eq:intro-2}_6\) are the initial conditions for the \emph{height}, velocity, and domain.
The initial \emph{height} profile we are interested in this paper connects to vacuum as follows:
\begin{align}\label{eq:intro-3}
\rho_0\in H^5(\bar{I}(0))\ \text{and}\ C_1d(x)\leq \rho_0(x)\leq C_2d(x) \quad \text{for\ all}\ x\in \bar{I}(0),
\end{align}
for some positive constants \(C_1\) and \(C_2\), where \(d(x)=\colon d(x,\Gamma(0))\) is the distant function from \(x\) to the initial boundary.
We also explain a little bit on the condition \eqref{eq:intro-3}. The condition \eqref{eq:intro-3} is equivalent to the following so-called "physical vacuum singularity". Let \(c(x,t)=\sqrt{\rho(x,t)}\) be the sound speed, and hence \(c_0=c(x,0)\) is the initial sound speed.
The physical vacuum singularity (see, for example, \cite{MR2779087, MR1485360}) is determined by the following condition
\begin{equation}\label{eq:intro-4}
\begin{aligned}
0<\bigg|\frac{\mathrm{d} c_0^2}{\mathrm{d} x}\bigg|<\infty \quad \mbox{on}\ \Gamma(0).
\end{aligned}
\end{equation}
It is straightforward to check that \eqref{eq:intro-3} is equivalent to \eqref{eq:intro-4} by assuming \(\rho_0(x)\) vanishes on the boundary \(\Gamma(0)\).
The study on the physical vacuum free boundary problem for the compressible Euler equations was first given by Jang-Masmoudi \cite{MR2547977} and Coutand-Lindblad-Shkoller \cite{MR2608125} with different methods handling the degeneracy near the free boundary. For other important progress on the vacuum free boundary problems in compressible fluids, one may also refer to \cite{MR2177323, MR2779087, MR2980528, MR3280249, MR3218831, MR3551252} and the references therein.
The physical vacuum free boundary problem of shallow waters was studied by both Duan \cite{MR2914266} and Ou-Zeng \cite{MR3397339}, with the external force "\(-\rho f\)" (imposed on the right hand side of the momentum equation \(\eqref{eq:intro-2}_2\)), for global theory. In \cite{MR2914266}, the author considered some kind initial density degenerated as \(d^{1/2}(x)\) near the vacuum boundary and showed the global well-posedness of weak solutions by establishing certain global space-time square estimates using Lagrangian mass coordinates. In \cite{MR3397339}, the authors considered some sort of initial density like \(d(x)\) near the vacuum boundary and showed the global well-posedness of strong solutions based on certain weighted energy estimates with both space and time weights using Hardy's inequality together with the particle path method.
We aim to present a detailed proof on the local well-posedness of classical solutions (see Definition \ref{Classical Solution} (b)) to the vacuum free boundary problem \eqref{eq:intro-2}-\eqref{eq:intro-3} in the present paper. Comparing with \cite{MR2914266, MR3397339}, our classical solution satisfies an additional Nuewmann boundary condition \(u_x=0\ \text{on}\ \Gamma(t)\), which is captured by the high regularity of the solution on the vacuum boundary (see Remark \ref{re:main-1}-\ref{re:main-3}).
To handle the degeneracy near the vacuum boundary and to capture
the feature \(u_x=0\ \text{on}\ \Gamma(t)\) of our classical solution, we first construct a higher-order energy functional associated to
the degeneracy near the vacuum boundary, and then develop some delicate weighted estimates to close the higher-order energy functional, in which the weighted Sobolev inequalities and some weighted interpolation inequality will play an important role. Our higher-order energy functional consists of the following four type terms:
\[\int_I \rho_0(\partial_t^{k_1}v)^2\,\mathrm{d} x,\
\int_I \rho_0(\partial_t^{k_2}v_x)^2\,\mathrm{d} x,\
\int_I \rho_0^{k_3}(\partial_t\partial_x^{k_3}v)^2\,\mathrm{d} x,\
\int_I \rho_0^{k_4}(\partial_x^{k_4}v)^2\,\mathrm{d} x,\]
for some non-negative integers \(k_1, k_2, k_3, k_4\) to be chosen. The first two type terms come from the time-differentiated energy estimates, which are essentially the estimates of the derivatives
in the tangential direction of the moving boundary.
While the last two type terms are from the elliptic estimates, which depend highly on the degenerate parabolic structure of the momentum equation in \eqref{eq:main-2} and make it possible for us to gain more regularities through the estimates of the derivatives in the normal direction of the moving boundary.
Constructing approximate solutions usually is not a trivial process in showing well-posedness of the physical vacuum free boundary problem of compressible fluids since the system degenerates on the boundary, see \cite{MR2547977, MR2779087, MR2980528, MR3280249}.
In \cite{MR2779087}, in order to get the regular solution to the compressible Euler equations, Coutand-Shkoller considered a degenerate parabolic regularization well matched with the compressible Euler equations, more precisely where the viscosity has a structure \(\kappa(\rho_0^2v_x)_x\). To show the existence of weak solutions to this degenerate parabolic equation by the Galerkin's scheme, the authors introduced a new variable \(X=\rho_0v_x\) which satisfies a Dirichlet boundary condition
\(X=0\ \text{on}\ \partial I\times [0,T]\) since \(\rho_0\) vanishes on the boundary and \(v_x\) is bounded and then studied the equation for \(X\) instead of \(v\).
(Note that \(v\) itself does not satisfy any boundary condition.) On the other hand, to tackle the strong degeneracy of the viscosity, the authors had to divide a weight \(\rho_0\) on both sides of the degenerate parabolic equation to lower the degeneracy (but there is no singularity in the new equation), where a new higher-order Hardy-type inequality necessitates.
It seems difficult to apply the idea of \cite{MR2779087} straightforwardly to construct approximate solutions of the viscous Saint-Venant system for shallow waters \eqref{eq:intro-2} (see Remark \ref{re:main-5}).
In this paper, we will construct a classical solution to the vacuum free boundary problem \eqref{eq:intro-2}-\eqref{eq:intro-3} satisfying the Nuewmann boundary condition \eqref{Nuewmann boundary condition-2} (see Remark \ref{re:main-1}-\ref{re:main-3}), so this boundary condition will play an important role in constructing approximate solutions in the Hilbert space \(\mathcal{H}(I)=\{h\in H^3(I): h_x=0 \ \text{on}\ \Gamma \}\). We will first use the Galerkin's scheme to
construct a unique weak solution to the linearized problem, and then improve the regularity of this weak solution based on some key higher order a priori estimates, and finally show that the approximate solutions converge to a unique classical solution to the degenerate parabolic problem by a contraction mapping method.
It should be pointed out that, on the one hand, in deducing a priori estimates on higher order derivatives here, one can not manipulate as \cite{MR2779087} to divide a \(\rho_0\) on both sides of the degenerate parabolic equation to lower the degeneracy since it will introduce some singularity in the new equation which prevents the analysis to work. Hence we will keep the original structure of the degenerate parabolic equation, and use mainly the weighted Sobolev inequalities to handle the degeneracy which depends heavily on the degenerate parabolic structure of the momentum equation in \eqref{eq:main-2}. On the other hand, due to the degeneracy, the energy estimates on the
approximate solutions are insufficient for us to pass limit in \(n\) on the iteration problem for time pointwisely.
Therefore we need use some weighted interpolation inequality that can help us to obtain a pointwise convergence for time
on the approximate solutions to the iteration problem (see Section \ref{classical solution}).
In \cite{MR2864798}, Guo-Li-Xin studied the multi-dimensional
viscous Saint-Venat system for the shallow waters and showed the global existence of a spherically symmetric
weak solution to its free boundary value problem, in which detailed regularity and Lagrangian structure
of this solution was presented. It is interesting to extend our classical solutions' result to the multi-dimensional (spherically symmetric) viscous Saint-Venat system for the shallow waters, which is left for future.
The paper is organized as follows. In Section \ref{main results}, we will first formulate the vacuum free boundary problem into a fixed boundary problem and then state our main results. Section \ref{Some Preliminaries} lists some preliminaries.
In Section \ref{Energy Estimates} and \ref{Elliptic estimates}, respectively,
we will focus on the a priori estimates that constitute the energy estimates and elliptic estimates. Section \ref{Existence Part} and \ref{Uniqueness Part} are devoted to showing the existence and uniqueness of a classical solution to our degenerate parabolic problem, respectively.
\section{Reformulation and main results}\label{main results}
\subsection{Fixing the domain}
The initial domain (the reference domain) in one-dimension is given by \(I(0)=(0,1)\). Afterwards, we will use the short notation \(I\) to replace \(I(0)\) for convenience, and also denote by \(\Gamma=\partial I\) the boundary of the reference domain.
Denote by \(\eta\) the position of the fluid particle \(x\) at time \(t\)
\begin{equation}\label{fluid particle}
\begin{cases}
\partial_t\eta(x,t)=u(\eta(x,t),t),\\
\eta(x,0)=x,
\end{cases}
\end{equation}
and also by \(f(x,t)\) and \(v(x,t)\) the Lagrangian \emph{height} and velocity
\begin{equation}\label{Lagrangian variable}
\begin{cases}
f(x,t)=\rho(\eta(x,t),t),\\
v(x,t)=u(\eta(x,t),t).
\end{cases}
\end{equation}
Then \eqref{eq:intro-2} is transformed to the following problem on the fixed reference interval \(I\):
\begin{equation}\label{eq:main-1}
\begin{cases}
f_t+{\frac{fv_x}{\eta_x}}=0 &\quad \text{in}\ I\times (0,T],\\
\eta_xfv_t+(f^2)_x=(f{\frac{v_x}{\eta_x}})_x &\quad \text{in}\ I\times (0,T],\\
f>0 &\quad \text{in}\ I\times (0,T],\\
f=0 &\quad \text{on}\ \Gamma\times (0,T],\\
(f,v,\eta)=(\rho_0,u_0,e) &\quad \text{on}\ I\times \{t=0\},
\end{cases}
\end{equation}
where \(e(x)=x\) denotes the identity map on \(I\).
Solving \(f\) from Equation \(\eqref{eq:main-1}_1\) yields
\begin{align}\label{eq:Lag-initial}
f(x,t)=\rho_0(x)\eta_x^{-1}(x,t),
\end{align}
one inserts \eqref{eq:Lag-initial} back to Equation \(\eqref{eq:main-1}_2\) to transfer the problem \eqref{eq:main-1} into
\begin{equation}\label{eq:main-2}
\begin{cases}
\rho_0v_t+\big({\frac{\rho_0^2}{\eta_x^2}}\big)_x
=\big(\frac{\rho_0v_x}{\eta_x^2}\big)_x &\quad \mbox{in}\ I\times (0,T],\\
(v,\eta)=(u_0,e) &\quad \mbox{on}\ I\times \{t=0\}.
\end{cases}
\end{equation}
The problem \eqref{eq:main-2} is a degenerate parabolic problem.
\begin{definition}[Classical Solution]\label{Classical Solution} {\rm{(a)}} We say a function \(v\)
is a classical solution to the problem \eqref{eq:main-2} provided \(v\) satisfies \(\eqref{eq:main-2}_1\) in \(\bar{I}\times (0,T]\) pointwisely and is continuous to the initial data \(u_0\).
{\rm{(b)}} We say the pair \((\rho(x,t), u(x,t), \Gamma(t))\) for \(t\in[0,T]\) and \(x\in I(t)\)
is a classical solution to the problem \eqref{eq:intro-2} provided \((\rho(x,t), u(x,t), \Gamma(t))\) satisfies \(\eqref{eq:intro-2}_1-\eqref{eq:intro-2}_5\) pointwisely and is continuous to the initial data \((\rho_0, u_0, \Gamma)\), additionally, \(\eqref{eq:intro-2}_1\) and \(\eqref{eq:intro-2}_2\) hold on the spatial boundary of \(I(t)\) pointwisely.
\end{definition}
\subsection{The higher-order energy functional}
Our main purpose is to study the local well-posedness of the degenerate parabolic problem \eqref{eq:main-2} in certain weighted Sobolev space with high regularity. For this, we will consider the following higher-order energy functional:
\begin{equation}\label{higher-order energy function}
\begin{aligned}
E(t,v)&=\sum_{k=0}^3\|\sqrt{\rho_0}\partial_t^kv(\cdot,t)\|_{L^2(I)}^2
+\sum_{k=0}^2\|\sqrt{\rho_0}\partial_t^kv_x(\cdot,t)\|_{L^2(I)}^2\\
&\quad+\sum_{k=2}^4\big\|\sqrt{\rho_0^k}\partial_t\partial_x^kv(\cdot,t)\big\|_{L^2(I)}^2
+\sum_{k=2}^6\big\|\sqrt{\rho_0^k}\partial_x^kv(\cdot,t)\big\|_{L^2(I)}^2.
\end{aligned}
\end{equation}
We define the polynomial function \(M_0\) by
\begin{align*}
M_0=P(E(0,v_0)),
\end{align*}
where \(P\) denotes a generic polynomial function of its arguments.
\subsection{Main result on the problem \eqref{eq:main-2}}
The main result in the paper can be stated as follows:
\begin{theorem}\label{th:main-1} Assume the initial data \((\rho_0,v_0)\) satisfy \eqref{eq:intro-3} and \(M_0<\infty\), then
there exist a suitably small \(T>0\) and a unique classical solution
\begin{align}\label{regularity}
v\in C([0,T]; H^3(I))\cap C^1([0,T]; H^1(I))
\end{align}
to the problem \eqref{eq:main-2} on \([0,T]\) such that
\begin{align}\label{eq:inequality-1}
\sup_{0\leq t\leq T} E(t,v)\leq 2M_0.
\end{align}
\end{theorem}
Moreover, \(v\) satisfies the Nuewmann boundary condition
\begin{align}\label{Nuewmann boundary condition}
v_x=0 \quad \mbox{on}\ \Gamma\times (0,T].
\end{align}
\subsection{Main result on the vacuum free
boundary problem \eqref{eq:intro-2}-\eqref{eq:intro-3}}
Due to \eqref{eta-bound}, the flow map \(\eta(\cdot,t)\colon I\rightarrow I(t)\) is inverse for any \(t\in[0,T]\) and we denote its inverse by \(\tilde{\eta}(\cdot,t)\colon I(t)\rightarrow I\), where \(T\) is determined in Theorem \ref{th:main-1}. Let \((\eta,v)\) be the unique classical solution in Theorem \ref{th:main-1}. For \(t\in[0,T]\) and \(y\in I(t)\), set
\begin{equation*}
\begin{aligned}
&\rho(y,t)=\rho_0(\tilde{\eta}(y,t))\eta_x^{-1}(\tilde{\eta}(y,t),t),\\
&u(y,t)=v(\tilde{\eta}(y,t),t).
\end{aligned}
\end{equation*}
Then the triple \((\rho(y,t), u(y,t), \Gamma(t))\)) (\(t\in[0,T]\)) defines a unique classical solution to the vacuum free
boundary problem \eqref{eq:intro-2}-\eqref{eq:intro-3}. More precisely, Theorem \ref{th:main-1} can be transferred into the following:
\begin{theorem}\label{th:main-2} Assume the initial data \((\rho_0,u_0)\) satisfy \eqref{eq:intro-3} and \(M_0<\infty\), then
there exist a \(T>0\) and a unique classical solution \((\rho(y,t), u(y,t), \Gamma(t))\) for \(t\in[0,T]\) and \(y\in I(t)\) to the vacuum free
boundary problem \eqref{eq:intro-2}-\eqref{eq:intro-3}. Moreover, \(\Gamma(t)\in C^2([0,T])\), and for \(t\in[0,T]\) and \(y\in I(t)\), we have
\begin{equation}\label{regularity-2}
\begin{aligned}
&\rho(y,t)\in C([0,T];H^3(I(t)))\cap C^1([0,T];H^2(I(t)));\\
&u(y,t)\in C([0,T];H^3(I(t)))\cap C^1([0,T];H^1(I(t))).
\end{aligned}
\end{equation}
Moreover, \(u\) satisfies the Nuewmann boundary condition
\begin{align}\label{Nuewmann boundary condition-2}
u_x=0 \quad \rm{on}\ \Gamma(t).
\end{align}
\end{theorem}
\subsection{Some remarks}
The following remarks are helpful for understanding our main results.
\begin{remark}\label{re:main-1}
By the trace theorem \(H^3(I)\hookrightarrow H^{5/2}(\Gamma)\) (see \cite{MR2597943} for instance) and \(v(\cdot,t)\in H^3(I)\) for each \(t\in(0,T]\), one may define the Nuewmann boundary condition \eqref{Nuewmann boundary condition} pointwisely due to \eqref{regularity}. Similarly, one can also define \eqref{Nuewmann boundary condition-2} by \(u(y,t)\in C([0,T];H^3(I(t)))\) for \(t\in[0,T]\) and \(y\in I(t)\) pointwisely due to \eqref{regularity-2}.
\end{remark}
\begin{remark}\label{re:main-2} It follows from Remark \ref{re:main-1} that \eqref{Nuewmann boundary condition} is well-defined if the solution to the problem \eqref{eq:main-2} possesses the regularity \eqref{regularity}. In fact, \eqref{Nuewmann boundary condition} holds naturally for the classical solution in the sense of Definition \ref{Classical Solution} (a), however, with a higher regularity \eqref{eq:inequality-1}. In the following, we show how to derive \eqref{Nuewmann boundary condition} from Definition \ref{Classical Solution} (a) together with \eqref{eq:inequality-1}.
First note from Equation \(\eqref{eq:main-2}_1\) that
\begin{align}\label{r1}
\rho_0v_t+{\frac{2\rho_0(\rho_0)_x}{\eta_x^2}}-{\frac{2\rho_0^2\eta_{xx}}{\eta_x^3}}
=\frac{(\rho_0)_xv_x}{\eta_x^2}+\rho_0\bigg(\frac{v_{xx}}{\eta_x^2}-\frac{2v_x\eta_{xx}}{\eta_x^3}\bigg),
\end{align}
for \((x,t)\in \ I\times (0,T]\).
It follows from \eqref{eq:inequality-1}, Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2} that
\begin{align*}
\rho_0v_t(\cdot, t),\ v_x(\cdot, t),\ \eta_x(\cdot, t),\ \rho_0v_{xx}(\cdot, t),\ \rho_0\eta_{xx}(\cdot, t)\in H^2(I)\quad \text{for}\ t\in (0,T],
\end{align*}
which combines the trace theorem \(H^2(I)\hookrightarrow H^{3/2}(\Gamma)\) yields
\begin{align*}
\rho_0v_t(\cdot, t),\ v_x(\cdot, t),\ \eta_x(\cdot, t),\ \rho_0v_{xx}(\cdot, t),\ \rho_0\eta_{xx}(\cdot, t)\in H^{3/2}(\Gamma)\quad \text{for}\ t\in (0,T].
\end{align*}
This implies that each term in \eqref{r1} is well-defined pointwisely on \( \Gamma\times (0,T]\).
Using \eqref{eq:intro-3}, \eqref{eta-bound}, and letting \(x\) go to the vacuum boundary \(\Gamma(t)\), then one obtains
\begin{align}\label{r2}
(\rho_0)_xv_x=0 \quad \rm{on}\ \Gamma\times (0,T].
\end{align}
By \eqref{eq:intro-3} again, one sees \((\rho_0)_x\neq 0\) on \(\Gamma\), hence \eqref{Nuewmann boundary condition} follows from \eqref{r2}.
On the other hand side, to construct a classical solution to the problem \eqref{eq:main-2}, we will use a Galerkin's scheme to study its linearized problem, in which the Nuewmann boundary condition \eqref{Nuewmann boundary condition} will play a crucial role.
\end{remark}
\begin{remark}\label{re:main-3}
For the problem \eqref{eq:intro-2}-\eqref{eq:intro-3}, since \(\rho\) vanishes on \(\Gamma(t)\),
the usual stress free condition
\begin{align}\label{r3}
S=\rho^2-\rho u_x=0\quad \rm{on}\ \Gamma(t)
\end{align}
holds automatically.
\end{remark}
\begin{remark}\label{re:main-4}
In \cite{MR2779087}, Coutand-Shkoller studied the well-posedness of the physical vacuum free boundary problem of the compressible Euler equations, which may be written in Lagrangian coordinates as
\begin{align}\label{r3.5}
\rho_0v_t+\bigg({\frac{\rho_0^\gamma}{\eta_x^\gamma}}\bigg)_x
=0.
\end{align}
For \(1<\gamma\leq 2\), the authors constructed the following energy functional (see Section 8 in \cite{MR2779087}):
\begin{equation}\label{CS'energy function}
\begin{aligned}
E_\gamma(t,v)&=\sum_{s=0}^4\|\partial_t^sv(\cdot, t)\|_{H^{2-s/2}}^2+\sum_{s=0}^2\|\rho_0\partial_t^{2s}v(\cdot, t)\|_{H^{3-s}}^2+\|\sqrt{\rho_0}\partial_t\partial_x^2v(\cdot, t)\|_{L^2}^2\\
&\quad+\|\sqrt{\rho_0}\partial_t^3\partial_xv(\cdot, t)\|_{L^2}^2
+\sum_{a=0}^{a_0}\|\sqrt{\rho_0}^{1+\frac{1}{\gamma-1}-a}\partial_t^{4+a_0-a}\partial_xv(\cdot,t)\|_{L^2}^2,
\end{aligned}
\end{equation}
where \(a_0\) satisfies \(1<1+\frac{1}{\gamma-1}-a_0\leq 2\). Note that the last sum in \(E_\gamma\) appears whenever \(1<\gamma< 2\), and the order of the time-derivative increases to infinity as \(\gamma\to 1^+\).
But the energy functional \eqref{CS'energy function} fails for \(\gamma=1\) whose equation corresponds to the isothermal Euler equation:
\begin{align}\label{r4}
\rho_0v_t+\bigg({\frac{\rho_0}{\eta_x}}\bigg)_x
=0.
\end{align}
Next, we will compare the isothermal Euler model with the shallow water model in the following two aspects. On the one hand, applying
\(\partial_t\) to Equation \eqref{r4} yields
\begin{align}\label{r5}
\rho_0\partial_t^2v=\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)_x.
\end{align}
The term \(\big({\frac{\rho_0v_x}{\eta_x^2}}\big)_x\) in Equation \eqref{r5} also appears in Equation \(\eqref{eq:main-2}_1\), which contributes the main difficulties in the elliptic estimates (see Section \ref{Elliptic estimates}).
One the other hand, it follows from \eqref{r4} that
\begin{align}\label{rIB}
\rho_0v_t+{\frac{(\rho_0)_x}{\eta_x}}-{\frac{\rho_0\eta_{xx}}{\eta_x^2}}=0.
\end{align}
One can claim that there is no classical solution to \eqref{r4} living in some weighted Sobolev space with high regularity such that
\begin{align*}
\rho_0v_t(\cdot, t),\ \rho_0\eta_{xx}(\cdot, t)\in H^2(I)\quad \text{for}\ t\in (0,T].
\end{align*}
Otherwise, one may argue as Remark \ref{re:main-2} for \eqref{rIB} to deduce
\begin{align*}
(\rho_0)_x=0,
\end{align*}
which contradicts \eqref{eq:intro-3}.
\end{remark}
\begin{remark}\label{re:main-5}
In \cite{MR2779087}, to construct the approximate solutions of \eqref{r3.5} with \(\gamma=2\), Coutand-Shkoller used the following parabolic \(\kappa\)-problem:
\begin{equation}\label{r6}
\begin{cases}
\rho_0v_t+\big({\frac{\rho_0^2}{\eta_x^2}}\big)_x
=\kappa(\rho_0^2v_x)_x &\quad \mbox{in}\ I\times (0,T],\\
(v,\eta)=(u_0,e) &\quad \mbox{on}\ I\times \{t=0\}
\end{cases}
\end{equation}
for small \( \kappa>0\). To show the existence of solutions to the problem \eqref{r6}, the authors considered its linearized problem
\begin{equation}\label{r7}
\begin{cases}
\rho_0v_t+\big({\frac{\rho_0^2}{\bar{\eta}_x^2}}\big)_x
=\kappa(\rho_0^2v_x)_x &\quad \mbox{in}\ I\times (0,T],\\
(v,\eta)=(u_0,e) &\quad \mbox{on}\ I\times \{t=0\},
\end{cases}
\end{equation}
where
\[\bar{\eta}(x,s)=x+\int_0^t\bar{v}(x,s)\,\mathrm{d} s\]
for \(\bar{v}\) in some Hilbert space \(\mathcal{C}_T(M)\). The solution to the parabolic \(\kappa\)-problem \eqref{r6} will then be obtained as a fixed point of the map \(\bar{v}\mapsto v\) (\(v\) is a unique solution to the problem \eqref{r7}) in \(\mathcal{C}_T(M)\) for small \(T>0\) via the Tychonoff fixed-point theorem (which requires that the solution space is a reflexive separable Banach space).
To show the existence of solutions to the problem \eqref{eq:main-2}, we also need to consider its linearized problem \eqref{existence-3}. However, the solution space (defined by \eqref{solution space}) for the problem \eqref{existence-3} (which is the same one with the problem \eqref{eq:main-2}) is a non-reflexive Banach space, which prevents us applying the Tychonoff fixed-point theorem straightforwardly to obtain the existence of solutions to the problem \eqref{eq:main-2}. To get around the difficulty, we will design a contraction mapping for the approximate solutions to the iteration problem \eqref{existence-21} and show its approximate solutions converge uniformly to a classical solution to the problem \eqref{eq:main-2}, in which some weighted interpolation inequality is needed to overcome the difficulty of passing limit in \(n\) on the approximate solutions to the iteration problem \eqref{existence-21}
for time pointwisely, which is caused by the degeneracy in the energy estimates (see Section \ref{classical solution}).
\end{remark}
\section{Some Preliminaries}\label{Some Preliminaries}
\subsection{Weighted Sobolev inequalities}
To handle the degeneracy near the vacuum boundary, we will need the following weighted Sobolev inequalities, whose proof can be found for instance in \cite{MR0802206}. Let \(d(x)=\colon d(x,\Gamma)\) be the distant function to the boundary \(\Gamma\). Then the following weighted Sobolev inequalities hold:
\begin{equation}\label{ineq:weighted Sobolev-0}
\begin{aligned}
\|w\|_{H^{1/2}(I)}^2\lesssim \int_Id(x)(w^2+w_x^2)(x)\,\mathrm{d} x,
\end{aligned}
\end{equation}
\begin{equation}\label{ineq:weighted Sobolev}
\begin{aligned}
\int_Id^k(x)w^2(x)\,\mathrm{d} x\lesssim \int_Id^{k+2}(x)(w^2+w_x^2)(x)\,\mathrm{d} x\quad \mbox{for}\ k=0,1,2,...,
\end{aligned}
\end{equation}
here and thereafter the convention \(\cdot\lesssim\cdot\) denotes \(\cdot\leq C\cdot \), and \(C\) always denotes a nonnegative universal constant which may be different from line to line.
Recall that the initial \emph{height} profile \(\rho_0(x)\) connects to vacuum as \eqref{eq:intro-3}, so the distance function \(d(x)\) can be replaced by \(\rho_0(x)\) in the weighted Sobolev inequalities \eqref{ineq:weighted Sobolev-0} and \eqref{ineq:weighted Sobolev}.
\subsection{Sobolev embedding}
The standard Sobolev embedding inequality
\begin{equation}\label{Sobolev inequaty}
\begin{aligned}
\|w\|_{L^{2/(1-2s)}(I)}\lesssim \|w\|_{H^s(I)}\quad \mbox{for}\ 0<s<1/2,
\end{aligned}
\end{equation}
will also be used.
\subsection{Consequences of \eqref{higher-order energy function}}\label{consequency of higher-order energy function}
As a prerequisite for later use, we will use the weighted Sobolev inequality \eqref{ineq:weighted Sobolev} to deduce some useful consequences of the boundness of the energy functional defined in \eqref{higher-order energy function}.
\begin{lemma}\label{le:Preliminary-1} It holds that
\begin{align}\label{Preliminary-1}
\|v(\cdot,t)\|_{H^3(I)}\lesssim E^{1/2}(t,v).
\end{align}
As a consequence, if \eqref{fluid particle} and \eqref{Lagrangian variable} hold, then
\begin{align}
&\|\eta_{xx}(\cdot,t)\|_{L^2(I)}+\|\partial_x^3\eta(\cdot,t)\|
_{L^2(I)}\lesssim t\sup_{0\leq s\leq t}E^{1/2}(t,v),\label{Preliminary-2}\\
&\|v_x(\cdot,t)\|_{L^\infty(I)}
+\|v_{xx}(\cdot,t)\|_{L^\infty(I)}\lesssim E^{1/2}(t,v),\label{Preliminary-3}\\
&\|\eta_{xx}(\cdot,t)\|_{L^\infty(I)}\lesssim t\sup_{0\leq s\leq t}E^{1/2}(s,v)\label{Preliminary-5}.
\end{align}
\end{lemma}
\begin{proof}
Indeed, it follows from the weighted Sobolev inequality \eqref{ineq:weighted Sobolev} that
\begin{equation*}
\begin{aligned}
\int_I v^2\,\mathrm{d} x\lesssim \int_I\rho_0^2(v^2+v_x^2)\,\mathrm{d} x\lesssim E(t,v),
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\int_I v_x^2\,\mathrm{d} x\lesssim \int_I\rho_0^2(v_x^2+v_{xx}^2)\,\mathrm{d} x\lesssim E(t,v),
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
&\int_I v_{xx}^2\,\mathrm{d} x\lesssim \int_I\rho_0^2[(v_{xx}^2+(\partial_x^3v)^2]\,\mathrm{d} x\\
&\lesssim E(t,v)+\int_I\rho_0^4[(\partial_x^3v)^2+(\partial_x^4v)^2]\,\mathrm{d} x
\lesssim E(t,v),
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
&\int_I (\partial_x^3v)^2\,\mathrm{d} x\lesssim \int_I\rho_0^2[(\partial_x^3v)^2+(\partial_x^4v)^2]\,\mathrm{d} x\\
&\lesssim \int_I\rho_0^4[(\partial_x^3v)^2+(\partial_x^4v)^2]\,\mathrm{d} x
+\int_I\rho_0^4[(\partial_x^4v)^2+(\partial_x^5v)^2]\,\mathrm{d} x\\
&\lesssim E(t,v)+\int_I\rho_0^6[(\partial_x^5v)^2+(\partial_x^6v)^2]\,\mathrm{d} x\lesssim E(t,v).
\end{aligned}
\end{equation*}
Hence \eqref{Preliminary-1} follows.
For \eqref{Preliminary-2}, it follows from \eqref{Preliminary-1} that
\begin{equation*}
\begin{aligned}
\|\partial_x^k\eta(\cdot,t)\|_{L^2(I)}&\leq \int_0^t\left(\int_I (\partial_x^kv)^2\,\mathrm{d} x\right)^{1/2}\mathrm{d} s
&\lesssim t \sup_{0\leq s\leq t}E^{1/2}(t,v)),\ k=2,3,
\end{aligned}
\end{equation*}
where one has used Minkowski's inequality in the first inequality.
The inequality \eqref{Preliminary-3} is a consequence of \eqref{Preliminary-1} and the Sobolev embedding \(H^1(I)\hookrightarrow L^\infty(I)\). Then the inequality \eqref{Preliminary-5} may be shown as
\begin{align*}
\|\eta_{xx}(\cdot,t)\|_{L^\infty(I)}\leq \int_0^t\|v_{xx}(\cdot,s)\|_{L^\infty(I)}\mathrm{d} s\leq t\sup_{0\leq s\leq t}E^{1/2}(s,v).
\end{align*}
\end{proof}
Similarly, one also has
\begin{lemma}\label{le:Preliminary-2} It holds that
\begin{align}\label{Preliminary-6}
\|\rho_0\partial_x^4v(\cdot,t)\|_{L^2(I)}+\|\rho_0^2\partial_x^5v(\cdot,t)\|_{L^2(I)}+\|\rho_0^3\partial_x^6v(\cdot,t)\|_{L^2(I)}\lesssim E^{1/2}(t,v).
\end{align}
As a consequence, if \eqref{fluid particle} and \eqref{Lagrangian variable} hold, then
\begin{align}
&\|\rho_0\partial_x^4\eta(\cdot,t)\|_{L^2(I)}+\|\rho_0^2\partial_x^5\eta(\cdot,t)\|_{L^2(I)}
+\|\rho_0^3\partial_x^6\eta(\cdot,t)\|_{L^2(I)}
\lesssim t\sup_{0\leq s\leq t}E^{1/2}(t,v),\label{Preliminary-7}\\
&\|\rho_0\partial_x^3v(\cdot,t)\|_{L^\infty(I)}+\|\rho_0^2\partial_x^4v(\cdot,t)\|_{L^\infty(I)}+\|\rho_0^3\partial_x^5v(\cdot,t)\|_{L^\infty(I)}\lesssim E^{1/2}(t,v),\label{Preliminary-8}\\
&\|\rho_0\partial_x^3\eta(t,\cdot)\|_{L^\infty(I)}
+\|\rho_0^2\partial_x^4\eta(\cdot,t)\|_{L^\infty(I)}+\|\rho_0^3\partial_x^5\eta(\cdot,t)\|_{L^\infty(I)}\lesssim t\sup_{0\leq s\leq t}E^{1/2}(s,v)\label{Preliminary-9}.
\end{align}
\end{lemma}
\begin{proof}
The proof follows a similar procedure as in that of Lemma \ref{le:Preliminary-1} by repeating using the weighted Sobolev inequality \eqref{ineq:weighted Sobolev}.
\end{proof}
\subsection{The a priori assumption.}\label{The a priori assumption} Let \(c_1\) be the Sobolev embedding \(H^1(I)\hookrightarrow L^\infty(I)\) constant, and \(c_2\) be the constant in the inequality \eqref{Preliminary-1}. Set \(M_1=2M_0\).
Let \((v,\eta)\) satisfy \eqref{fluid particle} and \eqref{Lagrangian variable}. Assume that there exists some suitably small \(T\in(0,1/(2c_1c_2\sqrt{M_1})]\cap (0,1)\) such that
\begin{align}\label{a priori assumption}
\sup_{0\leq t\leq T}E(t,v)\leq M_1.
\end{align}
Then one has
\begin{align}\label{eta-bound}
1/2\leq \eta_x(x,t)\leq 3/2, \quad (x,t)\in I\times [0,T].
\end{align}
Indeed, it follows from \eqref{fluid particle} that
\begin{align*}
\eta(x,t)=x+\int_0^tv(x,s)\,\mathrm{d} s,\quad (x,t)\in I\times [0,T],
\end{align*}
which leads to
\begin{equation*}
\begin{aligned}
|\eta_x(x,t)-1|&\leq\int_0^t\|v_x(\cdot,s)\|_{L^\infty(I)}\,\mathrm{d} s\leq T\sup_{0\leq t\leq T}\|v_x(\cdot,t)\|_{L^\infty(I)}\\
&\leq c_1T\sup_{0\leq t\leq T}\|v_x(\cdot,t)\|_{H^1(I)}\leq c_1c_2T\sup_{0\leq t\leq T}E^{1/2}(t,v)\\
&\leq c_1c_2\sqrt{M_1}T\leq 1/2, \quad (x,t)\in I\times [0,T].
\end{aligned}
\end{equation*}
Hence \eqref{eta-bound} follows.\\
\begin{remark}\label{determin M_1 and T} The a priori assumption \eqref{a priori assumption} will be closed by the a priori bound \eqref{I-Priori-ellip-33}.
\end{remark}
\section{Energy Estimates}\label{Energy Estimates}
This section is devoted to deducing some basic energy estimates on time-derivatives. Let \((v,\eta)\) be a solution to the problem \eqref{eq:main-2} satisfying \eqref{a priori assumption}.\\
\noindent{\bf{Estimate of \(\sum_{k=0}^3\|\sqrt{\rho_0}\partial_t^kv\|_{L^2(I)}\).}}
We first estimate \(\|\sqrt{\rho_0}\partial_t^3v\|_{L^2(I)}\). To this end,
one can apply \(\partial_t^3\) to Equation \(\eqref{eq:main-2}_1\), multiplying it by \(\partial_t^3v\), after some elementary computations, to obtain that
\begin{equation}\label{II-Priori-time-1}
\begin{aligned}
&\frac{1}{2}\int_I \rho_0(\partial_t^3v)^2\,\mathrm{d} x
+\int_0^t\int_I\frac{\rho_0(\partial_t^3v_x)^2}{\eta_x^2}\,\mathrm{d} x\mathrm{d} s\\
&\quad=\frac{1}{2}\int_I \rho_0(\partial_t^3v)^2(x,0)\,\mathrm{d} x+\int_0^t\int_I\partial_t^3\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\partial_t^3v_x\,\mathrm{d} x\mathrm{d} s\\
&\qquad-\int_0^t\int_I\bigg[\partial_t^3\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)\partial_t^3v_x
-\frac{\rho_0(\partial_t^3v_x)^2}{\eta_x^2}\bigg]\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
Using \eqref{eta-bound}, one finds that
\begin{equation}\label{Add-1}
\begin{aligned}
\bigg|\partial_t^3\bigg(\frac{1}{\eta_x^2}\bigg)\bigg|\lesssim |\partial_t^2v_x|+|v_x\partial_tv_x|+|v_x|^3,
\end{aligned}
\end{equation}
and
\begin{equation}\label{Add-2}
\begin{aligned}
\bigg|\partial_t^3\bigg(\frac{v_x}{\eta_x^2}\bigg)\partial_t^3v_x
-\frac{(\partial_t^3v_x)^2}{\eta_x^2}\bigg|\lesssim \big[|v_x\partial_t^2v_x|+|\partial_tv_x|(v_x^2+|\partial_tv_x|)+|v_x|^4\big]|\partial_t^3v_x|.
\end{aligned}
\end{equation}
Then one may use Cauchy's inequality to get
\begin{equation}\label{II-Priori-time-2}
\begin{aligned}
&\bigg|\int_0^t\int_I\partial_t^3\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\partial_t^3v_x\,\mathrm{d} x\mathrm{d} s\bigg|\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s
+C\int_0^t\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\qquad+C\int_0^t\|v_x\|_{L^\infty}^2\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s
+C\int_0^t\|v_x\|_{L^\infty}^4\int_I\rho_0v_x^2\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
and
\begin{equation}\label{II-Priori-time-3}
\begin{aligned}
&\bigg|\int_0^t\int_I\bigg[\partial_t^3(\frac{\rho_0v_x}{\eta_x^2})\partial_t^3v_x
-{\frac{\rho_0(\partial_t^3v_x)^2}{\eta_x^2}}\bigg]\,\mathrm{d} x\mathrm{d} s\bigg|\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s
+C\int_0^t\|v_x\|_{L^\infty}^2\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\qquad+C\int_0^t\|v_x\|_{L^\infty}^4\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s
+C\int_0^t\|\partial_tv_x\|_{L^\infty}^2\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\qquad+C\int_0^t\|v_x\|_{L^\infty}^6\int_I\rho_0v_x^2\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where \eqref{Preliminary-3} was used in \eqref{II-Priori-time-2} and \eqref{II-Priori-time-3}, while,
\begin{equation*}
\begin{aligned}
\|\partial_tv_x\|_{L^\infty}&\lesssim \|\partial_tv_x\|_{L^2}+\|\partial_tv_{xx}\|_{L^2}\\
&\lesssim \|\rho_0\partial_tv_x\|_{L^2}+\|\rho_0\partial_tv_{xx}\|_{L^2}+\|\rho_0\partial_t\partial_x^3v\|_{L^2}\\
&\lesssim E^{1/2}(s,v)+\|\rho_0^2\partial_t\partial_x^3v\|_{L^2}+\|\rho_0^2\partial_t\partial_x^4v\|_{L^2}
\lesssim E^{1/2}(s,v)
\end{aligned}
\end{equation*}
was used in \eqref{II-Priori-time-3}, here the weighted Sobolev inequality \eqref{ineq:weighted Sobolev} was utilized. Here and thereafter \(P(\cdot)\) denotes a generic polynomial function of its arguments.
Due to the bound \eqref{eta-bound}, and noting that the term \(\int_0^t\int_I\frac{\rho_0(\partial_t^3v_x)^2}{\eta_x^2}\,\mathrm{d} x\mathrm{d} s\) on the left hand side (which will be abbreviated as LHS from now on) of \eqref{II-Priori-time-1} is bounded from below by \(\frac{4}{9}\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s\),
hence one inserts \eqref{II-Priori-time-2} and \eqref{II-Priori-time-3} into \eqref{II-Priori-time-1} to obtain
\begin{equation}\label{II-Priori-time-4}
\begin{aligned}
\int_I \rho_0(\partial_t^3v)^2\,\mathrm{d} x
+\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Next, we estimate \(\|\sqrt{\rho_0}\partial_t^2v\|_{L^2(I)}\). Since
\begin{equation*}
\begin{aligned}
\partial_t^2v(x,t)=\partial_t^2v(x,0)+\int_0^t\partial_t^3v(x,s)\,\mathrm{d} s,
\end{aligned}
\end{equation*}
it then follows from Cauchy's inequality and Fubini's theorem that
\begin{equation}\label{I-Priori-time-12}
\begin{aligned}
\int_I \rho_0(\partial_t^2v)^2\,\mathrm{d} x
&\lesssim \int_I \rho_0(\partial_t^2v)^2(x,0)\,\mathrm{d} x
+t\int_0^t\int_I \rho_0(\partial_t^3v)^2\,\mathrm{d} x\mathrm{d} s\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where \eqref{II-Priori-time-4} has been used in the last line. Similarly, by \eqref{I-Priori-time-12}, one can get
\begin{equation}\label{I-Priori-time-8}
\begin{aligned}
\int_I \rho_0(\partial_tv)^2\,\mathrm{d} x
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
and
\begin{equation}\label{I-Priori-time-3}
\begin{aligned}
\int_I \rho_0v^2\,\mathrm{d} x
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\sum_{k=0}^2\|\sqrt{\rho_0}\partial_t^kv_x\|_{L^2(I)}\).}}
We start with \(\|\sqrt{\rho_0}\partial_t^2v_x\|_{L^2(I)}\).
Applying \(\partial_t^2\) to Equation \(\eqref{eq:main-2}_1\),
and multiplying it by \(\partial_t^3v\), one gets by some direct calculations that
\begin{equation}\label{II-Priori-time-5}
\begin{aligned}
&\int_0^t\int_I \rho_0(\partial_t^3v)^2\,\mathrm{d} x\mathrm{d} s
+\frac{1}{2}\int_I\frac{\rho_0(\partial_t^2v_x)^2}{\eta_x^2}\,\mathrm{d} x\\
&\quad=\frac{1}{2}\int_I\rho_0(\partial_t^2v_x)^2(x,0)\,\mathrm{d} x-2\int_0^t\int_I\bigg(-3\frac{\rho_0^2v_x^2}{\eta_x^4}
+\frac{\rho_0^2\partial_tv_x}{\eta_x^3}\bigg)\partial_t^3v_x\,\mathrm{d} x\mathrm{d} s\\
&\qquad-\int_0^t\int_I\frac{\rho_0v_x(\partial_t^2v_x)^2}{\eta_x^3}\,\mathrm{d} x\mathrm{d} s
-6\int_0^t\int_I\frac{\rho_0v_x^3\partial_t^3v_x}{\eta_x^4}\,\mathrm{d} x\mathrm{d} s \\
&\qquad +6\int_0^t\int_I\frac{\rho_0v_x\partial_tv_x\partial_t^3v_x}{\eta_x^3}\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
The above three terms on the right hand side (which will be abbreviated as RHS from now on) of \eqref{II-Priori-time-5} can be estimated as follows:
\begin{equation}\label{II-Priori-time-6}
\begin{aligned}
&\bigg|\int_0^t\int_I\bigg(-3\frac{\rho_0^2v_x^2}{\eta_x^4}
+\frac{\rho_0^2\partial_tv_x}{\eta_x^3}\bigg)\partial_t^3v_x\,\mathrm{d} x\mathrm{d} s\bigg|\\
&\quad\lesssim \int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s+\int_0^t\|v_x\|_{L^\infty}^2\int_I\rho_0v_x^2\,\mathrm{d} x\mathrm{d} s\\
&\qquad+\int_0^t\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
\begin{equation}\label{II-Priori-time-7}
\begin{aligned}
\bigg|\int_0^t\int_I\frac{\rho_0v_x(\partial_t^2v_x)^2}{\eta_x^3}\,\mathrm{d} x\mathrm{d} s\bigg|
&\lesssim \int_0^t\|v_x\|_{L^\infty}\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
\begin{equation}\label{II-Priori-time-8}
\begin{aligned}
\bigg|\int_0^t\int_I\frac{\rho_0v_x^3\partial_t^3v_x}{\eta_x^4}\,\mathrm{d} x\mathrm{d} s\bigg|
&\lesssim \int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\quad+\int_0^t\|v_x\|_{L^\infty}^4\int_I\rho_0v_x^2\,\mathrm{d} x\mathrm{d} s\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
and
\begin{equation}\label{II-Priori-time-9}
\begin{aligned}
\bigg|\int_0^t\int_I\frac{\rho_0v_x\partial_tv_x\partial_t^3v_x}{\eta_x^3}\,\mathrm{d} x\mathrm{d} s\bigg|
&\lesssim \int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\quad+\int_0^t\|v_x\|_{L^\infty}^2\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Here \eqref{Preliminary-3} has been used in the last line of \eqref{II-Priori-time-6}- \eqref{II-Priori-time-9}.
Hence substituting \eqref{II-Priori-time-6}-\eqref{II-Priori-time-9} into \eqref{II-Priori-time-5} yields
\begin{equation}\label{II-Priori-time-10}
\begin{aligned}
\int_0^t\int_I \rho_0(\partial_t^3v)^2\,\mathrm{d} x\mathrm{d} s
+\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
We now consider \(\|\sqrt{\rho_0}\partial_tv_x\|_{L^2(I)}\). Since
\begin{equation*}
\begin{aligned}
\partial_tv_x(x,t)=\partial_tv_x(x,0)+\int_0^t\partial_t^2v_x(x,s)\,\mathrm{d} s,
\end{aligned}
\end{equation*}
it then follows from Cauchy's inequality and Fubini's theorem that
\begin{equation}\label{I-Priori-time-21}
\begin{aligned}
\int_I \rho_0(\partial_tv_x)^2\,\mathrm{d} x
&\lesssim \int_I \rho_0(\partial_tv_x)^2(x,0)\,\mathrm{d} x
+t\int_0^t\int_I \rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))\quad \text{for\ small}\ t>0,
\end{aligned}
\end{equation}
where \eqref{II-Priori-time-10} has been used in the last line.
In view of \eqref{I-Priori-time-21}, one can derive similarly that
\begin{equation}\label{I-Priori-time-16}
\begin{aligned}
\int_I\rho_0v_x^2\,\mathrm{d} x
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\section{Elliptic estimates}\label{Elliptic estimates}
Having the estimates on time-derivatives in Section \ref{Energy Estimates}, we will use the elliptic theory to gain the spatial regularity of the solutions in this section.\\
\noindent{\bf{Estimate of \(\|\rho_0v_{xx}\|_{L^2(I)}\).}}
It follows from Equation \(\eqref{eq:main-2}_1\) that
\begin{equation}\label{I-Priori-ellip-1}
\begin{aligned}
\bigg\|\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)_{x}\bigg\|_{L^2}^2
&\leq \|\rho_0\partial_tv\|_{L^2}^2+\bigg\|\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_x\bigg\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
in which \(\|\rho_0\partial_tv\|_{L^2}\) is bounded by \eqref{I-Priori-time-8}, and the bound on \(\big\|\big({\frac{\rho_0^2}{\eta_x^2}}\big)_x\big\|_{L^2}\) relies on
\begin{equation*}
\begin{aligned}
\bigg|\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_{x}\bigg|\lesssim 1+\rho_0|\eta_{xx}|,
\end{aligned}
\end{equation*}
and hence
\begin{equation}\label{I-Priori-ellip-2}
\begin{aligned}
\bigg\|\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_x\bigg\|_{L^2}
&\lesssim 1+\|\eta_{xx}\|_{L^2}\lesssim 1+t P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
where one has used \eqref{Preliminary-2}.
Next, we estimate \(\|\rho_0v_{xx}\|_{L^2(I)}\). Note that
\begin{equation}\label{I-Priori-ellip-2.2}
\begin{aligned}
\rho_0\eta_x^{-2}v_{xx}+(\rho_0)_x\eta_x^{-2}v_{x}
=\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{x}-
\rho_0(\eta_x^{-2})_{x}v_x.
\end{aligned}
\end{equation}
The last term in \eqref{I-Priori-ellip-2.2} can be estimated as follows:
\begin{equation}\label{I-Priori-ellip-2.4}
\begin{aligned}
\|\rho_0(\eta_x^{-2})_{x}v_x\|_{L^2}
\lesssim \|v_x\|_{L^2}\|\eta_{xx}\|_{L^\infty}
\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where \eqref{Preliminary-1} and \eqref{Preliminary-5} were used.
We then insert \eqref{I-Priori-ellip-1} and \eqref{I-Priori-ellip-2.4} into \eqref{I-Priori-ellip-2.2} to get
\begin{equation}\label{I-Priori-ellip-3}
\begin{aligned}
\|\rho_0\eta_x^{-2}v_{xx}+(\rho_0)_x\eta_x^{-2}v_{x}\|_{L^2}^2
\leq M_0+CtP(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Integration by parts yields
\begin{equation}\label{I-Priori-ellip-4}
\begin{aligned}
&\|\rho_0\eta_x^{-2}v_{xx}\|_{L^2}^2\\
&\quad= \|\rho_0\eta_x^{-2}v_{xx}+(\rho_0)_x\eta_x^{-2}v_{x}\|_{L^2}^2\\
&\qquad-\|(\rho_0)_x\eta_x^{-2}v_{x}\|_{L^2}^2
-\int_I\rho_0(\rho_0)_x\eta_x^{-4}(v_x^2)_x\,\mathrm{d} x\\
&\quad=\|\rho_0\eta_x^{-2}v_{xx}+(\rho_0)_x\eta_x^{-2}v_{x}\|_{L^2}^2+
\int_I\rho_0[(\rho_0)_x\eta_x^{-4}]_{x}v_x^2\,\mathrm{d} x\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where one has used \eqref{I-Priori-ellip-3} and the estimate
\begin{equation}\label{additial estimate}
\begin{aligned}
&\bigg|\int_I\rho_0[(\rho_0)_x\eta_x^{-4}]_xv_x^2\,\mathrm{d} x\bigg|\\
&\quad\lesssim \bigg|\int_I\rho_0(\rho_0)_{xx}\eta_x^{-4}v_x^2\,\mathrm{d} x\bigg|
+\bigg|\int_I\rho_0(\rho_0)_x\eta_x^{-5}\eta_{xx}v_x^2\,\mathrm{d} x\bigg|\\
&\quad\lesssim
(1+\|\eta_{xx}\|_{L^\infty})\int_I\rho_0v_x^2\,\mathrm{d} x\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Here \eqref{Preliminary-5} and \eqref{I-Priori-time-16} have been used.
It follows from \eqref{eta-bound} and \eqref{I-Priori-ellip-4} that
\begin{equation}\label{I-Priori-ellip-5}
\begin{aligned}
\|\rho_0v_{xx}\|_{L^2}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0^{3/2}\partial_x^3v\|_{L^2(I)}\).}}
First, it follows from \eqref{ineq:weighted Sobolev}, \eqref{I-Priori-time-8} and \eqref{I-Priori-time-21} that
\begin{equation}\label{I-Priori-ellip-6}
\begin{aligned}
\|(\rho_0\partial_tv)_x\|_{L^2}^2
&\lesssim \|\partial_tv\|_{L^2}^2+\|\rho_0\partial_tv_x\|_{L^2}^2\\
&\lesssim \|\rho_0\partial_tv\|_{L^2}^2+\|\rho_0\partial_tv_x\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Since
\begin{equation*}
\begin{aligned}
\bigg|\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_{xx}\bigg|\lesssim 1+\rho_0|\eta_{xx}|+\rho_0^2(|\eta_{xx}|^2+|\partial_x^3\eta|),
\end{aligned}
\end{equation*}
one may estimate by Lemma \ref{le:Preliminary-1} that
\footnote{We can throw away the weight \(\rho_0\) in \(\|\rho_0\eta_{xx}\|_{L^2}\), \(\|\rho_0\eta_{xx}\|_{L^\infty}\), \(\|\rho_0^2\eta_{xxx}\|_{L^2}\) and similar terms later on since we work with the energy functional \(E(t,v)\), see the difference when one works with a lower-order energy functional in Subsection \ref{lower-order energy function}.}
\begin{equation}\label{I-Priori-ellip-7}
\begin{aligned}
\bigg\|\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_{xx}\bigg\|_{L^2}
&\lesssim 1+\|\eta_{xx}\|_{L^2}+\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|\partial_x^3\eta\|_{L^2}\\
&\leq 1+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
It then follows from \eqref{I-Priori-ellip-6} and \eqref{I-Priori-ellip-7} that
\begin{equation}\label{I-Priori-ellip-10}
\begin{aligned}
\bigg\|\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)_{xx}\bigg\|_{L^2}^2
&\leq \|(\rho_0\partial_tv)_x\|_{L^2}^2+\bigg\|\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_{xx}\bigg\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
To estimate \(\|\rho_0^{3/2}\partial_x^3v\|_{L^2(I)}\), we first write
\begin{equation}\label{I-Priori-ellip-11}
\begin{aligned}
\rho_0\eta_x^{-2}\partial_x^3v+2(\rho_0)_x\eta_x^{-2}v_{xx}=\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)_{xx}
-2\rho_0(\eta_x^{-2})_{x}v_{xx}-(\rho_0\eta_x^{-2})_{xx}v_x.
\end{aligned}
\end{equation}
Considering the second term on the RHS of \eqref{I-Priori-ellip-11}, one use \eqref{Preliminary-1} and \eqref{Preliminary-5} to estimate
\begin{equation}\label{I-Priori-ellip-12}
\begin{aligned}
\|\rho_0(\eta_x^{-2})_{x}v_{xx}\|_{L^\infty}
\leq \|v_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}
\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Since
\begin{equation}\label{I-Priori-ellip-13}
\begin{aligned}
|(\rho_0\eta_x^{-2})_{xx}|\lesssim 1+|\eta_{xx}|+\rho_0(|\eta_{xx}|^2+|\partial_x^3\eta|),
\end{aligned}
\end{equation}
the last term on the RHS of \eqref{I-Priori-ellip-11} may be estimated as follows:
\begin{equation}\label{I-Priori-ellip-14}
\begin{aligned}
\|(\rho_0\eta_x^{-2})_{xx}v_x\|_{L^2}
&\lesssim \|v_x\|_{L^2}+
\|v_x\|_{L^\infty}(\|\eta_{xx}\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}
+\|\partial_x^3\eta\|_{L^2})\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2}.
\end{aligned}
\end{equation}
In the last line of \eqref{I-Priori-ellip-14}, one has used \eqref{Preliminary-2}, \eqref{Preliminary-3}, \eqref{Preliminary-5} and the estimate
\begin{equation}\label{I-Priori-ellip-15}
\begin{aligned}
\|v_x\|_{L^2}
&\lesssim \|\rho_0v_x\|_{L^2}+\|\rho_0v_{xx}\|_{L^2}\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation}
which follows from \eqref{I-Priori-time-16} and \eqref{I-Priori-ellip-5}.
Inserting \eqref{I-Priori-ellip-10}, \eqref{I-Priori-ellip-12} and \eqref{I-Priori-ellip-14} into \eqref{I-Priori-ellip-11} yields
\begin{equation}\label{I-Priori-ellip-16}
\begin{aligned}
\|\rho_0\eta_x^{-2}\partial_x^3v+2(\rho_0)_x\eta_x^{-2}v_{xx}\|_{L^2}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
We then compensate a weight \(\rho_0^{1/2}\) and integrate by parts to deduce that
\begin{equation}\label{I-Priori-ellip-17}
\begin{aligned}
&\|\rho_0^{3/2}\eta_x^{-2}\partial_x^3v\|_L^2\\
&\quad= \|\rho_0^{3/2}\eta_x^{-2}\partial_x^3v+2\rho_0^{1/2}(\rho_0)_x\eta_x^{-2}v_{xx}\|_{L^2}^2
-4\|\rho_0^{1/2}(\rho_0)_x\eta_x^{-2}v_{xx}\|_{L^2}^2\\
&\qquad-2\int_I\rho_0^2(\rho_0)_x\eta_x^{-4}[(v_{xx})^2]_x\,\mathrm{d} x\\
&\quad= \|\rho_0^{3/2}\eta_x^{-2}\partial_x^3v+2\rho_0^{1/2}(\rho_0)_x\eta_x^{-2}v_{xx}\|_{L^2}^2
+\int_I\rho_0^2[(\rho_0)_x\eta_x^{-4}]_{x}v_{xx}^2\,\mathrm{d} x\\
&\quad\lesssim \|\rho_0^{3/2}\eta_x^{-2}\partial_x^3v+2\rho_0^{1/2}(\rho_0)_x\eta_x^{-2}v_{xx}\|_{L^2}^2
+(1+\|\eta_{xx}\|_{L^\infty})\int_I\rho_0^2v_{xx}^2\,\mathrm{d} x\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where \eqref{Preliminary-5}, \eqref{I-Priori-ellip-5} and \eqref{I-Priori-ellip-16} have been used.
The inequality \eqref{I-Priori-ellip-17} and the bound \eqref{eta-bound} give
\begin{equation}\label{I-Priori-ellip-third}
\begin{aligned}
\|\rho_0^{3/2}\partial_x^3v\|_{L^2}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0\partial_tv_{xx}\|_{L^2(I)}\).}} We first claim that
\begin{equation}\label{I-Priori-ellip-claim}
\begin{aligned}
\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x\bigg\|_{L^2}^2
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
To verify \eqref{I-Priori-ellip-claim}, we note that
\begin{equation}\label{I-Priori-ellip-20}
\begin{aligned}
\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x=\rho_0\partial_t^2v
+\partial_t\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)_x.
\end{aligned}
\end{equation}
Since
\begin{equation*}
\begin{aligned}
\bigg|\partial_t\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_x\bigg|
\lesssim \rho_0|v_x|+\rho_0^2(|v_x\eta_{xx}|+|v_{xx}|),
\end{aligned}
\end{equation*}
one obtains
\begin{equation}\label{I-Priori-ellip-21}
\begin{aligned}
\bigg\|\partial_t\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_x\bigg\|_{L^2}
&\lesssim \|\rho_0v_x\|_{L^2}+\|v_x\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|\rho_0v_{xx}\|_{L^2}\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation}
where one has used \eqref{Preliminary-2}, \eqref{Preliminary-3}, \eqref{I-Priori-time-16} and \eqref{I-Priori-ellip-5} in the last inequality. Then \eqref{I-Priori-ellip-claim} follows from \eqref{I-Priori-ellip-20}, \eqref{I-Priori-ellip-21} and \eqref{I-Priori-time-12}.
Direct calculations give
\begin{equation}\label{I-Priori-ellip-21.5}
\begin{aligned}
(\rho_0\partial_tv_x)_x
&=\partial_t\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)_x\eta_x^2+2\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)_x
\eta_xv_x\\
&\quad+2\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)\eta_{x}\eta_{xx}
+2\frac{\rho_0v_x}{\eta_x^2}(v_x\eta_{xx}+\eta_{x}v_{xx}).
\end{aligned}
\end{equation}
It follows from \eqref{I-Priori-ellip-claim} and \eqref{I-Priori-ellip-1} that the \(L^2-\) norm of the first two terms on the RHS of \eqref{I-Priori-ellip-21.5} has the desired bound. It suffices to handle the last two terms on the RHS of \eqref{I-Priori-ellip-21.5}. Considering the third term on the RHS of \eqref{I-Priori-ellip-21.5}, by \(H^1(I)\hookrightarrow L^\infty(I)\), one has
\begin{equation*}
\begin{aligned}
\bigg\|\partial_t\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)\eta_{x}\eta_{xx}\bigg\|_{L^2}&\lesssim
\|\eta_{xx}\|_{L^2}\left(\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)\bigg\|_{L^2}
+\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x\bigg\|_{L^2}\right)\\
&\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
where in the last line one has used \eqref{I-Priori-ellip-claim} and the estimate
\begin{equation}\label{I-Priori-ellip-21.6}
\begin{aligned}
\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)\bigg\|_{L^2}
&\lesssim \|\rho_0\partial_tv_x\|_{L^2}+\|v_x\|_{L^\infty}\|\rho_0v_x\|_{L^2}\\
&\leq M_0+C(t+1) P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
which together with \(\|\eta_{xx}\|_{L^2}\) yields the bound \(Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))\) since \(\|\eta_{xx}\|_{L^2}\) contributes a factor \(t\) due to \eqref{Preliminary-2}.
In the last term on the RHS of \eqref{I-Priori-ellip-21.5}, the \(L^2-\) norm of the first part is bounded by \(\|v_x\|_{L^\infty}^2\|\eta_{xx}\|_{L^2}\) which contributes the bound \(Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))\), and the second part can be estimated by \eqref{Sobolev inequaty} as follows:
\begin{equation*}
\begin{aligned}
\bigg\|\frac{\rho_0v_x}{\eta_x^2}\eta_{x}v_{xx}\bigg\|_{L^2}
&\lesssim \|v_x\|_{L^4}\|\rho_0v_{xx}\|_{L^4}\lesssim \|v_x\|_{H^{1/2}}\|\rho_0v_{xx}\|_{H^{1/2}}\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
since each factor in the second inequality enjoys the same bound \([M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2}\). Indeed, one can apply \eqref{ineq:weighted Sobolev-0}, \eqref{ineq:weighted Sobolev}, \eqref{I-Priori-time-16}, \eqref{I-Priori-ellip-5} and \eqref{I-Priori-ellip-third} to deduce
\begin{equation*}
\begin{aligned}
\|v_x\|_{H^{1/2}}&\lesssim \|\rho_0^{1/2}v_x\|_{L^2}+\|\rho_0^{1/2}v_{xx}\|_{L^2}\\
&\lesssim \|\rho_0^{1/2}v_x\|_{L^2}+(\|\rho_0^{3/2}v_{xx}\|_{L^2}+\|\rho_0^{3/2}v_{xxx}\|_{L^2})\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation*}
and similarly,
\begin{equation*}
\begin{aligned}
\|\rho_0v_{xx}\|_{H^{1/2}}&\lesssim \|\rho_0^{1/2}(\rho_0v_{xx})\|_{L^2}+\|\rho_0^{1/2}(\rho_0v_{xx})_x\|_{L^2}\\
&\lesssim \|\rho_0^{1/2}v_{xx}\|_{L^2}+\|\rho_0^{3/2}v_{xxx}\|_{L^2}\\
&\lesssim \|\rho_0^{3/2}v_{xx}\|_{L^2}+\|\rho_0^{3/2}v_{xxx}\|_{L^2}\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2}.
\end{aligned}
\end{equation*}
Taking all the cases into account and noticing
\begin{equation*}
\begin{aligned}
(\rho_0\partial_tv_x)_x=\rho_0\partial_tv_{xx}+(\rho_0)_x\partial_tv_{x},
\end{aligned}
\end{equation*}
one obtains
\begin{equation}\label{I-Priori-ellip-21.7}
\begin{aligned}
\|\rho_0\partial_tv_{xx}+(\rho_0)_x\partial_tv_{x}\|_{L^2}^2
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Then integration by parts yields
\begin{equation}\label{I-Priori-ellip-21.8}
\begin{aligned}
\|\rho_0\partial_tv_{xx}\|_{L^2}^2
&= \|\rho_0\partial_tv_{xx}+(\rho_0)_x\partial_tv_{x}\|_{L^2}^2
-\|(\rho_0)_x\partial_tv_{x}\|_{L^2}^2\\
&\quad-\int_I\rho_0(\rho_0)_x[(\partial_tv_{x})^2]_x\,\mathrm{d} x\\
&= \|\rho_0\partial_tv_{xx}+(\rho_0)_x\partial_tv_{x}\|_{L^2}^2
+\int_I\rho_0(\rho_0)_{xx}(\partial_tv_{x})^2\,\mathrm{d} x\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where \eqref{I-Priori-time-21} and \eqref{I-Priori-ellip-21.7} have been used. Therefore it follows from \eqref{I-Priori-ellip-21.8} that
\begin{equation}\label{I-Priori-ellip-22}
\begin{aligned}
\|\rho_0\partial_tv_{xx}\|_{L^2}^2
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0^2\partial_x^4v\|_{L^2(I)}\).}} Applying \(\partial_x^2\) to Equation \(\eqref{eq:main-2}_1\) gives
\begin{equation}\label{I-Priori-ellip-23}
\begin{aligned}
\partial_x^3\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)=(\rho_0\partial_tv)_{xx}
+\partial_x^3\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg).
\end{aligned}
\end{equation}
A direct calculation shows that
\begin{equation*}
\begin{aligned}
\bigg|\partial_x^3\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\bigg|
&\lesssim 1+|\eta_{xx}|+\rho_0(\eta_{xx}^2+|\partial_x^3\eta|)\\
&\quad+\rho_0^2(|\eta_{xx}|^3+|\eta_{xx}\partial_x^3\eta|+|\partial_x^4\eta|).
\end{aligned}
\end{equation*}
We then may apply Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2} to estimate
\begin{equation}\label{I-Priori-ellip-25}
\begin{aligned}
\bigg\|\partial_x^3\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\bigg\|_{L^2}
&\lesssim 1+\|\eta_{xx}\|_{L^2}+(\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|\partial_x^3\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^2\|\eta_{xx}\|_{L^2}
+\|\eta_{xx}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}
+\|\rho_0\partial_x^4\eta\|_{L^2})\\
&\leq 1+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
On the other hand, it holds that
\begin{equation}\label{I-Priori-ellip-25.5}
\begin{aligned}
\|(\rho_0\partial_tv)_{xx}\|_{L^2}^2
&\leq \|(\rho_0)_{xx}\partial_tv\|_{L^2}^2+2\|(\rho_0)_x\partial_tv_{x}\|_{L^2}^2+\|\rho_0\partial_tv_{xx}\|_{L^2}^2\\
&\lesssim \|\rho_0\partial_tv\|_{L^2}^2+\|\rho_0\partial_tv_{x}\|_{L^2}^2+\|\rho_0\partial_tv_{xx}\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where one has used \eqref{ineq:weighted Sobolev} for \(\|\partial_tv\|_{L^2}\) and \(\|\partial_tv_{x}\|_{L^2}\) in the second inequality, and \eqref{I-Priori-time-8}, \eqref{I-Priori-time-21} and \eqref{I-Priori-ellip-22} in the last inequality.
In view of \eqref{I-Priori-ellip-23}, \eqref{I-Priori-ellip-25} and \eqref{I-Priori-ellip-25.5}, we deduce
\begin{equation}\label{I-Priori-ellip-26}
\begin{aligned}
\bigg\|\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)_{xxx}\bigg\|_{L^2}^2
&\leq \|(\rho_0\partial_tv)_{xx}\|_{L^2}^2
+\bigg\|\partial_x^3\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)\bigg\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
To bound \(\|\rho_0^2\partial_x^4v\big\|_{L^2(I)}\), one notes that
\begin{equation}\label{I-Priori-ellip-27}
\begin{aligned}
\rho_0\eta_x^{-2}\partial_x^4v+3(\rho_0)_x\eta_x^{-2}\partial_x^3v
&=\partial_x^3\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)-3\rho_0(\eta_x^{-2})_x\partial_x^3v\\
&\quad-3(\rho_0\eta_x^{-2})_{xx}v_{xx}-\partial_x^3(\rho_0\eta_x^{-2})v_x.
\end{aligned}
\end{equation}
Considering the second term on the RHS of \eqref{I-Priori-ellip-27}, one may estimate
\begin{equation}\label{I-Priori-ellip-27.2}
\begin{aligned}
\|\rho_0^2(\eta_x^{-2})_{x}\partial_x^3v\|_{L^2}
\lesssim \|\partial_x^3v\|_{L^2}\|\eta_{xx}\|_{L^\infty}
\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where \eqref{Preliminary-1} and \eqref{Preliminary-5} have been utilized.
Recalling \eqref{I-Priori-ellip-13}, we may estimate the third term on the RHS of \eqref{I-Priori-ellip-27} as follows:
\begin{equation}\label{I-Priori-ellip-27.5}
\begin{aligned}
\|\rho_0(\rho_0\eta_x^{-2})_{xx}v_{xx}\|_{L^2}
&\lesssim \|\rho_0v_{xx}\|_{L^2}+
\|v_{xx}\|_{L^\infty}\big(\|\eta_{xx}\|_{L^2}+\\
&\quad(\|\eta_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}
+\|\partial_x^3\eta\|_{L^2})\big)\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2}.
\end{aligned}
\end{equation}
Since
\begin{equation}\label{I-Priori-ellip-28}
\begin{aligned}
|\partial_x^3(\rho_0\eta_x^{-2})|
&\lesssim 1+|\eta_{xx}|+(\eta_{xx}^2+|\partial_x^3\eta|)\\
&\quad+\rho_0(|\eta_{xx}|^3+|\eta_{xx}\partial_x^3\eta|+|\partial_x^4\eta|),
\end{aligned}
\end{equation}
the last term on the RHS of \eqref{I-Priori-ellip-27} can be estimated as follows:
\begin{equation}\label{I-Priori-ellip-28.5}
\begin{aligned}
&\|\rho_0\partial_x^3(\rho_0\eta_x^{-2})v_{x}\|_{L^2}\\
&\quad\lesssim \|\rho_0v_{x}\|_{L^2}+
\|v_{x}\|_{L^\infty}\big(\|\eta_{xx}\|_{L^2}
+(\|\eta_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}
+\|\partial_x^3\eta\|_{L^2})\\
&\qquad+(\|\eta_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}^2
+\|\eta_{xx}\|_{L^2}\|\rho_0\partial_x^3\eta\|_{L^\infty}
+\|\rho_0\partial_x^4\eta\|_{L^2})\big)\\
&\quad\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation}
where \eqref{I-Priori-time-16}, Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2} have been used.
Thus inserting \eqref{I-Priori-ellip-26}, \eqref{I-Priori-ellip-27.2}, \eqref{I-Priori-ellip-27.5} and \eqref{I-Priori-ellip-28.5} into \eqref{I-Priori-ellip-27} yields
\begin{equation}\label{I-Priori-ellip-29}
\begin{aligned}
\|\rho_0^2\eta_x^{-2}\partial_x^4v+3\rho_0(\rho_0)_x\eta_x^{-2}\partial_x^3v\|_{L^2}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
We then use integration by parts and invoke \eqref{Preliminary-5}, \eqref{I-Priori-ellip-third} and \eqref{I-Priori-ellip-29} to find
\begin{equation}\label{I-Priori-ellip-30}
\begin{aligned}
&\|\rho_0^2\eta_x^{-2}\partial_x^4v\|_{L^2}^2\\
&\quad=\|\rho_0^2\eta_x^{-2}\partial_x^4v+3\rho_0(\rho_0)_x\eta_x^{-2}\partial_x^3v\|_{L^2}^2
-9\|\rho_0(\rho_0)_x\eta_x^{-2}\partial_x^3v\|_{L^2}^2\\
&\qquad-3\int_I\rho_0^3(\rho_0)_x\eta_x^{-4}[(\partial_x^3v)^2]_x\,\mathrm{d} x\\
&\quad= \|\rho_0^2\eta_x^{-2}\partial_x^4v+3\rho_0(\rho_0)_x\eta_x^{-2}\partial_x^3v\|_{L^2}^2+\int_I\rho_0^3[(\rho_0)_x\eta_x^{-4}]_x(\partial_x^3v)^2\,\mathrm{d} x\\
&\quad\lesssim \|\rho_0^2\eta_x^{-2}\partial_x^4v+3\rho_0(\rho_0)_x\eta_x^{-2}\partial_x^3v\|_{L^2}^2+(1+\|\eta_{xx}\|_{L^\infty})
\int_I\rho_0^3(\partial_x^3v)^2\,\mathrm{d} x\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Hence \eqref{I-Priori-ellip-30} and \eqref{eta-bound} imply
\begin{equation}\label{I-Priori-ellip-31}
\begin{aligned}
\|\rho_0^2\partial_x^4v\|_{L^2}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0^{3/2}\partial_t\partial_x^3v\|_{L^2(I)}\).}}
Since
\begin{equation*}
\begin{aligned}
\bigg|\partial_t\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_{xx}\bigg|
&\lesssim |v_x|+\rho_0(|v_x\eta_{xx}|+|v_{xx}|)\\
&\quad+\rho_0^2(|v_x\eta_{xx}^2|+|v_x\partial_x^3\eta|+|v_{xx}\eta_{xx}|+|\partial_x^3v|),
\end{aligned}
\end{equation*}
it holds that
\begin{equation}\label{II-Priori-ellip-1}
\begin{aligned}
\bigg\|\partial_t\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_x\bigg\|_{L^2}
&\lesssim \|v_x\|_{L^2}+\|v_x\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|\rho_0v_{xx}\|_{L^2}\\
&\quad+\|v_x\|_{L^\infty}\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|v_x\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|v_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|\rho_0^2\partial_x^3v\|_{L^2}\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation}
where one has used \eqref{I-Priori-ellip-5}, \eqref{I-Priori-ellip-15}, \eqref{I-Priori-ellip-third}, Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2} in the last inequality.
Applying \(\partial_{tx}^2\) to Equation \(\eqref{eq:main-2}_1\) gives
\begin{equation}\label{II-Priori-ellip-2}
\begin{aligned}
\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{xx}=(\rho_0\partial_t^2v)_x
+\partial_t\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)_{xx},
\end{aligned}
\end{equation}
we then utilize \eqref{I-Priori-time-12}, \eqref{II-Priori-time-10}, \eqref{II-Priori-ellip-1} and \eqref{II-Priori-ellip-2} to estimate
\begin{equation}\label{II-Priori-ellip-3}
\begin{aligned}
\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{xx}\bigg\|_{L^2}^2
&\leq \|(\rho_0)_x\partial_t^2v\|_{L^2}^2+\|\rho_0\partial_t^2v_x\|_{L^2}^2
+\bigg\|\partial_t\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)_{xx}\bigg\|_{L^2}^2\\
&\lesssim \|\rho_0\partial_t^2v\|_{L^2}^2+\|\rho_0\partial_t^2v_x\|_{L^2}^2
+\bigg\|\partial_t\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)_{xx}\bigg\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where \eqref{ineq:weighted Sobolev} has been used for \(\|\partial_t^2v\|_{L^2}\) in the second inequality.
Write
\begin{equation*}
\begin{aligned}
&(\rho_0\partial_tv_x)_{xx}\\
&=\partial_t\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)_{xx}\eta_x^2
+4\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{x}\eta_{x}\eta_{xx}
+2\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)(\eta_{xx}^2+\eta_{x}\partial_x^3\eta)\\
&\quad+2\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)_{xx}\eta_xv_x
+4\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{x}(v_x\eta_{xx}+\eta_{x}v_{xx})\\
&\quad+2\frac{\rho_0v_x}{\eta_x^2}(2v_{xx}\eta_{xx}+v_x\partial_x^3\eta+\eta_{x}\partial_x^3v)=\colon \sum_{k=1}^6I_k.
\end{aligned}
\end{equation*}
It is clear that \(\|I_1\|_{L^2}\) satisfies the desired bound due to \eqref{II-Priori-ellip-3}.
In view of \eqref{Preliminary-5} and \eqref{I-Priori-ellip-21}, one may estimate
\begin{equation*}
\begin{aligned}
\|I_2\|_{L^2}
\lesssim
\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x\bigg\|_{L^2}
\|\eta_{xx}\|_{L^\infty}
\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation*}
The estimates \eqref{I-Priori-ellip-21} and \eqref{I-Priori-ellip-21.6} together with \eqref{Preliminary-2} and
\eqref{Preliminary-5} yield
\begin{equation*}
\begin{aligned}
\|I_3\|_{L^2}
&\lesssim
\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)\bigg\|_{L^\infty}(\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|\partial_x^3\eta\|_{L^2})\\
&\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation*}
It follows from \eqref{I-Priori-ellip-1} and \eqref{I-Priori-ellip-10} that
\begin{equation*}
\begin{aligned}
\|I_4\|_{L^2}\lesssim
\|v_{x}\|_{L^\infty}\bigg\|\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{xx}\bigg\|_{L^2}
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\|I_5\|_{L^2}&\lesssim
\bigg\|\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{x}\bigg\|_{L^\infty}(\|v_{x}\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|v_{xx}\|_{L^2})\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
where one has used \eqref{ineq:weighted Sobolev} to find
\begin{equation}\label{II-Priori-ellip-3.2}
\begin{aligned}
\|v_{xx}\|_{L^2}&\lesssim
\|\rho_0v_{xx}\|_{L^2}+\|\rho_0\partial_x^3v\|_{L^2}\\
&\lesssim
\|\rho_0v_{xx}\|_{L^2}+\|\rho_0^2\partial_x^3v\|_{L^2}+\|\rho_0^2\partial_x^4v\|_{L^2}\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation}
due to \eqref{I-Priori-ellip-5}, \eqref{I-Priori-ellip-third} and \eqref{I-Priori-ellip-31}, and thus
\begin{equation}\label{II-Priori-ellip-3.3}
\begin{aligned}
\|v_x\|_{L^\infty}
&\lesssim
\|v_x\|_{L^2}+\|v_{xx}\|_{L^2}\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation}
which follows from \eqref{I-Priori-ellip-15} and \eqref{II-Priori-ellip-3.2}.
\(I_6\) can be estimated as
\begin{equation*}
\begin{aligned}
\|I_6\|_{L^2}
&\lesssim
\|v_x\|_{L^\infty}(\|v_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|v_{x}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}+\|\rho_0\partial_x^3v\|_{L^2})\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
where, to estimate the term \(\|v_x\|_{L^\infty}\|\rho_0\partial_x^3v\|_{L^2}\), one has used the fact that each term enjoys the bound \([M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2}\), which follows from \eqref{II-Priori-ellip-3.3} and
\begin{equation}\label{II-Priori-ellip-3.4}
\begin{aligned}
\|\rho_0\partial_x^3v\|_{L^2}
&\lesssim
\|\rho_0^2\partial_x^3v\|_{L^2}+\|\rho_0^2\partial_x^4v\|_{L^2}\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation}
due to \eqref{I-Priori-ellip-third} and \eqref{I-Priori-ellip-31}.
Collecting all the cases, we finally get
\begin{equation}\label{II-Priori-ellip-3.5}
\begin{aligned}
\|(\rho_0\partial_tv_x)_{xx}\|_{L^2}^2
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Since
\begin{equation*}
\begin{aligned}
\rho_0\partial_t\partial_x^3v+2(\rho_0)_{x}\partial_tv_{xx}=
(\rho_0\partial_tv_x)_{xx}-(\rho_0)_{xx}\partial_tv_x,
\end{aligned}
\end{equation*}
it follows that
\begin{equation}\label{II-Priori-ellip-3.6}
\begin{aligned}
&\|\rho_0^{3/2}\partial_t\partial_x^3v+2\rho_0^{1/2}(\rho_0)_{x}\partial_tv_{xx}\|_{L^2}^2\\
&\quad\lesssim \|\rho_0^{1/2}(\rho_0\partial_tv_x)_{xx}\|_{L^2}^2+\|\rho_0^{1/2}\partial_tv_x\|_{L^2}^2\\
&\quad\lesssim \|(\rho_0\partial_tv_x)_{xx}\|_{L^2}^2+\|\rho_0^{1/2}\partial_tv_x\|_{L^2}^2\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where one has used \eqref{I-Priori-time-21} and \eqref{II-Priori-ellip-3.5}.
Integration by parts gives
\begin{equation}\label{II-Priori-ellip-3.7}
\begin{aligned}
&\|\rho_0^{3/2}\partial_t\partial_x^3v\|_{L^2}^2\\
&\quad=\|\rho_0^{3/2}\partial_t\partial_x^3v+2\rho_0^{1/2}(\rho_0)_{x}\partial_tv_{xx}\|_{L^2}^2
-4\|\rho_0^{1/2}(\rho_0)_{x}\partial_tv_{xx}\|_{L^2}^2\\
&\qquad-2\int_I\rho_0^2(\rho_0)_x[(\partial_tv_{xx})^2]_x\,\mathrm{d} x\\
&\quad=\|\rho_0^{3/2}\partial_t\partial_x^3v+2\rho_0^{1/2}(\rho_0)_{x}\partial_tv_{xx}\|_{L^2}^2
+2\int_I\rho_0^2(\rho_0)_{xx}(\partial_tv_{xx})^2\,\mathrm{d} x\\
&\quad\lesssim \|\rho_0^{3/2}\partial_t\partial_x^3v+2\rho_0^{1/2}(\rho_0)_{x}\partial_tv_{xx}\|_{L^2}^2
+
\int_I\rho_0^2(\partial_tv_{xx})^2\,\mathrm{d} x\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where one has used \eqref{I-Priori-ellip-22} and \eqref{II-Priori-ellip-3.6}.
Hence it follows from \eqref{II-Priori-ellip-3.7} that
\begin{equation}\label{II-Priori-ellip-4}
\begin{aligned}
\|\rho_0^{3/2}\partial_t\partial_x^3v\|_{L^2}^2
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0^{5/2}\partial_x^5v\|_{L^2(I)}\).}} Applying \(\partial_x^3\) to Equation \(\eqref{eq:main-2}_1\) gives
\begin{equation}\label{II-Priori-ellip-4.5}
\begin{aligned}
\partial_x^4\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)=\partial_x^3(\rho_0\partial_tv)
+\partial_x^4\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg).
\end{aligned}
\end{equation}
We will estimate the \(L^2-\) norm of \(\partial_x^4\big(\frac{\rho_0v_x}{\eta_x^2}\big)\) with {suitable weight using \eqref{II-Priori-ellip-4.5}. We start with the term \(\partial_x^3(\rho_0\partial_tv)\). For this, due to \eqref{II-Priori-ellip-4}, one shall compensate a weight \(\rho_0^{1/2}\) to estimate
\begin{equation}\label{II-Priori-ellip-5}
\begin{aligned}
\|\rho_0^{1/2}\partial_x^3(\rho_0\partial_tv)\|_{L^2}^2
&\lesssim \|\rho_0^{1/2}\partial_tv\|_{L^2}^2+\|\rho_0^{1/2}\partial_tv_x\|_{L^2}^2\\
&\quad+\|\rho_0^{1/2}\partial_tv_{xx}\|_{L^2}^2
+\|\rho_0^{3/2}\partial_t\partial_x^3v\|_{L^2}^2\\
&\lesssim \|\rho_0^{1/2}\partial_tv\|_{L^2}^2+\|\rho_0^{1/2}\partial_tv_x\|_{L^2}^2\\
&\quad+\|\rho_0^{3/2}\partial_tv_{xx}\|_{L^2}^2
+\|\rho_0^{3/2}\partial_t\partial_x^3v\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Here one has used \eqref{ineq:weighted Sobolev} for \(\|\rho_0^{1/2}\partial_tv_{xx}\|_{L^2}\) in the second inequality, and \eqref{I-Priori-ellip-22} and \eqref{II-Priori-ellip-4} in the last equality.
Next, we deal with the term \(\partial_x^4\big({\frac{\rho_0^2}{\eta_x^2}}\big)\).
Direct calculations give
\begin{equation}\label{II-Priori-ellip-6}
\begin{aligned}
\bigg|\partial_x^4\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\bigg|
&\lesssim 1+|\eta_{xx}|+(\eta_{xx}^2+|\partial_x^3\eta|)
+\rho_0(|\eta_{xx}|^3+|\eta_{xx}\partial_x^3\eta|+|\partial_x^4\eta|)\\
&\quad+\rho_0^2(|\eta_{xx}|^4+\eta_{xx}^2|\partial_x^3\eta|+(\partial_x^3\eta)^2+|\eta_{xx}\partial_x^4\eta|+|\partial_x^5\eta|).
\end{aligned}
\end{equation}
Thus, one can get
\begin{equation}\label{II-Priori-ellip-7}
\begin{aligned}
\bigg\|\partial_x^4\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\bigg\|_{L^2}
&\lesssim 1+\|\eta_{xx}\|_{L^2}+(\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|\partial_x^3\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^2\|\eta_{xx}\|_{L^2}
+\|\eta_{xx}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}
+\|\rho_0\partial_x^4\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^3\|\eta_{xx}\|_{L^2}
+\|\eta_{xx}\|_{L^\infty}^2\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}
+\|\eta_{xx}\|_{L^2}\|\rho_0\partial_x^4\eta\|_{L^2}
+\|\rho_0^2\partial_x^5\eta\|_{L^2})\\
&\leq 1+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where in the last inequality Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2} have been utilized.
By compensating a weight \(\rho_0^{1/2}\), we deduce from \eqref{II-Priori-ellip-4.5}, \eqref{II-Priori-ellip-5} and \eqref{II-Priori-ellip-7} that
\begin{equation}\label{II-Priori-ellip-8}
\begin{aligned}
\bigg\|\rho_0^{1/2}\partial_x^4\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)\bigg\|_{L^2}^2
&\leq \|\rho_0^{1/2}\partial_x^3(\rho_0\partial_tv)\|_{L^2}^2
+\bigg\|\rho_0^{1/2}\partial_x^4\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)\bigg\|_{L^2}^2\\
&\lesssim \|\rho_0^{1/2}\partial_x^3(\rho_0\partial_tv)\|_{L^2}^2
+\bigg\|\partial_x^4\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)\bigg\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
We next control \(\|\rho_0^{5/2}\partial_x^5v\|_{L^2}\).
Note that
\begin{equation*}
\begin{aligned}
&\rho_0\eta_x^{-2}\partial_x^5v+4(\rho_0)_x\eta_x^{-2}\partial_x^4v\\
&=\partial_x^4\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)-4\rho_0(\eta_x^{-2})_x\partial_x^4v-6(\rho_0\eta_x^{-2})_{xx}\partial_x^3v\\
&\quad-4\partial_x^3(\rho_0\eta_x^{-2})v_{xx}-\partial_x^4(\rho_0\eta_x^{-2})v_x=\colon \sum_{k=1}^5I_k.
\end{aligned}
\end{equation*}
The term \(I_1\) has been handled by compensating a weight \(\rho_0^{1/2}\) due to \eqref{II-Priori-ellip-8}.
It follows from Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2} that \(I_2\) and \(I_4\) may be estimated as follows:
\begin{equation*}
\begin{aligned}
\|I_2\|_{L^2}
\leq \|\eta_{xx}\|_{L^\infty}\|\rho_0\partial_x^4v\|_{L^2}
\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\|I_4\|_{L^2}
&\lesssim \|v_{xx}\|_{L^2}+
\|v_{xx}\|_{L^\infty}\big(\|\eta_{xx}\|_{L^2}
+(\|\eta_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}
+\|\partial_x^3\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}^2
+\|\eta_{xx}\|_{L^2}\|\rho_0\partial_x^3\eta\|_{L^\infty}
+\|\rho_0\partial_x^4\eta\|_{L^2})\big)\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation*}
where \eqref{I-Priori-ellip-28} and \eqref{II-Priori-ellip-3.2} have been used in estimating \(I_4\).
For \(I_3\) and \(I_5\), one can use a weight \(\rho_0\) and apply Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2} to get
{\small \begin{equation*}
\begin{aligned}
\|\rho_0I_3\|_{L^2}
&\lesssim \|\rho_0\partial_x^3v\|_{L^2}+
\|\rho_0\partial_x^3v\|_{L^\infty}\big(\|\eta_{xx}\|_{L^2}+
(\|\eta_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}
+\|\partial_x^3\eta\|_{L^2})\big)\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation*}}
and
{\small \begin{equation*}
\begin{aligned}
\|\rho_0I_5\|_{L^2}
&\lesssim \|\rho_0v_{x}\|_{L^2}+
\|v_{x}\|_{L^\infty}\big(\|\eta_{xx}\|_{L^2}
+(\|\eta_{xx}\|_{L^2}\|\rho_0\eta_{xx}\|_{L^\infty}
+\|\rho_0\partial_x^3\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}^2
+\|\eta_{xx}\|_{L^2}\|\rho_0\partial_x^3\eta\|_{L^\infty}
+\|\rho_0\partial_x^4\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^2}\|\eta_{xx}\|_{L^\infty}^3
+\|\partial_x^3\eta\|_{L^2}\|\eta_{xx}\|_{L^\infty}^2
+\|\partial_x^3\eta\|_{L^2}\|\rho_0\partial_x^3\eta\|_{L^\infty}\\
&\quad+\|\eta_{xx}\|_{L^2}\|\rho_0^2\partial_x^4\eta\|_{L^\infty}
+\|\rho_0^2\partial_x^5\eta\|_{L^2})\big)\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation*}}
here one has invoked \eqref{I-Priori-ellip-13} and \eqref{II-Priori-ellip-3.4} in estimating \(I_3\), and used
\begin{equation*}
\begin{aligned}
|\partial_x^4(\rho_0\eta_x^{-2})|
&\lesssim 1+|\eta_{xx}|+(\eta_{xx}^2+|\partial_x^3\eta|)
+(|\eta_{xx}|^3+|\eta_{xx}\partial_x^3\eta|+|\partial_x^4\eta|)\\
&\quad+\rho_0(|\eta_{xx}|^4+\eta_{xx}^2|\partial_x^3\eta|+\eta_{xxx}^2+|\eta_{xx}\partial_x^4\eta|+|\partial_x^5\eta|),
\end{aligned}
\end{equation*}
in estimating \(I_5\).
It follows from theses estimates and using a weight \(\rho_0\) that
\begin{equation}\label{II-Priori-ellip-9}
\begin{aligned}
\|\rho_0^2\eta_x^{-2}\partial_x^5v+4\rho_0(\rho_0)_x\eta_x^{-2}\partial_x^4v\|_{L^2}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Then integration by parts leads to
\begin{equation}\label{II-Priori-ellip-10}
\begin{aligned}
&\|\rho_0^{5/2}\eta_x^{-2}\partial_x^5v\|_{L^2}^2\\
&\quad=\|\rho_0^{5/2}\eta_x^{-2}\partial_x^5v
+4\rho_0^{3/2}(\rho_0)_x\eta_x^{-2}\partial_x^4v\|_{L^2}^2\\
&\qquad-16\|\rho_0^{3/2}(\rho_0)_x\eta_x^{-2}\partial_x^4v\|_{L^2}^2-4\int_I\rho_0^4(\rho_0)_x\eta_x^{-4}[(\partial_x^4v)^2]_x\,\mathrm{d} x\\
&\quad= \|\rho_0^{5/2}\eta_x^{-2}\partial_x^5v
+4\rho_0^{3/2}(\rho_0)_x\eta_x^{-2}\partial_x^4v\|_{L^2}^2
+4\int_I\rho_0^4[(\rho_0)_x\eta_x^{-4}]_x(\partial_x^4v)^2\,\mathrm{d} x\\
&\quad\lesssim \|\rho_0^{5/2}\eta_x^{-2}\partial_x^5v
+4\rho_0^{3/2}(\rho_0)_x\eta_x^{-2}\partial_x^4v\|_{L^2}^2
+(1+\|\eta_{xx}\|_{L^\infty})\int_I\rho_0^4(\partial_x^4v)^2\,\mathrm{d} x\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where \eqref{Preliminary-5}, \eqref{I-Priori-ellip-31} and \eqref{II-Priori-ellip-9} have been used.
The inequality \eqref{II-Priori-ellip-10} and \eqref{eta-bound} yield
\begin{equation}\label{II-Priori-ellip-11}
\begin{aligned}
\|\rho_0^{5/2}\partial_x^5v\|_{L^2}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0^2\partial_t\partial_x^4v\|_{L^2(I)}\).}} Applying \(\partial_t\partial_x^2\) to Equation \(\eqref{eq:main-2}_1\) gives
\begin{equation}\label{II-Priori-ellip-11.5}
\begin{aligned}
\partial_t\partial_x^3\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)=(\rho_0\partial_t^2v)_{xx}
+\partial_t\partial_x^3\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg).
\end{aligned}
\end{equation}
Thus, to estimate the \(L^2-\) norm of \(\partial_t\partial_x^3\big(\frac{\rho_0v_x}{\eta_x^2}\big)\), it suffices to estimate \(L^2-\) norm of \(\partial_t\partial_x^3\big({\frac{\rho_0^2}{\eta_x^2}}\big)\) and \((\rho_0\partial_t^2v)_{xx}\).
We start with \(\partial_t\partial_x^3\big({\frac{\rho_0^2}{\eta_x^2}}\big)\).
Since
\begin{equation*}
\begin{aligned}
\bigg|\partial_t\partial_x^3\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\bigg|
&\lesssim |v_x|+(|v_x\eta_{xx}|+|v_{xx}|)
+\rho_0(|v_x\eta_{xx}^2|+|v_x\partial_x^3\eta|\\
&\quad+|v_{xx}\eta_{xx}|+|\partial_x^3v|)
+\rho_0^2(|v_x\eta_{xx}^3|+|v_x\eta_{xx}\partial_x^3\eta|\\
&\quad+|v_x\partial_x^4\eta|+|v_{xx}\eta_{xx}^2|
+|v_{xx}\partial_x^3\eta|+|\partial_x^3v\eta_{xx}|+|\partial_x^4v|),
\end{aligned}
\end{equation*}
one gets
\begin{equation}\label{II-Priori-ellip-12}
\begin{aligned}
\bigg\|\partial_t\partial_x^3\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\bigg\|_{L^2}
&\lesssim \|v_x\|_{L^2}+(\|v_x\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|v_{xx}\|_{L^2})\\
&\quad+(\|v_x\|_{L^\infty}\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|v_x\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|v_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}+
\|\rho_0\partial_x^3v\|_{L^2})\\
&\quad+(\|v_x\|_{L^\infty}\|\eta_{xx}\|_{L^\infty}^2\|\eta_{xx}\|_{L^2}+\|v_x\|_{L^\infty}\|\eta_{xx}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|v_x\|_{L^\infty}\|\rho_0\partial_x^4\eta\|_{L^2}
+\|v_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}\\
&\quad+\|v_{xx}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}+\|\eta_{xx}\|_{L^\infty}\|\partial_x^3v\|_{L^2}+\|\rho_0^2\partial_x^4v\|_{L^2})\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation}
where one has used \eqref{I-Priori-ellip-15}, \eqref{II-Priori-ellip-3.2}, Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2}.
Next, we deal with \((\rho_0\partial_t^2v)_{xx}\).
Applying \(\partial_t^2\) to Equation \(\eqref{eq:main-2}_1\) yields
\begin{equation*}
\begin{aligned}
\partial_t^2\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{x}=\rho_0\partial_t^3v
+\partial_t^2\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)_{x}.
\end{aligned}
\end{equation*}
Since
\begin{equation*}
\begin{aligned}
\bigg|\partial_t^2\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_{x}\bigg|
\lesssim \rho_0(v_x^2+|\partial_tv_x|)+\rho_0^2(v_x^2|\eta_{xx}|
+|\partial_tv_x\eta_{xx}|+|v_{x}v_{xx}|+|\partial_tv_{xx}|),
\end{aligned}
\end{equation*}
one gets
\begin{equation}\label{II-Priori-ellip-13}
\begin{aligned}
\bigg\|\partial_t^2\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_{x}\bigg\|_{L^2}
&\lesssim (\|\rho_0v_x\|_{L^\infty}\|v_x\|_{L^2}+\|\rho_0\partial_tv_x\|_{L^2})\\
&\quad+(\|v_x\|_{L^\infty}^2\|\eta_{xx}\|_{L^2}+\|\rho_0\partial_tv_x\|_{L^2}\|\eta_{xx}\|_{L^\infty}\\
&\quad+\|v_x\|_{L^\infty}\|\rho_0v_{xx}\|_{L^2}+\|\rho_0\partial_tv_{xx}\|_{L^2})\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation}
where one has used the fact that in \(\|\rho_0v_x\|_{L^\infty}\|v_x\|_{L^2}\) and \(\|v_x\|_{L^\infty}\|\rho_0v_{xx}\|_{L^2}\), each factor enjoys the same bound \([M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2}\), due to \eqref{I-Priori-ellip-15}, \eqref{II-Priori-ellip-3.3} and \eqref{I-Priori-ellip-5}.
In view of \eqref{II-Priori-time-4} and \eqref{II-Priori-ellip-13}, we obtain
\begin{equation}\label{II-Priori-ellip-14}
\begin{aligned}
\bigg\|\partial_t^2\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{x}\bigg\|_{L^2}^2
&\lesssim \|\rho_0\partial_t^3v\|_{L^2}^2+\bigg\|\partial_t^2\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_{x}\bigg\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Note that
\begin{equation*}
\begin{aligned}
\rho_0\partial_t^2v_{xx}&=\partial_t^2\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{x}\eta_x^2
-\partial_t^2\bigg[\bigg(\frac{\rho_0}{\eta_x^2}\bigg)_xv_{x}\bigg]\eta_x^2\\
&\quad-\partial_t^2\bigg(\frac{\rho_0}{\eta_x^2}\bigg)v_{xx}\eta_x^2
-\partial_t\bigg(\frac{\rho_0}{\eta_x^2}\bigg)\partial_tv_{xx}\eta_x^2
=\colon\sum_{k=1}^4I_k.
\end{aligned}
\end{equation*}
The estimate on the \(L^2-\) norm of \(I_1\) follows from \eqref{II-Priori-ellip-14}.
The terms \(I_3\) and \(I_4\) can be estimated straightforwardly as follows:
\begin{equation*}
\begin{aligned}
\|I_3\|_{L^2}
&\lesssim \|\rho_0(v_x^2+\partial_tv_x)v_{xx}\|_{L^2}
\lesssim \|v_{x}\|_{L^\infty}^2\|\rho_0v_{xx}\|_{L^2}
+\|\rho_0v_{xx}\|_{L^\infty}\|\partial_tv_x\|_{L^2}\\
&\lesssim \|v_{x}\|_{L^\infty}^2\|\rho_0v_{xx}\|_{L^2}
+\|\rho_0v_{xx}\|_{L^\infty}(\|\rho_0\partial_tv_x\|_{L^2}
+\|\rho_0\partial_tv_{xx}\|_{L^2})\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\|I_4\|_{L^2}
&\lesssim \|\rho_0v_x\partial_tv_{xx}\|_{L^2}
\lesssim \|v_{x}\|_{L^\infty}\|\rho_0\partial_tv_{xx}\|_{L^2}\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
since each factor on the RHS of \(I_3\) and \(I_4\) enjoys the same bound \([M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2}\).
Here one has used \eqref{ineq:weighted Sobolev} for \(\|\partial_tv_x\|_{L^2}\) in the third inequality of \(I_3\), and bounded \(\|\rho_0v_{xx}\|_{L^\infty}\) by \eqref{II-Priori-ellip-3.2} and \eqref{II-Priori-ellip-3.4} in the forth inequality of \(I_3\).
For \(I_2\), we first calculate
{\small \begin{equation*}
\begin{aligned}
\bigg|\partial_t^2\bigg[\bigg(\frac{\rho_0}{\eta_x^2}\bigg)_xv_{x}\bigg]\bigg|
&\lesssim |v_x|\big(v_x^2+|\partial_tv_x|
+\rho_0(|\eta_{xx}|v_x^2
+|v_xv_{xx}|+|\eta_{xx}\partial_tv_x|+|\partial_tv_{xx}|)\big)\\
&\quad+|\partial_tv_x|\big(|v_x|+\rho_0(|\eta_{xx}v_x|+|v_{xx}|)\big)+|\partial_t^2v_x|(1+\rho_0|\eta_{xx}|),
\end{aligned}
\end{equation*}}
and then compensate a weight \(\rho_0\) to estimate
\begin{equation*}
\begin{aligned}
\|\rho_0I_2\|_{L^2}
&\lesssim (\|v_{x}\|_{L^\infty}^2\|v_{x}\|_{L^2}+\|v_{x}\|_{L^\infty}\|\rho_0\partial_tv_x\|_{L^2}
+\|v_{x}\|_{L^\infty}^3\|\eta_{xx}\|_{L^2}\\
&\quad+\|v_{x}\|_{L^\infty}^2\|v_{xx}\|_{L^2}
+\|v_{x}\|_{L^\infty}\|\eta_{xx}\|_{L^\infty}\|\rho_0\partial_tv_x\|_{L^2}\\
&\quad+\|v_{x}\|_{L^\infty}\|\rho_0\partial_tv_{xx}\|_{L^2})\\
&\quad+(\|v_{x}\|_{L^\infty}\|\rho_0\partial_tv_x\|_{L^2}
+\|v_{x}\|_{L^\infty}\|\eta_{xx}\|_{L^\infty}\|\rho_0\partial_tv_x\|_{L^2}\\
&\quad+\|\rho_0v_{xx}\|_{L^\infty}\|\rho_0\partial_tv_x\|_{L^2})\\
&\quad+(\|\rho_0\partial_t^2v_x\|_{L^2}
+\|\eta_{xx}\|_{L^\infty}\|\rho_0\partial_t^2v_x\|_{L^2})\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2}.
\end{aligned}
\end{equation*}
It follows from the estimates in \(I_i,\ i=1,2,3,4\) that
\begin{equation}\label{II-Priori-ellip-15}
\begin{aligned}
\|\rho_0^2\partial_t^2v_{xx}\|_{L^2}^2
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Consequently, \eqref{II-Priori-ellip-11.5}, \eqref{II-Priori-ellip-12} and \eqref{II-Priori-ellip-15} yield
\begin{equation}\label{II-Priori-ellip-16}
\begin{aligned}
\bigg\|\rho_0\partial_t\partial_x^3\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)\bigg\|_{L^2}^2
&\leq \|\rho_0(\rho_0\partial_t^2v)_{xx}\|_{L^2}^2
+\bigg\|\rho_0\partial_t\partial_x^3\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)\bigg\|_{L^2}^2\\
&\lesssim \|\rho_0\partial_t^2v\|_{L^2}^2+\|\rho_0\partial_t^2v_{x}\|_{L^2}^2
+\|\rho_0^2\partial_t^2v_{xx}\|_{L^2}^2\\
&\quad+\bigg\|\partial_t\partial_x^3\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)\bigg\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Next, we derive the weighted \(L^2\) estimate of \(\partial_x^3(\rho_0\partial_tv_x)\).
Note that
\begin{equation*}
\begin{aligned}
&\partial_x^3(\rho_0\partial_tv_x)\\
&=\partial_t\partial_x^3\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)\eta_x^2
+6\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{xx}\eta_{x}\eta_{xx}\\
&\quad+6\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x(\eta_{xx}^2+\eta_{x}\partial_x^3\eta)\\
&\quad+2\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)(3\eta_{xx}\partial_x^3\eta+\eta_{x}\partial_x^4\eta)\\
&\quad+2\partial_x^3\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)\eta_xv_x
+6\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{xx}(v_x\eta_{xx}+\eta_{x}v_{xx})\\
&\quad+6\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{x}(2v_{xx}\eta_{xx}+v_x\partial_x^3\eta+\eta_{x}\partial_x^3v)\\
&\quad+2\frac{\rho_0v_x}{\eta_x^2}(3v_{xx}\partial_x^3\eta+3v_{xxx}\eta_{xx}+v_x\partial_x^4\eta+\eta_{x}\partial_x^4v)
=\colon\sum_{k=1}^8I_k.
\end{aligned}
\end{equation*}
For the terms \(I_k\) when \(k=2,3,4,5,6\), one may get directly
\begin{equation*}
\begin{aligned}
\|I_2\|_{L^2}
\lesssim
\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{xx}\bigg\|_{L^2}\|\eta_{xx}\|_{L^\infty}
\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\|I_3\|_{L^2}&\lesssim
\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x\bigg\|_{L^\infty}(\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|\partial_x^3\eta\|_{L^2})\\
&\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\|I_4\|_{L^2}&\lesssim
\bigg\|\partial_t\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)\bigg\|_{L^\infty}(\|\eta_{xx}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}
+\|\partial_x^4\eta\|_{L^2})\\
&\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\|I_5\|_{L^2}
\lesssim
\|v_{x}\|_{L^\infty}\bigg\|\partial_x^3\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)\bigg\|_{L^2}
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\|I_6\|_{L^2}&\lesssim
\bigg\|\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_{xx}\bigg\|_{L^\infty}(\|v_{x}\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|v_{xx}\|_{L^2})\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation*}
To estimate \(I_1\), \(I_7\) and \(I_8\), we need a weight \(\rho_0\). The estimate on \(\rho_0I_1\) has been done due to \eqref{II-Priori-ellip-16}.
For \(I_7\) and \(I_8\), one can get
{\small \begin{equation*}
\begin{aligned}
\|\rho_0I_7\|_{L^2}
&\lesssim
\bigg\|\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x\bigg\|_{L^\infty}(\|v_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|v_{x}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}+\|\rho_0\partial_x^3v\|_{L^2})\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}}
and
\begin{equation*}
\begin{aligned}
\|\rho_0I_8\|_{L^2}
&\lesssim
\|v_x\|_{L^\infty}(\|v_{xx}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}
+\|v_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}\\
&\quad+\|v_{x}\|_{L^\infty}\|\rho_0\partial_x^4\eta\|_{L^2}+\|\rho_0^2\partial_x^4v\|_{L^2})\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
where one has used \eqref{I-Priori-ellip-31}, \eqref{II-Priori-ellip-3.4}, Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2}.
Collecting all the cases leads to
\begin{equation}\label{II-Priori-ellip-17}
\begin{aligned}
\|\rho_0\partial_x^3(\rho_0\partial_tv_x)\|_{L^2}^2
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Since
\begin{equation*}
\begin{aligned}
\rho_0\partial_t\partial_x^4v+3(\rho_0)_{x}\partial_t\partial_x^3v=
\partial_x^3(\rho_0\partial_tv_x)-\partial_x^3\rho_0\partial_tv_x-3(\rho_0)_{xx}\partial_tv_{xx},
\end{aligned}
\end{equation*}
one can get
\begin{equation}\label{II-Priori-ellip-17.2}
\begin{aligned}
&\|\rho_0^2\partial_t\partial_x^4v+3\rho_0(\rho_0)_{x}\partial_t\partial_x^3v\|_{L^2}^2\\
&\quad\lesssim \|\rho_0\partial_x^3(\rho_0\partial_tv_x)\|_{L^2}^2+\|\rho_0\partial_tv_x\|_{L^2}^2+\|\rho_0\partial_tv_{xx}\|_{L^2}^2\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where \eqref{I-Priori-time-21}, \eqref{I-Priori-ellip-22} and \eqref{II-Priori-ellip-17} have been utilized.
Integration by parts gives
\begin{equation}\label{II-Priori-ellip-17.5}
\begin{aligned}
&\|\rho_0^2\partial_t\partial_x^4v\|_{L^2}^2\\
&\quad=\|\rho_0^2\partial_t\partial_x^4v+3\rho_0(\rho_0)_{x}\partial_t\partial_x^3v\|_{L^2}^2
-9\|\rho_0(\rho_0)_{x}\partial_t\partial_x^3v\|_{L^2}^2\\
&\qquad-3\int_I\rho_0^3(\rho_0)_x[(\partial_t\partial_x^3v)^2]_x\,\mathrm{d} x\\
&\quad=\|\rho_0^2\partial_t\partial_x^4v+3\rho_0(\rho_0)_{x}\partial_t\partial_x^3v\|_{L^2}^2
+3\int_I\rho_0^3(\rho_0)_{xx}(\partial_t\partial_x^3v)^2\,\mathrm{d} x\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where one has used \eqref{II-Priori-ellip-4} and \eqref{II-Priori-ellip-17.2}.
Hence we obtain from \eqref{II-Priori-ellip-17.5} that
\begin{equation}\label{II-Priori-ellip-18}
\begin{aligned}
\|\rho_0^2\partial_t\partial_x^4v\|_{L^2}^2
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0^3\partial_x^6v\|_{L^2(I)}\).}}
We first claim that
\begin{equation}\label{II-Priori-ellip-19}
\begin{aligned}
\bigg\|\rho_0\partial_x^5\bigg({\frac{\rho_0v_x}{\eta_x^2}}\bigg)\bigg\|_{L^2}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Applying \(\partial_x^4\) to Equation \(\eqref{eq:main-2}_1\) gives
\begin{equation}\label{II-Priori-ellip-19.5}
\begin{aligned}
\partial_x^5\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)=\partial_x^4(\rho_0\partial_tv)
+\partial_x^5\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg).
\end{aligned}
\end{equation}
A direct calculation shows that
\begin{equation*}
\begin{aligned}
\bigg|\partial_x^5\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\bigg|
&\lesssim 1+|\eta_{xx}|+(\eta_{xx}^2+|\partial_x^3\eta|)
+(|\eta_{xx}|^3+|\eta_{xx}\partial_x^3\eta|+|\partial_x^4\eta|)\\
&\quad+\rho_0(|\eta_{xx}|^4+\eta_{xx}^2|\partial_x^3\eta|+(\partial_x^3\eta)^2+|\eta_{xx}\partial_x^4\eta|+|\partial_x^5\eta|)\\
&\quad+\rho_0^2(|\eta_{xx}|^5+|\eta_{xx}^3\partial_x^3\eta|+|\eta_{xx}|(\partial_x^3\eta)^2+\eta_{xx}^2|\partial_x^4\eta|\\
&\quad+|\partial_x^3\eta\partial_x^4\eta|+|\eta_{xx}\partial_x^5\eta|+|\partial_x^6\eta|).
\end{aligned}
\end{equation*}
Due to \eqref{Preliminary-7}, the estimate of the last term \(\rho_0^2\partial_x^6\eta\) requires a weight \(\rho_0\) ,
hence one may estimate as follows:
\begin{equation}\label{II-Priori-ellip-20}
\begin{aligned}
\bigg\|\rho_0\partial_x^5\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)\bigg\|_{L^2}
&\lesssim 1+\|\eta_{xx}\|_{L^2}+(\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}+\|\partial_x^3\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^2\|\eta_{xx}\|_{L^2}
+\|\eta_{xx}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|\rho_0\partial_x^4\eta\|_{L^2})
+(\|\eta_{xx}\|_{L^\infty}^3\|\eta_{xx}\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^\infty}^2\|\partial_x^3\eta\|_{L^2}+\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^\infty}\|\rho_0\partial_x^4\eta\|_{L^2}+\|\rho_0^2\partial_x^5\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^4\|\eta_{xx}\|_{L^2}+\|\eta_{xx}\|_{L^\infty}^3\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^\infty}\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^\infty}^2\|\rho_0\partial_x^4\eta\|_{L^2}+\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\rho_0\partial_x^4\eta\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^\infty}\|\rho_0^2\partial_x^5\eta\|_{L^2}
+\|\rho_0^3\partial_x^6\eta\|_{L^2})\\
&\leq 1+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}
where Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2} have been used.
One the other hand, it holds that
\begin{equation}\label{II-Priori-ellip-21}
\begin{aligned}
\|\rho_0\partial_x^4(\rho_0\partial_tv)\|_{L^2}^2
&\lesssim \|\rho_0\partial_tv\|_{L^2}^2+\|\rho_0\partial_tv_x\|_{L^2}^2
+\|\rho_0\partial_tv_{xx}\|_{L^2}^2\\
&\quad+\|\rho_0\partial_t\partial_x^3v\|_{L^2}^2+\|\rho_0^2\partial_t\partial_x^4v\|_{L^2}^2\\
&\lesssim \|\rho_0\partial_tv\|_{L^2}^2+\|\rho_0\partial_tv_x\|_{L^2}^2
+\|\rho_0\partial_tv_{xx}\|_{L^2}^2\\
&\quad+\|\rho_0^2\partial_t\partial_x^3v\|_{L^2}^2+\|\rho_0^2\partial_t\partial_x^4v\|_{L^2}^2\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
Here one has used \eqref{ineq:weighted Sobolev} for \(\|\rho_0\partial_t\partial_x^3v\|_{L^2}\) in the second inequality, and \eqref{II-Priori-ellip-18} in the last inequality.
Then \eqref{II-Priori-ellip-19} follows from \eqref{II-Priori-ellip-19.5}, \eqref{II-Priori-ellip-20} and \eqref{II-Priori-ellip-21}.
Next, we estimate \(\|\rho_0^3\partial_x^6v\|_{L^2}\).
To this end, one notes that
\begin{equation*}
\begin{aligned}
&\rho_0\eta_x^{-2}\partial_x^6v+5(\rho_0)_x\eta_x^{-2}\partial_x^5v\\
&=\partial_x^5\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)-5\rho_0(\eta_x^{-2})_x\partial_x^5v-10(\rho_0\eta_x^{-2})_{xx}\partial_x^4v\\
&\quad-10\partial_x^3(\rho_0\eta_x^{-2})\partial_x^3v-5\partial_x^4(\rho_0\eta_x^{-2})v_{xx}-\partial_x^5(\rho_0\eta_x^{-2})v_{x}=\colon\sum_{k=1}^6I_k.
\end{aligned}
\end{equation*}
First, it follows from \eqref{II-Priori-ellip-19} that \(\|\rho_0I_1\|_{L^2}\) has the desired bound.
For the terms \(I_k\) when \(k=2,4,5\), in view of \eqref{I-Priori-time-16}, \eqref{I-Priori-ellip-5}, \eqref{II-Priori-ellip-3.4}, Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2},
we may choose a weight \(\rho_0\) to estimate each term as follows:
\begin{equation*}
\begin{aligned}
\|\rho_0I_2\|_{L^2}
\lesssim \|\eta_{xx}\|_{L^\infty}\|\rho_0^2\partial_x^5v\|_{L^2}
\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation*}
{\small \begin{equation*}
\begin{aligned}
\|\rho_0I_4\|_{L^2}
&\lesssim \|\rho_0\partial_x^3v\|_{L^2}+
\|\rho_0\partial_x^3v\|_{L^\infty}\big(\|\eta_{xx}\|_{L^2}
+(\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|\partial_x^3\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^2\|\eta_{xx}\|_{L^2}
+\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|\rho_0\partial_x^4\eta\|_{L^2})\big)\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation*}}
{\small \begin{equation*}
\begin{aligned}
\|\rho_0I_5\|_{L^2}
&\lesssim \|\rho_0v_{xx}\|_{L^2}+
\|v_{xx}\|_{L^\infty}\big(\|\eta_{xx}\|_{L^2}
+(\|\rho_0\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|\rho_0\partial_x^3\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^2\|\eta_{xx}\|_{L^2}
+\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|\rho_0\partial_x^4\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^3\|\eta_{xx}\|_{L^2}
+\|\eta_{xx}\|_{L^\infty}^2\|\partial_x^3\eta\|_{L^2}
+\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|\rho_0^2\partial_x^4\eta\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|\rho_0^2\partial_x^5\eta\|_{L^2})\big)\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2}.
\end{aligned}
\end{equation*}}
For \(I_3\) and \(I_6\), we choose a weight \(\rho_0^2\) to estimate them as follows:
\begin{equation*}
\begin{aligned}
\|\rho_0^2I_3\|_{L^2}
&\lesssim \|\rho_0^2\partial_x^4v\|_{L^2}(1+\|\eta_{xx}\|_{L^\infty}
+\|\eta_{xx}\|_{L^\infty}^2
+\|\rho_0\partial_x^3\eta\|_{L^\infty})\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation*}
and
{\small \begin{equation*}
\begin{aligned}
\|\rho_0^2I_6\|_{L^2}
&\lesssim \|\rho_0v_{x}\|_{L^2}+
\|v_{x}\|_{L^\infty}\big(\|\eta_{xx}\|_{L^2}
+(\|\eta_{xx}\|_{L^\infty}\|\eta_{xx}\|_{L^2}
+\|\partial_x^3\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^2\|\eta_{xx}\|_{L^2}
+\|\eta_{xx}\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}
+\|\rho_0\partial_x^4\eta\|_{L^2})\\
&\quad+(\|\eta_{xx}\|_{L^\infty}^3\|\eta_{xx}\|_{L^2}
+\|\eta_{xx}\|_{L^\infty}^2\|\partial_x^3\eta\|_{L^2}
+\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^\infty}\|\rho_0\partial_x^4\eta\|_{L^2}
+\|\rho_0^2\partial_x^5\eta\|_{L^2})
+(\|\eta_{xx}\|_{L^\infty}^4\|\eta_{xx}\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^\infty}^3\|\partial_x^3\eta\|_{L^2}
+\|\eta_{xx}\|_{L^\infty}\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\partial_x^3\eta\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^\infty}^2\|\rho_0\partial_x^4\eta\|_{L^2}+\|\rho_0\partial_x^3\eta\|_{L^\infty}\|\rho_0\partial_x^4\eta\|_{L^2}\\
&\quad+\|\eta_{xx}\|_{L^\infty}\|\rho_0^2\partial_x^5\eta\|_{L^2}
+\|\rho_0^3\partial_x^6\eta\|_{L^2})\big)\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v))]^{1/2},
\end{aligned}
\end{equation*}}
where in estimating \(I_6\) one has used
{\small \begin{equation*}
\begin{aligned}
|\partial_x^5(\rho_0\eta_x^{-2})|
&\lesssim 1+|\eta_{xx}|+(\eta_{xx}^2+|\partial_x^3\eta|)
+(|\eta_{xx}|^3+|\eta_{xx}\partial_x^3\eta|+|\partial_x^4\eta|)\\
&\quad+(|\eta_{xx}|^4+\eta_{xx}^2|\partial_x^3\eta|+(\partial_x^3\eta)^2+|\eta_{xx}\partial_x^4\eta|+|\partial_x^5\eta|)\\
&\quad+\rho_0(|\eta_{xx}|^5+|\eta_{xx}^3\partial_x^3\eta|+|\eta_{xx}|(\partial_x^3\eta)^2+\eta_{xx}^2|\partial_x^4\eta|\\
&\quad+|\partial_x^3\eta\partial_x^4\eta|+|\eta_{xx}\partial_x^5\eta|+|\partial_x^6\eta|).
\end{aligned}
\end{equation*}}
Consequently,
{\small \begin{equation}\label{II-Priori-ellip-23}
\begin{aligned}
&\|\rho_0^3\eta_x^{-2}\partial_x^5v+5\rho_0^2(\rho_0)_x\eta_x^{-2}\partial_x^4v\|_{L^2}^2\\
&\quad\lesssim \|\rho_0^2I_1\|_{L^2}^2+\|\rho_0^2I_2\|_{L^2}^2+\|\rho_0^2I_3\|_{L^2}^2+\|\rho_0^2I_4\|_{L^2}^2+\|\rho_0^2I_5\|_{L^2}^2+\|\rho_0^2I_6\|_{L^2}^2\\
&\quad\lesssim \|\rho_0I_1\|_{L^2}^2+\|\rho_0I_2\|_{L^2}+\|\rho_0^2I_3\|_{L^2}^2+\|\rho_0I_4\|_{L^2}^2+\|\rho_0I_5\|_{L^2}^2+\|\rho_0^2I_6\|_{L^2}^2\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}}
Then integration by parts yields
{\small \begin{equation}\label{II-Priori-ellip-24}
\begin{aligned}
&\|\rho_0^3\eta_x^{-2}\partial_x^6v\|_{L^2}^2\\
&\quad=\|\rho_0^3\eta_x^{-2}\partial_x^6v
+5\rho_0^2(\rho_0)_x\eta_x^{-2}\partial_x^5v\|_{L^2}^2
-25\|\rho_0^2(\rho_0)_x\eta_x^{-2}\partial_x^5v\|_{L^2}^2\\
&\qquad-5\int_I\rho_0^5(\rho_0)_x\eta_x^{-4}[(\partial_x^5v)^2]_x\,\mathrm{d} x\\
&\quad=\|\rho_0^3\eta_x^{-2}\partial_x^6v
+5\rho_0^2(\rho_0)_x\eta_x^{-2}\partial_x^5v\|_{L^2}^2
+5\int_I\rho_0^5[(\rho_0)_x\eta_x^{-4}]_{x}(\partial_x^5v)^2\,\mathrm{d} x\\
&\quad\lesssim\|\rho_0^3\eta_x^{-2}\partial_x^6v
+5\rho_0^2(\rho_0)_x\eta_x^{-2}\partial_x^5v\|_{L^2}^2
+(1+\|\eta_{xx}\|_{L^\infty})
\int_I\rho_0^5(\partial_x^5v)^2\,\mathrm{d} x\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)),
\end{aligned}
\end{equation}}
where \eqref{Preliminary-5}, \eqref{II-Priori-ellip-11} and \eqref{II-Priori-ellip-23} have been used.
Hence we get from \eqref{II-Priori-ellip-24} and \eqref{eta-bound} that
\begin{equation}\label{II-Priori-ellip-25}
\begin{aligned}
\|\rho_0^3\partial_x^6v\|_{L^2}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,v)).
\end{aligned}
\end{equation}
\section{An a priori Bound}\label{A priori Bound}
Collecting all inequalities \eqref{II-Priori-time-4}-\eqref{I-Priori-time-3} and \eqref{II-Priori-time-10}-\eqref{I-Priori-time-16} in Section \ref{Energy Estimates}, \eqref{I-Priori-ellip-5}, \eqref{I-Priori-ellip-third}, \eqref{I-Priori-ellip-22}, \eqref{I-Priori-ellip-31}, \eqref{II-Priori-ellip-4}, \eqref{II-Priori-ellip-11}, \eqref{II-Priori-ellip-18} and \eqref{II-Priori-ellip-25} in Section \ref{Elliptic estimates}, we obtain
\begin{equation}\label{I-Priori-ellip-32}
\begin{aligned}
E(t,v)
\leq M_0+CtP(\sup_{0\leq s\leq t}E^{1/2}(s,v))\quad \mathrm{for\ all}\ t\in [0,T],
\end{aligned}
\end{equation}
where \(P\) denotes a generic polynomial function of its arguments, and \(C\) is an absolutely constant only depending on \(\|\partial_x^l\rho_0\|_{L^\infty(I)}\ (l=0,1,...,5)\). The inequality
\eqref{I-Priori-ellip-32} implies for sufficiently small \(T>0\),
\begin{equation}\label{I-Priori-ellip-33}
\begin{aligned}
\sup_{0\leq t\leq T}E(t,v)
\leq 2M_0.
\end{aligned}
\end{equation}
\section{Proof of Theorem \ref{th:main-1}: Existence}\label{Existence Part}
In this section, we will show the existence of a classical solution to the problem \eqref{eq:main-2}. For given \(T>0\), let \(\mathcal{X}_T\) be a Banach space defined by
\begin{equation*}
\begin{aligned}
\mathcal{X}_T=\{ v\in L^\infty([0,T]; H^3(I)):\ \sup_{0\leq t\leq T}E(t,v)<\infty\},
\end{aligned}
\end{equation*}
endowed with its natural norm
\begin{equation*}
\begin{aligned}
\|v\|_{\mathcal{X}_T}^2=\sup_{0\leq t\leq T}E(t,v).
\end{aligned}
\end{equation*}
For given \(M_1\), we define \(\mathcal{C}_T(M_1)\) to be a closed, bounded, and
convex subset of \(\mathcal{X}_T\) given by
\begin{equation}\label{solution space}
\begin{aligned}
\mathcal{C}_T(M_1)&=\{ v\in \mathcal{X}_T:\|v\|_{\mathcal{X}_T}^2\leq M_1
,\ \partial_t^kv|_{t=0}=g_k
\ \mbox{for}\ k=0,1,2,3,\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \mbox{and}\ \partial_t^kv_x|_{t=0}=h_k\ \mbox{for}\ k=0,1,2\},
\end{aligned}
\end{equation}
where \(g_k\) and \(h_k\) are defined as follows:
\begin{equation*}
\begin{aligned}
&g_0=v|_{t=0}=u_0,\\
&g_1=\partial_tv|_{t=0}=\rho_0^{-1}\bigg[\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x
-\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_x\bigg]\bigg|_{t=0}
=\rho_0^{-1}[(\rho_0(u_0)_x)_x-(\rho_0^2)_x],\\
&g_k:=\partial_t^kv|_{t=0}=\rho_0^{-1}\partial_t^{k-1}\bigg[\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x
-\bigg({\frac{\rho_0^2}{\eta_x^2}}\bigg)_x\bigg]\bigg|_{t=0}\quad \text{for}\ k=2,3,\\
&h_0:=v_x|_{t=0}=(u_0)_x,\\
& h_k:=\partial_t^kv_x|_{t=0}=(g_k)_x\quad \text{for}\ k=1,2.
\end{aligned}
\end{equation*}
Note that each \(g_k\) (\(k=0,1,2,3\)) and \(h_k\) (\(k=1,2\)) is a function of spatial derivatives of \(\rho_0\) and \(u_0\).
For any given \(\bar{v}\in\mathcal{C}_T(M_1)\), define
\begin{align}\label{existence-1}
\bar{\eta}(x,t)=x+\int_0^t\bar{v}(x,s)\,\mathrm{d} s.
\end{align}
Arguing as for \eqref{eta-bound}, by choosing \(T>0\) suitably small, one also has
\begin{align}\label{eta-bound-2}
1/2\leq \bar{\eta}_x(x,t)\leq 3/2,\quad (x,t)\in I\times [0,T].
\end{align}
The choice of \(M_1\) and \(T\) is given in Subsection \ref{The a priori assumption}.
We then consider the following
linearized problem for \(v\):
\begin{equation}\label{existence-3}
\begin{cases}
\rho_0v_t+\big({\frac{\rho_0^2}{\bar{\eta}_x^2}}\big)_x
=\big(\frac{\rho_0v_x}{\bar{\eta}_x^2}\big)_x &\quad \mbox{in}\ I\times (0,T],\\
v=u_0 &\quad \mbox{on}\ I\times \{t=0\}.
\end{cases}
\end{equation}
In order to construct classical solutions to the problem \eqref{existence-3}, we first study its weak solutions.
\subsection{Existence and uniqueness of a weak solution to the problem \eqref{existence-3}.}\label{weak solution}
Let \(\langle\cdot, \cdot\rangle\) be the pairing of \(H^{-1}(I)\) and \(H^1(I)\), and \((\cdot,\cdot)\) stand for the inner product of \(L^2(I)\). Then we give the following definition:
\begin{definition}[Weak Solution]\label{Weak Solution} A function \(v\), satisfying
\begin{equation*}
\begin{aligned}
\rho_0^{1/2}v_x\in L^2([0,T];L^2(I))\quad {\rm{and}}\quad \rho_0 v_t\in L^2([0,T];H^{-1}(I)),
\end{aligned}
\end{equation*}
is said to be a weak solution to the problem \eqref{existence-3} provided\\
{\rm{(a)}}
\begin{equation*}
\begin{aligned}
\langle\rho_0v_t, \phi\rangle+\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}, \phi_x\bigg)=\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, \phi_x\bigg)
\end{aligned}
\end{equation*}
for each \(\phi\in H^1(I)\) and a.e. \(0< t\leq T\), and\\
{\rm{(b)}} \(\|\rho_0v(t,\cdot)-\rho_0v(0, \cdot)\|_{L^2(I)}\to 0\ \text{as}\ t\to 0^+\), and $v(0,\cdot)=u_0(\cdot) \ \mbox{a.e.\ on}\ I$.
\end{definition}
We will use the Galerkin's scheme (see \cite{MR2597943}) to construct weak solutions to the problem \eqref{existence-3}. Set
\[\mathcal{H}(I)=\{h\in H^3(I): h_x=0 \ \text{on}\ \Gamma\}.\]
Let \(\{e_n\}_{n=1}^\infty\) be a Hilbert basis of \(\mathcal{H}(I)\), with each \(e_n\) being of class \(H^k(I)\) for any \(k\geq 1\). Such a choice of basis indeed exists since one can take for instance
the eigenfunctions of the Laplace operator on \(I\) with the Nuewmann boundary
condition \(h_x=0\) for \(x\in \Gamma\).
Given a positive integer \(n\), we set
\begin{equation}\label{existence-4}
\begin{aligned}
X^n(t,x)=\sum_{i=1}^n\lambda_i^n(t)e_i(x),
\end{aligned}
\end{equation}
in which the coefficients \(\lambda_i^n(t)\) are chosen such that
\begin{eqnarray}\label{existence-5}
\begin{cases}
\big(\rho_0\partial_tX^n,e_j\big)
+\bigg(\frac{\rho_0X_x^n}{\bar{\eta}_x^2},(e_j)_x\bigg)
=\bigg(\frac{\rho_0^2}{\bar{\eta}_x^2},(e_j)_x\bigg)
&\quad \mbox{in}\ (0,T],\\
\lambda_j^n=(u_0,e_j) &\quad \mbox{on}\ \{t=0\},
\end{cases}
\end{eqnarray}
where \(j=1,2,...,n\).
Inserting \eqref{existence-4} into \eqref{existence-5} leads to
\begin{equation}\label{existence-6}
\begin{cases}
\sum_{i=1}^n\int_I \rho_0 e_ie_j\,\mathrm{d} x\cdot [\lambda_j^n(t)\big]_t\\
\quad+\sum_{i=1}^n\int_I\frac{\rho_0(e_i)_x(e_j)_x}{\bar{\eta}_x^2}\,\mathrm{d} x\cdot \lambda_j^n(t)
=\int_I \frac{\rho_0^2 (e_j)_x}{\bar{\eta}_x^2}\,\mathrm{d} x &\quad \mbox{in}\ (0,T],\\
\lambda_j^n=(u_0,e_j) &\quad \mbox{on}\ \{t=0\},
\end{cases}
\end{equation}
where \(j=1,2,...,n\).
It is clear that each integral in \eqref{existence-6} is well-defined since each \(e_i\) lives in \(H^k(I)\cap \mathcal{H}(I)\) for all \(k\geq 1\). On the one hand, the \(\{e_n\}_{n=1}^\infty\) are linearly independent, so are the \(\{\sqrt{\rho_0}e_n\}_{n=1}^\infty\). Hence the determinant of the matrix
\begin{align*}
[\sqrt{\rho_0}e_i,\sqrt{\rho_0}e_j]_{i,j\in\{1,\dots,n\}}
\end{align*}
is nonzero. On the other hand, it follows from \(\bar{v}\in \mathcal{C}_T(M_1)\) and \eqref{eta-bound-2} that \(1/\bar{\eta}_x\) is continuous for \(t\in[0,T]\), which implies
\begin{align*}
\int_I\frac{\rho_0(e_i)_x(e_j)_x}{\bar{\eta}_x^2}\,\mathrm{d} x
\end{align*}
is continuous, and
\begin{align*}
\int_I \frac{\rho_0^2 (e_j)_x}{\bar{\eta}_x^2}\,\mathrm{d} x
\end{align*}
is Lipschitz continuous for \(t\in[0,T]\). By the standard ODEs' theory, one can find solutions \(\lambda_i^n(t)\in C^1([0,T_n])\) \ \((i=1,...,n)\) to \eqref{existence-6}, which
means there exist approximate solutions \(X^n(t,x)\in C^1([0,T_n],\mathcal{H}(I))\)\ \((n=1,2,...)\) to \eqref{existence-5}.\\
We next show that \(\{X^n\}_{n=1}^\infty\) satisfy some uniform estimates in \(n\geq 1\).
\begin{lemma}\label{uniform estimates} The approximate solutions \(\{X^n\}_{n=1}^\infty\) satisfy the following uniform estimates in \(n\geq 1\):
\begin{equation}\label{first uniform estimates}
\begin{aligned}
&\underset{t\in[0,T]}{\sup}\|\rho_0^{1/2}X^n\|_{L^2(I)}^2
+\|\rho_0^{1/2}X_x^n\|_{L^2([0,T];L^2(I))}^2+\|\rho_0 X_t^n\|_{L^2([0,T];H^{-1}(I))}^2\\
&\quad\leq C\|\rho_0^{1/2}u_0\|_{L^2(I)}^2+CT.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
It follows from \eqref{existence-4} and \(\eqref{existence-5}_1\) that
\begin{align*}
\big(\rho_0\partial_tX^n,X^n\big)
+\bigg(\frac{\rho_0\partial_xX^n}{\bar{\eta}_x^2},\partial_xX^n\bigg)
=\bigg(\frac{\rho_0^2}{\bar{\eta}_x^2},\partial_xX^n\bigg).
\end{align*}
Integrating it over \(I\times[0,T_n]\) and integration by parts yield
\begin{equation}\label{existence-7}
\begin{aligned}
&\frac{1}{2}\int_I \rho_0(X^n)^2\,\mathrm{d} x
+\int_0^{T_n}\int_I\frac{\rho_0(\partial_xX^n)^2}{\bar{\eta}_x^2}\,\mathrm{d} x\mathrm{d} s\\
&=\frac{1}{2}\int_I \rho_0(X^n)^2(x,0)\,\mathrm{d} x
+\int_0^{T_n}\int_I\frac{\rho_0^2\partial_xX^n}{\bar{\eta}_x^2}\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
\eqref{eta-bound-2} and Cauchy's inequality imply
\begin{equation}\label{existence-8}
\begin{aligned}
\int_0^{T_n}\int_I\frac{\rho_0(\partial_xX^n)^2}{\bar{\eta}_x^2}\,\mathrm{d} x\mathrm{d} s
\geq \frac{4}{9} \int_0^{T_n}\int_I\rho_0(\partial_xX^n)^2\,\mathrm{d} x\mathrm{d} s,
\end{aligned}
\end{equation}
and
\begin{equation}\label{existence-9}
\begin{aligned}
\bigg|\int_0^{T_n}\int_I\frac{\rho_0^2\partial_xX^n}{\bar{\eta}_x^2}\,\mathrm{d} x\mathrm{d} s\bigg|
\leq CT_n+\frac{1}{100}\int_0^{T_n}\int_I\rho_0(\partial_xX^n)^2\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
Hence it follows from \eqref{existence-7}-\eqref{existence-9} that
\begin{equation}\label{existence-10}
\begin{aligned}
\int_I \rho_0(X^n)^2\,\mathrm{d} x
+\int_0^{T_n}\int_I\rho_0(\partial_xX^n)^2\,\mathrm{d} x\mathrm{d} s
\leq C\|\rho_0^{1/2}u_0\|_{L^2(I)}^2+CT_n.
\end{aligned}
\end{equation}
Fix any \(\phi\in H^1(I)\) with \(\|\phi\|_{H^1(I)}\leq 1\), and write \(\phi=\phi_1+\phi_2\), where
\begin{equation*}
\begin{aligned}
\phi_1\in \text{span}\{e_i\}_{i=1}^n\quad {\rm{and}}\quad (\phi_2,e_i)=0\ (i=1,...,n).
\end{aligned}
\end{equation*}
Recalling that the functions \(\{e_i\}_{i=1}^n\) are orthogonal in \(H^1(I)\), one has
\begin{equation*}
\begin{aligned}
\|\phi_1\|_{H^1(I)}\leq \|\phi\|_{H^1(I)}\leq 1.
\end{aligned}
\end{equation*}
It follows from \(\eqref{existence-5}_1\) that
\begin{equation}\label{existence-11}
\begin{aligned}
(\rho_0X_t^n, \phi_1)+\bigg(\frac{\rho_0X_x^n}{\bar{\eta}_x^2}, (\phi_1)_x\bigg)=\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, (\phi_1)_x\bigg),
\end{aligned}
\end{equation}
for a.e. \(0\leq t\leq T\).
Hence \eqref{existence-11} yields
\begin{equation*}
\begin{aligned}
\langle\rho_0X_t^n, \phi\rangle
&=(\rho_0X_t^n, \phi)=(\rho_0X_t^n, \phi_1)\\
&=\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, (\phi_1)_x\bigg)-\bigg(\frac{\rho_0X_x^n}{\bar{\eta}_x^2}, (\phi_1)_x\bigg),
\end{aligned}
\end{equation*}
which furthermore implies
\begin{equation*}
\begin{aligned}
|\langle\rho_0X_t^n, \phi\rangle|&\leq C(1+\|\rho_0^{1/2}X_x^n\|_{L^2})\|\phi_1\|_{H^1(I)}\\
&\leq C(1+\|\rho_0^{1/2}X_x^n\|_{L^2}).
\end{aligned}
\end{equation*}
This results in
\begin{equation*}
\begin{aligned}
\|\rho_0X_t^n\|_{H^{-1}(I)}\leq C(1+\|\rho_0^{1/2}X_x^n\|_{L^2}),
\end{aligned}
\end{equation*}
and therefore
\begin{equation}\label{existence-12}
\begin{aligned}
\int_0^{T_n}\|\rho_0X_t^n\|_{H^{-1}(I)}^2\,\mathrm{d} t&\leq C\int_0^{T_n}\int_I\rho_0(X_x^n)^2\,\mathrm{d} x\,\mathrm{d} s+CT_n\\
&\leq C\|\rho_0^{1/2}u_0\|_{L^2(I)}^2+CT_n,
\end{aligned}
\end{equation}
due to \eqref{existence-10}.
It follows from \eqref{existence-10} and \eqref{existence-12} that
\begin{equation}\label{existence-13}
\begin{aligned}
&\underset{t\in[0,T_n]}{\sup}\|\rho_0^{1/2}X^n\|_{L^2(I)}^2
+\|\rho_0^{1/2}X_x^n\|_{L^2([0,T_n];L^2(I))}^2+\|\rho_0 X_t^n\|_{L^2([0,T_n];H^{-1}(I))}^2\\
&\quad\leq C\|\rho_0^{1/2}u_0\|_{L^2(I)}^2+CT_n.
\end{aligned}
\end{equation}
Note that \eqref{eta-bound-2} holds on \(I\times[0,T]\), hence \(T_n\) can reach \(T\). Consequently \eqref{first uniform estimates} follows from \eqref{existence-13}.
\end{proof}
Finally, we show the existence of a weak solution to the problem \eqref{existence-3}.
\begin{lemma} There exists a unique weak solution \(v\) to the problem \eqref{existence-3} with
\begin{equation*}
\begin{aligned}
&\rho_0^{1/2}v\in L^\infty([0,T],L^2(I)),\ \rho_0^{1/2}v_x\in L^2([0,T],L^2(I)),\\
&\rho_0\partial_tv\in L^2([0,T];H^{-1}(I)).
\end{aligned}
\end{equation*}
Moreover, the solution \(v\) satisfies the following estimate
\begin{equation}\label{solution bound}
\begin{aligned}
&\underset{t\in[0,T]}{\sup}\|\rho_0^{1/2}v\|_{L^2(I)}^2
+\|\rho_0^{1/2}v_x\|_{L^2([0,T];L^2(I))}^2
+\|\rho_0v_t\|_{L^2([0,T];H^{-1}(I))}^2\\
&\quad\leq C|\rho_0^{1/2}u_0\|_{L^2(I)}^2+C(1+T).
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
It follows from Lemma \ref{uniform estimates} that
\begin{equation*}
\begin{aligned}
\|\rho_0^{1/2}X_x^n\|_{L^2([0,T];L^2(I))}\quad {\rm{and}}\quad \|\rho_0 X_t^n\|_{L^2([0,T];H^{-1}(I))}
\end{aligned}
\end{equation*}
are uniformly bounded in \(n\geq 1\). So there exist a subsequence of \(\{X^n\}_{n=1}^\infty\) (which is still denoted by \(\{X^n\}_{n=1}^\infty\) for convenience) and a function \(v\) satisfying \(\rho_0^{1/2}v_x \in\ L^2([0,T];L^2(I))\) and \(\rho_0v_t \in\ L^2([0,T];H^{-1}(I))\) such that as \(n\to\infty\)
\begin{equation*}
\begin{cases}
\rho_0^{1/2}X_x^n\rightharpoonup \rho_0^{1/2}v_x &\quad \text{in}\ L^2([0,T];L^2(I)),\\
\rho_0X_t^n\rightharpoonup \rho_0v_t &\quad \text{in}\ L^2([0,T];H^{-1}(I)).
\end{cases}
\end{equation*}
Then, the estimate \eqref{solution bound} follows easily from the energy estimates \eqref{first uniform estimates} by the lower semi-continuity of the norms.
We claim that \(v\) is a weak solution to the problem \eqref{existence-3}. Fix any positive integer \(m\geq 1\), and choose a function \(\Phi\in C^1([0,T]; H^1(I))\)
of the form
\begin{equation}\label{existence-14}
\begin{aligned}
\Phi=\sum_{i=1}^m\mu_i(t)e_i(x),
\end{aligned}
\end{equation}
where \(\{\mu_i(t)\}_{i=1}^m\) are any given smooth functions. Choosing \(n\geq m\), multiplying \(\eqref{existence-5}_1\) by \(\mu_i(t)\), summing up for \(i=1,...,m\), and integrating with respect to \(t\) over \([0,T]\), we get
\begin{equation}\label{existence-15}
\begin{aligned}
\int_0^T\langle\rho_0X_t^n, \Phi\rangle+\bigg(\frac{\rho_0X_x^n}{\bar{\eta}_x^2}, \Phi_x\bigg)\,\mathrm{d} t=\int_0^T\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, \Phi_x\bigg)\,\mathrm{d} t.
\end{aligned}
\end{equation}
Taking the limit \(n\rightarrow\infty\) yields
\begin{equation}\label{existence-16}
\begin{aligned}
\int_0^T\langle\rho_0v_t, \Phi\rangle+\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}, \Phi_x\bigg)\,\mathrm{d} t=\int_0^T\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, \Phi_x\bigg)\,\mathrm{d} t.
\end{aligned}
\end{equation}
Since functions of the form \eqref{existence-14} are dense in \(C([0,T]; H^1(I))\),
\eqref{existence-16} holds for all \(\Phi\in C^1([0,T]; H^1(I))\).
In particular, it holds that
\begin{equation}\label{existence-weak solution}
\begin{aligned}
\langle\rho_0v_t, \phi\rangle+\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}, \phi_x\bigg)=\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, \phi_x\bigg)
\end{aligned}
\end{equation}
for each \(\phi\in H^1(I)\) and a.e. \(0< t\leq T\).
By Definition \ref{Weak Solution}, it remains to check that
\begin{equation}\label{initial-add-1}
\begin{aligned}
\|\rho_0v(t,\cdot)-\rho_0v(0, \cdot)\|_{L^2(I)}\to 0\quad \text{as}\ t\to 0^+,
\end{aligned}
\end{equation}
and
\begin{equation}\label{initial-add-2}
\begin{aligned}
v(0):=v(0, \cdot)=u_0(0, \cdot) \quad \text{a.e.\ in}\ I.
\end{aligned}
\end{equation}
First note that
\begin{equation*}
\begin{aligned}
&\|\rho_0v\|_{L^2([0,T],H^1(I))}^2\\
&\quad\lesssim \|\rho_0^{1/2}v\|_{L^2([0,T],L^2(I))}^2+\|\rho_0^{1/2}v_x\|_{L^2([0,T],L^2(I))}^2
+\|v\|_{L^2([0,T],L^2(I))}^2\\
&\quad\lesssim \|\rho_0^{1/2}v\|_{L^2([0,T],L^2(I))}^2+\|\rho_0^{1/2}v_x\|_{L^2([0,T],L^2(I))}^2
+\|v\|_{L^2([0,T],H^{1/2}(I))}^2\\
&\quad\lesssim \|\rho_0^{1/2}v\|_{L^2([0,T],L^2(I))}^2+\|\rho_0^{1/2}v_x\|_{L^2([0,T],L^2(I))}^2\\
&\quad\lesssim \|\rho_0^{1/2}v\|_{L^\infty([0,T],L^2(I))}^2+\|\rho_0^{1/2}v_x\|_{L^2([0,T],L^2(I))}^2,
\end{aligned}
\end{equation*}
where \eqref{ineq:weighted Sobolev-0} has been used in the third inequality. Hence
\begin{equation*}
\begin{aligned}
\rho_0v\in L^2([0,T],H^1(I)),
\end{aligned}
\end{equation*}
which together with \(\rho_0\partial_tv\in L^2([0,T];H^{-1}(I))\) yields
\begin{equation}\label{initial-add-3}
\begin{aligned}
\rho_0v\in C([0,T],L^2(I)).
\end{aligned}
\end{equation}
Thus \eqref{initial-add-1} follows.
Then one may deduce from \eqref{existence-16} and \eqref{initial-add-3} that
\begin{equation}\label{existence-17}
\begin{aligned}
\int_0^T-\langle\Phi_t, \rho_0v\rangle+\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}, \Phi_x\bigg)\,\mathrm{d} t=\int_0^T\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, \Phi_x\bigg)\,\mathrm{d} t+(\rho_0v(0), \Phi(0))
\end{aligned}
\end{equation}
for each \(\Phi\in C^1([0,T]; H^1(I))\) with \(\Phi(T)=0\). For this \(\Phi\), it follows from \eqref{existence-15} that
\begin{equation}\label{existence-18}
\begin{aligned}
\int_0^T-\langle\Phi_t, \rho_0X^n\rangle+\bigg(\frac{\rho_0X_x^n}{\bar{\eta}_x^2}, \Phi_x\bigg)\,\mathrm{d} t=\int_0^T\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, \Phi_x\bigg)\,\mathrm{d} t+(\rho_0X^n(0), \Phi(0)).
\end{aligned}
\end{equation}
Passing limits in \(n\to \infty\) in \eqref{existence-18} gives
\begin{equation}\label{existence-19}
\begin{aligned}
\int_0^T-\langle\Phi_t, \rho_0v\rangle+\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}, \Phi_x\bigg)\,\mathrm{d} t=\int_0^T\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, \Phi_x\bigg)\,\mathrm{d} t+(\rho_0u_0, \Phi(0)),
\end{aligned}
\end{equation}
where one has used the fact \(\|\rho_0^{1/2}X^n(0)-\rho_0^{1/2}u_0\|_{L^2(I)}\to 0\) as \(n\to\infty\). As \(\Phi(0)\) is arbitrary, comparing \eqref{existence-17} and \eqref{existence-19}, one gets
\begin{equation*}
\begin{aligned}
\|\rho_0v(0)-\rho_0u_0\|_{L^2(I)}=0,
\end{aligned}
\end{equation*}
which yields
\begin{equation*}
\begin{aligned}
\rho_0v(0)=\rho_0u_0\quad a.e.\ \text{in}\ I.
\end{aligned}
\end{equation*}
Hence \eqref{initial-add-2} follows due to \eqref{eq:intro-3}.
The uniqueness of weak solutions of the problem \eqref{existence-3} is easy to check since \eqref{existence-3} is a linear problem.
\end{proof}
\subsection{Regularity.}\label{Regularity}
We have the following regularity result:
\begin{lemma}\label{Regularity lemma} The weak solution \(v\) to the problem \eqref{existence-3} has the following regularity:
\begin{equation}\label{existence-20}
\begin{aligned}
\sup_{0\leq t\leq T}E(t,v)
\leq M_1.
\end{aligned}
\end{equation}
Consequently the solution map \(\bar{v}\mapsto v:\mathcal{C}_T(M_1)\rightarrow \mathcal{C}_T(M_1)\) is well-defined.
\end{lemma}
\begin{proof} To prove \eqref{existence-20}, it suffices to show
\begin{equation}\label{existence-add-0}
\begin{aligned}
E(t,v)
\leq M_0+CtP(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v}))\quad \mathrm{for\ all}\ t\in [0,T]
\end{aligned}
\end{equation}
whose proof is similar to that of \eqref{I-Priori-ellip-32} in Section \ref{Energy Estimates} and Section \ref{Elliptic estimates}. So we only sketch the proof of \eqref{existence-add-0} and point out the main modifications.
\noindent{\bf{Estimate of \(\|\sqrt{\rho_0}\partial_tv\|_{L^2(I)}\).}} We start with estimating \(\|\sqrt{\rho_0}\partial_tv\|_{L^2(I)}\) based on \eqref{first uniform estimates} by some basic energy estimates.
To this end,
one can apply \(\partial_t\) to \(\eqref{existence-5}_1\), multiply it by \(\partial_t\lambda_i^n(t)\), and sum \(j=1,2,...,n\), to obtain that
\begin{equation*}
\big(\rho_0\partial_t^2X^n,\partial_tX^n\big)
+\bigg(\partial_t\bigg(\frac{\rho_0X_x^n}{\bar{\eta}_x^2}\bigg),\partial_tX_x^n\bigg)
=\bigg(\partial_t\bigg(\frac{\rho_0^2}{\bar{\eta}_x^2}\bigg),\partial_tX_x^n\bigg),
\end{equation*}
which gives
\begin{equation}\label{existence-add-1}
\begin{aligned}
&\frac{1}{2}\int_I \rho_0(\partial_tX^n)^2\,\mathrm{d} x
+\int_0^t\int_I\frac{\rho_0(\partial_tX^n_x)^2}{\bar{\eta}_x^2}\,\mathrm{d} x\mathrm{d} s\\
&\quad=\frac{1}{2}\int_I \rho_0(\partial_tX^n)^2(x,0)\,\mathrm{d} x+\int_0^t\int_I\partial_t\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}\bigg)\partial_tX^n_x\,\mathrm{d} x\mathrm{d} s\\
&\qquad-\int_0^t\int_I\bigg[\partial_t\bigg(\frac{\rho_0X^n_x}{\bar{\eta}_x^2}\bigg)\partial_tX^n_x
-\frac{\rho_0(\partial_tX^n_x)^2}{\bar{\eta}_x^2}\bigg]\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
Then one uses Cauchy's inequality to obtain
\begin{equation}\label{existence-add-2}
\begin{aligned}
&\bigg|\int_0^t\int_I\partial_t\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}\bigg)\partial_tX^n_x\,\mathrm{d} x\mathrm{d} s\bigg|\lesssim \int_0^t\int_I\rho_0^2|\bar{v}_x\partial_tX^n_x|\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_tX^n_x)^2\,\mathrm{d} x\mathrm{d} s
+C\int_0^t\int_I\rho_0\bar{v}_x^2\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_tX^n_x)^2\,\mathrm{d} x\mathrm{d} s+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
and
\begin{equation}\label{existence-add-3}
\begin{aligned}
&\bigg|\int_0^t\int_I\bigg[\partial_t\bigg(\frac{\rho_0X^n_x}{\bar{\eta}_x^2}\bigg)\partial_tX^n_x
-\frac{\rho_0(\partial_tX^n_x)^2}{\bar{\eta}_x^2}\bigg]\,\mathrm{d} x\mathrm{d} s\bigg|\\
&\quad\lesssim \int_0^t\int_I\rho_0|\bar{v}_xX^n_x\partial_tX^n_x|\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_tX^n_x)^2\,\mathrm{d} x\mathrm{d} s
+C\sup_{0\leq s\leq t}\|\bar{v}_x\|_{L^\infty}^2\int_0^t\int_I\rho_0(X^n_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_tX^n_x)^2\,\mathrm{d} x\mathrm{d} s+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
where \eqref{first uniform estimates} has been used in the last inequality.
It follows from \eqref{existence-add-1}-\eqref{existence-add-3} that
\begin{equation}\label{existence-add-4}
\begin{aligned}
\int_I \rho_0(\partial_tX^n)^2\,\mathrm{d} x
+\int_0^t\int_I\rho_0(\partial_tX^n_x)^2\,\mathrm{d} x\mathrm{d} s
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
By the lower semi-continuity of the norms, it follows from \eqref{existence-add-4} by taking limit \(n\to\infty\) that
\begin{equation}\label{existence-add-5}
\begin{aligned}
\int_I \rho_0(\partial_tv)^2\,\mathrm{d} x
+\int_0^t\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\sqrt{\rho_0}v_x\|_{L^2(I)}\).}} Next, we estimate \(\|\sqrt{\rho_0}v_x\|_{L^2(I)}\) using \eqref{existence-add-4}.
Multiplying Equation \(\eqref{existence-5}_1\) by \(\partial_t\lambda_i^n(t)\), and summing \(j=1,2,...,n\), one obtains
\begin{equation*}
\big(\rho_0\partial_tX^n,\partial_tX^n\big)
+\bigg(\frac{\rho_0X_x^n}{\bar{\eta}_x^2},\partial_tX_x^n\bigg)
=\bigg(\frac{\rho_0^2}{\bar{\eta}_x^2},\partial_tX_x^n\bigg),
\end{equation*}
which yields
\begin{equation}\label{existence-add-6}
\begin{aligned}
&\int_0^t\int_I \rho_0(\partial_tX^n)^2\,\mathrm{d} x\mathrm{d} s
+\frac{1}{2}\int_I\frac{\rho_0(X_x^n)^2}{\bar{\eta}_x^2}\,\mathrm{d} x\\
&\quad=\frac{1}{2}\int_I\rho_0(X_x^n)^2(x,0)\,\mathrm{d} x+\int_0^t\int_I{\frac{\rho_0^2\partial_tX_x^n}{\bar{\eta}_x^2}}\,\mathrm{d} x\mathrm{d} s
+\int_0^t\int_I\frac{\rho_0(X_x^n)^2\bar{v}_x}{\bar{\eta}_x^3}\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
\eqref{first uniform estimates}, \eqref{existence-add-4} and Cauchy's inequality imply
\begin{equation}\label{existence-add-6.1}
\begin{aligned}
\bigg|\int_0^t\int_I\frac{\rho_0^2\partial_tX_x^n}{\bar{\eta}_x^2}\,\mathrm{d} x\mathrm{d} s\bigg|
&\lesssim t+\int_0^t\int_I\rho_0(\partial_tX_x^n)^2\,\mathrm{d} x\mathrm{d} s\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
and
\begin{equation}\label{existence-add-6.2}
\begin{aligned}
\bigg|\int_0^t\int_I\frac{\rho_0(X_x^n)^2\bar{v}_x}{\bar{\eta}_x^3}\,\mathrm{d} x\mathrm{d} s\bigg|
&\lesssim \int_0^t\int_I\rho_0\bar{v}_x^2\,\mathrm{d} x\mathrm{d} s+\int_0^t\int_I\rho_0(X_x^n)^2\,\mathrm{d} x\mathrm{d} s\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
It follows from \eqref{existence-add-6}-\eqref{existence-add-6.2} that
\begin{equation}\label{existence-add-7}
\begin{aligned}
\int_0^t\int_I \rho_0(\partial_tX^n)^2\,\mathrm{d} x\mathrm{d} s
+\int_I\rho_0(X_x^n)^2\,\mathrm{d} x
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
By the lower semi-continuity of the norms again, one gets from \eqref{existence-add-7} by taking limit \(n\to\infty\) that
\begin{equation}\label{existence-add-8}
\begin{aligned}
\int_0^t\int_I \rho_0(\partial_tv)^2\,\mathrm{d} x\mathrm{d} s
+\int_I\rho_0v_x^2\,\mathrm{d} x
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0v_{xx}\|_{L^2(I)}\).}} Now, we estimate \(\|\rho_0v_{xx}\|_{L^2(I)}\) based on \eqref{existence-add-5} and \eqref{existence-add-8} by carrying out some elliptic estimates.
We start with the following equality:
\begin{equation}\label{existence-add-9}
\begin{aligned}
(\rho_0v_t, \phi)+\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}, \phi_x\bigg)=\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}, \phi_x\bigg)
\end{aligned}
\end{equation}
for each \(\phi\in H^1(I)\) and a.e. \(0< t\leq T\), which follows from \eqref{existence-weak solution} and \eqref{existence-add-5}. Indeed, \eqref{existence-add-5} implies \(\rho_0^{1/2}v_t\in L^\infty([0,T],L^2(I))\), and thus \(\rho_0v_t\in L^\infty([0,T],L^2(I))\), which leads to
\[\langle\rho_0v_t, \phi\rangle=(\rho_0v_t, \phi).\]
Since \(\rho_0\) satisfies the assumption \eqref{eq:intro-3}, one can obtain the interior \(H^2(I)\)-regularity \(v\in H^2_{\text{loc}}(I)\) from \eqref{existence-add-9} by a standard argument (see \cite{MR2597943}). Hence
\begin{equation}\label{existence-add-10}
\begin{aligned}
\rho_0v_t+\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}\bigg)_x
=\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}\bigg)_x &\quad \mbox{a.e.\ in}\ I\times (0,T].
\end{aligned}
\end{equation}
Now one can repeat the argument in estimating \eqref{I-Priori-ellip-5} from Equation \eqref{existence-add-10} to obtain the boundary regularity
\begin{equation}\label{existence-add-12}
\begin{aligned}
\|\rho_0v_{xx}\|_{L^2(I)}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
Indeed, it is easy to check that the only key estimate in this assignment is \eqref{I-Priori-ellip-2.4}{\color{red}{,}} which should be replaced by
\begin{equation*}
\begin{aligned}
\|\rho_0(\bar{\eta}_x^{-2})_{x}v_x\|_{L^2}
\lesssim \|\rho_0^{1/2}v_x\|_{L^2}\|\bar{\eta}_{xx}\|_{L^\infty}
\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation*}
where \eqref{existence-add-8} has been used.
In the following, making using \eqref{existence-add-10}, we can show that the remaining terms in \(E(t,v)\) have the desired bound \(M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v}))\).\\
\noindent{\bf{Estimate of \(\|\sqrt{\rho_0}\partial_t^2v\|_{L^2(I)}\).}} Applying \(\partial_t^2\) to \eqref{existence-add-10} and multiplying it by \(\partial_t^2v\) yield
\begin{equation}\label{existence-add-13}
\begin{aligned}
&\frac{1}{2}\int_I \rho_0(\partial_t^2v)^2\,\mathrm{d} x
+\int_0^t\int_I\frac{\rho_0(\partial_t^2v_x)^2}{\bar{\eta}_x^2}\,\mathrm{d} x\mathrm{d} s\\
&\quad=\frac{1}{2}\int_I \rho_0(\partial_t^2v)^2(x,0)\,\mathrm{d} x+\int_0^t\int_I\partial_t^2\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}\bigg)\partial_t^2v_x\,\mathrm{d} x\mathrm{d} s\\
&\qquad-\int_0^t\int_I\bigg[\partial_t^2\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}\bigg)\partial_t^2v_x
-\frac{\rho_0(\partial_t^2v_x)^2}{\bar{\eta}_x^2}\bigg]\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
Note that
\begin{equation*}
\begin{aligned}
\bigg|\partial_t^2\bigg(\frac{1}{\bar{\eta}_x^2}\bigg)\bigg|\lesssim |\partial_t\bar{v}_x|+\bar{v}_x^2,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\bigg|\partial_t^2\bigg(\frac{v_x}{\bar{\eta}_x^2}\bigg)\partial_t^2v_x
-\frac{(\partial_t^2v_x)^2}{\bar{\eta}_x^2}\bigg|\lesssim (|v_x\partial_t\bar{v}_x|+|v_x\bar{v}_x^2|+|\bar{v}_x\partial_tv_x|)|\partial_t^2v_x|.
\end{aligned}
\end{equation*}
Then one may use Cauchy's inequality to obtain
\begin{equation}\label{existence-add-14}
\begin{aligned}
&\bigg|\int_0^t\int_I\partial_t^2\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}\bigg)\partial_t^2v_x\,\mathrm{d} x\mathrm{d} s\bigg|\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s+C\int_0^t\int_I\big[\rho_0(\partial_t\bar{v}_x)^2+\rho_0\bar{v}_x^2\big]\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
and
\begin{equation}\label{existence-add-15}
\begin{aligned}
&\bigg|\int_0^t\int_I\bigg[\partial_t^2(\frac{\rho_0v_x}{\bar{\eta}_x^2})\partial_t^2v_x
-{\frac{\rho_0(\partial_t^2v_x)^2}{\bar{\eta}_x^2}}\bigg]\,\mathrm{d} x\mathrm{d} s\bigg|\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s
+C\int_0^t\|\partial_t\bar{v}_x\|_{L^\infty}^2\int_I\rho_0v_x^2\,\mathrm{d} x\mathrm{d} s\\
&\quad\quad+C\int_0^t\|\bar{v}_x\|_{L^\infty}^4\int_I\rho_0v_x^2\,\mathrm{d} x\mathrm{d} s
+C\sup_{0\leq s\leq t}\|\bar{v}_x\|_{L^\infty}^2\int_0^t\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
where \eqref{existence-add-5} and \eqref{existence-add-8} have been used.
It follows from \eqref{existence-add-13}-\eqref{existence-add-15} that
\begin{equation}\label{existence-add-16}
\begin{aligned}
\int_I \rho_0(\partial_t^2v)^2\,\mathrm{d} x
+\int_0^t\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\sqrt{\rho_0}\partial_tv_x\|_{L^2(I)}\).}} Applying \(\partial_t\) to \eqref{existence-add-10},
and multiplying it by \(\partial_t^2v\), one obtains by some direct calculations that
\begin{equation}\label{existence-add-17}
\begin{aligned}
&\int_0^t\int_I \rho_0(\partial_t^2v)^2\,\mathrm{d} x\mathrm{d} s
+\frac{1}{2}\int_I\frac{\rho_0(\partial_tv_x)^2}{\bar{\eta}_x^2}\,\mathrm{d} x\\
&=\frac{1}{2}\int_I\rho_0(\partial_tv_x)^2(x,0)\,\mathrm{d} x+\int_0^t\int_I\partial_t\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}\bigg)\partial_t^2v_x\,\mathrm{d} x\mathrm{d} s\\
&\quad+\frac{1}{2}\int_0^t\int_I\partial_t\bigg({\frac{\rho_0}{\bar{\eta}_x^2}}\bigg)(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s
-\int_0^t\int_I\partial_t\bigg({\frac{\rho_0}{\bar{\eta}_x^2}}\bigg)v_x\partial_t^2v_x\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
The last three terms on the RHS of \eqref{existence-add-17} can be estimated as follows:
\begin{equation}\label{existence-add-18}
\begin{aligned}
\bigg|\int_0^t\int_I\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}\bigg)\partial_t^2v_x\,\mathrm{d} x\mathrm{d} s\bigg|
&\lesssim \int_0^t\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s+\int_0^t\int_I\rho_0\bar{v}_x^2\,\mathrm{d} x\mathrm{d} s\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
\begin{equation}\label{existence-add-19}
\begin{aligned}
\bigg|\int_0^t\int_I\partial_t\bigg({\frac{\rho_0}{\bar{\eta}_x^2}}\bigg)(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s\bigg|
&\lesssim \sup_{0\leq s\leq t}\|\bar{v}_x\|_{L^\infty}\int_0^t\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
and
\begin{equation}\label{existence-add-20}
\begin{aligned}
\bigg|\int_0^t\int_I\partial_t\bigg({\frac{\rho_0}{\bar{\eta}_x^2}}\bigg)v_x\partial_t^2v_x\,\mathrm{d} x\mathrm{d} s\bigg|&\lesssim \int_0^t\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\quad+\int_0^t\|\bar{v}_x\|_{L^\infty}^2\int_I\rho_0v_x^2\,\mathrm{d} x\mathrm{d} s\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
where one has used \eqref{existence-add-16} in \eqref{existence-add-18} and \eqref{existence-add-20}.
It then follows from \eqref{existence-add-17}-\eqref{existence-add-20} that
\begin{equation}\label{existence-add-21}
\begin{aligned}
\int_0^t\int_I \rho_0(\partial_t^2v)^2\,\mathrm{d} x\mathrm{d} s
+\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0^{3/2}\partial_x^3v\|_{L^2(I)}, \|\rho_0\partial_tv_{xx}\|_{L^2},\|\rho_0^2\partial_x^4v\|_{L^2}\).}} Now, one can estimate \(\|\rho_0^{3/2}\partial_x^3v\|_{L^2(I)}\) by using \eqref{existence-add-21}. Indeed, one just needs to replace \eqref{I-Priori-ellip-12} by
\begin{equation*}
\begin{aligned}
\|\rho_0(\bar{\eta}_x^{-2})_{x}v_{xx}\|_{L^2}
\lesssim \|\rho_0v_{xx}\|_{L^2}\|\bar{\eta}_{xx}\|_{L^\infty}
\leq Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation*}
due to \eqref{existence-add-12}, and replace \eqref{I-Priori-ellip-14} by
\begin{equation*}
\begin{aligned}
\|(\rho_0\bar{\eta}_x^{-2})_{xx}v_x\|_{L^2}
&\lesssim \|v_x\|_{L^2}(1+\|\bar{\eta}_{xx}\|_{L^\infty}
+\|\bar{\eta}_{xx}\|_{L^\infty}^2
+\|\rho_0\partial_x^3\bar{\eta}\|_{L^\infty})\\
&\lesssim (\|\rho_0v_x\|_{L^2}+\|\rho_0v_{xx}\|_{L^2})\\
&\quad\times(1+\|\bar{\eta}_{xx}\|_{L^\infty}
+\|\bar{\eta}_{xx}\|_{L^\infty}^2
+\|\rho_0\partial_x^3\bar{\eta}\|_{L^\infty})\\
&\leq [M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v}))]^{1/2},
\end{aligned}
\end{equation*}
due to \eqref{existence-add-8} and \eqref{existence-add-12},
and then repeat the argument for \eqref{I-Priori-ellip-third} to get
\begin{equation}\label{existence-add-23}
\begin{aligned}
\|\rho_0^{3/2}\partial_x^3v\|_{L^2(I)}^2\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
Similarly, one may repeat the arguments for \eqref{I-Priori-ellip-22} and \eqref{I-Priori-ellip-31} to obtain
\begin{equation}\label{existence-add-24}
\begin{aligned}
\|\rho_0\partial_tv_{xx}\|_{L^2}^2+\|\rho_0^2\partial_x^4v\|_{L^2}^2
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\sqrt{\rho_0}\partial_t^3v\|_{L^2(I)}\).}} Next, \(\|\sqrt{\rho_0}\partial_t^3v\|_{L^2(I)}\) can be estimated due to \eqref{existence-add-23} and \eqref{existence-add-24}. Indeed, one can apply \(\partial_t^3\) to \eqref{existence-add-10} and multiply it by \(\partial_t^3v\), after some elementary computations, to obtain that
\begin{equation}\label{existence-add-26}
\begin{aligned}
&\frac{1}{2}\int_I \rho_0(\partial_t^3v)^2\,\mathrm{d} x
+\int_0^t\int_I\frac{\rho_0(\partial_t^3v_x)^2}{\bar{\eta}_x^2}\,\mathrm{d} x\mathrm{d} s\\
&\quad=\frac{1}{2}\int_I \rho_0(\partial_t^3v)^2(x,0)\,\mathrm{d} x+\int_0^t\int_I\partial_t^3\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}\bigg)\partial_t^3v_x\,\mathrm{d} x\mathrm{d} s\\
&\qquad-\int_0^t\int_I\bigg[\partial_t^3\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}\bigg)\partial_t^3v_x
-\frac{\rho_0(\partial_t^3v_x)^2}{\bar{\eta}_x^2}\bigg]\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
Similar to \eqref{Add-1} and \eqref{Add-2}, one gets from \eqref{eta-bound-2} that
\begin{equation*}
\begin{aligned}
\bigg|\partial_t^3\bigg(\frac{1}{\bar{\eta}_x^2}\bigg)\bigg|\lesssim |\partial_t^2\bar{v}_x|+|\bar{v}_x\partial_t\bar{v}_x|+|\bar{v}_x^3|,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\bigg|\partial_t^3\bigg(\frac{v_x}{\bar{\eta}_x^2}\bigg)\partial_t^3v_x
-\frac{(\partial_t^3v_x)^2}{\bar{\eta}_x^2}\bigg|
&\lesssim \big[|\bar{v}_x\partial_t^2v_x|+|\partial_tv_x|(|\bar{v}_x|^2+|\partial_t\bar{v}_x|)+|v_x||\bar{v}_x|^3\\
&\quad+|v_x\partial_t^2\bar{v}_x|+|\partial_t\bar{v}_x||\bar{v}_xv_x|\big]|\partial_t^3v_x|.
\end{aligned}
\end{equation*}
Then one uses Cauchy's inequality to obtain
\begin{equation}\label{existence-add-27}
\begin{aligned}
&\bigg|\int_0^t\int_I\partial_t^3\bigg({\frac{\rho_0^2}{\bar{\eta}_x^2}}\bigg)\partial_t^3v_x\,\mathrm{d} x\mathrm{d} s\bigg|\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s
+C\int_0^t\int_I\rho_0(\partial_t^2\bar{v}_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\qquad+C\int_0^t\|\bar{v}_x\|_{L^\infty}^2\int_I\rho_0(\partial_t\bar{v}_x)^2\,\mathrm{d} x\mathrm{d} s
+C\int_0^t\|\bar{v}_x\|_{L^\infty}^4\int_I\rho_0\bar{v}_x^2\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
and
\begin{equation}\label{existence-add-28}
\begin{aligned}
&\bigg|\int_0^t\int_I\bigg[\partial_t^3\bigg(\frac{\rho_0v_x}{\bar{\eta}_x^2}\bigg)\partial_t^3v_x
-\frac{\rho_0(\partial_t^3v_x)^2}{\bar{\eta}_x^2}\bigg]\,\mathrm{d} x\mathrm{d} s\bigg|\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s
+C\sup_{0\leq s\leq t}\|\bar{v}_x\|_{L^\infty}^2\int_0^t\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\qquad+C\int_0^t\|\bar{v}_x\|_{L^\infty}^4\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s+C\int_0^t\|\partial_t\bar{v}_x\|_{L^\infty}^2\int_I\rho_0(\partial_tv_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\qquad+C\int_0^t\|\bar{v}_x\|_{L^\infty}^6\int_I\rho_0v_x^2\,\mathrm{d} x\mathrm{d} s+C\int_0^t\|v_x\|_{L^\infty}^2\int_I\rho_0(\partial_t^2\bar{v}_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\qquad+C\int_0^t\|\bar{v}_x\|_{L^\infty}^2\|v_x\|_{L^\infty}^2\int_I\rho_0(\partial_t\bar{v}_x)^2\,\mathrm{d} x\mathrm{d} s\\
&\quad\leq \frac{1}{100}\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})),
\end{aligned}
\end{equation}
where one has used \eqref{existence-add-8}, \eqref{existence-add-12}, \eqref{existence-add-23} and \eqref{existence-add-24} to estimate
\begin{equation*}
\begin{aligned}
\|v_x\|_{L^\infty}&\lesssim \|v_x\|_{L^2}+\|v_{xx}\|_{L^2}\\
&\lesssim (\|\rho_0v_x\|_{L^2}+\|\rho_0v_{xx}\|_{L^2})+(\|\rho_0v_{xx}\|_{L^2}+\|\rho_0\partial_x^3v\|_{L^2})\\
&\lesssim \|\rho_0v_x\|_{L^2}+\|\rho_0v_{xx}\|_{L^2}+\|\rho_0^2\partial_x^3v\|_{L^2}+\|\rho_0^2\partial_x^4v\|_{L^2}\\
&\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation*}
It follows from \eqref{existence-add-26}-\eqref{existence-add-28} that
\begin{equation}\label{existence-add-29}
\begin{aligned}
\int_I \rho_0(\partial_t^3v)^2\,\mathrm{d} x
+\int_0^t\int_I\rho_0(\partial_t^3v_x)^2\,\mathrm{d} x\mathrm{d} s
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\sqrt{\rho_0}\partial_t^2v_x\|_{L^2(I)}\).}} In view of \eqref{existence-add-29}, similar to \eqref{II-Priori-time-10}, one can derive that
\begin{equation}\label{existence-add-30}
\begin{aligned}
\int_0^t\int_I \rho_0(\partial_t^3v)^2\,\mathrm{d} x\mathrm{d} s
+\int_I\rho_0(\partial_t^2v_x)^2\,\mathrm{d} x
\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
\noindent{\bf{Estimate of \(\|\rho_0^{3/2}\partial_t\partial_x^3v\|_{L^2},\|\rho_0^{5/2}\partial_x^5v\|_{L^2}, \|\rho_0^2\partial_t\partial_x^4v\|_{L^2(I)}, \|\rho_0^3\partial_x^6v\|_{L^2(I)}\).}}
By \eqref{existence-add-29} and \eqref{existence-add-30}, one may repeat the arguments for \eqref{II-Priori-ellip-4}, \eqref{II-Priori-ellip-11}, \eqref{II-Priori-ellip-18} and \eqref{II-Priori-ellip-25} to obtain
\begin{equation}\label{existence-add-31}
\begin{aligned}
&\|\rho_0^{3/2}\partial_t\partial_x^3v\|_{L^2}^2+\|\rho_0^{5/2}\partial_x^5v\|_{L^2}^2+\|\rho_0^2\partial_t\partial_x^4v\|_{L^2(I)}^2+\|\rho_0^3\partial_x^6v\|_{L^2(I)}^2\\
&\quad\leq M_0+Ct P(\sup_{0\leq s\leq t}E^{1/2}(s,\bar{v})).
\end{aligned}
\end{equation}
Finally, \eqref{existence-add-0} follows from \eqref{solution bound}, \eqref{existence-add-5}, \eqref{existence-add-8}, \eqref{existence-add-12}, \eqref{existence-add-16}, \eqref{existence-add-21}, \eqref{existence-add-23}, \eqref{existence-add-24}, \eqref{existence-add-29}, \eqref{existence-add-30} and \eqref{existence-add-31}.
\end{proof}
\subsection{Existence of a classical solution to the problem \eqref{existence-3}.}\label{classical solution}
In order to show that
there exists a classical solution to the problem \eqref{existence-3}, we will construct its approximate solutions and show the approximate solutions converge uniformly by a contraction mapping method. Therefore we consider the following iteration problem:
\begin{equation}\label{existence-21}
\begin{cases}
\rho_0v_t^{(n)}+\bigg[\frac{\rho_0^2}{(\eta_x^{(n-1)})^2}\bigg]_x
=\bigg[\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n-1)})^2}\bigg]_x &\quad \mbox{in}\ I\times (0,T],\\
v^{(n)}=u_0 &\quad \mbox{on}\ I\times \{t=0\},\\
v_x^{(n)}=0 &\quad \mbox{on}\ \Gamma\times (0,T].
\end{cases}
\end{equation}
For \(n=1\), we impose
\(\eta^{(0)}(t,x)=x+tu_0(x)\). We then solve the problem \eqref{existence-21} for \(n=1,2,...\) iteratively.
Given \(T>0\) sufficiently small,
in view of Lemma \ref{Regularity lemma}, one can obtain \(\{v^{(n)}\}_{n=1}^\infty\subset \mathcal{C}_T(M_1)\) for any \(n\geq 1\).
In the following, we will show that the approximate solutions \(\{v^{(n)}\}_{n=1}^\infty\)
are contractive in some appropriate energy space. To this end,
setting \(\sigma(v^{(n)}):=v^{(n+1)}-v^{(n)}\), one deduces
\begin{equation}\label{existence-22}
\begin{cases}
\rho_0\partial_t\sigma(v^{(n)})+\bigg[\frac{\rho_0^2}{(\eta_x^{(n)})^2}\bigg]_x-\bigg[\frac{\rho_0^2}{(\eta_x^{(n-1)})^2}\bigg]_x\\
\qquad\qquad\qquad=\bigg[\frac{\rho_0v_x^{(n+1)}}{(\eta_x^{(n)})^2}\bigg]_x
-\bigg[\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n-1)})^2}\bigg]_x &\quad \mbox{in}\ I\times (0,T],\\
\sigma(v^{(n)})=0 &\quad \mbox{on}\ I\times \{t=0\},\\
\sigma_x(v^{(n)})=0 &\quad \mbox{on}\ \Gamma\times (0,T].
\end{cases}
\end{equation}
\begin{lemma} It holds that
\begin{equation}\label{existence-23}
\begin{aligned}
&\frac{\mathrm{d}}{\mathrm{d} t}\int_I\rho_0[\sigma(v^{(n)})]^2\,\mathrm{d} x
+\int_I\rho_0[\sigma_x(v^{(n)})]^2\,\mathrm{d} x\\
&\quad\leq C(M_1^{1/2}+1)t\int_0^t\int_I\rho_0[\sigma_x(v^{(n-1)})]^2\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof} Multiplying Equation \(\eqref{existence-22}_1\) by \(\sigma(v^{(n)})\) and integrating by parts with respect to \(x\) yield
\begin{equation}\label{existence-24}
\begin{aligned}
&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\int_I\rho_0[\sigma(v^{(n)})]^2\,\mathrm{d} x
+\int_I\bigg[\frac{\rho_0v_x^{(n+1)}}{(\eta_x^{(n)})^2}
-\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n-1)})^2}\bigg]\sigma_x(v^{(n)})\,\mathrm{d} x\\
&\quad=\int_I\bigg[\frac{\rho_0^2}{(\eta_x^{(n)})^2}-\frac{\rho_0^2}{(\eta_x^{(n-1)})^2}\bigg]\sigma_x(v^{(n)})\,\mathrm{d} x.
\end{aligned}
\end{equation}
Note that
\begin{equation}\label{existence-25}
\begin{aligned}
&\int_I\bigg[\frac{\rho_0v_x^{(n+1)}}{(\eta_x^{(n)})^2}
-\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n-1)})^2}\bigg]\sigma_x(v^{(n)})\,\mathrm{d} x\\
&\quad=\int_I\bigg[\frac{\rho_0v_x^{(n+1)}}{(\eta_x^{(n)})^2}
-\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n)})^2}\bigg]\sigma_x(v^{(n)})\,\mathrm{d} x\\
&\qquad+\int_I\bigg[\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n)})^2}
-\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n-1)})^2}\bigg]\sigma_x(v^{(n)})\,\mathrm{d} x\\
&\quad=\colon I_1+I_2.
\end{aligned}
\end{equation}
Direct estimates yield
\begin{equation}\label{existence-26}
\begin{aligned}
I_1=\int_I\frac{\rho_0}{(\eta_x^{(n)})^2}
[\sigma_x(v^{(n)})]^2\,\mathrm{d} x\geq \frac{4}{9}\int_I\rho_0[\sigma_x(v^{(n)})]^2\,\mathrm{d} x,
\end{aligned}
\end{equation}
and
\begin{equation}\label{existence-27}
\begin{aligned}
|I_2|&=\bigg|\int_I\bigg[\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n)})^2(\eta_x^{(n-1)})^2}(\eta_x^{(n)}+\eta_x^{(n-1)})
\int_0^t\sigma_x(v^{(n-1)})\,\mathrm{d} s\bigg]\sigma_x(v^{(n)})\,\mathrm{d} x\bigg|\\
&\leq \frac{1}{100}\int_I\rho_0[\sigma_x(v^{(n)})]^2\,\mathrm{d} x+Ct\|v_x^{(n)}\|_{L^\infty}\int_0^t\int_I\rho_0[\sigma_x(v^{(n-1)})]^2\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
Hence it follows from \eqref{existence-25}-\eqref{existence-27} that
\begin{equation}\label{existence-28}
\begin{aligned}
&-\int_I\bigg[\bigg(\frac{\rho_0v_x^{(n+1)}}{(\eta_x^{(n)})^2}\bigg)_x
-\bigg(\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n-1)})^2}\bigg)_x\bigg]\sigma(v^{(n)})\,\mathrm{d} x\\
&\quad\geq \frac{1}{3}\int_I\rho_0[\sigma_x(v^{(n)})]^2\,\mathrm{d} x-Ct\|v_x^{(n)}\|_{L^\infty}\int_0^t\int_I\rho_0[\sigma_x(v^{(n-1)})]^2\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
Similar to \eqref{existence-27}, one has
\begin{equation}\label{existence-29}
\begin{aligned}
&\int_I\bigg[\frac{\rho_0^2}{(\eta_x^{(n)})^2}-\frac{\rho_0^2}{(\eta_x^{(n-1)})^2}\bigg]\sigma_x(v^{(n)})\,\mathrm{d} x\\
&\quad=\int_I\bigg[\frac{\rho_0^2}{(\eta_x^{(n)})^2(\eta_x^{(n-1)})^2}(\eta_x^{(n)}+\eta_x^{(n-1)})
\int_0^t\sigma_x(v^{(n-1)})\,\mathrm{d} s\bigg]\sigma_x(v^{(n)})\,\mathrm{d} x\\
&\quad\leq\frac{1}{100}\int_I\rho_0[\sigma_x(v^{(n)})]^2\,\mathrm{d} x+Ct\int_0^t\int_I\rho_0[\sigma_x(v^{(n-1)})]^2\,\mathrm{d} x\mathrm{d} s.
\end{aligned}
\end{equation}
Substituting \eqref{existence-28} and \eqref{existence-29} into \eqref{existence-24} gives
\begin{equation*}
\begin{aligned}
&\frac{\mathrm{d}}{\mathrm{d} t}\int_I\rho_0[\sigma(v^{(n)})]^2\,\mathrm{d} x
+\int_I\rho_0[\sigma_x(v^{(n)})]^2\,\mathrm{d} x\\
&\quad\leq C(\|v_x^{(n)}\|_{L^\infty(I)}+1)t\int_0^t\int_I\rho_0[\sigma_x(v^{(n-1)})]^2\,\mathrm{d} x\mathrm{d} s
\\
&\quad\leq C(M_1^{1/2}+1)t\int_0^t\int_I\rho_0[\sigma_x(v^{(n-1)})]^2\,\mathrm{d} x\mathrm{d} s,
\end{aligned}
\end{equation*}
where one has used \(\{v^{(n)}\}_{n=1}^\infty\subset \mathcal{C}_T(M_1)\) in the last line. Hence \eqref{existence-23} follows.
\end{proof}
Integrating \eqref{existence-23} with respect to \(t\) on \([0,T]\), we deduce
\begin{equation*}
\begin{aligned}
&\underset{0\leq t\leq T}{\sup}\|\rho_0^{1/2}\sigma(v^{(n)})\|_{L^2(I)}^2+\|\rho_0^{1/2}\sigma_x(v^{(n)})\|_{L^2([0,T];L^2(I))}^2\\
&\quad\leq \|\rho_0^{1/2}\sigma(v^{(n)})(0)\|_{L^2(I)}^2+C(M_1^{1/2}+1)T\|\rho_0^{1/2}\sigma_x(v^{(n-1)})\|_{L^2([0,T];L^2(I))}^2\\
&\quad=C(M_1^{1/2}+1)T\|\rho_0^{1/2}\sigma_x(v^{(n-1)})\|_{L^2([0,T];L^2(I))}^2\\
&\quad\leq \frac{1}{4}\big(\underset{0\leq t\leq T}{\sup}\|\rho_0^{1/2}\sigma(v^{(n-1)})\|_{L^2(I)}^2+
\|\rho_0^{1/2}\sigma_x(v^{(n-1)})\|_{L^2([0,T];L^2(I))}^2\big),
\end{aligned}
\end{equation*}
since \(T>0\) is sufficiently small.
Hence for any \(n\geq 1\)
\begin{equation}\label{existence-30}
\begin{aligned}
&\underset{0\leq t\leq T}{\sup}\|\rho_0^{1/2}\sigma(v^{(n)})\|_{L^2(I)}+\|\rho_0^{1/2}\sigma_x(v^{(n)})\|_{L^2([0,T];L^2(I))}\\
&\quad\leq \frac{1}{2}\big(\underset{0\leq t\leq T}{\sup}\|\rho_0^{1/2}\sigma(v^{(n-1)})\|_{L^2(I)}+
\|\rho_0^{1/2}\sigma_x(v^{(n-1)})\|_{L^2([0,T];L^2(I))}\big).
\end{aligned}
\end{equation}
The estimates \eqref{existence-30} imply that \(\{v^{(n)}\}_{n=1}^\infty\) is a Cauchy sequence in the space \(L^2([0,T],L^2(I))\) by using the weighted Sobolev inequality \eqref{ineq:weighted Sobolev}. According to this fact and the a priori bound \eqref{I-Priori-ellip-33} (see \eqref{Preliminary-1} that this a priori bound \eqref{I-Priori-ellip-33} controls \(H^3(I)\)-bound of \(v\)), one may furthermore deduce that \(\{v^{(n)}\}_{n=1}^\infty\) is a Cauchy sequence in the space \(L^2([0,T],H^s(I))\) \((0<s<3)\) by using the standard Gagliardo-Nirenberg interpolation inequality for functions in spatial variables (see \cite{MR3813967}).
However this is insufficient for us to pass limit in \(n\) in Equation \(\eqref{existence-3}_1\) for time pointwisely. To get around this difficulty, we need the following weighted interpolation inequality:
\begin{lemma}[Weighted Interpolation Inequality]\label{le:Preliminary-5} The following weighted interpolation holds
\begin{align}\label{Weighted Interpolation}
\|g\|_{L^2(I)}\lesssim \|g\|_{L_{\rho_0}^2(I)}^{1/2}\|g\|_{H_{\rho_0}^1(I)}^{1/2},
\end{align}
where
\begin{equation*}
\begin{aligned}
\|g\|_{L_{\rho_0}^2(I)}^2=\int_I\rho_0g^2\,\mathrm{d} x\quad {\rm{and}}\quad \|g\|_{H_{\rho_0}^1(I)}^2=\int_I\rho_0(g^2+g_x^2)\,\mathrm{d} x.
\end{aligned}
\end{equation*}
\end{lemma}
\begin{proof} Due to the assumption \eqref{eq:intro-3} on \(\rho_0\), it suffices to prove \eqref{Weighted Interpolation} for \(\rho_0\) with \(\rho_0(x)=x\) on \([0,1/2]\) and \(1-x\) on \([1/2,1]\). Note that
\begin{equation*}
\begin{aligned}
\int_Ig^2\,\mathrm{d} x= \int_0^{1/2}g^2\,\mathrm{d} x+\int_{1/2}^1g^2\,\mathrm{d} x.
\end{aligned}
\end{equation*}
Integration by parts yields
\begin{equation}\label{existence-31}
\begin{aligned}
\int_0^{1/2}g^2\,\mathrm{d} x&=xg^2(x)\big|_{x=0}^{x=1/2}-2\int_0^{1/2}xgg_x\,\mathrm{d} x\\
&=\frac{1}{2}g^2(\frac{1}{2})-2\int_0^{1/2}\rho_0gg_x\,\mathrm{d} x.
\end{aligned}
\end{equation}
To estimate \(g(\frac{1}{2})\), one has
\begin{equation}\label{existence-32}
\begin{aligned}
&\int_0^{1/2}\rho_0g^2\,\mathrm{d} x=\int_0^{1/2}xg^2\,\mathrm{d} x\\
&\quad=\frac{1}{2}x^2g^2(x)\big|_{x=0}^{x=1/2}-\int_0^{1/2}x^2gg_x\,\mathrm{d} x\\
&\quad=\frac{1}{8}g^2(\frac{1}{2})-\int_0^{1/2}\rho_0^2gg_x\,\mathrm{d} x.
\end{aligned}
\end{equation}
It follows from \eqref{existence-31} and \eqref{existence-32} that
\begin{equation}\label{existence-33}
\begin{aligned}
\int_0^{1/2}g^2\,\mathrm{d} x
&=4\int_0^{1/2}\rho_0g^2\,\mathrm{d} x+4\int_0^{1/2}\rho_0^2gg_x\,\mathrm{d} x-2\int_0^{1/2}\rho_0gg_x\,\mathrm{d} x\\
&\lesssim \int_0^{1/2}\rho_0g^2\,\mathrm{d} x+\bigg(\int_0^{1/2}\rho_0^2g^2\,\mathrm{d} x\bigg)^{1/2}\bigg(\int_0^{1/2}\rho_0^2g_x^2\,\mathrm{d} x\bigg)^{1/2}\\
&\quad+\bigg(\int_0^{1/2}\rho_0g^2\,\mathrm{d} x\bigg)^{1/2}\bigg(\int_0^{1/2}\rho_0g_x^2\,\mathrm{d} x\bigg)^{1/2}\\
&\lesssim \bigg(\int_I\rho_0g^2\,\mathrm{d} x\bigg)^{1/2}\bigg(\int_I\rho_0(g^2+g_x^2)\,\mathrm{d} x\bigg)^{1/2}.
\end{aligned}
\end{equation}
Similarly, one can obtain
\begin{equation}\label{existence-34}
\begin{aligned}
\int_{1/2}^1g^2\,\mathrm{d} x
\lesssim \bigg(\int_I\rho_0g^2\,\mathrm{d} x\bigg)^{1/2}\bigg(\int_I\rho_0(g^2+g_x^2)\,\mathrm{d} x\bigg)^{1/2}.
\end{aligned}
\end{equation}
Finally, \eqref{Weighted Interpolation} follows from \eqref{existence-33} and \eqref{existence-34}.
\end{proof}
Taking \(g(\cdot)=\sigma(v^{(n)})(\cdot,t)\) in \eqref{Weighted Interpolation}, one has that for each \(t\in[0,T]\)
\begin{equation}\label{existence-35}
\begin{aligned}
\|\sigma(v^{(n)})(\cdot,t)\|_{L^2(I)}\lesssim \|\sigma(v^{(n)})(\cdot,t)\|_{L_{\rho_0}^2(I)}^{1/2}\|\sigma(v^{(n)})(\cdot,t)\|_{H_{\rho_0}^1(I)}^{1/2}.
\end{aligned}
\end{equation}
It follows from \eqref{existence-30} that \(\|\sigma(v^{(n)})(\cdot,t)\|_{L_{\rho_0}^2(I)}\to 0\) as \(n\to \infty\).
And \eqref{I-Priori-ellip-33} implies that \(\|\sigma(v^{(n)})(\cdot,t)\|_{H_{\rho_0}^1(I)}\) is uniformly bounded in \(n\geq 1\). Hence \eqref{existence-35} implies that as \(n\to \infty\)
\begin{align}\label{existence-36}
v^{(n)}\rightarrow v\quad \text{in}\ C([0,T];L^2(I)).
\end{align}
Then the standard Gagliardo-Nirenberg interpolation inequality on a bounded domain (see \cite{MR3813967}) shows that for any \(s\in(0,3)\)
\begin{equation}\label{existence-37}
\begin{aligned}
\|\sigma(v^{(n)})(\cdot,t)\|_{H^s(I)}\lesssim \|\sigma(v^{(n)})(\cdot,t)\|_{L^2(I)}^{1-\frac{s}{3}}
\|\sigma(v^{(n)})(\cdot,t)\|_{H^3(I)}^{\frac{s}{3}}.
\end{aligned}
\end{equation}
Since \(\|\sigma(v^{(n)})(\cdot,t)\|_{H^3(I)}\) is uniformly bounded in \(n\geq 1\), it follows from \eqref{existence-36} and \eqref{existence-37} that as \(n\to \infty\)
\begin{align*}
v^{(n)}\rightarrow v\quad \text{in}\ C([0,T];H^s(I)), \quad \forall\ s\in(0,3),
\end{align*}
which furthermore implies by Sobolev embedding that as \(n\to \infty\)
\begin{align}\label{existence-38}
v^{(n)}\rightarrow v\quad \text{in}\ C([0,T];C^2(I)).
\end{align}
According to \(\eqref{existence-21}_1\), one has
\begin{equation*}
\begin{aligned}
\rho_0v_t^{(n)}=-\bigg[\frac{\rho_0^2}{(\eta_x^{(n-1)})^2}\bigg]_x
+\bigg[\frac{\rho_0v_x^{(n)}}{(\eta_x^{(n-1)})^2}\bigg]_x,
\end{aligned}
\end{equation*}
which, together with \eqref{existence-38}, yields that as \(n\to \infty\)
\begin{align}\label{existence-39}
\rho_0v_t^{(n)}\rightarrow -\bigg(\frac{\rho_0^2}{\eta_x^2}\bigg)_x
+\bigg(\frac{\rho_0v_x}{\eta_x^2}\bigg)_x\quad \text{in}\ C([0,T];C(I)).
\end{align}
Due to \eqref{existence-39}, the
distribution limit of \(v_t^{(n)}\) must be \(v_t\) as \(n\to \infty\), so, in particular, \(v\) is a classical solution to the problem \eqref{eq:main-2}.
Moreover, following the standard argument (see \cite{MR1867882}), one may show \(v\in C([0,T]; H^3(I))\cap C^1([0,T]; H^1(I))\).
\section{Proof of Theorem \ref{th:main-1}: Uniqueness}\label{Uniqueness Part}
The following observation will be useful in showing the uniqueness of the classical solution to the problem \eqref{eq:main-2}.
\subsection{A lower-order energy function}\label{lower-order energy function}
Define the following lower-order energy functional:
\begin{equation}\label{energy function-2}
\begin{aligned}
\mathcal{E}(t,v)&=\sum_{k=0}^2\|\sqrt{\rho_0}\partial_t^kv\|_{L^2(I)}^2
+\sum_{k=0}^1\|\sqrt{\rho_0}\partial_t^kv_x\|_{L^2(I)}^2\\
&+\|\rho_0\partial_tv_{xx}\|_{L^2(I)}^2+\sum_{k=2}^4\big\|\sqrt{\rho_0^k}\partial_x^kv\big\|_{L^2(I)}^2.
\end{aligned}
\end{equation}
Then one can also close the energy estimates, namely \(\mathcal{E}(t,v)\) satisfies
\begin{equation}\label{a priori bound-2}
\begin{aligned}
\mathcal{E}(t,v)
\leq \mathcal{M}_0+CtP(\sup_{0\leq s\leq t}\mathcal{E}^{1/2}(s,v))\quad \mathrm{for\ all}\ t\in [0,T]
\end{aligned}
\end{equation}
with \(\mathcal{M}_0\) given by
\begin{align*}
\mathcal{M}_0=P(\mathcal{E}(0,v_0)),
\end{align*}
where \(P\) denotes a generic polynomial of its arguments, and \(C\) is an absolutely constant depending only on \(\|\partial_x^l\rho_0\|_{L^\infty(I)}\ (l=0,1,2,3)\).\\
In fact, \eqref{a priori bound-2} follows from \eqref{I-Priori-time-8}, \eqref{I-Priori-time-12}, \eqref{II-Priori-time-4} in Section \ref{Energy Estimates}, \eqref{I-Priori-time-21}, \eqref{II-Priori-time-10}, \eqref{I-Priori-ellip-5}, \eqref{I-Priori-ellip-third}, \eqref{I-Priori-ellip-22}, and \eqref{I-Priori-ellip-31} in Section \ref{Elliptic estimates}. Indeed, \eqref{a priori bound-2} can be proved in a similar way as \eqref{I-Priori-ellip-32} with some modifications as follows.
In this case, the estimates on highest order
time-derivatives are \eqref{I-Priori-time-8} and
\eqref{I-Priori-time-21}, which can be obtained straightforwardly as \eqref{I-Priori-time-3} and \eqref{I-Priori-time-16}, respectively. Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2} should be replaced by
\begin{lemma}\label{le:Preliminary-3} It holds that
\begin{align}\label{Preliminary-10}
\|v(\cdot,t)\|_{H^2(I)}\lesssim \mathcal{E}^{1/2}(t,v).
\end{align}
Hence,
\begin{align}
&\|\eta_{xx}(\cdot,t)\|_{L^2(I)}\lesssim t\sup_{0\leq s\leq t}\mathcal{E}^{1/2}(t,v),\label{Preliminary-11}\\
&\|v_x(\cdot,t)\|_{L^\infty(I)}
\lesssim \mathcal{E}^{1/2}(t,v).\label{Preliminary-12}
\end{align}
\end{lemma}
\begin{lemma}\label{le:Preliminary-4} It holds that
{\small \begin{align}\label{Preliminary-13}
\|\rho_0\partial_x^3v(\cdot,t)\|_{L^2(I)}\lesssim \mathcal{E}^{1/2}(t,v).
\end{align}}
Consequently,
{\small \begin{align}
&\|\rho_0\partial_x^3\eta(\cdot,t)\|_{L^2(I)}
\lesssim t\sup_{0\leq s\leq t}\mathcal{E}^{1/2}(t,v),\label{Preliminary-14}\\
&\|\rho_0v_{xx}(\cdot,t)\|_{L^\infty(I)}\lesssim \mathcal{E}^{1/2}(t,v),\label{Preliminary-15}\\
&\|\rho_0\eta_{xx}(\cdot,t)\|_{L^\infty(I)}
\lesssim t\sup_{0\leq s\leq t}\mathcal{E}^{1/2}(s,v)\label{Preliminary-16}.
\end{align}}
\end{lemma}
In elliptic estimates, one can use Lemma \ref{le:Preliminary-3} and Lemma \ref{le:Preliminary-4} to replace Lemma \ref{le:Preliminary-1} and Lemma \ref{le:Preliminary-2}. On the one hand, the second term on the RHS of \eqref{additial estimate} can be estimated as follows:
\begin{equation*}
\begin{aligned}
\bigg|\int_I\rho_0(\rho_0)_x\eta_x^{-5}\eta_{xx}v_x^2\,\mathrm{d} x\bigg|
\lesssim
\|\rho_0\eta_{xx}\|_{L^\infty}\|v_x\|_{L^2}^2
\leq Ct P(\sup_{0\leq s\leq t}\mathcal{E}^{1/2}(s,v)).
\end{aligned}
\end{equation*}
One may also handle the similar term in \eqref{I-Priori-ellip-17} as
\begin{equation*}
\begin{aligned}
\bigg|\int_I\rho_0^2(\rho_0)_x\eta_x^{-5}\eta_{xx}v_{xx}^2\,\mathrm{d} x\bigg|
\lesssim
\|\rho_0\eta_{xx}\|_{L^\infty}\|v_{xx}\|_{L^2}^2
\leq Ct P(\sup_{0\leq s\leq t}\mathcal{E}^{1/2}(s,v)),
\end{aligned}
\end{equation*}
and the one in \eqref{I-Priori-ellip-30} as
\begin{equation*}
\begin{aligned}
\bigg|\int_I\rho_0^3(\rho_0)_x\eta_x^{-5}\eta_{xx}(\partial_x^3v)^2\,\mathrm{d} x\bigg|
\lesssim
\|\rho_0\eta_{xx}\|_{L^\infty}\|\rho_0\partial_x^3v\|_{L^2}^2
\leq Ct P(\sup_{0\leq s\leq t}\mathcal{E}^{1/2}(s,v)).
\end{aligned}
\end{equation*}
On the other hand, one can use \(\|\rho_0\eta_{xx}\|_{L^\infty}\) to replace \(\|\eta_{xx}\|_{L^\infty}\) in \eqref{I-Priori-ellip-7}, \eqref{I-Priori-ellip-12}, \eqref{I-Priori-ellip-14}, \eqref{I-Priori-ellip-25}, \eqref{I-Priori-ellip-27.2}, \eqref{I-Priori-ellip-27.5} and \eqref{I-Priori-ellip-28.5}; and use
\(\|\rho_0v_{xx}\|_{L^\infty}\) to replace \(\|v_{xx}\|_{L^\infty}\) in \eqref{I-Priori-ellip-27.5}; and use
\(\|\rho_0\partial_x^3\eta\|_{L^2}\) to replace \(\|\partial_x^3\eta\|_{L^2}\) in \eqref{I-Priori-ellip-7}, \eqref{I-Priori-ellip-25}, \eqref{I-Priori-ellip-27.5} and \eqref{I-Priori-ellip-28.5}; and use \(\|\rho_0\partial_x^3v\|_{L^2}\) to replace \(\|\partial_x^3v\|_{L^2}\) in \eqref{I-Priori-ellip-27.2}; and use \(\|\rho_0^2\partial_x^4\eta\|_{L^2}\) to replace \(\|\rho_0\partial_x^4\eta\|_{L^2}\) in \eqref{I-Priori-ellip-25} and \eqref{I-Priori-ellip-28.5}. All of these replacements are possible due to the suitable choice of weights in the corresponding formulae.
\begin{remark} The main reason that we use \(E(t,v)\) instead of \(\mathcal{E}(t,v)\) to define the solution space is to achieve the regularity \(v\in L^\infty([0,T]; H^3(I))\) which is needed to define the classical solutions. The energy functional \(\mathcal{E}(t,v)\) only gives us the regularity \(v\in L^\infty([0,T]; H^2(I))\), however, which will play an important role in showing the uniqueness of the classical solution to the problem \eqref{eq:main-2} in the next section.
\end{remark}
\subsection{Uniqueness of the classical solution to the problem \eqref{eq:main-2}}\label{Uniqueness of the classical solution}
Let \(v\) and \(w\) be two solutions to the problem \eqref{eq:main-2} on \([0,T]\) with initial data \((\rho_0,u_0)\) satisfying the same estimate. Their corresponding flow maps are:
\begin{equation*}
\begin{aligned}
\eta(x,t)=x+\int_0^tv(x,s)\,\mathrm{d} s,\\
\zeta(x,t)=x+\int_0^tw(x,s)\,\mathrm{d} s.
\end{aligned}
\end{equation*}
Set
\begin{equation*}
\begin{aligned}
\delta_{vw}=v-w.
\end{aligned}
\end{equation*}
Then \(\delta_{vw}\) satisfies:
\begin{equation*}
\begin{cases}
\rho_0(\delta_{vw})_t+\big[\rho_0^2\big(\frac{1}{\eta_x^2}
-\frac{1}{\zeta_x^2}\big)\big]_x
=\big[\rho_0\big(\frac{v_x}{\eta_x^2}-\frac{w_x}{\zeta_x^2}\big)\big]_x&\quad \mbox{in}\ I\times (0,T],\\
(\delta_{vw},\eta)=(0,e) &\quad \mbox{on}\ I\times \{t=0\},\\
(\delta_{vw})_x=0 &\quad \mbox{on}\ \Gamma\times (0,T].
\end{cases}
\end{equation*}
Note that
\begin{equation*}
\begin{aligned}
\bigg[\rho_0^2\bigg(\frac{1}{\eta_x^2}-\frac{1}{\zeta_x^2}\bigg)\bigg]_x
=-\bigg[\frac{\rho_0^2}{\eta_x^2\zeta_x^2}
\bigg(\int_0^t(\delta_{vw})_x\,\mathrm{d} s\int_0^t(v_x+w_x)\,\mathrm{d} s\bigg)\bigg]_x,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\bigg[\rho_0\bigg(\frac{v_x}{\eta_x^2}-\frac{w_x}{\zeta_x^2}\bigg)\bigg]_x
&=\bigg[\frac{\rho_0}{\eta_x^2\zeta_x^2}
\bigg((\delta_{vw})_x+2(\delta_{vw})_x\int_0^tw_x\,\mathrm{d} s
-2w_x\int_0^t(\delta_{vw})_x\,\mathrm{d} s\\
&\quad
-\int_0^t(\delta_{vw})_x\,\mathrm{d} s\int_0^t(v_x+w_x)\,\mathrm{d} s\bigg)\bigg]_x,
\end{aligned}
\end{equation*}
which contain some additional error terms:
\begin{equation*}
\begin{aligned}
(\delta_{vw})_x\int_0^tw_x\,\mathrm{d} s,\quad w_x\int_0^t(\delta_{vw})_x\,\mathrm{d} s\quad {\rm{and}}\quad \int_0^t(\delta_{vw})_x\,\mathrm{d} s\int_0^t(v_x+w_x)\,\mathrm{d} s.
\end{aligned}
\end{equation*}
Unfortunately, it can be checked that these additional error terms make it difficult to derive an inequality as \eqref{I-Priori-ellip-32} for
\(E(t,\delta_{vw})\). In other words, it needs some higher-order energy functionals than \(E(t,v)\) and \(E(t,w)\) to control these error terms if one works with \(E(t,\delta_{vw})\).
So we instead work with the lower-order energy functional \(\mathcal{E}(t,\delta_{vw})\) defined by \eqref{energy function-2}, and find that all the error terms can be easily controlled by the energy functionals \(E(t,v)\) and \(E(t,w)\). Therefore we may get finally (see Subsection \ref{lower-order energy function} for more details) that
\begin{equation}\label{uniques}
\begin{aligned}
\sup_{0\leq s\leq t}\mathcal{E}(t,\delta_{vw})
\leq CtP(\sup_{0\leq s\leq t}\mathcal{E}^{1/2}(s,\delta_{vw}))\quad \mathrm{for\ all}\ t\in [0,T],
\end{aligned}
\end{equation}
where \(C\) depends on \(E(t,v)\) and \(E(t,w)\). Hence \(\delta_{vw}=0\) follows from \eqref{uniques}.
\section*{Acknowledgment}
Li's research was supported by the National Natural Science Foundation of China (Grant Nos. 11931010 and 11871047), and by the key research project of Academy for Multidisciplinary Studies, Capital Normal University, and by the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds 007/20530290068. Wang's research was supported by the Grant No. 830018 from China.
Xin's research was supported by the Zheng Ge Ru Foundation and by Hong Kong RGC Earmarked Research Grants CUHK14301421, CUHK14300819, CUHK14302819, CUHK14300917, Basic and Applied Basic Research Foundations of Guangdong Province 20201\
31515310002, and the Key Project of National Nature Science Foundation of China (Grant No. 12131010).
\end{document} |
\begin{document}
\title{On the feasibility of a nuclear exciton laser}
\author{Nicolai ten Brinke}
\author{Ralf Sch\"utzhold}
\email[e-mail:\,]{[email protected]}
\affiliation{Fakult\"at f\"ur Physik, Universit\"at Duisburg-Essen,
Lotharstrasse 1, D-47057 Duisburg, Germany}
\author{Dietrich Habs}
\affiliation{Fakult\"at f\"ur Physik, Ludwig-Maximilians-Universit\"at
M\"unchen, Am Coulombwall 1, D-85748 Garching, Germany}
\date{\today}
\begin{abstract}
Nuclear excitons known from M\"ossbauer spectroscopy describe coherent
excitations of a large number of nuclei -- analogous to Dicke states
(or Dicke super-radiance) in quantum optics.
In this paper, we study the possibility of constructing a laser
based on these coherent excitations.
In contrast to the free electron laser (in its usual design),
such a device would be based on stimulated emission and thus might
offer certain advantages, e.g., regarding energy-momentum accuracy.
Unfortunately, inserting realistic parameters, the window of
operability is probably not open (yet) to present-day technology --
but our design should be feasible in the UV regime, for example.
\end{abstract}
\pacs{
42.50.-p,
42.55.Ah,
42.50.Gy,
33.25.+k.
}
\maketitle
\section{Introduction}
The invention of the laser lead to a giant leap in the field of classical
and quantum optics.
This light source offers unprecedented possibilities regarding features
such as coherence, intensity, and brilliance etc.
Unfortunately, however, it is not easy to transfer this successful concept
beyond the optical or near-optical regime,
cf.~\cite{Baldwin:1997ve,Tkalya:2011dq}.
Free-electron lasers, for example, work at much higher energies -- but their
principle of operation (in their usual design) is more similar to classical
emission instead of stimulated emission.
As a result, their properties (e.g., regarding coherence) are not quite
comparable to optical lasers.
There is another phenomenon in this energy range $\,{\cal O}(\rm keV)$ in which
coherence plays a crucial role -- nuclear excitons known from M\"ossbauer
spectroscopy \cite{Mossbauer:1958fk,Mossbauer:1958kx}.
These coherent excitations of a large number of nuclei
\cite{Hannon:1999fk,Smirnov:2005uq,Habs:2009uq} are analogous to
Dicke states \cite{Dicke:1954kx} (also known as Dicke super-radiance
\cite{Scully:2006fk,Scully:2007fk,Sete:2010fu}) in quantum optics.
The coherence results in constructive interference of the emission
amplitudes from many nuclei \cite{Burnham:1969uq} and is facilitated
by the fact that the
photon recoil is absorbed by the whole lattice
\cite{Mossbauer:1958fk,Mossbauer:1958kx} instead of the individual
nuclei (which would destroy the coherence).
For example, the coherent nature of the propagation of nuclear
excitons through resonant media, showing quantum beats, was observed in
\cite{Frohlich:1991kx,Burck:1999fk}.
Other cooperative effects of coherently excited nuclei have been studied,
such as the collective Lamb shift \cite{Rohlsberger:2010fk},
coherent control of nuclear x-ray pumping \cite{Palffy:2011vn}, and
electromagnetically induced transparency \cite{Rohlsberger:2012uq}.
In the following, we study the possibility of constructing a laser-type
device employing these nuclear excitons,
which is based on stimulated emission \cite{ELIwhitebook}.
Such a device could combine the advantages of the free-electron laser
with the coherence and brilliance of nuclear excitons.
\section{Hamiltonian}
First, we describe a single nucleus as a two-level system with transition
frequency $\omega$ interacting resonantly with a single-mode field.
In rotating-wave and dipole approximation, the Hamiltonian can then
be cast into the standard form ($\hbar = c = \varepsilon_0 = 1$)
\begin{eqnarray}
\label{eq:hsingle}
\hat{H}_{\rm single}
=
\left(
g \hat{a} \sigma_\ell^+ e^{i \fk{\kappa} \cdot \fk{r}_\ell} + {\rm H.c.}
\right)
+ \frac{\omega}{2} \left( \hat{\sigma}^z_\ell + 1 \right)
+ \omega \hat{a}^\dagger \hat{a}
\,.
\end{eqnarray}
As usual, the ladder operators
$\sigma_\ell^{\pm} = ( \sigma_\ell^x \pm i \sigma_\ell^y ) / 2$
and the Pauli matrix $\sigma_\ell^z$ describe the two-level system.
The first term governs the interaction (with coupling constant~$g$)
with the electromagnetic field and
thus contains photonic annihilation and creation operators
$\hat{a} / \hat{a}^\dagger$ and phase factors
$e^{i \fk{\kappa} \cdot \fk{r}_\ell}$ depending on the location
of the nucleus, $\f{r}_\ell$, and the wavenumber $\f{k}=\f{\kappa}$
of the photon mode with $|\f{\kappa}| = \omega$.
The second and third term account for the energy stored in the two-level
nucleus and in the single-mode field, respectively.
When dealing with many $S \gg 1$ two-level nuclei instead of one,
we can sum up the individual-nucleus Hamiltonians and arrive at
\begin{eqnarray}
\label{eq:hmany}
\hat{H}
=
\left( g \hat{a} \hat{\Sigma}^+ + {\rm H.c.} \right)
+ \omega \left( \hat{\Sigma}^z + \frac{S}{2} \right)
+ \omega \hat{a}^\dagger \hat{a}
\,,
\end{eqnarray}
where quasispin-$S$-operators have been introduced
\begin{eqnarray}
\hat{\Sigma}^\pm
=
\sum_{\ell=1}^S \sigma_\ell^\pm \exp\{\pm i \f{\kappa} \cdot \f{r}_\ell\}
\,,\quad
\hat{\Sigma}^z = \frac{1}{2} \sum_{\ell=1}^S \sigma_\ell^z
\,.
\end{eqnarray}
In the interaction picture, the perturbation Hamiltonian,
originating from the first term in Eq.~(\ref{eq:hmany}), reads
\begin{eqnarray}
\label{eq:hinteract}
\hat{V} = g \hat{a} \hat{\Sigma}^+ + {\rm H.c.}
\,.
\end{eqnarray}
The quasispin-$S$-operators
$\hat{\Sigma}^\pm=\hat{\Sigma}^x\pm i\hat{\Sigma}^y$
and
$\hat{\Sigma}^z$ generate an $SU(2)$ algebra \cite{Lipkin:2002fk}.
Thus, the transition matrix elements for collective transitions not
only depend on the number of nuclei involved, but also on the number
of excitations $s$
\begin{eqnarray}
\label{eq:matrix_elements}
\hat{\Sigma}^+\ket{s}&=&\sqrt{(S-s)(s+1)}\ket{s+1}
\,,\nonumber\\
\hat{\Sigma}^-\ket{s}&=&\sqrt{(S-s+1)s}\ket{s-1}
\,,
\end{eqnarray}
where $\ket{s} \propto (\hat{\Sigma}^+)^s \ket{0}$ denotes a coherent
state with $s$ excitons, often referred to as Dicke states
\cite{Dicke:1954kx}.
\section{Coherent emission}
In contrast to the spontaneous decay of a single excited nucleus,
where the resulting photon can be emitted in all directions, exciton
states as in Eq.~(\ref{eq:matrix_elements}) predominantly emit
photons in forward direction $\f{\kappa}$.
Only in this case, all the phases $e^{i \fk{\kappa} \cdot \fk{r}_\ell}$
add up coherently (we assume random locations $\f{r}_\ell$),
see Fig.~\ref{fig:exciton_and_dicke}.
We will now investigate spontaneous and stimulated emission from an
ensemble of $S$ coherently excited nuclei in more detail.
\begin{figure}
\caption{Sketch of the coherent properties of nuclear excitons.
An incident photon with wave-vector $\f{k}
\label{fig:exciton_and_dicke}
\end{figure}
\subsection{Spontaneous emission}
We start with the case of collective spontaneous emission
(a.k.a.\ Dicke super-radiance \cite{Scully:2006fk,Scully:2007fk,Sete:2010fu})
from a coherent state $\ket{s}$.
First of all, as the $S$ nuclei are not enclosed by a resonator
or a cavity in our set-up, we have to consider all $\f{k}$-modes.
Thus, the Hamiltonian~(\ref{eq:hinteract}) changes into
\begin{eqnarray}
\label{eq:hmanymodes}
\hat{V}_{\rm sp} \left( \tau \right)
=
\int d^3k\; g_{\fk{k}} \hat{a}_{\fk{k}} \,
e^{-i \left( \omega_{\fk{k}} - \omega \right) \tau} \hat{\Sigma}^+
\left( \f{k} \right) + {\rm H.c.}
\,,
\end{eqnarray}
where $\hat{a}_{\fk{k}}$ is the photonic annihilation operator for the
mode $\f{k}$ with frequency $\omega_{\fk{k}}$ and $g_{\fk{k}}$ the
associated coupling strength.
Note that we neglect polarization effects, i.e., we assume that
the polarization vectors are directed along the same axis as the dipole
moments of the absorbing nuclei.
Furthermore, $\hat{\Sigma}^+\left(\f{k}\right)$ denotes the
quasispin-$S$-operators with the wavenumber $\f{k}$ instead of $\f{\kappa}$.
However, when $\f{k}$ is not close to $\f{\kappa}$, the phase factors
of $\hat{\Sigma}^\pm \left( \f{k} \right)$ and $\ket{s}$ do not match,
and the transition is not coherent, i.e., not enhanced by a factor
$S$ according to Eq.~(\ref{eq:matrix_elements}),
and can thus be neglected.
Note that this is the reason why collectively emitted photons are
directed along (almost) the same axis as previously absorbed
photons \cite{Scully:2006fk,Scully:2007fk,Sete:2010fu},
see also Fig.~\ref{fig:exciton_and_dicke}.
For simplicity, the quasispin-$S$-operators are therefore approximated
by introducing a cut-off function $g \left( \f{\kappa} - \f{k} \right)$
that is only non-zero for small deviations $\f{\kappa} - \f{k}$
of the supported direction $\f{\kappa}$, i.e.,
$\hat{\Sigma}^\pm \left( \f{k} \right) \approx
g \left( \f{\kappa} - \f{k} \right)\,\hat{\Sigma}^\pm$.
For large $S$, we may approximate the quasispin-$S$-operators
classically, i.e.,
$\hat{\Sigma}^- \approx \Sigma^- = \sqrt{\left( S - s +1 \right) s}$,
and thus the effect of the Hamiltonian Eq.~(\ref{eq:hmanymodes}) acting on
the vacuum state can be expressed by a coherent state
\begin{eqnarray}
\hat{U}_{\rm sp} ( t ) \ket{0}
\approx
\exp\left(
\int d^3k \, \alpha_{\fk{k}} \hat{a}_{\fk{k}}^\dagger - {\rm H.c.}
\right)
\ket{0}
\,,
\end{eqnarray}
with the amplitudes
\begin{eqnarray}
i\alpha_{\fk{k}}
=
g_{\fk{k}}^* g ( \f{\kappa} - \f{k} )
\sqrt{\left( S - s +1 \right) s}
\int_{0}^t
d\tau \,
e^{i \left( \omega_{\fk{k}} - \omega \right) \tau}
\,.
\end{eqnarray}
The number of emitted photons per mode is given by $|\alpha_{\fk{k}}|^2$
and the total photon number grows linearly with $t$
\begin{eqnarray}
\mathcal{N}_\gamma
=
\int d^3k \, |\alpha_{\fk{k}}|^2
\approx
2 \pi^2 \left( S - s +1 \right) s |g_{\fk{\kappa}}|^2 \frac{t}{L_\bot^2}
\,,
\end{eqnarray}
where $L_\bot^2$ denotes the transversal cross-section area of the ensemble,
which determines the transverse area in $\f{k}$-space where
$g ( \f{\kappa} - \f{k} )$ is non-zero.
In addition to this spatial resonance condition, the temporal resonance
was incorporated via approximating the squared time-integral by $t^2$ for
$| \omega_k - \omega | < 1/t$ and zero otherwise.
Strictly speaking, this relation is only valid for a fixed number of
excitations $s$, i.e., the time-dependence of $s(t)$ due to the emission
of photons (energy conservation) is neglected.
Assuming that this time-dependence $s(t)$ is slow compared to $\omega$
(i.e., that the coupling strength is small enough), we may take it into
account approximately via defining the instantaneous spontaneous emission
rate
\begin{eqnarray}
\Gamma_{\rm sp}(t)
= \frac{d\mathcal{N}_\gamma}{dt}
= \gamma \left[ S - s(t) +1 \right] s(t)
\,,
\end{eqnarray}
with the abbreviation $\gamma = 2 \pi^2 |g_{\fk{\kappa}}|^2 / L_\bot^2$.
The change of $s(t)$ in the time interval $dt$ is then governed by
$\Gamma_{\rm sp}(t)$
\begin{eqnarray}
\label{eq:dgl}
\frac{ds(t)}{dt}
=
- \Gamma_{\rm sp}(t)
=
-\gamma \left[ S - s(t) + 1 \right] s(t)
\,.
\end{eqnarray}
For the initial condition $s(0) = S/2$ (see below),
the solution for $S \gg 1$ is given by
\begin{eqnarray}
s(t) = \frac{S}{1 + e^{\gamma S t}}
\,.
\end{eqnarray}
This yields the intensity due to spontaneous emission
\begin{eqnarray}
\label{eq:ispon}
I_{\rm sp}(t)
&=&
-\frac{dE}{dt} \frac{1}{L_\bot^2}
=
-\frac{ds(t)}{dt} \frac{\omega}{L_\bot^2}
\nonumber\\
&=&
\frac{1}{4} \gamma S^2 {\rm sech}^2 \left( \gamma S t /2 \right)
\frac{\omega}{L_\bot^2}
\,,
\end{eqnarray}
where $L_\bot^2$ is the cross-section area of the emitted beam,
see also \cite{Rehler:1971fk}.
The time-dependence in the ${\rm sech}$-function can be used to define
an effective time constant via
\begin{eqnarray}
\label{eq:tauspon}
\tau_{\rm sp} = \frac{4}{\gamma S}
\,,
\end{eqnarray}
after which $I_{\rm sp}(t)$ has dropped to 7\% of its initial value.
Let us now briefly compare this time-scale $\tau_{\rm sp}$ for the coherent
spontaneous emission process with the time-scale in the incoherent case.
For incoherent emission from $s$ excited nuclei, we can regard each
nucleus independently.
According to standard Weisskopf-Wigner theory \cite{Scully:1997fk},
the life-time of an excited nucleus is given by
\begin{eqnarray}
\label{eq:tausingle}
\tau_{\rm single}
=
\frac{1}{\Gamma_{\rm single}}
=
\frac{1}{8 \pi^2} \frac{1}{|g_{\fk{\kappa}}|^2 \omega^2}
\,.
\end{eqnarray}
Comparing the two time-scales Eq.~(\ref{eq:tauspon}) and (\ref{eq:tausingle})
\begin{eqnarray}
\frac{\tau_{\rm sp}}{\tau_{\rm single}}
=
64\pi^2 \frac{L_\bot^2}{\lambda^2} \frac{1}{S}
\,,
\end{eqnarray}
we find that for large $S$, the coherent spontaneous emission process is
much faster than the incoherent process
(see also \cite{Junker:2012fk}).
Taking for example $S = 10^{10}$ $^{57}$Fe-nuclei with resonance at
$\Delta E_\gamma = 14.4\;{\rm keV}$, lifetime
$\tau_{\rm single} = 141\;{\rm ns}$ and $L_\bot = 0.1\;\mu{\rm m}$,
the quotient evaluates to $\tau_{\rm sp}/\tau_{\rm single} \approx 0.09$,
i.e. the coherent emission runs over ten times faster than the incoherent
emission.
Note that there are also competing processes, such as decay via
electron conversion -- but they are incoherent and thus can be suppressed
for large $S$, i.e., small $\tau_{\rm sp}$.
\subsection{Stimulated emission}
In order to study stimulated emission from a coherently excited $S$-nuclei
ensemble, we regard the incoming field
$A_{\rm in}(t) = \sqrt{I_{\rm in}(t)}/\omega$ classically.
That is, we use the Hamiltonian Eq.~(\ref{eq:hinteract}), but replace
$g \hat{a}$ by $\tilde{g} A_{\rm in}(t)$.
For simplicity, we assume the transition matrix element $\tilde{g}$
of the nucleus to be real
\begin{eqnarray}
\label{eq:hstim}
\hat{V}_{\rm st}
=
\tilde{g} A_{\rm in}(t) \left( \hat{\Sigma}^+ + \hat{\Sigma}^- \right)
=
2 \tilde{g} A_{\rm in}(t) \hat{\Sigma}^x
\,.
\end{eqnarray}
Applying Heisenberg picture and employing the properties of the
$SU(2)$-algebra yields
\begin{eqnarray}
\label{eq:sigmaz}
\hat{U}_{\rm st}^\dagger (t) \hat{\Sigma}^z \hat{U}_{\rm st} (t)
&=&
\cos \left( 2 \tilde{g} \int_0^t d\tau\, A_{\rm in}(\tau)\right)
\hat{\Sigma}^z
\nonumber\\
&+& \sin \left( 2 \tilde{g} \int_0^t d\tau\, A_{\rm in}(\tau)\right)
\hat{\Sigma}^y
\,.
\end{eqnarray}
As envisaged for laser application (see below), we choose $s(0) = S$ here,
that is all $S$ nuclei are in the coherently excited state.
The time-dependent number of excitations is given by
$\bra{S}\hat{U}_{\rm st}^\dagger(t)\hat{\Sigma}^z\hat{U}_{\rm st}+S/2\ket{S}$
and thus the energy stored in the $S$ nuclei at time $t$ is
\begin{eqnarray}
E(t)
=
\frac{S \omega}{2} \left[ \cos
\left( 2 \tilde{g} \int_0^t d\tau\, A_{\rm in}(\tau)\right)
+ 1\right]
\,.
\end{eqnarray}
This yields the emitted intensity $I_{\rm st}(t)$ stimulated by the incoming
intensity $I_{\rm in}(t)$
\begin{eqnarray}
\label{eq:istim}
I_{\rm st}(t)
=
\frac{\tilde{g} S}{L_\bot^2}
\sin \left(
\frac{2 \tilde{g}}{\omega} \int_0^t d\tau\, \sqrt{I_{\rm in}(\tau)}
\right)
\sqrt{I_{\rm in}(t)}
\,,
\end{eqnarray}
where we have assumed that both beams have the same cross-section area
$L_\bot^2$.
We define the time-scale of the stimulated emission as the time
$\tau_{\rm st}$, after which all the energy initially stored in the $S$
nuclei has been emitted, i.e.,
\begin{eqnarray}
\label{eq:taustim}
\int_0^{\tau_{\rm st}} d\tau\,\sqrt{I_{\rm in}(\tau)}
=
\frac{\pi \omega}{2 \tilde{g}}
\,.
\end{eqnarray}
Now let us imagine that we have two separate ensembles (e.g., foils)
of coherently excited nuclei, such that the first foil spontaneously
emits the intensity $I_{\rm in}(t)$ as in Eq.~(\ref{eq:ispon}) which
causes stimulated emission according to Eq.~(\ref{eq:istim})
in the second foil.
In this case, we can insert $ I_{\rm in}(t) = I_{\rm sp}(t)$,
and Eq.~(\ref{eq:taustim}) can be solved for $\tau_{\rm st}$
\begin{eqnarray}
\tau_{\rm st}
=
\frac{4}{\gamma S}\,{\rm ArTanh}
\left[ {\rm Tan} \left( \frac{1}{8}
\sqrt{\frac{\pi}{2}} \right) \right]
\approx 0.16\times\tau_{\rm sp}
\,.
\end{eqnarray}
Since both foils contain the same nuclei (with the same coupling strengths),
the time-scale for the stimulated emission of the second foil,
$\tau_{\rm st}$, is completely determined by the time-scale of the
spontaneous emission process of the first foil, $\tau_{\rm sp}$.
\section{Pumping}
After having discussed coherent spontaneous emission as well as coherent
stimulated emission, let us investigate the pumping process for a single
foil of $S$ nuclei, which are initially in the state $s=0$, i.e.,
$\bra{0} \hat{\Sigma}^z \ket{0} = -S/2$.
Note that it is very easy to over- or under-estimate the efficiency of the
pumping process by using too simplified pictures.
On the one hand, one might expect that the number of excitons in the foil
grows linearly with the number of photons incident and thus linearly with
the interaction time $t$.
However, this is only true for pumping with incoherent light
(for further details, see the appendix), but not for coherent pumping,
which is the case considered here.
On the other hand, since the transition matrix elements in
Eq.~(\ref{eq:matrix_elements}) scale with $\sqrt{s}$ and thus the
effective line-width increases with $s$, one might expect a behavior
like $\dot s\propto s$, which would imply an exponential growth
$s(t)\propto e^{\kappa t}$, at least for small $s\ll S$.
This picture is also wrong, since -- in view of the unitarity of the
time-evolution -- not just the absorption rate but also the emission
rate increase with $s$.
Thus, the correct answer is that $s(t)$ grows quadratically
$s(t)\propto t^2$ for small $s$, i.e., somewhere in between
linear and exponential.
To show this, let us consider pumping with one coherent pump-pulse
$A_{\rm pump}(t)$ for the whole interaction time.
We can employ the Hamiltonian Eq.~(\ref{eq:hstim}) again with the sole
difference that the incoming field $A_{\rm in}(t)$ is now given by the
pump-field $A_{\rm in}(t) = A_{\rm pump}(t)$.
Thus, Eq.~(\ref{eq:sigmaz}) again holds, and the exciton number is given by
\begin{eqnarray}
s(t)
&=&
\frac{S}{2}
\left[
1 - \cos \left( 2 \tilde{g} \int_0^t d\tau\, A_{\rm pump}(\tau)\right)
\right]
\nonumber\\
&=&
S \tilde{g}^2 \left( \int_0^t d\tau\, A_{\rm pump}(\tau) \right)^2
+ \,{\cal O} \left( \tilde{g}^4 t^4 \right)
\,,
\end{eqnarray}
i.e., the exciton number grows quadratically for small $t$
(for an alternative approach, see the appendix).
We moreover find that a full cycle
(i.e., a sign flip of $\hat{\Sigma}^z\to-\hat{\Sigma}^z$)
occurs after the pump time $\tau_{\rm pump}$ where
\begin{eqnarray}
\label{eq:pump_constraint}
\int_0^{\tau_{\rm pump}} d\tau\, A_{\rm pump}(\tau) = \frac{\pi}{2 \tilde{g}}
\,.
\end{eqnarray}
The simplest example would be a constant pump pulse
$A_{\rm pump}=A_0$ with $\tau_{\rm pump} = \pi / ( 2 \tilde{g} A_0 )$.
In order to see if such a pump-field is feasible in general, we calculate
a rough estimate for the required intensity of the pump-field.
For simplicity, we assume that the intensity is constant over the pulse,
i.e., $A_{\rm pump} (t) = \sqrt{I_{\rm pump}}/\omega$.
Then we find
$I_{\rm pump} = \pi^2 \omega^2 / (4 \tilde{g}^2 \tau_{\rm pump}^2)$.
Now $\tilde{g}$ can be expressed in terms of the (single-nucleus) decay
rate $\Gamma_{\rm single}$ and the frequency $\omega$ of the considered
nuclear excitation (see the appendix).
Moreover, if we replace the coherent pulse-length
$\tau_{\rm pump} = {\mathfrak N} \lambda = 2 \pi {\mathfrak N} / \omega$
by the number ${\mathfrak N}$ of (coherent) wave-cycles, we find
\begin{eqnarray}
\label{eq:ipump}
I_{\rm pump}
=
\frac{1}{32 \pi {\mathfrak N}^2}\,\frac{\omega^5}{\Gamma_{\rm single}}
\,.
\end{eqnarray}
Thus, nuclear resonances with low energies $\omega$ but high decay-rates
$\Gamma_{\rm single}$ require low pump intensities.
Concrete examples will be discussed at the end of this article.
\section{Laser}
Now we have gathered all the tools required to understand the set-up
of the proposed nuclear exciton laser.
The envisaged set-up consists of a series of $N \gg 1$ foils
$n = 1,2,...,N$, each foil containing $S_n$ nuclei
(two-level systems) with a nuclear resonance at frequency $\omega$.
At the beginning, we assume that all $n = 1,2,...,N$ foils are in the
ground state, corresponding to a quasi-spin $\Sigma^z_n = -S_n/2$,
see Fig.~\ref{fig:sequence}(a).
To prepare the emission of a laser pulse, the foils need to be pumped
to suitable coherent states.
Let us distinguish between the first foil and all later foils $n = 2,...,N$.
While the latter all should be pumped to the maximum of $\Sigma^z_n = S_n/2$,
i.e., $s_n = S_n$, the first foil should only be pumped such that half of the
nuclei are in the excited state, i.e., $\Sigma^z_1 = 0$ and $s_1 = S_1/2$.
For simplicity, we envisage the whole pumping process to be achieved by only
one coherent pump-pulse, which goes through all the foils one after another
and is only weakly changed by absorption.
\begin{figure}
\caption{
Sketch of the operation sequence of the proposed nuclear exciton laser.
Initially, all foils (here $N=3$) are in the ground state
$\Sigma^z_n = -S_n/2$ (a).
The pump-pulse then rotates the quasispin of the first foil to
$\Sigma^z_1 = 0$
and the quasispin of all subsequent foils to $\Sigma^z_n = +S_n/2$ (b).
Then, the ``half-filled'' first foil spontaneously emits a pulse
$I_{\rm sp}
\label{fig:sequence}
\end{figure}
The pump-pulse should satisfy Eq.~(\ref{eq:pump_constraint}) in order to
rotate the quasispin $\Sigma^z_n$ of each foil from $\Sigma^z_n = -S_n/2$
to $\Sigma^z_n = +S_n/2$.
Additional measures need to be taken to ensure that the first foil is only
pumped to $s_1 = S_1/2$.
One option could be to have a different kind of nuclei in the first foil,
which have the same resonance frequency as those in the other foils, while
the coupling strengths differ by a factor of two (approximately).
Another option could be to switch the first foil
(mechanically or magnetically
\cite{Shvydko:1996fk,Rohlsberger:2000fk,Coussement:2002zr,
Palffy:2009kx,Adams:2011ys})
during the pumping process.
When the set-up is prepared as shown in Fig.~\ref{fig:sequence}(b),
the emission process automatically starts, as the first foil immediately
begins with the spontaneous emission discussed above, Eq.~(\ref{eq:ispon}).
The idea is that, due to the ``half-filled'' coherent state $\Sigma^z_1 = 0$,
the emission process of the first foil happens much faster than the
spontaneous emission of the subsequent foils.
Taking, e.g., the second foil ($s_2 = S_2$), the time-scale for the emission
of a single photon would be $1 / \Gamma_{\rm sp} = 1 / (\gamma S_2)$.
For the first foil, the time-scale for the whole emission process
(of nearly all photons, not only one) is given by
$\tau_{\rm sp} = 4 / (\gamma S_1)$.
So by choosing $S_1 \gg S_2$, e.g., by making the first foil ten times
thicker than the subsequent foils, it is assured that the second foil
is still in the state $\Sigma^z_2 = S_2/2$, when the intensity emitted
from the first foil is incident.
Stimulated emission then occurs at the second foil according to
Eq.~(\ref{eq:istim}) and the second foil has emitted all its energy
after $\tau _{\rm st} \approx 0.16\times\tau_{\rm sp}$, i.e.,
before the stimulating pulse coming from the first foil declines.
After the second foil, the overall intensity thus adds up to
$I_{\rm total}^{(2)}(t) = I_{\rm sp}(t) + I_{\rm st}^{(2)}(t)$.
This overall intensity then causes stimulated emission at the third foil,
resulting in an even bigger intensity
$I_{\rm total}^{(3)}(t) = I_{\rm total}^{(2)}(t) + I_{\rm st}^{(3)}(t)$, etc.
In this way, the intensity of the light pulse grows stepwise with each
passed foil.
Numerical analysis has been done for the case of $N = 50$ foils.
Iteratively, $I_{\rm st}^{(n)} (t)$ was calculated from
$I_{\rm total}^{(n-1)} (t)$, where
$I_{\rm total}^{(n)} (t) = I_{\rm total}^{(n-1)} (t) + I_{\rm st}^{(n)} (t)$,
starting with $I_{\rm total}^{(1)} (t) = I_{\rm sp}(t)$.
It was assumed that the first foil consists of $S_1 = 10^{10}$
$^{57}$Fe-nuclei
(with $\Delta E_\gamma = 14.4\;{\rm keV}$ and $\tau = 141\;{\rm ns}$)
while all other foils are ten times thinner, i.e., $S_n = 10^{9}$.
Transversal dimensions of the foils and the laser beam are chosen as
$L_\bot^2 = (0.1\;\mu{\rm m})^2$.
Note that the useful part of the laser pulse $I_{\rm total}^{(n)}(t)$
is determined by the time after which the last foil has emitted all its
excitations, $\tau^{(n)}_{\rm st}$, because afterwards re-absorption
takes place.
This time $\tau^{(n)}_{\rm st}$ becomes shorter with rising $n$,
as the intensity which causes the stimulated emission grows with $n$.
As a result, the average intensity of the useful part of the laser pulse,
\begin{eqnarray}
\overline{I_{\rm total}^{(n)}}
=
\frac{1}{\tau^{(n)}_{\rm st}}
\int_0^{\tau^{(n)}_{\rm st}} d\tau\,
I_{\rm total}^{(n)} (\tau)
\,,
\end{eqnarray}
increases with a power law. In the concrete example given above,
$\overline{I_{\rm total}^{(n)}}$ grows roughly $\propto n^{3/2}$,
see Fig.~\ref{fig:powerlaw}.
\begin{figure}
\caption{Average intensity $\overline{I_{\rm total}
\label{fig:powerlaw}
\end{figure}
\section{Conclusions}
In summary, we described a proposal for a laser in the $\,{\cal O}({\rm keV})$
regime which is based on stimulated emission and works with nuclear
excitons.
The pumping could be achieved with a free-electron laser, for example.
Note that the pump pulse $A_{\rm pump}$ and the generated laser pulse
$A_{\rm laser}$ both correspond to a 180$^\circ$-rotation of the last
foil according to Eq.~(\ref{eq:sigmaz}) and thus are related via
\begin{eqnarray}
\int_0^{\tau_{\rm pump}} d\tau\, A_{\rm pump}(\tau)
=
\int_0^{\tau^{(N)}_{\rm st}} d\tau \, A_{\rm laser}(\tau)
=
\frac{\pi}{2\tilde g}
\,.
\end{eqnarray}
However, the intensity of the pump pulse $\propto|A_{\rm pump}^2|$ is much
larger than that of the laser pulse $\propto|A_{\rm laser}^2|$.
On the other hand, the duration ${\tau^{(N)}_{\rm st}}$ of the laser pulse
is much larger and thus its frequency accuracy is much higher
(see also \cite{Kim:2008qf} for a different approach).
This could be important for spectroscopy etc.
Let us discuss some example data for the required intensity of the
pump-pulse.
First, we consider $^{57}$Fe-nuclei with a resonance at
$\Delta E_\gamma = 14.4\;{\rm keV}$ with a mean lifetime of
$\tau = 141\;{\rm ns}$.
If we assume that the pump pulse consists of ${\mathfrak N} = 10^6$
coherent wave-trains, we would need a pump intensity of
$I_{\rm pump} \approx 8.3 \cdot 10^{20} \;{\rm W}/{\rm cm^2}$
according to Eq.~(\ref{eq:ipump}).
(Comparable or even higher intensities have already been
considered in e.g.\ \cite{Burvenich:2006bh,Palffy:2008uq,Liao:2011cr}.)
This is probably beyond the capabilities of present
free-electron lasers, see, e.g., \cite{felbasics}.
However, future light sources such as seeded free-electron lasers
should achieve improved coherence times and higher intensities
(especially after focussing with X-ray lenses).
On the other hand, when considering other nuclear resonances beyond
the well-known $^{57}$Fe-example, we find that the requirements are
somewhat easier to fulfill.
For example, considering the $^{201}$Hg resonance at
$\Delta E_\gamma = 1.6\;{\rm keV}$ with $\tau = 81\;{\rm ns}$
and again assuming ${\mathfrak N} = 10^6$ coherent wave-trains,
we would ``only'' need a pump intensity of
$I_{\rm pump} \approx 8.0 \cdot 10^{15}\;{\rm W}/{\rm cm^2}$
according to Eq.~(\ref{eq:ipump}).
Unfortunately, this intensity is probably still too large:
After inserting typical values for the absorption cross section of
$1.6\;{\rm keV}$-photons in metals (or other solid materials), we find
that the pump beam deposits enough energy in the foil to evaporate it.
Even though the thermalization dynamics following the illumination
with such a $8.0 \cdot 10^{15}\;{\rm W}/{\rm cm^2}$-beam of
$1.6\;{\rm keV}$-photons is not well studied yet, one would expect that
the foil starts to disintegrate after a few pico-seconds \cite{Siwick:2003fk}
and hence does not survive long enough for our purposes.
In summary, the major difficulty of our set-up is that it requires
extremely large pump intensities.
As we may infer from Eq.~(\ref{eq:ipump}), the pump intensity scales with
the fifth power of the transition energy $\omega$.
Thus, our scheme should be much easier to realize at lower energies.
As one possible example, let us envisage a UV-laser.
In this case, the nuclear transitions could be replaced by
suitable electronic transitions in atoms or molecules.
The pumping could be achieved either directly via a free-electron
laser in the low-energy regime or indirectly via a two-photon transition
generated by two optical lasers, for example.
\begin{figure}
\caption{Sketch (not to scale) of the level scheme.
The pumping process from 2s to 4p is induced by a detuned two-photon
transition, i.e., both photons together are in resonance
$E_4-E_2=\omega_1+\omega_2$ while one-photon absorption is
suppressed by the detuning $\Delta$ where $E_3-E_2=\omega_1-\Delta$.
The laser operates via the one-photon transition from 4p back to 2s and
emits photons of the energy $\omega_{\rm uv}
\label{fig:three_level}
\end{figure}
Let us discuss the latter case using a three-level system as
depicted in Fig.~\ref{fig:three_level}.
Assuming two pump-lasers with optical frequencies $\omega_1$ and
$\omega_2$, the laser could operate in the ultra-violet regime
$\omega_{\rm uv}$.
In this case, the expression $\tilde g A_{\rm pump}$ for one-photon
pumping in Eq.~(\ref{eq:pump_constraint}) should be replaced by
$\tilde g_{23}A^{\rm pump}_1\tilde g_{34}A^{\rm pump}_2/\Delta$.
Note that the coupling constant $\tilde g_{23}$ of the
``dipole-forbidden'' 2s-3d transition
is typically much smaller than $\tilde g_{34}$.
Assuming typical values, such as a dipole coupling length of three Bohr
radii,
we would need pump-laser intensities of about
$I_{\rm pump} = \,{\cal O}(10^{10}\;{\rm W}/{\rm cm}^2)$
over a length of ${\mathfrak N} = 10^4$ coherent wave-trains with a
detuning of $\Delta = \,{\cal O}(10^{13}\;{\rm Hz})$ in order to
prevent unwanted excitations of the middle 3d level.
The condition for dominant coherent emission,
$\tau_{\rm sp}/\tau_{\rm single} \ll 1$,
can be fulfilled for $S_1/L_\bot^2 = \,{\cal O}(10^{5}\;\mu{\rm m}^{-2})$,
which is quite reasonable.
In this scenario, the duration of the laser pulse ${\tau^{(N)}_{\rm st}}$
is comparable to the length of the pump pulse
$\tau_{\rm pump} \approx {\tau^{(N)}_{\rm st}} = \,{\cal O}(10\;{\rm ps})$
and its intensity is well above $\,{\cal O}(10^5\;{\rm W}/{\rm cm}^2)$,
depending on the number of foils.
Again, the main idea would be that the coherent emission is strongly
enhanced for $S\gg1$ in comparison to competing non-coherent decay
channels.
To this end, the two pump lasers must be parallel to ensure the
spatial phase matching.
\section*{Appendix}
\subsection{Coherent versus incoherent pumping}
Let us review the pumping process by applying the
Holstein-Primakoff \cite{Holstein:1940vn} transformation
\begin{eqnarray}
\hat{\Sigma}^+
=
\hat{b}^\dagger \sqrt{S - \hat{b}^\dagger \hat{b}}
=
(\hat{\Sigma}^-)^\dagger
\,,\quad
\hat{\Sigma}^z = \hat{b}^\dagger \hat{b} - S/2
\,,
\end{eqnarray}
to the Hamiltonian Eq.~(\ref{eq:hstim}) and considering the limit $S\gg s$,
i.e., the beginning of the pumping process
\begin{eqnarray}
\hat{V}_{\rm st}
\approx
\tilde{g} A_{\rm pump}(t)
\left( \sqrt{S} \hat{b}^\dagger + {\rm H.c.} \right)
\,.
\end{eqnarray}
We first analyze the case of coherent pumping, that is pumping with a
coherent pulse $A_{\rm pump}(t)$, i.e., the same time-evolution operator
for the whole pumping process
\begin{eqnarray}
\hat U_{\rm st}(t) = \exp\left(
\beta(t)\hat b^\dagger-{\rm H.c.}
\right)
\,.
\end{eqnarray}
A time-dependent coherent state of excitons is created
\begin{eqnarray}
\beta(t)=-i \tilde{g} \sqrt{S} \int_0^t d\tau A_{\rm pump}(\tau)
\,,
\end{eqnarray}
whose exciton number grows quadratically with time $t$
\begin{eqnarray}
n(t)=|\beta(t)|^2
=\,{\cal O}\left( \tilde{g}^2 S A_{\rm pump}^2 t^2 \right)
\,.
\end{eqnarray}
An incoherent pump-pulse, in contrast, can be approximated as a
succession of many uncorrelated coherent pulses $A_{\rm pump}^{(i)}(t)$
incident on the target.
The time-evolution operator then is a product of many coherent
displacement operators
\begin{eqnarray}
\hat U_{\rm eff}
\approx
\prod_i
\hat{U}_{\rm st}^{(i)}
=
\exp\left(\sum_i\beta_i \hat b^\dagger-{\rm H.c.}\right)
\,.
\end{eqnarray}
For uncorrelated pulses, the $\beta_i$ have random phases,
such that the sum corresponds to a random walk
\begin{eqnarray}
\beta_{\rm eff}(j)
=\sum_{i=1}^j\beta_i\propto\sqrt{j}\propto\sqrt{t}
\,,
\end{eqnarray}
such that the exciton number $n = |\beta_{\rm eff}|^2$
grows merely linearly with time in this case.
\subsection{Expressing $\tilde{g}$ in terms of $\Gamma_{\rm single}$ and $\omega$}
The coupling constant $|g_{\fk{\kappa}}|$ can be expressed via the decay
rate $\Gamma_{\rm single}$ and frequency $\omega$
\begin{eqnarray}
\Gamma_{\rm single}
=
2 \pi \int d^3k\, |g_{\fk{k}}|^2 \delta
\left( \omega_k - \omega \right)
\approx
8 \pi^2 |g_{\fk{\kappa}}|^2 \omega^2
\,.
\end{eqnarray}
The dimensionless coupling constant $\tilde{g}$ can be obtained via
$\tilde{g} = |g_{\fk{\kappa}}| \sqrt{2 (2 \pi)^3 \omega}$ since our
Hamiltonian contains the classical field $A_{\rm pump}(t)$.
As a result we arrive at
\begin{eqnarray}
\tilde{g} = \sqrt{\frac{2 \pi \Gamma_{\rm single}}{\omega}}
\,.
\end{eqnarray}
\end{document} |
\begin{document}
\title{Smooth bang-bang shortcuts to adiabaticity for atomic transport in a moving harmonic trap}
\author{Yongcheng Ding}
\email{[email protected]}
\affiliation{International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) \\ and Department of Physics, Shanghai University, 200444 Shanghai, China}
\affiliation{Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain}
\author{Tang-You Huang}
\affiliation{International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) \\ and Department of Physics, Shanghai University, 200444 Shanghai, China}
\author{Koushik Paul}
\affiliation{International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) \\ and Department of Physics, Shanghai University, 200444 Shanghai, China}
\author{Minjia Hao}
\affiliation{International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) \\ and Department of Physics, Shanghai University, 200444 Shanghai, China}
\author{Xi Chen}
\email{[email protected]}
\affiliation{International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) \\ and Department of Physics, Shanghai University, 200444 Shanghai, China}
\affiliation{Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain}
\date{\today}
\begin{abstract}
Bang-bang control is often used to implement a minimal-time shortcut to adiabaticity for efficient transport of atoms in a moving harmonic trap. However, drastic changes of the on-off controller, leading to high transport-mode excitation and energy consumption, become infeasible under realistic experimental conditions. To circumvent these problems, we propose smooth bang-bang protocols with near-minimal time, by setting the physical constraints on the relative displacement, speed, and acceleration between the mass center of the atom and the trap center. We adopt Pontryagin's maximum principle to obtain the analytical solutions of smooth bang-bang protocol for near-time-minimal control. More importantly, it is found that the energy excitation and sloshing amplitude are significantly reduced at the expense of operation time. We also present a multiple shooting method for the self-consistent numerical analysis. Finally, this method is applied to other tasks, e.g., energy minimization, where obtaining smooth analytical form is complicated.
\end{abstract}
\maketitle
\section{introduction}
\label{sec:intro}
Precise control and manipulation of ultracold atomic systems without excitation or loss are challenging and important for the practical applications in atom interferometry, quantum-limited metrology, and quantum information processing ~\cite{benkish2002ion,hansel2001bose,hansel2001prl,gustavson2001prl,reichle,spaceborneBEC,schwartz,klitzingnature2019}. For example, protocols with existing adiabatic methods have been well developed to transport cold atoms by various traps ~\cite{denschlagnjp2006,davidpra2006,wangapl2010,jamespra2011,winelandprl2012,poschingerprl2012,sterrnjp2012,homenjp2013,gaaloulnjp2018}. However, the operation time required for approximating adiabatic processes is much longer than decoherence time, which may ruin the desired results in practice. To remedy it, several approaches, including but not limited to Fourier method \cite{davidepl2008,davidandmuga}, optimal control theory~\cite{calarcopra2009,calarocqip2013,hessmosp2017} and machine learning~\cite{shersonnature2016,sels}, have been attempted to reduce the timescales beyond the adiabatic limits.
Over the last decade, the concept of ``shortcuts to adiabaticity" (STA)~\cite{chenprl2010} provides an alternative approach for speeding up adiabatic processes without residual energy excitation in various quantum systems (see review articles~\cite{review1,review2}). The most common methods include fast-forwarding scaling~\cite{masuda2010,masudapra2012}, counter-diabatic driving~\cite{prx,kimnc2016}, and invariant-based inverse engineering~\cite{erikpra2011,eriknjpbec,mikelpra2013,mikelpra2014,tobalina}, in which the adiabatic transport is accelerated by modifying the trap trajectory or introducing auxiliary interaction to compensate the inertial force. Among them, inverse engineering, combined with perturbation theory or/and optimal control, is capable of designing the optimal shortcuts with transport time~\cite{chenpra2011,kochnjp}, energy excitation \cite{charronsp2019}, anharmonic effect~\cite{qipra2015,qijpb2016,jing}, fluctuating trap frequency and position~\cite{xiaojing14,xiaojing15,xiaojing18}. As expected from Pontryagin's maximum principle, bang-bang control is indeed the time-optimal solution of atomic transport with harmonic traps ~\cite{chenpra2011,kochnjp}. However, it has been observed that the abrupt change of the control function at the switching points has severe consequences for practical implications. For instance, as seen in Ref.~\cite{nessnjp2018}, this leads to the excitation of dynamical modes around the switching points which violates the fundamental assumption of STA methods of constraining the system in a particular mode during the evolution. In addition, the onset of step function entails sudden control over position, velocity, and acceleration of the trap by a spatial light modulator, which makes the experiment complicated. Therefore an effective and continuous control of the physical constraints of trap velocity ~\cite{stefanatosieee} or acceleration is essential that suppresses the energy excitation by protracting the process as a trade-off.
In this paper, we present a study on near-minimal-time transport of cold atoms with a moving trap by combining inverse engineering and optimal control theory. Previous research~\cite{chenpra2011} suggests that the bounded controller for time-optimal transport should be of bang-bang type, which maximizes the control Hamiltonian following Pontryagin's maximum principle.
Here we focus on the smooth bang-bang trajectories by setting up more constraints that bound the first- and second-order derivatives of the control input, describing the relative velocity and acceleration. We verify that the energy excitation and sloshing amplitude can be significantly reduced by smooth bang-bang protocols, while the minimal timescale is slightly increased. Since the analytical expressions of trap trajectories become more complicated when the higher-order derivatives of the controller are bounded, we introduce a multiple shooting method~\cite{bassam}, as a numerical approach, to confirm the analytical results. Additionally, this numerical method can be further exploited to minimize other target functionals, e.g., time-averaged potential energy, where finding an analytical solution might pose difficulties. Finally, we emphasize that our results can be extended to other scenarios~\cite{stefanatospra2010,xiaojingpra2014,kosloff,freericksbangbang,freericksbangbang2} without loss of generality.
\section{Hamiltonian and Model}
\label{sec:model}
\begin{figure}
\caption{Schematic diagram of STA transport of a $^{87}
\label{Fig.scheme}
\end{figure}
For simplicity, we consider the time-dependent Hamiltonian that describes the transport of a single atom trapped in a rigid harmonic trap (see Fig.~\ref{Fig.scheme}),
with center $q_0(t) \equiv q_0$ and trap frequency $\omega_0$, which reads as
\begin{equation}
\label{Ht}
H(t)=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega_0^2[\hat{q}-q_0(t)]^2,
\end{equation}
where $\hat{p}$ and $\hat{q}$ are momentum and position operators.
Eq. (\ref{Ht}) provides a good approximation for optical dipole interaction in low temperatures as one could easily neglect the effect of the anharmonic terms \cite{qijpb2016,nessnjp2018}.
This single-particle Hamiltonian
possesses a quadratic-in-momentum Lewis-Riesenfield invariant \cite{erikpra2011,LRO,LR1,LR2},
\begin{equation}
\label{It}
I(t)=\frac{1}{2m}(\hat{p}-m\dot{q}_c)^2+\frac{1}{2}m\omega_0^2[\hat{q}-q_c(t)]^2,
\end{equation}
where the parameter $q_c(t) \equiv q_c$ satisfies the auxiliary equation
\begin{equation}
\label{aeq}
\ddot{q}_c+\omega_0^2(q_c-q_0)=0,
\end{equation}
to guarantee self-consistency because of the invariant condition
\begin{equation}
\label{ic}
\frac{dI(t)}{dt}\equiv\frac{\partial I(t)}{\partial t}+\frac{1}{i\hbar}[I(t),H(t)]=0.
\end{equation}
Coincidentally, Eq.~\eqref{aeq} has the same structure of Newton's equation that governs the dynamics of a classical harmonic oscillator. Transport modes are described as
\begin{eqnarray}
\psi_n(q,t)=e^{i\frac{m\dot{q}_cq}{\hbar}} \phi_n(q-q_c),
\label{ev}
\end{eqnarray}
where $\phi_n$ are the eigenstates of a static harmonic oscillator. The solution of the time-dependent Schr\"odinger equation, $i \hbar \partial_t \Psi(q,t)= H(t) \Psi(q,t)$, is constructed as the superposition of transport modes,
$\Psi(q,t)=\sum_n c_n \exp(i \alpha_n)\psi_n (q,t)$,
where $c_n$ are the time-independent coefficients, and
$\psi_n(q,t)$ are the eigenstates of dynamical invariant $I(t)$. Here the eigenvalues $\lambda_n$, satisfying $I(t) \psi_n (t) = \lambda_n \psi_n(t)$, are constants and the Lewis-Riesenfield phase $\alpha_n$ is calculated as
\begin{equation}
\alpha_n(t)=-\frac{1}{\hbar}\int_0^t [(n+\frac{1}{2})\hbar\omega_0+\frac{1}{2}m\dot{q}_c^2] dt',\label{LRphaseS}
\end{equation}
It is noted that all transport modes are orthogonal to each other at any time, being centered at $q_c(t)$.
For a transport mode, the instantaneous average energy, $E(t)= \langle \Psi(t)|H(t)|\Psi(t)\rangle$, is calculated as~\cite{chenpra2011},
\begin{equation}
E(t)=\hbar\omega_0(n+\frac{1}{2})+\frac{m}{2} \dot{q}_c^2+ \frac{m}{2} \omega_0^2(q_c-q_0)^2,
\label{iaE}
\end{equation}
where the instantaneous average potential energy reads,
\begin{equation}
V(t)=\hbar\omega_0(n+\frac{1}{2})+ E_p.
\label{iaE}
\end{equation}
The first term refers to constant ``internal" contribution. The second term $E_p= m \omega_0^2(q_c-q_0)^2/2$ shares the form of a potential energy for a classical particle. Intuitively, high potential energy results in easy escape of a cold atom from the anharmonic trap in practice, reducing the effectiveness of STA
\cite{qijpb2016}. In order to characterize the energy excitation for the whole process, we finally write down the time-averaged potential energy,
\begin{equation}
\label{energy1}
\bar{E}_p \equiv \frac{1}{t_f} \int_{0}^{t_f} E_p dt = \frac{1}{t_f}\int_{0}^{t_f} \frac{m}{2} \omega_0^2(q_c-q_0)^2 dt,
\end{equation}
as a consequence.
In addition, we are also interested in sloshing amplitude $\mathcal{A}$,
\begin{equation}
\label{energy2}
\mathcal{A} (t_f)= \left|\int_{0}^{t_f} \dot{q}_0(t) e^{- i \omega_0 t'} dt'\right|,
\end{equation}
which is the Fourier component at the trap frequency of the trap velocity trajectory. Nullifying the sloshing amplitude provides the optimal trajectory in the anharmonic case, thus improving the performance of STA in a realistic experiment \cite{nessnjp2018}.
In order to design the optimal trajectory of the harmonic trap by inverse engineering as usual, we suppose that the harmonic trap moves from $q_0(0)=0$ to $q_0(t_f)=d$ at finite shortened time $t_f$. To avoid final energy excitation, boundary conditions
\begin{eqnarray}
q_c(0)=0;~ \dot{q}_c(0)=0;~ \ddot{q}_c(0)=0,
\label{bc1}
\\
q_c(t_f)=d;~ \dot{q}_c(t_f)=0;~ \ddot{q}_c(t_f)=0,
\label{bc2}
\end{eqnarray}
are imposed along with Eq.~\eqref{aeq}. In addition, the boundary conditions
\begin{eqnarray}
\dddot{q}_c(0)=0;~ ~ \dddot{q}_c(t_f)=0,
\label{bc3}
\end{eqnarray}
are introduced to eliminate sloshing amplitude $\mathcal{A} (t_f)$ for encapsulating the energy in transport modes. Here we give an example of a simple polynomial \textit{Ans\"atz}, interpolating the center of transport modes,
\begin{equation}
\label{poly}
q_c(t) = d\left[35 s^4-84s^5+70s^6-20s^7\right],
\end{equation}
originally proposed by Ref.~\cite{nessnjp2018},
with $s=t/t_f$. Once $q_c(t)$ and transport time $t_f$ are fixed, the optimal trajectory of the harmonic trap can be given by Eq.~\eqref{aeq}. However, we notice that this \textit{Ans\"atz} is not optimized enough, which will be analyzed by numerical results below.
\section{Smooth bang-Bang Control with Near-minimal-time}
In this section, we use Pontryagin's maximum principle~\cite{book} for solving the near-minimal-time transport problem, leading to smooth bang-bang control. In general, the time-dependent control function $u(t)$ for minimizing the cost functional,
\begin{equation}
J(u)=\int_0^{t_f}g[\textbf{x}(t),u]dt,
\end{equation}
can be solved by constructing the following control Hamiltonian
\begin{equation}
H_c[\textbf{p}(t),\textbf{x}(t),u]=p_0g[\textbf{x}(t),u]+\textbf{p}^T\cdot\textbf{f}[\textbf{x}(t),u],
\end{equation}
where for the dynamical system $\dot{\textbf{x}}=\textbf{f}[\textbf{x}(t),u]$, the extremal solutions satisfy the canonical equations
\begin{equation}a
\dot{\textbf{x}}=\frac{\partial H_c}{\partial \textbf{p}},~~~
\dot{\textbf{p}}=-\frac{\partial H_c}{\partial \textbf{x}}.
\label{canonicaleqp}
\end{equation}a
Here the corresponding adjoint state $\textbf{p}$ formed by Lagrange multipliers, where $p_0 < 0$ can be chosen for convenience, and all components are nonzero and continuous, is such that $H_c$ reaches its maximum at $u\equiv u(t)$ for almost all $0\leq t \leq t_f$. More specifically,
to find the time-optimal problem, we define the cost functional
\begin{equation}
J_T =\int_0^{t_f} 1 dt,
\label{J}
\end{equation}
and a control Hamiltonian,
\begin{equation}
H_c[\textbf{p}(t),\textbf{x}(t),u]=p_0+\textbf{p}^T\cdot\textbf{f}[\textbf{x}(t),u],
\label{Hc}
\end{equation}
where the dynamical system $\dot{\textbf{x}}=\textbf{f}[\textbf{x}(t),u]$ is governed by Eq.~\eqref{aeq}.
\subsection{bang-bang time-optimal control}
Let us first review the time-optimal control with bounded relative displacement to establish the background for analyzing smooth bang-bang control. By introducing a new notation,
\begin{equation}
x_1=q_c, x_2=\dot{q}_c, u(t)=q_c-q_0,
\label{newnotation-bangbang}
\end{equation}
we reformulate Eq.~\eqref{aeq} in the language of optimal control theory as follows,
\begin{eqnarray}
\dot{x}_1&=&x_2,
\\
\dot{x}_2&=&-\omega_0^2 u(t),
\label{aeqnn-bangbang}
\end{eqnarray}
where $x_{1,2}$ are the components of state vector $\textbf{x}$, and $u(t)$ is the scalar control function. Due to the anharmonicity of traps \cite{qijpb2016}, relative displacement $u(t)$ should be bounded by $|u(t)|\leq \delta$. Hence, the time optimization problem essentially comes down to the cost functional $J_T$ [see Eq.~\eqref{J}],
under the constraint $|u(t)|\leq\delta$. With the boundary conditions, $u(0)=u(t_f)=0$, the transport process occurs between $x_1(0)=0$ and $x_1(t_f)=d$ while $x_2(0)=x_2(t_f)=0$. The control Hamiltonian (\ref{Hc}) for such choices can be written as
\begin{equation}
\label{Htime}
H_c (\textbf{p}, \textbf{x}, u)= p_0 + p_1 x_2 - p_2 \omega_0^2 u(t),
\end{equation}
translating canonical equations~\eqref{canonicaleqp} into a set of costate equations
\begin{eqnarray}
\label{costate time-1}
\dot{p}_1 &=& 0,
\\
\label{costate time-2}
\dot{p}_2 &=& -p_1.
\end{eqnarray}
Once we solve the costate functions mentioned above, the time-optimal control function $u(t)$ of bang-bang type is obtained as
\begin{eqnarray}
\label{control function-time bang-ang}
u(t) = \left\{\begin{array}{ll}
0,& t \leq 0
\\
-\delta, & 0 <t <t_1
\\
\delta, & t_1 < t <t_f
\\
0, & t_f \leq t
\end{array}\right.,
\end{eqnarray}
where the minimal time is found to be
\begin{equation}
\label{mint}
t^{\min}_f = \frac{2}{\omega_0}\sqrt{\frac{d}{\delta}},
\end{equation}
with switching point $t_1 =t_f/2$. Fig.~\ref{fig2}(a) illustrates the bang-bang controller $u(t)$, where the parameters are chosen to correspond to the transport experiment of cold atoms~\cite{davidepl2008}, with trap frequency $\omega_0=2\pi\times20$ Hz, transport distance $d=1 \times 10^{-2}$ m, and the mass of $^{87} \mbox{Rb}$ atoms $m=1.44269 \times 10^{-25}$ kg. Here the constraint on relative displacement $\delta/d=0.1$ is fixed, therefore the minimal time $t^{\min}_f =50.3$ ms.
However, there exist three sudden jumps in the control function $u(t)$, leading to infinite relative speed of the trap at switching points, which could be problematic in the experimental implementation.
\begin{figure}
\caption{(a) Bang-bang-type controller $u(t)$ (solid red) and smooth bang-bang controller $u(t)$ (dashed blue) with a constrained relative displacement $\delta$ and velocity $\epsilon$. (b) Smooth trajectories of the trap center $q_0(t)$ (solid red) and the mass center $q_c(t)$ of a cold atom (dashed blue). Parameters: trap frequency $\omega_0=2\pi\times20$ Hz; distance of transporting $d=0.01$ m; $m=1.44269 \times 10^{-25}
\label{fig2}
\end{figure}
\subsection{smooth bang-bang control with constrained relative velocity and acceleration}
Motivated by the problem arising from bang-bang control, we introduce more constraints to cancel the sudden jumps, ensuring feasibility in experiments as well. A new component $x_3$ is added into the state vector $\textbf{x}$, with the relations between two nearby components being
\begin{equation}
x_1=q_c,\ x_2=\dot{q}_c,\ x_3=-\omega_0^2 \dot{u}(t),\ u(t)=q_c-q_0.
\label{newnotation}
\end{equation}
Thus, Eq.~\eqref{aeq} can be rewritten into the form for solving the time-optimal control problem as
\begin{eqnarray}
\dot{x}_1&=&x_2,
\\
\dot{x}_2&=&x_3,
\\
\dot{x}_3&=&-\omega_0^2 \dot{u}(t).
\label{aeqnn}
\end{eqnarray}
The new control Hamiltonian $H_c$ (\ref{Hc}) can be updated with the cost functional $J_T$ in Eq.~\eqref{J},
\begin{equation}
\label{Htime}
H_c (\textbf{p}, \textbf{x}, u, \dot{u})= p_0 + p_1 x_2 + p_2 x_3 - p_3 \omega_0^2 \dot{u}(t),
\end{equation}
giving new costate equations as
\begin{eqnarray}
\label{costate time-1}
\dot{p}_1 &=& 0,
\\
\label{costate time-2}
\dot{p}_2 &=& -p_1,
\\
\label{costate time-3}
\dot{p}_3 &=& -p_2.
\end{eqnarray}
which can be solved easily as $p_1 = c_1$ , $p_2 = -c_1 t +c_2$, and $p_3 = -c_1 t^2/2+c_2 t + c_3$ with constants $c_1$, $c_2$, and $c_3$. Based on Pontryagin's maximum principle~\cite{book}, the time-optimal controller $u(t)$ maximizes the control Hamiltonian (\ref{Htime}) with the new constraint on the relative velocity, $|\dot{u}(t)|\leq\epsilon$. In order to smooth out the bang-bang control, $\dot{u}(t)$ can be taken as
\begin{eqnarray}
\label{control functiond-time}
\dot{u} (t) = \left\{\begin{array}{ll}
-\epsilon, & 0\ \leq t<t_1
\\
0, & t_1<t <t_2
\\
\epsilon, & t_2< t <t_3
\\
0, & t_3 <t <t_4
\\
-\epsilon, & t_4<t\ {\leq}\ t_f
\end{array}\right..
\end{eqnarray}
After combining the previous constraint on the relative displacement, $|u(t)|\leq\delta$, the ``sudden-jump-free" controller $u(t)$ becomes
\begin{eqnarray}
\label{control function-time}
u(t) = \left\{\begin{array}{ll}
-\epsilon t + c_1, & 0\ \ {\leq}\ t<t_1
\\
c_2, & t_1<t <t_2
\\
\epsilon t + c_3, & t_2< t <t_3
\\
c_4, & t_3 <t <t_4
\\
-\epsilon t + c_5, & t_4<t\ {\leq}\ t_f
\end{array}\right.,
\end{eqnarray}
where $c_2=-c_4=-\delta$, $c_1=0$, $c_3=-(\delta+\epsilon t_2)$, and $c_5=\epsilon t_f$.
\begin{figure}
\caption{Phase diagram of smooth bang-bang control of fast transport with different relative velocity constraints, where the relative displacement is bounded by $\delta/d=0.1$, keeping the constraint on the relative velocity variable. The trajectories with $\epsilon/(d\omega_0)=0.05$ (solid red) and $\epsilon/(d\omega_0)=0.1$ (dotted black) become much smoother with larger time $t_f=68.7$ and $58.9$ ms, calculated by Eq.~\eqref{eqtfv}
\label{fig3}
\end{figure}
According to the boundary conditions, the symmetry, and continuity conditions, one can find four switching points $t_1$, $t_2$, $t_3$ and $t_4$ with the values of $\delta/\epsilon$, $t_f/2-\delta/\epsilon$, $t_f/2+\delta/\epsilon$, and $t_f-\delta/\epsilon$, respectively.
Substituting $u(t)$ into Eq.~\eqref{aeq}, and with boundary conditions [see Eqs.~\eqref{bc1} and (\ref{bc2})], we find
the solution of $q_c(t)$ in different time intervals as follows
\begin{equation}
q_c(t) = \left\{\begin{array}{ll}
\frac{1}{6}\omega_0^2\epsilon t^3
\\
\frac{1}{2}\omega_0^2\delta (t^2 - \frac{\delta}{\epsilon}t + \frac{1}{3}\frac{\delta^2}{\epsilon^2})
\\
-\frac{1}{6}\omega_0^2\epsilon(t-\frac{t_f}{2})^3+\omega_0^2\delta(\frac{t_f}{2}-\frac{\delta}{\epsilon})t-\frac{1}{4}\omega_0^2\delta t_f(\frac{t_f}{2}-\frac{\delta}{\epsilon})
\\
-\frac{1}{2}\omega_0^2\delta[t^2 - (\frac{t_f}{2} - \frac{\delta}{\epsilon})t-\frac{1}{3}\frac{\delta^2}{\epsilon^2}+\frac{t_f^2}{2}]
\\
d- \frac{1}{6}\omega_0^2\epsilon (t_f-t)^3
\end{array}\right.,
\end{equation}
from which the trajectory of the trap center $q_0(t)$
can be easily obtained through Eq.~\eqref{aeq}.
After straightforward calculation, we obtain the near-minimal time as follows,
\begin{equation}
\label{eqtfv}
t_f =\frac{\delta}{\epsilon}
+\frac{2}{\omega_0}\sqrt{\frac{d}{\delta}+\frac{\omega_0^2\delta^2}{4\epsilon^2}},
\end{equation}
which tends to the minimal time in Eq.~\eqref{mint}, when relative velocity is no longer limited, i.e., $\epsilon \rightarrow \infty$. Fig.~\ref{fig2} demonstrates the trajectories of trap center and mass center of the atom with a smoother controller $u(t)$ at switching points, when the relative velocity is bounded. Apparently, the constraint in relative velocity prolongs the time-optimal transport, as shown in a phase diagram (see Fig.~\ref{fig3}),
where the trajectory becomes smoother. To be precise, the transport time increases from $t_f= 58.9$ to $68.7$ ms,
when the constraint on the relative velocity decreases from $\epsilon/(d \omega_0) =0.1$ to $0.05$,
with the same bounded relative displacement, $\delta/d=0.1$.
\begin{figure}
\caption{(a) Controller $u(t)$ with different constraints, where $\epsilon/(d\omega_0)=0.1$, $\zeta/(d\omega_0^2)=0.5$ (solid red), $\epsilon/(d\omega)=0.2$, $\zeta/(d\omega_0^2)=1$ (dashed blue), $\epsilon/(d\omega_0)=0.5$, $\zeta/(d\omega_0^2)=2$ (dotted black), and other parameters are the same as those in Fig.~\ref{fig2}
\label{fig4}
\end{figure}
Next, we find the near-minimal-time protocol with an extra constraint condition on the relative acceleration, i.e., $|\ddot{u}(t)|\leq \zeta$, since the discontinuity of trap speed, leads to infinite acceleration in previous protocols. Therefore, the new notation $x_4=\dot{x}_3=-\omega_0^2 \ddot{u}(t)$ is added to equations for defining the control Hamiltonian as
\begin{eqnarray}
\label{newHtime}
H_c (\textbf{p}, \textbf{x}, u, \dot{u}, \ddot{u})= p_0 + p_1 x_2 + p_2 x_3 + p_3 x_4 - p_4 \omega_0^2 \ddot{u},
\end{eqnarray}
from which we use canonical equation (\ref{canonicaleqp}) to obtain the following costate functions:
\begin{eqnarray}
\label{newcostate time-1}
\dot{p}_1 &=& 0,
\\
\label{newcostate time-2}
\dot{p}_2 &=& -p_1,
\\
\label{newcostate time-3}
\dot{p}_3 &=& -p_2,
\\
\label{newcostate time-4}
\dot{p}_4 &=& -p_3.
\end{eqnarray}
Accordingly, the optimal control that maximizes $H_c$ in Eq.~\eqref{newHtime} is determined by the sign of $p_4$, when $\ddot{u}(t)$ is bounded by $|\ddot{u}(t)| \leq \zeta$. Here we apply three constraints simultaneously, with the other two being $|u(t)| \leq \delta$ and $|\dot{u}(t)| \leq \epsilon$. The near-minimal-time protocol meets the limitations of the relative acceleration, velocity, and displacement simultaneously. As a consequence, the second derivative of the controller, $\ddot{u}(t)$, has the following form of bang-bang type,
\begin{eqnarray}
\ddot{u}(t) = \left\{\begin{array}{ll}
-\zeta, & 0\ \ \ {\leq}\ t<t_1
\\
0, & t_1\ <t <t_2
\\
\zeta, & t_2\ < t <t_3
\\
0, & t_3\ <t <t_4
\\
\zeta, & t_4\ <t<t_5
\\
0, & t_5\ <t <t_6
\\
-\zeta, & t_6\ <t <t_7
\\
0, & t_7\ <t <t_8
\\
-\zeta, & t_8\ <t <t_9
\\
0, & t_9\ <t <t_{10}
\\
\zeta, & t_{10} <t \leq t_f
\end{array}\right.,
\end{eqnarray}
With boundary conditions at switching points, after a simple integration, $\dot{u}(t)$ can be given by
\begin{eqnarray}
\dot{u}(t) = \left\{\begin{array}{ll}
-\zeta t, & 0\ \ \ {\leq}\ t<t_1
\\
-\epsilon, & t_1\ <t <t_2
\\
\zeta(t-t_2)-\epsilon, & t_2\ < t <t_3
\\
0, & t_3 \ <t <t_4
\\
\zeta(t-t_4), & t_4\ <t<t_5
\\
\epsilon, & t_5 \ <t <t_6
\\
-\zeta(t-t_6)+\epsilon, & t_6 \ <t <t_7
\\
0, & t_7 \ <t <t_8
\\
-\zeta(t-t_8), & t_8 \ <t <t_9
\\
-\epsilon, & t_9 \ <t <t_{10}
\\
\zeta(t-t_{10})-\epsilon, & t_{10} <t \leq t_f
\end{array}
\right.,
\end{eqnarray}
from which the switching points can be calculated as
$t_1=\epsilon/\zeta$,
$t_2=\delta/\epsilon$,
$t_3=\delta/\epsilon+\epsilon/\zeta$,
$t_{4,5}=\frac{1}{2}(t_f-2\delta/\epsilon \mp \epsilon/\zeta)$,
$t_{6,7}=\frac{1}{2}(t_f+2\delta/\epsilon \mp \epsilon/\zeta)$,
$t_8=t_f-t_3$,
$t_9=t_f-t_2$,
and $t_{10}=t_f-t_1$.
With these switching points, the controller [see Fig.~\ref{fig4}(a)] can be finally expressed by
\begin{eqnarray}
\label{newcontrol function-time}
u(t) = \left\{\begin{array}{ll}
-\frac{1}{2}\zeta t^2
\\
-\epsilon(t-\frac{\epsilon}{\zeta})-\frac{\epsilon^2}{2\zeta}
\\
\frac{1}{2}(\zeta t^2-2\epsilon t+\frac{\epsilon^2}{\zeta}-\frac{2\delta\zeta t}{\epsilon}+\frac{\delta^2\zeta}{\epsilon^2})
\\
-\delta
\\
-\delta+\frac{[\epsilon^2+2\delta\zeta-\epsilon(t_f-2t)\zeta]^2}{8\epsilon^2\zeta}
\\
\epsilon(t-\frac{t_f}{2})
\\
\frac{1}{8}\left[-\frac{4\delta^2\zeta}{\epsilon^2}-\frac{(\epsilon+\zeta(t_f-2t))^2}{\zeta}+\frac{4\delta(\epsilon-\zeta t_f+2\zeta t)}{\epsilon}\right]
\\
\delta
\\
\delta-\frac{(\epsilon^2+\delta\zeta+\epsilon\zeta(t-t_f))^2}{2\epsilon^2\zeta}
-\frac{\epsilon[\epsilon+2(t-t_f)\zeta]}{2\zeta}
\\
\frac{1}{2}\zeta(t_f-t)^2
\end{array}\right.,
\end{eqnarray}
\begin{figure}
\caption{ Dependence of near-minimal time $t_f$ on differently bounded relative velocities and accelerations: $\zeta/(d\omega_0^2)=0.8$ (solid red), $\zeta/(d\omega_0^2)=1.2$ (dashed blue), and $\zeta/(d\omega_0^2)=1.6$ (dotted black), where other parameters are the same as those in Fig.~\ref{fig2}
\label{fig5}
\end{figure}
Trajectories of trap center $q_0(t)$ and mass center of a cold atom $q_c(t)$ can be easily calculated through Eq.~\eqref{aeq} [see Fig.~\ref{fig4}(b) and (c)].
Obviously, this shows a feasible way to realize smooth transport, only taking a little more time as cost than the previous cases. Thus, the final expression of near-minimal time in this case is given by
\begin{equation}
\label{eqtfa}
t_f =\frac{\delta}{\epsilon} + \frac{\delta \epsilon}{\zeta}
+\frac{2}{\omega_0}\sqrt{\frac{d}{\delta}+\frac{\omega_0^2\delta^2}{4\epsilon^2}}.
\end{equation}
The minimal time given here is just increased a little by $\delta\epsilon/\zeta$, which is the exact price for smooth bang-bang control by bounding relative acceleration. For instance, we choose three different
constraints in Fig.~\ref{fig4}, where $\epsilon/(d\omega_0)=0.1$,
$\zeta/(d\omega_0^2)=0.5$ (solid red),
$\epsilon/(d\omega)=0.2$, $\zeta/(d\omega_0^2)=1$ (dashed blue), $\epsilon/(d\omega_0)=0.5$, $\zeta/(d\omega_0^2)=2$ (dotted black), and other parameters are the same as those in Fig.~\ref{fig2}. It is obvious that the larger the constraint, the more similar the control. Experimental realization without energy excitation also becomes harder for larger constraints despite near-minimal times; here we emphasize that one can further smooth the protocol by introducing more constraints on the higher-order derivatives of the controller. However, it might be unnecessary to do so, since numerical studies given below convince us.
Figure \ref{fig5} clarifies how much price one should pay for smoothing the bang-bang control out. In general, the influence of the constraint on the relative velocity, $\epsilon$, is more pronounced, as compared to the constraint on the relative acceleration, $\zeta$. Setting more constraints on its first and second derivatives of the controller can smooth out bang-bang time-optimal control more, with extra cost of transport time as a trade-off.
\begin{figure}
\caption{Time-averaged potential energy $\bar{E}
\label{fig6}
\end{figure}
Moreover, we shed light on the energy excitation, characterized by time-averaged potential energy (\ref{energy1}), for smooth bang-bang protocols and a polynomial trajectory ~(\ref{poly}) that is used in the experiment \cite{nessnjp2018}. In general, the energy excitation can be suppressed by smooth bang-bang protocols (see Fig.~\ref{fig6}) since it is proportional to $u^2$. In spite of the fact that the excitation energy increases with the upper bounds of velocity and acceleration, a fair comparison would be to calculate the transport time of polynomial trajectory (\ref{poly}) corresponding to each smooth bang-bang protocol with different upper limits. Clearly, the polynomial trajectory produces larger energy excitation than smooth bang-bang protocol. Also, one can calculate the sloshing amplitudes, to quantify the performance of STA. The ultimate sloshing can be suppressed from $\mathcal{A} (t_f) =4 \times 10^{-3}$ to $\mathcal{A}(t_f)\simeq10^{-12}$ by smooth bang-bang controls. As mentioned above, the polynomial \textit{Ans\"atz} \eqref{poly} carried out in the experiment \cite{nessnjp2018} is not optimized enough with respect to time or time-averaged potential energy, though the corresponding sloshing is $\mathcal{A}(t_f)\simeq10^{-16}$. In this case, the resulting relative displacement during the process is continuous, but exceeds the upper limit, $|u|\leq\delta$, used in the time-optimal solution. This might be problematic in practice when the anharmonic effect is taken into account in an optical Gaussian trap \cite{qipra2015,qijpb2016,jing}.
\section{Numerical multiple shooting algorithm}
In this section, we present the numerical multiple shooting method to solve such near-time-optimal control with two-fold reasons. On one hand, the analytical expressions become too complicated to solve, when high-order derivatives of controller are considered. Thus, the numerical algorithm is required to calculate automatically the switching points and minimal time with different constraints for double checking and simplicity. On the other hand, a shooting method may encounter numerical difficulties for solving the optimal control, since the shooting function is not smooth when the control is bang-bang \cite{bassam}. Beyond that, the reason for applying multiple shooting method, as a tool of our numerical studies, is that, it can be parallelized for certain problems, which can have a non-negligible advantage in efficiency, comparing with other algorithms. In what follows, we shall formulate the boundary-value problem, and solve the smoothing procedure by using multiple shooting method. The detailed steps of our algorithm are as follows.
\begin{figure}
\caption{With updating rate $\rho=0.5$, the incorrect initial guess (upper faded lines) of switching points $\{5, 10,...,55\}
\label{fig7}
\end{figure}
(\romannumeral1) We get the expression of $q_c(t)$ with ten switching points and the minimal transport time, which are unknown, by solving the classical equation with boundary conditions and continuous conditions.
(\romannumeral2) Then we can write a column vector $f=(q_c(t_f)-d,\dot{q_c}(t_f),u(t_f),\dot{u}(t_f),u(t_3)+\delta,u(t_7)-\delta,\dot{u}(t_1)+\epsilon,\dot{u}(t_3),\dot{u}(t_5)-\epsilon,\dot{u}(t_7),\dot{u}(t_9)+\epsilon)^{T}$. Its norm, as the objective function, should be optimized to zero when all the switching points and minimal time are corrected.
(\romannumeral3) A Jacobian matrix $J_{ij}=\partial f_i/\partial t_j$ is defined to calculate the modifications of switching points and minimal time.
(\romannumeral4) We set another column vector $g$ to be $g=(t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,t_9,t_{10},t_f)^T$. All the elements' initial values are our assumptions of switching points and minimal transport time, which will be updated by the algorithm iteratively.
(\romannumeral5) Calculate the values of $f$ and $J$ with times given by $g$. Gradient Del is defined as $\text{Del}=J^{-1}f$. In this way, the modified $g$ will be $g=g-\rho \text{Del}$, where $\rho$ is a constant deciding the speed of convergence between zero and one. After that, calculate the norm of the new $f$.
(\romannumeral6) Repeat step (\romannumeral5) until the norm of $f$ is smaller than an acceptable fixed tolerance value.
To demonstrate the algorithm, we give an example of smooth transport calculated with multiple shooting method and plot the incorrect initial guess, middle (epoch=3), and final trajectories (epoch=13), for showing how the protocol converges to the near-time-optimal solution (see Fig.~\ref{fig7}), where the constraints on the relative displacement, velocity, and acceleration---$\delta/d=0.1$, $\epsilon/(d\omega_0)=0.1$, and $\zeta/(d\omega_0^2)=0.5$, respectively---hold.
\begin{figure}
\caption{Controller $u(t)$ that minimizes the time-averaged potential energy $\bar{E}
\label{fig8}
\end{figure}
In addition to time-optimal control, the minimization of energy excitation is dealt with using the same numerical algorithm mentioned above. From Eq.~(\ref{energy1}), the cost functional reads
\begin{equation}
J_E=\int_0^{t_f} E_pdt=\int_0^{t_f}\frac{1}{2}m\omega_0^2u^2dt,
\end{equation}
where transport time $t_f$ is fixed. The control Hamiltonian can be written as
\begin{equation}
H_c=-p_0\frac{1}{2}m\omega_0^2u^2+p_1x_2-p_2\omega_0^2u,
\end{equation}
leading to new costate equations. Following Pontryagin's maximum principle, the unbounded control, i.e., without any constraints on $u$, gives the lowest bound~\cite{chenpra2011}
\begin{equation}
\bar{E}^{\min}_p = 6md^2/\omega^2_0 t^4_f,
\end{equation}
with linear time-varying controller
\begin{equation}
u(t) =\frac{6d}{\omega^2_0 t^2_f} \left(2\frac{t}{t_f}-1\right).
\end{equation}
Similarly, the controller $u(t)$ is not zero at $t=0$ and $t=t_f$, implying an infinite speed of the moving trap. The more complicated case with bounded controller $u(t)$ can be calculated as well in Ref.~\cite{chenpra2011}. However, it is impossible to achieve the analytical expression, when higher-order derivatives of controller $u(t)$ are bounded. For this task, we apply multiple shooting method again to numerically design STA with arbitrary dimension of states and constraints. Transport interval $[0,t_f]$ is partitioned by $N$ grid points, where the control function consists of $N-1$ subintervals with length $t_f/(N-1)$. In order to find the solution of boundary-value problems, we define a $D$-dimensional state $\textbf{x}$ and its derivative $\dot{\textbf{x}}$, initializing it by guessing. We define a time step $h=t_f/(N-1)(M-1)$ for applying fourth-order Runge-Kutta method as an ordinary differential equation (ODE) solver. In each subinterval, we calculate the following four terms
\begin{eqnarray}
\textbf{k}_1&=&\dot{\textbf{x}}_j,\\
\textbf{k}_2&=&\dot{\textbf{x}}_j+\frac{h}{2}\dot{\textbf{k}}_1,\\
\textbf{k}_3&=&\dot{\textbf{x}}_j+\frac{h}{2}\dot{\textbf{k}}_2,\\
\textbf{k}_4&=&\dot{\textbf{x}}_j+h\dot{\textbf{k}}_3,
\end{eqnarray}
for updating the state of the next time step by
\begin{equation}
\textbf{x}_{j+1}=\textbf{x}_{j}+\frac{h}{6}(\textbf{k}_1+2\textbf{k}_2+2\textbf{k}_3+\textbf{k}_4),
\end{equation}
where $j\in\{1,2,...,M-1\}$. Thus, we obtain $\textbf{x}_{i+1}$ for all $i\in\{1,2,...,N-1\}$ with a given $\textbf{x}_{i}$. Combining it with an optimizer, we can optimize any objective function, satisfying constraint conditions at the same time. In Fig.~\ref{fig8}, we use multiple shooting method for solving ODEs, minimizing potential energies with \textsc{matlab} optimizer \textsc{fmincon} under different constraint conditions. Again, the linear time-varying controller $u(t)$, initially with drastic changes at initial and final times, becomes smoother at the cost of potential-energy increase.
\begin{figure}
\caption{Phase diagram of smooth bang-bang control of fast transport within fixed $t_f$, minimizing the time-averaged potential energy $\bar{E}
\label{fig9}
\end{figure}
Moreover, we show the phase diagram of smooth transport protocols in Fig.~\ref{fig9}. We notice that when higher-order constraints are introduced the phase diagram becomes asymmetric around $t=t_f/2$, resulting in a local minimum of potential energy. It is hard to obtain a global optimal solution because the gradient algorithm depends on its initial input as trial solution. However, with a reasonable range of guesses, this numerical algorithm converges to sub-optimal solutions, which are friendly enough for experimental implementations.
\section{Conclusion and Outlook}
In summary, we present analytical and numerical methods for
smooth bang-bang shortcuts to adiabaticity for atomic transport. Preceding researches provide the time-optimal solution, as typical bang-bang control, which contains drastic changes of controllers, resulting in high residual energy and difficulty in experimental implementation. Here we propose smooth bang-bang controls, corresponding to the near-minimal time, by bounding the first and second-order derivatives of the controller. Further comparison between our smooth bang-bang and simple polynomial protocols shows that both energy excitation and sloshing amplitude are significantly suppressed with increasing slightly the transport time as a tradeoff. To make our results more applicable, the numerical multiple shooting algorithm is developed for time or energy minimization, where an analytical solution might not be feasible or solvable. Within this framework, different \textit{Ans\"atze}, including high-order polynomial and trigonometric functions, can be compared, and they are suitable for obtaining sub-optimal solutions~\cite{compare} and enhanced STA~\cite{qcesta} in further work.
Finally, we emphasize that our analytical and numerical methods, supplemented by machine learning~\cite{shersonnature2016,sels,machinelearning}, will provide a versatile toolbox for quantum control since time-optimal bang-bang solutions are ubiquitous with applications including atom cooling~\cite{stefanatospra2010,kosloff,xiaojingpra2014}, transport of trapped-ion qubits \cite{mikelpra2013,mikelpra2014,xiaojing14,xiaojing15,xiaojing18}, ground-state preparation~\cite{freericksbangbang,freericksbangbang2}, and long-distance transport in an optical lattice~\cite{alberti,xiaojing,andreas}. These results can be further extended to other problems, including compact interferometers with spin-dependent force~\cite{socBEC,armsguiding},
load manipulation by cranes in a classical system~\cite{mugacrane}, and Brownian motion in statistical physics~\cite{prados}.
\end{document} |
\begin{document}
\preprint{}
\title{Quantum secure direct communication based on order rearrangement of single photons}
\author{Jian Wang}
\email{[email protected]}
\affiliation{School of Electronic Science and Engineering,
\\National University of Defense Technology, Changsha, 410073, China }
\author{Quan Zhang}
\affiliation{School of Electronic Science and Engineering,
\\National University of Defense Technology, Changsha, 410073, China }
\author{Chao-jing Tang}
\affiliation{School of Electronic Science and Engineering,
\\National University of Defense Technology, Changsha, 410073, China }
\begin{abstract}
Based on the ideal of order rearrangement and block transmission of
photons, we present a quantum secure direct communication scheme
using single photons. The security of the present scheme is ensured
by quantum no-cloning theory and the secret transmitting order of
photons. The present scheme is efficient in that all of the
polarized photons are used to transmit the sender's secret message
except those chosen for eavesdropping check. We also generalize this
scheme to a multiparty controlled quantum secret direct
communication scheme which the sender's secret message can only be
recovered by the receiver under the permission of all the
controllers.
\end{abstract}
\pacs{03.67.Dd, 03.67.Hk}
\keywords{Quantum key distribution; Quantum teleportation}
\maketitle
Quantum cryptography has been one of the most promising applications
of quantum information science. It utilizes quantum effects to
provide unconditionally secure information exchange. Since the first
QKD protocol was proposed by Benneett and Brassard in 1984
\cite{bb84}, quantum key distribution (QKD) which provides
unconditionally secure key exchange has progressed quickly. In
recent years, a good many of other quantum cryphtography schemes
have also been proposed and pursued, such as quantum secret sharing
(QSS)\cite{hbb99,kki99,zhang,gg03,zlm05,xldp04}, quantum secure
direct communication (QSDC)
\cite{beige,Bostrom,Deng,denglong,cai1,cai4,jwang1,jwang2,cw1,cw2,tg,zjz}.
QSS is the generalization of classical secret sharing to quantum
scenario and can share both classical and quantum messages among
sharers. QSDC's object is to transmit the secret message directly
without first establishing a key to encrypt them, which is different
to QKD. QSDC can be used in some special environments which has been
shown by Bostr\"{o}em and Deng et al.\cite{Bostrom,Deng}. Many
researches have been carried out in QSDC. We can divide these works
into two types, one utilizes single photons \cite{denglong,cai1},
the other utilizes entangled state
\cite{Bostrom,Deng,cai1,cai4,jwang1,jwang2,cw1,cw2,tg,zjz}. Deng et
al. proposed a QSDC scheme using batches of single photons which
serves as one-time pad \cite{denglong}. Cai et al. presented a
deterministic secure direct communication scheme using single qubit
in a mixed state \cite{cai1}. The QSDC scheme using entanglement
state is certainly the mainstream. Bostr\"{o}m and Felbinger
proposed a "Ping-Pong" QSDC protocol which is quasi-secure for
secure direct communication if perfect quantum channel is used
\cite{Bostrom}. Cai et al. pointed out that the "Ping-Pong" Protocol
is vulnerable to denial of service attack or joint horse attack with
invisible photon \cite{cai2,cai3}. They also presented an improved
protocol which doubled the capacity of the "Ping-Pong" protocol
\cite{cai4}. Deng et al. put forward a two-step QSDC protocol using
Einstein-Podolsky-Rosen (EPR) pairs \cite{Deng}. We presented a QSDC
scheme using EPR pairs and teleportation \cite{jwang1} and a
multiparty controlled QSDC scheme using Greenberger-Horne-Zeilinger
states \cite{jwang2}.
In Ref. \cite{deng-core}, Deng et al. utilize controlled order
rearrangement encryption to realize a QKD scheme. In their scheme,
the communication parties share a control key used to control the
order rearrangement operation. Very recently, A. D. Zhu et al.
proposed a QSDC scheme based on secret transmitting order of
entangled particles \cite{zhu}. The security of their scheme is
based on entanglement and the secret transmitting order of
particles. In this Letters, we present a QSDC scheme using single
photons based on the ideal of the order rearrangement and qubit
transmission in batches \cite{deng-core,zhu,long}. The initial state
of the transmitting photon is prepared randomly in one of the four
states belonging to two conjugate basis, which is similar to the
BB84 QKD protocol. The security of the scheme is based on quantum
no-cloning theory and the secret transmitting order of single
photons. All of the single photons are used to generate secret
message except those used for eavesdropping check. It is not
necessary for the communication parties to choose a random measuring
basis for eavesdropping check. Compared with schemes using EPR
pairs, this scheme is more realizable. We also generalize this
scheme to a multiparty controlled quantum secret direct
communication (MCQSDC) scheme. In the MCQSDC scheme, the sender's
secret message is transmitted directly to the receiver and can only
be reconstructed by the receiver with the permission of all the
controllers. We also discuss the security of the two schemes, which
is unconditionally secure.
Here We first describe the details of our QSDC scheme using single
photons. Suppose the sender Bob wants to transmit his secret message
directly to the receiver Alice.
(S1) Alice prepares $N$ single photons each of which is randomly in
one of the following states
\begin{eqnarray}
& &\ket{H}=\ket{0},\nonumber\\
& &\ket{V}=\ket{1},\nonumber\\
& &\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1}),\nonumber\\
& &\ket{-}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1}).
\end{eqnarray}
She then send the $N$ photons $[P_1,P_2,\cdots,P_n]$, called
$P$-sequence to Bob.
(S2) Bob selects randomly a sufficiently large subset from the
$P$-sequence for eavesdropping check, called checking sequence
($C$-sequence). The remaining photons of the $P$-sequence form a
message sequence ($M$-sequence). Bob performs randomly one of the
two unitary operations
\begin{eqnarray}
& &I=\ket{0}\bra{0}+\ket{1}\bra{1},\nonumber\\
& &U=i\sigma_y=\ket{0}\bra{1}-\ket{1}\bra{0}.
\end{eqnarray}
on each of the photons in the $C$-sequence. He also encode his
secret message on the $M$-sequence by performing one of the unitary
operations $I$ and $U$, according to his secret message. If his
secret message is ``0'' (``1''), Bob performs operation $I$ ($U$).
The operation $U$ flips the state in both $Z$-basis
(\ket{0},\ket{1}) and $X$-basis (\ket{+},\ket{-}), as
\begin{eqnarray}
& &U\ket{0}=-\ket{1}, U\ket{1}=\ket{0},\nonumber\\
& &U\ket{+}=\ket{-}, U\ket{-}=-\ket{+}.
\end{eqnarray}
(S3) Bob disturbs the order of the photons in the $P$-sequence and
generates a rearranged photon sequence, called $P'$-sequence [$P_1',
P_2',\cdots,P_n'$]. He then sends the $P'$-sequence to Alice. The
order of $P'$-sequence is completely secret to others but Bob
himself, which ensures the security of the present scheme.
(S4) Alice tells Bob she has received the $P'$-sequence. After
hearing from Alice, Bob announces the position of the $C$-sequence
and the secret rearranged order in it. He also publishes his
corresponding operations on the photons in the $C$-sequence.
(S5) Alice has the initial state information and the position of
each checking photons. She then performs von Neumann measurement on
each of the checking photons. If the initial state of the checking
photon is \ket{H} or \ket{V} (\ket{+} or \ket{-}), Alice performs
$Z$-basis ($X$-basis) measurement on it. Alice has the initial
states information of the checking photons, Bob's operation
information on the checking photons and her measurement results on
them. She can then evaluate the error rate of the transmission of
the $P$-sequence. If the error rate exceeds the threshold, they
abort the scheme. Otherwise, they continue to the next step.
(S6) Bob publishes the secret order of the $M$-sequence. According
to the initial states information of the $M$ sequence, Alice
performs $Z$-basis or $X$-basis measurement on the $M$-sequence. She
can then obtain Bob's secret message.
We now discuss the unconditional security of the present scheme. The
security of the scheme is based on quantum no-cloning theory and the
secret transmitting order of the photons. Quantum no-cloning theory
ensures that an eavesdropper, Eve cannot make certain the initial
states of the transmitting photons prepared by Alice, which is
similar to the BB84 QKD protocol. The difference between the BB84
QKD protocol and the present scheme is that the communication
parties perform $Z$-basis or $X$-basis measurement randomly for
preventing eavesdropping in the former, but the order rearrangement
is used to prevent Eve from obtaining the sender's secret message in
the latter. Suppose Eve intercepts the $P$-sequence and resends
another photon sequence prepared by Eve to Bob. After Bob has
performed his operations on the photon sequence, he then sends it to
Alice. Eve can also intercepts this photon sequence. However, Eve
cannot obtain Bob's operation information because Bob disturbs the
order of the photon sequence and Eve's attack will be detected
during the eavesdropping check. Without the correct order of the
photon sequence, Eve can only obtain a batch of meaningless data.
Obviously, the present scheme is also safe against collective attack
due to the secret order of rearranged photon sequence. As we
described above, the present scheme is unconditionally secure.
We then generalize this QSDC scheme to a MCQSDC scheme. Suppose the
sender Bob wants to transmit his secret message directly to the
receiver Alice under the control of the controllers Charlie,
Dick,$\cdots$, York and Zach.
(S1$'$) Alice prepares a batch of N single photons randomly in one
of the four states \ket{H}, \ket{V}, \ket{+}, \ket{-} and sends this
batch of photons to Charlie.
(S2$'$) Charlie performs randomly one of the three unitary
operations $I$, $U$, $H$ on each photon, where
\begin{eqnarray}
H=\frac{1}{\sqrt{2}}(\ket{0}\bra{0}-\ket{1}\bra{1}+\ket{0}\bra{1}+\ket{1}\bra{0})
\end{eqnarray}
is a Hadamada operation. $H$ can realize the transformation between
$Z$-basis and $X$-basis,
\begin{eqnarray}
& &H\ket{0}=\ket{+}, H\ket{1}=\ket{-},\nonumber\\
& &H\ket{+}=\ket{H}, H\ket{-}=\ket{V}.
\end{eqnarray}
He then sends the $N$ photons to the next controller, say Dick. Dick
and the remaining controllers repeat the similar operations as
Charlie until Zach finishes his operations on the $N$ photons. Zach
then sends the $N$ photons to Bob.
(S3$'$) Similar to (S2), Bob encodes his random message and secret
message on the $C$-sequence and $M$-sequence, respectively. He also
disturbs the order of the $P$-sequence and send the rearranged
$P$-sequence to Alice.
(S4$'$) After hearing from Alice, Bob announces the $C$-sequence and
the secret order in it. He then let Alice publish the initial states
of the sampling photons. To prevent Alice's intercept-resend attack,
for each of the sampling photons Bob selects randomly a controller
to announce his or her $H$ operation information on the sampling
photon firstly and then the others do in turn. That is to say, the
controllers do not publish their $I$ or $U$ operation information
but only publish their $H$ operation information on the sampling
photons. According to these information, Alice can measure the
sampling photons in a correct measuring basis. She tells Bob his
measurement results. Bob chooses randomly a controller to publish
his or her $I$, $U$ operation information on each of the sampling
photons firstly and then the others do one by one. Thus Bob can
determine the error rate of the transmission of the $P$ sequence. If
he confirms there is no eavesdropping, the process is continued.
Otherwise, the process is stopped.
(S5$'$) Bob publishes the secret order of the $M$-sequence. If the
controllers permit Alice to reconstruct Bob's secret message, they
tell Alice their operation information. Thus Alice can obtain Bob's
secret message under the permission of the controllers Charlie,
$\cdots$, York, Zach.
The $H$ operation performed by the controllers is very important for
the security of the scheme. The nice feature of the $H$ operation
which can realize the transformation between $Z$-basis and $X$-basis
can prevent Eve or a dishonest controller from obtaining the control
information of the controllers. If the controllers only performs $I$
or $i\sigma_y$ operations on the $P$ sequence, Eve or a dishonest
Alice can obtain the control information by taking intercept-resend
attack. Eve intercepts the $P$-sequence and resends a fake photon
sequence to the controller. After the controller has performed his
or her operations on the $P$-sequence and sent it to the next
controller, Eve can also intercept the photon sequence and then
obtain the controller's information by measuring it. Bob firstly let
the controllers publish their $H$ operation information and then
Alice can choose a correct measuring basis. Only after Alice has
published her measurement results could the controllers announce
their $I$, $U$ operation information. It can prevent Alice from
obtaining Bob's secret message without the control of the
controllers. If the controllers publish all of their operation
information firstly, Alice can break the control of the controllers
by taking intercept-resend attack. In this attack, she sends the
$P$-sequence to Bob directly and a fake photon sequence to Charlie.
Certainly, she should intercept the photon sequence which Zach sends
to Bob. With their controller's operation information, during the
eavesdropping check Alice can successfully deceive Bob and then
obtain his secret message without the permission of the controllers.
Bob chooses randomly a controller to publish his or her operation
information on each of the sampling photons firstly and then the
others do one by one during the eavesdropping check. It ensures each
controller can really act as a controller. Suppose Bob let Charlie,
Dick, $\cdots$, York publish their operation information firstly and
Zach do finally. In other words, Bob does not select randomly a
controller to publish his or her information. Alice can then
collaborate with Zach to acquire Bob's secret message without the
control of other controllers. Alice sends the $P$-sequence to Zach
directly and a fake sequence to Charlie. Zach resends the
$P$-sequence to Bob without doing any operation. Zach can know what
operation information he should publish according to the operation
information of the controllers Charlie, Dick, $\cdots$, York. The
attack of Alice and Zach will not be detected by Bob during the
eavesdropping check. On the basis of the above analysis, the present
MCQSDC scheme is secure.
So far we have presented a QSDC scheme and a MCQSDC scheme using
single photons. Quantum no-cloning theory and the secret
transmitting order of photons ensure the security of these schemes.
In these schemes, all of the polarized photons are used to transmit
the sender's secret message directly to the receiver except those
chosen for checking eavesdropping. During the process of the scheme,
it only needs once eavesdropping check. Compared with the schemes
using entangled state, these schemes are practical within the
present technology.
\begin{acknowledgments}
This work is supported by the National Natural Science Foundation of
China under Grant No. 60472032.
\end{acknowledgments}
\end{document} |
\begin{document}
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\frontmatter
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\begin{center}
{\Large \bf Generation and Detection of Quantum Correlations and Entanglement on a Spin-Based Quantum Information Processor} \\
\vspace*{1cm}
{\large{\bf Thesis}} \\
\vspace*{0.5cm}
{For the award of the degree of}\\
{\large{\bf DOCTOR OF PHILOSOPHY}} \\
\end{center}
\vspace*{3cm}
\begin{tabular}{lp{6cm}l}
{{\it Supervised by:}} &&
{{\it Submitted by:}} \\
\\
{\bf Prof. Kavita Dorai} &&
{\bf Amandeep Singh} \\
{\bf Prof. Arvind} &&\\
\end{tabular}
\begin{center}
\vspace*{1.5cm}
\hspace*{0cm}
\end{center}
\vspace*{-1cm}
\begin{center}
\includegraphics[scale=0.24]{iiser_logo.jpg}\\
\vspace*{1.5cm}
{\bf Indian Institute of Science Education \& Research Mohali\\
Mohali - 140 306\\
India\\ (February 2019)
}
\end{center}
\thispagestyle{empty}
\begin{center}
\end{center}
\thispagestyle{empty}
\centerline{\Large \bf Declaration}
\par
\noindent The work presented in this thesis has been carried out by me under the guidance of
Prof. Kavita Dorai and Prof. Arvind at the Indian Institute of Science Education and Research Mohali.\\
\noindent This work has not been submitted in part or in full for a degree, diploma or a fellowship to any other University or Institute. Whenever contributions of others are involved, every effort has been made to indicate this clearly, with due acknowledgment of collaborative research and discussions. This thesis is a bonafide record of original work done by me and all sources listed within have been detailed in the bibliography.
\vspace*{1in}
\hspace*{-0.25in}
\parbox{8in}{
\noindent {{\bf Amandeep Singh}}
}
\vspace*{0.125in}
\hspace*{-0.25in}
\noindent
\parbox{2.5in}{
\noindent Place~: \\
\noindent Date~:
}
\vspace*{0.5in}
\noindent
In our capacity as supervisors of the candidate's PhD thesis work, we certify that the above statements by the candidate are true to the best of our knowledge.
\vspace*{2.5cm}
\hspace*{-0.25in}
\parbox{8in}{
\noindent {{\bf Dr. Kavita Dorai} \hspace*{5.5cm} {\bf Dr. Arvind}} \\
\noindent Professor of Physics \hspace*{5.1cm} Professor of Physics\\
\noindent Department of Physical Sciences \hspace*{3cm} Department of Physical Sciences\\
\noindent IISER Mohali \hspace*{6.2cm} IISER Mohali
}
\vspace*{0.125in}
\hspace*{-0.25in}
\noindent
\parbox{8in}{
\noindent Place~: \hspace*{7.4cm} Place~: \\
\noindent Date~: \hspace*{7.55cm} Date~:
}
\thispagestyle{empty}
\begin{center}
\end{center}
\thispagestyle{empty}
\centerline{\Large \bf Acknowledgments}
First and foremost I would like to express my sincere gratitude to my thesis supervisors Prof. Kavita Dorai and Prof. Arvind for their relentless support, guidance and motivation throughout the course of PhD. I am, and will always be, short of words to express my thanks for their resolute encouragement. More than teaching and guiding, on various projects in this thesis, they inculcated the attitude to understand, formulate and accomplish any given task for which I'll always be thankful to them.\\
I would also like to thank my PhD doctoral committee member Prof. Ramandeep Singh Johal for his continued support and guidance. I am also grateful to all my excellent PhD course instructors Prof. Jasjeet Singh Bagla, Prof. Sudeshna Sinha, Dr. Rajeev Kapri, Dr. Abhishek Chaudhuri and Head of the Physics Department Dr. Sanjeev Kumar for all the support they have given during and after the course work.\\
I am also thankful to all the members of journal club meetings of QCQI and NMR research groups. I have benefited a lot from the discussions of these meeting. I would like to thank my group members Dr. Shruti Dogra, Dr. Harpreet Singh, Dr. Satnam Singh, Dr. Navdeep Gogna, Rakesh Sharma, Jyotsana Ojha, Akshay Gaikwad and Dileep Singh. I am also grateful to Dr. Gopal Verma, Dr. Archana Sangwan, Varinder Singh and Dr. Kavita Mehlawat for their cheerful company. I cherish the memories of discussions during routine tea breaks with Dr. Raju Nanda, Sumit Mishra and Akanksha Gautam.\\
I owe special thank to scientific officer Dr. Paramdeep Singh Chandi for his help and support with software troubleshooting. I am also thankful to Bruker application engineer Dr. Bhawani Shanker Joshi for his help in solving problems and rectifying errors during NMR spectrometer operations. I am thankful to NMR lab scientific staff Mr. Balbir Singh for his generous support.\\
I would like to thankfully acknowledge all the support provided by the NMR research facility, IISER Mohali to carry out my experimental work. I also owe thank to IISER Mohali for the research fellowship as well as financial support to attend an international conference. I am also thankful to Perimeter Institute for Theoretical Physics, Institute for Quantum Computing and the German Physical Society for the travel grants and financial support to attend conferences in Canada and Germany.\\
And finally, last but by no means least, I would like to express my heartfelt gratitude to my family for their continuous encouragement and belief in me!
\baselineskip=15pt
\rightline{\bf \large{Amandeep Singh}}
\chapter{Abstract}
This thesis focuses on the experimental creation and detection of different types of quantum correlations using nuclear magnetic resonance (NMR) hardware. The idea of encoding computational problems into physical quantum system and then harnessing the quantum evolution to perform information processing is at the core of quantum computing. Quantum entanglement is a striking feature exhibited by composite quantum systems which has no classical analog. It has been shown that quantum entanglement is a key resource to achieve computational speedup in quantum information processing and for quantum communication related tasks. Creation and detection of such correlations experimentally is a major thrust area in experimental quantum computing. Main goals of the studies undertaken in this thesis were to design experimental strategies to detect the entanglement in a `state-independent' way and with fewer experimental resources. Experimental schemes have been devised which enables the measurement of desired observable with high accuracy and these schemes were utilized in all the investigations. Experimental protocols were successfully implemented to detect the entanglement of random two-qubit states. Further, the schemes for the experimental detection as well as classification of generic and general three-qubit pure states have also been devised and implemented successfully. Detection of quantum correlations possessed by mixed separable states, bound-entanglement for states of $2\otimes4$ systems and non-local nature of quantum systems were also investigated. In all the investigations, results were verified by one or more alternative ways \textit{e.g.~} full quantum state tomography, quantum discord, negativity and \textit{n}-tangle.\\
\noindent Content of the thesis has been distributed in
seven chapters and the chapter-wise abstract is as follows.
\subsubsection*{Chapter 1}
This chapter briefly introduces the field of quantum
computation followed by the main features of NMR quantum
processor architecture. Latter part of the chapter describes
the theory of entanglement detection and experimental
realization on various hardware. Chapter concludes with
goals and motivations for the work undertaken in this
thesis.
\subsubsection*{Chapter 2} This chapter focuses on the
entanglement detection of random two-qubit states. Random
local measurements have recently been proposed to construct
entanglement witnesses and thereby detect the presence of
bipartite entanglement. We experimentally demonstrate the
efficacy of one such scheme on a two-qubit NMR
quantum-information processor. We show that a set of three
random local measurements suffices to detect the
entanglement of a general two-qubit state. We experimentally
generate states with different amounts of entanglement and
show that the scheme is able to clearly witness
entanglement. We perform complete quantum state tomography
for each state and compute state fidelity to validate our
results. Further, we extend previous results and perform a
simulation using random local measurements to optimally
detect bipartite entanglement in a hybrid system of 2$
\otimes $3 dimensionality.
\subsubsection*{Chapter 3} In this chapter the focus is on a
more general kind of quantum correlation possessed by
separable states. A bipartite quantum system in a mixed
state can exhibit non-classical correlations, which can go
beyond quantum entanglement. While quantum discord is the
standard measure of quantifying such general quantum
correlations, the non-classicality can be determined by
simpler means via the measurement of witness operators. We
experimentally construct a positive map to witness
non-classicality of two-qubits in an NMR system. The map
can be decomposed in terms of measurable spin magnetization
so that a single run of an experiment on an ensemble of
spins suffices to detect the non-classicality in the state,
if present. We let the state evolve in time and use the map
to detect non-classicality as a function of time. To
evaluate the efficacy of the witness operator as a means to
detect non-classicality, quantum discord was measured by
performing full quantum state tomography at each time
instant and obtain a fairly good match between the two
methods.
\subsubsection*{Chapter 4} This chapter details the
experimental detection of the entanglement present in
arbitrary three-qubit pure quantum states on an NMR quantum
information processor. Measurements of only four observables
suffice to experimentally differentiate between the six
classes of states which are inequivalent under stochastic
local operation and classical communication (SLOCC). The
experimental realization is achieved by mapping the desired
observables onto Pauli $z$-operators of a single qubit,
which is directly amenable to measurement. The detection
scheme is applied to known entangled states as well as to
states randomly generated using a generic scheme that can
construct all possible three-qubit states. The results are
substantiated via direct full quantum state tomography as
well as via negativity calculations and the comparison
suggests that the protocol is indeed successful in detecting
tripartite entanglement without requiring any {\it a priori}
information about the states.
\subsubsection*{Chapter 5} This chapter details the
experimental creation and characterization of a class of
qubit-ququart PPT (positive under partial transpose)
entangled states using three nuclear spins on an NMR quantum
information processor. Entanglement detection and
characterization for systems with a Hilbert space dimension
$ > 2 \otimes 3$ is nontrivial since there are states in
such systems which are both PPT as well as entangled. The
experimental detection scheme that we employed for the
detection of this qubit-ququart PPT entanglement was based
on the measurement of three Pauli operators. The class of
states considered, in the current study, is an incoherent
mixture of five pure states. Measuring three Pauli
operators, with high precision using our recently devised
method, is crucial to detect entanglement. All the five
states were prepared with high fidelities and the resulting
PPT entangled states were prepared with mean fidelity $ \geq
$ 0.944 using temporal averaging technique.
\subsubsection*{Chapter 6} This chapter presents the
experimental investigations of non-local nature of quantum
correlations possessed by multipartite quantum states. It
has been shown that fewer body correlations can reveal the
non-local nature of the correlations arising from quantum
mechanical description of the nature. Such tests on the
correlations can be transformed to a semi-definite-program
(SDP). This study presents the experimental implementation
of Navascu\'es-Pironio-Ac\'{\i}n (NPA) hierarchy on NMR
hardware utilizing three nuclear spins. The protocol has
been tested on two types of genuine tripartite entangled
states. In both the cases the experimentally measured
correlations were used to formulate the SDP under linear
constraints on the entries of the moment matrix. It has been
observed that in both the cases SDP failed to find a
semi-definite-positive moment matrix consistent with the
experimental data which is indeed the signature that the
observed correlations can not arise from local measurements
on a separable state and hence are non-local in nature. This
also confirms that both the states under test are indeed
entangled. Results were verified by direct full quantum
state tomography in each case.
\subsubsection*{Chapter 7} This chapter summarizes the
results of all the projects constituting this thesis, and
the key findings, with possible future directions of work.
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{\LARGE \bf List of Publications}
\vspace*{12pt}
\begin{enumerate}
\addtolength{\itemsep}{12pt}
\item \textbf{Amandeep Singh}, Arvind and Kavita Dorai. \textit{ Entanglement detection on an NMR quantum information processor using random local measurements}, \href{http://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.062309}{\rm Phys. Rev. A \textbf{94}, 062309 (2016)}.
\item \textbf{Amandeep Singh}, Arvind and Kavita Dorai. \textit{Witnessing nonclassical correlations via a single-shot experiment on an ensemble of spins using NMR}, \href{http://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.062318}{\rm Phys. Rev. A \textbf{95}, 062318 (2017)}.
\item Akshay Gaikwad, Diksha Rehal, \textbf{Amandeep Singh}, Arvind and Kavita Dorai. \textit{Experimental demonstration of selective quantum process tomography on an NMR quantum information processor}, \href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.97.022311}{\rm Phys. Rev. A \textbf{97}, 022311 (2018)}.
\item \textbf{Amandeep Singh}, Harpreet Singh, Kavita Dorai and Arvind. \textit{ Experimental Classification of Entanglement in Arbitrary Three-Qubit Pure States on an NMR Quantum Information Processor}, \href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.032301}{\rm Phys. Rev. A \textbf{98}, 032301 (2018)}.
\item \textbf{Amandeep Singh}, Kavita Dorai and Arvind. \textit{ Experimentally identifying the entanglement class of pure tripartite states}, \href{https://doi.org/10.1007/s11128-018-2105-5}{\rm Quant. Info. Proc. \textbf{17}, 334 (2018)}.
\item \textbf{Amandeep Singh}, Akanksha Gautam, Kavita Dorai and Arvind. \textit{Experimental Detection of Qubit-Ququart Pseudo-Bound Entanglement using Three Nuclear Spins}, \href{https://doi.org/10.1016/j.physleta.2019.02.027}{\rm Phys. Lett. A \textbf{383}(14), 1549-1554 (2019)}.
\item \textbf{Amandeep Singh}, Kavita Dorai and Arvind. \textit{Detection of Non-local Quantum Correlations via Experimental Implementation of NPA Hierarchy on NMR}, (Manuscript in preparation)
\end{enumerate}
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\chapter{Abbreviations used in the Thesis}
\begin{table}[H]
\begin{center}
\begin{tabular}{l l l }
BE &:& Bound Entangled \\
BS &:& Bi-Separable \\
CCNR &:& Computable Cross Norm or Realignment \\
CP &:& Completely Positive \\
CHSH &:& Clauser-Horne-Shimony-Holt \\
EM &:& Entanglement Measure(s) \\
EPR &:& Einstein-Podolsky-Rosen \\
EW &:& Entanglement Witness \\
FID &:& Free Induction Decay \\
FT &:& Fourier Transform \\
GHZ &:& Greenberger-Horne-Zeilinger \\
LOCC &:& Local Operations and Classical Communications \\
NCC &:& Non Classical Correlations \\
NMR &:& Nuclear Magnetic Resonance \\
NMRQC &:& NMR Quantum Computing \\
NPT &:& Negative under Partial Transpose \\
PCC &:& Properly Classically Correlated \\
PNCP &:& Positive but Not Completely Positive\\
POVM &:& Positive Operator Valued Measure\\
PPT &:& Positive under Partial Transpose \\
PT &:& Partial Transposition \\
QC &:& Quantum Computing \\
QD &:& Quantum Discord \\
QIP &:& Quantum Information Processing \\
QST &:& Quantum State Tomography \\
RF &:& Radio Frequency \\
SDP &:& Semi Definite Program/Programming \\
SLOCC &:& Stochastic LOCC \\
\end{tabular}
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\mainmatter
\chapter{Introduction}
Being an intelligent species, modern human beings, \textit{i.e.~} \textit{Homosapiens}, (meaning a `wise man' in Latin), started using the first computing machine named \textit{the `Abacus'}. Records show that the earliest users of abacus were the Sumerians and the Egyptians back in 2000 BC. The principle is as: a frame holding a series of rods, with ten sliding beads on each. When all the beads had been slid across the first rod, it was time to move one across on the next, showing the number of tens, and thence to the next rod, showing hundreds, and so on (with the ten beads on the initial row returned to the original position), (Fig.(\ref{abacus})).
\begin{figure}
\caption{Abacus: The First Computer \cite{firkin-online-13}
\label{abacus}
\end{figure}
That is where technology of computation was stuck for nearly 3600 years until the beginning of the 17$^\mathrm{th}$ century AD, when mechanical calculators started appearing in Europe. Most notably, after John Napier invented logarithms, and Edmund Gunter created the logarithmic scales (lines, or rules) upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division; and he is credited as the inventor of the slide rule in 1622, Fig.(\ref{sliderule}). William Oughtred (1574-1660) was an English mathematician born in Eton and he was the person who introduced the symbol `$\times$' for multiplication as well as $sin$ and $cos$ for trigonometric functions \textit{Sine} and \textit{Cosine}, respectively. The slide rule is basically a sliding stick that uses logarithmic scales to allow rapid multiplication and division, (Fig.(\ref{sliderule})). Slide rules evolved to allow advanced trigonometry and logarithms, exponential and square roots.
\begin{figure}
\caption{The Slide Ruler used in 17$^\mathrm{th}
\label{sliderule}
\end{figure}
A further step forward in computation occurred in 1887 when Dorr. E. Felt's US-patented key driven 'Comptometer' took calculating into the push button age, (Fig(\ref{comptometer})).
\begin{figure}
\caption{Comptometer: The Modern Day Calculator \cite{comptometer-online-06}
\label{comptometer}
\end{figure}
Further, the story of electronic calculator began in 1930 when world was preparing for war. Such calculator were much in demand due calculation of trajectories of bombs. During the Second World War, the challenges of code-breaking produced the first all-electronic computer, \textit{Colossus}. But this was a specialized machine that basically performed ``exclusive or'' (XOR) Boolean algorithms. With the advent of semiconductor-devices, in the mid of the 20$^\mathrm{th}$ century, the first generation of modern days computers came into the existence which was a huge leap as compared the huge-inefficient computation machines based on hundred of thermionic valves. Since then the power of computing machines grows exponentially following the then proposed Moore's law. The law was described as early as 1965 by the Intel co-founder Gordon E. Moore after whom it is named \cite{moore-e-65}.\\
There have been a large number of classical algorithms/problems posed which seem unsolvable on these computing machines although the computational power was expanding at an exponential rate. In computer science, the computational complexity of an algorithm is usually ascribed to the time complexity which estimates the time to run the algorithm as a function of order of input strings. Order of the input string is generally denoted by $n$ while the complexity of the computation is represented represented using big $O$. For example, an algorithm with time complexity $O(n)$ is a linear time algorithm, $O(n^{\alpha})$ with $\alpha>1$ represent a polynomial time algorithm while $O(2^{p(n)})$ represents the time complexity of an exponential time algorithm with $p(n)$ being polynomial of order $n$. Two extreme cases are the constant time (a sub-class of polynomial time) and exponential time, termed as EXPTIME, complexity classes. It is understandable that if an algorithm has its time complexity in EXPTIME class then in order to run such an algorithm the required time scales exponentially with the size of input string $n$. One may not wish to wait too long, \textit{e.g.~} few years, to run an algorithm. Some of EXPTIME class of problems are the prime factorization problem, optimization problems with large number of variables, matrix chain multiplication via brute-force search and simulation of quantum systems using classical models. Then in the early eighties Feynman proposed the idea of exploring quantum systems to simulate quantum systems \cite{feynman-ijtp-82}. The idea of encoding computational problems into physical quantum system and then harnessing the quantum evolution to perform information processing is at the core of quantum computing and quantum information (QCQI) processing \cite{nielsen-book-02}. There are various features, \textit{e.g.~} quantum superposition and quantum entanglement, which enable computation utilizing quantum systems to outperform any classical computing machine. These concepts will be briefly introduced in subsequent sections after introducing the basic ideas of QCQI.
\section{Quantum Computing and Quantum Information Processing}
Classical information processing is solely realized by encoding the bit-strings into the classical states of physical systems \cite{cover-book-91}. For example high or low voltages in a digital electronic circuit are used to represent the classical Boolean states `0' or `1', respectively or light or no-light can be used to encode `0' or `1' in an optical computer. Most of the digital information processing hardware available today performs classical information processing by encoding the problems into binary strings and performing logical operations governed by Boolean algebra \cite{boole-book-09}. Classical physical systems can be either in `0' or `1' state at an instant of time, and this limits the information processing achievable on such classical information processors. On the other hand the states of quantum physical systems can exist in superposition of `0' and `1'. allowing new possibilities for computation \cite{nielsen-book-02}. Later subsection of this chapter will address the issue of physical realization of such quantum states.
There a striking similarity between what a classical computer does and how a physical system evolves. A computer performs a computation on some input bit string under certain logical operations to yield the output. Analogously, a physical system evolves from an initial state following the laws of motion to give the final state. The idea of simulating classical as well as quantum systems by encoding the problem as an initial state of the quantum system was put forward by Feynman\cite{feynman-ijtp-82}. Computation can be achieved by the quantum evolution and the results get encoded in the final state of the quantum system which can be read. This was radically a new way of performing the computation.
\subsection{The Quantum Bit}
Information can be encoded in the physical state of a quantum system and the minimum dimension of the involved Hilbert space, to represent the states of such system, is two. Such encoding can be achieved by using a two-level quantum system, \textit{e.g.~} a spin-half system, generally termed as a quantum-bit or \textit{Qubit}. The two eigenstates of such two-level quantum systems representing the logical states `0' and `1' are $\vert 0 \rangle$ and $\vert 1 \rangle$ which represent the eigenvectors $\begin{bmatrix} 1 \\
0
\end{bmatrix}
$ and $\begin{bmatrix} 0 \\
1
\end{bmatrix}
$ respectively. The most general state $\vert \psi \rangle$ of a qubit can be a superposition of basis vectors and can assume the polar form as
\begin{equation}\label{1qubitstate}
\vert \psi \rangle=cos\left(\frac{\theta}{2}\right) \vert 0 \rangle + e^{i\phi}sin\left(\frac{\theta}{2}\right) \vert 1 \rangle
\end{equation}
where the global phase is ignored in writing the above polar form as it does not have any observable effect during quantum evolution or on measurement outcomes. The most contrasting feature of the qubit from a classical bit is that a qubit can simultaneously exist in the basis states $\vert 0 \rangle$ and $\vert 1 \rangle$ and this quantum parallelism of quantum systems gives them tremendous computational power which a classical computer may never match \cite{nielsen-book-02}. Each value of the pair ($\theta$,\;$\phi$) represents a valid quantum state on the surface of a three-dimensional unit radius sphere, shown in Fig.(\ref{blochrep}), called the Bloch sphere. The radius of the sphere is indeed related to the quantity $(\vert a_0 \vert ^2 + \vert a_1 \vert ^2)$ which is unit due to normalization of state vector $\vert \psi \rangle$. One can observe that the North ($\theta=0$) and South ($\theta=\pi$) poles of the Bloch sphere represent the basis states $\vert 0 \rangle$ and $\vert 1 \rangle$ respectively.\\
\begin{figure}
\caption{The Bloch-sphere representation of a single quantum-bit.}
\label{blochrep}
\end{figure}
Similarly one may have multiqubit quantum register. For the general case of an $N$-qubit quantum register, the basis vectors of Hilbert space have dimension $2^N$ and can be obtained from the tensor product of individual qubit states as
\begin{equation}\label{nqubitsep}
\vert \Psi \rangle= \vert \psi_1 \rangle \otimes \vert \psi_2 \rangle\otimes... \otimes \vert \psi_N \rangle
\end{equation}
It will be seen later that the most general $N$-qubit state can be put in the form
\begin{equation} \label{nqubitstate}
\vert \Psi \rangle= \sum_{i=1}^{2^N} \alpha_i \vert \alpha_i \rangle
\end{equation}
here $\vert \alpha_i \rangle$ is a $N$-qubit quantum register of form $\vert b^i_1b^i_2.....b^i_N \rangle$ with $b_j\in [0,1]$ and $\sum_i \vert \alpha_i \vert ^2=1$. There exist multiqubit states which can be cast in the form of Eq.(\ref{nqubitstate}) and may not assume the form given by Eq.(\ref{nqubitsep}). Such states are called entangled states and they play a major role in QCQI.
\subsection{The Density Matrix Formalism}
As discussed in the previous section, the state of a quantum system has one-to-one correspondence with the vectors in Hilbert space. Consider a quantum system prepared in a state $\vert \psi_1 \rangle$ and we have $n_1$ such systems constituting a pure ensemble. Similarly consider another ensemble composed of $ n_2 $ quantum systems, each of which is in the state $\vert \psi_2 \rangle $. If one mixes these two ensembles then how can one write the quantum state of the resulting ensemble? The total number of quantum systems are $N_T=n_1+n_2$. Another important question is, if we now pick a quantum system from this ensemble and measure it, what is the result? There are two probabilities for the action: (i) The probability with which the chosen quantum system can come from ensemble $\vert \psi_1 \rangle$ or $\vert \psi_2 \rangle$ \textit{i.e.~} $p_1=p=\frac{n_1}{N_T}$ and $p_2=1-p=\frac{n_2}{N_T}$ and (ii) the probability with which, the chosen quantum system after measurement collapses to $\vert 0 \rangle$ or $\vert 1 \rangle$. One thing is clear that the state description of form Eq.(\ref{nqubitstate}) is not appropriate for this situation \textit{i.e.~} such an ensemble can not be represented by vectors in a Hilbert space.\\
It has been shown that a more suitable state representation is the density operator formalism \cite{reif-book-65,sakurai-book-94,nielsen-book-02,oliveira-book-07}. For a pure state the density operator can be written as
\begin{equation}\label{pure_ch1}
\rho=\vert \psi \rangle \langle \psi \vert
\end{equation}
In order to write the density matrix one may choose a set of orthogonal basis vectors ${\vert b_i \rangle}$ and the matrix elements can be computed as $\rho_{ij}=\langle b_i \vert \rho \vert b_j \rangle$. It can be shown that
\begin{equation}
\sum_i\rho_{ii}=\sum_i\langle b_i \vert \rho \vert b_i \rangle=1
\end{equation}
This will lead to the condition Tr$(\rho)=1$ independent of chosen basis. For pure states $\rho^2=\vert \psi \rangle \langle \psi \vert.\vert \psi \rangle \langle \psi \vert=\vert \psi \rangle \langle \psi \vert=\rho$ and hence Tr$(\rho^2)=$Tr$(\rho)=1$. For a mixed ensemble, the density operator of the ensemble can be written as
\begin{equation}
\rho_{\mathrm{ensemble}}=p\vert \psi_1 \rangle \langle \psi_1 \vert+(1-p)\vert \psi_2 \rangle \langle \psi_2 \vert
\end{equation}
and this correctly incorporates both the probabilities mentioned earlier. The most general density operator for a single-qubit system can be written as
\begin{equation}\label{mixedrho}
\rho=\sum_j p_j\vert \psi_j \rangle \langle \psi_j \vert; \;\;\;\;\;\; p_j\geq 0; \;\;\;\;\;\; \sum_j p_j=1
\end{equation}
It may be noted that one-qubit mixed states \textit{i.e.~} states with Tr$(\rho^2)<1$ can be represented with points inside the Bloch sphere in Fig.(\ref{blochrep}) and the center of the sphere represents the maximally mixed state
\begin{equation}
\rho^{}_{\mathrm{mixed}_{max}}=\frac{\mathbb{I}}{2^1}=\begin{pmatrix} 1/2 & 0 \\
0 & 1/2 \end{pmatrix}
\end{equation}
For a general $N$-qubit state the condition $\frac{1}{2^N} \leq \mathrm{Tr}(\rho^2) \leq 1$ is valid. The lower and upper bounds on Tr$(\rho^2)$ are achieved by maximally mixed and pure states respectively.
\subsection{Quantum Evolution}
In quantum mechanics there are broadly two kinds of time evolution (a) the continuous time evolution of the states of a closed quantum system as governed by Schr{\"o}dinger equation and (b) a discontinuous time evolution during a quantum measurement following Born's rule for the probabilities of the possible outcomes. Following subsections briefly describe both of these time evolutions. A detailed description of quantum evolution is given in Refs. \cite{shankar-book-80,sakurai-book-94}.
\subsubsection{Continuous Time Evolution: Unitary Evolution}
\textbf{Closed quantum system:} The continuous time evolution of closed quantum systems is unitary \textit{i.e.~} the time evolution operator can be represented by a unitary matrix $U$ obeying $UU^{\dagger}=U^{\dagger}U=\mathbb{I}$. Equivalently one can say that the state of a quantum system at time `$t_o$' transforms to the state at a later time `$t_o+t$' via unitary transformation as
\begin{equation}
\vert \psi ' (t_o+t)\rangle =U\vert \psi(t_o) \rangle
\end{equation}
For a closed quantum system, described by the Hamiltonian $\mathit{H}$, the equation for state evolution can be written as
\begin{equation}\label{TDSEq}
i\hbar\frac{\partial\vert \psi \rangle}{\partial t}=\mathit{H}\vert \psi \rangle
\end{equation}
With minimal assumptions and considering the case of a time independent Hamiltonian one can solve the above differential equation. The result leads to a time evolution operator of form $U\sim e^{-i\mathit{H} t/\hbar}$. Important point to note here is that, once $\mathit{H}$ is defined and the resulting $U$ is obtained, the state $\vert \psi ' (t+t_o)\rangle$ at all later times evolves continuously in a predictable fashion via $U$. The advantage of the unitary evolution is that such evolutions are always reversible.\\
\noindent \textbf{Open quantum system:} There can be situation that a system under consideration interacts with its environment and usually termed as open quantum systems. Such a composite system can be assumed to be in a separable state $\rho_{comp}=\rho_{sys}\otimes\rho_{env}$. There can be three possible energy operators in this scenario (i) system Hamiltonian, $\textit{H}_{sys}$, (ii) environment Hamiltonian, $\textit{H}_{env}$, and (iii) the interaction Hamiltonian, $\textit{H}_{int}$ due to the interaction between system and its environment. Hence the Hamiltonian of the composite system can be written as
\begin{equation}
\textit{H}_{comp}=\textit{H}_{sys}+\textit{H}_{env}+\textit{H}_{int}
\end{equation}
The time evolution of density operator can be obtained from time-dependent Schr{\"o}dinger Eq.(\ref{TDSEq}) as follows:
\begin{equation}\label{TDSE_ch1}
\frac{\partial\vert \psi \rangle}{\partial t}=-\frac{i}{\hbar}\mathit{H}\vert \psi \rangle \;\;\; \Leftrightarrow \;\;\; \frac{\partial \langle \psi \vert}{\partial t}=\frac{i}{\hbar}\langle \psi \vert\mathit{H}
\end{equation}
For a pure state of form Eq.(\ref{pure_ch1})
\begin{eqnarray}
\frac{\partial\rho}{\partial t}&=&\frac{\partial \left[ \vert \psi \rangle\langle \psi \vert \right]}{\partial t}\nonumber\\
&=& \left[\frac{\partial\vert \psi \rangle}{\partial t}\right] \langle \psi \vert + \vert \psi \rangle \frac{\partial\langle \psi \vert}{\partial t}
\end{eqnarray}
and on using Eq.(\ref{TDSE_ch1})
\begin{eqnarray}
\frac{\partial\rho}{\partial t}&=&-\frac{i}{\hbar}\mathit{H}\vert \psi \rangle\langle \psi \vert + \frac{i}{\hbar}\vert \psi \rangle\langle \psi \vert\mathit{H}\nonumber\\
\frac{\partial\rho}{\partial t}&=& -\frac{i}{\hbar}\left[\mathit{H},\rho\right]
\end{eqnarray}
Above is the Liouville-Von Neumann equation. Although, the above equation is derived using pure state density operator but it can be shown that it is also valid for mixed states.\\
For the composite system $\rho_{comp}$ the equation of motion can be written as
\begin{equation}
\frac{\partial\rho_{comp}(t)}{\partial t} = -\frac{i}{\hbar}\left[\mathit{H}_{comp},\rho_{comp}(t)\right]
\end{equation}
The solution to the above equation is of form $\rho_{comp}(t)=U(t)\rho_{comp}(0)U^{\dagger}(t)$. It is worth mentioning here that the unitary time evolution operator $U(t)$ acts on the composite system $\rho_{comp}$ and one may be interested in the evolution of the system state $\rho_{sys}(t)$ only. The way out is that one may trace out the environment to get $\rho_{sys}(t)=\mathrm{Tr}_{env}(\rho_{comp})$. Evolution of $\rho_{sys}(t)$ can formally be derived using Lindblad master equation formalism and we will not expand on this here. Another important aspect here is that although the evolution of $\rho_{comp}$ is unitary but the state of the system $\rho_{sys}$ may evolve non-unitarily and irreversibly.
\subsubsection{Discontinuous Time Evolution: Quantum Measurement}
During a quantum measurement process, the state of a quantum system, \textit{e.g.~} Eq.(\ref{nqubitstate}), abruptly collapses to one of the eigenstates, of the observable being measured, in an unpredictable way. Quantum measurement is typically described by a set of measurement operators $\{ M_m \}$. Here the index `$m$' is the label of the measurement outcome after quantum measurement $M$. If the state of the system before measurement is $\vert \psi \rangle$ then the probability of getting the measurement outcome $m$, \textit{i.e.~} $p(m)$, is given by
\begin{equation}
p(m)=\langle \psi \vert M^{\dagger}_m M_m \vert \psi \rangle
\end{equation}
while the renormalized state after obtaining the measurement outcome $m$ can be written as
\begin{equation}
\frac{M_m\vert \psi \rangle}{\sqrt{\langle \psi \vert M^{\dagger}_m M_m \vert \psi \rangle}}
\end{equation}
The sum of probabilities \textit{i.e.~} $\sum_m p(m)=1$, is equivalently represented by the condition that all the measurement operators sum to identity and usually referred as \textit{The Completeness Condition}:
\begin{equation}
\sum_m M^{\dagger}_m M_m =\mathbb{I}
\end{equation}
One of the commonly used measurement basis is the computation basis $\{ \vert 0 \rangle,\;\vert 1 \rangle \}$ and in such a scenario $M_0=\vert 0 \rangle \langle 0 \vert$ and $M_1=\vert 1 \rangle \langle 1 \vert$. Hence if one measure the observable \textit{e.g.~} $\sigma_z$ on a quantum state given by Eq.(\ref{1qubitstate}), the probability of getting `0' or `1' is given by $p(0)=\langle \psi \vert M^{\dagger}_0 M_0 \vert \psi \rangle=\langle \psi \vert M_0 \vert \psi \rangle=\vert a_0 \vert ^2$ and $p(1)=\langle \psi \vert M^{\dagger}_1 M_1 \vert \psi \rangle=\langle \psi \vert M_1 \vert \psi \rangle=\vert a_1 \vert ^2$ respectively. The most unsettling thing here is that there is \textit{no-way} to predict that after a measurement in which eigenstate the quantum system will collapse to! The only thing quantum mechanics predicts is the probability with which a quantum system, after measurement, will collapse to a certain eigenstate of the observable being measured. This is the standard measurement scenario in quantum mechanics and is one of the postulates of the theory. However in functional analysis of quantum measurement theory, the quantum measurements are usually associated with a positive-operator valued measures (POVMs). POVMs are positive operator on Hilbert space and for a given measurement they sum to identity. Projective measurements on a large system \textit{i.e.~}, measurements that are performed mathematically by a projection-valued measure (PVM) will act on a subsystem in ways that cannot be described by a PVM on the subsystem alone, the POVM formalism becomes necessary.
\subsection{Expectation Values}
In quantum formalism, every observable is represented by a Hermitian operator, say $\mathcal{A}$. One may be interested in writing the average of an observable resulted from a large number of measurements of such an observable on a state, say $\vert \psi \rangle$, and can be written as
\begin{equation}\label{expecvalue}
\langle \mathcal{A} \rangle=\langle \psi \vert \mathcal{A} \vert \psi \rangle
\end{equation}
One may choose some orthonormal basis $\lbrace \vert \alpha_i \rangle \rbrace $ to expand the state $\vert \psi \rangle$ as $\vert \psi \rangle =\sum_i \alpha_i \vert \alpha_i \rangle $. On using this expansion, Eq.(\ref{expecvalue}) yields
\begin{eqnarray}\label{expecvalue1}
\langle \mathcal{A} \rangle=\langle \psi \vert \mathcal{A} \vert \psi \rangle &=& \left( \alpha_1^{\ast} \langle \alpha_1 \vert+\alpha_2^{\ast} \langle \alpha_2 \vert+... \right)\mathcal{A}\left( \alpha_1^{} \vert \alpha_1 \rangle+\alpha_2^{} \vert \alpha_2 \rangle+... \right)\nonumber\\
&=&\sum_{i,j}\alpha^{\ast}_i\alpha^{}_j\langle \alpha^{}_i \vert \mathcal{A} \vert \alpha^{}_j \rangle\nonumber\\
&=&\sum_{i,j} \alpha^{\ast}_i\alpha^{}_j \mathcal{A}_{ij}
\end{eqnarray}
Here $\mathcal{A}_{ij}$ is the matrix representation of the Hermitian operator $\mathcal{A}$ in the basis $\lbrace \vert \alpha_i \rangle \rbrace $ and the expansion coefficients can be obtained as $\alpha_i=\langle \alpha_i \vert \psi \rangle$ and hence
\begin{equation}
\alpha^{\ast}_i\alpha^{}_j=\langle \psi \vert \alpha_i \rangle \langle \alpha_j \vert \psi \rangle=\langle \alpha_j \vert \psi \rangle \langle \psi \vert \alpha_i \rangle=\langle \alpha_j \vert \rho \vert \alpha_i \rangle
\end{equation}
Using above expression, Eq.(\ref{expecvalue1}) further takes the form
\begin{eqnarray}
\langle \mathcal{A} \rangle &=&\sum_{i,j} \alpha^{\ast}_i\alpha^{}_j \mathcal{A}_{ij}=\sum_{i,j} \langle \alpha_j \vert \psi \rangle \langle \psi \vert \alpha_i \rangle \mathcal{A}_{ij}=\sum_{i,j} \langle \alpha_j \vert \rho \vert \alpha_i \rangle \mathcal{A}_{ij}\nonumber\\
&=& \sum_{i,j}\langle \alpha_j \vert \rho \vert \alpha_i \rangle \langle \alpha^{}_i \vert \mathcal{A} \vert \alpha^{}_j \rangle=\sum_{j}\langle \alpha_j \vert \rho \left\lbrace \sum_{i} \vert \alpha_i \rangle \langle \alpha^{}_i \vert \right\rbrace \mathcal{A} \vert \alpha^{}_j \rangle\nonumber\\
&=& \sum_{j}\langle \alpha_j \vert \rho \mathcal{A} \vert \alpha^{}_j \rangle=\mathrm{Tr}(\rho\mathcal{A})
\end{eqnarray}
while the fact that an orthonormal basis follows the completeness property, \textit{i.e.~}
\linebreak
$\sum_{i} \vert \alpha_i \rangle \langle \alpha^{}_i \vert=\mathbb{I}$, is used in writing the last line of the above equation. Similarly, for a mixed state of form Eq.(\ref{mixedrho}), it can also be shown that $\langle \mathcal{A} \rangle=\mathrm{Tr}(\rho\mathcal{A})$.
\subsection{Quantum Gates}
In classical computation, there are logical gates to realize the logical Boolean operations \textit{e.g.~} OR, AND and NOT gate. There are other gates, composed of the three basic logic gates, \textit{e.g.~} Exclusive-OR (XOR), NOR, NAND, bubbled-AND ($\sim$ OR) and bubbled-OR ($\sim$ AND) gates. \textit{Universal logic gates} are those gates from which any arbitrary logic gate can be realized. There are many universal gates available to achieve arbitrary Boolean logic operations, with NAND and NOR being common examples. Most of the multi-bit gates are irreversible in nature \textit{i.e.~} given the output of the logic gate one may not always predict the input with certainty. Analogously, similar logic gates can be constructed using quantum systems, via unitary evolution. Due to the unitary nature, in principle, one can always retrieve the input state of the quantum system if the output state after gate implementation is known. Similar to universal gates in classical computation there are universal quantum gates. It was shown that a set of gates that consists of all one-bit quantum gates $[U(2)]$ and the two-bit exclusive-OR gate which maps Boolean values $(x,y)$ to $(x,x\oplus y)$, is universal in the sense that all unitary operations on arbitrarily many bits $n$ $[U(2^n)]$ can be expressed as compositions of one-bit and two-bit XOR quantum unitary gates \cite{barenco-pra-95}.
\subsubsection*{The Pauli $X$-Gate}
The Pauli $X$-gate or NOT gate is the quantum analog of the classical NOT gate. This single-qubit gate inverts the state of the logical qubit. The matrix representation of NOT gate is
\begin{equation}
X=\begin{bmatrix} 0 & 1\\
1 & 0 \end{bmatrix}
\end{equation}
One may observe the action of the NOT gate on the single qubit state of Eq.(\ref{1qubitstate}) as
\begin{equation}
X\vert \psi \rangle=\begin{bmatrix} 0 & 1\\
1 & 0 \end{bmatrix}
\begin{bmatrix} a_0\\
a_1 \end{bmatrix} = \begin{bmatrix} a_1\\
a_0 \end{bmatrix}
\end{equation}
Hence the resulting state is $(a_0\vert 1 \rangle+a_1\vert 0 \rangle)$ which on comparison with Eq.(\ref{1qubitstate}) clearly reflects the inverting effect of the NOT gate.
\subsubsection*{The Pauli $Y$-Gate and $Z$-Gate}
Pauli $Y$ and $Z$ gates have the following matrix representation.
\begin{equation}
Y=\begin{bmatrix} 0 & -1\\
1 & 0 \end{bmatrix} \;\;\;\;\;\;\; Z=\begin{bmatrix} 1 & 0\\
0 & -1 \end{bmatrix}
\end{equation}
The Pauli $Z$-gate acting on $\vert \psi \rangle$ results in $(a_0\vert 0 \rangle-a_1\vert 1 \rangle)$ while the action of $Y$-gate yields $(-a_0\vert 1 \rangle+a_1\vert 0 \rangle)$. Hence the $Z$-gate introduces a relative phase between the basis states while the $Y$-gate introduces a relative phase as well as inverts the basis states.
\subsubsection*{The Hadamard Gate} Another very important single-qubit quantum gate is the Hadamard gate or $ H $-gate. The matrix representation of $H$-gate is
\begin{equation}
H=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1\\
1 & -1 \end{bmatrix}
\end{equation}
The action of $H$-gate on the computational basis states produces an equal superposition of all the basis states \textit{e.g.~}
\begin{equation}
H\vert 0 \rangle = \frac{\vert 0 \rangle + \vert 1 \rangle}{\sqrt{2}}\;\;\;\;\; \mathrm{and} \;\;\;\;\; H\vert 1 \rangle = \frac{\vert 0 \rangle - \vert 1 \rangle}{\sqrt{2}}
\end{equation}
\subsubsection*{The Most General Single Qubit Gate}
The action of any arbitrary single qubit can be simulated using only three unitary operations parametrized by four real numbers $\alpha,\;\beta,\;\gamma$ and $\delta$ as following \cite{nielsen-book-02}
\begin{equation}
U(2)=e^{i\alpha}\begin{bmatrix} e^{-i\beta/2} & 0\\
0 & e^{i\beta/2} \end{bmatrix}\begin{bmatrix} cos(\frac{\gamma}{2}) & -sin(\frac{\gamma}{2})\\
sin(\frac{\gamma}{2}) & cos(\frac{\gamma}{2}) \end{bmatrix}\begin{bmatrix} e^{-i\delta/2} & 0\\
0 & e^{i\delta/2} \end{bmatrix}
\end{equation}
Here the parameter $\alpha$ introduces a global phase and has no observable effect on the state of the quantum system.
\subsubsection*{The Controlled Not Gate}
The controlled-$\mathrm{NOT}$ or $\mathrm{CNOT}$ gate is a two-qubit quantum gate. Action of this gate is to perform a $\mathrm{NOT}$ operation on the target qubit depending upon the logical state of the control qubit. The matrix representation of the $\mathrm{CNOT}$ gate is:
\begin{equation}
\mathrm{CNOT}=\begin{bmatrix} 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \end{bmatrix}
\end{equation}
One can verify that the action of $\mathrm{CNOT}$ gate on two-qubit basis states, $\vert 0 \rangle \otimes \vert 0 \rangle \sim \vert 00 \rangle$ and $\vert 01 \rangle$, is to leave them unaltered while interconvert the basis states $\vert 10 \rangle$ and $\vert 11 \rangle$. Another example of a two-qubit gate is the SWAP gate which maps the state $\vert 01 \rangle \Leftrightarrow \vert 10 \rangle $ onto each other while leaving the states $\vert 00 \rangle$ and $\vert 11 \rangle$ unaltered. It can be shown that the SWAP gate can be achieved using three $\mathrm{CNOT}$ gates as $\mathrm{SWAP\equiv CNOT_{12}.}$ $\mathrm{CNOT_{21}.CNOT_{12}}$, where $\mathrm{CNOT}_{ij}$ represents a $\mathrm{CNOT}$ gate with $i$ being the control qubit and $j$ being the target.
\subsection{Quantum Computing}
Quantum computing is a radical way of computation that utilizes the quantum mechanical phenomena, such as entanglement and superposition, for computation. Classical computation, in principle, can be thought of in terms of circuits built from universal logic gates and one can analogously construct quantum circuit using various quantum gates to process quantum information and hence quantum computation. Information can be encoded in the physical states of the quantum systems. The requirement to perform quantum computation is to implement the quantum gates efficiently. For the physical realization of quantum computation there are certain benchmarks laid down \cite{DiVincenzo-fdp-00} and are briefly discussed as follows.
\subsubsection{The DiVincenzo Criterion} \label{DiVincenzo_criterion}
The DiVincenzo criterion were devised as a set of requirements for any physical realization of a quantum information processor \cite{DiVincenzo-fdp-00}.
\begin{itemize}
\item[(i)] A scalable physical system with well characterized qubits.
\item[(ii)] The ability to initialize the state of the qubits to a fiducial state, such as $\vert 000... \rangle$.
\item[(iii)] Long relevant decoherence times, much longer than the gate operation time.
\item[(iv)] A ``universal'' set of quantum gates.
\item[(v)] A qubit-specific measurement capability.
\end{itemize}
There are two additional requirements specifically on the physical realization of quantum information processor to be utilized for quantum communications and quantum cryptography.
\begin{itemize}
\item[(vi)] The ability to interconvert stationary and flying qubits and
\item[(vii)] The ability to faithfully transmit flying qubits between specified locations.
\end{itemize}
\subsubsection{Physical Realization}
For the physical implementation of a quantum information processor, various hardware have been tried. None of them satisfies DiVincenzo criterion completely. Following is the list of a few physically realized quantum processor.
\begin{itemize}
\item[$\bullet$] Superconducting Josephson junctions as qubits \cite{clarke-nature-08}.
\item[$\bullet$] Quantum dot computer, spin-based: qubit given by the spin states of trapped electrons \cite{imamoglu-prl-99}.
\item[$\bullet$] Quantum dot computer, spatial-based: qubit given by electron position in double quantum dot \cite{fedichkin-qcc-00}.
\item[$\bullet$] Optical lattices: qubit implemented by internal states of neutral atoms trapped in an optical lattice.
\item[$\bullet$] Trapped ion quantum computer: qubit implemented by the internal state of trapped ions.
\item[$\bullet$] Coupled Quantum Wire: qubit implemented by a pair of Quantum Wires coupled by a quantum point contact \cite{bertoni-prl-00}.
\item[$\bullet$] Nuclear magnetic resonance quantum computer: (NMRQC) implemented with the nuclear magnetic resonance of molecules in solution, where qubits are provided by nuclear spins \cite{cory-pnas-97,chuang-sc-97,vandersypen-rmp-05}.
\item[$\bullet$] Solid-state NMR quantum computers: qubit realized by the nuclear spin state of phosphorus donors in silicon.
\item[$\bullet$] Diamond-based quantum computer: qubit realized by the electronic or nuclear spin of nitrogen-vacancy centers in diamond \cite{neumann-sc-08}.
\item[$\bullet$] Fullerene-based ESR quantum computer: qubit based on the electronic spin of atoms or molecules encased in fullerenes.
\item[$\bullet$] Cavity quantum electrodynamics: qubit provided by the state of trapped atoms coupled to high-finesse cavities.
\item[$\bullet$] Molecular magnet: qubit given by spin states \cite{leuenberger-nature-01}.
\item[$\bullet$] Electrons-on-helium quantum computers: qubit is the electron spin.
\item[$\bullet$] Linear optical quantum computer: qubits realized by processing states of different modes of light through linear elements \textit{e.g.~} mirrors, beam splitters and phase shifters \cite{knill-nature-01}.
\item[$\bullet$] Bose-Einstein condensate-based quantum computer.
\end{itemize}
In this thesis the physical realization of the quantum information processor is achieved utilizing nuclear spins of a molecule on an NMR hardware.
\section{Basics of NMR Spectroscopy}\label{nmrbasics}
The magnetic properties of an atomic nucleus forms the basis of NMR spectroscopy. A nuclear spin as well as magnetic moment are quantized \textit{i.e.~} they exhibit discrete values when measured. The total spin angular momentum of the nucleus is the vector sum of all the spin and orbital angular momentum, of the constituting nucleons, and in general complex to compute \cite{krane-book-88}. Similar to the electronic configuration used to distribute electrons in atomic shells one can utilize the nuclear shell model to arrange the nucleons (\textit{i.e.~} protons and neutrons) in nuclear energy levels \cite{gapon-dn-32}. From the knowledge of the nucleon arrangement in the nucleus one can obtain the nuclear spin angular momentum $ \hbar \mathbf{I} $. In addition to nuclear spin, the nuclei also possess magnetic moment $\mathbf{\mu}$ which is related to the spin angular momentum as
\begin{equation}
\mathbf{\mu} = \gamma \hbar \mathbf{I}
\end{equation}
where $ \gamma $ is the \textit{nuclear gyromagnetic ratio} and is a characteristic feature of the nucleus. For an intuitive picture, such nuclei can be considered as tiny magnets and interact with external magnetic fields in quite a similar way. The Hamiltonian for a magnetic moment $ \mathbf{\mu} $ placed in a magnetic field applied in $z$-direction, \textit{i.e.~} $ \mathbf{B}=B_o\hat{z} $ can be written as
\begin{equation}\label{hamil_Ch1}
\mathit{H}=-\mathbf{\mu}.\mathbf{B}=-\gamma \hbar \mathbf{I}.\mathbf{B}=-\gamma \hbar B_o I_z= -\hbar \omega_L I_z
\end{equation}
$I_z$ being the $z$-component of the spin angular momentum. The nuclear spin experiences a torque due to the interactions of magnetic moment with the $ B_o $ and dictates the precession of the spin angular momentum, about $\hat{z}$, at a characteristic \textit{Larmor frequency} \textit{i.e.~} $ \omega_L = \gamma B_o $. z-direction is defined by $ \mathbf{B} $ and all the operators acts on the vector space spanned by $ \vert m \rangle $ where $m=-I,\;-I+1,...,0,...,\;I-1,\;I$ is the magnetic spin quantum number. The Cartesian components of the spin angular momentum in the transverse direction \textit{i.e.~} $\langle I_x \rangle$ and $\langle I_y \rangle$ exhibit oscillatory motion with frequency $\omega_L$ while longitudinal component, \textit{i.e.~} $\langle I_z \rangle$, stays stationary \cite{oliveira-book-07}. In this sense the nuclear magnetic moment is analogous to the classical magnetic dipole. See Fig.(\ref{lar}) for the analogy \cite{oliveira-book-07}.
\begin{figure}
\caption{The precession of a top spinning in the gravitational field analogous to the nuclear spin precession in a magnetic field.}
\label{lar}
\end{figure}
The energy eigenvalues of the Hamiltonian, Eq.(\ref{hamil_Ch1}), acting on the state space $\vert m \rangle$ can be computed as
\begin{equation}\label{zeeman}
\mathit{H}\vert m \rangle = E_m \vert m \rangle=-m\hbar \omega \vert m \rangle
\end{equation}
One may observe from the quantum formalism of angular momentum and above energy eigen equation that for a nucleus, with $I \neq 0$, the nuclear energy spectrum is composed of $(2I+1)$ equally spaced energy levels and the energy gap between two consecutive levels is $\hbar\omega$. The lowest energy level is given by $m=I$ while the highest is given by $m=-I$. The population distribution for an ensemble of identical nuclear spin at high temperature is governed by the Boltzmann distribution \cite{reif-book-65}. For example, for $I=\frac{1}{2}$ there are two energy levels correspond to $m=\pm\frac{1}{2}$ and the population of the energy levels characterized by $m=\frac{1}{2}$ and $m=-\frac{1}{2}$, denoted by $n_+$ and $n_-$ respectively, is governed by Boltzmann factor as
\begin{equation}
\frac{n_-}{n_+}=e^{-(E_{-I}-E_{+I})/k_BT}=e^{-\hbar\omega_L/k_BT}
\end{equation}
$k_B$ is the Boltzmann constant and $T$ is the absolute temperature of the spin ensemble. For $^1H$ ensemble placed inside a magnetic field of 14.1 Tesla, the Boltzmann factor is $\approx 10^{-5}$ which implies only one in $10^5$ spins is aligned in the external field direction which makes the ensemble weakly paramagnetic in nature, (Fig.(\ref{z_mag})). Nevertheless, this slight difference between the populations of the energy levels gives rise to a total magnetization in $z$-direction given by
\begin{equation}
M_z=\frac{\mu_o\gamma^2\hbar^2B_o}{4k_BT}
\end{equation}
\noindent where $\mu_o$ is the magnetic susceptibility and not to be confused with magnetic moment $\mu$.
\begin{figure}
\caption{In an external magnetic field, more spins will be precessing around the direction parallel to the field than against it. This imbalance creates a macroscopic magnetization which points in the direction of the field.}
\label{z_mag}
\end{figure}
\subsection{Interaction of the Nuclear Spin with Radio Frequency: The Nuclear Magnetic Resonance Phenomenon }\label{rfint}
The undisturbed spin ensemble in the presence of an external static magnetic field will stay in thermal equilibrium with population of various energy levels following Boltzmann distribution. However, transition between the energy eigenstates of the Hamiltonian defined by Eq.(\ref{hamil_Ch1}) can be induced using an oscillating magnetic field of appropriate Larmor frequency. For nuclear spins in a static magnetic field of few Tesla, the Larmor frequency is of the order of MHz, and hence to induce the transition between various energy levels a radio frequency (RF) field is required. In order to excite the population one can consider the transverse RF magnetic field $\mathbf{B}_1(t)$ perpendicular to the static magnetic field $\mathbf{B}$ as
\begin{equation}
\mathbf{B}_1(t)=2B_1cos(\Omega t+\phi)\hat{i}
\end{equation}
where $\Omega$ and $\phi$ are the frequency and phase of the RF field and $\hat{i}$ is the unit vector in $x$-direction. The interaction Hamiltonian between nuclear spin and the RF field can be written as
\begin{equation}\label{rf_hamil}
\mathit{H}_{RF}=-\mathbf{\mu}.\mathbf{B}_1(t)=-\gamma\hbar I_x [2B_1cos(\Omega t+\phi)]
\end{equation}
The Hamiltonian due to RF field can be considered as a perturbation to the Zeeman Hamiltonian, Eq.(\ref{hamil_Ch1}), as the magnitude of $\mathbf{B}_1(t)$ field is a few Gauss as compared to the $\mathbf{B}$ field magnitude. Under this consideration the effect of $\mathit{H}_{RF}$ can be investigated using time-dependent perturbation theory \cite{sakurai-book-94}. To understand the key features of the results of time-dependent perturbation theory one can assume that the linearly oscillating magnetic field $\mathbf{B}_1(t)$ is composed of two circularly polarized fields, with same amplitude and phase as that of $\mathbf{B}_1(t)$, precessing about $z$-axis in opposite direction \textit{i.e.~}
\begin{eqnarray}
\mathbf{B}_1(t) &=& \mathbf{B}^+_1(t)+\mathbf{B}^-_1(t) \nonumber\\
\mathbf{B}^+_1(t) &=& B_1[cos(\Omega t+\phi)\hat{i}+sin(\Omega t+\phi)\hat{j}] \nonumber\\
\mathbf{B}^-_1(t) &=& B_1[cos(\Omega t+\phi)\hat{i}-sin(\Omega t+\phi)\hat{j}]
\end{eqnarray}
For $\Omega=\omega$, \textit{i.e.~} on resonance, the $\mathbf{B}^-_1(t)$ component rotates around $z$-axis in sync with the nuclear spin. In a coordinate system rotating with angular velocity $\mathbf{\Omega}=-\Omega \hat{k}$, \textit{i.e.~} \textit{rotating frame} the component $\mathbf{B}^-_1(t)$ will appear stationary to the nuclear spins and spins experience a torque. By controlling the RF exposure time to the spins they can be excited from low energy eigenstate to higher energy eigenstates and this forms the basis of the NMR signal \cite{ernst-book-90}.
\section{NMR Quantum Information Processing}\label{NMRQIP}
At the turn of twentieth century, NMR was proposed as a potential platform for the physical realization of quantum information processor \cite{cory-pnas-97,chuang-sc-97,vandersypen-rmp-05}. The NMR quantum information processor utilizes the spin ensemble to encode and process quantum information and the results of the computation are obtained via expectation values of the observables. Since then NMR has been a useful testbed for the experimental demonstrations of quantum algorithms as well as quantum information processing. NMR has been utilized for the experimental demonstration of Grover's search algorithm \cite{jones-nature-98}, Shor's algorithm \cite{vandersypen-nature-01}, Deutsch-Jozsa algorithm utilizing non-commuting selective pulses \cite{dorai-pra-00}, Order-Finding algorithm \cite{vandersypen-prl-00}, adiabatic quantum-optimization algorithm \cite{steffen-prl-03} and many more \cite{lu-bookchapter-16}.\\
The following subsections reviews the capabilities of NMR as demanded by the DiVincenzo criterion \cite{DiVincenzo-fdp-00} discussed in Sec-(\ref{DiVincenzo_criterion}).
\subsection{Nuclear Spins as Qubits}
As discussed earlier, the atomic nuclei with non vanishing nuclear spin placed in a static magnetic field exhibit nuclear Zeeman effect, see Eq.(\ref{zeeman}) \textit{i.e.~} degeneracy in various energy eigenstates of the spin Hamiltonian is lifted in the presence of static magnetic field. This generates an energy spectrum of $(2I+1)$ levels. Minimum possible nuclear spin angular momentum \textit{i.e.~} $I$ is one-half. Examples of spin-1/2 nuclei are $^{1}H$, $^{13}C$, $^{15}N$, $^{19}F$ and $^{31}P$. All such spin-1/2 nuclei are two-level quantum systems and can encode the quantum information as a qubit. Although nuclear spins with spin $>$ 1/2 were utilized for NMR quantum information processing but in general controlling higher-dimensional quantum systems in liquid state NMR is much more complicated due to their very low coherence times. Nevertheless, NMR qubits have been utilized extensively in the physical realization of quantum information \cite{cory-pnas-97,chuang-sc-97,jones-nature-98,vandersypen-rmp-05,dorai-pra-00,vandersypen-prl-00,vandersypen-nature-01,steffen-prl-03,lu-bookchapter-16}. \\
\begin{figure}
\caption{Energy level diagram of a single spin-1/2 nucleus as a two-level quantum system.}
\label{NMR_qubit}
\end{figure}
The energy eigenvalues of the spin Hamiltonian, given by Eq.(\ref{hamil_Ch1}), are $-\hbar\omega/2$ and $\hbar\omega/2$ and corresponding eigenvectors are $\vert I_z : (m=+\frac{1}{2}) \rangle$ and $\vert I_z : (m=-\frac{1}{2}) \rangle$ respectively. The eigenvectors, $\vert I_z : (m=+\frac{1}{2}) \rangle$ and $\vert I_z : (m=-\frac{1}{2}) \rangle$, of operator $\sigma_z=2I_z$ serve as computational basis and usually denoted by $\vert 0 \rangle$ and $\vert 1 \rangle$ respectively. See Fig.(\ref{NMR_qubit}) for a schematic of NMR qubit. Further, there can be more than one spin-1/2 nuclei in a molecule. Such spins can interact via direct \textit{magnetic dipole-dipole} interaction or indirectly via covalent bonds termed as \textit{scalar-coupling} or \textit{J-coupling} interactions. Dipole-dipole interactions are direct interactions of the nuclear magnetism and need no medium while J-coupling is through the interaction of nucleus with the electronic environment of the bonded electron cloud to the other nuclei \cite{ernst-book-90}. The Hamiltonian for $n$ such weakly interacting spin-1/2 systems is given by
\begin{equation}\label{NMR_hamil_Ch1}
\mathit{H}=-\sum_{i=1}^{n}\hbar \omega^i I^i_{z}+\sum_{\underset{i<j}{i,j=1}}^{n}2\pi\hbar J_{ij}I^i_{z}I^j_{z}
\end{equation}
where $J_{ij}$ is the scalar coupling constant between $i^{\rm th}$ and $j^{\rm th}$ spins. Usually the NMR Hamiltonian, Eq.(\ref{NMR_hamil_Ch1}), is written in frequency units by letting Plank constant $h=1$. Intuitively one can interpret the second term of the Hamiltonian, Eq.(\ref{NMR_hamil_Ch1}), as additional magnetic field created by surrounding spins which further shifts the energy levels of $i^{\rm th}$ spin by $-\frac{J_{ij}}{2}$ if $j^{\rm th}$ spin is in $\vert 0 \rangle$ state or by $\frac{J_{ij}}{2}$ if $j^{\rm th}$ spin is in $\vert 1 \rangle$ state. Fig.(\ref{two_coupled_spins_ELD}) depicts the modification of energy levels in the presence of scalar J-coupling between two spins.
\begin{figure}
\caption{Energy level diagram for two $J$-coupled spins. Dashed lines are the energy levels in the absence of $J$-coupling while solid lines are the energy levels modified by $J$-coupling.}
\label{two_coupled_spins_ELD}
\end{figure}
In this scenario, each spin transition splits up into two transitions at frequencies $\omega^i \pm \frac{J}{2}$ and results in doublet in NMR frequency spectrum. For a given system the exact energy level diagram can be obtained by diagonalizing the Hamiltonian given in Eq.(\ref{NMR_hamil_Ch1}).
\subsection{Ensemble State Initialization}\label{NMR_ensemble_state}
The next requirement of a quantum information processor is to initialize the quantum register in a fiducial state \textit{i.e.~} a pure state. NMR deals with a large ensemble and inherently the ensemble is in a mixed state. Although mixed states are inadequate for QIP, elegant procedures were devised independently by Cory \textit{et al.} \cite{cory-pnas-97} and Chuang \textit{et al.} \cite{chuang-sc-97} whereby they have demonstrated that the spin magnetization can be manipulated to prepare spin ensemble in an effective pure state termed as \textit{pseudo pure state} (PPS). The motivation behind such a construct is that in NMR, one can interact with the deviation part of the ensemble density operator by means of RF fields. So once such deviation density part of the ensemble is initialized similar to the deviation part of pure state $\vert\psi\rangle\langle\psi\vert$ then such an PPS ensemble can mimic the pure state evolution under unitary transformations achieved via RF fields.\\
As discussed earlier in Sec-(\ref{nmrbasics}), the NMR spin ensemble follows Boltzmann distribution law and thermal equilibrium state of the spin ensemble at temperature $T$ in the presence of magnetic field $B$ can be written as follows
\begin{equation}
\rho_{th}=\frac{e^{-\textit{H}/k_BT}}{\sum_m e^{-E_m/k_BT}}
\end{equation}
The term in the denominator, \textit{i.e.~} $Z=\sum_m e^{-E_m/k_BT}$ is the spin ensemble partition function. For the Zeeman Hamiltonian Eq.(\ref{zeeman}) and basis formed by eigenstates of $I_z$ the diagonal entries, which are proportional to the energy level population, of the thermal state can be simplified as
\begin{equation}
[ \rho_{th} ]_{mm}=\frac{e^{m\hbar\omega/k_BT}}{\sum_{s=-I}^{I} e^{s\hbar\omega/k_BT}}
\end{equation}
Under high temperature limit \textit{i.e.~} $\hbar\omega << k_BT$ following approximation can be used to simplify the expression of $\rho_{th}$
\begin{eqnarray}
e^{m\Delta} &\approx& 1+m\Delta \nonumber\\
\mathrm{and}\;\;\; \sum_{s=-I}^{I} e^{s\Delta} &\approx& 2I+1
\end{eqnarray}
where $\Delta=\frac{\hbar\omega}{k_BT}<<1$ is a measure of thermal magnetization of spin ensemble at temperature $T$ in the presence of magnetic field $B$. So in high temperature limit the thermal equilibrium state can be recast as
\begin{equation}\label{nmrthrm}
\rho_{th}=\frac{1}{2I+1}\mathbb{I}+\frac{\Delta}{2I+1}I_z
\end{equation}
One can observe that first the term on the right hand side of Eq.(\ref{nmrthrm}) is the uniform background represented by identity operator $\mathbb{I}$ and only a tiny part ($\Delta\approx 10^{-5}$) is in the state having deviation part $I_z$. Similarly the NMR ensemble can be initialized in PPS of form Eq.(\ref{nmrthrm}) and the state can be put in the form
\begin{equation}
\rho_{\mathrm{PPS}} = \frac{(1-\Delta)}{2^n}+\frac{\Delta}{2^n} \vert\psi\rangle\langle\psi\vert
\end{equation}
There are a number of techniques to prepare PPS in NMR \textit{e.g.~} temporal averaging \cite{knill-pra-98}, spatial averaging \cite{cory-physD-98,oliveira-book-07}, logical labeling \cite{chuang-sc-97}, state initialization utilizing long-lived singlet states \cite{roy-pra-10} and NMR line-selective pulses\cite{peng-cpl-01}. Generally such methods of PPS preparation suffer magnetization loss due to non-unitary evolution achieved by gradient pulses and remedies have been proposed to circumvent such difficulties \cite{kawamura-ijqc-04}. Nevertheless, it is well established that the NMR ensemble can be initialized in the PPS which mimics the pure state behavior and can be used for QCQI \cite{cory-pnas-97,chuang-sc-97,vandersypen-rmp-05}. The next subsection details the type of evolution feasible with NMR and methods to implement unitary operation utilizing RF fields.
\subsection{NMR Unitary Gate Implementation}\label{nmrgates}
In the computational basis the three spin angular momentum operators for spin 1/2 , in $\hbar$ units, can be written as
\begin{equation}
I_x=\frac{\sigma_x}{2}=\frac{1}{2}
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\;\;\;\;
I_y=\frac{\sigma_y}{2}=\frac{1}{2}
\begin{bmatrix}
0 & -i \\
i & 0
\end{bmatrix}
\;\;\;\;
I_z=\frac{\sigma_z}{2}=\frac{1}{2}
\begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
\end{equation}
where $\sigma$'z are the Pauli spin operators. From Eq.(\ref{nmrthrm}), one can write the deviation density matrix as
\begin{equation}\label{deviation_rho}
\Delta \rho_{th}=\frac{\hbar \omega}{4k_BT}\begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
\end{equation}
As discussed in Sec.(\ref{rfint}), the interaction of RF fields can be understood utilizing rotating frame considerations. The lab frame density operator, $\rho^{\mathrm{lab}}$ can be transformed to rotating frame density operator using rotation operator $e^{-i\Omega t I_z}$ and can be written as
\begin{equation}\label{rottherm}
\rho^{\mathrm{rot}}= e^{-i\Omega t I_z}.\rho^{\mathrm{lab}}.e^{i\Omega t I_z}
\end{equation}
The deviation density operator $\Delta \rho_{th}$ is invariant under the above transformation and the superscript ``rot'' can be dropped for convenience. The total Hamiltonian for spin-1/2 ensemble in the presence of magnetic field $\mathbf{B}$ being acted upon RF field $\mathbf{B_1}$ is the sum of Zeeman Hamiltonian in Eq.(\ref{hamil_Ch1}) and RF Hamiltonian in Eq.(\ref{rf_hamil}) which can be further transformed to rotating frame using the rotation operator. The resulting effective Hamiltonian in rotating frame can be computed as
\begin{equation}\label{H_eff}
\mathit{H}_{\mathrm{eff}}=-\hbar(\omega-\Omega)I_z-\hbar\omega_1I_x
\end{equation}
The striking feature of the above rotating frame Hamiltonian is its time independence. For the case when RF frequency, (also termed as nutation frequency), $\omega_1>>(\omega-\Omega)$, rotating frame Hamiltonian can be approximated as $H_{\mathrm{eff}}\approx -\hbar\omega_1I_x$. This approximation becomes true for the on resonance excitation \textit{i.e.~} for $\omega=\Omega$. This approximation becomes particularly suitable for small resonance offsets, \textit{i.e.~} $\vert\omega-\Omega\vert\approx 0$, and strong RF pulse with irradiation time say $t_p$. One can write explicitly the evolution operator for the RF pulse as
\begin{equation}
U_p=e^{-i\mathit{H}_{\mathrm{eff}}t_p/\hbar}=e^{i\omega_1 t_p I_x}=R_x(-\theta_p)
\end{equation}
Here $R_x(-\theta_p)$ is the rotation operator, in rotating frame, about $x$-axis through an angle $-\theta_p$ dictated by RF irradiation time $t_p$. One can write explicit forms of rotation operators achievable in NMR as
\begin{equation}
R_x(\theta_p)=
\begin{bmatrix}
cos\left( \frac{\theta_p}{2}\right) & -i\;sin\left( \frac{\theta_p}{2}\right) \\
-i\;sin\left( \frac{\theta_p}{2}\right) & cos\left( \frac{\theta_p}{2}\right)
\end{bmatrix}
\end{equation}
\begin{equation}
R_y(\theta_p)=
\begin{bmatrix}
cos\left( \frac{\theta_p}{2}\right) & sin\left(\frac{\theta_p}{2} \right) \\
-sin\left( \frac{\theta_p}{2}\right) & cos\left( \frac{\theta_p}{2}\right)
\end{bmatrix}
\end{equation}
\begin{equation}
R_{\phi_p}(\theta_p)=
\begin{bmatrix}
cos\left( \frac{\theta_p}{2}\right) & -i\;sin\left( \frac{\theta_p}{2}\right)e^{-i\phi_p} \\
-i\;sin\left( \frac{\theta_p}{2}\right)e^{i\phi_p} & cos\left( \frac{\theta_p}{2}\right)
\end{bmatrix}
\end{equation}
The last rotation operator is for RF pulse having a phase $\phi_p$ with $x$-axis in the rotating frame. Using above formulated rotation operators one can compute the resulting state of the ensemble after the action of $\mathit{H}_{\mathrm{eff}}$, Eq.(\ref{H_eff}) on deviation density operator, Eq(\ref{deviation_rho}). Considering $\omega_1 t_p=\theta_p=\frac{\pi}{2}$ it can be shown that
\begin{equation}
\Delta\rho(t_p)=R_x\left(-\frac{\pi}{2} \right).\Delta\rho_{th}.R_x\left(\frac{\pi}{2} \right)=\frac{\hbar\omega}{2k_BT}I_y
\end{equation}
A semi-classical description gives an intuitive picture that the initial deviation density operator proportional to $I_z$ evolve to a density operator proportional to $I_y$ under a rotation by an angle $-\pi/2$ about $x$-axis in rotating frame which strikingly appears to be a classical behavior!\\
Its interesting to note that a $R_y(\pi/2)$ achieves the effect of Hadamard(H)-gate while the NOT(X)-gate can be achieved by $R_y(\pi)$ rotation. Similarly a $\mathrm{CNOT}$ gate can be achieved by exploiting the scalar $J$-couplings of the spins and a typical NMR pulse sequence of $\mathrm{CNOT}$ gate, for two coupled spins, is as follows
\begin{equation}
U_{\mathrm{CNOT}}=R^1_z\left( -\frac{\pi}{2} \right)R^2_x\left(\frac{\pi}{2} \right)R^2_y\left(- \frac{\pi}{2} \right)\frac{1}{2J}R^2_y\left( \frac{\pi}{2} \right)
\end{equation}
Here the time period $1/2J$ is the free evolution for which spin system evolve under NMR weak field Hamiltonian, Eq.(\ref{NMR_hamil_Ch1}), to effectively achieve the non-local operation of $\mathrm{CNOT}$ gate. Superscript on various rotations denote spin label. Also the $z$-rotation can be achieved by cascading three $x$, $y$ rotations. In nutshell the NMR technique is equipped to achieve, in principle, any arbitrary unitary operator.
\subsection{Measurements in NMR and State Tompgraphy}
As mentioned earlier, NMR generally deals with the ensemble of spin-1/2 nuclei and a typical NMR liquid state sample, having volume $\sim$ 500-600 $\mu l$, contains $\sim 10^{18}$ spins. As governed by the Boltzmann distribution, these spins generate a bulk magnetization in the presence of an external magnetic field, Fig-(\ref{z_mag}). External RF field can be used to manipulate this bulk magnetization. The net magnetization in the thermal equilibrium state, \textit{i.e.~} $\rho_{th}$, is along $z$-direction and can be brought in $xy$-plane by applying RF field of appropriate duration, using a rotation operator $R_x (-\pi/2)$.
\begin{figure}
\caption{Precession of bulk magnetization in the presence of an external static magnetic field induces current in the pick-up coils which further amplified and stored as a time domain signal termed as free induction decay (FID) \cite{fid-online-12}
\label{nmrsignal}
\end{figure}
In the transverse plane, the net magnetization undergoes Larmor precession about the $-z$-direction and can be detected by induction coils. Changing magnetic flux induces an electromotive force in the pick-up coils which in turn produces a detectable current. This induced current is then digitized and stored as time-domain NMR signal and termed as free induction decay (FID). FID typically has an oscillatory decaying nature due to various NMR relaxation processes and one can obtain the frequency-domain NMR signal by performing discrete Fourier transform (FT) on the digitized time-domain signal. Such processing in NMR results in \textit{Lorentzian} peaks correspond to transitions, (Fig-(\ref{two_coupled_spins_ELD})), between various energy eigenstate of NMR Hamiltonian, (Eq.(\ref{hamil_Ch1})). Schematic of NMR signal acquisition and processing is depicted in the Fig.(\ref{nmrsignal}).\\
The normalized intensities/amplitudes of NMR peaks are proportional to the respective spin ensemble magnetization which in turn is proportional to the expectation value of operator $I_z$ in the state of spin ensemble \cite{ernst-book-90} \textit{i.e.~} $\langle I_z \rangle _{\rho}$. Its worth noting here that NMR enables the measurement of ensemble average in a single experiment as the net effect of the NMR measurement is equivalent of measuring, \textit{e.g.~} $I_z$, on individual spins one-by-one and take the average.\\
Any single-qubit density matrix, in the computaional basis, can be brought into the form \cite{sakurai-book-94}
\begin{equation}\label{1Qdm}
\rho=\frac{\mathbb{I}}{2}+a\sigma_x+b\sigma_y+c\sigma_z
\end{equation}
which can further be written as
\begin{equation}
\rho=
\begin{bmatrix}
\frac{1}{2}-c & a-i\;b \\
a+i\;b & \frac{1}{2}+c
\end{bmatrix}
\end{equation}
One may observe that $\langle \sigma_z \rangle _{\rho}=\mathrm{Tr}(\rho\sigma_z)=c$. Further, with an appropriate choice of rotation operators one can measure the unknown parameters $a$ and $b$ as well and hence can reconstruct the density operator $\rho$ in Eq.(\ref{1Qdm}). The process of reconstructing the density operator from several experimental settings is known as quantum state tomography (QST) and there have been numerous studies on developing schemes for QST \cite{long-job-01,leskowitz-pra-04}. In this thesis, methods have been developed for accurately measuring the expectation values of two- and three-qubit Pauli operators in a given ensemble state which can be generalized to higher-dimensional Hilbert spaces \cite{singh-pra-16,singh-pra-18,singh-qip-18}.\\
Typical hallmark of QCQI, the projective measurements, generally is not possible in NMR, although there have been few experimental studies reporting projective measurements in NMR \cite{lee-apl-06,khitrin-qip-11} utilizing non-unitary evolutions by means of gradient pulses.
\section{Quantum Entanglement}\label{entanglementtheory}
Quantum entanglement first described by Erwin Schr{\"o}dinger \cite{schrodinger-Naturwissenschaften-35,schrodinger-mpc-35} in 1935. Quantum entanglement \cite{schrodinger-mpc-35} is a counter-intuitive feature exhibited by quantum particles which has no analog in classical mechanics. Quantum entanglement arrises when for a composite quantum system we are not able to describe the state of the quantum system in terms of quantum states of the parts. It has been shown \cite{aditi-cs-17} that quantum entanglement is a key resource to achieve computational speedup in quantum information processing (QIP) \cite{horodecki-rmp-09} and for quantum communication related tasks \cite{sibasish-njp-02,aditi-pra-09, pankaj-epd-15}. Being a fragile resource, prone to decoherence \cite{gedik-pla-06}, there have been proposals and demonstrations to store and protect entanglement \cite{mahesh-pra-11,aditi-pra-15} as well as environment-assisted enhancement \cite{archan-pra-06} of the entanglement.\\
Entanglement detection and characterization is of utmost importance for the physical realization of quantum information processors \cite{guhne-pr-09}. There have been a large number of measures proposed for the detection of quantum correlations \cite{aditi-pra-07} and in particular quantum entanglement \cite{guhne-pr-09}. In recent years, enormous experimental efforts have gone in the creation of entanglement. Typically in such an experimental scenario one would always be interested in queries like does entanglement actually get created in the experiment and can one detect and quantify the entanglement. In general, these questions are difficult to answer. There are many proposals to address such queries such as positivity under partial transposition (PPT)\cite{horodecki-pla-96} criterion, permutation based measures of quantum correlations \cite{arul-pra-16}, correlations in successive spin measurements \cite{sibasish-ijqi-07}, entanglement measures based on no-local-cloning and deleting\cite{aditi-pra-04} and isotropic spin lattice entanglement characterization \cite{aditi-prl-13}. Entanglement detection utilizing entanglement witnesses \cite{archan-qip-13,archan-pra-14a} is also a well developed field where such detections are explored from teleportation capabilities \cite{archan-prl-11}. Tripartite quantum states were characterized \cite{aditi-pra-12} using monogamy scores as well as mutual information in permutation symmetric states \cite{arul-pre-18}. Universal bipartite entanglement detection using two copies \cite{sibasish-arxiv-18} as well as extent of entanglement by sequential observers \cite{aditi-pra-18a} have recently been explored in a measurement-device-independent way \cite{sibasish-pra-17}.\\
Further, some of the experiments create entanglement between more than two subsystems and there are different classes \cite{dur-pra-00, verstraete-pra-02} of entanglement that exist in such cases. So entanglement characterization should be capable of distinguishing different classes. It should be noted that entanglement characterization is much more challenging rather than mere detection \cite{guhne-pr-09}. The situation is even more complex \cite{pankaj-epd-18a} in case of mixed states \cite{aditi-jmo-03} where geometric measures \cite{aditi-pra-16} have been resorted to for quantification of quantum correlations in multipartite and mutidimensional \cite{pankaj-pra-17} cases.\\
Entanglement witnesses \cite{lewenstein-pra-00,bourennane-prl-04} and approximation of positive maps \cite{rahimi-jpamg-06, rahimi-pra-07} proved to be experiment friendly but lack generalization, since most of the experiments focus on creation of a specific entangled state and witness-based detection protocols usually require the state information beforehand. In this thesis, the goals are to explore the entanglement detection as well as characterization protocols and experimentally implement them in a \textit{state-independent} manner using nuclear magnetic resonance (NMR). NMR has been proposed as a promising candidate for realizing quantum processors \cite{jones-jcp-98,laflamme-qic-02}. NMR has been the testbed for the demonstration of the Deutsch$-$Jozsa algorithm \cite{anil-pra-01, mahesh-jmr-01}, quantum No-hiding theorem \cite{anil-prl-11} and parallel search algorithm \cite{rangeet+anil-pra-05} as well as of foundational aspects such as delayed choice experiments \cite{mahesh-pra-12} and querying Franck-Condon factor \cite{mahesh-pra-14}. Control of 5 to 8 qubits for quantum information processing was achieved \cite{rangeet-jmr-04,rangeet-aipcp-06} and bench-marking of quantum controls on a 12 qubit quantum processor was demonstrated \cite{mahesh-prl-06} using NMR systems. Highly accurate control via radio frequency pulses made initialization of NMR system \cite{cory-physD-98,anil-pra-00} and read out using quantum state tomography \cite{leskowitz-pra-04,mahesh-pra-13a} accessible, in contrast to other hardware.\\
As discussed above, quantum entanglement is a striking feature of quantum mechanical description of nature and was quickly followed by the demand of a physical and more intuitive interpretation by Einstein-Podolsky-Rosen \cite{EPR-pr-35}. Quantum entanglement proved to be a physical resource\cite{nielsen-book-02} which can be utilized to accomplish quantum computational tasks \cite{horodecki-rmp-09}, which are impossible to perform using classical resources.
\subsection{Bipartite Entanglement}
Consider a quantum system consisting of two subsystems $A$ and $B$. Quantum states of $A$ and $B$ can be defined in respective Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$ having dimension $d_A$ and $d_B$ respectively. The states of the composite system are defined by vectors in tensor-product of the Hilbert spaces $\mathcal{H}_A \otimes \mathcal{H}_B$ having dimension $d_Ad_B$. Any vector in the joint Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$ can be expressed as
\begin{equation}
\vert \psi \rangle = \sum_{i,j=1}^{d_A,d_B}{c_{ij} \vert a_i \rangle \otimes \vert b_i \rangle } \;\; \in \mathcal{H}_A \otimes \mathcal{H}_B
\end{equation}
where $\lbrace \vert a_i \rangle \rbrace $ is basis in $\mathcal{H}_A$, $\lbrace \vert b_i \rangle \rbrace $ is basis in $\mathcal{H}_B$ while $ c_{ij} \in \mathbb{C} $ and normalization of $ \vert \psi \rangle $ requires $ \sum_{i,j=1}^{d_A,d_B}{\vert c_{ij} \vert ^2=1} $. Now if $ \vert \psi \rangle \in \mathcal{H} $ represents any general pure state of the composite system $AB$ and one can express it as
\begin{equation}
\vert \psi \rangle = \vert \phi^A \rangle \otimes \vert \phi^B \rangle
\end{equation}
where states $\vert \phi^A \rangle \in \mathcal{H}_A$ and $\vert \phi^B \rangle \in \mathcal{H}_B$, then the state $\vert \psi \rangle $ is separable else it is entangled. Physically, the separable states are uncorrelated from statistics of measurement outcomes perspective. In a more general case when the system can be in any one of the states $ \vert \phi_i \rangle \in \mathcal{H}$ with probability $p_i$ then mixed state of the system can be expressed as
\begin{equation}
\rho=\sum_i{p_i \vert \phi_i \rangle \langle \phi_i \vert}
\end{equation}
with $ \sum_i{p_i}=1$ and $p_i\geq 0$. If the state of a composite system can expressed as a convex mixture, of the product states $\rho^A \otimes \rho^B$, as
\begin{equation}\label{mixedsep_ch1}
\rho=\sum_i{w_i\rho_i^A \otimes \rho_i^B}
\end{equation}
then $\rho$ is separable otherwise it is entangled. These definitions of bipartite entanglement can be generalized to multipartite cases as well.
\subsection{Entanglement Detection and Characterization}
We review here the commonly used entanglement detection criteria for bipartite systems.
\subsubsection{The Positive Under Partial Transposition (PPT) Criterion}
The density operator of a composite bipartite system can be expanded in a chosen product basis as
\begin{equation}
\rho=\sum_{i,j=1}^{N}\sum_{k,l=1}^{M} \rho_{ij,kl}\vert i \rangle \langle j \vert \otimes \vert k \rangle \langle l \vert
\end{equation}
here $N$ and $M$ are the dimensions of local Hilbert spaces of the bipartite state. Having defined the above decomposition, one can write the partial transposition of the density operator with respect to subsystem A as
\begin{equation}
\rho^{T_A}=\sum_{i,j=1}^{N}\sum_{k,l=1}^{M} \rho_{ji,kl}\vert i \rangle \langle j \vert \otimes \vert k \rangle \langle l \vert
\end{equation}
Partial transposition with respect to subsystem B \textit{i.e.~} $\rho^{T_B}$ can be obtained by interchanging the indices $k$ and $l$ instead $i$ and $j$. One may also use the fact that the usual transposition $\rho^{T}=(\rho^{T_A})^{T_B}=(\rho^{T_B})^{T_A}$ to obtain $\rho^{T_B}=(\rho^{T_A})^T$. Partial transposition depends upon the basis in which it is performed but the spectrum is independent of the basis which is also true for matrix transposition. The density matrix in a basis has positive partial transposition \textit{i.e.~} $\rho$ is PPT, if its partial transposed density matrix does not have negative eigenvalues and hence is positive semi-definite. If a density operator is not PPT then it is NPT. Based on PPT there are two strong conditions satisfied by separable states described as follows:\\
\noindent\textbf{PPT Criterion:} If $\rho$ is a bipartite separable state then $\rho$ is PPT.\\
\noindent PPT criterion has an intuitive description. As mentioned earlier, the information regarding the state of an entangled composite quantum system is stored in the joint state of the system rather than its parts. State of a bipartite separable composite system can be cast in the form of Eq.(\ref{mixedsep_ch1}). Hence and partial transposition operation will independently transpose the state of either subsystem $A$ or $B$ and always result in PPT $\rho$. Here the fact, that the transpose of a positive semi-definite density operator will also be a positive semi-definite, is used. Hence any state which is NPT is always entangled \cite{horodecki-pla-96} but a PPT state may be separable or entangled.\\
\noindent\textbf{Horodecki Theorem:} If $\rho$ is the density operator acting on a 2$\otimes$2 or 2$\otimes$3 Hilbert space, then $\rho^{T_A}\geq 0$ implies $\rho$ is separable.
\begin{equation}
\rho=\sum_{i,j=1}^{N}\sum_{k,l=1}^{M} \rho_{ij,kl}\vert i \rangle \langle j \vert \otimes \vert k \rangle \langle l \vert
\end{equation}
For higher dimensional Hilbert spaces, \textit{i.e.~} $>$ 2$\otimes$3, this may not be the case \cite{horodecki-pla-96}. There are entangled states which do not violate Horodecki's theorem \textit{i.e.~} PPT entangled states. Such PPT entangled states makes an important class of states falls in the class of bound entanglement, Sec-\ref{boundEnt}.
\subsubsection{The Computable Cross Norm or Realignment (CCNR) criterion}
The PPT criterion in not a necessary condition for the density matrices acting on Hilbert spaces having dimension greater than six. Although many stronger criteria were proposed but its worth mentioning the CCNR criterion \cite{rudolph-qip-05}. To describe the CCNR criterion the concept of Schmidt decomposition is utilized. For a density matrix $\rho$, the Schmidt decomposition can be written as
\begin{equation}\label{sch-decom}
\rho=\sum_{k} \lambda_k G^A_k\otimes G^B_k
\end{equation}
where $\lambda_k\geq0$ and $G^A_k$ and $G^B_k$ are orthonormal bases of the observable spaces $ H_A $ and $ H_B $. Such a basis consists of $d^2$ Hermitian observables satisfying Tr$(G^A_iG^A_j)$=Tr$(G^B_iG^B_j)$ =$\delta_{ij}$.\\
\noindent\textbf{CCNR Criterion:} If the state $\rho$ is separable, then the sum of all $\lambda_k$ in Eq.(\ref{sch-decom}) is smaller than 1 \textit{i.e.~}
\begin{equation}\label{ccnr}
\sum_k \lambda_k \leq 1
\end{equation}
Hence if $ \sum_k \lambda_k > 1 $ then $ \rho $ is entangled.
\subsubsection{The Positive Map Method}
PPT criterion is an example of positive but not completely positive (PNCP) maps. One may define PNCP as follows: Let the $\mathcal{B}(\mathcal{H}_i)$ be linear operators acting on the Hilbert spaces $\mathcal{H}_B$ and $\mathcal{H}_C$. A positive linear map, \textit{i.e.~} $\Lambda:\mathcal{B}(\mathcal{H}_B)\rightarrow \mathcal{B}(\mathcal{H}_C) $, will map the Hermitian operators onto the Hermitian operators and satisfies $\Lambda (X^{\dagger})=\Lambda (X)^{\dagger}$ and $ \Lambda(X) \geq 0 $ for $X\geq 0$. A map $\Lambda$ is completely positive (CP) if for an arbitrary Hilbert space $\mathcal{H}_A$, the map $\mathbb{I}_A\otimes \Lambda$ is also positive otherwise $\Lambda$ is PNCP. For example, transposition is PNCP map: as transpose of a positive operator is positive but partial transposition (equivalent to transposition after including extended operator space) may result in negative operator. Having defined PNCP maps, the separability criterion can be described as follows: For any separable state $\rho$ and any positive map $\Lambda$ following is always satisfied \cite{horodecki-pla-96}
\begin{equation}
\mathbb{I}_A\otimes \Lambda \geq 0
\end{equation}
Above condition is a sufficient criterion \textit{i.e.~} a state violating above criterion is entangled but a state which doesn't show violation may also be entangled.
\subsubsection{The Majorization Criterion}
For a general bipartite state $\rho$, one can obtain the reduced state of subsystem B by tracing out the state of subsystem A \textit{i.e.~} $\rho_B=\mathrm{Tr}_B(\rho)$. Let $\mathcal{P}=(p_1,\;p_2,...)$ and $\mathcal{Q}=(q_1,\;q_2,...)$ denote the sets of decreasingly ordered eigenvalues of $\rho$ and $\rho_B$ respectively. The majorization criterion states that if $\rho$ is separable then
\begin{equation}
\sum_{i=1}^k p_i \leq \sum_{i=1}^k q_i
\end{equation}
holds for all $k$ \cite{nielsen-prl-01}. For separable $\rho$ above condition also holds for the reduced state of subsystem A \textit{i.e.~} for $\rho_A=\mathrm{Tr}_A(\rho)$.
\subsection{Bound Entanglement}\label{boundEnt}
Generally maximally entangled two qubit states, \textit{i.e.~} singlet states, are needed to accomplish many tasks in quantum information theory \textit{e.g.~} teleportation, superdense coding and cryptography. But generally in an experiment, due to inadvertent noise, one ends up with mixed states. It is a practical question that how one can create singlet state from some given mixed states. This process of creating singlet or maximally entangled state from given mixed states is called \textit{entanglement distillation}. Entanglement distillation can be described as follows: Consider two parties Alice and Bob, sharing an arbitrary, but finite, number of copies of the entangled state $\rho$. Entanglement distillation is the process of transforming available quantum resources/states, by performing local operations and classical communications (LOCC), to a singlet state \textit{i.e.~}
\begin{equation}
\underbrace{\rho\otimes \rho\otimes ... \otimes \rho}_{k-\rm copies}\stackrel{\rm LOCC}{\longrightarrow} \frac{1}{\sqrt{2}}(\vert 00 \rangle - \vert 11 \rangle)
\end{equation}
If Alice and Bob can achieve the above task with $k$-copies of $\rho$ then $\rho$ is distillable else $\rho$ is \textit{bound entangled}. Although there is no protocol which ensures entanglement distillability but the sufficient conditions for undistillability \cite{horodecki-pra-99, tohya-prl-03} as well as distillability \cite{deutsch-prl-96, horodecki-prl-98} have already been proposed. A special kind of undistillable entanglement is PPT entanglement. It has been shown that \\
\textit{ If a bipartite state is PPT, then the state is undistillable. If a state violates the reduction criterion (\textit{e.g.~}, due to a violation of the majorization criterion) then the state is distillable.}\\
One of the first PPT entangled class of states was proposed in Ref. \cite{horodecki-pla-97} and later more classes were discovered \cite{bennet-prl-99, dagmar-pra-00, piani-pra-06, piani-pra-07}. Entangled states that are undistillable are called bound entangled states and PPT entangled states is the most important class of such states. Characterization of bound entanglement is an interesting as well as challenging task in entanglement theory.
\subsection{Entanglement Witnesses}
All of the entanglement detection criteria discussed above require knowledge of the density operator. However there is a sufficient entanglement criterion in terms of a measurable observable termed as \textit{Entanglement Witness} (EW) \cite{horodecki-pla-96,tehral-pla-00,lewenstein-pra-00,dagmar-jmo-02}.\\
An observable $\mathcal{W}$ is an \textit{Entanglement Witness} iff:
\begin{itemize}
\item $\mathrm{Tr}(\rho_s\mathcal{W})\geq 0$ for all separable states $\rho_s$ and
\item $\mathrm{Tr}(\rho_e\mathcal{W})< 0$ for at least one entangled state $\rho_e$
\end{itemize}
holds. Thus entanglement of $\rho_e$ is witnessed by measuring $\mathcal{W}$ and establishing $\mathrm{Tr}(\rho_e\mathcal{W})$
\linebreak
$<0$. It is worth mentioning here that constructing an EW is, in general, a difficult task. There may be the cases when a given EW unable to witness the entanglement. $\mathrm{Tr}(\rho\mathcal{W})< 0$ confirms the presence of entanglement but for the case when $\mathrm{Tr}(\rho\mathcal{W})\geq 0$, $\rho$ may be separable or entangled. EW is one of the most utilized concept for the entanglement detection in experiments. Following has been proved \cite{horodecki-pla-96} as a strong criterion for entanglement detection in experiments.\\
\textit{Completeness of Witnesses: For each entangled state $\rho_e$ there exists an entanglement witness detecting it}.\\
A few methods to construct an EW are:\\
\begin{itemize}
\item Consider an entangled NPT state $\rho_e$ whose partial transpose, \textit{i.e.~} $\rho_e^{T_A}$, has at least one negative eigenvalue $\lambda_{-}$ and let $\vert \eta \rangle $ be the corresponding eigenvector. It can be shown that
\begin{equation}
\mathcal{W}=\vert \eta \rangle \langle \eta \vert ^{T_A}
\end{equation}
can act as an EW for the detection of entanglement of $\rho_e$. It can be proved as Tr$(\rho_e\mathcal{W})$=Tr$(\rho_e \vert \eta \rangle \langle \eta \vert ^{T_A})$=Tr$(\rho_e^{T_A}\vert \eta \rangle \langle \eta \vert)$=$\lambda_{-}<0$ hence $\rho_e$ is entangled.
\item Consider a state $\rho_e$ violating CCNR criterion. Then by definition, there exists a Schmidt decomposition given by Eq.(\ref{sch-decom}) with $\lambda_k\geq 0$. In such cases, the EW can be formulated \cite{yu-prl-05,guhne-pra-06} as
\begin{equation}
\mathcal{W}=\mathbb{I}-\sum_{k} \lambda_k G^A_k\otimes G^B_k
\end{equation}
with $G^{A/B}_k$ are the local observables in Schmidt decomposition, (Eq. (\ref{sch-decom})). One can see that Tr$(\rho_r\mathcal{W})$=1-$\sum_{k} \lambda_k<0$ and hence detect the entanglement in $\rho_e$.
\item To construct the entanglement witnesses one can consider the states close to an entangled state, which must also be entangled depending upon their overlap with the original entangled state. For a pure entangled state $\vert \psi \rangle $ the projector based EW can be written as
\begin{equation}
\mathcal{W}=\alpha\mathbb{I}-\vert \psi \rangle \langle \psi \vert
\end{equation}
Motivation for the above construct is that the quantity $\mathrm{Tr}(\rho \vert \psi \rangle \langle \psi \vert) $ = $ \langle \psi \vert \rho \vert \psi \rangle $ is the fidelity of the state $ \vert \psi \rangle $ in the mixed state $ \rho $ and if this fidelity exceeds the threshold value $ \alpha $ then above EW $ \mathcal{W} $ detects the entanglement in $\rho$. $\alpha$ can be computed \cite{bourennane-prl-04} such that expectation value of $\mathcal{W}$ is non-negative for all the separable states and given as follows:
\begin{equation}
\alpha=\underset{\rho\;\rm{is\;separable}}{\mathrm{max}} \mathrm{Tr}(\rho \vert \psi \rangle \langle \psi \vert) = \underset{\vert \phi \rangle = \vert a \rangle \otimes \vert b \rangle}{\mathrm{max}} \vert \langle \psi \vert \phi \rangle \vert^2
\end{equation}
The fact that a linear function takes its maximum on a convex set in one of the extremal points has been used, and for the convex set of the separable states these extremal points are just the pure product states. It has been shown \cite{bourennane-prl-04} that the above maximum can be directly computed and is given by the square of the maximal Schmidt coefficient of $\vert \psi \rangle$.
\end{itemize}
\subsection{Entanglement Measures}
Above discussed methods enables the detection of entanglement in a given state however one may be interested in quantifying the entanglement in the state. In order to do so there exist a number of entanglement measures (entanglement monotone) \cite{horodecki-rmp-09}. Its worth mentioning the requirements of an entanglement measure (EM). First and basic requirement for EM is that it should quantify the entanglement present in a given state \cite{vedral-prl-97}:
\begin{itemize}
\item[(i)] An entanglement measure $E(\rho)$ should vanish for all separable states.
\item [(ii)] An entanglement measure should be invariant under local change of basis \textit{i.e.~} it should be invariant under local unitary transformation of form:
\begin{equation}
E(\rho)=E(U_A\otimes U_B \rho U_A^{\dagger}\otimes U_B^{\dagger}).
\end{equation}
\item[(iii)] Entanglement cannot be created or increased under LOCC so $E(\rho)$ should not increase under LOCC. If $\Lambda^{\rm LOCC}$ is positive map that can be implemented using only LOCC then
\begin{equation}\label{EM_3}
E(\Lambda^{\rm LOCC}(\rho)) \leq E(\rho)
\end{equation}
A stronger version of the above requirement is that $E(\rho)$ should not increase on an average under LOCC \textit{i.e.~} if LOCC operations maps $\rho$ to $\rho_k$ with probabilities $p_k$ then
\begin{equation}
\sum_kp_k E(\rho_k)\leq E(\rho)
\end{equation}
The monotonicity under LOCC in Eq. (\ref{EM_3}), implies invariance under local unitary transformations.
\item[(iv)] Entanglement decreases on mixing two or more states \textit{i.e.~}
\begin{equation}
E\left(\sum_k p_k \rho_k\right) \leq \sum_kp_k E(\rho_k)
\end{equation}
This condition requires that if one starts with an ensemble of states $\rho_k$, and loses the information about the single instance of $\rho_k$, then entanglement should decrease.
\item[(v)] For the case when one have access to two or more copies of the states then additivity of EM should obey
\begin{equation}
E(\rho^{\otimes n})=n E(\rho)
\end{equation}
Here Alice and Bob shares $n$-copies of the same state $\rho$. In case Alice and Bob share different states, say $\rho_A$ and $\rho_B$ then even a stronger requirement of additivity requirement can be written as
\begin{equation}
E(\rho_A\otimes\rho_B)=E(\rho_A)+E(\rho_B)
\end{equation}
Above additivity requirement is in general difficult to prove and satisfied by few EMs \cite{plenio-qic-07}.\\
\end{itemize}
Various EMs have been proposed which satisfy partially the above listed requirements. Commonly used EMs are entanglement cost \cite{bennett-pra-96a}, entanglement of formation \cite{wootters-prl-98,chen-prl-05}, concurrence \cite{wootters-prl-98}, Negativity \cite{zyczkowski-pra-98,vidal-pra-02}, relative entropy of entanglement \cite{vedral-pra-98} and $n$-tangle \cite{coffman-pra-00}. Many of these measures are used in this thesis and their details are given in the subsequent chapters where they have been introduced first.\\
Quantum discord (QD) captures the fact that even separable states can possess quantum correlations \cite{zurek-adp-00,ollivier-prl-02}. There have been intense theoretical and experimental advancements \cite{aditi-rpp-18, aditi-pre-16} utilizing QD to capture nonclassicality \cite{rahimi-pra-10} and quantum-to-classical transition \cite{sibasish-pra-10}. Dynamics of QD was studied \cite{mahesh-pra-12} and used as a quantifier of nonclassicality \cite{singh-pra-17}. Quantum correlation dynamics in a hybrid qubit-qutrirt system was explored \cite{gedik-pla-11} utilizing QD. The interplay between entanglement and nonclassicality was explored \cite{ivan-pra-11} in multimode radiation states. Invariant QD in hybrid qubit-qutrit quantum system and tradeoff with entanglement display interesting features \cite{gedik-ps-13}. This thesis also explores the detection of nonclassical correlations possessed by separable mixed states using positive maps and QD. Certain quantum states possess non-local nature of quantum correlations and violation \cite{aditi-pra-01} of a Bell type inequality \cite{ bell-ppf-64} may reveal such non-localities. It has been shown that W class of states possess stronger non-locality than GHZ class of states\cite{aditi-pra-03}. Such non-local correlations need to be investigated from \textit{ease of experimental implementation} as well as \textit{state independence} perspective.\\
\subsection{Entanglement in NMR}
As discussed in Sec-{\ref{NMR_ensemble_state}}, typically the state of NMR ensemble at room temperature remains in the vicinity of maximally mixed state and hence it is not possible to create a genuine entangled state of the nuclear spins in small thermally polarized molecules in liquid state \cite{braunstein-prl-99}. This has initiated a debate on the quantumness of the states in a typical NMRQC experiments which argued that all the states produced by NMR are classical. On the contrary any simulation of the dynamics of coupled nuclear spins using any classical model has been proved unsuccessful and it is conjectured that although the states in NMR may be in the vicinity of maximally mixed state but the dynamics is truly quantum mechanical \cite{schack-pra-99}. This can be well observed from the discussion in Sec-(\ref{nmrbasics}) that dynamics of NMR ensemble follows the laws of quantum evolution and as conjectured that the PPS perfectly mimics the behaviors of pure state and indeed generates the observable NMR signal. The the state of the sub-ensemble, \textit{e.g.~} in NMR, truly possesses all the quantum features like superposition and entanglement. So with this understanding ensemble can be prepared in any desired state and is generally termed as \textit{pseudo} to make a distinction from \textit{pure} state.
\section{Motivations and Organization of the Thesis}
This thesis focuses on the experimental creation and detection of different types of quantum correlations using nuclear spins and NMR hardware. Quantum entanglement, being the most important and counter-intuitive, is one of the main types of correlation considered in this thesis. One of the main goals of the studies undertaken in this thesis was to design experimental strategies to detect the entanglement in a \textit{state-independent} way that are low on experimental resources. Core of all the detection protocols is a novel method which enables the measurement of any observable with high accuracy. Although these methods have been implemented on NMR hardware but they were developed in a hardware-independent manner and hence can be utilized on other QCQI hardware. Experimental protocols have been successfully implemented to detect the entanglement of random two-qubit states utilizing semi definite programming to construct an entanglement witness and thereby detect the entanglement. This random measurement based scheme to detect entanglement is also extended to a bipartite hybrid qubit-qutrit quantum system. It is shown via simulations that a two parameter class of qubit-qutrit entangled states get detected using only four local observables. Further, schemes for the experimental detection as well as classification of generic and general three-qubit pure states have also been devised and implemented successfully. Protocol to detect and classify three-qubit generic entangled states utilized only four observables and scheme require no prior state information. Three-qubit entanglement detection scheme is further extended to the general case of three-qubit pure states and the effect of mixedness present in the state is also investigated. This thesis also explores the quantum correlations possessed by mixed and/or separable states \textit{e.g.~} non-classical correlations. A positive map-based witness is used to detect the non-classicality in the experimentally created non-classically correlated states. Results of non-classicality detection are compared with the quantum correlation measure QD while the state is evolving under free NMR Hamiltonian. The salient feature of the developed experimental scheme is that a single-shot experiment is able to detect the non-classicality present in the state under investigation. Schemes to detect pseudo-bound entanglement in a qubit-ququart system are also explored. Only three observables suffice to detect the entanglement in such PPT entangled states. In all the investigations, results were verified by one or more alternative ways \textit{e.g.~} full quantum state tomography, QD, negativity and \textit{n}-tangle.\\
Thesis is organized as follows: Chapter \ref{chap2} describes the creation and detection of entanglement in arbitrary two-qubit states. Experimental creation and detection of non-classical correlations possessed by mixed separable states is discussed in Chapter \ref{chapter_ncc}. Chapter \ref{chapter_3QEntDet} details the experimental entanglement detection as well as characterization of three-qubit random states. In Chapter \ref{BoundEnt} a more subtle type of entanglement possessed by mixed states undetectable by PPT criterion is discussed. The experimental explorations of quantum non-local nature of the quantum correlations are reported in Chapter \ref{NPA}. Thesis concludes with Chapter \ref{Summary} briefing the main results and future directions of work.
\chapter{Bipartite Entanglement Detection on an NMR Quantum Processor}\label{chap2}
\section{Introduction}
Quantum entanglement \cite{schrodinger-mpc-35} is a striking feature exhibited by quantum systems and have no analog in classical mechanics. It has been shown \cite{aditi-cs-17} that quantum entanglement is a key resource to achieve computational speedup in quantum information processing (QIP) \cite{horodecki-rmp-09} and for other quantum communication related tasks \cite{sibasish-njp-02,aditi-pra-09,pankaj-epd-15}. In general, the detection of entanglement is a hard problem in quantum mechanics \citep{horodecki-rmp-09}. It has been demonstrated that the entanglement classification and
detection are daunting tasks \citep{guhne-pr-09}. There have
been attempts to do so utilizing methods based on Bell-type
inequalities
\citep{marcus-pra-11,bastian-prl-11,wallman-pra-12}, quantum
state tomography \citep{white-prl-99,thew-pra-02}, dynamic
learning tools and numerical schemes \citep{behrman-qic-13},
entanglement witnesses
\citep{lewenstein-pra-00,guhne-jmo-03,arrazola-pra-12},
positive-partial-transpose mixtures
\citep{novo-pra-13,bartkiewicz-pra-15}, and expectation
values of Pauli operators
\citep{zhao-pra-13,miranowicz-pra-14}. The negativity under
partial transpose (NPT) of the density operator is a
necessary and sufficient condition for the existence of
entanglement in 2$ \otimes $2 and 2$ \otimes $3
dimensional quantum systems
\citep{horodecki-pla-96,peres-prl-96}. For quantum states in
higher dimensional Hilbert spaces there are sufficient
conditions available but complete entanglement
characterization is still an open problem
\citep{guhne-pr-09}.
The creation of entanglement in an experiment and then
detecting the same is an important theme in experimental quantum computing. Experimental
detection and characterization of entanglement was
demonstrated using optical hardware
\citep{qin-lsa-15,wang-sb-16, gou-s-china-15}. Entanglement
was explored in an NMR scenario using an entanglement
witness \citep{filgueiras-qip-12}, by measuring quantum
correlations of an unknown quantum state,
\citep{silva-prl-13} as well as by a multiple-quantum
coherence based entanglement witness
\citep{feldman-jetpl-08}. It was shown that tomography is
necessary for universal entanglement detection using
single-copy observables in a system of two qubits
\citep{lu-prl-16}. Single-copy here implies that not more
than one copy of the state was used in a single run of the
experiment designed for entanglement detection. Three NMR
qubits were used to simulate the entanglement dynamics of
two interacting fermions \citep{lu-sb-15}. Three-qubit
entanglement was characterized on an NMR quantum-information
processor \citep{dogra-pra-15,das-pra015}, and the evolution
of multiqubit entangled states was studied with a view to
control their decoherence
\citep{kawamura-ijqc-06,singh-pra-14}.
To study entanglement, at least two quantum systems are
required which can be entangled, and to begin with one can
choose two two-level quantum systems. The resultant Hilbert space, of this 2$ \otimes $2 system, is 4 dimensional. Although the NPT criterion is
necessary and sufficient to detect entanglement in this case, the pre-requisite is that the full density operator
is known \citep{horodecki-pla-96,peres-prl-96}.
A promising direction of research in the detection of
quantum entanglement is the use of local observables to find
an optimal decomposition of entanglement witnesses
\citep{guhne-pra-02}. Although the method assumes some prior
knowledge of the density matrix, it can detect entanglement
by performing only a few local measurements
\citep{guhne-ijtp-03,toth-prl-05,guhne-njp-10}. Entanglement
detection schemes were designed for pure states with totally
uncorrelated measurement settings that use only two copies
of the state \citep{tran-pra-15}. These entanglement
detection schemes were recently extended
\citep{szangolies-njp-15} to the case of completely unknown
states with no prior information. This scheme uses a set of
random local measurements and optimizes over the space of
possible entanglement witnesses that can be constructed
thereof \citep{szangolies-njp-15}.
In the current chapter, focus is on experimental use of a
set of random local measurements to detect bipartite
entanglement of unknown pure entangled states. Particularly
interest from an experimental point of view is that can one
be able to detect entanglement using a minimum number of
experimental settings. Current experiments demonstrate the
optimality of using random local measurements to detect
entanglement in a system of two qubits on an NMR quantum
information processor. The expectation values of a set of
local measurement operators are obtained and used in
semi-definite programming to thereby construct the witness
operator to detect the presence of entanglement. It is shown
that a set of three local measurements is sufficient to
unequivocally detect entanglement of most entangled states
of two qubits. States with different amounts of entanglement
are generated experimentally and their entangled (or
separable) nature is evaluated by performing this optimal
set of local measurements. Further, results are validated by
constructing experimental tomographs of each state and
negativity is computed as a measure of entanglement from
them. With a view to generalize these methods to larger
Hilbert spaces, simulations were performed to detect
bipartite entanglement of unknown pure entangled states in a
2$ \otimes $3 dimensional system, using a set of random
measurements acting locally on the qubit and the qutrit. It
is observed that by performing a few measurements, the
entanglement of most states gets characterized.
\section{Entanglement Detection in a $ 2\otimes2 $ Dimensional Quantum System by Sub-System Measurements}\label{det_protocol2Q}
Several protocols~\citep{guhne-pr-09, dagmar-jmp-02} have been put forward to detect the entanglement and most of them are based on full knowledge of the quantum state. Most of the proposed protocols are not readily implementable \textit{i.e.~} they can not be measured directly in an experiment. Entanglement witnesses are the observables which can give a `yes/no' answer on the entanglement present in a state when measured in an experiment. For witness-based experimental entanglement detection, knowledge about the state is required beforehand. One may argue that if the state is already known or has been tomographed, then one can calculate its entanglement properties by using the witness. In the present study a different approach is followed, where \textit{a priori} state information is not required but instead local measurements were strategically chosen. Semidefinite programming (SDP) is used to obtain the relative weights of the expectation values of these local measurements which are then used to build the entanglement witness for the unknown state. Procedure outlined by Szangolies \textit{et al} \citep{szangolies-njp-15} is followed to construct a class of decomposable entanglement witness operators for an unknown state using random local measurements. In this protocol, once the set of measurements got fixed, the witness is optimized to increase the possibility of detecting entanglement.
Consider a composite system in a Hilbert space $
\mathcal{H}_{AB}= \mathcal{H}_A \otimes \mathcal{H}_B $. A
witness operator is a Hermitian operator $ W $ acting on the
composite Hilbert space, such that Tr$ (W\rho) > $0 for all
separable $ \rho $ and Tr$ (W\rho) \leq $0 for at least one
entangled $ \rho $. A witness is called decomposable if it can be written as linear a combination of two positive operators $P$ and $Q$ such that
\begin{eqnarray}
W = P + Q^{T_A}
\end{eqnarray}
where the operation $ T_A $ represents the partial transposition with
respect to subsystem $A$. Further, since one would like to
build the witness operator out of local measurements,
consider local Hermitian operators $ A_i $ and $ B_j $
acting on $ \mathcal{H}_A $ and $ \mathcal{H}_B $
respectively. Indices $ i,\; j $ are the measurement labels
that one wish to carry out for each local system. Range of
the indices $ i $ and $ j $ depends upon the number of
orthogonal operators spanning an arbitrary operator acting
locally on the respective Hilbert spaces, (see
Sec-\ref{Exp_Measure_NMR}). One would therefore like the
witness operator to be given as
\begin{equation}\label{Wdecomp}
W=\sum_{i,j}c_{ij}A_i\otimes
B_j
\end{equation}
with $ \rm c_{ij}\in \mathbb{R} $. It
should be noted here that if one were to allow $\rm A_i $ and $ \rm
B_j $ to run over a complete set of bases in the local
operator spaces, then by Bloch decomposition, every
Hermitian operator can be written in the form given in Eq.~(\ref{Wdecomp}) \citep{szangolies-njp-15}. However, in
present case first the measurements will be chosen which one
wants to experimentally perform and then witness operator
optimized in such a way that the chances of entanglement
detection get maximized. \subsection{Semi Definite Program
(SDP) for Entanglement Detection}
Finding the expectation value of the entanglement witness
operator W (given the set of local observables $ A_i $ and
$ B_j $) is equivalent to finding the coefficients $ c_{ij}
$ subject to the trace constraints on the witness operator.
Let us define a column vector $ \boldsymbol{c} $ where one
can take the columns of $ c_{ij} $ and stack them one below
the other and similarly define a vector $ \boldsymbol{m} $
in which experimentally measured expectation values $
\langle A_i\otimes B_j \rangle $ are stacked to form a long
column vector such that \begin{eqnarray} \rm Tr(W\rho) =
\sum_{i,j} c_{ij}\langle A_i\otimes B_j\rangle =
\boldsymbol{\rm c^{T}.m} \end{eqnarray} The SDP looks for
the class of entanglement witness operators with unit trace
that are decomposable as $ P + Q^{T_A} $. This is the most
general witness capable of detecting states with
non-positive partial transpose as Tr$(\rho W) < 0$ implies
$ \rho^{T_A}\ngeqq 0$, since Tr$(\rho
W)=$Tr$(P\rho)+$Tr$(Q\rho^{T_A}) $, which can only be
smaller than zero if $ \rho^{T_A} $ is not positive. Hence
this decomposition ensures the detection of bipartite NPT
states. The corresponding SDP can be constructed as
\citep{szangolies-njp-15} \begin{eqnarray}\label{SDP_Def}
\rm Minimize &:& \boldsymbol{c^{T}.m} \nonumber\\ \rm s.t.
\nonumber\\ W &=& \rm P+Q^{T_A} \nonumber\\ P &\geq & 0
\nonumber\\ Q &\geq & 0 \nonumber\\ Tr(W) &=& 1
\end{eqnarray} SDP is implemented using MATLAB
\citep{MATLAB} subroutines that employed SEDUMI
\citep{strum-oms-99} and YALMIP \citep{Lofberg2004} as SDP
solvers. MATLAB script and detailed code explanation, using
data in Sec-\ref{SDP_example} as an example, is given in
Appendix-\ref{Append-A}.
\subsection{Measuring Expectation Values via NMR Experiments}\label{Exp_Measure_NMR}
This section describe the procedure followed to
experimentally measuring the expectation values of various observables using NMR for a system of two weakly interacting spin-1/2 particles. The density operator for this system can be decomposed as a linear combination of products of Cartesian spin angular momentum operators $ I_{ni} $, with $n=1,2$ labeling the spin and $i = x,y$ or $z$ \citep{ernst-book-90}. A total of 16 product
operators completely span the space of all 4$ \times $4
Hermitian matrices. The four maximally entangled Bell states
for two qubits and their corresponding entanglement witness
operators can always be written as a linear combination of
the three product operators $ 2I_{1x}I_{2x} $, $
2I_{1y}I_{2y} $, $ 2I_{1z}I_{2z} $ and the identity
operator. The symbols $ O_i $ (1 $ \leq i \leq $ 15) is
used to represent product operators, with the first three
symbols $ O_1 $, $ O_2 $ and $ O_3 $ representing the
operators $ 2I_{1x}I_{2x} $, $ 2I_{1y}I_{2y} $, and $
2I_{1z}I_{2z} $ respectively. Need is to experimentally
determine the expectation values of these operators $ O_i $
in state $ \rho $ whose entanglement is to be characterized.
The expectation values of these operators are mapped to the
local $z$ magnetization of either of the two qubits by
specially crafted unitary operator implemented and are
summarized in Table~\ref{mapping2Q}. $CNOT$ is the
controlled-NOT gate with first qubit as the control qubit
and second qubit as the target qubit. $X$, $\overline{X}$,
$Y$ and $\overline{Y}$ represent local $\frac{\pi}{2}$
unitary rotations with phase $x$, $-x$, $y$ and $-y$
respectively. Subscript on $ \frac{\pi}{2} $ local unitary
rotations denotes the qubit number. The expectation values
are obtained by measuring the $z$ magnetization of the
corresponding qubit. The unitary operations given in
Table~\ref{mapping2Q}, implemented via NMR, transform the
state via a single measurement, which is completely
equivalent to the originally intended measurement of local
operators, and considerably simplifies the experimental
protocol.
\begin{table}[t]
\caption{\label{mapping2Q} All 15
observables for two qubits, mapped to the local z
magnetization of one of the qubits. This mapping allows a
simpler method to measure the expectation values of the
operators $ O_i $ and is completely equivalent to the
measurement of the original local operators.} \centering
\begin{tabular}{l l} \hline Observable & Initial state
mapped to\\
\hline
\hline
$ \langle O_{1} \rangle$ = Tr[$
\rho_{1}.I_{2z}$] & $
\rho_1=CNOT.Y_2.Y_1.\rho_0.Y_1^{\dagger}.Y_2^{\dagger}.CNOT^{\dagger}$
\\
$ \langle O_{2} \rangle$ = Tr[$ \rho_{2}.I_{2z}$] & $
\rho_2=CNOT.\overline{X}_2.\overline{X}_1.\rho_0.\overline{X}_1^{\dagger}.\overline{X}_2^{\dagger}.CNOT^{\dagger}$
\\
$ \langle O_{3} \rangle$ = Tr[$ \rho_{3}.I_{2z}$] & $\rm
\rho_{3}=CNOT.\rho_0.CNOT^{\dagger}$ \\
$ \langle O_{4} \rangle$ = Tr[$ \rho_{4}.I_{2z}$] & $\rm
\rho_4=CNOT.\overline{X}_2.Y_1.\rho_0.Y_1^{\dagger}.\overline{X}_2^{\dagger}.CNOT^{\dagger}$
\\
$ \langle O_{5} \rangle$ = Tr[$ \rho_{5}.I_{2z}$] & $\rm
\rho_5=CNOT.Y_1.\rho_0.Y_1^{\dagger}.CNOT^{\dagger}$ \\
$ \langle O_{6} \rangle$ = Tr[$ \rho_{6}.I_{2z}$] & $\rm
\rho_6=CNOT.\overline{Y}_2.X_1.\rho_0.X_1^{\dagger}.\overline{Y}_2^{\dagger}.CNOT^{\dagger}$
\\
$ \langle O_{7} \rangle$ = Tr[$ \rho_{7}.I_{2z}$] & $\rm
\rho_{7}=CNOT.X_1.\rho_0.X_1^{\dagger}.CNOT^{\dagger}$ \\
$ \langle O_{8} \rangle$ = Tr[$ \rho_{8}.I_{2z}$] & $\rm
\rho_{8}=CNOT.\overline{Y}_2.\rho_0.\overline{Y}_2^{\dagger}.CNOT^{\dagger}$
\\
$ \langle O_{9} \rangle$ = Tr[$ \rho_{9}.I_{2z}$] & $\rm
\rho_{9}=CNOT.X_2.\rho_0.X_2^{\dagger}.CNOT^{\dagger}$ \\
$ \langle O_{10} \rangle$ = Tr[$ \rho_{10}.I_{1z}$] & $\rm
\rho_{10}=\overline{Y}_1.\rho_0.\overline{Y}_1^{\dagger}$ \\
$ \langle O_{11} \rangle$ = Tr[$ \rho_{11}.I_{1z}$] & $\rm
\rho_{11}=X_1.\rho_0.X_1^{\dagger}$ \\
$ \langle O_{12} \rangle$ = Tr[$ \rho_0.I_{1z}$] & $\rm
\rho_0$ is initial state \\
$ \langle O_{13} \rangle$ = Tr[$ \rho_{13}.I_{2z}$] & $\rm
\rho_{13}=\overline{Y}_2.\rho_0.\overline{Y}_2^{\dagger}$
\\
$ \langle O_{14} \rangle$ = Tr[$ \rho_{14}.I_{2z}$] & $\rm
\rho_{14}=X_2.\rho_0.X_2^{\dagger}$ \\
$ \langle O_{15} \rangle$ = Tr[$ \rho_{0}.I_{2z}$] & $\rm
\rho_0$ is initial state \\ \hline \hline \end{tabular}
\end{table}
Description of the quantum system used for experimental
demonstration of the entanglement detection protocol is as
follows. The two NMR qubits were encoded in a molecule of $
^{13} $C enriched chloroform, with the $ ^1 $H and $ ^{13}
$C nuclei and encodes the first and second qubits,
respectively. The molecular structure, experimental
parameters and NMR spectrum of the thermal initial state are
shown in Fig~\ref{mol2Q}.
\begin{figure}
\caption{(a)
Structure of the $ ^{13}
\label{mol2Q}
\end{figure}
Experiments were performed at room temperature (293K) on a
Bruker Avance III 600 MHz NMR spectrometer equipped with a
quadruple resonance inverse (QXI) probe. The Hamiltonian of
this weakly interacting two-qubit system in the rotating
frame \citep{ernst-book-90} is \begin{equation} H = \nu_H
I_z^H + \nu_C I_z^C + J_{CH}I_z^HI_z^C \end{equation} where
$ \nu_H $, $ \nu_C $ are the Larmor resonance frequencies;
$ I_z^H $, $ I_z^C $ are the $z$ components of the spin
angular momentum operators for the proton and carbon nuclei,
respectively; and $ J_{CH} $ is the spin-spin coupling
constant.
The two-qubit system was initially prepared in the
pseudo-pure state $ \vert 00 \rangle $ using the spatial
averaging technique \citep{cory-physD-98}, with the density
operator given by
\begin{equation}\label{pps2Q_eq}
\rho_{00}=\frac{1}{4}(1-\epsilon)\mathbb{I}+\epsilon\vert
00\rangle\langle00\vert
\end{equation}
where $ \epsilon
\approx 10^{-5} $ is an estimate of the thermal
polarization. One may note here that NMR is an ensemble
technique that can experimentally observe only deviation
density matrices (with zero trace). The state fidelity was
calculated from the Uhlmann-Jozsa relation
\citep{uhlmann-rpmp-76, jozsa-jmo-94}
\begin{equation}\label{fidelity_eq} F=\left[Tr\left(
\sqrt{\sqrt{\rho_{{\rm th}}}\rho_{{\rm ex}} \sqrt{\rho_{{\rm
th}}}}\right)\right]^2 \end{equation} where $ \rho_{th} $
and $ \rho_{ex} $ represent the theoretical and
experimentally measured density operators, respectively. The
experimentally prepared pseudo-pure state was tomographed
using full quantum state tomography
\citep{leskowitz-pra-04}, and the state fidelity was
computed to be 0.98 $ \pm $ 0.01.
\begin{figure}
\caption{(a) Quantum circuit to implement the entanglement detection
protocol. The first red box creates states with different
amounts of entanglement. The second red box maps the
observables $ O_i $ to the $z$ magnetization of either
qubit. Only one $z$ magnetization is finally measured in an
experiment (inner green box). (b) NMR pulse sequence for the
quantum circuit. Unfilled rectangles represent $
\frac{\pi}
\label{ckt2Q}
\end{figure}
The quantum circuit to implement the two-qubit entanglement
detection protocol is shown in Fig. \ref{ckt2Q}(a). The
first block in the circuit (enclosed in a dashed red box)
transforms the $ \vert 00 \rangle $ pseudo-pure state to an
entangled state with a desired amount of entanglement.
Control of the entanglement present in the state was
achieved by controlling the time evolution under the
nonlocal interaction Hamiltonian. A controlled-NOT (CNOT)
gate that achieves this control is represented by a dashed
line. The next block of the circuit (enclosed in a dashed
red box) maps any one of the observables $ O_i $ (1 $ \leq i
\leq $ 15) to the local $z$ magnetization of one of the
qubits, with $ U_1^i $ and $ U_2^j $ representing local
unitaries (as represented in Table~\ref{mapping2Q}). The
dashed green box represents the measurement. Only one
measurement is performed in a single experiment. The NMR
pulse sequence to implement the quantum circuit for
entanglement detection using random local measurements,
starting from the pseudo-pure state $ \vert 00 \rangle $, is
shown in Fig.~\ref{ckt2Q}(b). Unfilled rectangles represent
$ \frac{\pi}{2} $ pulses, while solid rectangles denote $
\pi $ pulses. Refocusing pulses were used in the middle of
all J-evolution periods to compensate for undesired chemical
shift evolution. Composite pulses are represented by z in
the pulse sequence, where each composite z rotation is a
sandwich of three pulses: $ xy\overline{x} $. The CNOT gate
represented by the dashed line in Fig.~\ref{ckt2Q}(a) was
achieved experimentally by controlling the evolution time $
\tau_i $ and the angle of $z$ rotation (the gray shaded
rectangle); $ \phi_1^i $ and $ \phi_2^j $ are local
rotations and depend upon which $ \langle O_i \rangle $
value is being measured, and the $ \tau $ time interval was
set to $ \tau=\frac{1}{2J_{CH}} $.
\subsection{An Example To Demonstrate Entanglement Detection via SDP}\label{SDP_example}
Following is an explicit example to demonstrate how the SDP
can be used to construct an entanglement witness. Consider
the Bell state $ \vert
\phi^-\rangle=\frac{1}{\sqrt{2}}(\vert 00 \rangle - \vert 11
\rangle) $. The corresponding density matrix can be written
as a linear superposition of two spin product
\begin{figure}
\caption{NMR spectra of $ ^1 $H and $ ^{13}
\label{oiz}
\end{figure}
\noindent operators \citep{ernst-book-90} as
\begin{equation} \rho =
\frac{\mathbb{I}}{4}+a2I_{1x}I_{2x}+b2I_{1y}I_{2y}+c2I_{1z}I_{2z}
\end{equation} where $b\; =\; c\; =\; -a\; =\; \frac{1}{1}$.
Since it is known that the given state is entangled, the
corresponding entanglement witness can be constructed as
\citep{guhne-pr-09} \begin{eqnarray} W_{\phi^-} &=&
c_{opt}\mathbb{I}-\rho \nonumber\\ &=&
\frac{\mathbb{I}}{4}-a2I_{1x}I_{2x}-b2I_{1y}I_{2y}-c2I_{1z}I_{2z}
\end{eqnarray} where $ c_{opt} $ is the smallest possible
value such that the witness is positive on all separable
states; for Bell states $ c_{opt}=\frac{1}{2} $. Noting that
Tr($ \rho W_{\phi^-} ) = -\frac{1}{2} < 0$ , by definition
W$ \phi^- $ detects the presence of entanglement in $ \rho
$. However, the detection protocol discussed in
Sec-\ref{det_protocol2Q} has to deal with the situation
where the state is unknown. The question now arises whether
the SDP method is able to find the minimum value of $
\boldsymbol{c^T.m} $ such that the correct $W _{\phi^-} $ is
constructed.
For the Bell state $ \vert \phi^-\rangle $, the expectation
values $ \langle O_1 \rangle $, $ \langle O_2 \rangle $ and
$ \langle O_3 \rangle $ yield $-$ 1/2, 1/2, and 1/2 ,
respectively. The experimental NMR spectra obtained after
measuring $ \langle O_1 \rangle $, $ \langle O_2 \rangle $
and $ \langle O_3 \rangle $ in state $ \vert \phi^-\rangle $
are shown in Fig.~\ref{oiz}, with measured expectation
values of $-$0.490 $ \pm $ 0.021, 0.487 $ \pm $ 0.030, and
0.479 $ \pm $ 0.015, respectively (these values correspond
to the area under the absorptive peaks normalized with
respect to the pseudo-pure state). These experimental
expectation values are used to construct the vector
\textbf{m}. The SDP protocol performs minimization under the
given constraints and, for this Bell state, is indeed able
to construct W$ _{\phi^-} $ as well as the exact values of
$a$, $b$, and $c$ which make up the vector \textbf{c}. Since
the minimum value of $ \boldsymbol{c^T.m} <$ 0 is achieved,
it confirms the presence of entanglement in the state. See
Appendix-\ref{Append-A} for MATLAB code used for
entanglement detection and the SDP result.
\subsection{Entanglement Detection in Unknown 2$\otimes$2 States} This section details the detection of entanglement
in states with varying amounts of entanglement. The
entanglement detection protocol is implemented
experimentally on several different states: four maximally
entangled states (labeled as B1, B2, etc.), two separable
states (labeled as S1 and S2), and 14 non-maximally
entangled states (labeled as E1, E2, E3, . . . ).
To prepare the 14 entangled states E1 to E14 (having
different amounts of entanglement), the control on the
amount of entanglement in the state was achieved by varying
the time interval $ \tau_i $ and the angle $ \theta $ of the
$z$ rotation [Fig. \ref{ckt2Q}(b)]. $ \theta
=n\frac{\pi}{30} $ and $ \tau_i = n\frac{30}{J_{CH}} $ was
used, with 1 $ \leq n \leq $ 14. These choices for $ \theta
$ and $ \tau_i $ represent a variation of the rotation angle
in a two-qubit controlled-rotation NMR gate and led to a
wide range of entanglement in the generated states (as
tabulated in Table~\ref{negTab}).
To characterize the amount of entanglement, the entanglement
measure negativity $ \mathcal{N} $ \citep{horodecki-pla-96}
is used and is given as:
\begin{equation} \mathcal{N}=\Vert \rho^{PT} -1 \Vert
\end{equation}
where $ \rho^{PT} $ denotes partial
transposition with respect to one of the qubits and $ \Vert
\cdot \Vert $ represents the trace norm. A nonzero
negativity confirms the presence of entanglement in $ \rho $
and can be used as a quantitative measure of entanglement.
The states prepared ranged from nearly separable (E1, E2
with a low value of negativity) to nearly maximally
entangled (E13, E14 with high negativity values). The
experimental results of the entanglement detection protocol
for two qubits are tabulated in Table~\ref{negTab}.
\begin{figure}
\caption{Real part of the tomographed density matrix for the states
described in Table~\ref{negTab}
\label{tomo}
\end{figure}
For some of the non-maximally entangled states, more than
three local measurements had to be used to detect
entanglement. For instance, SDP required six local
measurement to build the vector $ \boldsymbol{m} $ for the
E$ _8 $ state in Table~\ref{negTab} and to establish that
min($ \boldsymbol{c^T.m}) <$ 0.
As is evident from Table~\ref{negTab}, this method of making
random local measurements on an unknown state followed by
SDP to construct an entanglement witness is able to
successfully detect the presence of quantum entanglement in
almost all the experimentally created states.
\begin{table}[h]
\caption{\label{negTab} Results of entanglement detection via local measurements followed by SDP. States are labeled as B, S, and E, indicating maximally entangled, separable, and non-maximally entangled, respectively. The second and third columns contain the theoretically expected and experimentally obtained values of the entanglement parameter negativity $ \mathcal{N} $. The $ \surd $ in the last column indicates the success of the experimental protocol in detecting entanglement.}
\centering
\begin{tabular}{c | c c c}
\hline \scriptsize & \multicolumn{2}{c}{$ \mathcal{N} $} & \\
& \multicolumn{2}{c}{$ \mathclap{\rule{3.2cm}{0.4pt}} $}&
Entanglement \\ State & Theo. & Expt. & Detected \\
\hline
\hline
B$_1$ & 0.500 & 0.486 $ \pm $ 0.011 & $\surd $ \\
B$_2$ & 0.500 & 0.480 $ \pm $ 0.013 & $\surd $ \\
B$_3$ & 0.500 & 0.471 $ \pm $ 0.021 & $\surd $ \\
B$_4$ & 0.500 & 0.466 $ \pm $ 0.025 & $\surd $ \\
S$_1$ & 0.000 & 0.000 $ \pm $ 0.000 & $\surd $ \\
S$_2$ & 0.000 & 0.000 $ \pm $ 0.000 & $\surd $ \\
E$_1$ & 0.052 & 0.081 $ \pm $ 0.005 & $\times$ \\
E$_2$ & 0.104 & 0.088 $ \pm $ 0.024 & $\times$ \\
E$_3$ & 0.155 & 0.177 $ \pm $ 0.015 & $\surd $ \\
E$_4$ & 0.203 & 0.182 $ \pm $ 0.031 & $\surd $ \\
E$_5$ & 0.250 & 0.212 $ \pm $ 0.029 & $\surd $ \\
E$_6$ & 0.294 & 0.255 $ \pm $ 0.033 & $\surd $ \\
E$_7$ & 0.335 & 0.297 $ \pm $ 0.045 & $\surd $ \\
E$_8$ & 0.372 & 0.351 $ \pm $ 0.039 & $\surd $ \\
E$_9$ & 0.405 & 0.400 $ \pm $ 0.033 & $\surd $ \\
E$_{10}$ & 0.433 & 0.410 $ \pm $ 0.040 & $\surd $ \\
E$_{11}$ & 0.457 & 0.430 $ \pm $ 0.037 & $\surd $ \\
E$_{12}$ & 0.476 & 0.444 $ \pm $ 0.029 & $\surd $ \\
E$_{13}$ & 0.489 & 0.462 $ \pm $ 0.022 & $\surd $ \\
E$_{14}$ & 0.497 & 0.473 $ \pm $ 0.025 & $\surd $ \\
\hline
\hline
\end{tabular}
\end{table}
The protocol failed to detect entanglement in states E$ _1 $
and E$ _2 $, a possible reason for this being that these
states have a very low negativity value (very little
entanglement), which is of the order of the experimental
error. In order to validate the experimental results full
quantum state tomography of all experimentally prepared
states is also performed. The resulting tomographs and
respective fidelities are shown in Fig.~\ref{tomo}, and the
negativity parameter obtained from the experimental
tomographs in each case is tabulated in Table~\ref{negTab}.
Figures ~\ref{tomo}(a) to ~\ref{tomo}(d) correspond to
the maximally entangled Bell states B$ _1 $ to B$ _4 $,
respectively, while Figs.~\ref{tomo}(e) and
~\ref{tomo}(f) are tomographs for the separable states S$
_1 $ and S$ _2 $, respectively, and Figs. ~\ref{tomo}(g)
to ~\ref{tomo}(t) correspond to states E$ _1 $ to E$ _{14}
$, respectively. The fidelity of each experimentally
prepared state is given above its tomograph in the figure.
Only the real parts of the experimental tomographs are
shown, as the imaginary parts of the experimental tomographs
turned out to be negligible.
\section{Entanglement Detection in a 2$ \otimes $3 Dimensional Quantum System}
The orthonormal basis states for a 2$ \otimes $3 dimensional
qubit-qutrit system $ \lbrace \vert ij \rangle: \; i\; =\;
0,\; 1 ,\; j\; =\; 0,\; 1,\; 2 \rbrace $ can be written in
the computational basis for the qubit $\lbrace \vert 0
\rangle $, $ \vert 1 \rangle \rbrace$ and the qutrit
$\lbrace \vert 0 \rangle $, $ \vert 1 \rangle $ and $ \vert
2 \rangle \rbrace$, respectively. It has been previously
shown that any arbitrary pure state of a hybrid qubit-qutrit
2$ \otimes $3 system can be transformed to one of the states
of a two-parameter class (with two real parameters) via
local operations and classical communication (LOCC) and that
states in this class are invariant under unitary operations
of the form U$ \otimes $U on the 2$ \otimes $3 system
\citep{chi-jpamg-03}. The state for such a bipartite 2$
\otimes $3 dimensional system can be written as
\citep{chi-jpamg-03} \begin{equation}\label{qubitqutritEqn}
\rho=\alpha \left[ \vert 02 \rangle\langle 02 \vert + \vert
12 \rangle\langle 12 \vert \right] + \beta\left[\vert \phi^+
\rangle\langle \phi^+ \vert + \vert \phi^- \rangle\langle
\phi^- \vert +\vert \psi^+ \rangle\langle \psi^+ \vert
\right] + \gamma \vert \psi^- \rangle\langle \psi^- \vert
\end{equation} where $ \vert \phi^{\pm}
\rangle=\frac{1}{\sqrt{2}}(\vert 00 \rangle \pm \vert 11
\rangle ) $ and $ \vert \psi^{\pm}
\rangle=\frac{1}{\sqrt{2}}(\vert 01 \rangle \pm \vert 10
\rangle ) $ are the maximally entangled Bell states. The
requirement of unit trace places a constraint on the real
parameters $ \alpha $, $ \beta $ and $ \gamma $ as
\begin{equation}
2\alpha + 3\beta + \gamma = 1
\end{equation}
This constraint implies that one can
eliminate one of the three parameters, and one can rewrite $
\beta $ in terms of $ \alpha $ and $ \gamma $ ; however, the
entire analysis is valid for the other choices as well. The
domains for $ \alpha $ and $ \gamma $ can be calculated from
the unit trace condition and turn out to be 0 $ \leq $ $
\alpha $ $ \leq $ 1/2 and 0 $ \leq $ $ \gamma $ $ \leq $ 1.
The Peres-Horodecki positive-partial-transposition (PPT)
criterion is a necessary and sufficient condition for 2$
\otimes $3 dimensional systems and can hence be used to
characterize the entanglement of $ \rho $ via the
entanglement measure negativity $ \mathcal{N} $ . The
partial transpose with respect to the qubit for the
two-parameter class of states defined in
Eq.~(\ref{qubitqutritEqn}) can be written as \begin{eqnarray}
\rho^{PT} = \alpha \left[ \vert 02 \rangle\langle 02 \vert
+ \vert 12 \rangle\langle 12 \vert \right] &+& \frac{(\beta
+ \gamma)}{2}\left[ \vert \phi^- \rangle\langle \phi^- \vert
+\vert 10 \rangle\langle 10 \vert +\vert 01 \rangle\langle
01 \vert \right] \nonumber\\ &+& \frac{(3\beta - \gamma)}{2}
\vert \phi^+ \rangle\langle \phi^+ \vert \end{eqnarray} The
negativity $ \mathcal{N}(\rho) $ for the two-parameter class
of states can be calculated from its partial transpose and
is given by \citep{chi-jpamg-03}
\begin{equation}
\rm \mathcal{N}(\rho)=max\lbrace (2\alpha+2\gamma-1),0\rbrace
\end{equation}
Clearly, states with 0.5 $ < \alpha+\gamma
\leq $ 1 have nonzero negativity (\textit{i.e.~}, are NPT) and are
hence entangled. To extend the Bloch representation for
qubits to a qubit-qutrit system described by a 2$ \otimes $3
dimensional hybrid linear vector space an operator O
operating on this joint Hilbert space can be written as
\citep{jami-ijp-05}
\begin{eqnarray}\label{qubitqutritDecmpEqn}
O=\frac{1}{6}\left[
\mathbb{I}_2\otimes\mathbb{I}_3+\sigma^A.\overrightarrow{u}\otimes\mathbb{I}_3+\sqrt{3}\mathbb{I}_2\otimes\lambda^B.\overrightarrow{v}+\sum_{i=1}^3\sum_{j=1}^8\beta_{ij}(\sigma_i^A\otimes\lambda_j^B)\right]
\end{eqnarray} where $ \overrightarrow{u} $ and $
\overrightarrow{v} $ are vectors belonging to linear vector
spaces of dimension 3 and 8 respectively, $ \mathbb{I}_2 $
and $ \mathbb{I}_2 $ are identity matrices of dimensions 2
and 3 respectively and $ \sigma_i $ are the Pauli spin matrices used to span operators acting on the Hilbert space of the qubit. $ \rm \lambda_j $ are the Gell-Mann matrices \citep{gelmann-osti-61}, used to span operators acting on the Hilbert space of the qutrit; other isomorphic choices are equally valid. A Hermitian witness operator can be constructed for every entangled quantum state, and the
expectation value of the witness operator can be locally
measured by decomposing the operator as a weighted sum of
projectors onto product-state vectors
\citep{horodecki-pla-96,bourennane-prl-04,brando-pra-05}.
The $ \rho $ for the 2$ \otimes $3 system given in
Eq.~(\ref{qubitqutritEqn}) is NPT for 0.5 $ < (\alphaα +
\gamma ) \leq $ 1. The eigenvalues for $ \rho^{PT} $ (where
PT represents partial transposition with respect to the
qubit) are $ \alpha $, $ \frac{1}{2}(1 - 2 \alpha - 2
\gamma ) $, and $\frac{1}{6}(1 - 2 \alpha + 2 \gamma )$.
The eigenvalue $ \frac{1}{2}(1 - 2 \alpha - 2 \gamma ) $
remains negative for NPT states, and the corresponding
eigenvector is denoted by $ \vert \eta \rangle $. The
corresponding entanglement witness operator can be written
as $ \rm W = (\vert \eta \rangle \langle \eta \vert )^{PT} $
with it's matrix representation
\begin{gather} W=\frac{1}{2} \begin{bmatrix} 1 & 0 & 0 & 0 &
0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0
& 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 &
0 & 0 & 1 \end{bmatrix}
\end{gather}
The entanglement
witness $W$ is capable of detecting entanglement of the 2$ \otimes $3 dimensional $ \rho $ given in
Eq.~(\ref{qubitqutritEqn}). Once can explore the decomposition of the entanglement witness $W$ in terms of local observables, so that it can used to detect entanglement of the two-parameter class of states of the 2$ \otimes $3 dimensional $ \rho $. The explicit decomposition of $W$ as
per Eq.~(\ref{qubitqutritDecmpEqn}) results in the following:
\begin{gather}
\label{uvBetaEqn}
\overrightarrow{u}=
\begin{bmatrix}
0 \\ 0 \\ 0
\end{bmatrix} ,
\;\;\;
\overrightarrow{v}=
\begin{bmatrix}
0 \\ 0 \\ 0 \\ 0 \\ 0 \\
0 \\ 0 \\ 1
\end{bmatrix}
\;\;\;
\rm and,\;\;\;
\beta=\frac{1}{2}
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\end{gather}
\noindent The components of $ \overrightarrow{u} $ and $
\overrightarrow{v} $, \textit{i.e.~}, $ u_i(i = 1,2,3) $ and $ v_j(j
=1,2,3, . . . ,8) $ can be obtained from $ u_i =Tr[
W(\sigma_i \otimes \mathbb{I}_3)] $ and $ v_j = Tr [
W(\mathbb{I}_2\otimes \lambda_j)] $. Similarly, the elements
of the matrix $ \beta $ can be obtained from $ \beta_{ij} =
Tr[ W(\sigma_i \otimes \lambda_j)] $. There are 35 real
coefficients in the expansion in
Eq.~(\ref{qubitqutritDecmpEqn}), of which 3 coefficients
constitute $ \overrightarrow{u} $, 8 coefficients constitute
$ \overrightarrow{v} $, and the remaining 24 are contained
in the $ \beta $ matrix. Each nonzero entry in $
\overrightarrow{u} $, $ \overrightarrow{v} $, or $ \beta $
matrix is the contribution of the corresponding qubit-qutrit
product operator \citep{ernst-book-90} used in the
construction of operator $W$. Hence one can infer by
inspection of the nonzero matrix entries in
Eq.~(\ref{uvBetaEqn}) that one requires the expectation values
of at least four operators in a given state in order to
experimentally construct the witness operator W. While the
maximum number of expectation values required to be measured
is four, the question remains if this is an optimal set or
if one can find a smaller set which will still be able to
detect entanglement.
Fraction of entanglement was computationally detected by gradually increasing the number of local observations, and the results
of the simulation are depicted in Fig. 5(a) as a bar chart.
One may note here that even if only one observable [one element of the $ \beta $ matrix in Eq.~(\ref{qubitqutritDecmpEqn})] is measured, half of the
randomly generated entangled states are detected. As the
number of measured observables is increased, the fraction of
detected entangled states improves, as shown in
Fig.~\ref{qubitQutritResFig}(a). To generate the bar plots
in Fig.~\ref{qubitQutritResFig}(a), only one random local
measurement is selected out of the maximum 35 possible
measurements. Only those choices which will establish a
decomposable entanglement witness of unit trace are valid.
For one such choice (denoted by $ W_I $), $ Tr(W_I\rho) $ is
plotted in Fig.~\ref{qubitQutritResFig}(b) in the range 0 $
\leq \alpha \leq $ 0.5 and 0$ \leq \gamma \leq $1. As is
evident, this $ \rm W_I $ (based on only one random local
measurement) does not detect all the entangled states which
were detected by W. The fraction of entangled states
detected by $ W_I $ can be computed from geometry, \textit{i.e.~}\;,
how much area that is spanned by the parameters $ \alpha $
and $ \gamma $ represents entangled states and how much of
that area is detected by the corresponding entanglement
witness operator.
\begin{figure}
\caption{(a) Bar
graph showing the fraction of the detected entangled states
plotted as a function of the number of local measurements
from the simulation on the qubit-qutrit system. Plots of (a)
$ Tr(W_{I}
\label{qubitQutritResFig}
\end{figure}
If one consider two random local measurements at a time to
construct a valid entanglement witness [$ W_{II} $ in
Fig.~\ref{qubitQutritResFig}(c)], the detected fraction of
entangled states improves from 0.50 to 0.67 [the second bar
in Fig.~\ref{qubitQutritResFig}(a)]. One can observe from
the geometry that $ \rm W_{II} $ detects more entangled
states than $ \rm W_I $, but this fraction is still smaller
than those detected by W. The result of choosing three
random local measurements (denoted by witness operator $ \rm
W_{III} $) is plotted in Fig.~\ref{qubitQutritResFig}(d),
and it detects 83.3 \% of the total entangled states [the
third bar in Fig.~\ref{qubitQutritResFig}(a)]. Increasing
the set of random local measurements hence increases the
probability of detecting entanglement. The worst-case
detection fraction is shown in
Fig.~\ref{qubitQutritResFig}(a) when choosing random local
measurements. A fraction of 1 in
Fig.~\ref{qubitQutritResFig}(a) implies that the
corresponding set of four random local measurements will
always be able to detect entanglement in the state if it
exists.
\section{Conclusions}
This chapter was aimed at detecting the entanglement of a two-qubit state without any prior state information and with minimum experimental efforts. It has been successfully demonstrated in an actual experiment that the scheme based on local random measurements is able to detect the presence of entanglement, on a two-qubit NMR quantum-information processor. An optimal set of random local measurements was arrived at via semi-definite programming to construct entanglement witnesses that detect bipartite entanglement. The local measurements on each qubit were converted into a single measurement on one of the qubits by transforming the state. This was done to simplify the experimental scheme and is completely equivalent to the originally intended local measurements. Scheme based on random local measurements have been extended to hybrid systems, where qudits of different dimensionality are involved. For the particular case of a qubit-qutrit system, a simulation is performed to demonstrate the optimality of the detection scheme. Characterization of entangled states of qudits is a daunting task, and this work holds promise for further research in this direction. Results of this chapter are published in \href{http://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.062309}{\rm Phys. Rev. A \textbf{94}, 062309 (2016)}.
\chapter{Non-Classical Correlations and Detection in a Single-Shot Experiment}\label{chapter_ncc}
\section{Introduction}
In this chapter, the focus is on more generalized quantum correlations possessed
even by separable states. Such quantum correlations are best captured by a nonclassicality quantifier ``quantum discord''(QD)~\cite{ollivier-prl-02}.
Quantum correlations are those correlations which are not present in classical
systems, and in bipartite quantum systems are associated with the presence of
QD~\cite{nielsen-book-02,ollivier-prl-02,modi-rmp-12}. In quantum information
theory, QD is a measure of nonclassical correlations between the subsystems of a
quantum system. It includes correlations that are due to quantum physical
effects and may not involve quantum entanglement. QD utilizes the concept of
mutual information. In a bipartite system the mutual information is a measure of
knowledge gained, by measuring one variable, about the other and involves entropy in the observed statistics. In classical information, theory there are two inequivalent expressions for the mutual information which are equivalent for all classical probability distributions or statistics observed by measuring
systems possessing only classical correlations. In case, the system involved possesses quantum correlations then one gets different mutual information, from two expressions defined in the classical information theory, by virtue of the way they are defined. This difference is indeed the QD as it arises due to the quantum nature of the involved system to generate the statistics which is further used to compute QD. Formal definition is given in Sec-\ref{qd-dyn}.
Quantum correlations captured by QD in a bipartite mixed state can go beyond quantum entanglement and therefore can be present even in separable states~\cite{ferraro-pra-10}. The threshold between classical and quantum
correlations was investigated in linear-optical systems by observing the emergence of QD~\cite{karo-pra-13}. QD was experimentally measured in systems such as NMR, that are described by a deviation density
matrix~\cite{pinto-pra-10,maziero-bjp-13,passante-pra-11}. Further,
environment-induced sudden transitions in QD dynamics and their preservation
were investigated using NMR~\cite{auccaise-prl-11-2, harpreet-discord}.
It has already been demonstrated that even with very low (or no) entanglement,
quantum information processing can still be performed using nonclassical
correlations~\cite{datta-pra-05,fahmy-pra-08}, which are typically characterized
by the presence of QD. However, computing and measuring QD typically involves
complicated numerical optimization and furthermore it has been shown that
computing QD is NP-hard~\cite{bryan-arx-10,huang-njp-14,cable-njp-15}. It is
hence of prime interest to find other means such as witnesses to detect the
presence of quantum correlations captured by QD~\cite{saitoh-qip-11}. While there have been
several experimental implementations of entanglement
witnesses~\cite{rahimi-jpamg-06,rahimi-pra-07,filgueiras-qip-11}, there have
been fewer proposals to witness nonclassicality. A nonlinear classicality
witness was constructed for a class of two-qubit systems~\cite{maziero-ijqi-12}
and experimentally implemented using NMR~\cite{auccaise-prl-11,pinto-ptrsa-12}
and was estimated in a linear optics system via statistics from a single
measurement~\cite{aguilar-prl-12}. It is to be noted that as the state space
for classical correlated systems is not convex, a witness for nonclassicality is
more complicated to construct than a witness for entanglement and is necessarily
nonlinear~\cite{saitoh-pra-08}.
In the this chapter two qubits were used to demonstrate the experimental
detection of nonclassicality through a recently proposed positive map
method~\cite{rahimi-pra-10}. Two NMR qubits have been recently used to
demonstrate very interesting QIP phenomena such as the quantum simulation of the
ground state of a molecular Hamiltonian~\cite{du-prl-10}, the quantum simulation
of the Avian compass~\cite{pearson-sc-2016}, observing time-invariant coherence
at room temperature~\cite{silva-prl-16} and preserving QD~\cite{silva-prl-13}.
The map is able to witness nonclassical correlations going beyond entanglement,
in a mixed state of a bipartite quantum system. The method requires much less
experimental resources as compared to measurement of QD using full state
tomography and therefore is an attractive alternative to demonstrating the
nonclassicality of a separable state. The map implementation involves two-qubit
gates and single-qubit magnetization measurements and can be achieved in a
single experimental run using NMR. Our implementation of the nonclassicality
witness involves the sequential measurement of different free induction decays
(FIDs, corresponding to the NMR signal) in a single run of the same experiment.
This is possible since NMR measurements are nondestructive, thus allowing
sequential measurements on the same ensemble. This feature was exploited to
implement the single-shot measurement of the map value. The NMR pulse sequence
used on Bruker Avance-III spectrometer is given Appendix-\ref{SeqFID}.
Experiments were performed on a two-qubit separable state (non-entangled) which
contains nonclassical correlations. Further, the state was allowed to freely
evolve in time under natural NMR decohering channels, and the amount of
nonclassicality present was evaluated at each time instant by calculating the
map value. Results were compare using the positive map witness with those
obtained by computing the QD via full state tomography, and a good match was
obtained.
Further, it was observed that beyond a certain time, the map was not able to
detect nonclassicality, although the QD measure indicated that nonclassicality
was present in the state. This implies that while the positive map
nonclassicality witness is easy to implement experimentally in a single
experiment and is a good indicator of nonclassicality in a separable state, it
is not able to characterize nonclassicality completely. In our case this is
typified by the presence of a small amount of QD when the state has almost
decohered or when the amount of nonclassicality present is small. This of course
leaves open the possibility of constructing a more optimal witness.
\section{Experimental Detection of Non-Classical Correlation (NCC)} \label{fullexptl}
\subsection{Nonclassicality Witness Map Construction} \label{mapvalue} For pure quantum states of a bipartite quantum system which are
represented by one-dimensional projectors $\vert \psi\rangle \langle \psi\vert$
in a tensor product Hilbert space $ \rm {\cal H}_A \otimes {\cal H}_B $, the
only type of quantum correlation is
entanglement~\cite{guhne-pr-09,oppenheim-prl-02}. However, for mixed states the
situation is more complex and quantum correlations can be present even if the
state is separable \textit{i.e.~} it is a classical mixture of separable pure states and
can be written as
\begin{equation}
\rm \rho_{sep}=\sum_i{w_i\rho_i^A\otimes\rho_i^B}
\label{sep}
\end{equation}
where $ \rm w_i $ are positive weights and $\rm \rho_i^A,\rho_i^B $ are pure states in
Hilbert spaces ${\cal H}_A$ and ${\cal H}_B$ respectively~\cite{peres-prl-96}.
A separable state is called a properly classically correlated state(PCC) if it
can be written in the form~\cite{horodecki-rmp-09}
\begin{equation}
\rm \rho_{PCC}=\sum_{i,j}{p_{ij}\vert e_i\rangle^{A}\langle e_i\vert\otimes\vert e_j\rangle^{B}\langle e_j\vert}
\label{pccform}
\end{equation}
where $ \rm p_{ij}$ is a joint probability distribution and $\rm \vert e_i\rangle^{A}$ and $
\rm \vert e_j\rangle^{B}$ are local orthogonal eigenbasis in local spaces ${\cal
H}_{\rm A}$ and ${\cal H}_{\rm B}$ respectively. A state that cannot be
written in the form given by Eq.~(\ref{pccform}) is called a nonclassically
correlated (NCC) state. An NCC state can be entangled or separable. The correlations in NCC states can go beyond those present in PCC states and are due to the fact that the eigenbasis for the respective subsystems may not be orthogonal~\cite{vedral-found-10}. A typical example of a bipartite two-qubit NCC state has been discussed in reference~\cite{streltsov-prl-11} and is given by: \begin{equation}\label{sigma}
\sigma=\frac{1}{2}\left[\vert00\rangle\langle
00\vert+\vert1+\rangle\langle1+\vert\right]
\end{equation} with $\vert + \rangle = \frac{1}{\sqrt{2}} \left(\vert 0 \rangle+\vert 1
\rangle\right)$. In this case the state has no product eigenbasis as the eigenbasis for subsystem B, since $\vert 0 \rangle$ and $\vert + \rangle$ are
not orthogonal to each other. The state is separable (not entangled) as it can
be written in the form given by Eq.~(\ref{sep}); however since it is an NCC
state, it has non-trivial quantum correlations and has non-zero QD. How to pin
down the nonclassical nature of such a state with minimal experimental effort
and without actually computing QD is something that is desirable. It has been
shown that such nonclassicality witnesses can be constructed using a positive
map~\cite{rahimi-pra-10}.
The map $\mathcal{W}$ over the state space ${\cal H}={\cal H}_{\rm A} \otimes
\cal{H}_{\rm B}$ takes a state to a real number $\mathbb{R}$ \begin{equation}
\mathcal{W}:{\cal H}\longrightarrow \mathbb{R} \label{map} \end{equation} This
map is a nonclassicality witness map \textit{i.e.~} it is capable of detecting NCC states
in ${\cal H}$ state space if and only if~\cite{rahimi-pra-10}: \begin{itemize}
\item[(a)] For every bipartite state $\rho_{{\rm PCC}}$ having a product
eigenbasis, $\mathcal{W}(\rho_{{\rm PCC}}) \geq0$. \item[(b)] There exists at
least one bipartite state $\rho_{{\rm NCC}}$ (having no product eigenbasis) such
that $\mathcal{W}(\rho_{{\rm NCC}})<0$. \end{itemize} A specific non-linear
nonclassicality witness map proposed by~\cite{rahimi-pra-10} is defined in terms
of expectation values of positive Hermitian operators $ \rm \hat{A}_1$, $ \rm
\hat{A}_2\ldots\hat{A}_m $: \begin{equation} \mathcal{W}(\rho)= \rm
c-\left(Tr(\rho\hat{A}_1)\right)\left(Tr(\rho\hat{A}_2)\right)\ldots\ldots\left(Tr(\rho\hat{A}_m)\right)
\end{equation} where $ \rm c \geq 0 $ is a real number. For the case of
two-qubit systems using the operators $ \rm A_1=\vert 00 \rangle\langle 00 \vert
$ and $ \rm A_2 = \vert 1+\rangle\langle 1+\vert $ a nonclassicality witness map
can be obtained for state in Eq.~(\ref{sigma}) as: \begin{equation}
\mathcal{W}_{\sigma}(\rho) = \rm c-\left(Tr(\rho\vert00\rangle\langle
00\vert)\right)\left(Tr(\rho\vert1+\rangle\langle 1+\vert)\right)
\label{sigmamap} \end{equation} The value of the constant $ \rm c$ in the above
witness map has to be optimized such that for any PCC state $\rho$ having a
product eigenbasis, the condition $\mathcal{W}_{\sigma}(\rho) \ge 0 $ holds
\cite{rahimi-pra-10}. In order to optimize value of $ c $ in Eq.~\ref{sigmamap}
let us set $ \tau= \rm \vert s \rangle \langle s \vert \otimes \rho^B $ with $
\rm \vert s \rangle = \frac{1}{\sqrt{2}}( \vert 0 \rangle + e^{i \theta} \vert 1
\rangle)$; $ \theta $ be any angle and $ \rm \rho^B $ is a single qubit state.
Then, $ \rm f(\rho_{PCC}) = \left(Tr(\rho \vert 00 \rangle \langle 00 \vert)
\right)\left(Tr(\rho\vert 1+\rangle\langle 1+\vert )\right)$ is maximized for a
state written in the form of $\tau $ \textit{i.e.~} \begin{center} $ \rm
c_{opt}=\displaystyle\ \max_{\substack{\tau}} f(\tau)= \displaystyle\
\max_{\substack{\rho^B}}
\frac{1}{4}\langle0|\rho^B|0\rangle\langle+|\rho^B|+\rangle $ \end{center}
Further using \[ \rm \rho^B= \left[ \begin{array}{cc} \rm a & b \\ \rm b^* &
1-a \end{array} \right]\] with $ \rm 0\leq a\leq 1$ and $\rm b$ being a complex
number. $ \rm \rho^B$ is positive requiring $ \rm |b|\leq \sqrt{a(1-a)}$. So\\
$$
{\rm c}_{\rm opt}=\displaystyle\ \max_{\substack{a,b}}
\frac{a[2Re(b)+1]}{8}=\displaystyle\ \max_{\substack{a}}
\frac{a[1+2\sqrt{a(1-a)}]}{8}
$$
After this step it is only a
maximization process and the above expression takes its maximum value when $ \rm
a$ is $ \rm \hat{a}=\frac{2+\sqrt{2}}{4}$, which results in
$$
{\rm
c}_{\rm opt}=\frac{\hat{a}[1+2\sqrt{\hat{a}(1-\hat{a})}]}{8}=0.182138...
$$
The map given by Eq.~(\ref{sigmamap}) does indeed witness the
nonclassical nature of the state $\sigma$ as $ \rm
\left(Tr(\rho\vert00\rangle\langle 00\vert)\right)\left(Tr(\rho\vert
1+\rangle\langle 1+\vert)\right)$ for $\rho\equiv\sigma$ has the value 0.25,
which suggests that the state $\sigma$ is an NCC state~\cite{rahimi-pra-10}. The
value of a nonclassicality map, which when negative implicates the nonclassical
nature of the state, is defined as its map value (MV).
\subsection{NMR Experimental Setup For NCC Detection} \label{expt}
In order to implemented the nonclassicality witness map $\mathcal{W}_\sigma$ on an NMR
sample of $ ^{13} $C-enriched chloroform dissolved in acetone-D6; the $ ^{1} $H
and $ ^{13} $C nuclear spins were used to encode the two qubits (see
Fig.~\ref{molecule} for experimental parameters).
Unitary operations were implemented by specially crafted transverse radio
frequency pulses of suitable amplitude, phase and duration. Since a
heteronuclear $ ^{1}$H-$^{13}$C spin system was used to encode the qubits,
standard pulse calibration methods available on the dedicated NMR spectrometer
software were used for pulse optimization and gave accurate results. A sequence
of spin-selective pulses interspersed with tailored free evolution periods were
used to prepare the system in an NCC state as described below, written using
spin-angular momentum operators:
\begin{eqnarray*}
&& \rm I_{1z}+I_{2z} \stackrel{(\pi/2)^1_x}{\longrightarrow}-I_{1y}+I_{2z}\stackrel{Sp. Av.}{\longrightarrow} I_{2z} \stackrel{(\pi/2)^2_y}{\longrightarrow} I_{2x} \stackrel{\frac{1}{4J}}{\longrightarrow} \nonumber\\ && \quad\quad \rm \frac{I_{2x}+2I_{1z}I_{2y}}{\sqrt{2}}\stackrel{(\pi/2)^2_x}{\longrightarrow} \frac{I_{2x}+2I_{1z}I_{2z}}{\sqrt{2}} \stackrel{(-\pi/4)^2_y}{\longrightarrow} \nonumber \\ && \quad\quad\quad\quad \rm \frac{\left(I_{2z}+I_{2x}+2I_{1z}I_{2z}-2I_{1z}I_{2x}\right)}{2}
\end{eqnarray*}
One can begin with the system in thermal equilibrium and ignore the identity
part of the density matrix, which does not evolve under RF (radio frequency)
pulses. The RF pulses $( \rm \alpha)^{i}_{j}$ are written above each arrow, with
$\alpha $ denoting the pulse flip angle, $ \rm i=1,2 $ denoting the qubit on
which the pulse is being applied and $ \rm j=x,y,z $ being the axis along
\begin{figure}
\caption{(a) Molecular structure of $^{13}
\label{molecule}
\end{figure}
which the pulse is applied. Spatial averaging (denoted by Sp. Av.) is achieved via a
dephasing z-gradient. The NMR spectra of the thermal state and the prepared NCC
state are shown in Fig.~\ref{molecule}(b), and the corresponding pulse sequence
is depicted in Fig.~\ref{ckt}(b).
\begin{figure}
\caption{(a) Quantum circuit and (b) NMR pulse sequence to create and detect an
NCC state. Unfilled rectangles depict $\frac{\pi}
\label{ckt}
\end{figure}
The quantum circuit to implement the nonclassicality witness map is shown in
Fig.~\ref{ckt}(a). The first module represents NCC state preparation using the
pulses as already described. The circuit to capture nonclassicality of the
prepared state consists of a controlled-Hadamard (CH) gate, followed by
measurement on both qubits, a CNOT gate and finally detection
on `Qubit 2'. The CH gate is analogous to a CNOT gate, with a Hadamard gate
being implemented on the target qubit if the control qubit is in the state
$\vert 1 \rangle$ and a `no-operation' if the control qubit is in the state
$\vert 0 \rangle$. The NMR pulse sequence corresponding to the quantum circuit
is depicted in Fig.~\ref{ckt}(b). The set of pulses grouped under the label
`State prep.' convert the thermal equilibrium state to the desired NCC state. A
dephasing z-gradient is applied on the gradient channel to kill undesired
coherences. After a delay $\tau$ followed by the pulse sequence to implement
the CH gate, the magnetizations of both qubits were measured with
$\frac{\pi}{2}$ readout pulses (not shown in the figure). In the last part of
detection circuit a CNOT gate is applied followed by a magnetization measurement
of `Qubit 2'; the scalar coupling time interval was set to $ \rm
\tau_{12}=\frac{1}{4J}$ where J is the strength of the scalar coupling between
the qubits. Refocusing pulses were used during all J-evolution to compensate for
unwanted chemical shift evolution during the selective pulses. State fidelity
was computed using the Uhlmann-Jozsa
measure~\cite{uhlmann-rpmp-76,jozsa-jmo-94}(also see Eq.~(\ref{fidelity_eq})), and
the NCC state was prepared with a fidelity of 0.97 $\pm$ 0.02.
To detect the nonclassicality in the prepared NCC state via the map
$\mathcal{W}_\sigma $, the expectation values of the operators $\vert
00\rangle\langle 00\vert$ and $\vert 1+\rangle\langle 1+\vert$ are required.
Re-working the map brings it to the following form~\cite{rahimi-pra-10}
\begin{eqnarray}\label{SeqFID_eq} \rm \mathcal{W}_{\sigma}(\rho) =
c_{opt}-\frac{1}{16}\left(1+\langle Z_1 \rangle+\langle Z_2\rangle+\langle
Z_2'\rangle\right)\times \left(1-\langle Z_1 \rangle+\langle Z_2\rangle-\langle
Z_2'\rangle\right) \end{eqnarray} where $ \rm \langle Z_1\rangle$ and $ \rm
\langle Z_2\rangle$ are the magnetizations of `Qubit 1' and `Qubit 2' after a CH
gate on the input state $\rho$, while $ \rm \langle Z_2'\rangle$ is the
magnetization of `Qubit 2' after a CNOT gate. The theoretically expected
normalized values of $ \rm \langle Z_1\rangle$, $ \rm \langle Z_2\rangle$ and $
\rm \langle Z_2'\rangle$ for state $\rho\equiv\sigma$ are $0$, $1$ and $0$
respectively. Map value (MV) is $-0.067862<0$ and as desired this map does
indeed witness the presence of nonclassicality. The experimentally computed MV
for the prepared NCC state turns out to be $-0.0406 \pm 0.0056$, proving that
the map is indeed able to witness the nonclassicality present in the state.
\subsection{Map Value Dynamics} \label{mv-dyn}
The prepared NCC state was allowed to evolve freely in time and the MV calculated at each time instant, in
order to characterize the decoherence dynamics of the nonclassicality witness
map. As theoretically expected, one should get a negative MV for states which
are NCC. MV was measured at time instants which were integral multiples of $ \rm
\frac{2}{J} $ \textit{i.e.~} $ \rm \frac{2n}{J} $ (with $ \rm n $ = 0, 1, 3, 5, 7, 9, 11,
13, 15, 20, 25, 30, 35, 40, 45 and 50), in order to avoid experimental errors
due to J-evolution. The results of experimental MV dynamics as a function of
time are shown in Fig.\ref{MV-NCC}(a). Experiments were repeated eight times to
estimate the errors as depicted in the figure. As seen from
Fig.~\ref{MV-NCC}(a), the MV remains negative (indicating the state is NCC) for
up-to 120 ms, which is approximately the $^{1}$H transverse relaxation time. The
standard NMR decoherence mechanisms are denoted by T$_2$ the spin-spin
relaxation time which causes dephasing among the energy eigenstates and T$_1$
the spin-lattice relaxation time, which causes energy exchange between the spins
and their environment. For comparison, the MV was also calculated directly using
Eq.~(\ref{sigmamap}) with $ \rm c = c_{opt}$, from the state which was
tomographically reconstructed at each time instant via full state
tomography~\cite{leskowitz-pra-04}. The results are shown in
Fig.~\ref{MV-NCC}(b), which are in good agreement with direct experimental MV
measurements. The state fidelity was also computed at the different time
instants and the results are shown in Fig.~\ref{MV-NCC}(c). The red squares in
Fig.~\ref{MV-NCC}(c) represent state fidelity of the experimental state $
\sigma_{{\rm exp}(t)}$ evolving in time, w.r.t. the theoretical NCC state
$\sigma_{{\rm theo}}(0) $ at time $ \rm t=0$ given in Eq.~(\ref{sigma}).
\begin{figure}
\caption{(a) Experimental
map value (in $\times10^{-2}
\label{MV-NCC}
\end{figure}
\subsection{Quantum Discord Dynamics}
\label{qd-dyn}
Map value evaluation of nonclassicality was also compared with the standard measure of nonclassicality, namely QD~\cite{ollivier-prl-02,luo-pra-08}. The state was reconstructed by
peRForming full quantum state tomography and the QD measure was computed from
the experimental data. Quantum mutual information can be quantified by the
equations: \begin{eqnarray} \rm I(\rho_{AB})&=& \rm
S(\rho_A)+S(\rho_B)-S(\rho_{AB}) \label{quantum-I} \nonumber \\ \rm
J_A(\rho_{AB})&=& \rm S(\rho_B)-S(\rho_B\vert \rho_A) \label{quantum-J}
\label{mutual} \end{eqnarray} where $ \rm S(\rho_B\vert\rho_A)$ is the
conditional von Neumann entropy of subsystem $B$ when $A$ has already been
measured. QD is defined as the minimum difference between the two formulations
of mutual information in Eq.~(\ref{mutual}): \begin{equation} \rm
D_A(\rho_{AB})=S(\rho_A)-S(\rho_{AB})+S(\rho_B\vert\lbrace\Pi^A_j\rbrace)
\label{discord} \end{equation} QD hence depends on projectors $ \rm
\lbrace\Pi^A_j\rbrace$. The state of the system, after the outcome corresponding
to projector $ \rm \lbrace\Pi^A_j\rbrace$ has been detected, is \begin{equation}
\rm \tilde{\rho}_{AB}\vert \lbrace\Pi^A_j\rbrace=\frac{\left(\Pi^A_j\otimes
I_B\right)\rho_{AB}\left(\Pi^A_j\otimes I_B\right)}{p_j} \label{B}
\end{equation} with the probability $ \rm p_j=Tr\left((\Pi^A_j\otimes
I_B)\rho_{AB}(\Pi^A_j\otimes I_B)\right)$; $ \rm I_B$ is identity operator on
subsystem B. The state of the system B, after this measurement is
\begin{equation} \rm \rho_B\vert
\lbrace\Pi^A_j\rbrace=Tr_A\left(\tilde{\rho}_{AB}\vert
\lbrace\Pi^A_j\rbrace\right) \label{A} \end{equation} $ \rm
S\left(\rho_B\vert\lbrace\Pi^A_j\rbrace\right)$ is the missing information about
B before measurement $ \rm \lbrace\Pi^A_j\rbrace$. The expression
\begin{equation} \rm S(\rho_B\vert\lbrace\Pi^A_j\rbrace)=\sum_{j}{p_jS
\left(\rho_B\vert\lbrace\Pi^A_j\rbrace\right)} \label{cond-entropy}
\end{equation} is the conditional entropy appearing in Eq.~(\ref{discord}). In
order to capture the true quantumness of the correlation one needs to peRForm an
optimization over all sets of von-Neumann type measurements represented by the
projectors $ \rm \lbrace\Pi^A_j\rbrace$. One can define two orthogonal vectors
(for spin half quantum subsystems), characterized by two real parameters
$\theta$ and $\phi$, on the Bloch sphere as: \begin{eqnarray} && \rm
\cos{\theta}\vert 0\rangle+e^{\textit{i}\phi}\sin{\theta}\vert 1\rangle
\nonumber \\ && \rm e^{-\textit{i}\phi}\sin{\theta}\vert
0\rangle-\cos{\theta}\vert 1\rangle \end{eqnarray} These vectors can be used to
construct the projectors $\rm \Pi^{A,B}_{1,2}$, which are then used to find out
the state of B after an arbitrary measurement was made on subsystem A. The
definition of conditional entropy (Eq.~(\ref{cond-entropy})) can be used to
obtain an expression which is parameterized by $\theta$ and $\phi$ for a given
state $\rm \rho_{AB}$. This expression is finally minimized by varying $\theta$
and $\phi$ and the results fed back into Eq.~(\ref{discord}), which yields a
measure of QD independent of the basis chosen for the measurement of the
subsystem.
To compare the detection via the positive map method with the standard QD
measure, the state was let evolve for a time $\tau$ and then reconstructed the
experimentally prepared via full quantum state tomography and calculated the QD
at all time instants where the MV was determined experimentally (the results are
shown in Fig.~\ref{MV-NCC}(d)). At $\tau$ = 0 s, a non-zero QD confirms the
presence of NCC and verifies the results given by MV. As the state evolves with
time, the QD parameter starts decreasing rapidly, in accordance with increasing
MV. Beyond 120 ms, while the MV becomes positive and hence fails to detect
nonclassicality, the QD parameter remains non-zero, indicating the presence of
some amount of nonclassicality (although by this time the state fidelity has
decreased to 0.7). However, value of QD is very close to zero and in fact cannot
be distinguished from contributions due to noise. One can hence conclude that
the positive map suffices to detect nonclassicality when decoherence processes
have not set in and while the fidelity of the prepared state is good. Once the
state has decohered however, a measure such as QD has to be used to verify if
the degraded state retains some amount of nonclassical correlations or not.
While the constructed nonclassicality witness is not optimal and hence cannot be
quantitatively compared with a stricter measure of nonclassicality such as a
measurement of the QD parameter, in most cases the witness suffices to detect
the presence of nonclassicality in a quantum state without having to resort to
more complicated experimental schemes.
\section{Conclusions} In the work described in this
chapter, nonclassical correlations were detected experimentally in a separable
two-qubit quantum state, using a nonlinear positive map as a nonclassicality
witness. The witness is able to detect nonclassicality and its obvious
advantage lies in its using much fewer experimental resources as compared to
quantifying nonclassicality by measuring quantum discord via full quantum state
tomography. It will be interesting to construct and utilize this map in
higher-dimensional quantum systems and for multi-qubits, where it is more challenging to distinguish between classical and quantum correlations. It has
been posited that quantum correlations, which can go beyond quantum entanglement
(and are captured by quantum discord and can thus be present even in separable
states), are responsible for achieving computational speedup in quantum
algorithms. It is important from the point of view of quantum information processing, to confirm the presence of such correlations in a quantum state,
without having to expend too much experimental effort.
The work described in this chapter is a step forward in this direction. Results of this chapter are contained in \href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.062318}{\rm Phys. Rev. A \textbf{95}, 062318 (2017)}.
\chapter{ Experimental Classification of Entanglement in Arbitrary Three-Qubit States}\label{chapter_3QEntDet}
\section{Introduction}
This chapter extends the goal of entanglement detection from a bipartite scenario, reported in Chapter \ref{chap2}, to multipartite case. Experimental characterization of arbitrary three-qubit pure states is undertaken. The three-qubit states can be classified into six inequivalent classes~\cite{dur-pra-00} under SLOCC~\citep{bennett-pra-00}. Protocols have been invented to carry out the classification of three-qubit states into the SLOCC classes~\cite{chi-pra-2010,zhao-pra-2013}. A recent proposal~\citep{datta-epd-18} aims to classify any three-qubit pure entangled state into these six inequivalent classes by measuring only four observables. Previously constructed scheme~\cite{dogra-pra-15} to experimentally realize a canonical form for general three-qubit states, is used to prepare arbitrary three-qubit states with an unknown amount of entanglement. Experimental implementation of the entanglement detection protocol is such that a single-shot (using only four experimental settings) is able to determine if a state belongs to the $\rm W $ class or to the ${\rm GHZ}$ class. Schemes are devised to map the desired observables onto the $z$-magnetization of one of the subsystems, making it possible to experimentally measure its expectation value on NMR systems~\citep{singh-pra-16}. Mapping of the observables onto Pauli $z$-operators of a single qubit eases the experimental determination of the desired expectation value, since the NMR signal is proportional to the ensemble average of the Pauli $z$-operator.
The protocol has been tested on known three-qubit entangled states such as the ${\rm GHZ}$ state and the $W$ state as well as on randomly generated arbitrary three-qubit states with an unknown amount of entanglement. Seven representative states belonging to the six SLOCC inequivalent classes as well as twenty random states were prepared experimentally, with state fidelities ranging between 89\% to 99\%. To decide the entanglement class of a state, the expectation values of four observables were experimentally measured in the state under investigation. All the seven representative states (namely, GHZ, W, $\rm W\overline{W}$, three bi-separable states and a separable state) were successfully detected within the experimental error limits. Using this protocol, the experimentally randomly generated arbitrary three-qubit states were correctly identified as belonging to either the GHZ, the W, the bi-separable or the separable class of states. Full quantum state tomography is performed to directly compute the observable value. Reconstructed density matrices were used to calculate the entanglement by computing negativity in each case, and the results compared well with those of the current protocol.
\section{Detecting Tripartite Entanglement} \label{Theory} There are six SLOCC inequivalent classes of entanglement in three-qubit systems, namely, the GHZ, W, three different biseparable classes and the separable class~\cite{dur-pra-00}. A widely used measure of entanglement is the $n$-tangle~\cite{wong-pra-01,li-qip-12} and a non-vanishing three-tangle is a signature of the GHZ entangled class and can hence be used for their detection. For three parties A, B and C, the three-tangle $\tau$ is defined as
\begin{equation}
\label{3tangle}
\tau=C^{2}_{ \rm {A(BC)}}-C^{2}_{\rm{AB}}-C^{2}_{\rm{AC}}
\end{equation}
with $C^{}_{\rm{AB}}$ and $C^{}_{\rm{AC}}$ being the concurrence that characterizes entanglement between A and B, and between A and C respectively; $C^{}_{\rm{A(BC)}}$ denotes the concurrence between A and the joint state of the subsystem comprising B and C~\cite{coffman-pra-00}. It was shown~\cite{coffman-pra-00} that for a pure state of form $ \vert \xi \rangle= \sum _{i,j,k=0}^{1} b_{ijk}\vert ijk \rangle $ with $ \sum _{i,j,k=0}^{1} \vert b_{ijk} \vert ^2 =1 $ the quadratic expression in concurrence can be written as
\begin{eqnarray}\label{concu}
C^{2}_{\rm {A(BC)}}-C^{2}_{\rm{AB}}-C^{2}_{\rm {AC}}=4\vert d_1 -2d_2+4d_3\vert
\end{eqnarray}
with
\begin{eqnarray}\label{dls}
d_1 &=& b_{000}^2b_{111}^2 + b_{001}^2b_{110}^2 + b_{010}^2b{101}^2 + b_{100}^2b_{011}^2 \nonumber\\
d_2 &=& b_{000}b_{111}b_{011}b_{100} + b_{000}b_{111}b_{101}b_{010} + b_{000}b_{111}b_{110}b_{001} \nonumber\\
&\textcolor[rgb]{1,1,1}{=}& + b_{011}b_{100}b_{101}b_{010} + b_{011}b_{100}b_{110}b_{001} + b_{101}b_{010}b_{110}b_{001} \nonumber\\
d_3 &=& b_{000}b_{110}b_{101}b_{011} + b_{111}b_{001}b_{010}b_{100}
\end{eqnarray}
The idea of using the three-tangle to investigate entanglement in three-qubit generic states is particularly interesting and general, as any three-qubit pure state can be written in the canonical form~\cite{acin-prl-01}\begin{equation}
\label{generic}
\vert\psi\rangle=a_0\vert 000 \rangle + a_1e^{\textit{i}
\theta}\vert 100 \rangle + a_2\vert 101 \rangle + a_3\vert
110 \rangle + a_4\vert 111 \rangle
\end{equation}
where $a_i\geq 0$, $\sum_i a^2_i=1$ and $\theta \in [0,\pi]$, and the class of states is written in the computational basis $\{ \vert 0 \rangle, \vert 1 \rangle \}$ of the qubits. On comparing the coefficients of general three-qubit pure state, $ \vert \xi \rangle $ with the generic state $ \vert \psi \rangle $ one can observe that $b_{000}=a_0$, $b_{100}=a_1e^{i\theta}$, $b_{101}=a_2$, $b_{110}=a_3$ and $b_{111}=a_4$ and hence can compute $d_l$ in Eq.~(\ref{dls}). On using resulting $d_l$ in Eq.~(\ref{concu}) the three-tangle for the generic state Eq.~(\ref{generic}) turns out to be~\citep{datta-epd-18}
\begin{equation}
\tau_{\psi}=4a^2_0a^2_4
\end{equation}
Three-tangle can be measured experimentally by measuring the expectation value of the operator $O=2\sigma^{}_{1x}\sigma^{}_{2x}\sigma^{}_{3x}$, in the three-qubit state $\vert \psi \rangle$. Here $\sigma^{}_{x,y,z}$ are the Pauli matrices and $i=1,2,3$ denotes qubits label and the tensor product symbol between the Pauli operators has been omitted for brevity. One can compute
\begin{gather}\label{three-tang}
\langle \Psi \vert O \vert \Psi \rangle=
\begin{bmatrix}
a_0 & 0 & 0 & 0 & a_1e^{-i\theta} & a_2 & a_3 & a_4
\end{bmatrix}
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\begin{bmatrix}
a_0 \\
0 \\
0 \\
0 \\
a_1e^{i\theta} \\
a_2 \\
a_3 \\
a_4
\end{bmatrix}
\end{gather}
and this results in $\langle \psi \vert O \vert \psi \rangle^2= \langle O \rangle^{2}_{\psi}= 4\tau_\psi$. A non-zero expectation value of $O$ implies that the state under investigation is in the GHZ class~\citep{dur-pra-00}. In order to further categorize the classes of three-qubit generic states three more observables are defined as $O^{}_1=2\sigma^{}_{1x}\sigma^{}_{2x}\sigma^{}_{3z}$, $O^{}_2=2\sigma^{}_{1x}\sigma^{}_{2z}\sigma^{}_{3x}$, $O^{}_3=2\sigma^{}_{1z}\sigma^{}_{2x}\sigma^{}_{3x}$. Experimentally measuring the expectation values of the operators $O$, $O^{}_1$, $O^{}_2$ and $O^{}_3$ can reveal the entanglement class of every three-qubit pure state~\cite{zhao-pra-2013,datta-epd-18}. Table~\ref{classification table} summarizes the classification of the six SLOCC inequivalent classes of entangled states based on the expectation values of the observables $O$, $O^{}_1$, $O^{}_2$, $O^{}_3$.
\begin{table}[h]
\begin{center}
\caption{\label{classification table}
Decision table for the classification of three-qubit pure
entangled states based on the expectation values of
operators $O$, $O^{}_1$, $O^{}_2$ and $O^{}_3$ in state $
\vert \psi \rangle $. Each class in the row is shown with
the expected values of the observables.}
\begin{tabular}{c | r r r r }
\hline
\textrm{Class} &
\textrm{$\langle O \rangle$}&
\textrm{$\langle O^{}_1 \rangle$} &
\textrm{$\langle O^{}_2 \rangle$} &
\textrm{$\langle O^{}_3 \rangle$} \\
\hline
\hline
GHZ & $\neq 0$ & $*$ & $*$ & $*$ \\
W & 0 & $\neq 0$ & $\neq 0$ & $\neq 0$ \\
BS$_1$ & 0 & 0 & 0 & $\neq 0$ \\
BS$_2$ & 0 & 0 & $\neq 0$ & 0 \\
BS$_3$ & 0 & $\neq 0$ & 0 & 0 \\
Separable & 0 & 0 & 0 & 0 \\
\hline
\hline
\end{tabular}
\end{center}
\begin{center}
$*$ May or may not be zero.
\end{center}
\end{table}
The six SLOCC inequivalent classes of three-qubit entangled states are GHZ, W, BS${}_1$, BS${}_2$, BS${}_3$ and separable. While GHZ and W classes are well known, BS$_1$ denotes a biseparable class having B and C subsystems entangled, the BS$_2$ class has subsystems A and C entangled, while the BS$_3$ class has subsystems A and B entangled. As has been summarized in Table~\ref{classification table} a non-zero value of $\langle O \rangle$ indicates that the state is in the GHZ class and this expectation value is zero for all other classes. For the W class of states all $\langle O_j\rangle$ are non-zero except $\langle O\rangle$. For the BS${}_1$ class only $\langle O_3\rangle$ is non-zero while only $\langle O_2\rangle$, and $\langle O_1\rangle$ are non-zero for the classes BS${}_2$ and BS${}_3$, respectively. For separable states all expectations are zero.
In order to experimentally realize the entanglement characterization protocol, one has to determine the expectation values $\langle O \rangle$, $\langle O_1 \rangle$, $\langle O_2 \rangle$ and $\langle O_3 \rangle$ for an experimentally prepared state $\vert \psi \rangle$. Next section describes the method to experimentally realize these expectation values based on subsystem measurement of the Pauli $z$-operator~\cite{singh-pra-16} and scheme for generating arbitrary three-qubit states~\cite{dogra-pra-15}.
\subsection{Mapping Pauli basis operators to single-qubit $z$-operators}
\label{Mapping}
A standard way to determine the expectation value of a desired observable in an experiment is to decompose the observable as a linear superposition of the observables accessible in the experiment~\cite{nielsen-book-02}. This task becomes particularly accessible while dealing with the Pauli basis. Any observable for a three-qubit system, acting on an eight-dimensional Hilbert space can be decomposed as a linear superposition of 64 basis operators, and the Pauli basis is one possible basis for this decomposition. Let the set of Pauli basis operators be denoted as $\mathbb{B}=\{ \rm{B}_i; 0\leq i \leq 63\}$. For example, $O^{}_2$ has the form $\sigma^{}_{1x}\sigma^{}_{2z}\sigma^{}_{3x}$ and it is the element B$_{29}$ of the basis set $\mathbb{B}$. The four observables $O$, $O^{}_1$, $O^{}_2$ and $O^{}_3$ are represented by the elements B$_{21}$, B$_{23}$, B$_{29}$ and B$_{53}$ respectively of the Pauli basis set $\mathbb{B}$. Also by this convention the single-qubit $z$-operators for the first, second and third qubit \textit{i.e.~} $\sigma^{}_{1z}$, $\sigma^{}_{2z}$ and $\sigma^{}_{3z}$ are the elements B$_{48}$, B$_{12}$ and B$_{3}$ respectively.
\begin{figure}
\caption{(a) Quantum circuit to achieve mapping of the state $ \rho $ to either of the states $ \rho^{}
\label{ckt+seq3Q}
\end{figure}
Table~\ref{Mapping Table} details the mapping of all 63 Pauli basis operators (excluding the 8$\otimes$8 identity operator) to the single-qubit Pauli $z$-operator. This mapping is particularly useful in an experimental setup where the expectation values of Pauli local $z$-operators are easily accessible. In NMR experiments, the $z$-magnetization of a nuclear spin in a state is proportional to the expectation value of Pauli $z$-operator of that spin in the state.
As an example of the mapping given in Table~\ref{Mapping Table}, the operator $O^{}_2$ has the form $\sigma^{}_{1x}\sigma^{}_{2z}\sigma^{}_{3x}$ and is the element B$_{29}$ of basis set $\mathbb{B}$. In order to determine $\langle O^{}_2 \rangle $ in the state $\rho=\vert \psi \rangle \langle \psi \vert$, one can map the state $\rho \rightarrow \rho^{}_{29}=U^{}_{29}.\rho.U^{\dagger}_{29} $ with $U^{}_{29}= {\rm CNOT}_{23}.\overline{Y}_3.$ ${\rm CNOT}_{12}.\overline{Y}_1 $. This is followed by finding $\langle \sigma^{}_{3z} \rangle $ in the state $\rho^{}_{29}$. The expectation value $\langle \sigma^{}_{3z} \rangle $ in the state $\rho^{}_{29}$ is equivalent to the expectation value of $\langle O^{}_2 \rangle $ in the state $\rho=\vert \psi \rangle \langle \psi \vert $(Table~\ref{Mapping Table}); the operation ${\rm CNOT}_{kl}$ is a controlled-NOT gate with $k$ as the control qubit and $l$ as the target qubit, and $X$, $\overline{X}$, $Y$ and $\overline{Y}$ represent local $\frac{\pi}{2}$ unitary rotations with phases $x$, $-x$, $y$ and $-y$ respectively. The subscript on $\pi/2$ local unitary rotations denotes qubit number. The quantum circuit to achieve such a mapping is shown in Fig.~\ref{ckt+seq3Q}(a).
It should be noted that while measuring the expectation values of $O$, $O^{}_1$, $O^{}_2$ or $O^{}_3$, all the $\overline{Y}$ local rotations may not act in all these four cases. The mapping given in Table~\ref{Mapping Table} is used to decide which $\overline{Y}$ local rotation in the circuit~\ref{ckt+seq3Q}(a) will act. All the basis operators in set $\mathbb{B}$ can be mapped to single-qubit $z$-operators in a similar fashion. The mapping given in Table~\ref{Mapping Table} is not unique and there are several equivalent mappings which can be worked out as per the experimental requirements.
\section{NMR Implementation of Three-Qubit Entanglement Detection Protocol}
\label{NMR Implementation}
The Hamiltonian~\citep{ernst-book-90} for a three-qubit system in the rotating frame is given by
\begin{equation}\label{Hamiltonian}
\mathcal{H}= -\sum_{i=1}^{3} \nu_i I_{iz} + \sum_{i>j,i=1}^{3} J_{ij}I_{iz}I_{jz}
\end{equation}
where the indices $i,j=$ 1,2 or 3 represent the qubit number and $\nu_i$ is the respective chemical shift in rotating frame, $J_{ij}$ is the scalar coupling constant and $I_{iz}$ is the Pauli $z$-spin angular momentum operator of the $i^{\rm{th}}$ qubit. To implement the entanglement detection protocol experimentally, $^{13}$C labeled diethylfluoromalonate dissolved in acetone-D6 sample was used. $^{1}$H, $^{19}$F and $^{13}$C spin-half nuclei were encoded as qubit 1, qubit 2 and qubit 3 respectively. The system was initialized in the pseudopure (PPS) state \textit{i.e.~} $\vert 000 \rangle$ using the spatial averaging~\cite{cory-physD-98,mitra-jmr-07} with the density operator being
\begin{equation}
\rho_{000}=\frac{1-\epsilon}{2^3}\mathbb{I}_8 +\epsilon
\vert 000 \rangle \langle 000 \vert
\end{equation}
where $\epsilon \approx 10^{-5}$ is the thermal polarization at room temperature and $ \mathbb{I}_8 $ is the 8 $ \times $ 8 identity operator. The experimentally determined NMR parameters (chemical shifts, T$_1$ and T$_2$ relaxation times and scalar couplings $\rm{J}_{ij}$) as well as the NMR spectra of the PPS state are shown in Fig.~\ref{molecule3Q}. Each spectral transition is labeled with the logical states of the passive qubits (\textit{i.e.~} qubits not undergoing any transition) in the computational basis. The state fidelity of the experimentally prepared PPS (Fig.~\ref{molecule3Q}(c)) was compute to be 0.98$\pm$0.01 and was calculated using the fidelity measure \citep{uhlmann-rpmp-76,jozsa-jmo-94} (also see Eqn.~\ref{fidelity_eq}). For the experimental reconstruction of density operator, full quantum state tomography (QST)\citep{leskowitz-pra-04,singh-pla-16} was performed using a preparatory pulse set $\left\lbrace III, XXX, IIY, XYX, YII, XXY, IYY \right\rbrace$, where $I$ implies ``no operation''. In NMR a $\frac{\pi}{2} $ local unitary rotation $X$($Y$) can be achieved using spin-selective transverse radio frequency (RF) pulses having phase $x$($y$).
\begin{figure}
\caption{(a) Molecular structure of $^{13}
\label{molecule3Q}
\end{figure}
Experiments were performed at room temperature ($293$K) on a Bruker Avance III 600-MHz FT-NMR spectrometer equipped with a QXI probe. Local unitary operations were achieved using highly accurate and calibrated spin selective transverse RF pulses of suitable amplitude, phase and duration. Non-local unitary operation were achieved by free evolution under the system Hamiltonian Eq.~(\ref{Hamiltonian}), of suitable duration under the desired scalar coupling with the help of embedded $\pi$ refocusing pulses. In the current study, the durations of $\frac{\pi}{2}$ pulses for $^{1}$H, $^{19}$F and $^{13}$C were 9.55 $\mu$s at 18.14 W power level, 22.80 $\mu$s at a power level of 42.27 W and 15.50 $\mu$s at a power level of 179.47 W, respectively.
\subsection{Measuring Observables by Mapping to Local $z$-Magnetization}\label{NMR Mapping}
As discussed in Sec.~\ref{Mapping}, the observables required to differentiate between six inequivalent classes of three-qubit pure entangled states can be mapped to the Pauli $z$-operator of one of the qubits. Further, in NMR the observed $z$-magnetization of a nuclear spin in a quantum state is proportional to the expectation value of $\sigma_{z} $-operator~\citep{ernst-book-90} of the spin in that state. The time-domain NMR signal \textit{i.e.~} the free induction decay with appropriate phase gives Lorentzian peaks when Fourier transformed. These normalized experimental intensities give an estimate of the expectation value of $\sigma_{z}$ of the quantum state.
\begin{table}
\caption{\label{Mapping Table}
All sixty three product operators, for a three spin (half) system, mapped to the Pauli $z$-operators (of either spin 1, spin 2 or spin 3) by mapping initial state $ \rho \rightarrow \rho_i=U_i.\rho.U_i^{\dagger} $.}
\scriptsize
\begin{tabular}{l l l l}
\hline
\textrm{Observable}&
\textrm{Initial State Mapped via}&
\textrm{Observable}&
\textrm{Initial State Mapped via}\\
\hline
\hline
$\langle B_{1} \rangle$ = Tr[$\rho_{1}.I_{3z}$] & $ U_{1}=\overline{Y}_3 $ & $\langle B_{33} \rangle$ = Tr[$\rho_{33}.I_{3z}$] & $ U_{33}={\rm CNOT}_{13}.\overline{Y}_3.X_1 $ \\
$\langle B_{2} \rangle$ = Tr[$\rho_{2}.I_{3z}$] & $ U_{2}=X_3 $ & $\langle B_{34} \rangle$ = Tr[$\rho_{34}.I_{3z}$] & $ U_{34}={\rm CNOT}_{13}.X_3.X_1 $ \\
$\langle B_{3} \rangle$ = Tr[$\rho_{3}.I_{3z}$] & $ U_{3}=\mathbb{I}_8 $ & $\langle B_{35} \rangle$ = Tr[$\rho_{35}.I_{3z}$] & $ U_{35}={\rm CNOT}_{13}.X_1 $ \\
$\langle B_{4} \rangle$ = Tr[$\rho_{4}.I_{2z}$] & $ U_{4}=\overline{Y}_2 $ & $\langle B_{36} \rangle$ = Tr[$\rho_{36}.I_{2z}$] & $ U_{36}={\rm CNOT}_{12}.\overline{Y}_2.X_1 $ \\
$\langle B_{5} \rangle$ = Tr[$\rho_{5}.I_{3z}$] & $ U_{5}={\rm CNOT}_{23}.\overline{Y}_3.\overline{Y}_2 $ & $\langle B_{37} \rangle$ = Tr[$\rho_{37}.I_{3z}$] & $ U_{37}={\rm CNOT}_{23}.\overline{Y}_3.{\rm CNOT}_{12}.\overline{Y}_2.X_1 $ \\
$\langle B_{6} \rangle$ = Tr[$\rho_{6}.I_{3z}$] & $ U_{6}={\rm CNOT}_{23}.X_3.\overline{Y}_2 $ & $\langle B_{38} \rangle$ = Tr[$\rho_{38}.I_{3z}$] & $ U_{38}={\rm CNOT}_{23}.X_3.{\rm CNOT}_{12}.\overline{Y}_2.X_1 $ \\
$\langle B_{7} \rangle$ = Tr[$\rho_{7}.I_{3z}$] & $ U_{7}={\rm CNOT}_{23}.\overline{Y}_2 $ & $\langle B_{39} \rangle$ = Tr[$\rho_{39}.I_{3z}$] & $ U_{39}={\rm CNOT}_{23}.{\rm CNOT}_{12}.\overline{Y}_2.X_1 $ \\
$\langle B_{8} \rangle$ = Tr[$\rho_{8}.I_{2z}$] & $ U_{8}=X_2 $ & $\langle B_{40} \rangle$ = Tr[$\rho_{40}.I_{2z}$] & $ U_{40}={\rm CNOT}_{12}.X_2.X_1 $ \\
$\langle B_{9} \rangle$ = Tr[$\rho_{9}.I_{3z}$] & $ U_{9}={\rm CNOT}_{23}.\overline{Y}_3.X_2 $ & $\langle B_{41} \rangle$ = Tr[$\rho_{41}.I_{3z}$] & $ U_{41}={\rm CNOT}_{23}.\overline{Y}_3.{\rm CNOT}_{12}.X_2.X_1 $ \\
$\langle B_{10} \rangle$ = Tr[$\rho_{10}.I_{3z}$] & $ U_{10}={\rm CNOT}_{23}.X_3.X_2 $ & $\langle B_{42} \rangle$ = Tr[$\rho_{42}.I_{3z}$] & $ U_{42}={\rm CNOT}_{23}.X_3.{\rm CNOT}_{12}.X_2.X_1 $ \\
$\langle B_{11} \rangle$ = Tr[$\rho_{11}.I_{3z}$] & $ U_{11}={\rm CNOT}_{23}.X_2 $ & $\langle B_{43} \rangle$ = Tr[$\rho_{43}.I_{3z}$] & $ U_{43}={\rm CNOT}_{23}.{\rm CNOT}_{12}.X_2.X_1 $ \\
$\langle B_{12} \rangle$ = Tr[$\rho_{12}.I_{3z}$] & $ U_{12}=\mathbb{I}_8 $ & $\langle B_{44} \rangle$ = Tr[$\rho_{44}.I_{2z}$] & $ U_{44}={\rm CNOT}_{12}.X_1 $ \\
$\langle B_{13} \rangle$ = Tr[$\rho_{13}.I_{3z}$] & $ U_{13}={\rm CNOT}_{23}.\overline{Y}_3 $ & $\langle B_{45} \rangle$ = Tr[$\rho_{45}.I_{3z}$] & $ U_{45}={\rm CNOT}_{23}.\overline{Y}_3.{\rm CNOT}_{12}.X_1 $ \\
$\langle B_{14} \rangle$ = Tr[$\rho_{14}.I_{3z}$] & $ U_{14}={\rm CNOT}_{23}.X_3 $ & $\langle B_{46} \rangle$ = Tr[$\rho_{46}.I_{3z}$] & $ U_{46}={\rm CNOT}_{23}.X_3.{\rm CNOT}_{12}.X_1 $ \\
$\langle B_{15} \rangle$ = Tr[$\rho_{15}.I_{3z}$] & $ U_{15}={\rm CNOT}_{23} $ & $\langle B_{47} \rangle$ = Tr[$\rho_{47}.I_{3z}$] & $ U_{47}={\rm CNOT}_{23}.{\rm CNOT}_{12}.X_1 $ \\
$\langle B_{16} \rangle$ = Tr[$\rho_{16}.I_{1z}$] & $ U_{16}=X_1 $ & $\langle B_{48} \rangle$ = Tr[$\rho_{48}.I_{1z}$] & $ U_{48}=\mathbb{I}_8 $ \\
$\langle B_{17} \rangle$ = Tr[$\rho_{17}.I_{3z}$] & $ U_{17}={\rm CNOT}_{13}.\overline{Y}_3.\overline{Y}_1 $ & $\langle
B_{49} \rangle$ = Tr[$\rho_{49}.I_{3z}$] & $ U_{49}={\rm CNOT}_{13}.\overline{Y}_3 $ \\
$\langle B_{18} \rangle$ = Tr[$\rho_{18}.I_{3z}$] & $ U_{18}={\rm CNOT}_{13}.X_3.\overline{Y}_1 $ & $\langle B_{50} \rangle$ = Tr[$\rho_{50}.I_{3z}$] & $ U_{50}={\rm CNOT}_{13}.X_3 $ \\
$\langle B_{19} \rangle$ = Tr[$\rho_{19}.I_{3z}$] & $ U_{19}={\rm CNOT}_{13}.\overline{Y}_1 $ & $\langle B_{51} \rangle$ = Tr[$\rho_{51}.I_{3z}$] & $ U_{51}={\rm CNOT}_{13} $ \\
$\langle B_{20} \rangle$ = Tr[$\rho_{20}.I_{2z}$] & $ U_{20}={\rm CNOT}_{12}.\overline{Y}_2.\overline{Y}_1 $ & $\langle B_{52} \rangle$ = Tr[$\rho_{52}.I_{2z}$] & $ U_{52}={\rm CNOT}_{12}.\overline{Y}_2 $ \\
$\langle B_{21} \rangle$ = Tr[$\rho_{21}.I_{3z}$] & $ U_{21}={\rm CNOT}_{23}.\overline{Y}_3.{\rm CNOT}_{12}.\overline{Y}_2.\overline{Y}_1 $ & $\langle B_{53} \rangle$ = Tr[$\rho_{53}.I_{3z}$] & $ U_{53}={\rm CNOT}_{23}.\overline{Y}_3.{\rm CNOT}_{12}.\overline{Y}_2 $ \\
$\langle B_{22} \rangle$ = Tr[$\rho_{22}.I_{3z}$] & $ U_{22}={\rm CNOT}_{23}.X_3.{\rm CNOT}_{12}.\overline{Y}_2.\overline{Y}_1 $ & $\langle B_{54} \rangle$ = Tr[$\rho_{54}.I_{3z}$] & $ U_{54}={\rm CNOT}_{23}.X_3.{\rm CNOT}_{12}.\overline{Y}_2 $ \\
$\langle B_{23} \rangle$ = Tr[$\rho_{23}.I_{3z}$] & $ U_{23}={\rm CNOT}_{23}.{\rm CNOT}_{12}.\overline{Y}_2.\overline{Y}_1 $ & $\langle B_{55} \rangle$ = Tr[$\rho_{55}.I_{3z}$] & $ U_{55}={\rm CNOT}_{23}.{\rm CNOT}_{12}.\overline{Y}_2 $ \\
$\langle B_{24} \rangle$ = Tr[$\rho_{24}.I_{2z}$] & $ U_{24}={\rm CNOT}_{12}.X_2.\overline{Y}_1 $ & $\langle B_{56} \rangle$ = Tr[$\rho_{56}.I_{2z}$] & $ U_{56}={\rm CNOT}_{12}.X_2 $ \\
$\langle B_{25} \rangle$ = Tr[$\rho_{25}.I_{3z}$] & $ U_{25}={\rm CNOT}_{23}.\overline{Y}_3.{\rm CNOT}_{12}.X_2.\overline{Y}_1 $ & $\langle B_{57} \rangle$ = Tr[$\rho_{57}.I_{3z}$] & $ U_{57}={\rm CNOT}_{23}.\overline{Y}_3.{\rm CNOT}_{12}.X_2 $ \\
$\langle B_{26} \rangle$ = Tr[$\rho_{26}.I_{3z}$] & $ U_{26}={\rm CNOT}_{23}.X_3.{\rm CNOT}_{12}.X_2.\overline{Y}_1 $ & $\langle B_{58} \rangle$ = Tr[$\rho_{58}.I_{3z}$] & $ U_{58}={\rm CNOT}_{23}.X_3.{\rm CNOT}_{12}.X_2 $ \\
$\langle B_{27} \rangle$ = Tr[$\rho_{27}.I_{3z}$] & $ U_{27}={\rm CNOT}_{23}.{\rm CNOT}_{12}.X_2.\overline{Y}_1 $ & $\langle B_{59} \rangle$ = Tr[$\rho_{59}.I_{3z}$] & $ U_{59}={\rm CNOT}_{23}.{\rm CNOT}_{12}.X_2 $ \\
$\langle B_{28} \rangle$ = Tr[$\rho_{28}.I_{2z}$] & $ U_{28}={\rm CNOT}_{12}.\overline{Y}_1 $ & $\langle B_{60} \rangle$ = Tr[$\rho_{60}.I_{2z}$] & $ U_{60}={\rm CNOT}_{12} $ \\
$\langle B_{29} \rangle$ = Tr[$\rho_{29}.I_{3z}$] & $ U_{29}={\rm CNOT}_{23}.\overline{Y}_3.{\rm CNOT}_{12}.\overline{Y}_1 $ & $\langle B_{61} \rangle$ = Tr[$\rho_{61}.I_{3z}$] & $ U_{61}={\rm CNOT}_{23}.\overline{Y}_3.{\rm CNOT}_{12} $ \\
$\langle B_{30} \rangle$ = Tr[$\rho_{30}.I_{3z}$] & $ U_{30}={\rm CNOT}_{23}.X_3.{\rm CNOT}_{12}.\overline{Y}_1 $ & $\langle B_{62} \rangle$ = Tr[$\rho_{62}.I_{3z}$] & $ U_{62}={\rm CNOT}_{23}.X_3.{\rm CNOT}_{12} $ \\
$\langle B_{31} \rangle$ = Tr[$\rho_{31}.I_{3z}$] & $ U_{31}={\rm CNOT}_{12}.{\rm CNOT}_{23}.\overline{Y}_1 $ & $\langle B_{63} \rangle$ = Tr[$\rho_{63}.I_{3z}$] & $ U_{63}={\rm CNOT}_{23}.{\rm CNOT}_{12} $ \\
$\langle B_{32} \rangle$ = Tr[$\rho_{32}.I_{1z}$] & $ U_{32}=X_1 $ & & \\
\hline
\hline
\end{tabular}
\end{table}
Let $\mathcal{\hat{O}}$ be the observable whose expectation value is to be measured in a state $ \rho =\vert \psi \rangle \langle \psi \vert $. Instead of measuring $\langle \mathcal{\hat{O}}\rangle _{\rho}$, the state $ \rho $ can be mapped to $ \rho_i $ using $ \rho_i=U_i.\rho.U^{\dagger}_i $ followed by $ z $-magnetization measurement of one of the qubits. Table~\ref{Mapping Table} lists the explicit forms of $U_i$ for all the basis elements of the Pauli basis set $\mathbb{B}$. In the present study, the observables of interest are $O$, $O^{}_1$, $O^{}_2$ and $O^{}_3$ as described in Sec.~\ref{Mapping} and Table~\ref{classification table}. The quantum circuit to achieve the required mapping is shown in Fig.~\ref{ckt+seq3Q}(a). The circuit is designed to map the state $ \rho $ to either of the states $ \rho^{}_{21} $, $ \rho^{}_{23} $, $ \rho^{}_{29} $ or $ \rho^{}_{53} $ followed by a $\sigma_z$ measurement on the third qubit in the mapped state. Depending upon the experimental settings, $ \langle B_3 \rangle $ in the mapped states is indeed the expectation values of $O$, $O^{}_1$, $O^{}_2$ or $O^{}_3$ in the initial state $ \rho $. The NMR pulse sequence to achieve the quantum mapping of circuit in Fig.~\ref{ckt+seq3Q}(a) is shown in Fig.~\ref{ckt+seq3Q}(b). The unfilled rectangles represent $\frac{\pi}{2}$ spin-selective pulses while the filled rectangles represent $\pi$ pulses. Evolution under chemical shifts has been refocused during all the free evolution periods (denoted by $ \tau_{ij}=\frac{1}{8J_{ij}}$) and $\pi$ pulses are embedded in between the free evolution periods in such a way that the system evolves only under the desired scalar coupling $J_{ij}$.
\subsection{Implementing the Entanglement Detection Protocol}
\label{demo states}
The three-qubit system was prepared in twenty seven different states in order to experimentally demonstrate the efficacy of the entanglement detection protocol. Seven representative states were prepared from the six inequivalent entanglement classes \textit{i.e.~} GHZ (GHZ and $\rm W\overline{W}$ states), W, three bi-separable and a separable class of states. In addition, twenty generic states were randomly generated (labeled as R$_1$, R$_2$, R$_3$,......., R$_{20}$). To prepare the random states the MATLAB\textsuperscript{\textregistered}-2016a random number generator was used. A recent experimental scheme~\citep{dogra-pra-15} was utilized to prepare the generic three-qubit states. For the details of quantum circuits as well as NMR pulse sequences used for state preparation see \citep{dogra-pra-15}. All the prepared states had state fidelities ranging between 0.89 to 0.99. Each prepared state $\rho$ was passed through the detection circuit \ref{ckt+seq3Q}(a) to yield the expectation values of the observables $O$,$O^{}_1$, $O^{}_2$ and $O^{}_3$ as described in Sec.~\ref{NMR Mapping}. Further, full QST \citep{cory-physD-98} was performed to directly estimate the expectation value of $O$, $O^{}_1$, $O^{}_2$ and $O^{}_3$ for all the twenty seven states. The results of the experimental implementation of the three-qubit entanglement detection protocol are tabulated in Table~\ref{result table}. For a visual representation of the data in Table~\ref{result table}, bar charts have been plotted and are shown in Fig.\ref{ResultPlot}. The seven known states were numbered as 1-7 while twenty random states were numbered as 8-27 in accordance with Table~\ref{result table}. Horizontal axes in plots of Fig.~\ref{ResultPlot}denote the state number while vertical axes represent the value of the respective observable. Black, cross-hatched and unfilled bars represent theoretical (The.), direct (Dir.) experimental and QST based expectation values, respectively. To further quantify the entanglement quotient, the entanglement measure, negativity~\citep{weinstein-pra-10,vidal-pra-02} was also computed theoretically as well as experimentally in all the cases (Table~\ref{negativity table}). Experiments were repeated several times for error estimation and to validate the reproducibility of the experimental results. All the seven representative states belonging to the six inequivalent entanglement classes were detected successfully within the experimental error limits, as suggested by the experimental results in first seven rows of Table~\ref{result table} in comparison with Table~\ref{classification table}. The errors in the experimental expectation values reported in the Table~\ref{result table} were in the range 3.1\%-8.5\%. The entanglement detection protocol with only four observables is further supported by negativity measurements (Table~\ref{negativity table}). It is to be noted here that one will never be able to conclude that the result of an experimental observation is exactly zero. However it can be established that the result is non-zero. This has to be kept in mind while interpreting the experimentally obtained values of the operators involved via the decision Table~\ref{classification table}.
\begin{table}
\begin{center}
\caption{\label{negativity table} Theoretically calculated and experimentally measured values of negativity.}
\scriptsize
\begin{tabular}{c | c c|| c| c c}
\hline
Negativity $\rightarrow$ & Theoretical & Experimental &Negativity $\rightarrow$ & Theoretical & Experimental \\
State $\downarrow$ & & & State $\downarrow$ & & \\
\hline
\hline
GHZ & 0.5 & 0.46 $\pm$ 0.03 & R$_{8}$ & 0 &0.02 $\pm$ 0.02 \\
$\rm W\overline{W}$ & 0.37 & 0.35 $\pm$ 0.03 & R$_{9}$ & 0.07&0.06 $\pm$ 0.03 \\
W & 0.47 & 0.41 $\pm$ 0.02 & R$_{10}$ &0.38&0.35$\pm$ 0.08 \\
BS$_1$ & 0 & 0.03 $\pm$ 0.02 & R$_{11}$ &0.32&0.28$\pm$ 0.06 \\
BS$_2$ & 0 & 0.05 $\pm$ 0.02 & R$_{12}$ &0.05&0.04$\pm$ 0.02 \\
BS$_3$ & 0 & 0.03 $\pm$ 0.03 & R$_{13}$ &0.18&0.15$\pm$ 0.03 \\
Sep & 0 & 0.02 $\pm$ 0.01 & R$_{14}$ &0.08&0.07$\pm$ 0.02 \\
R$_{1}$ & 0.02&0.04 $\pm$ 0.02 & R$_{15}$ &0.34&0.32$\pm$ 0.06 \\
R$_{2}$ & 0.16&0.12 $\pm$ 0.04 & R$_{16}$ &0.30&0.28$\pm$ 0.06 \\
R$_{3}$ & 0.38&0.35 $\pm$ 0.07 & R$_{17}$ &0 & 0.03$\pm$ 0.02 \\
R$_{4}$ & 0.38&0.34 $\pm$ 0.06 & R$_{18}$ &0 & 0.02$\pm$ 0.02 \\
R$_{5}$ & 0.03&0.04 $\pm$ 0.02 & R$_{19}$ &0.39&0.36$\pm$ 0.09 \\
R$_{6}$ & 0.21&0.18 $\pm$ 0.04 & R$_{20}$ &0 & 0.02$\pm$ 0.02 \\
R$_{7}$ & 0.09&0.08 $\pm$ 0.03 & & & \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\caption{\label{result table}
Results of the three-qubit entanglement detection protocol for twenty seven states. Label BS is for biseparable states while R is for random states. First column depicts the state label, top row lists the observable (Obs.) while second row specify if the observable value is theoretical (The.), direct experimental (Dir.) or from QST.}
\scriptsize
\begin{tabular}{c | c c c | c c c | c c c | c c c}
\hline
Obs. $\rightarrow$ & \multicolumn{3}{c}{$\langle O
\rangle$} & \multicolumn{3}{c}{$\langle O^{}_{1} \rangle$} &
\multicolumn{3}{c}{$\langle O^{}_{2} \rangle$} &
\multicolumn{3}{c}{$\langle O^{}_{3} \rangle$} \\
State($F$) $\downarrow$ & The. & Dir. & QST & The. & Dir. &
QST & The. & Dir. & QST & The. & Dir. & QST \\
\hline
\hline
GHZ(0.95 $\pm$ 0.03) & 1.00 & 0.91 & 0.95 & 0 & -0.04 & 0.03 & 0 & -0.07 & 0.05 & 0 & 0.07 & -0.02 \\
$\rm W\overline{W}$(0.98 $\pm$ 0.01) & 1.00 & 0.94 & 0.96 & 0 & 0.02 & 0.03 & 0 & 0.05 & -0.02 & 0 & -0.03 & 0.05 \\
W(0.96 $\pm$ 0.02) & 0 & 0.05 & 0.04 & 0.67 & 0.60 & 0.62 & 0.67 & 0.61 & 0.69 & 0.67 & 0.59 & 0.63 \\
BS$_1$(0.95 $\pm$ 0.02) & 0 & -0.03 & 0.02 & 0 & -0.07 & 0.06 & 0 & 0.09 & 0.03 & 1.00 & 0.93 & 0.95 \\
BS$_2$(0.96 $\pm$ 0.03) & 0 & 0.04 & 0.04 & 0 & 0.06 & -0.05 & 1.00 & 0.90 & 0.95 & 0 & 0.05 & 0.05 \\
BS$_3$(0.95 $\pm$ 0.04) & 0 & 0.08 & -0.06 & 1.00 & 0.89 & 0.94 & 0 & 0.09 & 0.07 & 0 & -0.04 & 0.02 \\
Sep(0.98 $\pm$ 0.01) & 0 & -0.05 & 0.02 & 0 & 0.09 & -0.04 & 0 & 0.04 & 0.03 & 0 & 0.08 & 0.07 \\
R$_{1}$ ( 0.91 $\pm$ 0.02 ) & -0.02 & -0.05 & -0.05 & 0.04 & 0.06 & 0.05 & 0.00 & 0.03 & 0.01 & 0.00 & 0.09 & 0.03 \\
R$_{2}$ ( 0.94 $\pm$ 0.03 ) & 0.06 & 0.09 & 0.08 & -0.22 & -0.32 & -0.33 & -0.25 & -0.46 & -0.41 & -0.09 & -0.13 & -0.16 \\
R$_{3}$ ( 0.93 $\pm$ 0.03 ) & -0.66 & -0.76 & -0.80 & 0.17 & 0.19 & 0.23 & -0.41 & -0.63 & -0.42 & -0.16 & -0.23 & -0.20 \\
R$_{4}$ ( 0.91 $\pm$ 0.01 ) & -0.17 & -0.25 & -0.31 & -0.15 & -0.25 & -0.21 & -0.29 & -0.37 & -0.48 & 0.46 & 0.55 & 0.60 \\
R$_{5}$ ( 0.94 $\pm$ 0.03 ) & -0.05 & -0.08 & -0.08 & 0.00 & 0.02 & 0.05 & 0.04 & 0.06 & 0.04 & 0.00 & 0.05 & 0.07 \\
R$_{6}$ ( 0.90 $\pm$ 0.02 ) & -0.34 & -0.65 & -0.48 & 0.10 & 0.16 & 0.19 & -0.21 & -0.29 & -0.24 & -0.12 & -0.19 & -0.20 \\
R$_{7}$ ( 0.93 $\pm$ 0.03 ) & -0.08 & -0.14 & -0.10 & 0.19 & 0.22 & 0.28 & 0.05 & 0.08 & 0.08 & -0.01 & -0.09 & -0.11 \\
R$_{8}$ ( 0.94 $\pm$ 0.01 ) & 0.00 & 0.03 & 0.04 & 0.00 & 0.04 & 0.04 & 0.00 & 0.06 & 0.05 & 0.01 & 0.04 & -0.02 \\
R$_{9}$ ( 0.95 $\pm$ 0.02 ) & -0.13 & -0.14 & -0.17 & -0.02 & -0.06 & 0.03 & -0.02 & 0.05 & -0.03 & 0.03 & 0.06 & 0.04 \\
R$_{10}$ ( 0.92 $\pm$ 0.03 ) & 0.64 & 0.84 & 0.73 & 0.03 & 0.06 & 0.05 & 0.00 & 0.07 & -0.03 & -0.23 & -0.41 & -0.25 \\
R$_{11}$ ( 0.93 $\pm$ 0.03 ) & 0.00 & 0.04 & -0.06 & 0.26 & 0.47 & 0.38 & 0.16 & 0.18 & 0.31 & 0.89 & 1.01 & 0.97 \\
R$_{12}$ ( 0.89 $\pm$ 0.02 ) &-0.02 & -0.08 & 0.03 & 0.12 & 0.19 & 0.13 & 0.02 & 0.04 & 0.03 & 0.04 & 0.07 & 0.07 \\
R$_{13}$ ( 0.92 $\pm$ 0.03 ) & -0.07 & -0.09 & -0.10 & -0.17 & -0.26 & -0.20 & 0.32 & 0.44 & 0.43 & -0.33 & -0.64 & -0.53 \\
R$_{14}$ ( 0.94$\pm$ 0.04 ) & -0.15 & -0.17 & -0.19 & 0.02 & 0.01 & -0.08 & -0.01 & -0.05 & 0.03 & -0.02 & -0.05 & -0.06 \\
R$_{15}$ ( 0.94 $\pm$ 0.03 ) & 0.08 & 0.16 & 0.12 & 0.12 & 0.16 & 0.15 & 0.48 & 0.51 & 0.68 & -0.37 & -0.46 & -0.61 \\
R$_{16}$ ( 0.93 $\pm$ 0.02 ) & -0.12 & -0.17 & -0.22 & -0.08 & -0.12 & -0.06 & -0.62 & -0.77 & -0.71 & 0.13 & 0.18 & 0.22 \\
R$_{17}$ ( 0.93 $\pm$ 0.04 ) & 0.00 & 0.07 & 0.04 & 0.00 & 0.02 & 0.05 & 0.00 & 0.05 & 0.05 & 0.00 & 0.09 & -0.03 \\
R$_{18}$ ( 0.90 $\pm$ 0.02 ) & -0.01 & -0.08 & 0.02 & 0.00 & 0.04 & -0.02 & 0.00 & 0.09 & 0.11 & 0.00 & 0.05 & 0.09 \\
R$_{19}$ ( 0.94 $\pm$ 0.02 ) & -0.19 & -0.22 & -0.27 & -0.63 & -0.82 & -0.86 & -0.48 & -0.73 & -0.54 & 0.13 & 0.20 & 0.16 \\
R$_{20}$ ( 0.93 $\pm$ 0.03 ) & 0.00 & -0.07 & -0.01 & 0.00 & 0.05 & 0.04 & 0.00 & -0.04 & 0.06 & 0.00 & 0.07 & -0.02 \\
\hline
\hline
\end{tabular}
\end{table}
The results for the twenty randomly generated generic states, numbered from 8-27 (R$_1$-R$_{20}$), are interesting. For instance, states R$_{10}$ and R$_{11}$ have a negativity of approximately 0.35 which implies that these states have genuine tripartite entanglement. On the other hand the experimental results of current detection protocol (Table~\ref{result table}) suggest that R$_{10}$ has a nonzero 3-tangle, which is a signature of the GHZ class. The states R$_{3}$, R$_{4}$, R$_{6}$, R$_{7}$, R$_{14}$, R$_{16}$ and R$_{19}$ also belong to the GHZ class as they all have non-zero 3-tangle as well as finite negativity. On the other hand, the state R$_{11}$ has a vanishing 3-tangle with non-vanishing expectation values of $O_1$, $O_2$ and $O_3$ which indicates that this state belongs to the W class. The states R$_{2}$, R$_{13}$ and R$_{15}$ were also identified as members of the W class using the detection protocol. These results demonstrate the fine-grained state discrimination power of the entanglement detection protocol as compared to procedures that rely on QST.
\begin{figure}
\caption{Bar plots of the expectation values of the observables $O$, $O_1$, $O_2$ and $O_3$ for states numbered from 1-27 (Table~\ref{result table}
\label{ResultPlot}
\end{figure}
Furthermore, all vanishing expectation values as well as a near-zero negativity, in the case of R$_8$ state, imply that it belongs to the separable class. The randomly generated states R$_{1}$, R$_{5}$, R$_{17}$, R$_{18}$ and R$_{20}$ have also been identified as belonging to the separable class of states. Interestingly, R$_{12}$ has vanishing values of 3-tangle, negativity, $ \langle O_2 \rangle $ and $ \langle O_3 \rangle $ but has a finite value of $ \langle O_1 \rangle $, from which one can conclude that this state belongs to the bi-separable BS$_3$ class.
\section{Generalized Three-Qubit Pure State Entanglement Classification}\label{Theory3Q New}
Above experimental demonstration \cite{singh-pra-18} of the three-qubit entanglement classification scheme~\citep{datta-epd-18} is limited to five-parameter generic state, Eq.~(\ref{generic})~\citep{acin-prl-01}. As two different three-qubit pure states of form $ \vert \xi \rangle= \sum _{i,j,k=0}^{1} b_{ijk}\vert ijk \rangle $ can have same generic state representation~\citep{acin-prl-01}. In order to detect and classify the entanglement in generalized three-qubit pure states the proposal in Ref.~\citep{zhao-pra-13} is explored. For the experimental demonstration concurrence~\citep{wootters-prl-98, wootters-qic-01,rungta-pra-01} based entanglement classification protocol~\citep{zhao-pra-13} is investigated. Core of the experimental procedure followed here depends upon finding the expectation value of a desired Pauli operators efficiently and to achieve this, the schemes described in Refs.~\citep{singh-pra-16, singh-arxiv-18,singh-pra-18} were utilized. Further, experimental procedures~\citep{dogra-pra-15} were developed to prepare any desired three-qubit generic states and used it in the current study to prepare states with arbitrary entanglement quotient.
Consider a three-qubit pure state $ \vert \Psi \rangle $. The state is fully separable if one can write $ \vert \Psi \rangle = \vert \psi_1 \rangle \otimes \vert \psi_2 \rangle \otimes \vert \psi_3 \rangle $. In case $ \vert \Psi \rangle $ is biseparable under bipartition $ 1 \vert 23 $ then it is always possible to write $ \vert \Psi \rangle = \vert \psi_1 \rangle \otimes \vert \psi_{23} \rangle $ where second and third qubits are in an entangled state $ \vert \psi_{23} \rangle $. Similarly one many have other two bipartitions as $ 2 \vert 13 $ and $ 3 \vert 12 $. In case $ \vert \Psi \rangle $ can not be written as a fully separable or biseparable then the state is genuinely entangled. There are two SLOCC inequivalent classes of genuine three-qubit entanglement \citep{dur-pra-00} namely, GHZ and W class. Hence any three-qubit pure state can belong to either of the six SLOCC inequivalent classes \textit{i.e.~} GHZ, W, three different biseparable classes or separable \citep{dur-pra-00}.
Following is the outline of the procedure \cite{zhao-pra-13} for generalized three-qubit pure state entanglement classification. Entanglement measure concurrence \citep{wootters-prl-98, wootters-qic-01,rungta-pra-01} is utilized to identify the above mentioned biseparable states in three-qubit pure states. The most general three-qubit pure state can be written as $ \vert \Psi \rangle= \sum _{i,j,k=0}^{1} a_{ijk}\vert ijk \rangle $ with $ \sum _{i,j,k=0}^{1} \vert a_{ijk} \vert ^2 =1 $. Concurrence for state $ \rho=\vert \Psi \rangle \langle \Psi \vert $ is given by $ C(\rho)=\sqrt{1-(tr\rho_1)^2} $ where $ \rho_1=tr_2(\rho) $ being the reduced density operator of first party. Squared concurrence for a three-qubit pure state under bipartiton $ 1 \vert 23 $ is given by
\begin{eqnarray}\label{conr}
C^2_{1 \vert 23}(\rho)=\left( \sum \limits _{j,k=0}^{1} \vert a_{0jk} \vert^2 \right)\left( \sum \limits _{j,k=0}^{1} \vert a_{1jk} \vert^2 \right) -\Big\vert \sum \limits _{j,k=0}^{1} a_{0jk}a_{1jk}^* \Big\vert^2
\end{eqnarray}
Further it was shown in \citep{zhao-pra-13} that after a lengthy calculation the squared concurrence, Eq.~(\ref{conr}), can be written as a quadratic polynomial of the expectation values of Pauli operators for three spin system. Let us symbolize $ C^2_{1 \vert 23}(\rho) $ as $ G_1(\rho) $ and it takes the form
\begin{eqnarray}
\label{G1}
G_1(\rho)=&\frac{1}{16}&(3 - \langle\sigma_0\sigma_0\sigma_3 \rangle^2
- \langle\sigma_0\sigma_3\sigma_0 \rangle^2
+ \langle\sigma_3\sigma_3\sigma_0 \rangle^2 \nonumber \\
&-3& \langle\sigma_3\sigma_0 \sigma_0 \rangle^2 + \langle\sigma_3\sigma_0\sigma_3 \rangle^2 - \langle\sigma_0\sigma_3\sigma_3 \rangle^2 + \langle\sigma_3\sigma_3\sigma_3 \rangle^2 \nonumber\\
&-3&\langle\sigma_1\sigma_0 \sigma_0 \rangle^2 + \langle\sigma_1\sigma_0\sigma_3 \rangle^2 + \langle\sigma_1\sigma_3\sigma_0 \rangle^2 + \langle\sigma_1\sigma_3\sigma_3 \rangle^2 \nonumber\\
&-3&\langle\sigma_2\sigma_0 \sigma_0 \rangle^2 + \langle\sigma_2\sigma_0\sigma_3 \rangle^2 + \langle\sigma_2\sigma_3\sigma_0 \rangle^2 + \langle\sigma_2\sigma_3\sigma_3 \rangle^2)
\end{eqnarray}
with $ \sigma_0=\vert 0 \rangle\langle 0 \vert+\vert 1 \rangle\langle 1 \vert $, $ \sigma_1=\vert 0 \rangle\langle 1 \vert+\vert 1 \rangle\langle 0 \vert $, $ \sigma_2=i(\vert 1 \rangle\langle 0 \vert-\vert 0 \rangle\langle 1 \vert) $ and $ \sigma_3=\vert 0 \rangle\langle 0 \vert-\vert 1 \rangle\langle 1 \vert $ being Pauli spin matrices in computational basis. Similar expressions for squared concurrences under other two bipartitions \textit{i.e.~} $ C^2_{2 \vert 13}(\rho) $ and $ C^2_{3 \vert 12}(\rho) $ can also be written by permutation and symbolized by $ G_2(\rho) $ and $ G_3(\rho) $ respectively \citep{zhao-pra-13}.
\label{th1}
As described in \textit{Theorem 1} of \citep{zhao-pra-13}, for any three-qubit pure state $ \rho= \vert \Psi \rangle \langle \Psi \vert $,
(i) $ \vert \Psi \rangle $ is fully separable iff $ G_l(\rho)=0 $, for $ l=2,3 $ or $ l=1,2 $ or $ l=1,3 $.
(ii) $ \vert \Psi \rangle $ is separable between \textit{l}$\rm ^{th}$ qubit and rest iff $ G_l(\rho)=0 $ and $ G_m(\rho)>0$ with $ l,m\in \{ 1,2,3 \} $ and $ l\neq m $.
(iii) $ \vert \Psi \rangle $ is genuinely entangled iff $ G_l(\rho)>0 $, for $ l=2,3 $ or $ l=1,2 $ or $ l=1,3 $.\\
Hence computing the nonlinear entanglement witnesses $ G_l(\rho) $, through experimentally measured expectation values of Pauli operators in an arbitrary three-qubit pure state $ \rho= \vert \Psi \rangle \langle \Psi \vert $, can immediately reveal the entanglement class of the state.
\subsection{Framework for Experimental Implementation}
Experimental creation of arbitrary general three-qubit pure states is a non-trivial task but one can resort to generic state \citep{acin-prl-01} for the demonstration of above discussed entanglement classification protocol.
It has been established \citep{acin-prl-01} that any three-qubit pure state can be transformed to generic state of canonical form \ref{generic}. The idea of five parameter generic state representation of three-qubit states was motivated from two-qubit generic states utilizing Schmidt decomposition \cite{schmidt-ma-1907, ekert-ajp-95} where any two-qubit state was shown to have form $\vert \Psi \rangle = cos\theta \vert 00 \rangle + sin\theta \vert 11 \rangle $ with $0 \leq \theta \leq \pi/2$, the relative phase has been absorbed into any of the local bases. Although the three-qubit generic representation doesn't follow from Schmidt decomposition but it was shown that combining adequately the local changes of bases corresponding to U(1)$\times$U(3)$\times$SU(2)$\times$SU(2)$\times$SU(2) transformations, one can always do with five terms, which precisely can carry only five entanglement parameters, leading thus to a non-superfluous unique representation.
It should be noted that the entanglement classification procedure outlined in Sec~\ref{Theory} works for any three-qubit pure state but arbitrary generic states were chosen for experimental demonstration. Two or more different states may have same generic canonical representation \citep{acin-prl-01}. Entanglement properties for the class of all such states can be fully characterized resorting only to the SLOCC equivalent generic state representative of that class. Such choice of states further ease the experimental efforts as nearly 40\% of the expectation values of Pauli operators appearing in the expressions of $ G_l(\rho) $ (\textit{e.g.} Eq.~(\ref{G1})) vanish in the case of generic states (Eq.~(\ref{generic})). In the recent work \citep{singh-arxiv-18} discussed in the previous sections, the entanglement classification of arbitrary three-qubit pure states was experimentally demonstrated. To do so only four observables suffice to classify the entanglement class. In contrast, the current classification works not only for generic states but also for any arbitrary three-qubit pure state of form $ \vert \Psi \rangle= \sum _{i,j,k=0}^{1} a_{ijk}\vert ijk \rangle $ and not limited to the canonical form \ref{generic}.
Further, as per \textit{Theorem 1}-(iii) the current entanglement classification protocol enables us to decide if a given pure state has genuine three-qubit entanglement or not but doesn't say anything if state belongs to GHZ or W class. To overcome this an observable is defined as $O=2\sigma^{}_{1}\sigma^{}_{1}\sigma^{}_{1}$ and use $n$-tangle \cite{wong-pra-01,li-qip-12} as an entanglement measure. For a three-qubit system, a non-vanishing 3-tangle, $ \tau $, implies it belongs to GHZ class. One may easily verify that for a given generic state $ \vert \Psi \rangle $, the 3-tangle \textit{i.e.~}\; $ \tau_{\Psi}=\langle \Psi \vert O \vert \Psi \rangle ^2/4$, Eq.~(\ref{three-tang}). Having defined $ O $ in addition to $ G_l(\rho) $, the protocol is equipped to experimentally classify any three-qubit pure state.
\section{NMR Implementation of Generalized Three-Qubit Entanglement Classification Protocol}
To experimentally implement the entanglement classification protocol discussed in Sec.~\ref{Theory3Q New} the experimental setup discussed in Sec.~\ref{Mapping}-\ref{NMR Implementation} is used. Three-qubit system is prepared in twenty seven different states. Seven states were prepared from the six SLOCC inequivalent entanglement classes \textit{i.e.~} GHZ (GHZ and $\rm W\overline{W}$ states), W, three bi-separable and a separable class of states. Three biseparable class states under partitions $ 1\vert 23 $, $ 2\vert 13 $ and $ 3\vert 12 $ are labeled as BS$ _1 $, BS$ _2 $ and BS$ _3 $ respectively. Additionally, twenty random generic states were prepared and labeled as R$_1$, R$_2$, R$_3$,......., R$_{20}$. To prepare the random states the random number generator available at \textit{www.random.org} was used. To experimentally prepare the desired three-qubit generic state, procedure outlined in \citep{dogra-pra-15} is followed. Ref.~\citep{dogra-pra-15} details the quantum circuits as well as NMR pulse sequences required to prepare all the desired quantum states in the current study. All such prepared states were found to have the fidelity (F) in the range 0.88 to 0.99. For each such prepared state the expectation values of the Pauli operators were found as described in Sec.~\ref{NMR Mapping} which in turn were used to compute $ G_l(\rho) $ using Eq.~(\ref{G1}). $ \langle O \rangle $ was also found in all the cases as it serves as an entanglement witness of the GHZ class.
\begin{table}
\caption{\label{result table 1} Results of the three-qubit entanglement classification protocol for twenty seven states. Label BS is for biseparable states while R is for random states. First column depicts the state label, top row lists the observable (Obs.) while the second row specifies if the observable value obtained is theoretical (The.), from QST or direct experimental (Dir.).}
\scriptsize
\begin{tabular}{c | c c c | c c c | c c c | c c c}
\hline
Obs. $\rightarrow$ & \multicolumn{3}{c}{$ \langle O \rangle $} & \multicolumn{3}{c}{$G_1$} & \multicolumn{3}{c}{$G_2$} &
\multicolumn{3}{c}{$G_3$} \\
State (F) $\downarrow$ & The. & QST & Dir. & The. & QST & Dir. & The. & QST & Dir. & The. & QST & Dir. \\
\hline
\hline
GHZ(0.96$ \pm $0.01) & 1.00 & 0.96 & 0.91 & 0.25 & 0.23 & 0.22 & 0.25 & 0.24 & 0.21 & 0.25 & 0.22 & 0.24 \\
$\rm W\overline{W}$(0.95$ \pm $0.02) & 1.00 & 0.95 & 0.94 & 0.14 & 0.11 & 0.13 & 0.14 & 0.13 & 0.12 & 0.14 & 0.15 & 0.13 \\
W(0.96$ \pm $0.02) & 0 & 0.03 & 0.02 & 0.22 & 0.19 & 0.21 & 0.22 & 0.24 & 0.25 & 0.22 & 0.25 & 0.23 \\
BS$_1$(0.98$ \pm $0.01) & 0 & 0.04 & 0.02 & 0 & 0.04 & 0.03 & 0.25 & 0.22 & 0.20 & 0.25 & 0.21 & 0.23 \\
BS$_2$(0.94$ \pm $0.03) & 0 & 0.04 & 0.03 & 0.25 & 0.21 & 0.24 & 0 & 0.02 & 0.03 & 0.25 & 0.24 & 0.27 \\
BS$_3$(0.95$ \pm $0.02) & 0 & 0.01 & 0.02 & 0.25 & 0.27 & 0.21 & 0.25 & 0.26 & 0.22 & 0 & 0.02 & 0.03 \\
Sep(0.98$ \pm $0.01) & 0 & 0.01 & 0.02 & 0 & 0.03 & 0.01 & 0 & 0.02 & 0.02 & 0 & 0.03 & 0.01 \\
R$_{ 1}$(0.92$ \pm $0.03) & 0 & 0.02 & 0.02 & 0 & 0.01 & 0.02 & 0 & 0.01 & 0.03 & 0 & 0.01 & 0.02 \\
R$_{ 2}$(0.93$ \pm $0.02) & -0.43 & -0.45 & -0.40 & 0.17 & 0.15 & 0.18 & 0.23 & 0.25 & 0.22 & 0.24 & 0.26 & 0.27 \\
R$_{ 3}$(0.96$ \pm $0.02) & -0.27 & -0.25 & -0.25 & 0.08 & 0.07 & 0.09 & 0.07 & 0.08 & 0.09 & 0.06 & 0.08 & 0.09 \\
R$_{ 4}$(0.94$ \pm $0.03) & -0.13 & -0.15 & -0.17 & 0.12 & 0.11 & 0.15 & 0.13 & 0.13 & 0.15 & 0.14 & 0.16 & 0.12 \\
R$_{ 5}$(0.93$ \pm $0.02) & 0.56 & 0.60 & 0.55 & 0.15 & 0.15 & 0.18 & 0.15 & 0.14 & 0.17 & 0.14 & 0.17 & 0.16 \\
R$_{ 6}$(0.89$ \pm $0.01) & 0 & 0.03 & 0.02 & 0.22 & 0.28 & 0.025 & 0.17 & 0.19 & 0.20 & 0.11 & 0.10 & 0.13 \\
R$_{ 7}$(0.96$ \pm $0.02) & -0.16 & -0.19 & -0.18 & 0.09 & 0.10 & 0.12 & 0.08 & 0.10 & 0.10 & 0.09 & 0.11 & 0.12 \\
R$_{ 8}$(0.93$ \pm $0.02) & 0 & 0.02 & 0.03 & 0.08 & 0.10 & 0.11 & 0.02 & 0.01 & 0.03 & 0.08 & 0.10 & 0.11 \\
R$_{ 9}$(0.97$ \pm $0.03) & 0.23 & 0.20 & 0.25 & 0.16 & 0.14 & 0.19 & 0.16 & 0.15 & 0.16 & 0.14 & 0.13 & 0.13 \\
R$_{10}$(0.93$ \pm $0.02) & -0.01 & 0.01 & 0.02 & 0 & 0.01 & 0.02 & 0.12 & 0.14 & 0.10 & 0.10 & 0.12 & 0.13 \\
R$_{11}$(0.94$ \pm $0.01) & 0.18 & 0.20 & 0.21 & 0.02 & 0.01 & 0.01 & 0.04 & 0.02 & 0.02 & 0.03 & 0.01 & 0.02 \\
R$_{12}$(0.95$ \pm $0.02) & 0.41 & 0.50 & 0.48 & 0.08 & 0.11 & 0.10 & 0.07 & 0.10 & 0.10 & 0.08 & 0.11 & 0.10 \\
R$_{13}$(0.93$ \pm $0.01) & 0.09 & 0.12 & 0.13 & 0.13 & 0.10 & 0.10 & 0.14 & 0.17 & 0.15 & 0.12 & 0.13 & 0.11 \\
R$_{14}$(0.94$ \pm $0.02) & 0.05 & 0.03 & 0.02 & 0.15 & 0.17 & 0.18 & 0.20 & 0.22 & 0.21 & 0.20 & 0.19 & 0.17 \\
R$_{15}$(0.98$ \pm $0.01) & 0.04 & 0.02 & 0.02 & 0.02 & 0.02 & 0.01 & 0.04 & 0.03 & 0.03 & 0.03 & 0.01 & 0.02 \\
R$_{16}$(0.96$ \pm $0.01) & 0 & -0.02 & -0.02 & 0 & 0.01 & 0.02 & 0 & 0.01 & 0.03 & 0 & 0.01 & 0.01 \\
R$_{17}$(0.95$ \pm $0.02) & 0 & 0.01 & 0.02 & 0.05 & 0.08 & 0.08 & 0.10 & 0.08 & 0.11 & 0.08 & 0.10 & 0.09 \\
R$_{18}$(0.90$ \pm $0.02) & -0.18 & -0.20 & -0.21 & 0.22 & 0.25 & 0.25 & 0.22 & 0.20 & 0.21 & 0.23 & 0.22 & 0.25 \\
R$_{19}$(0.94$ \pm $0.02) & 0 & 0.01 & 0.01 & 0 & 0.03 & 0.01 & 0.23 & 0.25 & 0.22 & 0.23 & 0.21 & 0.20 \\
R$_{20}$(0.96$ \pm $0.02) & 0 & 0.02 & 0 & 0 & 0.01 & 0.02 & 0 & 0.02 & 0.02 & 0 & 0.01 & 0.02 \\
\hline
\hline
\end{tabular}
\end{table}
Experimental results of the three-qubit entanglement classification and detection protocol are shown in Table~\ref{result table 1}. A bar chart has been plotted in Fig.~\ref{ResultPlot1} for a visual representation of the experimental results of Table~\ref{result table 1}. To obtain the bar plots of Fig.~\ref{ResultPlot1}, the experimentally prepared states were numbered from 1 to 27 as per the order shown in Table~\ref{result table 1}. As detailed in Sec.~\ref{Theory}, the concurrence $ G_l(\rho) $ acts as the entanglement witness and the additional observable $ O $ helps in the experimental discrimination of GHZ class states from rest of the states. In order to further validate the results negativity \citep{weinstein-pra-10,vidal-pra-02} has been computed from experimentally reconstructed state via QST \citep{leskowitz-pra-04} and the results are shown in the Table~\ref{negativity table 1}. In each case experiments were repeated several times for experimental error estimates. Experimental errors were in the range of 2.2\% - 5.7\% for the values reported in the Table~\ref{result table 1}.
\begin{figure}
\caption{Bar plots of the expectation value of the observable $O$ and the squared concurrences $G_1$, $G_2$ and $G_3$ for states numbered from 1-27 (Table~\ref{result table}
\label{ResultPlot1}
\end{figure}
\begin{table}
\caption{\label{negativity table 1}
Theoretically calculated and experimentally measured negativity values for all twenty seven states under investigation.
}
\begin{center}
\scriptsize
\begin{tabular}{c | c c || c | c c}
\hline
Negativity $\rightarrow$ & Theoretical & Experimental & Negativity $\rightarrow$ & Theoretical & Experimental \\
State $\downarrow$ & & & State $\downarrow$ & & \\
\hline
\hline
GHZ & 0.5 & 0.47 $\pm$ 0.02 & R$_{8}$ & 0.22 & 0.21 $\pm$ 0.02 \\
$\rm W\overline{W}$ & 0.37 & 0.39 $\pm$ 0.02 & R$_{9}$ & 0.39 & 0.37 $\pm$ 0.04 \\
W & 0.47 & 0.44 $\pm$ 0.01 & R$_{10}$ &0.03 & 0.01 $\pm$ 0.01 \\
BS$_1$ & 0 & 0.02 $\pm$ 0.02 & R$_{11}$ &0.17 & 0.14 $\pm$ 0.02 \\
BS$_2$ & 0 & 0.03 $\pm$ 0.01 & R$_{12}$ &0.27 & 0.30 $\pm$ 0.03 \\
BS$_3$ & 0 & 0.02 $\pm$ 0.02 & R$_{13}$ &0.16 & 0.12 $\pm$ 0.04 \\
Sep & 0 & 0.02 $\pm$ 0.02 & R$_{14}$ &0.42 & 0.37 $\pm$ 0.04 \\
R$_{1}$ & 0 & 0.01 $\pm$ 0.01 & R$_{15}$ &0.02 & 0.03 $\pm$ 0.01 \\
R$_{2}$ & 0.46 & 0.43 $\pm$ 0.04 & R$_{16}$ &0 & 0.01 $\pm$ 0.01 \\
R$_{3}$ & 0.26 & 0.24 $\pm$ 0.03 & R$_{17}$ &0.26 & 0.22 $\pm$ 0.03 \\
R$_{4}$ & 0.18 & 0.17 $\pm$ 0.03 & R$_{18}$ &0.47 & 0.41 $\pm$ 0.04 \\
R$_{5}$ & 0.38 & 0.35 $\pm$ 0.02 & R$_{19}$ &0 & 0.02 $\pm$ 0.02 \\
R$_{6}$ & 0.40 & 0.37 $\pm$ 0.04 & R$_{20}$ &0 & 0.03 $\pm$ 0.02\\
R$_{7}$ & 0.29 & 0.31 $\pm$ 0.03 & \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
One may observe from Table~\ref{result table 1} that the
seven states, from six SLOCC inequivalent classes, were
prepared with experimental fidelity $ \geq $ 0.95. The
entanglement classes of all these seven states were
correctly identified with the current protocol. It may
further be noted that the states R$ _2 $, R$ _3 $, R$ _4 $,
R$ _5 $, R$ _6 $, R$ _7 $, R$ _8 $, R$ _9 $, R$ _{11} $, R$
_{12} $, R$ _{13} $, R$ _{14} $, R$ _{17} $ and R$ _{18} $
have two non-zero concurrences and hence are all genuinely
entangled states. This fact is further supported by
negativity of these states reported in Table~\ref{negativity
table 1}. As discussed earlier, in order to discriminate GHZ
class from the rest one can resort to the observable $ O $.
Non-vanishing values of $ \langle O \rangle $ in
Table~\ref{result table 1} imply that the states R$ _2 $, R$
_3 $, R$ _4 $, R$ _5 $, R$ _7 $, R$ _9 $, R$ _{11} $, R$
_{12} $, R$ _{13} $ and R$ _{18} $ belong to GHZ class. In
contrast genuinely entangled states R$ _6 $, R$ _8 $, R$
_{14} $ and R$ _{17} $ have vanishing values of $ \langle O
\rangle $ and hence have vanishing 3-tangle as well so they
were identified as W class members. States R$ _{10} $ and R$
_{19} $ have the vanishing concurrence $ G_1 $ implying that
state belong to BS$ _1 $ class. Also states R$ _{1} $, R$
_{15} $, R$ _{16} $ and R$ _{20} $ were identified as
separable as all the observables have near zero values as
well as zero negativity.
\section{Effect of Mixedness in the Prepared States}
While the proposed entanglement classification protocol assumes the state under investigation to be pure, the experimentally prepared states are invariably mixed. The experimentally prepared density operator $\rho_e $ can be expanded in terms of its eigenvalues $ \lambda_j $ and corresponding eigenvectors $ \vert \lambda_j \rangle $ as $\rho_e=\sum_{j=1}^8 \lambda_j \vert \lambda_j \rangle \langle \lambda_j \vert $, obeying the normalization condition $\sum_{j=1}^8 \lambda_j=1$. For a pure state $\rho_p$, only one of the eigenvalue can be non-zero, hence one can take $\lambda_1^p=1 $ and other eigenvalues to be zero. The expectation value of the observable $ \hat{\mathcal{O}} $ can then be written as
\begin{equation}\label{idealexpectation}
\langle \hat{\mathcal{O}} \rangle _p=\langle \lambda_1^p \vert \hat{\mathcal{O}} \vert \lambda_1^p \rangle = Tr[\rho_p.\hat{\mathcal{O}}]
\end{equation}
In an actual experiment the situation is different and several eigenvalues of the density operator may be non-zero. The errors can arise either from the mixedness present in the experimentally prepared state $ \rho_e $ or in the experimental measurement of $ \langle \hat{\mathcal{O}} \rangle $. These errors are dominantly caused by imperfections in the unitary rotations used in state
preparation, rf inhomogeneity of the applied magnetic field, as well as T$ _2 $ and T$ _1 $ decoherence processes.
Let $\lambda_1$ be the maximum eigenvalue of the experimentally prepared state $\rho_e$. Mixedness is indicated by non-zero eigenvalues $\lambda_j$ for $j\neq 1$. The expectation value of $\hat{\mathcal{O}}$ can be written as an equation similar to Eq.~(\ref{idealexpectation}).
\begin{equation}
\langle \hat{\mathcal{O}} \rangle e=Tr[\rho_e.\hat{\mathcal{O}}]=\sum_{j=1}^8 \lambda_i Tr[P_i .\hat{\mathcal{O}}]=\sum_{j=1}^8 \lambda_i o_i
\end{equation}
The question is that if one approximate the state to be a pure state corresponding to the largest eigenvalue
$\lambda_1$ and take $ \langle \hat{\mathcal{O}} \rangle_p=\langle \lambda_1 \vert \hat{\mathcal{O}} \vert \lambda_1 \rangle
$, how much error is introduced and how do these errors affect the results.
In order to estimate the error in the value of $\langle\hat{\mathcal{O}} \rangle $ due to the mixedness, one can define the fractional error as \begin{equation}
\Delta=\frac{ \langle\hat{\mathcal{O}} \rangle_p -\langle\hat{\mathcal{O}} \rangle _e }{\langle\hat{\mathcal{O}} \rangle _p} \cong
(1-\lambda_1)-\frac{\sum_{j=2}^8 \lambda_j o_j}{o_1}
\end{equation}
where $o_j$s depend upon the operator involved. The experimental states have a minimum $ \lambda_1=0.88$ while $ \lambda_1 \geq 0.92 $ in other cases. In case of all the four observables $ O $, $ O_1 $, $ O_2 $ and $ O_3 $, $ \Delta $ was computed for all the 27 experimentally prepared states and the obtained values as percentage error were in the range 1.1\% $ \leq \Delta \leq $ 9.3\%.
In the light of the errors introduced by the mixedness
present in the experimentally prepared states the detection
protocol has to take $ \Delta $ error values into
consideration in addition to the experimental errors
reported in the Table~\ref{result table} for deciding the
class of three-qubit entanglement. As is evident from the
above analysis, in the worst-case scenario the protocol
works 90\% of the time. To further increase the fidelity of
the protocol, one can repeat the entire scheme on the same
prepared state, a number of times.
\section{Conclusions}
This chapter is aimed at the experimental classification of
arbitrary three-qubit pure states. To accomplish the goal
two different strategies were followed. In the first
classification protocol only four observable were defined
and measured experimentally for the classification. In the
second case, the entanglement measure concurrence was
measured experimentally to detect the class of three-qubit
pure states. Former protocol has the limitation that it can
only classify the entanglement class of the states in
generic form while the later is capable of detecting the
entanglement class of any arbitrary three-qubit pure state.
All the representative states from six SLOCC inequivalent
classes were detected by both the protocols.
Detection/classification protocols were further used to detect the entanglement class of twenty randomly generated
three-qubit pure states and both the protocols correctly
identified the entanglement class within the experimental
error limits. Results were further verified and
substantiated by full quantum state tomography as well as
negativity calculations which indeed confirms the results
obtained by both the protocols. Results of this chapter are
contained in
\href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.032301}{\rm
Phys. Rev. A \textbf{98}, 032301 (2018)} and
\href{https://doi.org/10.1007/s11128-018-2105-5}{\rm Quant.
Info. Proc. \textbf{17}, 334 (2018)}.
\chapter{Detection of Qubit-Ququart Pseudo-Bound Entanglement}\label{BoundEnt}
\section{Introduction}
This chapter details the experimental investigations of a special type of entanglement which is fundamentally different from the entanglement we usually encounter in QIP. Entanglement exists in two fundamentally different forms~\citep{horodecki-pla-97,horodecki-prl-98}: ``free'' entanglement which can be distilled into EPR pairs using local operations and classical communications (LOCC)~\citep{nielsen-prl-99} and ``bound'' entanglement which cannot be distilled into EPR pairs via LOCC, as discussed in Sec-(\ref{boundEnt}) . The term ``bound'' essentially implies that although
correlations were established during the state preparation, they cannot brought into a ``free'' form in terms of EPR pairs by a distillation process, and used wherever EPR pairs can be used as a resource. PPT entangled states are a prime example of bound entangled states and have been shown to be useful to establish a secret key~\citep{horodecki-prl-05}, in the conversion of pure entangled states~\citep{ishizaka-prl-04} and for quantum secure communication~\citep{zhou-prl-18}. While the existence of ``bound'' entangled states has been proved beyond doubt, there are still only a few known classes of such states~\cite{horodecki-pla-97,smolin-pra-01,acin-prl-01,sengupta-pra-11}. The problem of finding all such PPT entangled states is
still unsolved at the theoretical level.\\
Experimentally, bound entanglement has been created using four-qubit polarization states~\citep{amselem-natureP-09,kaneda-prl-12} and entanglement unlocking of a four-qubit bound entangled state was also demonstrated~\citep{lavoie-prl-10}. Entanglement was characterized in bit-flip and phase-flip lossless quantum channel and the experiments were able to differentiate between free entangled, bound-entangled and separable states~\citep{amselem-sr-13}. Continuous variable photonic bound-state entanglement has been created and detected in various experiments~\citep{dobek-lp-13,diGuglielmo-prl-11,steinhoff-pra-14}. Two photon qutrit Bound-entangled states of two qutrits were investigated utilizing orbital angular momentum degrees of freedom~\citep{hiesmayr-njp-13}. In NMR, a three-qubit system was used to prepare a three-parameter pseudo-bound entangled state~\citep{suter-pra-10}.\\
In the work described in this chapter, experiments were
performed to create and characterize a one-parameter family of qubit-ququart PPT entangled states using three nuclear spins on an NMR quantum information processor. There are a few proposals to detect PPT entanglement in the class of states introduced in Reference~\citep{horodecki-pla-97} by exploring local sum uncertainties~\citep{zhao-pra-13} and by measuring individual spin magnetisation along different directions~\citep{akbari-ijqi-17}. The proposal of Ref.~\citep{akbari-ijqi-17} is implemented to experimentally detect PPT entanglement in states prepared on an NMR quantum information processor. The family of states considered in the current study is an incoherent mixture of five pure states and the relative strengths of the components of the mixture is controlled by a real parameter. Different PPT-entangled states were prepared experimentally, parameterized by a real parameter. These states represent five points on the one-parameter family of states. Discrete values of the real parameter were used which were uniformly distributed over the range for which the current detection protocol detects the entanglement. In order to experimentally detect entanglement in these states, three Pauli operators need to be measured in each case. Previously developed schemes~\citep{singh-pra-16,singh-pra-18,singh-qip-18} were utilized to measure the required observables, which unitarily map the desired state followed by NMR ensemble average measurements. In each case full quantum state
tomography (QST)~\citep{leskowitz-pra-04,singh-pla-16} was
also performed to verify the success of the detection
protocol as well as to establish that the experimentally
created states are indeed PPT entangled. This work is
important both in the context of preparing and
characterizing bound entangled states and in devising new
experimental schemes to detect PPT entangled states which
use much fewer resources than are required by full quantum
state tomography schemes. It should be noted here that we
prepare the PPT entangled states using NMR in the sense that
the total density operator for the spin ensemble always
remains close to the maximally mixed state and at any given
instance one is dealing with pseudo-entangled
states~\citep{laflemme-ptsca-98}.
\section{Bound Entanglement in a Qubit-Ququart System}
\label{theory}
Consider a 3-qubit quantum system with an 8-dimensional
Hilbert space ${\mathcal H}={\mathcal H}_1\otimes{\mathcal
H}_2{}\otimes{\mathcal H}_3$, where ${\mathcal H}_i$
represent qubit Hilbert spaces. If we choose to club the
last two qubits into a single system with a four-dimensional
Hilbert space ${\mathcal H}_q={\mathcal
H}_2{}\otimes{\mathcal H}_3$, the three-qubit system can be
reinterpreted as a qubit-ququart bipartite system with
Hilbert space ${\mathcal H}={\mathcal H}_1\otimes{\mathcal
H}_q$.\\
Formally one can say that the four ququart basis vectors
$\vert e_i \rangle$ are mapped to the logical state vectors
of the second and third qubits as $ \vert e_1 \rangle
\leftrightarrow \vert 00 \rangle ,\; \vert e_2
\rangle\leftrightarrow \vert 01 \rangle ,\; \vert e_3
\rangle\leftrightarrow \vert 10 \rangle$ and $\vert e_4
\rangle\leftrightarrow \vert 11 \rangle $ in the
computational basis. With this understanding, we will freely
use the three-qubit computational basis for this
qubit-ququart system, where all along it is understood that
the last two qubits form a ququart. For this system consider
a family of PPT bound entangled states parametrized by a
real parameter $ b \in (0,1)$ introduced by
Horodecki~\citep{horodecki-pla-97}. \begin{eqnarray}
\label{BE} \sigma_b=\frac{7b}{7b+1}\sigma_{\rm insep}+
\frac{1}{7b+1}\vert \phi_b \rangle \langle \phi_b\vert
\end{eqnarray} with \begin{eqnarray}\label{BE-1} \sigma_{\rm
insep}&=&\frac{2}{7}\sum_{i=1}^3 \vert \psi_i\rangle\langle
\psi_i \vert+\frac{1}{7}\vert 011 \rangle\langle 011 \vert,
\nonumber\\ \vert \phi_b\rangle &=& \vert 1 \rangle
\otimes\frac{1}{\sqrt{2}}\left(\sqrt{1+b}\vert 00
\rangle+\sqrt{1-b}\vert 11 \rangle \right), \nonumber\\
\vert \psi_1 \rangle &=& \frac{1}{\sqrt{2}}(\vert 000
\rangle + \vert 101 \rangle),\nonumber\\ \vert \psi_2
\rangle &=& \frac{1}{\sqrt{2}}(\vert 001 \rangle + \vert 110
\rangle),\nonumber\\ \vert \psi_3 \rangle &=&
\frac{1}{\sqrt{2}}(\vert 010 \rangle + \vert 111 \rangle)
\end{eqnarray} It has been shown in~\citep{horodecki-pla-97}
that the states in the family $\sigma_b$ defined above are
entangled for $ 0<b<1 $ and is separable in the limiting
cases $ b=0 \;\; \rm or \;\; 1 $. One can explicitly write
the density operator for the mixed PPT entangled states
defined in Equation~(\ref{BE}) in the computational basis as
\begin{equation} \label{BE-mtrx}
\sigma_b=\frac{1}{1+7b}\left[ \begin{array}{cccccccc}
b&0&0&0&0&b&0&0\\ 0&b&0&0&0&0&b&0\\ 0&0&b&0&0&0&0&b\\
0&0&0&b&0&0&0&0\\
0&0&0&0&\frac{(1+b)}{2}&0&0&{\frac{\sqrt{1-b^2}}{2}}\\
b&0&0&0&0&b&0&0\\ 0&b&0&0&0&0&b&0\\
0&0&b&0&{\frac{\sqrt{1-b^2}}{2}}&0&0&\frac{(1+b)}{2}\\
\end{array} \right] \end{equation} It is interesting to
observe that for $ b=0 $, this family of states reduce to a
separable state in 2$ \otimes $4 dimensions while it is
still entangled in the three-qubit space and the
entanglement is restricted to the two qubits forming the
ququart.\\
Having defined the family of PPT entangled states in
Eq.(\ref{BE-mtrx}) parameterized by `$ b $', the method that
used to experimentally detect their entanglement using a
protocol proposed in Reference~\citep{akbari-ijqi-17} is
described as follows. Although the family of states in Eq.~(\ref{BE-mtrx}) is PPT entangled in $2 \otimes
4$-dimensional Hilbert space, it is useful to exploit the
underlying three-qubit structure. For the detection
protocol, we define three observables $B_i$, with $i=1,2,3$
(here $B_1$ acting on the qubit space and $B_2$ and $B_3$
act in the state space of qubits 2 and 3 forming the
ququart)
\begin{equation}\label{BE-obs}
B_1=\mathbb{I}_2 \otimes \sigma_x \otimes \sigma_x, \;\; B_2=\mathbb{I}_2 \otimes \sigma_y \otimes \sigma_y, \;\; B_3=\sigma_z \otimes \sigma_z \otimes \sigma_z
\end{equation}
where $ \sigma_{x,y,z} $ are the Pauli operators and $
\mathbb{I}_2 $ is the 2$ \times $2 identity operator.
Although the observables $B_j$ defined above are written in
the three qubit notation, they are bonafide observables of
the qubit-ququart system. The main result of
Reference~\citep{akbari-ijqi-17} is that any three-qubit
separable state, $ \rho_s $, obeys the four inequalities
given by
\begin{equation}\label{BE-inequality}
\vert \langle B_1 \rangle_{\rho_s} \pm \langle B_2 \rangle_{\rho_s} \pm \langle B_3 \rangle_{\rho_s} \vert \leq 1
\end{equation}
Therefore, if a states violates even one of the four
inequalities given in Eq.~(\ref{BE-inequality}), it has to
be entangled. It was shown numerically
in~\cite{akbari-ijqi-17} that the inequalities defined in
Equation~(\ref{BE-inequality}) can be used to detect the
entanglement present in the states $\sigma_b$ defined in
Eq.~(\ref{BE-mtrx}) for $ 0<b<\frac{1}{\sqrt{17}}$. Hence,
the protocol briefed above is able to detect the
entanglement of this family of PPT entangled states in $2
\otimes 4$ dimensions.
\begin{figure}
\caption{(a) Quantum circuit to prepare $ \vert
\phi_b\rangle $ from $ \vert 000 \rangle$ pseudopure state.
(b) NMR pulse sequence for quantum circuit given in (a).
Blank rectangles represent $ \frac{\pi}
\label{ckt_ch5}
\end{figure}
\section{Experimental Detection of 2$ \otimes $4 Bound Entanglement}
\label{NMR-Implementation} Next is to proceed toward
procedure followed to experimentally prepare several
different states from the family of states given by
Eq.~(\ref{BE-mtrx}), detect them by measuring the
observables defined in Eq.~(\ref{BE-obs}) and check if we
observe a violation of the inequalities defined in
Eq.~(\ref{BE-inequality}). i
\begin{figure}
\caption{(a) Quantum circuit to map $ \sigma_b $ to the state $ \sigma^{\prime}
\label{mapping_ckt_ch5}
\end{figure}
In order to prepare the PPT entangled family of states in $2
\otimes 4$ dimensions using three qubits, three spin-1/2
nuclei ($^1$H, $^{19}$F and $^{13}$C) were chosen to encode
the three qubits in a $^{13}$C-labeled sample of
diethylfluoromalonate dissolved in acetone-D6. See
Sec-\ref{Mapping} for NMR parameters, PPS preparation and
state mapping details. Three dedicated channels for $ ^1
$H, $ ^{19} $F and $ ^{13} $C nuclei were employed having $
\frac{\pi}{2} $ RF pulse durations of 9.33 $\mu $s, 22.55
$\mu $s and 15.90 $\mu$s at the power levels of 18.14 W,
42.27 W and 179.47 W respectively.
The experimentally prepared bound entangled states in the current study were directly detected using the protocol discussed in Sec.~\ref{theory} and full QST \citep{leskowitz-pra-04} was also performed in each case to verify the results.\\
The next step was to experimentally prepare the PPT entangled family of states given in Eq.~(\ref{BE-mtrx}) (each with a fixed value of the parameter $b$) and to achieve this we utilized the method of temporal averaging~\citep{cory-physD-98}. The family of states $\sigma_b$ is an incoherent mixture of several pure states as given in Eq.(\ref{BE-1}), and the quantum circuit to prepare one such nontrivial state ($ \vert \phi_b \rangle $) is given in Fig.\ref{ckt_ch5}(a), where $ R_x(\pi) $ represents a local unitary rotation through an angle $ \pi $ with a phase $x$. After experimentally preparing the state, one can measure the desired observable in Eq.~(\ref{BE-obs}), by mapping the state onto the Pauli basis operators.\\
The quantum circuit to achieve this is shown in Fig.\ref{mapping_ckt_ch5}(a), and this circuit maps the state $ \sigma_b \rightarrow \sigma_b^{\prime}$ such that $ \langle B_1 \rangle_{\sigma_b} = \langle I_{3z} \rangle_{\sigma_b^{\prime}} $. The motivation for such a mapping~\citep{singh-pra-16,singh-pra-18,singh-qip-18} relies on the fact that in an NMR scenario, the expectation value $ \langle I_{z} \rangle $, can be readily measured \citep{ernst-book-90}. The crux of the temporal averaging technique relies on the fact that the five states composing the PPT entangled state are generated via five different experiments. The states of these experiments are then added with appropriate probabilities to achieve the desired PPT entangled state. All the five states, appearing in Eq.(\ref{BE-1}), \textit{i.e.~}\; $ \vert \phi_b \rangle $, $ \vert \psi_1 \rangle $, $ \vert \psi_2 \rangle $, $ \vert \psi_3 \rangle $ and the separable PPS state $ \vert 011 \rangle $ were experimentally prepared with state fidelities $ \geq $ 0.96. It is worthwhile to note here that $ \vert \phi_b \rangle $ is a generalized biseparable state while $ \vert \psi_1 \rangle $ and $ \vert \psi_3 \rangle $ are LOCC equivalent biseparable states with maximal entanglement between the first and third qubits and $ \vert \psi_2 \rangle $ is a state belonging to the GHZ class. For the experimental demonstration of the
\begin{figure}
\caption{Bars represent theoretically expected values, red circles are the values obtained via QST and blue triangles are the direct experimental values for the inequality appearing in Eq.(\ref{BE-obs}
\label{resultplot_ch5}
\end{figure}
\noindent detection protocol discussed in Sec.\ref{theory} values of $ b=0.04,\; 0.08,\; 0.12,\; 0.16 \; \rm and \; 0.20 $ were chosen and thereby prepared five different PPT entangled states. The quantum circuit as well as the NMR pulse sequence to prepare $ \vert \phi_b \rangle $ is shown in Fig.(\ref{ckt_ch5}). Other states in Eq.(\ref{BE-1}) have similar circuits as well as pulse sequences and are not shown here. The tomograph for one such experimentally prepared PPT entangled state, with $ b=0.04 $ and fidelity $ \rm F=0.968 $, is shown in Fig.(\ref{tomo_ch5}).\\
In order to measure the expectation values of the observables appearing in Eq.(\ref{BE-obs}) our earlier work~\citep{singh-pra-16,singh-pra-18} was utilized. The idea is to unitarily map the state $ \sigma_b $ to a state say $ \sigma_b^{\prime} $, such that $ \langle \mathbb{O} \rangle_{\sigma_b}= \langle I_{iz} \rangle _{\sigma_b^{\prime}} $ where $ \mathbb{O} $ is one of the observables to be measured in the state $ \sigma_b $. This is achieved by measuring $ I_{iz} $ on $ \sigma_b^{\prime} $.
\begin{table} [h]
\caption{Experimentally measured values of
the inequality in Eq.~(\ref{BE-inequality})
showing maximum violation for five different PPT entangled states.}
\label{table_ch5}
\vspace*{12pt}
\centering
\renewcommand{1.5}{1.5}
\begin{tabular}{c | c | c | c | c }
\hline
Obs. $\rightarrow$ & &\multicolumn{3}{c}{Inequality value from: } \\
State(F) $\downarrow$ & $b$ & Theory & QST & Experiment\\
\hline
$\sigma_{b_1}$(0.946$\pm$0.019) & 0.04 & 2.311 &
2.061$\pm$0.046 & 2.269$\pm$0.118 \\
$\sigma_{b_2}$(0.947$\pm$0.022) & 0.08 & 1.876 &
1.660$\pm$0.027 & 1.784$\pm$0.090 \\
$\sigma_{b_3}$(0.949$\pm$0.009) & 0.12 & 1.557 &
1.382$\pm$0.028 & 1.451$\pm$0.086 \\
$\sigma_{b_4}$(0.953$\pm$0.007) & 0.16 & 1.327 &
1.179$\pm$0.028 & 1.213$\pm$0.065 \\
$\sigma_{b_5}$(0.925$\pm$0.009) & 0.20 & 1.150 &
0.807$\pm$0.029 & 1.007$\pm$0.061 \\
\hline
\end{tabular}
\end{table}
\begin{figure}
\caption{Real and imaginary parts of the tomograph of the (a) theoretically expected and (b) experimentally reconstructed density operator for PPT entangled state with $b=0.04$ and state fidelity F=0.968.}
\label{tomo_ch5}
\end{figure}
As an example, one can find the expectation value $ \langle B_1 \rangle_{\sigma_b} $ using the quantum circuit given in Fig.\ref{mapping_ckt_ch5}(a) and the NMR pulse sequence given in Fig.\ref{mapping_ckt_ch5}(b) is implemented, followed by a measurement of the spin magnetization of the third qubit. Such a normalized magnetization of a qubit in the mapped state is indeed proportional to the expectation value of the $z$-spin angular momentum of the qubit~\citep{ernst-book-90}.\\
Experimentally measured values of the inequality given in
Eq.~(\ref{BE-inequality}) with maximum violation are reported
in Table-\ref{table_ch5}. For all five states with
different $b$ values, full QST was also performed and the
observables \ref{BE-obs} were analytically computed from the reconstructed density operators. All the experimental results, tabulated in Table-\ref{table_ch5}, are plotted in Fig.~(\ref{resultplot_ch5}). All the experiments were performed several times to ensure the reproducibility of the experimental results as well as to estimate the errors reported in Table~\ref{table_ch5}. It was observed that the experimental values, within experimental error limits, agree well with theoretically expected values and validate the success of the detection protocol in identifying the PPT entangled family of states. The direct QST based measurements of the state also validate our experimental results.
\section{Conclusions}\label{concludingRemarks}
The characterization of bound entangled states is useful
since it sheds light on the relation between intrinsically
quantum phenomena such as entanglement and nonlocality. The
detection of bound entangled states is theoretically a hard
task and there are as yet no simple methods to characterize
all such states for arbitrary composite quantum systems. The
structure of PPT entangled states is rather complicated and
does not easily lead to a simple parametrization in terms of
a noise parameter. Work described in this chapter reports
the experimental creation of a family of PPT entangled
states of a qubit-ququart system and the implementation of
a detection protocol involving local measurements to detect
their bound entanglement. Five different states which were
parametrized by a real parameter `$ b $', were
experimentally prepared (with state fidelities $ \geq $
0.95) to represent the PPT entangled family of states. All
the experiments were repeated several times to ensure the
reproducibility of the experimental results and error
estimation. In each case it was observed that the detection
protocol successfully detected the PPT entanglement of the
state in question within experimental error limits. The
results were further substantiated via full QST for each
prepared state. It would be interesting to create the PPT
entangled family of states using different pseudopure
creation techniques and in higher dimensions. The results of
this chapter are contained in \href{https://doi.org/10.1016/j.physleta.2019.02.027}{\rm Phys. Lett. A}, (2019), doi:10.1016/j.physleta.2019.02.027.
\chapter{Experimental Implementation of
Navascu\'es-Pironio-Ac\'{\i}n Hierarchy to Detect Quantum
Non-Locality}\label{NPA}
\section{Introduction}
This chapter reports the experimental investigations of the
non-local character of quantum correlations in a
hierarchy-based protocol. It is well established that
quantum computation has a computational advantage over its
classical counterpart and the main resources utilized for
quantum computation are superposition, entanglement and
other quantum correlations \cite{nielsen-book-02}.
Entanglement plays a key role in several quantum
computational tasks \textit{e.g.} quantum cryptography
protocols \citep{ekert-prl-91}, quantum teleportation
\citep{bennett-prl-93}, quantum super dense coding
\citep{bennett-prl-92}, measurement-based quantum
computation \citep{briege-nature-09} and quantum key
distribution schemes \citep{barrett-prl-05, acin-njp-06}.
Creating entangled states in an experiment and certifying
the presence of entanglement in such states is of utmost
interest \citep{guhne-pr-09, horodecki-rmp-09} and
importance from the practical as well as the foundational
aspects of quantum physics. Most of the known entanglement
detection schemes rely on experimental quantum state
reconstruction \citep{horodecki-rmp-09}. It has been shown
that quantum state reconstruction is not cost-effective with
respect to experimental and computational resources
\citep{haffner-nature-05} and further, the detection of
entanglement of a known state is computationally a hard
problem \citep{brandao-prl-12} and scales exponentially with
the number of qubits. Methods to detect entanglement have
used violation of Bell-type inequalities
\citep{huber-pra-11, jungnitsch-prl-11}, entanglement
witnesses \citep{lewenstein-pra-00,guhne-jmo-03},
expectation values of the Pauli operators
\citep{zhao-pra-13, miranowicz-pra-13} as well as dynamical
learning techniques \citep{behrman-qic-08}. Although a
number of schemes exist for entanglement detection, but most
of them lack generality.\\
The motivation of the investigation is to experimentally
implement a non-local correlation and thereby devise an
entanglement detection protocol which can readily be
generalized to higher numbers of qubits as well as to
multi-dimensional quantum systems. A promising direction is
to experimentally observe the violation of Bell-type
inequalities \citep{bell-book, bell-ppf-64, chsh-prl-69}.
This is particularly suitable as such inequalities have
recently been proposed as a general method for certifying
the non-local nature of the experimentally observed
correlations in a device-independent manner
\citep{npa-prl-07, pironio-siam-10,flavio-prx-17}. Work
described in this chapter details demonstration of the
experimental implementation of the
Navascu\'es-Pironio-Ac\'{\i}n (NPA) hierarchy to certify the
non-local correlations arising from local measurements
\citep{npa-prl-07, pironio-siam-10,flavio-prx-17} on a
three-spin system.\\
Consider a joint probability distribution $ P_{\alpha\beta} $. The question addressed in Ref.\citep{npa-prl-07} is that can there be a quantum description of $ P_{\alpha\beta} $ \textit{i.e.~} can one have a quantum state $ \rho $, acting on the joint Hilbert space $ \mathcal{H}^{}_A \otimes \mathcal{H}^{}_B $, and the local measurement operators $ E_{\alpha}=\tilde{E}_{\alpha}\otimes I $ and $ E_{\beta}=I\otimes\tilde{E}_{\beta} $ such that
\begin{equation}\label{probabilitydistr}
P_{\alpha\beta}=\mathrm{Tr}(E_{\alpha}^{}E_{\beta}^{}.\rho)
\end{equation}
Here $ \tilde{E}_{\alpha} $ and $ \tilde{E}_{\beta} $ are local projection operators. This question can be used to design a test for the detection of non-locality from the actual probability distribution $ P_{\alpha\beta} $. In order to answer the above question, in general, one may need to search over all physical $ \rho $ and projection operators $ E_{\mu} $ which makes the problem computationally hard. A few attempts have been made to solve this problem and the first one to find the maximum violation of Clauser-Horne-Shimony-Holt (CHSH) inequality \citep{chsh-prl-69} by a quantum description was Tsirelson \citep{tsirelson-lpm-80}. Such attempts were limited to the simplest scenarios and were nowhere near generalization. Notable work has been done by Landau \citep{landau-fp-88} and Wehner \citep{wehner-pra-06}, where they have shown that the test of whether the experimental correlations arise from quantum mechanical description of nature or not, can be transformed to a semi-definite program (SDP). Solving such an SDP can reveal the local or non-local nature of the observed correlations.
\section{Brief Review of NPA Hierarchy}
\label{Theory_ch6}
In order to define SDP, consider projectors corresponding to
outcomes belonging to same measurement $M$ as E$ _{\nu} $
and E$_{\mu} $. The projectors :
\begin{enumerate}[(i)]
\item are orthogonal \textit{i.e.~} \; $ E_{\nu}E_{\mu}=0$ for $ \nu $,
$ \mu \in$ M, $ \mu\neq\nu $
\item sum to identity \textit{i.e.~} \; $ \sum_{\mu\in M} E_{\mu}=I $
\item obey $ E_{\mu}^2=E_{\mu}^{\dagger}=E_{\mu} $
\item obey the commutation rule (for projectors on
subsystems A and B) as:
\begin{equation}
\label{projector_rules}
[ E_{\alpha},E_{\beta}]=0
\end{equation}
\end{enumerate}
It was assumed in Ref. \citep{npa-prl-07} that such a $ \rho
$ exists that satisfies Eq.(\ref{probabilitydistr}) and
(\ref{projector_rules}), and then they looked for the
implications as follows: It was observed that by taking
products of projection operators $E_{\mu} $ and linear
superposition of such products, one may define new operators
which may neither be projectors anymore nor Hermitian. Let $
S=\lbrace S_1,S_2,....,S_n \rbrace$ be a set of $n$ such
operators. There exists an $ n \times n $ matrix associated
with every such set $S$ and defined as
\begin{equation}\label{moment-matrix}
\Gamma_{ij}=Tr(S_i^{\dagger}S_j \rho)
\end{equation}
$\Gamma $ is Hermitian and satisfies
\begin{equation}\label{sdpCons1}
\sum_{i,j} c_{ij}\Gamma_{ij}=0 \;\;\; if \;\;\;\sum_{i,j} c_{ij}S_i^{\dagger}S_j=0
\end{equation}
\begin{equation}\label{sdpCons2}
\sum_{i,j}
c_{ij}\Gamma_{ij}=\sum_{\alpha,\beta}d_{\alpha\beta}P_{\alpha\beta}
\;\;\; if \;\;\; \sum_{i,j}
c_{ij}S_i^{\dagger}S_j=\sum_{\alpha,\beta}d_{\alpha\beta}E_{\alpha}E_{\beta}
\end{equation}
Further, it can easily be proved that $ \Gamma $ is positive
semi-definite \textit{i.e.~} $ \Gamma \geq 0 $ \citep{npa-prl-07}. So,
if a joint probability distribution P$ _{\alpha\beta} $ has
a quantum description \textit{i.e.~} there exists a state $ \rho $ and
local measurement operators satisfying
Eq.(\ref{probabilitydistr}) and (\ref{projector_rules})
respectively, then finding such a state is equivalent to
finding the matrix $ \Gamma \geq 0 $ satisfying linear
constraints similar to Eq.~(\ref{sdpCons1}) and
Eq.~(\ref{sdpCons2}) and this amounts to solving an SDP
problem.\\
Having introduced the NPA hierarchy, one can move on to the protocol outlined in Ref. \citep{flavio-prx-17} that has been exploited in the current experimental study. The joint probability distribution considered is assumed to arise from local measurements on a separate state $ \rho_N $. The state $ \rho_N $ is shared among $N$ parties, each of them can
perform `$m$' measurements and each such measurement can
have `$d$' outcomes. Measurement by i$ ^{th}$ party is
represented by $ M_{x_i}^{a_i} $ with $ x_i\in \lbrace
0,...,m-1 \rbrace $ being the measurement choice and $
a_i\in \lbrace 0,...,d-1 \rbrace $ being the corresponding
outcome. By observing the statistics generated by measuring
all possible $ M_{x_i}^{a_i} $, one may write the empirical
values for the joint probability distributions
\begin{equation}\label{cond-Prob} p(a_1,...,a_N\vert
x_1,...,x_N)=Tr(M_{x_1}^{a_1}\otimes ...\otimes
M_{x_N}^{a_N}\rho_N)
\end{equation}
The correlations observed by measuring $ M_{x_i}^{a_i} $
locally, get encoded in the conditional probability
distributions having the form (\ref{cond-Prob}). Similar
expressions can be written for the reduced state probability
distribution which may arise from local measurements on a
reduced system.
\begin{equation}\label{cond-Red-Prob}
p(a_{i_1},...,a_{i_k}\vert
x_{i_1},...,x_{i_k})=Tr(M_{x_{i_1}}^{a_{i_1}}\otimes...\otimes
M_{x_{i_k}}^{a_{i_k}}\rho_{i_1.....i_k}) \nonumber
\end{equation}
with $ 0\leq i_1 <...<i_k<N $, \;$ 1\leq k <N $ and $
\rho_{i_1.....i_k} $ is the reduced density operator
obtained from $\rho_N$ by tracing out an appropriate subsystem. Since we are dealing with dichotomic measurements
on qubits, it will be useful to introduce the concept of
correlators and their expectation values as follows
\begin{equation}
\langle M_{x_{i_1}}^{a_{i_1}}\otimes...\otimes
M_{x_{i_k}}^{a_{i_k}} \rangle = \sum(-1)^{\sum_{l=1}^k
a_{i_l}}p(a_{i_1},...,a_{i_k}\vert x_{i_1},...,x_{i_k})
\end{equation}
The index $k$ here dictates the order of the correlator
while $0 \leq i_1 < ... < i_k < N$ with $x_{i_j}\in \{
0,m-1\}$ and $1\leq k \leq N$. For $ k=2 $ the correlator
will be of second order of form $\langle
M_{x_{i_1}}^{(i_1)} M_{x_{i_2}}^{(i_2)}\rangle$ while for $
k=N $ one can have the full body correlator. It will be seen
later that these correlators in the simplest case turn out
to be multi-qubit Pauli operators entering the moment matrix
(Eq.(\ref{2QGamma})).
\subsection{Modified NPA Hierarchy}\label{m-NPA}
Having discussed the main features of NPA hierarchy
\citep{npa-prl-07}, the method for the detection of
non-local correlation is described as follows. Consider a
set $ O=\lbrace O_i \rbrace $ with $ 1 \leq i \leq k $ and $
O_i $ are some product of the measurement operators $
\lbrace \rm M_{x_i}^{a_i} \rbrace $ or their linear
combinations. One can associate a $k \times k$ matrix with
$O$ defined by Eq.(\ref{moment-matrix}) as $
\Gamma_{ij}=Tr(O_i^{\dagger}O_j.\rho_N ) $. For a given
choice of measurements on a separable state (a) $ \Gamma $
will be a positive semi-definite matrix, (b) Matrix elements
of $ \Gamma $ satisfy the linear constraints similar to
Eq.(\ref{sdpCons1})-(\ref{sdpCons2}), (c) Some of the matrix
element of $ \Gamma $ can be obtained by experimentally
measuring the probability distribution and (d) Some of the $
\Gamma $ matrix entries corresponds to unobservables.
Keeping these facts in mind, one can design a hierarchy
based test to see if a given set of correlations can arise
from an actual quantum realization by performing local
measurements on a separable state. One can define a set $
O_{\nu} $ consisting of products of `$ \nu $' local
measurement operators or linear superpositions of such
products. Once $ O_{\nu} $ is defined, one can look for
associated $ \Gamma \geq 0 $ satisfying constraints similar
to Eq.(\ref{sdpCons1})-(\ref{sdpCons2}) to see if a given
set of correlations can arise from actual local measurements
on a separable state. If no solution is obtained to such an
SDP then this would imply that the given set of correlations
cannot arise by local measurements on a separable quantum
state and hence the correlations are non-local. One can
always find a stricter set of constraints by increasing the
value of $ \nu $ \;\textit{i.e.~}\; testing the nature of correlations
at the next level of the hierarchy.\\
In the experimental demonstration, as suggested in
Ref.\citep{flavio-prx-17}, the set of commutating
measurements have been used to design the SDP \textit{i.e.~} an
additional constraint is introduced on the entries of $
\Gamma $ such that local measurements also commutate. This
additional constraint considerably reduces the original
computationally-hard problem \citep{flavio-prx-17}. All the
ideas developed till now can be understood with an example.
Consider $N=2$, two dichotomic measurements per party at the
hierarchy level $\nu =2 $. Let the measurement be labeled as
$ A_x $ and $ B_y$ with $ x,y=0,1 $. Set of operators is $
O_2=\lbrace I, A_0, A_1, B_0, B_1, A_0A_1, A_0B_0, A_0B_1,
A_1B_0, A_1B_1, B_0B_1 \rbrace $. One can see the
corresponding moment matrix $\Gamma $ can be written as
\begin{footnotesize}
\begin{equation}
\label{2QGamma}
\hspace{-2cm}
\Gamma=\left(
\begin{array}{lllllllllll}
1 & \textcolor{red}{\langle A_0 \rangle} &
\textcolor{black}{\langle A_1 \rangle} &
\textcolor{blue}{\langle B_0 \rangle} &
\textcolor{black}{\langle B_1 \rangle} &
\textcolor{yellow}{v_1} & \color[rgb]{0,0.58,0}{\langle A_0
B_0 \rangle} & \textcolor{purple}{\langle A_0 B_1 \rangle} &
\textcolor{orange}{\langle A_1 B_0 \rangle} &
\textcolor{black}{\langle A_1 B_1 \rangle} &
\textcolor[rgb]{0.49,0.62,0.75}{v_2} \\
\textcolor{red}{\langle A_0 \rangle} & 1 &
\textcolor{yellow}{v_1} & \color[rgb]{0,0.58,0}{\langle A_0
B_0 \rangle} & \textcolor{purple}{\langle A_0 B_1 \rangle} &
\textcolor{black}{\langle A_1 \rangle} &
\textcolor{blue}{\langle B_0 \rangle} &
\textcolor{black}{\langle B_1 \rangle} &
\color[rgb]{0.55,0,0}{v_3} & \textcolor{green}{v_4} &
\textcolor{magenta}{v_5}\\
\textcolor{black}{\langle A_1 \rangle} &
\textcolor{yellow}{v_1^{\ast}} & 1 &
\textcolor{orange}{\langle A_1 B_0 \rangle} &
\textcolor{black}{\langle A_1 B_1 \rangle} &
\textcolor{red}{v_6} & \color[rgb]{0.55,0,0}{v_3^{\ast}} &
\textcolor{green}{v_4^{\ast}} & \textcolor{blue}{\langle B_0
\rangle} & \textcolor{black}{\langle B_1 \rangle} &
\color[rgb]{0.55,0.55,0}{v_7}\\
\textcolor{blue}{\langle B_0 \rangle} &
\color[rgb]{0,0.58,0}{\langle A_0 B_0 \rangle} &
\textcolor{orange}{\langle A_1 B_0 \rangle} & 1 &
\textcolor[rgb]{0.49,0.62,0.75}{v_2} &
\color[rgb]{0.55,0,0}{v_3} & \textcolor{red}{\langle A_0
\rangle} & \textcolor{magenta}{v_5} &
\textcolor{black}{\langle A_1 \rangle} &
\color[rgb]{0.55,0.55,0}{v_7} & \textcolor{black}{\langle
B_1 \rangle}\\
\textcolor{black}{\langle B_1 \rangle} &
\textcolor{purple}{\langle A_0 B_1 \rangle} &
\textcolor{black}{\langle A_1 B_1 \rangle} &
\textcolor[rgb]{0.49,0.62,0.75}{v_2^{\ast}} & 1 &
\textcolor{green}{v_4} & \textcolor{magenta}{v_5^{\ast}} &
\textcolor{red}{\langle A_0 \rangle} &
\color[rgb]{0.55,0.55,0}{v_7^{\ast}} &
\textcolor{black}{\langle A_1 \rangle} &
\textcolor{blue}{v_8}\\
\textcolor{yellow}{v_1^{\ast}} & \textcolor{black}{\langle
A_1 \rangle} & \textcolor{red}{v_6^{\ast}} &
\color[rgb]{0.55,0,0}{v_3^{\ast}} &
\textcolor{green}{v_4^{\ast}} & 1 &
\textcolor{orange}{\langle A_1 B_0 \rangle} &
\textcolor{black}{\langle A_1 B_1 \rangle} &
\textcolor[rgb]{0.24,0.7,0.44}{v_9} &
\textcolor{purple}{v_{10}} & \color[rgb]{0,1,1}{v_{11}} \\
\color[rgb]{0,0.58,0}{\langle A_0 B_0 \rangle} &
\textcolor{blue}{\langle B_0 \rangle} &
\color[rgb]{0.55,0,0}{v_3} & \textcolor{red}{\langle A_0
\rangle} & \textcolor{magenta}{v_5} &
\textcolor{orange}{\langle A_1 B_0 \rangle} & 1 &
\textcolor[rgb]{0.49,0.62,0.75}{v_2} &
\textcolor{yellow}{v_1} & \color[rgb]{0,1,1}{v_{12}} &
\textcolor{purple}{\langle A_0 B_1 \rangle} \\
\textcolor{purple}{\langle A_0 B_1 \rangle} &
\textcolor{black}{\langle B_1 \rangle} &
\textcolor{green}{v_4} & \textcolor{magenta}{v_5^{\ast}} &
\textcolor{red}{\langle A_0 \rangle} &
\textcolor{black}{\langle A_1 B_1 \rangle} &
\textcolor[rgb]{0.49,0.62,0.75}{v_2^{}\ast} & 1 &
\color[rgb]{0,1,1}{v_{13}} & \textcolor{yellow}{v_1} &
\color[rgb]{0,0.58,0}{v_{14}} \\
\textcolor{orange}{\langle A_1 B_0 \rangle} &
\color[rgb]{0.55,0,0}{v_3^{\ast}} & \textcolor{blue}{\langle
B_0 \rangle} & \textcolor{black}{\langle A_1 \rangle} &
\color[rgb]{0.55,0.55,0}{v_7} &
\textcolor[rgb]{0.24,0.7,0.44}{v_9^{\ast}} &
\textcolor{yellow}{v_1^{\ast}} &
\color[rgb]{0,1,1}{v_{13}^{\ast}} & 1 &
\textcolor[rgb]{0.49,0.62,0.75}{v_2} &
\textcolor{black}{\langle A_1 B_1 \rangle} \\
\textcolor{black}{\langle A_1 B_1 \rangle} &
\textcolor{green}{v_4^{\ast}} & \textcolor{black}{\langle
B_1 \rangle} & \color[rgb]{0.55,0.55,0}{v_7^{\ast}} &
\textcolor{black}{\langle A_1 \rangle} &
\textcolor{purple}{v_{10}^{\ast}} &
\color[rgb]{0,1,1}{v_{12}^{\ast}}
&\textcolor{yellow}{v_1^{\ast}} &
\textcolor[rgb]{0.49,0.62,0.75}{v_2^{}\ast} & 1 &
\textcolor{orange}{v_{15}}\\
\textcolor[rgb]{0.49,0.62,0.75}{v_2^{\ast}}&
\textcolor{magenta}{v_5^{}\ast} &
\color[rgb]{0.55,0.55,0}{v_7^{\ast}} &
\textcolor{black}{\langle B_1 \rangle} &
\textcolor{blue}{v_8^{\ast}} &
\color[rgb]{0,1,1}{v_{11}^{\ast}} &
\textcolor{purple}{\langle A_0 B_1 \rangle} &
\color[rgb]{0,0.58,0}{v_{14}^{\ast}} &
\textcolor{black}{\langle A_1 B_1 \rangle} &
\textcolor{orange}{v_{15}^{\ast}} & 1
\end{array}
\right)
\end{equation}
\end{footnotesize}
\noindent while following are the unassigned variables\\
\linebreak
\begin{footnotesize}
$
\begin{array} {llll}
v_1= \langle A_0A_1 \rangle, & v_2=\langle B_0B_1 \rangle,&
v_3=\langle A_0A_1B_0 \rangle,& v_4=\langle A_0A_1B_1
\rangle,\\
v_5= \langle A_0B_0B_1 \rangle, & v_6=\langle A_1A_0A_1
\rangle,& v_7=\langle A_1B_0B_1 \rangle,& v_8=\langle
B_1B_0B_1 \rangle,\\
v_9= \langle A_1A_0A_1B_0 \rangle, & v_{10}=\langle
A_1A_0A_1B_1 \rangle,& v_{11}=\langle A_1A_0B_0B_1 \rangle,&
v_{12}=\langle A_0A_1B_0B_1 \rangle, \\
v_{13}= \langle A_0A_1B_1B_0 \rangle, & v_{14}=\langle
A_0B_1B_0B_1 \rangle, & v_{15}=\langle A_1B_1B_0B_1 \rangle
&
\end{array}
$
\end{footnotesize}
\\
One may note that by introducing local measurements
commutativity \textit{i.e.~} $ [A_0,A_1]=[B_0,B_1]=0 $ the matrix
elements, of the $ \Gamma $ matrix given by
Eq.(\ref{2QGamma}) were simplified. Particularly, the
following reduction in the number of variables can be
noticed : $v_i=v_i^{\ast}$ for $i\in [1,15]$ and
$v_6=\langle A_0 \rangle$, $v_8=\langle B_0 \rangle$,
$v_9=v_14=\langle A_0B_0 \rangle$, $v_{10}=\langle A_0B_1
\rangle$, $v_{15}=\langle A_1B_0 \rangle$ and also
$v_{11}=v_{12}=v_{13}$. For a visual representation, the
variables that become identical because of the commutativity
constraints are represented by the same color in
Eq.(\ref{2QGamma}). The hence generated SDP will check if
the set of observed correlations $ \rm \lbrace \langle A_x
\rangle, \langle B_y \rangle, \langle A_xB_y \rangle \rbrace
$ are local. This can be achieved by substituting the
experimental values of the correlators in $ \Gamma $ matrix
and leaving the unobservables as variables. SDP will
optimize over such variables to see if a given set of
correlations are local or non-local. It has been shown
\citep{navascues-njp-08, pironio-siam-10} that this method
converges \textit{i.e.~}\; if a given set of correlations are non-local
then the SDP will fail at a finite number of steps $\nu $.\\
\section{Tripartite Non-Local Correlation Detection}\label{tripartite-sdp}
To experimentally demonstrate the detection of correlations which can not arise from local measurements on a separable state, a
three-qubit system was used. It has been shown
\cite{dur-pra-00} that a genuine three-qubit system can be
entangled in two inequivalent ways. CHSH scenario
\citep{chsh-prl-69} deals with (2, 2, 2) case \textit{i.e.~} $N=2$,
$m=2$ and $d=2$. Any correlation violating CHSH inequality
exhibits non-local nature in a sense that in principle one
cannot write a local hidden variable theory which can
reproduce the observed statistics. In the current
experimental study, the scenario is (3,2,2) \textit{i.e.~}\; three
parties with two dichotomic observables per party. The
measurements of three parties are labeled as $ A_x $ , $ B_y
$ and $ C_z $ with $ x,\;y,\;z \in [0,1] $. One can
construct set $ O_2 $ for three parties the way it was done
in the previous section for $ N=2 $. As detailed in Ref.
\citep{flavio-prx-17} to detect non-local correlations
arising from W state one needs to perform local measurements
$ M_0^{(i)}=\sigma_x $ and $ M_1^{(i)}=\sigma_z $ for all
three parties for the observables entering the moment matrix
associated with $ O_2 $ defined above. Here $ \sigma_{x/y/z}
$ are the spin-half Pauli operators. Also for GHZ type state
the measurements to generate the statistics were chosen as $
M_0^{(i)}=\sigma_x $ and $ M_1^{(i)}=\frac{\sigma_z +
\sigma_x}{\sqrt{2}} $ . A full body correlator is also
introduced while detecting non-local correlations generated
by GHZ state as such states are not suitable for detection
of non-local correlation using fewer body correlators
\citep{flavio-prx-17}.
\subsection{NMR Implementation of Non-Local Correlations Detection Scheme}\label{NMRImplementation-Ch6}
In order to experimentally demonstrate the detection of
non-local correlations, NMR hardware was used. Further, a
three nuclear spin-$\frac{1}{2}$ ensemble was utilized to
initialize the quantum system in prerequisite state on which
local measurements were performed. As already stated there
are only two inequivalent classes \citep{dur-pra-00}, under
local operations and classical communications (LOCC), of
genuine tripartite entanglement viz W-class and GHZ-class.
So the system was initialized in the representative states
of both these classes, to be tested experimentally. In order
to test the non-locality present in experimental expectation
values of the correlators following steps were followed for
a given state:
\begin{itemize}
\item Quantum system was initialized in one of the genuine
tripartite pure states.
\item It was assumed that the correlations observed from
local measurements on such states will fail SDP formulated
in Sec.-\ref{Theory_ch6} at the second level of the modified
NPA hierarchy.
\item At the second level ($ \nu=2 $) of the hierarchy, the
expectation values of all the correlators were measured
experimentally in the state under investigation.
\item Once all the observables of the moment matrix $ \Gamma
$ Eq. (\ref{moment-matrix}) were measured experimentally,
they were fed in the matrix $ \Gamma $, then the rest of the
unobservable entries were left as variables to be optimized
via SDP to achieve $ \Gamma \geq 0 $ under linear
constraints similar to Eq.(\ref{sdpCons1})-(\ref{sdpCons2})
as well as commutativity relaxation constraints $
[A_0,A_1]=[B_0,B_1]=[C_0,C_1]=0 $ to NPA hierarchy.
\item Above formulated SDP was solved using codes available
at \cite{NPACode-github} by modifying them for (3,2,2)
scenario.
\end{itemize}
\subsection{NMR Experimental Set-up and System Initialization}
For the experimental realization $ ^{13}C $ labeled
diethylflouromalonate sample dissolved in acetone-D6 in
liquid state NMR is used. Three spin-$\frac{1}{2}$ nuclei
\textit{i.e.~}\; $ ^1H $, $ ^{19}F $ and $ ^{13}C $ encode the qubit 1, qubit 2 and qubit 3 respectively. The free Hamiltonian of three qubit system in the rotating frame is given by
\citep{ernst-book-90}
\begin{equation}\label{NMR-Hamiltonian}
H=-\sum_{i=1}^3 \omega_i I_{iz}+2\pi\sum_{i,j=1}^3 J_{ij}I_{iz}I_{jz}
\end{equation}
with indices $i,j $=1, 2 or 3 represent the qubit
number, $ \omega_i $ is the respective chemical shift, $
I_{iz} $ being the $z$-component of spin angular momentum
and $ J_{ij} $ is the scalar coupling constant. System was initialized in the pseudopure state (PPS) $ \vert 000
\rangle $ using spatial averaging technique
\citep{cory-physD-98, mitra-jmr-07}
\begin{equation}
\rho_{_{PPS}}^{}=\frac{1-\epsilon}{2^3}\mathbb{I}_8+\epsilon\vert 000 \rangle\langle 000 \vert \nonumber
\end{equation}
where $ \epsilon\sim 10^{-5} $ is the room temperature
thermal magnetization and $ \mathbb{I}_8 $ is 8$ \times $8
identity operator. Details of the experimental parameters,
state preparation and state mapping can be found in
Sec.-\ref{Mapping}. State mapping is used to measure the
desired correlators which happens to be Pauli operators in
the current demonstration. Quantum circuits and NMR pulse
sequences to prepare the W and GHZ states are given in
Ref.\citep{dogra-pra-15}.
\subsection{Non-Locality Detection by Experimentally Measuring the Moments/Correlators}
At the second level of the modified NPA hierarchy in (3, 2, 2) scenario the set $ O_2=\lbrace \mathbb{I}_8, A_0, A_1, B_0, B_1, C_0, C_1, A_0A_1, A_0B_0, A_0B_1, A_0C_0, A_0C_1, A_1B_0, A_1B_1, A_1C_0, A_1C_1,$
\linebreak
$ B_0B_1, B_0C_0, B_0C_1, B_1C_0, B_1C_1, C_0C_1 \rbrace $.
\begin{figure}
\caption{Bar plots for the observable moments of the moment
matrix $\Gamma$ for (a) W-state and (b) GHZ-states. Bars
represent theoretically expected values while green squares
are the experimentally observed values.}
\label{gammaplots}
\end{figure}
\noindent The moment matrix in this case is a 22 $ \times $ 22 matrix
with all diagonal entries as 1. Further, the matrix has 26
observable moments while rest of the moments enter the
moment matrix as unobservables and were left as variables to
be optimized in SDP as detailed in Sec.-\ref{NMRImplementation-Ch6}. As an example, the moment/correlator $ \Gamma_{4,12} $, in the case of W state, is an observable $ \sigma_{1x} \sigma_{2x} \sigma_{3z} $ while the moment/correlator $ \Gamma_{1,18} $ is $ -i\sigma_{2y} $ which is not an observable and hence entered the moment matrix as a SDP variable. The next task was to find the expectation values of the correlators in the state under investigation. In NMR experiments the observed signal is proportional to the $z$-magnetization of the ensemble which indeed is proportional to the expectation value of the Pauli $z$-spin angular momentum operator in the given state. Hence the direct observable in typical NMR experiments is the Pauli $z$-operator expectation values of the nuclear spins. In recent works \citep{singh-pra-16,singh-pra-18}, schemes were developed to find the expectation values of any desired Pauli operators in the given state. This was achieved by mapping the state $ \rho\rightarrow \rho_l=U_l.\rho.U_{l}^{\dagger} $ followed by $z$-magnetization measurement. It has been shown in \citep{singh-pra-16,singh-pra-18} that the expectation value of $ \sigma_{iz} $ in state $ \rho_l $ is indeed the expectation value of the desired Pauli operator in the state $ \rho $. The explicit forms of the unitary operators $ U_l $, as well as quantum circuits and NMR pulse sequences, for two and three qubit Pauli spin operators are given in Refs. \citep{singh-pra-16} \& \citep{singh-pra-18} respectively.\\
As stated earlier the information regarding local/non-local nature of the observed correlations gets encoded in the measured correlators $ \lbrace \langle A_x \rangle,\;\langle B_y \rangle, \;\langle C_z \rangle,\; \langle A_x B_y \rangle,
\linebreak
\langle A_x C_z \rangle,\; \langle B_y C_z \rangle,\;
\langle A_x B_y C_z \rangle \rbrace $. The hence formulated SDP in both the cases, \textit{i.e.~} W as well GHZ state, failed to find $ \Gamma \geq 0 $ at the second level of the modified NPA hierarchy. This confirmed that the observed correlations can not arise from the local measurements on a separable state and hence the states are genuinely entangled. A bar plot for the observable moments of the moment matrix $\Gamma$ for W-state and GHZ-states is depicted in Fig(\ref{gammaplots}). In both the cases the SDP was also formulated directly from experimentally reconstructed density matrices using full QST. This further verified and supported the results of modified NPA protocol obtained by the direct measurements of the correlators. It is interesting to note here the experimental protocol demonstrated here was on pure states but the scheme is also capable of detecting non-locality of states which are convex sum of white noise and pure states up to a certain degree of mixedness \citep{flavio-prx-17}.
\section{Conclusions} \label{remarks}
Modified NPA hierarchy was used to detect the non-local
nature of the correlations performing local measurements by
means of a semi-definite program. A set comprising of
products and/or linear superpositions of such products was
defined and an associated positive semi-definite moment
matrix was also defined. Non-local correlation detection
protocols require measuring some correlator experimentally,
to generate the statistics to be tested. Once the moment
matrix embedded with the empirical data is obtained, the
semi-definite program optimizes, under some linear
constraints on the entries of the moment matrix, to see if
the observed correlations can arise from local measurements
on a separable state. The protocol has been tested
experimentally on three-qubit W and GHZ states utilizing NMR
hardware. In both the cases, the SDP successfully detected
the non-local nature of the observed correlation, as the
resulting SDP was unfeasible at the second level of the
modified hierarchy. These results were also verified by
direct full quantum state tomography. It would be
interesting to see the performance of the protocol in higher
dimension as well as more number of parties on an actual
physical system, as the structure of the entanglement
classes is much more complex in such cases.\\
The subsystems involved in our experiments reside on the same molecule and therefore, strictly speaking it is not possible to achieve a space-like separation between the events occurring in the different subsystem spaces. Therefore, the term ``local'' here pertains to subsystems and non-local implies something that goes across subsystems \textit{i.e.~} involves operators that are go beyond subsystems and refer to joint measurements. This word of caution is important and therefore we explicitly mention it here.
\chapter{Summary and Future Outlook}\label{Summary}
This thesis is a step further in the direction of experimental detection of quantum correlations including entanglement and discord. Most of the existing quantum correlation detection protocols require the state information beforehand to yield the detection results. Also, most such protocols are cost-intensive on experimental as well as computational resources. The thesis begins with the experimental investigation of the quantum entanglement in arbitrary bipartite states. In order to witness the entanglement the concept of entanglement witness (EW) was utilized. The key feature of the detection protocol is that it does not require any prior state information. A set of local measurements was chosen in such a way that, after measurement, they helped in the construction of EW and thereby entanglement detection utilizing semi-definite programming (SDP). It was demonstrated that only a three measurement setting sufficed to detect the entanglement in the case of maximally entangled Bell states. The protocol was also tested on a two-parameter class of qubit-qutrit entangled states and simulations suggest that only four measurement settings can successfully detect the entanglement. This work is an extension of the simulations on qutrit-qutrit entanglement detection \cite{szangolies-njp-15} and is a promising candidate for higher-dimensional bipartite entanglement detection utilizing EW through SDP. Its worth mentioning that the a critical step is to strategically choose the set of
measurement settings and this in-turn ensure the optimal
entanglement detection.
The next experimental investigation was focused on the detection of quantum correlations possessed by separable mixed states \textit{e.g.~} non-classical correlations (NCC). A non-linear positive map was successfully implemented to detect the NCC present in a two-qubit state. The key feature of the experimental implementation was that the detection was achieved in a \textit{single-shot} NMR experiment due to the non-destructive nature of NMR measurements. The detection capabilities of the positive-map were also explored by letting the state evolve. It was observed that at the transverse relaxation time, \textit{i.e.~} $T_2$ scale, the map was unable to detect NCC which otherwise was detected by direct quantum discord (QD) calculations. Nevertheless, this appeared to be attributed to the fact that the positive map utilized very low state information to yield `\textit{yes}/\textit{no}' answer on the status of NCC while QD requires full quantum state tomography (QST). Scheme looks promising for the experimental exploration of mixed states quantum correlations in higher-dimensional as well as multipartite quantum systems.
Entanglement detection as well as characterization in random three-qubit states was also investigated experimentally. The detection protocol was tested on seven representative states of six SLOCC-inequivalent classes and twenty randomly generated states. The entanglement measure 3-tangle was utilized to differentiate between genuine three-qubit entangled states, \textit{i.e.~} GHZ and W class of states, in a single experiment. Only four experimental settings proved to be sufficient for the successful classification of the entanglement in the three-qubit pure generic states. Further, concurrence based three-qubit entanglement classification protocol for the most general three-qubit states was also implemented successfully. It would be interesting to explore the entanglement classification in more than three-qubit states, utilizing Pauli witness operators, as the entanglement characterization is a challenging and non-trivial task.
Mixed states possess a more subtle type of entanglement, called the bound entanglement(BE), which is undistillable into EPR pairs via LOCC operations. Quantum states possessing BE doesn't violate PPT (positive under partial transposition) criterion. The thesis also explored a single-parameter class of qubit-ququart BE states with the aim of their detection using minimum experimental settings. The qubit-ququart states were mapped on to three-qubit states as in both the cases the underlying Hilbert space dimension is eight. BE detection method used the measurement of only three Pauli observables to see the violation of standard quantum limit (SQL). BE was successfully detected in several representative states from the single-parameter BE class of states and QST was used to establish the PPT nature of the experimentally created states. Key feature was that the BE was detected in the states lying near the entangled/separable boundary by directly measuring the Pauli observables while QST failed to detect BE.
There are entangled states whose measurement statistics can be simulated by a local hidden variable model. Bell-type inequalities \textit{e.g.~} Clauser-Horne-Shimony-Holt (CHSH) inequality in particular is suitable for experimental entanglement detection by observing the violation of SQL. CHSH inequality is limited to $(N=2,m=2,d=2)$ scenario \textit{i.e.~} two-parties, two-measurement settings per party with two outcomes for each measurement setting. Experimental investigation for a general case $(N,m,d)$ was considered utilizing Navascu\'es-Pironio-Ac\'{\i}n (NPA) hierarchy. Protocol was tested on genuine three-qubit entangled states \textit{i.e.~} GHZ and W states. It was demonstrated that the non-local nature of the quantum correlations was detected at the second level of NPA hierarchy. This method is particularly useful as it can easily be implemented in multipartite as well as higher dimensional cases.
\appendix
\chapter{Semi-Definite Program to Detect Entanglement in Random Two-Qubit States}\label{Append-A}
Following is the MATLAB script, used to detect entanglement using pre-chosen set of measurements, utilizing SDP defined in Eq.(\ref{SDP_Def}) and data given in Sec-\ref{SDP_example}. YALMIP \citep{Lofberg2004} and SeDuMi \citep{strum-oms-99} packages are required to be installed before running the following code. Following code as well as the \textbf{``mkstate''} function code are available at\\
\href{https://sites.google.com/site/amandeepsidhuiiserm/codes?authuser=0}{https://sites.google.com/site/amandeepsidhuiiserm/codes?authuser=0}.
\lstinputlisting[style=Matlab-editor, basicstyle=\mlttfamily\scriptsize]{SDPscript.m}
Following is the output and SDP runtime parameters yielded by above written MATLAB code.
\lstinputlisting[style=Matlab-editor, basicstyle=\mlttfamily\scriptsize]{SDPresult.m}
\chapter{NMR Pulse Program for Sequential Measurements}\label{SeqFID}
Below is the 2D NMR pulse program used in sequential measurements, in order to detect NCC using Eqn.(\ref{SeqFID_eq}), introduced in Sec-\ref{chapter_ncc} and Sec-\ref{mapvalue}.
\lstinputlisting[style=Matlab-editor, basicstyle=\mlttfamily\scriptsize]{SeqFID.m}
\renewcommand{References}{References}
\end{document} |
\begin{document}
\setlength{\textheight}{8.0truein}
\runninghead{Majorana fermions and non-locality}
{E. T. Campbell, M. J. Hoban and J. Eisert}
\normalsize\baselineskip=13pt
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\centerline{\bf MAJORANA FERMIONS AND NON-LOCALITY}
\vspace*{0.37truein}
\centerline{\footnotesize
EARL T.\ CAMPBELL\footnote{[email protected].}}
\vspace*{0.015truein}
\centerline{\footnotesize\it Dahlem Center for Complex Quantum Systems, Freie Universit{\"a}t Berlin,}
\baselineskip=10pt
\centerline{\footnotesize\it Berlin, Germany }
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\centerline{\footnotesize
MATTY J.\ HOBAN}
\vspace*{0.015truein}
\centerline{\footnotesize\it ICFO-Institut de Ci\`{e}ncies Fot\`{o}niques, Mediterranean Technology Park,}
\baselineskip=10pt
\centerline{\footnotesize\it Castelldefels (Barcelona), Spain}
\vspace*{10PT}
\centerline{\footnotesize JENS EISERT}
\vspace*{0.015truein}
\centerline{\footnotesize\it Dahlem Center for Complex Quantum Systems, Freie Universit{\"a}t Berlin,}
\baselineskip=10pt
\centerline{\footnotesize\it Berlin, Germany }
\vspace*{0.225truein}
\vspace*{0.21truein}
\abstracts{Localized Majorana fermions emerge in many topologically ordered systems and exhibit exchange statistics of Ising anyons. This enables noise-resistant implementation of a limited set of operations by braiding and fusing Majorana fermions. Unfortunately, these operations are incapable of implementing universal quantum computation. We show that, regardless of these limitations, Majorana fermions could be used to demonstrate non-locality (correlations incompatible with a local hidden variable theory) in experiments using only topologically protected operations. We also demonstrate that our proposal is optimal in terms of resources, with 10 Majorana fermions shown to be both necessary and sufficient for demonstrating bipartite non-locality. Furthermore, we identify severe restrictions on the possibility of tripartite non-locality. We comment on the potential of such entangled systems to be used in quantum information protocols.}{}{}
\vspace*{10pt}
\vspace*{3pt}
\vspace*{1pt} \baselineskip=13pt
\section{Introduction}
\noindent
Fermions that are their own anti-particle are known as Majorana, as opposed to Dirac, fermions. While presently there is no evidence that any fundamental particles are Majorana fermions, they frequently emerge as localised quasi-particles in models of condensed matter systems~\cite{Kitaev01,FuKane08,Alicea10,Lutchyn10,Oreg10}. Recent years have seen a race to experimentally confirm their existence, with some evidence already found~\cite{Mourik12,Das12,Deng12,Rokhinson12,Churchill13,Finck13}. Research into these systems is driven, at least partially, by their potential applications in topological quantum computing. For localised and well-separated Majorana fermions with zero energy, the system is protected from noise effects that could otherwise prove devastating in quantum computers.
Suitable Majorana fermions emerge in many two-dimensional (2D) systems including the Kitaev honeycomb lattice~\cite{Kitaev2006}, fractional quantum hall systems~\cite{Moore91,Nayak96}, topological insulators~\cite{FuKane08,FuKane09}, and a variety of other systems~\cite{Alicea12}. Adiabatically exchanging these fermions, and so braiding their world-lines, gives rise to the non-abelian exchange statistics of Ising anyons. Majorana fermions can also emerge as edge modes in one-dimensional (1D) systems~\cite{Alicea12}, such as the Kiteav wire~\cite{Kitaev01}. While braiding is not a meaningful concept in strict 1D systems, networks of wires also enable braiding with Ising statistics~\cite{Alicea11}. The unitary evolution from braiding is geometric in origin, and so robust against small experimental imperfections. However, these topologically protected braiding operations are not computationally powerful enough for universal quantum computing. Indeed, any free fermionic system can be efficiently classically simulated~\cite{Bravyi05b,Piotr}. Access to some non-topological operations --- which may be noisy, but not too noisy~\cite{VirPlen,Howard11,Howard12b,deMelo13} --- can be used to promote the system to full universality~\cite{BraKit05,Bravyi06,Meier13,Bravyi12,Jones12}. However, here we are interested in understanding the purely topologically protected capabilities of Majorana fermions, and in particular their capacity for demonstrating non-locality \cite{WernerWolf}.
Non-locality is the inability for a local hidden variable (LHV) theory to reproduce the correlations of space-like separated measurements. Until now, non-locality has only been investigated in more exotic topological systems by Brennen \emph{et al.}~\cite{Brennen09}. Having only considered systems capable of universal quantum computation, Brennen \emph{et al.} concluded their work saying, ``\emph{it is intriguing to ask whether one could find intermediate anyonic theories which have the power to generate Bell violating states by topologically protected gates, but are not universal for topological quantum computation}". We resolve this mystery by showing that Majorana fermions, and equivalently Ising anyons, could be used in an experiment demonstrating the non-locality of quantum mechanics. It appears that the standard and ubiquitous Clauser-Horne-Shimony-Holt (CHSH) inequality cannot be violated with only topological operations, and the non-topological resources sufficient for a CHSH violation have been investigated~\cite{Howard12}. We turn instead to a non-local experiment proposed by Cabello~\cite{Cabello01a,Cabello01} where 2 parties each select from 3 possible measurements, with each measurement producing 3 bits of classical information as outcomes. This non-locality proposal was built on the idea of the Mermin-Peres ``magic square", which was originally used to show the contextuality of quantum mechanics~\cite{Mermin90b,Peres90}. We find that a variant of these experiments could be implemented with each of two parties holding 5 Majorana fermions, and present a Bell inequality based on the magic square. We also present no-go results showing for two parties holding 4 Majoranas each, all experimental statistics can be produced by a local hidden variable theory. If two parties unequally share Majoranas, say Alice holds $n$ and Bob holds $m$ with $n<m$, we find that the resource is locally equivalent to both parties holding just $n$ Majorana fermions. Hence, 10 Majorana fermions are
both necessary and sufficient for the phenomenon of bipartite non-locality to be topologically demonstrated. Furthermore, we find that the correlations required to demonstrate the three-party Greenberger-Horne-Zeilinger (GHZ) paradox~\cite{Mermin90,Anders09,Hoban11} cannot be implemented with any number of Majoranas. This indicates that our proposal is the simplest possible non-locality experiment with Majorana fermions.
Bell experiments were intended, initially, to falsify alternative theories that claim the world to be local in Nature. Now these Bell-type non-locality experiments are also known to have practical applications. Such experiments can form the basis of quantum cryptography when the devices used are faulty or even untrusted~\cite{Acin07,Barrett12}. Typically, such proposals are envisaged for photonic systems, since they provide an easier means of accomplishing space-like separation of measurement events (a requirement for a non-local experiment). Polarisation-entangled photons have, for many years, been used to test Bell's proposal, proving successful if one ignores the detector loophole~\cite{Aspect82}. Experiments with trapped ions have efficient enough measurements to avoid the detector loophole, but do not satisfy space-like measurement separation~\cite{Rowe01}. Designing better experiments, hopefully closing all loopholes, is an active area of research~\cite{Buhrman03,Brunner12}. We shall not argue that our topological proposal is more promising than the aforementioned approaches. Rather, we take the first step by showing theoretical feasibility under ideal conditions. Implicit in our approach is the assumption that the physical system can be shared between two parties over sufficient distances that measurements are space-like separated; this is a similar technical difficulty faced by ion trap designs~\cite{Rowe01}.
\section{Majorana fermions and braiding}
\noindent
\subsection{Majorana fermions}
\noindent
We begin with a review of Majorana fermions and their available dynamics, largely following
Refs.~\cite{Bravyi06,Alicea12}. Generally, Majorana fermions are described by a set of Hermitian operators $ c_1,\dots, c_{2n}$ satisfying
\begin{equation}
\{ c_j, c_k\} = 2\delta_{j,k}
\end{equation}
and $ c_j^\dagger = c_j$ for all $j$, acting on the physical Hilbert space ${\cal H}= {\cal H}_0\oplus {\cal H}_1$ constituting a direct sum between the even and odd parity sectors. The algebra of physical operators ${\cal F} = {\cal L}({\cal H})$ is
spanned by products of an even number of Majorana fermion operators.
Taking two such fermions and clockwise braiding their world lines results in a unitary
$U({j,k})$
that maps the operators as
\begin{equation}
U({j,k}) c_a U({j,k})^\dagger =
\left\{
\begin{array}{ll}
c_a & {\rm if }\, a\neq \{j,k\},\\
c_k & {\rm if }\, a =j,\\
- c_j & {\rm if }\, a =k,
\end{array}
\right.
\end{equation}
for $k>j$.
Composing these braid operations results in a permutation $P\in S_{2n}$, with possible phase change $Q$, so that
${c}_{j}\mapsto (-1)^{Q_{j}} {c}_{P_{j}}$. The phases are constrained so that the global parity is preserved,
leaving $\prod_{j=1}^{2n} {c}_{j}$ unchanged.
Such braidings are a special case of unitary transformations $U$ acting on ${\cal H}$ that reflect linear mode transformation
\begin{equation}\label{TF}
c_j\mapsto U c_j' U^{\dagger} = \sum_{k=1}^{2n} V_{j,k} c_k
\end{equation}
for $V\in SO(2n)$.
Such unitary transformations are the ones commonly
considered in the context of fermionic linear optics.
All states encountered in this work are Gaussian fermionic states \cite{Bravyi05b,Piotr,Kastoryano}. They are
entirely described by their anti-symmetric covariance matrix $\gamma\in \mathbb{R}^{2n\times 2n}$, $\gamma=-\gamma^T$,
which has entries
$\gamma_{j,k} = i{\rm tr}(\rho[ c_j, c_k])/2$.
\subsection{Stabiliser language}
\noindent
We primarily describe quantum states in the Heisenberg picture by specifying a sufficient number of eigenvalue equations.
We say an operator $s\in {\cal F}$ stabilises a state vector $\ket{\psi}$ if
\begin{equation}
s \ket{\psi} = \ket{\psi}.
\end{equation}
We assume that initialisation of the system prepares a state vector $\ket{\psi_{0}}$ stabilised by $g_{j}=i {c}_{2j-1}{c}_{2j}$ for all $j\in\{1,\dots,n\}$.
Therefore, any state prepared from the initialisation $\ket{\psi_{0}}$ by braiding will be stabilised by $g_{j}=\pm i {c}_{P_{2j-1}}{c}_{P_{2j}}$ for some
permutation $P\in S_{2n}$.
Note that, since permutations are one-to-one mappings, if $j\neq k$, then $P_{j}\neq P_{k}$. Products of stabilisers are again stabilisers, and so $g_{j}$ generate a group $\mathcal{S}$ associated with the state. Similarly, topologically protected measurements, also called charge measurements, are those of the form $i{c}_{j}{c}_{k}$. Throughout, we say a state is \textit{accessible} if and only if it can be prepared with the above described topological operations. Again, states obtained by performing such measurements on
Gaussian states are Gaussian.
Collective charge measurements, such as of ${c}_{1}{c}_{2}{c}_{3}{c}_{4}$, cannot be measured non-destructively and in a topological manner. The collective charge observable can, however, be inferred by measuring $i{c}_{1}{c}_{2}$ and $i{c}_{3}{c}_{4}$ and multiplying the outcomes. However, such a process is destructive in that it will always disentangle these Majoranas from all other systems. We labour this point because the capacity to make non-destructive charge measurements increases the computational power of the system~\cite{Bravyi06} beyond that assumed in our later no-go theorems.
\subsection{Anyonic formalism}
\noindent
We shall restate some of the above in the anyonic formalism, which may be of benefit to some readers. Using, $\sigma$ to denote an Ising anyon, $\psi$ for another fermion, and $\mathds{1}$ for the vacuum, we have the fusion channel $\sigma \times \sigma \rightarrow \mathds{1} + \Psi$. Fusing Ising anyons $j$ and $k$ is equivalent to measuring $i c_{j}c_{k}$, where producing a $\Psi$ particle is equivalent to the $-1$ measurement outcome (eigenvalue) and producing the vacuum $\mathds{1}$ outcome denotes the $+1$ eigenvalue. Conversely, if we begin with a vacuum and create $n$ pairs of Ising anyons, we have the initialisation state described above if anyons from the $j^{\mathrm{th}}$ pair are labeled as $2j-1$ and $2j$.
\subsection{Entanglement properties}
\noindent
We proceed by identifying the equivalence classes of entangled states that are accessible under the operations described above. Consider two parties, Alice and Bob, holding respective sets of Majorana fermions $A= \{1,2,\dots,n \}$ and $B=\{n+1,\dots, n+m\}$. Clearly, any state stabilised by $\pm i{c}_{j}{c}_{k}$ can be locally prepared by Alice for $j,k \in A$ labeling a pair of Majorana fermions. A similar statement holds for Bob, and so the interesting stabilisers have $j \in A$ and $k \in B$ with each party holding one half of a pair of Majorana fermions (henceforth referred to as Majorana pairs). Assuming they hold $N$ such pairs, they can always locally braid such that the state is stabilised by $i{c}_{j}{c}_{j+N}$ for all $1 \leq j \leq N$. We see that the entanglement is entirely captured by the number of such Majorana pairs shared by Alice and Bob, and throughout we assume this canonical form for the stabilisers. For notational simplicity, it is beneficial to use ${a}_{j}={c}_{j}$ and ${b}_{j}={c}_{j+n}$, so Majorana pairs are stabilised by $i {a}_{j} {b}_{j}$.
Crucially, the number of Majorana pairs is distinct from the number of Bell pairs that give rise to useful entanglement. To investigate the useful correlations between Alice and Bob, we must consider how such pairs respond to measurements. Consider a pair of stabilisers $i{a}_{j}{b}_{j}$ and $i{a}_{k}{b}_{k}$ for $j\neq k$,
the state is also an eigenstate of their product $(i {a}_{j} {b}_{j})(i {a}_{k} {b}_{k})$ which using anti-commutation of fermions equals $-(i {a}_{j} {a}_{k})(i {b}_{j} {b}_{k})$. The factors $i {a}_{j} {a}_{k}$ and $i {b}_{j} {b}_{k}$ are locally measurable, and so their outcomes must be anti-correlated. Hence, the state has the flavor of a singlet state.
To make measurements Alice and Bob must share at least two Majorana pairs, but two alone is trivial since there is only one possible local measurement and with only one measurement we cannot construct a test of non-locality. With three Majorana pairs, Alice has three possible measurement observables that mutually anti-commute and are isomorphic
to the Pauli operators. An encoding is a collection of maps ${\cal F}\rightarrow {\cal B}({\cal H})$, identifying the set of fermionic modes with qubit or
spin systems. The identification of products of Majorana fermions with Pauli operators of an associated qubit system can be taken as
\begin{eqnarray}
X=i{a}_{1}{a}_{2} , Y=i{a}_{1}{a}_{3} , Z=i{a}_{2}{a}_{3}.
\end{eqnarray}
From the anti-commutation relations of the Majorana fermions, one can readily verify that these operators generate the Pauli group of a single qubit.
Bob similarly has measurement options isomorphic to the Pauli operators. The correlations between Alice and Bob resulting from the above measurements on three Majorana pairs match those of Pauli measurements on the two-qubit singlet state (for Alice and Bob each having a single qubit). Hence, three Majorana pairs can be said to reproduce the entanglement of a Bell pair. However, it is well-known that the measurement statistics under the operations allowed here can be reproduced by a local hidden variable (LHV) theory. Later we present a strengthened proof that even \emph{four} Majorana pairs are incapable of demonstrating non-locality.
\section{Non-locality}
\noindent
Before proceeding we refine the concepts of non-locality. Assume Alice and Bob hold some quantum state $\rho$ and are able to freely choose from a set of measurements $\{ \mathcal{A}_{j} :j\in I\}$ and $\{ \mathcal{B}_{k}: k \in I \}$ respectively, where $I$ denotes the set of different kinds of measurement settings.
The measurements will have possible outcomes that we denote $\alpha,\beta\in R$ for a suitable $r$ that occur with probability
\begin{equation}
P(\alpha, \beta | j , k) = \ensuremath{\mathrm{tr}} [ ( \Pi_{j,\alpha} \otimes \Pi_{k,\beta} ) \rho ]
\end{equation}
where $\Pi_{j,\alpha} \geq 0 $ ($\Pi_{k,\beta}\geq 0$) is the positive-operator valued measure for setting $j$ ($k$) and outcome $\alpha$ ($\beta$). There are many different experiments that can achieve the same probability distributions, and each of these is a realisation of $P$. Conversely, we say a probability distribution $P$ is quantum if there exists a choice of measurements and a quantum state $\rho$ that realises $P$.
When are these observations a proof of non-locality? Imagine that Alice and Bob are space-like separated and are allowed to make their choice of measurement freely, they now want to know whether any classical model can produce these observations. Since they cannot communicate, this classical model has some pre-determined instructions that, given Alice and Bob's choice of measurements, will output some value. Such an instruction set is called a LHV theory. An arbitrary hidden variable $\lambda$ takes values in some space $\Lambda$ equipped with a probability measure, and which determines local probability functions
\begin{equation}
\lambda \mapsto p_A( {\alpha} |j,\lambda) ,
\,
\lambda \mapsto p_B( {\beta} | k,\lambda).
\end{equation}
The probability of obtaining the outcome pair $(\alpha,\beta)$, given that $j,k$ have been chosen by Alice and Bob, is then
\begin{equation}
P( {\alpha}, {\beta} | j, k) = \int dM(\lambda) p_A( {\alpha} |j,\lambda) p_B( {\beta} | k,\lambda).
\end{equation}
A probability distribution is local (also called a LHV) if and only if there exists a decomposition of the above form, and we denote this as $P \in \mathcal {P}_{L}$. If no such local model exists, $P \notin \mathcal {P}_{L}$, we say the probability distribution is non-local. Furthermore, any experiment realizing a non-local probability distribution is a non-locality experiment.
Typically, we confirm non-locality by checking whether a probability distribution violates a Bell inequality. Let us fix the number of measurement settings and outcomes, and denote the entire set of possible probability distributions by $\mathcal {P}$. The Bell inequalities follow by first defining a real-valued linear function, $G: \mathcal {P} \rightarrow \mathbb{R}$, which is sometimes called a non-local game~\cite{Brassard,Toner,Cabello01a}, the respective parties are referred to as players and the
action taken are strategies. The non-local game, as the term is used here, is described by a real-valued function,
$V: R \times R \times I\times I\rightarrow \mathbb{R}$ such that
\begin{equation}
\label{non-localGame}
G(P) = \sum_{\alpha, \beta, j, k} V(\alpha, \beta, j, k) P(\alpha, \beta | j, k).
\end{equation}
For any such non-local game, there exists a Bell inequality that holds for all local $P$
\begin{equation}
\label{generalBell}
G(P) \leq \Omega_{c}(G) := \sup \{ G(P) : P \in \mathcal {P}_{L} \} .
\end{equation}
The above is true simply by definition. However, for any $P$ and $G$ where $G(P)>\Omega_{c}(G)$ we can conclude $P$ is non-local.
Let us consider a particular non-local game $G$, and its classical limit $\Omega_{c}(G)$. Since non-local games are linear functions and $\mathcal {P}_{L}$ is a convex set, the classical limit can always be achieved by an extremal point in $\mathcal {P}_{L}$. That is, the value of $\Omega_c(G)$ can always be achieved by a probability distribution of the form
\begin{equation}
P( {\alpha}, {\beta} | j, k) = c_A( {\alpha} |j ) c_B( {\beta} | k ),
\end{equation}
where Alice and Bob deterministically assign measurement outcomes. There are only finitely many such distributions,
so this greatly simplifies the evaluation of $\Omega_{c}(G)$.
\section{Magic squares with Majoranas}
\noindent
To demonstrate that these Majorana fermions exhibit non-locality we adapt a specific proof of ``non-locality without inequalities" first devised by Cabello~\cite{Cabello01a,Cabello01}. There exist other proofs of non-locality for qubits without the use of a Bell inequality, such as those by Hardy~\cite{Hardy92} and the
GHZ paradox~\cite{Mermin90,Anders09,Hoban11}.
Crucially, Hardy's argument does not work for maximally entangled states and so it cannot be applied. Furthermore we show later that the GHZ argument does not apply here either. Cabello's proof works for two Bell states, a four-qubit state, shared by Alice and Bob each of which can make a measurement on two qubits of this state. The analysis shares much of its character with the proof that quantum mechanics is non-contextual derived by Peres and Mermin, often referred to as the ``Magic Square Game". We will follow the simplification of Cabello's approach as presented by Aravind~\cite{Aravind02} so that the result may be of more general appeal, but optimally tailored to systems of Majorana fermions.
\begin{table}
\tcaption{Squares describing measurements made by Alice and Bob on their respective parts of their quantum state (made up of two Bell pairs with one half of each pair held by each party). Measuring these operators on five Majorana pairs, Alice and Bob yield identical measurement outcomes for the entry they have in common in their respective squares. Notice that, up to a phase accounting for correlation vs. anti-correlation, the tables are equivalent under interchanging ${a}_{j}$ with ${b}_{j}$.
Furthermore, for each column the product of the three entries yields $+\mathds{1}$, whereas for each row the product of all entries gives $-\mathds{1}$.}
\centering
\begin{tabular}{c}
\begin{tabular}{r|c|c|c|}
\multicolumn{1}{c}{} & \multicolumn{3}{c}{{Alice}} \\
\multicolumn{1}{c}{} & \multicolumn{1}{c}{$\mathcal{A}_{1}$} & \multicolumn{1}{c}{$\mathcal{A}_{2}$} & \multicolumn{1}{c}{$\mathcal{A}_{3}$} \\ \cline{2-4}
& $-a_1 a_3 a_4 a_5 $ & $ i a_1 a_4 $ & $ i a_3 a_5$ \\ \cline{2-4}
& $i a_3 a_4 $ & $- a_1 a_2 a_3 a_4 $ & $-i a_{1}a_{2}$ \\ \cline{2-4}
& $i a_1 a_5 $ & $ i a_2 a_3 $ & $ a_1 a_2 a_3 a_5$ \\ \cline{2-4}
\multicolumn{1}{c}{} & \multicolumn{3}{c}{} \\
\multicolumn{1}{c}{} & \multicolumn{3}{c}{{Bob}} \\ \cline{2-4}
$\mathcal{B}_{1}$ &$-b_1 b_3 b_4 b_5 $ & $- i b_1 b_4 $ & $- i b_3 b_5$ \\ \cline{2-4}
$\mathcal{B}_{2}$ & $-i b_3 b_4 $ & $-b_1 b_2 b_3 b_4 $ & $i b_{1}b_{2}$ \\ \cline{2-4}
$\mathcal{B}_{3}$ & $-i b_1 b_5 $ & $ -i b_2 b_3 $ & $ b_1 b_2 b_3 b_5$ \\ \cline{2-4}
\end{tabular}
\end{tabular}
\label{MagicSquare}
\end{table}
\subsection{The set-up}
\noindent
We now describe the set-up. We assume that Alice and Bob share five Majorana pairs, prepared in the canonical form outlined earlier.
Each party then makes a choice of three different measurement settings.
Each measurement setting relates to three measurements involving a subset of the Majorana modes held by the respective party. Hence,
for each measurement setting, a string of three values of $\pm 1$ is obtained, so $R=\{\pm1\}^{\times 3}$.
For Alice we label the choice of measurement as
$\mathcal{A}_{j}=(\mathcal{A}_{1,j},\mathcal{A}_{2,j}, \mathcal{A}_{3,j})$
for $j\in\{1,2,3\}= I$ and for Bob the choice of measurement is written as $\mathcal{B}_{k}= (\mathcal{B}_{k,1},\mathcal{B}_{k,2}, \mathcal{B}_{3,k}$)
for $k\in I$. Alice and Bob then obtain the output strings
${\alpha}=(\alpha_{1}, \alpha_{2}, \alpha_{3})$,
${\beta}=(\beta_{1}, \beta_{2}, \beta_{3})\in \{\pm 1\}^{\times 3}$, respectively, with $j,k\in I$ labeling the choice of measurements $\mathcal{A}_{j}$ and $\mathcal{B}_{k}$. The measurements are fixed by the columns (for Alice) and rows (for Bob), of square tables (so-called ``magic squares"), as shown in Table~\ensuremath{\mathrm{Re}}f{MagicSquare}. In these magic squares, Alice and Bob's respective square tables contain, up-to a phase, the same measurements performed by the respective parties. It can easily be verified that an element of Alice's table multiplied with the same element of Bob's table (i.e. the same $j^{\mathrm{th}}$ row and $k^{\mathrm{th}}$ column in each table) gives a stabiliser of their shared quantum state. Formally, this enforces that $\alpha_{k}=\beta_{j}$ for all $j,k\in I$. Furthermore, the product of observables in any column of both tables gives $+\mathds{1}$, and so for any measurement setting $\alpha_{1} \alpha_{2} \alpha_{3}=1$. Whereas the product of observables in any row of both tables gives $-\mathds{1}$, and so $\beta_{1} \beta_{2} \beta_{3}=-1$. Notice that any triple of observables, for either Alice or Bob, always contains a single measurement observable acting on 4 Majorana modes. As remarked earlier, such measurements cannot be directly measured but can be inferred from other measurement outcomes. Here this poses no problem as the required information is provided by the remaining pair of observables. For instance, the first column for Alice corresponds to measuring $\{ -a_1 a_3 a_4 a_5, i a_3 a_4 , i a_1 a_5 \}$, and while $ -a_1 a_3 a_4 a_5$ cannot be directly measured, we see that $( i a_3 a_4) ( i a_1 a_5 ) = -a_1 a_3 a_4 a_5$ and so we can simply infer the outcome from $\alpha_{1}=\alpha_{2}\alpha_{3}$. In any actual experiment, Alice and Bob only measure a pair of observables, but for clarity of exposition it is convenient to speak of each party measuring a whole triple of observables.
Aside from these constraints, the measurement outcomes are entirely random and so the experiment realises
\begin{equation}
\label{majoranaProb}
P^{*}({\alpha}, {\beta} | j, k) = \bigg\{ \begin{array}{l}
\frac{1}{8} \mbox{ if } ( \alpha_{k}=\beta_{j} ) \wedge ( \alpha_{1}\alpha_{2}=\alpha_{3} ) \wedge ( \beta_{1}\beta_{2}=-\beta_{3} ), \\
0 \mbox{ otherwise.}
\end{array}
\end{equation}
and in the next section we confirm this to be non-local and robust against some experimental noise.
\subsection{The magic square game without inequalities}
\noindent
Earlier, we saw that for any non-local game the best local strategy can be achieved by Alice and Bob deterministically assigning measurement outcomes. In the problem at hand, it is convenient to describe these local probability distributions by a $3\times 3$ table with entries in $\{\pm 1\}$.
We use $T^{A}$ to denote Alice's table, which has entries such that
\begin{equation}
c_{A}({\alpha}|j) = \left\{\begin{array}{l} 1 \mbox{ if } (T_{1,j}^{A},T_{2,j}^{A},T_{3,j}^{A}) = {\alpha}, \\
0 \mbox{ otherwise}.
\end{array}
\right.
\end{equation}
Whereas for Bob, we take
\begin{equation}
c_{B}({\beta}|k) = \left\{
\begin{array}{l} 1 \mbox{ if } (T_{k,1}^{B},T_{k,2}^{B},T_{k,2}^{B}) = {\beta}, \\
0 \mbox{ otherwise}.
\end{array}
\right.
\end{equation}
Clearly, there is a one-to-one correspondence between these tables and deterministic probability distributions. Note also the slight difference in definitions for Alice and Bob. For Alice, a single measurement setting will output a column of her table, whereas for Bob, a single measurement setting will output a row of his table. This convention is justified because the quantum correlations satisfy $\alpha_{k}=\beta_{j}$ for all $j,k\in I$, and so if the LHV theory replicates this correlation we have that
\begin{equation}
T^{A}_{k,j}=T^{B}_{k,j}.
\end{equation}
Therefore, Alice and Bob must share identical tables to reproduce this feature of the quantum correlations. However, the quantum correlations have an additional feature, they satisfy parity constraints. Alice's parity constraint entails that for all $j$, we have $\prod_{l\in I} T^{A}_{l,j}=1$, and hence the parity of the entire table is $\prod_{l,j\in I} T^{A}_{l,j}=1$. In contrast, Bob's parity constraint entails that for all $k$, we have $\prod_{l\in I} T^{B}_{k,l}=-1$, and so $\prod_{l,k\in I} T^{B}_{k,l}=-1$. We conclude that Alice and Bob cannot hold identical classical tables and also satisfy all the parity constraints, and so they can never perfectly reproduce the quantum correlations. Since quantum observables do not commute, finding the parity of the table of operators can change depending on the order we multiply the entries.
\subsection{The magic square game with Bell inequalities}
\noindent
We have seen that classical experiments can never reproduce the quantum correlations demonstrated in the previous section. However, we are interested in what level of imperfection can be tolerated by any quantum experiment while still violating locality. To quantify this we must specify a particular non-local game, see Eq.~(\ensuremath{\mathrm{Re}}f{non-localGame}),
by specifying a function $V$. The goal is to have a larger value when the correlations match those of the quantum predications, and smaller when they fail. The simplest choice is a function that takes two values, and it is conventional to take these as $\pm 1$, and so we have
\begin{equation}
V({\alpha}, {\beta}, j, k) = \left\{ \begin{array}{l}
1 \mbox{ if } ( \alpha_{k}=\beta_{j} ) \wedge ( \alpha_{1}\alpha_{2}=\alpha_{3} ) \wedge ( \beta_{1}\beta_{2}=-\beta_{3} ), \\
-1 \mbox{ otherwise.}
\end{array}
\right.
\end{equation}
This non-local game we call the magic square game. It is easy to verify that using Majorana fermions, which realise the probability distribution $P^{*}$ (see Eq.~\ensuremath{\mathrm{Re}}f{majoranaProb}), we find $G(P^{*})=9$.
We are now in a position to identify $\Omega_c(G)$, the maximum classical value. Alice and Bob could use identical tables, but they then contravene many parity constraints, and we find this results in at most $G(P)=3$. However, the maximum is achieved if Alice and Bob use classical tables that differ only in a single entry and satisfy all the parity constraints, such as in Table~\ensuremath{\mathrm{Re}}f{classical}. This yields $G(P)=7=\Omega_{c}(G)$ since for 8 of the measurement settings we acquire a contribution of $1$, but for one setting we have $-1$. Hence, any imperfect experiment with Majorana fermions that attains $G(P)>7$ is sufficient to demonstrate non-locality.
\begin{table}
\centering
\tcaption{An attempted classical strategy for the magic square game, where Alice and Bob satisfy all of their row/column constraints. To achieve this they must differ in at least 1 entry, which here is the bottom right entry. When the referee specifies the bottom row for Bob and right-most column for Alice, this strategy will fail. However, it will win in the other 8 choices of row and columns.}
\begin{tabular}{ccc}
{Alice} & & {Bob} \\
\begin{tabular}{|c|c|c|}
\hline
+1 & +1 & -1 \\ \hline
+1 & -1 & +1 \\ \hline
+1 & -1 & \emph{-1} \\ \hline
\end{tabular}
& &
\begin{tabular}{|c|c|c|}
\hline
+1 & +1 & -1 \\ \hline
+1 & -1 & +1 \\ \hline
+1 & -1 & \emph{+1} \\ \hline
\end{tabular}
\end{tabular}
\label{classical}
\end{table}
We also comment on how the imperfect quantum setting can, sometimes, economically be described. For perfect operations of the type considered here, the covariance matrix $\gamma$ will only contain entries
contained in $\{0,-1,1\}$ and satisfies $\gamma^2=-\mathds{1}$.
If errors and imperfections are present, the entries of $\gamma$ will also attain values different from those,
while it is still true that
\begin{equation}
-\gamma^2\leq \mathds{1}.
\end{equation}
Under braiding transformations of the form (\ensuremath{\mathrm{Re}}f{TF}), covariance matrices transform as congruences
$\gamma\mapsto V\gamma V^T$ with orthogonal matrices $V$,
which can easily be kept track of. The statistics of measurements can still be determined based on the covariance
matrix only, even if errors are taken into account in the preparation step.
\section{Too few Majorana pairs}
\subsection{Setting}
\noindent
Earlier we saw that if Alice and Bob share three Majorana pairs, the available measurements are isomorphic to Pauli measurements on a single Bell pair. It is well-known folklore that such a system can be modeled by a local-hidden variable theory. Here we present a stronger argument covering up to 4 Majorana pairs. This shows that five or more Majorana pairs are necessary, as well as sufficient, to demonstrate non-locality. We begin by characterizing the full set of possible measurements. For Alice, there are three pairs of commuting measurements she can perform
\begin{eqnarray}
\mathcal{A}_{1} &=& ( i a_{1}a_{2} , i a_{3}a_{4} ) , \\
\mathcal{A}_{2} &=& ( i a_{1}a_{3} , i a_{4}a_{2} ) , \\
\mathcal{A}_{3} &=& ( i a_{1}a_{4} , i a_{2}a_{3} ) .
\end{eqnarray}
If Alice measures a pair of observables $\mathcal{A}_{j}$, she get two random bits, which we denote $\alpha_{1}, \alpha_{2} \in \{ +1, -1 \}$. Similar measurement sets, labeled $\mathcal{B}_{k}$, are available to Bob, with $b_{j}$ replacing $a_{j}$. If Bob measures the analogous set, so $k=j$, he gets perfectly anticorrelated outcomes, such that $\beta_{1}=-\alpha_{1}$ and $\beta_{2}'=-\alpha_{2}'$. Next we consider when Bob measures a different set of operators, so $k \neq j$. We need to determine which products of measurements correspond to stabilisers of the 4 Majorana pairs. We observe that for Alice and any measurement set, the product of the observables is $-a_{1}a_{2}a_{3}a_{4}$, and similarly for Bob the product is $-b_{1}b_{2}b_{3}b_{4}$. Since the observables are matching and contain an even number of fermionic operators, these observables will have correlated outcomes. Formally, these correlations entail that $\alpha_{1} \alpha_{2} = \beta_{1} \beta_{2}$ for all measurement settings $j$ and $k$. However, when $j \neq k$, this is the only correlation and otherwise the measurement outcomes are entirely random. From these observations we can deduce that the outcome probabilities, as a function of measurement settings $j$ and $k$, are
\begin{equation}
\label{Pconditions}
P({\alpha}, {\beta} | j, k) = \left\{ \begin{array}{ll} \frac{1}{4}
& \mbox{if } (j=k) \wedge (\alpha_{1}=-\beta_{1}) \wedge ( \alpha_{2}=-\beta_{2} ) ,\\
0 & \mbox{if } ( j=k) \wedge ( \alpha_{1} \neq-\beta_{1} \mbox{ or } \alpha_{2} \neq-\beta_{2} ), \\
\frac{1}{8} & \mbox{if } (j \neq k ) \wedge ( \alpha_{1} \alpha_{2}= \beta_{1} \beta_{2} ) ,\\
0 & \mbox{if } ( j \neq k) \wedge ( \alpha_{1} \alpha_{2} \neq \beta_{1} \beta_{2} ).
\end{array}\right.
\end{equation}
The structure is richer than for Pauli measurements on a Bell pair. However, it can still be explained by an LHV theory, which we now turn to.
\subsection{Local hidden variable model}
\noindent
For our LHV theory we use a set of four hidden variables, which we label as $\lambda= \{ \nu_{1}, \nu_{2}, \nu_{3}, \mu \}$, each of which takes values $\{ +1, -1\}$.
Hence, we can write
\begin{equation}
\int M(d\lambda) = \sum_\lambda p(\lambda).
\end{equation}
We distinguish $\mu$ from the other variables as it plays a unique role. Next we fix Alice's probabilities to depend deterministically on the hidden variables as follows
\begin{eqnarray}
p_{A}({\alpha} | j, \lambda) =
\left\{\begin{array}{ll} 1 & \mbox{if } \alpha_{1}=\nu_{j}, \alpha_{2}=\mu \nu_{j} ,\\
0 & \mbox{otherwise},
\end{array}
\right.
\end{eqnarray}
and for Bob we take
\begin{eqnarray}
p_{B}({\beta} | k, \lambda) =
\left\{
\begin{array}{ll} 1 & \mbox{if } \beta_{1}=-\nu_{k}, \beta_{2}=-\mu \nu_{k} , \\
0 & \mbox{ otherwise}.
\end{array}
\right.
\end{eqnarray}
Notice that, for all measurement settings, the measurement outcomes obey
\begin{equation}
\alpha_{1} \alpha_{2} = \mu,\,\beta_{1} \beta_{2} = \mu.
\end{equation}
Hence, for all choices of the hidden variables
$\lambda$, the measurements outcomes satisfy $\alpha_{1} \alpha_{2} = \beta_{1} \beta_{2}$. Furthermore, when $j=k$ we have the added constraint that measurements satisfy $\alpha_{1}=-\beta_{1}$ and $\alpha_{2}=-\beta_{2}$. This tells us that for all choices of hidden variables, the distributions satisfy the second and fourth lines of Eqs.~(\ensuremath{\mathrm{Re}}f{Pconditions}). To achieve the correct weighting of the non-zero probabilities we simply take a uniform distribution over the hidden variables so that $p(\lambda) =1/16$ for all $\mu$ and $\nu_{j}$. This completes our account of a LHV theory for all possible measurements on 4 Majorana pairs.
\section{Impossibility of GHZ state preparation}
\label{NOghz}
\noindent
Here we show that the correlations of 3-party GHZ states cannot be prepared using topologically protected operations and Majorana fermions, blocking attempts to violate locality using the Mermin-GHZ paradox. The proof technique can be easily extended to larger GHZ states, and the associated generalizations of the GHZ paradox~\cite{Hoban11}. We begin with some definitions and general observations. We consider products of fermionic operators, of the form
\begin{equation}
S = i^{q} \prod c_{j}^{w_{j}} ,
\end{equation}
where $w_{j} \in \{ 0,1 \}$ and $q \in \{0, 1,2,3\}$. For two such operators, $S$ and $S'$, denote with
$S \star S'$ the overlap in fermionic operators they hold in common, so that $S \star S' = \sum_{j} w_{j} w_{j}'$. Furthermore, let $S \star S' \star S''$ denote the overlap shared by all three operators, which is $S \star S' \star S''= \sum_{j} w_{j}w_{j}'w_{j}''$. We will show that, for any accessible state with stabiliser $\mathcal{S}$, both the following hold
\begin{enumerate}
\item[(i)] for all $S,S' \in \mathcal{S}$, we have that $S \star S'$ is even;
\item[(ii)] for all $S,S', S'' \in \mathcal{S}$, we have that $S \star S' \star S''$ is even.
\end{enumerate}
Accessible states possess a stabiliser $\mathcal{S}$ generated by $g_{j}= \pm i c_{P_{2j-1}}c_{P_{2j}}$, where $P\in S_{2n}$.
The structure of the generators imposes a structure on the whole group, which we use to prove the above. First, note that distinct generators $g_{j}$ share no fermionic operators in common, and so $g_{j} \star g_{k} = 2 \delta_{j,k}$. Expressing the stabilisers of an accessible state in terms of the generators, we have
\begin{equation}
S = \prod_{k} g_{k}^{u_{k}} ,
\end{equation}
for some $u_{k}\in \{0,1\}$.
For two such operators, each generator they share contributes a pair of fermionic operators in common, and so
\begin{equation}
S \star S' = 2 \sum_{k} u_{k} u_{k}' .
\end{equation}
Clearly, this is always even. Furthermore,
\begin{equation}
S \star S' \star S'' = 2 \sum_{k} u_{k} u_{k}' u_{k}'' ,
\end{equation}
which is again even. More generally, overlaps between larger collections of operators must, for accessible states, again be even. Note also that, property (i) also follows from the necessary commutativity of the group $\mathcal{S}$, and is true for all quantum states. Whereas property (ii) is a genuine constraint on the whole set of quantum states.
The correlations required for the Mermin paradox can be achieved by measuring Pauli operators on a GHZ state stabilised by
\begin{eqnarray}
S&=& X_{1}Z_{2}Z_{3} , \\
S'&=& Z_{1}X_{2}Z_{3} , \\
S''&=&Z_{1}Z_{2}X_{3} , \\
S'''&=&-X_{1}X_{2}X_{3} ,
\end{eqnarray}
using some encoding ${\cal F}\rightarrow {\cal B}({\cal H})$.
From the anti-commutation of the Pauli operators, that is $X_{j}Z_{j}=-Z_{j}X_{j}$, we see that the last stabiliser follows from the first three, such that $S'''=S''S'S$. We consider all possibilities where the Pauli operators are replaced by appropriate products of Majorana operators. To be measurable, every $X_{j}$ and $Z_{j}$ must consist of a product of an even number of Majorana operators. As remarked earlier, it is essential $X_{j}$ anti-commutes with $Z_{j}$ and so $X_{j} \star Z_{j} = \gamma_{j}$ where $\gamma_{j}$ is odd. Finally, they must be local operators, so operators associated with different parties must be supported on distinct subsets of Majorana modes. From locality we deduce that
\begin{eqnarray}
S \star S' \star S'' &=& (X_{1} \star Z_{1})+(X_{2} \star Z_{2})+(X_{3} \star Z_{3}) \nonumber \\
&=& \gamma_1 + \gamma_2 + \gamma_3 .
\end{eqnarray}
From anti-commutivity, all of the $\gamma_{j}$ are odd, and the sum of 3 odd numbers is again odd. We conclude that such a set of stabilisers contradicts property (ii) of accessible states. Under very general assumptions we have shown that no accessible states are isomorphic to the 3-qubit GHZ state. In particular, the most natural encoding would be to take $Z_{j}=i c_{3j}c_{3j+1}$ and $X_{j}=i c_{3j}c_{3j+2}$, for which $\gamma_{j}=1$ and $S' \star S'' \star S'''=3$. However, more exotic encoding beyond this more canonical choice are also covered by the above argument. Alternative proofs based on the fermionic Gaussian nature of the states generated by topological operations are conceivable, and could potentially make use of Wick's theorem.
\section{Quantum information protocols with Majorana fermions}
\noindent
We finally comment on the perspective of using Majorana fermions in basic protocols of quantum information processing involving entanglement. Surely, all of the consequences of braiding operations can be classically efficiently simulated, by virtue of the observation that they constitute a subset of those operations in fermionic linear optics. Still, a number of interesting quantum information protocols involving entanglement can readily be conceived which are sketched here.
This analysis complements the findings of Ref.\ \cite{Freedman}, in which measurement-only topological quantum computation has been considered.
\subsection{Teleportation}
\noindent
Notably, instances of {\it teleportation} \cite{BBCJPW01a}
are possible. Meaningful variants of teleportation involving Majorana fermions
should share the features that (i) an unknown state is considered the input, taken from a set of at least two non-orthogonal quantum
states, and (ii) the output should be statistically indistinguishable from the input to the protocol.
Here, Alice and Bob not only share $n$ Majorana pairs, so as usual, the initial state is stabilised by $i a_{j}b_{j}$ for $ j=1,\dots, n$. But furthermore, Alice holds further
$n$ Majorana modes that are not entangled with Bob. These local ancilla are stabilised by
\begin{equation}
i a_{j}a_{j+1},\,\, j={n+1, n+3,\dots , 2n-1}.
\end{equation}
and represent the modes that will be teleported, herein called the input modes. Such a scheme can indeed be devised by making use of 4 Majorana pairs, so $n=4$. The input register consists of 4 Majorana modes, initially stabilised by $ia_5 a_6$ and $ia_7 a_8$. Many different inputs could be prepared by braiding the input modes. For instance, the input can be set to one of two non-orthogonal inputs by either (I) braiding $a_6$ and $a_7$ or (II) not. It is not difficult to see that these two situations (I) and (II) cannot be perfectly distinguished with unit probability.
The remainder of the protocol does not depend on this input, and neither is knowledge of it required. In the next step,
$i a_5 a_1$,
$i a_6 a_2$,
$i a_7 a_3$,
$i a_8 a_4$
are measured and the results, elements of $\{\pm 1\}^{\times 4}$,
classically communicated to Bob. One can then show that
the output statistics of measurements performed by Bob, $i b_1 b_2$ and $i b_3 b_4$, appropriately interpreted using the received classical bits, is indistinguishable from
the respective measurements on $i a_5 a_6$ and $i a_7 a_8$. For example, in scenario (II), if the outcomes $(-1,-1,-1,-1)$ are communicated, the
outcomes of measurements $i b_1 b_2$ and $i b_3 b_4$| are completely determined to be $ +1$.
\subsection{Dense coding}
\noindent
In a similar fashion, {\it dense coding} \cite{Dense}
can be performed with Majorana fermions and the above specified operations. Again, let us be specific what a fair analogue of such a scheme would be.
A valid dense coding scheme is one in which the entanglement-assisted single shot classical capacity of a quantum channel is higher compared to
the corresponding capacity in the absence of entanglement. We hence would like to introduce a protocol
where starting from entangled shared resources,
with local braiding operations and a subsequent transmission of some modes, one can encode more bits of
classical information than is possible without having shared entangled resources available,
but transmitting the same number of modes. For qudits, it is known that the entanglement-assisted capacity is exactly double the one
reachable without
assistance \cite{WernerDenseCoding}. For Majorana fermions, it turns out,
the same holds true.
This is readily possible with 3 Majorana pairs, again partially shared by Alice and Bob and partially held by Alice.
Both parties share 2 Majorana pairs, stabilised by
$i a_1 b_1$,
$i a_2 b_2$. In addition, Alice holds 2 Majorana modes in a state stabilised by $i a_3 a_4$. By local braiding, Alice can achieve a state
stabilised by
\begin{equation}
\biggl\{i (-1)^{\gamma_1} a_1 b_1,
i (-1)^{\gamma_2} a_2 b_2,
i (-1)^{\gamma_1+\gamma_2} a_3 a_4: \gamma_1,\gamma_2\in \{0,1\}
\biggr\},
\end{equation}
in a scheme that maps
\begin{equation}
a_j \mapsto (-1)^{\gamma_j} a_j,
\end{equation}
$\gamma_j\in\{0,1\}$ for $j=1,\dots, 4$, and fixing $\gamma_3=\gamma_1+\gamma_2$ and $\gamma_4=1$.
It is easy to see that with a suitable choice of $\gamma_1,\gamma_2\in\{0,1\}$, 2 bits of information can be encoded. Furthermore, after Alice transmits Majorana modes $a_{1}$ and $a_{2}$ to Bob, he can reliably retrieve $\gamma_1$ and $\gamma_2$ by locally measuring $ia_{1}b_{1}$ and $ia_{2}b_{2}$. At the same time, with two Majorana modes and no shared entanglement, a single bit of information can be encoded only. This constitutes a
valid dense coding scheme based on Majorana fermions. It also resembles the situation of transmitting two bits of classical information with a single
transmitted qubit, if entangled resources are initially available.
\section{Closing remarks}
\noindent
We have seen that locality can be violated using only the topologically protected operations of Majorana fermions or Ising anyons. These operations are a subclass of fermionic linear optics~\cite{Bravyi05b}, and so we conclude that these systems can also violate locality. The model of fermionic linear optics was proposed as an analog of bosonic linear optics, where a well-known LHV theory prevents any non-locality experiment~\cite{Bell}. Seen in this light, our result is quite surprising. Furthermore, we remark that that fractionalized analogs of Majorana fermions, known as {\it parafermions}~\cite{Fendley12}, may face similar obstacles from the existence of certain hidden variable theories~\cite{Gross06,Veitch12,Mari12}. It is natural to ask more generally what other anyonic systems are non-local, again without the requirement of universal quantum computing capabilities.
We know of only one related study~\cite{Wootton}, where a six-level classical spin model was used to simulate the charge submodel of $D(S_{3})$ (the quantum double model based on the symmetric group of 3 objects). Although non-locality was not explicitly discussed, it is clear that the classical spin model amounts to a LHV theory. Finally, we speculate that these experiments could prove useful as a probe to certify the presence of Majorana fermions and potentially help dispel some of the present ambiguity surrounding experiments.
In the final stages of completing this research we became aware of Ref.~\cite{Deng13}, which addresses to some extent similar questions and concerns the non-locality of Majorana fermions when nondestructive collective-charge measurements are available. This represents an additional experimental capability, posing additional challenges, beyond braiding and standard measurements. However, without this capability, the GHZ correlations required by Ref.~\cite{Deng13} cannot be implemented by virtue of our no-go result of Sec.~\ensuremath{\mathrm{Re}}f{NOghz}.
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\end{document} |
\begin{document}
\title{When quantum tomography goes wrong: drift of quantum sources and other errors}
\author{S.J.~van Enk$^{1,2}$ and Robin Blume-Kohout$^3$}
\affiliation{$^1$Department of Physics and
Oregon Center for Optics\\
University of Oregon, Eugene, OR 97403\\
$^2$ Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125\\
$^3$ Sandia National Laboratories, Albuquerque, NM 87123}
\begin{abstract}
The principle behind quantum tomography is that a large set of observations -- many samples from a ``quorum'' of distinct observables -- can all be explained satisfactorily as measurements on a \emph{single} underlying quantum state or process. Unfortunately, this principle may not hold. When it fails, any standard tomographic estimate should be viewed skeptically. Here we propose a simple way to test for this kind of failure using Akaike's Information Criterion (AIC).
We point out that the application of this criterion in a quantum context, while still powerful, is not as straightforward as it is in classical physics. This is especially the case when future observables differ from those constituting the quorum.
\end{abstract}
\maketitle
\section{Introduction}
\subsection{General remarks}
The goal of quantum-state tomography \cite{Paris2004}
is to give a statistically reliable estimate of a quantum state $\rho$.
Two further questions may come to mind: (i) what is the purpose of that estimate $\rho$? And (ii), why or when are we correct in giving an estimate of {\em just one} quantum state?
There are at least two answers to the first question: our experiment may be aimed at producing a particular state, say, a cluster state, and we may just want to verify how close $\rho$ is to the desired state. But that answer provides really just an intermediate goal. The ultimate goal is always to use the desired state for some particular quantum information processing task. So we could say that the goal of producing an estimate $\rho$ is to be able to predict the future performance in a particular protocol of one or more unmeasured quantum system(s) produced by the same source.
Now there is a nice statistical method
for ranking different models according to their ability to predict future measurement results ({\em not} on how well they fit the past data!), based on the Akaike Information Criterion (AIC) \cite{Akaike}. That criterion was developed entirely within a classical context, but it ought to apply to quantum-state estimation, too. We show this is true, even though we will point out some interesting differences between classical and quantum statistics.
The motivation behind the second question is as follows.
Since we do not have full control over all physical quantities relevant
to the quantum-state generation process (for example, even the best laser suffers from phase diffusion; and there are always spatially and temporally fluctuating magnetic and electric fields), the quantum states produced by a quantum source are not all identical. A possible description of the individual states of $M$ systems $k=1\ldots M$ would be a sequence $\{\rho_k,k=1\ldots M\}$ where each $\rho_{k+1}$ is a little different from the previous one (even with entanglement or correlation between the different systems, we can define $\rho_k$ by tracing out all the other systems).
So, why would we use just a single estimate $\rho$ in this case? One aspect of the answer is, of course, that we have no way of estimating each individual $\rho_k$. A more positive answer is that multiple measurements of a given observable $\hat{O}$ only yield estimates of {\em average} quantities such as $\expect{O}=\overline{\mathrm{Tr}\rho_k\hat{O}}$ or $p_n=\overline{\mathrm{Tr}\rho_k\proj{O_n}}$, where the average is over those $k$ on which $\hat{O}$ was measured, and where $\ket{O_n}$ denotes an eigenstate of $\hat{O}$. These averages being linear in $\rho_k$ are determined by a {\em single} density matrix, namely the average density matrix $\rho=\overline{\rho_k}$.
This simple picture has been made much more rigorous by Renner in \cite{Renner2007}.
He showed that the crucial ingredient (missing in the simple picture) is permutation invariance.
That is, if we randomly permute the sequence of quantum systems, and then trace out some subset, the joint state of the remaining systems is to a good approximation independently and identically distributed (i.i.d.).
In our context this means that as long as the quorum of observables is measured in a random order, then to a good approximation any one of the remaining unmeasured systems can be described by a single density matrix $\rho$. We now discuss what may go wrong if we measure the observables constituting a quorum in a nonrandom order.
\subsection{Possible errors in standard quantum state tomography}
It is much easier to measure a given observable from the quorum many times in a row, before switching to measurement of the next observable. Such a procedure is standard practice, but it voids Renner's proof, and so it may be that there is not a single density matrix that can be validly assigned to the remaining unmeasured quantum systems.
Let us introduce this problem with a simple example. Given an ensemble of $3N\gg1$ qubits that -- we assume! -- are identically and independently prepared, we want to estimate their density matrix. So we divide them into three equal \emph{and sequential} groups, and measure $\sigma_x$ on samples $1\ldots N$, $\sigma_y$ on samples $N+1\ldots 2N$, and $\sigma_z$ on the last $N$. Now, if the samples are indeed identically prepared in some state $\rho$, then we can safely perform the measurements in this order -- the state $\rho^{\otimes 3N}$ is invariant under permutations, so all orderings are equivalent. But if the source is drifting over time, the first $N$ copies are best described by a mean density matrix $\bro{1}$, while the second and third sets of $N$ qubits are best described by (possibly different) average states $\bro{2}$ and $\bro{3}$, respectively.
For an amusing (albeit extreme) example, consider a situation where the first $N$ copies are best described by $\bro{1} = \proj{+}$, the second group by $\bro{2}=\proj{+i}$, and the third by $\bro{3}=\proj{0}$. The measurement outcomes in this case are not random at all: every single measurement (of $\sigma_x$, $\sigma_y$, $\sigma_z$) will yield eigenvalue $+1$. Linear inversion tomography will yield a radically non-positive state
\begin{equation}
\rhohat_{\scriptscriptstyle\mathrm{tomo}} = \left(\begin{array}{cc} 1 & \frac{1+i}{2} \\ \frac{1-i}{2} & 0 \end{array}\right),
\end{equation}
and maximum likelihood estimation (MLE) yields the projector onto $\rhohat_{\scriptscriptstyle\mathrm{tomo}}$'s positive eigenspace. Although both estimates are plausible answers to "What single matrix best fits the observed data?", neither one of them is of any predictive use at all! The source is drifting so rapidly and drastically that \emph{this} set of 3N samples really tells us almost nothing about future observations. This is the simplest and best conclusion at which our data analysis should arrive.
This is a rather extreme and contrived example of experimental drift [below we will discuss a more common type of nonrandom experiment where the above cycle of measurements is repeated once: so we measure $\sigma_x$ on the first $N/2$ copies, then $\sigma_y$, then $\sigma_z$, and then $\sigma_x, \sigma_y,\sigma_z$ again, each on $N/2$ sequential copies]. More realistic examples show similar behavior, though. The statistics given above are actually more consistent with a different (and still plausible) mechanism: When the measurement apparatus is ``rotated'' to perform a different measurement, the experimenter inadvertently ``rotates'' the samples as well. A particularly na\"ive version of this could occur with photon polarization, where one way to physically rotate a polarizer is for the experimentalist to simply rotate his own frame of reference (e.g., by lying down). Such a passive rotation obviously fails to change the relative orientation of samples and apparatus. More realistic examples occur when similar quantum gate devices are used to (1) prepare states (e.g. EPR states) and (2) implement measurements. In quantum process tomography, this sort of pitfall is well known; it violates the conditions for complete positivity of processes, and causes negative eigenvalues just as in our example above \cite{WeinsteinJCP04}.
All of these failures are examples of a single phenomenon: \emph{sample-apparatus correlation}. In process tomography, this is usually explained by correlation between the system and its environment. In state tomography, there is no environment per se, but if the state of the $k$th sample is (in any way) correlated with the behavior of the measurement apparatus (e.g., with what measurement it is oriented to perform), then tomography goes wrong. Experimental drift is a simple and easy to understand example: the sample state is correlated with time, and if the apparatus setting is also allowed to vary with time, then there will be sample-apparatus correlation. As noted above, this can be eliminated by explicitly randomizing the order of measurements, so that while the samples are still time-dependent, the apparatus is not. Other kinds of sample-apparatus correlation are not so easy to remedy.
In the example given above, the extremity of the data -- and the fact that the linear inversion estimate is radically negative -- are a dead giveaway. On the other hand, linear inversion can produce negative estimates even with ideal data \cite{BlumeKohoutNJP10,JamesPRA01} because of statistical fluctuations. The \emph{raison d'etre} of MLE is to fix this negativity, but by constraining the estimate to positive states, MLE also hides the tell-tale signature of failed tomography. Moreover, negative estimates are not (in general) a reliable symptom even of drastic experimental drift. If the drifting states in the example above were a bit more mixed -- e.g. $\bro{k}' = \frac12\bro{k} + \frac141\!\mathrm{l}$ -- then linear inversion and MLE would yield identical and positive density matrices. But, just as in the original example, those estimates would be useless and not predictive.
Fortunately, there is a general solution to this problem. It elegantly generalizes the observation (made above) that a radically negative $\rhohat_{\scriptscriptstyle\mathrm{tomo}}$ should trigger skepticism. It can also diagnose drift in the absence of negativity if the data are sufficiently rich. It is called \emph{model selection}.
The core principle is that, when tomography fails:
\begin{enumerate}
\item The standard model for tomography -- i.i.d. samples described by a single density matrix -- is bad.
\item Some other model will be better.
\item We can quantify ``bad'' and ``better'', and use the results to decide whether our tomography went wrong.
\end{enumerate}
Clearly, putting this into practice requires that we come up with alternative models to describe the data. Model \emph{design} is more of an art than a science. Here, we demonstrate alternative models for some simple and relevant problems, and leave the rich problems of general and optimal alternative-model design to future work. Instead, we focus on model \emph{selection}, which means determining whether (i) the standard tomographic model is pretty good, or (ii) some other model (e.g. a drifting source model) is better.
\subsection{Akaike to the rescue}
To accomplish this, we propose, as we mentioned above, to use the \emph{Akaike Information Criterion} (AIC) \cite{Akaike}. Widely used outside of physics \cite{BookAIC1,BookAIC2}, the AIC is relatively unknown within the physics community. However, it has been applied in astrophysics \cite{AIC2007}, entanglement verification \cite{Pavel2009}, and quantum state estimation \cite{Usami,Jun4,Guta}. Its function is to quantify (by assigning a real number) how well a given model describes the data from a given experiment. The AIC's absolute value is not meaningful, but the \emph{relative} AIC values for multiple different models have a deep and useful meaning (see following section for a more detailed discussion of the AIC, its meaning, and its derivation). Their simplest use is to rank all the different models, and thus to identify (a) which is the best, and (b) how significantly ``worse'' the others are.
The AIC assigns a number $\Omega_k$ to each model $k$, given by \footnote{Usually an overall minus sign and an extra factor of 2 appear on the right-hand side of (\ref{AIC}), but for our purposes it is more convenient not to include those.}
\begin{equation}\label{AIC}
\Omega_k:=\ln {\cal L}_k -K_k,
\end{equation}
where ${\cal L}_k$ is likelihood of model $k$ -- or, if model $k$ has adjustable parameters (as is usually the case), the \emph{maximum} of the likelihood over all those parameters -- and $K_k$ is the number of independent model parameters used in model $k$ to
fit the data \footnote{Actually, when the number of measurements (e.g. $3N$ in the example given) is small, there is a correction term. The \emph{corrected AIC} (AIC$_c$) is given by $\Omega^c_k = \Omega_k - K_k(K_k+1)/(3N-K_k-1)$, with $\Omega_k$ given by \ref{AIC}. Hereafter, we will simply assume $3N\gg K_k$.}. The larger the AIC ($\Omega_k$) is, the higher the model is ranked. While $\Omega_k$'s absolute value is meaningless, the difference $\Delta = \Omega_k-\Omega_{k'}$ represents (roughly speaking) the weight of evidence in favor of $k$ over $k'$, measured in bits. So, for example, if we want to report a weighted average of the two models, the ratio of the weights assigned to models $k$ and $k'$ should be $w_k/w_{k'} = \exp(\Omega_k-\Omega_{k'})$.
The AIC's simple form admits a simple interpretation: fitting the data better (higher likelihood) is good, but extra parameters are bad. Additional parameters must justify their existence by improving the likelihood (a measure of goodness-of-fit) by at least a factor of $e$. This helps to prevent overfitting. Adding adjustable parameters will always improve a model's fit -- but a good fit to past data is \emph{not} a guarantee that the model will accurately predicting future measurements. \emph{Example}: If we measure each of $3N$ qubits, measuring $\hat{O}_j$ on qubits $j$ for $j=1\ldots 3N$, then the best possible fit to the data is to assume that each qubit $j$ just happened to be in the appropriate eigenstate of $\hat{O}_j$ so that the probability of the observed data is $\mathcal{L} = 1$! Intuitively, this ``explanation'' is absurd. The AIC quantifies that intuition; that model requires a huge ($O(N)$) number of parameters, and the resulting penalty will overwhelm its higher likelihood, ensuring that its AIC is far worse than that of simpler models.
To apply the AIC to our example, we need an alternative model (the ``standard model'' just uses a single density matrix for all 3N qubits). A simple alternative that describes experimental drift (as well as some other forms of sample-apparatus correlation) is to use one density matrix for each of the 3 groups of samples. This alternative model will always fit the data at least as well, but it may use more parameters \footnote{Intuitively, the alternative model is more contrived, and this should be reflected in its ranking. However, it is not immediately obvious how to quantify this complexity.} The AIC ranks both models, and quantifies how much better one is than the other. We perform and analyze this calculation for our single-qubit example (where just two models are sufficient) in Section \ref{qubit}, and
address more complicated variations on this theme -- with multiple alternative models -- in Section \ref{two}.
To conclude this (long) Introduction, we note that the the appearance of maximum likelihoods in the AIC does \emph{not} imply any privileged role for MLE estimation of states or any other physical quantities. The likelihood is a central concept in statistics, and appears in almost every method. In the AIC, it is used specifically to quantify goodness-of-fit, and (obviously) the AIC balances this quantity against another (model complexity). Moreover, the AIC is used only to rank different models. There is no implicit requirement that the highest-ranked model must be chosen exclusively (in fact, a common strategy is to average over high-ranked models), and even if the ``best'' model is chosen, we remain free to analyze that model without MLE (e.g., via Bayesian averaging).
\section{Examples}
In this Section we first treat the example from the Introduction, tomography on single qubits, in more detail (Sec.~\ref{qubit}). In this example, inconsistencies can arise only when the observed average values of $\sigma_x, \sigma_y, \sigma_z$ are inconsistent with each other, which in turn can only happen if the density matrix obtained by linear inversion is unphysical.
The next example, discussed in \ref{One2}, also concerns single qubits, but now measurements of $\sigma_x,\sigma_y,\sigma_z$ are each repeated once. In this (experimentally more relevant) case inconsistencies can arise when two estimates of the same quantity are statistically different. Ad-hoc methods that just consider this particularly simple type of inconsistencies work just as well as the AIC.
In the last subsection, \ref{two}, we will consider the case of multiple qubits, in which the validity of ad-hoc methods is much harder to verify, but the AIC still works in the same manner, thus showing the universality of that method.
\subsection{One qubit, part 1}\label{qubit}
We return to tomography of single qubits, where we measure $\sigma_x$ on the first $N$ qubits, then $\sigma_y$ on the next $N$, and $\sigma_z$ on the last $N$ qubits.
Denote the three thusly observed averages by $X:=\expect{\sigma_x}_{{\rm obs}}$,
$Y:=\expect{\sigma_y}_{{\rm obs}}$, and
$Z:=\expect{\sigma_z}_{{\rm obs}}$.
In order to calculate likelihoods, we need the frequencies of having observed spin up ($+$) and down ($-$), respectively. They are
given in terms of these averages by
\begin{subequations}
\begin{eqnarray}
f_{\pm}^{(x)}&=&\frac{1\pm X}{2},\\
f_{\pm}^{(y)}&=&\frac{1\pm Y}{2},\\
f_{\pm}^{(z)}&=&\frac{1\pm Z}{2}.
\end{eqnarray}
\end{subequations}
A density matrix describing just the first set of $N$ measurements really uses or needs only one parameter, $X$ (the other two parameters are, obviously, not at all determined by those data). And no matter what $X$ is, there is always a perfect fit to the data. The logarithm of the (maximum) likelihood of such a density matrix is, therefore,
\begin{equation}
\ln{\cal L}^{(x)}=Nf_+^{(x)}\ln(f_+^{(x)})+Nf_-^{(x)}\ln(f_-^{(x)})
=-NH(\tfrac{1+X}{2}),
\end{equation}
with $H(.)$ the Shannon entropy.
The same story holds for the next two sets of measurements, and so there is always a perfect fit to the data when we use the ``alternative model'' with three density matrices, and that model needs three independent parameters. We conclude that the AIC assigns the following ranking to the alternative model:
\begin{equation}
\Omega_a=-N\left\{
H(\tfrac{1+X}{2})
+H(\tfrac{1+Y}{2})+H(\tfrac{1+Z}{2})\right\}-3.
\end{equation}
The performance of the ``standard model'' depends on the value of just one number. If
\begin{equation}\label{condqub}
R^2:=X^2+Y^2+Z^2\leq 1,
\end{equation}
there is a single maximum likelihood density matrix $\bar{\rho}$ (with purity $\mathrm{Tr}\bar{\rho}^2=(R^2+1)/2$) that describes the whole measurement perfectly, just as the alternative model does. The standard model also needs three parameters in this case, and the maximum likelihood is also the same as for the alternative model. So, in this case there is no real difference between the two models---we could pick $\bro{1}=\bro{2}=\bro{3}=\bar{\rho}$---and we have $\Omega_s=\Omega_a$. There is no reason to reject the standard model when $R\leq 1$.
Now let us suppose that $R>1$. We have then the choice between two descriptions:
\begin{enumerate}
\item \underline{Alternative model}: We describe
each of the three measurements by their own density matrix. The maximum likelihood estimates of those three states satisfy
\begin{subequations}
\begin{eqnarray}
\mathrm{Tr}\bro{1}\sigma_x&=&X,\\
\mathrm{Tr}\bro{2}\sigma_y&=&Y,\\
\mathrm{Tr}\bro{3}\sigma_z&=&Z.
\end{eqnarray}
\end{subequations}
{\em Three} independent parameters are needed for this model. ($\bro{1},\bro{2},\bro{3}$ are underdetermined, of course, but for the purpose of finding the maximum likelihood ${\cal L}_a$ the information suffices.)
\item \underline{Standard model}: We use one density matrix to describe all three measurements together.
The maximum likelihood estimate of that state will be {\em pure}. There is no known method to compute it exactly, but a generally good approximation is given by
\begin{subequations}\label{a}
\begin{eqnarray}
\mathrm{Tr}\bro{s}\sigma_x&=&X/R,\\
\mathrm{Tr}\bro{s}\sigma_y&=&Y/R,\\
\mathrm{Tr}\bro{s}\sigma_z&=&Z/R,
\end{eqnarray}
\end{subequations}
and this state's likelihood is a strict (but generally pretty tight) lower bound on the maximum likelihood for the standard model.
{\em Two} independent parameters are needed in this model \footnote{One of the authors remains bothered by the decision to separate the standard model, effectively, into two distinct models that contain (i) all the mixed states ($K=3$) and (ii) the pure states ($K=2$), respectively. However, this protocol is specifically discussed and justified by Burnham and Anderson (\cite{BookAIC1}, section 6.9.6). They express a similar concern, but also provide some preliminary justification for it. So, while further thought and research seems warranted, so does this choice.}.
\end{enumerate}
The reason we end up with a {\em pure} maximum likelihood state in the standard model is that the
single matrix fitting the data perfectly lies outside the set of physical states (it has a negative eigenvalue), and the closest physical state lies {\em on} the boundary \cite{BlumeKohoutNJP10}. In the case of qubits, this means a pure state. More precisely,
if the unphysical best-fit matrix $\tilde{\rho}$ is written in its diagonal form,
$\tilde{\rho}=\sum_{k=+,-}\lambda_k\proj{\psi_k}$, with $\lambda_+>1$ and $\lambda_-<0$, then the maximum likelihood estimate would be
$\bro{s}=\proj{\psi_+}$. The latter state has the properties (\ref{a}), as can be easily verified by explicit calculation.
Thus, when $R>1$ the
alternative model fits the data better but uses one more parameter than does the standard model.
We can calculate the maximum likelihoods analytically in each of the two models, and thus obtain the relative AIC score of the two models:
\begin{eqnarray}\label{Oms}
\Omega_s-\Omega_a=1+N\sum_{M=X,Y,Z}
\frac{1}{2}\ln
\frac{1-M^2/R^2}{1-M^2}
+\nonumber\\
\frac{M}{2}\ln\frac{(R+M)(1-M)}{(R-M)(1+M)}.
\end{eqnarray}
We accept the standard model as consistent iff $\Omega_s\geq\Omega_a$. This will happen only if $R$ is sufficiently close to 1. If we expand $R$ around 1, we can Taylor expand the right-hand side of (\ref{Oms}) as
\begin{equation}
\Omega_s-\Omega_a\approx 1-N\sum_{M=X,Y,Z}\frac{(R-1)^2M^2}{2(1-M^2)},
\end{equation}
provided $(R-1)^2\ll (1-M^2)$ for $M=X,Y,Z$. That is, with this proviso, the standard model is consistent only when
\begin{equation}\label{cond}
(R-1)\leq
\frac{C}{\sqrt{N}},
\end{equation}
with the constant $C$ given by
\begin{equation}
C=\frac{1}{\sqrt{\sum_M M^2/2(1-M^2)}}.
\end{equation}
The dependence of the condition (\ref{cond}) on $N$ agrees with the simple idea that it is sufficient for $R$ to be less than about a standard deviation or two above 1 for the standard model to still apply, and that standard deviation, of course,
decays like $1/\sqrt{N}$ for $N\rightarrow\infty$.
\subsection{One qubit, part 2}\label{One2}
The implementation of tomography in the previous example is probably too simple and too obviously wrong for it to have been applied in an actual experiment. The straightforward improvement to measure each of $\sigma_x, \sigma_y, \sigma_z$ in two separate blocks will allow one to detect drift. Let us denote the 6 observed averages by $X_{1,2}:=\expect{\sigma_x}_{{\rm obs 1,2}}$,
$Y_{1,2}:=\expect{\sigma_y}_{{\rm obs 1,2}}$, and
$Z_{1,2}:=\expect{\sigma_z}_{{\rm obs 1,2}}$. Drift can be detected by comparing the pairs of estimates $X_{1,2}$ with each other, $Y_{1,2}$ with each other, and $Z_{1,2}$ with each other.
The AIC works as follows:
We need again at least two different models for describing the data. One will be the standard model, with one density matrix describing all 6 measurements. This density matrix will be determined by the three averages $(X_1+X_2)/2$ etc.
The alternative model may consist of two independent density matrices (with 6 parameters in total) or of two density matrices that are not independent with either 4 or 5 parameters in total. Let us test the AIC in a simulation of data generated by single-qubit states of the form
\begin{equation}
\rho_{{\rm actual}}=p\proj{\psi_\phi}
+(1-p)\openone/2,
\end{equation}
where the pure state $\ket{\psi_\phi}$ depends on an angle $\phi$, which we assume to undergo a random walk, and with the following meaning:
\begin{eqnarray}
\bra{\psi_\phi}\sigma_x\ket{\psi_\phi}&=&\cos\phi;\nonumber\\
\bra{\psi_\phi}\sigma_y\ket{\psi_\phi}&=&\sin\phi;\nonumber\\
\bra{\psi_\phi}\sigma_z\ket{\psi_\phi}&=&0.
\end{eqnarray}
For $p$ we take the value $p=0.9$.
We perform in total 3000 measurements, divided into 6 groups of 500, in which we measure $\sigma_x,\sigma_y,\sigma_z,\sigma_x,\sigma_y,\sigma_z$ in that order.
\begin{figure}
\caption{Top:
Simulation of diffusion of the angle $\phi$ in the state $\ket{\psi_\phi}
\end{figure}
\begin{figure}
\caption{Same as Fig.~1, but for a case where the drift is much smaller over the course of 3000 measurements. Here
$\Omega_s-\Omega_a=1.38$, so that the
standard model is better than the best alternative model (which has two extra parameters). Tomography succeeded.}
\end{figure}
In Figs 1 and 2 we plot two qualitatively different cases. In the first case the diffusion of $\phi$ is so fast that it leads to noticeably different values of $X_{1,2}$ and $Y_{1,2}$. The AIC in this case gives a very clear preference for the alternative model of using two density matrices with 5 parameters in total (only the expectation value of $\sigma_z$ does not change over the course of the experiment).
In the second case the drift over the course of the experiment is small enough so that the standard model is still the best, even though there is some drift, and even though the more complicated model does, of course, fit the data slightly better.
\subsection{Two or more qubits}\label{two}
Consider now a tomographically complete measurement on $9N$ copies of two qubits, where on the first $N$ pairs of qubits we measure
$\sigma_x$ on both qubits independently, then on the next $N$ pairs we measure $\sigma_x$ on one qubit and $\sigma_y$ on the other, then on the third set of $N$ pairs we measure $\sigma_x$ on the one and $\sigma_z$ on the other $\ldots$ until on the last (9$^{{\rm th}}$) set of $N$ pairs we measure $\sigma_z$ on both qubits. The first measurement is described by {\em three} independent averages that are obtained from measuring $\sigma_x$ on both qubits independently:
\begin{subequations}
\begin{eqnarray}
XX&:=&\expect{\sigma_x\otimes\sigma_x}_{{\rm obs,1}},\\
IX&:=&\expect{\openone\otimes\sigma_x}_{{\rm obs, 1}},\\
XI&:=&\expect{\sigma_x\otimes\openone}_{{\rm obs, 1}}.\label{XI1}
\end{eqnarray}
\end{subequations}
Thus, a two-qubit density matrix
perfectly fitting the data of the first measurements needs three parameters. The description of the second measurement of
$\sigma_x$ on one qubit and $\sigma_y$ on the other is likewise determined by three observed averages
\begin{subequations}
\begin{eqnarray}
XY&:=&\expect{\sigma_x\otimes\sigma_y}_{{\rm obs, 2}},\\
IY&:=&\expect{\openone\otimes\sigma_y}_{{\rm obs, 2}},\\
XI'&:=&\expect{\sigma_x\otimes\openone}_{{\rm obs, 2}}.\label{XI2}
\end{eqnarray}
\end{subequations}
The new feature arising here is that we get a second estimate of the same parameter, $XI$ in this case. That is, if there were only a single two-qubit state in the experiment, the estimates (\ref{XI1}) and (\ref{XI2}) would have to agree (within error bars). Conversely, if they do not agree, we have encountered a new diagnosis of inconsistent tomography.
Writing down all different averages
obtained from this particular experiment, we find
9 quantities that are measured once, and 6 other quantities that are measured thrice.
It becomes now much harder to judge when all the differences between those different estimates of the same quantities are, in total, statistically significant or not. That is, the generalization of the ad-hoc method that worked fine for a single qubit, becomes troublesome. This, of course, becomes exponentially worse for more than two qubits.
On the other hand, the AIC can be applied straightforwardly to various alternative models. It is sufficient to find just one alternative model superior to the standard model in order to
have succeeded in diagnosing an inconsistency in our tomographic experiment. Of course there is a large multitude of alternative models, but one can be guided in searching for such models by looking for those estimates of the same quantities that are the least consistent.
\section{Model selection, the AIC, and quantum quirks}
Data are generally assumed to be generated by some stochastic process \footnote{A semi-philosophical note: it's not necessary to invoke intrinsic randomness. The underlying process might be deterministic but so complex that its description is hopeless or just not worth the time. The Bayesian view of probabilities is entirely compatible with this view, and permits us to \emph{describe} data and the processes that generate them without needing to ``believe in'' randomness.} -- e.g., a probability distribution $f(x)$ (where $x$ denotes the sample space containing all possible events). Unfortunately, these ``true'' probabilities are unknown to us. All we have are some data. So, in order to (i) describe the data; (ii) approximate the underlying process $f$; and (iii) most importantly, predict \emph{future} observations, we use \emph{models}.
A model is just another probability distribution $g(x)$. Almost always, the model contains a whole family of \emph{parameterized} distributions $g_\theta(x)$, where $\theta$ comprises the values of $K$ distinct [real-valued] parameters. One obvious model is the universal one where each of the probabilities $g(x)$ -- for every possible value of $x$ -- is itself a free parameter. This is the richest possible model, with the most parameters. If $x$ takes on uncountably many values, this model is utterly intractable (and the AIC penalizes it infinitely for its richness). The ubiquity of this problem in statistics motivates the use of restricted parameterized models (e.g., Gaussian distributions) where finitely many parameters can specify $g(x)$ for every possible $x$.
Quantum tomography applications usually involve finitely many parameters, but few-parameter models are still important. This is partly because of the simplification obtained by eliminating many parameters (e.g., when a quantum state in $2^N$ dimensions is approximated by a matrix product state with $\mathrm{poly}(N)$ parameters), but even more importantly because it guards against overfitting. This is precisely where well-designed model selection techniques come in, and the AIC is a canonical example. When there is a choice between different candidate models describing one and the same experiment, the AIC provides a numerical ranking of the different models.
The AIC (as given in Eq.~(\ref{AIC})) appears very simple. Moreover, it bears a strong resemblance to quantities that appear in likelihood-ratio (LR) hypothesis testing (see, e.g.~\cite{BlumeKohoutPRL10}). But in fact, the AIC's theoretical underpinnings are rather different, and remarkably elegant (see \cite{BookAIC1} for extensive discussion). Likelihood ratios are a fundamentally frequentist technique: given two competing models, we calculate ahead of time the probability that various values of the LR statistic will be observed \emph{if} one model or the other is ``correct'', and then we formulate a rule for what to announce upon seeing any given value of the LR statistic. Many canonical results on LR tests require that the models be nested -- i.e., that one be a subset of the other. In particular, given this and a few other conditions, it is possible to derive expectation values of the LR statistic that look identical to Eq.~(\ref{AIC}) because the loglikelihood ratio is $\chi^2_K$ distributed, and has mean value $K$.
But despite this similarity, the AIC is derived differently. Akaike began by postulating that ``goodness'' of a model is quantified by the Kullback-Leibler divergence \cite{KL51} between the model and the ``true model'' that actually underlies the data. Then, rather remarkably, he showed that it is possible to estimate this divergence \footnote{More precisely, this estimate is in fact determined only up to a constant, but that constant is the same for all models, and hence drops out when comparing different models.} -- \emph{even when the true model is unknown!} The AIC is the expected value of the [unknown] Kullback-Leibler divergence between a specified model and the [unknown] true model, conditional upon the data in our possession. So the AIC (i) has a powerful and universal interpretation, and (ii) can be used to compare arbitrary models, without any requirement for nesting.
This is not to say that the AIC is the acme of model selection, nor that it is perfectly adapted to quantum tomography problems. First, there are competing derivations of other model ranking statistics, such as the Bayesian Information Criterion (BIC -- again, see \cite{BookAIC1}). Moreover, the AIC is inherently an asymptotic result -- much like, for example, the efficiency of MLE. So, even though there is a finite sample size correction (the AIC$_c$), this correction is part of an asymptotic expansion and may be unreliable for any fixed $N$.
One significant consequence of this is that, for finite samples, an event $x$ whose true probability is nonzero may not be observed -- in which case a model might assign zero probability to it. (The MLE within the full model, where each probability $g(x)$ is a parameter, behaves this way). This results inevitably in an \emph{infinite} Kullback-Leibler divergence. Asymptotically, the probability of such a pathology occurring goes to zero almost certainly. But for any finite sample size it is a concern. So, beware of rank-deficient estimates in tomography!
A related phenomenon is [almost] unique to quantum tomography. Akaike's derivation assumes that a very good (if not the best) measure of predictive power is the Kullback-Leibler divergence between the true model $f(x)$ for the observed process $x$ and the assigned model $g(x)$. But in quantum tomography, the observed process ``$x$'' is some particular (and rather arbitrary) quorum of measurements that the tomographer has performed. We don't necessarily care about predicting \emph{those} measurements! Instead, we care about the underlying quantum state -- or, to put it more operationally, we care about a large and unknown set of \emph{other} measurements that might be performed on samples of that state in the future. Quite frequently, we care about measurements of that state's diagonal basis. This completely undermines Akaike's assumption (that predicting $x$ is the goal). This does not mean that the AIC should not be used -- but it does strongly suggest that:
\begin{enumerate}
\item Conclusions drawn from the AIC, or any other classical statistical method, should be treated with thoughtful care,
\item Better methods may still be derived (e.g., a ``quantum AIC'')
\item Estimates obtained via the AIC should \emph{not} be expected to have good properties with respect to quantum relative entropy (the quantum version of Kullback-Leibler divergence).
\end{enumerate}
Importantly, however, there are cases where our future measurements will be the same as those used for our preliminary quantum tomography experiment. For instance, in the case of quantum computing, where error correction is implemented by CSS codes, all measurements will be Pauli measurements. In such a case, the conclusions of the AIC, applied to a tomography experiment that used Pauli measurements as well, should be trustworthy.
\section{Summary and Discussion}\label{discuss}
Our central message here is that when the assumptions of tomography fail, it is often due to some sort of sample-apparatus correlation, and that this can be detected with statistical reliability by model selection using the AIC. One particular example, the drifting source, clearly voids the single-density-matrix model, but can be described naturally (and more accurately!) by multiple density matrices associated with different times and/or measurement settings. The AIC is a particularly good and elegant tool for identifying whether the added complexity of this model is justified. Ultimately, the point of model selection (especially using the AIC) is to get better predictions of future measurement outcomes -- \emph{not} just better fits to observed data.
While the AIC ranks competing models, by assigning each model $k$ a number $\Omega_k$, through Eq.~(\ref{AIC}), we have great flexibility in what to do with that ranking. Small differences in AIC are not significant; if $|\Omega_k - \Omega_{k'}| << 1$, then both models are equally good. But even when significant differences exist, we may choose to use the ``best'' model exclusively, or to hedge by mixing it with lower-ranked models (with weights determined by their respective AICs). We could apply Bayesian methods to the highest-ranked model, or use maximum likelihood estimates to choose model parameters. Choosing between these alternatives is beyond the scope of this paper.
If a model-selection (e.g., AIC) analysis finds overwhelming evidence of sample-apparatus correlation (e.g. source drift), it is often possible to go beyond the conclusion ``tomography has failed!'' What has really failed is the i.i.d. assumption -- we have convincing evidence that the samples are not identically distributed. The joint state is therefore not (with high confidence) of De Finetti form (see \cite{Caves2002}). But it may be possible to assign states with a relaxed De Finetti form, and thereafter to do tomography with this in mind. For example, if the AIC declares the alternative three-state model much superior to the single-state model, one could assign a state of the form
\begin{equation}\label{DF}
\rho^{(3N)}=
\int d\bro{1}\int d\bro{2}
\int d\bro{3}\, P_a(\bro{1},\bro{2},\bro{3})\,\bro{1}^{\otimes N}
\otimes
\bro{2}^{\otimes N}
\otimes
\bro{3}^{\otimes N}
\end{equation}
to the $3N$ qubits, where $P_a(.,.,.)$ is a joint probability distribution over three 2D density matrices. This form itself needs to be tested and validated, by comparison to a richer model (e.g., a model with 6, 9, or more different states). In general, validating a model requires more sophisticated model design -- e.g., to describe more arbitrary forms of source drift -- and perhaps different measurements or experiments specifically aimed at detecting those models, as proposed in \cite{Lucia}. But once a given model \emph{is} validated, if it implies a relaxed De Finetti form as in Eq.~(\ref{DF}), then we can in principle perform tomography independently on each of the i.i.d. subsets of the whole sample.
In the simplest case of tomography on single qubits, we discussed two competing models. Either one uses just a single density matrix $\bar{\rho}$ to describe the experiment [the standard model], or one uses three--$\bro{1},\bro{2},\bro{3}$--one for each set of $N$ qubits used to measure $\sigma_x$, $\sigma_y$, and $\sigma_z$, respectively [the alternative model].
But what does it mean to use {\em three} density matrices for predicting future measurement outcomes?
The answer is that the predictions
refer to measurements on qubits that have not been measured yet (of course). Consider one unmeasured qubit taken from, say, a set of $N+n$ qubits, from which $N$ qubits were randomly picked to be measured in the $\sigma_x$ basis and $n$ were not measured. In this case, those $n$ qubits would be assigned a state of the form
\begin{equation}
\rho^{(n)}=\int d\bro{1}\int d\bro{2}\int d\bro{3}\,
P_a(\bro{1},\bro{2},\bro{3})\,\bro{1}^{\otimes n},
\end{equation}
valid for any $n$, including $n=1$.
The mixed model, as mentioned above, would combine the standard and alternative models and assign an even more mixed state. For example, in the case $n=1$ it would assign the estimate
\begin{eqnarray}
\rho_{{\rm mixed}}&=&w_a\int d\bro{1}\int d\bro{2}\int d\bro{3}\,
P_a(\bro{1},\bro{2},\bro{3})\,\bro{1}\,\nonumber\\
&+&w_s\int d\bar{\rho}\,P_s(\bar{\rho})\,\bar{\rho},
\end{eqnarray}
with $w_a=\exp(\Omega_a)/(\exp(\Omega_a)+\exp(\Omega_s))$ and $w_s=1-w_a$ the relative weights of the two models, as assigned by the AIC, and with $P_s(.)$ the standard De Finetti probability distribution over single density matrices.
Although we have avoided discussion of model design here, one simple but powerful technique deserves mention. In the example at the beginning of the paper, we introduced an alternative model wherein each measurement setting is associated with a different density matrix. When the measurements are informationally complete, this alternative model has precisely as many parameters as the standard model. But if they are \emph{overcomplete}, then the alternative model has more parameters. As long as the samples really are i.i.d., we expect the alternative model to fit slightly better, and the AIC to declare them (on average) equally good. However, in the presence of experimental drift, we will find inconsistencies \emph{within} the overcomplete measurement set -- i.e., we will \emph{not} be able to fit all the measurements well with a single density matrix! This is a simple test for experimental drift that does not rely on negativity of $\rhohat_{\scriptscriptstyle\mathrm{tomo}}$.
For the main point of this paper, however, all these complications are unnecessary. All that matters is whether assigning a single density matrix to our tomography experiment constitutes the best model or not. If not, something is amiss, but at least we have diagnosed the problem.
The main issue we left open is the following: is there a sense in which the AIC works reliably if future measurements are different than those used in our tomography experiment?
If not, is there a ``quantum'' version of the AIC that, e.g., takes into account the quorum of observables that have been measured, as well as the set of observables that will be measured?
(Upon completion of this paper Ref.~\cite{Langford} appeared, which
is similar in spirit to our paper, but which uses $\chi^2$ tests to detect errors in tomography.
It points out, too, the problem with pure-state assignments for those tests.)
(After submission of the page proofs we became aware of two more relevant papers: \cite{Rosset2012,Moroder2012}.)
\section*{Acknowledgments}
This work was supported by NSF Grant No.~PHY-1004219. Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of EnergyÕs National Nuclear Security Administration under contract DE-AC04-94AL85000.
\end{document} |
\begin{document}
\title{
Degree bounds for projective division fields
associated to elliptic modules with a trivial endomorphism ring
}
\date{February 19, 2020}
\author{
Alina Carmen Cojocaru and Nathan Jones
}
\address[Alina Carmen Cojocaru]{
\begin{itemize}
\item[-]
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S Morgan St, 322
SEO, Chicago, 60607, IL, USA;
\item[-]
Institute of Mathematics ``Simion Stoilow'' of the Romanian Academy, 21 Calea Grivitei St, Bucharest, 010702,
Sector 1, Romania
\varepsilonnd{itemize}
} \varepsilonmail[Alina Carmen Cojocaru]{[email protected]}
\address[Nathan Jones]{
\begin{itemize}
\item[-]
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S Morgan St, 322
SEO, Chicago, 60607, IL, USA;
\varepsilonnd{itemize}
} \varepsilonmail[Nathan Jones]{[email protected]}
\renewcommand{\arabic{footnote}}{\fnsymbol{footnote}}
\footnotetext{\varepsilonmph{Key words and phrases:} Elliptic curves, Drinfeld modules, division fields, Galois representations}
\renewcommand{\arabic{footnote}}{\arabic{footnote}}
\renewcommand{\arabic{footnote}}{\fnsymbol{footnote}}
\footnotetext{\varepsilonmph{2010 Mathematics Subject Classification:} Primary 11G05, 11G09, 11F80}
\renewcommand{\arabic{footnote}}{\arabic{footnote}}
\thanks{A.C.C. was partially supported by a Collaboration Grant for Mathematicians from the Simons Foundation
under Award No. 318454.}
\begin{abstract}
Let $k$ be a global field,
let $A$ be a Dedekind domain with $\mathbb Quot(A) = k$,
and
let $K$ be a finitely generated field.
Using a unified approach for both elliptic curves and Drinfeld modules $M$ defined over $K$ and
having a trivial endomorphism ring,
with $k= \mathbb Q$, $A = \mathbb Z$ in the former case
and $k$ a global function field, $A$ its ring of functions regular away from a fixed prime in the latter case,
for any nonzero ideal $\mathfrak{a} \lhd A$
we prove best possible estimates
in the norm $|\mathfrak{a}|$ for the degrees over $K$ of the subfields of the $\mathfrak{a}$-division fields of $M$
fixed by scalars.
\varepsilonnd{abstract}
\maketitle
\section{Introduction}
In the theory of elliptic modules -- elliptic curves and Drinfeld modules -- division fields play a fundamental
role; their algebraic properties (e.g., ramification, degree, and Galois group structure) are intimately related
to properties of Galois representations and are essential to global and local questions about elliptic modules themselves. Among the subfields of the division fields of an elliptic module, those fixed by the scalars are of special
significance. For example, as highlighted in \cite[Chapter 5]{Ad01} and \cite[Section 3]{CoDu04},
in the case of an elliptic curve $E$ defined over $\mathbb Q$ and a positive integer $a$,
the subfield $J_a$ of the $a$-division field $\mathbb Q(E[a])$ fixed by the scalars of $\mathbb Gal(\mathbb Q(E[a]/\mathbb Q)) \leq \mathbb GL_2(\mathbb Z/a \mathbb Z)$ is closely related to the modular curve
$X_0(a)$ which parametrizes cyclic isogenies of degree $a$ between elliptic curves;
indeed, $J_a$ may be interpreted as the splitting field of the modular polynomial $\mathbb Phi_a(X, j(E))$ (see \cite[Section 69]{We08} and \cite[Section 11.C]{Co89} for the properties of the modular polynomials $\mathbb Phi_a(X, Y)$).
The arithmetic properties of the family of fields $(J_a)_{a \geq 1}$ are closely related
to properties of the reductions $E(\operatorname{o}peratorname{mod} p)$ of $E$ modulo primes $p$,
including
the growth of the order of the Tate-Shafarevich group of the curve $E(\operatorname{o}peratorname{mod} p)$ when viewed as constant over its own function field
(see \cite{CoDu04})
and
the growth of the absolute discriminant of the endomorphism ring of the curve $E(\operatorname{o}peratorname{mod} p)$ when viewed over the finite field $\mathbb F_p$
(see \cite{CoFi19}).
An essential ingredient when deriving properties about $E(\operatorname{o}peratorname{mod} p)$ from the fields $J_a$ is the growth of the degrees
$[J_a :\mathbb Q]$. The goal of this article is to prove best possible estimates in $a$ for the degrees of such fields in the unified setting of elliptic curves and Drinfeld modules with a trivial endomorphism ring.
To state our main result, we proceed as in \cite{Br10} and fix:
$k$ a global field,
$A$ a Dedekind domain with $\mathbb Quot(A) = k$,
$K$ a finitely generated field,
and
$M$ a $(G_K, A)$-module of rank $r \geq 2$,
where $G_K = \mathbb Gal(K^{\operatorname{sep}}/K)$ denotes the absolute Galois group of $K$.
Specifically,
$M$ is an $A$-module
with a continuous $G_K$-action that commutes with the $A$-action
and
with the property that, for any ideal $0 \neq \mathfrak{a} \lhd A$, the $\mathfrak{a}$-division submodule
$$
M[\mathfrak{a}] := \left\{x \in M: \alpha x = 0 \ \forall \alpha \in \mathfrak{a}\right\}
$$
has $A$-module structure
$$
M[\mathfrak{a}] \simeq_A (A/\mathfrak{a})^r.
$$
The $(G_K, A)$-module structure on $M$ gives rise to a compatible system of Galois representations
$\rho_{\mathfrak{a}}: G_K \longrightarrow \mathbb GL_r(A/\mathfrak{a})$
and to a continuous representation
$$
\rho : G_K \longrightarrow \mathbb GL_r(\operatorname{h}at{A}),
$$
where $\operatorname{h}at{A} := \displaystyle\operatorname{o}peratorname{li}m_{ \leftarrow \atop{\mathfrak{a} \lhd A}} A/\mathfrak{a}$.
Associated to these representations we have the $\mathfrak{a}$-division fields
$K_{\mathfrak{a}} := (K^{\operatorname{sep}})^{\operatorname{o}peratorname{Ker} \rho_{\mathfrak{a}}}$,
for which we distinguish the subfields
$J_{\mathfrak{a}}$
fixed by the scalars
$\{\lambda I_r : \lambda \in (A/\mathfrak{a})^\times \} \cap \mathbb Gal(K_{\mathfrak{a}}/K)\}$
(with $\mathbb Gal(K_{\mathfrak{a}}/K)$ viewed as a subgroup of $\mathbb GL_r(A/\mathfrak{a})$).
Denoting by
$\operatorname{h}at{\rho}_{\mathfrak{a}}: G_K \longrightarrow \mathbb PGL_r(A/\mathfrak{a})$
the composition of the representation
$\rho_{\mathfrak{a}}$
with the canonical projection $\mathbb GL_r(A/\mathfrak{a}) \longrightarrow \mathbb PGL_r(A/\mathfrak{a})$,
we observe that
$J_{\mathfrak{a}} = (K^{\operatorname{sep}})^{\operatorname{o}peratorname{Ker} \operatorname{h}at{\rho}_{\mathfrak{a}}}$
and we deduce that $[J_{\mathfrak{a}} :K] \leq \left|\mathbb PGL_r(A/\mathfrak{a})\right|$. Our main result provides a lower bound
for $[J_{\mathfrak{a}} :K]$ of the same order of growth as $\left|\mathbb PGL_r(A/\mathfrak{a})\right|$, as follows:
\begin{theorem}\label{Thm1}
We keep the above setting and assume that
\begin{equation}\label{assumption}
\left|\mathbb GL_r(\operatorname{h}at{A}) : \rho(G_K)\right| < \infty.
\varepsilonnd{equation}
Then, for any ideal $0 \neq \mathfrak{a} \lhd A$,
\begin{equation}\label{final-bound}
|\mathfrak{a}|^{r^2 - 1}
\ll_{M, K}
[J_{\mathfrak{a}} :K]
\leq
|\mathfrak{a}|^{r^2 - 1},
\varepsilonnd{equation}
where $|\mathfrak{a}| := |A/\mathfrak{a}|$.
\varepsilonnd{theorem}
By specializing the above general setting to elliptic curves and to Drinfeld modules, we obtain:
\begin{corollary}\label{Cor2}
Let $K$ be a finitely generated field with $\operatorname{o}peratorname{char} K = 0$
and let $E/K$ be an elliptic curve over $K$ with
$\mathbb End_{\operatorname{o}verline{K}}(E) \simeq \mathbb Z$.
Then, for any integer $a \geq 1$, the degree $[J_a : K]$ of
the subfield $J_a$ of the $a$-division field $K_a := K(E[a])$ fixed by the scalars of $\mathbb Gal(K(E[a])/K)$ satisfies
\begin{equation}\label{final-bound-ec}
a^3
\ll_{E, K}
[J_a :K]
\leq
a^3.
\varepsilonnd{equation}
\varepsilonnd{corollary}
\begin{corollary}\label{Cor3}
Let $k$ be a global function field,
let $\infty$ be a fixed place of $k$,
let $A$ be the ring of elements of $k$ regular away from $\infty$,
let $K$ be a finitely generated $A$-field with
$A$-$\operatorname{o}peratorname{char} K = 0$
(i.e. $k \subseteq K$),
and
let $\psi: A \longrightarrow K\{\tau\}$ be a Drinfeld $A$-module over $K$ of rank $r \geq 2$
with $\mathbb End_{\operatorname{o}verline{K}}(\psi) \simeq A$.
Then, for any ideal $0 \neq \mathfrak{a} \lhd A$,
the degree $[J_{\mathfrak{a}} : K]$ of the subfield $J_{\mathfrak{a}}$ of the $\mathfrak{a}$-division field
$K_{\mathfrak{a}} := K(\psi[\mathfrak{a}])$ fixed by the scalars of
$\mathbb Gal(K(\psi[\mathfrak{a}])/K)$
satisfies
\begin{equation}\label{final-bound-dm}
|\mathfrak{a}|^{r^2 - 1}
\ll_{\psi, K}
[J_{\mathfrak{a}} :K]
\leq
|\mathfrak{a}|^{r^2 - 1}.
\varepsilonnd{equation}
\varepsilonnd{corollary}
The proof of Theorem \ref{Thm1} relies
on consequences of assumption (\ref{assumption}),
on applications of Goursat's Lemma,
as well as
on vertical growth estimates for open subgroups of $\mathbb GL_r$.
Specializing to elliptic curves and to Drinfeld modules,
assumption (\ref{assumption}) is essentially Serre's Open Image Theorem \cite{Se72}
and, respectively, Pink-R\"{u}tsche's Open Image Theorem \cite{PiRu09}.
Variations of these open image theorems also hold for elliptic curves and Drinfeld modules with nontrivial endomorphism rings.
While these complementary cases are treated unitarily in \cite{Br10} when investigating the growth of torsion,
when investigating the growth of $[J_{\mathfrak{a}}:K]$
they face particularities whose treatment we relegate to future work.
We emphasize that the upper bound in Theorem \ref{Thm1} always holds and does not necessitate assumption (\ref{assumption}). In contrast, the lower bound in Theorem \ref{Thm1} is intimately related to assumption (\ref{assumption}).
Indeed, one consequence of (\ref{assumption}) is that there exists an ideal
${\mathfrak{a}}(M, K) \unlhd A$,
which (a priori) depends on $M$ and $K$ and which has the property that, for any prime ideal $\mathfrak{l} \nmid \mathfrak{a}(M, K)$,
$\mathbb Gal(J_{\mathfrak{l}}/K) \simeq \mathbb PGL_r(A/\mathfrak{l})$. Then, for such an ideal $\mathfrak{l}$,
the lower bound in (\ref{final-bound}) follows immediately.
The purpose of Theorem \ref{Thm1} is to prove similar lower bounds for {\it{all}} ideals $\mathfrak{a} \lhd A$.
The dependence of the lower bound in (\ref{final-bound}) on $M$ (which also includes dependence on $r$)
and on $K$ is an important topic related to the uniform boundedness of the torsion of $M$;
while we do not address it in the present paper, we refer the reader to \cite{Br10} and \cite{Jo19}
for related discussions and for additional references.
The fields $J_\mathfrak{a}$ play a prominent role in a multitude of problems, such as
in deriving
non-trivial upper bounds for the number of non-isomorphic Frobenius fields
associated to an elliptic curve and, respectively, to a Drinfeld module
(see \cite{CoDa08ec} and \cite{CoDa08dm});
in investigating the discriminants of the endomorphism rings of the reduction of an elliptic curve and, respectively, of a Drinfeld module
(see \cite{CoFi19} and \cite{CoPa20});
and
in proving non-abelian reciprocity laws for primes and, respectively, for irreducible polynomials
(see \cite{DuTo02}, \cite{CoPa15}, and \cite{GaPa19}).
For such applications, an essential piece of information is the growth of the degree $[J_{\mathfrak{a}} : K]$
as a function of the norm $|\mathfrak{a}|$.
For example, Corollary \ref{Cor2} is a key ingredient in proving that,
for any elliptic curve $E/\mathbb Q$ with $\mathbb End_{\operatorname{o}verline{\mathbb Q}}(E) \simeq \mathbb Z$, provided the Generalized Riemann Hypothesis holds for the division fields of $E$,
there exists a set of primes $p$ of natural density 1 with the property that the absolute discriminant of the imaginary quadratic order $\mathbb End_{\mathbb F_p}(E)$ is as close as possible to its natural upper bound;
see \cite[Theorem 1]{CoFi19}.
Similarly, Corollary \ref{Cor3} is a key ingredient in proving that,
denoting by $\mathbb F_q$ the finite field with $q$ elements and assuming that $q$ is odd,
for any generic Drinfeld module $\psi: \mathbb F_q[T] \longrightarrow \mathbb F_q(T)\{\tau\}$ of rank 2 and with
$\mathbb End_{\operatorname{o}verline{\mathbb F_q(T)}}(\psi) \simeq \mathbb F_q[T]$,
there exists a set of prime ideals $\mathfrak{p} \lhd \mathbb F_q[T]$ of Dirichlet density 1 with the property that the norm of the discriminant of the imaginary quadratic order $\mathbb End_{\mathbb F_{\mathfrak{p}}}(\psi)$ is as close as possible to its natural upper bound;
see \cite[Theorem 6]{CoPa20}. We expect that Theorem \ref{Thm1} will be of use to other arithmetic studies of elliptic modules.
{\bf{Notation}}.
In what follows, we use the standard $\ll$, $\gg$, and $\asymp$ notation:
given suitably defined real functions $h_1, h_2$,
we say that
$h_1 \ll h_2$ or $h_2 \gg h_1$
if
$h_2$ is positive valued
and
there exists a positive constant $C$ such that
$|h_1(x)| \leq C h_2(x)$ for all $x$ in the domain of $h_1$;
we say that
$h_1 \asymp h_2$
if
$h_1$, $h_2$ are positive valued
and
$h_1 \ll h_2 \ll h_1$;
we say that
$h_1 \ll_D h_2$ or $h_2 \gg_D h_1$
if
$h_1 \ll h_2$
and
the implied $\ll$-constant $C$ depends on priorly given data $D$;
similarly,
we say that
$h_1 \asymp_D h_2$
if the implied constant in either one of the $\ll$-bounds
$h_1 \ll h_2 \ll h_1$
depends on priorly given data $D$.
We also use the standard divisibility notation for ideals in a Dedekind domain. In particular,
given two ideals $\mathfrak{a}$, $\mathfrak{b}$,
we write $\mathfrak{a} \mid \mathfrak{b}^{\infty}$ if
all the prime ideal factors of $\mathfrak{a}$ are among the prime ideal factors of $\mathfrak{b}$
(with possibly different exponents).
Further notation will be introduced over the course of the paper as needed.
\section{Goursat's Lemma and variations}
In this section we recall Goursat's Lemma on fibered products of groups (whose definition we also recall shortly)
and detail the behavior of such fibered products under intersection.
\begin{lemma} \label{goursat} (Goursat's Lemma)
Let $G_1$, $G_2$ be groups
and
for $i \in \{1, 2 \}$ denote by $\pi_i : G_1 \times G_2 \longrightarrow G_i$ the projection map onto the $i$-th factor.
Let $G \leq G_1 \times G_2$ be a subgroup and
assume that $\pi_1(G) = G_1$, $\pi_2(G) = G_2$.
Then there exist a group $\mathbb Gamma$ and a pair of surjective group homomorphisms
$\psi_1 : G_1 \longrightarrow \mathbb Gamma$,
$\psi_2 : G_2 \longrightarrow \mathbb Gamma$
such that
$$G = G_1 \times_\psi G_2 := \{ (g_1,g_2) \in G_1 \times G_2 : \psi_1(g_1) = \psi_2(g_2) \}.$$
\varepsilonnd{lemma}
\begin{proof}
See \cite[Lemma 5.2.1]{Ri76}.
\varepsilonnd{proof}
We call $G_1 \times_\psi G_2$ the \varepsilonmph{fibered product of $G_1$ and $G_2$ over $\psi := (\psi_1, \psi_2)$}.
The next lemma details what happens when we intersect such a fibered product with a subgroup of the form $H_1 \times H_2$ defined by subgroups $H_1 \leq G_1$ and $H_2 \leq G_2$.
It is clear that
$$
\left( H_1 \times H_2 \right) \cap \left( G_1 \times_\psi G_2 \right) = H_1 \times_{\psi} H_2 := \{ (h_1, h_2) \in H_1 \times H_2 : \psi_1(h_1) = \psi_2(h_2) \}.
$$
However, this representation does not specify the restricted common quotient inside $\mathbb Gamma$.
In particular, it can be the case that the fibered product $H := H_1 \times_{\psi} H_2$
does \varepsilonmph{not} satisfy $\pi_i(H) = H_i$ for each $i \in \{1, 2\}$. The following lemma clarifies this situation.
\begin{lemma} \label{goursat-variation}
Let $G_1$, $G_2$ be groups,
let $\psi_1 : G_1 \rightarrow \mathbb Gamma$, $\psi_2 : G_2 \rightarrow \mathbb Gamma$ be surjective group homomorphisms onto a group $\mathbb Gamma$,
and
let $G_1 \times_\psi G_2$ be the associated fibered product.
Furthermore, let $H_1 \leq G_1$, $H_2 \leq G_2$ be subgroups.
Define the subgroup
$$\mathbb Gamma_H := \psi_1(H_1) \cap \psi_2(H_2) \leq \mathbb Gamma.$$
Then
\begin{equation} \label{fiberedintersectionequality}
\left( H_1 \times H_2 \right) \cap \left( G_1 \times_\psi G_2 \right)
=
\left( H_1 \cap \psi_1^{-1}(\mathbb Gamma_H) \right) \times_{\psi} \left( H_2 \cap \psi_2^{-1}(\mathbb Gamma_H) \right)
\varepsilonnd{equation}
and the canonical projection maps
\[
\begin{split}
\pi_1 : &\left( H_1 \cap \psi_1^{-1}(\mathbb Gamma_H) \right) \times_\psi \left( H_2 \cap \psi_2^{-1}(\mathbb Gamma_H) \right) \longrightarrow H_1 \cap \psi_1^{-1}(\mathbb Gamma_H), \\
\pi_2 : &\left( H_1 \cap \psi_1^{-1}(\mathbb Gamma_H) \right) \times_\psi \left( H_2 \cap \psi_2^{-1}(\mathbb Gamma_H) \right) \longrightarrow H_2 \cap \psi_2^{-1}(\mathbb Gamma_H)
\varepsilonnd{split}
\]
are surjective.
\varepsilonnd{lemma}
\begin{proof}
We first establish \varepsilonqref{fiberedintersectionequality}.
Since the containment ``$\supseteq$'' is immediate, we only need to establish ``$\subseteq$.'' Let $(h_1,h_2) \in \left( H_1 \times H_2 \right) \cap \left( G_1 \times_\psi G_2 \right)$, i.e. $h_1 \in H_1$, $h_2 \in H_2$, and $\psi_1(h_1) = \psi_2(h_2)$. From the definition of $\mathbb Gamma_H$, it follows that $\psi_1(h_1) = \psi_2(h_2) \in \mathbb Gamma_H$.
Thus $(h_1,h_2) \in \left( H_1 \cap \psi_1^{-1}(\mathbb Gamma_H) \right) \times_{\psi} \left( H_2 \cap \psi_2^{-1}(\mathbb Gamma_H) \right)$, establishing \varepsilonqref{fiberedintersectionequality}.
To see why the projection map
\begin{equation} \label{surjectivityofpi1}
\pi_1 : H_1 \cap \psi_i^{-1}(\mathbb Gamma_H) \longrightarrow H_1 \cap \psi_1^{-1}(\mathbb Gamma_H)
\varepsilonnd{equation}
is surjective, fix $h_1 \in H_1 \cap \psi_1^{-1}(\mathbb Gamma_H)$ and set $\alphamma := \psi_1(h_1) \in \mathbb Gamma_H$. By the definition of $\mathbb Gamma_H$, we find $h_2 \in H_2$ with $\psi_2(h_2) = \alphamma$.
Thus $(h_1,h_2) \in \left( H_1 \cap \psi_1^{-1}(\mathbb Gamma_H) \right) \times_\psi \left( H_2 \cap \psi_2^{-1}(\mathbb Gamma_H) \right)$ and $\pi_1(h_1,h_2) = h_1$, proving the surjectivity of $\pi_1$ in \varepsilonqref{surjectivityofpi1}.
The surjectivity of $\pi_2$ is proved similarly.
\varepsilonnd{proof}
\section{Proof of Theorem \ref{Thm1}}
In this section we prove Theorem \ref{Thm1}.
We will make use of the following notation:
$$G := \rho(G_K) \leq \mathbb GL_r(\operatorname{h}at{A});$$
for any ideal $0 \neq \mathfrak{a} \lhd A$,
we write
$$G(\mathfrak{a}) := \rho_{\mathfrak{a}}(G_K)
\leq
\mathbb GL_r(A/\mathfrak{a});$$
for any subgroup $H \leq \mathbb GL_r(A/\mathfrak{a})$, we write
$$\operatorname{Scal}_H := H \cap \{\alpha I_r: \alpha \in (A/\mathfrak{a})^{\times} \}.$$
With this notation, we see that
$J_{\mathfrak{a}} = K(E[\mathfrak{a}])^{\operatorname{Scal}_{G(\mathfrak{a})}}$.
To prove the theorem,
let $0 \neq \mathfrak{a} \lhd A$ be a fixed arbitrary ideal.
The proof of the upper bound is an immediate consequence to the
injection
$\mathbb Gal(J_{\mathfrak{a}}/K) \operatorname{h}ookrightarrow \mathbb PGL_r(A/\mathfrak{a})$
defined by $\operatorname{h}at{\rho}_{\mathfrak{a}}$.
Indeed,
using that
$$
\left|\mathbb PGL_r(A/\mathfrak{a})\right|
=
\frac{1}{\left|(A/\mathfrak{a})^{\times}\right|} \left|\mathbb GL_r(A/\mathfrak{a})\right|,
$$
$$
|(A/\mathfrak{a})^{\times}|
=
|\mathfrak{a}| \displaystyle\operatorname{pr}od_{\mathfrak{p} \mid \mathfrak{a}} \left(1 - \frac{1}{|\mathfrak{p}|}\right),
$$
and
$$
\left|\mathbb GL_r(A/\mathfrak{a})\right|
=
|\mathfrak{a}|^{r^2 }
\displaystyle\operatorname{pr}od_{
\mathfrak{p} \mid \mathfrak{a}
\atop{\mathfrak{p} \ \text{prime}}
}
\left(1 - \frac{1}{\left|\mathfrak{p}\right|}\right)
\left(1 - \frac{1}{\left|\mathfrak{p}\right|^2}\right)
\ldots
\left(1 - \frac{1}{\left|\mathfrak{p}\right|^r}\right)
$$
(see \cite[Lemma 2.3, p. 1244]{Br10} for the latter),
we obtain that
\begin{eqnarray*}
[J_{\mathfrak{a}} : K]
\leq
|\mathbb PGL_r(A/\mathfrak{a})|
=
|\mathfrak{a}|^{r^2 - 1}
\displaystyle\operatorname{pr}od_{
\mathfrak{p} \mid \mathfrak{a}
\atop{\mathfrak{p} \ \text{prime}}
}
\left(1 - \frac{1}{\left|\mathfrak{p}\right|^2}\right)
\ldots
\left(1 - \frac{1}{\left|\mathfrak{p}\right|^r}\right)
\leq
|\mathfrak{a}|^{r^2 - 1}.
\varepsilonnd{eqnarray*}
The proof of the lower bound relies on several consequences to assumption (\ref{assumption}),
as well as on applications of Goursat's Lemma \ref{goursat} and its variation Lemma \ref{goursat-variation}, as detailed below.
Thanks to (\ref{assumption}), there exists an ideal $\mathfrak{m} = \mathfrak{m}_{M, K} \unlhd A$ such that
\begin{equation}\label{def-m}
G = \pi^{-1}(G(\mathfrak{m})),
\varepsilonnd{equation}
where
$\pi: \mathbb GL_r(\operatorname{h}at{A}) \longrightarrow \mathbb GL_r(A/\mathfrak{m})$ is the canonical projection.
We take $\mathfrak{m}$ to be the smallest such ideal with respect to divisibility
and we write its unique prime ideal factorization as
$\mathfrak{m}
=
\displaystyle\operatorname{pr}od_{\mathfrak{p}^{v_{\mathfrak{p}}(\mathfrak{m}) } || \mathfrak{m}}
\mathfrak{p}^{v_{\mathfrak{p}}(\mathfrak{m}) },$
where each exponent satisfies $v_{\mathfrak{p}}(\mathfrak{m}) \geq 1$.
With the ideal $\mathfrak{m}$ in mind,
we write the arbitrary ideal $\mathfrak{a}$ uniquely as
\begin{equation}\label{factor-a}
\mathfrak{a} = \mathfrak{a}_1 \mathfrak{a}_2,
\varepsilonnd{equation}
where
\begin{equation}\label{def-a1}
\mathfrak{a}_1 \mid \mathfrak{m}^{\infty},
\varepsilonnd{equation}
\begin{equation}\label{def-a2}
\operatorname{o}peratorname{gcd}(\mathfrak{a}_2, \mathfrak{m}) = 1.
\varepsilonnd{equation}
For future use, we record that
\begin{equation}\label{gcd-a1-a2}
\operatorname{o}peratorname{gcd}(\mathfrak{a}_1, \mathfrak{a}_2) = 1.
\varepsilonnd{equation}
We also write the ideal $\mathfrak{a}_1$ uniquely as
\begin{equation*}
\mathfrak{a}_1 = \mathfrak{a}_{1, 1} \ \mathfrak{a}_{1, 2},
\varepsilonnd{equation*}
where
$\mathfrak{a}_{1, 1} =
\displaystyle\operatorname{pr}od_{
\mathfrak{p}^{e_{\mathfrak{p}}} || {\mathfrak{a}}_{1, 1}
\atop{
e_{\mathfrak{p}} > v_{\mathfrak{p}}(\mathfrak{m})
}
}
\mathfrak{p}^{e_{\mathfrak{p}}}$
and
$\mathfrak{a}_{1, 2} =
\displaystyle\operatorname{pr}od_{
\mathfrak{p}^{f_{\mathfrak{p}}} || {\mathfrak{a}}_{1, 1}
\atop{
f_{\mathfrak{p}} \leq v_{\mathfrak{p}}(\mathfrak{m})
}
}
\mathfrak{p}^{f_{\mathfrak{p}}}$.
Note that
\begin{equation}\label{gcd-a11-a12}
\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1, 1}, \mathfrak{a}_{1, 2}) = 1,
\varepsilonnd{equation}
\begin{equation}\label{def-a11}
\mathfrak{a}_{1, 1} \mid \mathfrak{m}^\infty
\varepsilonnd{equation}
and
\begin{equation}\label{def-a12}
\mathfrak{a}_{1, 2} \mid \mathfrak{m}.
\varepsilonnd{equation}
Under the isomorphism of the Chinese Remainder Theorem,
we deduce from (\ref{def-m}) that
\begin{equation}\label{G-G1-G2}
G(\mathfrak{a}) \simeq G(\mathfrak{a}_1) \times \mathbb GL_r(A/\mathfrak{a}_2)
\varepsilonnd{equation}
and, consequently, that there exist group isomorphisms
\begin{equation}\label{Scal1-Scal2}
\operatorname{Scal}_{G(\mathfrak{a})}
\simeq
{\operatorname{Scal}_{G(\mathfrak{a}_1)} \times {\operatorname{Scal}_{\mathbb GL_r(A/\mathfrak{a}_2)}}},
\varepsilonnd{equation}
\begin{equation}\label{Ga/Scala}
G(\mathfrak{a})/\operatorname{Scal}_{G(\mathfrak{a})}
\simeq
\left(G(\mathfrak{a}_1)/\operatorname{Scal}_{G(\mathfrak{a}_1)}\right) \times \mathbb PGL_r(A/\mathfrak{a}_2).
\varepsilonnd{equation}
Next, applying Lemma \ref{goursat} to the groups
$G = G(\mathfrak{a}_1)$,
$G_1 = G(\mathfrak{a}_{1, 1})$, and $G_2 = G(\mathfrak{a}_{1, 2})$,
we deduce that
there exist
a group $\mathbb Gamma$
and
surjective group homomorphisms
$\psi_1: G(\mathfrak{a}_{1, 1}) \longrightarrow \mathbb Gamma$,
$\psi_2: G(\mathfrak{a}_{1, 2}) \longrightarrow \mathbb Gamma$,
which give rise to a group isomorphism
\begin{equation}\label{G-a1}
G(\mathfrak{a}_1) \simeq G(\mathfrak{a}_{1, 1}) \times_{\psi} G(\mathfrak{a}_{1, 2}).
\varepsilonnd{equation}
Furthermore, applying Lemma \ref{goursat-variation} to the subgroups
$H_1 = \operatorname{Scal}_{G(\mathfrak{a}_{1, 1})}$ and $H_2 = \operatorname{Scal}_{G(\mathfrak{a}_{1, 2})}$,
we deduce that
there exists a group isomorphism
\begin{equation}\label{scal-products}
\left(
\operatorname{Scal}_{G(\mathfrak{a}_{1, 1})} \times \operatorname{Scal}_{G(\mathfrak{a}_{1, 2})}
\right)
\cap
\left(
G(\mathfrak{a}_{1, 1}) \times_{\psi} G(\mathfrak{a}_{1, 2})
\right)
\simeq
\left(
\operatorname{Scal}_{G(\mathfrak{a}_{1, 1})} \cap \psi_1^{-1} (\mathbb Gamma_{\operatorname{Scal}})
\right)
\times_{\psi}
\left(
\operatorname{Scal}_{G(\mathfrak{a}_{1, 2})} \cap \psi_2^{-1} (\mathbb Gamma_{\operatorname{Scal}})
\right),
\varepsilonnd{equation}
where
$$
\mathbb Gamma_{\operatorname{Scal}}
:=
\psi_1\left(\operatorname{Scal}_{G(\mathfrak{a}_{1, 1})}\right) \cap \psi_1\left(\operatorname{Scal}_{G(\mathfrak{a}_{1, 2})}\right)
\leq
\mathbb Gamma.
$$
From (\ref{Ga/Scala}) we derive that
\begin{equation}\label{Ja-degree-first}
[J_{\mathfrak{a}} :K]
=
\left|G(\mathfrak{a})/\operatorname{Scal}_{G(\mathfrak{a})}\right|
=
\left|G(\mathfrak{a}_1)/\operatorname{Scal}_{G(\mathfrak{a}_1)}\right| \cdot \left|\mathbb PGL_r(A/\mathfrak{a}_2)\right|.
\varepsilonnd{equation}
Then, using
(\ref{G-a1}) and (\ref{scal-products}),
we derive that
\begin{eqnarray}\label{Ga1/Scala1}
\left|
G(\mathfrak{a}_1)/\operatorname{Scal}_{G(\mathfrak{a}_1)}
\right|
&=&
\frac{
\left|
G(\mathfrak{a}_{1, 1}) \times_{\psi} G(\mathfrak{a}_{1, 2})
\right|
}
{
\left|
\left(
\operatorname{Scal}_{G(\mathfrak{a}_{1, 1})} \times \operatorname{Scal}_{G(\mathfrak{a}_{1, 2})}
\right)
\cap
G(\mathfrak{a}_1)
\right|
}
\nonumber
\\
&=&
\frac{
\left|
G(\mathfrak{a}_{1, 1}) \times_{\psi} G(\mathfrak{a}_{1, 2})
\right|
}{
\left|
\left(
\operatorname{Scal}_{G(\mathfrak{a}_{1, 1})} \cap \psi_1^{-1}(\mathbb Gamma_{\operatorname{Scal}})
\right)
\times_{\psi}
\left(
\operatorname{Scal}_{G(\mathfrak{a}_{1, 2})} \cap \psi_2^{-1}(\mathbb Gamma_{\operatorname{Scal}})
\right)
\right|
}
\nonumber
\\
&=&
\frac{
\left|G(\mathfrak{a}_{1, 1})\right|
}{
\left|
\operatorname{Scal}_{G(\mathfrak{a}_{1, 1})} \cap \psi_1^{-1}(\mathbb Gamma_{\operatorname{Scal}})
\right|
}
\cdot
\frac{
\left|\mathbb Gamma_{\operatorname{Scal}}\right|
}{
|\mathbb Gamma|
}
\cdot
\frac{
\left|
G(\mathfrak{a}_{1, 2})
\right|
}{
\left|
\operatorname{Scal}_{G(\mathfrak{a}_{1, 2})} \cap \psi_2^{-1}(\mathbb Gamma_{\operatorname{Scal}})
\right|
}
\nonumber
\\
&=&
\frac{
\left|G(\mathfrak{a}_{1, 1})\right|
}{
\left|
\operatorname{Scal}_{G(\mathfrak{a}_{1, 1})} \cap \psi_1^{-1}(\mathbb Gamma_{\operatorname{Scal}})
\right|
}
\cdot
\frac{
\left|
\psi_1\left(\operatorname{Scal}_{G(\mathfrak{a}_{1, 1})}\right)
\cap
\psi_2\left(\operatorname{Scal}_{G(\mathfrak{a}_{1, 2})}\right)
\right|
}{
|\mathbb Gamma|
}
\cdot
\frac{
\left|
G(\mathfrak{a}_{1, 2})
\right|
}{
\left|
\operatorname{Scal}_{G(\mathfrak{a}_{1, 2})} \cap \psi_2^{-1}(\mathbb Gamma_{\operatorname{Scal}})
\right|
}.
\varepsilonnd{eqnarray}
Recalling (\ref{def-a12}), we deduce that the last two factors above are bounded, from above and below,
by constants depending on $\mathfrak{m}$, hence on $M$ and $K$:
\begin{equation}\label{bound-a12-part}
\frac{
\left|
\psi_1\left(\operatorname{Scal}_{G(\mathfrak{a}_{1, 1})}\right)
\cap
\psi_2\left(\operatorname{Scal}_{G(\mathfrak{a}_{1, 2})}\right)
\right|
}{
|\mathbb Gamma|
}
\cdot
\frac{
\left|
G(\mathfrak{a}_{1, 2})
\right|
}{
\left|
\operatorname{Scal}_{G(\mathfrak{a}_{1, 2})} \cap \psi_2^{-1}(\mathbb Gamma_{\operatorname{Scal}})
\right|
}
\asymp_{M, K}
1.
\varepsilonnd{equation}
It remains to analyze the first factor in (\ref{Ga1/Scala1}).
For this, consider the canonical projection
$$
\pi_{1, 1}: \mathbb GL_r(A/\mathfrak{a}_{1, 1}) \longrightarrow \mathbb GL_r(A/\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1, 1}, \mathfrak{m}))
$$
and, upon recalling \varepsilonqref{def-m}, observe that
\begin{equation}\label{Ga11}
G(\mathfrak{a}_{1, 1}) = \pi_{1, 1}^{-1}(G(\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1, 1}, \mathfrak{m})))
\varepsilonnd{equation}
and
\begin{equation}\label{Ker-pi11}
\operatorname{o}peratorname{Ker} \pi_{1, 1} \subseteq \operatorname{o}peratorname{Ker} \psi_1.
\varepsilonnd{equation}
Thus the subgroups
$G(\mathfrak{a}_{1, 1}) \leq \mathbb GL_r(A/\mathfrak{a}_{1, 1})$
and
$G(\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1, 1}, \mathfrak{m})) \leq \mathbb GL_r(A/\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1,1}, \mathfrak{m}))$,
together with the group $\mathbb Gamma$,
fit into a commutative diagram
$$
\xymatrix{
G(\mathfrak{a}_{1, 1})
\; \; \;
\ar@{->>}[rd]_{\psi_1}
\ar@{->>}[r]^{\operatorname{h}space*{-0.7cm} \pi_{1,1}}
& \; \; G(\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1, 1}, \mathfrak{m}))
\ar@{-->>}[d]^{\rho}
\\
& \mathbb Gamma
}
$$
in which
the vertical map $\rho$ is some surjective group homomorphism
and
the horizontal map $\left.\pi_{1,1}\right|_{G(\mathfrak{a}_{1, 1})}$ is
$\left(\frac{|\mathfrak{a}_{1, 1}|}{|\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1,1}, \mathfrak{m})|}\right)^{r^2}$ to $1$.
Furthermore,
the subgroups
$\psi_1^{-1}(\mathbb Gamma_{\operatorname{Scal}}) \cap \operatorname{Scal}_{G(\mathfrak{a}_{1,1})} \leq \operatorname{Scal}_{\mathbb GL_r(A/\mathfrak{a}_{1, 1})} \simeq (A/\mathfrak{a}_{1, 1})^{\times}$
and
$\rho^{-1}(\mathbb Gamma_{\operatorname{Scal}}) \cap \operatorname{Scal}_{G\left(\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1,1},\mathfrak{m})\right)} \leq \operatorname{Scal}_{\mathbb GL_r(A/\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1, 1}, \mathfrak{m}))} \simeq (A/\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1, 1}, \mathfrak{m}))^{\times}$,
together with the group $\mathbb Gamma_{\operatorname{Scal}}$,
fit into the commutative diagram
$$
\xymatrix{
\psi_1^{-1}(\mathbb Gamma_{\operatorname{Scal}}) \cap \operatorname{Scal}_{G(\mathfrak{a}_{1,1})}
\; \; \;
\ar@{->>}[rd]_{\psi_1}
\ar@{->>}[r]^{\operatorname{h}space*{-0.1cm} \pi_{1,1}}
& \; \; \rho^{-1}(\mathbb Gamma_{\operatorname{Scal}}) \cap \operatorname{Scal}_{G\left(\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1,1},\mathfrak{m})\right)}
\ar@{->>}[d]^{\rho}
\\
& \mathbb Gamma_{\operatorname{Scal}}
}
$$
in which
the horizontal map $\left.\pi_{1,1}\right|_{\psi_1^{-1}(\mathbb Gamma_{\operatorname{Scal}})\cap \operatorname{Scal}_{G(\mathfrak{a}_{1,1})}}$ is
$\frac{|\mathfrak{a}_{1, 1}|}{|\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1,1}, \mathfrak{m})|}$ to $1$.
We deduce that
\begin{equation}\label{Ga11-order}
\left|G(\mathfrak{a}_{1, 1})\right|
=
\left(\frac{|\mathfrak{a}_{1, 1}|}{|\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1,1}, \mathfrak{m})|}\right)^{r^2}
\left|
G(\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1, 1}, \mathfrak{m}))
\right|
\asymp_{M, K}
|\mathfrak{a}_{1, 1}|^{r^2}
\varepsilonnd{equation}
and
\begin{equation}\label{Scala11-order}
\left| \psi_1^{-1}(\mathbb Gamma_{\operatorname{Scal}}) \cap \operatorname{Scal}_{G(\mathfrak{a}_{1, 1})} \right|
=
\frac{|\mathfrak{a}_{1, 1}|}{|\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1,1}, \mathfrak{m})|}
\left| \rho^{-1}(\mathbb Gamma_{\operatorname{Scal}}) \cap \operatorname{Scal}_{G(\operatorname{o}peratorname{gcd}(\mathfrak{a}_{1, 1}, \mathfrak{m}))} \right|
\asymp_{M, K}
|\mathfrak{a}_{1,1}|.
\varepsilonnd{equation}
Putting together
(\ref{Ja-degree-first}),
(\ref{Ga1/Scala1}),
(\ref{bound-a12-part}),
(\ref{Ga11-order}),
and
(\ref{Scala11-order}),
we obtain that
\begin{equation}\label{Ja-degree-second}
[J_{\mathfrak{a}} :K]
\asymp_K
|\mathfrak{a}_{1, 1}|^{r^2 - 1} \left|\mathbb PGL_r(A/\mathfrak{a}_2)\right|.
\varepsilonnd{equation}
To conclude the proof, observe that
\begin{eqnarray*}\label{PGL-a2}
\left|\mathbb PGL_r(A/\mathfrak{a}_2)\right|
&=&
|\mathfrak{a}_2|^{r^2 - 1}
\displaystyle\operatorname{pr}od_{
\mathfrak{p} \mid \mathfrak{a}_2
\atop{\mathfrak{p} \ \text{prime}}
}
\left(1 - \frac{1}{\left|\mathfrak{p}\right|^2}\right)
\ldots
\left(1 - \frac{1}{\left|\mathfrak{p}\right|^r}\right)
\\
&\geq&
|\mathfrak{a}_2|^{r^2 - 1}
\displaystyle\operatorname{pr}od_{
\mathfrak{p}
\atop{\mathfrak{p} \ \text{prime}}
}
\left(1 - \frac{1}{\left|\mathfrak{p}\right|^2}\right)
\ldots
\left(1 - \frac{1}{\left|\mathfrak{p}\right|^r}\right)
\\
&\gg_{r, K}&
|\mathfrak{a}_2|^{r^2 - 1},
\varepsilonnd{eqnarray*}
which, combined with (\ref{Ja-degree-second}), (\ref{factor-a}) and (\ref{def-a12}),
gives
\begin{eqnarray*}
[J_{\mathfrak{a}} :K]
\asymp_{K}
\frac{|\mathfrak{a}_{1}|^{r^2 - 1}}{|\mathfrak{a}_{1, 2}|^{r^2 - 1}}
\left|\mathbb PGL_r(A/\mathfrak{a}_2)\right|
\gg_{r, K}
\frac{|\mathfrak{a}_{1} \mathfrak{a}_2|^{r^2 - 1}}{|\mathfrak{a}_{1, 2}|^{r^2 - 1}}
\gg_{M, K}
|\mathfrak{a}|^{r^2 -1}.
\varepsilonnd{eqnarray*}
\section{Proof of Corollaries \ref{Cor2} and \ref{Cor3}}
First consider the setting of Corollary \ref{Cor2}:
$K$ a finitely generated field with $\operatorname{o}peratorname{char} K = 0$
and
$E/K$ an elliptic curve over $K$ with $\mathbb End_{\operatorname{o}verline{K}}(E) \simeq \mathbb Z$.
This is the specialization to the setting of Theorem \ref{Thm1} to
$k = \mathbb Q$, $A = \mathbb Z$, $K$ as above, and $M = E$.
In this case, $r = 2$
and assumption (\ref{assumption}) holds thanks to an extension of Serre's Open Image Theorem for elliptic curves
over number fields \cite[Th\'{e}or\`{e}me 3, p. 299]{Se72} as explained in \cite[Theorem 3.2, p. 1248]{Br10}.
Corollary \ref{Cor2} follows.
Next consider the setting of Corollary \ref{Cor3}:
$k$ a global function field,
$\infty$ a fixed place of $k$,
$A$ the ring of elements of $k$ regular away from $\infty$,
$K$ a finitely generated $A$-field with $\operatorname{o}peratorname{char} K = \operatorname{o}peratorname{char} k$ and $A$-$\operatorname{o}peratorname{char} K = 0$,
and
$\psi: A \longrightarrow K\{\tau\}$ a Drinfeld $A$-module over $K$ of rank $r \geq 2$
with $\mathbb End_{\operatorname{o}verline{K}}(\psi) \simeq A$.
This is the specialization to the setting of Theorem \ref{Thm1} to
$k$, $A$, $K$ as above, and $M = \psi$.
In this case, assumption (\ref{assumption}) holds thanks to Pink-R\"{u}tshe's Open Image Theorem for Drinfeld modules \cite[Theorem 0.1, p. 883]{PiRu09}.
Corollary \ref{Cor3} follows.
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}}
\varepsilonnd{document} |
\begin{document}
\title{Ricci flow coupled with harmonic map flow\
Flot de Ricci coupl\'e avec le flot harmonique}
\pagenumbering{arabic}
\begin{abstract}
We investigate a coupled system of the Ricci flow on a closed
manifold $M$ with the harmonic map flow of a map $\phi$ from $M$ to
some closed target manifold $N$,
\begin{equation*}
\dt g = -2\Rc{} + 2\alpha \nabla\phi \otimes \nabla\phi,\qquad \dt \phi =
\tau_g \phi,
\end{equation*}
where $\alpha$ is a (possibly time-dependent) positive coupling
constant. Surprisingly, the coupled system may be less singular than
the Ricci flow or the harmonic map flow alone. In particular, we can
always rule out energy concentration of $\phi$ a-priori by choosing
$\alpha$ large enough. Moreover, it suffices to bound the curvature
of $(M,g(t))$ to also obtain control of $\phi$ and all its
derivatives if $\alpha \geq \underaccent{\bar}\alpha>0$. Besides
these new phenomena, the flow shares many good properties with the
Ricci flow. In particular, we can derive the monotonicity of an
\emph{energy}, an \emph{entropy} and a \emph{reduced volume}
functional. We then apply these monotonicity results to rule out
non-trivial breathers and geometric collapsing at finite times.\\
\begin{center}\textbf{R\'esum\'e}\end{center}
Nous \'etudions un syst\`eme d'\'equations consistant en un couplage
entre le flot de Ricci et le flot harmonique d'une fonction $\phi$
allant de $M$ dans une vari\'et\'e cible $N$,
\begin{equation*}
\dt g = -2\Rc{} + 2\alpha \nabla\phi \otimes \nabla\phi,\qquad \dt \phi =
\tau_g \phi,
\end{equation*}
o\`u $\alpha$ est une constante de couplage strictement positive (et
pouvant d\'ependre du temps). De mani\`ere surprenante, ce syst\`eme
coupl\'e peut \^etre moins singulier que le flot de Ricci ou le flot
harmonique si ceux-ci sont consid\'er\'es de mani\`ere isol\'ee. En
particulier, on
peut toujours montrer que la fonction $\phi$ ne se concentre pas le long
de ce syst\`eme \`a condition de prendre $\alpha$ assez grand. De plus,
il est suffisant de borner la courbure de $(M,g(t))$ le long du flot
pour obtenir le contr\^ole de $\phi$ et de toutes ses d\'eriv\'ees si
$\alpha \geq \underaccent{\bar}\alpha>0$. A part ces ph\'enom\`enes
nouveaux, ce flot poss\`ede certaines propri\'et\'es analogues \`a
celles du flot de Ricci. En particulier, il est possible de montrer la
monotonie d'une \emph{\'energie}, d'une \emph{entropie} et d'une
fonctionnelle \emph{volume r\'eduit}. On utilise la monotonie de ces
quantit\'es pour montrer l'absence de solutions en "accord\'eon" et
l'absence d'effondrement en temps fini le long du flot.
\end{abstract}
\section{Introduction and main results}
Let $(M^m\!,g)$ and $(N^n\!,\gamma)$ be smooth Riemannian manifolds
without boundary. According to Nash's embedding theorem
\cite{Nash:embedding} we can assume that $N$ is isometrically
embedded into Euclidean space $(N^n,\gamma) \hookrightarrow
\mathbb{R}^d$ for a sufficiently large $d$. If $e_N:N \to \mathbb{R}^d$ denotes
this embedding, we identify maps $\phi:M\to N$ with $e_N \circ
\phi:M\to\mathbb{R}^d$, such maps may thus be written as $\phi
=(\phi^\lambda)_{1\leq\lambda\leq d}$. Harmonic maps $\phi:M\to N$
are critical points of the energy functional
\begin{equation}\label{0.eq1}
E(\phi) = \int_M \abs{\nabla\phi}^2 dV.
\end{equation}
Here, $\abs{\nabla\phi}^2 := 2e(\phi) =
g^{ij}\nabla_i\phi^\lambda\nabla_j\phi^\lambda$ denotes the local energy
density, where we use the convention that repeated Latin indices are
summed over from $1$ to $m$ and repeated Greek indices are summed
over from $1$ to $d$. We often drop the summation indices for $\phi$
when clear from the context. Harmonic maps generalize the concept of
harmonic functions and in particular include closed geodesics and minimal
surfaces.
\newline
To study the existence of a harmonic map $\phi$ homotopic to a given
map $\phi_0:M\to N$, Eells and Sampson \cite{EellsSampson} proposed
to study the $L^2$-gradient flow of the energy functional
(\ref{0.eq1}),
\begin{equation}\label{0.eq2}
\dt \phi = \tau_g\phi,\qquad \phi(0)=\phi_0,
\end{equation}
where $\tau_g\phi$ denotes the intrinsic Laplacian of $\phi$, often
called the tension field of $\phi$. They proved that if $N$ has
non-positive sectional curvature there always exists a unique,
global, smooth solution of (\ref{0.eq2}) which converges smoothly to
a harmonic map $\phi_\infty:M\to N$ homotopic to $\phi_0$ as
$t\to\infty$ suitably. On the other hand, without an assumption on
the curvature of $N$, the solution might blow up in finite or
infinite time. Comprehensive surveys about harmonic maps and the harmonic map
flow are given in Eells-Lemaire \cite{EellsLemaireI,EellsLemaireII},
Jost \cite{Jost:Riemannian} and Struwe \cite{Struwe:EvProblems}. The
harmonic map flow was the first appearance of a nonlinear heat flow
in Riemannian geometry. Today, geometric heat flows have become an
intensely studied topic in
geometric analysis.
\newline
Another fundamental problem in differential geometry is to find
canonical metrics on Riemannian manifolds, for example metrics with
constant curvature in some sense. Using the idea of evolving an object to
such an ideal state by a nonlinear heat flow, Richard Hamilton
\cite{Hamilton:3folds} introduced the Ricci flow in 1982. His
idea was to smooth out irregularities of the curvature by evolving a
given Riemannian metric $g$ on a manifold $M$ with respect to the
nonlinear weakly parabolic equation
\begin{equation}\label{0.eq3}
\dt g = -2 \Rc{},\qquad g(0)=g_0,
\end{equation}
where $\Rc{}$ denotes the Ricci curvature of $(M,g)$.
Strictly speaking, the Ricci flow is not
the gradient flow of a functional $\mathcal{F}(g)=\int_M F(\partial^2
g,\partial g,g)dV$, but in 2002, Perelman \cite{Perelman:entropy}
showed that it is gradient-like nevertheless. He presented a
new functional which may be regarded as an improved version of the
Einstein-Hilbert functional $E(g)=\int_M R\,dV$, namely
\begin{equation}\label{0.eq4}
\mathcal{F}(g,f) := \int_M \Big(R + \abs{\nabla f}^2 \Big)e^{-f} dV.
\end{equation}
The Ricci flow can be interpreted as the gradient flow of $\mathcal{F}$ modulo
a pull-back by a family of diffeomorphisms. Hamilton's Ricci flow has a
successful history. Most importanty, Perelman's
work \cite{Perelman:entropy,Perelman:surgery} led to a completion of
Hamilton's program \cite{Hamilton:survey} and a complete proof
of Thurston's geometrization conjecture \cite{Thurston} and (using
a finite extinction result from Perelman \cite{Perelman:extinction}
or Colding and Minicozzi \cite{ColdingMinicozziI,ColdingMinicozziII})
of the Poincar\'{e} conjecture \cite{Poincare}. Introductory surveys on
the Ricci flow and Perelman's functionals
can be found in the books by Chow and Knopf \cite{RF:intro}, Chow,
Lu and Ni \cite{ChowLuNi}, M\"{u}ller \cite{Muller:Harnack} and Topping
\cite{Topping:Lectures}. More advanced explanations of Perelman's
proof of the two conjectures are given in Cao and Zhu \cite{CaoZhu}
Chow et al. \cite{RF:TAI,RF:TAII}, Kleiner and Lott
\cite{KleinerLott} and Morgan and Tian
\cite{MorganTian,MT:completion}. A good survey on Perelman's work is
also given in Tao \cite{Tao:Perelman}.
\newline
The goal of this article is to study a coupled system of the two
flows (\ref{0.eq2}) and (\ref{0.eq3}). Again, we let $(M^m\!,g)$ and
$(N^n\!,\gamma)$ be smooth manifolds without boundary and with
$(N^n\!,\gamma) \hookrightarrow \mathbb{R}^d$. Throughout this article, we
will assume in addition that $M$ and $N$ are compact, hence closed.
However, many of our results hold for more general manifolds.
\newline
Let $g(t)$ be a family of Riemannian metrics on $M$ and $\phi(t)$ a
family of smooth maps from $M$ to $N$. We call $(g(t),\phi(t))_{t
\in [0,T)}$ a solution to the coupled system of Ricci flow and
harmonic map heat flow with coupling constant $\alpha(t)$, the
$(RH)_\alpha$ flow for short, if it satisfies
\begin{equation}
\left\{\begin{aligned}\dt g &= -2\Rc{} + 2\alpha \nabla\phi \otimes \nabla\phi,\\%
\dt \phi &= \tau_g \phi. \end{aligned}\right. \tag*{$(RH)_\alpha$}
\end{equation}
Here, $\tau_g \phi$ denotes the tension field of the map $\phi$ with
respect to the evolving metric $g$, and $\alpha(t)\geq 0$ denotes a
(time-dependent) coupling constant. Finally, $\nabla\phi\otimes\nabla\phi$
has the components $(\nabla\phi\otimes\nabla\phi)_{ij}
=\nabla_i\phi^\lambda\nabla_j\phi^\lambda$. In particular, $\abs{\nabla\phi}^2$
as defined above is the trace of $\nabla\phi\otimes\nabla\phi$ with respect
to $g$.
\newline
The special case where $N\subseteq\mathbb{R}$ and $\alpha\equiv 2$ was studied by List \cite{List:diss}, his motivation coming from general relativity and the study of Einstein vacuum equations. Moreover, List's flow also arises as the Ricci flow of a warped product, see \cite[Lemma A.3]{Muller:diss}. After completion of this work, we learned that another special case of $(RH)_\alpha$ with $N\subseteq SL(k\mathbb{R})/SO(k)$ arises in the study of the long-time behaviour of certain Type III Ricci flows, see Lott \cite{Lott:TypeIII} and a recent paper of Williams \cite{Williams} for details and explicit examples.
\newline
The paper is organized as follows. In order to get a feeling for the
flow, we first study explicit examples of solutions of $(RH)_\alpha$
as well as soliton solutions which are generalized fixed points modulo
diffeomorphisms and scaling. The stationary solutions of
$(RH)_\alpha$ satisfy $\Rc{} = \alpha \nabla \phi \otimes \nabla \phi$,
where $\phi$ is a harmonic map. To prevent
$(M,g(t))$ from shrinking to a point or blowing up, it is convenient
to introduce a volume-preserving version of the flow.
\newline
In Section 3, we prove that for constant coupling functions
$\alpha(t) \equiv \alpha > 0$ the $(RH)_\alpha$ flow can be interpreted
as a gradient flow for an energy functional $\mathcal{F}_\alpha(g,\phi,f)$
modified by a family of diffeomorphisms generated by $\nabla f$. If
$(g(t),\phi(t))$ solves $(RH)_\alpha$ and $e^{-f}$ is a
solution to the adjoint heat equation under the flow, then
$\mathcal{F}_\alpha$ is non-decreasing and constant if and only if
$(g(t),\phi(t))$ is a steady gradient soliton. In the more general
case where $\alpha(t)$ is a positive function, the monotonicity
result still holds whenever $\alpha(t)$ is non-increasing. This section
is based on techniques of Perelman \cite[Section 1]{Perelman:entropy}
for the Ricci flow.
\newline
In the fourth section, we prove short-time existence for the flow
using again a method from Ricci flow theory known as DeTurck's trick
(cf.~\cite{DeTurck:trick}), i.e.~we transform the weakly parabolic
system $(RH)_\alpha$ into a strictly parabolic one by pushing it
forward with a family of diffeomorphisms. Moreover, we
compute the evolution equations for the Ricci and scalar curvature,
the gradient of $\phi$ and combinations thereof. In particular, the
evolution equations for the symmetric tensor $S_{ij}:= R_{ij}-\alpha
\nabla_i\phi\nabla_j\phi$ and its trace $S=R-\alpha\abs{\nabla\phi}^2$ will be
very useful.
\newline
In Section 5, we study first consequences of the evolution equations
for the existence or non-existence of certain types of
singularities. Using the maximum principle, we show that $\min_{x\in
M}S(x,t)$ is non-decreasing along the flow. This has the rather
surprising consequence that if $\abs{\nabla\phi}^2(x_k,t_k)\to\infty$
for $t_k\nearrow T$, then $R(x_k,t_k)$ blows up as well,
i.e.~$g(t_k)$ must become singular as $t_k\nearrow T$. Conversely,
if $\abs{\Rm{}}$ stays bounded along the flow, $\abs{\nabla\phi}^2$ must
stay bounded, too. This leads to the conjecture that a uniform
Riemann-bound is enough to conclude long-time existence. This
conjecture is proved in Section 6. To this end, we first compute
estimates for the Riemannian curvature tensor, its derivatives and
the higher derivatives of $\phi$ and then follow Bando's
\cite{Bando:estimates} and Shi's \cite{Shi:complete} results for the
Ricci flow to derive interior-in-time gradient estimates.
\newline
In Section 7, we introduce an entropy functional $\mathcal{W}_\alpha(g,\phi,f,\tau)$
which corresponds to Perelman's shrinker entropy for the Ricci flow
\cite[Section 3]{Perelman:entropy}. Here $\tau=T-t$ denotes a
backwards time. For $\alpha(t)\equiv\alpha>0$, the entropy functional
is non-decreasing and constant exactly on shrinking solitons. Again, the entropy is
monotone if we allow non-increasing positive coupling functions
$\alpha(t)$ instead of constant ones. Using $\mathcal{F}_\alpha$ and
$\mathcal{W}_\alpha$ we can exclude nontrivial breathers, i.e.~we show that
a breather has to be a gradient soliton. In the case of a steady or
expanding breather the result is even stronger, namely we can show
that $\phi(t)$ has to be harmonic in these cases for all $t$.
\newline
Finally in the last section, we state the monotonicity of a backwards
reduced volume quantity for the $(RH)_\alpha$ flow
with positive non-increasing $\alpha(t)$. This follows from our more
general result from \cite{Muller:MonotoneVolumes}. We apply this
monotonicity to deduce a local non-collapsing theorem.
\newline
In the appendix, we collect the commutator identities on bundles like
$T^*M\otimes \phi^*TN$, which we need for the evolution equations in
Section 4 and 6.
\newline
This article originates from the authors PhD thesis \cite{Muller:diss}
from 2009, where some of the proofs and computations are carried out in more details.
The author likes to thank Klaus Ecker, Robert Haslhofer, Gerhard
Huisken, Tom Ilmanen, Peter Topping and in particular Michael Struwe
for stimulating discussions and valuable remarks and suggestions
while studying this new flow. Moreover, he thanks the Swiss National
Science Foundation that partially supported his research and Zindine Djadli
who translated the abstract into flawless French.
\section{Examples and special solutions}
In this section, we only consider time-independent coupling
constants $\alpha(t)\equiv \alpha$. First, we study two very simple
homogeneous examples for the $(RH)_\alpha$ flow system to illustrate
the different behavior of the flow for different coupling constants
$\alpha$. In particular, the existence or non-existence of
singularities will depend on the choice of $\alpha$. We study the
volume-preserving version of the flow as well. We say that
$(g(t),\phi(t))$ is a solution of the normalized $(RH)_\alpha$ flow,
if it satisfies
\begin{equation}\label{4.eq1}
\left\{\begin{aligned}\dt g &= -2\Rc{} + 2\alpha \nabla\phi \otimes
\nabla\phi + \tfrac{2}{m}\,g\;\Xint-_M \big(R-\alpha\abs{\nabla\phi}^2\big)dV,\\%
\dt \phi &= \tau_g \phi. \end{aligned}\right.
\end{equation}
\subsection{Two homogeneous examples with $\phi=\id$}
Assume that $(M,g(0))$ is a round two-sphere of constant Gauss
curvature $1$. Under the Ricci flow, the sphere shrinks to a point
in finite time. Let us now consider the $(RH)_\alpha$ flow, assuming
that $(N,\gamma)=(M,g(0))$ and $\phi(0)$ is the identity map. With the ansatz $g(t)=c(t)g(0)$, $c(0)=1$ and the fact that $\phi(t)=\phi(0)$ is
harmonic for all $g(t)$, the $(RH)_\alpha$ flow reduces to
\begin{equation*}
\dt c(t) = -2+2\alpha.
\end{equation*}
For $\alpha < 1$, $c(t)$ goes to zero in finite time, i.e.~$(M,g(t))$
shrinks to a point, while the scalar curvature $R$
and the energy density $\abs{\nabla\phi}^2$ both go to infinity. For
$\alpha=1$, the solution is stationary. For $\alpha>1$, $c(t)$ grows
linearly and the flow exists forever, while both the scalar
curvature $R$ and the energy density $\abs{\nabla\phi}^2$ vanish
asymptotically. Instead of changing $\alpha$, we can also scale the
metric $\gamma$ on the target manifold. Mapping into a larger sphere
has the same consequences as choosing a larger $\alpha$.
Note that the volume-preserving version (\ref{4.eq1})
of the flow is always stationary, as it is for the normalized Ricci flow, too.
\newline
A more interesting example is obtained if we let
$(M^4,g(t))=(\mathbb{S}^2\times L,c(t)g_{\mathbb{S}^2}\oplus
d(t)g_L)$, where $(\mathbb{S}^2,g_{\mathbb{S}^2})$ is again a round
sphere with Gauss curvature $1$ and $(L,g_L)$ is a surface (of
genus $\geq 2$) with constant Gauss curvature $-1$. Under the Ricci
flow, $\dt c(t)=-2$ and $\dt d(t)=+2$. In particular, $c(t)$ goes to
zero in finite time while $d(t)$ always expands. Under the
normalized Ricci flow $\dt g=-2\Rc{}+\tfrac{1}{2}\,g\;\Xint- R\;
dV$, we have
\begin{equation*}
\dt c = -2 + \tfrac{d-c}{d}=-1-c^2,\qquad \dt
d=+2+\tfrac{d-c}{c}=+1+d^2.
\end{equation*}
Again, $c(t)$ goes to zero in finite time. At the same time, $d(t)$
goes to infinity. Now, let us consider the $(RH)_\alpha$ flow for this example,
setting $(N,\gamma)=(M,g(0))$ and $\phi(0)=\id$. First,
note that $\phi(0)$ is always harmonic and thus $\phi(t)=\phi(0)$ is
unchanged. The identity map between the same manifold with
two different metrics is not necessarily harmonic in general, but here
it is. The flow equations reduce to $\dt c(t)=-2+2\alpha$
and $\dt d(t)=+2+2\alpha$. While $d(t)$ always grows, the behavior
of $c(t)$ is exactly the same as in the first example above, where
we only had a two-sphere. On the other hand, if we consider the
normalized flow (\ref{4.eq1}), we obtain
\begin{equation*}
\dt c = (\alpha-1) -(\alpha+1)c^2,\qquad \dt d=(\alpha+1) -
(\alpha-1)d^2.
\end{equation*}
In the case where $\alpha <1$, $c(t)$ goes to zero in finite time,
while $d(t)$ blows up at the same time, similar to the normalized
Ricci flow above. For $\alpha=1$, $c(t)=(1+2t)^{-1}$ goes to zero in
infinite time while $d(t)=(1+2t)$ grows linearly, i.e.~we have
long-time existence but no natural convergence (to a manifold with
the same topology). In the third case, where $\alpha
>1$, both $c(t)$ and $d(t)$ converge with
\begin{equation*}
c(t) \to \sqrt{\tfrac{\alpha-1}{\alpha+1}}, \qquad d(t) \to
\sqrt{\tfrac{\alpha+1}{\alpha-1}}, \qquad \textrm{as }t\to\infty.
\end{equation*}
These examples show that both the unnormalized and the normalized
version of our flow can behave very differently from the Ricci flow if $\alpha$
is chosen large. In particular, they may be more regular in special situations.
\subsection{Volume-preserving version of $(RH)_\alpha$}
Here, we show that the unnormalized $(RH)_\alpha$ flow and the
normalized version are related by rescaling the metric $g$ and the
time, while keeping the map $\phi$ unchanged. Indeed, assume that
$(g(t),\phi(t))_{t\in[0,T)}$ is a solution of $(RH)_\alpha$. Define
a family of rescaling factors $\lambda(t)$ by
\begin{equation}\label{4.eq9}
\lambda(t) := \bigg(\int_M dV_{g(t)}\bigg)^{\!-2/m}, \quad t\in
[0,T),
\end{equation}
and let $\bar{g}(t)$ be the family of rescaled metrics
$\bar{g}(t)=\lambda(t)g(t)$ having constant unit volume
\begin{equation*}
\int_M dV_{\bar{g}(t)} = \int_M \lambda^{m/2}(t)\; dV_{g(t)} =1,
\quad \forall t\in [0,T).
\end{equation*}
Write $\mathcal{S}=\Rc{}-\alpha\nabla\phi\otimes\nabla\phi$ with trace
$S=R-\alpha\abs{\nabla\phi}^2$. It is an immediate consequence of the
scaling behavior of $\Rc{}$, $R$, $\nabla\phi\otimes\nabla\phi$ and
$\abs{\nabla\phi}^2$ that $\bar{\mathcal{S}}=\mathcal{S}$ and $\bar{S}=\lambda^{-1}S$
for $\bar{g}=\lambda g$. Note that $\lambda(t)$ is a smooth function
of time, with
\begin{equation*}
\frac{d}{dt} \lambda(t) = -\tfrac{2}{m}\bigg(\int_M
dV_g\bigg)^{\!-\frac{2+m}{m}}\!\int_M (-S) dV_g=
\tfrac{2}{m}\,\lambda\; \Xint-_M S\; dV_g =
\tfrac{2}{m}\,\lambda^2\; \Xint-_M \bar{S}\; dV_{\bar{g}}.
\end{equation*}
Now, we rescale the time. Put $\bar{t}(t):=\int_0^t\lambda(s) ds$,
so that $\tfrac{d\bar{t}}{dt}=\lambda(t)$. Then, we obtain
\begin{align*}
\tfrac{\partial}{\partial \bar{t}}\bar{g} &= \lambda^{-1}\dt(\lambda
g ) = \dt g + \big(\lambda^{-2}\dt\lambda\big)\bar{g} = -2\bar{\mathcal{S}}
+\tfrac{2}{m}\,\bar{g}\;\Xint-_M \bar{S}\; dV_{\bar{g}},\\%
\tfrac{\partial}{\partial \bar{t}}\phi &= \lambda^{-1}\dt \phi =
\lambda^{-1}\tau_g\phi = \tau_{\bar{g}}\phi.
\end{align*}
This means that $(\bar{g}(\bar{t}),\phi(\bar{t}))$ solves the
volume-preserving $(RH)_\alpha$ flow (\ref{4.eq1}) on $[0,\bar{T})$,
where $\bar{T}=\int_0^T \lambda(s) ds$.
\subsection{Gradient solitons}
A solution to $(RH)_\alpha$ which changes under a one-parameter
family of diffeomorphisms on $M$ and scaling is called a soliton (or
a self-similar solution). These solutions correspond to fixed points
modulo diffeomorphisms and scaling. The more general class of
periodic solutions modulo diffeomorphisms and scaling, the so-called
breathers, will be defined (but also ruled out) in Section 7.
\begin{defn}\label{4.def1}
A solution $(g(t),\phi(t))_{t\in[0,T)}$ of\/ $(RH)_\alpha$ is called
a \emph{soliton} if there exists a one-parameter family of
diffeomorphisms $\psi_t:M\to M$ with $\psi_0=\id_M$ and a scaling
function $c:[0,T)\to\mathbb{R}_{+}$ such that
\begin{equation}\label{4.eq2}
\left\{\begin{aligned}g(t)&=c(t)\psi_t^*g(0),\\%
\phi(t)&=\psi_t^*\phi(0).\end{aligned}\right.
\end{equation}
The cases $\dt c=\dot{c}<0$, $\dot{c}=0$ and $\dot{c}>0$ correspond
to \emph{shrinking}, \emph{steady} and \emph{expanding} solitons,
respectively. If the diffeomorphisms $\psi_t$ are generated by a
vector field $X(t)$ that is the gradient
of some function $f(t)$ on $M$, then the soliton is called
\emph{gradient} soliton and $f$ is called the \emph{potential} of
the soliton.
\end{defn}
\begin{lemma}\label{4.lemma2}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a gradient soliton with
potential $f$. Then for any $t_0\in[0,T)$, this soliton satisfies
the coupled elliptic system
\begin{equation}\label{4.eq3}
\left\{\begin{aligned}0 &=\Rc{}-\alpha\nabla\phi\otimes\nabla\phi+\Hess(f)+\sigma g,\\%
0 &= \tau_g\phi-\scal{\nabla\phi,\nabla f},\end{aligned}\right.
\end{equation}
for some constant $\sigma(t_0)$. Conversely, given a function $f$ on
$M$ and a solution of (\ref{4.eq3}) at $t=0$, there exist one-parameter families
of constants $c(t)$ and diffeomorphisms $\psi_t:M\to M$ such that defining
$(g(t),\phi(t))$ as in (\ref{4.eq2}) yields a solution of\/ $(RH)_\alpha$.
Moreover, $c(t)$ can be chosen linear in $t$.
\end{lemma}
\begin{proof}
Suppose we have a soliton solution to $(RH)_\alpha$. Without loss of
generality, $c(0)=1$ and $\psi_0=\id_M$. Thus, the solution
satisfies
\begin{align*}
-2\Rc{g(0)}+2\alpha(\nabla\phi\otimes\nabla\phi)(0) &= \dt g(t)|_{t=0} =
\dt(c(t)\psi_t^*g(0))|_{t=0}\\%
&= \dot{c}(0)g(0)+\mathcal{L}_{X(0)}g(0) =
\dot{c}(0)g(0)+2\Hess(f(\cdot,0)),
\end{align*}
where $X(t)$ is the family of vector fields generating $\psi_t$.
Moreover, we compute
\begin{equation*}
(\tau_g\phi)(0) = \dt \phi(t)|_{t=0} = \mathcal{L}_{X(0)}\phi(0) =
\scal{\nabla\phi,\nabla f}.
\end{equation*}
Together, this proves (\ref{4.eq3}) with
$\sigma=\tfrac{1}{2}\dot{c}(0)$ for $t_0=0$. Hence, by a
time-shifting argument, (\ref{4.eq3}) must hold for any
$t_0\in[0,T)$. One can easily see that
$\sigma(t_0)=\dot{c}(t_0)/2c(t_0)$.
\newline
Conversely, let $(g(0),\phi(0))$ solve (\ref{4.eq3}) for some
function $f$ on $M$. Define $c(t):=1+2\sigma t$ and $X(t):=\nabla
f/c(t)$. Let $\psi_t$ be the diffeomorphisms generated by the family
of vector fields $X(t)$ (with $\psi_0=\id_M$) and define
$(g(t),\phi(t))$ as in (\ref{4.eq2}). For $\sigma<0$ this is
possible on the time interval $t\in[0,\tfrac{-1}{2\sigma})$, in the
case $\sigma\geq 0$ it is possible for $t\in[0,\infty)$. Then
\begin{align*}
\dt g(t) &= \dot{c}(t)\psi_t^*(g(0)) + c(t)\psi_t^*(\mathcal{L}_{X(t)}g(0))
= \psi_t^*(2\sigma g(0) +\mathcal{L}_{\nabla f}g(0))\\%
&= \psi_t^*(2\sigma g(0) + 2\Hess(f)) =
\psi_t^*\big({-2}\Rc{g(0)}+2\alpha(\nabla\phi\otimes\nabla\phi)(0)\big)\\%
&= -2\Rc{g(t)}+2\alpha(\nabla\phi\otimes\nabla\phi)(t),
\end{align*}
as well as
\begin{align*}
\dt\phi(t)&=\psi_t^*(\mathcal{L}_{X(t)}\phi(0))=\psi_t^*\scal{\nabla\phi(0),\nabla
f/c(t)} = c(t)^{-1}\psi_t^*(\tau_{g(0)}\phi(0))\\%
&= c(t)^{-1}\tau_{\psi_t^*g(0)}\phi(t) = \tau_{g(t)}\phi(t).
\end{align*}
This means that $(g(t),\phi(t))$ is a solution of $(RH)_\alpha$ and
thus a soliton solution.
\end{proof}
By Lemma \ref{4.lemma2} and rescaling, we may assume that $c(t)=T-t$
for shrinking solitons (here $T$ is the maximal time of existence
for the flow), $c(t)=1$ for steady solitons and $c(t)=t-T$ for
expanding solitons (where $T$ defines a \emph{birth} time). An
example for a soliton solution is the very first example from this
section, where $(M,g(0))=(N,\gamma)=(\mathbb{S}^2,g_{\mathbb{S}^2})$
and $\phi(0)=\id$. For $\alpha<1$, the soliton is shrinking, for
$\alpha=1$ steady and for $\alpha>1$ expanding. Since $\phi_t=\id_M$
for all $t$ in all three cases, these are gradient solitons with
potential $f=0$.
\newline
Taking the trace of the first equation in (\ref{4.eq3}), we see
that a soliton must satisfy
\begin{equation}\label{4.eq4}
R-\alpha\abs{\nabla\phi}^2+\triangle f + \sigma m = 0.
\end{equation}
Taking covariant derivatives in (\ref{4.eq3}) and using the
twice traced second Bianchi identity $\nabla_jR_{ij}=\tfrac{1}{2}\nabla_iR$, we obtain
(analogous to the corresponding equation for soliton solutions of
the Ricci flow)
\begin{equation}\label{4.eq5}
R-\alpha\abs{\nabla\phi}^2+\abs{\nabla f}^2 + 2\sigma f = const.
\end{equation}
Finally, with $f(\cdot,t)=\psi_t^*(f(\cdot,0))$, we get
\begin{equation}\label{4.eq6}
\dt f = \mathcal{L}_Xf = \abs{\nabla f}^2.
\end{equation}
Combining this with (\ref{4.eq4}), we obtain the evolution equation
\begin{equation}\label{4.eq7}
(\dt + \triangle)f = \abs{\nabla f}^2-R+\alpha\abs{\nabla\phi}^2-\sigma m.
\end{equation}
For steady solitons, for which the formulas
(\ref{4.eq3})--(\ref{4.eq7}) hold with $\sigma=0$, equation (\ref{4.eq7})
is equivalent to $u= e^{-f}$ solving the adjoint heat equation
\begin{equation}\label{4.eq8}
\Box^*u=-\dt u-\triangle u +Ru-\alpha \abs{\nabla \phi}^2 u = 0.
\end{equation}
For shrinking solitons, (\ref{4.eq3})--(\ref{4.eq7}) hold with
$\sigma(t)=-\tfrac{1}{2}(T-t)^{-1}$ and (\ref{4.eq7}) is equivalent
to $u= (4\pi(T-t))^{-m/2}e^{-f}$ solving the adjoint heat equation.
Finally, for expanding solitons, $\sigma(t)=+\tfrac{1}{2}(t-T)^{-1}$
and (\ref{4.eq7}) is equivalent to the fact that $u=
(4\pi(t-T))^{-m/2}e^{-f}$ solves the adjoint heat equation
(\ref{4.eq8}).
\section{The $(RH)_\alpha$ flow as a gradient flow}
In this section, we introduce an energy functional $\mathcal{F}_\alpha$ for
the $(RH)_\alpha$ flow, which corresponds to Perelman's $\mathcal{F}$-energy
for the Ricci flow introduced in \cite[Section 1]{Perelman:entropy}.
For a detailed study of Perelman's functional, we refer to Chow et
al. \cite[Chapter 5]{RF:TAI}, M\"{u}ller \cite[Chapter
3]{Muller:Harnack}, or Topping \cite[Chapter 6]{Topping:Lectures}.
In the special case $N\subseteq\mathbb{R}$, the corresponding functional was
introduced by List \cite{List:diss}. We follow his work closely in the first
part of this section.
\subsection{The energy functional and its first variation}
Let $g=g_{ij}\in \Gamma\big(\Sym^2_{+}(T^*M)\big)$ be a Riemannian
metric on a closed manifold $M$, $f:M\to\mathbb{R}$ a smooth function and
$\phi\in C^\infty(M,N):=\{\phi\in C^\infty(M,\mathbb{R}^d)\mid \phi(M)
\subseteq N\}$. For a constant $\alpha(t)\equiv \alpha>0$, we set
\begin{equation}\label{1.eq1}
\mathcal{F}_\alpha(g,\phi,f) := \int_M \Big(R_g + \abs{\nabla f}^2_g -
\alpha\abs{\nabla\phi}^2_g\Big)e^{-f}dV_g.
\end{equation}
Take variations
\begin{align*}
g_{ij}^\varepsilon &= g_{ij}+\varepsilon h_{ij}, &h_{ij} &\in\Gamma
\big(\Sym^2(T^*M)\big),\\%
f^\varepsilon &= f+\varepsilon \ell, &\ell &\in C^\infty(M),\\%
\phi^\varepsilon &= \pi_N(\phi + \varepsilon \vartheta), &\vartheta &\in
C^\infty(M,\mathbb{R}^d) \text{ with } \vartheta(x) \in T_{\phi(x)}N,
\end{align*}
where $\pi_N$ is the smooth nearest-neighbour projection defined on
a tubular neighbourhood of $N \subset \mathbb{R}^d$. Note that we used the
identification $T_pN \subset T_p\mathbb{R}^d \cong \mathbb{R}^d$. We denote by
$\delta$ the derivative $\delta =\frac{d}{d\varepsilon}\big|_{\varepsilon=0}$,
i.e.~we have $\delta g=h$, $\delta f = \ell$ and $\delta \phi =
(d\pi_N \circ \phi)\vartheta = \vartheta$. Our goal is to compute
\begin{align*}
\delta \mathcal{F}_{\alpha,g,\phi,f}(h,\vartheta,\ell)&:=
\frac{d}{d\varepsilon}\Big|_{\varepsilon=0} \mathcal{F}_\alpha(g+\varepsilon h,\pi_N(\phi +
\varepsilon\vartheta), f+\varepsilon\ell)\\%
&\phantom{:}= \underbrace{\delta\int_M \big(R + \abs{\nabla f}^2\big)
e^{-f}dV}_{=:\;I} - \;\alpha \cdot \underbrace{\delta\int_M
\abs{\nabla\phi}^2 e^{-f}dV}_{=:\;I\!I}.
\end{align*}
For the variation of the first integral, we know from Ricci flow
theory that
\begin{equation*}
I = \int_M \Big({-h^{ij}}\big(R_{ij} + \nabla_i\nabla_j f\big) +
\big(\tfrac{1}{2} \tr_g h - \ell\big)\big(2\triangle f - \abs{\nabla f}^2
+R\big)\Big) e^{-f}dV,
\end{equation*}
see Perelman \cite[Section 1]{Perelman:entropy}, or M\"{u}ller
\cite[Lemma 3.3]{Muller:Harnack}, for a detailed proof. For the
variation of the second integral, we compute with a partial
integration
\begin{align*}
I\!I &= \int_M 2g^{ij}\nabla_i \phi^\lambda \nabla_j \vartheta^\lambda e^{-f}dV +
\int_M \Big(-h^{ij}\nabla_i\phi^\lambda \nabla_j \phi^\lambda +
\abs{\nabla\phi}^2_g \big(\tfrac{1}{2}\tr_g h - \ell\big)\Big)
e^{-f}dV,\\%
&= \int_M \Big({-2}\vartheta^\lambda\big(\triangle_g \phi^\lambda
-\Scal{\nabla\phi^\lambda,\nabla f}_g\big) -h^{ij}\nabla_i\phi^\lambda \nabla_j
\phi^\lambda + \abs{\nabla\phi}^2_g \big(\tfrac{1}{2}\tr_g h -
\ell\big)\Big) e^{-f}dV.
\end{align*}
Hence, by putting everything together, we find
\begin{align}
\delta \mathcal{F}_{\alpha,g,\phi,f}(h,\vartheta,\ell) &= \int_M
-h^{ij}\big(R_{ij} + \nabla_i\nabla_j f-\alpha \nabla_i \phi \nabla_j
\phi\big)e^{-f}dV\notag\\%
&\quad+ \int_M \big(\tfrac{1}{2}\tr_g h - \ell\big)\big(2\triangle f -
\abs{\nabla f}^2 + R - \alpha \abs{\nabla \phi}^2\big)e^{-f}dV\label{1.eq2}\\%
&\quad+\int_M 2\alpha\vartheta \big(\tau_g \phi - \Scal{\nabla\phi,\nabla
f}\big) e^{-f}dV,\notag
\end{align}
since $\vartheta\triangle_g \phi = \vartheta\tau_g \phi$, where $\tau_g
\phi := \triangle_g \phi - A(\phi)(\nabla\phi,\nabla\phi)_M$ denotes the tension
field of $\phi$.
\subsection{Gradient flow for fixed background measure}
Now, we fix the measure $d\mu = e^{-f}dV$, i.e.~let $f=
-\log\big(\frac{d\mu}{dV}\big)$, where $\frac{d\mu}{dV}$ denotes the
Radon-Nikodym differential of measures. Then, from $0= \delta d\mu =
\left(\frac{1}{2}\tr_g h -\ell\right)d\mu$ we get $\ell =
\frac{1}{2}\tr_g h$. Thus, for a fixed measure $\mu$ the functional
$\mathcal{F}_\alpha$ and its variation $\delta \mathcal{F}_\alpha$ depend only on
$g$ and~$\phi$ and their variations $\delta g = h$ and $\delta \phi
= \vartheta$. In the following we write
\begin{equation}\label{1.eq3}
\mathcal{F}^\mu_\alpha(g,\phi) :=
\mathcal{F}_\alpha\big(g,\phi,-\log\big(\tfrac{d\mu}{dV}\big)\big)
\end{equation}
and
\begin{equation*}
\delta \mathcal{F}^\mu_{\alpha,g,\phi}(h,\vartheta) := \delta
\mathcal{F}_{\alpha,g,\phi,-\log(\frac{d\mu}{dV})}
\left(h,\vartheta,\tfrac{1}{2}\tr_g h\right).
\end{equation*}
Equation (\ref{1.eq2}) reduces to
\begin{equation}\label{1.eq4}
\delta \mathcal{F}^\mu_{\alpha,g,\phi}(h,\vartheta) = \int_M
\Big({-h^{ij}}\big(R_{ij}+\nabla_i\nabla_j f - \alpha \nabla_i \phi
\nabla_j\phi\big)+ 2\alpha\vartheta \big(\tau_g \phi-\Scal{\nabla\phi,\nabla
f}\big) d\mu.
\end{equation}
Let $(g,\phi)\in \Gamma(\Sym^2_{+}(T^*M))\times C^{\infty}(M,N)$ and
define on $H:=H_{g,\phi}=\Gamma(\Sym^2(T^*M))\times T_\phi
C^\infty(M,N)$ an inner product depending on $\alpha$ and the
measure $\mu$ by
\begin{equation*}
\Scal{(k_{ij},\psi),(h_{ij},\vartheta)}_{H,\alpha,\mu}:= \int_M
\big(\tfrac{1}{2} h^{ij}k_{ij} + 2\alpha\psi\vartheta\big)d\mu.
\end{equation*}
From $\delta \mathcal{F}^\mu_{\alpha,g,\phi}(h,\vartheta) = \Scal{\grad
\mathcal{F}^\mu_\alpha(g,\phi),(h,\vartheta)}_{H,\alpha,\mu}$ we then deduce
\begin{equation}\label{1.eq5}
\grad \mathcal{F}^\mu_\alpha(g,\phi) = \big(-2(R_{ij}+\nabla_i\nabla_j f -\alpha\nabla_i
\phi \nabla_j \phi),\; \tau_g \phi -\Scal{\nabla\phi,\nabla f}\big).
\end{equation}
Let $\pi_1$, $\pi_2$ denote the natural projections of $H$ onto its
first and second factors, respectively. Then, the gradient flow of
$\mathcal{F}^\mu_\alpha$ is
\begin{equation*}
\left\{\begin{aligned}\dt g_{ij} &= \pi_1(\grad
\mathcal{F}^\mu_\alpha(g,\phi)),\\%
\dt\phi &=\pi_2(\grad \mathcal{F}^\mu_\alpha(g,\phi)).\end{aligned}\right.
\end{equation*}
Thus, recalling the equation $\dt f = \ell = \frac{1}{2}\tr_g
\!\big(\dt g_{ij}\big)$, we obtain the gradient flow system
\begin{equation}\label{1.eq6}
\left\{\begin{aligned}\dt g_{ij} &= -2(R_{ij}+\nabla_i\nabla_j f -\alpha
\nabla_i
\phi \nabla_j \phi),\\%
\dt \phi &= \tau_g \phi - \Scal{\nabla\phi,\nabla f},\\%
\dt f &= -R -\triangle f + \alpha \abs{\nabla\phi}^2.\end{aligned}\right.
\end{equation}
\subsection{Pulling back with diffeomorphisms}
As one can do for the Ricci flow (see Perelman \cite[Section
1]{Perelman:entropy} or M\"{u}ller \cite[page 52]{Muller:Harnack}), we
now pull back a solution $(g,\phi,f)$ of (\ref{1.eq6}) with a family
of diffeomorphisms generated by $X=\nabla f$. Indeed, recalling the
formulas for the Lie derivatives $(\mathcal{L}_{\nabla f}g)_{ij}= 2
\nabla_i\nabla_j f$, $\mathcal{L}_{\nabla f}\phi = \Scal{\nabla\phi,\nabla f}$ and
$\mathcal{L}_{\nabla f}f = \abs{\nabla f}^2$, we can rewrite (\ref{1.eq6})
in the form
\begin{equation*}
\left\{\begin{aligned}\dt g &= -2\Rc{}+2\alpha\nabla \phi \otimes\nabla
\phi - \big(\mathcal{L}_{\nabla f}g\big),\\%
\dt \phi &= \tau_g \phi - \big(\mathcal{L}_{\nabla f}\phi\big),\\%
\dt f &= -\triangle f + \abs{\nabla f}^2 -R + \alpha \abs{\nabla\phi}^2 -
\big(\mathcal{L}_{\nabla f}f\big).\end{aligned}\right.
\end{equation*}
Hence, if $\psi_t$ is the one-parameter family of diffeomorphisms
induced by the vector field $X(t)=\nabla f(t) \in \Gamma(TM)$, $t \in
[0,T)$, i.e.~if $\dt \psi_t = X(t) \circ \psi_t$, $\psi_0 = \id$,
then the pulled-back quantities $\tilde{g}=\psi^*_t g$,
$\tilde{\phi}=\psi^*_t \phi$, $\tilde{f}=\psi^*_t f$ satisfy
\begin{equation*}
\left\{\begin{aligned}\dt \tilde{g} &= -2\tilde{\Rc{}}+2\alpha
\nabla \tilde{\phi} \otimes \nabla \tilde{\phi},\\%
\dt \tilde{\phi} &= \tau_{\tilde{g}} \tilde{\phi},\\%
\dt \tilde{f} &= -\triangle_{\tilde{g}} \tilde{f} + \abs{\nabla
\tilde{f}}^2_{\tilde{g}} -\tilde{R} + \alpha
\abs{\nabla\tilde{\phi}}^2_{\tilde{g}}.\end{aligned}\right.
\end{equation*}
Here, $\tilde{\Rc{}}$ and $\tilde{R}$ denote the Ricci and scalar
curvature of $\tilde{g}$ and $\triangle$, $\tau$ and the norms are also
computed with respect to $\tilde{g}$. In the following, we will
usually consider the pulled-back gradient flow system and therefore
drop the tildes for convenience of notation.
\newline
Note that the formal adjoint of the heat operator $\Box=\dt-\triangle$
under the flow $\dt g = h$ is $\Box^*=-\dt-\triangle-\frac{1}{2}\tr_g h$.
Indeed, for functions $v,w\colon M\times[0,T]\to \mathbb{R}$, a
straightforward computation yields
\begin{equation*}
\int_0^T \int_M (\Box v)w \; dV\,dt = \bigg[\int_M vw \; dV
\bigg]_0^T + \int_0^T \int_M v(\Box^*w) \; dV \,dt.
\end{equation*}
In our case where $h_{ij}=-2R_{ij}+2\alpha \nabla_i\phi \nabla_j\phi$, this
is $\Box^*=-\dt-\triangle +R-\alpha \abs{\nabla \phi}^2$ and thus the
evolution equation for $f$ is equivalent to $e^{-f}$ solving the
adjoint heat equation $\Box^*e^{-f}=0$. The system now reads
\begin{equation}\label{1.eq7}
\left\{\begin{aligned}\dt g &= -2\Rc{} + 2\alpha \nabla\phi \otimes \nabla\phi,\\%
\dt \phi &= \tau_g \phi,\\%
0 &= \Box^*e^{-f}. \end{aligned}\right.
\end{equation}
This means that $(RH)_\alpha$ can be interpreted as the
gradient flow of $\mathcal{F}^\mu_\alpha$ for any fixed background measure
$\mu$. Moreover, using (\ref{1.eq4}), (\ref{1.eq5}) and the
diffeomorphism invariance of $\mathcal{F}_\alpha$, we get the following.
\begin{prop}\label{1.prop1}
Let $(g(t),\phi(t))_{t \in [0,T)}$ be a solution of the
$(RH)_\alpha$ flow with coupling constant $\alpha(t)\equiv \alpha>0$
and let\/ $e^{-f}$ solve the adjoint heat equation under this flow.
Then the energy functional $\mathcal{F}_\alpha(g,\phi,f)$ defined in
(\ref{1.eq1}) is non-decreasing with
\begin{equation}\label{1.eq8}
\frac{d}{dt} \mathcal{F}_\alpha = \int_M \Big(2\Abs{\Rc{}-\alpha \nabla\phi
\otimes \nabla\phi+\Hess(f)}^2 +2\alpha \Abs{\tau_g \phi-\Scal{\nabla\phi,\nabla
f}}^2\Big) e^{-f}dV \geq 0.
\end{equation}
Moreover, $\mathcal{F}_\alpha$ is constant if and only if $(g(t),\phi(t))$
is a steady soliton.
\end{prop}
Allowing also time-dependent coupling constants $\alpha(t)$, we
obtain the following.
\begin{cor}\label{1.cor2}
Let $(g(t),\phi(t))_{t \in [0,T)}$ solve $(RH)_\alpha$ for a
positive coupling function $\alpha(t)$ and let\/ $e^{-f}$ solve the
adjoint heat equation under this flow. Then
$\mathcal{F}_{\alpha(t)}(g(t),\phi(t),f(t))$ satisfies
\begin{equation*}
\frac{d}{dt} \mathcal{F}_\alpha = \int_M \Big(2\Abs{\Rc{}-\alpha \nabla\phi
\otimes \nabla\phi+\Hess(f)}^2 +2\alpha \Abs{\tau_g \phi-\Scal{\nabla\phi,\nabla
f}}^2 - \dot{\alpha} \abs{\nabla\phi}^2 \Big) e^{-f}dV,
\end{equation*}
in particular, it is non-decreasing if $\alpha(t)$ is a
non-increasing function.
\end{cor}
\subsection{Minimizing over all probability measures}
Following Perelman \cite{Perelman:entropy}, we define
\begin{equation}\label{1.eq13}
\lambda_\alpha(g,\phi):= \inf\big\{\mathcal{F}^\mu_\alpha(g,\phi)\;\big|\;
\mu(M)=1\big\} = \inf\left\{\mathcal{F}_\alpha(g,\phi,f)\;\bigg|\; \int_M
e^{-f}dV =1\right\}.
\end{equation}
The first task is to show that the infimum is always achieved.
Indeed, if we set $v=e^{-f/2}$, we can write the energy as
\begin{equation*}
\mathcal{F}_\alpha(g,\phi,v)=\int_M \Big(Rv^2 + 4\abs{\nabla
v}^2-\alpha\abs{\nabla\phi}^2v^2\Big)dV = \int_M v\Big(Rv - 4\triangle v -
\alpha\abs{\nabla\phi}^2v\Big)dV.
\end{equation*}
Hence
\begin{equation*}
\lambda_\alpha(g,\phi)=\inf\left\{\int_M v\Big(Rv - 4\triangle v -
\alpha\abs{\nabla\phi}^2v\Big)dV \;\bigg| \; \int_M v^2 dV=1\right\}
\end{equation*}
is the smallest eigenvalue of the operator $-4\triangle{} +R
-\alpha\abs{\nabla\phi}^2$ and $v$ is a corresponding normalized
eigenvector. Since the operator (for any time $t$ and map $\phi(t)$)
is a Schr\"{o}dinger operator, there exists a unique positive and
normalized eigenvector $v_{min}(t)$, see for example Reed and Simon
\cite{ReedSimon} and Rothaus \cite{Rothaus:Schroedinger}. From
eigenvalue perturbation theory, we see that if $g(t)$ and $\phi(t)$
depend smoothly on $t$, then so do $\lambda_\alpha(g(t),\phi(t))$
and $v_{min}(t)$.
\begin{prop}\label{1.prop3}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a smooth solution of the
$(RH)_\alpha$ flow with constant $\alpha(t)\equiv\alpha>0$. Then
$\lambda_\alpha(g,\phi)$ as defined in (\ref{1.eq13}) is monotone
non-decreasing in time and it is constant if and only if
\begin{equation}\label{1.eq14}
\left\{\begin{aligned}0 &= \Rc{}-\alpha\nabla\phi\otimes\nabla\phi+\Hess(f),\\%
0 &= \tau_g \phi -\scal{\nabla\phi,\nabla f}, \end{aligned}\right.
\end{equation}
for the minimizing function $f=-2\log v_{min}$.
\end{prop}
\begin{proof}
Pick arbitrary times $t_1$, $t_2 \in [0,T)$ and let $v_{min}(t_2)$
be the unique positive minimizer for $\lambda_\alpha
(g(t_2),\phi(t_2))$. Put $u(t_2)=v_{min}^2(t_2)>0$ and solve the
adjoint heat equation $\Box^*u=0$ backwards on $[t_1,t_2]$. Note
that $u(x',t')>0$ for all $x'\in M$ and $t'\in [t_1,t_2]$ by the
maximum principle and the constraint $\int_M u \;dV=\int_M v^2 \;dV
=1$ is preserved since
\begin{equation*}
\frac{d}{dt} \int_M u\;dV = \int_M \big(\dt u\big)dV + \int_M
u\big(\dt dV\big) = -\int_M\triangle u\; dV = 0.
\end{equation*}
Here, we used $\dt dV = -\tfrac{1}{2}\tr_g(\dt g)dV =
(-R+\alpha\abs{\nabla\phi}^2)dV$ and $\dt u = (-\triangle +R-\alpha \abs{\nabla
\phi}^2)u$, the latter following from $\Box^*u=0$. Thus, with $u(t)=
e^{-\bar{f}(t)}$ for all $t\in [t_1,t_2]$, we obtain with
Proposition \ref{1.prop1}
\begin{equation*}
\lambda_\alpha(g(t_1),\phi(t_1))\leq
\mathcal{F}_\alpha(g(t_1),\phi(t_1),\bar{f}(t_1)) \leq
\mathcal{F}_\alpha(g(t_2),\phi(t_2),\bar{f}(t_2)) =
\lambda_\alpha(g(t_2),\phi(t_2)).
\end{equation*}
The condition (\ref{1.eq14}) in the equality case follows directly
from (\ref{1.eq8}).
\end{proof}
Again the monotonicity of $\lambda_\alpha(g,\phi)$ is preserved if
we allow a positive non-increasing coupling function $\alpha(t)$
instead of a time-independent positive constant $\alpha$.
\section{Short-time existence and evolution equations}
Due to diffeomorphism invariance, our flow is only weakly parabolic.
In fact, the principal symbol for the first equation is the same as
for the Ricci flow since the additional term is of lower order.
Thus, one cannot directly apply the standard parabolic existence
theory. Fortunately, shortly after Hamilton's first proof of
short-time existence for the Ricci flow in \cite{Hamilton:3folds}
which was based on the Nash-Moser implicit function theorem, DeTurck
\cite{DeTurck:trick} found a substantially simpler proof which can
easily be modified to get an existence proof for our system
$(RH)_\alpha$. Note that we only consider the case where $M$ is
closed, but following Shi's short-time existence proof in
\cite{Shi:complete} for the Ricci flow on complete noncompact
manifolds, one can also prove more general short-time existence
results for our flow. We first recall some results for the Ricci
flow, following the presentation of Hamilton \cite{Hamilton:survey}
very closely.
\newline
Since some results strongly depend on the curvature of $(N,\gamma)$,
it is more convenient to work with $\phi:M\to N$ itself instead of
$e_N\circ\phi:M\to \mathbb{R}^d$ as in the last section. Therefore,
repeated Greek indices are summed over from $1$ to $n = \dim N$ in
this section.
\subsection{Dual Ricci-Harmonic and Ricci-DeTurck flow}
Let $g(t)$ be a solution of the Ricci flow and
$\psi(t)\colon(M,g)\to (M,h)$ a one parameter family of smooth maps
satisfying the harmonic map flow $\dt \psi = \tau_g
\psi$ with respect to the evolving metric $g$. Note that this is $(RH)_{\alpha\equiv 0}$.
If $\psi(t)$ is a diffeomorphism at time
$t=0$, it will stay a diffeomorphism for at least a short time. Now,
we consider the push-forward $\tilde{g}:=\psi_*g$ of the metric $g$
under $\psi$. The evolution equation for $\tilde{g}$ reads
\begin{equation}\label{2.eq1}
\dt \tilde{g}_{ij}= -2\tilde{R}_{ij} + (\mathcal{L}_V
\tilde{g})_{ij} = -2\tilde{R}_{ij} + \tilde{\nabla}_iV_j +
\tilde{\nabla}_jV_i,
\end{equation}
where $\tilde{\nabla}$ denotes the Levi-Civita connection of $\tilde{g}$
and $\dt \psi = -V \circ \psi$. One calls this the dual
Ricci-Harmonic flow or also the $h$-flow. An easy computation (see
DeTurck \cite{DeTurck:trick} or Chow and Knopf \cite[Chapter
3]{RF:intro}) shows that $V$ is given by
\begin{equation}\label{2.eq2}
V^\ell=\tilde{g}^{ij}\big(\!\phantom{l}^{\tilde{g}}\Gamma^\ell_{ij}-
\!\phantom{l}^h\Gamma^\ell_{ij}\big),
\end{equation}
the trace of the tensor which is the difference between the
Christoffel symbols of the connections of $\tilde{g}$
and of $h$, respectively. Note that the evolution equation of
$\tilde{g}$ involves only the metrics $\tilde{g}$ and $h$ and not
the metric $g$, and since it involves $\!\phantom{l}^h\Gamma$ for
the fixed background metric $h$ it is no longer diffeomorphism
invariant. Indeed, one can show (see Hamilton \cite[Section
6]{Hamilton:survey}) that
\begin{equation}\label{2.eq3}
\dt \tilde{g}_{ij} = \tilde{g}^{k\ell}\tilde{\nabla}_k \hat{\nabla}_\ell
\tilde{g}_{ij},
\end{equation}
where $\hat{\nabla}$ denotes the connection of the
background metric $h$. Since $\hat{\nabla}$ is independent of
$\tilde{g}$ and $\tilde{\nabla}$ only involves first derivatives of
$\tilde{g}$ this is a quasilinear equation. Its principal symbol is
$\sigma(\xi)=\tilde{g}^{ij}\xi_i \xi_j\cdot \id$, where $\id$ is the
identity on tensors $\tilde{g}$. Hence this flow equation is
strictly parabolic and we get short-time existence from the standard
parabolic theory for quasilinear equations, see e.g.~\cite{ManMar}
for a recent and detailed proof. If we additionally
assume that $(M,h)=(M,g_0)$ and $\psi(0)=\id_M$,
the flow $\tilde{g}$ which has the same initial data $\tilde{g}(0)
=g_0$ as $g$ is called the Ricci-DeTurck flow \cite{DeTurck:trick}.
\newline
Now, one can find a solution to the Ricci flow with smooth initial
metric $g(0)=g_0$ as follows. Chose any diffeomorphism
$\psi(0)\colon M\to M$. Since $g(0)$ is smooth, its push-forward
$\tilde{g}(0)$ is also smooth and equation (\ref{2.eq3}) has a
smooth solution for a short time. Next, one computes the vector
field $V$ with (\ref{2.eq2}) and solves the ODE system
\begin{equation*}
\dt \psi = -V \circ \psi.
\end{equation*}
One then recovers $g$ as the pull-back $g=\psi^*\tilde{g}$. This
method also proves uniqueness of the Ricci flow. Indeed, let
$g^1(t)$ and $g^2(t)$ be two solutions of the Ricci flow equation
for $t\in [0,T)$ satisfying $g^1(0)=g^2(0)$. Then one can solve the
harmonic map heat flows $\dt \psi^i=\tau_{g^i}\psi^i$, $i\in\{1,2\}$
with $\psi^1(0)=\psi^2(0)$. This yields two solutions $\tilde{g}^i =
\psi^i_*g^i$ of the dual Ricci-Harmonic map flow with the same
initial values, hence they must agree. Then the corresponding vector
fields $V^i$ agree and the two ODE systems $\dt \psi^i=-V^i
\circ\psi^i$ with the same initial data must have the same solutions
$\psi^1\equiv\psi^2$. Hence also the pull-back metrics $g^1$ and
$g^2$ must agree for all $t\in [0,T)$.
\newline
For the dual Ricci-Harmonic flow, the evolution equations in
coordinate form, using only the fixed connection $\hat{\nabla}$ of
the background metric $h$, are (see e.g.~Simon \cite{Simon:C0metrics})
\begin{equation}\label{2.eq4}
\begin{aligned}\dt \tilde{g}_{ij}&=\tilde{g}^{k\ell}
\hat{\nabla}_k\hat{\nabla}_\ell\tilde{g}_{ij}-\tilde{g}^{k\ell}\tilde{g}_{ip}
h^{pq}\hat{R}_{jkq\ell}-\tilde{g}^{k\ell}\tilde{g}_{jp}h^{pq}
\hat{R}_{ikq\ell}\\%
&\quad + \tfrac{1}{2}\tilde{g}^{k\ell}\tilde{g}^{pq}
\big(\hat{\nabla}_i\tilde{g}_{pk}\hat{\nabla}_j\tilde{g}_{q\ell}
+2\hat{\nabla}_k\tilde{g}_{ip}\hat{\nabla}_q\tilde{g}_{j\ell}
-2\hat{\nabla}_k\tilde{g}_{ip}\hat{\nabla}_\ell\tilde{g}_{jq}
-4\hat{\nabla}_i\tilde{g}_{pk}\hat{\nabla}_\ell\tilde{g}_{jq}\big),
\end{aligned}
\end{equation}
where $\hat{R}_{ijkl}=(\Rm{h})_{ijkl}$ denotes the Riemannian curvature
tensor of $h$.
\newline
Recent work of Isenberg, Guenther and Knopf
\cite{IsenbergGuentherKnopf}, Schn\"{u}rer, Schulze and Simon
\cite{SchnuererSchulzeSimon} and others shows that DeTurck's trick
is not only useful to prove short-time existence for the Ricci flow,
but is also useful for convergence and stability results. Their results show that
the $h$-flow itself is also interesting to study. In this article
however, we only use it as a technical tool.
\subsection{Short-time existence and uniqueness for $(RH)_\alpha$}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a solution of the $(RH)_\alpha$
flow with initial data $(g(0),\phi(0))=(g_0,\phi_0)$. As for the
Ricci-DeTurck flow above, we now let $\psi(t)\colon(M,g(t))
\to(M,g_0)$ be a solution of the harmonic map heat flow $\dt
\psi=\tau_g\psi$ with $\psi(0)=\id_M$ and denote by
$(\tilde{g}(t),\tilde{\phi}(t))$ the push-forward of
$(g(t),\phi(t))$ with $\psi$. Analogous to formula (\ref{2.eq1})
above, we find
\begin{equation}\label{2.eq5}
\begin{aligned}
\dt \tilde{g}_{ij} &= \psi_*(\dt g)_{ij}+ (\mathcal{L}_V
\tilde{g})_{ij} = -2\tilde{R}_{ij}+2\alpha
\nabla_i\tilde{\phi}\nabla_j\tilde{\phi}+ \tilde{\nabla}_iV_j
+ \tilde{\nabla}_jV_i,\\%
\dt \tilde{\phi} &= \psi_*(\dt \phi)+ \mathcal{L}_V\tilde{\phi}=
\tau_{\tilde{g}}\tilde{\phi}+\scal{\nabla\tilde{\phi},V},\end{aligned}
\end{equation}
where $V^\ell=\tilde{g}^{ij}
(\!\phantom{l}^{\tilde{g}}\Gamma^\ell_{ij}-
\!\phantom{l}^{g_0}\Gamma^\ell_{ij})$ and $\tilde{\nabla}$ denotes the
covariant derivative with respect to $\tilde{g}$. Note that
$\nabla_i\tilde{\phi}\nabla_j\tilde{\phi}=(d\tilde{\phi}\otimes
d\tilde{\phi})_{ij}$ as well as
$\scal{\nabla\tilde{\phi},V}=d\tilde{\phi}(V)$ are independent of the
choice of the metric. Using (\ref{2.eq3}), we find
\begin{equation}\label{2.eq6}
\dt \tilde{g}_{ij} = \tilde{g}^{k\ell}\tilde{\nabla}_k \hat{\nabla}_\ell
\tilde{g}_{ij}+2\alpha(\nabla\tilde{\phi}\otimes\nabla\tilde{\phi})_{ij},
\end{equation}
which is again quasilinear strictly parabolic. The explicit
evolution equation involving only the fixed Levi-Civita connection
$\hat{\nabla}$ of $g_0$ can be found from (\ref{2.eq4}) by adding
$2\alpha\hat{\nabla}_i\tilde{\phi}\hat{\nabla}_j\tilde{\phi}$ on the right
and replacing $h$ by $g_0$.
\newline
The evolution equation for $\tilde{\phi}(t)$ in terms of $\hat{\nabla}$
can be computed as follows. Using normal coordinates on $(N,\gamma)$
we have $\!\phantom{l}^N\!\Gamma^\lambda_{\mu\nu}=0$ at the base point and thus
$\tau_{\tilde{g}}\tilde{\phi}=\triangle_{\tilde{g}}\tilde{\phi}$. We find
\begin{equation}\label{2.eq7}
\begin{aligned}
\dt \tilde{\phi}^\lambda &= \triangle_{\tilde{g}}\tilde{\phi}^\lambda +
\scal{\nabla\tilde{\phi}^\lambda,V} =
\tilde{g}^{k\ell}\big(\partial_k\partial_\ell\tilde{\phi}^\lambda-
\!\phantom{l}^{\tilde{g}}\Gamma^j_{k\ell}\nabla_j\tilde{\phi}^\lambda\big)
+\nabla_j\tilde{\phi}^\lambda V^j\\%
&= \tilde{g}^{k\ell}\big(\hat{\nabla}_k\hat{\nabla}_\ell\tilde{\phi}^\lambda
+\!\phantom{l}^{g_0}\Gamma^j_{k\ell}\nabla_j\tilde{\phi}^\lambda
-\!\phantom{l}^{\tilde{g}}\Gamma^j_{k\ell}\nabla_j\tilde{\phi}^\lambda\big)+
\nabla_j\tilde{\phi}^\lambda\cdot\tilde{g}^{k\ell}
\big(\!\phantom{l}^{\tilde{g}}\Gamma^j_{k\ell}-
\!\phantom{l}^{g_0}\Gamma^j_{k\ell}\big)\\%
&= \tilde{g}^{k\ell}\hat{\nabla}_k\hat{\nabla}_\ell\tilde{\phi}^\lambda.
\end{aligned}
\end{equation}
Putting these results together, we have proved the following.
\begin{prop}\label{2.prop1}
Let $(g(t),\phi(t))$ be a solution of the $(RH)_\alpha$ flow with
initial data $(g(0),\phi(0))=(g_0,\phi_0)$. Let
$\psi(t)\colon(M,g(t)) \to(M,g_0)$ solve the harmonic map heat flow
$\dt \psi=\tau_g\psi$ with $\psi(0)=\id_M$ and let
$(\tilde{g}(t),\tilde{\phi}(t))$ denote the push-forward of
$(g(t),\phi(t))$ with $\psi$. Let $\hat{\nabla}$ be the (fixed)
Levi-Civita connection with respect to $g_0$. Then the dual
$(RH)_\alpha$ flow $(\tilde{g}(t),\tilde{\phi}(t))$ satisfies
\begin{align*}
\dt \tilde{g}_{ij}&=\tilde{g}^{k\ell}
\hat{\nabla}_k\hat{\nabla}_\ell\tilde{g}_{ij}-\tilde{g}^{k\ell}\tilde{g}_{ip}
g_0^{pq}\hat{R}_{jkq\ell}-\tilde{g}^{k\ell}\tilde{g}_{jp}g_0^{pq}
\hat{R}_{ikq\ell}+2\alpha\hat{\nabla}_i\tilde{\phi}\hat{\nabla}_j\tilde{\phi}\\%
&\quad + \tfrac{1}{2}\tilde{g}^{k\ell}\tilde{g}^{pq}
\big(\hat{\nabla}_i\tilde{g}_{pk}\hat{\nabla}_j\tilde{g}_{q\ell}
+2\hat{\nabla}_k\tilde{g}_{ip}\hat{\nabla}_q\tilde{g}_{j\ell}
-2\hat{\nabla}_k\tilde{g}_{ip}\hat{\nabla}_\ell\tilde{g}_{jq}
-4\hat{\nabla}_i\tilde{g}_{pk}\hat{\nabla}_\ell\tilde{g}_{jq}\big),\\%
\dt \tilde{\phi}^\lambda &= \tilde{g}^{k\ell}\hat{\nabla}_k\hat{\nabla}_\ell
\tilde{\phi}^\lambda +
\tilde{g}^{k\ell}\big(\!\phantom{l}^N\!\Gamma^\lambda_{\mu\nu}\circ\phi\big)
\hat{\nabla}_k\tilde{\phi}^\mu\hat{\nabla}_\ell\tilde{\phi}^\nu.
\end{align*}
In particular, the principal symbol for both equations is
$\sigma(\xi)=\tilde{g}^{ij}\xi_i \xi_j\cdot \id$, i.e.~the
push-forward flow is a solution to a system of strictly parabolic
equations.
\end{prop}
Short-time existence and uniqueness for the dual flow (and hence
also for the $(RH)_\alpha$ flow itself) now follow exactly as in the
simpler case of the Ricci and the dual Ricci-Harmonic flow described
above.
\subsection{Evolution equations for $R$, $\mathrm{Rc}$,
$\abs{\nabla\phi}^2$ and $\nabla\phi\otimes\nabla\phi$}
In the following, we often use commutator identities on bundles like
$T^*M \otimes \phi^*TN$. The necessary formulas are collected in the
appendix. We denote the Riemannian curvature tensor on $(N,\gamma)$
by $\NRm{}$ and let $\Rm{}$, $\Rc{}$ and $R$ be the Riemannian, Ricci
and scalar curvature on $(M,g)$. Moreover, we write
\begin{align*}
\scal{\Rc{},\nabla\phi\otimes\nabla\phi}&:=R_{ij}\nabla_i\phi^\kappa\nabla_j\phi^\kappa,\\%
\Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi}\nabla_j\phi, \nabla_i\phi} &:= \!\phantom{l}^N\!
\!R_{\kappa\mu\lambda\nu}\nabla_i
\phi^\kappa\nabla_j\phi^\mu\nabla_i\phi^\lambda\nabla_j\phi^\nu.
\end{align*}
Finally, we use the fact that $\tau_g\phi=\nabla_p\nabla_p\phi$ for the
covariant derivative $\nabla$ on $T^*M \otimes \phi^*TN$, cf.~Jost
\cite[Section 8.1]{Jost:Riemannian}. From the commutator identities
in the appendix, we immediately obtain for $\phi \in C^\infty(M,N)$
\begin{equation}\label{4.Lap}
\begin{split}
\triangle_g(\nabla_i\phi\nabla_j\phi) &=
\nabla_i\tau_g\phi\nabla_j\phi+\nabla_i\phi\nabla_j\tau_g\phi +
2\nabla_i\nabla_p\phi\nabla_j\nabla_p\phi\\%
&\quad\,+ R_{ip}\nabla_p\phi\nabla_j\phi + R_{jp}\nabla_p\phi\nabla_i\phi -
2\Scal{\!\NRm{\nabla_i\phi,\nabla_p\phi}\nabla_p\phi,\nabla_j\phi}.
\end{split}
\end{equation}
\begin{rem}
Taking the trace and using $\tau_g\phi=0$, we find the well-known
Bochner identity for harmonic maps, cf.~Jost \cite[Section
8.7]{Jost:Riemannian},
\begin{equation*}
-\triangle_g \abs{\nabla\phi}^2 + 2\abs{\nabla^2\phi}^2 +
2\scal{\Rc{},\nabla\phi\otimes\nabla\phi} =
2\Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi}\nabla_j\phi,\nabla_i\phi}.
\end{equation*}
\end{rem}
Now, we compute the evolution equations for the scalar and Ricci
curvature on $M$.
\begin{prop}\label{2.prop3}
Let $(g(t),\phi(t))$ be a solution to the $(RH)_\alpha$ flow
equation. Then the scalar curvature evolves according to
\begin{equation}\label{2.eq9}
\begin{aligned}
\dt R &= \triangle R + 2\abs{\Rc{}}^2-4\alpha
\scal{\Rc{},\nabla\phi\otimes\nabla\phi}
+2\alpha\abs{\tau_g\phi}^2-2\alpha\abs{\nabla^2\phi}^2\\
&\quad\,+ 2\alpha \Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi} \nabla_j\phi,\nabla_i\phi}
\end{aligned}
\end{equation}
and the Ricci curvature evolves by
\begin{equation}\label{2.eq10}
\begin{aligned}
\dt R_{ij} &= \triangle_L R_{ij}-2R_{iq}R_{jq}+2R_{ipjq}R_{pq}
+2\alpha\,\tau_g\phi\nabla_i\nabla_j\phi-2\alpha\nabla_p\nabla_i\phi\nabla_p\nabla_j\phi\\
&\quad\,+2\alpha R_{pijq}\nabla_p\phi\nabla_q\phi +2\alpha
\Scal{\!\NRm{\nabla_i\phi,\nabla_p\phi}\nabla_p\phi,\nabla_j\phi}.
\end{aligned}
\end{equation}
Here, $\triangle_L$ denotes the Lichnerowicz Laplacian, introduced in
\cite{Lichnerowicz}, which is defined on symmetric two-tensors
$t_{ij}$ by
\begin{equation*}
\triangle_L t_{ij} := \triangle
t_{ij}+2R_{ipjq}t_{pq}-R_{ip}t_{pj}-R_{jp}t_{pi}.
\end{equation*}
\end{prop}
\begin{proof}
We know that for $\dt g_{ij} = h_{ij}$ the evolution equation for
the Ricci tensor is given by
\begin{equation}\label{2.eq11}
\dt R_{ij} = -\tfrac{1}{2}\triangle_L h_{ij}+\tfrac{1}{2}\nabla_i\nabla_p
h_{pj}+\tfrac{1}{2}\nabla_j\nabla_p h_{pi}-\tfrac{1}{2}\nabla_i\nabla_j(\tr_g h),
\end{equation}
see for example \cite[Proposition 1.4]{Muller:Harnack} for a
proof of this general variation formula. For $h_{ij}=-2R_{ij}$, we
obtain (with the twice contracted second Bianchi identity)
$\dt R_{ij}=\triangle_L R_{ij}$. For $h_{ij}=2\nabla_i\phi\nabla_j\phi$, we
compute, using (\ref{4.Lap})
\begin{align*}
\dt R_{ij}&=-\triangle_L(\nabla_i\phi\nabla_j\phi)+\nabla_i\nabla_p(\nabla_p\phi\nabla_j\phi)+
\nabla_j\nabla_p(\nabla_p\phi\nabla_i\phi)-\nabla_i\nabla_j(\nabla_p\phi\nabla_p\phi)\\%
&= 2\tau_g\phi\nabla_i\nabla_j\phi-2\nabla_i\nabla_p\phi\nabla_j\nabla_p\phi
-2R_{ipjq}\nabla_p\phi\nabla_q\phi
+2\Scal{\!\NRm{\nabla_i\phi,\nabla_p\phi}\nabla_p\phi,\nabla_j\phi}.
\end{align*}
Linearity then yields (\ref{2.eq10}). For the evolution equation for
$R$, use
\begin{equation*}
\dt R =\dt (g^{ij}R_{ij})=g^{ij}(\dt
R_{ij})+(2R_{ij}-2\alpha\nabla_i\phi\nabla_j\phi)R_{ij}.
\end{equation*}
The desired evolution equation (\ref{2.eq9}) follows. An alternative
and more detailed proof can be found in the author's thesis
\cite[Proposition 2.3]{Muller:diss}.
\end{proof}
Next, we compute the evolution equations for $\abs{\nabla\phi}^2$ and
$\nabla\phi\otimes\nabla\phi$.
\begin{prop}\label{2.prop4}
Let $(g(t),\phi(t))$ be a solution of\/ $(RH)_\alpha$. Then the
energy density of $\phi$ satisfies the evolution equation
\begin{equation}\label{2.eq12}
\dt\abs{\nabla\phi}^2=\triangle\abs{\nabla\phi}^2-2\alpha\abs{\nabla\phi\otimes\nabla\phi}^2
-2\abs{\nabla^2\phi}^2 +2\Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi}
\nabla_j\phi,\nabla_i\phi}.
\end{equation}
Furthermore, we have
\begin{equation}\label{2.eq13}
\begin{aligned}
\dt (\nabla_i\phi\nabla_j\phi) &=\triangle(\nabla_i\phi\nabla_j\phi)-
2\nabla_p\nabla_i\phi\nabla_p\nabla_j\phi-R_{ip}\nabla_p\phi\nabla_j\phi-
R_{jp}\nabla_p\phi\nabla_i\phi\\
&\quad\,+ 2\Scal{\!\NRm{\nabla_i\phi,\nabla_p\phi}\nabla_p\phi,\nabla_j\phi}.
\end{aligned}
\end{equation}
\end{prop}
\begin{proof}
We start with the second statement. We have
\begin{align*}
\dt(\nabla_i\phi\nabla_j\phi) &= (\nabla_t\nabla_i\phi)\nabla_j\phi
+(\nabla_t\nabla_j\phi)\nabla_i\phi\\%
&=\nabla_i(\dt\phi)\nabla_j\phi + \nabla_j(\dt\phi)\nabla_i\phi\\%
&=\nabla_i\tau_g\phi\nabla_j\phi+\nabla_j\tau_g\phi\nabla_i\phi,
\end{align*}
where the meaning of the covariant time derivative $\nabla_t$ is
explained in the appendix. The desired evolution equation
(\ref{2.eq13}) now follows directly from (\ref{4.Lap}). We
obtain (\ref{2.eq12}) from (\ref{2.eq13}) by taking the trace,
\begin{align*}
\dt\abs{\nabla\phi}^2 &= (2R_{ij}-2\alpha\nabla_i\phi\nabla_j\phi)
\nabla_i\phi\nabla_j\phi + g^{ij}\dt(\nabla_i\phi\nabla_j\phi)\\
&=-2\alpha\abs{\nabla\phi\otimes\nabla\phi}^2+ \triangle\abs{\nabla\phi}^2
-2\abs{\nabla^2\phi}^2 +2\Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi}
\nabla_j\phi,\nabla_i\phi}.\qedhere
\end{align*}
\end{proof}
\subsection{Evolution of $\mathcal{S}=\mathrm{Rc}-\alpha\nabla\phi\otimes\nabla\phi$
and its trace}
We write again $\mathcal{S}:=\Rc{}-\alpha\nabla\phi\otimes\nabla\phi$ with
components $S_{ij}=R_{ij}- \alpha\nabla_i\phi\nabla_j\phi$ and let
$S= R-\alpha\abs{\nabla\phi}^2$ be its trace. Then, we can write
the $(RH)_\alpha$ flow as
\begin{equation*}
\left\{\begin{aligned} \dt g_{ij} &= -2S_{ij},\\
\dt \phi &= \tau_g\phi,\end{aligned}\right.
\end{equation*}
and the energy from Section 3 as $\mathcal{F}_\alpha(g,\phi,f) := \int_M \big(S+\abs{\nabla f}^2_g\big)e^{-f}dV_g$. It is thus convenient to study the
evolution equations for $\mathcal{S}$ and
$S$. Indeed, many terms cancel and we get much nicer equations than
in the previous subsection.
\begin{thm}\label{2.thm5}
Let $(g(t),\phi(t))$ solve $(RH)_\alpha$ with $\alpha(t) \equiv
\alpha > 0$. Then $\mathcal{S}$ and\/ $S$ defined as above satisfy the
following evolution equations
\begin{equation}\label{2.eq16}
\begin{aligned}
\dt S&= \triangle S + 2\abs{S_{ij}}^2+2\alpha \abs{\tau_g\phi}^2,\\
\dt S_{ij}&= \triangle_L S_{ij} + 2\alpha \,\tau_g\phi\nabla_i\nabla_j\phi.
\end{aligned}
\end{equation}
\end{thm}
\begin{proof}
This follows directly by combining the evolution equations from
Proposition \ref{2.prop3} with those from Proposition \ref{2.prop4}.
\end{proof}
\begin{rem}
Note that in contrast to the evolution of $\Rc{}$, $R$,
$\nabla\phi\otimes \nabla\phi$ and $\abs{\nabla\phi}^2$ the evolution equations
in Theorem \ref{2.thm5} for the combinations
$\Rc{}-\alpha\nabla\phi\otimes\nabla\phi$ and $R-\alpha\abs{\nabla\phi}^2$ do
\emph{not} depend on the intrinsic curvature of $N$.
\end{rem}
\begin{cor}\label{2.cor6}
For a solution $(g(t),\phi(t))$ of\/ $(RH)_\alpha$ with a
time-dependent coupling function $\alpha(t)$, we get
\begin{equation}\label{2.eq17}
\begin{aligned}
\dt S&= \triangle S + 2\abs{S_{ij}}^2+2\alpha \abs{\tau_g\phi}^2
-\dot{\alpha}\abs{\nabla\phi}^2,\\
\dt S_{ij}&= \triangle_L S_{ij} + 2\alpha \,\tau_g\phi\nabla_i\nabla_j\phi
-\dot{\alpha}\nabla_i\phi\nabla_j\phi.
\end{aligned}
\end{equation}
\end{cor}
\section{First results about singularities}
In this section, we often use the weak maximum principle which
states that for parabolic partial differential equations with a
reaction term a solution of the corresponding ODE yields
pointwise bounds for the solutions of the PDE. Since
we work on an evolving manifold, we need a slightly generalized
version. The following result is proved in \cite[Theorem
4.4]{RF:intro}.
\begin{prop}\label{app.prop1}
Let $u:M\times[0,T)\to \mathbb{R}$ be a smooth function satisfying
\begin{equation}\label{app.eq15}
\dt u \geq \triangle_{g(t)}u + \scal{X(t),\nabla u}_{g(t)}+F(u),
\end{equation}
where $g(t)$ is a smooth 1-parameter family of metrics on $M$,
$X(t)$ a smooth 1-parameter family of vector fields on $M$, and
$F:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz function. Suppose that
$u(\cdot,0)$ is bounded below by a constant $C_0\in\mathbb{R}$ and let
$\phi(t)$ be a solution to
\begin{equation*}
\dt \phi = F(\phi), \quad \phi(0)=C_0.
\end{equation*}
Then $u(x,t)\geq \phi(t)$ for all $x\in M$ and all $t\in[0,T)$ for
which $\phi(t)$ exists.
\end{prop}
Similarly, if (\ref{app.eq15}) is replaced by $\dt u \leq \triangle_{g(t)}u
+\scal{X(t),\nabla u}_{g(t)}+F(u)$ and $u(\cdot,0)$ is bounded from above
by $C_0$, then $u(x,t)\leq \phi(t)$ for all $x\in M$ and $t\in[0,T)$
for which the solution $\phi(t)$ of the corresponding ODE exists.
\newline
Using this, an immediate consequence of Corollary \ref{2.cor6} is
the following.
\begin{cor}\label{2.cor7}
Let $(g(t),\phi(t))$ be a solution to the $(RH)_\alpha$ flow with a
nonnegative, non-increasing coupling function $\alpha(t)$. Let
$S(t)=R(g(t))-\alpha(t)\abs{\nabla\phi(t)}^2_{g(t)}$ as above, with
initial data $S(0)>0$ on $M$. Then $R_{min}(t):=\min_{x\in M}R(x,t)
\to \infty$ in finite time and thus $g(t)$ must become singular in
finite time $T_{sing} \leq \frac{m}{2S_{min}(0)} < \infty$.
\end{cor}
\begin{proof}
Since $\alpha(t) \geq 0$ and $\dot{\alpha}(t) \leq 0$ for all $t\geq
0$, Corollary \ref{2.cor6} yields
\begin{equation}\label{2.eq18}
\dt S \geq \triangle S +2\abs{S_{ij}}^2 \geq \triangle S + \tfrac{2}{m}S^2
\end{equation}
and thus by comparing with solutions of the ODE $\tfrac{d}{dt} a(t)
= \frac{2}{m}a(t)^2$ which are
\begin{equation*}
a(t) = \frac{a(0)}{1-\tfrac{2t}{m}a(0)},
\end{equation*}
the maximum principle above, yields
\begin{equation}\label{2.eq19}
S_{min}(t) \geq \frac{S_{min}(0)}{1-\tfrac{2t}{m}S_{min}(0)}
\end{equation}
for all $t \geq 0$ as long as the flow exists. In particular, if
$S_{min}(0)>0$ this implies that $S_{min}(t)\to\infty$ in finite
time $T_0 \leq \frac{m}{2S_{min}(0)}<\infty$. Since
$R=S+\alpha\abs{\nabla\phi}^2\geq S$, we find that also $R_{min}(t)
\to\infty$ before $T_0$ and thus $g(t)$ has to become singular in
finite time $T_{sing}\leq T_0 \leq \frac{m}{2S_{min}(0)}<\infty$.
\end{proof}
As a second consequence, we see that if the energy density $e(\phi)
= \frac{1}{2}\abs{\nabla\phi}^2$ blows up at some point in space-time
while $\alpha(t)$ is bounded away from zero, then also $g(t)$ must
become singular at this point.
\begin{cor}\label{2.cor8}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a smooth solution of\/
$(RH)_\alpha$ with a non-increasing coupling function $\alpha(t)$
satisfying $\alpha(t)\geq \underaccent{\bar}\alpha >0$ for all
$t\in[0,T)$. Suppose that $\abs{\nabla\phi}^2(x_k,t_k) \to \infty$ for a
sequence $(x_k,t_k)_{k\in\mathbb{N}}$ with $t_k \nearrow T$. Then
also $R(x_k,t_k)\to \infty$ for this sequence and thus $g(t_k)$ must
become singular as $t_k$ approaches $T$.
\end{cor}
\begin{proof}
From (\ref{2.eq18}) we obtain $S\geq S_{min}(0)$ and thus
\begin{equation}\label{2.eq20}
R = \alpha\abs{\nabla\phi}^2 + S \geq
\underaccent{\bar}\alpha\abs{\nabla\phi}^2 + S_{min}(0), \quad \forall
(x,t)\in M\times[0,T).
\end{equation}
Hence, if $\abs{\nabla\phi}^2(x_k,t_k)\to\infty$ for a sequence
$(x_k,t_k)_{k\in\mathbb{N}}\subset M\times[0,T)$ with $t_k\nearrow
T$ then also $R(x_k,t_k)\to\infty$ for this sequence and $g(t_k)$
must become singular as $t_k\nearrow T$.
\end{proof}
\begin{rem}
The proof shows that Corollary \ref{2.cor8} stays true if $\alpha(t)
\searrow 0$ as $t \nearrow T$ as long as $\abs{\nabla\phi}^2(x_k,t_k)
\to \infty$ fast enough such that $\alpha(t_k)\abs{\nabla\phi}^2
(x_k,t_k) \to \infty$ still holds true.
\end{rem}
Now, we derive for $t>0$ an improved version of (\ref{2.eq20}) which
does not depend on the initial data $S(0)$. Using (\ref{2.eq18}) and
the maximum principle, we see that if $S_{min}(0)\geq C\in \mathbb{R}$ we
obtain
\begin{equation*}
S_{min}(t)\geq \frac{C}{1-\tfrac{2t}{m}C} \longrightarrow -
\frac{m}{2t} \qquad(C\to-\infty)
\end{equation*}
and thus $S(t) \geq -\frac{m}{2t}$ for all $t> 0$ as long as the
flow exists, independent of $S(0)$. More rigorously, this is
obtained as follows. The inequality (\ref{2.eq18}) implies
\begin{equation*}
\dt (tS) = S + t\big(\dt S\big) \geq \triangle(tS) +
S\big(1+\tfrac{2t}{m}S\big).
\end{equation*}
If $(x_0,t_0)$ is a point where $tS$ first reaches its minimum over
$M\times[0,T-\delta]$, $\delta>0$ arbitrarily small, we get
$S(x_0,t_0)\big(1+\tfrac{2t_0}{m}S(x_0,t_0))\leq 0$, which is only
possible for $t_0S(x_0,t_0)\geq -\tfrac{m}{2}$. Hence
$tS\geq-\frac{m}{2}$ on all of $M\times[0,T-\delta]$. Since $\delta$
was arbitrary, we obtain the desired inequality $S(t) \geq
-\frac{m}{2t}$ everywhere on $M\times(0,T)$. This yields
\begin{equation*}
R\geq \alpha\abs{\nabla\phi}^2 -\tfrac{m}{2t} \geq
\underaccent{\bar}\alpha\abs{\nabla\phi}^2 -\tfrac{m}{2t}, \quad \forall
(x,t)\in M\times(0,T),
\end{equation*}
which immediately implies the following converse of Corollary
\ref{2.cor8}.
\begin{cor}\label{2.cor9}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a smooth solution of\/
$(RH)_\alpha$ with a non-increasing coupling function $\alpha(t)\geq
\underaccent{\bar}\alpha>0$ for all\/ $t\in[0,T)$. Assume that
$R\leq R_0$ on $M\times[0,T)$. Then
\begin{equation}\label{2.eq21}
\abs{\nabla\phi}^2 \leq
\frac{R_0}{\underaccent{\bar}\alpha}+\frac{m}{2\underaccent{\bar}\alpha
t}, \quad \forall (x,t)\in M\times(0,T).
\end{equation}
\end{cor}
Singularities of the type as in Corollary \ref{2.cor8}, where the
energy density of $\phi$ blows up, can not only be ruled out if the
curvature of $M$ stays bounded. There is also a way to rule them out
\emph{a-priori}. Namely, such singularities cannot form if either $N$ has
non-positive sectional curvatures or if we choose the coupling
constants $\alpha(t)$ large enough such that
\begin{equation*}
\max_{y \in N}\!\!\phantom{l}^N\! K(y) \leq \frac{\alpha}{m}.
\end{equation*}
Here $\!\phantom{l}^N\! K$ denotes the sectional curvature of $N$. More precisely, we
have the following estimates for the energy density
$\abs{\nabla\phi}^2$.
\begin{prop}\label{2.prop10}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a solution of\/ $(RH)_\alpha$
with a non-increasing $\alpha(t) \geq 0$ and let
the sectional curvature of\/ $N$ be bounded above by $\!\!\phantom{l}^N\! K\leq
c_0$. Then
\begin{itemize}
\item[i)] if $N$ has non-positive sectional curvatures or more
generally if\/ $c_0 - \frac{\alpha(t)}{m} \leq 0$, the energy
density of $\phi$ is bounded by its initial data,
\begin{equation}\label{2.eq22}
\abs{\nabla\phi(x,t)}^2 \leq \max_{y\in M}\abs{\nabla\phi(y,0)}^2, \quad
\forall (x,t)\in M\times[0,T).
\end{equation}
\item[ii)] if $N$ has non-positive sectional curvatures and
$\alpha(t)\geq \underaccent{\bar}\alpha>0$, we have in addition to
(\ref{2.eq22}) the estimate
\begin{equation}\label{2.eq23}
\abs{\nabla\phi(x,t)}^2 \leq \frac{m}{2\underaccent{\bar}\alpha t},
\quad \forall (x,t)\in M\times(0,T).
\end{equation}
\item[iii)] in general, the energy density satisfies
\begin{equation}\label{2.eq24}
\abs{\nabla\phi(x,t)}^2 \leq 2\max_{y\in M}\abs{\nabla\phi(y,0)}^2, \quad
\forall (x,t)\in M\times[0,T^*),
\end{equation}
where $T^*:= \min\big\{T,\big(4c_0\max_{y\in
M}\abs{\nabla\phi(y,0)}^2\big)\!\!\phantom{.}^{-1}\big\}$.
\end{itemize}
\end{prop}
\begin{proof}
This is a consequence of the evolution equation (\ref{2.eq12}) and
the Cauchy-Schwarz inequality
\begin{equation}\label{2.eq25}
\tfrac{1}{m}\abs{\nabla\phi}^4 =
\tfrac{1}{m}\abs{g^{ij}\nabla_i\phi\nabla_j\phi}^2 \leq
\abs{\nabla_i\phi\nabla_j\phi}^2 \leq \abs{\nabla\phi}^4.
\end{equation}
\begin{itemize}
\item[i)] If $N$ has non-positive sectional curvatures,
$\Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi}\nabla_j\phi,\nabla_i\phi} \leq 0$, the
evolution equation (\ref{2.eq12}) implies
\begin{equation}\label{2.eq26}
\dt \abs{\nabla\phi}^2 \leq \triangle \abs{\nabla\phi}^2.
\end{equation}
If $c_0 -\frac{\alpha(t)}{m}\leq 0$, we have
$2\Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi}\nabla_j\phi,\nabla_i\phi} \leq
2c_0\abs{\nabla\phi}^4\leq 2\tfrac{\alpha}{m}\abs{\nabla\phi}^4\leq
2\alpha\abs{\nabla_i\phi\nabla_j\phi}^2$, and we get again (\ref{2.eq26})
from (\ref{2.eq12}). The claim now follows from the maximum
principle applied to (\ref{2.eq26}).
\item[ii)] If $\Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi}\nabla_j\phi,\nabla_i\phi} \leq
0$, (\ref{2.eq12}) and (\ref{2.eq25}) imply
\begin{equation*}
\dt\abs{\nabla\phi}^2\leq \triangle
\abs{\nabla\phi}^2-2\alpha\abs{\nabla_i\phi\nabla_j\phi}^2\leq
\triangle\abs{\nabla\phi}^2-2\tfrac{\alpha}{m}\abs{\nabla\phi}^4.
\end{equation*}
We obtain
\begin{equation*}
\dt \big(t\abs{\nabla\phi}^2\big)=\abs{\nabla\phi}^2 +t\big(\dt
\abs{\nabla\phi}^2\big) \leq \triangle\big(t\abs{\nabla\phi}^2\big) +
\abs{\nabla\phi}^2\big(1-2t\tfrac{\alpha}{m}\abs{\nabla\phi}^2\big).
\end{equation*}
At the first point $(x_0,t_0)$ where $t\abs{\nabla\phi}^2$ reaches its
maximum over $M\times[0,T-\delta]$, $\delta>0$ arbitrary, we find
$1-2t_0\frac{\alpha}{m}\abs{\nabla\phi}^2(x_0,t_0)\geq 0$, i.e.
\begin{equation*}
t_0\abs{\nabla\phi}^2(x_0,t_0) \leq \frac{m}{2\alpha}\leq
\frac{m}{2\underaccent{\bar}\alpha},
\end{equation*}
which implies that
$t\abs{\nabla\phi}^2\leq\frac{m}{2\underaccent{\bar}\alpha}$ for
every $(x,t)\in M\times[0,T-\delta]$. The claim follows.
\item[iii)] From (\ref{2.eq12}), we get
\begin{equation*}
\dt \abs{\nabla\phi}^2 \leq \triangle\abs{\nabla\phi}^2 + 2c_0\abs{\nabla\phi}^4.
\end{equation*}
By comparing with solutions of the ODE $\tfrac{d}{dt} a(t) = 2c_0
a(t)^2$, which are
\begin{equation*}
a(t) = \frac{a(0)}{1-2c_0a(0)t}, \quad t \leq \frac{1}{2c_0a(0)},
\end{equation*}
the maximum principle from Proposition \ref{app.prop1} implies
\begin{equation}\label{2.eq27}
\abs{\nabla\phi(x,t)}^2 \leq \frac{\max_{y\in M}\abs{\nabla\phi(y,0)}^2}{1
-2c_0\max_{y\in M}\abs{\nabla\phi(y,0)}^2\, t},
\end{equation}
for all $x\in M$ and $t\leq \min\big\{T,\big(2c_0\max_{y\in
M}\abs{\nabla\phi(y,0)}^2\big)\!\!\phantom{.}^{-1}\big\}$. In
particular, this proves the doubling-time estimate that we
claimed.\qedhere
\end{itemize}
\end{proof}
\section{Gradient estimates and long-time existence}
For solutions $(g(t),\phi(t))$ of the $(RH)_\alpha$ flow with
non-increasing $\alpha(t)\geq\underaccent{\bar}\alpha>0$, we have
seen in Corollary \ref{2.cor9} that a uniform bound on the curvature
of $(M,g(t))$ implies a uniform bound on $\abs{\nabla\phi}^2$.
Therefore, one expects that a uniform curvature bound suffices to
show long-time existence for our flow. The proof of this result is
the main goal of this section.
\subsection{Evolution equations for $\mathrm{Rm}$ and $\nabla^2\phi$}
With $\dt g_{ij} = h_{ij} :=
-2R_{ij}+2\alpha\nabla_i\phi^\mu\nabla_j\phi^\mu$, we find the evolution
equation for the Christoffel symbols
\begin{equation}\label{3.eq1}
\begin{aligned}
\dt\Gamma_{ij}^p &=\tfrac{1}{2}g^{pq}(\nabla_ih_{jq}+\nabla_jh_{iq}
-\nabla_qh_{ij})\\%
&=g^{pq}(-\nabla_iR_{jq}-\nabla_jR_{iq}+\nabla_qR_{ij})
+2\alpha\nabla_i\nabla_j\phi\nabla^p\phi
\end{aligned}
\end{equation}
With this, an elementary computation yields the following evolution equation
for the Riemannian curvature tensor (see \cite[Proposition 3.2]{Muller:diss}
for a detailed proof).
\begin{prop}\label{3.prop2}
Let $(g(t),\phi(t))_{t\in [0,T)}$ be a solution of\/ $(RH)_\alpha$.
Then the Riemann tensor satisfies
\begin{equation}\label{3.eq4}
\begin{aligned}
\dt R_{ijk\ell} &= \nabla_i\nabla_kR_{j\ell} -\nabla_i\nabla_\ell R_{jk}
-\nabla_j\nabla_kR_{i\ell} + \nabla_j\nabla_\ell R_{ik}-R_{ijq\ell}R_{kq}
-R_{ijkq}R_{\ell q}\\%
&\quad\,+2\alpha\big(\nabla_i\nabla_k\phi\nabla_j\nabla_\ell\phi
-\nabla_i\nabla_\ell\phi\nabla_j\nabla_k\phi -\Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi}
\nabla_k\phi,\nabla_\ell\phi}\big).
\end{aligned}
\end{equation}
\end{prop}
\begin{rem}
Taking the trace of (\ref{3.eq4}), we obtain (\ref{2.eq10}), using
the twice traced second Bianchi identity. This gives
an alternative proof of Proposition \ref{2.prop3}.
\end{rem}
If we set $\alpha = 0$ in (\ref{3.eq4}), we obtain the evolution
equation for the curvature tensor under the Ricci flow. It is
well-known that this evolution equation can be written in a nicer
form, in which its parabolic nature is more apparent. In \cite[Lemma
7.2]{Hamilton:3folds}, Hamilton proved
\begin{equation}\label{3.eq5}
\begin{aligned}
\nabla_i\nabla_kR_{j\ell}-\nabla_i&\nabla_\ell R_{jk}-\nabla_j\nabla_kR_{i\ell}+\nabla_j\nabla_\ell
R_{ik}\\%
&= \triangle R_{ijk\ell}+2(B_{ijk\ell}-B_{ij\ell k}-B_{i\ell jk}
+B_{ikj\ell}) -R_{pjk\ell}R_{pi}-R_{ipk\ell}R_{pj},
\end{aligned}
\end{equation}
where $B_{ijk\ell} := R_{ipjq}R_{kp\ell q}$. Plugging this into
(\ref{3.eq4}) yields the following corollary.
\begin{cor}\label{3.cor3}
Along the $(RH)_\alpha$ flow, the Riemannian curvature tensor
evolves by
\begin{equation}\label{3.eq6}
\begin{aligned}
\dt R_{ijk\ell} &= \triangle R_{ijk\ell} + 2(B_{ijk\ell}-B_{ij\ell
k}-B_{i\ell jk} +B_{ikj\ell})\\%
&\quad -(R_{pjk\ell}R_{pi}+R_{ipk\ell}R_{pj}
+R_{ijp\ell}R_{pk}+R_{ijkp}R_{p\ell})\\%
&\quad +2\alpha\big(\nabla_i\nabla_k\phi\nabla_j\nabla_\ell\phi-
\nabla_i\nabla_\ell\phi\nabla_j\nabla_k\phi -\Scal{\!\NRm{\nabla_i\phi,\nabla_j\phi}
\nabla_k\phi,\nabla_\ell\phi}\big).
\end{aligned}
\end{equation}
\end{cor}
There is a useful convention for writing such equations in a short
form.
\begin{defn}\label{3.defn1}
For two quantities $A$ and $B$, we denote by $A*B$ any quantity
obtained from $A \otimes B$ by summation over pairs of matching
(Latin and Greek) indices, contractions with the metrics $g$ and
$\gamma$ and their inverses, and multiplication with constants
depending only on $m=\dim M$, $n=\dim N$ and the ranks of $A$ and
$B$. We also write $(A)^{*1}:=1*A$, $(A)^{*2}=A*A$, etc.
\end{defn}
This notation allows us to write (\ref{3.eq1}) and (\ref{3.eq6}) in
the short forms
\begin{equation}\label{3.eq3}
\dt \Gamma = (\nabla\Rm{})^{*1} + \alpha\nabla^2\phi*\nabla\phi.
\end{equation}
and
\begin{equation}\label{3.eq7}
\dt \Rm{} = \triangle \Rm{} + (\Rm{})^{*2} + \alpha (\nabla^2\phi)^{*2} +
\alpha \NRm{}*(\nabla\phi)^{*4}.
\end{equation}
It is now easy to compute the evolution of the length of the Riemann
tensor. Together with $\dt g^{-1} = (\Rm{})^{*1}+
\alpha(\nabla\phi)^{*2}$, the above formula yields
\begin{equation}\label{3.eq8}
\begin{aligned}
\dt \abs{\Rm{}}^2 &= \big(\dt g^{-1}\big)*\Rm{}*\Rm{}
+ 2R_{ijk\ell}\big(\dt R_{ijk\ell}\big)\\%
&= \triangle \abs{\Rm{}}^2 - 2\abs{\nabla\Rm{}}^2 + (\Rm{})^{*3} +
\alpha(\Rm{})^{*2}*(\nabla\phi)^{*2}\\%
&\quad + \alpha \Rm{}*(\nabla^2\phi)^{*2} + \alpha
\Rm{}*\NRm{}*(\nabla\phi)^{*4}.
\end{aligned}
\end{equation}
\begin{cor}\label{3.cor4}
Along the $(RH)_\alpha$ flow, the Riemannian curvature tensor
satisfies
\begin{equation}\label{3.eq9}
\begin{aligned}
\dt \abs{\Rm{}}^2 &\leq \triangle \abs{\Rm{}}^2 - 2\abs{\nabla\Rm{}}^2
+C \abs{\Rm{}}^3 + \alpha C \abs{\nabla\phi}^2\abs{\Rm{}}^2\\%
&\quad + \alpha C \abs{\nabla^2\phi}^2\abs{\Rm{}} + \alpha Cc_0
\abs{\nabla\phi}^4\abs{\Rm{}},
\end{aligned}
\end{equation}
for constants $C\geq 0$ depending only on the dimension of $M$ and
$c_0=c_0(N)\geq 0$ depending only on the curvature of $N$. If $N$ is
flat, we can choose $c_0 = 0$.
\end{cor}
\begin{proof}
Follows directly from (\ref{3.eq8}) and the fact that $\abs{\NRm{}}$
is bounded on compact $N$.
\end{proof}
For the evolution equation for the Hessian of $\phi$, it is
important that we do not use the $*$-notation directly. Indeed, we
will see that all the terms containing derivatives of the curvature
of $M$ cancel each other (using the second Bianchi identity), a
phenomenon which cannot be seen when working with the $*$-notation.
\newline
A short computation using (\ref{app.eq7}) and (\ref{app.eq8}) shows that
the commutator $[\nabla_i\nabla_j,\triangle]\phi^\lambda =
\nabla_i\nabla_j\tau_g\phi^\lambda-\triangle\nabla_i\nabla_j\phi^\lambda$ is given by
\begin{equation}\label{3.eq10}
\begin{aligned}
{[\nabla_i\nabla_j,\triangle]}\phi^\lambda &= \nabla_kR_{jpik}\nabla_p\phi^\lambda
+2R_{ikjp}\nabla_k\nabla_p\phi^\lambda\\%
&\quad\,-R_{ip}\nabla_j\nabla_p\phi^\lambda -\nabla_iR_{jp}\nabla_p\phi^\lambda
-R_{jp}\nabla_i\nabla_p\phi^\lambda\\%
&\quad\,+\big(\NRm{}*\nabla^2\phi*(\nabla\phi)^{*2} +
(\partial\NRm{})*(\nabla\phi)^{*4}\big)_{ij},
\end{aligned}
\end{equation}
see \cite[equation (3.10)]{Muller:diss} for details. With (\ref{3.eq1}) we continue
\begin{align*}
[\nabla_i\nabla_j,\triangle]\phi^\lambda -
\big(\dt\Gamma_{ij}^k\big)\nabla_k\phi^\lambda
&=\big(\Rm{}*\nabla^2\phi^\lambda\big)_{ij}
-2\alpha\nabla_i\nabla_j\phi\nabla_k\phi\nabla_k\phi^\lambda\\%
&\quad\,+\big(\NRm{}*\nabla^2\phi*(\nabla\phi)^{*2} +
(\partial\NRm{})*(\nabla\phi)^{*4}\big)_{ij},
\end{align*}
where we used the second Bianchi identity
$(\nabla_kR_{jpik}+\nabla_jR_{ip}-\nabla_pR_{ij})\nabla_p\phi^\lambda=0$ to cancel
all terms containing derivatives of the curvature of $(M,g)$. Since
the $\nabla^2\phi$ live in different bundles for different times, we
work again with the covariant time derivative $\nabla_t$ (and with
the interpretation of $\nabla^2\phi$ as a $2$-linear $TN$-valued map
along $\tilde{\phi}$), as we already did in Section 4, see appendix for
details. At the base point of coordinates satisfying
(\ref{app.eq13}), we find with (\ref{app.eq14}) and the remark
following it
\begin{equation}\label{3.eq11}
\begin{aligned}
\nabla_t(\nabla_i\nabla_j\phi^\lambda) &=\nabla_i\nabla_j\dt\phi^\lambda
-\big(\dt\Gamma_{ij}^k\big)\nabla_k\phi^\lambda +
\NRm{\dt\phi,\nabla_i\phi}\nabla_j\phi^\lambda\\%
&= \triangle\nabla_i\nabla_j\phi^\lambda +
(\Rm{}*\nabla^2\phi^\lambda)_{ij} + \alpha \nabla_i\nabla_j\phi*\nabla\phi*\nabla\phi^\lambda\\%
&\quad\,+\big(\NRm{}*\nabla^2\phi*(\nabla\phi)^{*2} +
(\partial\NRm{})*(\nabla\phi)^{*4}\big)_{ij}.
\end{aligned}
\end{equation}
With $\triangle\abs{\nabla^2\phi}^2
=2\triangle(\nabla_i\nabla_j\phi^\lambda)\nabla_i\nabla_j\phi^\lambda
+2\abs{\nabla^3\phi}^2$, we finally obtain
\begin{equation}\label{3.eq12}
\begin{aligned}
\dt\abs{\nabla^2\phi}^2 &=(\dt g^{-1})*(\nabla^2\phi)^{*2}
+2\nabla_t(\nabla_i\nabla_j\phi^\lambda)\nabla_i\nabla_j\phi^\lambda\\%
&= \Rm{}*(\nabla^2\phi)^{*2}+\alpha(\nabla\phi)^{*2}*(\nabla^2\phi)^{*2}
+\triangle\abs{\nabla^2\phi}^2-2\abs{\nabla^3\phi}^2\\%
&\quad\,+\NRm{}*(\nabla^2\phi)^{*2}*(\nabla\phi)^{*2} +
(\partial\NRm{})*(\nabla^2\phi)*(\nabla\phi)^{*4}
\end{aligned}
\end{equation}
Since $\abs{\NRm{}}$ and $\abs{\partial \NRm{}}$ are bounded on
compact manifolds $N$, say by a constant $c_1$, this proves the
following proposition.
\begin{prop}\label{3.prop5}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a solution of\/ $(RH)_\alpha$.
Then the norm of the Hessian of $\phi$ satisfies the estimate
\begin{equation}\label{3.eq13}
\begin{aligned}
\dt\abs{\nabla^2\phi}^2 &\leq \triangle\abs{\nabla^2\phi}^2 -2\abs{\nabla^3\phi}^2
+C\abs{\Rm{}}\abs{\nabla^2\phi}^2\\%
&\quad + \alpha C\abs{\nabla\phi}^2\abs{\nabla^2\phi}^2
+Cc_1\abs{\nabla\phi}^4\abs{\nabla^2\phi} + Cc_1
\abs{\nabla\phi}^2\abs{\nabla^2\phi}^2
\end{aligned}
\end{equation}
along the flow for some constants $C=C(m)\geq 0$ and $c_1=c_1(N)\geq
0$ depending on the dimension $m$ of $M$ and the curvature of $N$,
respectively. If $N$ is flat, we may choose $c_1=0$.
\end{prop}
\begin{rem}
If we set $\alpha \equiv 0$, Corollary \ref{3.cor4} and Proposition
\ref{3.prop5} yield the formulas for the Ricci-DeTurck flow
$(RH)_0$, in particular (\ref{3.eq9}) reduces to the well-known
evolution inequality
\begin{equation*}
\dt \abs{\Rm{}}^2 \leq \triangle
\abs{\Rm{}}^2-2\abs{\nabla\Rm{}}^2+C\abs{\Rm{}}^3
\end{equation*}
for the Ricci flow, first derived by Hamilton \cite[Corollary
13.3]{Hamilton:3folds}. Moreover, if $\alpha \equiv 2$ and $N\subseteq\mathbb{R}$
(and thus $c_0=c_1=0$), the estimates (\ref{3.eq9}) and
(\ref{3.eq13}) reduce to the estimates found by List (cf.
\cite[Lemma 2.15 and 2.16]{List:diss}).
\end{rem}
\subsection{Interior-in-time higher order gradient estimates}
Using the evolution equations for the curvature tensor and the
Hessian of $\phi$, we get evolution equations for higher order
derivatives by induction.
\begin{defn}\label{3.defn6}
To keep the notation short, we define for $k\geq 0$
\begin{equation}\label{3.eq14}
\begin{aligned}
I_k &:=\sum_{i+j=k}\nabla^i\Rm{}*\nabla^j\Rm{} +
\alpha\sum_{A_k}(\partial^i\NRm{}+1)*\nabla^{j_1}\phi*\ldots*\nabla^{j_\ell}\phi\\%
&\quad\;+\alpha\sum_{B_k}\nabla^{j_1}\phi*\ldots*
\nabla^{j_{\ell-1}}\phi*\nabla^{j_\ell}\Rm{},
\end{aligned}
\end{equation}
where the last two sums are taken over all elements of the index
sets defined by
\begin{align*}
A_k &:= \{(i,j_1,\ldots,j_\ell)\mid 0\leq i\leq k+1,\, 1\leq
j_s\leq k+2\;\forall s \textrm{ and } j_1+\ldots +j_\ell=k+4\},\\%
B_k &:= \{(j_1,\dots,j_\ell)\mid 1\leq j_s< k+2\;\forall s<\ell,\,
0\leq j_\ell\leq k \textrm{ and } j_1+\ldots + j_\ell=k+2\}.
\end{align*}
\end{defn}
\begin{lemma}\label{3.lamma7}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a solution to the $(RH)_\alpha$
flow. Then for $k\geq 0$, with $I_k$ defined as in (\ref{3.eq14}),
we obtain
\begin{equation}\label{3.eq15}
\dt \nabla^k \Rm{} = \triangle \nabla^k\Rm{} + I_k.
\end{equation}
\end{lemma}
\begin{rem}
If $(N,\gamma)=(\mathbb{R},\delta)$, all terms in $I_k$ containing
$\partial^i\NRm{}$ vanish, and the result reduces to (a slightly
weaker version of) List's result \cite[Lemma 2.19]{List:diss}. Note
that we do not need all the elements of $A_k$ here, but defining
$A_k$ this way allows us to use the same index set again in
Definition \ref{3.defn8}.
\end{rem}
\begin{proof}
From (\ref{3.eq7}), we see that (\ref{3.eq15}) holds for $k=0$. For
the induction step, assume that (\ref{3.eq15}) holds for some
$k\geq0$ and compute
\begin{equation*}
\dt\nabla^{k+1}\Rm{}=\dt(\partial\nabla^k\Rm{}+\Gamma*\nabla^k\Rm{})
=\nabla(\triangle\nabla^k\Rm{}) +\nabla I_k +\dt\Gamma*\nabla^k\Rm{}.
\end{equation*}
Since $\nabla I_k$ is of the form $I_{k+1}$ and also
$\dt\Gamma*\nabla^k\Rm{}=\big(\nabla\Rm{}+\alpha\nabla^2\phi*\nabla\phi\big)*\nabla^k\Rm{}$
appears in $I_{k+1}$, it remains to compute the very first term.
With the commutator rule (\ref{app.eq7}), we get
\begin{align*}
\nabla(\triangle\nabla^k\Rm{})&=\triangle\nabla^{k+1}\Rm{}+\nabla\Rm{}*\nabla^k\Rm{}+\Rm{}
*\nabla^{k+1}\Rm{}\\%
&=\triangle\nabla^{k+1}\Rm{}+I_{k+1}.\qedhere
\end{align*}
\end{proof}
Similar to (\ref{3.eq8}), we obtain
\begin{align*}
\dt\abs{\nabla^k\Rm{}}^2 &= \big(\dt g^{-1}\big)*\nabla^k\Rm{}*\nabla^k\Rm{}
+2\nabla^k\Rm{}\big(\dt\nabla^k\Rm{}\big)\\%
&=\Rm{}*(\nabla^k\Rm{})^{*2}+\alpha\,(\nabla\phi)^{*2}*(\nabla^k\Rm{})^{*2}
+2\nabla^k\Rm{}(\triangle\nabla^k\Rm{})+\nabla^k\Rm{}*I_k.
\end{align*}
Hence, using the fact that $\Rm{}*\nabla^k\Rm{}$ as well as
$\alpha\,(\nabla\phi)^{*2}*\nabla^k\Rm{}$ are already contained in $I_k$, we
find
\begin{equation}\label{3.eq16}
\dt\abs{\nabla^k\Rm{}}^2=\triangle\abs{\nabla^k\Rm{}}^2
-2\abs{\nabla^{k+1}\Rm{}}^2+\nabla^k\Rm{}*I_k.
\end{equation}
\begin{defn}\label{3.defn8}
To compute the higher order derivatives of $\phi$, we define
\begin{equation}\label{3.eq17}
\begin{aligned}
J_k &:=\sum_{i+j=k}\nabla^i\Rm{}*\nabla^{j+2}\phi +
\sum_{A_k}(\partial^i\NRm{}+1)*\nabla^{j_1}\phi*\ldots*\nabla^{j_\ell}\phi\\%
&\quad\;+\alpha\sum_{B_k}\nabla^{j_1}\phi*\ldots*
\nabla^{j_{\ell-1}}\phi*\nabla^{j_\ell+2}\phi,
\end{aligned}
\end{equation}
with $A_k$ and $B_k$ defined as in Definition \ref{3.defn6}.
\end{defn}
\begin{lemma}\label{3.lemma9}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a solution to the $(RH)_\alpha$
flow. Then for $k\geq 0$, with $J_k$ defined as in (\ref{3.eq17}),
we have
\begin{equation}\label{3.eq18}
\nabla_t(\nabla^{k+2}\phi) = \triangle \nabla^{k+2} \phi + J_k.
\end{equation}
\end{lemma}
\begin{proof}
For $k=0$, the statement holds by (\ref{3.eq11}). For the induction
step, we use again the interpretation of $\nabla^k\phi$ as a $k$-linear
$TN$-valued map along $\tilde{\phi}$ and compute analogously to
(\ref{app.eq14}) and the remark following it
\begin{align*}
\nabla_t(\nabla^{k+3}\phi) &=\nabla\nabla_t(\nabla^{k+2}\phi) +
\dt\Gamma*\nabla^{k+2}\phi + \NRm{\dt\phi,\nabla\phi}\nabla^{k+2}\phi\\
&= \nabla(\triangle\nabla^{k+2}\phi)+\nabla J_k +\dt\Gamma*\nabla^{k+2}\phi +
\NRm{}*\nabla^2\phi*\nabla\phi*\nabla^{k+2}\phi.
\end{align*}
Again, we only have to look at the first term, since $\nabla J_k$,
$\NRm{}*\nabla^2\phi*\nabla\phi*\nabla^{k+2}\phi$ and $\dt\Gamma*\nabla^{k+2}\phi =
\big(\nabla\Rm{}+\alpha\nabla^2\phi*\nabla\phi\big)
*\nabla^{k+2}\phi$ are obviously of the form $J_{k+1}$. With a higher
order analog to (\ref{app.eq10}), we obtain
\begin{align*}
\nabla(\triangle\nabla^{k+2}\phi)&=\nabla_p\nabla\nabla_p\nabla^{k+2}\phi +\Rm{}*\nabla^{k+3}\phi +
\NRm{}*\nabla^{k+3}\phi*\nabla\phi*\nabla\phi\\%
&=\nabla_p\nabla_p\nabla^{k+3}\phi + \nabla\Rm{}*\nabla^{k+2}\phi+ \Rm{}*\nabla^{k+3}\phi\\%
&\quad\,+(\partial \NRm{})*\nabla^{k+2}\phi*(\nabla\phi)^{*3} +
\NRm{}*\nabla^{k+3}\phi*(\nabla\phi)^{*2}\\%
&\quad\,+\NRm{}*\nabla^{k+3}\phi*\nabla^2\phi*\nabla\phi\\%
&=\triangle\nabla^{k+3}\phi+J_{k+1},
\end{align*}
and the claim follows.
\end{proof}
As in (\ref{3.eq12}), we compute
\begin{align*}
\dt\abs{\nabla^{k+2}\phi}^2&=\big(\dt g^{-1}\big)
*\nabla^{k+2}\phi*\nabla^{k+2}\phi+2\nabla^{k+2}\phi^\lambda
\nabla_t\big(\nabla^{k+2}\phi^\lambda\big)\\
&= \Rm{}*(\nabla^{k+2}\phi)^{*2}+\alpha\,
(\nabla\phi)^{*2}*(\nabla^{k+2}\phi)^{*2}\\
&\quad+2\nabla^{k+2}\phi^\lambda(\triangle\nabla^{k+2}\phi^\lambda) +
\nabla^{k+2}\phi*J_k\\
&= 2\nabla^{k+2}\phi^\lambda(\triangle\nabla^{k+2}\phi^\lambda)+\nabla^{k+2}\phi*J_k.
\end{align*}
With $\triangle\abs{\nabla^{k+2}\phi}^2=2\nabla^{k+2}\phi^\lambda(\triangle\nabla^{k+2}\phi^\lambda)
+2\abs{\nabla^{k+3}\phi}^2$, we finally find
\begin{equation}\label{3.eq19}
\dt\abs{\nabla^{k+2}\phi}^2=\triangle\abs{\nabla^{k+2}\phi}^2-2\abs{\nabla^{k+3}\phi}^2
+ \nabla^{k+2}\phi*J_k.
\end{equation}
The next trick is to combine the two equations (\ref{3.eq16}) and
(\ref{3.eq19}) to a single equation. Remember that we already used a
similar idea in Section 4, where we combined the evolution
equations of $\Rc{}$ and $\nabla\phi\otimes\nabla\phi$ (respectively $R$ and
$\abs{\nabla\phi}^2$) to a single equation for a combined quantity
$S_{ij}$ (respectively $S$), which was much more convenient to deal
with. Here, we define the ``vector''
\begin{equation}\label{3.eq20}
\mathcal{T}=(\Rm{},\nabla^2\phi) \in \Gamma\big((T^*M)^{\otimes 4}\big) \times
\Gamma\big((T^*M)^{\otimes 2}\otimes\phi^*TN\big)
\end{equation}
with norm $\abs{\mathcal{T}}^2 = \abs{\Rm{}}^2 + \abs{\nabla^2\phi}^2$ and
derivatives $\nabla^k\mathcal{T}=(\nabla^k\Rm{},\nabla^{k+2}\phi)$. Combining the
evolution equations (\ref{3.eq16}) and (\ref{3.eq19}), we get
\begin{equation}\label{3.eq21}
\dt\abs{\nabla^k\mathcal{T}}^2 = \triangle\abs{\nabla^k\mathcal{T}}^2-2\abs{\nabla^{k+1}\mathcal{T}}^2 + L_k,
\end{equation}
where $L_k := \nabla^k\Rm{}*I_k + \nabla^{k+2}\phi*J_k$. We can now apply
Bernstein's ideas \cite{Bernstein:estimates} to obtain
interior-in-time estimates for all derivatives $\abs{\nabla^k\mathcal{T}}^2$ via
an induction argument. For the Ricci flow, this was independently
done by Bando \cite{Bando:estimates} and Shi \cite{Shi:complete}.
\begin{thm}\label{3.thm10}
Let $(g(t),\phi(t))_{t\in[0,T)}$ solve $(RH)_\alpha$ with
non-increasing $\alpha(t)\in[\underaccent{\bar}\alpha,\bar\alpha]$,
$0<\underaccent{\bar}\alpha\leq\bar\alpha<\infty$ and $T<\infty$.
Let the Riemannian curvature tensor of $M$ be uniformly bounded
along the flow, $\abs{\Rm{}}\leq R_0$. Then there exists a constant
$K=K(\underaccent{\bar}\alpha,\bar\alpha,R_0,T,m,N)<\infty$ such
that the following two estimates hold
\begin{align}
\abs{\nabla\phi}^2 &\leq \tfrac{K}{t}, \quad \forall (x,t)\in
M\times(0,T),\label{3.eq22}\\
\abs{\mathcal{T}}^2 &=\abs{\Rm{}}^2+\abs{\nabla^2\phi}^2 \leq \tfrac{K^2}{t^2},
\quad \forall (x,t)\in M\times(0,T).\label{3.eq23}
\end{align}
Moreover, there exist constants $C_k$ depending on $k$,
$\bar\alpha$, $m$ and $N$, such that
\begin{equation}\label{3.eq24}
\abs{\nabla^k\mathcal{T}}^2 = \abs{\nabla^k\Rm{}}^2+\abs{\nabla^{k+2}\phi}^2 \leq C_k
\big(\tfrac{K}{t}\big)^{k+2}, \quad\forall (x,t)\in M\times(0,T).
\end{equation}
\end{thm}
\begin{proof}
Since the method of proof is quite standard, we only give a brief sketch of the argument and
refer to the authors thesis \cite[Theorem 3.10]{Muller:diss} for more details. Setting $1\leq K_1 :=
\max\big\{\frac{2m^2R_0T+m}{2\underaccent{\bar}\alpha}, R_0T,
1\big\}<\infty$, we obtain
\begin{equation*}
\abs{\nabla\phi}^2\leq \frac{m^2R_0}{\underaccent{\bar}\alpha}
+\frac{m}{2\underaccent{\bar}\alpha t} \leq \frac{K_1}{t} \quad
\textrm{ and }\quad \abs{\Rm{}}\leq \frac{K_1}{t}, \quad \forall
(x,t)\in M\times(0,T)
\end{equation*}
from Corollary \ref{2.cor9}. In the following, $C$ denotes a
constant depending on $K_1$, $\bar\alpha$, $m$ and the geometry
of $N$, possibly changing from line to line. With the estimates for
$\abs{\Rm{}}$ and $\abs{\nabla\phi}^2$, and using $\abs{\nabla^2\phi}\leq
\frac{1}{t} + t\abs{\nabla^2\phi}^2$, we obtain for $f(x,t):=
t^2\abs{\nabla^2\phi}^2\big(8K_1 + t\abs{\nabla\phi}^2\big)$
\begin{align*}
\big(\dt-\triangle\big)f
&\leq -2t^2\abs{\nabla^3\phi}^2\big(8K_1+t\abs{\nabla\phi}^2\big) +
\tfrac{C}{t}f + \tfrac{C}{t}\cdot 9K_1 -2t^3\abs{\nabla^2\phi}^4 +
\tfrac{C}{t}f\\
&\quad\, + 8t^3\abs{\nabla^3\phi}\abs{\nabla^2\phi}
\cdot\abs{\nabla^2\phi}\abs{\nabla\phi}
\end{align*}
on $M\times(0,T)$. The last term can be absorbed by the two negative terms,
\begin{align*}
8t^3\abs{\nabla^3\phi}\abs{\nabla^2\phi}^2\abs{\nabla\phi}
&\leq \tfrac{1}{2}(8K_1)\big(4t^2\abs{\nabla^3\phi}^2\big) +
\tfrac{1}{2}(8K_1)^{-1}\big(16t^4\abs{\nabla^2\phi}^4\abs{\nabla\phi}^2\big)\\
&=2t^2\abs{\nabla^3\phi}^2\cdot 8K_1 +
\tfrac{8t\abs{\nabla\phi}^2}{8K_1}\cdot t^3\abs{\nabla^2\phi}^4\\
&\leq 2t^2\abs{\nabla^3\phi}^2\big(8K_1+t\abs{\nabla\phi}^2\big) +
t^3\abs{\nabla^2\phi}^4.
\end{align*}
Here, we used $\tfrac{8t\abs{\nabla\phi}^2}{8K_1}\leq 1$ which motivates
our choice of the constant $8K_1$ in the definition of $f$. From
$\big(\dt-\triangle\big)f \leq\tfrac{C}{t}f+\tfrac{C}{t}-t^3\abs{\nabla^2\phi}^4 \leq
\tfrac{1}{(9K_1)^2t}\big(Cf + C-f^2\big)$, we conclude, using $f(\cdot,0)=0$ and the maximum principle, that $-f^2+Cf+C\geq 0$. Equivalently, $f \leq D:=\frac{1}{2}\big(C + \sqrt{C^2+4C}\big)$ on $M\times[0,T)$. For
positive $t$, this implies
\begin{equation}\label{3.eq28}
\abs{\nabla^2\phi}^2 = \frac{f}{t^2(8K_1+t\abs{\nabla\phi}^2)} \leq
\frac{D}{8K_1t^2}\leq \Big(\frac{K_2}{t}\Big)^2,
\end{equation}
where $K_2:= \sqrt{D/8K_1}<\infty$. Setting $K:=K_1+K_2$, we get
(\ref{3.eq22}) and (\ref{3.eq23}). Using a similar argument, one can then prove (\ref{3.eq24}) inductively. The crucial estimates are $L_k \leq C\big(\tfrac{K}{t}\big)^{k+3}$ and
\begin{equation*}
L_{k+1} \leq C\big(\tfrac{K}{t}\big)^{k+4}+C\tfrac{K}{t}\abs{\nabla^{k+1}\mathcal{T}}^2,
\end{equation*}
where $C$ now denotes a constant depending only on $\bar\alpha$, $m$, $N$ and $k$
(but not on $K$ or $T$). Defining $h(x,t):=t^{k+3}\abs{\nabla^{k+1}\mathcal{T}}^2\big(\lambda +
t^{k+2}\abs{\nabla^k\mathcal{T}}^2\big)$ with $\lambda = 8C_kK^{k+2}$, these estimates give
\begin{align*}
\big(\dt-\triangle\big)h &\leq -2t^{k+3}\abs{\nabla^{k+2}\mathcal{T}}^2\big(\lambda + t^{k+2}
\abs{\nabla^k\mathcal{T}}^2\big)+\tfrac{CK}{t}h+\tfrac{C}{t}K^{k+4}\big(\lambda
+ t^{k+2}\abs{\nabla^k\mathcal{T}}^2\big)\\
&\quad\,-2t^{2k+5}\abs{\nabla^{k+1}\mathcal{T}}^4+\tfrac{C}{t}h+\tfrac{C}{t}K^{k+3}
t^{k+3}\abs{\nabla^{k+1}\mathcal{T}}^2\\
&\quad\,+8t^{2k+5}\abs{\nabla^{k+2}\mathcal{T}}\abs{\nabla^{k+1}\mathcal{T}}\cdot
\abs{\nabla^{k+1}\mathcal{T}}\abs{\nabla^k\mathcal{T}}.
\end{align*}
Using $K\geq 1$, the inductive assumption and Cauchy-Schwarz, we rewrite this as
\begin{align*}
\big(\dt-\triangle\big)h &\leq -2t^{k+3}\abs{\nabla^{k+2}\mathcal{T}}^2\big(\lambda +
t^{k+2} \abs{\nabla^k\mathcal{T}}^2\big) -\tfrac{3}{2}t^{2k+5}
\abs{\nabla^{k+1}\mathcal{T}}^4\\
&\quad\,+\tfrac{CK}{t}h+\tfrac{C}{t}K^{2k+6}
+8t^{2k+5}\abs{\nabla^{k+2}\mathcal{T}}\abs{\nabla^{k+1}\mathcal{T}}^2\abs{\nabla^k\mathcal{T}}.
\end{align*}
Again, the last term can be absorbed by the negative terms
\begin{align*}
8t^{2k+5}\abs{\nabla^{k+2}\mathcal{T}}\abs{\nabla^{k+1}\mathcal{T}}^2\abs{\nabla^k\mathcal{T}}
&\leq\tfrac{1}{2}\lambda\big(4t^{k+3}\abs{\nabla^{k+2}\mathcal{T}}^2\big) +
\tfrac{1}{2}\lambda^{-1}\big(16t^{3k+7}\abs{\nabla^{k+1}\mathcal{T}}^4
\abs{\nabla^k\mathcal{T}}^2\big)\\
&=2t^{k+3}\abs{\nabla^{k+2}\mathcal{T}}^2\cdot \lambda +
\tfrac{8t^{k+2}\abs{\nabla^k\mathcal{T}}^2}{\lambda}\cdot
t^{2k+5}\abs{\nabla^{k+1}\mathcal{T}}^4\\
&\leq 2t^{k+3}\abs{\nabla^{k+2}\mathcal{T}}^2\big(\lambda
+t^{k+2}\abs{\nabla^k\mathcal{T}}^2\big)+t^{2k+5}\abs{\nabla^{k+1}\mathcal{T}}^4,
\end{align*}
which explains our choice of $\lambda$. A maximum principle argument like the one for $f$ above then yields $-h^2+CK^{2k+5}h+CK^{4k+10} \geq 0$, i.e.~$h \leq\tfrac{1}{2}(C+\sqrt{C^2+4C})K^{2k+5}=:DK^{2k+5}$ on $M\times[0,T)$. For $t>0$,
\begin{equation*}
\abs{\nabla^{k+1}\mathcal{T}}^2 =
\frac{h}{t^{k+3}(\lambda+t^{k+2}\abs{\nabla^k\mathcal{T}}^2)}\leq
\frac{DK^{2k+5}}{t^{k+3}8C_kK^{k+2}} =
C_{k+1}\Big(\frac{K}{t}\Big)^{k+3},
\end{equation*}
where $C_{k+1}:=D/(8C_k)$. This proves the induction step and hence
the theorem.
\end{proof}
In the following corollary, we state a local version of the gradient estimates.
The setting is made in such a way to perfectly fit the proof of the non-collapsing
result in Section 8.
\begin{cor}\label{app.prop5}
Let $(g(t),\phi(t))_{t\in[0,T)}$ solve $(RH)_\alpha$ with
non-increasing $\alpha(t)\in[\underaccent{\bar}\alpha,\bar\alpha]$,
$0<\underaccent{\bar}\alpha\leq\bar\alpha<\infty$ and $T'<T<\infty$.
Let $B:=B_{g(T')}(x,r)$ be a ball around $x$ with radius $r$,
measured with respect to the metric at time $T'$. Assume that
$\abs{\Rm{}}\leq R_0$ on the set $B\times[0,T')$. Then there exist
constants $K=K(\underaccent{\bar}\alpha,\bar\alpha,
R_0,T,m,N)<\infty$ and $C_k=C_k(k,\bar\alpha,m,N)$ for $k\in\mathbb{N}$,
$C_0=1$, such that the following estimates hold for $k\geq0$
\begin{align}
\abs{\nabla\phi}^2 &\leq
\tfrac{K}{t}\quad\text{and}\quad\abs{\Rm{}}\leq\tfrac{K}{t},
\quad\forall (x,t)\in B^{1/2}\times(0,T'),\label{app.eq26}\\
\abs{\nabla^k\mathcal{T}}^2 &= \abs{\nabla^k\Rm{}}^2+\abs{\nabla^{k+2}\phi}^2 \leq C_k
\big(\tfrac{K}{t}\big)^{k+2}, \quad\forall(x,t)\in
B^{1/2}\times(0,T'),\label{app.eq27}
\end{align}
where $B^{1/2}:=B_{g(T')}(x,r/2)$ is the ball of half the radius and
the same center as $B$.
\end{cor}
\begin{proof}
The statement (\ref{app.eq26}) follows exactly as in Theorem
\ref{3.thm10}. The induction step is carried out using a cut-off function
to ensure that the maxima are attained in the interior of the set $B$.
More details can be found in the authors thesis
\cite[Proposition A.5]{Muller:diss}.
\end{proof}
\subsection{Long-time existence}
This subsection follows Section 6.7 about long-time existence for
the Ricci flow from Chow and Knopf's book \cite{RF:intro}. We first
need a technical lemma.
\begin{lemma}\label{3.lemma11}
Let $(g(t),\phi(t))_{t\in[0,T)}$ solve $(RH)_\alpha$ with a
non-increasing $\alpha(t)\in[\underaccent{\bar}\alpha,\bar\alpha]$,
$0<\underaccent{\bar}\alpha\leq\bar\alpha<\infty$ and $T<\infty$.
Let the Riemannian curvature tensor of $M$ be uniformly bounded
along the flow, $\abs{\Rm{}}\leq R_0$, and fix a background metric
$\tilde{g}$. Then for each $k\geq 0$ there exists a constant $C_k$
depending on $k$, $m$, $N$, $T$, $\underaccent{\bar}\alpha$,
$\bar\alpha$, $R_0$ and the initial data $(g(0),\phi(0))$ such that
\begin{equation}\label{3.eq32}
\abs{\tilde{\nabla}^kg(x,t)}_{\tilde{g}}^2 +
\abs{\tilde{\nabla}^k\Rm{x,t}}_{\tilde{g}}^2 +
\abs{\tilde{\nabla}^k\phi(x,t)}_{\tilde{g}}^2 \leq C_k
\end{equation}
for all $(x,t)\in M\times[0,T)$. Here, $\tilde{\nabla}=
\!\phantom{l}^{\tilde{g}}\nabla$ denotes the Levi-Civita connection with
respect to the background metric $\tilde{g}$.
\end{lemma}
\begin{proof}
With Theorem \ref{3.eq10}, the proof becomes a straight forward
computation, and we therefore only give a sketch. In \cite[Section
6.7]{RF:intro}, all the details are carried out in the case of the
Ricci flow and they can easily be adopted to our flow. Since $M$ is
closed, there exists a finite atlas for which we have uniform bounds
on the derivatives of the local charts. Working in such a chart
$\psi:U\to\mathbb{R}^m$, it suffices to derive the desired estimates for
the Euclidean metric $\delta$ in $U$ and the ordinary derivatives,
since $\tilde{g}$ and $\tilde{\nabla}$ are fixed. In particular, we can
interpret $\Gamma$ as a tensor, namely
$\Gamma=\Gamma-\!\phantom{l}^\delta\Gamma$. On the compact interval
$[0,T/2]$, all the derivatives $\abs{\nabla^k\phi}^2_g$ and
$\abs{\nabla^k\Rm{}}^2_g$ are uniformly bounded. On the interval
$[T/2,T)$, Theorem \ref{3.thm10} above yields uniform bounds for
these derivatives. Hence
\begin{equation}\label{3.eq33}
\abs{\nabla^k\phi}^2_g + \abs{\nabla^k\Rm{}}^2_g \leq \bar{C}_k
\end{equation}
for some $\bar{C}_k<\infty$. In particular,
$\mathcal{S}=\Rc{}-\alpha\nabla\phi\otimes\nabla\phi$ is uniformly bounded on
$[0,T)$. From \cite[Lemma 6.49]{RF:intro}, we infer that all $g(t)$
are uniformly equivalent on $[0,T)$, and thus for some constant $C$
\begin{equation}\label{3.eq34}
C^{-1}\delta\leq g(x,t)\leq C\delta, \quad \forall (x,t)\in
U\times[0,T).
\end{equation}
With $\dt(\partial g)=\partial(\dt g)= -2\partial\mathcal{S} = -2(\nabla\mathcal{S} +
\Gamma*\mathcal{S})$, we compute
\begin{equation}\label{3.eq35}
\abs{\dt\partial g}_{\delta}\leq C\abs{\dt\partial g}\leq
C\abs{\nabla\mathcal{S}}+C\abs{\Gamma}\abs{\mathcal{S}}.
\end{equation}
Thus, we have
\begin{equation*}
\abs{\dt\Gamma}\leq
C\abs{\nabla\Rc{}}+2\bar{\alpha}\abs{\nabla\phi}\abs{\nabla^2\phi}
\end{equation*}
which yields a bound for $\abs{\Gamma}$ by integration. Together
with the bounds for $\abs{\mathcal{S}}$ and $\abs{\nabla\mathcal{S}}$ that we obtain
from (\ref{3.eq33}), we conclude that $\abs{\dt\partial g}_{\delta}$
is uniformly bounded, and hence -- again by integration --
$\abs{\partial g}_{\delta}$ is uniformly bounded on $U\times[0,T)$.
Finally, using a partition of unity for the chosen atlas, we obtain
a uniform bound for $\abs{\tilde{\nabla}g}_{\tilde{g}}$ on $M\times
[0,T)$. A short computation as for (\ref{3.eq35}) yields
\begin{equation}\label{3.eq36}
\abs{\dt\tilde{\nabla}^kg}_{\tilde{g}} \leq C \abs{\dt\tilde{\nabla}^kg}
\leq \sum_{i=0}^k c_i\abs{\Gamma}^i\abs{\nabla^{k-i}\mathcal{S}} +
\sum_{i=1}^{k-1}c_i'\abs{\partial^i\Gamma}\abs{\tilde{\nabla}^{k-1-i}\mathcal{S}},
\end{equation}
where the constants $c_i$, $c_i'$ only depend on $m$ and $k$. From
this formula, we inductively obtain the desired bounds for
$\abs{\tilde{\nabla}^kg}_{\tilde{g}}$. Similarly, the estimates for
$\abs{\tilde{\nabla}^k\Rm{x,t}}_{\tilde{g}}$ and
$\abs{\tilde{\nabla}^k\phi(x,t)}_{\tilde{g}}$ are obtained from
(\ref{3.eq33}) with a transformation analogous to (\ref{3.eq36}).
The lemma then follows by plugging everything together.
\end{proof}
Finally, we obtain our desired criterion for long-time existence.
\begin{thm}\label{3.thm12}
Let $(g(t),\phi(t))_{t\in[0,T)}$ solve $(RH)_\alpha$ with
non-increasing $\alpha(t)\in[\underaccent{\bar}\alpha,\bar\alpha]$,
$0<\underaccent{\bar}\alpha\leq\bar\alpha<\infty$ and $T<\infty$.
Suppose that $T<\infty$ is maximally chosen, i.e.~the solution
cannot be extended beyond $T$ in a smooth way. Then the curvature of
$(M,g(t))$ has to become unbounded for $t\nearrow T$ in the sense
that
\begin{equation}\label{3.eq37}
\limsup_{t\nearrow T}\Big(\max_{x\in M}\abs{\Rm{x,t}}^2\Big) =
\infty.
\end{equation}
\end{thm}
\begin{proof}
The proof is by contradiction. Suppose that the curvature stays
bounded on $[0,T)$, say $\abs{\Rm{}}\leq R_0$. For any point $x\in
M$ and vector $X\in T_xM$, define $g(x,T)(X,X):=\lim_{t\to
T}g(x,t)(X,X)$. We estimate
\begin{equation*}
\abs{g(x,T)(X,X)-g(x,t)(X,X)}\leq \int_t^T 2\abs{\mathcal{S}(x,\tau)(X,X)}d\tau
\leq C\abs{X}^2(T-t),
\end{equation*}
where we used again the fact that $\mathcal{S}$ is uniformly bounded on
$M\times[0,T)$. This shows that the limit $g(x,T)(X,X)$ is well
defined and continuous in $x$. Hence, we obtain a continuous limit
$g(\cdot,T)\in \Gamma(\Sym^2(T^*M))$ by polarization. From
\cite[Lemma 6.49]{RF:intro}, all metrics $g(\cdot,t)$ are uniformly
equivalent -- as in (\ref{3.eq34}) -- which implies that this limit
must be a (continuous) Riemannian metric. Moreover, we define
$\phi(x,T):=\lim_{t\to T}\phi(x,t)$, where we use again the
embedding $e_N:N\hookrightarrow\mathbb{R}^d$ to interpret $\phi$ as a map
into $\mathbb{R}^d$. We estimate
\begin{equation*}
\abs{\phi(x,T)-\phi(x,t)}\leq \int_t^T \abs{\dt\phi(x,\tau)}d\tau
\leq C(T-t),
\end{equation*}
since the bound on $\abs{\nabla^2\phi}$ yields a bound on
$\abs{\dt\phi}=\abs{\tau_g\phi}$. This implies that $\phi(\cdot,T)$
is well defined and continuous in $x$. The uniform bounds
(\ref{3.eq32}) from Lemma \ref{3.lemma11} then also hold for the
limit $(g(T),\phi(T))$ and hence $g(T)$ and $\phi(T)$ are smooth.
Indeed, for an arbitrary background metric $\tilde{g}$, we have
\begin{equation*}
\abs{\tilde{\nabla}^kg(T)-\tilde{\nabla}^kg(t)}_{\tilde{g}} \leq \int_t^T
\abs{\dt\tilde{\nabla}^kg(\tau)}_{\tilde{g}}d\tau \leq C(T-t),
\end{equation*}
which follows from the uniform bound for
$\abs{\dt\tilde{\nabla}g}_{\tilde{g}}$ that we have derived in the lemma
above. This means, the convergence $g(t)\to g(T)$ is smooth. With
\begin{equation*}
\abs{\tilde{\nabla}^k\phi(T)-\tilde{\nabla}^k\phi(t)}_{\tilde{g}} \leq
\int_t^T \abs{\dt\tilde{\nabla}^k\phi(\tau)}_{\tilde{g}}d\tau \leq
C(T-t)
\end{equation*}
we see that also $\phi(t)\to\phi(T)$ uniformly in any $C^k$-norm.
Finally, restarting the flow with $(g(T),\phi(T))$ as new initial
data, we obtain a solution $(g(t),\phi(t))_{t\in[T,T+\varepsilon)}$ by the
short-time existence result from Chapter 2. This yields an extension
of our solution beyond time $T$ which is smooth in space for each
time. From the flow equations and the uniform bounds on
$\abs{\nabla^k\Rm{}}$ as well as $\abs{\nabla^k\phi}$, the time derivatives
(and hence also the mixed derivatives) are smooth too, in particular
near $t=T$. This means, the extension of the flow is smooth in space
\emph{and} time, contradicting the maximality of $T$.
\end{proof}
\section{Monotonicity formula and no breathers theorem}
The entropy functional $\mathcal{W}_\alpha$ introduced in this section is
the analogue of Perelman's shrinker entropy for the Ricci flow from
\cite[Section 3]{Perelman:entropy}. It is obtained from the energy
functional $\mathcal{F}_\alpha$ from (\ref{1.eq1}), by
introducing a positive scale factor $\tau$ (later interpreted as a
backwards time) and some correction terms. For detailed explanations
of Perelman's result, we again refer to Chow et al. \cite[Chapter
6]{RF:TAI} and M\"{u}ller \cite[Chapter 3]{Muller:Harnack}. Moreover for
the special case $N\subseteq\mathbb{R}$, the entropy functional $\mathcal{W}_\alpha$
can be found in List's dissertation \cite{List:diss}.
\subsection{The entropy functional and its first variation}
Let again $g=g_{ij} \in \Gamma\big(\Sym^2_{+}(T^*M)\big)$, $\phi \in
C^\infty(M,N)$, $f:M\to\mathbb{R}$ and $\tau >0$. For a time-independent
coupling constant $\alpha(t) \equiv \alpha > 0$, we set
\begin{equation}\label{5.eq1}
\mathcal{W}_\alpha(g,\phi,f,\tau) := \int_M \Big( \tau \big(R_g + \abs{\nabla
f}^2_g - \alpha \abs{\nabla\phi}^2_g\big) +
f-m\Big)(4\pi\tau)^{-m/2}e^{-f}dV_g.
\end{equation}
As in Section 3, we take variations $g^\varepsilon=g+\varepsilon h$, $f^\varepsilon=f+\varepsilon \ell$,
$\phi^\varepsilon=\pi_N(\phi + \varepsilon \vartheta)$,
such that $\delta g = h$, $\delta f = \ell$ and $\delta \phi =
\vartheta$. Additionally, set $\tau^\varepsilon = \tau + \varepsilon\sigma$ for some $\sigma
\in \mathbb{R}$, i.e.~$\delta \tau = \sigma$. The variation
\begin{equation*}
\delta\mathcal{W}_\alpha :=
\delta\mathcal{W}_{\alpha,g,\phi,f,\tau}(h,\vartheta,\ell,\sigma):=
\frac{d}{d\varepsilon}\Big|_{\varepsilon=0} \mathcal{W}_\alpha(g+\varepsilon h, \pi_N(\phi +
\varepsilon\vartheta), f+\varepsilon\ell, \tau + \varepsilon \sigma)
\end{equation*}
is easiest computed using the variation of $\mathcal{F}_\alpha$ and
\begin{equation}\label{5.eq2}
\mathcal{W}_\alpha(g,\phi,f,\tau) = (4\pi\tau)^{-m/2}\Big(
\tau\,\mathcal{F}_\alpha(g,\phi,f) + \int_M (f-m)e^{-f}dV \Big).
\end{equation}
Using
\begin{equation*}
\delta \int_M (f-m)e^{-f}dV = \int_M \Big(\ell +
(f-m)\big(\tfrac{1}{2}\tr_g h -\ell\big)\Big)e^{-f}dV
\end{equation*}
and equation (\ref{1.eq2}) for the variation of
$\mathcal{F}_\alpha(g,\phi,f)$, we get from (\ref{5.eq2})
\begin{align*}
\delta\mathcal{W}_\alpha &= \int_M -\tau h_{ij}\big(R_{ij} + \nabla_i\nabla_j
f-\alpha \nabla_i \phi \nabla_j
\phi\big)d\mu\\%
&\quad+ \int_M \tau \big(\tfrac{1}{2}\tr_g h - \ell\big)\big(2\triangle f
- \abs{\nabla f}^2 + R - \alpha \abs{\nabla \phi}^2 +
\tfrac{f-m}{\tau}\big)d\mu\\%
&\quad+ \int_M \big(\ell + \sigma(1-\tfrac{m}{2})(R+\abs{\nabla
f}^2-\alpha\abs{\nabla\phi}^2)-\tfrac{m\sigma}{2\tau}(f-m)\big)
d\mu\\%
&\quad+ \int_M 2\tau\alpha\vartheta\big(\tau_g \phi -
\Scal{\nabla\phi,\nabla f}\big)d\mu,
\end{align*}
where we used the abbreviation $d\mu := (4\pi\tau)^{-m/2}e^{-f}dV$.
Rearranging the terms, writing
\begin{equation*}
\ell = (-\tau h_{ij} +\sigma g_{ij})\big(\tfrac{-1}{2\tau}
g_{ij}\big) + \tau \big(\tfrac{1}{2}\tr_g h -\ell -
\tfrac{m\sigma}{2\tau}\big)\big(\tfrac{-1}{\tau}\big),
\end{equation*}
and using $\int_M (\triangle f-\abs{\nabla f}^2)d\mu =
-(4\pi\tau)^{-m/2}\int_M \triangle(e^{-f})dV = 0$, we get
\begin{align*}
\delta\mathcal{W}_\alpha &= \int_M (-\tau h_{ij}+\sigma g_{ij})\big(R_{ij} +
\nabla_i\nabla_j f-\alpha \nabla_i \phi \nabla_j \phi-
\tfrac{1}{2\tau}g_{ij}\big)d\mu\\%
&\quad+ \int_M \tau \big(\tfrac{1}{2}\tr_g h -
\ell-\tfrac{m\sigma}{2\tau}\big)\big(2\triangle f - \abs{\nabla f}^2 + R -
\alpha \abs{\nabla \phi}^2 + \tfrac{f-m-1}{\tau}\big)d\mu\\%
&\quad+ \int_M 2\tau\alpha\vartheta\big(\tau_g \phi -
\Scal{\nabla\phi,\nabla f}\big)d\mu.
\end{align*}
\subsection{Fixing the background measure}
Similar to Section 3, we now fix the measure $d\mu =
(4\pi\tau)^{-m/2}e^{-f}dV$. This means, we have $f=
-\log((4\pi\tau)^{m/2}\frac{d\mu}{dV})$ and from $0=\delta d\mu =
(\frac{1}{2}\tr_g h -\ell -\frac{m\sigma}{2\tau})d\mu$, we deduce
$\ell = \frac{1}{2}\tr_g h -\frac{m\sigma}{2\tau}$. Moreover, we
require the variation of $\tau$ to satisfy $\delta\tau = \sigma=-1$.
This allows us to interpret $\tau$ as backwards time later. We then
write
\begin{equation}\label{5.eq4}
\mathcal{W}_\alpha^\mu(g,\phi,\tau) := \mathcal{W}_\alpha\big(g,\phi,
-\log\big((4\pi\tau)^{m/2}\tfrac{d\mu}{dV}\big),\tau\big)
\end{equation}
and
\begin{equation*}
\delta \mathcal{W}^\mu_{\alpha,g,\phi,\tau}(h,\vartheta) := \delta
\mathcal{W}_{\alpha,g,\phi,-\log((4\pi\tau)^{m/2}\frac{d\mu}{dV}),\tau}
\left(h,\vartheta, \tfrac{1}{2}\tr_g h + \tfrac{m}{2\tau},-1\right).
\end{equation*}
The variation formula above reduces to
\begin{equation}\label{5.eq5}
\begin{aligned}
\delta \mathcal{W}^\mu_{\alpha,g,\phi,\tau}(h,\vartheta) &= \int_M (-\tau
h_{ij}- g_{ij})\big(R_{ij} + \nabla_i\nabla_j f-\alpha \nabla_i \phi \nabla_j \phi-
\tfrac{1}{2\tau}g_{ij}\big)d\mu\\%
&\quad+ \int_M 2\tau\alpha\vartheta\big(\tau_g \phi -
\Scal{\nabla\phi,\nabla f}\big)d\mu,
\end{aligned}
\end{equation}
which is monotone under the gradient-like system of evolution
equations given by
\begin{equation}\label{5.eq6}
\left\{\begin{aligned}\dt g_{ij} &= -2(R_{ij}+\nabla_i\nabla_j f -\alpha
\nabla_i
\phi \nabla_j\phi),\\%
\dt \phi &= \tau_g \phi - \Scal{\nabla\phi,\nabla f},\\%
\dt f &= -R -\triangle f + \alpha \abs{\nabla\phi}^2 +
\tfrac{m}{2\tau},\\%
\dt \tau &= -1.\end{aligned}\right.
\end{equation}
As in Section 3, pulling back the solutions of (\ref{5.eq6}) with
the family of diffeomorphisms generated by $\nabla f$, we get a solution
of
\begin{equation}\label{5.eq7}
\left\{\begin{aligned}\dt g &= -2\Rc{} + 2\alpha \nabla\phi \otimes \nabla\phi,\\%
\dt \phi &= \tau_g \phi,\\%
\dt \tau &= -1,\\%
0 &= \Box^*((4\pi\tau)^{-m/2}e^{-f}). \end{aligned}\right.
\end{equation}
Since $\mathcal{W}_\alpha$ is diffeomorphism invariant, we find the analogue
to Proposition \ref{1.prop1}.
\begin{prop}\label{5.prop1}
Let $(g(t),\phi(t))_{t \in [0,T)}$ be a solution of\/ $(RH)_\alpha$
with $\alpha(t)\equiv \alpha>0$, $\tau$ a backwards time with $\dt
\tau = -1$ and $(4\pi\tau)^{-m/2}e^{-f}$ a solution of the adjoint
heat equation under the flow. Then the entropy functional
$\mathcal{W}_\alpha(g,\phi,f,\tau)$ is non-decreasing with
\begin{equation}\label{5.eq8}
\begin{aligned}
\frac{d}{dt} \mathcal{W}_\alpha &= \int_M 2\tau\Abs{\Rc{}-\alpha \nabla\phi
\otimes \nabla\phi+\Hess(f)-\tfrac{g}{2\tau}}^2
(4\pi\tau)^{-m/2}e^{-f}dV\\%
&\quad +\int_M 2\tau\alpha \Abs{\tau_g \phi-\Scal{\nabla\phi,\nabla f}}^2
(4\pi\tau)^{-m/2}e^{-f}dV.
\end{aligned}
\end{equation}
\end{prop}
\begin{rem}
As seen in Corollary \ref{1.cor2} for the energy $\mathcal{F}_\alpha$, the
monotonicity of the entropy $\mathcal{W}_\alpha$ also holds true for
non-increasing positive coupling functions $\alpha(t)$
instead of a constant $\alpha>0$.
\end{rem}
\subsection{Minimizing over all probability measures}
Similar to $\lambda_\alpha(g,\phi)$ defined in (\ref{1.eq13}), we
set
\begin{equation}\label{5.eq9}
\begin{aligned}
\mu_\alpha(g,\phi,\tau) &:= \inf \big\{\mathcal{W}_\alpha^\mu(g,\phi,\tau)
\mid \mu(M)=1 \big\}\\
&\phantom{:}= \inf \left\{\mathcal{W}_\alpha(g,\phi,f,\tau) \; \bigg| \;
\int_M (4\pi\tau)^{-m/2}e^{-f}dV =1 \right\}.
\end{aligned}
\end{equation}
Our goal is again to show that the infimum is always achieved. Note
that for $\tilde{g}=\tau g$ we have $R_{\tilde{g}} =
\frac{1}{\tau}R_g$, $\abs{\nabla f}^2_{\tilde{g}} =
\frac{1}{\tau}\abs{\nabla f}^2_g$, $\abs{\nabla \phi}^2_{\tilde{g}} =
\frac{1}{\tau}\abs{\nabla \phi}^2_g$, $dV_{\tilde{g}}=\tau^{m/2}dV_g$
and thus
\begin{equation*}
\mu_\alpha(\tau g,\phi, \tau) = \mu_\alpha(g,\phi,1).
\end{equation*}
We can hence reduce the problem to the special case where $\tau =1$.
Set $v= (4\pi)^{-m/4}e^{-f/2}$. This yields
\begin{equation*}
\mathcal{W}_\alpha(g,\phi,v,1) =\int_M v\Big(Rv -4\triangle v - \alpha\abs{\nabla\phi}^2v -2v\log v
-\tfrac{mv}{2}\log(4\pi)-mv\Big)dV,
\end{equation*}
and hence $\mu_\alpha(g,\phi,1) = \inf\{\mathcal{W}_\alpha(g,\phi,v,1)
\;|\; \int_M v^2 dV =1\}$ is the smallest eigenvalue of $L(v) = -4\triangle v +
(R-\alpha\abs{\nabla\phi}^2 -\frac{m}{2}\log(4\pi)-m)v -2v\log v$ and
$v$ is a corresponding normalized eigenvector. As in Section 3, a
unique smooth positive normalized eigenvector $v_{min}$ exists (cf.
Rothaus \cite{Rothaus:Schroedinger} or List \cite{List:diss}) and we
get the following.
\begin{prop}\label{5.prop2}
Let $(g(t),\phi(t))$ solve $(RH)_\alpha$ for a constant $\alpha > 0$
and let $\dt \tau = -1$. Then $\mu_\alpha(g,\phi,\tau)$ is monotone
non-decreasing in time. Moreover, it is constant if and only if
\begin{equation}\label{5.eq10}
\left\{\begin{aligned}0 &= \Rc{}-\alpha\nabla\phi\otimes\nabla\phi
+\Hess(f)-\tfrac{g}{2\tau},\\%
0 &= \tau_g \phi -\scal{\nabla\phi,\nabla f}. \end{aligned}\right.
\end{equation}
for the minimizer $f$ that corresponds to $v_{min}$. As always, the
monotonicity result stays true if we allow $\alpha(t)$ to be a
positive non-increasing function instead of a constant.
\end{prop}
\begin{proof}
The proof is completely analogous to the proof of Proposition
\ref{1.prop3}, using the monotonicity of $\mathcal{W}_\alpha$ instead on the
monotonicity of $\mathcal{F}_\alpha$.
\end{proof}
\subsection{Non-existence of nontrivial breathers}
Breathers correspond to periodic solutions modulo
diffeomorphisms and scaling, generalizing the notion of solitons
defined in Definition \ref{4.def1}.
\begin{defn}\label{5.def3}
A solution $(g(t),\phi(t))_{t\in[0,T)}$ of\/ $(RH)_\alpha$ is called
a \emph{breather} if there exists $t_1,t_2\in[0,T)$, $t_1<t_2$, a
diffeomorphism $\psi:M\to M$ and a constant $c\in\mathbb{R}_{+}$ such that
\begin{equation}\label{5.eq11}
\left\{\begin{aligned}g(t_2)&=c\,\psi^*g(t_1),\\%
\phi(t_2)&=\psi^*\phi(t_1).\end{aligned}\right.
\end{equation}
The cases $c<1$, $c=1$ and $c>1$ correspond to \emph{shrinking},
\emph{steady} and \emph{expanding} breathers.
\end{defn}
\begin{thm}\label{5.thm4}
Let $M$ and $N$ be closed and let $(g(t),\phi(t))_{t\in[0,T)}$ be a
solution of\/ $(RH)_\alpha$ with $\alpha(t)\equiv\alpha$.
\begin{itemize}
\item[i)] If this solution is a steady breather, then it necessarily is a
steady gradient soliton. Moreover, $\phi(t)$ is harmonic
and $\Rc{}=\alpha\nabla\phi\otimes\nabla\phi$, i.e.~the solution is
stationary.
\item[ii)] If the solution is an expanding breather, then it necessarily
is an expanding gradient soliton. Again $\phi(t)$ must be harmonic
(and thus stationary, $\phi(t)=\phi(0)$), while $g(t)$ changes only
by scaling.
\item[iii)] If the solution is a shrinking breather, then it has to be a
shrinking gradient soliton.
\end{itemize}
If we assume in addition that $\dim M=2$ or that $(M,g(0))$ is
Einstein, then in the first two cases above, $\phi(t)$ is not only
harmonic but also conformal, hence a minimal branched immersion,
provided that it is non-constant.
\end{thm}
\begin{proof}
This is an application of the monotonicity results for
$\lambda_\alpha(g,\phi)$ from Proposition \ref{1.prop3} and for
$\mu_\alpha(g,\phi,\tau)$ from Proposition \ref{5.prop2}. Since the
proof is very similar to the Ricci flow case solved by Perelman in
\cite{Perelman:entropy}, we closely follow the notes from Kleiner
and Lott \cite{KleinerLott} on Perelman's paper.
\begin{itemize}
\item[i)] Assume $(g(t),\phi(t))_{t\in[0,T)}$ is a steady breather.
Then there exist two times $t_1$, $t_2$ such that (\ref{5.eq11})
holds with $c=1$. From diffeomorphism invariance of
$\lambda_\alpha(g,\phi)$ defined in (\ref{1.eq13}), we obtain
$\lambda_\alpha(g,\phi)(t_1)=\lambda_\alpha(g,\phi)(t_2)$. From
Proposition \ref{1.prop3}, we get condition (\ref{1.eq14}) on
$[t_1,t_2]$, which means that $(g(t),\phi(t))$ must be a gradient
steady soliton according to Lemma \ref{4.lemma2} and uniqueness of
solutions. Moreover, the minimizer $f=-2\log v_{min}$ which realizes
$\lambda_\alpha(g,\phi)$ is the soliton potential. From
$(-4\triangle+R-\alpha \abs{\nabla\phi}^2)v_{min}
=\lambda_\alpha(g,\phi)v_{min}=:\lambda_\alpha v_{min}$, we obtain
\begin{equation}\label{5.eq12}
2\triangle f -\abs{\nabla f}^2+R-\alpha\abs{\nabla\phi}^2=\lambda_\alpha.
\end{equation}
Since $(g(t),\phi(t))$ is a steady soliton, (\ref{4.eq4}) holds with
$\sigma=0$. Plugging this into (\ref{5.eq12}) yields $\triangle f-\abs{\nabla
f}^2=\lambda_\alpha$, and we obtain from $\int_Me^{-f}dV=1$
\begin{equation*}
\lambda_\alpha=\int_M\lambda_\alpha e^{-f}dV = \int_M(\triangle f-\abs{\nabla
f}^2)e^{-f}dV = -\int_M\triangle(e^{-f})dV =0,
\end{equation*}
i.e.~$\triangle f=\abs{\nabla f}^2$. Another integration yields
\begin{equation*}
\int_M \abs{\nabla f}^2dV=\int_M\triangle f\;dV=0,
\end{equation*}
and thus $\nabla f\equiv 0$, $\Hess(f)\equiv 0$ on $M\times[0,T)$ and
(\ref{1.eq14}) becomes
\begin{equation*}
\left\{\begin{aligned}0 &= \Rc{}-\alpha\nabla\phi\otimes\nabla\phi,\\%
0 &= \tau_g \phi. \end{aligned}\right.
\end{equation*}
In particular, $\phi(t)$ is harmonic and $(g(t),\phi(t))$ is
stationary.
\item[ii)] The proof here is analogous to the case of steady
breathers, but we first need to construct a scaling invariant
version of $\lambda_\alpha(g,\phi)$. We define
\begin{equation}\label{5.eq13}
\bar{\lambda}_\alpha(g,\phi):=\lambda_\alpha(g,\phi)\bigg(\int_M
dV_g\bigg)^{\!2/m}.
\end{equation}
This quantity is invariant under rescaling $\tilde{g}=c g$. A proof
of this fact is given in the appendix of the author's dissertation
\cite{Muller:diss}. Moreover, we claim that at times where
$\bar{\lambda}_\alpha(t):=\bar{\lambda}_\alpha(g,\phi)(t)\leq 0$, we
have $\dt\bar{\lambda}_\alpha(t)\geq 0$. Indeed, note that
$\bar{f}=\log V(t)$ satisfies $\int_M
e^{-\bar{f}}dV=1$ and is thus an admissible test function in the
definition of $\lambda_\alpha$, hence
\begin{equation}\label{5.eq15}
\lambda_\alpha(g,\phi)\leq \mathcal{F}_\alpha(g,\phi,\log V(t))=\int_M
S\,e^{-\log V(t)}dV=V(t)^{-1}\int_M S\; dV.
\end{equation}
With the assumption $\lambda_\alpha(t)\leq 0$, we find
\begin{equation}\label{5.eq17}
\begin{aligned}
\dt\bar{\lambda}_\alpha &\geq 2V^{2/m}\int_M
\Big(\abs{\mathcal{S}+\Hess(f)-\tfrac{1}{m}(S+\triangle f)g}^2+\alpha \abs{\tau_g
\phi-\scal{\nabla\phi,\nabla f}}^2\Big)d\mu\\%
&\quad\,+\tfrac{2}{m}V^{2/m}\bigg(\int_M(S+\triangle f)^2d\mu
-\Big(\int_M(S+\triangle f)d\mu\Big)^2\bigg),
\end{aligned}
\end{equation}
the right hand side being nonnegative by H\"{o}lder's inequality (see again
\cite{Muller:diss} for a more detailed computation). Now, assume that
$(g(t),\phi(t))$ is an expanding breather. Since
$\bar{\lambda}_\alpha(g,\phi)$ is invariant under diffeomorphisms
and scaling, we have
$\bar{\lambda}_\alpha(t_1)=\bar{\lambda}_\alpha(t_2)$ for the two
times $t_1$, $t_2$ that satisfy (\ref{5.eq11}). Since
$V(t_1)<V(t_2)$, there must be a time $t_0\in[t_1,t_2]$ with $\dt
V(t_0)>0$ and hence with (\ref{5.eq15})
\begin{equation*}
\lambda_\alpha(t_0)\leq V(t_0)^{-1}\int_M S\; dV =-V(t_0)^{-1}\dt
V(t_0) <0.
\end{equation*}
The claim applies and we obtain $\bar{\lambda}_\alpha(t_1)
\leq\bar{\lambda}_\alpha(t_0)<0$ and since
$\bar{\lambda}_\alpha(t_2)=\bar{\lambda}_\alpha(t_1)$, we see that
$\bar{\lambda}_\alpha(t)$ must be a negative constant. Hence, both
lines on the right hand side of (\ref{5.eq17}) have to vanish. This
means that $(S+\triangle f)$ has to be constant in space for all $t$ and
because $\lambda_\alpha(t)=\int_M(S+\triangle f)d\mu$, this constant has to be
$\lambda_\alpha(t)$. From the first line of (\ref{5.eq17}) we obtain
\begin{equation}\label{5.eq18}
\left\{\begin{aligned}0 &= \Rc{}-\alpha\nabla\phi\otimes\nabla\phi+\Hess(f)
-\tfrac{\lambda_\alpha}{m}g,\\%
0 &= \tau_g \phi-\scal{\nabla\phi,\nabla f}. \end{aligned}\right.
\end{equation}
By Lemma \ref{4.lemma2}, $(g(t),\phi(t))_{t\in[0,T)}$ is an
expanding soliton with potential $f=-2\log v_{min}$. This means that
we can use (\ref{5.eq12}), which implies
\begin{equation}\label{5.eq19}
0= 2\triangle f-\abs{\nabla f}+S-\lambda_\alpha = 2\triangle f -\abs{\nabla f}+S-(\triangle
f + S) = \triangle f - \abs{\nabla f}
\end{equation}
and thus by integration $\nabla f \equiv 0$, $\Hess(f)\equiv 0$, as
above. Plugging this into (\ref{5.eq18}), the second equation tells
us that $\phi(t)$ is harmonic and the first equation yields
\begin{equation*}
\dt g = -2\Rc{}+2\alpha\nabla\phi\otimes\nabla\phi =
-2\tfrac{\lambda_\alpha}{m}g,
\end{equation*}
i.e.~$(M,g(t))$ simply expands without changing its
shape.
\item[iii)] If $(g(t),\phi(t))$ is a shrinking breather, there exist
$t_1$, $t_2$ and $c<1$ which such that (\ref{5.eq11}) is satisfied.
We define
\begin{equation*}
\tau_0:= \frac{t_2-c\,t_1}{1-c}>t_2, \qquad\text{and}\qquad
\tau(t)=\tau_0-t.
\end{equation*}
Note that $\tau(t)$ is always positive on $[t_1,t_2]$. Moreover,
$c=(\tau_0-t_2)/(\tau_0-t_1)=\tau(t_2)/\tau(t_1)$. Then, from the
scaling behavior of $\mu_\alpha(g,\phi,\tau)$ and diffeomorphism
invariance we obtain
\begin{equation}\label{5.eq20}
\begin{aligned}
\mu_\alpha(g(t_2),\phi(t_2),\tau(t_2))&=
\mu_\alpha(c\,\psi^*g(t_1),\psi^*\phi(t_1),c\tau(t_1))\\
&=\mu_\alpha(\psi^*g(t_1),\psi^*\phi(t_1),\tau(t_1))\\
&=\mu_\alpha(g(t_1),\phi(t_1),\tau(t_1)).
\end{aligned}
\end{equation}
By the equality case of the monotonicity result in Proposition
\ref{5.prop2}, $(g(t),\phi(t))$ must satisfy (\ref{5.eq10}) and
according to Lemma \ref{4.lemma2} thus has to be a gradient
shrinking soliton.
\end{itemize}
It remains to prove the additional statement in the cases where
$\dim M=2$ or $(M,g(0))$ is Einstein. If $(g(t),\phi(t))$ is a
steady or expanding breather, we have seen that $\dt g= cg$. In
particular, if $(M,g(t))$ is Einstein at $t=0$ it remains Einstein
under the flow. Moreover, since $\Rc{}= \tfrac{R}{m}g$ in these two
cases, we get
\begin{equation*}
(\phi^*\gamma)_{ij}=\nabla_i\phi\nabla_j\phi=\tfrac{1}{2\alpha}\big(\dt
g_{ij}+2R_{ij}\big)=\tfrac{1}{2\alpha}\big(2\tfrac{R}{m}+c\big)g_{ij},
\end{equation*}
i.e.~$\phi$ is conformal. It is a well-known fact that conformal
harmonic maps have to be minimal branched immersions
(cf.~Hartman-Wintner \cite{HartmanWintner}).
\end{proof}
\section{Reduced volume and non-collapsing theorem}
Let us briefly restate the main result from our previous article
\cite{Muller:MonotoneVolumes} about the monotonicity of reduced
volumes for flows of the form $\dt g_{ij}=-2S_{ij}$, where $S_{ij}$
is a symmetric tensor with trace $S=g^{ij}S_{ij}$. (The resuts can also be
found in the authors thesis \cite{Muller:diss}.)
\subsection{Monotonicity of backwards reduced volume}
In order to define the backwards reduced distance and volume, we
need a backwards time $\tau(t)$ with $\dt\tau(t) = -1$. Without loss
of generality, one may assume (possibly after a time shift) that
$\tau = -t$.
\begin{defn}\label{6.def2}
Assume $\dtau g_{ij}=2S_{ij}$ has a solution for $\tau \in [0,\bar{\tau}]$ and
$0\leq \tau_1<\tau_2\leq \bar{\tau}$, we define the $\mathcal{L}_b$-length
of a curve $\eta:[\tau_0,\tau_1]\to M$ by
\begin{equation*}
\mathcal{L}_b(\eta) := \int_{\tau_0}^{\tau_1}\sqrt{\tau}\left(S(\eta(\tau))
+ \Abs{\tfrac{d}{d\tau}\eta(\tau)}^2 \right)d\tau.
\end{equation*}
Fix the point $p\in M$ and $\tau_0=0$ and define the backwards
reduced distance by
\begin{equation}\label{6.eq4}
\ell_b(q,\tau_1):= \inf_{\eta \in
\Gamma}\left\{\frac{1}{2\sqrt{\tau_1}}\int_0^{\tau_1}\sqrt{\tau}
\left(S+\Abs{\tfrac{d}{d\tau}\eta}^2\right)d\tau\right\},
\end{equation}
where $\Gamma= \{\eta:[0,\tau_1]\to M \mid
\eta(0)=p,\,\eta(\tau_1)=q\}$. The backwards reduced volume is
defined by
\begin{equation}\label{6.eq5}
\tilde{V}_b(\tau) := \int_M
(4\pi\tau)^{-m/2}e^{-\ell_b(q,\tau)}dV(q).
\end{equation}
\end{defn}
The following is proved in \cite[Theorem
1.4]{Muller:MonotoneVolumes}.
\begin{thm}\label{6.thm4}
Suppose that $g(t)$ evolves by $\dt g_{ij}=-2S_{ij}$ and the quantity
\begin{equation}\label{6.eq6}
\begin{aligned}
\mathcal{D}(\mathcal{S},X) &:= \dt S-\triangle S -2\Abs{S_{ij}}^2
+4(\nabla_i S_{ij})X_j -2(\nabla_j S)X_j\\%
&\phantom{:}\quad + 2R_{ij}X_iX_j - 2S_{ij}X_iX_j,
\end{aligned}
\end{equation} is nonnegative for all vector fields $X \in \Gamma(TM)$
and all times $t$ for which the flow exists. Then the backwards
reduced volume $\tilde{V}_b(\tau)$ is non-increasing in $\tau$,
i.e.~non-decreasing in $t$.
\end{thm}
In our case where $S_{ij}$ is given by $R_{ij}- \alpha\nabla_i\phi\nabla_j\phi$, the evolution
equation (\ref{2.eq16}) for $S_{ij}$ together with $4(\nabla_i
S_{ij})X_j - 2(\nabla_j S)X_j = -4\alpha\, \tau_g \phi \nabla_j\phi X_j$
yields
\begin{equation*}
\mathcal{D}(S_{ij},X)=2\alpha \abs{\tau_g\phi-\nabla_X\phi}^2
-\dot{\alpha}\abs{\nabla\phi}^2
\end{equation*}
for all $X$ on $M$. This means, $\mathcal{D}(S_{ij},X)\geq 0$ is satisfied
for the $(RH)_\alpha$ flow with a positive non-increasing coupling
function $\alpha(t)$ and the monotonicity of the reduced volume
holds.
\subsection{No local collapsing theorem}
We have seen in Section 6 that the metrics $g(t)$ along the
$(RH)_\alpha$ flow are uniformly equivalent as long as the curvature
on $M$ stays uniformly bounded. But a-priori, it could happen that at a
singularity (i.e.~when $\Rm{}$ blows up) the solution collapses
geometrically in the following sense.
\begin{defn}
Let $(g(t),\phi(t))_{t\in[0,T)}$ be a maximal solution of\/
$(RH)_\alpha$, or more generally of any flow of the form $\dt
g_{ij}=-2S_{ij}$. We say that this solution is locally collapsing at
time $T$, if there is a sequence of times $t_k\nearrow T$ and a
sequence of balls $B_k:=B_{g(t_k)}(x_k,r_k)$ at time $t_k$, such
that the following holds. The ratio $r_k^2/t_k$ is bounded, the
curvature satisfies $\abs{\Rm{}}\leq r_k^{-2}$ on the parabolic
neighborhood $B_k\times[t_k-r_k^2,t_k]$ and $r_k^{-m}\vol(B_k)\to 0$
as $k\to\infty$.
\end{defn}
Using the monotonicity of the reduced volume, we obtain the
following result.
\begin{thm}\label{6.thm13}
Let $(g(t),\phi(t))$ be a solution of\/ $(RH)_\alpha$ with
non-increasing $\alpha(t)\in[\underaccent{\bar}\alpha,\bar\alpha]$,
$0<\underaccent{\bar}\alpha\leq\bar\alpha<\infty$ on a finite time
interval $[0,T)$. Then this solution is not locally collapsing at
$T$.
\end{thm}
The only ingredients of the proof are the interior
gradient estimates from Corollary \ref{app.prop5} and the
monotonicity of the backwards reduced volume stated above. Hence,
every flow $\dt g=-2\mathcal{S}$ that satisfies the assumption of Theorem
\ref{6.thm4} and some interior estimates for $\mathcal{S}$, $\nabla S$ in the
spirit of Corollary \ref{app.prop5} will also satisfy the
non-collapsing result. For the $(RH)_\alpha$ flow, it is possible to
obtain a slightly stronger result using the monotonicity of
$\mu_\alpha(g,\phi,\tau)$ from Section 7 instead of the monotonicity
of the backwards reduced volume. In the special case $N\subseteq\mathbb{R}$,
this can be found in List's dissertation \cite[Section 7]{List:diss}.
The proof in the case of $(RH)_\alpha$ is analogous. However, the
result here is more general in the sense that it may be adopted to
other flows $\dt g=-2\mathcal{S}$ in the way explained above.
\begin{proof}
The proof follows Perelman's results for the Ricci
flow \cite{Perelman:entropy} very closely, see also the notes on his
paper by Kleiner and Lott \cite{KleinerLott} and the book by Morgan
and Tian \cite{MorganTian}. However, we need the more general
results from \cite{Muller:MonotoneVolumes} that also hold for our
coupled flow system. We only give a sketch.
\newline
The proof is by contradiction. Assume that there is some sequence of
times $t_k\nearrow T$ and some sequence of balls
$B_k:=B_{g(t_k)}(x_k,r_k)$ at each time $t_k$, such that $r_k^2$ is
bounded, the curvature is bounded by $\abs{\Rm{}}\leq r_k^{-2}$ on
the parabolic neighborhood $B_k\times[t_k-r_k^2,t_k]$ and
$r_k^{-m}\vol(B_k)\to 0$ as $k\to\infty$. Define
$\varepsilon_k:=r_k^{-1}\vol(B_k)^{1/m}$, then $\varepsilon_k\to 0$ for
$k\to\infty$. For each $k$, we set $\tau_k(t)=t_k-t$ and define the
backwards reduced volume $\tilde{V}_k$ using curves going backward
in real time from the base point $(x_k,t_k)$, i.e.~forward in time
$\tau_k$ from $\tau_k=0$. The goal is to estimate the reduced
volumes $\tilde{V}_k(\varepsilon_kr_k^2)$, where $\tau_k=\varepsilon_kr_k^2$
corresponds to the real time $t=t_k-\varepsilon_k r_k^2$, which is very
close to $t_k$ and hence close to $T$.
\paragraph{Claim 1:} $\lim_{k\to\infty}\tilde{V}_k(\varepsilon_kr_k^2)=0$.
\begin{proof}
An $\mathcal{L}_b$-geodesic $\eta(\tau)$ starting at $\eta(0)=x_k$ is
uniquely defined through its initial vector $v=\lim_{\tau\to
0}2\sqrt{\tau}X=\lim_{\lambda\to0}\tilde{X}$. First, we show that if
$\abs{v}\leq \frac{1}{8}\varepsilon_k^{-1/2}$ with respect to the metric at
$(x_k,t_k)$, then $\eta(\tau)$ does not escape from
$B_k^{1/2}:=B_{g(t_k)}(x_k,r_k/2)$ in time $\tau=\varepsilon_kr_k^2$. Write
$\hat{t}_k=t_k-r_k^2$. Since $\abs{\Rm{}}\leq r_k^{-2}$ on
$B_k\times[\hat{t}_k,t_k]$ by assumption, we obtain from Corollary
\ref{app.prop5}
\begin{equation*}
\abs{\nabla\phi}^2 \leq \frac{C}{t-\hat{t}_k},\quad\abs{\nabla^2\phi}^2 \leq
\frac{C}{(t-\hat{t}_k)^2},\quad\abs{\nabla\Rm{}}^2\leq
\frac{C}{(t-\hat{t}_k)^3},\quad\textrm{on }B_k^{1/2}
\times(\hat{t}_k,t_k),
\end{equation*}
for some constant $C$ independent of $k$. Without loss of
generality, $\varepsilon_k\leq\tfrac{1}{2}$ so that
$t-\hat{t}_k\geq\tfrac{1}{2}r_k^2$ whenever $t\in
[t_k-\varepsilon_kr_k^2,t_k)$. This means that
\begin{equation*}
\abs{\nabla\phi}^2 \leq Cr_k^{-2},\quad\abs{\nabla^2\phi} \leq
Cr_k^{-2},\quad\abs{\nabla\Rm{}}\leq Cr_k^{-3},\quad\textrm{on
}B_k^{1/2}\times[t_k-\varepsilon_kr_k^2,t_k).
\end{equation*}
Together with the assumption $\abs{\Rm{}}\leq r_k^{-2}$, this yields
\begin{equation}\label{6.eq38}
\begin{aligned}
\abs{\mathcal{S}}&\leq\abs{\Rc{}}+\abs{\nabla\phi}^2\leq Cr_k^{-2},\\%
\abs{\nabla S}&\leq \abs{\nabla R}+\abs{\nabla\phi}\abs{\nabla^2\phi}\leq Cr_k^{-3},
\end{aligned}
\end{equation}
on $B_k^{1/2}\times[t_k-\varepsilon_kr_k^2,t_k)$. Plugging this into the
estimate (4.5) of \cite{Muller:MonotoneVolumes}, we get
\begin{equation}\label{6.eq39}
\dl\lvert\tilde{X}\rvert \leq \lambda C\lvert\tilde{X}\rvert
r_k^{-2} + \lambda^2 Cr_k^{-3}\leq C\lvert\tilde{X}\rvert
\varepsilon_k^{1/2}r_k^{-1}+C\varepsilon_kr_k^{-1},
\end{equation}
for $\lambda=\sqrt{\tau}\leq\sqrt{\varepsilon_kr_k^2}=\varepsilon_k^{1/2}r_k$.
Since $\lvert\tilde{X}(0)\rvert=\abs{v}\leq\frac{1}{8}\varepsilon_k^{-1/2}$
we obtain the estimate $\lvert\tilde{X}(\lambda)\rvert
\leq\frac{1}{4}\varepsilon_k^{-1/2}$ for all $\tau\in[0,\varepsilon_kr_k^2]$ if
$k$ is large enough, i.e.~$\varepsilon_k$ small enough. With an
integration, we find
\begin{equation*}
\int_0^{\varepsilon_kr_k^2}\abs{X(\tau)}d\tau = \int_0^{\sqrt{\varepsilon_k}\,r_k}
\lvert\tilde{X}(\lambda)\rvert d\lambda
\leq\int_0^{\sqrt{\varepsilon_k}\,r_k} \tfrac{1}{4}\varepsilon_k^{-1/2}d\lambda
\leq \tfrac{1}{4}r_k.
\end{equation*}
Since the metrics $g(\tau=0)$ and $g(\tau=\varepsilon_kr_k^2)$ are close to
each other, the length of the curve $\eta$ measured with respect to
$g(\tau=0)=g(t_k)$ will be at most $r_k/2$ for large enough $k$.
This means that indeed
\begin{equation}\label{6.eq40}
(\eta(\tau),t_k-\tau)\in
B_k^{1/2}\times[t_k-\varepsilon_kr_k^2,t_k),\qquad\forall 0<\tau\leq
\varepsilon_kr_k^2.
\end{equation}
With the bounds from (\ref{6.eq38}) and the lower bound in
\cite[Lemma 4.1]{Muller:MonotoneVolumes}, we obtain
\begin{equation*}
\mathcal{L}_b(\eta)\geq -Cr_k^{-2}(\varepsilon_kr_k^2)^{3/2} =
-C\varepsilon_k^{3/2}r_k,\quad\textrm{i.e.}\quad\ell_b(q,\varepsilon_kr_k^2)\geq
-C\varepsilon_k
\end{equation*}
Thus, the contribution to the reduced volume
$\tilde{V}_k(\varepsilon_kr_k^2)$ coming from $\mathcal{L}_b$-geodesics with
initial vector $\abs{v}\leq\frac{1}{8}\varepsilon_k^{-1/2}$ is bounded
above for large $k$ by
\begin{equation*}
\int_{B_k^{1/2}}(4\pi\varepsilon_kr_k^2)^{-m/2}e^{C\varepsilon_k}dV \leq
C\varepsilon_k^{-m/2}r_k^{-m}\vol(B_k^{1/2})\leq C\varepsilon_k^{m/2}\to
0\quad(k\to\infty).
\end{equation*}
Next, we estimate the contribution of geodesics with large initial
vector $\abs{v}> \frac{1}{8}\varepsilon_k^{-1/2}$ to the reduced volume
$\tilde{V}_k(\varepsilon_kr_k^2)$. Note that we can write the reduced
volume with base point $(x_k,t_k)$ as
\begin{equation*}
\tilde{V}_k(\tau_1)=\int_M
(4\pi\tau_1)^{-m/2}e^{-\ell(q,\tau_1)}dV(q) =
\int_{\Omega(\tau_1,k)}
(4\pi\tau_1)^{-m/2}e^{-\ell(\mathcal{L}_b\mathrm{exp}_{x_k}^{\tau_1}(v),\tau_1)}J(v,\tau_1) dv.
\end{equation*}
Here, $\mathcal{L}_b\mathrm{exp}_{x_k}^{\tau_1}$ is the $\mathcal{L}_b$-exponential map defined in
\cite{Muller:MonotoneVolumes}, taking $v$ to $\eta(\tau_1)$ with
$\eta$ being the $\mathcal{L}_b$-geodesic with initial vector $v$,
$J(v,\tau_1)=\det d(\mathcal{L}_b\mathrm{exp}_{x_k}^{\tau_1})$ denotes the Jacobian of $\mathcal{L}_b\mathrm{exp}_{x_k}^{\tau_1}$ and
$\Omega(\tau_1,k)\subset T_{x_k}M$ is a set which is mapped
bijectively to $M$ up to a set of measure zero under the map
$\mathcal{L}_b\mathrm{exp}_{x_k}^{\tau_1}$. In \cite{Muller:diss}, we prove that the integrand
\begin{equation}\label{6.eq41}
f(v,\tau_1):=(4\pi\tau_1)^{-m/2}e^{-\ell(\mathcal{L}_b\mathrm{exp}_{x_k}^{\tau_1}(v),\tau_1)}J(v,\tau_1)
\end{equation}
is non-increasing in $\tau_1$ for fixed $v$ and has the limit
$\lim_{\tau_1\to 0}f(v,\tau_1)=\pi^{-m/2}e^{-\abs{v}^2}$. Together
with $\Omega(\tau')\subset\Omega(\tau)$ for $\tau\leq\tau'$ this
yields an alternative proof of the monotonicity of the
reduced volumes obtained in \cite{Muller:MonotoneVolumes}. Moreover,
it implies that the contribution to the reduced volume
$\tilde{V}_k(\varepsilon_kr_k^2)$ coming from $\mathcal{L}_b$-geodesics with
initial vector $\abs{v}>\frac{1}{8}\varepsilon_k^{-1/2}$ can be bounded by
\begin{equation}\label{6.eq45}
\int_{\abs{v}>\frac{1}{8}\varepsilon_k^{-1/2}}\pi^{-m/2}e^{-\abs{v}^2}dv
\leq Ce^{-\frac{1}{64\varepsilon_k}} \to 0\quad(k\to\infty),
\end{equation}
which completes the proof of Claim 1.
\end{proof}
\paragraph{Claim 2:} $\tilde{V}_k(t_k)$ is bounded below away from
zero.
\begin{proof}
Let us remark that $\tau=t_k$ corresponds to real time $t=0$. We
assume that $k$ is large enough, so that $t_k\geq T/2$. The idea
behind the proof is to go from $(x_k,t_k)$ to some point $q_k$ at
the real time $T/2$ (i.e.~$\tau=t_k-T/2$) for which the reduced
$\mathcal{L}_b$-distance $\ell_b(q_k,t_k-T/2)$ is small. From the upper
bound on $L_b$ from \cite[Lemma 4.1]{Muller:MonotoneVolumes}, we see
that for small $\tau$ it is possible to find a point $q_k(\tau)$
such that $\ell_b(q_k(\tau),\tau)\leq \frac{m}{2}$. On the other
hand, combining the evolution equations for $\dtau \ell_b$ and $\triangle
\ell_b$, we obtain
\begin{equation}\label{6.eq46}
\dtau\big|_{\tau=\tau_1} \ell_b + \triangle \ell_b \leq
-\tfrac{1}{\tau_1}\ell_b + \tfrac{m}{2\tau_1}
\end{equation}
(in the barrier sense) and hence for the minimum of
$\ell_{min}(\tau)=\min_{q\in M}\ell_b(q,\tau)$
\begin{equation}\label{6.eq47}
\dtau\big|_{\tau=\tau_1} \ell_{min} \leq
-\tfrac{1}{\tau_1}\ell_{min} + \tfrac{m}{2\tau_1}
\end{equation}
in the sense of difference quotients. The latter is obtained by
applying the maximum principle to a smooth barrier. The inequality
(\ref{6.eq47}) shows that there is some point $q_k(\tau)$ with
$\ell_b(q_k(\tau),\tau)\leq \frac{m}{2}$ for every $\tau$. As
mentioned above, we choose $q_k$ at the real time $T/2$ with
$\ell_b(q_k,t_k-T/2)\leq \frac{m}{2}$. Let $\eta:[0,t_k-T/2]\to M$
be an $\mathcal{L}_b$-geodesic realizing this length. Moreover, let
$\eta_p:[t_k-T/2,t_k]\to M$ be $g(t=0)$-geodesics (i.e. a
$g(\tau=t_k)$-geodesic) from $q_k$ at time $\tau=t_k-T/2$ to $p\in
B^{q_k}:=B_{g(\tau=t_k)}(q_k,1)=B_{g(t=0)}(q_k,1)$ at time
$\tau=t_k$. Since $\abs{\Rm{}}$ is uniformly bounded for
$t\in[0,T/2]$, we get a uniform bound for $S$ along this family of curves.
Since all the metrics $g(\tau)$ with $\tau\in[t_k-T/2,t_k]$ are uniformly
equivalent, we get an uniform upper bound for the $\mathcal{L}_b$-length of
all $\eta_p$. From this, we see that the concatenations
$(\eta\!\smile\!\eta_p): [0,t_k]\to M$ connecting $x_k$ to $p\in
B^{q_k}$ have uniformly bounded $\mathcal{L}_b$-length, independent of $p$
and $k$. This gives a uniform bound $\ell_b(p,t_k)\leq C$, for all
$p\in B^{q_k}$ and $k\in\mathbb{N}$ large enough. We can then estimate
\begin{equation*}
\tilde{V}_k(t_k)=\int_M (4\pi t_k)^{-m/2}e^{-\ell_b(q,t_k)}dV(q)
\geq \int_{B^{q_k}} (4\pi t_k)^{-m/2}e^{-C}dV \geq C\inf_{q_k\in
M}\vol({B^{q_k}}),
\end{equation*}
which is bounded below away from zero, independently of $k$. This
proves Claim 2.
\end{proof}
Since the backwards reduced volumes $\tilde{V}_k$ are non-increasing
in $\tau$ (i.e.~non-decreasing in real time $t$) according to
Theorem \ref{6.thm4}, we obtain $\tilde{V}_k(t_k)\leq
\tilde{V}_k(\varepsilon_kr_k^2)$ for $k$ large enough. But since
$\tilde{V}_k(t_k)$ is bounded below away from zero by Claim 2 while
$\tilde{V}_k(\varepsilon_kr_k^2)$ converges to zero with $k\to\infty$ by
Claim 1, we obtain the desired contradiction that proves the
theorem.
\end{proof}
\begin{appendix}
\section{Commutator identities}
It is well-known that a $(p,q)$-tensor $B$ (i.e.~a smooth
section of the bundle $(T^*M)^{\otimes p}\otimes(TM)^{\otimes q}$)
satisfies the following commutator identity in local coordinates
$(x^1,\ldots,x^n)$ induced by a chart $\phi: U \to \mathbb{R}^n$, $U\subseteq M$,
\begin{equation}\label{app.eq7}
[\nabla_i,\nabla_j]B^{k_1\dots k_q}_{\ell_1\dots\ell_p} = \sum_{r=1}^q
R_{ijm}^{k_r}B^{k_1\dots k_{r-1}m k_{r+1}\dots
k_q}_{\ell_1\dots\ell_p} + \sum_{s=1}^p R_{ij\ell_s m}B^{k_1\dots
k_q}_{\ell_1\dots\ell_{s-1}m\ell_{s+1}\dots\ell_p}.
\end{equation}
Now, assume that we are given Levi-Civita connections for all
$(p,q)$ tensors over $(M,g)$ and over $(N,\gamma)$. For a map $\phi:(M,g)\to
(N,\gamma)$, there is a canonical notion of pull-back bundle
$\phi^*TN$ over $M$ with sections $\phi^*V = V\circ\phi$ for
$V\in\Gamma(TN)$. The Levi-Civita connection $\nabla^{TN}$ on $TN$ also
induces a connection $\nabla^{\phi^*TN}$ on this pull-back bundle via
\begin{equation*}
\nabla^{\phi^*TN}_X\phi^*V = \phi^*\big(\nabla^{TN}_{\phi_*X}V\big), \qquad
X\in\Gamma(TM),\; V\in\Gamma(TN).
\end{equation*}
We obtain connections on all product bundles over $M$ with
factors $TM$, $T^*M$, $\phi^*TN$ and $\phi^*T^*N$ via the product
rule and compatibility with contractions. Take coordinates $x^k$
on $M$, $k=1,\ldots,m=\dim M$, and $y^\mu$ on
$N$, $\mu=1,\ldots,n=\dim N$, and write $\partial_k$ for
$\tfrac{\partial}{\partial x^k}$ and $\partial_\mu$ for
$\tfrac{\partial}{\partial y^\mu}$. We get $\nabla_i\nabla_jV_\kappa-\nabla_j\nabla_iV_\kappa=R_{ij\kappa\lambda}V_\lambda$ with
\begin{align*}
R_{ij\kappa\lambda}(x)&=\big\langle\Rm{\partial_i,
\partial_j}\phi^*\big(\partial_\lambda\big),
\phi^*\big(\partial_\kappa\big)\big\rangle_{\phi^*TN}(x)\\%
&=\Scal{\NRm{}\big(\phi_*\partial_i, \phi_*\partial_j\big)
\partial_\lambda, \partial_\kappa}_{TN}(\phi(x))\\%
&= \!\phantom{l}^N\!\!R_{\mu\nu\kappa\lambda}(\phi(x))
\nabla_i\phi^\mu(x)\nabla_j\phi^\nu(x),
\end{align*}
where we used $\phi_*\partial_i =\nabla_i\phi^\mu\,\partial_\mu$. This
allows to extend (\ref{app.eq7}) to mixed tensors, for example
\begin{equation}\label{app.eq8}
[\nabla_i,\nabla_j]B^{k\kappa}_{\ell\lambda} =
R_{ijp}^kB^{p\kappa}_{\ell\lambda} + R_{ij\ell
p}B^{k\kappa}_{p\lambda} + R_{ij\varrho}^\kappa
B^{k\varrho}_{\ell\lambda} + R_{ij\lambda\varrho}
B^{k\kappa}_{\ell\varrho}.
\end{equation}
The standard example that will be used quite often is the following.
The derivative $\nabla\phi$ of $\phi:M\to N$ is a section of
$T^*M\otimes\phi^*TN$. Thus, the intrinsic second order derivative
is built with the connection on this bundle, i.e.~$\nabla_i\nabla_j\phi^\lambda = \partial_i\partial_j\phi^\lambda -
\Gamma_{ij}^k\partial_k\phi^\lambda + \!\phantom{l}^N\!\Gamma_{\mu\nu}^\lambda
\partial_i\phi^\mu\partial_j\phi^\nu$ and similar for higher derivatives.
Using (\ref{app.eq8}), we obtain
\begin{equation}\label{app.eq10}
\nabla_i\nabla_j\nabla_\ell\phi^\beta-\nabla_j\nabla_i\nabla_\ell\phi^\beta = R_{ij\ell p}\nabla_p\phi^\beta +
\!\phantom{l}^N\!\!R_{\mu\nu\lambda}^\beta\nabla_\ell\phi^\lambda\nabla_i\phi^\mu\nabla_j\phi^\nu.
\end{equation}
There is also a different way to obtain these formulas, which is
especially useful when $\phi$ is evolving and we also want to
include time derivatives. We learned this from \cite{Lamm:diss}.
Here, we interpret $\nabla^k\phi$ as a $k$-linear $TN$-valued map along
$\phi\in C^\infty(M,N)$ rather than as a section in $(T^*M)^{\otimes
k}\otimes \phi^*TN$. Leting $\omega$ be any such $k$-linear $TN$-valued
map along $\phi$, i.e.~$\omega(x):(T_xM)^{\times k}\to T_{\phi(x)}N$,
the covariant derivative $\nabla\omega$ is a $(k+1)$-linear $TN$-valued
map along $\phi$ etc. The curvature tensor $\!\phantom{l}^k\Rm{}$ for
$\omega$ can then be computed by
\begin{equation}\label{app.eq12}
\begin{aligned}
(\!\phantom{l}^k\!\Rm{X,Y}\omega)(X_1,\ldots,X_k)
&=\NRm{\nabla\phi(X),\nabla\phi(Y)}\omega(X_1,\ldots,X_k)\\%
&\quad\,-\sum_{s=1}^k \omega(X_1,\ldots,\Rm{X,Y}X_s,\ldots,X_k).
\end{aligned}
\end{equation}
Of course, this agrees with the definition above, where we used the
bundle interpretation. Now, if $\phi$ is time-dependent, we simply
interpret it as a map $\tilde{\phi}:M\times I\to N$ and interpret
$\nabla^k\phi$ as $k$-linear $TN$-valued maps on $M\times I$ along
$\tilde{\phi}$. The formalism stays exactly the same.
\newline
Note that $\dt$ induces a covariant time derivative $\nabla_t$ (on all
bundles over $M\times I$) that agrees with $\dt$ for time-dependent
functions. Choose coordinates $x^i$ for $M$ with
\begin{equation}\label{app.eq13}
\nabla_t(\dt) = \nabla_i(\partial_j) = \nabla_t(\partial_i) = \nabla_i(\dt) = 0,
\quad \forall i,j=1,\ldots,m
\end{equation}
at some base point $(p,t)$ in $M\times I$. Then, using
(\ref{app.eq12}), we obtain for $\omega=\nabla\phi$
\begin{equation}\label{app.eq14}
\begin{aligned}
\nabla_t(\nabla_i\nabla_j\phi) &= \nabla_t((\nabla_i\omega)(\partial_j)) =
(\nabla_t\nabla_i\omega)(\partial_j) = (\nabla_i\nabla_t\omega +
\!\!\phantom{l}^1\!\Rm{\dt,\partial_i}\omega)(\partial_j)\\%
&=\nabla_i((\nabla_t\omega)(\partial_j))+\NRm{\dt\phi,\nabla_i\phi}\omega(\partial_j)
- \omega(\!\phantom{l}^{M\times I}\!\Rm{\dt,\partial_i}\partial_j)\\%
&=\nabla_i\nabla_j\dt\phi + \NRm{\dt\phi,\nabla_i\phi}\nabla_j\phi.
\end{aligned}
\end{equation}
\begin{rem}
If we also vary the metric $g$ on $M$ in time, we will get an
additional term from the evolution of $\nabla_i\nabla_j$, namely
$-(\dt\Gamma_{ij}^k)\nabla_k\phi$. Note that $\dt\Gamma$ is a tensor,
while $\Gamma$ itself is not.
\end{rem}
\end{appendix}
\makeatletter
\def\@listi{
\itemsep=0pt
\parsep=1pt
\topsep=1pt}
\makeatother
{\fontsize{10}{11}\selectfont
Reto M\"uller\\
{\sc Scuola Normale Superiore di Pisa, 56126 Pisa, Italy}
\end{document} |
\begin{document}
\title{Poisson Reduction}
\author{Chiara Esposito}
\address{Department of Mathematics, Universitat Autonoma de Barcelona, 08193 Bellaterra. Spain}
\maketitle
\begin{abstract}
In this paper we develope a theory of reduction for classical systems with Poisson Lie groups symmetries using the notion of momentum map introduced by Lu. The local description of Poisson manifolds and Poisson Lie groups and the properties of Lu's momentum map allow us to define a Poisson reduced space.
\end{abstract}
\section{Introduction}\label{intro}
In this paper we prove a generalization of the Marsden-Weinstein reduction to the general case of an arbitrary Poisson Lie group action on a Poisson manifold. Reduction procedures are known in many different settings. In particular, a reduction theory is known in the case of Poisson Lie groups acting on symplectic manifolds \cite{Lu3} and in the case of Lie groups acting on Poisson manifolds \cite{RO}, \cite{MR}. An important generalization to the Dirac setting has been studied in \cite{BC}.
The theory of symplectic reduction plays a key role in classical mechanics. The phase
space of a system of $n$ particles is described by a symplectic or more generally Poisson manifold. Given a symmetry group of dimension $k$ acting on a mechanical system, the dimension of the phase space can be reduced by $2k$. Marsden-Weinstein reduction formalizes this feature. Recall roughly the notion of Hamiltonian actions in this setting. Given a Poisson manifold $M$ there are natural Hamiltonian vector fields $\{f, \cdot \}$ on $M$. Let $G$ be a Lie group acting on $M$ by $\Phi$; the action is Hamiltonian if the vector fields defined by the infinitesimal generator of $\Phi$ are Hamiltonian. More precisely, let $G$ be a Lie group acting on a Poisson manifold $(M, \pi)$. The action $\Phi:G\times M \to M$ is canonical if it preserves the Poisson structure $\pi$. Suppose that there exists a linear map $H: \mathfrak{g} \to C^{\infty}(M)$ such that the infinitesimal generator $\Phi_{X}$ for $X\in \mathfrak{g}$ of the canonical action is induced by $H$ by
$$
\Phi_{X}= \{H_{X}, \cdot\}.
$$
A canonical action induced by $H$ is said Hamiltonian if $H$ is a Lie algebra homomorphism. We can define a map $\boldsymbol{\mu}: M \to \mathfrak{g}^*$, called momentum map, by $H_{X}(m)=\langle \boldsymbol{\mu}(m), X\rangle$ for $m\in M$. It is equivariant if the corresponding $H$ is a Lie algebra homomorphism. Given an Hamiltonian action, under certain assumptions, the reduced space has been defined as $M//G:=\boldsymbol{\mu}^{-1}(u)/G_{u}$ and it has been proved that it is a Poisson manifold \cite{MsWe}.
In this paper we are interested in analyzing the case in which one has an extra structure on the Lie group, a Poisson structure making it a Poisson Lie group. Poisson Lie groups are very interesting objects in mathematical physics. They may be regarded as classical limit of quantum groups \cite{Dr1} and they have been studied as carrier spaces of dynamical systems \cite{LMS}. It is believed that actions of Poisson Lie groups on Poisson manifolds should be used to understand the ``hidden symmetries'' of certain integrable systems \cite{STS}. Moreover, the study of classical systems with Poisson Lie group symmetries may give information about the corresponding quantum group invariant system (an attempt can be found in \cite{me}, \cite{me1}).
The purpose of this paper is to prove that, given a Poisson manifold acted by a Poisson Lie group, under certain conditions, we can also reduce this phase space to another Poisson manifold.
The paper is organized as follows. In Section \ref{sec_pg} we recall some basic elements of Poisson geometry: Poisson manifolds and their local description, Lie bialgebras and Poisson Lie groups. A nice review of these results can be found in \cite{V} and \cite{YK}. The Section \ref{sec_mm} is devoted to Poisson actions and associated momentum maps and we discuss dressing actions and their properties. In Section \ref{sec: pr} we present the main result of this paper, the Poisson reduction, and we discuss an example.
\noindent{\bf Acknowledgments:} I would like to thank my advisor Ryszard Nest and Eva Miranda for many interesting discussions about Poisson reduction and its possible developments. I also wish to thank George M. Napolitano for his help and his useful suggestions and Rui L. Fernandes for his comments regarding Dirac reduction theory.
\section{Poisson manifolds, Poisson Lie groups and Lie bialgebras}\label{sec_pg}
In this section we introduce the notion of Poisson manifolds and their local description, we give some background about Poisson Lie groups and Lie bialgebras which will be used in the paper. For more details on this subject, see \cite{Lu3}, \cite{Dr1}, \cite{YK}, \cite{V}, \cite{We1}.
\subsection{Poisson manifolds and symplectic foliation}\label{sec_1.1}
A Poisson structure on a smooth manifold $M$ is a Lie bracket $\{\cdot, \cdot\}$ on the space $C^{\infty}(M)$ of smooth functions on $M$ which satisfies the Leibniz rule. This bracket is called Poisson bracket and a manifold $M$ equipped with such a bracket is called Poisson manifold.
Therefore, a bivector field $\pi$ on $M$ such that the bracket
$$
\{ f, g\}:= \langle \pi, df\wedge dg\rangle
$$
is a Poisson bracket is called Poisson tensor or Poisson bivector field. A Poisson tensor can be regarded as a bundle map $\pi^{^{\sharp}arp}: T^*M\to TM$:
$$
\langle \alpha, \pi^{^{\sharp}arp}(\beta)\rangle = \pi(\alpha,\beta)
$$
\begin{definition}
A mapping $\phi: (M_1,\pi_1)\rightarrow (M_2,\pi_2)$ between two Poisson manifolds is called a Poisson mapping if $\forall f,g\in C^{\infty}(M_2)$ one has
\begin{equation}
\lbrace f\circ\phi, g\circ\phi\rbrace_1=\lbrace f,g\rbrace_2\circ \phi
\end{equation}
\end{definition}
The structure of a Poisson manifold is described by the splitting theorem of Alan Weinstein \cite{We1}, which shows that locally a Poisson manifold is a direct product of a symplectic manifold with another Poisson manifold whose Poisson tensor vanishes at a point.
\begin{theorem}[Weinstein]\label{thm: split}
On a Poisson manifold $(M,\pi)$, any point $m\in M$ has a coordinate neighborhood with coordinates
$(q_1,\dots,q_k,p_1,\dots,p_k,\allowbreak y_1,\dots,y_l)$ centered at $m$, such that
\begin{equation}\label{eq: splitp}
\pi=\sum_i \frac{\partial}{\partial q_i}\wedge\frac{\partial}{\partial p_i}+\frac{1}{2}\sum_{i,j}\phi_{ij}(y) \frac{\partial}{\partial y_i}\wedge\frac{\partial}{\partial y_j}\qquad \phi_{ij}(0)=0.
\end{equation}
The rank of $\pi$ at $m$ is $2k$. Since $\phi$ depends only on the $y_i$s, this theorem gives
a decomposition of the neighborhood of $m$ as a product of two Poisson manifolds: one with rank $2k$, and the other with rank 0 at $m$.
\end{theorem}
The term
\begin{equation}
\frac{1}{2}\sum_{i,j}\phi_{ij}(y)\frac{\partial}{\partial y_i}\wedge \frac{\partial}{\partial y_j}
\end{equation}
is called transverse Poisson structure and it is evident that the equations $y_{i}=0$ determine the symplectic leaf through $m$.
\subsection{Lie bialgebras and Poisson Lie groups}
\label{sec:lie bialgebras}
\begin{definition}
A Poisson Lie group $(G,\pi_G)$ is a Lie group equipped with a multiplicative Poisson structure $\pi_G$, i.e. such that the multiplication map $G\times G \to G$ is a Poisson map.
\end{definition}
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. The linearization $\delta:= d_{e}\pi_{G}: \mathfrak{g}\to \mathfrak{g}\wedge \mathfrak{g}$ of $\pi_{G}$ at $e$ defines a Lie algebra structure on the dual $\mathfrak{g}^*$ of $\mathfrak{g}$ and, for this reason, it is called cobracket. The pair $(\mathfrak{g},\mathfrak{g}^*)$ is called Lie bialgebra. The relation between Poisson Lie groups and Lie bialgebras has been proved by Drinfeld \cite{Dr1}:
\begin{theorem}\label{thm: dr}
If $(G,\pi_G)$ is a Poisson Lie group, then the linearization of $\pi_G$ at $e$ defines a Lie algebra structure on $\mathfrak{g}^*$ such that $(\mathfrak{g},\mathfrak{g}^*)$ form a Lie bialgebra over $\mathfrak{g}$,
called the tangent Lie bialgebra to $(G,\pi_G)$. Conversely, if $G$ is connected and simply connected, then every Lie bialgebra $(\mathfrak{g},\mathfrak{g}^*)$ over $\mathfrak{g}$ defines a unique multiplicative Poisson
structure $\pi_G$ on $G$ such that $(\mathfrak{g},\mathfrak{g}^*)$ is the tangent Lie bialgebra to the Poisson Lie group $(G,\pi_G)$.
\end{theorem}
From this theorem it follows that there is a unique connected and simply connected Poisson Lie group $(G^*,\pi_{G^*})$, called the dual of $(G,\pi_G)$, associated to the Lie bialgebra $(\mathfrak{g}^*,\delta)$. If $G$ is connected and simply connected, then the dual of $G^*$ is $G$.
\begin{example}[$\mathfrak{g}=ax+b$]\label{ex: 1}
Consider the Lie algebra $\mathfrak{g}$ spanned by $X$ and $Y$ with commutator
\begin{equation}
[X,Y]=Y
\end{equation}
and cobracket given by
\begin{equation}
\delta(X)=0 \quad \delta(Y)= X\wedge Y.
\end{equation}
The Lie bracket on $\mathfrak{g}^*$ is given by
$$
[X^*,Y^*]=Y^*.
$$
A matrix representation of $\mathfrak{g}$ is the Lie algebra $\mathfrak{gl}(2,\mathbb{R})$ via
$$
X = \left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right) \quad Y = \left(\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}\right)
$$
and
$$
X^* = \left(\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}\right) \quad Y^* = \left(\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}\right)
$$
with the metric $\gamma(a,b)= tr(aJbJ)$ and $J=\left(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right)$.
The corresponding Poisson Lie group $G$ and dual $G^*$
are subgroups
of $GL(2,\mathbb{R})$ of matrices with positive determinant are given by
\begin{equation}
G=\left\lbrace \left(\begin{matrix} 1 & 0 \\ \xi & \eta \end{matrix}\right)\; :\eta>0\right\rbrace \qquad G^*=\left\lbrace \left(\begin{matrix} s & t \\ 0 & 1 \end{matrix}\right)\; :s>0\right\rbrace
\end{equation}
\end{example}
\section{Poisson actions and Momentum maps}\label{sec_mm}
In this section we first introduce the concept of Poisson action of a Poisson Lie group on a Poisson manifold, which generalizes the canonical action of a Lie group on a symplectic manifold. We define momentum maps associated to such actions and finally we consider the particular case of a Poisson Lie group $G$ acting on its dual $G^*$ by dressing transformations. This allows us to study the symplectic leaves of $G$ that are exactly the orbits of the dressing action. These topics can be found e.g. in \cite{Lu3}, \cite{Lu1} and \cite{STS}.
From now on we assume that $G$ is connected and simply connected.
\begin{definition}
The action $\Phi:G\times M\rightarrow M$ of a Poisson Lie group $(G,\pi_G)$ on a Poisson manifold $(M,\pi)$ is called Poisson action if $\Phi$ is a Poisson map, where $G\times M$ is a Poisson manifold with structure $\pi_G\oplus\pi$.
\end{definition}
This definition generalizes the notion of canonical action; indeed, if
$G$ carries the trivial Poisson structure $\pi_G=0$, the action $\Phi$ is Poisson if and only if it preserves $\pi$, i.e. if it is canonical. In general, the structure $\pi$ is not invariant with respect to the action $\Phi$. The easiest examples of Poisson actions are given by the left and right actions of $G$ on itself.
For an action $\Phi: G\times M \to M$ we use $\Phi: \mathfrak{g} \to Vect\, M: X \mapsto \Phi_{X}$ to denote the Lie algebra anti-homomorphism which defines the infinitesimal generators of this action. The proof of the following Theorem can be found in \cite{LuWe1}.
\begin{theorem}
The action $\Phi: G\times M\rightarrow M$ is a Poisson action if and
only if
\begin{equation}\label{eq: pa}
L_{\Phi_{X}}(\pi)=(\Phi\wedge\Phi)\delta(X)
\end{equation}
for any $X\in\mathfrak{g}$, where $L$ denotes the Lie derivative and $\delta$ is the derivative of $\pi_{G}$ at $e$.
\end{theorem}
Let $\Phi:G\times M \to M$ be a Poisson action of $(G, \pi_{G})$ on $(M,\pi)$. Let $G^*$ be the dual Poisson Lie group of $G$ and let $\Phi_{X}$ be the vector field on $M$ which generates the action $\Phi$.
In this formalism the definition of momentum map reads (Lu, \cite{Lu3}, \cite{Lu1}):
\begin{definition}\label{def: mm}
A momentum map for the Poisson action $\Phi:G\times M\rightarrow M$ is a map $\boldsymbol{\mu}: M\rightarrow G^*$ such that
\begin{equation}\label{eq: mmp}
\Phi_{X}=\pi^{^{\sharp}arp}(\boldsymbol{\mu}^*(\theta_{X}))
\end{equation}
where $\theta_{X}$ is the left invariant 1-form on $G^*$ defined by the element $X\in\mathfrak{g}=(T_eG^*)^*$ and $\boldsymbol{\mu}^*$ is the cotangent lift $T^* G^*\rightarrow T^*M$.
\end{definition}
In other words, the momentum map generates the vector field $\Phi_{X}$ via the construction
$$
X\in \mathfrak{g} \to \theta_{X}\in T^*G^* \to \alpha_{X}= \boldsymbol{\mu}^*(\theta_{X})\in T^*M \to \pi^{^{\sharp}arp}(\alpha_{X})\in TM
$$
It is important to remark that Noether's theorem still holds in this general context.
\begin{theorem}
Let $\Phi:G\times M \to M$ a Poisson action with momentum map $\boldsymbol{\mu}: M\rightarrow G^*$. If $H\in C^{\infty}(M)$ is $G$-invariant, then $\boldsymbol{\mu}$ is an integral of the Hamiltonian vector field associated to $H$.
\end{theorem}
It is important to point out that in this setting the vector field $\Phi_{X}$ is not Hamiltonian, unless the Poisson structure on $G$ is trivial. In this case $G^*=\mathfrak{g}^*$, the differential 1-form $\theta_{X}$ is the constant 1-form $X$ on $\mathfrak{g}^*$, and
\begin{equation}
\boldsymbol{\mu}^*(\theta_{X})=d(H_{X}),\quad\text{where}\quad H_{X}(m)=\langle\boldsymbol{\mu}(m),X \rangle.
\end{equation}
This implies that the momentum map is the canonical one and
\begin{equation}
\Phi_{X}=\pi^{^{\sharp}arp}(dH_{X})=\{H_{X}, \cdot\}.
\end{equation}
In other words, $\Phi_{X}$ is the Hamiltonian vector field with Hamiltonian $H_{X}\in C^{\infty}(M)$. We observe that,
when $\pi_G$ is not trivial, $\theta_{X}$ is a Maurer-Cartan form, hence $\boldsymbol{\mu}^*(\theta_{X})$ can not be written as a differential of a Hamiltonian function.
In the following we give an example for the infinitesimal generator in this general case.
\subsection{Dressing Transformations}\label{sec: dressing}
One of the most important example of Poisson action is the dressing action of $G$ on $G^*$. The name ``dressing'' comes from the theory of integrable systems and was introduced in this context in \cite{STS}. Interesting examples can be found in \cite{AM}.
We remark that, given a Poisson Lie group $(G, \pi_{G})$, the left (right) invariant 1-forms on $G^*$ form a Lie algebra with respect to the bracket:
$$
[\alpha,\beta] = L_{\pi^{^{\sharp}arp}(\alpha)}\beta-L_{\pi^{^{\sharp}arp}(\beta)}\alpha - d(\pi(\alpha, \beta)).
$$
For $X\in\mathfrak{g}$, let $\theta_{X}$ be the left invariant 1-form on $G^*$ with value $X$ at $e$. Let us define the vector field on $G^*$
\begin{equation}\label{eq: idr}
l(X)=\pi_{G^*}^{\sharp}(\theta_{X}).
\end{equation}
The map $l: \mathfrak{g}\to TG^*: X\mapsto l(X)$ is a Lie algebra anti-homomorphism. We call $l$ the left infinitesimal dressing action of $\mathfrak{g}$ on $G^*$; its linearization at $e$ is the coadjoint action of $\mathfrak{g}$ on $\mathfrak{g}^*$. Similarly we can define the right infinitesimal dressing action.
Let $l(X)$ (resp. $r(X)$) a left (resp. right) dressing vector field on $G^*$. If all the dressing vector fields are complete, we can integrate the $\mathfrak{g}$-action into an action of $G$ on $G^*$ called the
dressing action and we say that the dressing actions consist of dressing transformations.
\begin{definition}
A multiplicative Poisson tensor $\pi_G$ on $G$ is complete if each left (equiv. right) dressing vector field is complete on $G$.
\end{definition}
From the definition of dressing action follows (the proof can be found in \cite{STS}) that the orbits of the right or left dressing action of $G^*$ (resp. $G$) are
the symplectic leaves of $G$ (resp. $G^*$).
It can be proved (see \cite{Lu3}) that if $\pi_{G}$ is complete, both left and right dressing actions are Poisson actions with momentum map given by the identity.
Assume that $G$ is a complete Poisson Lie group. We denote respectively the left (resp. right) dressing action of $G$ on its dual $G^*$ by $g\mapsto l_g$ (resp. $g\mapsto r_g$).
\begin{definition}
A momentum map $\boldsymbol{\mu}:M\rightarrow G^*$ for a left (resp. right) Poisson action $\Phi$ is called G-equivariant if it is such with respect to the left dressing action of $G$ on $G^*$, that is,
$\boldsymbol{\mu}\circ \Phi_g=\lambda_g\circ \boldsymbol{\mu}$ (resp. $\boldsymbol{\mu}\circ \Phi_g=\rho_g\circ \boldsymbol{\mu}$)
\end{definition}
It is important to remark that a momentum map is $G$-equivariant if and only if it is a Poisson map, i.e. $\boldsymbol{\mu}_*\pi=\pi_{G^*}$.
\begin{definition}
An action $\Phi: G\times M \to M$ of a Poisson Lie group $(G, \pi_{G})$ on a Poisson manifold $(M, \pi)$ is said Hamiltonian if it is a Poisson action generated by an equivariant momentum map.
\end{definition}
\section{Poisson Reduction}\label{sec: pr}
\label{sec: poisson reduction}
In this section we present the main result of this paper. We show that, given a Hamiltonian action $\Phi$, as defined above, we can define a reduced manifold in terms of momentum map and prove that it is a Poisson manifold. The approach used is a generalization of the orbit reduction \cite{Ml} in symplectic geometry. Recall that, under certain conditions, the orbit space of $\Phi$ is a smooth manifold and it carries a Poisson structure. First, we give an alternate proof of this claim. Then, we consider a generic orbit $\mathcal{O}_{u}$ of the dressing action of $G$ on $G^*$, for $u\in G^*$, and we prove that the set $\boldsymbol{\mu}^{-1}(\mathcal{O}_{u})/G$ is a regular quotient manifold with Poisson structure induced by the Poisson structure on $M$. Similarly to the symplectic case, this reduced space is isomorphic to the space $\boldsymbol{\mu}^{-1}(u)/G_{u}$ which will be regarded as the Poisson reduced space.
\subsection{Poisson structure on $M/G$}
Consider a Hamiltonian action of a connected and simply connected Poisson Lie group
$(G,\pi_{G})$ on a Poisson manifold $(M,\pi)$. It is known that, if the action is proper and free, the orbit space $M/G$ is a smooth manifold, it carries a Poisson structure such that the natural projection $M \to M/G$ is a Poisson map (a proof of this result can be found in \cite{STS}).
In this section we give an alternate proof of this result, by introducing an explicit formulation for the infinitesimal generator of the Hamiltonian action, in terms of local coordinates.
As discussed in the previous section, a Hamiltonian action is a Poisson action induced by an equivariant momentum map $\boldsymbol{\mu}: M \to G^*$ by formula (\ref{eq: mmp}). In other words, the map
$$
\alpha: \mathfrak{g} \to \Omega^1(M): X \mapsto \alpha_{X}=\boldsymbol{\mu}^*(\theta_{X})
$$
is a Lie algebra homomorphism such that
$$
\Phi_{X}=\pi^{^{\sharp}arp}(\alpha_{X})
$$
The dual map of $\alpha$ defines a $\mathfrak{g}^*$-valued 1-form on $M$, still denoted by $\alpha$, satisfying Maurer-Cartan equation (as proved in \cite{Lu3})
$$
d\alpha+\frac{1}{2}[\alpha,\alpha]_{\mathfrak{g}^*}=0.
$$
In particular,
$$
\{\alpha_{X}: X\in\mathfrak{g}\}
$$
defines a foliation $\mathcal{F}$ on $M$.
\begin{lemma}\label{thm: m/g}
The space of $G$-invariant functions on $M$ is closed under Poisson bracket. Hence $\pi$ defines a Poisson structure on $M/G$
\end{lemma}
\begin{proof}
Let $H_{i}$, $i=1, \dots n$ be local coordinates on $M$ such that
$$
\mathcal{F}= Ker\{dH_{1}, \dots, dH_{n}\}
$$
Then
\begin{equation}\label{eq: al}
\alpha_{X}=\sum_{i}c_{i}(X)dH_i
\end{equation}
and
\begin{equation}\label{eq: xis}
\Phi_{X}[f]=\pi^{^{\sharp}arp}(\alpha_{X})=\sum_{i}c_{i}(X)\lbrace H_j,f\rbrace_{M}.
\end{equation}
This implies that a function $f\in C^{\infty}(M)$ is $G$-invariant ($\Phi_{X}[f]=0$) if and only if
$\lbrace H_i,f\rbrace=0$ for any $i$. If $f,g$ are $G$-invariant functions on $M$, we have $\lbrace H_i,f\rbrace=\lbrace H_i,g\rbrace=0$ for any $i$. Then, using the Jacobi identity we get
$\lbrace H_i,\lbrace f,g\rbrace\rbrace=0$. Since $G$ is connected, the result follows.
\end{proof}
\subsection{Poisson reduced space}
Assume that $G$ is connected, simply connected and complete. In order to define a reduced space and to prove that it is a Poisson manifold we consider a generic orbit $\mathcal{O}_u$ of the dressing orbit of $G$ on $G^*$ passing through $u\in G^*$. First, we prove the following:
\begin{lemma}
Let $\Phi:G\times M \to M$ be a free and Hamiltonian action of a compact Poisson Lie group $(G,\pi_{G})$ on a Poisson manifold $(M,\pi)$. Then:
\begin{itemize}
\item[(i)] $\mathcal{O}_u$ is closed and the Poisson structure $\pi_{G^*}$ does not depend on the transversal coordinates on $\mathcal{O}_u$.
\item[(ii)] $\boldsymbol{\mu}^{-1}(\mathcal{O}_u)/G$ is a smooth manifold.
\end{itemize}
\end{lemma}
\begin{proof}
\begin{itemize}
\item[(i)] If $G$ is compact, any $G$-action is automatically proper. This implies that, given $u\in G^*$ the generic orbit $\mathcal{O}_u$ of the dressing action is closed. From section (\ref{sec: dressing}) we know that $\mathcal{O}_u$ is the symplectic leaf through $u$. Using the local description of Poisson manifolds introduced in Theorem (\ref{thm: split}) it is evident that $\pi_{G^*}$ restricted to $\mathcal{O}_u$ does not depend on the transversal coordinates $y_{i}$.
\item[(ii)] If the action $\Phi$ is free, the momentum map $\boldsymbol{\mu}:M \to G^*$ is a submersion onto some open subset of $G^*$. This implies that $\boldsymbol{\mu}^{-1}(u)$ is a closed submanifold of $M$. As $\boldsymbol{\mu}$ is equivariant, it follows that $\boldsymbol{\mu}^{-1}(u)$ is $G$-invariant. Free and proper actions of $G$ on $M$ restrict to free and proper $G$-actions on $G$-invariant submanifolds. In particular, the action of $G$ on $\boldsymbol{\mu}^{-1}(u)$ is still proper, then $G\cdot \boldsymbol{\mu}^{-1}(u)$ is closed. Using the equivariance we have that $G\cdot \boldsymbol{\mu}^{-1}(u)= \boldsymbol{\mu}^{-1}(\mathcal{O}_u)$, which is still $G$-invariant. The action of $G$ on $\boldsymbol{\mu}^{-1}(\mathcal{O}_u)$ is proper and free, so we can conclude that the orbit space $\boldsymbol{\mu}^{-1}(\mathcal{O}_u)/G$ is a smooth manifold.
\end{itemize}
\end{proof}
We aim to prove that the manifold $N/G:=\boldsymbol{\mu}^{-1}(\mathcal{O}_u)/G$ carries a Poisson structure.
In the previous Lemma we stated that $\pi_{G^{*}}$ restricted to $\mathcal{O}_u$ does not depend on the transversal coordinates $y_i$'s; if $x_{i}$ are local coordinates along $N=\boldsymbol{\mu}^{-1}(\mathcal{O}_u)$ and $H_{i}$ are pullback of the transversal coordinates $y_{i}$'s by
\begin{equation}
H_{i}:= y_{i}\circ \boldsymbol{\mu}
\end{equation}
we can easily deduce that the Poisson structure $\pi$ on $M$ involves derivatives in $H_{i}$ only in the combination
$$
\partial_{x_i}\wedge\partial_{H_i}
$$
This is evident because the differential $d\boldsymbol{\mu}$ between $TM\vert_N/TN$ and $TG^{*}/T\mathcal{O}_u$ is a bijective map. Moreover, since $\{y_{i},y_{j}\}$ vanishes on the orbit $\mathcal{O}_u$, $\{H_{i},H_{j}\}$ vanishes on the preimage $N$ and $dH_{i}$'s are in the span of $\{\alpha_{X}:X\in\mathfrak{g}\}$.
Now we introduce the ideal $\mathcal{I}$ generated by $H_i$ and prove some properties.
\begin{lemma}
Let $\mathcal{I}=\{f\in C^{\infty}(M): f\vert_{N}=0\}$.
\begin{itemize}
\item[(i)] $\mathcal{I}$ is defined in an open $G$-invariant neighborhood $U$ of $N$.
\item [(ii)] $\mathcal{I}$ is closed under Poisson bracket.
\end{itemize}
\end{lemma}
\begin{proof}
\begin{itemize}
\item[(i)] The coordinates $H_i$ are locally defined but we can show that $\mathcal{I}$ is globally defined.
Considering a different neighborhood on the orbit of $G^{*}$ we have transversal coordinates $y_i^{\prime}$ and their pullback to $M$ will be $H_i^{\prime}=y_i^{\prime}\circ \boldsymbol{\mu}$.
The coordinates $H_i^{\prime}$ are defined in a different open neighborhood $V$ of $N$, but we can see that the ideal $\mathcal{I}$ generated by $H_i$ coincides with $\mathcal{I}^{\prime}$ generated by $H_i^{\prime}$
on the intersection of $U$ and $V$, then it is globally defined.
\item [(ii)] Since $\boldsymbol{\mu}$ is a Poisson map we have:
$$
\{ H_i,H_j\}_{M}=\{ y_i\circ \boldsymbol{\mu},y_j\circ \boldsymbol{\mu}\}_{M}=\{ y_i,y_j\}_{G^*}\circ \boldsymbol{\mu}.
$$
Hence the ideal $\mathcal{I}$ is closed under Poisson brackets.
\end{itemize}
\end{proof}
Motivated by this Lemma we use the following identification
$$
C^{\infty}(N/G)\simeq(C^{\infty}(U)/\mathcal{I})^G.
$$
\begin{lemma}\label{lem: id2}
Suppose that $N/G$ is an embedded submanifold of the smooth manifold $M/G$, then
\begin{equation}
(C^{\infty}(U)/\mathcal{I})^G \simeq(C^{\infty}(U)^G + \mathcal{I})/\mathcal{I}
\end{equation}
\end{lemma}
\begin{proof}
Let $f$ be a smooth function on $U$ satisfying $[f]\in (C^{\infty}(U)/\mathcal{I})^G$. As the equivalence class $[f]$ is $G$-invariant, we have
\begin{equation}
f(G\cdot m)=f(m)+i(m),
\end{equation}
where $i\in \mathcal{I}$ and $G\cdot m$ is a generic orbit of the Hamiltonian action of $G$ on $M$. It is clear that $f\vert_N$ is $G$-invariant and hence it defines a smooth function $\bar{f}\in C^{\infty}(N/G)$.
Since $N/G$ is a $k$-dimensional embedded submanifold of the $n$-dimensional smooth manifold $M/G$, the inclusion map $\iota: N/G\rightarrow M/G$ has local coordinates representation:
\begin{equation}
(x_1,\dots,x_k)\mapsto (x_1,\dots,x_k,c_{k+1},\dots,c_n)
\end{equation}
where $c_i$ are constants. Hence we can extend $\bar{f}$ to a smooth function $\phi$ on $M/G$ by setting $\bar{f}(x_1,\dots,x_k)=\phi(x_1,\dots,x_k,0,\dots,0)$.
The pullback $\tilde{f}$ of $\phi$ by $\text{pr}:M\rightarrow M/G$ is $G$-invariant and satisfies
\begin{equation}
\tilde{f}-f\vert_N=0,
\end{equation}
hence $\tilde{f}-f\in \mathcal{I}$.
\end{proof}
Using these results we can prove the following:
\begin{theorem}\label{thm: pred}
Let $\Phi:G\times M\rightarrow M$ be a free Hamiltonian action of a compact Poisson Lie group $(G,\pi_G)$ on a Poisson manifold $(M,\pi)$ with momentum map $\boldsymbol{\mu}:M\rightarrow G^*$. The orbit space $N/G$ has a Poisson structure induced by $\pi$.
\end{theorem}
\begin{proof}
First we prove that the Poisson bracket of $M$ induces a well defined Poisson bracket on $(C^{\infty}(U)^G+\mathcal{I})/\mathcal{I}$.
In fact, for any $f+i\in C^{\infty}(U)^{G}/\mathcal{I}$ and $j\in \mathcal{I}$ the Poisson bracket $\{ f+i,j\}$ still belongs to the ideal $\mathcal{I}$.
Since the ideal $\mathcal{I}$ is closed under Poisson brackets, $\{ i,j\}$ belongs to $\mathcal{I}$.
The function $j$, by definition on the ideal $\mathcal{I}$, can be written as a linear combination of $H_i$, so $\{ f,j\}=\sum_i a_i\{ f,H_i\}$.
By Lemma \ref{thm: m/g}, we have $\{ f,H_i\}=0$, hence $\{ f+i,j\}\in \mathcal{I}$ as stated.
Finally, using the isomorphism proved in the Lemma (\ref{lem: id2}) and the identification $
C^{\infty}(N/G)\simeq(C^{\infty}(U)/\mathcal{I})^G$, the claim is proved.
\end{proof}
Finally, we observe that there is a natural isomorphism
\begin{equation}
\boldsymbol{\mu}^{-1}(u)/G_{u}\simeq \boldsymbol{\mu}^{-1}(\mathcal{O}_u)/G.
\end{equation}
We refer to $\boldsymbol{\mu}^{-1}(u)/G_{u}$ as the Poisson reduced space.
\section{An example}
\label{sec: ex}
In this section we discuss a concrete example of Poisson reduction. Consider the Lie bialgebra $\mathfrak{g}=ax+b$ discussed in Example (\ref{ex: 1}). The Poisson tensor on the dual Poisson Lie group $G^*$ is given, in the coordinates $(s,t)$ introduced in the matrix representation, by
\begin{equation}
\pi_{G^*}=st\partial_s\wedge\partial_t.
\end{equation}
It is clear that $(s,t)$ are global coordinates on $G^*$.
First, we need to study the orbits of the dressing action.
Remember that the dressing orbits $\mathcal{O}_u$ through a point $u\in G^*$ are the same as the symplectic leaves, hence it
is clear that they are determined by the equation $t=0$. The symplectic foliation of the manifold $G^*$ in this case is given by two open orbits, determined by the conditions
$t>0$ and $t<0$ respectively, and a closed orbit given by $t=0$ and $a\in\mathbb{R}^+$.
Consider a Hamiltonian action $\Phi:G\times M \to M$ of $G$ on a generic Poisson manifold $M$ induced by the equivariant momentum map $\boldsymbol{\mu}:M\rightarrow G^*$.
Its pullback
\begin{equation}
\boldsymbol{\mu}^*: C^{\infty}(G^*)\longrightarrow C^{\infty}(M)
\end{equation}
maps the coordinates $s$ and $t$ on $G^*$ to
$$
x(u)=s(\boldsymbol{\mu}(u)) \qquad y(u)=t(\boldsymbol{\mu}(u)).
$$
It is important to underline that we have no information on the dimension of $M$, so $x$ and $y$ are just a pair of the possible coordinates. Nevertheless, since $\boldsymbol{\mu}$ is a Poisson map, we have
\begin{equation}
\lbrace x,y\rbrace=xy
\end{equation}
on $M$. The infinitesimal generators of the action $\Phi$ can be written in terms of these
coordinates $(x,y)$ as
\begin{equation}
\Phi(X)=x\{y,\cdot \}\quad \Phi(Y)=x\{x^{-1},\cdot \}.
\end{equation}
In the following, we discuss the Poisson reduction case by case, by considering the different dressing orbits studied above.
\paragraph{Case 1: $(t>0)$} Consider the dressing orbit $\mathcal{O}_u$ determined by the condition $t>0$. Since $s$ and $t$ are both positive, we can put
\begin{equation}
x=e^p,\quad y=e^q.
\end{equation}
Since $\lbrace x,y\rbrace=xy$ we have
\begin{equation}
\lbrace p,q\rbrace=1.
\end{equation}
For this reason the preimage of the dressing orbit can be split as $N=\mathbb{R}^2\times M_1$ and $C^{\infty}(N)$ is given explicitly by the set of functions generated by $y^{-1}$. The infinitesimal generators are given by
\begin{equation}
\Phi(X)=e^p\{e^q,\cdot \}\qquad \Phi(Y)=e^p\{e^{-p},\cdot \}
\end{equation}
which is the action of $G$ on the plane. Hence the Poisson reduction in this case is given by
\begin{equation}
(C^{\infty}(M)[y^{-1}])^G.
\end{equation}
\paragraph{Case 2: $(t<0)$} This case is similar, with the only difference that $y=-e^q$.
\paragraph{Case 3: $(t=0)$} The orbit $\mathcal{O}_u$ is given by fixed points on the line $t=0$, then we choose the point $s=1$. Consider the ideal $\mathcal{I}=\langle x-1,y\rangle$ of functions vanishing on $N$. It is easy to check that it is $G$-invariant, hence the Poisson reduction in this case is simply given by
\begin{equation}
(C^{\infty}(M)/\mathcal{I})^G.
\end{equation}
\subsection{Questions and future directions}
The theory of Poisson reduction can be further developed, as it has been obtained under the assumption that the orbit space $M/G$ is a smooth manifold. This result could be proved under weaker hypothesis, for instance requiring that $M/G$ is an orbifold.
As stated in the introduction, the idea of momentum map and Poisson reduction can be also used for the study of symmetries in quantum mechanics. In particular, the approach of deformation quantization would provide a relation between classical and quantum symmetries. A notion of quantum momentum map has been defined in \cite{me}, \cite{me1} and it can be used to define the quantization of the Poisson reduction.
At classical level, Poisson reduction could be generalized to actions of Dirac Lie groups \cite{MJ} on Dirac manifolds \cite{Co}. Finally, a possible development of this theory is its integration to symplectic groupoids by means of the theories on the integrability of Poisson brackets \cite{CF} and Poisson Lie group actions \cite{FP}.
\end{document} |
\begin{document}
\title{Bounded harmonic maps}
à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}uthor{Yves Benoist \&
Dominique Hulin}
\date{}
\maketitle
\begin{abstract}
The classical Fatou theorem identifies
bounded harmonic functions on the unit disk with
bounded measurable functions on the boundary circle.
We extend this theorem to bounded harmonic maps.
\end{abstract}
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}}
\footnotetext{\emph{2020 Math. subject class.} {Primary 58E20~; Secondary 53C43, 31C12} }
\footnotetext{\emph{Key words} {Harmonic map,
fine limit, Fatou theorem,
subharmonic function, potential theory, spectral gap, non-positive curvature}
\renewcommand{\fnsymbol{footnote}}{à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}rabic{footnote}} }
{\footnotesize \tableofcontents}
\section{Introduction}
\begin{quotation}
The aim of this paper is to present
an extension to harmonic maps
of a classical theorem for harmonic functions
due to Fatou around 1905-1910.
\end{quotation}
\subsection{The Fatou theorem}
\label{secfatouherglotz}
\begin{quotation}
We first recall the classical theorem of Fatou for bounded harmonic functions on the Euclidean disk.
\end{quotation}
This theorem deals with the unit open ball $B$ and with the unit sphere $S=\partial B$ in the Euclidean space $\m R^k$ with $k=2$ or, more generally, with $k\geq 2$.
It identifies the space $\mc H_b(B,\m R)$ of bounded harmonic functions $h:B\rightarrow \m R$ on the ball
with the space $L^\infty(\partial B,\m R)$ of bounded measurable functions on the boundary $\partial B$.
We recall that a harmonic function on $B$ is a $C^2$-function $h$ that satisfies $\displaystyleelta_0 h=0$, where $\displaystyleelta_0$ is the Euclidean Laplacian. We denote by $\sigma} \def\ta{\tau} \def\up{\upsilon_0$ the rotationally invariant
probability measure on the sphere $\partial B$, and we refer to Section \ref{secmaindefinition} for the definition of a non-tangential limit.
\begin{Fact}
\label{facfatou}
{\bf (Fatou)}
$a)$ Let $h:B\to\m R$ be a bounded harmonic function. \\ For $\sigma} \def\ta{\tau} \def\up{\upsilon_0$-almost all $\xi$ in $\partial B$, the function $h$ admits a non-tangential limit $\varphi} \def\ch{\chi} \def\ps{\psi(\xi):=\underset{x\rightarrow\xi}{\rm NTlim}\,h(x)$ at the point $\xi$.\\
$b)$ The map $h\mapsto \varphi} \def\ch{\chi} \def\ps{\psi$ is a bijection $\beta : \mc H_b(B,\m R)\rightarrow L^\infty(\partial B,\m R)$ called the boundary transform.
\end{Fact}
The inverse of the map $\beta$ is given by an explicit formula, the Poisson formula.
For every $\varphi} \def\ch{\chi} \def\ps{\psi\in L^\infty(\partial B,\m R)$, one can indeed recover $h$ as $h=P_0\varphi} \def\ch{\chi} \def\ps{\psi$ where
$P_0\varphi} \def\ch{\chi} \def\ps{\psi$ is the bounded harmonic function defined on $B$ by
$$
P_{0}\varphi} \def\ch{\chi} \def\ps{\psi(x):=\int_{\partial B} \varphi} \def\ch{\chi} \def\ps{\psi(\xi)\,P_{0,\xi}(x)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_0(\xi),
\;\;{\rm where}\;\;
P_{0,\xi}(x)=\frac{1-|x|^2}{|x-\xi|^k }$$
is the Poisson kernel.
The proof of this fact
can be found in Rudin's book \cite[Chap. 11]{RudinRealComplex} when $k=2$,
or in Armitage and Gardiner's book \cite[Chap. 4]{ArmitageGardinerPotential}
for $k\geq 2$.
Proving extensions of this fact has
a long history that has already lasted for more than a century.
Indeed, an important goal of Potential Theory is to understand to what extent
this fact still holds for either harmonic or superharmonic functions, on more general spaces.
pace{0.5em}
The aim of this paper is to extend Fatou's theorem to bounded harmonic maps.
We will allow the target space $Y$ to be
any complete CAT(0) space, the first examples to have in mind being the hyperbolic spaces $\m H^k$.
We will also allow more general source spaces.
Since, in dimension $k=2$, the harmonicity condition
depends only on the conformal structure on the source space $B$,
we can think of $B$ as the hyperbolic plane.
We will explain in Theorem \ref{thmbijectiveboundary} how to replace $B$ by a
{\bf GGG } Riemannian manifold $X$.
Later on, we will also explain in Corollary \ref{corbijectiveboundary}
how to
replace $B$
by any bounded Riemannian domain
$\Om$ with Lipschitz boundary.
\subsection{Main result}
\label{secmainresult}
\begin{quotation}
We now state our main result, postponing the definitions to Section \ref{secmaindefinition}.
\end{quotation}
\begin{Def}
\label{defGGG}
We will say that a Riemannian manifold $X$ is {\bf GGG } as a shortcut for
{\bf G}romov Hyperbolic
with Bounded {\bf G}eometry and
Spectral {\bf G}ap.
\end{Def}
\begin{Thm}
\label{thmbijectiveboundary}
Let $X$ be a {\bf GGG } Riemannian manifold,\! and $Y$ be a complete {\rm CAT(0)}-space.\\
$a)$ Let $h\!:\! X\!\rightarrow\! Y$ be a bounded harmonic map.
Then, for $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost all $\xi\!\in\! \partial X$,
the map $h$ admits a non-tangential limit $\varphi} \def\ch{\chi} \def\ps{\psi(\xi):=\underset{x\rightarrow\xi}{\rm NTlim}\,h(x)$ at the point $\xi$.\\
$b)$ The map $h\mapsto \varphi} \def\ch{\chi} \def\ps{\psi$
is a bijection
$
\beta:\mc H_b( X,Y){\rightarrow} L^\infty(\partial X,Y).
$
\end{Thm}
The main examples of {\bf GGG } Riemannian manifolds $X$ are the pinched Hadamard manifolds ~: those have negative curvature.
The condition {\bf GGG } allows a little bit of positive curvature on $X$. It also allows $X$ to be non contractible. For instance, the quotient of
a pinched Hadamard manifold by a convex cocompact
group of isometries is {\bf GGG }.
\subsection{Main definitions}
\label{secmaindefinition}
Here are the definitions that are needed to understand our theorem \ref{thmbijectiveboundary}.
All our manifolds will be assumed to be connected and
with dimension $k\geq 2$.
\begin{Def}
\label{defboundedgeometry}
A Riemannian manifold $X$ has {\it bounded geometry}
if it is complete, with bounded sectional curvature $-K_{max}\leq K_X\leq K_{max}$, and if the
injectivity radius has a uniform lower bound
${\rm inj}_X\geq r_{min}>0$.
\end{Def}
As explained in \cite[Section 1.1]{KemperLohkamp}, one could replace
in Definition \ref{defboundedgeometry} the bound on the sectional curvature
by a bound on the Ricci curvature.
\begin{Def}
\label{defgromov}
The Riemannian manifold $X$ is Gromov hyperbolic
if there exists $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta>0$ such that, for all
$o$, $x$, $y$, $z$ in $X$ one has
\begin{equation}
\label{eqngromovproduct}
(x|z)_o\geq \min ( (x|y)_o,(y|z)_o )-\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta.
\end{equation}
Here $(x|y)_o:=\tfrac12(d(o,x)+d(o,y)-d(x,y))$
is the Gromov product of the points $x$ and $y$ seen from $o$.
In this case $\partial X$ will denote the ``Gromov boundary'' of $X$ and $\ol X=X\cup\partial X $ will be the ``Gromov compactification'' of $X$ where $\partial X$ is the set of geodesic rays on $X$, two geodesic rays being identified
if they remain within bounded distance from each other.
See \cite{GhysHarp90}.
\end{Def}
\begin{Def}
\label{defspectralgap}
A Riemannian manifold $X$ has a {\it spectral gap},
or a {\it coercive Laplacian},
if the Rayleigh quotients admit a uniform lower bound
\begin{equation}
\la_1:=\inf\limits_{\varphi} \def\ch{\chi} \def\ps{\psi\in C^\infty_c(X)}\frac{\int_X\|\nabla\varphi} \def\ch{\chi} \def\ps{\psi\|^2\,{\rm d}v_g}{\int_X\varphi} \def\ch{\chi} \def\ps{\psi^2\,{\rm d}v_g}
\; >\; 0.
\end{equation}
\end{Def}
Note that the spectral gap implies
that $X$ is non compact.
\begin{Def}
\label{defHadamard}
A pinched Hadamard manifold $X$ is
a complete simply-connected Riemannian manifold with dimension at least $2$
whose sectional curvature is pinched between two negative constants~:
$-b^2\leq K_X\leq -a^2<0$.
\end{Def}
Examples are~: hyperbolic spaces $\m H^k$,
rank one non compact Riemannian symmetric spaces,
any small perturbation of those...
\begin{Def}
A CAT(0) space $Y$ is a geodesic metric space such that,
for every geodesic triangle $T$ in $Y$,
there exists a $1$-Lipschitz map $j:T_0\to T$ where $T_0$ is the
triangle the Euclidean plane with same side lengths as $T$ and $j$
sends each vertex of $T_0$ to the corresponding vertex of $T$.
See \cite{BridsonHaefliger}.
\end{Def}
It is not restrictive to assume that $Y$ is complete, since the
metric completion of a CAT(0) space still is a CAT(0) space.
We will not assume that $Y$ is locally compact.
Examples are~: Hadamard manifolds (namely, complete and simply connected Riemannian manifolds with non positive curvature), Euclidean buidings,
$\m R$-trees, convex subsets in Hilbert spaces...
\begin{Def}
A map $h: X\rightarrow Y$ is (energy minimizing) harmonic if it is locally Lipschitz continuous
and
if it is a minimum for the Korevaar-Schoen energy $E(h)$ with respect to variations of $h$ with compact support $Z\subset X$.
\end{Def}
When $Y$ is a CAT$(0)$ Riemannian manifold, the Korevaar-Schoen energy on $Z$ coincide with the
Dirichlet energy $E(h)=\int_Z |Dh(x)|^2 dv_{g}(x)$.
In this case, the harmonicity condition can be expressed by a partial differential equation which is not linear any more,
see \cite{EellsSampson64}, \cite{Hamilton75} or \cite{Jost84}.
When $Y$ is only a {\rm CAT}(0) space, the energy of $h$ on $Z$ is the integral $E(h)=\int_Z e_h(x) dv_{g}(x)$ of the energy density $e_h$ where,
for a Lipschitz continuous map $h$, the energy density is given by
$e_h(x)=\limsup\limits_{\end{Prop}s\rightarrow 0}\end{Prop}s^{-2-k}v_k^{-1}\int_{B(x,\end{Prop}s)} d(h(x),h(x'))^2dv_{g}(x') $, where $v_k$ is the volume of the
unit Euclidean ball and
where this limit should be understood in a weak sense.
See \cite[Section 1.5]{KorevaarSchoen1} for a precise definition.
See also \cite{Jost95}.
The measure $\sigma} \def\ta{\tau} \def\up{\upsilon$ refers to any finite Borel measure on $\partial X$ which is equivalent to
the harmonic measures on $\partial X$.
The ``$\sigma} \def\ta{\tau} \def\up{\upsilon$-almost surely'' means that the property holds except
on a set of measure zero for the measure $\sigma} \def\ta{\tau} \def\up{\upsilon$ on $\partial X$.
Note that, when $X$ is a pinched Hadamard manifold, such a measure $\sigma} \def\ta{\tau} \def\up{\upsilon$ is often found to be singular with respect to the Lebesgue measure on the sphere $\partial X$.
The set
$\mc H_b( X,Y)$ is the set of bounded
harmonic maps $h: X\rightarrow Y$, and the set
$L^\infty(\partial X,Y)$ is the set of bounded measurable maps from $\partial X$ to $Y$ where two measurable maps are identified if they are $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost surely equal.
\begin{Def}
A function $h: X\rightarrow Y$ has a non-tangential limit
$y$ at a point $\xi\in \partial X$
(also called a conical limit), and we write
$y=\underset{x\rightarrow\xi}{\rm NTlim}\,h(x),$
if $\displaystyle y=\lim_{n\rightarrow\infty} h(x_n)$ holds for any sequence $(x_n)$ in $ X$
converging non tangentially to $\xi$, i.e. such that
$\;\displaystyle\sup\limits_{n\geq 1}d(x_n, o\xi)<\infty$
where $o\xi$ is any geodesic ray from a point $o\in X$ to $\xi$.
\end{Def}
\subsection{Previous results}
\label{secprevious}
When $Y=\m R$, we are dealing with harmonic functions.
As we have already seen in Section \ref{secfatouherglotz},
Theorem \ref{thmbijectiveboundary} for $X=B$ is the classical
Fatou theorem.
The extension to the case
where $X$ is a pinched Hadamard manifold appeared in the $80's$
and is due to Anderson and Schoen in \cite{AndersonSchoen}.
The extension to the case where $X$ is a {\bf GGG } Riemannian manifold
is due to Ancona in \cite{AnconaStFlour}.
When $Y$ is a CAT$(0)$ Riemannian manifold
and $X$ is a pinched Hadamard manifold, Theorem \ref{thmbijectiveboundary}.$a$ is due to
Aviles, Choi, Micallef in \cite[Thm 5.1]{AvilesChoiMicallef91},
and Theorem \ref{thmbijectiveboundary}.$b$ is expected to be true
by these authors
as a final observation
in \cite[Section 1]{AvilesChoiMicallef91} where they say that
such a theorem would be ``a consequence of the solvability
of the Dirichlet problem with $L^\infty$ boundary condition''. This solvability
is one of the main technical issues in our paper (Proposition \ref{prosurjectiveboundary}).
Note that the solvability of the Dirichlet problem with continuous boundary
condition is proven in \cite[Thm 3.2]{AvilesChoiMicallef91}.
The first case of Theorem \ref{thmbijectiveboundary}.$b$
that seems to be new is when both
$X$ and $Y$ are the hyperbolic plane $\m H^2$.
When $Y$ is a CAT(0) space, the proof of Theorem \ref{thmbijectiveboundary} will rely on the solution of the Dirichlet problem
for harmonic maps with values in a CAT(0) space under Lipschitz continuous boundary condition, due to Korevaar and Schoen in \cite{KorevaarSchoen1},
a result that extends the Hamilton theorem in \cite{Hamilton75}.
\begin{Rmq}
\label{remsimplyconnected}
Note that we cannot assume $Y$ to be only locally CAT(0). The fact that $Y$ is simply connected will be important here.
Indeed, it is not clear
how to parametrize the set of harmonic maps from the unit disk
to a compact hyperbolic surface. Similarly, it is not clear how to parametrize the set of all harmonic functions on the unit disk.
\end{Rmq}
\begin{Rmq}
Theorem \ref{thmbijectiveboundary} is an analog of the
theorems that parametrize unbounded harmonic maps between pinched Hadamard manifolds
by their ``quasi-symmetric'' boundary condition at infinity. See the successive papers \cite{Markovic17}, \cite{LemmMarkovic},
\cite{BH15}, \cite{BH18}, and \cite{SidlerWenger}.
that deal with an increasing level of generality.
\end{Rmq}
\subsection{Strategy of proof}
\label{secstrategy}
We will split the statement of Theorem \ref{thmbijectiveboundary}
into five propositions.
\begin{Prop}
\label{pronontangentiallimit}
{\bf (Construction of the boundary map) }
Let $ X$ be a {\bf GGG } Riemannian manifold
and let $Y$ be a complete {\rm CAT(0)}-space.
Let $h: X\rightarrow Y$ be a bounded harmonic map.
Then, for $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost all $\xi\in \partial X$,
the map $h$ admits a non-tangential limit $\varphi} \def\ch{\chi} \def\ps{\psi(\xi):=\underset{x\rightarrow\xi}{\rm NTlim}\,h(x)$ at the point $\xi$.
\end{Prop}
We denote by
\begin{equation}
\label{eqnboundarymap}
\beta h:=\varphi} \def\ch{\chi} \def\ps{\psi\in L^\infty(\partial X,Y)
\end{equation}
the bounded measurable map from $\partial X$ to $Y$
given by Proposition \ref{pronontangentiallimit}.
This map $\be h$ is called the boundary map of $h$, and the map
$$
\beta:\mc H_b( X,Y)\rightarrow L^\infty(\partial X,Y)
$$
is called the boundary transform.
\begin{Prop}
\label{proinjectiveboundary}
{\bf (Injectivity of the boundary transform) }
Same notation.
Two harmonic maps $h$, $h'$ from $ X$ to $ Y$
with $\be h=\be h'$ are equal.
\end{Prop}
In order to prove that the transformation $\be$ is onto, we will construct its inverse map $P$.
We first rely on the theorem
that solves the Dirichlet problem for harmonic maps with regular boundary data. It is due to Hamilton
in \cite{Hamilton75} when the target is a manifold, and to Korevaar-Schoen in \cite[Thm 2.2]{KorevaarSchoen1} when the target is a
{\rm CAT(0)} space.
\begin{Fact}
\label{fachamilton}
{\bf (Hamilton, Korevaar and Schoen)}\\ Let $ \Om$ be a bounded Lipschitz Riemannian domain
and $Y$ be a complete {\rm CAT(0)} space.
Let $\varphi} \def\ch{\chi} \def\ps{\psi:\partial \Om\rightarrow Y$ be a Lipschitz map.
Then, there exists a unique harmonic map $h=P\varphi} \def\ch{\chi} \def\ps{\psi$ from $ \Om$ to $ Y$ that extends
continuously $\varphi} \def\ch{\chi} \def\ps{\psi$.
\end{Fact}
We then need to extend Fact \ref{fachamilton} to continuous boundary data, and to deal with a boundary at infinity. This is included in the following
proposition wich will be proven in Section \ref{secdirichletproblem}.
Note that when $X$ is a pinched Hadamard manifold and $Y$ is a
CAT(0) Riemannian manifold, this proposition is
already in
\cite[Thm 3.2 and 4.7]{AvilesChoiMicallef91}.
\begin{Prop}
\label{prodirichlet}{\bf (Dirichlet problem with continuous data)}
Let $ X$ be a {\bf GGG } Riemannian manifold
and $Y$ be a complete {\rm CAT(0)}-space.
Let $\varphi} \def\ch{\chi} \def\ps{\psi:\partial X\rightarrow Y$ be a continuous map.
Then, there exists a unique harmonic map $h=P\varphi} \def\ch{\chi} \def\ps{\psi$ from $ X$ to $ Y$ that extends
continuously $\varphi} \def\ch{\chi} \def\ps{\psi$.
\end{Prop}
The conclusion in Proposition \ref{prodirichlet} means that the map $\ol h:\ol X\rightarrow Y$ that is equal to $h$ on $ X$ and to $\varphi} \def\ch{\chi} \def\ps{\psi$
on $\partial X$ is continuous. Idem for Fact \ref{fachamilton}.
The main result in this article,
Theorem \ref{thmbijectiveboundary}, extends
Proposition \ref{prodirichlet} to
more general boundary conditions $\varphi} \def\ch{\chi} \def\ps{\psi$. Indeed, it allows $\varphi} \def\ch{\chi} \def\ps{\psi$ to be
any bounded
measurable map from $\partial X$ to $Y$.
As will be very clear in the next proposition,
the proof of our main Theorem \ref{thmbijectiveboundary}
relies on the Hamilton, Korevaar, Schoen theorem.
We endow the space $L^\infty(\partial X,Y)$
with ``the topology of the convergence in probability'', see \end{quotation}ref{eqndphphp}.
The subspace $C(\partial X,Y)$ of continuous maps is then dense in $L^\infty(\partial X,Y)$, see Lemma \ref{lemcontinuousdense}.
We also endow the space $\mc H_b( X,Y)$ of bounded harmonic maps $h: X\rightarrow Y$ with the topology of uniform convergence on compact subsets of $ X$.
\begin{Prop}
\label{propoissontransform}
{\bf (Construction of the Poisson transform) } Let $ X$ be a {\bf GGG } Riemannian manifold
and $Y$ be a complete {\rm CAT(0)}-space.
The map
$$P:C(\partial X,Y)\rightarrow \mc H_b( X,Y)$$ given by Proposition $\ref{prodirichlet}$
has a unique continuous extension
$$P:L^\infty(\partial X,Y)\rightarrow \mc H_b( X,Y).$$
\end{Prop}
We still call the extended map $P$ the Poisson transform.
\begin{Prop}
\label{prosurjectiveboundary}
{\bf (Surjectivity of the boundary transform) } Same notation.
For all $\varphi} \def\ch{\chi} \def\ps{\psi\in L^\infty(\partial X,Y)$, one has $\varphi} \def\ch{\chi} \def\ps{\psi=\be P\varphi} \def\ch{\chi} \def\ps{\psi$.
\end{Prop}
\subsection{Overview}
\label{secplan}
\hspace{1em}
In Chapter \ref{secpreliminaryresult},
we recall preliminary facts about harmonic, subharmonic and superharmonic functions $u$
on a {\bf GGG } Riemannian manifold.
The key points that we will use are a control on the Poisson kernel in Proposition \ref{propoisson2},
upper bounds on the harmonic measures in Lemmas \ref{lemcontrolmeasure} and
\ref{lemsidoubling},
and the existence of non-tangential limit
for bounded Lipschitz superharmonic functions
on $ X$ in Proposition \ref{pronontangentiallimit2}.
In Chapter \ref{secboundarymap}, we recall two facts about harmonic maps.
The first one is the control, due to
Cheng, of the Lipschitz constant of a harmonic map
(Lemma \ref{lemcheng}).
The second one is the subharmonicity of the distance function between two harmonic maps (Lemma \ref{lemharmonicsubharmonic}). We use these two facts, together with Proposition \ref{pronontangentiallimit2}, to prove
the existence of non-tangential limit
for our bounded harmonic map $h: X\rightarrow Y$
(Proposition \ref{pronontangentiallimit}).
This provides the construction of the boundary map $\varphi} \def\ch{\chi} \def\ps{\psi=\be h:\partial X\rightarrow Y$. These arguments also prove that the boundary transform $\be:h\mapsto \varphi} \def\ch{\chi} \def\ps{\psi$ is injective (Proposition \ref{proinjectiveboundary}).
In Chapter \ref{secpoissontransform}, we first
construct the Poisson transform $P:\varphi} \def\ch{\chi} \def\ps{\psi\mapsto h$
when the boundary data $\varphi} \def\ch{\chi} \def\ps{\psi$ is continuous
(Proposition \ref{prodirichlet}) by
building on the Hamilton, Korevaar
and Schoen theorem (Fact \ref{fachamilton}).
We then extend this transform $P:\varphi} \def\ch{\chi} \def\ps{\psi\mapsto h$ to bounded measurable boundary data $\varphi} \def\ch{\chi} \def\ps{\psi$
(Proposition \ref{propoissontransform}). The key point is a
suitable uniform continuity property of this transform $\varphi} \def\ch{\chi} \def\ps{\psi\mapsto h$.
In Chapter \ref{secboundarypoisson}
one proves that the Poisson transform $P$
is a right inverse for the boundary transform $\be$, so that the boundary transform $\be$ is surjective (Proposition \ref{prosurjectiveboundary}).
The key point is an estimate on
sequen\-ces of subharmonic functions (Lemma \ref{lemradialconvergence}) that relies on
the control of the Poisson kernel $P_\xi$ in
Proposition \ref{propoisson2}
and on the Lebesgue density theorem for a doubling measure on a compact quasi-metric space
(Fact \ref{faclebesgue}).
\section{Harmonic and subharmonic functions}
\label{secpreliminaryresult}
\begin{quotation}
In this second chapter, we gather a few results concerning harmonic and subharmonic functions on a {\bf GGG } Riemannian manifold $X$ that will be used in the proof ot Theorem \ref{thmbijectiveboundary}.
\end{quotation}
We set $g$ for the Riemannian metric,
$d$ for the Riemannian distance,
$\displaystyleelta$ for the Laplace Beltrami operator and $k=\dim X$. For the Potential theory of a {\bf GGG } Riemannian manifold, we refer
to the seminal paper \cite{AnconaStFlour}
and to its recent update \cite{KemperLohkamp}.
\subsection{The Harnack inequality and the Green function}
\label{secHarnack}
\begin{quotation}
In this section we present three classical Harnack inequalities for positive harmonic functions.
\end{quotation}
We recall that a function $u: X\rightarrow \m R$ is superharmonic
if it is lower semicontinuous, locally integrable, and if $\displaystyleelta u\leq 0$ holds in the weak sense.
A function $u$ is subharmonic if $-u$ is superharmonic.
A function $u$ is harmonic if
it is both subharmonic and superharmonic
The Harnack inequality, which has been improved by Serrin and by S.T. Yau, gives a uniform control for positive harmonic functions
on compact sets.
See \cite[Lemma 2.1]{LiWang02} for a short proof, and also \cite[Cor. 8.21]{GilbargTrudinger}.
\begin{Fact}
\label{facHarnack1}{\bf (Harnack inequality)}
Let $X$ be a complete Riemannian manifold with bounded sectional curvature.
There exists a constant $c_0>0$ such that, for any positive harmonic function $u$ on a
ball
$
B(x_0, r)
$
with $r\leq 1$,
one has
$$
\|\nabla \log u(x)\|\;\leq\; c_0/r
\;\;\;\mbox{\rm for all $x$ in $B(x_0,r/2)$}.
$$
One then has
$$
u(y)\;\leq\; e ^{c_0}u(x)
\;\;\;\mbox{\rm for all $x$, $y$ in $B(x_0,r/2)$}.
$$
\end{Fact}
The Green operator $G$ is the ``inverse'' of
the Laplacian. The spectral gap assumption ensures that
the Green operator is bounded as an operator on $L^2(X)$.
The Green kernel $G(x,y)$ is the kernel of the Green
operator. It is is symmetric i.e.
$G(x,y)=G(y,x)$. It is a positive $C^\infty$-function
on $X\times X\smallsetminus \displaystylee_X$ and, for each $x$
in $X$, the function $G_x:=G(x,.)$ satisfies
$\displaystylee G_x=-\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_x$. In particular the function $G_x$ is harmonic
outside $\{x\}$.
The bounded geometry assumption ensures the following control on the Green function. We set $\log_*(t):={\rm max}(1, \log t)$.
\begin{Fact}
\label{facGreen}{\bf (Green function)}
Let $X$ be a Riemannian manifold with bounded geometry and with spectral gap. There exist $C_0>1$ and $\end{Prop}s_0>0$
such that~:\\
$a)$ For $x, y$ in $ X$ with
$d(x,y)\leq 1$, one has
\begin{eqnarray*}
\label{eqngreenproche}
C_0^{-1}d(x,y)^{2-k}\leq G(x,y)\leq C_0\,d(x,y)^{2-k}
\hspace{1.2em}
& &
\mbox{if $k\neq 2$,}\\
\nonumber
C_0^{-1}\log_*(1/d(x,y))\leq G(x,y)\leq C_0\,\log_*(1/d(x,y))
\hspace{-1em}
& &
\mbox{if $k= 2$.}
\end{eqnarray*}
$b)$ For $x, y$ in $ X$ with
$d(x,y)\geq 1$, one has
\begin{eqnarray}
\label{eqngreenloin}
G(x,y)&\leq &C_0 \, e^{-\end{Prop}s_0\, d(x,y)}
\end{eqnarray}
\end{Fact}
See for instance \cite[Prop. 2.7 and 2.12]{KemperLohkamp}.
\subsection{The Ancona Inequality}
\label{secAncona}
The Gromov hyperbolicity assumption ensures a much more precise control on the Green function due to Ancona.
\begin{Fact}
\label{facAncona}
{\bf (Ancona Inequality)}
Let $X$ be a {\bf GGG } Riemannian manifold.
Then, there exists $C_1>1$ such that for any point $y$
on a geodesic segment $[x,z]$ in $X$
such that $d(x,y)\geq 1$ and $d(y,z)\geq 1$
one has
\begin{equation}
\label{eqnAncona}
C_1^{-1}\,G(x,y)\,G(y,z)\;\leq\; G(x,z)\; \leq\;
C_1\,G(x,y)\,G(y,z)
\end{equation}
\end{Fact}
The boundary Harnack inequality
compares the behavior of two positive harmonic functions
near a piece of the boundary $\partial X$
where they both go to zero.
In order to state this inequality we need to
introduce some notation.
We first recall the definition of the Gromov product
for two points $\end{Thm}a_1$, $\end{Thm}a_2$ in $\ol X=X\cup \partial X$
seen from a point $o\in X$~:
$$
(\end{Thm}a_1|\end{Thm}a_2)_{o}:=
\limsup\limits_{\substack{x_1\rightarrow\end{Thm}a_1\\x_2\rightarrow\end{Thm}a_2}}
(x_1|x_2)_{o}.
$$
This quantity is equal, up to a uniformly bounded
error term, to the distance between
$o$ and a geodesic going from $\end{Thm}a_1$ to $\end{Thm}a_2$.
For $x$ in $\ol X$, we introduce the sets
\begin{equation*}
\mc H_o^{t}(x)=\{y\in X\mid (y|x)_o\geq t\},
\end{equation*}
\begin{equation*}
\ol{\mc H}_o^{t}(x)=\{y\in \ol X\mid (y|x)_o\geq t\}.
\end{equation*}
Note that these sets are empty when $d(o,x)<t$.
We recall that the topology of $\ol X$ is the topology that extends the topology of $X$
and such that a neighborhood basis of a point $\xi\in \partial X$
is given by the sets $\ol{\mc H}_o^{t}(\xi)$ with $t>0$.
See \cite{GhysHarp90}.
We will call them (rough) half-spaces.
We also recall that, since $X$ is Gromov hyperbolic,
we can choose $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta>1$ satisfying \end{quotation}ref{eqngromovproduct} and such that, for every
geodesic triangle $x$, $y$, $z$ in $X$,
every point $u$ on the edge $xz$
is at distance at most $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$ of the union $xy\cup yz$ of the other
edges.
We record a five properties of these half-spaces~:
\begin{equation}
\label{eqnhsxyhtyx}
\ol{\mc H}_x^{s}(y)\cup \ol{\mc H}_y^{t}(x)
\; =\; \ol X
\;\;
\mbox{\rm for all $x$, $y$ in $X$ with
$d(x,y)\geq s+t$,}
\end{equation}
\begin{equation}
\label{eqnhsxyhtyxbis}
\ol{\mc H}_x^{s}(y)\cap \ol{\mc H}_y^{t}(x)
\; =\; \emptyset
\;\;
\mbox{\rm for all $x$, $y$ in $X$ with
$d(x,y)< s+t$.}
\end{equation}
\begin{equation}
\label{eqnhtxyhtxz}
\ol{\mc H}_x^{t}(y)\subset \ol{\mc H}_x^{t-\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta}(z)
\;\;
\mbox{\rm for all $x$ in $X$, $y$, $z$ in $\ol X$ with
$z\in\ol{\mc H}_x^{t}(y)$,}
\end{equation}
\begin{equation}
\label{eqnhtoyhtmy1}
\ol{\mc H}_x^{t+s}(y)\subset \ol{\mc H}_m^{t}(y)
\;\;
\mbox{\rm for all $m$, $x$, $y$ in $X$ with $d(x,m)=s$ ,}
\end{equation}
\begin{equation}
\label{eqnhtoyhtmy}
\ol{\mc H}_m^{t}(y)
\subset \ol{\mc H}_x^{t+s-\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta}(y)
\;\;
\mbox{\rm for $m$ in $xy$ with $d(x,m)=s$ and $t>\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$.}
\end{equation}
These properties explain why these sets are called
half-spaces.
The first four ones are straightforward.
For the last one, notice that, since $t>\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$,
for any $z$ in $\ol{\mc H}_m^{t}(y)$, the distance between $m$
and a geodesic $yz$ is larger than $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$ and hence
there exists a point $m'$ on a geodesic $xz$ such that
$d(m,m')\leq \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$.
We now state the strong boundary Harnack inequality for
a {\bf GGG } Riemannian manifold $X$. This inequality
is actually equivalent to the Ancona Inequality
in Fact \ref{facAncona}.
\begin{Fact}
\label{facHarnack3}{\bf (Strong Boundary Harnack inequality)}
Let $X$ be a {\bf GGG } Riemannian manifold.
There exists $C_2>0$ and $t_0>0$ such that
for all $o\in X$, all $\xi\in \partial X$, all $t\geq 0$
and all positive continuous functions $u$, $v$ on the
half-space $\ol{\mc H}:=\ol{\mc H}^{t}_{o}(\xi)$
which are zero on $\partial X\cap \ol{\mc H}$ and harmonic on $X\cap \ol{\mc H}$,
one has
\begin{eqnarray}
\label{eqnHarnack3}
\frac{u(y)}{v(y)}\leq C_2\, \frac{u(x)}{v(x)}
&\mbox{\it for all}& x, y \in \mc H^{t+t_0}_o(\xi).
\end{eqnarray}
\end{Fact}
\begin{Rmq}
Fact \ref{facHarnack3} is due to Anderson and Schoen in \cite[Corollary 5.2]{AndersonSchoen} when $X$ is a pinched Hadamard manifold. It is due to Ancona
in \cite{AnconaStFlour} when $X$ is a
{\bf GGG } Riemannian manifold.
\end{Rmq}
The following statement is a Corollary
of Facts \ref{facHarnack1} and \ref{facHarnack3}.
\begin{Cor}
\label{corHarnack4}{\bf (Boundary Harnack inequality)}
Let $X$ be a {\bf GGG } Riemannian manifold.
Let $O\subset \ol X$ be a connected open subset and $K\subset O$ be a compact subset. There exists a constant $C=C_{K,O,X}>0$
such that for all continuous functions
$u$, $v$ on $O$ that are
harmonic and positive on $O\cap X$ and zero on $O\cap \partial X$,
one has
\begin{eqnarray}
\label{eqnHarnack4}
\frac{u(y)}{v(y)}\leq C\, \frac{u(x)}{v(x)}
&\mbox{\it for all}& x, y \in K\cap X.
\end{eqnarray}
\end{Cor}
Note that Fact \ref{facHarnack3} is stronger
than Corollary \ref{corHarnack4} since it requires the constant $C_2$
not to depend on the half-space $\ol{\mc H}$.
\subsection{The Poisson kernel}
\label{secpoissonfunction}
\begin{quotation}
We now recall the definition of the Poisson kernel,
also called the Martin kernel.
\end{quotation}
The following fact is due to
Anderson and Schoen
in \cite{AndersonSchoen} for a pinched Hadamard manifold,
and has been generalized by Ancona in \cite{AnconaStFlour}
for a {\bf GGG } Riemannian manifold $X$. See also \cite{KemperLohkamp} for more general measured metric spaces.
It describes all the positive harmonic functions on $X$
and, more precisely, it describes the Martin boundary of $X$.
The key point in the proof is the strong boundary Harnack inequality of Fact \ref{facHarnack3}.
\begin{Fact} {\bf (Martin Boundary)}
\label{facMartin}
Let $X$ be a {\bf GGG } Riemannian manifold, and fix a point $o\in X$.\\
$a)$ For all $\xi$ in $\partial X$,
there exists a unique non-negative continuous fonction
$$
x\mapsto P_\xi(x)=P_\xi(o,x)
$$
on $\ol X\smallsetminus\{\xi\}$
which is harmonic on $X$, zero on $\partial X \smallsetminus\{\xi\}$ with $P_\xi(o)=1$.\\
$b)$ For $x\in X$ and $\xi\in \partial X$, $P_\xi(x)$ is obtained as the limit
\begin{equation}
P_\xi(x)=\lim_{y\rightarrow\xi}\,\frac{G(x,y)}{G(o,y)}\, .
\end{equation}
$c)$ Any positive harmonic function $h$ on $X$ can be written as
$$
h(x)=\int_{\partial X}P_\xi(x)\,{\rm d}\mu(\xi)
$$
for a unique
positive finite Borel measure $\mu=\mu_h$ on $\partial X$.\\
$d)$ The function $(x,\xi)\mapsto P_\xi(x)$
is continuous on $\ol{X}\times\partial X\smallsetminus \displaystylee_{\partial X}$ where $\displaystylee_{\partial X}$ denotes the diagonal in $\partial X\times\partial X$.
\end{Fact}
By the very definition of the Poisson functions, the following holds~:
\begin{equation*}
\label{eqnpoissonsymmetric}
P_\xi(x,o)=P_\xi(o,x)^{-1}.
\end{equation*}
Those functions $P_\xi$ are exactly the minimal positive harmonic functions on $X$.
In the proof of Proposition \ref{prosurjectiveboundary},
we will need the following estimate for the Poisson functions.
\begin{Prop}
\label{propoisson2}
Let $X$ be a {\bf GGG } Riemannian manifold.
Then, there exists a constant $C_3>0$ such that for all
$o$ in $X$, all $\xi$ in $\partial X$, all $x$ on a geodesic ray $o\xi$, and
for all $\end{Thm}a_1$, $\end{Thm}a_2$ in $\partial X$
with
\begin{equation}
\label{eqnhyppoisson2}
|(\xi|\end{Thm}a_1)_{o}-(\xi|\end{Thm}a_2)_{o}|\leq 1\, ,
\end{equation}
one has
\begin{equation}
\label{eqnpoisson2}
C_3^{-1}\leq \frac{P_{\end{Thm}a_{_1}}(x)}{P_{\end{Thm}a_{_2}}(x)}\leq C_3\, .
\end{equation}
\end{Prop}
We begin by a lemma that extends Ancona's inequality to
Poisson functions.
Since $X$ is Gromov hyperbolic, for every
geodesic triangle with distinct vertices $x$, $y$, $z$ in $\ol X$, there exists a point $m$
whose distance to any of the three geodesic sides is at most $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$.
Such a point $m$ is called a center of the triangle $x,y,z$.
It is not unique, but the distance between two centers is
at most $8\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$. See \cite{GhysHarp90}.
\begin{Lem}
\label{lemAncona2}
Let $X$ be a {\bf GGG } Riemannian manifold.
Then, there exists a constant $C_4>1$ such that
for all $\xi$ in $\partial X$, all points $o$, $x$ in $X$
one has
\begin{equation}
\label{eqnAncona2}
C_4^{-1}\,\frac{G(m,x)}{G(m,o)}\;\leq\; P_\xi(o,x)\; \leq\;
C_4\,\frac{G(m,x)}{G(m,o)},
\end{equation}
where $m$ is a center of a geodesic triangle with vertices
$o$, $x$, $\xi$ such that $d(o,m)\geq 1$ and $d(x,m)\geq 1$.
\end{Lem}
Note that it is possible to choose such a center $m$ since $X$ is Gromov hyperbolic with constant $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta>1$.
A particular instance of \end{quotation}ref{eqnAncona2}, when $x$ is on a geodesic ray from $o$ to $\xi$ and $d(o,x)\geq 1$, reads as
\begin{equation}
\label{eqnAncona2bis}
C_4^{-1}G(o,x)^{-1}\;\leq\; P_\xi(o,x)\; \leq\;
C_4\, G(o,x)^{-1}.
\end{equation}
\begin{proof}
The center $m$ is at a distance at most $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$ from both a geodesic
ray $o\xi$ and a geodesic ray $x\xi$.
Therefore when a point $y\in X$ is sufficiently near $\xi$,
the point $m$ is at distance at most
$2\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$ from both the geodesic rays $oy$ and $xy$.
Therefore, using the Harnack inequality in Fact
\ref{facHarnack1} and the
Ancona inequality in Fact \ref{facAncona} one gets,
with a constant $c_4$ depending only on $X$~:
\begin{eqnarray*}
\label{eqnAncona3}
c_4^{-1}\,G(x,m)G(m,y)\;\leq\; G(x,y)\; \leq\;
c_4\,G(x,m)G(m,y),\\
c_4^{-1}\,G(o,m)G(m,y)\;\leq\; G(o,y)\; \leq\;
c_4\,G(o,m)G(m,y).
\end{eqnarray*}
Taking the ratio of these estimates yields, with $C_4=c_4^2$,
\begin{equation*}
\label{eqnAncona4}
C_4^{-1}\,\frac{G(x,m)}{G(o,m)}\;\leq\; \frac{G(x,y)}{G(o,y)}\; \leq\;
C_4\,\frac{G(x,m)}{G(o,m)}.
\end{equation*}
One then gets \end{quotation}ref{eqnAncona2} by letting $y$ converge to $\xi$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{propoisson2}]
Let $m_1$ be a center of the triangle $o$, $x$, $\end{Thm}a_1$
and $m_2$ be a center of the triangle $o$, $x$, $\end{Thm}a_2$.
Those points can be chosen so that $d(m_i,o)\geq 1$
and $d(m_i,x)\geq 1$.
The assumption \end{quotation}ref{eqnhyppoisson2} tells us that the distance $d(m_1,m_2)$ is bounded by a constant depending only on $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$.
Therefore by the Harnack principle in Fact \ref{facHarnack1},
there exists a constant $c_3>1$ depending only on $X$ such that
\begin{equation}
\label{eqnAncona5}
c_3^{-1}\leq \frac{G(x,m_2)}{G(x,m_1)}\leq c_3
\;\;{\rm and}\;\;
c_3^{-1}\leq \frac{G(o,m_2)}{G(o,m_1)}\leq c_3
\end{equation}
Taking the ratio of the bound \end{quotation}ref{eqnAncona2} with $\xi=\end{Thm}a_1$
by the bound \end{quotation}ref{eqnAncona2} with $\xi=\end{Thm}a_2$
one gets
\begin{equation}
\label{eqnpoisson3}
C_4^{-2}\,\frac{G(x,m_1)}{G(x,m_2)}\,\frac{G(o,m_2)}{G(o,m_1)}
\;\leq\;
\frac{P_{\end{Thm}a_{_1}}(x)}{P_{\end{Thm}a_{_2}}(x)}
\;\leq\;
C_4^2\,\frac{G(x,m_1)}{G(x,m_2)}\,\frac{G(o,m_2)}{G(o,m_1)}.
\end{equation}
Combining \end{quotation}ref{eqnAncona5} with \end{quotation}ref{eqnpoisson3}, one obtains
\end{quotation}ref{eqnpoisson2} with $C_3=c_3^2C_4^2$.
\end{proof}
\begin{Cor}
\label{corAncona2}
Let $X$ be a {\bf GGG } Riemannian manifold.
Then, there exists a constant $C_4>1$ such that
for all $\xi$ in $\partial X$, all points $x$ in $X$
and all point $y$ on a ray $x\xi$ with $d(x,y)\geq 1$,
one has
\begin{equation}
\label{eqnAncona6}
{C_4}^{-1}\, G(x,y)\,P_\xi(y)\;\leq\;
P_\xi(x)
\; \leq\;
C_4\,G(x,y)\,P_\xi(y).
\end{equation}
\end{Cor}
\begin{proof}
Since, by definition $P_\xi(x,y)=P_\xi(y)/P_\xi(x)$,
inequalities \end{quotation}ref{eqnAncona6} are nothing but a reformulation of \end{quotation}ref{eqnAncona2bis}.
\end{proof}
See also \cite[Cor. 6.4]{AndersonSchoen} and \cite{LedrappierLim} for other estimates on the
Poisson kernel when $X$ is a pinched Hadamard manifold.
We recall that, in Geometric Group Theory, the word ``geodesic'' means
``minimizing geodesic''. This is why the following lemma is non trivial.
\begin{Lem}
\label{lemnodeadend}
Let $X$ be a {\bf GGG } Riemannian manifold.
Then there exists $C_5>1$ such that,
for all $x$ in $X$, there exists $\xi$, $\end{Thm}a$ in $\partial X$
with $(\xi|\end{Thm}a)_x\leq C_5$.
\end{Lem}
We do not explicitely use this lemma.
It illustrates the influence of the spectral gap condition
on the geometry of $X$. It tells us there are no dead ends in $X$.
Here is a sketch of proof.
\begin{proof}
Since $X$ is Gromov hyperbolic,
if there were dead ends,
we would be able to find, for all $n\geq 1$,
a ball $B_n:=B(x_n, n)$ of radius $n$ in $X$
whose boundary $S_n:=S(x_n,n)$ has diameter at most $2\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$.
Since $X$ has bounded geometry, the volumes of the balls
$B_n$ go to infinity while the diameters
of $S_n$ are bounded.
This contradicts the spectral gap.
\end{proof}
\subsection{The harmonic measures}
\label{secharmonicmeasure}
\subsubsection{Harmonic measures for a bounded domain}
We first recall the solution of the Dirichlet problem
for harmonic functions on a Lipschitz bounded Riemannian domain
$\Om$ (see Section \ref{seclipschitzdomain} for a precise definition). This can be found in \cite[Chapter 6 and 8]{GilbargTrudinger}
when $\partial \Om$ is smooth and in \cite{Ancona78} when $\partial \Om$ is Lipschitz continuous. It says:
\begin{Fact} {\bf (Dirichlet problem for functions on a bounded domain)} Let $\Om$ be a Lipschitz bounded Riemannian domain.
For every continuous function $\varphi} \def\ch{\chi} \def\ps{\psi\in C(\partial \Om,\m R)$, there exists a unique continuous function $h:\ol\Om\rightarrow \m R$ which is harmonic on $\Om$ and equal to $\varphi} \def\ch{\chi} \def\ps{\psi$
on $\partial \Om$.
For $x\in \Om$, it is given by
\begin{eqnarray}
\label{eqnuintphiom}
h(x):=\int_{\partial \Om} \varphi} \def\ch{\chi} \def\ps{\psi(\xi)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_x(\xi),
\end{eqnarray}
where $\sigma} \def\ta{\tau} \def\up{\upsilon_x=\sigma} \def\ta{\tau} \def\up{\upsilon^\Om_x$ is the harmonic measure on $\partial \Om$ seen from $x$.
\end{Fact}
\noindent
By a theorem of Dahlberg in \cite{Dahlberg79},
the harmonic measure $\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om}_{x}$ at each point $x \in\Om$ is equivalent to the
Riemannian measure on $\partial \Om$.
The measure $\sigma} \def\ta{\tau} \def\up{\upsilon_x^\Om$ is a doubling measure on $\partial \Om$.
This means, see \cite[Section 11.3]{CaffarelliSalsa}, that there exists a constant
$c=c_{\Om,x}$ such that for all $r>0$ and
all $\xi$ in $\partial \Om$ one has
\begin{equation}
\label{eqndoubling}
\sigma} \def\ta{\tau} \def\up{\upsilon_{x}^\Om(B(\xi,2r))\leq c\, \sigma} \def\ta{\tau} \def\up{\upsilon_{x}^\Om(B(\xi,r)).
\end{equation}
In this notation, we think of $\sigma} \def\ta{\tau} \def\up{\upsilon_x^\Om$ as a measure on $X$
supported by $\partial \Om$.
\begin{Rmq}
From a probabilistic point of view, the harmonic measure $\sigma} \def\ta{\tau} \def\up{\upsilon^\Om_x$ on $\partial \Om$
is the exit probability measure
of a Brownian motion on $ \Om$ starting at point $x$.
\end{Rmq}
\subsubsection{Harmonic measures on {\bf GGG } Riemannian manifolds }
We now recall the solution of the Dirichlet problem
for harmonic functions on a {\bf GGG } Riemannian manifold $X$.
This is independently due to Anderson and Sullivan
when $X$ is a pinched Hadamard manifold,
see \cite{AndersonSchoen} for a nice account.
It is due to Ancona when $X$ is a {\bf GGG } Riemannian manifold,
as a consequence of the description of the Martin boundary of $X$.
\begin{Fact}
{\bf (Dirichlet problem for functions on {\bf GGG } manifolds)}\\
Let $X$ be a {\bf GGG } Riemannian manifold.
For every continuous function $\varphi} \def\ch{\chi} \def\ps{\psi\in C(\partial X,\m R)$, there exists a unique continuous function $h:\ol X\rightarrow \m R$ which is harmonic on $ X$ and is equal to $\varphi} \def\ch{\chi} \def\ps{\psi$
on $\partial X$.
For $x\in X$, it is given by
\begin{eqnarray}
\label{eqnuintphix}
h(x):=\int_{\partial X} \varphi} \def\ch{\chi} \def\ps{\psi(\xi)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_x(\xi),
\end{eqnarray}
where $\sigma} \def\ta{\tau} \def\up{\upsilon_x=\sigma} \def\ta{\tau} \def\up{\upsilon^X_x$ is the harmonic measure on $\partial X$ seen from $x$.
\end{Fact}
For $x=o$, the probability measure $\sigma} \def\ta{\tau} \def\up{\upsilon_o$ on $\partial X$ is the one
that appears in
the decomposition of the constant harmonic function $h=1$ in Fact \ref{facMartin}.$c$.
For every $x$ in $X$, the positive measure $\sigma} \def\ta{\tau} \def\up{\upsilon^X_x$
is given by the formula
\begin{equation*}
{{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon^X_x(\xi)= P_\xi(x)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon^X_{o}}(\xi)
\end{equation*}
so that Equation \end{quotation}ref{eqnuintphix} can be rewritten as
\begin{equation}
\label{eqnuintphi2}
h(x):=\int_{\partial X} \varphi} \def\ch{\chi} \def\ps{\psi(\xi)\,P_\xi(x)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon^X_{o}(\xi).
\end{equation}
When $\varphi} \def\ch{\chi} \def\ps{\psi$ is continuous, the function $h$
defined on $X$ by \end{quotation}ref{eqnuintphi2} is harmonic and extends continuously $\varphi} \def\ch{\chi} \def\ps{\psi$.
Indeed each function $\xi\mapsto P_\xi(x)$, for ${x\in X}$,
is positive and satisfies
$\int_{\partial X}P_\xi(x)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}^X(\xi) =1$ and, when a sequence $(x_n)$ converges to $\xi\in \partial X$, the sequence of probability measures $(P_\xi(x_n){\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon^X_{o}(\xi))$ converges weakly to $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_{\xi}$.
Note that, even when $X$ is a pinched Hadamard manifold,
the measure $\sigma} \def\ta{\tau} \def\up{\upsilon_{o}^X$ is not always equivalent
to the ``visual measure''.
In order to have shorter notation,
we will think of the harmonic measures $\sigma} \def\ta{\tau} \def\up{\upsilon_x^X$ as
measures on $\ol X$ supported by $\partial X$.
\subsubsection{Upper bound for the harmonic measures}
\label{secupperhar}
We will need the following
uniform control of the harmonic measures
$\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om}_{o}$
for bounded subdomains of $X$ with Lipschitz boundary. By definition, the
probability measure $\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om}_{o}$ is supported by the boundary $\partial \Om$.
This control tells us that, seen from $o$,
the measure of the part of $\partial \Om$
cut out by a half space far away from $o$ is uniformly small.
\begin{Lem}
\label{lemcontrolmeasure}
Let $X$ be a {\bf GGG } Riemannian manifold.
For all $\end{Prop}s >0$ there exists $\end{Lem}l=\end{Lem}l_\end{Prop}s>0$ such that
for all $o$ in $X$, $x$ in $\ol X$,
one has
\begin{equation}
\label{eqncontrolx}
\sigma} \def\ta{\tau} \def\up{\upsilon^{X}_{o}(\ol{\mc H}^{\end{Lem}l}_{o}(x))
\;\leq\; \end{Prop}s,
\end{equation}
and, for all bounded Lipschitz subdomain $\Om\subset X$ containing $o$, one has
\begin{equation}
\label{eqncontrolom}
\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om}_{o}(\mc H^{\end{Lem}l}_{o}(x))
\;\leq\; \end{Prop}s.
\end{equation}
\end{Lem}
\begin{proof} We first prove \end{quotation}ref{eqncontrolom}.
We introduce the set
$$
E:=\mc H^{\end{Lem}l}_{o}(x)\cap \partial \Om ,
$$
where $\end{Lem}l>\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$ will be chosen later, and the open $1$-neighborhood of $E$
$$U:=\{y\in X\mid d(y,E)<1\}.
$$
We introduce then the reduced function $u:=R_1^{U}$ of the constant function $1$ to this open set $U$.
By definition, $u$ is the smallest positive superharmonic function on $X$ which is larger than ${\bf 1}_U$. This function is equal to $1$ on $U$,
it is harmonic on $X\smallsetminus\ol U$
and one has $0\leq u\leq 1$. Since $U$ is relatively compact
this function $u$ is a potential on $X$ i.e. its largest harmonic minorant is $0$.
By the Riesz decomposition theorem, such a potential $u$ can be written in a unique way as
\begin{equation}
\label{eqnbalayage}
u(x)=\int_X G(x,y)\,{\rm d}\la(y)\, ,
\end{equation}
where $\la$ is the Riesz measure of $u$.
This measure $\la$ is a finite positive Borel measure supported by the boundary $\partial U$.
Since $u$ is a positive superharmonic function on $X$
which is equal to $1$ on $E$
one has, for all $z\in \Om$,
\begin{equation}
\label{eqnsibxux}
\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om}_{z}(E)\leq u(z).
\end{equation}
We can assume that $E$ is not empty, and hence that
$d(o,x)\geq \end{Lem}l$. Let $m$ be a point on a geodesic segment from $o$ to $x$
with $d(o,m)=\end{Lem}l$.
We claim that there exists a constant $C>0$
depending only on $X$ such that
\begin{eqnarray}
\label{eqnuxuxgx}
u(o)\leq C\, G(o,m) \, u(m).
\end{eqnarray}
Indeed,
for each $y$ in $\mc H^{\end{Lem}l}_{o}(x)$, any geodesic segment from $o$ to $y$ intersects the
ball $B(m,\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta)$.
Applying Harnack Inequality and Ancona Inequality,
one finds a constant $C_6>0$ depending only on $X$ such that, for all $t>1$ and all $y$ in $\partial U$, one has
$$
G(o,y)\leq C_6\, G(o,m)\, G(m,y).
$$
Applying this inequality to each of the Green functions
in the integral \end{quotation}ref{eqnbalayage},
one gets our claim \end{quotation}ref{eqnuxuxgx}.
Since $u$ is bounded by $1$, it follows from
\end{quotation}ref{eqnsibxux}, \end{quotation}ref{eqnuxuxgx} and \end{quotation}ref{eqngreenloin} that
$$
\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om}_{o}(E)
\leq u(o)\leq C_6\, G(o,m)
\leq C_6\, C_0\, e^{-\end{Prop}s_0 \end{Lem}l}\leq
\end{Prop}s
$$
if $\end{Lem}l =\end{Lem}l_\end{Prop}s$ is chosen large enough.
We now prove \end{quotation}ref{eqncontrolx} with the same $\end{Prop}s$
and $\end{Lem}l_\end{Prop}s$ as in \end{quotation}ref{eqncontrolom}.
If \end{quotation}ref{eqncontrolx} were not true, there would exist a point $x\in\ol X$, a small constant $à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l>0$ and a continuous function $\varphi} \def\ch{\chi} \def\ps{\psi:\partial X\rightarrow [0,1-à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l]$ supported by an open subset of $\partial X$ included in
$\ol{\mc H}^{\end{Lem}l}_{o}(x)$ whose harmonic extension
$h:X\rightarrow [0,1-à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l]$ satisfies $h(o)>\end{Prop}s$.
Since $h$ is continuous, if $\Om$ contains a sufficiently large ball $B(o,R)$, the restriction of $h$ to
the complement
$\partial \Om \smallsetminus\mc H^{\end{Lem}l}_{o}(x)$
is bounded by $à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l\end{Prop}s$.
Therefore, applying formula \end{quotation}ref{eqnuintphiom} with $\varphi} \def\ch{\chi} \def\ps{\psi=h$ and
using \end{quotation}ref{eqncontrolom}, we get
$$
\end{Prop}s
\;<\;
h(0)
\;\leq\;
(1-à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l)\,\sigma} \def\ta{\tau} \def\up{\upsilon_o^\Om({\mc H}^{\end{Lem}l}_{o}(x))+à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l\end{Prop}s
\;\leq\;
(1-à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l)\,\end{Prop}s +à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l \end{Prop}s
\;=\;
\end{Prop}s.
$$
This contradiction proves that \end{quotation}ref{eqncontrolx} is true.
\end{proof}
\subsubsection{Lower bound for the harmonic measure}
\label{seclowerhar}
In order to prove the doubling property of the harmonic measure on $X$, we will need the following uniform lower bound
on the harmonic measure
\begin{Lem}
\label{lemcontrold}
Let $X$ be a {\bf GGG } Riemannian manifold.
For all $\end{Lem}l\geq 0$, there exists $\end{Prop}s_\end{Lem}l>0$
such that
such that for all $o$ in $X$ and $\xi\in \partial X$, one has
\begin{equation}
\label{eqncontrold}
\sigma} \def\ta{\tau} \def\up{\upsilon_{o}^X(\ol{\mc H}^{\end{Lem}l}_o(\xi))\;\geq\; \end{Prop}s_\end{Lem}l.
\end{equation}
\end{Lem}
\begin{proof} Let $\end{Lem}l_0>0$ be the length given by \end{quotation}ref{eqncontrolx}
with $\end{Prop}s =1/2$.
Let $m$ be a point on a geodesic ray $o\xi$ such that
$d(o,m)=\end{Lem}l+\end{Lem}l_0+\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$, so that, by
\end{quotation}ref{eqnhtxyhtxz} and \end{quotation}ref{eqnhsxyhtyx},
$$
\ol{\mc H}^{\end{Lem}l}_o(\xi)\supset
\ol{\mc H}^{\end{Lem}l+\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta}_o(m)\supset
\ol X\smallsetminus \ol{\mc H}^{\end{Lem}l_{_0}}_m(o).
$$
By the Harnack inequality applied to the harmonic function
$x\mapsto \sigma} \def\ta{\tau} \def\up{\upsilon_x^X(\ol{\mc H}^\end{Lem}l_o(\xi))$, there exists
a constant $C_\end{Lem}l>0$ depending only on $X$ and $\end{Lem}l$ such that
\begin{equation*}
\sigma} \def\ta{\tau} \def\up{\upsilon_o^X(\ol{\mc H}^{\end{Lem}l}_o(\xi))
\;\geq\;
C_\end{Lem}l^{-1}\sigma} \def\ta{\tau} \def\up{\upsilon_m^X(\ol{\mc H}^{\end{Lem}l}_o(\xi))
\;\geq \; C_\end{Lem}l^{-1}\,(1-\sigma} \def\ta{\tau} \def\up{\upsilon_m^X(\ol{\mc H}^{\end{Lem}l_{_0}}_m(o))).
\end{equation*}
The choice of $\end{Lem}l_0$ implies
$\sigma} \def\ta{\tau} \def\up{\upsilon_m^X(\ol{\mc H}^{\end{Lem}l_{_0}}_m(o))\leq 1/2$.
This gives \end{quotation}ref{eqncontrold} with the constant $\end{Prop}s_\end{Lem}l=1/(2C_\end{Lem}l)$.
\end{proof}
\subsubsection{Doubling for the harmonic measure}
\label{secdoublinghar}
The following Lemma \ref{lemsidoubling} tells us that the measure $\sigma} \def\ta{\tau} \def\up{\upsilon_o^{X}$ satisfies a doubling property.
See \cite[Lemma 7.4]{AndersonSchoen}
when $X$ is a pinched Hadamard manifold.
\begin{Lem}
\label{lemsidoubling}
Let $X$ be a {\bf GGG } Riemannian manifold.
There exists a constant
$c=c_{X}$
such that for all $o$ in $X$, $\xi\in \partial X$ and $t\geq 0$, one has
\begin{equation}
\label{eqndoubling2}
\sigma} \def\ta{\tau} \def\up{\upsilon_{o}^X(\ol{\mc H}^{t}_o(\xi))\;\leq\;
c\, \sigma} \def\ta{\tau} \def\up{\upsilon_{o}^X(\ol{\mc H}^{t+1}_o(\xi)).
\end{equation}
\end{Lem}
\begin{proof} By Lemma \ref{lemcontrold}, we may assume that
$t\geq 1$.
We first claim that there exists a constant $C_7>1$ such that for all $o$ in $X$, all $\xi$ in $\partial X$, and all $x_t$ on a geodesic ray $o\xi$ with $d(o,x_t)=t\geq 1$, one has
\begin{equation}
\label{eqnsiohtoxi}
C_7^{-1}\, G(o,x_t)
\;\leq\;
\sigma} \def\ta{\tau} \def\up{\upsilon_{o}^X(\ol{\mc H}^{t}_o(\xi))
\;\leq\;
C_7 \, G(o,x_t)\, .
\end{equation}
In order to prove this claim, we introduce for each $t>0$ the harmonic function
$$
z\mapsto h_t(z):=\sigma} \def\ta{\tau} \def\up{\upsilon^X_z(\ol{\mc H}^{t}_o(\xi))=
\int_{\ol{\mc H}^{t}_o(\xi)\cap\partial X}P_\end{Thm}a(z)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_o^X(\end{Thm}a).
$$
Integrating
the inequalities \end{quotation}ref{eqnAncona6}, one finds a constant $C_4>1$ depending only on $X$
such that
\begin{equation}
\label{eqnhxthoxht}
{C_4}^{-1}G(o,x_t)h_t(x_t)
\;\leq\;
h_t(o)
\;\leq\;
{C_4}\,G(o,x_t)h_t(x_t).
\end{equation}
We recall from \end{quotation}ref{eqnhtoyhtmy}
that
\begin{equation*}
\label{eqnhotxi}
\ol{\mc H}^{2\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta}_{x_{_t}}(\xi)
\;\subset\;
\ol{\mc H}^{t}_o(\xi),
\end{equation*}
so that, using Lemma \ref{lemcontrold} with $\end{Lem}l=2\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta$,
one gets a constant $\end{Prop}s_{2\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta}>0$ such that
\begin{equation}
\label{eqnsixthxtd}
\end{Prop}s_{2\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta}
\;\leq\;
\sigma} \def\ta{\tau} \def\up{\upsilon_{x_{_t}}(\ol{\mc H}^{2\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta}_{x_{_t}}(\xi))
\;\leq\;
h_t(x_{t})
\;\leq\;
1.
\end{equation}
Combining \end{quotation}ref{eqnhxthoxht} with \end{quotation}ref{eqnsixthxtd},
we obtain our claim \end{quotation}ref{eqnsiohtoxi}.
Now, the
Harnack inequality in Fact \ref{facHarnack1}
provides a constant $C$
depending only on $X$ such that, for $t\geq 1$,
\begin{equation}
\label{eqngoxgoxgox}
G(o,x_{t})
\;\leq\;
C\, G(o,x_{t+1}).
\end{equation}
The bound \end{quotation}ref{eqndoubling2} follows from
\end{quotation}ref{eqnsiohtoxi} and \end{quotation}ref{eqngoxgoxgox}.
\end{proof}
\subsection{Non-tangential limits}
\label{secradiallimit}
According to Fatou's theorem, every bounded harmonic function on
the Euclidean ball $B\subset \m R^k$ admits
a non-tangential limit at $\sigma} \def\ta{\tau} \def\up{\upsilon_0$-almost all points
of the boundary sphere $\partial B$ (see \cite[Theorem 4.6.7]{ArmitageGardinerPotential}).
This is not always true for a bounded superharmonic function $u$, see \cite[p. 175]{Tsuji75}.
Yet, according to Littlewood's theorem,
every bounded superharmonic function $u$ on
$B$ admits a radial limit at $\sigma} \def\ta{\tau} \def\up{\upsilon_0$-almost all points of $\partial B$,
see \cite[Thm. 4.6.4, Cor. 4.6.8]{ArmitageGardinerPotential}.
One needs an extra assumption on $u$ to ensure that
this radial limit is also a non-tangential limit.
This condition is the ``Lipschitz continuity of $u$
for the hyperbolic metric on the ball $B$''.
\begin{Prop}
\label{pronontangentiallimit2} Let $X$ be
a {\bf GGG } Riemannian manifold, $o\in X$ and let $\sigma} \def\ta{\tau} \def\up{\upsilon=\sigma} \def\ta{\tau} \def\up{\upsilon_o^X$.
Let $u\!:\! X\rightarrow \m R$ be a bounded Lipschitz superharmonic function.\\
$a)$\! For $\!\,\sigma} \def\ta{\tau} \def\up{\upsilon$-almost all $\xi\!\in\!\partial X$, the non-tangential limit $\psi(\xi)\!\!:=\!\!\underset{x\rightarrow\xi}{\rm NTlim}\,u(x)$ exists.\\
$b)$ If this limit $\psi(\xi)$ is $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost surely null, then one has $u\geq 0$.
\end{Prop}
Note that the Lipschitz continuity of $u$
is true for all bounded harmonic functions, because of the Harnack inequality in Fact \ref{facHarnack1}.
\begin{proof}[Proof of Proposition \ref{pronontangentiallimit2}]~:
This follows from the Fatou--Na\"{\i}m--Doob theorem and the Brelot-Doob trick, as they are explained by Ancona in \cite{AnconaStFlour}.
$a)$
It is proven in
\cite[Thm 1.8]{AnconaStFlour}
that for any superharmonic function $u$ on $X$,
for $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost all $\xi$ in $\partial X$, the minimal fine limit $$
\psi(\xi)\!:=\!\underset{x\rightarrow\xi}{\rm MFlim}\,u(x)
$$
exists.
This means that the limit of $u(x)$ when $x\rightarrow \xi$ exists as soon as $x$ avoids a subset $E=E_\xi$ which is minimally thin at $\xi$. We recall that a subset $E\subset X$ is minimally thin if the function $P_\xi{\bf 1}_E$ is bounded by a potential on $X$. And we recall that a {\it potential}
is a positive superharmonic function whose largest harmonic minorant is zero.
Moreover, there is a formula for this limit~:
$$
\psi(\xi)=\frac{{\rm d}\mu_h}{{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon}(\xi)
$$
where $\mu_h$ is the trace measure on $\partial X$ of the harmonic
function $h$ in the Riesz decomposition of $u$
as a sum $u=p+h$ of a potential $p$ and a harmonic function $h$.
It is also proven in
\cite[p.99-100]{AnconaStFlour}
that for a Lipschitz continuous function $u$ on $X$ and a point $\xi$ in $\partial X$, if the minimal fine limit $\end{Lem}l\!:=\!\underset{x\rightarrow\xi}{\rm MFlim}\,u(x)$ exists
then the non-tangential limit
$\!\underset{x\rightarrow\xi}{\rm NTlim}\,u(x)$ exists
and is equal to $\end{Lem}l$.
$b)$ Since $u$ and hence its harmonic part $h$ are bounded on $X$, the measure
$\mu_h$ is absolutely continuous to $\sigma} \def\ta{\tau} \def\up{\upsilon$.
Hence, when the limit $\psi(\xi)$ is $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost surely zero,
the trace measure $\mu_h$ is zero, and the harmonic function $h$ is also $0$.
This tells us that $u$ is a potential, so that one has in particular
$u\geq 0$.
\end{proof}
\section{The boundary transform}
\label{secboundarymap}
\begin{quotation}
In this third chapter, we construct the boundary transform $\beta:\mc H_b( X,Y)\longrightarrow L^\infty(\partial X,Y)$ and prove that it is injective.
\end{quotation}
We recall that $X$ is a {\bf GGG } Riemannian manifold, that $Y$ is a complete CAT(0) space, and that $\ol X=X\cup\partial X$.
\subsection{Harmonic maps and subharmonic functions}
\label{secharmonicsubharmonic}
\begin{quotation}
The following lemmas relate harmonic maps $h: X\rightarrow Y$ with
Lipschitz subharmonic functions $u$ on $ X$.
They will allow us to apply the results on superharmonic functions
from Chapter \ref{secpreliminaryresult}.
\end{quotation}
We begin by a useful bound for the Lipschitz constant of a harmonic map due to Cheng.
\begin{Lem}
\label{lemcheng}
Let $X$ be a Riemannian manifold with bounded geometry, and let $Y$ be a bounded {\rm CAT(0)}-space.
There exists $L>0$ such that
for all $x_0$ in $X$, all $r\leq 1$ and any harmonic map $h:B(x_0,r)\rightarrow Y$,
the restriction of $h$ to the ball $B(x_0,r/2)$
is $L/r$-Lipschitz.
\end{Lem}
\begin{proof}
When $Y$ is a manifold,
this is a simplified version of \cite[Formula 2.9]{Cheng80}.
See also \cite[Theorem 6]{GiaquintaHildebrandt82}.
When $Y$ is a more general CAT(0) space, the extension of Cheng Lemma has been proven in \cite[Theorem 1.4]{ZhangZhongZhu}.
\end{proof}
\begin{Lem}
\label{lemharmonicsubharmonic}
Let $ X$ be a complete Riemannian manifold with bounded sectional curvature, and let $Y$ be a {\rm CAT(0)}-space.\\
$a)$ Let $h: X\rightarrow Y$ be a bounded harmonic map and $y_0\in Y$. Then the function
$x\mapsto d(y_0,h(x))$ is a bounded Lipschitz subharmonic function on $ X$.\\
$b)$ Let $h,h': X\rightarrow Y$ be two bounded harmonic maps. Then the function
$x\mapsto d(h(x),h'(x))$ is also a bounded Lipschitz subharmonic function on $ X$.
\end{Lem}
\begin{proof} When $Y$
is a manifold this is in\cite[Lemmas 3.8.1 and 3.8.2]{Jost06}.
$a)$ We can assume that the CAT$(0)$ space $Y$ is bounded.
Since $Y$ is CAT$(0)$, the function $à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l$ on $Y$ defined
by $ à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l(y):=d(y_0,y)$ is convex.
Therefore, by \cite[Lemma 1.7.1]{Jost84}
when $Y$ is a manifold
and \cite[Lemma 10.2]{EellsFuglede} in general, the function $u:=à} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\e{è} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\u {ù} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\E{é} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ii{{ï\be\alpha} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\be{\beta} \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zetaf\ga{\gamma}l\circ h$ is subharmonic on $ X$.
The Lipschitz continuity of $u$ follows from the Cheng bound
in Lemma \ref{lemcheng}.
$b)$ The proof is as in $a)$. Indeed, the map $(h,h'): X\rightarrow Y\times Y$ is harmonic, the product space
$Y\times Y$ is ${\rm CAT}(0)$,
and the function $(y,y')\to d(y,y')$ is a convex function.
\end{proof}
\subsection{Construction of the boundary map}
\label{secconstructionboundary}
\begin{quotation}
In this section, we prove Proposition \ref{pronontangentiallimit}.
\end{quotation}
\begin{proof}[Proof of Proposition \ref{pronontangentiallimit}]
Fix $o\in X$ and set $\sigma} \def\ta{\tau} \def\up{\upsilon=\sigma} \def\ta{\tau} \def\up{\upsilon_o^X$. Let $h: X\rightarrow Y$ be a harmonic map.
We want to prove that, for $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost all $\xi\in \partial X$,
the map $h$ has a non-tangential limit $\varphi} \def\ch{\chi} \def\ps{\psi(\xi)$ at the point $\xi$.
Let $y\in Y$. By Lemma \ref{lemharmonicsubharmonic}, the function $u_y:x\rightarrow d(y,h(x))$ is a bounded Lipschitz subharmonic function on $ X$.
Hence, by Proposition \ref{pronontangentiallimit2}.$a$, there exists a subset $F_y$
of full measure in $\partial X$ such that
the function $u_y$ admits a non-tangential limit $\psi_y(\xi)$ at each point $\xi\in F_y$.
Let $Y_1\subset Y$ be the closure of the convex hull of $h(X)$ in $Y$. This subspace
$Y_1$ is a bounded separable complete CAT(0) space.
Let $D\subset Y_1$ be a countable dense subset of $Y_1$.
The intersection $F\subset\partial X$ of all the sets $F_y$, for $y$ in $D$, still has full $\sigma} \def\ta{\tau} \def\up{\upsilon$-measure.
Note that, for all $y$, $y'$ in $Y_1$ and $x$ in $ X$, one has
$$
d(u_y(x),u_{y'}(x))\leq d(y,y').
$$
Therefore, for all $\xi\in F$ and all $y$ in $Y_1$,
the function $u_y$ has a non-tangential limit at the point $\xi$.
We introduce the map
\begin{eqnarray*}
\Phi:Y_1&\rightarrow &{\rm Lip_1}(Y_1,[0,\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_{Y_1}]) \\
y'&\mapsto& (d(y,y'))_{y\in Y_{_1}}.
\end{eqnarray*}
where $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_{Y_1}$ is the diameter of $Y_1$ and ${\rm Lip}_1$ refers to the set of $1$-Lipschitz functions endowed with the $\sup$ distance.
This map $\Phi$ is an
isometry onto its image $\Ph(Y_1)$ and, since $Y_1$ is complete, this image $\Phi(Y_1)$ is closed.
Let $\xi\in F$. Since we have just seen that the map $\Ph\circ h$
has a non-tangential limit at the point $\xi$,
the map $h$ also has a non-tangential limit $\varphi} \def\ch{\chi} \def\ps{\psi(\xi)\in Y_1$ at the point $\xi$.
\end{proof}
\subsection{Injectivity of the boundary transform}
\label{secinjectiveboundary}
\begin{quotation}
In this section, we prove Proposition \ref{proinjectiveboundary}.
\end{quotation}
\begin{proof}
Let $h$ and $h'$ be two harmonic maps from $ X$ to $ Y$
whose boundary maps $\be h$ and $\be h'$ are $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost surely equal.
We want to prove that $h=h'$.
By Lemma \ref{lemharmonicsubharmonic}, the function $u:x\to d(h(x),h'(x))$
is a bounded Lipschitz subharmonic function on $ X$.
By assumption the non-tangential limit $\underset{x\rightarrow\xi}{\rm NTlim}\,u(x)$ is zero for $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost all $\xi$ in $\partial X$.
Therefore, by Proposition \ref{pronontangentiallimit2}.$b$, the function $u$ must be non positive.
Since $u$ is already non negative, we must have $u=0$, and hence $h=h'$.
\end{proof}
\section{The Poisson transform}
\label{secpoissontransform}
\begin{quotation}
In this fourth chapter, we construct the Poisson transform.
\end{quotation}
\subsection{Density of the Lipschitz maps}
\label{seclipschitzdense}
We first need a lemma on the density of Lipschitz maps on a compact manifold $S$ inside the set of bounded measurable maps.
We will apply it to the boundary $S=\partial \Om$
of a bounded Lipschitz domain $\Om$. Such a boundary is bi-Lipschitz
homeomorphic to a compact smooth manifold.
\begin{Lem}
\label{lemlipschitzdense}
Let $S$ be a compact manifold and
$Y$ be a CAT(0) space. Then, every
continuous map $\varphi} \def\ch{\chi} \def\ps{\psi:S\rightarrow Y$ is a uniform limit of Lipschitz maps.
\end{Lem}
We begin by recalling the classical construction of the
weighted barycenter $\be=\be_\mu(y_0,\ldots,y_n)$ of $n\!+\!1$ points $(y_0,\ldots ,y_n)$ in a CAT(0)-space
$Y$. The weight $\mu$ belongs
to the standard $n$-simplex
$$
\Si_n:=\{\mu=(\mu_1,\ldots,\mu_n)\mid
\mu_i\geq 0\; \mbox{\rm for all $i$, and} \;\mu_0+\cdots+\mu_n=1\}.
$$
We endow this simplex with the $\end{Lem}l^1$-distance.
This barycenter $\be$ is the unique point where the strictly convex function on $Y$
$$
y\mapsto \psi_\mu(y):=\sum_{0\leq i\leq n} \mu_i\,d(y_i,y)^2
$$
achieves its minimum.
As a function of the weight, this barycenter map
$$
\mu\mapsto \be_\mu(y_0,\ldots,y_n)
$$
is $L$-Lipschitz
continuous where $L$ is the diameter
of the finite set $\{y_0,\ldots, y_n\}$.
We refer to
\cite[Lemma 4.2]{Kleiner99} for these properties.
\begin{proof}[Proof of Lemma \ref{lemlipschitzdense}]
Using a triangulation of $S$ we can assume that $S$
is a compact $CW$-complex.
We endow each $n$-simplex $\Si_0$ of $S$ with the $\end{Lem}l^1$-norm and we endow $S$ with the corresponding length metric. This new metric is Lipschitz equivalent to the Riemannian metric on $S$.
Each $n$-simplex $\Si_0$
of $S$ can be
decomposed as a union of $2^n$ half-size $n$-simplices.
Iterating $k$ times this process we obtain
a decomposition of $\Si_0$ as a union
of $2^{kn}$ $n$-simplices of level $k$ whose size is
$2^{-k}$ the size of $\Si_0$.
Fix $\end{Prop}s>0$ and $\varphi} \def\ch{\chi} \def\ps{\psi\in C(S,Y)$. There exists an integer $k$ such that,
for each simplex $\Si$ of level $k$, one can uniformly bound the diameter
$$
{\rm diam} (\varphi} \def\ch{\chi} \def\ps{\psi(\Si))\leq \end{Prop}s/2.
$$
For each simplex $\Si$ of level $k$,
we denote by $f_\Si:\Si\rightarrow Y$ the barycenter map
such that $f_\Si(s)=\varphi} \def\ch{\chi} \def\ps{\psi(s)$ for each vertex $s$ of $\Si$.
One then has
$$
{\rm diam} (f_\Si( \Si))\leq \end{Prop}s/2,
$$
and $d(f_\Si(s),\varphi} \def\ch{\chi} \def\ps{\psi(s))\leq \end{Prop}s$ holds for all $s$ in $\Si$.
These maps $f_\Si$ are $2^k\end{Prop}s$-Lipschitz continuous.
These maps $f_\Si$ being compatible, each $f_\Si$ is the restriction to $\Si$
of a map $f:S\rightarrow Y$. This map $f$ is also
$2^k\end{Prop}s$-Lipschitz continuous, and one has
$d(f(s),\varphi} \def\ch{\chi} \def\ps{\psi(s))\leq \end{Prop}s$ for all $s$ in $S$.
\end{proof}
\begin{Lem}
\label{lemcontinuousdense}
Let $S$ be a compact metric space,
$\sigma} \def\ta{\tau} \def\up{\upsilon$ be a Borel probability measure on $S$, and
$Y$ be a CAT(0) space.
Then the set $C(S,Y)$ of continuous maps
$f:S\rightarrow Y$ is dense in the set $L^\infty(S,Y)$
of bounded measurable maps $\varphi} \def\ch{\chi} \def\ps{\psi:S\rightarrow Y$.
\end{Lem}
We recall that $L^\infty(S,Y)$ is endowed with the ``topology of convergence in probability''. The distance between two maps $\varphi} \def\ch{\chi} \def\ps{\psi$, $\varphi} \def\ch{\chi} \def\ps{\psi'$ in $L^\infty(S,Y)$ is given by
\begin{equation}
\label{eqndphphp}
d(\varphi} \def\ch{\chi} \def\ps{\psi,\varphi} \def\ch{\chi} \def\ps{\psi'):=\inf\{\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta\geq 0\mid\sigma} \def\ta{\tau} \def\up{\upsilon (\{\xi\in S\mid d(\varphi} \def\ch{\chi} \def\ps{\psi(\xi),\varphi} \def\ch{\chi} \def\ps{\psi'(\xi))\geq \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta\} )\leq \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta\}\, .
\end{equation}
The space $L^\infty(S,Y)$ and its topology
do not depend on the choice of the measure $\sigma} \def\ta{\tau} \def\up{\upsilon$ inside its equivalence class of measures.
\begin{proof}[Proof of Lemma \ref{lemcontinuousdense}]
Let $\varepsilon >0$. By Lusin's theorem, there exists a compact subset $K\subset S$ such that the complement $K^c$ satisfies $\sigma} \def\ta{\tau} \def\up{\upsilon(K^c)\leq \end{Prop}s$ and such that the restriction $\varphi} \def\ch{\chi} \def\ps{\psi|_K$ is continuous. Since a CAT(0) space $Y$ is an absolute retract metric
space, see \cite[Lemma 1.1]{Ontaneda}, and since an absolute retract metric space
is an absolute extension metric space,
there exists a continuous function $f:S\rightarrow Y$ whose restriction to $K$
is equal to $\varphi} \def\ch{\chi} \def\ps{\psi|_K$, so that $d(\varphi,f)\leq\varepsilon$.
\end{proof}
\subsection{The continuous Dirichlet problem}
\label{secdirichletproblem}
\begin{quotation}
In this section we prove Proposition
\ref{prodirichlet}.
\end{quotation}
We first deal with bounded domains.
\begin{Prop}
\label{prodirichleti}
Let $\Om$ be a bounded Lipschitz Riemannian domain, $Y$ a complete ${\rm CAT}(0)$ space
and $\varphi} \def\ch{\chi} \def\ps{\psi: \partial \Om\rightarrow Y$ a continuous map, then there exists a unique harmonic map
$h: \Om \rightarrow Y$ which is a continuous extension of $\varphi} \def\ch{\chi} \def\ps{\psi$.
\end{Prop}
\begin{proof}
By Lemma \ref{lemlipschitzdense}, we can choose a sequence
$\varphi} \def\ch{\chi} \def\ps{\psi_n\in {\rm Lip}(\partial \Om,Y)$ that converges uniformly to $\varphi} \def\ch{\chi} \def\ps{\psi$.
It suffices to prove that the sequence of their harmonic extensions $h_n:=P\varphi} \def\ch{\chi} \def\ps{\psi_n$ given by Fact \ref{fachamilton}
converge uniformly.
We introduce the subharmonic functions
on $ \Om$ given by
$$
u_{m,n}(x):=d(h_m(x),h_n(x)).
$$
They extend the continuous functions
on $\partial \Om$ given by
$$
\psi_{m,n}(\xi)=d(\varphi} \def\ch{\chi} \def\ps{\psi_m(\xi),\varphi} \def\ch{\chi} \def\ps{\psi_n(\xi)).
$$
By the maximum principle, the supremum of $u_{m,n}$ on $ \Om$
is equal to the supremum of $\psi_{m,n}$ on $\partial \Om$.
Hence it goes to zero when $m,n\to\infty$.
Therefore the sequence $h_n$ converges uniformly to a map $h$
which is harmonic and which extends continuously $\varphi} \def\ch{\chi} \def\ps{\psi$.
This harmonic extension is unique because if $h'$ is another harmonic extension, the positive function
$$
x\mapsto u(x):=d(h(x),h'(x))
$$
is subharmonic on $\Om$ and goes to zero near the boundary.
Hence $u=0$ and $h=h'$.
\end{proof}
We now deal with a {\bf GGG } Riemannian manifold $X$.
\begin{proof}[Proof of Proposition \ref{prodirichlet} ]
We fix $o$ in $X$. We can assume that the diameter $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y$ of $Y$ is finite.
As we have seen in the proof of Lemma \ref{lemcontinuousdense},
since the compactification $\ol X$ is a metrizable compact space
and $Y$ is a ${\rm CAT}(0)$ space, there exists a continuous function
$$
\psi:\ol X\rightarrow Y
\;\;\mbox{\rm such that}\;\;
\psi|_{\partial X}=\varphi} \def\ch{\chi} \def\ps{\psi.
$$
We choose an increasing sequence of bounded Lipschitz domain
$\Om_N\subset X$ such that $o\in \Om_0$ and $\Om_N$ contains the
$1$ neighborhood of $\Om_{N-1}$.
We denote by $h_N:\Om_N\rightarrow Y$ the harmonic extension
of the function $\psi_N:=\psi|_{\partial \Om_{_N}}$
given by Proposition \ref{prodirichleti}.
We claim that
\begin{equation}
\label{eqnclaimdirichlet}
\forall\end{Prop}s>0,\;\exists n_0>0,\;\forall N>n>n_0,\;
\forall
x\in \Om_n,\;\; d(h_N(x),h_n(x))\leq \end{Prop}s \, .
\end{equation}
Since the function
$x\mapsto d(h_n(x),h_N(x))$ is subharmonic,
by the maximum principle
it is enough to check \end{quotation}ref{eqnclaimdirichlet}
for $x$ in $\partial \Om_n$, that is~:
\begin{equation}
\label{eqnclaimdirichlet1}
\forall\end{Prop}s>0,\exists n_0>0,\forall N>n>n_0,\;
\forall x\in \partial \Om_n,\;\;
d(h_N(x),\psi(x))\leq \end{Prop}s \, .
\end{equation}
Let $\end{Prop}s >0$. According to Lemma
\ref{lemcontrolmeasure},
there exists
$t_0>0$ such that
for all $x$ in $\Om_N$,
\begin{equation}
\label{eqnclaimdirichlet3}
\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om_N}_{x}(\mc H^{t_0}_{x}(o))
\leq \end{Prop}s/(2 \delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y).
\end{equation}
By uniform continuity of $\psi$ there exists $t_1>0$ such that,
for all $x$ in $\ol X$~:
\begin{equation}
\label{eqnclaimdirichlet2}
d(\psi(x),\psi(y))\leq \end{Prop}s/2
\;\;\mbox{\rm for all $y$ in $\mc H_o^{t_1}(x)$.}
\end{equation}
We choose $n_0\geq t_0\!+\!t_1$ and let $N\geq n\geq n_0$.
We fix $x$ in $\partial \Om_ n$
and introduce the subharmonic function
$z\mapsto u(z):=d(h_N(z),\psi (x))$
on $\Om_N$.
We want to prove that $u(x)\leq \end{Prop}s$.
We observe that
\begin{equation}
\label{eqnuxintuy}
u(x)\;\leq\; \int_{\partial \Om_N}u(y)\,
{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om_N}_{x}(y).
\end{equation}
Since $d(o,x)\geq n \geq t_0+t_1$, by \end{quotation}ref{eqnhsxyhtyx}, one has
$$
X=\mc H^{t_0}_x(o)\cup \mc H_o^{t_1}(x),
$$
and we can bound this integral \end{quotation}ref{eqnuxintuy} by the sum
$I'+I''$ where~:\\
- $I'$ is the integral on the half-space
$\mc H^{t_0}_x(o)$,
which by \end{quotation}ref{eqnclaimdirichlet3} has small volume,\\
- $I''$ is the integral
on $\mc H^{t_1}_o(x)$ on which by \end{quotation}ref{eqnclaimdirichlet2}
the function $u$ is small.
Hence
$$
u(x)\;\leq\;
\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y \;\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om_N}_{x}(\mc H^{t_0}_x(o))
\;+\;
\frac{\end{Prop}s}{2}\;
\sigma} \def\ta{\tau} \def\up{\upsilon^{\Om_N}_{x}(\mc H^{t_1}_o(x))
\;\leq\; \frac{\end{Prop}s}{2}+\frac{\end{Prop}s}{2}
\;=\; \end{Prop}s\, ,
$$
which proves our claim \end{quotation}ref{eqnclaimdirichlet}.
Now the claim \end{quotation}ref{eqnclaimdirichlet}
proves that the sequence of maps $(h_N)$ converges uniformly to a harmonic map $h:X\rightarrow Y$ that extends continuously
$\varphi} \def\ch{\chi} \def\ps{\psi$.
The proof of uniqueness is as for
Proposition \ref{prodirichleti}.
\end{proof}
\subsection{Construction of the Poisson transform}
\label{secconstructionpoisson}
The construction uses the following classical ``continuous extension theorem''.
\begin{Lem}
\label{lemuniconext}
Let $E$ be a metric space, $D\subset E$ a dense subset and $F$ a complete metric space.
Then every uniformly continuous map $P:D\rightarrow F$ admits a unique continuous extension $P:E\rightarrow F$.
\end{Lem}
\begin{proof}[Proof of Lemma \ref{lemuniconext}]
This is classical.
\end{proof}
\begin{proof}[Proof of Proposition \ref{propoissontransform}]
Without loss of generality, we may assume that $Y$ is bounded.
We use Lemma \ref{lemuniconext}
with $E=L^\infty(\partial X,Y)$, $D=C(\partial X,Y)$
and $F=\mc H_b( X,Y)$. Note that $F$ is a complete
metric space since a uniform limit of harmonic maps is harmonic.
We want to prove that the map $P:C(\partial X,Y)\rightarrow \mc H_b( X,Y)$ given by Proposition \ref{prodirichlet}
has a unique continuous extension to $L^\infty(\partial X,Y)$.
By Lemmas \ref{lemuniconext} and \ref{lemcontinuousdense}, it suffices to prove that this map
$P$ is uniformly continuous.
We fix a compact $K\subset X$ and a point $o\in K$, and we set
$$
C_K=\sup\limits_{\xi\in \partial X,\, x\in K}\! P_\xi(x)
\;<\; \infty\, ,
$$
where
$P_\xi(x)$ is the Poisson kernel.
Let $0<\end{Prop}s\leq 1$ and $\varphi} \def\ch{\chi} \def\ps{\psi$, $\varphi} \def\ch{\chi} \def\ps{\psi'$ be two continuous maps from $\partial X$ to $Y$
such that $d(\varphi} \def\ch{\chi} \def\ps{\psi,\varphi} \def\ch{\chi} \def\ps{\psi')\leq \end{Prop}s$.
This means that the function $\psi$ on $\partial X$
defined, for $\xi$ in $\partial X$, by $\psi(\xi):=d(\varphi} \def\ch{\chi} \def\ps{\psi(\xi),\varphi} \def\ch{\chi} \def\ps{\psi'(\xi))$ satisfies
\begin{equation}
\label{eqndistanceproba}
\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\{\xi\in \partial X\mid \psi(\xi)\geq \end{Prop}s\})\leq \end{Prop}s,
\;\;\mbox{\rm where $\sigma} \def\ta{\tau} \def\up{\upsilon_o=\sigma} \def\ta{\tau} \def\up{\upsilon_o^X$}.
\end{equation}
Note that this function $\psi$ is bounded by the diameter $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y$ of $Y$.
Let $h=P\varphi} \def\ch{\chi} \def\ps{\psi$ and $h'=P\varphi} \def\ch{\chi} \def\ps{\psi'$
be their harmonic extensions to $X$.
By Lemma \ref{lemharmonicsubharmonic}, the continuous function $u$ on $X$
given, for $x$ in $X$, by
$u(x)\!:=\! d(h(x),h'(x))$ is subharmonic on $ X$. This function is a continuous extension of $\psi$.
Therefore it satisfies, for all $x$ in $X$,
\begin{eqnarray*}
u(x)&\leq &\int_{\partial X} \psi(\xi)\,P_\xi(x)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_o(\xi)\, .
\end{eqnarray*}
Plugging \end{quotation}ref{eqndistanceproba} in this inequality, one gets for every $x$ in $K$~:
\begin{eqnarray*}
u(x)&\leq &\end{Prop}s\int_{\partial X} P_\xi(x)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_o(\xi) +\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y\int_{\{\psi(\xi)\geq \end{Prop}s\}} P_\xi(x)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_o(\xi)\\
&\leq& \end{Prop}s + C_K\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y\end{Prop}s.
\end{eqnarray*}
This proves, for any $\varphi,\varphi'$ in $C(\partial X,Y)$ and any compact subset $K\subset X$, the inequality~:
$$
\sup_{x\in K}d(h(x),h(x'))\leq (1+C_K\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y)d(\varphi} \def\ch{\chi} \def\ps{\psi,\varphi} \def\ch{\chi} \def\ps{\psi')
\, .
$$
This is the uniform continuity of the map $P$.
\end{proof}
\section{The boundary and the Poisson transform}
\label{secboundarypoisson}
\begin{quotation}
In this chapter, we prove that the Poisson transform $P$ is the inverse of the boundary transform $\be$.
\end{quotation}
\subsection{The Lebesgue density theorem}
\label{seclebesgue}
We recall here the generalized Lebesgue density theorem.
See \cite[Section 4.6]{BenoistFive} for a short and complete proof.
\begin{Def}
\label{defbdistance}
A quasi-distance on a space $S$ is a map
$d_0 : S\times S \rightarrow [0, \infty[$ for which there exists
$b>0$ \\
such that
$
d_0(\xi_1, \xi_3) \leq
b (d_0(\xi_1, \xi_2)+d_0(\xi_2, \xi_3))\; ,\; \;\forall \xi_1,\xi_2,\xi_3 \in S$,\\
such that $d_0(\xi_1, \xi_2) = d_0(\xi_2, \xi_1)$ and\\
such that
$d_0(\xi_1, \xi_2) = 0 \Leftrightarrow \xi_1 = \xi_2$.
\end{Def}
- In this case, one says that $S$ is a quasi-metric space. Then, there exists a topology on
$S$ for which the balls
$B(\xi, \end{Prop}s) := \{ \end{Thm}a \in S \mid d_0(\xi, \end{Thm}a) \leq \end{Prop}s \}$,
with $\xi\in S$ and $\end{Prop}s > 0$, form a basis of
neighborhood of the points $\xi$.\\
- One then has the inclusion $\ol{B(\xi, \end{Prop}s)} \subset B(\xi, b \end{Prop}s)$.
Let $(S, d_0)$ be a compact quasi-metric space and
$\sigma} \def\ta{\tau} \def\up{\upsilon$ be a finite Borel measure on $S$. One says that
$\sigma} \def\ta{\tau} \def\up{\upsilon$ is doubling
if there exists $C>0$ such that, for all $\xi\in S$ and
$r > 0$, one has $\sigma} \def\ta{\tau} \def\up{\upsilon(B(\xi, 2r)) \leq C\,\sigma} \def\ta{\tau} \def\up{\upsilon(B(\xi,r))$.
Let $F\subset S$ be a measurable subset. A point $\xi \in S$ is called a density point if
$$\lim\limits_{\end{Prop}s\rightarrow 0}\frac{\sigma} \def\ta{\tau} \def\up{\upsilon (B(\xi, \end{Prop}s) \cap F)}{\sigma} \def\ta{\tau} \def\up{\upsilon(
B(\xi, \end{Prop}s))}
= 1.
$$
\begin{Fact}
{\bf (Lebesgue)}
\label{faclebesgue}
Let $(S, d_0)$ be a compact quasi-metric space, $\sigma} \def\ta{\tau} \def\up{\upsilon$ be a doubling finite Borel measure on $S$, and let $F$ be a measurable subset of $S$.
Then $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost every point of $F$ is a density point.
\end{Fact}
In the next section, we will apply Fact \ref{faclebesgue} with
$S=\partial X$ and with $\sigma} \def\ta{\tau} \def\up{\upsilon=\sigma} \def\ta{\tau} \def\up{\upsilon_{o}$.
We will use the quasi-distance on $\partial X$ defined, for two points
$\end{Thm}a_1$ and $\end{Thm}a_2$, by
the exponential inverse of the Gromov product~:
\begin{equation}
\label{eqnbdistance}
d_0(\end{Thm}a_1,\end{Thm}a_2)= e^{-(\end{Thm}a_1|\end{Thm}a_2)_{o}}.
\end{equation}
This formula defines indeed a quasi-distance on $\partial X$,
because of \end{quotation}ref{eqngromovproduct}.
Note that one can modify this formula so that $d_0$ is actually a distance,
see \cite{GhysHarp90}.
The balls for this quasi-distance in $\partial X$ are
the
trace at infinity of the
half-spaces of $X$.
The doubling property for
$\sigma} \def\ta{\tau} \def\up{\upsilon_{o}$ is proven in Lemma \ref{lemsidoubling}.
\subsection{Limit of subharmonic functions}
\label{seclimitsubharmonic}
\begin{quotation}
In this section we prove the technical lemma \ref{lemradialconvergence}
that plays a crucial role in the proof of
the surjectivity of the boundary transform.
\end{quotation}
We fix a point $o$ in $X$. For all $\xi\in \partial X$ we define $N\xi$ as the union
of the geodesic rays $o\xi$
from $o$ to $\xi$.
We define then {\it the tube $NF$
over a compact set} $F\subset \partial X$ as
the union
\begin{eqnarray}
\label{eqntube}
NF:=\bigcup_{ \xi\in F}N\xi \subset X.
\end{eqnarray}
\begin{Lem}
\label{lemradialconvergence}
Let $ X$ be a {\bf GGG } Riemannian manifold, and fix a point $o\in X$.
Let $\psi_n\!:\!\partial X\!\rightarrow\! [0,1]$ be a sequence of con\-ti\-nuous functions that converges $\sigma} \def\ta{\tau} \def\up{\upsilon_{o}$-almost surely to $0$.\!\!
Let $u_n\!:\!X\!\rightarrow\! [0,1]$ be non negative subharmonic functions on $X$
that extend continuously $\psi_n$.
Then, for all $\end{Prop}s>0$, there exists a compact subset
$F_{\end{Prop}s}\subset\partial X$ with
$\sigma} \def\ta{\tau} \def\up{\upsilon_{o}( F_{\end{Prop}s}^c)\leq\end{Prop}s$,
such that the sequence $(u_n)$ converges uniformly to $0$ on
the tube $NF_\end{Prop}s$.
\end{Lem}
The arguments of Section \ref{secconstructionpoisson} tell us that
the sequence $(u_n)$ converges to $0$ uniformly on the compact subsets $K$
of $ X$. Lemma \ref{lemradialconvergence} tells us that this convergence
is still uniform on ``large radial subsets of $ X$''.
\begin{proof}
{\bf First step } {\it We control the Poisson kernel on tubes.}
For $\xi$ in $\partial X$ and $m\geq 0$ we
denote by
$$
B_m(\xi):=B(\xi,e^{-m})=
\{\end{Thm}a\in \partial X\mid (\end{Thm}a|\xi)_o\geq m\}
$$
the balls for the
quasidistance $d_0$ in \end{quotation}ref{eqnbdistance},
and we introduce the annuli
$$
A_m(\xi):=B_m(\xi)\smallsetminus B_{m+1}(\xi),
$$
so that one has
$\cup_{m\geq 0}A_m(\xi)=\partial X\!\smallsetminus\! \{\xi\}$.
By Proposition \ref{propoisson2}, there exists $C_3>0$ such that,
for $\xi\in \partial X$ and $m\geq 0$,
\begin{equation}
\label{eqnratiopoisson2}
\frac{P_{\end{Thm}a_{_1}}(x)}{P_{\end{Thm}a_{_2}}(x)}
\;\leq\; C_3
\;\;\mbox{\rm holds for all $x\in N\xi$ and $\end{Thm}a_1$, $\end{Thm}a_2$ in $A_m(\xi)$.}
\end{equation}
{\bf Second step} {\it We apply the Lebesgue density theorem.}
Let $\end{Prop}s>0$. Since the sequence $(\psi_n)$ converges $\sigma} \def\ta{\tau} \def\up{\upsilon_{o}$-almost surely to $0$,
there exist an integer $n_\end{Prop}s\geq 1$ and a compact subset $K_\end{Prop}s\subset \partial X$ with
$\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(K_\end{Prop}s^c)\leq \end{Prop}s/2$, and such that
$\psi_n(\xi)\leq \end{Prop}s$ for all $n\geq n_\end{Prop}s$ and $\xi\in K_\end{Prop}s$.
By the Lebesgue density theorem (Fact \ref{faclebesgue}), applied to the doubling measures $\sigma} \def\ta{\tau} \def\up{\upsilon_{o}=\sigma} \def\ta{\tau} \def\up{\upsilon_o^X$
and the family of balls $B_m(\xi)$,
the sequence of functions
$$
f_m(\xi):=\frac{\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(B_m(\xi)\cap K_\end{Prop}s^c )}{ \sigma} \def\ta{\tau} \def\up{\upsilon_{o}(B_m(\xi))}
$$
converges to zero for $\sigma} \def\ta{\tau} \def\up{\upsilon_{o}$-almost all $\xi\in K_\end{Prop}s$.
Since the measure $\sigma} \def\ta{\tau} \def\up{\upsilon_{o}$ is doubling, the ratios
$\displaystyle
\frac{\sigma} \def\ta{\tau} \def\up{\upsilon_{o}( B_m(\xi))}{\sigma} \def\ta{\tau} \def\up{\upsilon_{o}( B_{m+1}(\xi))}
$
are uniformly bounded,
and this implies that the ratios
$\displaystyle
\frac{\sigma} \def\ta{\tau} \def\up{\upsilon_{o}( B_m(\xi))}{\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(A_m(\xi) )}$
are also uniformly bounded.
Hence the sequence of functions
$\displaystyle
g_m(\xi):=\frac{\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(A_m(\xi)\cap K_\end{Prop}s^c)}{ \sigma} \def\ta{\tau} \def\up{\upsilon_{o}(A_m(\xi))}
$
also converges to zero for $\sigma} \def\ta{\tau} \def\up{\upsilon_{o}$-almost all $\xi\in K_\end{Prop}s$.
Therefore, by Egorov theorem, there exist a compact subset $L_\end{Prop}s\subset K_\end{Prop}s$
and an integer $m_\end{Prop}s\geq 1$ such that
$\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(L_\end{Prop}s^c)\leq \end{Prop}s$ and with
\begin{equation}
\label{eqnratiomassanuli}
\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(A_m(\xi)\cap K_\end{Prop}s^c)\leq \end{Prop}s\, \sigma} \def\ta{\tau} \def\up{\upsilon_{o}(A_m(\xi))\,
\;\;\mbox{\rm for all $m\geq m_\end{Prop}s$ and $\xi\in L_\end{Prop}s$.}
\end{equation}
{\bf Third step} {\it We bound the functions $u_n$ by using the Poisson kernel.}
Since each function $u_n$ is subharmonic with boundary value $\psi_n$, one has
\begin{eqnarray*}
u_n(x)&\leq &\int_{\partial X} \psi_n(\end{Thm}a)\,P_\end{Thm}a(x) \,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\end{Thm}a)
\;\;\mbox{\rm for all $x\in X$.}
\end{eqnarray*}
We now assume that $x$ belongs to a tube $N\xi$
with $\xi\in L_\end{Prop}s$. We write
\begin{eqnarray*}
u_n(x)&\leq &\sum_{m=0}^{\infty}I_{m,n}(x,\xi)
\;\; \mbox{\rm where
$I_{m,n}(x,\xi):=\displaystyle
\int_{A_m(\xi)} \psi_n(\end{Thm}a)\,P_\end{Thm}a(x) \,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\end{Thm}a)$.}
\end{eqnarray*}
We split this sum into two parts, according to whether $m< m_\end{Prop}s$ or $m\geq m_\end{Prop}s$.
First assume that $m<m_\end{Prop}s$.
The function $(x,\end{Thm}a)\mapsto P_\end{Thm}a(\xi)$
being continuous on $(\ol{X}\times\partial X)\smallsetminus \displaystylee_{\partial X}$, there exists a constant $C_8=C_8(m_\end{Prop}s)>0$
such that one has, for all $\xi\in\partial X$~:
\begin{eqnarray*}
\label{eqnpxeta}
P_{\end{Thm}a}(x)\leq C_8
\;\;\;\;\mbox{\rm for all $x\in N\xi$
and $\end{Thm}a\in \partial X\smallsetminus B_{m_{_\end{Prop}s}}(\xi)$.}
\end{eqnarray*}
This gives the bound
\begin{eqnarray}
\label{eqnksmall}
\sum_{m<m_\end{Prop}s}I_{m,n}(x,\xi)&\leq& C_8
\int_{\partial X} \psi_n(\end{Thm}a)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\end{Thm}a)\, ,
\end{eqnarray}
where the integral converges to $0$ when $n\to\infty$ by the Lebesgue dominated convergence theorem.
Now assume that $m\geq m_\end{Prop}s$. One splits the integral $I_{m,n}(x,\xi)$ as a sum
\begin{eqnarray*}
\label{eqnipkn0}
I_{m,n}(x,\xi)&=&I'_{m,n}(x,\xi)+I''_{m,n}(x,\xi)
\;\; \mbox{\rm where}\\
I'_{m,n}(x,\xi)&:=&
\int_{A_m(\xi)\cap K_\end{Prop}s^c} \psi_n(\end{Thm}a)\,P_\end{Thm}a(x) \,
{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\end{Thm}a),\\
I''_{m,n}(x,\xi)&:=&
\int_{A_m(\xi)\cap K_\end{Prop}s} \psi_n(\end{Thm}a)\,P_\end{Thm}a(x) \,
{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\end{Thm}a).
\end{eqnarray*}
Since $m\geq m_\end{Prop}s$ and $\xi\in L_\end{Prop}s$, we obtain by using \end{quotation}ref{eqnratiopoisson2} and \end{quotation}ref{eqnratiomassanuli} and the bound $\|\psi_n\|_\infty \leq 1$~:
\begin{eqnarray}
\label{eqnipkn}
\nonumber
I'_{m,n}(x,\xi)&\leq& \max_{\end{Thm}a\in A_m(\xi)}P_\end{Thm}a(x)\; \;
\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(A_m(\xi)\cap K_\end{Prop}s^c)\\
\nonumber
&\leq& C_3\;\min_{\end{Thm}a\in A_m(\xi)}P_\end{Thm}a(x)\;\; \end{Prop}s\,\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(A_m(\xi))\\
&\leq&
\end{Prop}s \, C_3\int_{A_m(\xi)} P_\end{Thm}a(x) \,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\end{Thm}a).
\end{eqnarray}
Assume moreover that $n\geq n_\end{Prop}s$. Using the definition of $K_\end{Prop}s$, we obtain~:
\begin{eqnarray}
\label{eqnippkn}
I''_{m,n}(x,\xi)
&\leq&
\end{Prop}s \int_{A_m(\xi)} P_\end{Thm}a(x) \,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\end{Thm}a).
\end{eqnarray}
Combining \end{quotation}ref{eqnipkn} and \end{quotation}ref{eqnippkn} and summing over $m\geq m_\end{Prop}s$, one gets
for $n\geq n_\end{Prop}s$~:
\begin{equation}
\label{eqnklarge}
\sum_{m\geq m_\end{Prop}s}I_{m,n}(x,\xi)
\;\;\leq\;\;
(1+C_3)\,\end{Prop}s
\int_{\partial X} P_\end{Thm}a(x)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\end{Thm}a)\\
\;\; =\;\;
(1+C_3)\,\end{Prop}s \, .
\end{equation}
We now define the compact $F_\end{Prop}s$ as the intersection
$F_\end{Prop}s:=\cap_{\end{Lem}l\geq 1}L_{\end{Prop}s_\end{Lem}l}$ with $\end{Prop}s_\end{Lem}l:=2^{-\end{Lem}l}\end{Prop}s$, so that $\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(F_\end{Prop}s^c)\leq \end{Prop}s$.
Combining \end{quotation}ref{eqnksmall} and \end{quotation}ref{eqnklarge} we observe that one has, for all $x$ in $N F_\end{Prop}s$ and every integers $\end{Lem}l\geq 1$ and $n\geq n_{\end{Prop}s_{_\end{Lem}l}}$~:
\begin{eqnarray*}
u_n(x)&\leq& C_8
\int_{\partial X} \psi_n(\end{Thm}a)\,{\rm d}\sigma} \def\ta{\tau} \def\up{\upsilon_{o}(\end{Thm}a)
\; + (1+C_3)\, \end{Prop}s_\end{Lem}l\, .
\end{eqnarray*}
If $\end{Lem}l$ is large enough the second term is small.
And, as we have already seen by using the Lebesgue dominated convergence theorem, the
first term is small if $n$ is large enough.
This proves that the sequence $(u_n)$ converges uniformly to $0$
on $NF_\end{Prop}s$.
\end{proof}
\subsection{Surjectivity of the boundary transform}
\label{secsurjectiveboundary}
\begin{proof}[Proof of Proposition \ref{prosurjectiveboundary}]
Let $\varphi} \def\ch{\chi} \def\ps{\psi\in L^\infty(\partial X,Y)$. We want to prove the equality $\varphi} \def\ch{\chi} \def\ps{\psi=\be P\varphi} \def\ch{\chi} \def\ps{\psi$. We may assume that $Y$ has finite diameter $\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y$.
Let $(\varphi} \def\ch{\chi} \def\ps{\psi_n)$ be a sequence in $ C^0(\partial X,Y)$ that converges almost surely to $\varphi} \def\ch{\chi} \def\ps{\psi$.
Such a sequence also converges to $\varphi} \def\ch{\chi} \def\ps{\psi$ in probability i.e. for the distance \end{quotation}ref{eqndphphp}.
Let $h_n=P\varphi} \def\ch{\chi} \def\ps{\psi_n$ and $h=P\varphi} \def\ch{\chi} \def\ps{\psi$. By construction, one has $\varphi} \def\ch{\chi} \def\ps{\psi_n=\be h_n$
and the harmonic map $h$ is the limit of the harmonic maps $h_n$, where
the convergence is uniform on compact sets of $ X$.
This is not enough to conclude. But we will prove below that,
for all $\end{Prop}s>0$,
this convergence is also uniform on the tube $NF_\end{Prop}s$
over a compact set
$F_\end{Prop}s\subset \partial X$ such that $\sigma} \def\ta{\tau} \def\up{\upsilon_{o}( F_\end{Prop}s^c)\leq \end{Prop}s$.
Since the sequence $(\varphi} \def\ch{\chi} \def\ps{\psi_n)$ converges almost surely to $\varphi} \def\ch{\chi} \def\ps{\psi$, the continuous functions
$
\psi_{m,n}:\partial X\rightarrow [0,\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y]$ defined, for $\xi$ in $\partial X$, by
$$
\psi_{m,n}(\xi)=d(\varphi} \def\ch{\chi} \def\ps{\psi_m(\xi),\varphi} \def\ch{\chi} \def\ps{\psi_n(\xi))
$$
converge almost surely to $0$
when $m,n$ go to $\infty$.
The functions
$u_{m,n}:X\rightarrow [0,\delta} \def\end{Prop}s{\varepsilon} \def\ze{\zeta_Y]$ defined by
$$
u_{m,n}(x)=d(h_m(x),h_n(x))
$$
for $x$ in $X$
extend continuously the functions $\psi_{m,n}$.
By Lemma \ref{lemharmonicsubharmonic}, these functions $u_{m,n}$
are subharmonic on $ X$.
Let $\end{Prop}s\!>\!0$. Lemma \ref{lemradialconvergence} ensures that
there exists a compact subset
$F_{\end{Prop}s}\!\subset\!\partial X$ such that
$\sigma} \def\ta{\tau} \def\up{\upsilon_{o}( F_{\end{Prop}s}^c)\leq\end{Prop}s$, and such that
the sequence $(u_{m,n})$ converges uniformly to $0$ on
the tube $NF_\end{Prop}s$.
This tells us that the convergence of the sequence $(h_n)$
to $h$ is uniform on the tube $NF_\end{Prop}s$.
By Egorov theorem, we may also assume that
the sequence $(\varphi} \def\ch{\chi} \def\ps{\psi_n)$ converges uniformly to $\varphi} \def\ch{\chi} \def\ps{\psi$ on $F_\end{Prop}s$.
Therefore the function $\ol h:\ol X\rightarrow Y$ equal to $h$ on $X$
and equal to $\varphi} \def\ch{\chi} \def\ps{\psi$ on $\partial X$ is continuous on
$NF_\end{Prop}s\cup F_\end{Prop}s$.
This proves that, when $\xi$ is in $F_\end{Prop}s$,
the non tangential limit
$
\underset{x\rightarrow\xi}{\rm NTlim}\,h(x)$
given by Proposition \ref{pronontangentiallimit}
is equal to $\varphi} \def\ch{\chi} \def\ps{\psi(\xi)$.
Since the measure of $F_\end{Prop}s^c$ is arbitrarily small,
the map $\varphi} \def\ch{\chi} \def\ps{\psi$ is the boundary map of $h$.
\end{proof}
\subsection{A concrete example}
We give a concrete example of application of
Theorem \ref{thmbijectiveboundary} in a situation
where the boundary map has finite image in the hyperbolic plane $\m H^2$.
\begin{Cor}
\label{corexample}
Let $X:=\m H^2$ and $x_1,\ldots,x_n$ be $n$ points on $\partial \m H^2$
cutting $\partial \m H^2$ into $n$ open arcs $I_1,\ldots, I_n$.
Let $y_1,\ldots,y_n$ be $n$ points on $Y:=\m H^2$.
Then there exists a unique bounded harmonic map $h:\m H^2\rightarrow \m H^2$ that extends continuously to the arcs $I_j$
with
$h(I_j)=\{y_j\}$, for all $j$.
\end{Cor}
\begin{proof}
The map $h$ has to be the Poisson transform of the map $\varphi} \def\ch{\chi} \def\ps{\psi:\partial\m H^2 \rightarrow \m H^2$
that sends the sides $I_j$ to the point $y_j$, for all $j$.
We only have to check that this map $h$ extends continuously $\varphi} \def\ch{\chi} \def\ps{\psi$
outside the points $x_j$. This is the content of Proposition \ref{procontinuityboundary}.
\end{proof}
\begin{Prop}
\label{procontinuityboundary}
Let $X$ be a {\bf GGG } Riemannian manifold,\! and $Y$ be a complete {\rm CAT(0)}-space.
Let $h\!:\! X\!\rightarrow\! Y$ be a bounded harmonic map
and $\varphi} \def\ch{\chi} \def\ps{\psi:\partial X\rightarrow Y$ be its boundary map.
Let $I\subset \partial X$ be an open set on which $\varphi} \def\ch{\chi} \def\ps{\psi$ is continuous.
Then, for all $\xi\!\in\! I$, one has $\varphi} \def\ch{\chi} \def\ps{\psi(\xi):=\underset{x\rightarrow\xi}{\rm lim}\,h(x)$.
\end{Prop}
\begin{proof}
This follows by the same argument as in the proof
of Proposition \ref{prodirichlet} in Section \ref{secdirichletproblem}.
\end{proof}
\subsection{Bounded Lipschitz domain}
\label{seclipschitzdomain}
A ``bounded Lipschitz Riemannian domain'' $ \Om$ means
a connected bounded open subset of a smooth Riemannian manifold $(M,g_0)$
such that $ \Om$ is the interior of a connected
compact submanifold $\ol \Om$ of $M$ whose boundary $\partial \Om$ is a non empty Lipschitz continuous codimension one submanifold.
The same argument as for Theorem \ref{thmbijectiveboundary}
will give the following corollary
\begin{Cor}
\label{corbijectiveboundary}
Let $\Om$ be a bounded Lipschitz Riemannian domain, and $Y$ be a complete {\rm CAT(0)}-space.\\
$a)$ Let $h\!:\! \Om\!\rightarrow\! Y$ be a bounded harmonic map.
Then, for $\sigma} \def\ta{\tau} \def\up{\upsilon$-almost all $\xi\!\in\! \partial \Om$,
the map $h$ admits a non-tangential limit $\varphi} \def\ch{\chi} \def\ps{\psi(\xi):=\underset{x\rightarrow\xi}{\rm NTlim}\,h(x)$ at the point $\xi$.\\
$b)$ The map $h\mapsto \be(h):=\varphi} \def\ch{\chi} \def\ps{\psi$
gives a bijection
$
\beta:\mc H_b( \Om,Y)\stackrel{\sigma} \def\ta{\tau} \def\up{\upsilonm}{\longrightarrow} L^\infty(\partial \Om,Y).
$
\end{Cor}
In this case, a non tangential limit means a limit along all sequences $x_n$ such that
$\displaystyle\sup\limits_{n\geq 1}\frac{d(x_n, \xi)}{d(x_n,\partial \Om)}<\infty$.
The measure $\sigma} \def\ta{\tau} \def\up{\upsilon$ is any finite Borel measure on $\partial \Om$ which is equivalent to any of
the harmonic measures of $\Om$.
Since $\Om$ is a bounded Lipschitz Riemannian domain, by Dahlberg's theorem in \cite{Dahlberg79}, one can
choose $\sigma} \def\ta{\tau} \def\up{\upsilon$ to be the Riemannian measure on $\partial \Om$.
As in Section \ref{secprevious},
when $Y$ is a Riemannian manifold, Corollary \ref{corbijectiveboundary}.$a$ is due to
Aviles, Choi, Micallef in \cite[Thm 5.1]{AvilesChoiMicallef91},
and Theorem \ref{corbijectiveboundary}.$b$ is expected to be true
as a final observation
in \cite[Section 1]{AvilesChoiMicallef91}.
The first cases of Corollary \ref{corbijectiveboundary}.$b$
that seem to be new is when
$X$ is the Euclidean unit ball in $\m R^k$ and
$Y$ is the hyperbolic space $\m H^\end{Lem}l$.
\begin{proof}
Corollary \ref{corbijectiveboundary}
is a corollary of the proof of
Theorem \ref{thmbijectiveboundary}.
The strategy is the same, relying on variations of
Propositions \ref{propoisson2} and \ref{pronontangentiallimit2}
for bounded Lipschitz Riemannian domains.
The proofs of these variations are very similar. The only difference is that they rely on \cite{Ancona78} instead of \cite{AnconaStFlour}.
\end{proof}
\begin{Rmq}
\label{remtrickBHK}
The fact that Corollary \ref{corbijectiveboundary} is a special case of Theorem \ref{thmbijectiveboundary} can also be explained
thanks to a trick due to Bonk, Heinonen and Koskela
in \cite[Chapter 8]{BonkHeinonenKoskela}. This trick
consists in replacing the Riemannian metric $g_0$ on $\Om$ by $g=\ol{d}(x)^{-2}g_0$ where $\ol d$ is a suitable $C^\infty$ function roughly equal to the distance to the boundary,
obtaining this way a {\bf GGG } Riemannian manifold $(\Om,g)$.
One then sees the harmonic and subharmonic functions on
$(\Om,g_0)$
as $\mc L$-harmonic and $\mc L$-subharmonic functions
on $X$ where $\mc L:= \ol{d}(x)^2\displaystylee_{g_0}$ is an elliptic
differential operator of order $2$ which is equal to the Laplacian
$\displaystylee_g$ up to terms of order $1$ and which also has spectral gap.
All the arguments we developped in this paper for the Laplace operator
$\displaystylee_g$ also apply to the operator $\mc L$.
This trick could be applied to a much wider class of bounded Riemannian domains called ``inner uniform domain''. See also Aikawa in \cite[Theorem 1.2]{Aikawa04}
for more on these domains.
\end{Rmq}
\small{
}
pace{2em}
{\small\noindent
Y. Benoist: CNRS \& Université Paris-Saclay
{\tt [email protected]}\\
D. Hulin:\;\;\; Université Paris-Saclay
{\tt [email protected]}}
\end{document} |
\begin{document}
\title{On the Distributions of Infinite Server Queues with Batch Arrivals}
\author{
Andrew Daw \\ School of Operations Research and Information Engineering \\ Cornell University
\\ 257 Rhodes Hall, Ithaca, NY 14853 \\ [email protected] \\
\and
Jamol Pender \\ School of Operations Research and Information Engineering \\ Cornell University
\\ 228 Rhodes Hall, Ithaca, NY 14853 \\ [email protected] \\
}
\maketitle
\abstract{
Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as ``batch'' (or, in some cases, ``bulk'') arrival queueing systems. In this work we study the affect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.
}
\section{Introduction}\label{Intro}
Queueing systems with batch arrivals have enjoyed a long and rich history of study, at least on the time scale of queueing theory. Researchers have been exploring models of this sort for no less than six decades, based on the April 1958 submission date of \citet{miller1959contribution}. Given this stretch of time, a wide variety of systems and settings have been considered under the banner of batch arrivals. Much of the earliest work focuses on single server models, including \citet{miller1959contribution, lucantoni1991new, masuyama2002analysis, liu1993autocorrelations} and \citet{foster1964batched}, although infinite server models followed soon after, such as work by \citet{shanbhag1966infinite} and \citet{brown1969some}. Later work has expanded the concept into a variety of related models, such as for priority queues \cite{takagi1991priority} and for handling server vacations \cite{lee1995batch}. Additionally, there is some work that proves heavy traffic limit theorems for queues with batch arrivals. Examples of this include \citet{chiamsiri1981diffusion, pang2012infinite, pender2013poisson}. These papers show that one can approximate the queue length process with Brownian motion and Ornstein-Uhlenbeck processes and also show that one can exploit the approximations even in multi-server and non-Markovian settings.
In this paper we consider queues with arrivals occurring at times following a Poisson process, with consideration given to both non-stationary and stationary rates. We analyze both both general and exponential service as conducted by infinitely many servers. Additionally, this work addresses both fixed and random batch sizes. Our analysis starts with the fixed batch size case. We begin by analyzing the transient behavior of the queue with Markovian service and time-varying arrival rates, providing explicit forms for the moment generating function, mean, and variance. Then, we show that if the arrival rate is stationary the resulting steady-state distribution can be written as a sum of independent, non-identical, scaled Poisson random variables. This leads us to uncover connections to the harmonic numbers and generalizations of the Hermite distribution. By viewing the batch arrival queue as a collection of infinite server sub-queues that receive solitary arrivals simultaneously, we are able to extend this Poisson sum construction to general service distributions. This perspective also provides an avenue for us to extend to random batch sizes. We also give fluid and diffusion scalings of the queue in the case of random batch sizes, as well as extending many of the results we found for fixed batch sizes.
One can note that the batch arrival queue may not always be given the name ``batch,'' as many authors choose to use the term ``bulk'' instead. Predominantly, this reflects two leading strands of applications, where ``bulk'' often gives a connotation of transportation settings whereas ``batch'' frequently implies applications in communications. Just as practical by any other name, this family of models has also been studied in a wide variety of applications beyond these two. Perhaps one most distinct from other types of queueing models is particle splitting in DNA caused by radiation, as discussed in \citet{sachs1992dna}. In this application, primary particles arrive at a cell nucleus and cause DNA double-strand breaks. These double-strand breaks occur in near simultaneity and are thus modeled as arriving in batches of random size, as it is possible that any number double-strand breaks will be induced. After they are induced, the double-strand breaks are then processed by cellular enzymes, corresponding to service in the queueing model. Another interesting and modern application of these models is in cloud-based data processing. In this case, the batches arriving to the system are collections of jobs submitted simultaneously. These jobs are then served by each being processed individually and returned. For more discussion, detailed models, and specific analysis for this setting, see works such as \citet{lu2011join, pender2016law, xie2017pandas, yekkehkhany2018gb} and references therein.
\subsection{Main Contributions of Paper}
Our contributions in this work can be summarized as follows:
\begin{enumerate}[i)]
\item We show that an infinite server queue with batch arrivals at Poisson process epochs is equivalent in steady state distribution to a sum of scaled independent Poisson random variables, including for generally distributed service and randomly distributed batch sizes. For exponential service, this reveals a connection to the harmonic numbers and generalized Hermite distributions.
\item We derive a limit of the process in which the batch size grows infinitely large and the number of entities in the system is scaled inverse proportionally, yielding a novel distribution characterized by the exponential integral functions. For distributions that meet a divisibility condition, we find that this also holds for random batch sizes.
\item In the case of time-varying arrival rates we give a transient moment generating function for fixed batch sizes as well as means and variance for both fixed and randomly sized batches.
\item We give fluid and diffusion limits of the queue for stationary arrival rates for batches of random size.
\end{enumerate}
\subsection{Organization of Paper}
The body of the remainder of this paper is organized in two main sections: Sections~\ref{deterSec} and~\ref{randomSec}. In Section~\ref{deterSec} we consider systems in which the size of the batches is fixed. Similarly, we devote Section~\ref{randomSec} to the case of randomly distributed batch sizes. At the beginning of each section we give a detailed overview of the contents within and provide context for the analysis in term of this project's scope. After these sections we conclude in Section~\ref{concSec}.
\section{Batches of Deterministic Size}\label{deterSec}
In this section we will consider infinite server queues with arrivals occurring in batches of a fixed size. We will assume that the arrival epochs occur according to a Poisson process, including both stationary and non-stationary models. We also will investigate both exponentially and generally distributed service.
This section starts with studying the case of Markovian arrivals and service in transient state in Subsection~\ref{transientSub}. For a time-varying arrival rate, we give the mean, variance, and moment generating function. We then use this in Subsection~\ref{markovSS} to find the steady-state distribution of the queue. Upon observing that this can be represented as a sum of scaled Poisson random variables, we establish connections to generalized Hermite distributions and to the harmonic numbers. Taking motivation from this, we derive the distribution of the limit of the scaled system as the batch size grows infinitely large. Finally, in Subsection~\ref{orderStatSubsec}, we examine the batch queue as a collection of infinite server sub-queues that simultaneously receive solitary arrivals. In doing so we extend our understanding of the steady-state distribution to the case of general service.
\subsection{Transient Analysis of the Markovian Setting}\label{transientSub}
We begin our analysis with the case of non-stationary Poisson arrival epochs and Markovian service. In Kendall notation, this is the $M_t^n/M/\infty$ queue. We let $Q_t$ represent the number of entities present in the queueing system at time $t \geq 0$, which we often refer to as the ``number in system.'' We will use this notation throughout the remainder of this work, where the precise setting of the queue will be implied by context. In this fully Markovian setting, we can use Dynkin's infinitesimal generator theorem to support our analysis. Specifically, we can note that for a sufficiently regular function $f:\mathbb{N} \to \mathbb{R}$, we have
\begin{align}\label{dynkin}
\frac{\mathrm{d}}{\mathrm{d}t}\E{f(Q_t)}
&
=
\E{\lambda(t) \left(f(Q_t + n) - f(Q_t)\right) + \mu Q_t \left(f(Q_t - 1) - f(Q_t)\right)}
,
\end{align}
for a batch arrival queue with arrival intensity $\lambda(t) > 0$. We will see in this subsection that this infinitesimal generator approach gives us a potent toolkit for exploring this model. Moreover, the insights we find in Markovian settings now and in Subsection~\ref{markovSS} will provide intuition that will guide our investigation of this system when the Markov property does not hold. To begin, we now derive the moment generating function of the number in system. We do so for a system with a non-stationary arrival rate given by a Fourier series, allowing these results to hold for all periodic arrival patterns.
\begin{proposition}\label{batchMGF}
For $\theta \in \mathbb{R}$, let $\mathcal{M}(\theta, t) = \E{e^{\theta Q_t}}$ be the moment generating function of the number in system of an infinite server queue with periodic arrival rate $\lambda + \sum_{k=1}^\infty a_k \cos(k t) + b_k \sin(k t) > 0$, arrival batch size $n \in \mathbb{Z}^+$, and exponential service rate $\mu > 0$. Then, $\mathcal{M}(\theta, t)$ is given by
\begin{align}
\mathcal{M}(\theta, t)
&
=
\left(e^{- \mu t}(e^{\theta} - 1) + 1\right)^{Q_0}
e^{
\sum_{j=1}^n {n \choose j} (e^\theta - 1)^j
\left(
\frac{\lambda}{j\mu}
\left(
1 - e^{-j \mu t}
\right)
+
\sum_{k=1}^\infty
\frac{(a_k j \mu - b_k k) }{k^2+j^2\mu^2}
\left(
\cos(kt) - e^{-j\mu t}
\right)
\right)
}
\nonumber
\\
&
\quad
\cdot
e^{
\sum_{j=1}^n {n \choose j} (e^\theta - 1)^j
\sum_{k=1}^\infty
\frac{(a_k k + b_k j \mu )\sin(k t)}{k^2+j^2\mu^2}
}
\end{align}
for all time $t \geq 0$, where $Q_0$ is the initial number in system.
\begin{proof}
From Equation~\ref{dynkin}, the MGF is given by the solution to the partial differential equation
\begin{align*}
\frac{\partial}{\partial t} \mathcal{M}(\theta,t)
&=
\left(\lambda + \sum_{k=1}^\infty a_k \cos(k t) + b_k \sin(k t)\right) \left(e^{n \theta } - 1\right)
\mathcal{M}(\theta,t)
+
\mu \left(e^{-\theta} - 1\right)
\frac{\partial}{\partial \theta}
\mathcal{M}(\theta,t)
\end{align*}
with initial solution $\mathcal{M}(\theta , 0) = e^{\theta Q_0}$. Because $\frac{\mathrm{d}\log(f(x))}{\mathrm{d}x} = \frac{1}{f(x)}\frac{\mathrm{d}f(x)}{\mathrm{d}x}$, we can observe that the partial differential equation for the cumulant generating function $G(\theta, t) = \log \left( \E{e^{\theta Q_t}}\right)$ is
$$
\mu(1 - e^{-\theta})\frac{\partial G(\theta, t)}{\partial \theta} + \frac{\partial G(\theta, t)}{\partial t} = \left(\lambda + \sum_{k=1}^\infty a_k \cos(k t) + b_k \sin(k t)\right)(e^{n\theta} - 1)
,
$$
with the initial condition
$
G(\theta, 0) = \log\left(\E{e^{\theta Q_0}}\right) = \theta Q_0
$.
We will now solve this PDE by the method of characteristics. We begin by establishing the characteristic ODE's and corresponding initial solutions as follows:
\begin{align*}
\frac{\mathrm{d}\theta}{\mathrm{d}s}(r,s) &= \mu(1 - e^{-\theta}), &\theta(r,0) = r,\\
\frac{\mathrm{d}t}{\mathrm{d}s}(r,s) &= 1, & t(r,0) = 0,\\
\frac{\mathrm{d}g}{\mathrm{d}s}(r,s) &= \left(\lambda + \sum_{k=1}^\infty a_k \cos(k t) + b_k \sin(k t)\right)(e^{n\theta} - 1), & g(r,0) = rQ_0.
\end{align*}
The first two of these initial value problems yield the following solutions.
\begin{align*}
\theta(r,s) &= \log(e^{c_1(r) + \mu s} + 1) &&\to\quad \theta(r,s) = \log\left((e^r - 1) e^{\mu s} + 1\right)\\
t(r,s) &= s + c_2(r) &&\to\quad t(r,s) = s
\end{align*}
Therefore we can simplify the remaining characteristic ODE to
\begin{align*}
\frac{\mathrm{d}g}{\mathrm{d}s}(r,s)
&
=
\left(\lambda + \sum_{k=1}^\infty a_k \cos(k s) + b_k \sin(k s)\right)\left(\left((e^r - 1) e^{\mu s}+1\right)^n -1 \right)
\\
&
=
\left(\lambda + \sum_{k=1}^\infty a_k \cos(k s) + b_k \sin(k s)\right) \sum_{j=1}^n {n \choose j} (e^r - 1)^j e^{j\mu s},
\end{align*}
and this produces the general solution of
\begin{align*}
g(r,s)
&
=
c_3(r)
+
\sum_{j=1}^n {n \choose j} (e^r - 1)^j
\left(
\frac{\lambda}{j\mu}
+
\sum_{k=1}^\infty
\frac{(a_k j \mu - b_k k) \cos(k s)}{k^2+j^2\mu^2}
+
\frac{(a_k k + b_k j \mu )\sin(k s)}{k^2+j^2\mu^2}
\right)
e^{j\mu s}
.
\intertext{This now equates to}
g(r,s)
&
=
r Q_0
+
\sum_{j=1}^n {n \choose j} (e^r - 1)^j
\Bigg(
\frac{\lambda}{j\mu}
\left(
e^{j \mu s} - 1
\right)
+
\sum_{k=1}^\infty
\frac{(a_k j \mu - b_k k) }{k^2+j^2\mu^2}
\left(
\cos(ks) e^{j \mu s} - 1
\right)
\\
&
\quad
+
\sum_{k=1}^\infty
\frac{(a_k k + b_k j \mu )\sin(k s)}{k^2+j^2\mu^2}e^{j\mu s}
\Bigg)
\end{align*}
as the solution to the initial value problem. We now find the solution to the original PDE by solving for each characteristic variable in terms of $t$ and $\theta$ and then substituting these expression into $g(r,s)$. That is, for $s = t$ and $r = \log\left(e^{- \mu t}(e^{\theta} - 1) + 1\right)$, we have that
\begin{align*}
G(\theta, t)
&=
g\left(\log\left(e^{- \mu t}(e^{\theta} - 1) + 1\right), t\right)
\\&=
\log\left(e^{- \mu t}(e^{\theta} - 1) + 1\right)Q_0
+
\sum_{j=1}^n {n \choose j} (e^\theta - 1)^j
\Bigg(
\frac{\lambda}{j\mu}
\left(
1 - e^{-j \mu t}
\right)
+
\sum_{k=1}^\infty
\frac{(a_k j \mu - b_k k) }{k^2+j^2\mu^2}
\\
&
\quad
\cdot
\left(
\cos(kt) - e^{-j\mu t}
\right)
+
\sum_{k=1}^\infty
\frac{(a_k k + b_k j \mu )}{k^2+j^2\mu^2} \sin(k t)
\Bigg)
.
\end{align*}
To conclude the proof, we note that $\mathcal{M}(\theta, t) = e^{G(\theta, t)}$.
\end{proof}
\end{proposition}
We now extend this analysis through two following corollaries. First, for systems with a stationary arrival rate, say $\lambda > 0$, we can further specify the moment generating function explicitly in Corollary~\ref{statCor}. This will be of use when we explore the distribution of the queue in steady-state, which we begin in Subsection~\ref{markovSS}. As with Proposition~\ref{batchMGF}, the uniqueness of moment generating functions will aid us in later exploration of the distributions within this model and within generalizations of it.
\begin{corollary}\label{statCor}
For $\theta \in \mathbb{R}$, let $\mathcal{M}(\theta, t) = \E{e^{\theta Q_t}}$ be the moment generating function of the number in system of an infinite server queue with stationary arrival rate $\lambda > 0$, arrival batch size $n \in \mathbb{Z}^+$, and exponential service rate $\mu > 0$. Then, $\mathcal{M}(\theta, t)$ is given by
\begin{align}
\mathcal{M}(\theta, t)
&
=
\left(e^{- \mu t}(e^{\theta} - 1) + 1\right)^{Q_0}
e^{
\lambda \sum_{j=1}^n {n \choose j} \frac{(e^\theta - 1)^j }{j\mu}
\left(
1 - e^{-j \mu t}
\right)
}
\end{align}
for all time $t \geq 0$, where $Q_0$ is the initial number in system.
\end{corollary}
For the second direct result of Proposition~\ref{batchMGF}, we can also give explicit expressions for the transient mean and variance of the queue. We derive these equations from the first and second derivatives, respectively, of the cumulant generating function $\log(\E{e^{Q_t}})$.
\begin{corollary}\label{meanvarcor}
Let $Q_t$ be an infinite server queue with periodic arrival rate $\lambda + \sum_{k=1}^\infty a_k \cos(k t) + b_k \sin(k t) > 0$, arrival batch size $n \in \mathbb{Z}^+$, and exponential service rate $\mu > 0$. Then, the mean and variance of the queue are given by
\begin{align}
\E{Q_t}
&
=
Q_0 e^{-\mu t}
+
\frac{n\lambda }{\mu}
\left(
1 - e^{- \mu t}
\right)
+
\sum_{k=1}^\infty
\frac{n(a_k \mu - b_k k) }{k^2+\mu^2}
\left(
\cos(kt) - e^{-\mu t}
\right)
\nonumber
\\
&
\quad
+
\sum_{k=1}^\infty
\frac{n(a_k k + b_k \mu )}{k^2+\mu^2} \sin(k t)
\\
\Var{Q_t}
&
=
Q_0\left(e^{-\mu t} - e^{-2\mu t}\right)
+
\frac{n\lambda }{\mu}
\left(
1 - e^{- \mu t}
\right)
+
\sum_{k=1}^\infty
\frac{n(a_k \mu - b_k k) }{k^2+\mu^2}
\left(
\cos(kt) - e^{-\mu t}
\right)
\nonumber
\\
&
\quad
+
\sum_{k=1}^\infty
\frac{n(a_k k + b_k \mu )}{k^2+\mu^2} \sin(k t)
+
\frac{n(n-1) \lambda }{2\mu}
\left(
1 - e^{-2 \mu t}
\right)
+
\sum_{k=1}^\infty
\frac{n(n-1)(2 a_k \mu - b_k k) }{k^2 + 4\mu^2}
\nonumber
\\
&
\quad
\cdot
\left(
\cos(kt) - e^{-2\mu t}
\right)
+
\sum_{k=1}^\infty
\frac{n(n-1)(a_k k + 2 b_k \mu )}{k^2+ 4\mu^2} \sin(k t)
\end{align}
for all time $t \geq 0$, where $Q_0$ is the initial number in system.
\end{corollary}
In the remainder of this work we will explore various modifications of this model, including general service and randomized batch sizes. The results of this subsection will serve as cornerstone throughout much of this upcoming analysis, both supporting the underlying derivation techniques and providing the intuition for new perspectives.
\subsection{The Markovian System with Stationary Arrival Rates}\label{markovSS}
Our first departure from our initial model will be modest: instead of studying the fully Markovian, non-stationary, fixed batch size system in transient time we will now move to addressing the stationary case, with much of our analysis focused on the system in steady-state. This simplified setting will allow us to extract greater intuition from our prior findings, which in turn will support generalization of the service distribution and randomization of the batch sizes. To begin, we find a representation of the steady-state distribution of the queue length in terms of a sum of independent, scaled Poisson random variables.
\begin{proposition}\label{ssDist}
In steady-state the distribution of the number in system of an infinite server queue with stationary arrival rate $\lambda > 0$, arrival batch size $n \in \mathbb{Z}^+$, and exponential service rate $\mu > 0$ is
\begin{align}
Q_\infty(n)
\stackrel{D}{=}
\sum_{j=1}^n
j Y_j
\end{align}
where $Y_j \sim \mathrm{Pois}\left(\frac{\lambda}{j\mu}\right)$ are independent.
\begin{proof}
From Proposition~\ref{batchMGF}, we have that the steady-state moment generating function of the queue is given by
\begin{align*}
\lim_{t\to \infty}\mathcal{M}(\theta, t)
&
=
e^{
\lambda \sum_{k=1}^n {n \choose k} \frac{\left(e^{\theta} - 1\right)^k}{k\mu}
}
.
\end{align*}
To satisfy our stated Poisson form, we are now left to show that $\sum_{k=1}^n {n \choose k}\frac{(e^\theta - 1)^k}{k} = \sum_{k=1}^n \frac{e^{k \theta} - 1}{k}$ for all $n \in \mathbb{Z}^+$. We proceed by induction. In the base case of $n=1$ we have $e^{\theta} - 1 = e^\theta - 1$ and so we are left to show the inductive step. We now assume $\sum_{k=1}^n {n \choose k}\frac{(e^\theta - 1)^k}{k} = \sum_{k=1}^n \frac{e^{k \theta} - 1}{k}$ holds at $n$. Then, by the Pascal triangle identity ${n \choose k} = {n+1 \choose k} - {n \choose k -1}$ and our inductive hypothesis we can observe
\begin{align*}
\sum_{k=1}^n \frac{e^{k \theta} - 1}{k}
&=
\sum_{k=1}^n {n \choose k}\frac{(e^\theta - 1)^k}{k}
=
\sum_{k=1}^n \left({n+1 \choose k} - {n \choose k -1}\right)\frac{(e^\theta - 1)^k}{k}.
\end{align*}
Now, by applying the identity ${n \choose k -1} = \frac{k}{n+1}{n+1 \choose k}$ and distributing the summation we can further note that
\begin{align*}
\sum_{k=1}^n \left({n+1 \choose k} - {n \choose k -1}\right)\frac{(e^\theta - 1)^k}{k}
&
=
\sum_{k=1}^n \left({n+1 \choose k} - \frac{k}{n+1}{n+1 \choose k}\right)\frac{(e^\theta - 1)^k}{k}
\\
&
=
\sum_{k=1}^n {n+1 \choose k}\frac{(e^\theta - 1)^k}{k} - \frac{\sum_{k=1}^n{n+1 \choose k}(e^\theta - 1)^k}{n+1}.
\end{align*}
Now, we can use the binomial theorem to see that
$$
\sum_{k=1}^n{n+1 \choose k}(e^\theta - 1)^k
=
(e^\theta - 1 + 1)^{n+1} - 1 - (e^{\theta}-1)^{n+1}
=
e^{(n+1)\theta} - 1 - (e^{\theta}-1)^{n+1}
,
$$
and so we can now simplify and find
\begin{align*}
\sum_{k=1}^n {n+1 \choose k}\frac{(e^\theta - 1)^k}{k}
-
\frac{\sum_{k=1}^n{n+1 \choose k}(e^\theta - 1)^k}{n+1}
&
=
\sum_{k=1}^n {n+1 \choose k}\frac{(e^\theta - 1)^k}{k}
+
\frac{(e^\theta - 1)^{n+1}}{n+1}
\\
&
\quad
-
\frac{e^{(n+1)\theta} - 1}{n+1}
.
\end{align*}
Hence, in conjunction with our initial equation, we have that
\begin{align*}
\sum_{k=1}^n \frac{e^{k \theta} - 1}{k}
&
=
\sum_{k=1}^n {n+1 \choose k}\frac{(e^\theta - 1)^k}{k}
+
\frac{(e^\theta - 1)^{n+1}}{n+1}
-
\frac{e^{(n+1)\theta} - 1}{n+1}
,
\end{align*}
and by rearranging terms we now complete the inductive approach:
\begin{align*}
\sum_{k=1}^{n+1} \frac{e^{k \theta} - 1}{k}
&
=
\sum_{k=1}^{n+1} {n+1 \choose k}\frac{(e^\theta - 1)^k}{k}
.
\end{align*}
We can now observe that we have a moment generating function that is a product of moment generating functions of scaled Poisson random variables, which yields the stated result.
\end{proof}
\end{proposition}
While we will continue to explore the stationary arrival rate setting throughout this subsection, we note that this Poisson sum representation will be a leading inspiration in the sequel. Specifically, in Subsection~\ref{orderStatSubsec} we will find intuition for this result by viewing the batch arrival queue as a collection of sub-systems.
\begin{remark}
In addition to this Poisson sum representation, we can also express the steady-state MGF in terms of the truncated polylogarithm function and harmonic numbers. From the MGF of the queue length in steady state for $\theta < 0$, we can observe that
\begin{align*}
\lim_{t\to \infty}\mathcal{M}(\theta, t)
&
=
e^{ \frac{\lambda}{\mu} \sum_{k=1}^n \frac{e^{k \theta} - 1}{k} }
= e^{ \frac{\lambda}{\mu} ( \mathrm{Li}(e^\theta,n,1) - H_n )}
\end{align*}
where we have $H_n$ as the $n^\text{th}$ harmonic number, given by $\sum^{n}_{k=1} \frac{1}{k}$, and where the truncated polylogarithm function $\mathrm{Li}(z,n,s)$ is defined as
\begin{equation*}
\mathrm{Li}(z,n,s) = \sum^{n}_{k=1} \frac{z^k}{k^s}.
\end{equation*}
\end{remark}
This decomposition into Poisson random variables can be quite useful from a computational standpoint. It allows us to simulate the steady state quite easily since we only need to simulate $n$ Poisson random variables instead of simulating an actual queue, which could be quite expensive. We can now observe that this construction also yields an interesting connection to both the harmonic number and Hermite distributions, as suggested in the remark above. To motivate our following analysis, suppose that $n=2$. Then, steady-state queue length has steady-state moment generating function given by
$$
\mathcal{M}_n(\theta, \infty)
=
e^{ \frac{\lambda}{\mu} \left(e^{ \theta} - 1\right) + \frac{\lambda}{2\mu} \left(e^{ 2\theta} - 1\right) }.
$$
We can now observe that this MGF corresponds to a Hermite distribution with parameters $\frac{\lambda}{\mu}$ and $\frac{\lambda}{2\mu}$. This implies that the steady-state CDF of the queue at $n=2$ is
\begin{align*}
P(Q_{\infty}(2) \leq k)
& = e^{-\frac{3\lambda}{2\mu} }\sum_{i=0}^{\lfloor k\rfloor} \sum_{j=0}^{\lfloor i/2\rfloor} \frac{\left(\frac{\lambda}{\mu}\right)^{i-2j}\left(\frac{\lambda}{2\mu}\right)^j}{(i-2j)!j!}
= e^{-\frac{3\lambda}{2\mu} } \sum_{i=0}^{\lfloor k\rfloor} \sum_{j=0}^{\lfloor i/2\rfloor} \frac{\left( \frac{\lambda}{\mu} \right)^{i-j} 2 ^{-j}}{(i-2j)!j!}
.
\end{align*}
Furthermore, the steady-state PMF of the queue length is given by
\begin{align*}
P(Q_{\infty}(2) = i ) & = e^{-\frac{3\lambda}{2\mu} } \sum_{j=0}^{\lfloor i/2\rfloor} \frac{\left( \frac{\lambda}{\mu} \right)^{i-j} 2 ^{-j}}{(i-2j)!j!} .
\end{align*}
This observation prompts us to ponder generalizations for $n \geq 3$. The term ``generalized Hermite distribution'' has taken on slightly varying (yet always interesting) definitions for different authors. For readers interested in the Hermite distribution and popular generalizations of it, we suggest \citet{kemp1965some,gupta1974generalized}, and \citet{milne1993generalized}. In our setting we note that the coefficients of $\frac{\lambda}{\mu}$ in the MGF for batch size $n$ will be $1$, $\frac{1}{2}$, $\frac{1}{3}$, \dots, $\frac{1}{n}$. For this reason, we think of this particular generalization of Hermite distributions to be the \textit{harmonic Hermite distribution}. We can now note that because of this harmonic structure we can instead fully characterize the distribution simply by $n$ and $\frac{\lambda}{\mu}$. In the following proposition we find a useful recursion for the probability mass function of this distribution at all $n \in \mathbb{Z}^+$.
\begin{proposition}\label{recurProp}
Let $Q_{t}(n)$ be an infinite server batch arrivals queue with arrival rate $\lambda > 0$, batch size $n \in \mathbb{Z}^+$, and service rate $\mu > 0$. Then, the steady-state distribution of the queue is given by the recursion
\begin{equation}
\mathbb{P}(Q_{\infty}(n) = j ) = p_j = \sum^{n}_{i=1} i p_{j-i} \frac{\lambda}{i j \mu} = \sum^{n}_{i=1} p_{j-i} \frac{\lambda}{ j \mu},
\end{equation}
where $p_0 = e^{-\frac{\lambda}{\mu}H_n}$ for $H_n$ as the $n^\text{th}$ harmonic number and $p_k = 0$ for all $k < 0$. Thus, we say that $Q_\infty(n)$ follows the ``harmonic Hermite distribution'' with parameter $n$.
\begin{proof}
We know from our Poisson representation of the steady state queue length that the steady-state moment generating function is
\begin{equation*}
M(\theta) = \sum^{\infty}_{j=0} \mathbb{P}(Q_{\infty}(n) = j ) \theta^j = \sum^{\infty}_{j=0} p_j \theta^j = \exp \left( \sum^{n}_{i=1} \frac{\lambda}{i \mu} \left( \theta^i - 1 \right) \right) .
\end{equation*}
If we take the logarithm of both sides we see that we have
\begin{equation*}
\log \left( \sum^{\infty}_{j=0} p_j \theta^j \right) = \sum^{n}_{i=1} \frac{\lambda}{i \mu} \left( \theta^i - 1 \right) .
\end{equation*}
Now we take the derivative of both sides with respect to the parameter $\theta$ and this yields the following expression
\begin{equation*}
\frac{ \sum^{\infty}_{j=1} j p_j \theta^{j-1} }{ \sum^{\infty}_{j=0} p_j \theta^j } = \sum^{n}_{i=1} \frac{\lambda}{ \mu} \theta^{i-1} .
\end{equation*}
By moving the denominator to the righthand side, we have that
\begin{equation*}
\sum^{\infty}_{j=1} j p_j \theta^{j-1} = \left( \sum^{\infty}_{j=0} p_j \theta^j \right) \left( \sum^{n}_{i=1} \frac{\lambda}{ \mu} \theta^{i-1} \right).
\end{equation*}
Finally, by matching similar powers of $\theta$ on the left and right sides, we complete the proof.
\end{proof}
\end{proposition}
From the above result, we see that for the steady state queue length $Q_{\infty}(n)$ we can derive the specific probabilities,
\begin{eqnarray*}
p_0 &=& e^{-\frac{\lambda}{\mu}H_n}, \\
p_1 &=& \frac{\lambda}{\mu} p_0 = \frac{\lambda}{\mu} e^{-\frac{\lambda}{\mu}H_n}, \\
p_2 &=& \frac{\lambda}{2\mu} ( p_0 + p_1 ) = \frac{\lambda}{2\mu} e^{-\frac{\lambda}{\mu}H_n} + \frac{\lambda^2}{2\mu^2} e^{-\frac{\lambda}{\mu}H_n} .
\end{eqnarray*}
We can repeat this process as needed for any desired probability. From Proposition~\ref{ssDist}, we can observe that the mean number in system grows linearly with the batch size, meaning that the mean of the $n^\text{th}$ harmonic Hermite distribution is
\begin{align}\label{mean1}
\E{Q_\infty(n)} = \sum_{j=1}^n j \E{Y_j} = \frac{n\lambda}{\mu}.
\end{align}
We can observe further that the second moment and variance are quadratic functions of $n$:
$$
\E{Q_\infty(n)^2} = \E{ \left( \sum_{j=1}^n j Y_j \right)^2} = \frac{n (n+1)\lambda}{2\mu} + n^2 \frac{\lambda^2}{\mu^2},
$$
$$
\mathrm{Var}[Q_\infty(n)] = \E{Q_\infty(n)^2} - \E{Q_\infty(n)}^2 = \frac{n (n+1)\lambda}{2\mu} .
$$
We note that from Proposition~\ref{ssDist} and the following remark, the moment generating function of this distribution is given by
\begin{align}\label{mgf1}
\lim_{t\to \infty}\mathcal{M}(\theta, t)
&
=
e^{ \frac{\lambda}{\mu} \sum_{k=1}^n \frac{e^{k \theta} - 1}{k} }
= e^{ \frac{\lambda}{\mu} ( \mathrm{Li}(e^\theta,n,1) - H_n )}.
\end{align}
If one is to consider this system as the batch size grows infinitely large we can see from Equations~\ref{mean1} and~\ref{mgf1} that the number in system will grow proportionally, tending to infinity as $n$ does. This leads us to ponder the limiting object of the scaled number in system $\frac{Q_t(n)}{n}$ as the batch size grows.
We begin by using Equation~\ref{mgf1} with $\theta$ replaced by $\frac{\theta}{n}$ to see that the steady-state moment generating function of this scaled queue length is
\begin{align}\label{scaleSS}
\lim_{t\to \infty}\mathcal{M}(\theta, t)
=
e^{ \frac{\lambda}{\mu} \sum_{k=1}^n \frac{e^{\frac{k}{n} \theta} - 1}{k} }
.
\end{align}
Furthermore, by replacing $\theta$ with $\frac{\theta}{n}$ and $Q_0(n)$ with $\frac{Q_0(n)}{n}$ in Proposition~\ref{batchMGF}, we can note that the transient moment generating function for this scaled system with constant arrival rate is given by
$$
\E{e^{\theta \cdot \frac{Q_{t}(n)}{n}}} \equiv \mathcal{M}_n(\theta, t)
=
\left(e^{- \mu t}(e^{\frac{\theta}{n}} - 1) + 1\right)^{\frac{Q_0}{n}}
e^{
\lambda \sum_{k=1}^n {n \choose k} \frac{\left(e^{\theta/n} - 1\right)^k}{k\mu} \left(1-e^{-k\mu t}\right)
}.
$$
Additionally, we can also observe that the steady-state distribution of the scaled queue can also be interpreted as a sum of Poisson random variables through direction application of Proposition~\ref{ssDist} or by inspection of Equation~\ref{scaleSS}. This representation is
\begin{align}\label{scalePois}
\frac{Q_\infty(n)}{n}
\stackrel{D}{=}
\sum_{j=1}^n \frac{j}{n} Y_j
,
\end{align}
where again $Y_j \sim \text{Pois}\left(\frac{\lambda}{j \mu}\right)$.
We now consider the limit as $n \to \infty$, in which we are both sending the size of batches of arrivals to infinity while also scaling the size of the queue inversely. We can use this construction to move beyond just the mean and variance and instead explicitly state every cumulant of the scaled queue. In Proposition~\ref{cumulants} we give exact expressions of all steady-state cumulants of the scaled queue as functions of the Bernoulli numbers. Further, we find a convenient form of every cumulant of the scaled queue as the batch size grows to infinity.
\begin{proposition}\label{cumulants}
Let $\lambda > 0$ be the arrival rate of batches of size $n \in \mathbb{Z}^+$ to an infinite server queue with exponential service rate $\mu > 0$. Then, the $k^\text{th}$ steady-state cumulant of the scaled queue $\mathcal{C}^k\left[ \frac{Q_\infty(n)}{n} \right]$ is given by
\begin{eqnarray}
\mathcal{C}^k\left[ \frac{Q_\infty(n)}{n} \right] &=& \frac{\frac{n^{k}}{k}+\frac{1}{2}n^{k-1}+\sum_{j=2}^{k-1} \frac{\mathrm{B}_{j}}{j!}(k-1)_{j-1}n^{k-j}}{n^k}.
\end{eqnarray}
where $(n)_{i} = \frac{n!}{(n-i)!} $ is the $i^\text{th}$ falling factorial of $n$ and $\mathrm{B}_i$ is the $i^\text{th}$ Bernoulli number, which is defined as
$$
B_i
=
\sum_{k=0}^i
\sum_{j=0}^k
(-1)^j
{k \choose j}
\frac{(j+1)^i}{k+1}
.
$$
Moreover, we have that
$
\lim_{n \to \infty} \mathcal{C}^k\left[ \frac{Q_\infty(n)}{n} \right] = \frac{\lambda}{k\mu}.
$
\begin{proof}
From our prior observation that
$
\frac{Q_\infty(n)}{n}
\stackrel{D}{=}
\sum_{j=1}^n
\frac{j}{n} Y_j
$
where $Y_j \sim \mathrm{Pois}\left(\frac{\lambda}{j\mu}\right)$, we have that
\begin{eqnarray*}
\mathcal{C}^k\left[ \frac{Q_\infty(n)}{n} \right]
=
\mathcal{C}^k\left[ \sum_{j=1}^n \frac{j}{n} Y_j \right]
=
\sum_{j=1}^n \mathcal{C}^k\left[ \frac{j}{n} Y_j \right]
=
\sum_{j=1}^n \frac{j^k}{n^k} \mathcal{C}^k\left[ Y_j \right]
=
\frac{\lambda}{\mu n^k} \sum_{j=1}^n j^{k-1},
\end{eqnarray*}
from the independence of these Poisson distributions.
Now, by using Faulhaber's formula as given in \citet{knuth1993johann}, we achieve the stated result.
\end{proof}
\end{proposition}
Just as we built from inherited expressions for the mean and variance to specify every cumulant in Proposition~\ref{cumulants}, we can also find the limit of the transient-state moment generating function for the scaled queue given in Equation~\ref{mgf1}.
\begin{proposition}\label{scaleMGF}
Let $Q_t$ be an infinite server queue with arrival rate $\lambda > 0$, arrival batch size $n \in \mathbb{Z}^+$, and exponential service rate $\mu > 0$. For $\theta \in \mathbb{R}$, let
$$
\mathcal{M}_\infty(\theta, t) = \lim_{n \to \infty} \E{e^{\frac{\theta Q_t(n)}{n}}}.
$$
Then, $\mathcal{M}_\infty(\theta, t)$ is given by
\begin{align}
\mathcal{M}_\infty(\theta, t)
&
=
\begin{cases}
e^{
\frac{\lambda}{\mu}
\left(
\mathrm{Ei}(\theta)
-
\mathrm{Ei}(\theta e^{-\mu t} )
-
\mu t
\right)
}
& \text{ if $\theta > 0$,}
\\
e^{
\frac{\lambda}{\mu}
\left(
E_1(-\theta e^{-\mu t})
-
E_1(-\theta)
-
\mu t
\right)
}
& \text{ if $\theta < 0$,}
\\
1
& \text{ if $\theta = 0$,}
\end{cases}
\end{align}
for all time $t \geq 0$, where the exponential integral functions $\mathrm{Ei}(x)$ and $E_1(x)$ are defined
$$
\mathrm{Ei}(x) = -\int_{-x}^\infty \frac{e^{-s}}{s} \mathrm{d}s,
\quad
E_1(x) = \int_{x}^\infty \frac{e^{-s}}{s} \mathrm{d}s,
$$
and are real-valued for $x > 0$.
\begin{proof}
While conventions may vary by application area, in this work we use the definition of exponential integral function given by
$$
\mathrm{Ei}(x) = -\int_{-x}^\infty \frac{e^{-s}}{s} \mathrm{d}s.
$$
By taking the limit of the MGF of the scaled queue, we have that
\begin{align*}
\frac{\partial}{\partial t} \mathcal{M}_\infty(\theta,t)
&=
\lambda \left(e^{ \theta } - 1\right)
\mathcal{M}_\infty(\theta,t)
-
\mu \theta \frac{\partial}{\partial \theta}
\mathcal{M}_\infty(\theta,t)
\end{align*}
with initial solution $\mathcal{M}_\infty(\theta , 0) = \lim_{n \to \infty} e^{\frac{\theta Q_0}{n}} = 1$. In the same manner as the proof of Theorem~\ref{batchMGF}, we solve the PDE of the cumulant generating function through use of the method of characteristics. We start by establishing the characteristic ODE's:
\begin{align*}
\frac{\mathrm{d}\theta}{\mathrm{d}s}(r,s) &= \mu\theta , &\theta(r,0) = r,\\
\frac{\mathrm{d}t}{\mathrm{d}s}(r,s) &= 1, & t(r,0) = 0,\\
\frac{\mathrm{d}g}{\mathrm{d}s}(r,s) &= \lambda(e^{\theta} - 1), & g(r,0) = 0.
\end{align*}
We now solve the first two initial value problems and find
\begin{align*}
\theta(r,s) &= c_1(r)e^{\mu s} &&\to\quad \theta(r,s) = r e^{\mu s},\\
t(r,s) &= s + c_2(r) &&\to\quad t(r,s) = s.
\end{align*}
This allows us to simplify the third characteristic equation to
$$
\frac{\mathrm{d}g}{\mathrm{d}s}(r,s)
=
\lambda(e^{r e^{\mu s}} - 1).
$$
Because $\theta = r e^{\mu s}$, we can note that $r$ and $\theta$ will match in sign: $r > 0$ if and only if $\theta > 0$. If $\theta > 0$, the general solution to this ODE is
\begin{align*}
g(r,s)
&
=
c_3(r)
+
\frac{\lambda}{\mu}
\left(
\mathrm{Ei}(r e^{\mu s})
-
\mu s
\right)
,
\end{align*}
whereas if $\theta < 0$, the solution is instead
\begin{align*}
g(r,s)
&
=
c_3(r)
-
\frac{\lambda}{\mu}
\left(
E_1(-r e^{\mu s})
+
\mu s
\right)
.
\end{align*}
This follows from the fact that for $x > 0$ the exponential integral functions are such that $\mathrm{Ei}(x) = -E_1(-x) - i \pi$; that is, the real parts of $E_1(-x)$ and $-\mathrm{Ei}(x)$ are the same. Moreover, for $x > 0$ one can consider $\mathrm{Ei}(x)$ as the real part of $-E_1(-x)$. Additionally, $E_1(x)$ is real for all $x > 0$. Hence, we use each definition of the exponential integral function when appropriate. As an alternative, we could replace each of these functions with $\texttt{real}(-E_1(-x))$ to have a single expression for both positive and negative $x$. For a collection of facts regarding the exponential integral functions, see Pages 228-237 of \citet{abramowitz1965handbook}.
Now, using this we have that the corresponding solutions to the initial value problems will be
\begin{align*}
g(r,s)
&
=
\begin{cases}
\frac{\lambda}{\mu}
\left(
\mathrm{Ei}(r e^{\mu s})
-
\mathrm{Ei}(r )
-
\mu s
\right)
&
\text{ if $r > 0$,}
\\
\frac{\lambda}{\mu}
\left(
E_1(-r)
-
E_1(-r e^{\mu s})
-
\mu s
\right)
&
\text{ if $r < 0$}
.
\end{cases}
\end{align*}
Hence, for $s = t$ and $r = \theta e^{-\mu t}$, this yields
\begin{align*}
G(\theta, t)
&=
g\left(\theta e^{-\mu t}, t\right)
=
\begin{cases}
\frac{\lambda}{\mu}
\left(
\mathrm{Ei}(\theta)
-
\mathrm{Ei}(\theta e^{-\mu t} )
-
\mu t
\right)
&
\text{ if $\theta > 0$,}
\\
\frac{\lambda}{\mu}
\left(
E_1(-\theta e^{-\mu t})
-
E_1(-\theta)
-
\mu t
\right)
&
\text{ if $\theta < 0$}
.
\end{cases}
\end{align*}
By $\mathcal{M}_\infty(\theta, t) = e^{G_\infty(\theta, t)}$, we complete the proof.
\end{proof}
\end{proposition}
By consequence, we can also give the moment generating function in steady-state.
\begin{corollary}\label{ssMGF}
The moment generating function of the scaled number in system in steady-state as $n \to \infty$ is given by
\begin{align}
\mathcal{M}_\infty(\theta)
&
=
\begin{cases}
\theta^{-\frac{\lambda}{\mu}}
e^{
\frac{\lambda}{\mu}
\left(
\mathrm{Ei}(\theta)
-
\gamma
\right)
}
& \text{ if $\theta > 0$,}
\\
(-\theta)^{-\frac{\lambda}{\mu}}
e^{
-
\frac{\lambda}{\mu}
\left(
E_1(-\theta)
+
\gamma
\right)
}
& \text{ if $\theta < 0$,}
\\
1
& \text{ if $\theta = 0$,}
\end{cases}
\end{align}
where $\gamma$ is the Euler-Mascheroni constant.
\begin{proof}
From \citet{abramowitz1965handbook}, for $x > 0$ we can expand the exponential integral functions as
\begin{align}
\mathrm{Ei}(x) &= \gamma + \log(x) + \sum_{k=1}^\infty \frac{x^k}{k k!} ,
\quad
E_1(x) = - \gamma - \log(x) - \sum_{k=1}^\infty \frac{(-x)^k}{k k!} , \label{e1expand}
\end{align}
where $\gamma$ is the Euler-Mascheroni constant. By expanding $\mathrm{Ei}(\theta e^{-\mu t})$ and $E_1(-\theta e^{-\mu t})$ in the respective cases of positive and negative $\theta$ and taking the limit as $t \to \infty$, we achieve the stated result.
\end{proof}
\end{corollary}
As a demonstration of the convergence of the steady-state moment generating functions of the batch scaled queues to the expression given in Corollary~\ref{ssMGF}, we plot the first four cases in comparison to the limiting scenario in Figure~\ref{limMGFscale}.
\begin{figure}
\caption{Steady-state MGF of the scaled queue for increasing batch size where $\frac{\lambda}
\label{limMGFscale}
\end{figure}
While it can be argued that even in steady-state the form of this moment generating function is unfamiliar, we can still observe interesting characteristics of it. In particular, for $\theta < 0$ we can uncover a connection back to the harmonic numbers. We now discuss this in the following remark.
\begin{remark}
Using Equation~\ref{e1expand}, we can note that for $\theta < 0$ the steady-state moment generating function of limit of the scaled queue can be expressed
$$
M(\theta)
=
(-\theta)^{-\frac{\lambda}{\mu}}
e^{
-
\frac{\lambda}{\mu}
\left(
E_1(-\theta)
+
\gamma
\right)
}
=
e^{
-\frac{\lambda}{\mu}\left(
E_1(-\theta)
+
\gamma
+
\log(-\theta)
\right)
}
=
e^{
-\frac{\lambda}{\mu}\left(
-
\sum_{k=1}^\infty \frac{\theta^k}{k k!}
\right)
}
.
$$
From \citet{dattoli2008note}, we have that $-e^{x} \sum_{k=1}^\infty \frac{(-x)^k}{k k!}$ is an exponential generating function for the harmonic numbers. That is,
$$
-e^{x} \sum_{k=1}^\infty \frac{(-x)^k}{k k!}
=
\sum_{n=1}^\infty
\frac{x^n}{n!}H_n
$$
where $H_n$ is the $n^\text{th}$ harmonic number. Thus, for $\theta < 0$ the steady-state moment generating function of this limiting object can be further simplified to
$$
M(\theta)
=
e^{
-\frac{\lambda}{\mu}\left(
-
\sum_{k=1}^\infty \frac{\theta^k}{k k!}
\right)
}
=
e^{
-\frac{\lambda}{\mu}
\sum_{n=1}^\infty
H_n
e^{\theta}
\frac{(-\theta)^n}{n!}
}
=
e^{
-\frac{\lambda}{\mu}
\E{
H_N
}
}
,
$$
where $N \sim \mathrm{Pois}(-\theta)$.
\end{remark}
In addition to this remark's connection of the moment generating function and the harmonic numbers, we can also gain insight into this limiting object through Monte Carlo methods. Using Equation~\ref{scalePois}, we have a simple and efficient approximate simulation method for this process through summing scaled Poisson random numbers. Furthermore, this approximation of course becomes increasingly precise as $n$ grows. As an example of this, we give the simulated steady-state densities across different relationships of $\lambda$ and $\mu$ in Figures~\ref{batchLim1}. In addition to the interesting shapes of the densities across the different settings, one can see the limiting form of the relationships given by the recursion in Proposition~\ref{recurProp} in these plots. We can note that one could also calculate these through a numerical inverse Laplace transform of the steady-state moment generating function in Corollary~\ref{ssMGF}, although this may likely incur significantly more computational costs than the simulation procedure.
\begin{figure}
\caption{Approximate steady-state density of the scaled queue limit for size where $\frac{\lambda}
\label{batchLim1}
\end{figure}
So far we have only considered exponentially distributed service. In the next subsection we will address this and extend this Poisson sum representation of the steady-state distribution to hold for general service. We do this through viewing the $n$-batch-size system as being composed of $n$ sub-systems that experience single arrivals simultaneously.
\subsection{Generalizing through Sub-System Perspectives}\label{orderStatSubsec}
Because of the infinite server construction of this model, we can also interpret this system as being a network of sub-systems that also feature infinitely many servers. However, this network's mutuality is not in its services but rather in its arrivals. Specifically, in this subsection we will think of infinite server queues with batch arrivals of size $n$ as being $n$ infinite server queues that all receive individual arrivals simultaneously. From this perspective, one can quickly observe that marginally each subsystem will be distributed as a standard infinite server queue.
For example, if the batch system is the $M_t^n/M/\infty$ that we first considered in Subsection~\ref{transientSub}, then each of these sub-queues are $M_t/M/\infty$ systems. These sub-systems are coupled through the coincidence of their arrival times but otherwise operate independently from one another. To quantify the relationship between these systems, in Proposition~\ref{covgen} we derive the transient covariance between two sub-systems for a general time-varying arrival rate.
\begin{proposition}\label{covgen}
Let the batch arrival queue $Q_t$ with batch size $n \in \mathbb{Z}^+$ be represented as a superposition of $n$ infinite server single arrival queues $\{Q_{t,i} \mid 1 \leq i \leq n \}$ that all receive arrivals simultaneously and each have independent exponentially distributed service, as described above. Let $\lambda(t) > 0$ be the non-stationary rate of simultaneous arrivals and let $\mu > 0$ be the rate of service. Then, for distinct $i , j \in \{1, \dots , n\}$, the covariance between $Q_{t,i}$ and $Q_{t,j}$ is given by
\begin{align}
&
\Cov{Q_{t,i},Q_{t,j}}
=
e^{-2\mu t}
\int_0^t
\lambda(s)
e^{2\mu s}
\mathrm{d}s
\end{align}
for all $t \geq 0$.
\begin{proof}
From Equation~\ref{dynkin}, we can solve for the product moment of the two sub-systems through the ODE
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t}\E{Q_{t,i}Q_{t,j}}
&
=
\lambda(t)\left(\E{Q_{t,i}} + \E{Q_{t,j}} + 1\right) - 2\mu \E{Q_{t,i}Q_{t,j}}.
\end{align*}
The solution to this differential equation is given by
\begin{align*}
\E{Q_{t,i}Q_{t,j}}
&
=
Q_{0,i}Q_{0,j}e^{-2\mu t}
+
e^{-2\mu t}
\int_0^t
\lambda(s)
\left(\E{Q_{s,i}}e^{2\mu s} + \E{Q_{s,j}}e^{2\mu s} + e^{2\mu s}\right)
\mathrm{d}s.
\intertext{By substituting the corresponding forms of $\E{Q_{s,k}} = Q_{0,k}e^{-\mu s} + e^{-\mu s} \int_0^s \lambda(u)e^{\mu u}\mathrm{d}u$ in for each of the two means, we have}
\E{Q_{t,i}Q_{t,j}}
&
=
Q_{0,i}Q_{0,j}e^{-2\mu t}
+
e^{-2\mu t}
\int_0^t
\lambda(s)
\bigg(
e^{2\mu s}
+
\left(Q_{0,i} + \int_0^s \lambda(u)e^{\mu u}\mathrm{d}u\right)e^{\mu s}
\\
&
\quad
+
\left(Q_{0,j} + \int_0^s \lambda(u)e^{\mu u}\mathrm{d}u\right)e^{\mu s}
\bigg)
\mathrm{d}s,
\intertext{and this simplifies to the following}
\E{Q_{t,i}Q_{t,j}}
&
=
Q_{0,i}Q_{0,j}e^{-2\mu t}
+
e^{-2\mu t}
\int_0^t
\lambda(s)
e^{2\mu s}
\mathrm{d}s
+
\left(Q_{0,i} + Q_{0,j}\right)
e^{-2\mu t}
\int_0^t
\lambda(s)
e^{\mu s}
\mathrm{d}s
\\
&
\quad
+
2 e^{-2\mu t}
\int_0^t
\lambda(s)
e^{\mu s} \int_0^s \lambda(u)e^{\mu u}\mathrm{d}u
\mathrm{d}s.
\intertext{We can now use the fact that for a function $F:\mathbb{R}^+ \to \mathbb{R}$ defined such that $F(t) = \int_0^t f(s) \mathrm{d}s$ for a given $f(\cdot)$, integration by parts implies
$$
\int_0^t f(s) F(s) \mathrm{d}s
=
F(t)^2
-
\int_0^t F(s) f(s) \mathrm{d}s
,
$$
and so $\int_0^t f(s) F(s) \mathrm{d}s = \frac{F(t)^2}{2}$. This allows us to simplify to}
\E{Q_{t,i}Q_{t,j}}
&
=
Q_{0,i}Q_{0,j}e^{-2\mu t}
+
e^{-2\mu t}
\int_0^t
\lambda(s)
e^{2\mu s}
\mathrm{d}s
+
\left(Q_{0,i} + Q_{0,j}\right)
e^{-2\mu t}
\int_0^t
\lambda(s)
e^{\mu s}
\mathrm{d}s
\\
&
\quad
+
e^{-2\mu t}
\left(
\int_0^t
\lambda(s)e^{\mu s}
\mathrm{d}s
\right)^2,
\end{align*}
and now we turn our focus to the product of the means. Here we distribute the multiplication to find that
\begin{align*}
\E{Q_{t,i}}\E{Q_{t,j}}
&
=
\left(
Q_{0,i}e^{-\mu t}
+
e^{-\mu t}
\int_0^t
\lambda(s)e^{\mu s}
\mathrm{d}s
\right)
\left(
Q_{0,j}e^{-\mu t}
+
e^{-\mu t}
\int_0^t
\lambda(s)e^{\mu s}
\mathrm{d}s
\right)
\\
&
=
Q_{0,i}Q_{0,j}e^{-2\mu t}
+
(Q_{0,i}+Q_{0,j})e^{-2\mu t}
\int_0^t
\lambda(s)e^{\mu s}
\mathrm{d}s
+
e^{-2\mu t}
\left(
\int_0^t
\lambda(s)e^{\mu s}
\mathrm{d}s
\right)^2
\end{align*}
and by subtracting this expression from that of the product moment, we complete the proof.
\end{proof}
\end{proposition}
As a consequence of this, we can specify the covariance between sub-systems in the non-stationary and stationary arrival settings we have considered thus far in this report. Further, for stationary arrival rates we capitalize on simplified expressions to also give an explicit expression for the correlation coefficient between two sub-systems.
\begin{corollary}\label{covariance}
Let $Q_t$ be an infinite server queue with arrival batch size $n \in \mathbb{Z}^+$ and exponential service rate $\mu > 0$. Further, let $Q_{t,k}$ for $k \in \{1, \dots, n\}$ be infinite server queues with solitary arrivals and exponential service rate $\mu > 0$, so that $\sum_{k=1}^n Q_{t,k} = Q_t$ for all $t \geq 0$. Let $i , j \in \{1, \dots , n\}$ be distinct. Then, if the arrival rate is given by $\lambda + \sum_{k=1}^\infty a_k \cos(k t) + b_k \sin(k t) > 0$, the covariance between $Q_{t,i}$ and $Q_{t,j}$ is
\begin{align}
\Cov{Q_{t,i},Q_{t,j}}
&
=
\frac{\lambda}{2\mu}\left(1 - e^{-2\mu t}\right)
+
\sum_{k=1}^\infty
\frac{a_k}{k^2 + 4\mu^2}
\left(
2\mu \cos(kt) + k \sin(kt) - 2\mu e^{-2\mu t}
\right)
\nonumber
\\
&
\quad
+
\sum_{k=1}^\infty
\frac{b_k}{k^2 + 4\mu^2}\left(2\mu \sin(kt) - k\cos(kt) + k e^{-2\mu t}\right)
,
\end{align}
and if the arrival rate is given by $\lambda > 0$, the covariance between $Q_{t,i}$ and $Q_{t,j}$ is
\begin{align}
&
\Cov{Q_{t,i},Q_{t,j}}
=
\frac{\lambda}{2\mu}\left(1 - e^{-2\mu t}\right)
,
\end{align}
where all $t \geq 0$. Finally, the correlation between two sub-systems in the stationary setting can be calculated as
\begin{align}
&
\mathrm{Corr}[Q_{t,i},Q_{t,j}]
=
\frac{
\frac{\lambda}{2\mu}\left(1 - e^{-2\mu t}\right)
}
{
\sqrt{
\left(
Q_{0,i}\left(e^{-\mu t} - e^{-2\mu t}\right)
+
\frac{\lambda }{\mu}
\left(
1 - e^{- \mu t}
\right)
\right)
\left(
Q_{0,j}\left(e^{-\mu t} - e^{-2\mu t}\right)
+
\frac{\lambda }{\mu}
\left(
1 - e^{- \mu t}
\right)
\right)
}
}
\nonumber
,
\end{align}
hence for stationary arrival rates, $\mathrm{Corr}[Q_{t,i},Q_{t,j}] \to \frac{1}{2}$ as $t \to \infty$.
\end{corollary}
Thus, we find that for a fully Markovian batch arrival queue with stationary arrival rate the correlation among any two sub-systems in steady-state is $\frac{1}{2}$, regardless of the arrival or service parameters. In some sense this seems to capture a balance between the effect of arrivals and of services on an infinite server system, with the latter being independent between these systems and the former being perfectly correlated.
Now, we can pause to note that we have actually made an implicit modeling choice by separating the batch into $n$ identical sub-systems. In this set-up we have decided to route all customers within one batch equivalently, but we are free to make other routing decisions and still maintain the $n$ sub-systems construction. With that in mind, it seems natural to wonder if we can uncover distributional structure of the full system if we choose our routing procedure carefully. We will now find that not only is this true, but we in fact already have already seen a suggestion on what type of routing to consider.
From Proposition~\ref{ssDist}, we have seen that the steady-state distribution of the $M^n/M/\infty$ system is equivalent to that of $\sum_{j=1}^n j Y_j$ where $Y_j \sim \mathrm{Pois}(\frac{\lambda}{j \mu })$ are independent. We can also note that just as the minimum of the independent sample $S_1, \dots, S_n \sim \mathrm{Exp}(\mu)$ will be exponentially distributed with rate $n\mu$, for $S_{(i)}$ as the $i^\text{th}$ ordered statistic of the $n$-sample we have that $S_{(i)} - S_{(i-1)} \sim \mathrm{Exp}((n-i+1)\mu)$. Of course, the sum of these differences will telescope so that $\sum_{j=1}^i S_{(j)} - S_{(j-1)} = S_{(i)}$.
Taking this as inspiration, we will now assume that upon the arrival of a batch we can now know the duration of each customer's service. We then take the sub-queues to be such that the first sub-system always receives the service with the shortest duration, the second sub-system receives the second shortest service, and so on. Thus, we will route each batch of customers according to the order statistics within each batch. For reference, we visualize this sub-system construction in Figure~\ref{orderDiagram}.
\begin{figure}
\caption{Queueing diagram for the batch arrival queue with infinite servers, in which the arriving entities are routed according to the ordering of their service durations.}
\label{orderDiagram}
\end{figure}
We can note that while the covariance structure we explored in Proposition~\ref{covgen} and Corollary~\ref{covariance} do not apply for this new routing, the sub-systems are certainly still correlated. Due to the order-statistics structuring of the service in each queue, we can note that now both the arrival processes and the service distributions will be dependent. However, we can in fact use our understanding of this dependence to not only understand how these systems relate to one another, but also to interpret how they form the structure of the full batch system as a whole. In this way, we will now consider a $M^n/G/\infty$ system. As follows in Theorem~\ref{orderStat}, we will find that the order-statistics-routing inspiration we have used from Proposition~\ref{ssDist} leads us to a generalized Poisson sum result for general service distributions.
\begin{theorem}\label{orderStat}
Let $Q_t(n)$ be an $M^n/G/\infty$ queue. That is, let $Q_t(n)$ be an infinite server queue with stationary arrival rate $\lambda > 0$, arrival batch size $n \in \mathbb{Z}^+$, and general service distribution $G$. Then, the steady-state distribution of the number in system $Q_\infty(n)$ is
\begin{align}
Q_\infty(n)
\stackrel{D}{=}
\sum_{j=1}^n
(n-j+1) Y_j
\end{align}
where $Y_j \sim \mathrm{Pois}\left(\lambda \E{S_{(j)} - S_{(j-1)}}\right)$ are independent, with $S_{(1)} \leq \dots \leq S_{(n)}$ as order statistics of the distribution $G$ and with $S_{(0)} = 0$.
\begin{proof}
As we have discussed in the paragraphs preceding this statement, we will consider the full queueing system as being composed of $n$ infinite server sub-systems to which we route the arriving customers in each batch. That is, let $Q_1$, \dots, $Q_n$ be infinite server queues of which we will consider the steady-state behavior. Upon the arrival of a batch, we order the customers according to the duration of their service. Then, we send the customer with the earliest service completion to $Q_1$, the customer with the second earliest to $Q_2$, and so on.
When viewing each sub-system on its own, we see that $Q_j$ is an infinite server queue with single arrivals according to a Poisson process with rate $\lambda$ and service distribution matching that of $S_{(j)}$, the $j^\text{th}$ order statistics of $G$. Thus, we can see that in steady-state $Q_j \sim \mathrm{Pois}\left(\lambda \E{S_{(j)}}\right)$ through the literature for $M/G/\infty$ queues, such as in \cite{eick1993physics}. While we can further observe that $Q_\infty(n) = \sum_{j=1}^n Q_j$, we must take care in re-assembling the sub-queues. In particular, we can note that $S_{(j)}$ shares a similar structure with $S_{(j-1)}$. Each order statistic can be viewed as a construction of the gaps between the lower ordered quantities:
$$
S_{(j)} = \sum_{k=1}^j S_{(k)} - S_{(k-1)} .
$$
Thus, from the thinning property of the Poisson distribution and the linearity of expectation, we can write the distribution of $Q_j$ as a sum of independent Poisson RV's, as given by
$$
Q_j
\sim
\sum_{k=1}^j
\mathrm{Pois}\left(\lambda \E{S_{(k)} - S_{(k-1)}}\right)
.
$$
We can note further that $j-1$ of the Poisson components of $Q_j$ are the exact components of $Q_{j-1}$, with $j-2$ of these components also shared with $Q_{j-2}$, $j-3$ with $Q_{j-3}$, and so on. Then, we see that the Poisson component $\mathrm{Pois}\left(\lambda \E{S_{(j)} - S_{(j-1)}}\right)$ is repeated $n-j+1$ times across this sub-system construction of $Q_\infty(n)$, as it appears in each of the Poisson sum expressions of $Q_{j}$, $Q_{j+1}$, \dots, $Q_{n-1}$, and $Q_n$. Assembling $Q_\infty(n)$ in this way, we complete the proof.
\end{proof}
\end{theorem}
One can also note that this order statistic sub-system structure also provides some motivation for the occurrence of the harmonic numbers that we observed in Subsection~\ref{markovSS} when viewing the largest order statistic, which we discuss now in the following remark.
\begin{remark}
For $S_i \sim \mathrm{Exp}(\mu)$, one can see through the telescoping construction of the order statistics that
$$
\E{S_{(n)}}
=
\sum_{i=1}^n \E{S_{(i)} - S_{(i-1)}}
=
\sum_{i=1}^n \frac{1}{(n-i+1) \mu}
=
\frac{1}{\mu} H_n
.
$$
\end{remark}
Now, throughout this section we have operated on the assumption that the batch size is a known, fixed constant. While this may be applicable in some settings there are certainly many settings where the batch size is unknown and varies between arrivals. Thus, we address this in Section~\ref{randomSec} and find that many of the results we have shown thus far can be replicated for models with random batch size.
\section{Random Batch Sizes}\label{randomSec}
We will now consider systems in which the size of an arriving batch is drawn from an independent and identically distributed sequence of random variables. We will treat the distribution of the batch size as general throughout this work. As in Section~\ref{deterSec}, we assume that the times of arrivals are given by a Poisson process, with consideration given to both stationary and non-stationary rates, and we will again analyze both exponential and general service distributions.
We start by giving the mean and variance of the system for time-varying arrival rates with exponential service in Subsection~\ref{meanvarRandSubsec}. Then, in Subsection~\ref{limitRandSubsec} we give three limiting results for the stationary arrivals model: a batch scaling, a fluid limit, and a diffusion limit. Finally in Subsection~\ref{orderStatRandSubsec} we extend the Poisson sum construction of the steady-state distribution to hold for random batch sizes.
One can note that many of these results are generalizations or extensions of findings from Section~\ref{deterSec}, thus implying them as a special case and perhaps even building a case for them to be omitted. Rather, these findings are critical to the narrative of this report. As we will see, the results for fixed batch size provide the analytic foundations and conceptual inspirations from which we derive much of the analysis in this section.
\subsection{Mean and Variance for Time-Varying, Markovian Case}\label{meanvarRandSubsec}
To begin our exploration into random batch size systems, we'll start simple: we'll look at a fully Markovian (albeit time-varying) system and find the mean and variance, using conditional probability and our results from Section~\ref{deterSec}. Specifically, in this subsection we will consider the $M_t^N/M/\infty$ queue. That is, take an infinite server queue with a general non-stationary arrival rate. We suppose that arrivals occur in batches of random size from a sequence of independent and identically distributed random variables. Furthermore, we suppose that service is exponentially distributed. We now give the mean and variance of this system in Proposition~\ref{meanvarRand}.
\begin{proposition}\label{meanvarRand}
Let $Q_t$ be an infinite server queue with finite, time-varying arrival rate $\lambda(t) > 0$, exponential service rate $\mu > 0$, and random batch size with finite mean, $\E{N}$. Then, the mean number in system is given by
\begin{align}
\E{Q_t}
&
=
Q_0 e^{-\mu t}
+
e^{-\mu t}\E{N}
\int_0^t \lambda(s)e^{\mu s}\mathrm{d}s
,
\end{align}
for all $t \geq 0$. Then, if the batch size distribution has finite second moment $\E{N^2}$, the variance of the number in system is given by
\begin{align}
\Var{Q_t}
&
=
Q_0\left(e^{-\mu t} - e^{-2\mu t}\right)
+
e^{-2\mu t}\left( \E{N^2} - \E{N} \right)
\int_0^t \lambda(s) e^{2\mu s} \mathrm{d}s
\nonumber
\\
&
\quad
+
e^{-\mu t}\E{N}\int_0^t \lambda(s) e^{\mu s} \mathrm{d}s
,
\end{align}
again for all $t \geq 0$.
\begin{proof}
Using the infinitesimal generator method, we have that the first and second moments of this system are given by the solutions to
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t}\E{Q_t}
&
=
\lambda(t) \E{N_1} - \mu \E{Q_t}
,
\\
\frac{\mathrm{d}}{\mathrm{d}t}\E{Q_t^2}
&
=
\lambda(t) \left(2\E{Q_t}\E{ N_1}+\E{N_1^2}\right) - 2\mu \E{Q_t^2} + \mu \E{Q_t}
,
\end{align*}
where $\{N_i \mid i \in \mathbb{Z}^+\}$ are the i.i.d. batch sizes that are also independent of the queue. Through noting that
$$
\frac{\mathrm{d}}{\mathrm{d}t}\Var{Q_t}
=
\frac{\mathrm{d}}{\mathrm{d}t}\E{Q_t^2}
-
2\E{Q_t} \frac{\mathrm{d}}{\mathrm{d}t}\E{Q_t}
=
\lambda(t)\E{N_1^2} + \mu \E{Q_t} - 2\mu \Var{Q_t}
,
$$
we can solve for the stated results.
\end{proof}
\end{proposition}
In addition to providing a direct comparison to the fixed batch size case in conjunction with Corollary~\ref{meanvarcor}, Proposition~\ref{meanvarRand} also provides a building block for the remainder of this section. In particular, in the following subsection we will develop a series of limiting results for this queueing system, including fluid and diffusion limits. In those cases, we will use this result for added interpretation. To expedite comparison in cases of stationary arrival rates, we now give the mean and variance for such systems in Corollary~\ref{meanvarRandStat}. Additionally, to also facilitate comparison to Corollary~\ref{meanvarcor}, we provide expressions for periodic arrival rates in Corollary~\ref{meanvarRandPer}.
\begin{corollary}\label{meanvarRandStat}
Let $Q_t$ be an infinite server queue with stationary arrival rate $\lambda > 0$, exponential service rate $\mu > 0$, and random batch size with mean $\E{N}$. Then, the mean number in system is given by
\begin{align}
\E{Q_t}
&
=
Q_0 e^{-\mu t}
+
\frac{\lambda \E{N}}{\mu}
\left(
1 - e^{- \mu t}
\right)
,
\end{align}
for all $t \geq 0$. Then, if the batch size distribution has finite second moment $\E{N^2}$, the variance of the number in system is given by
\begin{align}
\Var{Q_t}
&
=
Q_0\left(e^{-\mu t} - e^{-2\mu t}\right)
+
\frac{\lambda \E{N}}{\mu}
\left(
1 - e^{- \mu t}
\right)
+
\frac{ \lambda }{2\mu}
\left(
\E{N^2} - \E{N}
\right)
\left(
1 - e^{-2 \mu t}
\right)
,
\end{align}
again for all $t \geq 0$.
\end{corollary}
\begin{corollary}\label{meanvarRandPer}
Let $Q_t$ be an infinite server queue with periodic arrival rate $\lambda + \sum_{k=1}^\infty a_k \cos(k t) + b_k \sin(k t) > 0$, exponential service rate $\mu > 0$, and random batch size with finite mean, $\E{N}$. Then, the mean number in system is given by
\begin{align}
\E{Q_t}
&
=
Q_0 e^{-\mu t}
+
\frac{\lambda \E{N}}{\mu}
\left(
1 - e^{- \mu t}
\right)
+
\sum_{k=1}^\infty
\frac{\E{N}(a_k \mu - b_k k) }{k^2+\mu^2}
\left(
\cos(kt) - e^{-\mu t}
\right)
\nonumber
\\
&
\quad
+
\sum_{k=1}^\infty
\frac{\E{N}(a_k k + b_k \mu )}{k^2+\mu^2} \sin(k t)
,
\end{align}
for all $t \geq 0$. Then, if the batch size distribution has finite second moment $\E{N^2}$, the variance of the number in system is given by
\begin{align}
\Var{Q_t}
&
=
Q_0\left(e^{-\mu t} - e^{-2\mu t}\right)
+
\frac{\lambda \E{N}}{\mu}
\left(
1 - e^{- \mu t}
\right)
+
\sum_{k=1}^\infty
\frac{E{N}(a_k \mu - b_k k) }{k^2+\mu^2}
\left(
\cos(kt) - e^{-\mu t}
\right)
\nonumber
\\
&
\quad
+
\sum_{k=1}^\infty
\frac{\E{N}(a_k k + b_k \mu )}{k^2+\mu^2} \sin(k t)
+
\frac{ \lambda }{2\mu}
\left(
\E{N^2} - \E{N}
\right)
\left(
1 - e^{-2 \mu t}
\right)
\nonumber
\\
&
\quad
+
\left(
\E{N^2} - \E{N}
\right)
\left(
\sum_{k=1}^\infty
\frac{2 a_k \mu - b_k k }{k^2 + 4\mu^2}
\left(
\cos(kt) - e^{-2\mu t}
\right)
+
\sum_{k=1}^\infty
\frac{a_k k + 2 b_k \mu }{k^2+ 4\mu^2} \sin(k t)
\right)
,
\end{align}
again for all $t \geq 0$.
\end{corollary}
\subsection{Limiting Results for Stationary Arrival Rates}\label{limitRandSubsec}
We will now focus on systems with stationary arrival rates throughout the analysis in this subsection. In doing so, we derive limit theorems for various scalings of this process. To begin, we show a brief technical lemma for the limit of non-negative random variables that can be represented as sums of independent and identically distributed random variables.
\begin{lemma}\label{divisLemma}
Let $X(n)$ be any random variable that
$
X(n) = \sum_{k=1}^n Y_k
$
where $Y_k$ are i.i.d. non-negative, discrete random variables. Then, the moment generating function of $X(n)$ is such that
\begin{align*}
\E{e^{\frac{\theta X(n)}{n}}}
\to
e^{\E{Y_1} \theta}
\end{align*}
as $n \to \infty$.
\begin{proof}
By the strong law of large numbers, we have that
$$
\lim_{n\to\infty}\frac{X(n)}n = \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n Y_k \stackrel{\text{a.s.}}{=} \E{Y_1},
$$
and this implies convergence in distribution, which is equivalent to convergence of moment generating functions.
\end{proof}
\end{lemma}
We can note that this condition is a weaker form of infinite divisibility. Thus, in addition to holding for any infinitely divisible and non-negative random variables such as the Poisson, and negative binomial distributions, Lemma~\ref{divisLemma} also holds for some distributions that are not infinitely divisible, such as the binomial. Using this lemma we can now find our first limit theorem for random batch sizes, a batch scaling result akin to Proposition~\ref{scaleMGF}.
\begin{theorem}\label{scaleMGFrand}
For $n \in \mathbb{Z}^+$, let $Q_t(n)$ be an infinite server queue with batch arrivals where the batch size is drawn from the i.i.d. sequence $\{N_i(n) \mid i \in \mathbb{Z}^+\}$. Let $\lambda > 0$ be the arrival rate and let $\mu > 0$ be the rate of exponentially distributed service. Then, suppose that for any $i$ and $n$ there is a sequence of i.i.d. non-negative, discrete random variables $\{B_k \mid k \in \mathbb{Z}^+\}$ such that
$
N_i(n)
=
\sum_{k=1}^n
B_k
.
$
Then, the limiting moment generating function of the batch scaled object
\begin{align}
\lim_{n\to\infty}
\E{e^{\frac{\theta}{n}Q_t(n)}}
=
\begin{cases}
e^{
\frac{\lambda}{\mu}
\left(
\mathrm{Ei}\left(\theta\E{B_1}\right)
-
\mathrm{Ei}\left(\theta\E{B_1} e^{-\mu t} \right)
-
\mu t
\right)
}
&
\text{ if $\theta > 0$,}
\\
e^{
\frac{\lambda}{\mu}
\left(
E_1\left(-\theta\E{B_1} e^{-\mu t}\right)
-
E_1\left(-\theta\E{B_1} \right)
-
\mu t
\right)
}
&
\text{ if $\theta < 0$,}
\\
1
&
\text{ if $\theta = 0$,}
\end{cases}
\end{align}
for all $t \geq 0$.
\begin{proof}
Because this system is Markovian, we can calculate the time derivative of the moment generating function for a given $n$ as
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t}\E{e^{\frac{\theta}{n} Q_t(n)}}
&
=
\E{
\lambda \left(e^{\frac{\theta}{n}N_1(n)}-1\right) e^{\frac{\theta}{n} Q_t(n)}
+
\mu Q_t(n) \left(e^{-\frac{\theta}{n}}-1\right) e^{\frac{\theta}{n} Q_t(n)}
}
\\
&
=
\lambda \left(\E{e^{\frac{\theta}{n}N_1(n)}}-1\right) \E{e^{\frac{\theta}{n} Q_t(n)}}
+
n \mu \left(e^{-\frac{\theta}{n}}-1\right) \E{\frac{Q_t(n)}{n}e^{\frac{\theta}{n} Q_t(n)}}
.
\intertext{This can then be re-expressed in partial differential equation form as}
\frac{\partial \mathcal{M}^n(\theta, t)}{\partial t}
&
=
\lambda \left(\E{e^{\frac{\theta}{n}N_1(n)}}-1\right) \mathcal{M}^n(\theta, t)
+
n \mu \left(e^{-\frac{\theta}{n}}-1\right) \frac{\partial \mathcal{M}^n(\theta, t)}{\partial \theta},
\end{align*}
where $\mathcal{M}^n(\theta, t) = \E{e^{\frac{\theta}n Q_t(n)}}$. Now, through Lemma~\ref{divisLemma}, we see that the limit of this partial differential equation is given by
\begin{align*}
\frac{\partial \mathcal{M}^\infty(\theta, t)}{\partial t}
&
=
\lambda \left(e^{\theta \E{B_1}}-1\right) \mathcal{M}^\infty(\theta, t)
-
\mu \theta \frac{\partial \mathcal{M}^\infty(\theta, t)}{\partial \theta}
.
\end{align*}
We achieve the stated result through a straightforward update of the method of characteristics approach in Proposition~\ref{scaleMGF}.
\end{proof}
\end{theorem}
We can note that a similar batch scaling of infinite server queues is discussed in \citet{de2017shot}, in which the authors show that the limiting process can be interpreted as a shot noise process. However, that work considers a different class of batch size distributions, as the authors define their batch size distribution in terms of the distribution of the marks through use of a ceiling rounding function. In this way, that paper is more oriented around the distribution of the marks in the shot noise process rather than the size of the batches.
From this result, we can identify a relationship between the moment generating functions of the deterministic and random batch size queues under batch scalings. Let $\mathcal{M}_n^\infty(\theta, t)$ be the limiting moment generating function of the fixed batch size queue as given in Proposition~\ref{scaleMGF} and let $\mathcal{M}_N^\infty(\theta,t)$ be the same for the random batch size queue as we have now seen in Theorem~\ref{scaleMGFrand}. Then, we can observe that
$$
\mathcal{M}_N^\infty(\theta,t)
=
\mathcal{M}_n^\infty(\theta \E{B_1},t)
,
$$
whenever the distribution of the random batch sizes meets the ``finite divisibility'' condition as described in Lemma~\ref{divisLemma}. The relationship between these limiting objects provides a direct comparison between the two different batch types.
As two additional limiting results, we now provide fluid and diffusion limits for scaling the arrival rate in Theorems~\ref{batchFluid} and~\ref{batchDiff}, respectively. We did not give fluid or diffusion limits for the deterministic batch cases in Section~\ref{deterSec}, so these two limits are built from scratch within this section. Although we did not develop such limits explicitly for the $M^n/M/\infty$ system, we will find that these limits can still be used to draw comparisons between this system and the $M^N/M/\infty$ queue simply by treating the random batch size as deterministically distributed. We now begin with the fluid limit.
\begin{theorem}\label{batchFluid}
For $n \in \mathbb{Z}^+$, let $Q_t(n)$ be an infinite server queue with batch arrivals where the batch size is drawn from the i.i.d. sequence $\{N_i \mid i \in \mathbb{Z}^+\}$. Let $n \lambda > 0$ be the arrival rate and let $\mu > 0$ be the rate of exponentially distributed service. Then, the limiting moment generating function of the fluid scaling is given by
\begin{align}
\lim_{n \to \infty} \E{e^{\frac{\theta}{n}Q_t(n)}}
&
=
e^{ \frac{\lambda \E{N_1} \theta}{\mu} \left(1 - e^{-\mu t}\right) + Q_0 \theta e^{-\mu t}}
,
\end{align}
for all $t \geq 0$.
\begin{proof}
We begin with the infinitesimal generator equation for the time derivative of the moment generating function at a given $n$. This is
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t}\E{e^{\frac{\theta}{n}Q_t(n)}}
&
=
\E{
n \lambda \left(e^{\frac{\theta N_1}{n}} - 1\right) e^{\frac{\theta}{n} Q_t(n)}
+
\mu Q_t(n) \left(e^{-\frac{\theta}{n}} - 1\right) e^{\frac{\theta}{n} Q_t(n) }
}
\\
&
=
n \lambda \left(\E{e^{\frac{\theta N_1}{n}}} - 1\right) \E{e^{\frac{\theta}{n} Q_t(n)}}
+
\mu n \left(e^{-\frac{\theta}{n}} - 1\right) \E{\frac{Q_t(n)}{n} e^{\frac{\theta}{n} Q_t(n) }}
,
\intertext{which can also be expressed in partial differential equation form as}
\frac{\partial\mathcal{M}^n(\theta,t)}{\partial t}
&
=
n \lambda \left(\E{e^{\frac{\theta N_1}{n}}} - 1\right) \mathcal{M}^n(\theta, t)
+
\mu n \left(e^{-\frac{\theta}{n}} - 1\right) \frac{\partial\mathcal{M}^n(\theta,t)}{\partial \theta}
,
\end{align*}
where $M^n(\theta,t) = \E{e^{\frac{\theta}n Q_t(n)}}$. By a Taylor expansion of the function $e^{\frac{\theta N_1}n}$ and by taking the limit as $n \to \infty$, we can see that this yields
\begin{align*}
\frac{\partial\mathcal{M}^\infty(\theta,t)}{\partial t}
&
=
\lambda \theta \E{N_1} \mathcal{M}^\infty(\theta, t)
-
\mu \theta \frac{\partial\mathcal{M}^\infty(\theta,t)}{\partial \theta}
.
\end{align*}
Using the initial condition $\mathcal{M}^\infty(\theta,0) = e^{Q_0 \theta}$, we can see that the solution to this partial differential equation will be
$$
\mathcal{M}^\infty(\theta,t)
=
e^{ \frac{\lambda \E{N_1} \theta}{\mu} \left(1 - e^{-\mu t}\right) + Q_0 \theta e^{-\mu t}}
,
$$
and this completes the proof.
\end{proof}
\end{theorem}
From Corollary~\ref{meanvarRandStat}, we see that the mean number in system for the $M^N/M/\infty$ queue is $\frac{\lambda \E{N_1} }{\mu} \left(1 - e^{-\mu t}\right) + Q_0 e^{-\mu t}$. Thus, this fluid limit moment generating function is equivalent to $e^{\theta \E{Q_t}}$ for all $t \geq 0$ and all $\theta$, showing that the fluid limit converges to the mean. We now find a connection to both the mean and the variance through a diffusion limit in Theorem~\ref{batchDiff}.
\begin{theorem}\label{batchDiff}
For $n \in \mathbb{Z}^+$, let $Q_t(n)$ be an infinite server queue with batch arrivals where the batch size is drawn from the i.i.d. sequence $\{N_i \mid i \in \mathbb{Z}^+\}$. Let $n \lambda > 0$ be the arrival rate and let $\mu > 0$ be the rate of exponentially distributed service. Then, the limiting moment generating function of the diffusion scaling is given by
\begin{align}
\lim_{n \to \infty} \E{e^{\frac{\theta}{\sqrt{n}}\left(Q_t(n) - \frac{n\lambda \E{N_1}}{\mu}\right)}}
&
=
e^{\frac{\lambda \theta^2}{4\mu} \left(\E{N_1}+\E{N_1^2}\right) \left(1-e^{-\mu t}\right) + \theta Q_0 e^{-\mu t}}
\end{align}
which gives a steady-state approximation of $X \sim \mathrm{Norm}\left(\frac{\lambda \E{N_1}}{\mu}, \frac{\lambda }{2\mu} \left(\E{N_1}+\E{N_1^2}\right) \right)$.
\begin{proof}
Through use of the infinitesimal generator, we have that the time derivative of the moment generating function for a given $n$ can be expressed
\begin{align*}
&
\frac{\mathrm{d}}{\mathrm{d}t}\E{e^{\frac{\theta}{\sqrt{n}}\left(Q_t(n) - \frac{n\lambda \E{N_1}}{\mu}\right)}}
\\
&
=
\E{
n \lambda \left(e^{\frac{\theta N_1}{\sqrt{n}}} - 1\right) e^{\frac{\theta}{\sqrt{n}}\left(Q_t(n) - \frac{n\lambda \E{N_1}}{\mu}\right)}
+
\mu Q_t(n) \left(e^{-\frac{\theta}{\sqrt{n}}} - 1\right) e^{\frac{\theta}{\sqrt{n}}\left(Q_t(n) - \frac{n\lambda \E{N_1}}{\mu}\right)}
}
\\
&
=
\E{\sqrt{n} \lambda \left( \theta N_1 + \frac{\theta^2 N_1^2}{2\sqrt{n}} + \mathrm{O}\left( \frac{\theta^3 N_1^3}{6 n}\right) \right) e^{\frac{\theta}{\sqrt{n}}\left(Q_t(n) - \frac{n\lambda \E{N_1}}{\mu}\right)}}
\\
&
\quad
+
\E{\mu \sqrt{n} \left(\frac{Q_t(n)}{\sqrt{n}} - \frac{n \lambda \E{N_1}}{\sqrt{n} \mu} + \frac{n \lambda \E{N_1}}{\sqrt{n} \mu} \right) \left(e^{-\frac{\theta}{\sqrt{n}}} - 1\right) e^{\frac{\theta}{\sqrt{n}}\left(Q_t(n) - \frac{n\lambda \E{N_1}}{\mu}\right)}}
,
\end{align*}
where here we have used a Taylor expansion of the function $e^{\frac{\theta N_1}{\sqrt{n}}}$. Now, for $\mathcal{M}^n(\theta, t) = \E{e^{\frac{\theta}{\sqrt{n}}\left(Q_t(n) - \frac{n\lambda \E{N_1}}{\mu}\right)}}$, this equation can be written as a partial differential equation as follows:
\begin{align*}
\frac{\partial \mathcal{M}^n(\theta, t)}{\partial t}
&
=
\lambda \theta \sqrt{n} \E{N_1} \mathcal{M}^n(\theta, t)
+
\frac{\lambda \theta^2}{2} \E{N_1^2}\mathcal{M}^n(\theta, t)
+
\sqrt{n}\lambda \E{\mathrm{O}\left(\frac{\theta^3 N_1^3}{6 n}\right) e^{\frac{\theta}{\sqrt{n}}\left(Q_t(n) - \frac{n\lambda \E{N_1}}{\mu}\right)}}
\\
&
\quad
+
\sqrt{n}\mu \left(e^{-\frac{\theta}{\sqrt{n}}}-1\right) \frac{\partial \mathcal{M}^n(\theta, t)}{\partial \theta}
+
n \lambda \E{N_1} \left(e^{-\frac{\theta}{\sqrt{n}}}-1\right) \mathcal{M}^n(\theta, t)
.
\end{align*}
As we take $n \to \infty$ this PDE becomes
\begin{align*}
\frac{\partial \mathcal{M}^\infty(\theta, t)}{\partial t}
&
=
\frac{\lambda \theta^2}2 \E{N_1} \mathcal{M}^\infty(\theta, t)
+
\frac{\lambda \theta^2}{2} \E{N_1^2}\mathcal{M}^\infty(\theta, t)
-
\mu \theta \frac{\partial \mathcal{M}^\infty(\theta, t)}{\partial \theta}
,
\end{align*}
and this yields a solution of
$$
\mathcal{M}^\infty(\theta, t)
=
e^{\frac{\lambda \theta^2}{4\mu} \left(\E{N_1}+\E{N_1^2}\right) \left(1-e^{-\mu t}\right) + \theta Q_0 e^{-\mu t}}
.
$$
To observe the steady-state distribution, we take the limit as $t \to \infty$ and observe that this produces the moment generating function for a Gaussian.
\end{proof}
\end{theorem}
By comparison to the limits of the expresions in Corollary~\ref{meanvarRandStat} as $t \to \infty$, we can now observe that this steady-state approximation is equal in mean and variance to the steady-state queue.
\subsection{Extending the Order Statistics Sub-Systems}\label{orderStatRandSubsec}
In Subsection~\ref{orderStatSubsec} we found that the steady-state distribution of infinite server queues with fixed batch size and general service can be written as a sum of scaled Poisson random variables, providing a succinct interpretation of the process and an efficient simulation procedure for approximate calculations. The underlying observation that supported this approach was that we can think of an infinite server queue with batch arrivals as a collection of infinite server queues with solitary arrivals that occur simultaneously. Using the thinning property of Poisson processes, we now extend this result to queues with random batch sizes and general service.
\begin{theorem}\label{orderStatRand}
Let $Q_t$ be a $M^N/G/\infty$ queue. That is, let $Q_t$ an infinite server queue with stationary arrival rate $\lambda > 0$, arrival batch of random size according to the i.i.d.~sequence of non-negative integer valued random variables $\{N_i \mid i \in \mathbb{Z}^+\}$, and general service distribution $G$. Then, the steady-state distribution of the number in system $Q_\infty$ is
\begin{align}
Q_\infty
\stackrel{D}{=}
\sum_{n=1}^\infty \sum_{j=1}^n
(n-j+1) Y_{j,n}
\end{align}
where $Y_{j,n} \sim \mathrm{Pois}\left(\lambda p_n \E{S_{(j,n)} - S_{(j-1,n)}}\right)$ are independent, with $S_{(1,n)} \leq \dots \leq S_{(n,n)}$ as order statistics of the distribution $G$ when $N_i = n$, where $S_{(0,n)} = 0$ for all $n$ and $p_n = \PP{N_1 = n}$.
\begin{proof}
To begin, we suppose that there is some $m \in \mathbb{Z}^+$ such that $\PP{N_i \in \{0,\dots, m\}} = 1$. Then, using the thinning property of Poisson processes, we separate the arrival process into $m$ arrival streams where the $n^\text{th}$ arrival rate is $\lambda p_n$. Then, by Theorem~\ref{orderStat} the steady-state distribution of the number in system from the $n^\text{th}$ stream is
$$
\sum_{j=1}^n
(n-j+1) \mathrm{Pois}\left(\lambda p_n \E{S_{(j,n)} - S_{(j-1,n)}}\right)
.
$$
Then, since the $m$ thinned Poisson streams are independent, we have that the full combined system will be distributed as
$$
\sum_{n=1}^m
\sum_{j=1}^n
(n-j+1) \mathrm{Pois}\left(\lambda p_n \E{S_{(j,n)} - S_{(j-1,n)}}\right)
.
$$
Through taking the limit as $m \to \infty$, we achieve the stated result.
\end{proof}
\end{theorem}
We can note that Theorem~\ref{orderStatRand} also provides a method for approximate empirical calculation through simulation. This representation can also be simplified if more information is known about the distribution of the batch size or of the service, or both. As an example, we give the distribution for the fully Markovian system in the following corollary.
\begin{corollary}
Let $Q_t$ be a $M^N/M/\infty$ queue. That is, let $Q_t$ an infinite server queue with stationary arrival rate $\lambda > 0$, arrival batch of random size according to the i.i.d.~sequence of non-negative integer valued random variables $\{N_i \mid i \in \mathbb{Z}^+\}$, and exponentially distributed service at rate $\mu > 0$. Then, the steady-state distribution of the number in system $Q_\infty$ is
\begin{align}
Q_\infty
\stackrel{D}{=}
\sum_{j=1}^\infty
j Y_{j}
\end{align}
where $Y_{j} \sim \mathrm{Pois}\left(\frac{\lambda}{j \mu} \bar F_N(j) \right)$ are independent, where $\bar F_N(j) = \PP{N_1 \geq j}$.
\end{corollary}
One can note that the moment generating function for this system in steady-state is
$$
\E{e^{\theta Q_\infty}}
=
e^{\sum_{j=1}^\infty \frac{\lambda}{j\mu} \bar{F}_N(j)\left(e^{j \theta} - 1\right)}
,
$$
and that this also admits a connection to the generalized Hermite distributions we discussed in Subsection~\ref{markovSS}. In particular, this generalized Hermite distribution can be characterized by $\frac{\lambda}{\mu}$, which is again the mean of the distribution, and the complementary cumulative distribution function of the batch size distribution, which dictates the coefficients at each $j$. For this reason, it may be possible that the steady-state distribution of the queue may be simplified even further for particular batch size distributions.
Because Theorem~\ref{orderStatRand} is again built upon an order statistics sub-queue perspective, it is natural to wonder how the distribution of the batch size would affect those sub-systems. In particular, we now consider the following scenario: suppose that the batch size is bounded by some constant, say $k$, and that we have $k$ sub-systems. For each arriving batch, the customer with the shortest service duration will go to the first sub-system, the second shortest to the second sub-system, and so on, but only up to the number that have just arrived: if this batch is of size $k-1$, the $k^\text{th}$ sub-queue will not receive an arrival. In this way, the $i^\text{th}$ sub-queue represents the number in system that were the $i^\text{th}$ smallest in their batch. In the following proposition we find the conditions on the batch size distribution under which the distributions of the sub-queues will be equivalent.
\begin{proposition}
Consider a $M^B/G/\infty$ queueing system in which the distribution of $B$ has support on $\{1,\dots, k\}$. Let $\phi \in [0,1]^{k-1}$ be such that $\phi_i = \PP{B = i}$, yielding $\PP{B = k} = 1 - \sum_{i=1}^{k-1} \phi_i$. Let $S_{(i,j)}$ be the $i^\text{th}$ order statistics in a sample of size $j$ from the service distribution. Furthermore, let $Q_i$ be steady-state number in system of an infinite server sub-queue to which the customer with the $i^\text{th}$ smallest service duration in an arriving batch will be routed whenever there are at least $i$ customers in the batch. Let $M \in \mathbb{R}^{k -1 \times k-1}$ be an upper triangular matrix such that
$$
M_{i,j} = \frac{\E{S_{(i,j)}}}{\E{S_{(k,k)}} - \E{S_{(i,k)}}}
,
$$
for $i \leq j$ and $M_{i,j} = 0$ otherwise. For $\mathbf{v} \in \mathbb{R}^{k-1}$ as the all-ones column vector, if $\phi$ is such that
$$
\mathbf{v}
=
\left(M + \mathbf{v} \mathbf{v}^{\mathrm{T}} \right) \phi
,
$$
then $Q_i \stackrel{D}{=} Q_j$ for all sub-queues $i$ and $j$. Moreover, if $1 + \mathbf{v}^{\mathrm{T}} M^{-1}\mathbf{v} \ne 0$, then the distributions of the sub-queues are equivalent if and only if $\phi = (M + \mathbf{v}\mathbf{v}^{\mathrm{T}})^{-1}\mathbf{v}$.
\begin{proof}
We start by considering the mean of each queue and solving for $\phi$ such that all the means are equal. Let $\lambda$ be the batch arrival rate. Then, the mean of $Q_i$ is
\begin{align*}
\E{Q_i}
&
=
\sum_{j=i}^{k-1} \lambda \phi_j \E{S_{(i,j)}}
+
\lambda \left(1 - \sum_{j=1}^{k-1}\phi_j\right) \E{S_{(i,k)}}
,
\end{align*}
as entities only arrive to $Q_i$ when $B \geq i$. We can note that for $Q_k$ this is
$$
\E{Q_k}
=
\lambda \left(1 - \sum_{j=1}^{k-1}\phi_j\right) \E{S_{(k,k)}}
.
$$
Then, we can see that all the queue means will be equal if $\E{Q_i} = \E{Q_k}$ for all $i$. Thus, we want to solve for $\phi$ such that
$$
0
=
\sum_{j=i}^{k-1} \lambda \phi_j \E{S_{(i,j)}}
+
\lambda \left(1 - \sum_{j=1}^{k-1}\phi_j\right) \E{S_{(i,k)}}
-
\lambda \left(1 - \sum_{j=1}^{k-1}\phi_j\right) \E{S_{(k,k)}}
,
$$
for all $i$. Rearranging this equation and dividing by $\lambda(\E{S_{(k,k)}} - \E{S_{(i,k)}})$, we receive
$$
\sum_{j=i}^{k-1}
\frac{\E{S_{(i,j)}}}{\E{S_{(k,k)}} - \E{S_{(i,k)}}} \phi_j
+
\sum_{j=1}^{k-1}
\phi_j
=
1
.
$$
We can now observe that this forms the linear system $(M + \mathbf{v}\mathbf{v}^{\mathrm{T}})\phi = \mathbf{v}$, and so we have shown that if $\phi$ satisfies this system then the means of the sub-queues will be equal. We can note moreover that $M + \mathbf{v}\mathbf{v}^{\mathrm{T}}$ is a rank one update of the matrix $M$. Thus, it is known that $M + \mathbf{v}\mathbf{v}^{\mathrm{T}}$ will be invertible if $1 + \mathbf{v}^{\mathrm{T}} M^{-1} \mathbf{v} \ne 0$; see Lemma 1.1 of \citet{ding2007eigenvalues}. In that case, we know that the unique solution to this system is $\phi = (M + \mathbf{v}\mathbf{v}^{\mathrm{T}})^{-1}\mathbf{v}$.
As we noted in the proof of Theorem~\ref{orderStatRand}, the steady-state distribution of an $M/G/\infty$ queue is $\mathrm{Pois}(\lambda \E{S})$ when the arrival rate is $\lambda$ and service distribution is equivalent to the random variables $S$. We can now note further that $\lambda \E{S}$ is the steady-state mean of such a queueing system. The distribution of $Q_i$ is then given by $\mathrm{Pois}(\E{Q_i})$ for each $i \in \{1, \dots, k\}$, and thus is equivalent across all sub-queues.
\end{proof}
\end{proposition}
For added motivation, we now consider the two dimensional case in the following remark.
\begin{remark}
If $k = 2$, $M$ and $\phi$ are scalars, given by
$$
M
=
\frac{\E{S}}{\E{S_{2,2}} - \E{S_{1,2}}}
,
\quad
\phi = \frac{\E{S_{2,2}} - \E{S_{1,2}}}{\E{S} + \E{S_{2,2}} - \E{S_{1,2}}}
.
$$
In this case, we can note that if $\PP{B = 1} = \phi$, then in steady-state the distribution of the workload in the system from the easier jobs from all batches will be equivalent to that of the harder jobs. If $\PP{B = 1} > \phi$ the number of harder jobs will stochastically dominate the number of easier jobs, and vice-versa is $\PP{B=1} < \phi$.
\end{remark}
This result implies if we have the ability to choose the probability of batch sizes, we can construct each of the sub-systems which are organized by the order statitics to have the same queue length distribution. Thus, providing equal work to all of the queues.
\section{Conclusion and Final Remarks} \label{concSec}
In this paper, we have found parallels between infinite server queues with batch arrivals, sums of scaled Poisson random variables, and Hermite distributions. Moreover, we also connect the stochastic objects to analytic quantities and functions of external interest, such as the harmonic numbers, the exponential integral function, the Euler-Mascheroni constant, and the polylogarithm function. In addition to being interesting in their own right, these connections have helped us to specify exact forms of valuable quantities related to this queueing system, including generating functions for the queue and for the limit of the queue scaled by the batch size. Thus, we have gained both insight into the queue itself and perspective on the model's place in operations research and applied mathematics more broadly.
For this reason, we believe continued work on these fronts is merited. For example, while we have some intuition for the harmonic Hermite distribution discussed in Subsection~\ref{markovSS}, we have less of an understanding of the limiting distribution of the scaled queue in that subsection and extended for random batch sizes in Subsection~\ref{limitRandSubsec}. Having more knowledge of what distribution might produce a moment generating function comprised of exponential integral function. Finding such a distribution could not only teach us about this queueing system, it would also likely be worth studying entirely on its own. Additionally, providing further connections of this distribution back to the harmonic numbers and the associated Hermite distribution would also be of interest, such as in the connection of the limiting moment generating function to the expected value of a harmonic number evaluated at a Poisson random variable that we remarked in Subsection~\ref{markovSS}. One could also consider control problems for the routing of arrivals to sub-systems, like what we discuss for the case of random batch sizes in Subsection~\ref{orderStatRandSubsec}.
For future expansions of this work into other areas of queueing, we can group the main themes of potential further investigations in three categories. First, the extension of our batch model beyond infinite server queues to multi-server queues, queues with abandonment, and networks of infinite server queues, a la \citet{mandelbaum2007service, massey2013gaussian, engblom2014approximations, gurvich2013excursion, pender2014gram, daw2017new}. It would be interesting to explore our limit theorems in these cases to understand the impact of having a finite number of servers. Second, it would also be interesting to explore the impact of the batch arrivals in the context of queues with delayed information as in \citet{pender2017queues, pender2017strong, pender2018analysis}. It would be of interest to know whether or not the batch arrivals would influence the Hopf bifurcations or oscillations that occur in the delayed information queues. Additionally, one could explore findings of this work, like the steady-state distribution representation or the batch scaling, in contexts where there is dependence among the service durations within each batch of arrivals, such as those studied in \citet{pang2012infinite, falin1994m}. Finally, we are particularly interested in studying the impact of batch arrivals in the context of self-exciting arrival processes such as Hawkes processes like in the work of \citet{gao2016functional, koops2017infinite, daw2018queues}. We intend to pursue the ideas described here as well as other related concepts in our future work.
\end{document} |
\begin{document}
\title{Linear Convergence Rate Analysis of Proximal Generalized ADMM for Convex Composite Programming}
\begin{abstract}
The proximal generalized alternating direction method of multipliers (p-GADMM) is substantially efficient for solving convex composite programming problems of high-dimensional to moderate accuracy.
The global convergence of this method was established by Xiao, Chen \& Li [{\tt Math. Program. Comput., 2018}], but its convergence rate was not given.
One may take it for granted that the convergence rate could be proved easily by mimicking the proximal ADMM, but we find the relaxed points will certainly cause many difficulties for theoretical analysis.
In this paper, we devote to exploring its convergence behavior and show that the sequence generated by p-GADMM possesses Q-linear convergence rate under some mild conditions. We would like to note that the proximal terms at the subproblems are required to be positive definite, which is very common in most practical implementations although it seems to be a bit strong.
\end{abstract}
\textbf{Key words}: Generalized ADMM, proximal terms, calmness, linear convergence.
\section{Introduction}
Let ${\mathcal Y}:={\mathcal Y}_1\times \cdots\times{\mathcal Y}_m$ and ${\mathcal Z}:={\mathcal Z}_1\times \cdots\times{\mathcal Z}_n$ be finite-dimensional real Euclidean spaces, each endowed with an inner product $ \langle \cdot,\cdot \rangle$ as well as its induced norm $\|\cdot \|$.
In this paper, we consider a canonical class of convex composite minimization problem with a separable objective function and linear constraint
\begin{equation}\label{eq:1}
\begin{aligned}
\min&{\;\:f(y)+g(z)}\\
\textrm{s.t.}\:&\;\: {\mathcal A}^*y+{\mathcal B}^*z=c,
\end{aligned}
\end{equation}
where $f:{\mathcal Y}\rightarrow \left ( -\infty ,+\infty \right ]$ and $g:{\mathcal Z}\rightarrow \left ( -\infty ,+\infty \right ]$ are closed proper convex but not necessarily smooth functions, ${\mathcal A}:{\mathcal X}\rightarrow {\mathcal Y}$ and ${\mathcal B}:{\mathcal X}\rightarrow {\mathcal Z}$ are linear operators with their adjoints ${\mathcal A}^*$ and ${\mathcal B}^*$, respectively, $c\in {\mathcal X}$ is the given date.
Given this structure, problem (\ref{eq:1}) arises in a
wide variety of statistical
and machine learning problems of recent interest, including the lasso,
sparse logistic regression, basis pursuit, covariance selection, support
vector machines, and many others \cite{Boyd2011DistributedOA}.
To solve (\ref{eq:1}), a simple but powerful algorithm is the alternating direction method of multipliers (ADMM) designed originally by Glowinski \& Marroco \cite{glowinski1975approximation} and Gabay \& Mercier \cite{Gabay1976ADA}, whose construction was closely correlated with Rockafellar's work on proximal point algorithm (PPA) for solving a more general maximal monotone inclusion problem \cite{Rockafellar1976AugmentedLA,Rockafellar1976MonotoneOA}. The readers may refer to \cite{Glowinski2014OnAD,Boyd2011DistributedOA} for reviewing the historical development of ADMM. Starting from $(y^0,z^0,x^0)\in{\mathcal Y}\times{\mathcal Z}\times{\mathcal X}$, the iterative scheme of ADMM for solving (\ref{eq:1}) takes the following form, for $k=0,1,\ldots,$
\begin{equation}\label{eq:17}
\left\{
\begin{array}{l}
y^{k+1} = \underset{y\in{\mathcal Y}}{\arg\min}\;{\mathcal L}_\sigma (y,z^k;x^k),\\
z^{k+1} = \underset{z\in{\mathcal Z}}{\arg\min}\;{\mathcal L}_\sigma (y^{k+1},z;x^k),\\
x^{k+1} = x^k-\tau \sigma ({\mathcal A}^*y^{k+1}+{\mathcal B}^*z^{k+1}-c),
\end{array}
\right.
\end{equation}
where $\tau\in(0,(1+\sqrt{5})/2)$ is a step-length, and ${\mathcal L}_\sigma (y,z;x)$ is the augmented Lagrangian function of problem (\ref{eq:1}) defined as
\begin{equation*}
{\mathcal L}_\sigma (y,z;x):=f(y)+g(z)-\left\langle x,{\mathcal A}^*y+{\mathcal B}^*z-c\right\rangle +\frac{\sigma }{2}\left \| {\mathcal A}^*y+{\mathcal B}^*z-c \right \|^2\!,
\end{equation*}
where $x\in{\mathcal X}$ is a multiplier and $\sigma>0$ is a penalty parameter.
It is interesting to note that the iteration scheme (\ref{eq:17}) is not always well-defined, one can consult \cite{chen2017note} for a counter-example.
Under the existence condition of a solution to the Karush-Kuhn-Tucker (KKT) system of (\ref{eq:1}), Gabay \cite{gabay1983chapter} showed that the ADMM (\ref{eq:17}) with an unit steplength is actually equivalent to the well-known Douglas-Rachford splitting (DRs) method \cite{Lions1979SplittingAF} to find a zero point to the stationary system coming from the dual of (\ref{eq:1}). Also, DRs can be considered as an application of PPA, see \cite{Eckstein1992OnTD,cai2022developments}. As a result, Eckstein \& Bertsekas \cite{Eckstein1992OnTD} applied an accelerated technique to (\ref{eq:17}) so as to getting a generalized variant ADMM, that is
\begin{equation}\label{eq:18}
\left\{
\begin{array}{l}
y^{k+1} = \underset{y\in{\mathcal Y}}{\arg\min}\;{\mathcal L}_\sigma (y,z^k;x^k),\\
z^{k+1}= \underset{z\in{\mathcal Z}}{\arg\min}\;g(z)-\left \langle {\mathcal B} x^k,z\right \rangle+\frac{\sigma }{2}\left \| \rho ({\mathcal A}^*y^{k+1}+{\mathcal B}^*z^k-c )+{\mathcal B}^*(z-z^k)\right \|^2,\\
x^{k+1}=x^k-\sigma {\mathcal B}ig(\rho ({\mathcal A}^*y^{k+1}+{\mathcal B}^*z^{k}-c)+{\mathcal B}^*(z^{k+1}-z^k) {\mathcal B}ig),
\end{array}
\right.
\end{equation}
where $\rho\in\left( 0,2\right) $ is a relaxation factor. It is quite clear to see that the (\ref{eq:17}) with $\tau=1$ is consistent with (\ref{eq:18}) under the setting $\rho=1$.
The generalized ADMM retains the benefits of treating the objective functions $f$ and $g$ individually, and at the same time, it also enjoys the easiness to implement. Most importantly, a suitable $\rho$ may lead to better numerical performance.
For the empirical studies of the generalized ADMM, one can refer to \cite{bertsekas1982constrained,Cai2013APP,Eckstein1994ParallelAD,xiao2018generalized}.
To make the subproblems in (\ref{eq:17}) admit unique solutions without further assumptions on the objective functions and constraints, Eckstien \cite{Eckstein1994SomeSS} suggested add a proximal term to each subproblem. Later, Fazel et al. \cite{Fazel2013HankelMR} showed that these added proximal terms are not necessarily positive definite, and then proposed
a more powerful but convenient semi-proximal ADMM (abbr. sPADMM).
The sPADMM covers the classic ADMM as a special case and has the ability to deal with multi-block convex composite semidefinite programming problems of a low to moderate accuracy \cite{li2015two,chen2017efficient,sun2015convergent}. In recent years, Xiao, Chen \& Li \cite{xiao2018generalized} introduced a variant of generalized ADMM with semi-proximal terms (p-GADMM), that is, starting from $\tilde{\omega}^0:=(\tilde{x}^0,\tilde{y}^0,\tilde{z}^0)\in{\mathcal X}\times\textrm{dom}f\times\textrm{dom}g$, it generates a sequence $\{(y^k,z^k,x^k)\}$ using the following frameworks
\begin{equation}\label{eq:19}
\begin{cases}
y^k:=\underset{y\in{\mathcal Y}}{\arg\min}{\mathcal L}_\sigma (y,\tilde{z}^{k};\tilde{x}^{k})+\frac{1}{2}\left \| y-\tilde{y}^k \right \|^2_{\mathcal S},\\
x^k:=\tilde{x}^k-\sigma ({\mathcal A}^*y^k+{\mathcal B}^*\tilde{z}^{k}-c),\\
z^k:=\underset{z\in{\mathcal Z}}{\arg\min}{\mathcal L}_\sigma (y^{k},z;x^k)+\frac{1}{2}\left \| z-\tilde{z}^k \right \|^2_{\mathcal T},\\
\tilde{\omega}^{k+1}:=\tilde{\omega}^{k}+\rho (\omega^{k}-\tilde{\omega}^{k}).
\end{cases}
\end{equation}
In fact, this variant is based on an important observation of Chen \cite{chen2012numerical} that the generalized ADMM (\ref{eq:18}) can be reformulated as an ADMM with an extra relaxation step with factor lying in $(0,2)$.
By comparison with \cite{lu2018convergence}, the semi-proximal terms in (\ref{eq:19}) is more natural and pretty in sense that it used the most recent values of variables.
Moreover, extensive numerical experiments on a class of linearly doubly non-negative semidefinite programming problems illustrated that the variant of generalized ADMM (\ref{eq:19}) performed more effectively and efficiently \cite{xiao2018generalized}.
It has been proved that the sequence generated by p-GADMM converges globally to the KKT point of (\ref{eq:1}) under some mild conditions. However, its convergence rate was not given.
In fact, the convergence rate analysis on ADMM and its related variants have been studied by in different contexts. For example, under only an error bound condition, Han, Sun \& Zhang \cite{Han2018LinearRC} established the linear rate convergence rate of sPADMM of Fazel et al. \cite{Fazel2013HankelMR} with $\tau\in\left ( 0,\left ( 1+\sqrt{5} \right )/2 \right )$.
For another example, Fang et al. \cite{fang2015generalized} derived the linear convergence rate of a linearized variant of generalized ADMM and proved the worst-case ${\mathcal O}\left( 1/k\right)$ iteration complexity in both ergodic and nonergotic cases.
This result was further improved by Wang et al. \cite{wang2022linear}, in which they showed that the Q-linear convergence result for generalized ADMM (\ref{eq:18}) hold if the proximal terms are positive and semidefinite.
But despite these achievements, the convergence rate of p-GADMM with respect to (\ref{eq:19}) is not trivial because the relaxed iterative points $(\tilde{x}^k,\tilde{y}^k,\tilde{z}^k)$ would certainly cause many difficulties.
The paper concentrates on theoretical analysis to prove that the sequence $\{(y^k,z^k)\}$ generated by p-GADMM possesses Q-linear convergence rate under the condition of calmness.
The remaining parts of this paper are organized as follows. Section \ref{sec2} is divided into two subsections, Subsection \ref{sec2.1} presents some results on the optimimality conditions for problem (\ref{eq:1}), and Subsection \ref{subsec2.2} briefly overviews the definitions and properties associated with calmness in variational analysis. Section \ref{sec3} is the main part of this paper, in which, we derive the local linear convergence results for p-GADMM under some certain assumption conditions. Finally, we conclude the paper in Section \ref{sec4}.
\section{Preliminaries}\label{sec2}
The section presents some notations and basic concepts appeared in the context, and summarizes some useful preliminaries used for later analysis.
\subsection{Notations and basic concepts}\label{sec2.1}
For any two vectors $x\in \mathbb{R}^n$ and $y\in\mathbb{R}^m$, we use $(x,y)$ to denote their adjunction, i.e., $(x,y)\in\mathbb{R}^{m+n}$. For $p\geq1$, we use $\left \| x \right \|_p$ to denote an $\ell_p$-norm of a vector $x$, and for $p=2$, we simply denote it as $ \| x \|$. For any symmetric and positive definite matrix ${\mathcal O}$, the ${\mathcal O}$-norm of $x$ is denoted by $\|x\|_{\mathcal O}:=\sqrt{x^\top{\mathcal O} x}$. For a given closed convex set ${\mathcal C}$, the distance of $x$ to ${\mathcal C}$ regarding ${\mathcal O}$-norm is denoted as $\text{dist}_{\mathcal O}(x,{\mathcal C}):=\textrm{inf}_{y\in {\mathcal C}}\| x-y\|_{\mathcal O}$. Give $f: {\mathcal Y} \rightarrow(-\infty,+\infty]$ be a proper closed convex function.
We use $\text{dom}f$ to denote the domain of $f$, that is, $\text{dom}f=\{y \in {\mathcal Y}\mid f(y)<\infty\}$. The proximal mapping of $f$ with $t >0$ is defined by
$$
\operatorname{Prox}_{f}(x) :=\underset{y \in {\mathcal Y}}{\operatorname{argmin}}{\mathcal B}ig\{f(y)+\frac{1}{2}\|y-x\|_2^{2}{\mathcal B}ig\}, \quad \forall x \in {\mathcal Y}.
$$
The Lagrangian function of problem (\ref{eq:1}) is defined by
\begin{equation}\label{eq:2}
{\mathcal L}(y,z;x):=f(y)+g(z)-\left \langle x,{\mathcal A}^*y+{\mathcal B}^*z-c \right \rangle,\; \; \; \forall (y,z,x)\in{\mathcal Y}\times{\mathcal Z}\times {\mathcal X},
\end{equation}
which is convex in $(y,z)\in{\mathcal Y}\times{\mathcal Z}$ and concave in $x\in{\mathcal X}$. The Slater constraint qualification for problem (\ref{eq:1}) is said to be held if
$$
{\mathcal B}ig \{ (y,z)\, |\, y\in\textrm{ri}(\textrm{dom}f),\, z\in\textrm{ri}(\textrm{dom}g), \, {\mathcal A}^*y+{\mathcal B}^*z=c {\mathcal B}ig \}\neq \varnothing,
$$
where $\text{ri}(\cdot)$ denotes the relative interior of a convex set.
Under this constraint qualification, from \cite[Corollaries 28.2.2 and 28.3.1]{rockafellar1970convex}, we know that $( \bar{y},\bar{z})\in \textrm{ri}(\textrm{dom}f\times \textrm{dom}g)$ is an optimal solution to problem (\ref{eq:1}) if and only if there exists a Lagrange multiplier $\bar{x}\in {\mathcal X}$ such that $(\bar{y},\bar{z},\bar{x})$ is a solution to the following KKT system:
\begin{equation}\label{eq:3}
{\mathcal A}{x}\in \partial f({y}),\; \; {\mathcal B}{x}\in \partial g({z})\; \; \textrm{and}\; \; {\mathcal A}^*{y}+{\mathcal B}^*{z}-c=0,
\end{equation}
where $\partial f$ and $\partial g$ are the subdifferential mappings of $f$ and $g$ in (\ref{eq:1}), respectively. Noting that the subdifferential mappings $\partial f$ and $\partial g$ are maximal monotone \cite[Theorem 12.17]{Rockafellar1998VariationalA}, then for all $y, y'\in \textrm{dom} f$, $\xi \in \partial f(y) $ and $ {\xi }'\in \partial f({y}')$, it holds
\begin{equation*}
\langle {\xi }'-\xi, {y}'-y\rangle\geq \| {y}'-y\|_{{\mathcal S}igma _f}^2,
\end{equation*}
and for all $z, z'\in \textrm{dom} g$, $\zeta \in \partial g(z) $ and ${\zeta }'\in \partial g({z}')$, it holds
\begin{equation*}
\langle {\zeta }'-\zeta , {z}'-z\rangle \geq \| {z}'-z \|_{{\mathcal S}igma _g}^2,
\end{equation*}
where ${\mathcal S}igma _f:{\mathcal Y}\rightarrow{\mathcal Y}$ and ${\mathcal S}igma _g:{\mathcal Z}\rightarrow{\mathcal Z}$ are self-adjoint and positive semidefinite linear operators.
Let $\bar{{\mathcal O}mega }:=(\bar{y},\bar{z},\bar{x})$ be the solution set to the KKT system (\ref{eq:3}). The nonempty of $\bar{{\mathcal O}mega}$ can be guaranteed under the setting of a certain constraint qualification condition.
In this paper, we only assume the existence of KKT point, and do not particularly emphasize which qualification is used.
\begin{assumption}\label{assumption1}
The KKT system (\ref{eq:2}) has a nonempty solution set, i.e., $\bar{{\mathcal O}mega}\neq\varnothing$.
\end{assumption}
In order to facilitate the subsequent theoretical analysis, we let $\nu :=(x,y,\tilde{y}-y,z,\tilde{z}-z)\in {\mathcal V}:={\mathcal X} \times {\mathcal Y} \times {\mathcal Y} \times {\mathcal Z} \times {\mathcal Z}$ for given $\tilde{y}\in{\mathcal Y}$ and $\tilde{z}\in{\mathcal Z}$, and define a KKT mapping $\hat{{\mathcal R}}:{\mathcal V}\rightarrow {\mathcal V}$ in the form of
\begin{equation}\label{eq:22}
\hat{{\mathcal R}}(\nu):=\begin{pmatrix}
{\mathcal A}^*y+{\mathcal B}^*z-c\\
y-\textrm{Prox}_f(y+{\mathcal A} x)\\
0\\
z-\textrm{Prox}_g(z+{\mathcal B} x)\\
0
\end{pmatrix},\qquad\forall\nu\in {\mathcal V},
\end{equation}
where $\textrm{Prox}_f( \cdot)$ represents the proximal mapping of a closed convex proper function $f$. Define $\bar{\nu}:=(\bar{x}, \bar{y}, 0, \bar{z}, 0)$ such that $(\bar{x}, \bar{y}, \bar{z})\in\bar{{\mathcal O}mega}$. For simplicity, we denote a generalized optimal solutions set $\bar{{\mathcal T}heta}:=\left \{ (0,0) \right \}\cup \bar{{\mathcal O}mega }$, which means that $(\bar{x}, \bar{y}, \bar{z})\in\bar{{\mathcal O}mega}$ if and only if $\bar{\nu}\in\bar{{\mathcal T}heta}$. By optimization theory, we know that the proximal mappings $\textrm{Prox}_f( \cdot)$ and $\textrm{Prox}_g ( \cdot )$ are globally Lipschitz continuous with modulus one. Then, the mapping $\hat{{\mathcal R}}(\cdot)$ is continuous on ${\mathcal V}$ and $\hat{{\mathcal R}}(\bar\nu)=0$ if and only if $\bar{\nu}\in\bar{{\mathcal T}heta}$.
\subsection{Locally upper Lipschitzian and calmness}\label{subsec2.2}
Given a set-valued function, say ${\mathcal P}si$, from ${\mathcal X}$ to ${\mathcal Y}$,
the graph of ${\mathcal P}si$ is defined as
${\mathcal G}amma _{\mathcal P}si:=\left \{ (x,y)\in {\mathcal X}\times{\mathcal Y} \ | \ y\in{\mathcal P}si(x) \right \}$. For convenience, we denote $\mathbf{B}_{\mathcal Y}$ as an Euclidean unit ball, i.e., $\mathbf{B}_{\mathcal Y}:=\left \{ y\in{\mathcal Y}|\left \| y \right \|\leq 1 \right \}$.
\begin{definition}[\cite{robinson1981some}]
The set-valued mapping ${\mathcal P}si : {\mathcal X} \rightrightarrows{\mathcal Y}$ is said to be locally upper Lipschitzian at a point $x_0\in{\mathcal X}$ with modulus $\lambda$, if for some neighborhoods ${\mathcal N}$ of $x_0$ and for all $x\in {\mathcal N}$ such that
\[{\mathcal P}si (x)\subseteq {\mathcal P}si(x_0)+\lambda \left \| x-x_0 \right \|\mathbf{B}_{\mathcal Y}, \:\:\:\forall x\in {\mathcal N}.\]
\end{definition}
\noindent The set-valued mapping ${\mathcal P}si$ is called piecewise polyhedral if its graph ${\mathcal G}amma _{\mathcal P}si$ is the union of finitely many polyhedral sets. The elementary relationship between the locally upper Lipschitzian and the piecewise polyhedral for a set-valued mapping ${\mathcal P}si$ is stated as follows:
\begin{proposition}[\cite{robinson1981some}]
Let set-valued mapping ${\mathcal P}si$ be piecewise polyhedral from ${\mathcal X}$ into ${\mathcal Y}$, then there
exists a constant $\lambda$ such that ${\mathcal P}si$ is locally upper Lipschitzian at each $x_0\in {\mathcal X}$ independent of the choice of $x_0$.
\end{proposition}
It was known from \cite[definition 10.20]{Rockafellar1998VariationalA} that, if a set-valued function ${\mathcal P}si$ is called piecewise linear-quadratic, then dom${\mathcal P}si$ can be represented as the union of finitely many polyhedral sets, relative to each of which ${\mathcal P}si$ is either given by an expression of affine or quadratic function.
Meanwhile, ${\mathcal P}si$ is piecewise linear-quadratic if and only if the subdifferential mapping $\partial{\mathcal P}si$ is piecewise polyhedral. The proof and its extensions can be found at the monograph \cite[proposition 12.30]{Rockafellar1998VariationalA}.
We now ready to state the definition of calmness for ${\mathcal P}si : {\mathcal X} \rightrightarrows{\mathcal Y}$ at $x_0$ for $y_0$ with $(x_0,y_0)\in {\mathcal G}amma _{\mathcal P}si$. For more details, one can see the disquisition of Dontchev \& Rockafellar \cite{dontchev2009implicit} and Rockafellar \& Wets \cite{Rockafellar1998VariationalA}.
\begin{definition}[\cite{dontchev2009implicit}]\label{def1}
A set-valued mapping ${\mathcal P}si : {\mathcal X} \rightrightarrows{\mathcal Y}$ is called to be calm at $x_0$ for $y_0$ if $(x_0,y_0)\in {\mathcal G}amma _{\mathcal P}si$ and there exists a constant $\lambda$ along with neighborhoods ${\mathcal N}$ of $x_0$ and ${\mathcal M}$ of $y_0$ such that
\[{\mathcal P}si(x)\cap {\mathcal M}\subseteq {\mathcal P}si(x_0)+\lambda \left \| x-x_0 \right \|\mathbf{B}_{\mathcal Y},\:\:\:\forall x\in {\mathcal N}.\]
\end{definition}
\noindent As can be seen from this definition that, suppose ${\mathcal P}si$ be the subdifferential mapping of a piecewise linear-quadratic function, then ${\mathcal P}si$ is calm at $x^0$ for $y^0$ meeting $(x_0,y_0)\in{\mathcal G}amma _{\mathcal P}si$ with modulus $\lambda\geq 0$ independent of the selection of $(x_0,y_0)$. At last, a set-valued mapping ${\mathcal P}si : {\mathcal X} \rightrightarrows{\mathcal Y}$ is called metrically subregular at $x_0$ for $y_0$ if $(x_0,y_0)\in {\mathcal G}amma _{\mathcal P}si$ and there exists a constant $\iota\geq0$ along with neighborhoods ${\mathcal N}$ of $x_0$ and ${\mathcal M}$ of $y_0$ such that
$$
\textrm{dist} (x,{\mathcal P}si ^{-1}(y_0))\leq \iota \, \textrm{dist}(y_0,{\mathcal P}si(x)\cap {\mathcal M}),\:\:\:\forall x\in {\mathcal N},
$$
which is a.k.s. error bound condition.
To end this section, we list the following result to reveal the equivalence of metric subregularity of a set-valued mapping with calmness of its inverse. For its proof, one can refer to \cite[Theorem 3H.3]{dontchev2009implicit}.
\begin{proposition}[\cite{dontchev2009implicit}]
For a set-valued mapping ${\mathcal P}si : {\mathcal X} \rightrightarrows{\mathcal Y}$, let $(x_0,y_0)\in {\mathcal G}amma _{\mathcal P}si$. Then ${\mathcal P}si$ is metrically subregular at $x_0$ for $y_0$ with a constant $\lambda$ if and only if its inverse ${\mathcal P}si^{-1}: {\mathcal Y} \rightrightarrows{\mathcal X}$ is calm at $y_0$ for $x_0$ with the same constant $\lambda$.
\end{proposition}
\section{Linear convergence rate}\label{sec3}
In this section, we present a general convergence rate analysis on algorithm p-GADMM.
It should be noted that the steps of p-GADMM has been fully stated in
\cite{xiao2018generalized}, but for the convenience of the subsequent analysis, we adjust the updating order and the upper script here.
\begin{algorithm}
\renewcommand{1}{1}
\caption{p-GADMM}\label{alg:2}
\begin{itemize}
\item[Step 0.] Let $\sigma \in(0,+\infty)$ and $\rho \in (0,2)$ be given parameters. Let ${\mathcal S}$ and ${\mathcal T}$ be self-adjoint positive definite linear operators defined on ${\mathcal Y}$ and ${\mathcal Z}$, respectively. Choose $(\tilde{x}^0,\tilde{y}^0,\tilde{z}^1)\in{\mathcal X}\times\textrm{dom}f\times \textrm{dom}g$.
\item[Step 1.] Compute
\begin{equation*}
\begin{cases}
y^0:=\underset{y\in{\mathcal Y}}{\arg\min}{\mathcal L}_{\sigma }(y,\tilde{z}^1;\tilde{x}^0)+\frac{1}{2}\left \| y-\tilde{y}^0 \right \|^2_{\mathcal S},\\
x^0:=\tilde{x}^0-\sigma (A^*y^0+B^*\tilde{z}^1-c).
\end{cases}
\end{equation*}
\item[Step 2.] For $k=1,2,3,\ldots$, do the following steps iteratively:
\begin{subequations}
\begin{numcases}{}
z^k:=\underset{z\in{\mathcal Z}}{\arg\min}{\mathcal L}_\sigma (y^{k-1},z;x^{k-1})+\frac{1}{2}\left \| z-\tilde{z}^k \right \|^2_{\mathcal T},\label{eq:a}\\
\tilde{y}^k:=\tilde{y}^{k-1}+\rho (y^{k-1}-\tilde{y}^{k-1}),\label{eq:b}\\
\tilde{x}^k:=\tilde{x}^{k-1}+\rho (x^{k-1}-\tilde{x}^{k-1}),\label{eq:c}\\
\tilde{z}^{k+1}:=\tilde{z}^{k}+\rho (z^{k}-\tilde{z}^{k}),\label{eq:d}\\
y^k:=\underset{y\in{\mathcal Y}}{\arg\min}{\mathcal L}_\sigma (y,\tilde{z}^{k+1};\tilde{x}^{k})+\frac{1}{2}\left \| y-\tilde{y}^k \right \|^2_{\mathcal S},\label{eq:e}\\
x^k:=\tilde{x}^k-\sigma ({\mathcal A}^*y^k+{\mathcal B}^*\tilde{z}^{k+1}-c). \label{eq:f}
\end{numcases}
\end{subequations}
\item[Step 3.] If a termination criterion is not met, set $k:=k+1$ and go to Step 2.
\end{itemize}
\end{algorithm}
From \cite[Lemma 5.2, Theorem 5.1]{xiao2018generalized}, it is a trivial task to get the following result which provides some useful highlights for the further convergence rate analysis.
\begin{theorem}\label{th:1}
Suppose that the KKT system (\ref{eq:3}) is nonempty. Let the sequence $\{(x^k, y^k, z^k; \tilde{x}^k,\tilde{y}^k,\tilde{z}^k) \}$ be generated by Algorithm \ref{alg:2}. Then the following results hold:
\begin{itemize}
\item [(i)] For any $k\geq0$,
\begin{equation}\label{eq:4}
\begin{split}
&(\sigma \rho )^{-1}\left \| x_e^k+\sigma (1-\rho ) {\mathcal A}^*y_e^k\right \|^2+\rho ^{-1}\left \| \tilde{y}_e^{k+1} \right \|_{\mathcal S}^2+\rho ^{-1}\left \| \tilde{z}_e^{k+1} \right \|_{\mathcal T}^2\\[2mm]
&+(2-\rho )\left \| y^k-\tilde{y}^k \right \|_S^2+\left ( 2-\rho \right )\sigma \left \| {\mathcal A}^*y_e^k \right \|^2\\[2mm]
\geq &(\sigma \rho )^{-1}\left \| x_e^{k+1}+\sigma (1-\rho ) {\mathcal A}^*y_e^{k+1}\right \|^2+\rho ^{-1}\left \| \tilde{y}_e^{k+2} \right \|_S^2+\rho ^{-1}\left \| \tilde{z}_e^{k+2} \right \|_{\mathcal T}^2\\[2mm]
&+(2-\rho )\left \| y^{k+1}-\tilde{y}^{k+1} \right \|_{\mathcal S}^2+\left ( 2-\rho \right )\sigma \left \| {\mathcal A}^*y_e^{k+1} \right \|^2\\[2mm]
&+2\left \| y_e^{k+1} \right \|_{{\mathcal S}igma _f}^2+2\left \| z_e^{k+1} \right \|_{{\mathcal S}igma _g}^2+(2-\rho )\sigma \left \| {\mathcal A}^*y_e^{k+1} +{\mathcal B}^*z_e^{k+1}\right \|^2\\[2mm]
&+(2-\rho )\left \| \tilde{y}^{k+1}-y^{k+1} \right \|_S^2+(2-\rho )\left \| \tilde{z} ^{k+1}-z^{k+1}\right \|_{\mathcal T}^2+\sigma \rho ^{-1}(2-\rho )^2\left \| {\mathcal A}^*(y^{k+1}-y^k) \right \|^2,
\end{split}
\end{equation}
where $x_e=x-\bar{x}$, $y_e=y-\bar{y}$, and $z_e=z-\bar{z}$.
\item[(ii)] Assume that both ${\mathcal S}$ and ${\mathcal T}$ be chosen such that ${\mathcal S}igma _f+{\mathcal S}+\sigma{\mathcal A}{\mathcal A}^*\succ 0$ and ${\mathcal S}igma _g+{\mathcal T}+\sigma{\mathcal B}{\mathcal B}^*\succ 0$, then the sequence $\left \{ (x^k, y^k, \tilde{y}^k-y^k, z^k, \tilde{z}^k-z^k)\right \}$ is automatically well-defined, and it converges to $(\bar{x}, \bar{y}, 0, \bar{z}, 0)\in\bar{{\mathcal T}heta}$.
\end{itemize}
\end{theorem}
Theorem \ref{th:1} presents a global convergence result for p-GADMM under fairly general and mild conditions. Evidently, one can choose positive semidefinite (and even indefinite) linear operators ${\mathcal S}$ and ${\mathcal T}$ to ensure ${\mathcal S}igma _f+{\mathcal S}+\sigma{\mathcal A}{\mathcal A}^*\succ 0$ and ${\mathcal S}igma _g+{\mathcal T}+\sigma{\mathcal B}{\mathcal B}^*\succ 0$. But, due to the existences of some coupling terms in (\ref{eq:4}), we
must restrict ${\mathcal S}$ and ${\mathcal T}$ to be positive definite. In this case, the conditions ${\mathcal S}igma _f+{\mathcal S}+\sigma{\mathcal A}{\mathcal A}^*\succ 0$ and ${\mathcal S}igma _g+{\mathcal T}+\sigma{\mathcal B}{\mathcal B}^*\succ 0$ hold automatically.
We now present some notations to facilitate the later theoretical analysis. For any self-adjoint linear operator ${\mathcal G}: {\mathcal X}\rightarrow{\mathcal X}$, we use the symbol $\lambda_{\max}({\mathcal G})$ to denote its largest eigen-value.
Denote
\begin{equation}\label{eq:8}
{\mathcal U}psilon(\nu^{k+1}):=\kappa\left(\left \| \tilde{y}^{k+1}-y^{k+1} \right \| _{\mathcal S}^2+\left \| \tilde{z}^{k+1}-z^{k+1} \right \|_{\mathcal T}^2+\left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1} \right \|^2+\left \| {\mathcal A}^*(y^{k+1}-y^k) \right \|^2\right),
\end{equation}
where
\begin{equation}\label{kapa}
\kappa :=\max\bigg \{ \|{\mathcal S} \|, 3\|{\mathcal T} \|,3(2-\rho )^2\sigma ^2\lambda _{\max}({\mathcal B}^*{\mathcal B}), 3(1-\rho )^2\sigma ^2\lambda _{\max}({\mathcal B}^*{\mathcal B})+1\bigg \}.
\end{equation}
Moreover, denote ${\mathcal X}i:{\mathcal V}\rightarrow{\mathcal V}$ in the form of
\begin{equation*}
{\mathcal X}i:=\left(
\begin{smallmatrix}
(\sigma \rho)^{-1}{\mathcal I} & (1-\rho)\rho^{-1}{\mathcal A}^* & 0& 0 & 0\\
(1-\rho)\rho^{-1}{\mathcal A} &\rho^{-1}\sigma{\mathcal A}{\mathcal A}^*+\rho^{-1}S+2{\mathcal S}igma _f & (1-\rho)\rho^{-1}{\mathcal S} & 0 &0 \\
0& (1-\rho)\rho^{-1}{\mathcal S} & \rho^{-1}{\mathcal S} & 0 & 0\\
0 & 0& 0& \rho^{-1}{\mathcal T}+2{\mathcal S}igma _g & (1-\rho)\rho^{-1}{\mathcal T}\\
0 & 0 & 0 & (1-\rho)\rho^{-1}{\mathcal T}& (1-\rho)^2\rho^{-1}{\mathcal T}
\end{smallmatrix}
\right)
+\frac{1}{2}(2-\rho)\sigma\vartheta\vartheta^*,
\end{equation*}
where ${\mathcal I}$ is an identity operator, and $\vartheta :{\mathcal X}\rightarrow {\mathcal V}$ is a linear operator such that its adjoint $\vartheta^*$ satisfies $\vartheta^*(\nu)= {\mathcal A}^*y+{\mathcal B}^*z$ for any $\nu\in{\mathcal V}$.
In the subsequent analysis, we use the operator ${\mathcal X}i$ to measure the weighted distance from the current point to the generalized optimal solutions set $\bar{{\mathcal T}heta}$. Obviously, if $\rho\in(0,2)$ and ${\mathcal S}\succ 0$, ${\mathcal T}\succ 0$, then ${\mathcal X}i$ must be positive definite, i.e.,
$$
\{{\mathcal S}\succ 0 \ \& \ {\mathcal T}\succ 0\}\; {\mathcal L}eftrightarrow \; {\mathcal X}i \succ 0,
$$
which plays a key rule in the linear convergence rate result.
Additionally, to conduct the rate of the decrease of $\left \| \nu ^k-\bar{\nu} \right \|^2$, we take an interest in deducing an upper bound for $\hat{R}(\cdot)$ computed at the sequence generated by the p-GADMM in the subsequent developments.
\begin{lemma}\label{lemma1}
Let $\left \{ \nu^k \right \}$ be the infinite sequence generated by p-GADMM. Then for any $k\geq1$, we have
\begin{equation}\label{ineq34}
{\mathcal U}psilon(\nu^{k+1})\geq \| \hat{{\mathcal R}}(\nu ^{k+1}) \|^2.
\end{equation}
\end{lemma}
\begin{proof}
From (\ref{eq:c}) and (\ref{eq:f}), it is easy to see that
\begin{equation*}
\tilde{x}^{k+1}=x^k+(1-\rho )(\tilde{x}^k-x^k)=x^k+\sigma (1-\rho)({\mathcal A}^*y_e^k+{\mathcal B}^*\tilde{z}_e^{k+1}).
\end{equation*}
Then, substituting this equality into (\ref{eq:f}), we get
\begin{equation}\label{eq:7}
\begin{split}
x^{k+1}&=\tilde{x}^{k+1}-\sigma ({\mathcal A}^*y_e^{k+1}+{\mathcal B}^*\tilde{z}_e^{k+2})\\[2mm]
&=x^k-\sigma \rho ({\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1})+\sigma (1-\rho){\mathcal A}^*(y^{k}-y^{k+1}).
\end{split}
\end{equation}
By the first order optimality condition of (\ref{eq:a}), we have
\begin{equation*}
\begin{split}
0&\in\partial g(z^{k+1})-{\mathcal B} x^k+\sigma {\mathcal B}({\mathcal A}^*y_e^k+{\mathcal B}^*z_e^{k+1})+{\mathcal T}(z^{k+1}-\tilde{z}^{k+1})\\[2mm]
&=\partial g(z^{k+1})-{\mathcal B}\left ( x^k-\sigma ({\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1})+\sigma {\mathcal A}^*(y^{k+1}-y^k)\right )+{\mathcal T}(z^{k+1}-\tilde{z}^{k+1}),
\end{split}
\end{equation*}
which leads to an equivalent expression for $z^{k+1}$, that is,
\begin{equation}\label{eq:5}
z^{k+1}=\textrm{Prox}_g(z^{k+1}+{\mathcal B}\left ( x^k-\sigma ({\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1})+\sigma {\mathcal A}^*(y^{k+1}-y^k)\right )-{\mathcal T}(z^{k+1}-\tilde{z}^{k+1})).
\end{equation}
It follows from (\ref{eq:e}) we can easily get
\begin{equation*}
0\in \partial f(y^{k+1})-{\mathcal A}\tilde{x}^{k+1}+\sigma {\mathcal A}({\mathcal A}^*y_e^{k+1}+{\mathcal B}^*\tilde{z}_e^{k+2})+{\mathcal S}(y^{k+1}-\tilde{y}^{k+1}),
\end{equation*}
which, from (\ref{eq:f}), is equivalent to
\begin{equation*}
0\in \partial f(y^{k+1})-{\mathcal A} x^{k+1}+{\mathcal S}(y^{k+1}-\tilde{y}^{k+1}).
\end{equation*}
Thus, we obtain that
\begin{equation}\label{eq:6}
y^{k+1}=\textrm{Prox}_f{\mathcal B}ig(y^{k+1}+{\mathcal A} x^{k+1}-{\mathcal S}(y^{k+1}-\tilde{y}^{k+1}){\mathcal B}ig).
\end{equation}
Secondly, associating with the equations (\ref{eq:7}), (\ref{eq:5}) and (\ref{eq:6}) and using the Lipschitz continuity of Moreau-Yosida proximal mapping, we get from the definition of $\hat{{\mathcal R}}(\cdot)$ in (\ref{eq:22}) that
\begin{equation*}
\begin{split}
\left \| \hat{{\mathcal R}}(\nu ^{k+1}) \right \|^2\leq &\left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1} \right \|^2+\left \| {\mathcal S}(y^{k+1}-\tilde{y}^{k+1}) \right \|^2\\[2mm]
&+\left \| {\mathcal B}(x^k-x^{k+1})-\sigma {\mathcal B}({\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1})+\sigma {\mathcal B}{\mathcal A}^*(y^{k+1}-y^k)-{\mathcal T}(z^{k+1}-\tilde{z}^{k+1}) \right \|^2\\[2mm]
=&\left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1} \right \|^2+\left \| {\mathcal S}(y^{k+1}-\tilde{y}^{k+1}) \right \|^2\\[2mm]
&+\left \| (\rho -1)\sigma {\mathcal B}({\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1})+(2-\rho )\sigma {\mathcal B}{\mathcal A}^*(y^{k+1}-y^k)-{\mathcal T}(z^{k+1}-\tilde{z}^{k+1}) \right \|^2\\[2mm]
\leq&\left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1} \right \|^2+\left \| {\mathcal S} \right \|\left \| y^{k+1}-\tilde{y}^{k+1} \right \|_{\mathcal S}^2+3\left \| {\mathcal T} \right \|\left \| z^{k+1}-\tilde{z}^{k+1} \right \|_{\mathcal T}^2\\[2mm]
&+3(1-\rho )^2\sigma ^2\lambda_{\max}({\mathcal B}^*{\mathcal B})\left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1} \right \|^2+3(2-\rho )^2\sigma ^2\lambda_{\max}({\mathcal B}^*{\mathcal B})\left \| {\mathcal A}^*(y^{k+1}-y^k) \right \|^2\\[2mm]
\leq&\kappa(\left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1} \right \|^2+\left \| y^{k+1}-\tilde{y}^{k+1} \right \|_{\mathcal S}^2+\left \| z^{k+1}-\tilde{z}^{k+1} \right \|_{\mathcal T}^2+\left \| {\mathcal A}^*(y^{k+1}-y^k) \right \|^2).
\end{split}
\end{equation*}
Using the definition of ${\mathcal U}psilon(\cdot)$ in (\ref{eq:8}), we get the inequality (\ref{ineq34}).
\end{proof}
To get the local linear convergence rate of p-GADMM \ref{alg:2}, we need another assumption to control the distance from an iterate $\nu$ to the KKT solution set $\bar{{\mathcal T}heta}$.
\begin{assumption}\label{ass2}
The inverse of the mapping $\hat{{\mathcal R}}(\cdot)$ defined in (\ref{eq:22}) is calm at the $0\in{\mathcal V}$ for $\bar{\nu}$ with modulus $\lambda>0$ if there exists a constant $\varepsilon>0$ such that
$$
\text{dist}(\nu, \bar{ {\mathcal T}heta })\leq \lambda \| \hat{{\mathcal R}}(\nu) \|, \; \; \; \forall \nu\in {\mathcal B}ig\{ \nu\in{\mathcal V}| \| \nu-\bar{\nu} \|\leq \varepsilon {\mathcal B}ig\}.
$$
\end{assumption}
In light of above analysis, we are ready to establish the local linear convergence rate result of algorithm p-GADMM.
\begin{theorem}\label{th2}
Suppose that Assumptions \ref{assumption1} and \ref{ass2} hold. Besides, assume that ${\mathcal S}$ and ${\mathcal T}$ are positive definite. Let the sequence $\{(x^k, y^k, z^k; \tilde{x}^k,\tilde{y}^k,\tilde{z}^k) \}$ be generated by Algorithm p-GADMM.
Then $\{ \nu^k :=(x^k, y^k, \tilde{y}^k-y^k, z^k, \tilde{z}^k-z^k) \}$ converges to $\bar{\nu} =(\bar{x}, \bar{y}, 0, \bar{z}, 0)\in\bar{{\mathcal T}heta}$, and there exists a threshold $\bar{\kappa}\geq1$ such that for all $k\geq \bar{\kappa}$, it holds that
\begin{equation}\label{eq:14}
\textrm{dist}_{{\mathcal X}i}^2\left (\nu^{k+1} ,\bar{{\mathcal T}heta }\right )\leq \alpha\textrm{dist}_{{\mathcal X}i}^2\left (\nu^{k} ,\bar{{\mathcal T}heta }\right ),
\end{equation}
where
$$
\alpha:=(1+\beta)^{-1} \:\:\:\textrm{and}\:\:\:\beta:=(2-\rho)\min \left \{ 1,\frac{1}{2} \sigma ,\sigma \rho ^{-1}(2-\rho)\right \}\left(\lambda ^{2}\kappa\lambda_{\max}({\mathcal X}i) \right)^{-1},
$$
where $\kappa$ is defined in (\ref{kapa}).
Moreover, there exists a positive number $\zeta\in[\alpha,1)$ such that for all $k\geq\bar{\kappa}$
\begin{equation}\label{eq:15}
\textrm{dist}_{{\mathcal X}i}^2\left (\nu^{k+1} ,\bar{{\mathcal T}heta }\right )\leq \zeta\textrm{dist}_{{\mathcal X}i}^2\left (\nu^{k} ,\bar{{\mathcal T}heta }\right ).
\end{equation}
\end{theorem}
\begin{proof}
From the part $(i)$ of Theorem \ref{th:1}, it holds that
\begin{equation*}
\begin{split}
&(\sigma \rho )^{-1}\left \| x_e^k+\sigma (1-\rho ) {\mathcal A}^*y_e^k\right \|^2+\rho ^{-1}\left \| \tilde{y}_e^{k+1} \right \|_{\mathcal S}^2+\rho ^{-1}\left \| \tilde{z_e}^{k+1} \right \|_{\mathcal T}^2+\left ( 2-\rho \right )\sigma \left \| {\mathcal A}^*y_e^k \right \|^2\\[2mm]
&+(2-\rho )\left \| y^k-\tilde{y}^k \right \|_{\mathcal S}^2+\frac{1}{2}(2-\rho )\sigma \left \| \vartheta^*(\nu_e^k)\right \|^2+2\left \| y_e^{k} \right \|_{{\mathcal S}igma _f}^2+2\left \| z_e^{k} \right \|_{{\mathcal S}igma _g}^2\\[2mm]
\geq &(\sigma \rho )^{-1}\left \| x_e^{k+1}+\sigma (1-\rho ) {\mathcal A}^*y_e^{k+1}\right \|^2+\rho ^{-1}\left \| \tilde{y}_e^{k+2} \right \|_{\mathcal S}^2+\rho ^{-1}\left \| \tilde{z_e}^{k+2} \right \|_{\mathcal T}^2+(2-\rho )\left \| y^{k+1}-\tilde{y}^{k+1} \right \|_{\mathcal S}^2\\[2mm]
&+\left ( 2-\rho \right )\sigma \left \| {\mathcal A}^*y_e^{k+1} \right \|^2+2\left \| y_e^{k+1} \right \|_{{\mathcal S}igma _f}^2+2\left \| z_e^{k+1} \right \|_{{\mathcal S}igma _g}^2+\frac{1}{2}(2-\rho )\sigma \left \| \vartheta^*(\nu_e^{k+1})\right \|^2\\[2mm]
&+(2-\rho )\left \| \tilde{y}^{k+1}-y^{k+1} \right \|_{\mathcal S}^2+(2-\rho )\left \| \tilde{z} ^{k+1}-z^{k+1}\right \|_{\mathcal T}^2\\[2mm]
&+\sigma \rho ^{-1}(2-\rho )^2\left \| {\mathcal A}^*(y^{k+1}-y^k) \right \|^2+\frac{1}{2}(2-\rho )\sigma \left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1}\right \|^2,
\end{split}
\end{equation*}
which implies that
\begin{equation}\label{eq:13}
\begin{split}
\left \| \nu_e^k \right \|_{\mathcal X}i ^2\geq &\left \| \nu_e^{k+1} \right \|_{\mathcal X}i ^2+(2-\rho )\left (\left \| \tilde{y}^{k+1}-y^{k+1} \right \|_{\mathcal S}^2+ \left \| \tilde{z} ^{k+1}-z^{k+1}\right \|_{\mathcal T}^2\right.\\[2mm]
&\left.+\sigma \rho ^{-1}(2-\rho )\left \| {\mathcal A}^*(y^{k+1}-y^k) \right \|^2+\frac{1}{2}\sigma \left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1}\right \|^2 \right ).
\end{split}
\end{equation}
Because $\bar{ {\mathcal T}heta }$ is a nonempty closed convex set, we can immediately get (\ref{eq:13}) to the following required result
\begin{equation}\label{eq:9}
\begin{split}
\text{dist}^2_{\mathcal X}i(\nu^{k},\bar{{\mathcal T}heta})\geq &\:\text{dist}^2_{\mathcal X}i(\nu^{k+1},\bar{{\mathcal T}heta})+(2-\rho )\left (\left \| \tilde{y}^{k+1}-y^{k+1} \right \|_{\mathcal S}^2+ \left \| \tilde{z} ^{k+1}-z^{k+1}\right \|_{\mathcal T}^2\right.\\
&\left.+\sigma \rho ^{-1}(2-\rho )\left \| {\mathcal A}^*(y^{k+1}-y^k) \right \|^2+\frac{1}{2}\sigma \left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1}\right \|^2 \right ).
\end{split}
\end{equation}
Observing the structure of right hand side of (\ref{eq:9}) and the definition of ${\mathcal U}psilon(\nu^{k+1})$ in (\ref{eq:8}), we know for all $k\geq1$ and $\rho\in(0,2)$ that
\begin{equation}\label{eq:10}
\begin{split}
&\kappa\left(\left \| \tilde{y}^{k+1}-y^{k+1} \right \|_{\mathcal S}^2+ \left \| \tilde{z} ^{k+1}-z^{k+1}\right \|_{\mathcal T}^2+\sigma \rho ^{-1}(2-\rho )\left \| {\mathcal A}^*(y^{k+1}-y^k) \right \|^2+\frac{1}{2}\sigma \left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1}\right \|^2\right)\\
\geq&\min \left \{ 1,\frac{1}{2} \sigma,\sigma \rho ^{-1}(2-\rho)\right \}{\mathcal U}psilon(\nu^{k+1}).
\end{split}
\end{equation}
According to part $(ii)$ of Theorem \ref{th:1}, we know that the sequence $\left \{ (x^k, y^k, \tilde{y}^k-y^k, z^k, \tilde{z}^k-z^k)\right \}$ converges to $\bar{\nu}^k =(\bar{x}, \bar{y}, 0, \bar{z}, 0)$, which means that there exists $\bar{\kappa}\geq1$ and $\varepsilon>0$ such that for any $k\geq\bar{\kappa}$, it holds that
\[\left \| \nu^{k+1}-\bar{\nu} \right \|\leq \varepsilon. \]
Subsequently, from Assumption \ref{ass2} and Lemma \ref{lemma1}, it gets for all $k\geq \bar{\kappa}$ that
\begin{equation}\label{eq:11}
\text{dist}^2(\nu^{k+1},\bar{{\mathcal T}heta})\leq \lambda ^2\left \| \hat{{\mathcal R}}(\nu^{k+1}) \right \|^2\leq \lambda ^2{\mathcal U}psilon(\nu^{k+1}).
\end{equation}
Combining (\ref{eq:10}) with (\ref{eq:11}) and using the fact $\rho\in(0,2)$, it yields that
\begin{equation}\label{eq:12}
\begin{split}
&(2-\rho)\left(\left \| \tilde{y}^{k+1}-y^{k+1} \right \|_{\mathcal S}^2+ \left \| \tilde{z} ^{k+1}-z^{k+1}\right \|_{\mathcal T}^2\right.\\
&\left.+\sigma \rho ^{-1}(2-\rho )\left \| {\mathcal A}^*(y^{k+1}-y^k) \right \|^2+\frac{1}{2}\sigma \left \| {\mathcal A}^*y_e^{k+1}+{\mathcal B}^*z_e^{k+1}\right \|^2\right)\\
\geq&(2-\rho)\min \left \{ 1,\frac{1}{2} \sigma ,\sigma \rho ^{-1}(2-\rho)\right \}\lambda ^{-2}\kappa ^{-1}\text{dist}^2(\nu^{k+1},\bar{{\mathcal T}heta})\\
\geq&\beta\;\text{dist}^2_{\mathcal X}i(\nu^{k+1},\bar{{\mathcal T}heta}).
\end{split}
\end{equation}
From (\ref{eq:12}), we can derive the assertion (\ref{eq:14}).
Recalling that $\alpha<1$, this readily ensures the fulfillment of linear convergence rate (\ref{eq:15}).
\end{proof}
From Theorem \ref{th2}, we know that the conclusion of local linear convergence rate for Algorithm p-GADMM relies on the calmness property.
As discussed in section \ref{subsec2.2} that, for any set-valued mapping ${\mathcal P}si$, the property of calmness at a certain point holds automatically if ${\mathcal P}si$ is piecewise polyhedral, and particularly, ${\mathcal P}si$ is a subdifferential mapping of a convex piecewise linear-quadratic function.
Based on the fact that ${\mathcal P}si^{-1}$ is piecewise polyhedral if and only if ${\mathcal P}si$ itself is piecewise polyhedral,
we can get that when piecewise polyhedral condition is imposed on the mapping $\hat{{\mathcal R}}(\cdot)$, then the global linear convergence rate of Algorithm p-GADMM can be derived.
\begin{corollary}
Let the sequence $\{(x^k, y^k, z^k; \tilde{x}^k,\tilde{y}^k,\tilde{z}^k) \}$ be generated by Algorithm p-GADMM.
Let $\nu^k :=(x^k, y^k, \tilde{y}^k-y^k, z^k, \tilde{z}^k-z^k)$ and $\rho\in(0,2)$, and let $\bar{\nu} =(\bar{x}, \bar{y}, 0, \bar{z}, 0)\in\bar{{\mathcal T}heta}$ be any limiting point of $\left \{ \nu^{k} \right \}$. Suppose that the solution set to the KKT system (\ref{eq:3}) is nonempty and that ${\mathcal S}$ and ${\mathcal T}$ are positive definite. Besides, suppose that the mapping $\hat{{\mathcal R}}(\cdot)$ is piecewise polyhedral. Then, there exists a constant $\hat{\lambda }>0$ such that for all $k\geq 1$ ,
\begin{equation}\label{eq:37}
\textrm{dist}\left ( \nu^k,\bar{{\mathcal T}heta } \right )\leq \hat{\lambda } \| {\mathcal R}\left ( \nu^k \right ) \|,
\end{equation}
and
\begin{equation}\label{eq:38}
\textrm{dist}_{{\mathcal X}i}^2\left (\nu^{k+1} ,\bar{{\mathcal T}heta }\right )\leq \hat{\alpha}\textrm{dist}_{{\mathcal X}i}^2\left (\nu^{k} ,\bar{{\mathcal T}heta }\right ),
\end{equation}
where
$$
\hat{\alpha}:=(1+\hat{\beta})^{-1} \:\:\:\textrm{and}\:\:\:\hat{\beta}:=(2-\rho)\min \left \{ 1,\frac{1}{2} \sigma ,\sigma \rho ^{-1}(2-\rho)\right \}\left(\hat{\lambda} ^{2}{\kappa}\lambda_{\max}({\mathcal X}i) \right)^{-1}.
$$
\end{corollary}
\begin{proof}
On the one hand, from the nonempty set $\bar{{\mathcal T}heta}$ and the piecewise polyhedral condition, there exist fixed $\lambda>0$ and $\delta>0$ such that
$$
\textrm{dist}(\nu^k,\bar{{\mathcal T}heta})\leq\lambda \| \hat{{\mathcal R}}(\nu^k) \|
$$
if $ \| \hat{{\mathcal R}}(\nu^k)\| \leq \delta$ .
On the other hand, following the proof of Theorem \ref{th2}, for all $k\geq1$, we know that $ \{\nu^k:= (x^k, y^k, \tilde{y}^k-y^k, z^k, \tilde{z}^k-z^k) \}$ converges to $\bar{\nu} =(\bar{x}, \bar{y}, 0, \bar{z}, 0)$ while $\left \| \nu^k-\bar{\nu} \right \|\leq \varepsilon$ with constant $\varepsilon>0$. For the $\nu^k$ satisfying $\| \hat{{\mathcal R}}(\nu^k) \|>\delta$, we get
\[\textrm{dist}(\nu^k, \bar{{\mathcal T}heta })\leq\left \| \nu^k-\bar{\nu} \right \|\leq\varepsilon<\varepsilon\delta^{-1} \| \hat{{\mathcal R}}(\nu^k) \|.\]
We readily obtain that there exists a positive number $\hat{\lambda}:=\max\left \{ \lambda, \varepsilon\delta^{-1} \right \}$ such that (\ref{eq:37}) holds.
Employing a similar proof of Theorem \ref{th2} yields the inequality (\ref{eq:38}), thereby the global linear convergence rate holds.
\end{proof}
At the end of this section, we notice that the assumption on $\hat{{\mathcal R}}$ implies that the condition $\textrm{dist}\left ( \nu^k,\bar{{\mathcal T}heta } \right )\leq \hat{\lambda } \| {\mathcal R}\left ( \nu^k \right ) \|$ hold automatically. Therefore, the global linear convergence rate can be achieved.
\section{Conclusion}\label{sec4}
We know that the method of p-GADMM proposed by Xiao, Chen \& Li \cite{xiao2018generalized} is highly efficient for convex composite programming problems. The global convergence of p-GADMM is known, but its convergence rate is worthy of exploring.
This paper was devoting to provide a theoretical analysis and proved that p-GADMM has Q-linear convergence rate under the assumption that the KKP mapping is calm.
This conclusion is consistent with the theoretical result of Han et al. \cite{Han2018LinearRC} to the semi-proximal ADMM of Fazel et al. \cite{Fazel2013HankelMR}.
Nevertheless, it is still worth emphasizing that Theorem \ref{th2} requires ${\mathcal S}$ and ${\mathcal T}$ being positive definite, which is slightly stronger than the one in \cite{Han2018LinearRC}. Despite this, we believe that this condition will not affect the contribution of this paper because it actually common in vast majority of practical implementations.
\section{Acknowledgments}
The work of Y. Xiao is supported by the National Natural Science Foundation of China (Grant
No. 11971149).
\end{document} |
\begin{eqnarray}gin{document}
\title{Quantum weak coin flipping with arbitrarily small bias}
\author{Carlos Mochon\thanks{Perimeter Institute for Theoretical Physics,
[email protected]}}
\date{November 26, 2007}
\maketitle
\begin{eqnarray}gin{abstract}
``God does not play dice. He flips coins instead.'' And though for some
reason He has denied us quantum bit commitment. And though for some reason
he has even denied us strong coin flipping. He has, in His infinite mercy,
granted us quantum weak coin flipping so that we too may flip coins.
Instructions for the flipping of coins are contained herein. But be warned!
Only those who have mastered Kitaev's formalism relating coin flipping and
operator monotone functions may succeed. For those foolhardy enough to even
try, a complete tutorial is included.
\end{abstract}
\setcounter{tocdepth}{2}
{\small \tableofcontents}
\section{Introduction}
It is time again for a sacrifice to the gods. Alice and Bob are highly
pious and would both like the honor of being the victim. Flipping a coin to
choose among them allows the gods to pick the worthier candidate for this
life changing experience. To keep the unworthy candidate from desecrating
the winner, the coin flip must be carried out at a distance and in a manner
that prevents cheating. Very roughly speaking, this is the problem known as
\textit{coin flipping by telephone} \cite{Blum}. Quantum coin flipping is a
variant of the problem where the participants are allowed to communicate
using quantum information.
Why is quantum coin flipping interesting/important/useful? First of all, it
is conceivably possible that someday somewhere someone will want to
determine something by flipping a coin with a faraway partner, and that for
some reason both will have access to quantum computers. The location of QIP
2050 could very well be determined in this fashion.
Secondly, coin flipping belongs to a class of cryptographic protocols known
as secure two-party computations. These arise naturally when two people
wish to collaborate but don't completely trust one another. Sadly, the
impossibility of quantum bit commitment \cite{May96,Lo:1998pn} shows that
quantum information is incapable of solving many of the problems in this
area. On the other hand, the possibility of quantum weak coin flipping
shows that quantum information may yet have untapped potential. Among the
most promising open areas is secure computation with cheat detection
\cite{Aharonov00,Hardy99} which may be better explored with the techniques
in this paper. At a minimum, the standard implementation of bit commitment
with cheat detection uses quantum weak coin flipping as a subroutine, so
improvements in the latter offer (modest) improvements in the former.
Finally, coin flipping is interesting because it appears to be hard, at
least relative to other cryptographic tasks such as key distribution
\cite{Wie83,BB84}. Of course, it is not our intent to belittle
the discovery of key distribution, whose authors had to invent many of the
foundations of quantum information along the way. But a savvy student
today, familiar with the field of quantum information, would likely have no
trouble in constructing a key distribution protocol. Most reasonable
protocols appear to work. Not so with coin flipping, where most obvious
protocols appear to fail.
A cynical reader may argue that hardness is relative and may simply be a
consequence of having formulated the problem in the wrong language. But
this is exactly our third point: that the difficulty of coin flipping is
really an opportunity to develop a formalism in which such problems are
(relatively) easily solvable.
This new formalism, which is the cornerstone of the protocols in this
paper, was developed by Kitaev \cite{Kit04} and can be used to relate
coin flipping (and many other quantum games) to the theory of convex cones
and operator monotone functions. From this perspective, the value of the
present coin flipping result is that it provides the first demonstration
of the power of Kitaev's formalism.
We note that the formalism relating coin-flipping and operator monotone
functions is an extension of Kitaev's original formalism which was used in
proving a lower bound on strong coin flipping \cite{Kitaev}. When we need
to distinguish them, we shall refer to them respectively as Kitaev's second
and first coin flipping formalisms.
We will delay the formal definition of coin flipping to
Section~\ref{sec:def} and the history and prior work on the problem to
Section~\ref{sec:hist}. Instead, we shall give below an informal
description of the new formalism. We shall focus more on what the finished
formalism looks like rather than on how to relate it to the more
traditional notions of quantum states and unitaries (a topic which will be
covered at length later in the paper).
In its simplest form Kitaev's formalism can be described as a sequence of
configurations, each of which consists of a few marked points on the
plane. The points are restricted to the closure of the first quadrant
(i.e., have non-negative coordinates) and each point carries a positive
weight, which we call a probability.
Two successive configurations can only differ by points on a single
vertical or horizontal line. The rule is that the total probability on the
line must be conserved (though the total number of points can change) and
that for every $\lambda\in(0,\infty)$ we must satisfy
\begin{eqnarray}
\sum_z \frac{\lambda z}{\lambda + z}p_z\leq \sum_{z'}
\frac{\lambda z'}{\lambda + z'}p_{z'},
\label{eq:constraint}
\end{eqnarray}
\noindent
where the left hand side is a sum over points before the transition and the
right hand side is a sum over points after the transition. The variable $z$
is respectively the $x$ coordinate for transitions occurring on a
horizontal line or the $y$ coordinate for transitions occurring on a
vertical line. The numbers $p_z$ are just the probabilities associated to
each point. An example of such a sequence is given in Fig.~\ref{fig:intro}.
\begin{eqnarray}gin{figure}[tb]
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\qquad
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\qquad
\begin{eqnarray}gin{picture}(2.5,2.3)(-0.5,-0.5)
\put(0,0){\line(1,0){1.7}}
\put(0,0){\line(0,1){1.7}}
\put(0.5,0){\line(0,-1){0.05}}
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\put(0.65,1.15){\makebox(0,0)[l]{$1$}}
\thicklines
\put(0.5,1){\makebox(0,0){$\bullet$}}
\end{picture}
\caption{A point game sequence with three configurations.
Numbers outside the axes label location and numbers inside the axes label
probability. The sequence corresponds to a protocol with $P_A^*=1$ and
$P_B^*=1/2$, that is, a protocol where Alice flips a coin and announces the
outcome.}
\label{fig:intro}
\end{center}
\end{figure}
The boundary conditions of the sequence are as follows: The stating
configuration always contains two points, each carrying probability one
half, with one point located at $x=1$ and $y=0$ and the other point at
$x=0$ and $y=1$. The final configuration must contain a single point, which
carries unit probability (as required by conservation of probability).
Each of these sequences can be translated into coin-flipping protocols such
that the amount of cheating allowed is bounded by the location of the final
point. In particular, if the final point is located at $(x,y)$ then the
resulting protocol will satisfy $P_A^*\leq y$ and $P_B^*\leq x$, and hence
the bias is bounded by $\max(x,y)-1/2$.
We call these sequences ``point games'' and they are completely equivalent
to standard protocols described by unitaries. There exists constructive
mappings from point games to standard protocols and vice versa. The optimal
coin-flipping protocol can be constructed and tightly bounded by a point
game. Hence, rather than searching for optimal protocols, one can
equivalently search for optimal point games. These point games are
formalized in Section~\ref{sec:Kit} and examples are given in
Section~\ref{sec:examples}.
The configurations above are roughly related to standard semidefinite
programing objects as follows: The $x$ coordinates are the eigenvalues of
the dual SDP operators on Alice's Hilbert space, the $y$ coordinates are
eigenvalues of the dual SDP operators on Bob's Hilbert space, and the
weights are the probabilities assigned by the honest state to each of these
eigenspaces.
The obscure condition of Eq.~(\ref{eq:constraint}) can best be understood
if we describe the points on the line before and after the transition by
functions $p(z),p'(z):[0,\infty)\rightarrow[0,\infty)$ with finite support.
We then are essentially requiring that $p'(z)-p(z)$ belong to the cone dual
to the set of operator monotone functions with domain $[0,\infty)$. For
those unfamiliar with operator monotone functions, their definition and a
few properties are discussed later in the paper.
We note an unusual convention that was used above and throughout most of
the paper: the description of point games follows a \textbf{reverse time
convention} where the final measurement occurs at $t=0$ and the initial
state preparation occurs at $t=n>0$. The motivation for this will become
clear as the formalism is developed.
Kitaev further simplified his formalism so that an entire point game can be
described by a single pair of functions $h(x,y)$ and $v(x,y)$ that take
real values and have finite support. The main constraint is that on every
horizontal line of $h(x,y)$ and every vertical line of $v(x,y)$, the sum of
the weights must be zero, and the weighted average of $\frac{\lambda
z}{\lambda + z}$ must be non-negative for every
$\lambda\in(0,\infty)$. Furthermore, $h(x,y)+v(x,y)$ must be zero
everywhere except at three points: $(1,0)$ and $(0,1)$ where it has value
$-1/2$, and a third point $(x,y)$, where it has value $1$, and which is the
equivalent of the final point of the original point games. This variant of
point games is described in Section~\ref{sec:Kit2}.
A simple example can be constructed using Fig.~\ref{fig:introlad16}. The
labeled points with outgoing horizontal arrows appear in $h(x,y)$ with
negative sign, whereas those with incoming horizontal arrows appear in
$h(x,y)$ with positive sign. Similarly for $v(x,y)$ and the vertical
arrows. The final point at $(\frac{2}{3},\frac{2}{3})$ appears in both
functions with positive coefficient and magnitude $1/2$. That means that
the point game corresponds to a protocol with $P_A^*=P_B^*=2/3$, or bias
$1/6$, and is a variant of the author's previous best protocol
\cite{me2005}.
\begin{eqnarray}gin{figure}[tb]
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\put(7.6,7.6){\makebox(0,0){$\cdot$}}
\put(7.7,7.7){\makebox(0,0){$\cdot$}}
\put(7.8,7.8){\makebox(0,0){$\cdot$}}
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\end{picture}
\caption{A coin-flipping protocol with bias $1/6$.}
\label{fig:introlad16}
\end{center}
\end{figure}
The power of Kitaev's formalism is evident from the previous example as a
complete protocol can be described by a single
picture. Section~\ref{sec:zero} discusses in detail how to build and
analyze such structures. Among the issues addressed are how to truncate the
above infinite ladder so that the resulting figure has only a finite number
of points as required by our description of Kitaev's formalism.
To achieve zero bias in coin flipping, one can use similar constructions,
but with more complicated ladders heading off to infinity. In particular,
for every integer $k\geq 0$ we will build a protocol with
\begin{eqnarray}
P_A^*=P_B^*=\frac{k+1}{2k+1}
\end{eqnarray}
\noindent
(technically, for each $k$ we will have a family of protocols that will
converge to the above values). The case $k=0$ allows both players to
maximally cheat, the case $k=1$ is the author's bias $1/6$ protocol, and
the limit $k\rightarrow \infty$ achieves arbitrarily small bias. The
details of this construction can also be found in Section~\ref{sec:zero}.
All the new protocols are formulated in the language of Kitaev's
formalism. Sadly, mechanically transforming these protocols back into the
language of unitaries, while possible, does not lead to particularly simple
or efficient protocols (i.e., in terms of laboratory resources). Finding
easy to implement protocols with a small bias remains an interesting open
problem. A number of other open problems can be found at the end of
Section~\ref{sec:end}.
A first stab at finding good easy to implement protocols is given by
Appendix~\ref{sec:ddb}. The section transforms the author's original bias
$1/6$ protocol (which uses a number of qubits linear in the number of
messages) into a new form that uses constant space. In fact, the total
space needed is one qutrit for each of Alice and Bob, and one qubit used to
send messages.
The key idea is to use early measurements to prune states that are known to
be illegal. While in a theoretical sense measurements can always be delayed
to the last step, their frequent use can provide practical simplifications
(as is well know in key distribution). In fact, all the early measurements
are of the flying qubit in the computational basis.
The protocol in Appendix~\ref{sec:ddb} is described in the standard
language of unitaries, and an analysis is sketched using Kitaev's first
formalism. As a bonus, the resulting protocol is related to the ancient and
most holy game of Dip-Dip-Boom, also described therein.
Appendix~\ref{sec:strong} proves strong duality for coin flipping, which is
an important lemma needed for both Kitaev's first and second
formalisms. While strong duality does not hold in general semidefinite
programs, it does in most, and there exists a number of lemmas that provide
sufficient conditions. Unfortunately, some of the simplest lemmas do not
directly apply to coin-flipping. Instead, the appendix directly proves
strong duality using simple arguments from Euclidean geometry. While all
the ideas in this section are taken from standard textbooks, the
presentation is still somewhat clever and novel.
Finally, Appendix~\ref{sec:f2m} proves another mathematical lemma needed
for Kitaev's second formalism. It is the key step needed to turn the
functions that underlie the point games back into matrices out of which
states and unitaries can be constructed. Though some of the ideas are
potentially novel, mostly it deals with standard technical issues from the
theory of matrices.
As a final goody, Section~\ref{sec:cheat} includes a brief discussion and
example of how to extend the formalism to include cheat detection. Of
course, because weak coin-flipping can be achieved with arbitrarily small
bias, adding in cheat detection isn't particularly useful. However, similar
techniques may prove helpful in studying cheat detection for strong coin
flipping and other secure computation problems.
\textit{Author's note:} Sections~\ref{sec:Kit} and~\ref{sec:Kit2} are based
on my recollection of a couple of discussions with Kitaev and a subsequent
group meeting he gave (of which I sadly kept no written record). As I have
had to reconstruct some of the details, and as I have strived to move the
discussion to finite dimensional spaces, some of Kitaev's original elegance
has been replaced by a more pedantically constructive (and hopefully
pedagogical) approach. I claim no ownership of the main ideas in these
sections, though am happy to accept the blame for any errors in my write
up. You should also know that all the terms such as UBP, ``point game,''
and ``valid transitions'' are my own crazy invention, and are unlikely to
be familiar to those who have leaned Kitaev's formalism from other sources.
At this point, those familiar with the definition and history of coin
flipping may wish to skip ahead to Section~\ref{sec:Kit}. Good luck!
\subsection{\label{sec:def}Coin flipping defined}
Coin flipping is a formalization of the notion of flipping or tossing a
coin under the constraints that the participants are mutually distrustful
and far apart.
The two players involved in coin flipping, traditionally called Alice and
Bob, must agree on a single random bit which represents the outcome of the
coin flip. As Alice and Bob do not trust each other, nor anyone else, they
each want a protocol that prevents the other player from
cheating. Furthermore, because they are far apart, the protocol must be
implementable using only interaction over a communication device such as a
telephone. The problem is known in the classical literature as
``coin flipping by telephone'' and was first posed by Manuel Blum in 1981
\cite{Blum}.
There are two variants of coin flipping. In the first variant, called weak
coin flipping, Alice and Bob each have a priori a desired coin outcome.
The outcomes can be labeled as ``Alice wins'' and ``Bob wins,'' and we
do not care if the players cheat in order to increase their own probability
of losing. In the second variant of coin flipping, called strong coin
flipping, there are no a priori desired outcomes and we wish to prevent
either player from biasing the coin in either direction.
Obviously, strong coin flipping is at least as hard as weak coin flipping
and in general it is harder. However, this paper is mainly concerned with
weak coin flipping which we often simply refer to as ``coin flipping''.
To be more precise in our definition, weak coin flipping is a two-party
communication protocol that begins with a completely uncorrelated state and
ends with each of the participants outputting a single bit. We say that
Alice wins on outcome 0 and Bob wins on outcome 1. The requirements are:
\begin{eqnarray}gin{enumerate}
\item When both players are honest, Alice's output is uniformly random and
equal to Bob's output.
\item If Alice is honest but Bob deviates from the protocol, then no matter
what Bob does, the probability that Alice outputs one (i.e., Bob wins) is
no greater than $P_B^*$.
\item Similarly, if Bob is honest but Alice deviates from the protocol,
then the maximum probability for Bob to declare Alice the winner is $P_A^*$.
\end{enumerate}
\noindent
The parameters $P_A^*$ and $P_B^*$ define the protocol. Ideally we want
$P_A^*=P_B^*=1/2$. Unfortunately, this is not always possible. We therefore
introduce the bias $\max(P_A^*,P_B^*)-1/2$ as a measure of the security of
the protocol. Our goal is to find a protocol with the smallest bias
possible.
Note that the protocol places no restrictions on the output of a cheating
player, as these are impossible to enforce. In particular, when one player
is cheating the outputs do not have to agree, and when both players are
cheating the protocol is not required to satisfy any properties. This also
means that if an honest player ever detects that their opponent has
deviated from the protocol (i.e., the other player stops sending messages
or sends messages of the wrong format) then the honest player can simply
declare victory rather than aborting. This will be an implicit rule in all
our weak coin-flipping protocols.
Occasionally, it is worth extending the definition of coin flipping to case
where the output is not uniformly random even when both players are
honest. In such a case we denote by $P_A$ the honest probability for Alice
to win and by $P_B=1-P_A$ the honest probability for Bob to win.
\subsubsection{Communication model}
It is not hard to see, that in a classical world, and without any further
assumptions, at least one player can guarantee victory. For instance, if
one of the players were in charge of flipping the coin, the other player
would have no way of verifying via a telephone that the outcome of the coin
is the one reported by the first player.
Coin flipping can be achieved in a classical setting by adding in certain
computational assumptions \cite{Blum}. However, some of these assumptions
will no longer be true once quantum computers become available. Coin
flipping can also be achieved in a relativistic setting \cite{Ken99}
if Alice and Bob's laboratories are assumed to satisfy certain spatial
arrangements. However, these requirements may not be optimal for today's
on-the-go coin flippers.
In this paper we shall focus on the quantum setting, where Alice and Bob
each have a quantum computer with as much memory as needed, and are
connected by a noiseless quantum channel. They are each allowed to do
anything allowed by the laws of quantum mechanics other than directly
manipulate their opponents qubits. The resulting protocols will have
information theoretic security.
Although at the moment such a setting seems impractical, if ever
quantum computers are built and are as widely available as classical
computers are today, then quantum weak coin flipping may become viable.
\subsubsection{On the starting state}
The starting state of coin-flipping protocols is by definition completely
uncorrelated, which means that Alice and Bob initially share neither
classical randomness nor quantum entanglement (though they do share a
common description of the protocol).
There are two good reasons for this definition. First, it is easy to see
that given a known maximally entangled pair of qubits, Alice and Bob could
obtain a correlated bit without even using communication. But the same
result can be obtained when starting with a uniformly distributed shared
classical random bit. Such protocols are trivial, and certainly do not
require the power of quantum mechanics. The purpose of coin flipping,
though, is to create these correlations.
Still, at first glance it would appear that by starting with correlated
states we can put the acquisition of randomness, or equivalently the
interaction with a third party, in the distant past. In such a model Alice
and Bob would buy a set of correlated bits from their supermarket and then
used them when needed. The problem is that they can now figure out the
outcomes of the coin flips before committing to them. It is not hard to
imagine that a cheater would have the power to order the sequence of events
that require a coin flip so that he wins on the important ones and loses
the less important ones. We would now have to worry about protecting the
ordering of events and that is a completely different problem.
A good one-shot coin-flipping protocol should not allow the players to
predict the outcome of the coin flip before the protocol has begun, and
enforcing this is the second reason that we require an uncorrelated stating
state.
It might be interesting to explore what happens when the no-correlation
requirement is weakened to a no-prior-knowledge-of-outcome requirement, but
that is beyond the scope of this paper.
\subsubsection{On the security guarantees}
The security model of coin flipping divides the universe into three parts:
Alice's laboratory, Bob's laboratory and the rest of the universe. We
assume that Alice and Bob each have exclusive and complete control over
their laboratories. Other than as a conduit for information between Alice
and Bob, the rest of the universe will not be touched by honest
players. However, a dishonest player may take control of anything outside
their opponent's laboratory, including the communication channel.
The security of the protocols therefore depends on the inability of a
cheater to tamper with their opponent's laboratory. What does this mean?
Abstractly, it can be defined as
\begin{eqnarray}gin{enumerate}
\item
All quantum superoperators that a cheater can
apply must act as the identity on the part of the Hilbert space that is
located inside the laboratory (including the message space when
appropriate).
\item
All operations of honest players are performed flawlessly and without
interference by the cheater.
\item
An honest player can verify that an incoming message has the right
dimension and can abort otherwise.
\end{enumerate}
\noindent
In practice, however, this translates into requirements such as
\begin{eqnarray}gin{enumerate}
\item
The magnetic shielding on the laboratory is good enough to prevent your
opponent from affecting your qubits.
\item
The grad student operating your machinery cannot be bribed to apply the
wrong operations.
\item
A nanobot cannot enter your laboratory though the communication channel.
\end{enumerate}
\noindent
As usual, the fact that a protocol is secure does not mean that it will
protect against the preceding attacks. The purpose of the security analysis
is to prove that the only way to cheat is to attack an opponents
laboratory, thereby guaranteeing that one's security is as good as the
security of one's laboratory.
\subsubsection{On the restriction to unitary operations}
It is customary when studying coin flipping to consider only protocols that
involve unitary operations with a single measurement at the end. It is also
customary to only consider cheating strategies that can be implemented
using unitary operations.
Nevertheless, any bounds that are derived under such conditions apply to
the most general case which includes players that can use measurements,
superoperators and classical randomness, and protocols that employ
extra classical channels.
The above follows from two separate lemmas, which roughly can be stated as:
\begin{eqnarray}gin{enumerate}
\item Given any protocol $P$ in the most general setting (including
measurements, classical channels, etc.) that has a maximum
bias $\epsilon$ under the most general cheating strategy (including,
measurements, superoperators, etc.) then there exists a second protocol
$P'$ that is specified using only unitary operations with a single
pair of measurements at the end and that also has a maximum
bias $\epsilon$ under the most general cheating strategy (including,
measurements, superoperators, etc.).
\item Given any protocol $P$ specified using only unitary operations with a
single pair of measurements at the end, and given any cheating strategy for
$P$ (which may include measurements, superoperators, etc) that achieves a
bias $\epsilon$, we can find a second cheating strategy for $P$ that also
achieves a bias $\epsilon$ but can be implemented using only unitary
operations.
\end{enumerate}
\noindent
Unfortunately neither of the above statements will be proven here, and the
proofs for the above statements are distributed among a number of published
papers, but good stating points are \cite{Lo:1998pn} and \cite{May96}.
The first statement implies that we need only consider protocols where
the honest actions can be described as a sequence of unitaries. The result
is important for proofs of lower-bounds on the bias, but is not needed for
the main result of this paper.
The reduction of the first statement also applies to protocols that
potentially have an infinite number of rounds, such as rock-paper-scissors,
where measurements are carried out at intermediate steps to determine if a
winner can be declared or if the protocol must go on. Such protocols are
dealt with by proving that there exist truncations that approximate the
original protocol arbitrarily well. In such a case, though, the resulting
bias will only come arbitrarily close to the original bias.
The second statement implies that in our search for the optimal cheating
strategy (or equivalently, in our attempts to upper bound the bias) we need
only consider unitary cheaters. Note that measurements are not even needed
in the final stage as we do not care about the output of dishonest players.
The basic idea of the proof of the second statement is that any allowed
quantum mechanical operation can be expressed as a unitary followed by the
discarding of some Hilbert subspace. If the cheater carries out his
strategy without ever discarding any such subspaces he will not reduce his
probability of victory and will only need to use unitary operations.
The second statement also holds when the honest protocol includes certain
projective measurements such as those used in this paper. In fact, we will
sketch a proof for this second statement in Section~\ref{sec:primal} as we
translate coin flipping into the language of semidefinite programing.
\subsection{\label{sec:hist}A brief history muddled by hindsight}
Roughly speaking, cryptography is composed of two fundamental problems: two
honest players try to complete a task without being disrupted by a third
malicious party (i.e., encrypted communication), and two mutually
distrustful players try to cooperate in a way that prevents the opposing
player from cheating, effectively simulating a trusted third party (i.e.,
choosing a common meeting time while keeping their schedules private).
The second case is commonly known as two-party secure computation.
In the mid 1980s, quantum information had a resounding success in the first
category by enabling key distribution with information theoretic security
\cite{Wie83,BB84}. Optimism was high, and it seemed like quantum
information would be able to solve all problems in the second category as
well. But after many failed attempts at producing protocols for a task
known as bit commitment, it was finally proven by Mayers, Lo and Chau
\cite{May96, Lo:1998pn} that secure quantum bit commitment was impossible.
One of the reasons people had focused on bit commitment is that it is a
powerful primitive from which all other two-party secure computation
protocols can be constructed \cite{Yao95}. Its impossibility means that all
other universal primitives for two-party secure computation must also be
unrealizable using quantum information \cite{Lo97}.
We note that when we speak of possible and impossible with regards to
quantum information we always mean with information theoretic security
(i.e., without placing bounds on the computational capacity of the
adversary). Classically all two-party secure-computation tasks can be
realized under certain complexity assumptions \cite{Yao82} but become
impossible if we demand information theoretic security. Surprisingly, most
multiparty secure-computations tasks can be done classically with
information theoretic security so long as all parties share private
pairwise communication channels and the number of cheating players is
bounded by some constant fraction (dependent on the exact model) of the
total number of players \cite{CCD88,BGW88,RB89}. Similar results hold
in the multiparty quantum case \cite{GGS02}.
Given the impossibility of quantum bit commitment, and the fact that most
multiparty problems can already be solved with classical information, the
new goal in the late 1990s became to find any two-party task that is
modestly interesting and can be realized with information theoretic
security using quantum information.
One of the problems that the literature converged on \cite{Goldenberg:1998bx}
was a quantum version of the problem of flipping a coin over the
telephone \cite{Blum}. Initially the focus was on strong coin flipping and
Ambainis \cite{Ambainis2002} and Spekkens and Rudolph \cite{Spekkens2001}
independently proposed protocols that achieve a bias of
$1/4$. Unfortunately, shortly thereafter Kitaev \cite{Kitaev} (see also
\cite{Ambainis2003}) proved a lower bound of $1/\sqrt{2}-1/2$ on the bias.
Research continued on weak coin flipping \cite{ker-nayak, Spekkens2002,
Spekkens2003} and the best known bias prior to the author's own work was
$1/\sqrt{2}-1/2\simeq 0.207$ by Spekkens and Rudolph \cite{Spekkens2002}.
The best lower bound was proven by Ambainis \cite{Ambainis2002} and states
that the number of messages must grow at least as $\Omega(\log \log
\frac{1}{\epsilon})$. In particular, it implies that no protocol with a
fixed number of messages can achieve an arbitrarily small bias.
At this point most known protocols used at most a few rounds of
communication. The first non-trivial many-round coin-flipping protocol was
published in \cite{me2004} by the author and achieved a bias of $0.192$ in
the limit of arbitrarily many messages. In subsequent work \cite{me2005} it
was shown that this protocol (and many of the good protocols known at the
time) were part of a large family of quantized classical public-coin
protocols. Furthermore, an analytic expression was given for the bias of
each protocol in the family, and the optimal protocol for each number of
messages was identified. Sadly, the best bias that can be achieved in this
family is $1/6$, and this only in the limit of arbitrarily many messages (a
new formulation of this bias $1/6$ protocol can be found in
Appendix~\ref{sec:ddb}, wherein we use early measurements to reduce the
space needed to run the protocol to a qutrit per player and a qubit for
messages).
Independently, Kitaev created a new formalism for studying two player
adversarial games such as coin flipping
\cite{Kit04}, which built on his earlier work \cite{Kitaev}. The formalism
describes the set of possible protocols as the dual to the cone of two
variables functions that are independently operator monotone in each
variable. Though the result was never published, we include in
Sections~\ref{sec:Kit} and \ref{sec:Kit2} a description of the
formalism. This formalism, which we shall refer to as Kitaev's second
coin-flipping formalism, is the crucial idea behind the results in the
present paper. A different extension of Kitaev's original formalism was
also proposed by Gutoski and Watrous \cite{GW07}.
Looking beyond coin flipping, there is the intriguing possibility that we
can still achieve most of protocols of two-party secure computation if we
are willing to loosen our requirements: instead of requiring that cheating
be impossible, we require that a cheater be caught with some non-zero
probability. Quantum protocols that satisfy such requirements for bit
commitment have already been constructed \cite{Aharonov00, Hardy99} though
the amount of potential cheat detection is known to be bounded
\cite{me2003-2}.
The possibility of some interesting quantum two-party protocols with
information theoretic security, plus many more protocols built using cheat
detection, may mean that ultimately quantum information will fulfill its
potential in the area of secure computation. But more work needs to be done
in this direction, and we hope that the results and techniques of the
present paper will be helpful.
\section{\label{sec:Kit}Kitaev's second coin-flipping formalism}
The goal of this section is to describe Kitaev's formalism which relates
coin flipping to the dual cone of a certain set of operator monotone
functions \cite{Kit04}.
The first step in the construction involves formalizing the problem of
coin flipping and proving the existence of certain upper bound certificates
for $P_A^*$ and $P_B^*$. This is done in Section~\ref{sec:kit1} and we
refer to the result as Kitaev's first coin-flipping formalism as most of
the material was used in the construction of the lower bounds on strong
coin flipping \cite{Kitaev}.
The next step, carried out in Section~\ref{sec:UBP}, involves using these
certificates to change the maximization over cheating strategies to a
minimization over certificates, and overall to transform the problem of
finding the best coin-flipping protocol into a minimization over objects we
call upper-bounded protocols (UBPs).
The third step involves stripping away most of the irrelevant information
of the UBPs to end up with a sequence of points moving around in the
plane. These ``point games'' are the main object of study of Kitaev's
second formalism. They come in two varieties: time dependent (which are
studied in Section~\ref{sec:TDPG}) and time independent (whose description
is delayed to Section~\ref{sec:TIPG}).
\subsection{\label{sec:kit1}Kitaev's first coin-flipping formalism}
The first goal is to formalize coin-flipping protocols using the standard
quantum communication model of a sequence of unitaries with measurements
delayed to the end.
\begin{eqnarray}gin{definition}
A \textbf{coin-flipping protocol} consists of the following data
\begin{eqnarray}gin{itemize}
\item ${\mathcal{A}}$, ${\mathcal{M}}$ and ${\mathcal{B}}$, three finite-dimensional Hilbert spaces
corresponding to Alice's qubits, the message channel and Bob's qubits
respectively. We assume that each Hilbert space is equipped with a
orthonormal basis of the form $\ket{0},\ket{1},\ket{2},\dots$ called
the computational basis.
\item $n$, a positive integer describing the number of messages.\\
For simplicity we shall assume $n$ is even.
\item A tensor product initial state:
$\ket{\psi_0} =
\ket{\psi_{A,0}}\otimes\ket{\psi_{M,0}}\otimes\ket{\psi_{B,0}}
\in{\mathcal{A}}\otimes{\mathcal{M}}\otimes{\mathcal{B}}$.
\item A set of unitaries $U_1,\dots,U_n$ on ${\mathcal{A}}\otimes{\mathcal{M}}\otimes{\mathcal{B}}$
of the form
\begin{eqnarray}
U_i = \begin{eqnarray}gin{cases}
U_{A,i}\otimes I_{\mathcal{B}} & \text{for $i$ odd,}\\
I_{\mathcal{A}}\otimes U_{B,i} & \text{for $i$ even,}
\end{cases}
\end{eqnarray}
\noindent
where $U_{A,i}$ acts on ${\mathcal{A}}\otimes{\mathcal{M}}$ and $U_{B,i}$ acts on ${\mathcal{M}}\otimes{\mathcal{B}}$.
\item $\left\{\Pi_{A,0},\Pi_{A,1}\right\}$, a POVM on ${\mathcal{A}}$.
\item $\left\{\Pi_{B,0},\Pi_{B,1}\right\}$, a POVM on ${\mathcal{B}}$.
\end{itemize}
Furthermore, the above data must satisfy
\begin{eqnarray}
\Pi_{A,1} \otimes I_{{\mathcal{M}}} \otimes \Pi_{B,0} \ket{\psi_n} =
\Pi_{A,0} \otimes I_{{\mathcal{M}}} \otimes \Pi_{B,1} \ket{\psi_n} = 0,
\label{eq:povmreq}
\end{eqnarray}
\noindent
where $\ket{\psi_n} = U_{n} \cdots U_{1} \ket{\psi_0}$.
\end{definition}
Given the above data, the protocol is run as follows:
\begin{eqnarray}gin{enumerate}
\item Alice starts with ${\mathcal{A}}$ and Bob starts with ${\mathcal{M}}\otimes{\mathcal{B}}$.
They initialize their state to $\ket{\psi_0}$.
\item For $i=1$ to $n$:\\
If $i$ is odd Alice takes ${\mathcal{M}}$ and applies $U_{A,i}$.\\
If $i$ is even Bob takes ${\mathcal{M}}$ and applies $U_{B,i}$.
\item Alice measures ${\mathcal{A}}$ with $\left\{\Pi_{A,0},\Pi_{A,1}\right\}$
and Bob measures ${\mathcal{B}}$ with $\left\{\Pi_{B,0},\Pi_{B,1}\right\}$.
They each output zero or one based on the outcome of the measurement.
\end{enumerate}
\noindent
When both Alice and Bob are honest, the above protocol starts off with the
state $\ket{\psi_0}$ and proceeds through the states
\begin{eqnarray}
\ket{\psi_i} = U_{i} \cdots U_{1} \ket{\psi_0}.
\label{eq:psik}
\end{eqnarray}
The final probabilities of winning are given by
\begin{eqnarray}
P_A &=& \left | \Pi_{A,0} \otimes I_{{\mathcal{M}}} \otimes \Pi_{B,0} \ket{\psi_n} \right|^2
= \Tr\left[ \Pi_{B,0} \Tr_{{\mathcal{A}}\otimes{\mathcal{M}}} \ket{\psi_n}\bra{\psi_n} \right],
\nonumber\\
P_B &=& \left | \Pi_{A,1} \otimes I_{{\mathcal{M}}} \otimes \Pi_{B,1} \ket{\psi_n} \right|^2
= \Tr\left[ \Pi_{A,1} \Tr_{{\mathcal{M}}\otimes{\mathcal{B}}} \ket{\psi_n}\bra{\psi_n} \right],
\label{eq:papb}
\end{eqnarray}
\noindent
where Eq.~(\ref{eq:povmreq}) guarantees the second
equalities above and the condition $P_A+P_B=1$. In general, we will also
want to impose $P_A=P_B=1/2$ to obtain a standard coin flip.
How many messages does the above protocol require? Traditionally, we have
one message after each unitary. We also need an initial message before the
first unitary so that Alice can get ${\mathcal{M}}$. We will think of this as the
zeroth message. In total, we have $n+1$ messages. However, the first and
last message are somewhat odd: Alice never looks at the last message, so we
could have never sent it. Also, in principle, Alice could have started with
${\mathcal{M}}$ and initialized it herself, which would at most reduce Bob's cheating
power. So the whole protocol could be run with only $n-1$ messages.
However, the moments in time when the message qubits are flying between
Alice and Bob (all $n+1$ of them), mark particularly good times to examine
the state of our system. In particular, we are interest in the state of
${\mathcal{A}}$ and ${\mathcal{B}}$ at these times which, when both players are honest, will be
\begin{eqnarray}
\sigma_{A,i} = \Tr_{{\mathcal{M}}\otimes{\mathcal{B}}} \ket{\psi_i}\bra{\psi_i},
\qquad\qquad
\sigma_{B,i} = \Tr_{{\mathcal{A}}\otimes{\mathcal{M}}} \ket{\psi_i}\bra{\psi_i},
\end{eqnarray}
\noindent
for $i=0,\dots,n$.
\subsubsection{\label{sec:primal}Primal SDP}
Now that we have formalized the protocol, we proceed with the formalization
of the optimization problem needed to find the maximum probabilities with
which the players can win by cheating. The resulting problems will be
semidefinite programs (SDPs).
We will study the case of Alice honest and Bob cheating (the other case
being nearly identical). As usual, we do not want to make any assumptions
about the operations that Bob does, or even the number of qubits that he
may be using, therefore we must focus entirely on the state of Alice's
qubits.
As Alice initializes her qubits independently from Bob, we know what their
state must be during the zeroth message:
\begin{eqnarray}
\rho_{A,0} = \ket{\psi_{A,0}}\bra{\psi_{A,0}}.
\label{eq:prim0}
\end{eqnarray}
Subsequently we shall lose track of their exact state, but we know by the
laws of quantum mechanics that they must satisfy certain requirements. The
simplest is that, since Bob cannot affect Alice's qubits, then during the
steps when Alice does nothing the state of the qubits cannot change:
\begin{eqnarray}
\rho_{A,i} = \rho_{A,i-1} \qquad\qquad \text{for $i$ even.}
\label{eq:primeven}
\end{eqnarray}
Note that this is true even if Bob performs a measurement as Alice will
not know the outcome, and therefore her mixed state description will still
be correct.
The more complicated case is the steps when Alice performs a unitary.
Let $\tilde \rho_{A,i}$ be the state of ${\mathcal{A}}\otimes{\mathcal{M}}$ immediately after
Alice receives the $i$th message (for $i$ even, of course). The laws of
quantum mechanics again require the consistency condition
\begin{eqnarray}
\Tr_{\mathcal{M}} \tilde \rho_{A,i} = \rho_{A,i} \qquad\qquad \text{for $i$ even,}
\label{eq:primtilde}
\end{eqnarray}
\noindent
where $\tilde \rho_{A,i}$ is only restricted by the fact that Bob cannot
affect the state of ${\mathcal{A}}$. Note also that the above equation holds valid
even if Bob uses his message to tell Alice the outcome of a previous
measurement.
Now when Alice applies her unitary and sends off ${\mathcal{M}}$ she
will be left with the state
\begin{eqnarray}
\rho_{A,i} = \Tr_{\mathcal{M}}\left[ U_{A,i} \tilde\rho_{A,i-1} U_{A,i}^\dagger\right]
\qquad\qquad\text{for $i$ odd.}
\label{eq:primodd}
\end{eqnarray}
Finally, Alice's output is determined entirely by the measurement of
$\rho_{A,n}$. In particular, Bob wins with probability
\begin{eqnarray}
P_{win} = \Tr\left[ \Pi_{A,1} \rho_{A,n} \right].
\end{eqnarray}
Now consider the maximization of the above quantity over density operators
$\rho_{A,0},\dots,\rho_{A,n}$ and
$\tilde\rho_{A,0},\dots,\tilde\rho_{A,n-2}$ subject to
Eqs.~(\ref{eq:prim0},\ref{eq:primeven},\ref{eq:primtilde},\ref{eq:primodd}).
Because the optimal cheating strategy must satisfy the above conditions we
have
\begin{eqnarray}
P_B^* \leq \max \Tr\left[ \Pi_{A,1} \rho_{A,n} \right],
\end{eqnarray}
\noindent
where the maximum is taken subject to the above constraints.
The bound is also tight because any sequence of states consistent with the
above constraints can be achieved by Bob simply by maintaining the
purification of Alice's state. We sketch the proof: we inductively
construct a strategy for Bob that only uses unitaries so that the total
state will always be pure. Assume that Alice has $\rho_{A,i-1}$ (and the
total state is $\ket{\phi_{i-1}}$) and Bob wants to make her transition to
a given $\rho_{A,i}$ consistent with the above constraints. If $i$ is even
this is trivial. If $i$ is odd, he must make sure to send the right message
so that Alice ends up with the appropriate $\tilde
\rho_{A,i-1}$. But let $\ket{\tilde \phi_{i-1}}$ be any
purification of $\tilde \rho_{A,i-1}$ into ${\mathcal{B}}$. Because the reduced
density operators on ${\mathcal{A}}$ of both $\ket{\phi_{i-1}}$ and $\ket{\tilde
\phi_{i-1}}$ are the same, they are related by a unitary on ${\mathcal{M}}\otimes{\mathcal{B}}$
and by applying this unitary Bob will succeed in this step. By induction he
also succeeds in obtaining the entire sequence, as the base case for $i=0$
is trivial. We therefore have
\begin{eqnarray}
P_B^* = \max \Tr\left[ \Pi_{A,1} \rho_{A,n} \right].
\label{eq:primmax}
\end{eqnarray}
\noindent
\subsubsection{Dual SDP}
In the last section we found a mathematical description for the problem of
computing $P_B^*$. Unfortunately, it is formulated as a maximization
problem whose solution is often difficult to find. It would be sufficient
for our purposes, though, to find an upper bound on $P_B^*$. Such upper
bounds can be constructed from the dual SDP.
In particular, in this section we will describe a set of simple-to-verify
certificates that prove upper bounds on $P_B^*$. These certificates are
known as dual feasible points.
The certificates will be a set of $n+1$ positive semidefinite operators
$Z_{A,0},\dots,Z_{A,n}$ on ${\mathcal{A}}$ whose main property is
\begin{eqnarray}
\Tr[Z_{A,i-1} \rho_{A,i-1}] \geq \Tr[Z_{A,i} \rho_{A,i}]
\label{eq:dualprop}
\end{eqnarray}
for $i=1,\dots,n$ and for all $\rho_{A,0},\dots,\rho_{A,n}$ consistent
with the constraints of
Eqs.~(\ref{eq:prim0},\ref{eq:primeven},\ref{eq:primtilde},\ref{eq:primodd}).
Additionally, we require
\begin{eqnarray}
Z_{A,n}= \Pi_{A,1}.
\label{eq:zn}
\end{eqnarray}
Given a solution $\rho_{A,0}^*,\dots,\rho_{A,n}^*$ which attains the
maximum in Eq.~(\ref{eq:primmax}), we can use the above properties to write
\begin{eqnarray}
\bra{\psi_{A,0}} Z_{A,0} \ket{\psi_{A,0}} =
\Tr[Z_{A,0} \rho_{A,0}^*] \geq \Tr[Z_{A,n} \rho_{A,n}^*] = P_B^*,
\end{eqnarray}
\noindent
obtaining an upper bound on $P_B^*$. The crucial trick is that while we do
not know the complete optimal solution, we do know that
$\rho_{A,0}^*=\ket{\psi_{A,0}}\bra{\psi_{A,0}}$, which gives us a way of
computing the upper bound.
How do we enforce Eq.~(\ref{eq:dualprop})? We do it independently for each
transition: For $i$ odd, Eqs.~(\ref{eq:primtilde},\ref{eq:primodd}) give us
$\rho_{A,i-1} = \Tr_{\mathcal{M}} \tilde \rho_{A,i-1}$ and $\rho_{A,i} = \Tr_{\mathcal{M}}[
U_{A,i} \tilde\rho_{A,i-1} U_{A,i}^\dagger]$. We are therefore trying to
impose
\begin{eqnarray}
\Tr\left[ \left(Z_{A,i-1}\otimes I_{\mathcal{M}}\right) \tilde \rho_{A,i-1} \right]
\geq
\Tr\left[ \left(Z_{A,i}\otimes I_{\mathcal{M}}\right) U_{A,i} \tilde \rho_{A,i-1}
U_{A,i}^\dagger\right].
\end{eqnarray}
A sufficient condition (which is also necessary if $\tilde \rho_{A,i-1}$ is
arbitrary) is given by
\begin{eqnarray}
Z_{A,i-1}\otimes I_{\mathcal{M}} \geq
U_{A,i}^\dagger \left(Z_{A,i}\otimes I_{\mathcal{M}}\right) U_{A,i}
\qquad\qquad\text{for $i$ odd.}
\label{eq:zodd}
\end{eqnarray}
\noindent
In general, even when Bob is cheating, not all possible density operators
$\tilde \rho_{A,i-1}$ are attainable (otherwise he would have complete
control over Alice's qubits). Therefore, the above constraint could in
principle be overly stringent. However, we shall prove in the next section
that arbitrarily good certificates can be found even when when using the
above constraint.
Using a similar logic, when $i$ is even we have the relation $\rho_{A,i} =
\rho_{A,i-1}$ and so a sufficient condition on the dual variables is
$Z_{A,i-1} \geq Z_{A,i}$. However, from Alice's perspective these are just
dummy transitions. We introduced extra variables to mark the passage of
time during Bob's actions, but really we want to keep Alice's system
unchanged during these time steps. Therefore, we impose the more stringent
requirement on the dual variables
\begin{eqnarray}
Z_{A,i-1} = Z_{A,i}\qquad\qquad \text{for $i$ even.}
\label{eq:zeven}
\end{eqnarray}
We can summarize the above as follows:
\begin{eqnarray}gin{definition}
Fix a coin-flipping protocol $P$. A set of positive semidefinite operators
$Z_{A,0},\dots,Z_{A,n}$ satisfying
Eqs.~(\ref{eq:zn},\ref{eq:zodd},\ref{eq:zeven}) is known as a
\textbf{dual feasible point} (for the problem of cheating Bob given
a protocol $P$).
\end{definition}
Our arguments above prove:
\begin{eqnarray}gin{lemma}
A dual feasible point $Z_{A,0},\dots,Z_{A,n}$ for a coin-flipping protocol
$P$ constitutes a proof of the upper bound $\bra{\psi_{A,0}} Z_{A,0}
\ket{\psi_{A,0}}\geq P_B^*$.
\label{lemma:dual}
\end{lemma}
The importance of the above upper bounds is that the infimum over dual
feasible points actually equals $P_B^*$. In other words, there exist
arbitrarily good upper bound certificates. This result is known as strong
duality and is proven in Appendix~\ref{sec:strong}.
\subsection{\label{sec:UBP}Upper-Bounded Protocols}
Thus far we have studied Kitaev's first coin-flipping formalism. Given a
protocol it helps us find the optimal cheating strategies for Alice and Bob
by formulating these problems as convex optimizations. In general, however,
we do not have a fixed protocol that we want to study. Rather, we want to
identify the optimal protocol from the space of all possible
protocols. Kitaev's second coin-flipping formalism will help us formulate
this bigger problem as a convex optimization.
The goal is to compute the minimum (over all coin-flipping protocols) of
the maximum (over all cheating strategies for the given protocol) of the
bias. Alternating minimizations and maximizations are often tricky, but we
can get rid of this problem by dualizing the inner maximization, that is,
by replacing the maximum over cheating strategies with a minimum over the
upper-bound certificates discussed in the last section. The goal becomes to
compute the minimum (over all coin-flipping protocols) of the minimum (over
all upper-bound certificates for the given protocol) of the bias.
But we can go a step further and pair up the protocols and upper bounds to
get a single mathematical object which we call an upper-bounded
protocol. The space of upper-bounded protocols includes bad protocols with
tight upper bounds, good protocols with loose upper bounds and even bad
protocols with loose upper bounds. But somewhere in this space is the
optimal protocol together with its optimal upper bound and by minimizing
the bias in this space we can find it (though, strictly speaking, we
must carry out an infimum not a minimum and will only arrive arbitrarily
close to optimality).
\begin{eqnarray}gin{definition}
An \textbf{upper-bounded (coin-flipping) protocol}, or \textbf{UBP},
consists of a coin-flipping protocol together with two numbers $\begin{eqnarray}ta$ and
$\alpha$, a set of positive semidefinite operators $Z_{A,0},\dots,Z_{A,n}$
defined on ${\mathcal{A}}$ and a set of positive semidefinite operators
$Z_{B,0},\dots,Z_{B,n}$ defined on ${\mathcal{B}}$ which satisfy the equations
\begin{eqnarray}
Z_{A,0}\ket{\psi_{A,0}}= \begin{eqnarray}ta\ket{\psi_{A,0}}
&\qquad&
Z_{B,0}\ket{\psi_{B,0}}= \alpha\ket{\psi_{B,0}}
\nonumber\\
Z_{A,i-1}\otimes I_{\mathcal{M}} \geq
U_{A,i}^\dagger \left(Z_{A,i}\otimes I_{\mathcal{M}}\right) U_{A,i}
&&
Z_{B,i-1} = Z_{B,i}
\qquad\qquad\qquad\qquad\qquad\qquad
\text{($i$ odd)}
\nonumber\\
Z_{A,i-1} = Z_{A,i}
&&
I_{\mathcal{M}}\otimes Z_{B,i-1} \geq
U_{B,i}^\dagger \left(I_{\mathcal{M}}\otimes Z_{B,i}\right) U_{B,i}
\qquad\text{($i$ even)}
\nonumber\\
Z_{A,n}=\Pi_{A,1}
&&
Z_{B,n}=\Pi_{B,0}
\end{eqnarray}
We shall refer to the pair $(\begin{eqnarray}ta,\alpha)$ as the upper bound of the UBP.
\end{definition}
\begin{eqnarray}gin{theorem}
A UBP satisfies
\begin{eqnarray}
P_B^*\leq \begin{eqnarray}ta,\qquad\qquad
P_A^*\leq \alpha,
\end{eqnarray}
\noindent
where $P_B^*$ and $P_A^*$ are the optimal cheating probabilities of the
underlying protocol.
\label{thm:ubp}
\end{theorem}
Note the reverse order of $\begin{eqnarray}ta$ and $\alpha$. That is because $\begin{eqnarray}ta$,
which is the upper bound on Bob's cheating, must be computed from
quantities that involve operations on Alice's qubits. We will normally list
quantities defined or computed on ${\mathcal{A}}$ before those from ${\mathcal{B}}$.
The proof of the bound on $P_B^*$ from Theorem~\ref{thm:ubp} follows
directly from Lemma~\ref{lemma:dual}. We shall not prove the equivalent
bound on $P_A^*$ tough it follows from nearly identical arguments.
The main difference between the dual feasible points described in the
previous section and the one used in the definition of UBPs is that the
latter is more restrictive: rather than just setting $\begin{eqnarray}ta =
\bra{\psi_{A,0}}Z_{A,0}\ket{\psi_{A,0}}$ we additionally require that
$\ket{\psi_{A,0}}$ be an eigenvector of $Z_{A,0}$.
Clearly these more restricted dual feasible points still yield the desired
upper bounds, proving the theorem. What we shall argue below, though, is
that we have not sacrificed anything by imposing this additional constraint.
Any upper bound that can be proven with the original certificates can be
proven with these more restricted certificates as well.
We use the fact that for every $\epsilon>0$ there exists a $\Lambda>0$ such
that
\begin{eqnarray}
\Big(\bra{\psi_{A,0}}Z_{A,0}\ket{\psi_{A,0}} + \epsilon\Big)
\ket{\psi_{A,0}} \bra{\psi_{A,0}} +
\Lambda \Big(I - \ket{\psi_{A,0}} \bra{\psi_{A,0}}\Big)
\geq
Z_{A,0}.
\end{eqnarray}
\noindent
The left hand side has $\ket{\psi_{A,0}}$ as an eigenvector as desired,
and by transitivity of inequalities it can be used as the new $Z_{A,0}$.
On Bob's side a similar construction can be used to replace both $Z_{B,0}$
and $Z_{B,1}$ while maintaining their equality. By taking $\epsilon$
arbitrarily small we can get arbitrarily close to the old dual feasible
points, and so the infimum over both sets will be the same.
In summary, we have defined our UBPs and argued that finding the infimum
over this set is equivalent to seeking the optimal coin-flipping protocol.
In particular, we have shown that:
\begin{eqnarray}gin{theorem}
Let $f(\begin{eqnarray}ta,\alpha):\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$f(\alpha',\begin{eqnarray}ta')\geq f(\alpha,\begin{eqnarray}ta)$ whenever $\alpha'\geq\alpha$ and
$\begin{eqnarray}ta'\geq\begin{eqnarray}ta$, then
\begin{eqnarray}
\inf_{\text{proto}} f(P_B^*,P_A^*) = \inf_{UBP} f(\begin{eqnarray}ta,\alpha),
\end{eqnarray}
\noindent
where the left optimization is carried out over all coin-flipping protocols
and the right one is carried out over all upper-bounded protocols. In
particular, the optimal bias $f(\begin{eqnarray}ta,\alpha)=\max(\begin{eqnarray}ta,\alpha)-1/2$ can
be found by optimizing either side.
\label{thm:kitmain1}
\end{theorem}
\subsubsection{Lower bounds and operator monotone functions}
We begin our study of upper-bounded protocols by showing how to place lower
bounds on the set of UBPs. Though we will not prove any new lower
bounds, the ideas presented in this section will motivate the constructions
of the next sections.
The main tool tool that we will be using is the following inequality
\begin{eqnarray}
\bra{\psi_{i-1}} Z_{A,i-1}\otimes I_{\mathcal{M}}\otimes Z_{B,i-1} \ket{\psi_{i-1}}
\geq
\bra{\psi_{i}} Z_{A,i}\otimes I_{\mathcal{M}}\otimes Z_{B,i} \ket{\psi_{i}}
\label{eq:lowerb}
\end{eqnarray}
\noindent
which is the bipartite equivalent of Eq.~(\ref{eq:dualprop}). The proof for
$i$ odd is that
\begin{eqnarray}
\bra{\psi_{i-1}} Z_{A,i-1}\otimes I_{\mathcal{M}}\otimes Z_{B,i-1} \ket{\psi_{i-1}}
&\geq&
\bra{\psi_{i-1}}
\big(U_{A,i}^\dagger \otimes I_{\mathcal{B}}\big)
Z_{A,i}\otimes I_{\mathcal{M}}\otimes Z_{B,i}
\big(U_{A,i} \otimes I_{\mathcal{B}}\big)
\ket{\psi_{i-1}}
\nonumber\\
&&=
\bra{\psi_{i}} Z_{A,i}\otimes I_{\mathcal{M}}\otimes Z_{B,i} \ket{\psi_{i}}
\label{eq:lowerbproof}
\end{eqnarray}
\noindent
and the proof for $i$ even is nearly identical.
Iterating we obtain the inequality
\begin{eqnarray}
\begin{eqnarray}ta\alpha =
\bra{\psi_{0}} Z_{A,0}\otimes I_{\mathcal{M}}\otimes Z_{B,0} \ket{\psi_{0}}
&\geq&
\bra{\psi_{n}} Z_{A,n}\otimes I_{\mathcal{M}}\otimes Z_{B,n} \ket{\psi_{n}}
\nonumber\\
&&= \bra{\psi_{n}} \Pi_{A,1}\otimes I_{\mathcal{M}}\otimes \Pi_{B,0} \ket{\psi_{n}} =
0
\end{eqnarray}
\noindent
or equivalently $P_B^* P_A^* \geq 0$, which admittedly is rather
disappointing.
But there is hope. If we had been studying strong coin flipping and were
interested in the case when both Alice and Bob want to obtain the outcome
one, the above analysis would be correct given the minor change
$Z_{B,n}=\Pi_{B,1}$. In such case the above inequality would read $P_B^*
P_A^*\geq 1/2$ which is Kitaev's bound for strong coin flipping
\cite{Kitaev}.
For historical purposes we note that the results up to this point were
already part of Kitaev's first formalism. Pedagogically, though, it makes
more sense to call the optimizations given a fixed protocol the ``first
formalism'' and the optimizations over all protocols the ``second
formalism.''
Returning to the case of weak coin flipping, we can obtain better bounds
by inserting operator monotone functions into the above inequalities.
An operator monotone function $f:[0,\infty)\rightarrow [0,\infty)$ is a
function that preserves the ordering of matrices. That is
\begin{eqnarray}
X\geq Y \mathbb{R}ightarrow f(X)\geq f(Y)
\end{eqnarray}
for all positive semidefinite operators $X$ and $Y$. The simplest example
of an operator monotone function is $f(z)=z$. Another example is $f(z)=1$.
Note that not all monotone functions are operator monotone. The classic
example is $f(z)=z^2$ which is monotone on the domain $[0,\infty)$ but is
not operator monotone. A few more facts about operator monotone functions
are collected in the next section.
How do we use the operator monotone functions? A moment ago we were
studying the expression $\bra{\psi_{i}} Z_{A,i}\otimes I_{\mathcal{M}}\otimes Z_{B,i}
\ket{\psi_{i}}$. But we could just as well study the expression
$\bra{\psi_{i}} Z_{A,i}\otimes I_{\mathcal{M}}\otimes f(Z_{B,i}) \ket{\psi_{i}}$ for
any operator monotone function $f$. We could then prove an inequality
similar to Eq.~(\ref{eq:lowerb}), and iterating we would end up with the
condition $\begin{eqnarray}ta f(\alpha) \geq P_B f(0)$, where $P_B$ is the honest
probability of Bob wining. Choosing $f(z)=1$ we can derive the bound
$P_B^*\geq P_B$, which at least has the potential of being saturated.
The next obvious step is to put operator monotone functions on both sides
and study expressions of the form $\bra{\psi_{i}} f(Z_{A,i})\otimes
I_{\mathcal{M}}\otimes g(Z_{B,i}) \ket{\psi_{i}}$. But because at any time step only
one of $Z_{A,i}$ and $Z_{B,i}$ increases, we can do even better.
\begin{eqnarray}gin{definition}
A \textbf{bi-operator monotone function} is a function
$f(x,y):[0,\infty)\times[0,\infty)\rightarrow[0,\infty)$ such that when one
of the variables is fixed, it acts as an operator monotone function in the
other variable.
More specifically, given $c\in[0,\infty)$ define $f(\underline
c,z):[0,\infty)\rightarrow[0,\infty)$ to be the function $z\rightarrow
f(c,z)$ (i.e., where the first argument has been fixed). Similarly, let
$f(z,\underline c):[0,\infty)\rightarrow[0,\infty)$ be the function
obtained by fixing the second argument. We say $f(x,y)$ is bi-operator
monotone if both $f(\underline c,z)$ and $f(z,\underline c)$ are operator
monotone for every $c\in[0,\infty)$.
Furthermore, given $f(x,y):[0,\infty)\times[0,\infty)\rightarrow[0,\infty)$ we
extend its definition to act on pairs of positive semidefinite operators as
follows: let $X=\sum_{i}x_i\ket{x_i}\bra{x_i}$ and
$Y=\sum_{i}y_i\ket{y_i}\bra{y_i}$ then
\begin{eqnarray}
f(X,Y) = \sum_{i,j} f(x_i,y_j) \ket{x_i}\bra{x_i}\otimes \ket{y_j}\bra{y_j}.
\end{eqnarray}
\end{definition}
Bi-operator monotone functions satisfy a few simple properties:
\begin{eqnarray}gin{itemize}
\item If $Y'\geq Y$ then $f(X,Y')\geq f(X,Y)$.
\item If $U$ is unitary then $f(X,U Y U^\dagger)= \big(I\otimes U\big) f(X,Y)
\big(I\otimes U^\dagger\big)$.
\item When acting on a tripartite system $f(X,I\otimes Y)=f(X\otimes I,Y)$.
\end{itemize}
We are now ready to prove our most general inequality. Given a bi-operator
monotone function $f$ then we can write
\begin{eqnarray}
\bra{\psi_{i-1}} f(Z_{A,i-1},I_{\mathcal{M}}\otimes
Z_{B,i-1}) \ket{\psi_{i-1}} \geq
\bra{\psi_{i}} f(Z_{A,i},I_{\mathcal{M}}\otimes Z_{B,i}) \ket{\psi_{i}},
\end{eqnarray}
\noindent
whose proof is nearly identical to Eq.~(\ref{eq:lowerbproof}). Iterating,
we obtain the lemma:
\begin{eqnarray}gin{lemma}
Given any bi-operator
monotone function $f$, we obtain a bound on coin-flipping protocols given by
\begin{eqnarray}
f(P_B^*,P_A^*) \geq
P_B f(1,0) + P_A f(0,1).
\end{eqnarray}
\end{lemma}
Can we use bi-operator monotone functions to prove an interesting bound on
weak coin flipping? Certainly not if we believe that we can achieve
arbitrarily small bias. However we shall see that the spaces of bi-operator
monotone functions and coin-flipping protocols are essentially duals. If
there were no arbitrarily good protocols then we would be able to prove
that fact using the above lemma.
\subsubsection{Some more facts about operator monotone functions}
Operator monotone functions are well studied and a good reference on the
subject is \cite{Bhatia}. As they are also central to the task of
constructing coin-flipping protocols we collect here some of their most
important properties which will be used throughout the paper.
Our main interest are functions that map the set of positive semidefinite
operators to itself. Therefore, when not otherwise stated, we assume all
operator monotone functions have domain $[0,\infty)$ and range contained
in $[0,\infty)$. In general, though, operator monotone functions can be
defined on any real domain.
The space of operator monotone functions (on a fixed domain) forms a convex
cone. If $f(z)$ and $g(z)$ are operator monotone then so are
\begin{eqnarray}
a f(z) + b g(z)
\end{eqnarray}
\noindent
for any $a\geq0$ and $b\geq 0$. Another simple property is that if $f(z)$
is operator monotone on a domain $(a,b)$ then for any $c\in\mathbb{R}$ we have
$f(z-c)$ is operator monotone on $(a+c,b+c)$.
A very important function which is operator monotone on the domain
$(0,\infty)$ is $f(z)=-1/z$. The proof is given by $Y\geq X>0\mathbb{R}ightarrow
I\geq Y^{-1/2}X Y^{-1/2}\mathbb{R}ightarrow I\leq (Y^{-1/2}X
Y^{-1/2})^{-1}\mathbb{R}ightarrow I\leq Y^{1/2}X^{-1} Y^{1/2} \mathbb{R}ightarrow
-Y^{-1}\geq -X^{-1}$. By shifting and restricting the domain we get
\begin{eqnarray}
f(z)=-\frac{1}{\lambda+z}
\end{eqnarray}
which is operator monotone on $[0,\infty)$ for $\lambda\in(0,\infty)$,
though the range is negative. The range can be fixed by scaling and adding
in the constant function to get $1-\frac{\lambda}{\lambda+z} =
\frac{z}{\lambda+z}$.
In fact, the above functions together with $f(z)=z$ and $f(z)=1$ span the
extremal rays of the convex cone of operator monotone functions. More
precisely, every operator monotone function
$f:(0,\infty)\rightarrow[0,\infty)$ has a unique integral representation
\begin{eqnarray}
f(z) = c_1 + c_2 z + \int_{0}^{\infty} \frac{\lambda z}{\lambda + z} d
w(\lambda),
\end{eqnarray}
\noindent
where $c_1,c_2\in\mathbb{R}$ are non-negative and $d w(\lambda)$ is a positive measure
such that $\int_0^\infty \frac{\lambda}{1+\lambda}d w(\lambda) < \infty$.
In particular, they are infinitely differentiable.
The general case, where the domain is the interval $I=(a,b)\subset\mathbb{R}$, is
nearly identical but with the integral ranging over $-\lambda\in\mathbb{R}\setminus
I$. When the domain is the closed interval $[a,b]$ then functions must be
operator monotone on $(a,b)$ and monotone on $[a,b]$ so that $f(a)\leq
\lim_{z\rightarrow a^+} f(z)$ and $f(b)\geq
\lim_{z\rightarrow b^-} f(z)$.
\subsection{\label{sec:TDPG}Time Dependent Point Games}
In previous sections we have paired the honest probability distribution
$\sigma_{A,i}$ with the dual variable $Z_{A,i}$. We have also
paired the full honest state $\ket{\psi_i}$ with the operator
$Z_{A,i}\otimes I\otimes Z_{B,i}$. These pairings have led to interesting
results, but they can also become rapidly unwieldy because they contain too
much information, such as a choice of basis. The goal of this section
is to get rid of most of this excess information and strip the problem to a
bare minimum that still contains the essence of coin flipping.
The key idea for the following discussion is to use the honest state to
define a probability distribution over the eigenvalues of the dual SDP
variables. This idea is captured by the next definition.
\begin{eqnarray}gin{definition}
Given $Z=\sum_{z\in\eig(Z)} z \Pi^{[z]}$, a positive semidefinite matrix
expressed as a sum of its eigenspaces, and $\sigma$, a second positive
semidefinite matrix defined on the same space, we define the function
{\boldmath $\Prob(Z,\sigma):[0,\infty)\rightarrow[0,\infty)$}
as follows
\begin{eqnarray}
p(z)=\Prob(Z,\sigma)\quad\mathbb{R}ightarrow\quad
p(z) =
\begin{eqnarray}gin{cases}
\Tr[\Pi^{[z]} \sigma] & z\in \eig(Z),\\
0 & \text{otherwise.}
\end{cases}
\end{eqnarray}
Similarly, given a vector $\ket{\psi}$ instead of $\sigma$ we define
{\boldmath $\Prob(Z,\ket{\psi}):[0,\infty)\rightarrow[0,\infty)$} by
\begin{eqnarray}
p(z)=\Prob(Z,\ket{\psi})\quad\mathbb{R}ightarrow\quad
p(z) =
\begin{eqnarray}gin{cases}
\bra{\psi}\Pi^{[z]}\ket{\psi} & z\in \eig(Z),\\
0 & \text{otherwise.}
\end{cases}
\end{eqnarray}
\end{definition}
\noindent
Note that by construction
$\Prob(Z,\ket{\psi})\equiv\Prob(Z,\ket{\psi}\bra{\psi})$.
We think of the above functions $p(z)$ as belonging to the space of
functions $[0,\infty)\rightarrow[0,\infty)$ with finite support. The
motivation for the construction is that given any function with arbitrary
support $f(z):[0,\infty)\rightarrow\mathbb{R}$ we have
\begin{eqnarray}
p(z)=\Prob(Z,\sigma)\quad\mathbb{R}ightarrow\quad
\sum_z p(z) f(z) = \Tr[\sigma f(Z)],
\end{eqnarray}
\noindent
where the sum on the left is over the finite support of $p(z)$.
A similar construction can be used for the bipartite case. Take
$Z_A=\sum_{z_A} z_A \Pi_{A}^{[z_A]}$ on ${\mathcal{A}}$, $Z_B=\sum_{z_B} z_B
\Pi_{B}^{[z_B]}$ on ${\mathcal{B}}$ and $\ket{\psi}$ on ${\mathcal{A}}\otimes{\mathcal{M}}\otimes{\mathcal{B}}$ and
combine them to form the two-variable function
\begin{eqnarray}
p(z_A,z_B) =
\begin{eqnarray}gin{cases}
\bra{\psi}\Pi_{A}^{[z_A]}\otimes I_{{\mathcal{M}}} \otimes \Pi_{B}^{[z_B]} \ket{\psi} &
z_A\in\eig(Z_A)\text{ and }z_B\in\eig(Z_B),\\
0 & \text{otherwise,}
\end{cases}
\end{eqnarray}
\noindent
which we will denote by $\Prob(Z_A,Z_B,\ket{\psi})$.
Often it will be useful to use an algebraic notation when describing
functions with finite support. For single-variable functions $p(z)$ with
finite support we introduce a basis $\{[z_i]\}$ of functions that take
the value one at $z_i$ and are zero everywhere else. For instance, a function
with two nonzero values $p(z_1)=c_1$ and $p(z_2)=c_2$ can be written as
\begin{eqnarray}
p = c_1 [z_1] + c_2 [z_2].
\end{eqnarray}
\noindent
Similarly, for the bipartite we use $\{[x,y]\}$ as basis elements for the
functions $p(x,y)$ with finite support. For instance, the initial state
$\Prob(Z_{A,0},Z_{B,0},\ket{\psi_0})$ of a UBP always has the form
\begin{eqnarray}
1[\begin{eqnarray}ta,\alpha].
\label{eq:tn}
\end{eqnarray}
On the other hand, because $Z_{A,n}=\Pi_{A,1}$ and $Z_{B,n}=\Pi_{B,0}$ are
just the projections onto the opposing player's winning space, the final
state $\Prob(Z_{A,n},Z_{B,n},\ket{\psi_n})$ of a UBP always has the form
\begin{eqnarray}
P_B[1,0] + P_A[0,1]
\label{eq:t0}
\end{eqnarray}
as can be verified from Eq.~(\ref{eq:papb}). Note, however, that
the label $0$ on $\Pi_{B,0}$ refers to the coin outcome and not the
associated eigenvalue. Because $Z_{B,n}=1\Pi_{B,0}+0\Pi_{B,1}$, the
projector onto the one eigenvalue of $Z_{B,n}$ is given by
$\Pi_{B}^{[1]}=\Pi_{B,0}$ and similarly $\Pi_{B}^{[0]}=\Pi_{B,1}$.
In fact, given a UBP we can compute for every $i$ the functions
$\Prob(Z_{A,i},Z_{B,i},\ket{\psi_i})$, which allows us to visualize the UBP
as a movie of sorts where at each time step there is a finite set of points
in the plane. The points move around between time steps according to some
rules which we will determine in a moment. We call these sequences
\textit{point games} and we shall show that they contain all the important
information about UBPs.
One important convention that we introduce, though, is that point games are
always described in \textbf{reverse time order}. Henceforth, we will refer
to Eq.~(\ref{eq:t0}) as the first or $t=0$ point configuration whereas
Eq.~(\ref{eq:tn}) will be referred to as the last or $t=n$ point
configuration. More generally, the point configuration at $t=i$ will be
constructed from the operators $Z_{A,n-i}$, $Z_{B,n-i}$ and
$\ket{\psi_{n-i}}$.
The motivation for reversing the time order is as follows: first we get to
begin at a known starting configuration such as $0.5[1,0]+0.5[0,1]$ (for
the main case of interest $P_A=P_B=1/2$). From there, we can move the
points around following the rules of point games until they merge into a
single point at some location $[\begin{eqnarray}ta,\alpha]$. We can do this without
fixing in advance the number of steps, but rather using the existence of a
single point as an end condition. The sequence of moves will then encode a
UBP with upper bound $(\begin{eqnarray}ta,\alpha)$.
So what are these rules for moving points around? Let begin with the
one-variable case and examine a transition between $p_i(z)$, constructed
from $Z_{A,n-i}$ and $\sigma_{A,n-i}$, and $p_{i+1}(z)$, constructed from
$Z_{A,n-i-1}$ and $\sigma_{A,n-i-1}$. A necessary condition is
\begin{eqnarray}
\sum_z p_i(z) f(z) \leq \sum_z p_{i+1}(z) f(z)
\end{eqnarray}
\noindent
for every operator monotone function $f$, where again the sums range over the
finite supports of the respective probability distributions.
The condition is trivially necessary on the time transitions when Bob acts
and Alice does nothing because then $p_i=p_{i+1}$. To prove that the condition
is necessary for the non-trivial transitions recall the usual relation
$Z_{A,n-i-1}\otimes I_{\mathcal{M}} \geq U_{A,n-i}^\dagger (Z_{A,n-i}\otimes I_{\mathcal{M}} )
U_{A,n-i}$. Also let $\tilde \sigma_{A,n-i-1} =
\Tr_{\mathcal{B}}[\ket{\psi_{n-i-1}}\bra{\psi_{n-i-1}}]$ so that
$\sigma_{A,n-i-1}=\Tr_{\mathcal{M}}[\tilde \sigma_{A,n-i-1}]$ and
$\sigma_{A,n-i}=\Tr_{\mathcal{M}}[U_{A,n-i} \tilde \sigma_{A,n-i-1}U_{A,n-i}^\dagger]$
and therefore
\begin{eqnarray}
\sum_z p_i(z) f(z) &=& \Tr[\sigma_{A,n-i} f(Z_{A,n-i})]
= \Tr[\tilde \sigma_{A,n-i-1}
f(U_{A,n-i}^\dagger Z_{A,n-i} \otimes I_{{\mathcal{M}}} U_{A,n-i})]
\label{eq:validproof}
\\\nonumber
&\leq&
\Tr[\tilde \sigma_{A,n-i-1} f(Z_{A,n-i-1} \otimes I_{{\mathcal{M}}})]
= \Tr[\sigma_{A,n-i-1} f(Z_{A,n-i-1})] = \sum_z p_{i+1}(z) f(z).
\end{eqnarray}
Sadly, there are no sufficient conditions that can be derived just by
looking at one side of the problem. Nevertheless, it will be useful to
define the transitions that satisfy the above constraints as the valid set
of transitions, as this notion will be used as a building block when
studying the more complicated bipartite transitions.
\begin{eqnarray}gin{definition}
Let $p_i(z)$ and $p_{i+1}(z)$ be two functions $[0,\infty)\rightarrow
[0,\infty)$ with finite support. We say $p_i(z)\rightarrow p_{i+1}(z)$ is
a \textbf{valid transition} if $\sum_z p_i(z) = \sum_z p_{i+1}(z)$,
and for every operator monotone function $f$ we have
\begin{eqnarray}
\sum_z p_i(z) f(z) \leq \sum_z p_{i+1}(z) f(z).
\end{eqnarray}
\noindent
Furthermore, we say that the transition is \textbf{strictly valid} if
$\sum_z p_i(z) f(z) < \sum_z p_{i+1}(z) f(z)$ for every non-constant
operator monotone function $f$.
\end{definition}
\noindent
For reasons to become clear below, we do not restrict the functions
$p_i(z)$ and $p_{i+1}(z)$ to sum to one. However, we do require that their
sum be equal so that probability is conserved.
Because we have a characterization of the extremal rays of the cone of
operator monotone functions, it is sufficient when checking for valid
transitions to use the set of functions $f_\lambda(z)=\frac{\lambda
z}{\lambda+z}$ for $\lambda\in(0,\infty)$. As for the other two extremal
functions, the constraint from $f(z)=1$ is independently imposed as
conservation of probability, and the constraint from $f(z)=z$ follows from
the limit $\lambda\rightarrow\infty$ of $f_\lambda(z)$. It is worth stating
this explicitly:
\begin{eqnarray}gin{lemma}
Let $p_i(z)$ and $p_{i+1}(z)$ be two functions $[0,\infty)\rightarrow
[0,\infty)$ with finite support. The transition $p_i(z)\rightarrow
p_{i+1}(z)$ is valid if and only if $\sum_z \left(p_{i+1}(z)- p_i(z)\right)=0$
and for every $\lambda\in(0,\infty)$ we have $\sum_z \frac{\lambda
z}{\lambda+z}\left(p_{i+1}(z) - p_i(z) \right)\geq 0$.
\end{lemma}
Not surprisingly, a set of necessary conditions for bipartite transitions
constructed from UBPs is that
\begin{eqnarray}
\sum_{x,y} p_i(x,y) f(x,y) \leq \sum_{x,y} p_{i+1}(x,y)
f(x,y)
\end{eqnarray}
\noindent
for every bi-operator monotone functions $f$, where as usual the sums are
over the finite support of the respective probability distributions.
Unfortunately, the space of bi-operator monotone functions is not as well
characterized as the space of operator monotone functions, and therefore it
would be better to have a set of conditions that are constructed from the
latter:
\begin{eqnarray}gin{definition}
Let $p_i(x,y)$ and $p_{i+1}(x,y)$ be two functions
$[0,\infty)\otimes [0,\infty)\rightarrow [0,\infty)$ with finite support.
We say $p_i(x,y)\rightarrow p_{i+1}(x,y)$ is a \textbf{valid
transition} if either
\begin{eqnarray}gin{enumerate}
\item for every $c\in[0,\infty)$ the transition
$p_i(z,\underline c)\rightarrow p_{i+1}(z,\underline c)$ is valid, or
\item for every $c\in[0,\infty)$ the transition
$p_i(\underline c,z)\rightarrow p_{i+1}(\underline c,z)$ is valid,
\end{enumerate}
\noindent
where as before $p_i(z,\underline c)$ is the one-variable function obtained
by fixing the second input. We call the first case a \textbf{horizontal
transition} and the second case a \textbf{vertical transition}.
\end{definition}
\noindent
The first case occurs when Alice applies a unitary and the second case when
Bob applies a unitary. As opposed to the single variable transitions, the
bipartite condition of validity is not transitive. However, we can define
the notion of transitively valid for two functions if there is a sequence
of functions beginning with the first one and ending with the second one,
such that each transition is valid. The main object of study for this
section will be transitively valid transitions of the form
$P_B[1,0]+P_A[0,1]\rightarrow 1[\begin{eqnarray}ta,\alpha]$, where we always assume
$P_A,P_B\geq 0$ are some fixed numbers such that $P_A+P_B=1$.
\begin{eqnarray}gin{definition}
A \textbf{time dependent point game (TDPG)} is a sequence
$p_0(x,y),\dots,p_n(x,y)$ of functions
$[0,\infty)\otimes [0,\infty)\rightarrow
[0,\infty)$ with finite support, such that every transition
$p_i(x,y)\rightarrow p_{i+1}(x,y)$ is valid and such that the first
and last distributions have the form
\begin{eqnarray}
p_0 = P_B[1,0] + P_A[0,1],
\qquad\text{and}\qquad
p_n = 1[\begin{eqnarray}ta,\alpha].
\end{eqnarray}
We say that $[\begin{eqnarray}ta,\alpha]$ is the final point of the TDPG.
\end{definition}
\noindent
The above definition can be extended to games beyond coin-flipping which
have many possible outcomes. If outcome $i$ has honest probability $q_i$,
and pays $a_i\geq 0$ to Alice and $b_i\geq 0$ to Bob, the same formalism
applies if we use as starting state $p_0 =
\sum_i q_i [b_i,a_i]$. This paper will focus exclusively on weak
coin-flipping though.
Our first task will be to prove that given a UBP with bound
$(\begin{eqnarray}ta,\alpha)$ we can build a TDPG with final point $[\begin{eqnarray}ta,\alpha]$. We
have already done most of the work by constructing the probability
distributions out of the UBP and showing that they have the right initial
and final states. What remains to be shown is that the transitions
$p_i(x,y)\rightarrow p_{i+1}(x,y)$ are valid.
We focus on the transitions when Alice applies a unitary, the other case
being nearly identical. Given a UBP, we construct the distribution
$p_i=\Prob(Z_{A,n-i},Z_{B,n-i},\ket{\psi_{n-i}})$ and the distribution
$p_{i+1}=\Prob(Z_{A,n-i-1},Z_{B,n-i-1},\ket{\psi_{n-i-1}})$. The UBP
operators satisfy the usual relations $Z_{A,n-i-1}\otimes I_{\mathcal{M}} \geq
U_{A,n-i}^\dagger (Z_{A,n-i}\otimes I_{\mathcal{M}}) U_{A,n-i}$,
$Z_{B,n-i-1}=Z_{B,n-i}$ and $\ket{\psi_{n-i}}= U_{A,n-i}\otimes I_B
\ket{\psi_{n-i-1}}$. We expand the state $\ket{\psi_{n-i-1}}$ as
\begin{eqnarray}
\ket{\psi_{n-i-1}} = \sum_{y} \ket{\phi_{y}}\otimes \ket{y},
\end{eqnarray}
\noindent
where $\ket{y}$ are normalized eigenvectors of $Z_{B,n-i}$ and
$\ket{\phi_{y}}$ are non-normalized states on ${\mathcal{A}}\otimes{\mathcal{M}}$. Because
this is not necessarily a Schmidt decomposition, the vectors
$\ket{\phi_{y}}$ are not necessarily orthogonal, however this will not be a
problem. The key idea now is to note that given a fixed $y$, the function
$p_{i+1}(x,\underline{y})=\Prob(Z_{A,n-i-1},\rho_{n-i-1,y})$ where
$\rho_{n-i-1,y}\equiv\Tr_{{\mathcal{M}}}[\ket{\phi_{y}}\bra{\phi_{y}}]$. Similarly,
the function $p_{i}(x,\underline{y})=\Prob(Z_{A,n-i},\rho_{n-i,y})$ where
$\rho_{n-i,y}\equiv\Tr_{{\mathcal{M}}}[U_{A,n-i}\ket{\phi_{y}}\bra{\phi_{y}}
U_{A,n-i}^\dagger]$. The relationship between these quantities is the same
as it was the analysis of one-sided probability transitions, and the proof
of Eq.~(\ref{eq:validproof}) goes through with $\rho_{n-i-1,y}$,
$\rho_{n-i,y}$ and $\ket{\phi_{y}}\bra{\phi_{y}}$ taking the place of
$\sigma_{A,n-i-1}$, $\sigma_{A,n-i}$ and $\tilde
\sigma_{A,n-i-1}$. Therefore, for every $y\in[0,\infty)$ the transition
$p_i(x,\underline{y})\rightarrow p_{i+1}(x,\underline{y})$ is valid
and therefore the full transition $p_i(x,y)\rightarrow
p_{i+1}(x,y)$ is valid as well.
We have just proven that given a UBP with bound $(\begin{eqnarray}ta,\alpha)$ we can
construct a TDPG with final point $[\begin{eqnarray}ta,\alpha]$. The converse is also
true in the following sense: given any TDPG with final point
$[\begin{eqnarray}ta,\alpha]$ and an $\epsilon>0$ there exists a UBP with bound
$(\begin{eqnarray}ta+\epsilon,\alpha+\epsilon)$. This is enough because we are only
concerned with infimums, and the infimums over both sets will be
equal. Constructing UBPs from TDPGs will require a fair amount of work to
be done in the next couple of sections. Some readers may prefer to first
study the TDPG examples in Section~\ref{sec:examples}.
\subsubsection{\label{sec:proj}Coin-flipping protocols with projections}
The description of the coin-flipping protocols built from TDPGs will be
greatly simplified if we can use measurements at intermediate steps
throughout the protocol. Of course, there is nothing special about
protocols that involve measurements, as these can always be delayed to the
last step. However, doing so requires simulating the measurement with a
unitary and keeping the simulated outcomes in some extra qubits, which
adds extra complexity to the description of the protocol. Early
measurements can result in significant improvements in the description of a
protocol and the number of qubits employed. An example of this is given in
Appendix~\ref{sec:ddb} which takes the author's original bias $1/6$
protocol requiring arbitrarily many qubits and reduces the space used to a
single qutrit per player plus a single qubit for messages.
In fact, for such simplifications we need only allow a special kind of
projective measurement: at certain steps the players will use a two outcome
POVM of the form $\{E,I-E\}$, where E is a projector (i.e.,
$E^\dagger=E^2=E$). The protocol will be set up so that if both players
play honestly the first outcome will always be obtained, and if the second
outcome is observed the players will immediately abort (at which point they
can declare themselves the winner). We will place one such projection
immediately after each unitary.
The goal of this section is to formalize the needed notion of protocols
with projections, describe their dual feasible points, and show how they
are equivalent to regular protocols. It can be safely skipped by those
familiar with the result.
\begin{eqnarray}gin{definition}
A \textbf{coin-flipping protocol with projections} is a coin-flipping
protocol with the addition of $n$ projection operators $E_1,\dots,E_n$ of
the form
\begin{eqnarray}
E_i = \begin{eqnarray}gin{cases}
E_{A,i}\otimes I_{\mathcal{B}} & \text{for $i$ odd,}\\
I_{\mathcal{A}}\otimes E_{B,i} & \text{for $i$ even,}
\end{cases}
\end{eqnarray}
such that $E_i\ket{\psi_{i}}=\ket{\psi_{i}}$ for every $i=1,\dots,n$
\end{definition}
\noindent
The protocol is implemented as before, except that immediately after
implementing $U_i$ the acting player measures using $\{E_i,I-E_i\}$ and
aborts on the second outcome.
To prove the equivalence of protocols with and without measurements, not
only do we need to construct a canonical unitary-based protocol for every
protocol with measurements, but we also need to show that the new protocol
does not allow for any extra cheating. This is done by constructing a
canonical map from the dual feasible points of the protocol with
measurements to the dual feasible points of the unitary protocol such that
the upper bounds are preserved (or at least come arbitrarily close to each
other). Effectively, we aim to construct a map from ``UBPs with
measurements'' to regular UBPs. As most of the constructions are fairly
standard, we will only sketch the details.
As usual we focus on the case of honest Alice and cheating Bob. The primal
SDP requires $\rho_{A,0} = \ket{\psi_{A,0}}\bra{\psi_{A,0}}$, $\rho_{A,i} =
\rho_{A,i-1}$ for $i$ even, and $P_{win} = \Tr\left[ \Pi_{A,1} \rho_{A,n}
\right]$ as before. However, the new element is that for $i$ odd we have
\begin{eqnarray}
\rho_{A,i} = \Tr_{\mathcal{M}}\left[ E_{A,i} U_{A,i} \tilde\rho_{A,i-1}
U_{A,i}^\dagger E_{A,i}\right],
\qquad\text{for}\qquad
\Tr_{\mathcal{M}} \tilde \rho_{A,i-1} \leq \rho_{A,i-1}.
\end{eqnarray}
\noindent
The trace of $\rho_{A,i}$ is no longer unity but rather it encodes the
probability that we have reached step $i$ without aborting and
$\rho_{A,i}/\Tr[\rho_{A,i}]$ is the state at step $i$ given that no aborts
have occurred. The inequality on the right equation allows for Bob to abort,
though it is certainly never optimal for him to do so.
The dual SDP has as before $Z_{A,n}= \Pi_{A,1}$, $Z_{A,i-1} = Z_{A,i}$ for $i$
even, and $\begin{eqnarray}ta=\bra{\psi_{A,0}}Z_{A,0}\ket{\psi_{A,0}}$ but now
for $i$ odd we impose the condition
\begin{eqnarray}
Z_{A,i-1}\otimes I_{\mathcal{M}} \geq
U_{A,i}^\dagger E_{A,i} \left(Z_{A,i}\otimes I_{\mathcal{M}}\right) E_{A,i} U_{A,i}.
\end{eqnarray}
Given a protocol with projections and a dual feasible point let us build a
unitary protocol with a matching dual feasible point. We will put primes on
all expressions of the new protocol that differ from the one with
measurements.
The number of rounds and the message space will be the same, but we will
add $n$ qubits to both ${\mathcal{A}}$ and ${\mathcal{B}}$ so that ${\mathcal{A}}P =
\left(\mathbb{C}^2\right)^{\otimes n}\otimes{\mathcal{A}}$ and ${\mathcal{B}}P =
{\mathcal{B}}\otimes\left(\mathbb{C}^2\right)^{\otimes n}$. These extra qubits will store the
measurement outcomes (though technically we only need $n/2$ qubits on each
side). The new initial state will set all the new qubits to zero
$\ket{\psi_{A,0}'}=\ket{0}\otimes\ket{\psi_{A,0}}$ and
$\ket{\psi_{B,0}'}=\ket{\psi_{B,0}}\otimes\ket{0}$, and when playing
honestly they will always remain zero. The final projectors only give the
victory to the other player if all the extra qubits are zero so that
$\Pi_{A,1}'=\ket{0}\bra{0}\otimes \Pi_{A,1}$ and $\Pi_{A,0}'=I-\Pi_{A,1}'$
and similarly $\Pi_{B,0}'= \Pi_{B,0}\otimes\ket{0}\bra{0}$ and
$\Pi_{B,1}'=I-\Pi_{B,0}'$. Finally, the new unitaries simply simulate a
measurement after applying the regular operation
\begin{eqnarray}
U_{A,i}' = M_{A,i} \left(I\otimes U_{A,i}\right)&\qquad&\text{for $i$ odd,}\\
U_{B,i}' = M_{B,i} \left(U_{B,i}\otimes I\right)&\qquad&\text{for $i$ even,}
\end{eqnarray}
\noindent
where $M_{A,i}$ and $M_{B,i}$ are controlled unitaries with target given by
the original ${\mathcal{A}}$ (or ${\mathcal{B}}$) and new qubit number $i$ and control given
by the other $n-1$ new qubits. The matrices acts as the identity unless all
the $n-1$ control qubits are zero, in which case they apply the operation
\begin{eqnarray}
M_{A,i}\rightarrow \mypmatrix{E_{A,i}&I-E_{A,i}\cr I-E_{A,i}&E_{A,i}},
\qquad\qquad
M_{B,i}\rightarrow \mypmatrix{E_{B,i}&I-E_{B,i}\cr I-E_{B,i}&E_{B,i}}.
\end{eqnarray}
\noindent
where the blocks correspond to the computational basis of new qubit $i$.
It is not hard to check that the new protocol is indeed a valid
coin-flipping protocol and that the honest probabilities of winning $P_A$
and $P_B$ are the same as in the original protocol.
Now we take a dual feasible point for the original protocol and fix
$\epsilon>0$. We will construct a dual feasible point for the new protocol
with $\begin{eqnarray}ta'= \begin{eqnarray}ta+n\epsilon$. For $i$ even define
\begin{eqnarray}
Z_{A,i}' = \ket{0}\bra{0} \otimes (Z_{A,i}+(n-i)\epsilon I) + \Lambda_i F_i,
\end{eqnarray}
\noindent
where $F_i$ is a projector onto the space such that at least one of the new
qubits labeled $i+1$ through $n$ is non-zero, and $\Lambda_i\geq 0$ is a
constant to be determined in a moment. By construction
$Z_{A,n}'=\ket{0}\bra{0} \otimes Z_{A,n} = \Pi_{A,1}'$ and $\begin{eqnarray}ta' =
\bra{\psi_{A,0}'}Z_{A,0}'\ket{\psi_{A,0}'} =
\bra{\psi_{A,0}}Z_{A,0}+n\epsilon\ket{\psi_{A,0}}=\begin{eqnarray}ta+n\epsilon$ as required.
We can also set $Z_{A,i-1}'=Z_{A,i}'$ for $i$ even, so that the only
constraint that remains to be checked is
\begin{eqnarray}
Z_{A,i-2}'\otimes I_{\mathcal{M}} \geq
{U_{A,i-1}'}^\dagger \left(Z_{A,i}'\otimes I_{\mathcal{M}}\right) U_{A,i-1}'.
\end{eqnarray}
We will describe a decomposition
${\mathcal{A}}P={\mathcal{H}}_1\oplus{\mathcal{H}}_2\oplus{\mathcal{H}}_3\oplus{\mathcal{H}}_4$ such that both sides of the
above inequality are block diagonal with respect to it, and therefore we can
check the inequality on each block separately. The decomposition is
obtained by looking at the $n$ new qubits of ${\mathcal{A}}P$ from last (qubit $n$)
to first (qubit $1$) and picking out the first non-zero qubit.
\begin{eqnarray}gin{itemize}
\item ${\mathcal{H}}_1$ contains vectors where the first non-zero qubit is one of
$i+1,\dots,n$.
\item ${\mathcal{H}}_2$ contains vectors where the first non-zero qubit is either
$i-1$ or $i$, but excluding the vector where qubit $i-1$ is the only non-zero.
\item ${\mathcal{H}}_3$ contains vectors where the first non-zero qubit is one of
$1,\dots,i-2$.
\item ${\mathcal{H}}_4$ contains the space where all new qubits are zero or where
all qubits but qubit $i-1$ are zero.
\end{itemize}
On ${\mathcal{H}}_1$ we have $F_{i-2}=F_{i}=I$ so the inequality reads
$\Lambda_{i-2}I\geq\Lambda_{i} I$, and is satisfied so long as $\Lambda_i$
is a decreasing sequence. On ${\mathcal{H}}_2$ we have $F_{i-2}=I$ and $F_{i}=0$ so
the inequality reads $\Lambda_{i-2}I\geq 0$. On ${\mathcal{H}}_3$ we have
$F_{i-2}=F_{i}=0$ so the inequality reads $0\geq 0$. Finally, ${\mathcal{H}}_4$ is
the only space on which $M_{A,i-1}$ acts non-trivially. Writing
$X=Z_{A,i-2}\otimes I_{\mathcal{M}} + (n-i+2)\epsilon I$ and $Y=Z_{A,i}\otimes I_{\mathcal{M}} +
(n-i)\epsilon I$ and using $U\equiv U_{A,i-1}$, $E\equiv E_{A,i-1}$ we need
to check the block diagonal inequality
\begin{eqnarray}
\mypmatrix{X & 0 \cr 0 &
\Lambda_{i-2} I}
&\geq&
\mypmatrix{U^\dagger & 0 \cr 0 & U^\dagger}
\mypmatrix{E & I-E \cr I-E & E}
\mypmatrix{Y & 0 \cr 0 & 0}
\mypmatrix{E & I-E \cr I-E & E}
\mypmatrix{U & 0 \cr 0 & U}
\nonumber\\\nonumber
&& = \mypmatrix{U^\dagger E Y E U & U^\dagger E Y (I - E) U\cr
U^\dagger (I - E) Y E U & U^\dagger (I - E) Y (I - E) U}.
\end{eqnarray}
\noindent
From the constraints on the original dual feasible point we had
$Z_{A,i-2}\otimes I_{\mathcal{M}} \geq U^\dagger E (Z_{A,i}\otimes I_{\mathcal{M}}) E U =
U^\dagger E Y E U - (n-i)\epsilon E$ and therefore $X > U^\dagger E Y E
U$. In turn, this implies that for sufficiently large $\Lambda_{i-2}$ the
whole matrix inequality holds, and then we just need to make sure that
$\Lambda_{i-2}$ is also larger than $\Lambda_i$. We have not specified yet
$\Lambda_n$ but this one can be chosen to be zero as it effectively never
appears anywhere. That concludes the proof of the equivalence of protocols
with and without projective measurement.
\subsubsection{\label{sec:compiling}Compiling TDPGs into UBPs}
We had previously shown how to construct a TDPG out of a UBP. In this
section we will describe the reverse construction, thereby proving
TDPGs and UBPs equivalent.
We assume that all transitions in the given TDPG alternate between
horizontal and vertical (i.e., if we have a sequence of two valid
horizontal transitions we can combine them into a single valid transition
by removing the middle step). We also assume that the first transition $p_0
\rightarrow p_1$ is vertical and the last one $p_{n-1}\rightarrow p_n$ is
horizontal (which can be accomplished by adding trivial transitions at the
beginning or end). All TDPGs obtained from UBPs have this form.
We will also need to assume that in the given TDPG all non-trivial
one-variable transitions are strictly valid. This is justified by the
following lemma.
\begin{eqnarray}gin{lemma}
Given any TDPG $p_0,\dots,p_n$ with final point $[\begin{eqnarray}ta,\alpha]$ and an
$\epsilon>0$, there exists a second TDPG $q_0,\dots,q_m$ with final point
$[\begin{eqnarray}ta+\epsilon/2,\alpha+\epsilon/2]$ such that every
non-trivial one-variable transition is strictly valid.
\end{lemma}
\begin{eqnarray}gin{proof}
We construct the new TDPG by shifting up (or left) each set of points in
$q_i$ relative to its predecessor. More specifically let $\lceil i/2
\rceil$ and $\lfloor i/2 \rfloor$ be $i/2$ rounded up and down
respective. There are respectively the number of vertical and horizontal
transitions that have occurred to reach $p_i$. Now define the new TDPG by
\begin{eqnarray}
q_i(x,y) = p_i(x-\left\lfloor\frac{i}{2}\right\rfloor\frac{\epsilon}{n},
y-\left\lceil\frac{i}{2}\right\rceil\frac{\epsilon}{n}).
\end{eqnarray}
The final point is $[\begin{eqnarray}ta+\frac{\epsilon}{2},\alpha+\frac{\epsilon}{2}]$
as required. Also each non-trivial transition is now strictly valid: for
instance, if $q_i\rightarrow q_{i+1}$ is a horizontal transition and
$y\in[0,\infty)$ such that $\sum_z q_{i}(z,\underline y)\neq0$, then (using
$y'=y-\left\lceil\frac{i}{2}\right\rceil\frac{\epsilon}{n}$)
\begin{eqnarray}
\sum_z p_{i+1}'(z,\underline y) f(z)
&=& \sum_z p_{i+1}(z,\underline y') f(z +
\left\lfloor\frac{i}{2}\right\rfloor\frac{\epsilon}{n} + \frac{\epsilon}{n})
\\\nonumber
&>& \sum_z p_{i+1}(z,\underline y') f(z +
\left\lfloor\frac{i}{2}\right\rfloor\frac{\epsilon}{n})
\geq \sum_z p_{i}(z,\underline y')
f(z +\left\lfloor\frac{i}{2}\right\rfloor\frac{\epsilon}{n})
= \sum_z p_{i}'(z,\underline y) f(z)
\end{eqnarray}
\noindent
for every non-constant operator monotone $f$. The first inequality follows
because non-constant operator monotone functions are strictly monotone and
the second because $f(z +c)$ is operator monotone for $c\geq0$.
\end{proof}
\noindent
The next step in our argument relies on two lemmas which we state below
and prove in Appendix~\ref{sec:f2m}. They are the essential ingredient
which takes pairs of functions, such as those in a TDPG, and compiles them
back into the language of matrices.
\begin{eqnarray}gin{lemma}
Let $p(z)$ and $q(z)$ be functions $[0,\infty)\rightarrow[0,\infty)$ with
finite support. If $p(z)\rightarrow q(z)$ is strictly valid then there
exists positive semidefinite matrices $X$ and $Y$ and an (unnormalized)
vector $\ket{\psi}$ such that $X\leq Y$, $p=\Prob(X,\ket{\psi})$ and
$q=\Prob(Y,\ket{\psi})$.
\label{lemma:f2mmain}
\end{lemma}
\begin{eqnarray}gin{lemma}
The matrices $X$ and $Y$ in Lemma~\ref{lemma:f2mmain} can be chosen such that
\begin{eqnarray}gin{enumerate}
\item The spectrum of $X$ is equal to $\{0\}\cup S(p)$,
with all non-zero eigenvalues occurring once.
\item The spectrum of $Y$ is equal to $\{\Lambda\}\cup S(q)$,
for some large $\Lambda>0$, with all other eigenvalues occurring once.
\item The dimension of $X$ and $Y$ is no greater than
$|S(p)|+|S(q)|-1$.
\end{enumerate}
\noindent
where $S(p)$ and $S(q)$ are respectively the supports of $p$ and $q$.
\label{lemma:stdformmain}
\end{lemma}
Because we are now assuming that every non-trivial one-variable transition
is strictly valid, we can use Lemma~\ref{lemma:f2mmain} to turn the
transitions into matrices from which we will extract unitaries. However,
first we need to standardize the Hilbert space on which all of these
matrices are defined.
Let us fix a finite set $S$ of non-negative numbers, and assume we are
given a strictly valid transition $p\rightarrow q$ such that the supports
of both $p$ and $q$ are contained in $S$. What we want to argue is that we
can choose $X$ and $Y$ of Lemma~\ref{lemma:f2mmain} so that
their spectrum is exactly $S$ (union zero for $X$ or union some large value
$\Lambda$ for $Y$) and such that the only degenerate eigenvalues are zero
for $X$ and $\Lambda$ for $Y$.
The argument is simple, we start with $X$ and $Y$ satisfying the
requirements of Lemma~\ref{lemma:stdformmain}. If the $\Lambda$ appearing
in $Y$ is not larger than the maximum of $S$ we can simply increase it. Now
we just start appending in the missing eigenvalues from $S$ one at a time
(increasing the dimension of the matrices by using a direct sum). If some
value $c\in S$ is in $X$ but not in $Y$ we can append it to $Y$ if at the
same time we append a zero to $X$. Similarly if $c\in S$ is in $Y$ but not
in $X$ we can append it to $X$ if at the same time we append an extra
$\Lambda$ eigenvalue to $Y$. If $c\in S$ appears in neither matrix we
append it to both at the same time. The dimension of the new matrices so
constructed is no larger than $2|S|$. The dimension can be made exactly
equal to $2|S|$ by appending in extra zeros to $X$ and $\Lambda$s to $Y$.
To extract unitaries from these matrices, note that given any basis, we can
find a unitary $U$ such that $U X U^\dagger$ is diagonal and $U\ket{\psi}$
has non-negative coefficients with respect to this basis. In particular, if
we choose the basis that diagonalizes $Y$ and such that $\ket{\psi}$ has
non-negative coefficients, then we can find such a unitary $U$ and get
\begin{eqnarray}
Y_d \geq U^\dagger X_d U,
\end{eqnarray}
\noindent
where $X_d$ and $Y_d$ are diagonal in the computational basis with the same
spectrum as $X$ and $Y$ respectively. Additionally, by construction the
non-negative coefficients in the computational basis must have the form
$\ket{\psi}=\sum_i \sqrt{q_i}\ket{i}$ and
$U\ket{\psi}=\sum_i\sqrt{p_i}\ket{i}$. Putting everything together, we
obtain the following rather surprising lemma.
\begin{eqnarray}gin{lemma}
Let $S$ be a finite set of non-negative numbers. Let $p\rightarrow q$ be a
strictly valid transition such that the support of both $p(z)$ and $q(z)$ are
contained in $S$. Let ${\mathcal{H}}$ be the Hilbert space spanned by
$\{\ket{i,s}:i\in\{0,1\},s\in S\}$ and define
\begin{eqnarray}
Z = \sum_{s\in S} s \ket{0,s}\bra{0,s}.
\end{eqnarray}
Then there exists a sufficiently large number $\Lambda>0$ and a unitary $U$
such that
\begin{eqnarray}
U \sum_{s\in S} \sqrt{q(s)} \ket{0,s} = \sum_{s\in S} \sqrt{p(s)} \ket{0,s}
\qquad\text{and}\qquad
Z + \Lambda P_1 \geq U^\dagger Z U,
\label{eq:magicU}
\end{eqnarray}
\noindent
where $P_1=\sum_s\ket{1,s}\bra{1,s}$ is the projector onto the space where
the first qubit is one.
\label{lemma:magicL}
\end{lemma}
\noindent
Note that the result is non-trivial. If $p\rightarrow q$ is not valid
then no such unitary exists.
In the protocol below we will essentially be able to choose all our dual
operators equal to $\sum_{s\in S} s \ket{0,s}\bra{0,s} + \Lambda P_1$. We
then use our projective measurements to reset the eigenvalue $\Lambda$ to
zero on every transition so we can apply the above lemma. The unitaries,
though, require more care. During every bipartite transition
$p_i(x,y)\rightarrow p_{i+1}(x,y)$ we have many one-variable strictly valid
transitions $p_i(x,\underline y)\rightarrow p_{i+1}(x,\underline y)$ during
Alice's turn (or $p_i(\underline x,y)\rightarrow p_{i+1}(\underline x, y)$
during Bob's turn), each of which defines a different unitary in the above
lemma.What Alice needs to do on her turn is to apply a block diagonal
unitary with each block corresponding to a different strictly valid
transition. In other words she needs to be able to apply a controlled
unitary with control given by Bob's state. That is what the messages are
used for. Bob will store his state in an entangled subspace of
${\mathcal{M}}\otimes{\mathcal{B}}$ so that Alice can use it as control but not change it too
much. Similarly, Alice will store her state in an entangled subspace of
${\mathcal{A}}\otimes{\mathcal{M}}$ so that Bob can access it. This construction is similar to
one used in
\cite{KMP04}.
We are now ready to construct the protocol. Fix a TDPG by $p_0,\dots,p_n$
with strictly valid transitions and final point $[\begin{eqnarray}ta,\alpha]$. The
protocol will be defined with the same $n$ representing the number of
messages.
To define the relevant Hilbert spaces let $S_A$ (resp. $S_B$) be the finite
set of $x$ coordinates (resp. $y$ coordinates) of points that are assigned
non-zero probability by $p_i(x,y)$ for some $i$. By construction
$0,1,\begin{eqnarray}ta\in S_A$ and $0,1,\alpha \in S_B$. Set
\begin{eqnarray}
{\mathcal{A}} &=& \vspan\{\ket{i,s_a}:i\in\{0,1\},s_a\in S_A\},\\
{\mathcal{M}} &=& \vspan\{\ket{s_a,s_b}:s_a\in S_A,s_b\in S_B\},\\
{\mathcal{B}} &=& \vspan\{\ket{s_b,i}:s_b\in S_B,i\in\{0,1\}\}.
\end{eqnarray}
\noindent
It will ocassionally be useful to write ${\mathcal{M}}={\mathcal{A}}P\otimes{\mathcal{B}}P$ where
${\mathcal{A}}P = \vspan\{\ket{s_a}:s_a\in S_A\}$ and
${\mathcal{B}}P = \vspan\{\ket{s_b}:s_b\in S_B\}$.
The initial states will be
\begin{eqnarray}
\ket{\psi_{0,A}} = \ket{0,\begin{eqnarray}ta},\qquad
\ket{\psi_{0,M}} = \ket{\begin{eqnarray}ta,\alpha},\qquad
\ket{\psi_{0,B}} = \ket{\alpha,0}.
\end{eqnarray}
The projections that follow the unitaries will have the form
\begin{eqnarray}
E_{A,i} = E_A &\equiv& \sum_{s_a\in S_A} \ket{0,s_a}\bra{0,s_a}
\otimes \ket{s_a}\bra{s_a}\otimes I_{\mathcal{B}}P,\\
\qquad
E_{B,i} = E_B &\equiv& I_{{\mathcal{A}}P}\otimes \sum_{s_b\in S_B} \ket{s_b}\bra{s_b}
\otimes \ket{s_b,0}\bra{s_b,0},
\end{eqnarray}
\noindent
where $E_{A,i}$ is defined for $i$ odd and $E_{B,i}$ for $i$
even. Basically $E_A$ acts on ${\mathcal{A}}\otimes{\mathcal{M}}$ and ensures that
the first qubit is $0$ and the registers $s_a$ in ${\mathcal{A}}$ and ${\mathcal{M}}$ agree.
The final measurement operators will have the form
\begin{eqnarray}
\Pi_{A,1} = \ket{0,1}\bra{0,1},\qquad \Pi_{B,0} = \ket{1,0}\bra{1,0}
\end{eqnarray}
\noindent
with $\Pi_{A,0}=I-\Pi_{A,1}$ and $\Pi_{B,1}=I-\Pi_{B,0}$.
All that remains is to describe the unitaries, which will be done in a
moment. The unitaries are to be chosen so that the honest state during the
$i$th messages is
\begin{eqnarray}
\ket{\psi_i} = \sum_{s_a\in S_A, s_b\in S_B} \sqrt{p_{n-i}(s_a,s_b)}\,
\ket{0,s_a}\otimes\ket{s_a,s_b}\otimes\ket{s_b,0},
\label{eq:TDPGpsi}
\end{eqnarray}
\noindent
where we remind the reader of our ``reverse time'' convention of TDPGs
relative to protocols. The above definition agrees with our choice of
initial state $\ket{\psi_0}$. It is also easy to see that the projection
operations that follow the unitaries will always succeed if both players are
honest. Finally, we have $\ket{\psi_n}= P_B\ket{0,1,1,0,0,0} +
P_A\ket{0,0,0,1,1,0}$ so Alice and Bob will agree on the coin outcome,
which will have the required probability distribution.
Before defining the unitaries, we fix a single $\Lambda>0$ larger than all
elements of $S_A$ and $S_B$ and large enough so that all the strictly valid
transitions in the given TDPG can be turned into unitaries satisfying
the constraints of Eq.~(\ref{eq:magicU}).
To construct the unitaries for Alice it is useful to define the subspace
${\mathcal{A}}B\subset{\mathcal{A}}\otimes{\mathcal{A}}P$ given by
${\mathcal{A}}B=\{\ket{i,s_a,s_a}:i\in\{0,1\},s_a\in S_A\}$ and let $P_{\mathcal{A}}B^\perp$ be
the the projector onto the complement of ${\mathcal{A}}B$ in ${\mathcal{A}}\otimes{\mathcal{A}}P$. Let
$i$ be odd so that we can build $U_{A,i}$ out of the horizontal transition
$p_{n-i}\rightarrow p_{n-i+1}$. The unitary will be block diagonal of the
form
\begin{eqnarray}
U_{A,i} = \sum_{s_b\in S_B} \left(U_{A,i}^{(s_b)} + P_{\mathcal{A}}B^\perp \right)
\otimes \ket{s_b}\bra{s_b},
\end{eqnarray}
\noindent
where $U_{A,i}^{(s_b)}$ can be viewed as a unitary operator on ${\mathcal{A}}B$.
If $p_{n-i}(s_a,\underline{s_b})= p_{n-i+1}(s_a,\underline{s_b})$ then we
choose $U_{A,i}^{(s_b)}=I$ otherwise
$p_{n-i}(s_a,\underline{s_b})\rightarrow p_{n-i+1}(s_a,\underline{s_b})$
is strictly valid and by Lemma~\ref{lemma:magicL} we can choose
$U_{A,n-i}^{(s_b)}$ on ${\mathcal{A}}B$ such that
\begin{eqnarray}
\begin{array}r{Z}_A + \Lambda \begin{array}r{P}_1
\geq {U_{A,i}^{(s_b)}}^\dagger \begin{array}r{Z}_A U_{A,i}^{(s_b)}
\end{eqnarray}
for $\begin{array}r{Z}_A = \sum_{s_a} s_a \ket{0,s_a,s_a}\bra{0,s_a,s_a}$,
$\begin{array}r{P}_1 = \sum_{s_a} \ket{1,s_a,s_a}\bra{1,s_a,s_a}$ and such that
\begin{eqnarray}
U_{A,i}^{(s_b)}\sum_{s_a\in S_A}\sqrt{p_{n-i+1}(s_a,s_b)}\ket{0,s_a,s_a}
= \sum_{s_a\in S_A}\sqrt{p_{n-i}(s_a,s_b)}\ket{0,s_a,s_a}.
\end{eqnarray}
We can now directly verify the equation $U_{A,i}\otimes I_{\mathcal{B}}
\ket{\psi_{i-1}} = \ket{\psi_{i}}$ with states given by
Eq.~(\ref{eq:TDPGpsi}). The unitaries $U_{B,i}$ for Bob are defined
analogously, and otherwise we have completed the description of the
coin-flipping protocol associated to the TDPG.
What remains is to describe dual feasible points for the above protocol
that prove the bounds $P_A^*\leq\alpha$ and $P_B^*\leq\begin{eqnarray}ta$. If we define
the operators
\begin{eqnarray}
Z_A &=& \sum_{s_a} s_a \ket{0,s_a}\bra{0,s_a} + \Lambda \sum_{s_a}
\ket{1,s_a}\bra{1,s_a}\\
Z_B &=& \sum_{s_b} s_b \ket{0,s_b}\bra{0,s_b} + \Lambda \sum_{s_b}
\ket{1,s_b}\bra{1,s_b}
\end{eqnarray}
on ${\mathcal{A}}$ and ${\mathcal{B}}$ respectively. The desired dual feasible points are given
by $Z_{A,i}=Z_{A}$ and $Z_{B,i}=Z_{B}$ for all $i$ except that we must set
$Z_{A,n}=Z_{A,n-1}=\Pi_{A,1}$ and $Z_{B,n}=\Pi_{B,0}$ as required by our
slightly inflexible definitions. As usual, we will only verify the case of
Alice honest and Bob cheating as the other case is nearly identical.
We trivially have $Z_{A,0}\ket{\psi_{A,0}}=\begin{eqnarray}ta \ket{\psi_{A,0}}$.
The main constraint that we need to verify is
\begin{eqnarray}
Z_{A,i-1}\otimes I_{\mathcal{M}} \geq U_{A,i}^\dagger E_{A} \left(
Z_{A,i}\otimes I_{\mathcal{M}} \right)
E_{A} U_{A,i}
\label{eq:TDPGdual}
\end{eqnarray}
\noindent
for $i$ odd. The special case of $i=n-1$ will be proven if we show that the
above inequality holds with $Z_{A,n-1}=Z_A$ because $Z_A\geq \Pi_{A,1}$ and
inequalities are transitive.
First we note that in all cases
\begin{eqnarray}
E_{A} \left( Z_{A,i}\otimes I_{\mathcal{M}} \right) E_{A} = \begin{array}r{Z}_A \otimes I_{\mathcal{B}}P
\equiv \sum_{s_a} s_a \ket{0,s_a,s_a}\bra{0,s_a,s_a}\otimes I_{\mathcal{B}}P,
\end{eqnarray}
\noindent
where $\begin{array}r{Z}_A$ has support on ${\mathcal{A}}B$. The unitary $U_{A,i}$ maps
${\mathcal{A}}B\otimes{\mathcal{B}}P$ to itself, so the right hand side of
Eq.~(\ref{eq:TDPGdual}) has support on ${\mathcal{A}}B\otimes{\mathcal{B}}P$. The operator
$Z_{A,i-1}\otimes I_{\mathcal{M}}$ is block diagonal with respect to the
decomposition of ${\mathcal{A}}\otimes{\mathcal{M}}$ into ${\mathcal{A}}B\otimes{\mathcal{B}}P$ and its complement,
and so the inequality is trivially satisfied in the latter space. In
${\mathcal{A}}B\otimes{\mathcal{B}}P$ what remains to be shown is
\begin{eqnarray}q
\left(\begin{array}r{Z}_A + \Lambda \sum_{s_a\in S_A} \ket{1,s_a,s_a}\bra{1,s_a,s_a}\right)
\otimes I_{\mathcal{B}}P \geq U_{A,i}^\dagger
\left( \begin{array}r{Z}_A \otimes I_{\mathcal{B}}P \right) U_{A,i} =
\sum_{s_b\in S_B} {U_{A,i}^{(s_b)}}^\dagger \begin{array}r{Z}_A U_{A,i}^{(s_b)}
\otimes\ket{s_b}\bra{s_b}
\end{eqnarray}q
\noindent
and the inequality follows from the definition of $U_{A,i}^{(s_b)}$.
What we have proven is that we can take a TDPG with strictly valid
transitions and final point $[\begin{eqnarray}ta,\alpha]$ and turn into a UBP with
projections with bound $(\begin{eqnarray}ta,\alpha)$. However, in the last section we
proved that UBPs with projections come arbitrarily close to regular UBPs,
and at the top of this section we proved that TDPGs with strictly valid
transitions come arbitrarily close to any arbitrary TDPGs. Therefore, we
have proven that TDPGs are equivalent to UBPs, which we state formally as
an extended version of Theorem~\ref{thm:kitmain1}:
\begin{eqnarray}gin{theorem}
Let $f(\begin{eqnarray}ta,\alpha):\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$f(\alpha',\begin{eqnarray}ta')\geq f(\alpha,\begin{eqnarray}ta)$ whenever $\alpha'\geq\alpha$ and
$\begin{eqnarray}ta'\geq\begin{eqnarray}ta$, then
\begin{eqnarray}
\inf_{\text{proto}} f(P_B^*,P_A^*) = \inf_{UBP} f(\begin{eqnarray}ta,\alpha)
= \inf_{TDPG} f(\begin{eqnarray}ta,\alpha),
\end{eqnarray}
\noindent
where the left optimization is carried out over all coin-flipping protocols,
the middle one is carried out over all upper-bounded protocols, and the
right one is carried out over all time dependent point games.
\label{thm:kitmain2}
\end{theorem}
Note that the above construction of a UBP out of a TDPG is not optimal
in terms of resources. In particular, in most cases the number
of qubits needed could be drastically reduced. It is also not in general
true that if we start with a UBP, translate it into a TDPG and then
construct from it a UBP we will end up with the same one. In fact, given
two UBPs with the same underlying protocol but different dual feasible
points, if we converted into TDPGs and then back into UBPs, the resulting
protocols will be radically different!
\section{\label{sec:examples}The illustrated guide to point games}
The purpose of this section is to build an intuition about point games. In
the first part of the section we will classify a few of the simplest valid
transitions. These moves will be part of a basic toolbox of transitions
that will be used throughout the paper.
In the second part of this section we use our basic moves to build some
simple coin-flipping protocols, all described in the language of TDPGs. The
examples include the bias $1/\sqrt{2}-1/2$ protocol of Spekkens and Rudolph
\cite{Spekkens2002} and the author's bias $1/6$ protocol \cite{me2005}.
In this section we will make extensive use of the basis for functions with
finite support introduced in the last section. In particular, we use $[z]$
to denote a one variable function that evaluates to one at a fixed point
$z$, and is zero everywhere else. We also use $[x,y]$ to denote a
two-variable function that is one at $(x,y)$ and zero everywhere else.
\subsection{Basic moves}
We aim to systematically describe all non-trivial one-variable valid
transitions of the following forms: one points to one point, two points to
one point and one point to two points. The latter two respectively
generate all transitions of the form $n$ points to one point and one point
to $n$ points.
Nevertheless, these will not form a complete basis of all valid
transitions. Even two point to two point transitions contain moves that
cannot be generated by the above set. Ultimately, the most concise
description of the set of all valid transitions is as the dual to the cone
of operator monotone functions.
\subsubsection{Point raising}
All possible one point to one point transitions have the form
\begin{eqnarray}
p[z]\rightarrow p[z'],
\end{eqnarray}
\noindent
where we have already imposed the constraint of probability conservation,
and we assume $p>0$. We now need to impose the constraints $p f(z) \leq p
f(z')$ for all operator monotone functions. Using $f(z)=z$ we have obtain
the necessary condition
\begin{eqnarray}
z\leq z'.
\end{eqnarray}
It is also sufficient because all operator monotone functions are
monotonically increasing.
When working in a bipartite case we see that $p[x,y]\rightarrow p[x',y]$ is
valid if and only if $x\leq x'$ (and similarly with $p[x,y]\rightarrow
p[x,y']$ and $y\leq y'$). More generally, $p[x,y]\rightarrow p[x',y']$ is
transitively valid if and only if $x\leq x'$ and $y\leq y'$. In simpler
words: we can always move points upwards or rightwards but not downwards or
leftwards. We will call these moves point raising (even when we are moving
rightwards).
Note that the presence of extra unmoving points does not affect any of the
one variable transitions. The transition $p[z]\rightarrow p[z']$ is valid
if and only if $p[z] + \sum_i p_i[z_i] \rightarrow p[z'] + \sum_i p_i[z_i]$
is valid (where $\sum_i p_i[z_i]$ is any other set of points with positive
probability).
\subsubsection{Point merging}
All possible two point to one point transitions have the form
\begin{eqnarray}
p_1[z_1]+p_2[z_2]\rightarrow (p_1+p_2)[z'],
\end{eqnarray}
\noindent
where we have already imposed the constraint of probability conservation,
and we assume $p_1>0$ and $p_2>0$. We now need to impose the constraints
$p_1 f(z_1) + p_2 f(z_2) \leq (p_1+p_2) f(z')$ for all operator monotone
functions. Using $f(z)=z$ we have obtain the necessary condition
\begin{eqnarray}
\frac{p_1 z_1 + p_2 z_2}{p_1 + p_2} \leq z'.
\label{eq:pointmerge}
\end{eqnarray}
It is also sufficient because operator monotone functions are concave (a
property that can be checked directly on the extremal functions
$f(z)=\frac{\lambda z}{\lambda+z}$ for $\lambda\in(0,\infty)$).
When equality holds in Eq.~(\ref{eq:pointmerge}) we call the move point
merging. The more general case is simply generated by point merging
followed by point raising). In simpler words: point merging takes two
points and replaces them with a single point carrying their combined
probability and average $z$ value.
For $n$ points merging into one point it is easy to see that we must
conserve probability and average $z$ (or strictly speaking average $z$
cannot decrease). But this exact final configuration can be achieved using
a sequence of pairwise point merges with a possible point raising at the
end.
For the bipartite case, the transition
\begin{eqnarray}
p_1[x_1,y]+p_2[x_2,y]\rightarrow
(p_1+p_2) \left[\frac{p_1 x_1 + p_2 x_2}{p_1 + p_2},y\right]
\end{eqnarray}
is clearly valid, and similarly with $x$ and $y$ interchanged. However, it
does not follow that the transition $0.5[0,0]+0.5[1,1]\rightarrow
1[0.5,0.5]$ is transitively valid. In fact, the proof of the impossibility
of strong coin flipping is a proof that $0.5[0,0]+0.5[1,1]\rightarrow
1[z,z]$ is transitively invalid if $z<1/\sqrt{2}$.
\subsubsection{Point splitting}
All possible one point to two point transitions have the form
\begin{eqnarray}
(p_1+p_2)[z] \rightarrow p_1[z_1']+p_2[z_2'],
\end{eqnarray}
\noindent
where we have already imposed the constraint of probability conservation,
and we assume $p_1>0$ and $p_2>0$. We now need to impose the constraints
$(p_1+p_2) f(z) \leq p_1 f(z_1') + p_2 f(z_2')$ for all operator monotone
functions. In particular, for $\lambda\in(0,\infty)$ the inequality is
satisfied for $f(z)=\frac{\lambda z}{\lambda+z} =
\lambda(1-\frac{\lambda}{\lambda+z})$ if and only if it is satisfied for
$f(z)=-\frac{1}{\lambda+z}$. If none of the points are located at zero,
then we can take the limit $\lambda\rightarrow 0$ and the inequality must
still be satisfied. In other words, a necessary condition is
\begin{eqnarray}
-\frac{p_1+p_2}{z} \leq - \frac{p_1}{z_1'} - \frac{p_2}{z_2'}.
\label{eq:pointsplit}
\end{eqnarray}
It is also sufficient. Let $w=1/z$ and assume the above inequality holds,
then verifying the original constraint with $f(z)=-\frac{1}{\lambda+z}$ is
equivalent to verifying $(p_1+p_2) g(w)\leq (p_1+p_2) g(\frac{p_1 w_1 + p_2
w_2}{p_1 + p_2}) \leq p_1 g(w_1') + p_2 g(w_2')$ with
$g(w)=-\frac{w}{1+\lambda w}$. But the first inequality holds because
$g(w)$ is monotonically decreasing and the second inequality holds because
$g(w)$ is convex. The special case of $f(z)=z$ follows by considering
the limit $\lambda\rightarrow\infty$ of $f(z)=\frac{\lambda z}{\lambda+z}$.
When equality holds in Eq.~(\ref{eq:pointsplit}) we call the move point
splitting. In simpler words: point splitting takes a point and replaces it
with two points such that the total probability and average $1/z$ is
conserved. All one point to two points valid transitions can then be
generated by point raising followed by point splitting. Similarly, all one
point to $n$ point valid transitions can be generated by point-raising and
a sequence of one-to-two point splittings.
The above arguments hold provided none of the points is located at zero. If
either $z_1'=0$ or $z_2'=0$ it is not hard to verify that we must also have
$z=0$. If $z=0$ all valid moves can be generated first by splitting the
point into two points located at zero and then using point raising
to move them to the required destination. In fact, we allow this type of
point splitting anywhere: we can always replace a point at $z$ with
probability $p$ with two points at $z$ and probabilities that add to $p$.
\subsubsection{Summary}
\begin{eqnarray}gin{lemma}
The following are valid transitions:
\begin{eqnarray}gin{itemize}
\item Point raising
\begin{eqnarray}
p[z]\rightarrow p[z']\qquad\qquad\text{(for $z\leq z'$).}
\end{eqnarray}
\item Point merging
\begin{eqnarray}
p_1[z_1]+p_2[z_2]\rightarrow \left(p_1+p_2\right)
\left[\frac{p_1 z_1+ p_2 z_2}{p_1+p_2}\right].
\end{eqnarray}
\item Point splitting
\begin{eqnarray}
\left(p_1+p_2\right)\left[\frac{p_1+p_2}{p_1 w_1'+ p_2 w_2'}
\right]\rightarrow p_1\left[\frac{1}{w_1'}\right]+p_2\left[\frac{1}{w_2'}\right].
\end{eqnarray}
\end{itemize}
\end{lemma}
\subsection{Basic protocols}
The simplest of all protocols are the ones when one player flips a coin
and tells the outcome to the other player. If Alice is in charge of
flipping the coin we get the TDPG
\begin{eqnarray}
\frac{1}{2}[1,0]+\frac{1}{2}[0,1] \quad\rightarrow\quad
\frac{1}{2}[1,1]+\frac{1}{2}[0,1] \quad\rightarrow\quad
1\left[\frac{1}{2},1\right],
\end{eqnarray}
\noindent
where the first move is point raising and the second is point
merging. Similarly, if Bob is in charge of flipping the coin we get
\begin{eqnarray}
\frac{1}{2}[1,0]+\frac{1}{2}[0,1] \quad\rightarrow\quad
\frac{1}{2}[1,0]+\frac{1}{2}[1,1] \quad\rightarrow\quad
1\left[1,\frac{1}{2}\right].
\end{eqnarray}
\noindent
These are graphically illustrated in Fig.~\ref{fig:prototriv}.
\begin{eqnarray}gin{figure}[tb]
\begin{eqnarray}gin{center}
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\qquad
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\end{picture}
\caption{The two trivial protocols where Alice (left) or Bob (right) flip a
coin and announce the outcome.}
\label{fig:prototriv}
\end{center}
\end{figure}
Note that in the second protocol Bob sends the first non-trivial message
(equivalently, the final move is vertical). Whereas for UBPs we enforced
the constraint that Alice always sent the first message, for TDPGs we will
let the first message be sent by whomever it is convenient.
To assign some meaning to the abstract operations we can think of point
merging as occurring when a player flips a coin and announces the outcome
to their opponent (recall the reverse time convention, in regular time
point merging occurs first and makes two points out of one). Point raising
does not correspond to any physical operation, but rather to one player
trusting or at least accepting the state provided by the other player.
\subsubsection{The Spekkens and Rudolph protocol}
Fix $x\in(1/2,1)$. Consider
\begin{eqnarray}
\frac{1}{2}[1,0]+\frac{1}{2}[0,1]
\quad\rightarrow\ &
\frac{2x-1}{2x}\bigg[x,0\bigg] +
\frac{1-x}{2x}\bigg[\frac{x}{1-x},0\bigg] +
\frac{1}{2}[0,1]&
\\\nonumber
\quad\rightarrow\ &
\frac{2x-1}{2x}\bigg[x,0\bigg] +
\frac{1-x}{2x}\bigg[\frac{x}{1-x},1\bigg] +
\frac{1}{2}[0,1]&
\\\nonumber
\quad\rightarrow\ &
\frac{2x-1}{2x}\bigg[x,0\bigg] +
\frac{1}{2x}\bigg[x,1\bigg]&
\ \rightarrow\quad
1 \bigg[x,\frac{1}{2x} \bigg].
\end{eqnarray}
\noindent
The TDPG is the sequence: split, raise, merge, merge. It is illustrated in
Fig.~\ref{fig:protosnr} for the case $x=1/\sqrt{2}$. The resulting
protocol satisfies $P_B^*= x$ and $P_A^*= \frac{1}{2x}$, achieving
the tradeoff curve $P_A^* P_B^*=1/2$ from \cite{Spekkens2002}.
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\caption{The Spekkens and Rudolph protocol with $x=1/\sqrt{2}$.}
\label{fig:protosnr}
\end{center}
\end{figure}
From the point of view of point games, the clever step above is the initial
split which was chosen so that, after the first merge, the remaining points
would be vertically aligned and a second merge could immediately take
place. The initial split corresponds to the cheat detection carried out at
the end of the protocol.
Another interpretation of the compromises made in the above protocol can be
understood as follows: We know that in each move the average value of $x$
and $y$ cannot decrease because $f(z)=z$ is operator monotone (and
$f(x,y)=x+y$ is bi-operator monotone). A perfect zero-bias protocol would
never increase these averages. In a non-perfect protocol every such
increase gets added to the final bias. In particular, the above protocol
has two such ``bad'' steps: the split (which increases average $x$) and the
raise (which increases average $y$). The protocol with
$P_A^*=P_B^*=1/\sqrt{2}$ balances these two effects so that they are equal.
\subsubsection{\label{sec:TDPG16}Quantum public-coin protocols}
The Spekkens and Rudolph protocol can be improved by using the TDPG
depicted by Fig.~\ref{fig:protome} (left). The idea is that we begin by
splitting the initial point on the vertical axis and use the resulting top
point to help raise the rightmost point. The operation on the rightmost
line becomes a point merging which preserves average $y$ instead of the old
point raising which increased average $y$. The cost of doing this, though,
is the split on the vertical axis (which increases average $y$) and the
point raising of the top point (which increases average $x$). Nevertheless,
when the parameters (probabilities and coordinates) are chosen properly the
above pattern results in a improvement.
Closer inspection shows that the added structure of the above protocol
relative to the Spekkens and Rudolph protocol is very similar to the added
structure of the Spekkens and Rudolph relative to the trivial protocol
where Bob announces the coin outcome. In fact, the process can be iterated
as depicted in Fig.~\ref{fig:protome} (right). The process begins by
splitting the two initial points into many points on the axes. Then point
raising is used on the rightmost point so that it is aligned with the
topmost point. The two points are merged and the resulting point ends up
lined up with the second-rightmost point. These two are again merged
producing a point that is lined up with the second-topmost point. All point
are merged in this fashion until a single point remains.
Obviously, the initial splits must be chosen with care so that all the
merges end up properly lined up. We will not describe here the precise
parameters needed to achieve this, though the details can be found in
\cite{me2005}. In fact, the paper describes an even larger family of
coin-flipping protocols which consisted of classical public-coin protocol
with quantum cheat detection. In the language of TDPGs all the protocols in
the family can be characterized as follows: First the point $P_B[1,0]$
splits horizontally into as many points as needed. Similarly the point
$P_A[0,1]$ splits vertically as needed. These steps are the cheat
detection. After that only point raising and merging are allowed, though in
any order or pattern desired. Sadly, the optimal bias that can be achieved
with protocols of this form is $1/6$, and is realized by the pattern from
Fig.~\ref{fig:protome} (right) in the limit of an arbitrarily large number
of merges.
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\put(2.5,0){\vector(1,0){1}}
\put(3.5,0){\vector(0,1){3}}
\put(3.5,3){\vector(-1,0){1}}
\put(0,3){\vector(0,1){1}}
\put(0,4){\vector(1,0){3.5}}
\put(3.5,4){\vector(0,-1){1}}
\put(3.5,0){\vector(1,0){1}}
\put(4.5,0){\vector(0,1){4}}
\put(4.5,4){\vector(-1,0){1}}
\end{picture}
\caption{An improvement to the Spekkens and Rudolph protocol
(left) and further iterations of the improvement (right).
Figures not to scale.}
\label{fig:protome}
\end{center}
\end{figure}
An improved version of the above protocol is presented in
Appendix~\ref{sec:ddb}, where we effectively note that the initial splits
can be done gradually as the protocol progresses (or equivalently, that
cheat-detection can be done gradually). The advantage of this is a
reduction in the number of required qubits to a constant number. The bias,
though, remains fixed at $1/6$.
Below we intend to give an informal description of the bias $1/6$ protocol
in the limit of infinitely many messages, in which our standard ``finite
points with probability'' TDPGs get replaced by probability
densities. Though we will not formalize these TDPGs with probability
densities, they are occasionally useful in studying protocols. In fact, the
main result of the paper will have this form, though in the formal proof we
shall approximate the continuous distribution by a discrete finite set.
Let us imagine that we have carried out the initial point-splitting, and
that we have split into so many points that we effectively have a
continuous probability density on the axes:
\begin{eqnarray}
\frac{1}{2}\int_{z^*}^\infty p(z) [z,0] dz +
\frac{1}{2}\int_{z^*}^\infty p(z) [0,z] dz,
\end{eqnarray}
\noindent
where $p(z)$ is some probability distribution with $\int_{z^*}^\infty p(z)
dz=1$, and $z^*>0$ is some cutoff below which no points are located.
The continuum limit of the process depicted in Fig.~\ref{fig:protome}
(right) consists of a point moving along the diagonal and collecting the
probability density off of the axes. The point starts at $[\infty,\infty]$
with zero probability and ends at $[z^*,z^*]$ once it has collected all the
probability. What we are trying to determine is for what probability
distributions $p(z)$ is such as thing possible. In other words, for what
probability distributions $p(z)$ is
\begin{eqnarray}
\frac{1}{2}\int_{z^*}^\infty p(z) [z,0] dz +
\frac{1}{2}\int_{z^*}^\infty p(z) [0,z] dz
\quad\rightarrow\quad
1 [z^*,z^*]
\end{eqnarray}
transitively valid?
Let $Q(z)$ be the probability of the point that is traveling down the
diagonal. Given the point $Q(z)[z,z]$ we can move downwards and
rightwards in a two step process: first we merge with a ``point'' on the
$x$-axis (with effective probability $\frac{p(z)}{2}dz$) to get to $(Q(z) +
p(z)/2 dz)[z,z-dz]$, then we merge with a ``point'' on the $y$-axis (again
with effective probability $\frac{p(z)}{2}dz$) to get to $(Q(z)+p(z)
dz)[z-dz,z-dz]$. Conservation of probability tells us that $Q(z-dz) =
Q(z)+p(z) dz$ or
\begin{eqnarray}
\frac{dQ(z)}{dz} = -p(z).
\end{eqnarray}
\noindent
But these transitions are point merges and should additionally conserve
average height during the first merge (and average $x$ position during the
second merge). In particular, we get a constraint of the form $(Q(z)) z +
(\frac{p(z)}{2} dz) 0 = (Q(z)+\frac{p(z)}{2}dz) (z-dz)$. Canceling a few
terms we get $0=-Q(z)dz+\frac{p(z)}{2}z dz$ or
\begin{eqnarray}
Q(z)=\frac{z p(z)}{2}.
\end{eqnarray}
\noindent
Combining the two constraints we get a differential equation in $Q(z)$
\begin{eqnarray}
\frac{dQ(z)}{dz} = - \frac{2Q(z)}{z}
\end{eqnarray}
\noindent
solved by $Q(z)=c/z^2$ for some constant $c$. Our original probability
distribution must have the form $p(z)=2c/z^3$, and we can now fix the
constant by the requirement $\int_{z^*}^\infty p(z) dz=1$. We get
$c=(z^*)^2$.
What our arguments above have shown is that for any $z^*>0$ the transition
\begin{eqnarray}
\frac{1}{2}\int_{z^*}^\infty \frac{2(z^*)^2}{z^3} [z,0] dz +
\frac{1}{2}\int_{z^*}^\infty \frac{2(z^*)^2}{z^3}[0,z] dz
\quad\rightarrow\quad
1 [z^*,z^*]
\label{eq:ddbmerge}
\end{eqnarray}
\noindent
is transitively valid.
But, of course, the true starting state is
$\frac{1}{2}[1,0]+\frac{1}{2}[0,1]$. To reach the initial state above we
must use point splitting independently on each axis. The constraints are
conservation of probability (already imposed) and a non-increasing average
$1/z$:
\begin{eqnarray}
1 \geq \int_{z^*}^\infty \frac{p(z)}{z} dz
= \int_{z^*}^\infty \frac{2(z^*)^2}{z^4} dz = \frac{2}{3z^*}
\quad\mathbb{R}ightarrow\quad \frac{2}{3}\leq z^*.
\label{eq:ddbsplit}
\end{eqnarray}
\noindent
In other words, we can achieve bias $1/6$ but no better.
\subsubsection{\label{sec:cheat}Protocols with cheat detection}
As mentioned in the introduction, even bit commitment can be accomplished
using quantum information if we are willing to settle for a cheat-detecting
solution. These protocols with cheat detection may prove to be an important
component of quantum cryptography.
As an example of how Kitaev's formalism can be extended to study
cheat-detecting protocols we will describe in this section a simple
generalization of the above protocol.
One approach to cheat detection is as a payoff maximization
problem. Specifically, in coin flipping we would formulate the problem as
follows: winning the coin flip earns you \$1, loosing the coin flip nets
you \$0, but if you get caught cheating you will lose
\$$\Lambda$ (i.e., the cheating player wins $-\$\Lambda$
and we assume $\Lambda\geq 0$). Calculating the maximum expected earnings
for each $\Lambda$ is equivalent to finding the tradeoff curve between the
probability of winning by cheating and the probability of getting caught
cheating.
As most of our equations so far have been designed for positive
semidefinite matrices, it is simplest to begin by modifying our payouts so
that they are all non negative: \$$(\Lambda+1)$ for winning, \$$\Lambda$ for
losing, and \$0 for getting caught cheating. These two formulations are
equivalent in that an optimal payout for one can be found from the optimal
payout for the other by adding/subtracting $\Lambda$.
Accommodating multiple payouts only requires modifying the final projection
operators. For instance, $Z_{A,n}=\Pi_{A,1}$ (which could be though of as a
payout matrix for the non-cheat detecting problem), now needs to be replaced
with a matrix with three eigenvalues: $\Lambda+1$, $\Lambda$ and $0$ so
that the honest states in which Bob wins have the appropriate eigenvalue
\begin{eqnarray}
Z_{A,n} \left( \Pi_{A,1}\ket{\psi_{A,n}}\right) =
(\Lambda+1) \left( \Pi_{A,1}\ket{\psi_{A,n}}\right),
\qquad\qquad
Z_{A,n} \left( \Pi_{A,0}\ket{\psi_{A,n}}\right) =
\Lambda \left( \Pi_{A,0}\ket{\psi_{A,n}}\right),
\end{eqnarray}
\noindent
and the orthogonal subspace has eigenvalue zero. In particular, everything
we have done so far is still valid, but the transition of interest becomes
\begin{eqnarray}
\frac{1}{2}[\Lambda+1,\Lambda]+\frac{1}{2}[\Lambda,\Lambda+1]
\quad\rightarrow\quad
1[\begin{eqnarray}ta,\alpha],
\end{eqnarray}
\noindent
where now $\begin{eqnarray}ta$ and $\alpha$ are upper bounds on the expected payouts of
Bob and Alice respectively.
What would happen if we were now to shift back to the original payouts of
\$1, \$0 and \$$-\Lambda$? We would again return to looking for transitions
of the form $\frac{1}{2}[1,0]+\frac{1}{2}[0,1]\rightarrow 1[\begin{eqnarray}ta,\alpha]$,
however our constraints for valid transitions would need to change. Rather
than looking for transitions that live in the dual to the cone of operator
monotone functions with domain $[0,\infty)$ we would look for transitions
in the dual to the cone of operator monotone functions with domain
$[-\Lambda,\infty)$.
In other words, in a setting with a penalty of $\Lambda$ for cheating, a
transition $p_i(z)\rightarrow p_{i+1}(z)$ is valid if and only if
probability is conserved and
\begin{eqnarray}
\sum_z p_i(z) \frac{\lambda z}{\lambda+z} \leq
\sum_z p_{i+1}(z) \frac{\lambda z}{\lambda+z}
\end{eqnarray}
\noindent
for $\lambda\in(\Lambda,\infty)$. The old rule is simply the special case
$\Lambda=0$.
Note that for $\Lambda>0$ we are simply removing restrictions from valid
transitions. Therefore, not surprisingly, any protocol that was valid with
no cheat detection is still valid in a cheat detecting world. However,
one can often do better in the latter case.
Returning to the coin-flipping protocol we described in the previous
section, Eq.~(\ref{eq:ddbmerge}) is still valid and essentially
optimal. However, the constraint imposed by Eq.~(\ref{eq:ddbsplit}) is no
longer necessary. It is not hard to check that in a cheat detecting
setting, the dominant function that constrains point splitting is
$f(z)=\frac{\Lambda z}{\Lambda+z}$ (or $f(z)=-\frac{1}{\Lambda+z}$ which is
equivalent when probability is conserved). Eq.~(\ref{eq:ddbsplit}) gets
replaced by
\begin{eqnarray}
-\frac{1}{\Lambda+1} \leq -\int_{z^*}^\infty \frac{p(z)}{\Lambda+z} dz
= -\int_{z^*}^\infty \frac{2(z^*)^2}{z^3(\Lambda+z)} dz
= -\left(\frac{\Lambda-2z^*}{\Lambda^2} +
\frac{2(z^*)^2}{\Lambda^3}\log\frac{z^*+\Lambda}{z^*}\right).
\end{eqnarray}
\noindent
Asymptotically, as $\Lambda\rightarrow\infty$, one finds an optimal
expected winnings of $z^*\sim\frac{1}{2}+\frac{\log\Lambda}{4\Lambda}$.
\section{\label{sec:Kit2}Kitaev's second coin-flipping formalism (cont.)}
In this section we shall conclude the description, that begun in
Section~\ref{sec:Kit}, of Kitaev's second coin-flipping formalism
\cite{Kit04}. The last step is the transition from points games that are
ordered in time to point games with no explicit time ordering.
\subsection{\label{sec:TIPG}Time Independent Point Games}
The new ingredient for this section is catalyst states. Given a
transitively valid transition such as $P_B[1,0]+P_A[0,1]\rightarrow
1[\begin{eqnarray}ta,\alpha]$ it trivially follows that
\begin{eqnarray}
P_B[1,0]+P_A[0,1]+\sum_i w_i [x_i,y_i]
\rightarrow 1[\begin{eqnarray}ta,\alpha]+\sum_i w_i [x_i,y_i]
\end{eqnarray}
is also transitively valid, for any ``catalyst'' state $\sum_i w_i
[x_i,y_i]$ with $w_i,x_i,y_i\geq 0$. The question we consider here is
whether the converse is true.
For one-variable transitions the converse is trivially true. The goal of
this section is to prove that the converse is also true for bipartite
transitions (including transitively valid transitions). The proof will
basically show that we can use a small amount of probability to create the
catalyst state, and then run the catalyzed transition in small enough
steps so that by comparison the catalyst state appears large
enough.
Before diving into the proof let us examine some of the surprising
consequences that will follow. The first consequence is that previously all
our probability distributions had a range contained in $[0,\infty)$, but
now we can allow ``probability'' distributions with a range of
$(-\infty,\infty)$. Negative values are simply points were we need to add
in some more probability using a catalyst state. This leads to the
following definition
\begin{eqnarray}gin{definition}
A function with finite support $p:[0,\infty)\rightarrow\mathbb{R}$ is
\textbf{valid} if $\sum_z p(z)=0$ and $\sum_z \left(\frac{-1}{\lambda +
z}\right)p(z)\geq 0$ for all $\lambda>0$.
\end{definition}
The definition implies $p$ is in the dual to the cone of operator monotone
functions. Note that because $\frac{\lambda z}{\lambda + z}= \lambda -
\frac{\lambda^2}{\lambda + z}$, and because of conservation of probability,
checking $\sum_z \left(\frac{\lambda z}{\lambda + z}\right)p(z)\geq 0$ for all
$\lambda>0$ is equivalent to checking $\sum_z \left(\frac{-1}{\lambda +
z}\right)p(z)\geq 0$ for all $\lambda>0$. The latter condition will be easier
to analyze in later sections, though.
The validity relation is essentially a partial order on functions. Instead
of saying $p\rightarrow q$ is valid we could equally write $p\prec q$.
Similarly, a valid function $p$ could be written as $0\prec p$. We use the
earlier notation because it is easier to say ``a function $p$ is valid''
than ``a function $p$ belongs to the cone dual to the operator monotone
functions.''
By construction any valid transition $p\rightarrow q$ can be converted into
the valid function $q-p$. We therefore immediately obtain a number of
standard valid functions (which follow from the proofs in the previous
section):
\begin{eqnarray}gin{lemma}
The following are valid functions:
\begin{eqnarray}gin{itemize}
\item Point raising
\begin{eqnarray}
-p[z]+p[z']\qquad\qquad\text{(for $z\leq z'$).}
\end{eqnarray}
\item Point merging
\begin{eqnarray}
-p_1[z_1]-p_2[z_2]+ \left(p_1+p_2\right)
\left[\frac{p_1 z_1+ p_2 z_2}{p_1+p_2}\right].
\end{eqnarray}
\item Point splitting
\begin{eqnarray}
-\left(p_1+p_2\right)\left[\frac{p_1+p_2}{p_1 w_1'+ p_2 w_2'}
\right]+ p_1\left[\frac{1}{w_1'}\right]+p_2\left[\frac{1}{w_2'}\right].
\end{eqnarray}
\end{itemize}
\label{lemma:rms}
\end{lemma}
The second, and more surprising, consequence of catalyst states is that all
point games can be run using exactly two transitions: one vertical and one
horizontal. The idea is that given any point game we can move all the
horizontal transitions to the beginning and all the vertical transitions to
the end, combining each set into a single vertical or horizontal
transition. Of course, the state after the first transition but before the
second may have some negative probabilities, but as discussed above this
can be fixed with an appropriate catalyst state. This leads to the
following definition:
\begin{eqnarray}gin{definition}
A function with finite support $p:[0,\infty)\times[0,\infty)\rightarrow\mathbb{R}$ is
\textbf{valid} if either
\begin{eqnarray}gin{itemize}
\item for every $c\in[0,\infty)$ the function $p(z,\underline c)$ is valid,
or
\item for every $c\in[0,\infty)$ the function $p(\underline c,z)$ is valid.
\end{itemize}
where as before $p(z,\underline c)$ is the one-variable function obtained
by fixing the second input. We call the first case a \textbf{valid horizontal}
function and the second case a \textbf{valid vertical} functions.
\end{definition}
Of course, we don't even need to specify whether the horizontal or the
vertical transition occurred first, and therefore we obtain a fully
time independent point game:
\begin{eqnarray}gin{definition}
A \textbf{time independent point game (TIPG)} consists of a pair of
functions with finite support $h,v:[0,\infty)\times[0,\infty)\rightarrow
\mathbb{R}$ such that
\begin{eqnarray}gin{itemize}
\item $h$ is a valid horizontal function.
\item $v$ is a valid vertical function.
\item $h+v = 1[\begin{eqnarray}ta,\alpha] - P_B[1,0] - P_A[0,1]$.
\end{itemize}
We say that $[\begin{eqnarray}ta,\alpha]$ is the final point of the TIPG.
\end{definition}
\subsubsection{Relating TDPGs and TIPGs}
Given a TDPG specified as $p_0,\dots,p_n$ with final point $[\begin{eqnarray}ta,\alpha]$
we shall construct a TIPG with the same final point. Let $H$ (resp $V$) be
the set of indices $1,\dots,n$ such that $p_{i-1}\rightarrow p_i$ is a
valid horizontal (resp. vertical) transition. Define
\begin{eqnarray}
h = \sum_{i\in H} (p_i-p_{i-1}),
\qquad\qquad
v = \sum_{i\in V} (p_i-p_{i-1}),
\end{eqnarray}
then $h$ is a valid horizontal function, $v$ is a valid vertical function
and
\begin{eqnarray}
h+v = p_n-p_0 = 1[\begin{eqnarray}ta,\alpha] - P_B[1,0] - P_A[0,1]
\end{eqnarray}
\noindent
as required.
To go the other way we begin with a TIPG specified by $h,v$ with final
point $[\begin{eqnarray}ta,\alpha]$. We define $v^-(x,y) = -\min(v(x,y),0)\geq 0$ as the
magnitude of the negative part of $v$. Then consider
\begin{eqnarray}
P_B[1,0] + P_A[0,1] + v^- &\rightarrow& P_B[1,0] + P_A[0,1] + v^- +v
\\\nonumber
&\rightarrow& P_B[1,0] + P_A[0,1] + v^- + v + h \ =\ 1[\begin{eqnarray}ta,\alpha] + v^-.
\end{eqnarray}
The first is a valid vertical transition and the second is a valid
horizontal transition. Also, the intermediate state is non-negative with
finite support. Therefore, it is a proof that $P_B[1,0] + P_A[0,1] + v^-
\rightarrow 1[\begin{eqnarray}ta,\alpha] + v^-$ is transitively valid. Below we will
show that we can get rid of the catalyst $v^-$ and construct for every
$\epsilon>0$ a sequence such that $P_B[1,0] + P_A[0,1]
\rightarrow 1[\begin{eqnarray}ta+\epsilon,\alpha+\epsilon]$ is transitively valid. This
is the desired TDPG.
What remains to be proven is that we can discard catalyst states. We will
only prove this for the special case of coin flipping, though the general
case is also true. We first require two simple lemmas. The first lemma
shows we can construct arbitrary catalyst states (as long as we allow
extra junk) and the second lemma shows we can clean up the catalyst states
(and extra junk).
\begin{eqnarray}gin{lemma}
Given a function $r:[0,\infty)\times[0,\infty)\rightarrow [0,\infty)$ with
finite support such that $r(0,0)=0$, there exists $c>0$ and a function
$r':[0,\infty)\times[0,\infty)\rightarrow [0,\infty)$ with finite support
such that
\begin{eqnarray}
c P_B[1,0]+c P_A[0,1]\rightarrow r+r'
\end{eqnarray}
\noindent
is transitively valid, where we additionally assume both $P_A,P_B>0$.
\label{lemma:catbegin}
\end{lemma}
\begin{eqnarray}gin{proof}
First we prove the lemma for the special case when $r$ has support at a
single point. Let $r=q[x,y]$ with $q,x>0$ and $y\geq0$ (the case $y>0$ and
$x\geq0$ will follow by exchanging the axes). If $x\geq 1$ then
\begin{eqnarray}
\frac{q}{P_B}P_B[1,0]+ \frac{q}{P_B}P_A[0,1]\rightarrow q[x,y] +
\frac{q}{P_B}P_A[0,1]
\end{eqnarray}
is transitively valid using point raisings, and the lemma is satisfied with
$c=\frac{q}{P_B}$ and $r'=\frac{q P_A}{P_B}[0,1]$. If $x<1$ the sequence
\begin{eqnarray}
P_B[1,0]+P_A[0,1]\rightarrow P_B[1,y]+P_A[0,1] \rightarrow \frac{x}{2}P_B[x,y]
+ \left(1-\frac{x}{2}\right)P_B[2-x,y]+P_A[0,1]
\end{eqnarray}
is transitively valid by point raising followed by point splitting. Now we
use the fact that we can scale the probability in transitions. That is,
if $\sum_i w_i [x_i,y_i]\rightarrow\sum_j w_j' [x_j',y_j']$
is transitively valid then for $a>0$ so is $\sum_i a w_i
[x_i,y_i]\rightarrow\sum_j a w_j' [x_j',y_j']$. In particular, if we scale
the previous transition by $c=2q/(x P_B)$ we satisfy the lemma with
$r' = c(1-x/2)P_B[2-x,y]+c P_A[0,1]$.
Finally, for the general case where $r=\sum_{i=1}^k q_i [x_i,y_i]$ let
$c_i$ and $r'_i$ be chosen as above so that $c_i P_B[1,0]+c_i
P_A[0,1]\rightarrow q_i [x_i,y_i]+r_i'$ is transitively valid. Then
\begin{eqnarray}
\sum_{i=1}^k c_i P_B[1,0]+ \sum_{i=1}^k c_i P_A[0,1]&\rightarrow&
\sum_{i=2}^k c_i P_B[1,0]+
\sum_{i=2}^k c_i P_A[0,1] + q_1 [x_1,y_1] + r_1'
\nonumber\\
&\rightarrow&\cdots\rightarrow
\sum_{i=j}^k c_i P_B[1,0]+
\sum_{i=j}^k c_i P_A[0,1] + \sum_{i=1}^{j-1} q_i [x_i,y_i] +
\sum_{i=1}^{j-1} r_i'
\nonumber\\
&\rightarrow&\cdots\rightarrow
r + \sum_{i=1}^{k} r_i',
\end{eqnarray}
where by construction each transition is transitively valid and so the
proof is completed by setting $c=\sum_i c_i$ and $r'=\sum_i r_i'$.
\end{proof}
\begin{eqnarray}gin{lemma}
Given $\epsilon>0$ and a function
$r''\rightarrow:[0,\infty)\times[0,\infty)\rightarrow [0,\infty)$ with
finite support and $\sum_{x,y} r''(x,y)=1$, there exists $1>\delta>0$ such
that
\begin{eqnarray}
(1-\delta)[\begin{eqnarray}ta,\alpha]+\delta\, r'' \rightarrow
1[\begin{eqnarray}ta+\epsilon,\alpha+\epsilon]
\end{eqnarray}
\noindent
is transitively valid.
\label{lemma:catend}
\end{lemma}
\begin{eqnarray}gin{proof}
Let $x''$ be the largest $x$-coordinate over all points in $r''$ and
similarly let $y''$ be the largest $y$-coordinate over all points in $r''$.
By point raising $r\rightarrow 1[x'',y'']$ is transitively valid, so we
focus on proving that we can find $1>\delta>0$ such that
\begin{eqnarray}
(1-\delta)[\begin{eqnarray}ta,\alpha]+\delta[x'',y''] \rightarrow
1[\begin{eqnarray}ta+\epsilon,\alpha+\epsilon]
\end{eqnarray}
is transitively valid. We can also assume (possibly using further point
raisings) that $x''>\begin{eqnarray}ta+\epsilon$ and $y''>\alpha+\epsilon$. Consider the
following sequence
\begin{eqnarray}
(1-\delta)[\begin{eqnarray}ta,\alpha]+\delta[x'',y''] &\rightarrow&
(1-\delta')[\begin{eqnarray}ta,\alpha]+(\delta'-\delta)[\begin{eqnarray}ta,y'']+ \delta[x'',y'']
\nonumber\\ &\rightarrow&
(1-\delta')[\begin{eqnarray}ta+\epsilon,\alpha]+
\delta'[\begin{eqnarray}ta+\epsilon,y'']
\nonumber\\ &\rightarrow&
1[\begin{eqnarray}ta+\epsilon,\alpha+\epsilon]
\end{eqnarray}
\noindent
corresponding to raise, merge, merge. To make the first merge valid
we need $(\delta'-\delta)\begin{eqnarray}ta+\delta x''= \delta'(\begin{eqnarray}ta+\epsilon)$
which is equivalent to $\delta(x''-\begin{eqnarray}ta)=\delta'\epsilon$.
The second merge requires $\delta'(y''-\alpha)=\epsilon$. Both conditions
can be satisfied by constants such that $1>\delta'>\delta>0$.
\end{proof}
Putting the lemmas together we can prove the main result.
\begin{eqnarray}gin{lemma}
Given $r:[0,\infty)\times[0,\infty)\rightarrow
[0,\infty)$ with finite support such that $r(0,0)=0$ and
\begin{eqnarray}
P_B[1,0]+P_A[0,1]+r\rightarrow 1[\begin{eqnarray}ta,\alpha] +r
\end{eqnarray}
\noindent
is transitively valid then for every $\epsilon>0$
\begin{eqnarray}
P_B[1,0]+P_A[0,1]\rightarrow 1[\begin{eqnarray}ta+\epsilon,\alpha+\epsilon]
\end{eqnarray}
is transitively valid as well.
\end{lemma}
\begin{eqnarray}gin{proof}
Fix $\epsilon>0$. Note that by conservation of probability we must have
$P_A+P_B=1$. We also assume $P_A,P_B>0$ as otherwise the proof is trivial.
Therefore, we can use Lemma~\ref{lemma:catbegin} to get $c>0$ and $r'$
so that $c P_B[1,0]+c P_A[0,1]\rightarrow r+r'$ is transitively valid.
If we set $r''=(1/c)(r+r')$ than again by conservation of probability we
must have $\sum_{x,y}r''=1$. We can therefore use Lemma~\ref{lemma:catend}
to get $1>\delta>0$ so that $(1-\delta)[\begin{eqnarray}ta,\alpha]+\delta r''
\rightarrow 1[\begin{eqnarray}ta+\epsilon,\alpha_\epsilon]$ is transitively valid. Now
consider the sequence
\begin{eqnarray}
P_B[1,0]+P_A[0,1]
&\rightarrow&
(1-\delta)P_B[1,0]+(1-\delta)P_A[0,1]+\frac{\delta}{c}r + +\frac{\delta}{c}r'
\nonumber\\&\rightarrow&
(1-\delta)[\begin{eqnarray}ta,\alpha]+\frac{\delta}{c}r + +\frac{\delta}{c}r'
\nonumber\\&\rightarrow&
1[\begin{eqnarray}ta+\epsilon,\alpha+\epsilon].
\end{eqnarray}
The first transition follows from a scaled version of the transition
obtained from Lemma~\ref{lemma:catbegin}, and the third transition is the
one obtained from Lemma~\ref{lemma:catend}. The middle transition follows
from repeated applications of $a P_B[1,0]+a P_A[0,1]+a r\rightarrow
a[\begin{eqnarray}ta,\alpha] + a r$ for $a\leq \delta/c$. Therefore the whole transition
is transitively valid as required.
\end{proof}
From the conditions of validity, it is easy to verify that neither $h$ nor
$v$ can be positive at the point $(0,0)$. Since their sum must be zero at
this point, individually they must be zero as well. Hence, our catalyst
state is zero at $(0,0)$ and we can apply the above lemma to complete our
argument for the equivalence of TDPGs and TIPGs. We can state the result
formally as a further extension of Theorem~\ref{thm:kitmain2}.
\begin{eqnarray}gin{theorem}
Let $f(\begin{eqnarray}ta,\alpha):\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$f(\alpha',\begin{eqnarray}ta')\geq f(\alpha,\begin{eqnarray}ta)$ whenever $\alpha'\geq\alpha$ and
$\begin{eqnarray}ta'\geq\begin{eqnarray}ta$, then
\begin{eqnarray}
\inf_{\text{proto}} f(P_B^*,P_A^*) = \inf_{UBP} f(\begin{eqnarray}ta,\alpha)
= \inf_{TDPG} f(\begin{eqnarray}ta,\alpha) = \inf_{TIPG} f(\begin{eqnarray}ta,\alpha).
\end{eqnarray}
\label{thm:kitmain3}
\end{theorem}
\section{\label{sec:zero}Towards zero bias}
In this section we will finally describe a family of protocols for
coin-flipping that achieves arbitrarily small bias. The protocols will be
described in Kitaev's second formalism using the tools of the previous
sections.
In the first part of the section we will attempt to give some intuition for
our construction. Along the way we will prove a couple of important
lemmas. In Section~\ref{sec:formal} we will simply present the
corresponding pair of functions $h$ and $v$ and prove that they satisfy the
necessary properties.
For those who have skipped ahead to this section, we review some of the key
concepts that have been defined in previous sections: A valid function
$f(z)$ has finite support and satisfies $\sum_z f(z)=0$ and $\sum_z
\frac{-1}{\lambda+z}f(z)\geq 0$ for all $\lambda>0$. These constraints are
equivalent to those discussed in the introduction. Examples of valid
functions are point raises, point merges and point splits as defined in
Lemma~\ref{lemma:rms}. A valid horizontal function $h(x,y)$ is valid as a
function of $x$ for every $y\geq 0$. Similarly, a valid vertical function
$v(x,y)$ is valid as a function of $y$ for every $x\geq 0$. A TIPG is a
valid horizontal function plus a valid vertical function such that $h+v =
1[\begin{eqnarray}ta,\alpha] -P_B[1,0] -P_A[0,1]$, where $[x_0,y_0]$ denotes a function
that takes value one at $x=x_0$, $y=y_0$ and is zero everywhere else. Such
a TIPG leads to a coin-flipping protocol with $P_A^*\leq\alpha$ and
$P_B^*\leq\begin{eqnarray}ta$.
\subsection{Guiding principles}
We begin our discussion with a TIPG example. We will analyze the TIPG with
bias $1/6$ first introduced in the introduction (Fig.~\ref{fig:introlad16})
and reproduced here in Fig.~\ref{fig:lad16}. The two figures are intended
to denote the same TIPG, though the latter figure has a slightly different
labeling convention.
The new labeling convention provides some useful intuition for TIPGs:
because of probability conservation, one can associate a probability with
each arrow. The arrows carry the probability from their base to their
head. The probability associated to a point can be computed as the sum of
incoming probabilities, (which equals the sum of outgoing probabilities for
all points except the initial and final points). Furthermore, the
probability carried by arrows not associated with the initial or final
points must go around in loops, such as boxes or figure eights. It is often
easiest in a figure to label each loop with the probability that goes
around it, and this is the idea behind the labeling of
Fig.~\ref{fig:lad16}.
\begin{eqnarray}gin{figure}[tb]
\begin{eqnarray}gin{center}
\unitlength = 30pt
\begin{eqnarray}gin{picture}(11,10)(-1,-1)
\put(0,0){\line(1,0){9}}
\put(0,0){\line(0,1){9}}
\put(2,0){\line(0,-1){0.1}}
\put(4,0){\line(0,-1){0.1}}
\put(5,0){\line(0,-1){0.1}}
\put(6,0){\line(0,-1){0.1}}
\put(7,0){\line(0,-1){0.1}}
\put(8,0){\line(0,-1){0.1}}
\put(0,2){\line(-1,0){0.1}}
\put(0,4){\line(-1,0){0.1}}
\put(0,5){\line(-1,0){0.1}}
\put(0,6){\line(-1,0){0.1}}
\put(0,7){\line(-1,0){0.1}}
\put(0,8){\line(-1,0){0.1}}
\put(2,-0.2){\makebox(0,0)[t]{$\frac{2}{3}$}}
\put(3,-0.2){\makebox(0,0)[t]{$1$}}
\put(4,-0.2){\makebox(0,0)[t]{$\frac{4}{3}$}}
\put(5,-0.2){\makebox(0,0)[t]{$\frac{5}{3}$}}
\put(6,-0.2){\makebox(0,0)[t]{$2$}}
\put(7,-0.2){\makebox(0,0)[t]{$\frac{7}{3}$}}
\put(8,-0.2){\makebox(0,0)[t]{$\frac{8}{3}$}}
\put(-0.2,2){\makebox(0,0)[r]{$\frac{2}{3}$}}
\put(-0.2,3){\makebox(0,0)[r]{$1$}}
\put(-0.2,4){\makebox(0,0)[r]{$\frac{4}{3}$}}
\put(-0.2,5){\makebox(0,0)[r]{$\frac{5}{3}$}}
\put(-0.2,6){\makebox(0,0)[r]{$2$}}
\put(-0.2,7){\makebox(0,0)[r]{$\frac{7}{3}$}}
\put(-0.2,8){\makebox(0,0)[r]{$\frac{8}{3}$}}
\put(3.2,0.2){\makebox(0,0)[bl]{$\frac{1}{2}$}}
\put(0.2,3.2){\makebox(0,0)[bl]{$\frac{1}{2}$}}
\put(3.5,2.5){\makebox(0,0){$1$}}
\put(4.5,3.5){\makebox(0,0){$1$}}
\put(5.5,4.5){\makebox(0,0){$1$}}
\put(6.5,5.5){\makebox(0,0){$1$}}
\put(2.5,3.5){\makebox(0,0){$1$}}
\put(3.5,4.5){\makebox(0,0){$1$}}
\put(4.5,5.5){\makebox(0,0){$1$}}
\put(5.5,6.5){\makebox(0,0){$1$}}
\thicklines
\put(3,0){\vector(0,1){2}}
\put(0,3){\vector(1,0){2}}
\put(3,2){\vector(-1,0){1}}
\put(2,3){\vector(0,-1){1}}
\put(3,2){\vector(1,0){1}}
\put(2,3){\vector(0,1){1}}
\put(4,3){\vector(-1,0){2}}
\put(3,4){\vector(0,-1){2}}
\put(2,4){\vector(1,0){1}}
\put(4,2){\vector(0,1){1}}
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\put(3,4){\vector(0,1){1}}
\put(5,4){\vector(-1,0){2}}
\put(4,5){\vector(0,-1){2}}
\put(3,5){\vector(1,0){1}}
\put(5,3){\vector(0,1){1}}
\put(5,4){\vector(1,0){1}}
\put(4,5){\vector(0,1){1}}
\put(6,5){\vector(-1,0){2}}
\put(5,6){\vector(0,-1){2}}
\put(4,6){\vector(1,0){1}}
\put(6,4){\vector(0,1){1}}
\put(6,5){\vector(1,0){1}}
\put(5,6){\vector(0,1){1}}
\put(7,6){\vector(-1,0){2}}
\put(6,7){\vector(0,-1){2}}
\put(5,7){\vector(1,0){1}}
\put(7,5){\vector(0,1){1}}
\put(7,6){\vector(1,0){1}}
\put(6,7){\vector(0,1){1}}
\put(7,7){\vector(-1,0){1}}
\put(7,7){\vector(0,-1){1}}
\put(7.6,7.6){\makebox(0,0){$\cdot$}}
\put(7.7,7.7){\makebox(0,0){$\cdot$}}
\put(7.8,7.8){\makebox(0,0){$\cdot$}}
\put(3,0){\makebox(0,0){$\bullet$}}
\put(0,3){\makebox(0,0){$\bullet$}}
\end{picture}
\caption{A TIPG with bias $1/6$.}
\label{fig:lad16}
\end{center}
\end{figure}
We can also use our algebraic notation to express the TIPG. The horizontal
arrows encode the function
\begin{eqnarray}
h &=&
+\frac{1}{2}\bigg[\frac{2}{3},\frac{2}{3}\bigg]
-\frac{3}{2}\bigg[1,\frac{2}{3}\bigg]
+1\bigg[\frac{4}{3},\frac{2}{3}\bigg]
\\\nonumber&&
-\frac{1}{2}\bigg[0,1\bigg]
+\frac{3}{2}\bigg[\frac{2}{3},1\bigg]
-2\bigg[\frac{4}{3},1\bigg]
+1\bigg[\frac{5}{3},1\bigg]
\\\nonumber&&
+\sum_{k=4}^\infty\left(
-1\bigg[\frac{k-2}{3},\frac{k}{3}\bigg]
+2\bigg[\frac{k-1}{3},\frac{k}{3}\bigg]
-2\bigg[\frac{k+1}{3},\frac{k}{3}\bigg]
+1\bigg[\frac{k+2}{3},\frac{k}{3}\bigg]
\right).
\end{eqnarray}
The last line of $h$ denotes a pattern that we shall call a ladder. More
generally, we shall refer to any regular pattern that heads to infinity on
the diagonal as a ladder. The problem with ladders is that they involve an
infinite number of points, and we previously required that $h$ have support
only on a finite set of points. In Section~\ref{sec:trunc} we will study
how to properly truncate these ladders, but for the moment we shall ignore
this issue.
By symmetry we can define $v(x,y)=h(y,x)$. By construction all terms in the
sum $h+v$ cancel except for the three required
\begin{eqnarray}
h+v = 1\bigg[\frac{2}{3},\frac{2}{3}\bigg]-
\frac{1}{2}\bigg[1,0\bigg]-\frac{1}{2}\bigg[0,1\bigg].
\end{eqnarray}
Let us verify that $h$ is a valid horizontal functions (which by symmetry
proves that $v$ is a valid vertical function). It is easy to see that for
every $y$ we have $\sum_x h(x,y)=0$. The other constraint that needs to be
checked is $\sum_x \frac{-1}{\lambda+x} h(x,y)\geq 0$ for all $\lambda>0$
and all $y\geq 0$. We begin with the case $y=k/3\geq 4/3$.
\begin{eqnarray}
\sum_{x} \frac{-1}{\lambda+x} h\left(x,\,\frac{k}{3}\right) &=&
\frac{1}{\lambda + \frac{k-2}{3}}
-\frac{2}{\lambda + \frac{k-1}{3}}
+\frac{2}{\lambda + \frac{k+1}{3}}
-\frac{1}{\lambda + \frac{k+2}{3}}
\nonumber\\
&=&
\frac{\frac{1}{3}}{(\lambda + \frac{k-2}{3})(\lambda + \frac{k-1}{3})}
-\frac{\frac{2}{3}}{(\lambda + \frac{k-1}{3})(\lambda + \frac{k+1}{3})}
+\frac{\frac{1}{3}}{(\lambda + \frac{k+1}{3})(\lambda + \frac{k+2}{3})}
\nonumber\\
&=&
\frac{\frac{1}{3}}{(\lambda + \frac{k-2}{3})(\lambda + \frac{k-1}{3})
(\lambda + \frac{k-1}{3})}
-\frac{\frac{1}{3}}{(\lambda + \frac{k-1}{3})(\lambda + \frac{k+1}{3})
(\lambda + \frac{k+2}{3})}
\nonumber\\
&=&
\frac{(\frac{1}{3})(\frac{4}{3})}
{(\lambda + \frac{k-2}{3})(\lambda + \frac{k-1}{3})
(\lambda + \frac{k-1}{3})(\lambda + \frac{k+2}{3})}
\geq 0
\label{eq:sampcons}
\end{eqnarray}
\noindent
for $\lambda> 0$, where the successive simplifications involves splitting
the middle terms and using relations of the form
\begin{eqnarray}
\frac{1}{\lambda+x_1}-\frac{1}{\lambda+x_2}=
\frac{x_2-x_1}{(\lambda+x_1)(\lambda+x_2)}.
\end{eqnarray}
\noindent
The idea is that we can interpret successive lines as follows: The first
line is the standard sum over points with a numerator corresponding to
probabilities. The second line is a sum over arrows with a numerator
corresponding to probabilities times distance traveled (which we can think
of as momentum). Finally, the third line is a sum over pairs of arrows with
zero net momentum.
The constraint at $y=1$ is roughly the same, except that the leftmost arrow
travels twice the distance and carries half the probability. More
specifically, we can write
\begin{eqnarray}
\label{eq:ladnsplit}
&&-\frac{1}{2}\bigg[0\bigg]
+\frac{3}{2}\bigg[\frac{2}{3}\bigg]
-2\bigg[\frac{4}{3}\bigg]
+1\bigg[\frac{5}{3}\bigg]
\\\nonumber
&&\qquad\qquad\qquad\qquad=
\left(-\frac{1}{2}\bigg[0\bigg]+1\bigg[\frac{1}{3}\bigg]
- \frac{1}{2}\bigg[\frac{2}{3}\bigg]\right)
+\left(-1\bigg[\frac{1}{3}\bigg]+ 2\bigg[\frac{2}{3}\bigg]
-2\bigg[\frac{4}{3}\bigg]+1\bigg[\frac{5}{3}\bigg]\right).
\end{eqnarray}
The right term is exactly what would be there if the ladder had been
extended to $y=1$. It is valid by Eq.~(\ref{eq:sampcons}). The left term is
the difference between the long arrow carrying probability $1/2$ and the
short arrow carrying probability $1$. It is valid because it is a point
merge. Therefore, the original expression is a sum of two valid terms and
itself is valid, as can also be checked by direct computation.
The constraint for $y=\frac{2}{3}$ can also be directly checked, though in
fact it is just a point splitting, and hence valid because
$\frac{3}{2}/1=\frac{1}{2}/\frac{2}{3}+1/\frac{4}{3}$.
Except for the fact that the ladder involves an infinite number of points,
we have completed the proof that the resulting TIPG corresponds to a
protocol with bias $1/6$. The infinite number of points, though, is a
serious problem from the point of view of our constructive description of
Kitaev's formalism: for instance, the canonical catalyst state used to
convert the TIPG into a TDPG carries an infinite amount of probability.
To proceed we therefore must truncate the ladder at some large distance
$\Gamma$. The truncation will add small extra terms to the bias, which
will go to zero as $\Gamma\rightarrow\infty$. We can think of the
different values for $\Gamma$ as a family of protocols which converges to
a bias of $1/6$.
While the formal truncation is done in Section~\ref{sec:trunc} we will try
to paint an intuitive picture here as to why truncation is possible. Let us
imagine that we naively cut the ladder diagonally at some point in such a
way that the end looks like the top of Fig.~\ref{fig:lad16}. The ladder is
still valid (the top rung is just a point merge), however we are left with
an excess of probability in the edges and a deficit of probability in the
center. To correct the situation we need to add in a term of the form
$2[\frac{\Gamma}{3},\frac{\Gamma}{3}]
-1[\frac{\Gamma+1}{3},\frac{\Gamma-1}{3}]
-1[\frac{\Gamma-1}{3},\frac{\Gamma+1}{3}]$, which is a coin-flipping
problem!
Admittedly it is a coin-flipping problem with twice the probability and two
thirds of the distance between points but that does not make much of a
difference. The problem is located far away from the axes, though, so it
really is a coin flipping with cheat detection problem as described by
Section~\ref{sec:cheat}. Even better, the cheat detection is proportional
to $\Gamma$ which can be made arbitrarily large at no cost to us (us being
the designers of the protocols, of course there is a practical cost
involved in implementing protocols with large $\Gamma$).
As the amount of cheat detection becomes infinite, the rules of point games
become very simple: probability is conserved and average $x$ and $y$ cannot
decrease. As the problem we are carrying off to infinity has zero net
probability, and zero net average $x$ and $y$, it should be resolvable at
infinity. Sadly, even at infinity zero-bias coin flipping is impossible
(only arbitrarily small bias is allowed) so after resolving the problem at
infinity, we still need to bring back an error term down through the
ladder.
In practice, we still need to truncate the ladder at a finite distance,
resolve the coin-flipping problem at that height, and carry the error terms
back down through the ladder. There is a fairly automatic way of taking care
of all of this, but it involves more complicated ladders. The next section
will present the most important result used in building such complicated
ladders.
\subsubsection{Obtaining non-negative numerators}
Whenever we want to verify that a function $p(x)$ is valid, we need to
examine expressions of the form
\begin{eqnarray}
\sum_i \left(\frac{-1}{\lambda+x_i}\right) p(x_i) =
\frac{f(-\lambda)}{\prod_i(\lambda+x_i)},
\end{eqnarray}
\noindent
where $x_1,\dots,x_n\geq 0$ are the finite support of $p$, and
$f(-\lambda)$ is some polynomial whose coefficients depends on the non-zero
values of $p(x)$. The reasons for making $f$ a function of $-\lambda$
rather than $\lambda$ will become clear below.
The function $p(x)$ is valid only if the above expression is non-negative
for $\lambda>0$, which in turn is true if and only if $f(-\lambda)$ is
non-negative for $\lambda>0$. The problem is that combining the terms to
find $f(-\lambda)$ is often tedious, and verifying its non-negativity can be
fairly difficult.
On the other hand, constructing a non-negative polynomial is generally easy
(for instance we can specify it as a product of its zeros). Therefore, it
is often easier to start with $f(-\lambda)$, and use it to compute an
appropriate distribution $p(x)$ over some previously selected points
$x_1,\dots,x_n$. That is the approach that we will be developing in this
section. The next two lemmas will help us prove that the desired expression
is $p(x_i) = -f(x_i)/\prod_{\substack{j\neq i}}(x_j-x_i)$, which also
satisfies probability conservation so long as $f(-\lambda)$ has degree no
greater than $n-2$.
\begin{eqnarray}gin{lemma}
Let $n\geq 2$ and $x_1,\dots,x_n\in\mathbb{R}$ be distinct. Then
\begin{eqnarray}
\sum_{i=1}^n\prod_{\substack{j=1\cr j\neq i}}^n \frac{1}{(x_j-x_i)}=0.
\end{eqnarray}
\end{lemma}
\begin{eqnarray}gin{proof}
We proceed by induction. For $n=2$ we trivially have
\begin{eqnarray}
\frac{1}{(x_2-x_1)}+\frac{1}{(x_1-x_2)}=0.
\end{eqnarray}
For $n>2$ we use the identities for $1<i<n$
\begin{eqnarray}
\frac{1}{(x_1-x_i)(x_n-x_i)}=\frac{1}{(x_n-x_1)}
\left(\frac{1}{(x_1-x_i)}-\frac{1}{(x_n-x_i)}\right)
\end{eqnarray}
to expand
\begin{eqnarray}
\sum_{i=1}^n\prod_{\substack{j=1\cr j\neq i}}^n \frac{1}{(x_j-x_i)}=
\frac{1}{(x_n-x_1)}\left(
\sum_{i=1}^{n-1}\prod_{\substack{j=1\cr j\neq i}}^{n-1} \frac{1}{(x_j-x_i)}-
\sum_{i=2}^n\prod_{\substack{j=2\cr j\neq i}}^n \frac{1}{(x_j-x_i)}\right)
\end{eqnarray}
and by induction both terms inside the parenthesis are zero.
\end{proof}
\begin{eqnarray}gin{lemma}
Let $n\geq 2$ and $x_1,\dots,x_n\in\mathbb{R}$ be distinct. Let $f(x)$ be a
polynomial of degree $k\leq n-2$. Then
\begin{eqnarray}
\sum_{i=1}^n \frac{f(x_i)}{\prod_{\substack{j\neq i}}(x_j-x_i)}=0.
\end{eqnarray}
\end{lemma}
\begin{eqnarray}gin{proof}
We proceed by induction on $k$. For $k=0$ the result follows from the
previous lemma. If $k>0$ we can write $f(x)=c\prod_{j=1}^{k}(x_j-x)+g(x)$
for some scalar $c\in\mathbb{R}$ and a polynomial $g(x)$ of degree less than $k$.
Then
\begin{eqnarray}
\sum_{i=1}^n \frac{f(x_i)}{\prod_{\substack{j\neq i}}(x_j-x_i)}=c
\sum_{i=k+1}^n\prod_{\substack{j=k+1\cr j\neq i}}^n \frac{1}{(x_j-x_i)}+
\sum_{i=1}^n \frac{g(x_i)}{\prod_{\substack{j\neq i}}(x_j-x_i)}
\end{eqnarray}
\noindent
and both terms are zero by induction.
\end{proof}
\begin{eqnarray}gin{lemma}
Let $x_1,\dots,x_n$ be distinct non-negative numbers and let $f(-\lambda)$
be a polynomial in $\lambda$ of degree $k\leq n-2$ which is non-negative
for $\lambda>0$. Then
\begin{eqnarray}
p = \sum_i \left(\frac{-f(x_i)}{\prod_{\substack{j\neq i}}(x_j-x_i)} \right)
[x_i]
\end{eqnarray}
is a valid function.
\label{lemma:posnumer}
\end{lemma}
\begin{eqnarray}gin{proof}
Using the previous lemma with an appended point $x_{n+1}=-\lambda$, we get
\begin{eqnarray}
\sum_{i=1}^n\frac{-1}{\lambda+x_i}
\left( \frac{f(x_i)}{\prod_{\substack{j\neq i}}(x_j-x_i)} \right) +
\frac{f(-\lambda)}{\prod_{i}(\lambda+x_i)}=0
\end{eqnarray}
which proves the constraints for $\lambda>0$. In fact, the above relation
holds so long as $f$ has degree $k\leq (n+1)-2$. However, we must reduce
the allowed degree by one more to get probability conservation
\begin{eqnarray}
\sum_i p(x_i) = \lim_{\lambda\rightarrow\infty}
\sum_{i=1}^n\frac{\lambda}{\lambda+x_i}
\left( \frac{-f(x_i)}{\prod_{\substack{j\neq i}}(x_j-x_i)} \right)
= \lim_{\lambda\rightarrow\infty}
\frac{-\lambda f(-\lambda)}{\prod_{i}(\lambda+x_i)}
\end{eqnarray}
\noindent
which converges to zero if the degree is $k\leq n-2$.
\end{proof}
\subsubsection{\label{sec:trunc}Truncating the ladder}
A single rung of our ladder has the form
\begin{eqnarray}
a\bigg[\frac{k-2}{3},\frac{k}{3}\bigg]
+b\bigg[\frac{k-1}{3},\frac{k}{3}\bigg]
+c\bigg[\frac{k+1}{3},\frac{k}{3}\bigg]
+d\bigg[\frac{k+2}{3},\frac{k}{3}\bigg]
\end{eqnarray}
for some constants $a,b,c,d\in\mathbb{R}$. Following the discussion in the last
section we want to set
\begin{eqnarray}
a=\frac{-f(\frac{k-2}{3})}{\left(\frac{1}{3}\right)
\left(\frac{3}{3}\right)\left(\frac{4}{3}\right)}=
-\frac{9f(\frac{k-2}{3})}{4}
\end{eqnarray}
\noindent
and similarly
\begin{eqnarray}
b = +\frac{9f(\frac{k-1}{3})}{2},\qquad
c = -\frac{9f(\frac{k+1}{3})}{2},\qquad
d = +\frac{9f(\frac{k+2}{3})}{4}.
\end{eqnarray}
The ladder for the bias $1/6$ protocol can be described this way with
$f(-\lambda)= 4/9$, which is clearly positive for $\lambda\geq 0$. But, of
course, we can allow $f$ to be other quadratic functions. More importantly,
we can allow different quadratic functions at different heights of the
ladder. That would lead to a function $f(x,y)$ with the constraint that for
every $y$ (on which the ladder is non-zero) $f(-\lambda,y)$ is a quadratic
polynomial in $\lambda$ that is non-negative for $\lambda>0$.
But there is a catch. We want to keep the symmetry of the problem so that we
can choose $v(x,y)=h(y,x)$ and still get $h+v$ to cancel on the ladder. In
other words, we want to ensure $h(x,y)=-h(y,x)$. This leads to the
conditions
\begin{eqnarray}
a_{x=\frac{k-2}{3},y=\frac{k}{3}} = -d_{x=\frac{k}{3},y=\frac{k-2}{3}}
&\qquad\Longrightarrow\qquad&
-\frac{9f(\frac{k-2}{3},\frac{k}{3})}{4} =
-\frac{9f(\frac{k}{3},\frac{k-2}{3})}{4}
\\
b_{x=\frac{k-1}{3},y=\frac{k}{3}} = -c_{x=\frac{k}{3},y=\frac{k-1}{3}}
&\qquad\Longrightarrow\qquad&
\frac{9f(\frac{k-1}{3},\frac{k}{3})}{2} =
\frac{9f(\frac{k}{3},\frac{k-1}{3})}{2}
\end{eqnarray}
\noindent
which are both satisfied if we enforce $f(x,y)=f(y,x)$.
Now we can choose our function $f$ to stop the ladder at a certain height
$y=\Gamma/3$ by setting
\begin{eqnarray}
f(x,y) =
C\left(\frac{\Gamma+1}{3}-x\right)\left(\frac{\Gamma+2}{3}-x\right)
\left(\frac{\Gamma+1}{3}-y\right)\left(\frac{\Gamma+2}{3}-y\right)
\end{eqnarray}
for some large integer $\Gamma$ and positive constant $C$ to be determined
below. The ladder part of $h$ becomes
\begin{eqnarray}
h_{lad}&=&\sum_{k=3}^\Lambda\bigg(
-\frac{9f(\frac{k-2}{3},\frac{k}{3})}{4}
\bigg[\frac{k-2}{3},\frac{k}{3}\bigg]
+\frac{9f(\frac{k-1}{3},\frac{k}{3})}{2}
\bigg[\frac{k-1}{3},\frac{k}{3}\bigg]
\\\nonumber&&\qquad\qquad\qquad\qquad\qquad\qquad
-\frac{9f(\frac{k+1}{3},\frac{k}{3})}{2}
\bigg[\frac{k+1}{3},\frac{k}{3}\bigg]
+\frac{9f(\frac{k+2}{3},\frac{k}{3})}{4}
\bigg[\frac{k+2}{3},\frac{k}{3}\bigg]
\bigg).
\end{eqnarray}
We have stopped the ladder sum at height $\Lambda/3$. Of course, we could
also have simply stopped the original ladder with $f=4/9$ at a particular
height, but we would have lost the antisymmetry of $h$. The fact that
$f(\frac{\Gamma+2}{3},\frac{\Gamma}{3})=
f(\frac{\Gamma+1}{3},\frac{\Gamma}{3})=
f(\frac{\Gamma+1}{3},\frac{\Gamma-1}{3})=0$ are all zero ensures that we
can stop the above pattern and still retain $h(x,y)=-h(y,x)$.
The next step is to verify that $h_{lad}$ is horizontally valid, but that
follows from the fact that $f(-\lambda,y)\geq0$ for $\lambda>0$ and $y\leq
\Gamma/3$, and that it is quadratic in $\lambda$.
Finally, let us examine the bottom of the ladder. If $\Gamma$ is very
large, and $x$ and $y$ are small compared to $\Gamma$, then $f\simeq C
\Gamma^4/3^4$. If we further choose $C\simeq 36/\Gamma^4$ we end up
approximating the original constant $f=4/9$ ladder.
The rest of this section will work out the details of merging this
truncated ladder with the structure that needs to lie at the bottom. The
discussion contains no critical new ideas and can be skipped on a first
reading.
Putting everything together we end up with a structure of the form
\begin{eqnarray}
h &=& h_{lad}
\\\nonumber
&&- \frac{1}{2}\bigg[0,1\bigg] + 1\bigg[\frac{1}{3},1\bigg]
- \frac{1}{2}\bigg[\frac{2}{3},1\bigg]
\\\nonumber
&&+\frac{1}{2}\bigg[\frac{2+\delta}{3},\frac{2}{3}\bigg]
+\left(\frac{1}{2}-h_{lad}\left(\frac{2}{3},1\right)\right)
\bigg[1,\frac{2}{3}\bigg]
-h_{lad}\left(\frac{2}{3},\frac{4}{3}\right)\bigg[\frac{4}{3},\frac{2}{3}\bigg]
\\\nonumber
&&-\frac{1}{2}\bigg[\frac{2}{3},\frac{2+\delta}{3}\bigg]
+\frac{1}{2}\bigg[\frac{2+\delta}{3},\frac{2+\delta}{3}\bigg]
\end{eqnarray}
\noindent
for some small $\delta>0$ to be determined in a moment.
Note that $h_{lad}$ runs up to $y=1$, which produces a point at $x=1/3$,
$y=1$. We must exactly cancel the amplitude in this point with the term on
the second line of $h$. We therefore choose
\begin{eqnarray}
C= \frac{4}{9}\left(\frac{3}{\Gamma}\right) \left(\frac{3}{\Gamma+1}\right)
\left(\frac{3}{\Gamma-2}\right) \left(\frac{3}{\Gamma-1}\right)
\end{eqnarray}
\noindent
so that $h_{lad}(\frac{1}{3},1)=-1$. Note that this $C$ has the right
behavior as $\Gamma\rightarrow \infty$. The validity of $h$ at $y=1$ then
follows because it is a sum of two valid terms (one coming from $h_{lad}$),
just as it was in the original ladder in Eq.~(\ref{eq:ladnsplit}).
The difficult line is $y=2/3$ where we have coefficients that are constrained
by the symmetry $h(x,y)=-h(y,x)$. The coefficients are
\begin{eqnarray}
\frac{1}{2},\qquad\qquad
\frac{1}{2}-h_{lad}\left(\frac{2}{3},1\right)= \frac{1}{2}-2
\left(\frac{\Gamma-1}{\Gamma+1}\right)
\qquad\text{and}\qquad
-h_{lad}\left(\frac{2}{3},\frac{4}{3}\right)=\frac{\Gamma-3}{\Gamma+1}.
\end{eqnarray}
Conservation of probability follows trivially, and the line will be valid
if it satisfies the point splitting constraint $\sum p_i/x_i=0$
\begin{eqnarray}
\frac{3}{2(2+\delta)}+\frac{1}{2}-2
\left(\frac{\Gamma-1}{\Gamma+1}\right)
+\frac{3}{4}\left(\frac{\Gamma-3}{\Gamma+1}\right)=0
\end{eqnarray}
\noindent
which implicitly defines $\delta=8/(3\Gamma-1)$.
Finally, without the last line we would have a protocol of the form
\begin{eqnarray}
\frac{1}{2}[1,0]+\frac{1}{2}[0,1]\rightarrow
\frac{1}{2}\bigg[\frac{2+\delta}{3},\frac{2}{3}\bigg]+
\frac{1}{2}\bigg[\frac{2}{3},\frac{2+\delta}{3}\bigg].
\end{eqnarray}
The last line uses point raising to merge the two final points at
$[\frac{2+\delta}{3},\frac{2+\delta}{3}]$, giving us a protocol with
$P_A^*=P_B^*=\frac{2+\delta}{3}$ where $\delta\rightarrow 0$ as
$\Gamma\rightarrow \infty$.
\subsubsection{\label{sec:alternate}Building better ladders}
Having completed our analysis of the original bias $1/6$ ladder, our task
now is to apply the knowledge gained to the building of better ladders.
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\caption{A few simple ladder protocols: Symmetric (left) and asymmetric
(right), with initial point split (bottom) and without (top).}
\label{fig:simplads}
\end{center}
\end{figure}
We begin by studying a few simple variants of the $1/6$ ladder which are
depicted in Fig.~\ref{fig:simplads}. Whereas the ladders discussed thus far
have been symmetric (by reflection across the diagonal), all the TDPGs
discussed in Section~\ref{sec:examples} were asymmetric. Because the space
of TIPGs is a cone, an asymmetric TIPG can always be made symmetric by
taking a combination of itself and its reflection. The advantage of working
with symmetric TIPGs is that the validity of $h$ implies the validity of
$v$ so there are less constraints to check. The disadvantage is that the
expressions are generally more complicated (as we shall see below).
Nevertheless, in this paper we will use symmetric ladders to express the
main result and only study asymmetric ladders for comparison.
There is also the possibility of starting the ladders with a point split on
the axes. Numerical optimizations of the ladders depicted in
Fig.~\ref{fig:simplads} using a variable ladder spacing indicate that the
optimal TIPGs with no initial point split can achieve $P_A^*=P_B^*\approx
0.64$ whereas those with an initial point split can achieve
$P_A^*=P_B^*\approx 0.57$. From this perspective, constructing TIPGs with
an initial point split may be better. On the other hand, TIPGs with no
initial split tend to have simpler analytical expressions.
Unfortunately, one can also analytically prove that none of the forms
depicted in Fig.~\ref{fig:simplads} can achieve arbitrarily small bias. We
will not cover the proof here and instead directly proceed to studying more
complicated ladders that have more than four points across a horizontal
section.
A horizontal rung of an asymmetric ladder with $2k$ points across and
constant lattice spacing $\epsilon$ has the form
\begin{eqnarray}
\sum_{i=1}^{2k} \frac{-f(x+i\epsilon)}{\prod_{j\neq i}
(j \epsilon - i \epsilon)}
\bigg[x+i\epsilon\bigg]=
\sum_{i=1}^{2k} \frac{(-1)^i f(x+i\epsilon)}{\epsilon^{2k-1}(i-1)!(2k-i)!}
\bigg[x+i\epsilon\bigg].
\end{eqnarray}
A symmetric ladder is similar, except that the center point is always
missing. Therefore a rung with $2k$ points can be written as
\begin{eqnarray}
\sum_{\substack{i=-k\cr i\neq 0}}^{k} \frac{-f(x+i\epsilon)}{\prod_{j\neq
i,j\neq 0} (j \epsilon - i \epsilon)} \bigg[x+i\epsilon\bigg]=
\sum_{\substack{i=-k\cr i\neq 0}}^{k} \frac{(-1)^{k+i} (i) f(x+i\epsilon)}
{\epsilon^{2k-1}(k+i)!(k-i)!}
\bigg[x+i\epsilon\bigg].
\end{eqnarray}
A complete symmetric ladder has the form
\begin{eqnarray}
h_{lad}= \sum_{j=j_0}^{\Gamma}
\sum_{\substack{i=-k\cr i\neq 0}}^{k}
\frac{(-1)^{k+i}(i) f((j+i)\epsilon,j\epsilon)}
{\epsilon^{2k-1}(k+i)!(k-i)!}
\bigg[(j+i)\epsilon,j\epsilon\bigg].
\end{eqnarray}
\noindent
The ladder has been truncated at $y/\epsilon=\Gamma$ which can be done if
we pick
\begin{eqnarray}
f(x,y) = g(x,y) \left(\prod_{i=1}^{k} \left(\Gamma\epsilon+i\epsilon-x\right)\right)
\left(\prod_{i=1}^{k} \left(\Gamma\epsilon+i\epsilon-y\right)\right)
\end{eqnarray}
\noindent
and then we are still free to choose a symmetric polynomial $g(x,y)=g(y,x)$
so long as $g(-\lambda,y)$ is non-negative for $\lambda>0$ and $y>0$, and
has degree at most $k-2$ in $\lambda$.
Because we only have $k-2$ zeros to play with in $g$ we can't fully
truncate the bottom of the ladder as we did the top. But that is not a
problem. After all, our goal is to attach the bottom of the ladder to our
coin-flipping problem.
It is still useful to truncate as much of the bottom of the ladder as we
can in order to have fewer points to deal with at the bottom. We
can partially truncate at some height $y=j_0\epsilon$ by setting
\begin{eqnarray}
g(x,y) = C (-1)^k \left(\prod_{i=1}^{k-2} \left(j_0\epsilon-i\epsilon-x\right)\right)
\left(\prod_{i=1}^{k-2} \left(j_0\epsilon-i\epsilon-y\right)\right).
\end{eqnarray}
\noindent
The overall sign is chosen so that $g(-\lambda,y)$ is positive for
$\lambda>0$ and $j_0 \epsilon\leq y \leq \Gamma\epsilon$.
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\caption{A ladder with $2k=8$ points across terminated at top and partially
terminated at bottom (with an added point split). The dashed lines indicate
the zeros of $f(x,y)$.}
\label{fig:step}
\end{center}
\end{figure}
With this truncation $h_{lad}$ has exactly three points to the left of the
of the first truncation band $(j_0-k+2)\epsilon\leq x\leq
(j_0-1)\epsilon$. The three points are located at
$[(j_0-k+1)\epsilon,(j_0+1)\epsilon]$, $[(j_0-k+1)\epsilon,j_0\epsilon]$
and $[(j_0-k)\epsilon,j_0\epsilon]$. If we add in an extra point split at
$y=j_0-k+1$, and use $v(x,y)=h(y,x)$ we end up with the situation depicted
in Fig.~\ref{fig:step}.
With some further tuning we can end up with
\begin{eqnarray}
h+v = -\frac{1}{2}[1,0]- \frac{1}{2}[0,1]
+\frac{1}{2}[1-\epsilon'',\epsilon']+ \frac{1}{2}[\epsilon',1-\epsilon'']
\end{eqnarray}
for some $0<\epsilon''<\epsilon'$, which is effectively a step along the
diagonal connecting $[1,0]$ and $[0,1]$ towards the center
$[\frac{1}{2},\frac{1}{2}]$. The slope $\epsilon'/\epsilon''$ of the step
should converge to one as $k\rightarrow\infty$, and therefore one can use a
sequence of such steps to merge the two points in the center and obtain a
coin-flipping protocol with arbitrarily small bias. We shall not explore
this construction further, though, and instead will focus our efforts on a
procedure that works is a single big step.
\subsubsection{Mixing ladders with points on the axes}
In this section we will present a family of protocols that converges to
zero bias. The corresponding TIPGs will mix ladders with probability
located on the axes. These TIPGs will be generalizations of the protocol
from Section~\ref{sec:TDPG16}, which is the simplest example of the
constructions used in this section. The discussion in this section will be
informal and the formal proofs will be deferred to the next section.
The complete protocol can be thought of as a three step process
\begin{eqnarray}
\label{eq:threestep}
\frac{1}{2}[1,0]+\frac{1}{2}[0,1]&\rightarrow&
\frac{1}{2}\left(\sum_{j=z^*/\epsilon}^\Gamma p(j\epsilon) [j\epsilon,0]\right)
+ \frac{1}{2}\left(\sum_{j=z^*/\epsilon}^\Gamma p(j\epsilon) [0,j\epsilon]\right)
\\\nonumber
&\rightarrow& \frac{1}{2}[z^*,z^*-k\epsilon]+\frac{1}{2}[z^*-k\epsilon,z^*]
\rightarrow 1[z^*,z^*]
\end{eqnarray}
\noindent
and depends on the usual parameters: integer $k\geq 1$, small $\epsilon>0$,
large integer $\Gamma>0$ and a final point $1/2<z^*<1$. It also involves a
function $p(z)$ to be chosen below. There are a few obvious constraints
such as $k\epsilon<z^*$ and $z^*/\epsilon\in\mathbb{Z}$ which will all be resolved
in the limits $\epsilon\rightarrow 0$ and $\Gamma\rightarrow\infty$ for
fixed $k$. Our goal will be to prove that the above process is valid for
some $z^*$, and then to find the minimum valid $P_A^*=P_B^*=z^*$ for a
given $k$.
The first transition of the above process is intended to be a series of
point splits along the axes, the second transition is the difficult step
involving ladders, and the third transition is trivially valid by
point-raising. The first transition is also valid given the following
simple constraints
\begin{eqnarray}
1= \sum_{j=z^*/\epsilon}^\Gamma p(j\epsilon)\qquad\text{and}\qquad
1\geq \sum_{j=z^*/\epsilon}^\Gamma p(j\epsilon)/z
\label{eq:psdisc}
\end{eqnarray}
which correspond to probability conservation and the point-splitting
constraint (i.e., non-increasing average $1/z$). Note that, as opposed to
Section~\ref{sec:TDPG16}, we are using here a discrete function $p(z)$ with
finite support.
The second transition is the interesting step and will consist of a process
involving a ladder that slowly collects the amplitude on the axes and
deposits it near $x=z^*$ and $y=z^*$. The complete transition will be
described as usual by a valid horizontal function $h(x,y)$ and a valid vertical
function $v(x,y)=h(y,x)$ such that
\begin{eqnarray}q
h+v = - \frac{1}{2}\left(\sum_{j=z^*/\epsilon}^\Gamma p(j\epsilon)
[j\epsilon,0]\right) - \frac{1}{2}\left(\sum_{j=z^*/\epsilon}^\Gamma
p(j\epsilon) [0,j\epsilon]\right) +
\frac{1}{2}[z^*,z^*-k\epsilon]+\frac{1}{2}[z^*-k\epsilon,z^*].
\end{eqnarray}q
The new element is that when constructing the coefficients
$-f(x_i,y_i)/\prod_{j\neq i}(x_j-x_i)$ for the ladder, one of the
coordinates that appears in the product in the denominator is $x_j=0$. Why
is this different? So far we have exploited the fact that the product of
distances for a point was the same (up to sign) whether we computed its
vertical or horizontal neighbors. But now the expression includes an extra
factor of $(0-x_i)$ or $(0-y_i)$, which means the two computations will be
different. In other words, using a symmetric $f(x,y)=f(y,x)$ will not yield
an antisymmetric $h(x,y)=-h(y,x)$. The solution is to pull out an extra
factor of $1/y$ out of $f(x,y)$, which will once again allow us to use
symmetric functions $f(x,y)$, and will not affect the computation of
horizontal validity (since $1/y$ is positive and order zero as a polynomial
in $x$). We therefore set
\begin{eqnarray}
h= \sum_{j=z^*/\epsilon}^{\Gamma}\left(
- \frac{p(j\epsilon)}{2} [0,j\epsilon]
+\sum_{\substack{i=-k\cr i\neq 0}}^{k}
\frac{-f((j+i)\epsilon,j\epsilon)}
{(j\epsilon)\prod_{x_\ell\neq x_i} (x_\ell-x_i)}
\bigg[(j+i)\epsilon,j\epsilon\bigg]\right),
\end{eqnarray}
\noindent
where $\prod_{x_\ell\neq x_i} (x_\ell-x_i)$ still includes a factor of
$(0-x_i)=-(j+i)\epsilon$.
In order to use Lemma~\ref{lemma:posnumer}, though, we must
write the coefficient of $[0,j\epsilon]$ is the standard form, which
imposes relation between $p(z)$ and $f(x,y)$
\begin{eqnarray}
\frac{p(j\epsilon)}{2}= \frac{f(0,j\epsilon)}
{(j\epsilon)\prod_{x_\ell\neq 0} (x_\ell-0)}=
\frac{f(0,j\epsilon)}
{\epsilon^{2k+1} \prod_{\ell=j-k}^{j+k}\ell}.
\end{eqnarray}
We now pick $f$ as usual to fully truncate the ladder at the top and to
truncate as much as possible of the ladder at the bottom
\begin{eqnarray}
f(x,y) &=& C (-1)^{k-1}\left(\prod_{i=1}^{k-1} \left(z^*-i\epsilon-x\right)\right)
\left(\prod_{i=1}^{k-1} \left(z^*-i\epsilon-y\right)\right)
\\\nonumber && \qquad\qquad \times
\left(\prod_{i=1}^{k} \left(\Gamma\epsilon+i\epsilon-x\right)\right)
\left(\prod_{i=1}^{k} \left(\Gamma\epsilon+i\epsilon-y\right)\right).
\end{eqnarray}
We have the usual symmetry properties $f(x,y)=f(y,x)$. Furthermore,
$f(-\lambda,y)$ is positive for $\lambda>0$ and $z^*\leq y \leq
\Gamma\epsilon$ (provided $\epsilon$ is small enough so that
$k\epsilon<z^*$). Finally, $f(-\lambda,y)$ has degree $2k-1$ in $\lambda$,
which is allowed because we have $2k+1$ points across, once we include the
point on the axis.
Because the truncation at the bottom of the ladder uses $k-1$ zeros, $h$
will only have a single point (excluding those on the axis) to the left of
the first truncation band $z^*-(k-1)\epsilon\leq x\leq z^*-\epsilon$. The
point will be located at $[z^*-k\epsilon,z^*]$ and by probability
conservation must have exactly the same amount of probability as was
originally located on the axes.
All that remains undone is to figure out what values of $z^*$ are
allowed. For the moment we will only compute this in the limit of
$\epsilon\rightarrow0$ and $\Gamma\rightarrow\infty$ in which
\begin{eqnarray}
f(0,z)= C' \left(\prod_{i=1}^{k-1} \left(z^*-i\epsilon-z\right)\right)
\left(\prod_{i=1}^{k} \left(\Gamma\epsilon+i\epsilon-z\right)\right)
\rightarrow C'' (z-z^*)^{k-1}
\end{eqnarray}
\noindent
for some $k$-dependent constants $C'$ and $C''$ where we used
$(\Gamma-z)/\Gamma\rightarrow 1$. We then have
\begin{eqnarray}
p(z) = \frac{2f(0,j\epsilon)}
{z\prod_{l\neq 0} (x_l-0)}
\rightarrow C''' \frac{(z-z^*)^{k-1}}{z^{2k+1}}
\end{eqnarray}
\noindent
and the two constants $z^*$ and $C'''$ are fixed by the constraints of
Eq.~(\ref{eq:psdisc}), now in integral form
\begin{eqnarray}
1=\int_{z^*}^\infty p(z)dz, \qquad\qquad
1=\int_{z^*}^\infty \frac{p(z)}{z}dz,
\label{eq:betabeg}
\end{eqnarray}
\noindent
where we we have imposed equality in the second constraint to obtain the
smallest $z^*$ for a given $k$. Using $w=z^*/z$, we can transform the
following integral into a representation of the Beta function to get
\begin{eqnarray}
\int_{z^*}^\infty \frac{(z-z^*)^j}{z^\ell} dz
&=& \left(z^*\right)^{j-\ell+1} \int_0^1 w^{\ell-j-2}(1-w)^j d w
= \left(z^*\right)^{j-\ell+1} B(\ell-j-1,j+1)
\nonumber\\
&=& \left(z^*\right)^{j-\ell+1} \frac{\Gamma(\ell-j-1)\Gamma(j+1))}{\Gamma(\ell)}
= \left(z^*\right)^{j-\ell+1} \frac{(\ell-j-2)!(j)!}{(l-1)!}.
\end{eqnarray}
\noindent
We then set our two constraint integrals equal to each other to cancel $C'''$
and solve for $z^*$
\begin{eqnarray}
\left(z^*\right)^{-(k+1)} \frac{(k)!(k-1)!}{(2k)!}=
\left(z^*\right)^{-(k+2)} \frac{(k+1)!(k-1)!}{(2k+1)!}
\qquad\Longrightarrow\qquad z^* = \frac{k+1}{2k+1}.
\label{eq:betaend}
\end{eqnarray}
In other words, we have constructed a coin-flipping protocol with
$P_A^*=P_B^*=(k+1)/(2k+1)$. The construction is valid for all $k\geq 1$.
The next section will formalize the protocol. All the main ideas will be
the same, though some of the functions will be redefined by constant
factors.
\subsection{\label{sec:formal}Formal proof}
\begin{eqnarray}gin{definition}
Fix an integer $k>0$, a small $\epsilon>0$ satisfying $k\epsilon<1/2$, a
large integer $\Gamma>4k$ and a parameter $z^*\in(\frac{1}{2},1)$ such that
$z^*/\epsilon$ is an integer. Let $\Upsilon=\{k,\epsilon,\Gamma,z^*\}$ and
define
\begin{eqnarray}
g_{\Upsilon}(z) &=&
\left(\prod_{j=1}^{k-1} \left(\frac{z^*-j\epsilon-z}{z^*-j\epsilon}\right)\right)
\left(\prod_{j=1}^{k} \left(\frac{(\Gamma+j)\epsilon-z}{(\Gamma+j)\epsilon}\right)
\right),
\\
p_{\Upsilon}(z) &=& (-1)^{k-1}g_{\Upsilon}(z)
\prod_{j=-k}^{k}\left(\frac{1}{z+j\epsilon}\right),
\\
C_{\Upsilon} &=& 1\bigg/\sum_{j=z^*/\epsilon}^\Gamma
p_{\Upsilon}(j\epsilon),
\\
D_{\Upsilon}(i) &=& \epsilon^{2k-1}
\prod_{\substack{\ell=-k\cr\ell\neq i}}^{k} \left(\ell-i\right),
\\
2h_{\Upsilon} &=& -1[1,0] + C_{\Upsilon}\sum_{j=z^*/\epsilon}^\Gamma
p_{\Upsilon}(j\epsilon)[j\epsilon,0]
\label{eq:h}
\\\nonumber
&&-1[z^*-k\epsilon,z^*]+1[z^*,z^*]
\\\nonumber
&&+C_{\Upsilon} \sum_{j=z^*/\epsilon}^\Gamma\left(
-p_{\Upsilon}(j\epsilon)[0,j\epsilon]+
\sum_{\substack{i=-k\cr i\neq 0}}^{k}
\frac{(-1)^k g_{\Upsilon}((j+i)\epsilon)g_{\Upsilon}(j\epsilon)}
{(j\epsilon)((j+i)\epsilon)D_{\Upsilon}(i)}
[(j+i)\epsilon,j\epsilon]
\right)
\end{eqnarray}
and $v_{\Upsilon}(x,y)=h_{\Upsilon}(y,x)$.
\label{def:main}
\end{definition}
\begin{eqnarray}gin{lemma}
Given $\Upsilon$ as in Definition~\ref{def:main}, if
\begin{eqnarray}
1\geq C_\Upsilon
\sum_{j=z^*/\epsilon}^\Gamma
\frac{p_{\Upsilon}(j\epsilon)}{j\epsilon}
\end{eqnarray}
then $h_{\Upsilon}$ is a valid horizontal function and $v_{\Upsilon}$
is a valid vertical function.
\end{lemma}
\begin{eqnarray}gin{proof}
By symmetry, it suffices to prove that $h_{\Upsilon}$ is a valid horizontal
function. It is also sufficient to check the conditions independently on
each of the three lines of Eq.~(\ref{eq:h}). Given the constraint in the
definition of the lemma, the first line is a valid point split and the
second line is a valid point raise (see Lemma~\ref{lemma:rms}).
Also $p_\upsilon(z)\geq 0$ for $z^*/\epsilon\leq z\leq\Gamma$ so
$C_\Upsilon\geq 0$ and it can be canceled as we focus on the term in
parenthesis in the third line for each integer $j\in[z^*/\epsilon,\Gamma]$.
If we define
\begin{eqnarray}
f_j(x) = \frac{(-1)^{k-1}
g_{\Upsilon}(x)g_{\Upsilon}(j\epsilon)}
{(j\epsilon)}
\end{eqnarray}
then we can write the line at $y=j\epsilon$ as
\begin{eqnarray}
\sum_i \left(\frac{-f_j(x_i)}{\prod_{\substack{j\neq i}}(x_j-x_i)} \right)
[x_i,y],
\end{eqnarray}
\noindent
where $x_i\in\{0\}\cup
\{(j-k)\epsilon,\dots,(j-1)\epsilon,(j+1)\epsilon,\dots,(j+k)\epsilon\}$
and we used $g_{\Upsilon}(0)=1$ among other relations.
Now we can apply Lemma~\ref{lemma:posnumer} which tells us that the above
function is valid so long as $f_j(-\lambda)$ is a polynomial in $\lambda$
of order no greater than $2k-1$ and positive for all $\lambda>0$. The first
condition is trivial and the second follows because
$g_{\Upsilon}(-\lambda)>0$ for $\lambda>0$ and
$(-1)^{k-1}g_{\Upsilon}(j\epsilon)\geq 0$ for $z^*\leq
j\epsilon\leq\Gamma\epsilon$.
\end{proof}
\begin{eqnarray}gin{lemma}
Given $\Upsilon$ as in Definition~\ref{def:main} we have
\begin{eqnarray}
h_{\Upsilon}+v_{\Upsilon} =
-\frac{1}{2}[1,0]-\frac{1}{2}[0,1]+1[z^*,z^*].
\end{eqnarray}
\end{lemma}
\begin{eqnarray}gin{proof}
The only nontrivial cancellation is on the points $[z^*-k\epsilon,z^*]$ and
$[z^*,z^*-k\epsilon]$ which have by symmetry the same coefficient (with the
same sign). Because each line of $h_{\Upsilon}$ conserves probability,
$h_{\Upsilon}+v_{\Upsilon}$ must have a net zero probability, and so the
coefficients of these points must also be zero.
\end{proof}
\begin{eqnarray}gin{corollary}
Given $\Upsilon$ as in Definition~\ref{def:main}, if
\begin{eqnarray}
1\geq C_\Upsilon
\sum_{j=z^*/\epsilon}^\Gamma
\frac{p_{\Upsilon}(j\epsilon)}{j\epsilon}
\end{eqnarray}
then $h_{\Upsilon}$ and $v_{\Upsilon}$ is a TIPG with final point
$[z^*,z^*].$
\end{corollary}
\begin{eqnarray}gin{lemma}
Given $\Upsilon$ as in Definition~\ref{def:main}, there exists a family of
solutions to
\begin{eqnarray}
1\geq C_\Upsilon
\sum_{j=z^*/\epsilon}^\Gamma
\frac{p_{\Upsilon}(j\epsilon)}{j\epsilon}
\end{eqnarray}
\noindent
such that $\epsilon\rightarrow0$, $\Gamma\rightarrow\infty$ and
\begin{eqnarray}
z^*\rightarrow \frac{k+1}{2k+1}.
\end{eqnarray}
\end{lemma}
\begin{eqnarray}gin{proof}
For a given $\epsilon$ and $\Gamma$, the best $z^*$ is constrained by
$z^*/\epsilon\in\mathbb{Z}$ and
\begin{eqnarray}
\sum_{j=z^*/\epsilon}^\Gamma
p_{\Upsilon}(j\epsilon)
\geq
\sum_{j=z^*/\epsilon}^\Gamma
\frac{p_{\Upsilon}(j\epsilon)}{j\epsilon},
\end{eqnarray}
\noindent
where we have expanded the definition of $C_\Upsilon$. We then use
\begin{eqnarray}
\left(\frac{z-(z^*-k\epsilon)}{z^*-k\epsilon}\right)^{k-1}
\left(\frac{1}{z-k\epsilon}\right)^{2k+1} \geq &p_{\Upsilon}(z)&
\geq
\left(\frac{z-z^*}{z^*}\right)^{k-1}\left(\frac{\Gamma-z}{\Gamma}\right)^k
\left(\frac{1}{z+k\epsilon}\right)^{2k+1}
\end{eqnarray}
\noindent
to note that if we choose $z^*$ in accordance with the strict inequality
\begin{eqnarray}
\sum_{j=z^*/\epsilon}^\infty
\left(\frac{j\epsilon-z^*}{z^*}\right)^{k-1}
\left(\frac{1}{j\epsilon+k\epsilon}\right)^{2k+1}
>
\sum_{j=z^*/\epsilon}^\infty
\left(\frac{j\epsilon-(z^*-k\epsilon)}{z^*-k\epsilon}\right)^{k-1}
\left(\frac{1}{j\epsilon-k\epsilon}\right)^{2k+2}
\end{eqnarray}
then we can always find a large enough integer $\Lambda$ so that the
original inequality is satisfied (that is because the new right-hand side is
greater than or equal to the original right-hand side, whereas the original
left-hand side will converge as $\Lambda\rightarrow\infty$ to an
expression greater than or equal to the new left-hand side).
Similarly, if we choose $z^*$ in accordance with the strict inequality
\begin{eqnarray}
\int_{z^*}^\infty
\left(\frac{z-z^*}{z^*}\right)^{k-1}
\left(\frac{1}{z}\right)^{2k+1} dz
&>&
\int_{z^*}^\infty
\left(\frac{z-z^*}{z^*}\right)^{k-1}
\left(\frac{1}{z}\right)^{2k+2} dz
\end{eqnarray}
we can find an appropriate $\epsilon>0$ so that the original constraints
are satisfied. The argument for solving the above inequality is the same as
the one used for Eq.~(\ref{eq:betabeg}) through Eq.~(\ref{eq:betaend}). The
constraint becomes
\begin{eqnarray}
z^*> \frac{k+1}{2k+1}
\end{eqnarray}
and therefore for any such $z^*$ we can find appropriate $\epsilon$ and
$\Gamma$ that satisfy the original constraints.
\end{proof}
\begin{eqnarray}gin{corollary}
For every integer $k>0$ there is a family of coin-flipping protocols
that converges to
\begin{eqnarray}
P_A^*=P_B^*=\frac{k+1}{2k+1}.
\end{eqnarray}
\end{corollary}
\begin{eqnarray}gin{corollary}
There exists protocols for quantum weak coin flipping with
arbitrarily small bias.
\end{corollary}
\section{\label{sec:end}Conclusions}
We have constructively proven the existence of protocols for quantum weak
coin flipping with arbitrarily small bias. In the end, it appears that
quantum information has fulfilled at least a small part of its promise in
the area of two-party secure computation.
We have also tried to provide a primer on how to use Kitaev's formalism to
build interesting protocols. Hopefully the present result will be the
first of many to be obtained by viewing quantum games as dual to the cone
of bi-operator monotone functions.
\subsection{Open problems}
\begin{eqnarray}gin{enumerate}
\item Improvements and extensions to the proof:
\begin{eqnarray}gin{enumerate}
\item We did not explicitly complete the proof that if no coin-flipping
protocols with arbitrarily small bias exists, then the bound can be proven
using a bi-operator monotone function. Completing the proof would show that
the cone of TIPGs and the cone of bi-operator monotone functions are dual.
\item We have only constructed protocols for the case
$P_A=P_B=1/2$. Protocols for other cases can be constructed using serial
composition of this protocol. Nevertheless, it should also be
straightforward to explicitly describe a TIPG for such cases.
\item The alternative construction from Section~\ref{sec:alternate} needs
to be fleshed out. It may lead to some elegant TIPGs.
\item Appendix~\ref{sec:f2m} needs to be simplified/made more elegant.
\end{enumerate}
\item Improvements and extensions to the protocol:
\begin{eqnarray}gin{enumerate}
\item Can we find a simple unitary description (such as the one in
Appendix~\ref{sec:ddb}) of a family of protocols that achieves arbitrarily
small bias?
\item Can we optimize the resources (messages, storage qubits, complexity
of unitaries?) needed to implement such a protocol.
\begin{eqnarray}gin{enumerate}
\item What are the asymptotic costs of achieving arbitrarily small bias?
\item What are the practical costs of achieving small bias? How hard would
it be to make a protocol with bias $0.001$?
\end{enumerate}
\item What can be said about multiparty weak coin flipping?
\end{enumerate}
\item Beyond weak coin flipping
\begin{eqnarray}gin{enumerate}
\item Weak coin flipping with arbitrarily small bias leads trivially to a
new protocol for strong coin flipping with bias (arbitrarily close to)
$1/4$. Is this the best that can be done?
\item More generally, what is the optimal protocol for strong coin flipping
with cheat detection? While the formalism in this paper can be adapted to
strong coin flipping, it probably cannot be used unmodified. The difference
is that in strong coin flipping we need to simultaneously bound four
quantities (Alice and Bob's probabilities each of obtaining zero and
one). Even classically one can construct protocols that achieve
$P_{A0}^*P_{B0}^*=1/2$ (so long as one is willing to tolerate
$P_{A1}=P_{B1}=1$).
\item Strong coin flipping with cheat detection can be used to bound bit
commitment with cheat detection. Do they share an optimal protocol? Can it
be used as a building block for all secure two-party computations with
cheat detection?
\item Kitaev's first formalism can also be used in the study of specific
oracle problems. In this context the second formalism could be used to
study all oracles simultaneously, which may prove useful in identifying
optimal oracles in some sense (for instance for proving separations between
quantum and classical computation).
\item What else?
\end{enumerate}
\end{enumerate}
\section*{Acknowledgments}
I would like to thank Alexei Kitaev for teaching me his new formalism,
without which this result would not have been possible. I would also like
to thank Dave Feinberg and Debbie Leung for their help and useful
discussions. Research at Perimeter Institute for Theoretical Physics is
supported in part by the Government of Canada through NSERC and by the
Province of Ontario through MRI.
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\appendix
\section{\label{sec:ddb}Dip-Dip-Boom and the bias 1/6 protocol}
We present in this section a reformulation of the bias $1/6$ protocol from
\cite{me2005}. The new version of the protocol is simpler and uses
measurements throughout the protocol in order to keep the total storage
space small: one qutrit for each of the players and one qubit for sending
messages. In fact, the message qubit can be discarded and reinitialized
after each message, so that only the qutrits need to be kept coherent for
the length of the protocol.
The new protocol can be described as a quantum version of the ancient game
of Dip-Dip-Boom, whose classical version is played as follows: two players
sequentially say either ``Dip'' or ``Boom''. In the first case the game
proceeds whereas in the second case the game immediately ends and the
person who said ``Boom'' is declared the winner. There are no bonus points
for longer games and a player can begin and immediately win a game by
saying ``Boom''. Why this game is played at MIT, and how it was accepted as
a major component of the author's doctoral thesis, are questions beyond the
scope of this paper.
What we shall do is build a coin-flipping protocol out of the above game.
First, we introduce honest and cheating players. Honest players will have
to output ``Dip'' vs ``Boom'' according to some previously fixed
probability distribution, whereas cheating players are still free to say
whatever they want at each round. A game is now specified by a set of
numbers $p_1, p_2, p_3, \dots$ so that $p_i$ is the probability that the
$i$th word is ``Boom''. Note that $p_1,p_3,\dots$ apply to the first player
whom we call Alice and $p_2,p_4,\dots$ apply to the second player whom we
call Bob. For convenience we shall fix an $n\geq1$ and assume that the game
ends after $n$ messages by setting $p_n = 1$.
We now define the quantities $P_A(i)$, $P_B(i)$ and $P_U(i)$ which are
respectively the probabilities that after $i$ messages Alice has won the
game, Bob has won the game, or the game remains undecided. These quantities
can be inductively calculated by
\begin{eqnarray}
P_A(i) &=&
\begin{eqnarray}gin{cases}
P_A(i-1) & \text{for $i$ even,}\cr
P_A(i-1) + p_i P_U(i-1) & \text{for $i$ odd,}
\end{cases}
\\
P_B(i) &=&
\begin{eqnarray}gin{cases}
P_B(i-1) + p_i P_U(i-1) & \text{for $i$ even,}\cr
P_B(i-1) & \text{for $i$ odd,}
\end{cases}
\\
P_U(i) &=& (1-p_i) P_U(i-1),
\end{eqnarray}
with initial conditions of $P_U(0)=1$ and $P_A(0)=P_B(0)=0$.
The quantum version of the protocol is simply a coherent version of the
above, with an additional cheat detection step. Alice and Bob will each
hold a qutrit
\begin{eqnarray}
{\mathcal{A}} = {\mathcal{B}} = \vspan\{\ket{A}, \ket{B}, \ket{U}\}
\end{eqnarray}
which encodes the state of the game: ``Alice has won'', ``Bob has won'' and
``undecided'' respectively. The message space is just a qubit
\begin{eqnarray}
{\mathcal{M}} = \vspan\{\ket{\text{DIP}},\ket{\text{BOOM}}\}
\end{eqnarray}
comprising the two possible messages.
Amplitude will be moved coherently using the unitary operator $\mathbb{R}ot$
defined by
\begin{eqnarray}
\mathbb{R}ot(\ket{\alpha},\ket{\begin{eqnarray}ta},\epsilon)
= \mypmatrix{\ket{\alpha} & \ket{\begin{eqnarray}ta}}
\mypmatrix{\sqrt{1-\epsilon} & -\sqrt{\epsilon}\cr
\sqrt{\epsilon} & \sqrt{1-\epsilon}}
\mypmatrix{\bra{\alpha} \cr \bra{\begin{eqnarray}ta}}
+
\bigg(I - \ket{\alpha}\bra{\alpha} - \ket{\begin{eqnarray}ta}\bra{\begin{eqnarray}ta} \bigg),
\end{eqnarray}
\noindent
which simply effectuates a rotation in the $\ket{\alpha}$, $\ket{\begin{eqnarray}ta}$
plane. For instance, consider the classical step where Alice says ``Boom''
with probability $p$ given that her state is $\ket{U}$. In the quantum protocol
this is described by the rotation $\mathbb{R}ot(\ket{U}\otimes\ket{\text{DIP}},
\ket{A}\otimes\ket{\text{BOOM}}, p)$. Assuming that previously
$\ket{A}\otimes\ket{\text{BOOM}}$ had zero amplitude, the operation will
move into this state an amplitude of $\sqrt{p}$ times the prior amplitude
of $\ket{U}\otimes\ket{\text{DIP}}$. We are now ready to state the quantum
protocol:
\begin{eqnarray}gin{protocol}[Weak coin flipping with bias 1/6, simplified]\ \\
Fix $n\geq 1$ and numbers $p_1,\dots,p_n\in[0,1]$. Define ${\mathcal{A}}$, ${\mathcal{B}}$,
${\mathcal{M}}$, $P_A(i)$, $P_B(i)$ and $P_U(i)$ as above.
The protocol has the following steps
\begin{eqnarray}gin{enumerate}
\item Initialization: Alice prepares $\ket{U}\otimes\ket{\text{DIP}}$
in ${\mathcal{A}}\otimes{\mathcal{M}}$. Bob prepares $\ket{U}$ in ${\mathcal{B}}$.
\item For $i = 1$ to $n$ execute the following steps:\\
(where we use $X$ to denote the player that would choose the $i$th message
in the classical protocol, and $Y$ is the other player. That is
$X=A$ and $Y=B$ if $i$ is odd or $X=B$ and $Y=A$ if $i$ is even).
\label{item:loop}
\begin{eqnarray}gin{enumerate}
\item $X$ applies the operator
\begin{eqnarray}
R_i \equiv \mathbb{R}ot\left(\ket{U}\otimes\ket{\text{DIP}},\,
\ket{X}\otimes\ket{\text{BOOM}},\, p_i\right).
\end{eqnarray}
\item $X$ sends the qubit $H_M$ which is received by $Y$.
\label{item:mes}
\item $Y$ applies the operator
\begin{eqnarray}
\tilde R_i = \mathbb{R}ot\left(\ket{U}\otimes\ket{\text{BOOM}},\,
\ket{X}\otimes\ket{\text{DIP}},\, \frac{p_i\,P_U(i-1)}{P_X(i)}\right).
\end{eqnarray}
\item $Y$ measures the message qubit ${\mathcal{M}}$ in the computational basis.\\
If the outcome is $\ket{\text{BOOM}}$ then $Y$ aborts and outputs $Y$.
\label{item:meas}
\end{enumerate}
\item Alice and Bob each measure their qutrit in the computational basis.
If the outcome is $U$ they declare themselves the winner, otherwise they
output the measurement outcome as the winner.
\end{enumerate}
\end{protocol}
\noindent
It is not hard to see that when both players are honest the state of the
system at the end of the $i$th iteration of Step~\ref{item:loop} can be
written as
\begin{eqnarray}
\left(\sqrt{P_A(i)} \ket{A}\otimes\ket{A}
+ \sqrt{P_B(i)} \ket{B}\otimes\ket{B}
+ \sqrt{P_U(i)} \ket{U}\otimes\ket{U}\right) \otimes \ket{\text{DIP}}.
\end{eqnarray}
To obtain a standard coin-flipping protocol we must therefore restrict the
choices of $p_1,\dots,p_n$ to values such that $P_A(n)=P_B(n)=1/2$. For
simplicity, we will also assume that for $i<n$, the other probabilities
$p_i$ are neither zero nor one. In particular, this requires the previously
discussed condition $p_n=1$.
Step~\ref{item:meas} is the new element of the quantum protocol, and serves
as a cheat detecting step. Without this measurement the protocol is exactly
equivalent to the original classical protocol.
It is important to note that when both players are honest, neither will
ever abort in Step~\ref{item:meas}. However, when one player is cheating
then the honest player will abort at Step~\ref{item:meas} with some
non-zero probability. Roughly speaking, the measurement compares the ratio
of amplitude in ``Boom'' (tensored with $\ket{U}$) in the present message,
to the total amplitude of ``Boom'' from previous messages. The classical
strategy of declaring with probability one ``Boom'' on the first message is
thwarted because this ratio will effectively be zero for all subsequent
messages. The optimal quantum cheating strategy involves a small amount
of cheating on each message, but that gives an honest player an opportunity
to declare victory too.
In the next section we will sketch the computation of $P_A^*$ and $P_B^*$
for the above protocol and find that they match the expressions for the
optimal protocols from \cite{me2005}. This is the first step in proving the
equivalence of the two protocols. A complete proof of equivalence would
take us too far afield, though one possible route follows from the
discussion in Section~\ref{sec:TDPG16}.
In practice, the main difference between the protocols is that in the one
presented above the cheat detection is done gradually as the protocol
progresses rather than in one big measurement at the end. This simplifies
the description of the protocol, and reduces the resources required for a
physical implementation. Alas, it does not make it any more cheat
resistant. In the limit $n\rightarrow\infty$ and with suitably chosen
values for $p_1,\dots,p_n$ (almost any choice so long as they go smoothly
to zero as $n\rightarrow\infty$) the above protocol achieves a bias of
$1/6$.
\subsection*{Analysis}
In this section we will make use of Kitaev's first formalism to compute
$P_B^*$ ($P_A^*$ can be obtained by a similar argument). In other words we
will find a feasible point of the dual SDP. Of course, this only proves an
upper bound on $P_B^*$, but the resulting expressions are in fact optimal,
and a matching lower bound can also be constructed.
The final result for this section will be $P_B^*$ as a function of the
variables $p_1,\dots,p_n$. We will not attempt to find the optimal values
for these parameters as that task is already done in \cite{me2005}.
We will also not attempt to describe the result as a TDPG, though this is a
simple task using the dual feasible point constructed below.
Henceforth we will consider the case of honest Alice and cheating Bob. The
analysis will be done from the perspective of Alice's qubits which must
satisfy the following SDP:
\begin{eqnarray}
\rho_1 &=& \Tr_{{\mathcal{M}}}[R_1\left(\ket{U}\bra{U}\otimes\ket{\text{DIP}}
\bra{\text{DIP}} \right) R_1^\dagger],\\
\Tr_{{\mathcal{M}}}[\rho_i] &=& \rho_{i-1}
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\
\text{for $i$ even,}\\
\rho_i &=& \Tr_{{\mathcal{M}}}\left[ R_i \Pi_{\text{DIP}} \tilde R_{i-1}
\rho_{i-1} \tilde R_{i-1}^\dagger \Pi_{\text{DIP}} R_i^\dagger \right]
\qquad\qquad\text{for $i>1$ odd,}\\
\rho_\text{f} &=& \begin{eqnarray}gin{cases}
\rho_n & \text{$n$ odd,}\\
\Tr_{{\mathcal{M}}}\left[\tilde R_n\rho_n \tilde R_n\dagger \right] & \text{$n$ even,}
\end{cases}\\
P_{win} &=& \bra{B} \rho_{f} \ket{B},
\end{eqnarray}
\noindent
where $\rho_i$ is the state of Alice's qubits immediately after the $i$th
message (i.e., after Step~\ref{item:mes}). The state $\rho_i$ for even $i$
is an operator on ${\mathcal{A}}\otimes{\mathcal{M}}$ and is unknown but constrained by
$\rho_{i-1}$. The state $\rho_i$ for odd $i$ is an operator on ${\mathcal{A}}$ and
can be computed from $\rho_{i-1}$ by applying the operators that Alice
would use.
Note that $\Pi_{\text{DIP}}\equiv I\otimes\ket{\text{DIP}}\bra{\text{DIP}}$
is a projector corresponding to a successful measurement in
Step~\ref{item:meas}. The normalization of $\rho_i$ will therefore be the
probability that Alice has not aborted yet. The final state $\rho_f$ on
${\mathcal{A}}$ will have a similar normalization and $P_{win}$ will be the
probability that Alice outputs a victory for Bob. In particular, Bob's
maximum probability of winning by cheating, $P_B^*$, is given by the
maximization of $P_{win}$ over positive semidefinite matrices satisfying
the above constraints.
The dual to the above SDP can be written in the following form:
\begin{eqnarray}
P_B^* &\leq& \bra{U}Z_0\ket{U},\\
Z_{i-1} \otimes\ket{\text{DIP}}\bra{\text{DIP}} &=&
\Pi_{\text{DIP}} R_{i}^\dagger \left(Z_{i}\otimes I_{{\mathcal{M}}}\right)
R_{i} \Pi_{\text{DIP}} \qquad\qquad\ \, \text{for $i$ odd,}
\label{eq:zodd2}\\
Z_{i-1} \otimes I_{{\mathcal{M}}} &\geq& \tilde R_{i}^\dagger
\left( Z_{i} \otimes\ket{\text{DIP}}\bra{\text{DIP}} \right) \tilde R_{i}
\qquad\qquad\quad\
\text{for $i$ even,}
\label{eq:zeven2}\\
Z_n &\geq& \ket{B}\bra{B},
\label{eq:zlast}
\end{eqnarray}
\noindent
where the variables $Z_0,\dots,Z_n$ are semidefinite operators on
${\mathcal{A}}$. Any assignment of these variables consistent with the above
constraints provides an upper bound on $P_B^*$. The infimum of
$\bra{U}Z_0\ket{U}$ will in fact be equal to $P_B^*$.
Now comes the crucial bit of guesswork, where we choose a diagonalizing
basis for the operators $Z_i$. Because any assignment to $Z_0,\dots,Z_n$
consistent with the constraints provides an upper bound on $P_B^*$,
choosing a bad basis will at worse give us a non-tight upper bound. Based
on experience from past protocols, we choose to restrict ourselves to
studying matrices that are diagonal in the computational basis, and we
write
\begin{eqnarray}
Z_i = \mypmatrix{a_i & 0 & 0 \cr 0 & b_i & 0 \cr 0 & 0 & u_i}.
\end{eqnarray}
We can now express Eq.~(\ref{eq:zodd2}) as
\begin{eqnarray}
\mypmatrix{a_{i-1} & 0 & 0 \cr 0 & b_{i-1} & 0 \cr 0 & 0 & u_{i-1}}
= \mypmatrix{a_i & 0 & 0 \cr 0 & b_i & 0 \cr 0 & 0 &
(1-p_i) u_{i} + p_i a_i}
\end{eqnarray}
\noindent
valid for $i$ odd. We can also write Eq.~(\ref{eq:zeven2}) as
\begin{eqnarray}
\mypmatrix{
a_{i-1} & 0 & 0 & 0 & 0 & 0 \cr
0 & a_{i-1} & 0 & 0 & 0 & 0 \cr
0 & 0 & b_{i-1} & 0 & 0 & 0 \cr
0 & 0 & 0 & b_{i-1} & 0 & 0 \cr
0 & 0 & 0 & 0 & u_{i-1} & 0 \cr
0 & 0 & 0 & 0 & 0 & u_{i-1}}
\geq
\mypmatrix{
a_i & 0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 & 0 \cr
0 & 0 & (1-\tilde p_i) b_i & 0 & 0 &
\sqrt{\tilde p_i (1-\tilde p_i)} b_i \cr
0 & 0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & u_i & 0 \cr
0 & 0 & \sqrt{\tilde p_i (1-\tilde p_i)} b_i &
0 & 0 & \tilde p_i b_i}
\end{eqnarray}
\noindent
valid for $i$ even, where we introduced $\tilde p_i = p_i P_U(i-1)/P_B(i)$
as a notation for the rotation parameter of $\tilde R_i$. The above
inequality can be simplified to $a_{i-1}\geq a_i$, $u_{i-1}\geq u_i$, plus
the positivity of the $2\times2$ matrix
\begin{eqnarray}
\mypmatrix{ b_{i-1} - (1-\tilde p_i) b_i &
-\sqrt{\tilde p_i (1-\tilde p_i)} b_i \cr
-\sqrt{\tilde p_i (1-\tilde p_i)} b_i &
u_{i-1} - \tilde p_i b_i
}\geq 0
\label{eq:2x2cond}
\end{eqnarray}
\noindent
which is satisfied if its determinant and its diagonal elements are
non-negative.
It is not hard to see that we can choose $a_i=0$ for all $i$. We are after
all trying to minimize $\bra{U}Z_0\ket{U} = u_0$. We now get the relation
$u_{i-1}=(1-p_i) u_i$ for $i$ odd, and we know that the best that we can
hope for is $u_{i-1}=u_i$ for $i$ even. If this were true we would have
\begin{eqnarray}
u_i = u_0 \prod_{\substack{j=1\cr\text{$j$ odd}}}^i \frac{1}{1-p_i}.
\end{eqnarray}
\noindent
Let us be optimistic, and assume the above holds and then check whether the
remaining constraints can be satisfied. Surprisingly, we shall find that the
answer is yes.
The intuition for the $b_i$ variables is that $b_n=1$ and $b_i$ increases
as $i$ decreases. In fact, the larger $b_0$ is, the better the bound we
will find, and the infimum is attained for $b_0=\infty$. With a little
care, we can directly handle this infinity and get $b_1=b_0=\infty$ and
$b_2 = u_1/\tilde p_2$ which satisfies Eq.~(\ref{eq:2x2cond}) for $i=2$.
The remaining conditions for $b_i$ will no longer involve infinities and
are obtained by minimizing the determinant Eq.~(\ref{eq:2x2cond}), which
leads to
\begin{eqnarray}
\frac{1}{b_i} = \frac{\tilde p_i}{u_{i-1}} + \frac{1-\tilde p_i}{b_{i-1}}
\qquad\qquad\text{for $i$ even,}
\end{eqnarray}
\noindent
which also guarantees that the constraint on the diagonal elements of
Eq.~(\ref{eq:2x2cond}) is satisfied. By induction we can write
\begin{eqnarray}
\frac{1}{b_i} &=&
\sum_{\substack{j=2\cr\text{$j$ even}}}^i \frac{\tilde p_j}{u_{j-1}}
\prod_{\substack{k=j+2\cr\text{$k$ even}}}^i (1-\tilde p_k)
=\sum_{\substack{j=2\cr\text{$j$ even}}}^i \frac{\tilde p_j}{u_{j-1}}
\prod_{\substack{k=j+2\cr\text{$k$ even}}}^i \frac{P_B(k-1)}{P_B(k)}
=\sum_{\substack{j=2\cr\text{$j$ even}}}^i \frac{\tilde p_j}{u_{j-1}}
\frac{P_B(j+1)}{P_B(i)}\\
&=& \sum_{\substack{j=2\cr\text{$j$ even}}}^i
\frac{p_j P_U(j-1)}{u_{j-1} P_B(i)}
= \frac{1}{u_0 P_B(i)} \sum_{\substack{j=2\cr\text{$j$ even}}}^i
p_j P_U(j-1) \prod_{\substack{k=1\cr\text{$k$ odd}}}^{j-1} (1-p_k),
\end{eqnarray}
\noindent
where in the second equality we used $P_B(k)- p_k P_U(k-1) = P_B(k-1)$ for
$k$ even, and in the third equality we used $P_B(k+1)/P_B(k)=1$ for $k$
even.
What we have done is solve for all the dual variables in terms of $u_0$.
We have also satisfied all constraints except for those from
Eq.~(\ref{eq:zlast}), which tells to set $b_n=1$ and allows us to solve for
$u_0$ to obtain our upper bound:
\begin{eqnarray}
P_B^* \leq u_0 = 2
\sum_{\substack{j=2\cr\text{$j$ even}}}^n p_j
\left(\prod_{k=1}^{j-1} (1-p_k)\right)
\left(\prod_{\substack{k=1\cr\text{$k$ odd}}}^{j-1} (1-p_k)\right),
\end{eqnarray}
\noindent
where we used $P_B(n)=1/2$.
The above expression is equivalent to the result found in
\cite{me2005}. For a simple example, take the Spekkens and Rudolph
\cite{Spekkens2002} protocol with $P_A^*=P_B^*=1/\sqrt{2}$, which can be
described in the above formalism by setting $p_1 = 1-1/\sqrt{2}$,
$p_2=1/\sqrt{2}$ and $p_3=1$. The above bound then becomes $u_0 = 2 p_2
(1-p_1)^2 = 1/\sqrt{2}$ as expected.
\section{\label{sec:strong}Proof of strong duality}
We say strong duality holds when the maximum of the primal SDP and the
infimum of the dual SDP are equal. Though strong duality does not hold in
general, it does hold in most cases. A number of lemmas which provide
sufficient conditions for strong duality can be found in convex
optimization books such as \cite{convex} or \cite{convex2}.
In this section, however, we aim to give a direct proof of strong duality
for the particular case of the coin-flipping SDP. This appendix follows the
notation of Section~\ref{sec:kit1}
More specifically, the goal for this section is to prove the existence of
arbitrarily good upper bound certificates. Mathematically, we aim to show
that $\inf \bra{\psi_{A,0}} Z_{A,0} \ket{\psi_{A,0}} = P_B^*$, where the
infimum is taken over all dual feasible points.
Let us begin by defining $R_i$ as the set of density matrices $\rho_{A,i}$
on ${\mathcal{A}}$ that are attainable by a cheating Bob after $i$ messages. We will
focus only on even $i$. These sets satisfy the properties:
\begin{eqnarray}gin{itemize}
\item $R_0 = \left\{\ket{\psi_{A,0}}\bra{\psi_{A,0}}\right\}$.
\item $R_{i+2} = \left\{\Tr_{\mathcal{M}}[ U_{A,i+1} \tilde\rho_{A,i} U_{A,i+1}^\dagger]
: \tilde \rho_{A,i}\geq 0 \text{ and }
\Tr_{\mathcal{M}} \tilde \rho_{A,i}\in R_i\right\}$.
\item $R_i$ is convex for all $i$.
\end{itemize}
\noindent
The last property follows from the previous two by induction.
Though in general we cannot find a dual feasible point such that
$\bra{\psi_{A,0}} Z_{A,0} \ket{\psi_{A,0}} = P_B^*$, we can get arbitrarily
close. Specifically, we can prove that for every $\epsilon>0$ we can pick
the dual variables one by one, starting with $Z_{n-2}$ and working our way
backwards towards $Z_0$, such that
\begin{eqnarray}gin{itemize}
\item $Z_{A,i}\otimes I_{\mathcal{M}} \geq
U_{A,i+1}^\dagger \left(Z_{A,i+2}\otimes I_{\mathcal{M}}\right) U_{A,i+1}$
\item $\max_{\rho_{A,i}\in R_i} \Tr[Z_{A,i} \rho_{A,i}] \leq
\max_{ \rho_{A,i+2}\in R_{i+2}} \Tr[Z_{A,i+2} \rho_{A,i+2}] + 2 \epsilon$
\end{itemize}
\noindent
for even $i$, where as usual $Z_{A,n} = \Pi_{A,1}$. The first condition
guarantees that the constructed solution is indeed a dual feasible point
(the operators for $i$ odd can be found from the equality $Z_{A,i} =
Z_{A,i+1}$). The second condition gives us
\begin{eqnarray}
\bra{\psi_{A,0}} Z_{A,0} \ket{\psi_{A,0}} =
\max_{\rho_{A,0}\in R_0} \Tr[Z_{A,0} \rho_{A,0}] \leq
\max_{\rho_{A,n}\in R_n} \Tr[Z_{A,n} \rho_{A,n}] + n\epsilon
= P_B^* + n\epsilon
\end{eqnarray}
\noindent
which is the desired result since $\epsilon>0$ is arbitrary and we already
know by weak duality that $\bra{\psi_{A,0}} Z_{A,0} \ket{\psi_{A,0}}\geq
P_B^*$.
Let us assume that $\epsilon>0$ has been given and that
$Z_{A,i+2},\dots,Z_{A,n}$ have been constructed according to the above
criteria. We shall find a $Z_{A,i}$ satisfying the criteria as well.
Let $\Gamma = U_{A,i+1}^\dagger \left(Z_{A,i+2}\otimes I_{\mathcal{M}}\right) U_{A,i+1}$,
define $\Pos({\mathcal{A}})$ to be the set of positive semidefinite operators on ${\mathcal{A}}$
and define the function $f:\Pos({\mathcal{A}})\rightarrow \mathbb{R}$ by
\begin{eqnarray}
f(\rho) = \max_{\substack{\tilde \rho\in\Pos({\mathcal{A}}\otimes{\mathcal{M}})
\cr \Tr_{\mathcal{M}} [\tilde \rho]=\rho}} \Tr[\Gamma \tilde \rho],
\end{eqnarray}
\noindent
where we are maximizing over positive semidefinite operators $\tilde \rho$
on on ${\mathcal{A}}\otimes{\mathcal{M}}$ whose partial trace is $\rho$. The function $f$ has
a number of simple to verify properties
\begin{eqnarray}gin{itemize}
\item $\max_{\rho\in R_n} f(\rho) =
\max_{ \rho_{A,i+2}\in R_{i+2}} \Tr[Z_{A,i+2} \rho_{A,i+2}]\equiv \gamma$.
\item $f$ is continuous.
\item $f$ is concave (equivalently $f(\rho)+f(\rho')\leq f(\rho+\rho')$).
\end{itemize}
\noindent
where we used the first equality to define $\gamma$.
We now aim to construct a convex set out of $f$ by using the space of
points below its graph. More specifically let $V$ be the vector space of
Hermitian operators on ${\mathcal{A}}$. We want to think of $V$ as a real Hilbert
space of dimension $(\dim{\mathcal{A}})^2$ with inner product
$\braket{H}{H'}=\Tr[H^\dagger H']$. Let $W = V\times \mathbb{R}$, which can be
parametrized by ordered pairs $(H,a)$ where $H$ is a Hermitian operator on
${\mathcal{A}}$ and $a\in\mathbb{R}$. Now define the set $X\subset W$ by
\begin{eqnarray}
X = \Big\{(H,a) : H\in\Pos({\mathcal{A}}), \Tr H \leq 2 \,
\text{ and } -1\leq a \leq f(H)
\Big\}.
\end{eqnarray}
\noindent
The numbers $2$ and $-1$ are arbitrary, and are mainly there to make $X$
compact. It is easy to check that $X$ is a convex and has non-empty
interior.
Now we define a second compact convex set $Y$ which will be disjoint from
$X$ but will sit above it (see Fig.~\ref{fig:strongdual}). We use the fact
that $f$ is continuous to find a $\delta>0$ such that
\begin{eqnarray}
\max_{\rho\in\Pos({\mathcal{A}}), \dist(\rho,R_i)\leq \delta} f(\rho) \leq
\max_{\rho\in R_i} f(\rho) +
\epsilon \equiv \gamma + \epsilon,
\end{eqnarray}
\noindent
where $\dist(\rho,R_i)$ is the $\ell_2$ distance. The idea is that we want
$Y$ to sit atop the set $R_i$ but we also want $Y$ to have non-empty
interior. Therefore we expand $R_i$ to a small (closed) neighborhood of
$R_i$ still consisting of positive semidefinite matrices $R_i^\delta =
\{\rho\in\Pos({\mathcal{A}}):\dist(\rho,R_i)\leq \delta\}$. Now we define
$Y\subset W$ as
\begin{eqnarray}
Y = \Big\{(H,a): H\in R_i^\delta
\text{ and } \gamma+2\epsilon\leq a \leq \gamma+2\epsilon+1
\Big\}.
\end{eqnarray}
\noindent
The set $Y$ is convex because $R_i$ was convex and the distance function is
convex. $Y$ is also compact with non-empty interior and disjoint from $X$.
\begin{eqnarray}gin{figure}[tb]
\begin{eqnarray}gin{center}
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\put(687,0){$R_i$}
\put(1437,1875){$\rho^*$}
\put(6987,1575){$V$}
\put(5937,2400){$f(V)$}
\put(650,3800){$Y$}
\put(3012,1425){$X$}
\end{picture}
\caption{The sets $X$ and $Y$. The horizontal axis is a cross section
through the density operators given by $\rho =
x\ket{0}\bra{0}+(1-x)\ket{1}\bra{1}$. The curve is given by $f(x) =
(\sqrt{x}+\sqrt{1-x})^2/2$ corresponding to $\Gamma$ being a projector onto
a bell state. The optimal state $\rho^*$ maximizes $f$ on the feasible set
$R_i$. The ideal hyperplane is the tangent to $f$ at
$\rho^*$. Unfortunately, it is quite common that along some cross sections
$R_i$ corresponds to a single boundary point where the slope of $f$ is
infinite. The non-zero width of $Y$ guarantees a finite slope for our
(non-optimal) hyperplane even in this case.}
\label{fig:strongdual}
\end{center}
\end{figure}
Now we use the separating hyperplane theorem (see for instance
\cite{convex} \S 2.5) which says that given two disjoint compact
convex sets there exists a hyperplane such that each set is on one side
(a proof sketch is let $x\in X$ and $y\in Y$ attain the minimum distance
between $X$ and $Y$, then define the hyperplane by the equation
$(x-y)\cdot (w -(x+y)/2)) = 0$ for $w\in W$).
Because $R_i^\delta$ is of non-empty interior in $V$, and $X$ and $Y$
respectively provide lower and upper bounds on the hyperplane is this
region, the hyperplane cannot be vertical and its normal vector can be
written in the form $(M,1)$ where $M$ is Hermitian. The hyperplane itself
can be parametrized as $(M,1)\cdot(H,a) = c$ for $(H,a)\in W$ and some
parameter $c\in\mathbb{R}$. As the hyperplane separates the sets $X$ and $Y$ we can
write
\begin{eqnarray}gin{itemize}
\item $(H,a)\in X \ \mathbb{R}ightarrow\ c - \Tr M H \geq a,$
\item $(H,a)\in Y \ \mathbb{R}ightarrow\ c - \Tr M H \leq a.$
\end{itemize}
\noindent
We are now ready to choose $Z_{A,i} = c I - M$. The two properties above
tell us that
\begin{eqnarray}gin{itemize}
\item For all density operators $\rho$ on ${\mathcal{A}}$ we have
$\Tr[Z_{A,i}\rho]\geq f(\rho)$.\\
$\Longrightarrow$ For all density operators $\tilde \rho$ on
${\mathcal{A}}\otimes{\mathcal{M}}$ we have
$\Tr\left[(Z_{A,i}\otimes I) \tilde\rho\right]
\geq \Tr[\Gamma \tilde \rho]$.\\
$\Longrightarrow Z_{A,i}\otimes I\geq U_{A,i+1}^\dagger
\left(Z_{A,i+2}\otimes I_{\mathcal{M}}\right) U_{A,i+1}$.
\item For all $\rho\in R_i$ we have $\Tr[Z_{A,i}\rho]\leq \gamma+2\epsilon$.\\
$\Longrightarrow \max_{\rho_{A,i}\in R_i} \Tr[Z_{A,i} \rho_{A,i}] \leq
\max_{ \rho_{A,i+2}\in R_{i+2}} \Tr[Z_{A,i+2} \rho_{A,i+2}] + 2 \epsilon$.
\end{itemize}
Therefore our chosen $Z_{A,i}$ satisfies the two required properties and
we have completed the proof of strong duality for coin flipping.
\section{\label{sec:f2m}From functions to matrices}
TDPGs are specified in terms of pairs of functions constituting valid
transitions. In order to compile them back into the language of quantum
mechanics and semidefinite programming, we need to have a way of extracting
matrices (states and unitaries) out of these transitions. That is our goal
below. As a corollary we shall prove Lemmas~\ref{lemma:f2mmain}
and~\ref{lemma:stdformmain}. This section uses the notation from
Section~\ref{sec:TDPG}.
We begin our discussion by considering the following alternate condition on
transitions:
\begin{eqnarray}gin{definition}
Let $p(z)$ and $q(z)$ be two functions $[0,\infty)\rightarrow
[0,\infty)$ with finite support. We say $p(z)\rightarrow q(z)$ is
\textbf{expressible by matrices} if there exists positive semidefinite
operators $X$ and $Y$, and an (unnormalized) vector $\ket{\psi}$ such that
$X\leq Y$ and $p(z)=\Prob(X,\ket{\psi})$ and $q(z)=\Prob(Y,\ket{\psi})$.
\end{definition}
\noindent
It is not hard to verify that all transitions expressible by matrices are
also valid, justifying our notation. It is also true that all transitions
constructed from UBPs are expressible by matrices (the only non-trivial
step is to expand the Hilbert space so as to purify the density
operator). What we will end up showing by the end of this section is
essentially the converse: that all strictly valid transitions are
expressible by matrices. This is the content of Lemma~\ref{lemma:f2mmain}.
Given that valid transitions and transitions expressible by matrices are
essentially equivalent concepts, we could have used the latter in our
definition of TDPGs. However, that would diminish one of the main
accomplishments of TDPGs: moving the problem of coin flipping outside the
traditional realm of quantum-mechanics/matrices/SDPs.
We note that there do exist valid (but not strictly valid) transitions that
are not expressible by matrices unless we allow infinite
eigenvalues. However, all operators in this paper are assumed to have finite
dimension and finite eigenvalues.
The restriction to strictly valid transitions can be lifted if we work with
functions defined on a compact domain rather than our usual domain of
$[0,\infty)$. For this section it will be useful to work with such compact
domains, and we extend to them our definitions of valid and expressible by
matrices:
\begin{eqnarray}gin{definition}
Fix a compact interval $[a,b]$. Given two functions
$p(z),q(z):[a,b]\rightarrow[0,\infty)$ with finite support we say
$p\rightarrow q$
\begin{eqnarray}gin{itemize}
\item is \textbf{valid on \boldmath $[a,b]$} if $\sum_z p(z)=\sum_z q(z)$
and $\sum_z \frac{\lambda z}{\lambda+z}(q(z) - p(z))\geq 0$ for all
$-\lambda\in\mathbb{R}\setminus I$ .
\item is \textbf{expressible by matrices in \boldmath $[a,b]$} if there
exists matrices $X$, $Y$ with spectrum in $[a,b]$, and a vector
$\ket{\psi}$ such that $X\leq Y$ and $p(z)=\Prob(X,\ket{\psi})$ and
$q(z)=\Prob(Y,\ket{\psi})$.
\end{itemize}
\end{definition}
\noindent
These definitions become useful with the following lemma:
\begin{eqnarray}gin{lemma}
Given two functions $p(z),q(z):[0,\infty)\rightarrow[0,\infty)$ with finite
support such that $p\rightarrow q$ is strictly valid, there exists
$\Lambda>0$, larger than the maximum of the supports of $p$ and $q$, such
that $p\rightarrow q$ is valid on $[0,\Lambda]$.
\label{lemma:inf2lamb}
\end{lemma}
\begin{eqnarray}gin{proof}
Because $p\rightarrow q$ is strictly valid $\sum_z z (q(z) - p(z))>0$,
which implies
\begin{eqnarray}
\lim_{\lambda\rightarrow-\infty} \sum_z \frac{\lambda z}{\lambda+z}
(q(z) - p(z))>0,
\end{eqnarray}
\noindent
where the expression inside the limit is only defined for $|\lambda|$
larger than the maximum of the support of $p$ and $q$. The expression
inside the limit is continuous as a function of $\lambda$, so there must
exists a finite $\Lambda>0$ such that for $\lambda\leq-\Lambda$ we also
have satisfy the inequality $\sum_z \frac{\lambda z}{\lambda+z} (q(z) -
p(z))>0$.
\end{proof}
\noindent
We can now restrict our attention to probability distributions and operator
monotone functions with domain $[0,\Lambda]$ and matrices with eigenvalues
in $[0,\Lambda]$, for some large $\Lambda>0$.
One approach to proving that valid on $[0,\Lambda]$ implies expressible by
matrices in $[0,\Lambda]$ is as follows: define $K$ to be the set of
functions with finite support of the form $g(z)\equiv
q(z)-p(z):[0,\Lambda]\rightarrow \mathbb{R}$ where $p\rightarrow q$ is expressible
by matrices with eigenvalues in $[0,\Lambda]$. The set $K$ is a convex
cone, and we can define its dual cone $K^*$ in the space of functions with
arbitrary support $f(z):[0,\Lambda]\rightarrow\mathbb{R}$. The inner product
between the two spaces is defined by $\braket{f}{g}=\sum_z f(z) g(z)$ which
is well defined because $g$ has finite support. It is easy to check that
$K^*$ is exactly the set of operator monotone functions with support
$[0,\Lambda]$. The dual $K^{**}$ of $K^*$ is the set of functions with
finite support of the form $g(z)\equiv q(z)-p(z):[0,\Lambda]\rightarrow \mathbb{R}$
where $p\rightarrow q$ is valid. Proving $K^{**}=\closure(K)$ covers
most of what we want to prove. The two difficulties with this approach is
that it requires some fairly advanced analysis to properly place $K$ and
$K^*$ into a pair of locally convex dual topological vector spaces (see for
instance \cite{convex2} \S IV.4), and that in the end we prove something
slightly weaker than needed: mainly we find a transition expressible by
matrices $p'\rightarrow q'$ so that $q'(z)-p'(z)=q(z)-p(z)$ rather than the
stronger conditions $p'(z)=p(z)$ and $q'(z)=q(z)$.
Instead we will use a more constructive approach to completing the proof:
Given $p\rightarrow q$ valid on $[a,b]$, we will construct a
perturbation $p'$ of $p$ so that $p'\rightarrow q$ is still valid on
$[a,b]$ and such that proving that this new transition is expressible
by matrices in $[a,b]$ will also prove that the original transition
is expressible by matrices in $[a,b]$. Alternatively, we may find a
perturbation $q'$ of $q$ and reduce the problem to studying $p\rightarrow
q'$. A sequence of such transformations can be used until we end up with a
valid transition that is also trivially expressible by matrices.
To formalize the perturbations we need to introduce some notation. Given
$p:[a,b]\rightarrow[0,\infty)$ with finite support we define a
\textit{canonical representation} $p=\Prob(X,\ket{\psi})$ by choosing $X$
diagonal with a non-degenerate set of eigenvalues equal to $S(p)$, the
support of $p$, and then choosing $\ket{\psi}=\sum_{z\in S(p)}
\sqrt{p(z)}\ket{z}$. In particular, the dimension of $X$ is the size of the
support of $p$, denoted by $|S(p)|$. Similarly we can construct a
canonical representation for $q:[a,b]\rightarrow[0,\infty)$ as
$q=\Prob(Y,\ket{\xi})$ with the spectrum of $Y$ is equal to $S(q)$. Note,
however, that even if $p\rightarrow q$ is valid on $[a,b]$ the matrices $X$
and $Y$ so constructed have no a priori relation, and in general are of
different dimensions.
All perturbations will be of the following form: let $c>0$ be a constant
and $\ket{\phi}$ a non-zero vector. Set $X'=X+c\ket{\phi}\bra{\phi}$ and
(assuming $X'$ has eigenvalues in $[a,b]$) set
$p'=\Prob(X',\ket{\psi})$. Then $p\rightarrow p'$ is trivially
constructable by matrices in $[a,b]$. Similarly, we can set
$q'=\Prob(Y',\ket{\xi})$ for $Y'=Y+c\ket{\phi}\bra{\phi}$ where now we want
$c<0$. Ensuring that the perturbations can be chosen so that $p'\rightarrow
q$ or $p\rightarrow q'$ are valid will take up most of the rest of the
section.
Our first simple result bounds the dimension of the space needed when
studying general transitions that are expressible by matrices. It also
standardizes the spectra of the matrices involved.
\begin{eqnarray}gin{lemma}
Let $p\rightarrow q$ be expressible by matrices in $[a,b]$, then we can
find matrices $X\leq Y$ and a vector $\ket{\phi}$ such that $p=
\Prob(X,\ket{\psi})$ and $q=\Prob(Y,\ket{\psi})$ and additionally:
\begin{eqnarray}gin{enumerate}
\item The spectrum of $X$ is equal to $\{a\}\cup S(p)$,
with all eigenvalues (excluding $a$) occurring once.
\item The spectrum of $Y$ is equal to $\{b\}\cup S(q)$,
with all eigenvalues (excluding $b$) occurring once.
\item The dimension of $X$ and $Y$ is no greater than
$|S(p)|+|S(q)|-1$.
\end{enumerate}
\label{lemma:stdmattrans}
\end{lemma}
\begin{eqnarray}gin{proof}
The fact that $p\rightarrow q$ is expressible by matrices in $[a,b]$,
guarantees the existence of matrices $X\leq Y$ with spectrum in $[a,b]$
and a vector $\ket{\phi}$ such that $p= \Prob(X,\ket{\psi})$ and
$q=\Prob(Y,\ket{\psi})$. What we need to prove that we can modify the
given matrices to satisfy the additional properties.
We begin by working with $X$ and $\ket{\psi}$. We can write
$\ket{\psi}=\sum_z \sqrt{p(z)} \ket{z;X}$ where $z$ ranges over the support
of $p$ and $\ket{z;X}$ is a normalized eigenvector of $X$ with eigenvalue
$z$. Let $\Pi=\sum_z\ket{z;X}\bra{z;X}$ be the projector onto the spaced
spanned by these eigenvalues. If we define $X'=\Pi X \Pi + a (I-\Pi)$ we
obtain a new matrix with eigenvalues in $[a,b]$ that is diagonal in the
same basis as $X$, but may have some eigenvalues changed to $a$. Therefore,
$X' \leq X \leq Y$. Furthermore, $X'$ has the desired spectrum and
$p=\Prob(X',\ket{\psi})$.
We can do something similar with $Y$. We can again write $\ket{\psi}=\sum_z
\sqrt{q(z)} \ket{z;Y}$ where $z$ ranges over the support of $q$ and
$\ket{z;Y}$ is a normalized eigenvector of $Y$ with eigenvalue $z$. Note
that even if $p$ and $q$ both have support on $z$ we may have
$\ket{z;X}\neq\ket{z;Y}$. We now set $\Pi=\sum_z\ket{z;Y}\bra{z;Y}$ and
define $Y'=\Pi Y\Pi + b (I-\Pi)$. The new matrix satisfies $X\leq Y\leq
Y'$, $q=\Prob(Y',\ket{\psi})$, and has the desired spectrum.
Everything is correct except for the dimension. Now let $\Pi$ be the
projector onto the space spanned by both $\{\ket{z;X}\}$ and
$\{\ket{z;Y}\}$. The dimension of this space is no greater than
$|S(p)|+|S(q)|-1$ (the minus one occurring because $\ket{\psi}$ is in the
span of both sets of vectors). Clearly $\Pi\ket{\psi}=\ket{\psi}$ and both
$X'$ and $Y'$ are block diagonal with respect to $\Pi$ and
$I-\Pi$. Therefore, the required objects are $X'$, $Y'$ and $\ket{\psi'}$
restricted to the support of $\Pi$.
\end{proof}
\begin{eqnarray}gin{corollary}
Expressible by matrices in $[a,b]$ is a transitive relation.
\end{corollary}
\begin{eqnarray}gin{proof}
Let $p\rightarrow r$ and $r\rightarrow q$ be expressible by matrices in
$[a,b]$. We use $X_1$, $Y_1$ and $\ket{\psi_1}$ for the first transition
and $X_2$, $Y_2$ and $\ket{\psi_2}$ for the second transition, all chosen
in accordance with the conditions of Lemma~\ref{lemma:stdmattrans}. The
matrices $Y_1$ and $X_2$ have the same spectrum, except that the second one
has extra $a$ eigenvalues and the first one has extra $b$ eigenvalues. We
can append eigenvalues to both of them using a direct sum so that
\begin{eqnarray}
\mypmatrix{Y_1&0\cr0&a I}\qquad\text{and}\qquad
\mypmatrix{X_2&0\cr0&b I}
\end{eqnarray}
have the same spectrum and dimension, where the blocks may be different
sized. We can also map $\ket{\psi_1}$ and $\ket{\psi_2}$ into this
enlarged space by using a direct sum with a zero vector.
Because the two unitaries have the same spectrum, there exists a unitary
that maps the second into the first by conjugation. We can further ask
that the unitary satisfy $U\ket{\psi_2}=\ket{\psi_1}$ because up to a phases
they assign the same coefficient to each eigenvector. Then
\begin{eqnarray}
\mypmatrix{X_1&0\cr0&a I}\leq
\mypmatrix{Y_1&0\cr0&a I}=
U\mypmatrix{X_2&0\cr0&b I}U^\dagger\leq
U\mypmatrix{Y_2&0\cr0&b I}U^\dagger.
\end{eqnarray}
We can construct $p$ out of the first matrix and the extended
$\ket{\psi_1}\bra{\psi_1}$ and we can construct $q$ out of the last matrix
and $\ket{\psi_1}\bra{\psi_1}$ as well. Therefore $p\rightarrow q$ is
expressible by matrices in $[a,b]$.
\end{proof}
Before studying the perturbations, we need one final simplification. Though
the domain $[0,\Lambda]$ is good, the domain $[-1,1]$ is even better
because we can write $\frac{\lambda z}{\lambda+z}=\frac{z}{1+\gamma
z}$ and $-\lambda\in\mathbb{R}\setminus[-1,1]$ is equivalent to $\gamma\in(-1,1)$,
which is a connected set. Note that the value $\gamma=0$ corresponds to the
function $f(z)=z$, and by continuity in $\gamma$ we can always
include/exclude it among our conditions. The following lemma follows by a
simple rescaling argument.
\begin{eqnarray}gin{lemma}
If every valid transition on $[-1,1]$ is expressible by matrices in $[-1,1]$
then every valid transition on $[0,\Lambda]$ is expressible by matrices in
$[0,\Lambda]$.
\label{lemma:lamb21}
\end{lemma}
It will also be convenient to work with functions $p$ and $q$ with support
in $(-1,1)$, though we still keep a domain of $[-1,1]$.
\begin{eqnarray}gin{lemma}
If every valid transition on $[-1,1]$ involving functions with support on
$(-1,1)$ is expressible by matrices in $[-1,1]$ then every valid transition
on $[-1,1]$ is expressible by matrices in $[-1,1]$.
\end{lemma}
\begin{eqnarray}gin{proof}
Fix $p,q:[-1,1]\rightarrow[0,\infty)$ with finite support so that
$p\rightarrow q$ is valid in $[-1,1]$. For any $0<c<1$
we define functions $[-1,1]\rightarrow[0,\infty)$ by
\begin{eqnarray}
p_c(z) = \begin{eqnarray}gin{cases}
p(\frac{z}{c}) & z\in [-c,c],\cr
0 & \text{otherwise,}
\end{cases}
\qquad\qquad
q_c(z) = \begin{eqnarray}gin{cases}
q(\frac{z}{c}) & z\in [-c,c],\cr
0 & \text{otherwise,}
\end{cases}
\end{eqnarray}
which have support in $(-1,1)$ and furthermore $p_c\rightarrow q_c$ is
valid in $[-1,1]$. Therefore, the transition is expressible by matrices in
$[-1,1]$ and and we can choose $X_c$, $Y_c$ and $\ket{\psi_c}$ in
accordance with the constraints of
Lemma~\ref{lemma:stdmattrans}. Furthermore, by changes of basis we can
ensure that $X_c$ and $\ket{\psi_c}$ converge as $c\rightarrow 1$ to the
canonical representation of $p$ (appended with $-1$ eigenvalues). Any
limit point of $Y_c$ as $c\rightarrow 1$ will complete the proof.
\end{proof}
For the rest of this section we will be concerned with the interval
$[-1,1]$. When clear from context we will use the terms valid and
expressible by matrices to refer to valid on $[-1,1]$ and expressible by
matrices in $[-1,1]$.
We now begin studying perturbations on $p$ and $q$ which will have
constraints arising from rational functions in $\gamma$ of the form $\sum_z
\frac{z}{1+\gamma z}(q(z)-p(z))$. The main difficulty will be near the
zeros of the function, which we want to approximate by simple expressions
of the form $a|\gamma-\gamma_0|^{k}$. The following is a standard result.
\begin{eqnarray}gin{lemma}
Let $N(\gamma)$ and $D(\gamma)$ be two non-negative polynomials for
$\gamma\in[-1,1]$. If $N$ has a zero of order $k$ at
$\gamma=\gamma_0\in[-1,1]$ and $D(\gamma_0)\neq 0$ then for any $k'\geq k\geq
k''>0$ there exists $\epsilon,a,b>0$ such that
\begin{eqnarray}
a|\gamma-\gamma_0|^{k'} \leq \frac{N(\gamma)}{D(\gamma)} \leq
b|\gamma-\gamma_0|^{k''}\qquad\text{for $\gamma\in[-1,1]$ satisfying
$|\gamma-\gamma_0|<\epsilon$}.
\end{eqnarray}
\label{lemma:poly}
\end{lemma}
\begin{eqnarray}gin{proof}
Choose $\epsilon>0$ so that $N(\gamma)$ and $D(\gamma)$ are non-zero for
$0<|\gamma-\gamma_0|\leq \epsilon$. Let $a$ be the infimum of
$\frac{N(\gamma)}{|\gamma-\gamma_0|^{k'} D(\gamma)}$ and $b$ be the
maximum of $\frac{N(\gamma)}{|\gamma-\gamma_0|^{k''} D(\gamma)}$ for
$\gamma\in[-1,1]$ satisfying $0<|\gamma-\gamma_0|\leq \epsilon$. Note that
the cases $\gamma_0=-1$ and $\gamma_0=1$ are special in that $k$ can be
odd, and the inequalities will only hold inside the region $[-1,1]$.
\end{proof}
For the next set of lemmas let ${\mathcal{H}}$ be a real $n$-dimensional Hilbert
space, let $X$ be an operator on ${\mathcal{H}}$, let $\ket{\psi}$ and $\ket{\phi}$
be non-zero vectors in ${\mathcal{H}}$ and let $c\neq 0$ be a real constant. Also let
$f_\gamma(x) = \frac{x}{1+\gamma x}$ for $x\in[-1,1]$ and $\gamma\in(-1,1)$.
\begin{eqnarray}gin{lemma}
Let $X$, $\ket{\phi}$, $c$ and $f_\gamma(x)$ be as above. If both $X$ and
$X+c\ket{\phi}\bra{\phi}$ have eigenvalues in $[-1,1]$, then
\begin{eqnarray}
f_\gamma(X+c\ket{\phi}\bra{\phi}) - f_\gamma(X) =
\left(\frac{c}{1+\gamma c \bra{\phi}
\left(I+\gamma X\right)^{-1}\ket{\phi}}\right)
\left(I+\gamma X\right)^{-1}\ket{\phi}\bra{\phi}
\left(I+\gamma X\right)^{-1}.
\label{eq:fgbound}
\end{eqnarray}
\end{lemma}
\begin{eqnarray}gin{proof}
The proof for $\gamma=0$ is trivial. Otherwise $f_\gamma(x) = \frac{1}{\gamma}
\left(1-\frac{1}{1+\gamma x}\right)$. Let $Z=I+\gamma X$ and $c'=\gamma c$.
Because $Z$ is positive definite and
$Z+c'\ket{\phi}\bra{\phi}$ is also positive definite, then
\begin{eqnarray}
\big(Z + c'\ket{\phi}\bra{\phi}\big)^{-1}&=&
\left( Z^{1/2} \left( I + c' Z^{-1/2}\ket{\phi}\bra{\phi}Z^{-1/2}\right)
Z^{1/2}\right)^{-1}
\nonumber\\
&=& Z^{-1/2} \left( I + c' Z^{-1/2}\ket{\phi}\bra{\phi}Z^{-1/2}\right)^{-1}
Z^{-1/2}
\nonumber\\
&=& Z^{-1/2} \left( I + \left( -1 + \frac{1}{1+c'\bra{\phi}Z^{-1}\ket{\phi}}\right)
\frac{Z^{-1/2}\ket{\phi}\bra{\phi}Z^{-1/2}}{\bra{\phi}Z^{-1}\ket{\phi}}\right)
Z^{-1/2}
\nonumber\\
&=& Z^{-1/2} \left( I - \left(\frac{c'}{1+c'\bra{\phi}Z^{-1}\ket{\phi}}\right)
Z^{-1/2}\ket{\phi}\bra{\phi}Z^{-1/2}\right)
Z^{-1/2}
\nonumber\\
&=& Z^{-1} - \left(\frac{c'}{1+c'\bra{\phi}Z^{-1}\ket{\phi}}\right)
Z^{-1}\ket{\phi}\bra{\phi}Z^{-1},
\end{eqnarray}
\noindent
where the third equality follows because $I + c'
Z^{-1/2}\ket{\phi}\bra{\phi}Z^{-1/2}$ is a matrix with only two
eigenvalues. The main result follows because its LHS equals
$-\frac{1}{\gamma}((Z+c'\ket{\phi}\bra{\phi})^{-1}-Z^{-1})$.
\end{proof}
We are ready to start imposing constraints on the transitions $p\rightarrow
p'$ for $p=\Prob(X,\ket{\psi})$ and $p'=\Prob(X',\ket{\psi})$, where
$X'=X+c\ket{\phi}\bra{\phi}$. In particular, we want to place an upper
bound on $\left|\sum_z \frac{z}{1+\gamma z} (p'(z)-
p(z))\right|=\left|\bra{\psi}\Big(f_\gamma(X+c\ket{\phi}\bra{\phi}) -
f_\gamma(X)\Big)\ket{\psi}\right|$ in the form of a polynomial with a zero of
order $2k$. The next lemma will show that there are many choices for
$\ket{\phi}$ that satisfy the bound. In fact, there is an $n-k$ subspace of
such vectors.
\begin{eqnarray}gin{lemma}
Let ${\mathcal{H}}$, $n$, $X$, $\ket{\psi}$ and $f_\gamma(x)$ be as above, with the
eigenvalues of $X$ restricted to $(-1,1)$. Given a polynomial
$b(\gamma-\gamma_0)^{2k}$ with constants $b>0$, integer $k>0$ and
$\gamma_0\in[-1,1]$, then we can construct an $(n-k)$-dimensional subspace
${\mathcal{H}}P\subset{\mathcal{H}}$ such that for every $\ket{\phi}\in{\mathcal{H}}P$ there exists $c>0$
and $\epsilon>0$ satisfying
\begin{eqnarray}q
\left|\bra{\psi}\Big(f_\gamma(X+c\ket{\phi}\bra{\phi}) -
f_\gamma(X)\Big)\ket{\psi}\right|
\leq b(\gamma-\gamma_0)^{2k}
\qquad\text{for $\gamma\in(-1,1)$ satisfying $|\gamma-\gamma_0|<\epsilon$,}
\end{eqnarray}q
where additionally $|c|$ must be small enough so that
$X+c\ket{\phi}\bra{\phi}$ also has eigenvalues in $(-1,1)$.
The same conditions can also be satisfied while demanding that in every
case $c<0$.
\label{lemma:vecadd}
\end{lemma}
\begin{eqnarray}gin{proof}
Because we are only considering matrices $X$ with eigenvalues in $(-1,1)$,
the matrix $|X|$ has a maximum eigenvalue $\lambda_{max}<1$ and if for a
given $\ket{\phi}$ we restrict $|c|<(1-\lambda_{max})/(2\braket{\phi}{\phi})$
then Eq.~(\ref{eq:fgbound}) implies
\begin{eqnarray}
\left|\bra{\psi}\Big(f_\gamma(X+c\ket{\phi}\bra{\phi}) -
f_\gamma(X)\Big)\ket{\psi}\right|
\leq
2|c|\left| \bra{\psi}\left(I+\gamma X\right)^{-1} \ket{\phi}\right|^2.
\label{eq:fgbound1}
\end{eqnarray}
The expression $\bra{\psi}(I+\gamma X)^{-1} \ket{\phi}$ is a rational
function in $\gamma$ (recall we are working in a real vector space) with no
poles in $[-1,1]$. If we can choose $\ket{\phi}$ such that it has a zero of
order at least $k$ at $\gamma_0$ then by Lemma~\ref{lemma:poly} we can
choose small enough $c>0$ (or large enough $c<0$) and $\epsilon>0$ such
that
\begin{eqnarray}
2|c|\left| \bra{\psi}\left(I+\gamma X\right)^{-1} \ket{\phi}\right|^2
\leq b(\gamma-\gamma_0)^{2k}
\qquad\text{for $\gamma\in(-1,1)$ satisfying $|\gamma-\gamma_0|<\epsilon$,}
\end{eqnarray}
thereby proving the constraint. What remains to be shown is that there
exists an $n-k$ dimensional space ${\mathcal{H}}P$ such that $\ket{\phi}\in{\mathcal{H}}P$
implies that $\bra{\psi}(I+\gamma X)^{-1} \ket{\phi}$ has a zero of order
at least $k$ at $\gamma_0$.
If we choose $\delta>0$ such that $\delta|I+\gamma_0 X|^{-1}<I$ then we can
write
\begin{eqnarray}
\bra{\psi}(I+\gamma X)^{-1} &=&
\bra{\psi}(I+\gamma_0 X + (\gamma-\gamma_0) X)^{-1}
\\\nonumber
&=& \sum_{j=1}^{\infty}
(\gamma-\gamma_0)^j \bra{\psi} (I+\gamma_0 X)^{-1}
\left(\frac{X}{I+\gamma_0 X} \right)^j
\qquad\text{for $|\gamma-\gamma_0|<\delta$,}
\end{eqnarray}
\noindent
where all matrices are diagonal in the eigenbasis of $X$, which justifies
the unordered products. We have shown that the requirement that
$\bra{\psi}(I+\gamma X)^{-1} \ket{\phi}$ have a zero of order at least $k$
at $\gamma_0$ is equivalent to $k$ linear constraints on $\ket{\phi}$, and
therefore there exists a subspace of dimension $n-k$ that simultaneously
satisfies all of them. The lemma follows so long as we ensure to pick the
constants $\epsilon$ small enough so that $\epsilon\leq\delta$.
\end{proof}
The following fact will also be useful: given $c>0$ and $\epsilon>0$
satisfying the inequality in the above lemma (for a given $\ket{\phi}$),
then it is also satisfied by any $c'$ and $\epsilon'$ such that $0<c'<c$
and $0<\epsilon'<\epsilon$, the former property arising because $f_\gamma$
is operator monotone and hence $\bra{\psi}f_\gamma(X+c\ket{\phi}\bra{\phi})
\ket{\psi}$ is monotone as a function of $c$. In the next lemma we will use
this to deal with multiple simultaneous constraints.
\begin{eqnarray}gin{lemma}
Let $p$ and $q$ be functions with support in $(-1,1)$ such that
$p\rightarrow q$ is valid. Let $S(p)$ and $S(q)$ be respectively the
supports of $p$ and $q$, and let $m$ be the number of zeros (including
multiplicities) for $\gamma\in[-1,1]$ of the rational function $\sum_z
\frac{z}{1+\gamma z} (q(z)- p(z))$.
\begin{eqnarray}gin{itemize}
\item If $m+2< 2|S(p)|$ there exists a vector $\ket{\phi}\neq0$ and
constant $c>0$ such that $p'\rightarrow q$ is valid,
where $p'=\Prob(X',\ket{\psi})$, $X'= X+c\ket{\phi}\bra{\phi}$ and
$p$ is canonically represented by $\Prob(X,\ket{\psi})$.
\item If $m+2< 2|S(q)|$ there exists a vector $\ket{\phi}\neq0$ and
constant $c<0$ such that $p\rightarrow q'$ is valid,
where $q'=\Prob(Y',\ket{\xi})$, $Y'= Y+c\ket{\phi}\bra{\phi}$ and
$q$ is canonically represented by $\Prob(Y,\ket{\xi})$.
\end{itemize}
\label{lemma:ppq}
\end{lemma}
\begin{eqnarray}gin{proof}
We will prove the first case, the second case being nearly identical.
The key idea is that $q(z)-p'(z)= (q(z)-p(z))-(p'(z)-p(z))$, and therefore
$p'\rightarrow q$ will be valid if
\begin{eqnarray}
\bra{\psi}\Big(f_\gamma(X+c\ket{\phi}\bra{\phi}) -
f_\gamma(X)\Big)\ket{\psi}
\leq \sum_z \frac{z}{1+\gamma z} (q(z)- p(z))
\qquad\text{for all $\gamma\in[-1,1]$.}
\label{eq:ppq}
\end{eqnarray}
By construction the right-hand side has no poles and exactly $m$ zeros
(including multiplicities) for $\gamma\in[-1,1]$. For each zero we can
find, by Lemma~\ref{lemma:poly}, a lower bound for the right-hand side of
the form $a(\gamma-\gamma_0)^{2k}$ valid in some neighborhood of the zero.
Since all multiplicities for zeros in $(-1,1)$ are even we can chooses $2k$
to equal the multiplicity of the respective zero. Zeros occurring at the
end points $-1$ and $1$ can have odd multiplicities, in which case we can
choose $2k$ to be at most one plus the multiplicity of the zero. By
Lemma~\ref{lemma:vecadd} for each neighborhood there is an $S(p)-k$
subspace of vectors $\ket{\phi}$ that satisfies the constraint, possibly in
a smaller neighborhood, for some $c>0$. The sum of the constants $2k$ over
each of the zeros is therefore at most $m+2$ (and this can only occur if
there are zeros of odd order at both $\gamma_0=-1$ and $\gamma_0=1$).
Therefore, the number of linear constraints needed to specify all the
subspaces is at most $\lfloor\frac{m+2}{2}\rfloor<|S(p)|$. Therefore, the
intersection of all these subspaces is non-trivial and we can find a
non-zero $\ket{\phi}$ and a $c>0$ (chosen as the smallest of the given $c$
constants) such that the inequality of Eq.~(\ref{eq:ppq}) is satisfied in
the union of all the neighborhoods.
We also need to ensure that $c>0$ is chosen small enough so that
$X'=X+c\ket{\phi}\bra{\phi}$ has eigenvalues in $(-1,1)$, but because $X$
has eigenvalues in $(-1,1)$ that is always possible. What remains to be
checked is that the inequality is satisfied outside the neighborhoods
surrounding the zeros. But the complement of these neighborhoods in
$[-1,1]$ is a compact set on which the inequality becomes a strict
inequality. Therefore, for any vector $\ket{\phi}$ we can find a $c>0$ such
that the inequality holds. The lemma is proven by choosing $c>0$ small
enough to satisfy all the preceding conditions.
\end{proof}
To make use of the above lemma we note that the degree of the
numerator of the rational function of $\gamma$ given by
\begin{eqnarray}
\sum_z \frac{z}{1+\gamma z} (q(z)- p(z))
\end{eqnarray}
is at most $|S(p)|+|S(q)|-1$ because we are summing at most $|S(p)|+|S(q)|$
terms. In fact, the degree of the numerator is at most $|S(p)|+|S(q)|-2$
because the coefficient of $\gamma^{|S(p)|+|S(q)|-1}$ is $(\prod_z
z)\sum_z(q(z)-p(z))=0$.
In the following discussion \textit{the number of zeros of $p\rightarrow
q$} refers to the number of zeros (including multiplicities) for
$\gamma\in[0,1]$ of the numerator of the rational function $\sum_z
\frac{z}{1+\gamma z} (q(z)- p(z))$. The argument from the last paragraph
shows that the number of zeros of $p\rightarrow q$ is at most
$|S(p)|+|S(q)|-2$.
As a consequence, if $|S(p)|>|S(q)|$, then $2|S(p)|>|S(p)|+|S(q)|\geq m+2$
where $m$ is the number of zeros of $p\rightarrow q$. Therefore, we can
always find a $\ket{\phi}$ as in the above lemma to perturb $p$. Similarly,
if $|S(p)|<|S(q)|$ there exists a $\ket{\phi}$ that can be used as a
perturbation on $q$. Both perturbations can be found if $|S(p)|=|S(q)|$ and
the number of zeros of $p\rightarrow q$ is less than its maximal value of
$2|S(p)|-2$. The special case of $|S(p)|=|S(q)|$ with maximal zeros will
be dealt with separately later in the section
Let us now discuss what happens once we have found a valid $\ket{\phi}$ and
$c$, and start increasing the value of $c$. For concreteness, we take the
first case of the preceding lemma so that $p'\rightarrow q$ is valid,
$p'=\Prob(X',\ket{\psi})$, $p=\Prob(X,\ket{\psi})$ is a canonical
representation so that $X$ has dimension $|S(p)|$, and
$X'=X+c\ket{\phi}\bra{\phi}$.
As we increase $c$ there are two constraints that can force us to stop:
either $X'(c)\equiv X+c\ket{\phi}\bra{\phi}$ gets an eigenvalues larger
than $1$, or $p'\rightarrow q$ is no longer valid. Let $c_0$ be the largest
allowed value of $c$ and let $p'_0 =\Prob(X'(c_0),\ket{\psi})$. Note that
$p\rightarrow p_0'$ is still expressible by matrices and $p_0'\rightarrow
q$ is still valid because the respective constraints are defined using
non-strict inequalities.
If the stopping condition for increasing $c$ is that the eigenvalues get
too large, then $X'(c_0)\leq I$ but $X'(c_0)$ has $1$ as an eigenvalue.
Because $q$ has support in $(-1,1)$, the value of $\sum_z
\frac{z}{1+\gamma z} q(z)$ at $\gamma=-1$ is finite, so as $c\rightarrow
c_0$ the amplitude of $\ket{\psi}$ on the largest eigenvector of $X'(c)$
must be going to zero, and must be zero at $c=c_0$. The first consequence
of this is that $p'_0(z)$ ultimately does not have support on $z=1$, and
therefore its support is still contained in $(-1,1)$. The second
consequence is that the size of the support of $p'_0$ is smaller
than the dimension of the space on which $X$ is defined, which equals the
support of $p$. In other words $|S(p'_0)|<|S(p)|$.
On the other hand lets assume that for $c>c_0$ the transition to $q$ is no
longer valid. In such a case the rational function $\sum_z \frac{z}{1+\gamma
z} (q(z)- p'_0(z))$ must have an extra zero in $[-1,1]$ that $\sum_z
\frac{z}{1+\gamma z} (q(z)- p(z))$ did not have, or one of the existing
zeros must have a higher degree. That is, the number of zeros in
$p_0'\rightarrow q$ is greater than the number of zeros in $p\rightarrow
q$. Note that in this case, by construction $|S(p'_0)|\leq|S(p)|$, whereas
in the previous case (where we proved $|S(p'_0)|<|S(p)|$) the number of
zeros cannot decrease.
Let $p\rightarrow q$ be valid with $|S(p)|>|S(p)|$. We can repeatedly apply
the above perturbation process to obtain a sequence of valid transitions
$p\equiv p_0\rightarrow p_1\rightarrow\cdots\rightarrow p_l\rightarrow q$,
such that, as functions of $i$, the number of zeros of $p_i\rightarrow q$
is monotonically increasing along the chain and $|S(p_i)|$ is monotonically
decreasing along the chain. Furthermore, in each transition either the
number of zeros increases or $|S(p_i)|$ decreases. The number of zeros is
upper bounded by $|S(p)|+|S(q)|-2$, and therefore after a finite number of
steps we must get to a $p_\ell$ such that either $|S(p_\ell)|<|S(q)|$ or
$|S(p_\ell)|=|S(q)|$ and the number of zeros is maximal. In the former case,
we can continue the chain by perturbing $q$. Repeatedly working on both
sides we can end up with a chain
\begin{eqnarray}
p \equiv p_0\rightarrow p_1\rightarrow\cdots\rightarrow p_\ell\equiv p'
\rightarrow q'\equiv q_{\ell'}\rightarrow\cdots\rightarrow q_1
\rightarrow q_0\equiv q
\end{eqnarray}
\noindent
such that all transitions are valid, and all but $p'
\rightarrow q'$ are known to be expressible by matrices. Furthermore,
$|S(p')|=|S(q')|$ and $p'\rightarrow q'$ has $2|S(p')|-2$ zeros. If we
can prove that $p'\rightarrow q'$ is expressible by matrices then by
transitivity we will have also proven that $p\rightarrow q$ is expressible
by matrices.
To complete the result of this section, we now need to study the remaining
special case of $|S(p')|=|S(q')|$ with maximal zeros. From the proof of
Lemma~\ref{lemma:ppq} we can see that in fact the only case when we can't
find a perturbation is when there are zeros of odd order at both
$\gamma_0=-1$ and $\gamma_0=1$. In all other cases we can continue the
above chain until $p'=q'$.
To deal with zeros of odd order at $\gamma_0=-1$ and $\gamma_0=1$ we need
to enlarge the vector space on which we are working. Rather than using a
canonical representation $p=\Prob(X,\ket{\psi})$ we append to $X$ an extra
eigenvalue $-1$, so that we end up with a new matrix $\begin{array}r X$ having
dimension $|S(p)+1|$ and eigenvalues in $[-1,1)$. Similarly, we can request
a representation $q=\Prob(\begin{array}r Y,\ket{\xi})$ so that $\begin{array}r Y$ has dimension
$|S(q)|+1$ including a single eigenvalue $1$. Note that it is pointless to
append eigenvalues of $1$ to $X$ (or $-1$ to $Y$) as we can't add
(resp. subtract) in any vectors $\ket{\phi}\bra{\phi}$ to that subspace
without ending up with eigenvalues outside $[-1,1]$.
The following discussion will concern the first case, where the matrix
$\begin{array}r X$ has a single eigenvector $\ket{-1}$ with eigenvalue $-1$. It will
be useful to define ${\mathcal{H}}B$ as the space containing $\begin{array}r X$, and
${\mathcal{H}}\subset{\mathcal{H}}B$ as the subspace orthogonal to $\ket{-1}$. We want to keep
$n$ as the size of the support of $p$, so ${\mathcal{H}}$ will have dimension $n$ and
${\mathcal{H}}B$ will have dimension $n+1$. We still want $p=\Prob(\begin{array}r
X,\ket{\psi})$ to have support in $(-1,1)$, though, so
$\braket{-1}{\psi}=0$ or equivalently $\ket{\psi}\in{\mathcal{H}}$. We will consider
perturbations of the form $\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r \phi}$ where
$\ket{\begin{array}r\phi}=\ket{\phi}+a\ket{-1}$.
Our immediate goal is to extend Lemma~\ref{lemma:vecadd} to deal with
matrices $X$ with eigenvalues in $[-1,1)$ for $c>0$. For
$\gamma_0\in[-1,1)$ it will be a straightforward extension: We want to find
an $(n-k)$-dimensional subspace ${\mathcal{H}}P\subset{\mathcal{H}}$ of vectors $\ket{\phi}$ so
that for any $a\in\mathbb{R}$ the vector $\ket{\begin{array}r\phi}=\ket{\phi}+a\ket{-1}$
satisfies the inequality
\begin{eqnarray}q
\left|\bra{\psi}\Big(f_\gamma(\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r \phi}) -
f_\gamma(\begin{array}r X)\Big)\ket{\psi}\right|
\leq b(\gamma-\gamma_0)^{2k}
\qquad\text{for $\gamma\in(-1,1)$ satisfying $|\gamma-\gamma_0|<\epsilon$}
\end{eqnarray}q
for some $c>0$ and $\epsilon>0$.
As the proof of the above is nearly identical to the proof of
Lemma~\ref{lemma:vecadd}, we will only discuss the differences. Our main
tool is Eq.~(\ref{eq:fgbound}) which also applies to our extend matrix $\begin{array}r
X$. For convenience we rewrite it here
\begin{eqnarray}
f_\gamma(\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r \phi}) - f_\gamma(\begin{array}r X) =
\left(\frac{c}{1+\gamma c \bra{\begin{array}r \phi}
\left(I+\gamma \begin{array}r X\right)^{-1}\ket{\begin{array}r \phi}}\right)
\left(I+\gamma \begin{array}r X\right)^{-1}\ket{\begin{array}r \phi}\bra{\begin{array}r \phi}
\left(I+\gamma \begin{array}r X\right)^{-1}.
\label{eq:fgboundbis}
\end{eqnarray}
\noindent
Given $\ket{\begin{array}r \phi}=\ket{\phi}+a\ket{-1}$ we want to choose
$c>0$ small enough so that
\begin{eqnarray}
\left|\bra{\psi}\Big(f_\gamma(\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r \phi}) -
f_\gamma(\begin{array}r X)\Big)\ket{\psi}\right|
\leq
2|c|\left| \bra{\psi}\left(I+\gamma \begin{array}r X\right)^{-1} \ket{\begin{array}r \phi}\right|^2
\end{eqnarray}
\noindent
for $\gamma\in(-1,1)$ satisfying $|\gamma-\gamma_0|<\epsilon$. To
accomplish this let $\lambda_{max}$ be the largest eigenvalue of $\begin{array}r X$
and restrict $c<(1-\lambda_{max}')
/(2\braket{\begin{array}r\phi}{\begin{array}r\phi})$. We then have $\gamma c \bra{\begin{array}r \phi}
\left(I+\gamma \begin{array}r X\right)^{-1}\ket{\begin{array}r
\phi}>-1/2$ as required. We then note that $\bra{\psi}\left(I+\gamma \begin{array}r
X\right)^{-1} \ket{\begin{array}r \phi} = \bra{\psi}\left(I+\gamma X\right)^{-1}
\ket{\phi}$ where $X$ is the restriction of $\begin{array}r X$ to ${\mathcal{H}}$.
We end up with an equation identical to Eq.~(\ref{eq:fgbound1}) in the
proof of Lemma~\ref{lemma:vecadd}. The rest of the proof carries through.
The interesting extension of Lemma~\ref{lemma:vecadd} occurs at $\gamma_0=1$. For $\gamma\in(-1,1)$ we have
\begin{eqnarray}
|a|^2(1-\gamma)^{-1}
\leq \bra{\begin{array}r \phi}\left(I+\gamma \begin{array}r X\right)^{-1} \ket{\begin{array}r \phi}
\end{eqnarray}
\noindent
because $1+\gamma \begin{array}r X$ is positive definite and the left-hand side drops
some positive terms. If $a\neq 0$ we can further write for $\gamma\in(0,1)$
\begin{eqnarray}
\left(\frac{c}{1+\gamma c \bra{\begin{array}r\phi}
\left(I+\gamma \begin{array}r X\right)^{-1}\ket{\begin{array}r \phi}}\right)\leq
\frac{c}{1+\gamma c |a|^2(1-\gamma)^{-1}}
= \frac{c(1-\gamma)}{1-\gamma(1- c |a|^2)}
\leq
\frac{1-\gamma}{|a|^2},
\end{eqnarray}
\noindent
where in the last step we assumed $c<1/|a|^2$ so that the
denominator is minimized by $\gamma\rightarrow 1$. Combined with
Eq.~(\ref{eq:fgboundbis}) we get
\begin{eqnarray}
\left|\bra{\psi}\Big(f_\gamma(\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r \phi}) -
f_\gamma(\begin{array}r X)\Big)\ket{\psi}\right|
\leq
\frac{1-\gamma}{|a|^2}
\left| \bra{\psi}\left(I+\gamma \begin{array}r X\right)^{-1} \ket{\begin{array}r \phi}\right|^2
\label{eq:gamma01}
\end{eqnarray}
\noindent
for $\gamma\in(1-\epsilon,1)$ so long as we choose $\epsilon<1$. Therefore,
at $\gamma_0=1$ we can prove something stronger than
Lemma~\ref{lemma:vecadd}. The constructed $(n-k)$-dimensional subspace will
satisfy an upper bound of $b(\gamma-\gamma_0)^{2k+1}$ rather than just
$b(\gamma-\gamma_0)^{2k}$. The caveat is that the bound will only hold when
$|a|$ is large enough:
\begin{eqnarray}gin{lemma}
Let ${\mathcal{H}}B$, ${\mathcal{H}}$, $n$, $\begin{array}r X$, $\ket{\psi}$ and $f_\gamma(x)$ be as
above. In particular, $\begin{array}r X$ has eigenvalues in $[-1,1)$ with a unique
eigenvector $\ket{-1}$ with eigenvalue $-1$ and $\braket{-1}{\psi}=0$.
Given a polynomial $b(1-\gamma)^{2k+1}$ with constants $b>0$ and integer
$k\geq 0$, we can find an $(n-k)$-dimensional subspace ${\mathcal{H}}P\subset{\mathcal{H}}$ and
a number $\Theta>0$ such that for any $\ket{\phi}\in{\mathcal{H}}P$ and $a\geq \Theta
\braket{\phi}{\phi}$ there exists
$c>0$ and $\epsilon>0$ satisfying $\begin{array}r X+c\ket{\begin{array}r\phi}\bra{\begin{array}r\phi}<I$
and
\begin{eqnarray}
\left|\bra{\psi}\Big(f_\gamma(\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r \phi}) -
f_\gamma(\begin{array}r X)\Big)\ket{\psi}\right|
\leq b(1-\gamma)^{2k+1}
\qquad\qquad\text{for $\gamma\in(1-\epsilon,1)$,}
\end{eqnarray}
\noindent
where $\ket{\begin{array}r\phi}= \ket{\phi}+a\ket{\-1}$.
\label{lemma:oddpolybound}
\end{lemma}
\begin{eqnarray}gin{proof}
The proof starts from Eq.~(\ref{eq:gamma01}). As before,
$\braket{-1}{\psi}=0$ implies $\bra{\psi}\left(I+\gamma \begin{array}r
X\right)^{-1}\ket{\begin{array}r \phi}=\bra{\psi}\left(I+\gamma X\right)^{-1} \ket{\phi}$
where $X$ is the restriction of $\begin{array}r X$ to ${\mathcal{H}}$. The argument from the
proof of Lemma~\ref{lemma:vecadd} gives us an $(n-k)$-dimensional subspace
${\mathcal{H}}P\subset{\mathcal{H}}$ of vectors $\ket{\phi}\in{\mathcal{H}}P$ such that the numerator of
the rational function $\bra{\psi}\left(I+\gamma X\right)^{-1} \ket{\phi}$ has a
zero of order $k$ at $\gamma=1$. Given $\ket{\phi}\in{\mathcal{H}}P$ we can find, by
Lemma~\ref{lemma:poly}, a small enough $\epsilon>0$ and large enough $a>0$
so that
\begin{eqnarray}
\frac{1-\gamma}{|a|^2}
\left| \bra{\psi}\left(I+\gamma X\right)^{-1} \ket{\phi}\right|^2
\leq b(\gamma-\gamma_0)^{2k+1}
\qquad\qquad\text{for $\gamma\in(1-\epsilon,1)$.}
\end{eqnarray}
\noindent
To complete the proof choose $\Theta$ to be the maximum of such choices of
$|a|^2$ over the compact set of $\ket{\phi}\in{\mathcal{H}}P$ that additionally satisfy
$\braket{\phi}{\phi}=1$.
\end{proof}
A similar result holds for $\begin{array}r Y$ with eigenvalues in $(-1,1]$ and
$c<0$. We now prove a special version of Lemma~\ref{lemma:ppq}.
\begin{eqnarray}gin{lemma}
Let $p\neq q$ be functions with support in $(-1,1)$ such that
$p\rightarrow q$ is valid, $|S(p)|=|S(q)|$ and the number of zeros of
$p\rightarrow q$ is maximal with odd order at both $\gamma=-1$ and
$\gamma=1$. Then
\begin{eqnarray}gin{itemize}
\item There exists a vector $\ket{\begin{array}r \phi}=\ket{\phi}+a\ket{-1}$ and
constant $c>0$ such that $p'\rightarrow q$ is valid,\\ where
$p'=\Prob(\begin{array}r X',\ket{\psi})$, $\begin{array}r X'= \begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r
\phi}\leq I$, $p=\Prob(\begin{array}r X,\ket{\psi})$, the dimension of $\begin{array}r X$ is
$|S(p)|+1$ including a unique eigenvector $\ket{-1}$ with eigenvalue $-1$,
$\braket{\phi}{-1}=0$ and $\ket{\phi}\neq0$.
\item There exists a vector $\ket{\begin{array}r \phi}=\ket{\phi}+a\ket{1}$ and
constant $c<0$ such that $p\rightarrow q'$ is valid,\\ where
$q'=\Prob(\begin{array}r Y',\ket{\xi})$, $\begin{array}r Y'= \begin{array}r Y+c\ket{\begin{array}r \phi}\bra{\begin{array}r
\phi}\geq -I$, $q=\Prob(\begin{array}r Y,\ket{\xi})$, the dimension of $\begin{array}r Y$ is
$|S(q)|+1$ including a unique eigenvector $\ket{1}$ with eigenvalue $1$,
$\braket{\phi}{1}=0$ and $\ket{\phi}\neq0$.
\end{itemize}
\end{lemma}
\begin{eqnarray}gin{proof}
We prove the first case, the second being nearly identical. As in the proof
of Lemma~\ref{lemma:ppq} the main goal is to ensure
\begin{eqnarray}
\bra{\psi}\Big(f_\gamma(\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r \phi}) -
f_\gamma(\begin{array}r X)\Big)\ket{\psi}
\leq \sum_z \frac{z}{1+\gamma z} (q(z)- p(z)).
\qquad\text{for all $\gamma\in[-1,1].$}
\end{eqnarray}
By assumption, the number of zeros of the right-hand side is $2|S(p)|-2$,
with odd order at both $\gamma=-1$ and $\gamma=1$. The problem with the
original proof of Lemma~\ref{lemma:ppq} is that it would seek vectors
$\ket{\begin{array}r \phi}$ such that the left-hand side had an even number of zeros
at $\gamma=-1$ and $\gamma=1$ and the total number of zeros was $2|S(p)|$.
Therefore, the total number of linear constraints on $\ket{\begin{array}r\phi}$ would
be $|S(p)|$ and the only vector $\ket{\begin{array}r\phi}$ that satisfies all the
constraints is $\ket{-1}$.
However, Lemma~\ref{lemma:oddpolybound} allows us to satisfy the bound
while placing only an odd number of zeros at $\gamma=1$ (though still an
even number of zeros at $\gamma=-1$), and therefore requiring only
$|S(p)|-1$ constraints (so long as the coefficient of $\ket{-1}$ is large
enough). In particular, there exists a non-zero vector $\ket{\phi}$, a
large enough $a>0$ and small enough $c>0$ so that $\ket{\begin{array}r\phi}=
\ket{\phi}+a\ket{\-1}$ satisfies the above inequality in a neighborhood of
each of the zeros of the right-hand side.
The proof is then completed by picking a potentially smaller $c>0$ to
ensure that the inequality is satisfied outside the neighborhoods and
that $\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r\phi}\leq I$.
\end{proof}
\begin{eqnarray}gin{lemma}
Let $p\neq q$ be functions with support in $(-1,1)$ such that $p\rightarrow
q$ is valid, $|S(p)|=|S(q)|$ and the number of zeros of $p\rightarrow q$ is
maximal with odd order at both $\gamma=-1$ and $\gamma=1$. Then
\begin{eqnarray}gin{itemize}
\item There exists $p':(-1,1)\rightarrow[0,\infty)$ such that
$p'\neq p$, $|S(p')|=|S(p)|$,
$p\rightarrow p'$ is expressible by matrices and
$p'\rightarrow q$ is valid.
\item There exists $q':(-1,1)\rightarrow[0,\infty)$ such that
$q'\neq q$, $|S(q')|=|S(q)|$,
$q'\rightarrow q$ is expressible by matrices and
$p\rightarrow q'$ is valid.
\end{itemize}
\end{lemma}
\begin{eqnarray}gin{proof}
We shall prove the first case. Take $p'$ from the previous lemma and
increase $c$ until either $p'\rightarrow q$ gets a new zero or until
$\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r\phi}$ gets an eigenvalue of $1$.
In the first case, either we either end up with $p'=q$, or we have
$|S(p')|=|S(q)|+1$ and $p'\rightarrow q$ has maximal zeros, in which case
we can use Lemma~\ref{lemma:ppq} to create a second perturbation to get
$p''\rightarrow q$. Increasing this second $c>0$ cannot increase the number
of zeros or decrease the support size of the support of $p''$ unless we end
up with $p''=q$. In either case we have proven that $p\rightarrow q$ is
expressible by matrices and we can choose $p'=q$ to satisfy the lemma.
Alternatively if $\begin{array}r X+c\ket{\begin{array}r \phi}\bra{\begin{array}r\phi}$ gets an eigenvalue
of $1$, then we end up with $|S(p')|=|S(p)|$ and by construction
$p\rightarrow p'$ is expressible by matrices and $p'\rightarrow q$ is
valid. To prove that $p'\neq p$ we note that because $\ket{\begin{array}r\phi}$ is
not proportional to $\ket{-1}$ the maximum $c$ must be less than $2$, so
the trace of the canonical matrix expressing $p'$ is smaller than the
trace of the canonical matrix expressing $p$.
\end{proof}
\begin{eqnarray}gin{lemma}
Let $p$ and $q$ be functions with support in $(-1,1)$ such that
$p\rightarrow q$ is valid, $|S(p)|=|S(q)|$ and the number of zeros of
$p\rightarrow q$ is maximal with odd order at both $\gamma=-1$ and
$\gamma=1$. Then $p\rightarrow q$ is expressible by matrices.
\end{lemma}
\begin{eqnarray}gin{proof}
Let $p=\Prob(X,\ket{\psi})$ with $X$ a $2|S(p)|-1$ dimensional matrix which
includes $|S(p)|-1$ orthogonal eigenvectors with eigenvalue $-1$. We seek
the infimum of $\int_{-1}^1 \sum_z \frac{z}{1+\gamma z} (q(z)-p'(z))
d\gamma$ over $p'=\Prob(X',\ket{\psi})$ satisfying $p'\rightarrow q$ valid
and $X\leq X'\leq I$. Because we are optimizing $X'$ over a compact set the
infimum is achievable, and the resulting $p'$ will satisfy $p\rightarrow
p'$ is expressible by matrices and $p'\rightarrow q$ is valid.
If $p'=q$ the lemma is proven. Otherwise, by the preceding theorem there
exists $p''\neq p$ such that $p'\rightarrow p''$ is expressible by matrices
and $p''\rightarrow q$ is valid. Then $\int_{-1}^1 \sum_z \frac{z}{1+\gamma
z} (p''(z)-p'(z)) d\gamma>0$ and hence $\int_{-1}^1 \sum_z
\frac{z}{1+\gamma z} (q(z)-p'(z)) d\gamma>\int_{-1}^1 \sum_z
\frac{z}{1+\gamma z} (q(z)-p''(z)) d\gamma$. But also, by transitivity,
$p\rightarrow p''$ is expressible by matrices and in fact, because the
dimension of $X$ is large enough, we can write
$p''=\Prob(X'',\ket{\psi})$ for $X\leq X''\leq I$. That is a contradiction
with the optimality of $p'$.
\end{proof}
\begin{eqnarray}gin{corollary}
If $p\rightarrow q$ is valid in $[-1,1]$ then it is expressible by matrices
in $[-1,1]$.
\end{corollary}
The results used in Section~\ref{sec:compiling} can be proven as follows:
Lemma~\ref{lemma:f2mmain} follows from the above corollary,
Lemmas~\ref{lemma:inf2lamb} and~\ref{lemma:lamb21}, and the definition of
expressible by matrices. Lemma~\ref{lemma:stdformmain} follows from
Lemma~\ref{lemma:stdmattrans}.
\end{document} |
\begin{document}
\author{Christian Arenz}
\affiliation{Theoretische Physik, Universit\"at des Saarlandes, D 66123 Saarbr\"ucken, Germany}
\author{Cecilia Cormick}
\affiliation{Theoretische Physik, Universit\"at des Saarlandes, D 66123 Saarbr\"ucken, Germany}
\affiliation{Institute for Theoretical Physics, Universit\"at Ulm, D 89081 Ulm, Germany}
\author{David Vitali}
\affiliation{School of Science and Technology, Physics Division, University of Camerino, Camerino (MC), Italy}
\author{Giovanna Morigi}
\affiliation{Theoretische Physik, Universit\"at des Saarlandes, D 66123 Saarbr\"ucken, Germany}
\title{Generation of two-mode entangled states by quantum reservoir engineering}
\date{\today}
\begin{abstract}
A method for generating entangled cat states of two modes of a microwave cavity field is proposed.
Entanglement results from the interaction of the field with a beam of atoms crossing the microwave resonator, giving
rise to non-unitary dynamics of which the target entangled state is a fixed point. We analyse the robustness of the
generated two-mode photonic ``cat state'' against dephasing and losses by means of numerical simulation. This proposal
is an instance of quantum reservoir engineering of photonic systems.
\end{abstract}
\pacs{42.50.Dv, 03.67.Bg, 03.65.Ud, 42.50.Pq}
\maketitle
\section{Introduction}
Quantum reservoir engineering generally labels a strategy at the basis of protocols which make use of the non-unitary
evolution of a system in order to generate robust quantum coherent states and dynamics \cite{Diehl_etal_NPhys_2008}. The
idea is in some respect challenging the naive expectation, that in order to obtain quantum coherent dynamics one shall
warrant that the evolution is unitary at all stages. Due to the stochastic nature of the processes which generate the
target dynamics, strategies based on quantum reservoir engineering are in general more robust against variations of the
parameters than protocols solely based on unitary evolution \cite{Diehl_etal_NPhys_2008,
Verstraete_etal_NPhys_2009,Kraus_etal_PRA_2008}. A prominent example of quantum reservoir engineering is laser cooling,
achieving preparation of atoms and molecules at ultralow temperatures by means of an optical excitation followed by
radiative decay \cite{Ya.B.Zeldovich_Wineland}. The concept of quantum reservoir engineering and
its application for quantum information processing has been formulated in Refs. \cite{Cirac_PRL_1993,
Poyatos_etal_PRL_1996}, and further pursued in Refs. \cite{Plenio-PRA-1999, Carvalho_PRL_2001, Plenio-Huelga-PRL-2002}.
Proposals for quantum reservoir engineering of quantum states in cavity quantum electrodynamics
\cite{Pielawa_etal_PRL_2007,Susanne_et.al,Sarlette-PRL-2011,Kastoryano-PRL-2011,Jaksch-2012,
Everitt-2012} and many-body systems \cite{Diehl_etal_NPhys_2008,Verstraete_etal_NPhys_2009, Fogarty,Zippilli:2013}
have been recently discussed in the literature and first experimental realizations have been reported
\cite{Krauter_PRL2011,Rempe_Sience2008,Barreiro}. Applications for quantum technologies are being pursued
\cite{Vollbrecht_PRL2011,Barreiro,Goldstein,Huelga}.
\begin{figure}
\caption{\label{fig:system}
\label{fig:system}
\end{figure}
In this article we propose a protocol based on quantum reservoir engineering for preparing a cavity in a highly
nonclassical entangled ``cat-like'' state. This protocol is applicable to the experimental setup realized in
\cite{Raimond_RMP2001, Walther_RPP2006}, which is pumped by a
beam of atoms with random arrival times. In this setup the system dynamics intrinsically stochastic due to the
impossibility of controlling the arrival times of the atoms, but only their rate of injection, and the finite detection efficiency. The protocol we discuss allows one to generate and stabilize an entangled state of two modes of a microwave resonator,
by means of an effective environment constituted by the atoms. We show that when the internal state of the atoms
entering the cavity is suitably prepared and external classical fields couple the atomic transitions, then the
asymptotic state of the cavity modes takes the form
\begin{equation}
|\psi_\infty\rangle=(|\alpha\rangle_{\rm A}|\alpha\rangle_{\rm B} +|-\alpha\rangle_{\rm A}|-\alpha\rangle_{\rm
B})/\mathcal
N\,,
\label{eq:target state}
\end{equation}
where $|\alpha\rangle_j$ denotes a coherent state of mode $j=A,B$ with complex amplitude $\alpha$ and $\mathcal
N=\sqrt{2[1+\exp(-4|\alpha|^{2})]}$ is the normalization constant.
Our proposal extends previous works of some of us, which are focussed on generating two-mode squeezing in a microwave
cavity \cite{Pielawa_etal_PRL_2007} and entangling two distant cavities using a beam of atoms \cite{Susanne_et.al}.
The state of Eq.~(\ref{eq:target state}) whose robust generation is proposed here is not simply entangled but
possesses strongly nonclassical features, being a nonlocal macroscopic superposition state similar to those discussed
in Ref.~\cite{Davidovich1993}. The setup we consider is sketched in Fig. \ref{fig:system}, and is
similar to the one realized in Ref. \cite{Walther_RPP2006,Deleglise_Nat_2008}.
This work is structured as follows. In Sec. \ref{sec:proposal} we sketch the general features of our proposal.
Section \ref{sec:Lindblad} presents a method to engineer each of the target dynamics starting from the Hamiltonian of an
atom of the beam, which interacts with the cavity for a finite time. Results from numerical simulations are reported and
discussed in Sec. \ref{sec:results}. The conclusions are drawn in Sec. \ref{Sec:V}.
\section{Target master equation and asymptotic state} \label{sec:proposal}
Let $\rho$ be the density matrix for the degrees of freedom of the two cavity modes and
$\rho_\infty=|\psi_\infty\rangle\langle \psi_\infty|$ the target state we want to generate with $|\psi_\infty\rangle$ in
Eq. \eqref{eq:target state}. The purpose of this section is to derive the master equation
\begin{equation}
\frac{\partial}{\partial t}\rho=\mathcal L\rho\,,
\end{equation}
for which $\rho_\infty$ is a fixed point, namely,
\begin{equation}
\label{eq:L}
\mathcal L\rho_\infty=0\,.\end{equation}
In order to determine the form of the Lindbladian ${\mathcal L}$ we first introduce the operators $a$ and $b$ which
annihilate a photon of the cavity mode A and B, respectively. It is simple to show that $\rho_\infty$ is a
simultaneous right eigenoperator at eigenvalue zero of the Liouvillians
\begin{equation} \label{eq:Liouville operators}
\mathcal L_j\rho=\gamma_{j}(2C_{j}\rho C_{j}^{\dagger}-\{C_{j}^{\dagger}C_{j},\rho\}), \quad j=1,2
\end{equation}
with $\gamma_j$ rates which are model-dependent and where the operators $C_{j}$ read
\begin{equation} \label{eq: Lindblad operator}
C_{1}=\frac{a-b}{\sqrt{2}}, \quad
C_{2}=2(ab-\alpha^{2}).
\end{equation}
In fact, $|\psi_j\rangle$ is eigenstate of $C_1$ and $C_2$ with eigenvalue 0, $C_j|\psi_{\infty}\rangle=0$.
The procedure we will follow aims at constructing effective dynamics described by the Liouvillian
\begin{equation}
\label{L12}
\mathcal L=\mathcal L_1+\mathcal L_2
\end{equation}
by making use of the interaction with a beam of atoms.
Before we start, we shall remark on two important points. In first place, the state $\rho_{\infty}$ is not the unique
solution of Eq. \eqref{eq:L} when $\mathcal L=\mathcal L_1+\mathcal L_2$. Indeed, states $|\alpha\rangle_{\rm
A}|\alpha\rangle_{\rm B}$ and $|-\alpha\rangle_{\rm A}|-\alpha\rangle_{\rm B}$, and any superposition of these two
states, are also eigenstates of both $C_1$ and $C_2$ at eigenvalue zero. We
denote the corresponding eigenspace by $\mathcal H_d$, which is a subspace of the Hilbert space of all states of the two cavity modes. The most general stationary state of $\mathcal L$ can be written as a statistical mixture, $\rho^{ss}= \sum_d p_d \ketbra{\psi_d}{\psi_d}$ \cite{Footnote}, where the sum spans over all the states $\ket{\psi_d} \in \mathcal H_d$, and $p_d$ are real and positive scalars such that $\sum_d p_d=1$.
Nevertheless, for the evolution determined by the Lindbladian of Eq. \eqref{L12} the state $\rho_\infty$ is the unique
asymptotic state provided that the initial state is the vacuum state for both cavity modes,
$\rho_0=\ketbra{0_A,0_B}{0_A,0_B}$. This can be shown using the parity
operator defined as
\begin{equation}
\Pi_+=(-1)^{c^{\dagger}_+c_+}
\end{equation}
with $c_\pm = (a\pm b)/\sqrt{2}$. Operator $\Pi_+$ commutes with the operators $C_1$ and $C_2$, since
\begin{equation}
\label{c:pm}
C_1 = c_-\,, \quad
C_2 = c_+^2-c_-^2-2\alpha^2\,.
\end{equation}
Therefore, if the initial state can be written as statistical mixture of eigenstates of $\Pi_+$ with eigenvalue $+1$,
the time-evolved state will also be a statistical mixture of eigenstates with eigenvalue $+1$, and so will be the steady
state. In particular, $|\psi_\infty\rangle$ is the only state of subspace $\mathcal H_d$ which is eigenstate of $\Pi_+$
with eigenvalue $+1$, namely, $\Pi_+|\psi_\infty\rangle=|\psi_\infty\rangle$, and thus, under this condition, the
asymptotic state will be pure and given by $\rho_{\infty}$. Here we will assume just this situation, i.e., that
the cavity modes are initially prepared in the vacuum state, which is an even eigenvalue of operator $\Pi_+$, and which
represents a very natural initial condition.
These considerations are so far applied to the ideal case in which the dynamics of the cavity modes density matrix are
solely determined by Liouvillian ${\mathcal L}$ in Eq. \eqref{L12}. In this article we will construct the dynamics in
Eq. \eqref{L12} using a beam of atoms crossing with the resonator, as it is usual in microwave cavity quantum
electrodynamics. We will then analyze the efficiency of generating state $\rho_{\infty}$ at the asymptotics of
the interaction of the cavity with the beam of atoms, taking also into account experimental limitations.
\section{Engineering dissipative processes} \label{sec:Lindblad}
Our starting point is the Hamiltonian for the coherent dynamics of an atom whose selected Rydberg transitions
quasi-resonantly couple with the cavity modes. The atoms form a beam with statistical Poissonian distribution in the
arrival times. The mean velocity determines the average interaction time $\tau$ during which each
atom interacts with the cavity field, while the arrival rate $r$ is such to warrant that $r\tau\ll 1$, namely, the
probability that two atoms interact simultaneously with the cavity is strongly suppressed. The master equation for the
density matrix $\chi$ describing the dynamics of the cavity modes coupled with one atom reads
\begin{equation}
\label{master:0}
\frac{\partial}{\partial t}\chi=\frac{1}{{\rm i}\hbar}[H,\chi]+\kappa{\mathcal K}\chi\,,
\end{equation}
with $H$ the Hamiltonian governing the coherent dynamics and
\begin{eqnarray}
\mathcal K \chi=2 a \chi a^\dagger + 2 b \chi b^\dagger - \{a^\dagger a, \chi\}- \{b^\dagger b, \chi\}
\end{eqnarray}
the superoperator describing decay of the cavity modes at rate $\kappa$. The field density matrix is found after tracing
out the atomic degrees of freedom, and formally reads $\rho(t)={\rm Tr}_{\rm at}\{\chi(t)\}$.
In the following we will specify the form of Hamiltonian $H$ and derive an effective master equation for the density
matrix $\rho$ of the cavity field interacting with a beam of atoms, which approximates the dynamics governed the
Liouvillian $\mathcal L$ in Eq. \eqref{L12}.
In the following we shall analyze separately each of the processes corresponding to the two types of Lindblad
superoperators composing the sum in Eq. \eqref{L12}. Note that cavity losses are detrimental, as they do not
preserve the parity $\Pi_+$ of the state of the cavity. In the rest of this section they will be
neglected, their effect will be considered when calculating numerically the efficiency of the protocol.
\subsection{Realization of the Lindblad superoperator ${\mathcal L}_1$.}
We now show how to implement the dynamics described by the Lindblad superoperator $\mathcal L_1$. For this purpose, we
assume that the atomic transitions effectively coupling with the cavity modes form a $\Lambda$-type configuration of
levels, as schematically represented in Fig. \ref{fig:L1}. The interaction of a single atom with the cavity modes is
governed by the Hamiltonian
\begin{eqnarray}
H& =& \hbar\omega_a a^\dagger a + \hbar\omega_b b^\dagger b + \hbar\omega_2 \sigma_{2,2} + \hbar \omega_3
\sigma_{3,3} \\
& &+ \hbar( g_a a^{\dagger}\sigma_{1,3} + g_b b^{\dagger}\sigma_{2,3} +{\rm H.c.})\,,\nonumber
\end{eqnarray}
where $\omega_a$ and $\omega_b$ are the frequencies of the cavity modes, $\omega_2$ ($\omega_3$) is the energy of
level $\ket{2}$ ($\ket{3}$), here setting the energy of level $\ket{1}$ to zero, $g_a$ and $g_b$ are the vacuum Rabi
frequencies characterizing the strength of the coupling of the dipolar transitions $\ket{1}\to\ket{3}$ and
$\ket{2}\to\ket{3}$, respectively, with the corresponding cavity mode, and $\sigma_{j,k}=\ketbra{j}{k}$ is the spin-flip
operator. In the following we assume that the transitions are resonant, i.e. $\omega_a= \omega_3$ and $\omega_b =
\omega_3-\omega_2$.
\begin{figure}
\caption{Relevant atomic levels and couplings leading to the dynamics which realizes the Lindblad superoperator
$\mathcal L_1$. The atom is prepared in state $\ket{-}
\label{fig:L1}
\end{figure}
In the reference frame rotating with the cavity modes, the Hamiltonian can be rewritten as
\begin{eqnarray}
H_1
= \hbar \sqrt{2} \frac{g_ag_b}{g}(c_-^{\dagger}\sigma_{-,3} +c_+^{\prime\dagger} \sigma_{+,3}+{\rm H.c.})\,,
\label{H_1}
\end{eqnarray}
where $g=\sqrt{g_a^2+g_b^2}$, $c_-$ is defined in Eq. \eqref{c:pm} and $\sigma_{\pm,3}=\ketbra{\pm}{3}$, with
\begin{align}
|-\rangle=\frac{g_b|1\rangle-g_a |2\rangle}{g}\,,\quad |+\rangle=\frac{g_a|1\rangle+g_b |2\rangle}{g}\,,
\label{eq:basis}
\end{align}
while $c_+'$ is a superposition of modes $a$ and $b$. This representation clearly shows that, if the atoms are injected
in the state $\ket{-}$ and interact with the resonator for a time $\tau_1$ such that $g\tau_1\ll 1$, they may only
absorb photons of the ``odd'' mode $c_-$. More precisely, the condition to be fulfilled is $g\tau_1\sqrt{N_-+1/2}\ll1$,
where $N_-$ is the mean number of photons in the odd mode, $N_-=\langle c_-^\dagger c_-\rangle$. In this case, if
$\rho(t)$ is the state of the field at the instant in which an atom in state $\ket{-}$ is injected, the state of the
field $\rho$ at time $t+\tau_1$ reads \begin{equation}
\rho(t+\tau_1)=\rho(t) + \frac{g_a^2g_b^2}{g^2}\tau_1^{2}
\left[2c_-\rho(t)c_-^{\dagger}-\{c_-^{\dagger}c_-,\rho(t)\}\right]\,.
\label{eq:master1}
\end{equation}
This corresponds to the desired process, which drives the odd mode into the vacuum state. Here, we neglect corrections
that are smaller by a factor of order $g^2\tau_1^2 (N_- +1/2)$.
Assuming that the atoms in state $\ket{-}$ are injected at rate $r_1$ with $r_1\tau_1\ll 1$, the probability of having
two atoms simultaneously inside the cavity can be neglected. In this case the field evolution can be analysed on a
coarsed-grained time scale $\Delta t$ such that $\Delta t \gg \tau_1$ and $r_1\Delta t\ll 1$. After expressing the
differential quotient $[\rho(t+\Delta t)-\rho(t)]/\Delta t$ as a derivative with respect to time one recovers the master
equation \cite{Susanne_et.al}
\begin{align}
\label{Master:1}
\frac{\partial}{\partial t}\rho(t)\simeq\gamma_1 \left[ 2c_-\rho(t)c_-^{\dagger}-\{c_-^{\dagger}c_-,\rho_(t)\} \right],
\end{align}
which corresponds to the dynamics governed by superoperator $\mathcal L_1$ in Eq. \eqref{eq:Liouville operators}. Here,
\begin{equation}
\gamma_1=r_1\frac{g_a^2g_b^2}{g^2}\tau_1^{2}\,.
\end{equation}
We note that Eq. \eqref{Master:1} is valid as long as higher order corrections are negligible. This condition provides
an upper bound to the rate $\gamma_1$, i.e., $\gamma_1\ll r_1$. However, it is not strictly necessary that the dynamics
take place in this specific limit: One can indeed speed up the process of photon absorption from the odd mode taking
longer interaction times between the atom and the cavity. In this case, the form of the master equation is different,
but one could obtain absorption of photons from the odd mode. We refer the reader to Ref. \cite{Susanne_et.al}, where
the required time has been characterized for a similar proposal in the different regimes.
\subsection{Realization of the Lindblad superoperator ${\mathcal L}_2$.} \label{subsec:Lindblad2}
The dynamics described by the Lindblad operator $\mathcal L_2$, Eq. \eqref{L12}, can be realized using a level scheme as
shown in Fig. \ref{fig:L2}. We denote by $\omega_j'$ the frequency of the atomic state $\ket{j=2,3}$, such that
$\omega_3'>\omega_2'>\omega_1'=0$. The transition is such that $\omega_3'=\omega_a+\omega_b$.
\begin{figure}
\caption{Relevant atomic levels and couplings leading to the dynamics which approximates the Lindblad superoperator
$\mathcal L_2$. A classical field of amplitude $\Omega$ drives resonantly the transition $\ket{1'}
\label{fig:L2}
\end{figure}
A laser drives resonantly the transition $\ket{1'}\to\ket{3'}$, so that the frequency
$\omega_L=\omega_3'=\omega_a+\omega_b$. In the frame rotating at the
frequency of the cavity modes the Hamiltonian governing the coherent dynamics reads
\begin{multline}
\label{H_2}
H_2= \hbar \Delta \sigma_{2'2'} +\hbar ( g_a' a^\dagger \sigma_{1'2'} + g'_b b^\dagger \sigma_{2'3'}
+ \Omega \sigma_{1'3'} + {\rm H.c.}) \,,
\end{multline}
where $\Delta=\omega_2'-\omega_a$. We assume that $g_a'\sqrt{\langle n_a\rangle}, g_b'\sqrt{\langle n_a\rangle}\ll
|\Delta|$, with $\langle n_j\rangle$ the mean number of photons in the cavity mode $j=A,B$, and analyze the state of the
cavity field after it has interacted with an atom which is injected in state $\ket{1'}$. The interaction time is denoted
by $\tau$ and is chosen such that $|\Delta|\tau\gg1$ and $g_j'^2\langle n_j\rangle\tau/|\Delta|\ll 1$. The density
matrix for the cavity field at time $t+\tau$ can be cast in the form \cite{Susanne_et.al}
\begin{eqnarray}
\label{Master:L2}
\rho(t+\tau)=\rho(t)&+&\frac{1}{8} \left(\frac{g'_ag'_b\tau}{\Delta}\right)^2\Bigg[ 2C_2\rho
C_2^{\dagger}-\left\{C_2^{\dagger}C_2, \rho_f \right\}\Bigg]\nonumber\\
&&+i\frac{g_{a}^{\prime 2}}{\Delta^2}\left(\Delta\tau - \sin\Delta \tau\right)[a^{\dagger}a,\rho]\nonumber\\
&&+2\frac{{g'_a}^2}{\Delta^2}\sin^2\left(\frac{\Delta \tau}{2}\right)\left(2a\rho a^\dagger -\{a^\dagger a,
\rho\}\right)\nonumber\\
&&-\frac{1}{2}\left(\frac{g_{a}^{\prime 2}\tau}{\Delta}\right)^2[a^{\dagger}a,[a^{\dagger}a,\rho]] ,
\end{eqnarray}
where $\rho(t)$ is the density matrix before the interaction and $C_2=2(ab-\alpha^2)$, Eq. \eqref{eq: Lindblad
operator}. Here, $\alpha^2 = \Omega \Delta/ (g_a' g_b')$, showing that the number of photons at the asymptotics is
determined by $\Omega$. Equation \eqref{Master:L2} has been derived in perturbation theory and by tracing out the
degrees of freedom of the atom after the interaction. The first line of Eq. \eqref{Master:L2} describes two-photon
processes leading to the target dynamics at a rate determined by the frequency
$$\gamma_2^{(0)}=\frac{1}{8} \left(\frac{g_a^{\prime}g_b^{\prime}\tau}{\Delta}\right)^2\,,$$
while the terms in the other lines are unwanted processes, which occur at comparable rates and therefore lead to
significant deviations from the ideal behaviour. The second line of Eq. \eqref{Master:L2}, in particular, corresponds to
one-photon processes on the transition $\ket{1'}\to\ket{2'}$, leading to phase fluctuations of the cavity mode A. The
third line describes losses of mode A due to one-photon processes, and the last line gives dephasing effects of
cavity
mode A associated with two-photon processes. Other detrimental processes, leading to dephasing and amplification of the
field of cavity mode B, have been discarded under the assumption that the corresponding amplitude is of higher
order. This assumption is correct as long as the amplitude $\Omega$, determining the number of photons, is chosen to be
of the order of $g_j'^2/\Delta$ and fulfills the inequalities $(|\Delta|\tau)(\Omega\tau)\gg 1$ and $\Omega\tau\ll 1$.
This is therefore a restriction over the size of the cat state one can
realize by means of this procedure.
Let us now discuss possible strategies in order to compensate the effect of the unwanted terms in Eq. \eqref{Master:L2}.
We first consider the term in the second line. This term scales with $g_a^{\prime 2}\tau/\Delta$ and is larger than
$\gamma_2^{(0)}$. It can be compensated by means of a term of the same amplitude and opposite sign. This can be
realized by considering another atomic transition which is quasi resonant with the same cavity field, say, a third
transition $\ket{1_{\rm aux}}\to\ket{2_{\rm aux}}$ such that cavity mode A couples with strength $g_{\rm aux}$ and
detuning $\Delta_{\rm aux}$ with the dipolar transition with $|\Delta_{\rm aux}|\gg g_{\rm aux}$. If the atom is
prepared in the superposition $\cos(\varphi)\ket{1'}+\sin(\varphi)\ket{1_{\rm aux}}$ before being injected into the
cavity, then the coherent dynamics are governed by Hamiltonian $H_2'=H_2+h_{\rm aux}$, with
\begin{equation}
h_{\rm aux}=\hbar\Delta' \sigma_{2_{\rm aux}2_{\rm aux}} +\hbar g_{\rm aux} (a^\dagger \sigma_{1_{\rm aux}2_{\rm aux}} +
{\rm H.c.}) \,,
\end{equation}
which is reported apart for a global energy shift of the auxiliary levels. It is thus sufficient to select the
parameters so that the condition $\cos^2(\varphi)g_a^{\prime 2}/\Delta+\sin^2(\varphi)g_{\rm aux}^2/\Delta_{\rm aux}=0$
is
fulfilled, requiring that $\Delta$ and $\Delta_{\rm aux}$ have opposite signs.
This operation does cancel part of the dephasing due to the dynamical Stark shift of cavity mode A. It does not
compensate, however, the dephasing and dissipation terms due to one-photon processes and scaling with $g_a^{\prime
2}\/\Delta^2\sin\Delta\tau$ and $g_a^{\prime 2}/\Delta^2\sin^2(\Delta\tau/2)$, respectively. Nor does it cancel
the term due to two-photon processes in the last line of Eq. \eqref{Master:L2}, which scales with rate $(g_{a}^{\prime
2}\tau/\Delta)^2/2$. The remaining terms due to one-photon processes have a negligible effect for the choice of
parameters we perform, since $(g_a^{\prime 2}\/\Delta^2)/\gamma_2^{(0)}\sim (g_b'\tau)^{-2}$ and we choose $g_b'\tau\gg
1$ in order to warrant reasonably large rates (in other parameter regimes, where this is not fulfilled, these terms
could be set to zero by an appropriate selection of the velocity distribution of the injected atoms).
The last term can be made smaller than $\gamma_2^{(0)}$ when $(g_{b}^{\prime}/g_{a}^{\prime})^2\gg 1 $. Nevertheless,
this ratio cannot be increased arbitrarily, since the model we consider is valid as long as $\Omega\tau\ll 1$. This term
can be identically canceled out when specific configurations can be realized, like the one shown in Fig. \ref{fig:L2:1}:
In this configuration state $\ket{1'}$ couples simultaneously with the excited states $\ket{2'}$ and $\ket{e}$ by
absorption of a photon of mode A. The coherent dynamics are now described by Hamiltonian $H'=H_2+h'$ with
\begin{equation}
h'=\hbar\Delta' \sigma_{ee} +\hbar g_a''(a^\dagger \sigma_{1'e} + {\rm H.c.}) \,,
\end{equation}
If the coupling strengths and detunings are such that $g_a^{\prime 2}/\Delta=-g_a^{'' 2}/\Delta'$, then not only the
dynamical Stark shift cancels out, but interference in two-photon processes lead to the disappearance of the last line
in Eq. \eqref{Master:L2}. Under this condition, the resulting master equation is obtained in a coarse-grained time scale
$\Delta t$ assuming the atoms are injected in state $\ket{1'}$ at rate $r_2$ with a velocity distribution leading to a
normalized distribution $p(\tau)$ over the interaction times $\tau$, with mean value $\tau_2$ and variance $\delta\tau$
such that $\Delta t>\tau_2+\delta\tau$. For $r_2\Delta t\ll1$ the master equation reads
\begin{eqnarray}
\label{L2:ideal}
\frac{\partial}{\partial t}\rho&=&\gamma_2 \Bigg[ 2C_2\rho C_2^{\dagger}-\left\{C_2^{\dagger}C_2, \rho_f
\right\}\Bigg]\\
&&-{\rm i}f_1[a^\dagger a, \rho]
+f_2\left(2a\rho a^\dagger -\{a^\dagger a, \rho\}\right) , \nonumber
\end{eqnarray}
with $$\gamma_2 = (r_2/8) (g_a'g_b'/\Delta)^2 (\tau_2^2+\delta\tau^2)\,,$$ and
\begin{eqnarray}
&&f_1=r_2\frac{{g'_a}^2}{\Delta^2} \int_0^{\Delta t}{\rm d}\tau p(\tau)\sin(\Delta \tau)\,,\\
&&f_2=r_2\frac{{g'_a}^2}{\Delta^2} \int_0^{\Delta t}{\rm d}\tau p(\tau)\sin^2\left(\frac{\Delta \tau}{2}\right)\,.
\end{eqnarray}
When $p(\tau)$ is a Dirac-$\delta$ function, namely, $\delta\tau\to 0$, and $\tau_2\Delta=2n\pi$ with
$n\in\mathbb{N}$, then $f_1$ and $f_2 $ vanish identically and the
dynamics describes the target Liouville operator. Under the condition that $\delta\tau\neq 0$, but $\epsilon\equiv
\Delta\delta\tau\ll 2\pi$, then $f_1={\rm O}(\epsilon^3)$ while $f_2=\epsilon^2/4$. In the other limit, in which
$p(\tau)$ is a flat distribution over $[0,2\pi/\Delta]$, then $f_1$ vanishes while $f_2\to 1/2$.
\begin{figure}
\caption{Level scheme leading to the master equation \eqref{L2:ideal}
\label{fig:L2:1}
\end{figure}
\subsection{Discussion}
In this section we have shown how to generate the target dynamics by identifying atomic transitions and initial states
for which the desired multiphoton processes are driven. The level schemes we consider could be the effective transitions tailored by means of
lasers. If the cavity modes to entangle have the same polarization but different frequencies, the levels which are
coupled can be circular Rydberg states, while the coupling strengths $g_j$ can be effective transition amplitudes,
involving cavity and/or laser photons. The scheme then requires the ability to tune external fields so as to address
resonantly two or more levels, together with the ability to prepare the internal state of the atoms entering the
resonator. Depending on the initial atomic state, then, the dynamics can follow either the one described by
superoperator $\mathcal L_1$ or $\mathcal L_2$. An important condition is that no more than a
single atom is present inside the resonator, which sets the bound over the total injection rate, $(r_1+r_2)\Delta t\ll
1$. The other important condition is that the dynamics are faster than the decay rate of the cavity. For the
experimental
parameters we choose, this imposes a limit, among others, on the choice of the ratio $g_j/|\Delta|$, determining both
the rate for reaching the ideal steady state as well as the mean number of photons per each mode, i.e., the size of the
cat.
\section{Results} \label{sec:results}
We now evaluate the efficiency of the scheme, implementing the dynamics given by Eq. \eqref{master:0} with $H=H_1+H_2'$,
where $H_1$ is given in Eq. \eqref{H_1} and $H_2'=H_2+h$, with $H_2$ given in Eq. \eqref{H_2} while $h$ depends on the
additional levels which are included in the dynamics in order to optimize it. The initial state of the cavity is the
vacuum, and the atoms are injected with rate $r_1$ in state $\ket{1}$ (thus undergoing the coherent dynamics governed by
$H_1$) and with rate $r_2$ in state $\ket{\tilde{1}}$, which depending on the considered scheme can be either (i)
$\ket{1'}$ when $h=h'$, or (ii) $\cos(\varphi)\ket{1'}+\sin(\varphi)\ket{1''}$, when $h=h_{\rm aux}$. The case $h=0$ is
not reported,
since the corresponding efficiency is significantly smaller than the one achievable in the other two cases. In
order to determine the efficiency of the scheme we display the fidelity, namely, the overlap between the density matrix
$\chi(t)$ and the target state $\ket{\psi_\infty}$ as a function of
the elapsed time. This is defined as
$$\mathcal F(t) = \bra{\psi_\infty} {\rm Tr}_{\rm at}\{\chi(t)\} \ket{\psi_\infty}\,,$$
where $\chi(t)$ is the density matrix of the whole system, composed by cavity modes and atoms of the beam which have
interacted with the cavity at time $t$, and ${\rm Tr}_{\rm at}$ denotes the trace over all atomic degrees of freedom.
For the purpose of identifying the best parameter regimes, we first analyze the dynamics neglecting the effect of cavity
losses. Figure \ref{fig:comparison} displays the fidelity as a function of time when the dynamics are governed by
Hamiltonian $H=H_1+H_2'$ for different realizations of $H_2'$ and for different parameter choices, when the amplitude of
the coherent state $\alpha=1$. Values of $\mathcal F\simeq 0.99$ are reached when $H_2'=H_2+h'$ is implemented. The
fidelity then slowly decays due to higher order effects, which become relevant at longer times. The effect of two-photon
processes involving mode A (which identically vanish for $H_2'=H_2+h'$) is visible in the two other curves, which
correspond to the dynamics governed by $H_2'=H_2+h_{\rm aux}$ when $g_b'=10g_a'$ (blue curve) and $g_b'=3g_a'$ (red
curve). A comparison between these two curves shows that detrimental two-photon processes can be partially suppressed
by choosing the coupling rate $g_a'$ sufficiently smaller than $g_b'$.
\begin{figure}
\caption{\label{fig:comparison}
\label{fig:comparison}
\end{figure}
Figure \ref{fig:ideal} displays in detail the optimal case where $H_2'=H_2+h'$. The fidelity for the parameter choices
$g_{b}'/\Delta=10^{-3}$ and $g_{b}'/\Delta=10^{-2}$ are reported, showing that a smaller ratio leads to larger fidelity
in absence of cavity decay. The inset shows the corresponding fidelity when $\alpha=0.5$, which is notably
larger: Reaching this target state starting from the vacuum, in fact, requires a shorter time, for which higher-order
corrections are still irrelevant.
\begin{figure}
\caption{\label{fig:ideal}
\label{fig:ideal}
\end{figure}
The effect of cavity losses is accounted for in Fig. \ref{fig:losses}, where the full dynamics of master equation
\eqref{master:0} is simulated when $H_2'=H_2+h'$ and for different choices of the ratio $\kappa/r$. One clearly
observes
that the effect of cavity losses can be neglected over time scales of the order of $10^{-2}/\kappa$, so that
correspondingly larger rates $\gamma_1$ and $\gamma_2$ are required. Considered the parameter choice, this is possible
only by increasing the injection rate $r$. However, this comes at the price of increasing the probability that more than
one atom is simultaneously inside the resonator, thus giving rise to further sources of deviation from the ideal
dynamics.
\begin{figure}
\caption{\label{fig:losses}
\label{fig:losses}
\end{figure}
These results show that degradation due to photon losses poses in general a problem to attain the target state
(\ref{eq:target state}): the rate of photon losses sets a maximum achievable fidelity, and also determines a time window
during which the fidelity is close to the maximum, after which the entanglement is gradually lost. The effect of the
photon losses is twofold: it leads to a decrease in the mean photon number, and also breaks the symmetry preservation in
the evolution. The decrease in the mean photon number can be compensated by increasing the strength $\Omega$ of the
pumping in the implementation of the second Lindblad operator, as long as the approximations made in Section
\ref{subsec:Lindblad2} are still valid.
\section{Concluding remarks}
\label{Sec:V}
A strategy has been discussed which implements non-unitary dynamics for preparing a cavity in an entangled state. It is
based on injecting a beam of atoms into a cavity, where the coherent interaction of the atoms with the cavity is a
multiphoton process pumping in phase photons, so that the cavity modes approach asymptotically the entangled state of
Eq. \eqref{eq:target state}. The procedure is robust against fluctuations of the number of atoms and interaction times.
It is however sensitive against cavity losses: the protocol is efficient, in fact, as long as the time scale needed in
order to realize the target state is faster than cavity decay. The effect of the photon losses is twofold: it damps the
mean photon number and also changes the parity of the state. It could be possible to partially revert the process by
measuring the parity of the total photon number and then performing a feedback mechanism, similar to the one proposed in
Refs.~\cite{Zippilli2003,Zippilli2004} and which has been partially
implemented in Refs.~\cite{Sayrin2011,Zhou2012}. Alternatively, one can find a dissipative way to stabilize a unique
entangled target state without the need for feedback. This would require a process that can stabilize the parity of the
photon number in the even mode. First studies have been performed showing some increase in the final fidelity. We
finally note that these ideas could also find application in other systems, such as circuit quantum electrodynamics
setups \cite{E.Solano_PRA2002}.
\acknowledgements
We gratefully acknowledge discussions with Luiz Davidovich, Bruno Taketani, and Serge Haroche. This work was supported
by the European Commission (IP AQUTE, STREP PICC), by the BMBF QuORep, by the Alexander-von-Humboldt Foundation, and by
the German Research Foundation.
\end{document} |
\begin{document}
\begin{center}
\vskip 1cm{\LARGE\bf Some conjectures on the ratio of Hankel
transforms for sequences and series reversion} \vskip 1cm \large
Paul Barry\\
School of Science\\
Waterford Institute of Technology\\
Ireland\\
\href{mailto:[email protected]}{\tt [email protected]} \\
\end{center}
\vskip .2 in
\begin{abstract} For each element of certain families of sequences, we study
the term-wise ratios of the Hankel transforms of three sequences
related to that element by series reversion. In each case, the
ratios define well-known sequences, and in one case, we recover the
initial sequence.
\end{abstract}
\section{Introduction} The Hankel transform for sequences (defined below) has attracted an increasing
amount of attention in recent years. The paper \cite{Layman} situated its study within the mainstream of
research into integer sequences, while papers such as \cite{Hankel1} hinted at how the study of
certain Hankel transforms can lead to results concerning classical sequences. That paper exploited a link
between continued fractions and the Hankel transform, as explained by Krattenhaler \cite{Krat}. The best known example of a Hankel
transform for sequences is that of the Catalan numbers. One of the earlier contributors to our stock of
knowledge about the Hankel transform, Christian Radoux, had published several proofs of this result, along with other
interesting examples \cite{Rad1}, \cite{Rad2},\cite{Rad3},\cite{Rad4},\cite{Rad5}. One should also note the interesting umbral interpretation
of the Hankel transform given in \cite{Zeil}. In this paper we indicate that the term-wise ratio of Hankel transforms of
shifted sequences are noteworthy objects of study, giving us more insight into the processes involved in the Hankel transform.
\section{Integer Sequences and Transforms on them}
In this note, we shall consider integer sequences
$$a:\mathbf{N}_0 \to \mathbf{Z} $$ with general term $a_n=a(n)$. Normally, sequences will be
described by their ordinary generating function (o.g.f.), that is, the function $g(x)$ such that
$$g(x)=\sum_{n=0}^{\infty}a_nx^n.$$
We shall study the Hankel transform of sequences in this note. This
is a transformation on the set of integer sequences defined as
follows. Given a sequence $a_n=a(n)$ as described above, we form the
$(n+1)\times(n+1)$ matrix $H_n$ with general term $a(i+j)$, where
$0\le i,j \le n$. Then the Hankel transform $h_n$ of the sequence
$a_n$ is defined by $$ h_n=\det(H_n).$$ Since the elements of the matrix
$H_n$ are elements of an integer sequence, it is clear that $h_n$ is
again an integer sequence. We shall see later that this
transformation is not invertible.
\begin{example} The Catalan numbers $1,1,2,5,14,42\ldots$, defined by $C(n)=\frac{\binom{2n}{n}}{n+1}$ have o.g.f. $\frac{1-\sqrt{1-4x}}{2x}$.
The Hankel transform of the Catalan numbers is the
sequence of all $1$'s. Thus each of the determinants
$$ |1|, \qquad \left|\begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array}\right|,
\qquad \left|\begin{array}{ccc} 1 & 1 & 2\\1 & 2 & 5\\2 & 5 &
14\end{array}\right|, \qquad \ldots$$ has value $1$. A unique
feature of the Catalan numbers is that the shifted sequence $C(n+1)$
also has a Hankel transform of all $1$'s. An interesting feature of
the Catalan numbers is that the sequence $C(n)-0^n$, or
$0,1,2,5,14,42,\ldots$ has Hankel transform $n$. Of direct relevance
to this note is the fact that the Hankel transform of the sequence
$0,1,1,2,5,14,42,\ldots$ with o.g.f. $\frac{1-\sqrt{1-4x}}{2}$ is
$-n$. This sequence is defined by the series reversion of the
logistic function $x(1-x)$.
\end{example}
\begin{example} The central binomial coefficients $1,2,6,20,70,252,\ldots$, defined by $a_n=\binom{2n}{n}$, have o.g.f. $\frac{1}{\sqrt{1-4x}}$.
The Hankel transform of the central binomial coefficients is given by $h_n=2^n$. That is,
$$ |1|=1, \qquad \left|\begin{array}{cc} 1 & 2 \\ 6 & 20 \end{array}\right|=2,
\qquad \left|\begin{array}{ccc} 1 & 2 & 6\\2 & 6 & 20\\20 & 70 & 252\end{array}\right|=4, \qquad \ldots$$
The sequence $0,1,2,6,20,\ldots$ with o.g.f. $\frac{x}{\sqrt{1-4x}}$
has Hankel transform $-n2^{n-1}$. This is the negative of the
binomial transform (see below) of $n$.
\end{example}
An important transformation on integer sequences that is invertible
is the so-called \emph{Binomial transform}. Given an integer
sequence $a_n$, this transformation returns the sequence with
general term
$$b_n = \sum_{k=0}^n \binom{n}{k}a_k.$$ If we consider the sequence with general term $a_n$ to be the vector
$\mathbf{a}=(a_0,a_1,\ldots)$
then we obtain the binomial transform of the sequence by multiplying this (infinite)
vector by the lower-triangle matrix $\mathbf{B}$ whose $(n,k)$-th element is equal to
$\binom{n}{k}$:
\begin{displaymath}\mathbf{B}=\left(\begin{array}{ccccccc} 1 & 0 & 0
& 0 & 0 & 0 & \ldots \\1 & 1 & 0 & 0 & 0 & 0 & \ldots \\ 1 & 2 & 1
& 0 & 0 & 0 & \ldots \\ 1 & 3 & 3 & 1 & 0 & 0 & \ldots \\ 1 & 4 &
6 & 4 & 1 & 0 & \ldots
\\1 & 5 & 10 & 10 & 5 & 1 &\ldots\\ \vdots & \vdots & \vdots & \vdots & \vdots
& \vdots & \ddots\end{array}\right)\end{displaymath} The inverse transformation
is given by
$$a_n=\sum_{k=0}^n (-1)^{n-k}\binom{n}{k}b_k.$$
If we use ordinary generating functions to describe a sequence, then
the sequence with o.g.f. $g(x)$ will have a binomial transform whose
o.g.f. is given by $\frac{1}{1-x}g(\frac{x}{1-x})$. \newline\newline
It is shown in \cite{Layman} that if $b_n$ is the binomial transform
of the sequence $a_n$, then both sequences have the same Hankel
transform. Thus the Hankel transform is not invertible.
\section{On the series reversion of certain families of generating
functions of sequences} In this note we shall be concerned mainly
with the Hankel transform of sequences whose o.g.f. will be defined
as the series reversion of the o.g.f.'s of certain basic sequences.
Thus in this section, we will briefly recall facts about the
sequences with o.g.f.'s of the forms given by
$\frac{x}{1+{\alpha}x+{\beta}x^2}$, $\frac{x(1-\alpha x)}{1-\beta
x}$ and $x(1-\alpha x)$ as well as their reversions. The first two
families have been studied in \cite{PasTri}.
\begin{example} The family $\frac{x}{1+{\alpha}x+{\beta}x^2}$.
\newline\newline
The sequence with o.g.f. $\frac{x}{1+{\alpha}x+{\beta}x^2}$ has
general term given by $$\sum_{k=0}^{\frac{\lfloor n-1
\rfloor}{2}}\binom{n-k-1}{k}(-\alpha)^{n-2k}(-\beta)^k.$$ The
reversion of the series $\frac{x}{1+{\alpha}x+{\beta}x^2}$, that is,
the solution $u=u(x)$ to the equation
$$\frac{u}{1+\alpha u+\beta u^2}=x$$ is given by $$u(x)=\frac{1-\alpha{x}-\sqrt{1-2\alpha{x}+(\alpha^2-4\beta)x^2}}{2\beta{x}}.$$
The sequence $u_n$ with this o.g.f. has general term
$$u_n=\sum_{k=0}^{\frac{\lfloor n-1
\rfloor}{2}}\binom{n-1}{2k}C(k)\alpha^{n-2k-1}\beta^k.$$ In this
note, we shall be interested in the termwise ratios of the Hankel
transforms of the three sequences $u_n$, $u_n^*=u_{n+1}$ and $u_n^{**}=u_{n+2}$.
\newline\newline We will take the case $\alpha=-3$ and $\beta=-5$ to illustrate our results.
Thus let $a_n$ be the sequence with o.g.f. $\frac{x}{1-3x-5x^2}$. Then $a_n$ begins $0,1,3,14,57,241,\ldots$
with $$a_n=\sum_{k=0}^{\frac{\lfloor n-1
\rfloor}{2}}\binom{n-k-1}{k}3^{n-2k-1}5^k.$$ The series reversion of $\frac{x}{1-3x-5x^2}$ is $\frac{\sqrt{1+6x+29x^2}-3x-1}{10x}$ which generates
the sequence $u_n$ which begins $0,1,-3,4,18,-139,357,\ldots$ where
$$u_n=\sum_{k=0}^{\frac{\lfloor n-1
\rfloor}{2}}\binom{n-1}{2k}C(k)(-3)^{n-2k-1}(-5)^k.$$ We now form the shifted sequences
$$u_n^*=u_{n+1}=\sum_{k=0}^{\frac{\lfloor n
\rfloor}{2}}\binom{n}{2k}C(k)\alpha^{n-2k}\beta^k$$ and
$$u_n^{**}=u_{n+2}=\sum_{k=0}^{\frac{\lfloor n+1
\rfloor}{2}}\binom{n+1}{2k}C(k)\alpha^{n-2k+1}\beta^k.$$ We now let $h_n$, $h_n^*$ and $h_n^{**}$, respectively, be the Hankel transforms
of these sequences. Numerically, we find that the following:
\begin{center}
\begin{tabular}{|c|c|}\hline
Sequence & Hankel transform \\\hline
$u_n$ & $0,-1,-15,1750,890625,-2353515625,\ldots$\\\hline
$u_n*$ & $1,-5,-125,15625,9765625,-30517578125,\ldots$\\\hline
$u_n^{**}$ & $-3, -70, 7125, 3765625, -9843750000, -129058837890625,\ldots$\\
\hline
\end{tabular}\end{center}
These results suggest that $h_n^*=(-5)^{\binom{n+1}{2}}$, and
$$ \frac{(-1)^{n+1}h_{n+1}}{h_n^*}=a_{n+1}$$ along with
$$ \frac{(-1)^{n+1}h_n^{**}}{h_n^*}=a_{n+2}.$$
Thus in this case we obtain
$$h_{n+1}=(-1)^{n+1}(-5)^{\binom{n+1}{2}}a_{n+1}=(-1)^{n+1}(-5)^{\binom{n+1}{2}}\sum_{k=0}^{\frac{\lfloor n
\rfloor}{2}}\binom{n-k}{k}3^{n-2k}5^k$$ from which we infer that
$$h_n=(-1)^{n}(-5)^{\binom{n}{2}}\sum_{k=0}^{\frac{\lfloor n-1
\rfloor}{2}}\binom{n-k-1}{k}3^{n-2k-1}5^k.$$
Similarly, we find
$$h_n^{**}=(-1)^{n+1}(-5)^{\binom{n+1}{2}}a_{n+2}=(-1)^{n+1}(-5)^{\binom{n+1}{2}}\sum_{k=0}^{\frac{\lfloor n+1
\rfloor}{2}}\binom{n-k+1}{k}3^{n-2k+1}5^k.$$
Summarizing, we thus have
\begin{center}
\begin{tabular}{|c|c|}\hline
Sequence & Hankel transform \\\hline
$u_n$ & $h_n=(-1)^{n}(-5)^{\binom{n}{2}}\sum_{k=0}^{\frac{\lfloor n-1
\rfloor}{2}}\binom{n-k-1}{k}3^{n-2k-1}5^k$\\\hline
$u_n*$ & $h_n^*=(-5)^{\binom{n+1}{2}}$\\\hline
$u_n^{**}$ & $h_n^{**}=(-1)^{n+1}(-5)^{\binom{n+1}{2}}\sum_{k=0}^{\frac{\lfloor n+1
\rfloor}{2}}\binom{n-k+1}{k}3^{n-2k+1}5^k$\\
\hline
\end{tabular}\end{center}
We note that we have been able to recover the sequence $a_n$ in this example. Since the reversion of
the reversion of a series is the original series, we can now posit the
\begin{conjecture}
Let $u_n$ be the sequence
$$u_n=\sum_{k=0}^{\frac{\lfloor n-1
\rfloor}{2}}\binom{n-1}{2k}C(k)\alpha^{n-2k-1}\beta^k$$ with integer parameters
$\alpha$ and $\beta$, and o.g.f.
$$u(x)=\frac{1-\alpha{x}-\sqrt{1-2\alpha{x}+(\alpha^2-4\beta)x^2}}{2\beta{x}}.$$
Let $h_n$ be the Hankel transform of $u_n$, $h_n^*$ the Hankel transform of $u_{n+1}$, and
$h_n^{**}$ be the Hankel transform of $u_{n+2}$. Further, let $a_n$ be the sequence with o.g.f. the
series reversion of $u(x)$, with
$$a_n=\sum_{k=0}^{\frac{\lfloor n-1
\rfloor}{2}}\binom{n-k-1}{k}\alpha^{n-2k-1}\beta^k.$$ Then
\begin{enumerate}
\item
$h_n^*=\beta^{\binom{n+1}{2}}$
\item
$ \frac{(-1)^{n+1}h_{n+1}}{h_n^*}=a_{n+1} \Rightarrow
h_n=(-1)^{n}\beta^{\binom{n}{2}}\sum_{k=0}^{\frac{\lfloor n-1
\rfloor}{2}}\binom{n-k-1}{k}\alpha^{n-2k-1}\beta^k$
\item
$ \frac{(-1)^{n+1}h_n^{**}}{h_n^*}=a_{n+2} \Rightarrow
h_n^{**}=(-1)^{n+1}\beta^{\binom{n+1}{2}}\sum_{k=0}^{\frac{\lfloor
n+1 \rfloor}{2}}\binom{n-k+1}{k}\alpha^{n-2k+1}\beta^k \qquad \qquad \Diamond$
\end{enumerate}
\end{conjecture}
Note that the Hankel transform of $u_{n+1}$, $h_n^*$ is independent of $\alpha$. This is due to
\begin{enumerate}
\item The binomial transform does not change the Hankel transform, and
\item The binomial transform of
$$\frac{1-\alpha{x}-\sqrt{1-2\alpha{x}+(\alpha^2-4\beta)x^2}}{2\beta x^2}$$ is given by
$$\frac{1-(\alpha+1){x}-\sqrt{1-2(\alpha+1){x}+((\alpha+1)^2-4\beta)x^2}}{2\beta x^2}.$$
\end{enumerate}
Elements in this family are related to coloured Motzkin paths. For other links between lattice paths and
Hankel transforms, see \cite{WW}.
The recovery of the sequence $a_n$ is an interesting feature of this family of sequences. That this is not always the case
is illustrated by the next example.
\end{example}
\begin{example} The family $\frac{x(1-\alpha x)}{1-\beta
x}$ for $\beta \ne 0$.
\newline\newline
The sequence with o.g.f. $\frac{x(1-\alpha x)}{1-\beta
x}$ has general term given by
$$a_n=(\beta-\alpha)\beta^{n-1}+\frac{\alpha}{\beta}0^n.$$ Here, $0^n$ is used to denote the sequence
beginning $1,0,0,0,\ldots$ with o.g.f. $1$.
The reversion of the series $\frac{x(1-\alpha x)}{1-\beta
x}$, that is, the solution $u=u(x)$ of the equation
$$\frac{u(1-\alpha u)}{1-\beta
u}=x$$ is given by $$u(x)=\frac{1+\beta x-\sqrt{1-(2\alpha-\beta)2x+\beta^2x^2}}{2\alpha}.$$\end{example}
The sequence $u_n$ with this o.g.f. has general term
$$u_n=\sum_{k=0}^{n-1}\binom{n+k-1}{2k}C(k)\alpha^k(-\beta)^{n-k-1}.$$
Again, we shall be interested in the term-wise ratios of the Hankel transforms $h_n$, $h_n^*$ and $h_n^{**}$ respectively of the sequences
$u_n$, $u_n^*=u_{n+1}$ and $u_n^{**}=u_{n+2}$.
We obtain
\begin{conjecture} Using the notation above, we have
\begin{enumerate}
\item $h_n^*=(\alpha(\alpha-\beta))^{\binom{n+1}{2}}$
\item $h_{n+1}/h_n^*=\frac{(\alpha-\beta)^{n+1}-\alpha^{n+1}}{\beta} \Rightarrow h_n=\frac{(\alpha-\beta)^n-\alpha^n}{\beta}(\alpha(\alpha-\beta))^{\binom{n}{2}}$
\item $h_n^{**}/h_n^*=(\alpha-\beta)^{n+1} \Rightarrow h_n^{**}=(\alpha-\beta)^{n+1}(\alpha(\alpha-\beta))^{\binom{n+1}{2}} \qquad \qquad \Diamond$\end{enumerate}\end{conjecture}
We note that $h_{n+1}/h_n^*$ is the general term of the sequence with o.g.f. $\frac{-1}{(1-\alpha x)(1-(\alpha-\beta)x)}$ while
$h_n^{**}/h_n^*$ is the general term of the power sequence with o.g.f. $\frac{\alpha-\beta}{1-(\alpha-\beta)x}$. Thus in this case we do not recover terms of the
sequence with o.g.f. $\frac{x(1-\alpha x)}{1-\beta
x}$.
\begin{example} The family $x(1-\alpha x)$.
\newline\newline We note that this is in fact the case of $\frac{x(1-\alpha x)}{1-\beta x}$ where
$\beta=0$. The sequence with o.g.f. $x(1-\alpha x)$ is the sequence $0,1,-\alpha,0,0,0,\ldots$. The
reversion of the series $x(1-\alpha x)$, that is, the solution $u=u(x)$ of the equation
$$u(1-\alpha u)=x$$ is given by
$$u(x)=\frac{1-\sqrt{1-4\alpha x}}{2\alpha}.$$ For instance, the case $\alpha=1$ is that of the Catalan numbers
preceded by $0$. In general, $\frac{1-\sqrt{1-4\alpha x}}{2\alpha}$ is the o.g.f. of the sequence
$0,1,\alpha,2\alpha^2,5\alpha^3,14\alpha^4,\ldots$ with general term
$$ a_0=0, \qquad a_n=C(n-1)\alpha^{n-1}, n>0.$$
We obtain
\begin{conjecture} Using the notation above, we have
\begin{enumerate}
\item $h_n=-n\alpha^{n^2-1}, \qquad \frac{h_{n+1}}{h_n^*}=-(n+1)\alpha^n$
\item $h_n^*=\alpha^{n(n+1)}$
\item $h_n^{**}=\alpha^{(n+1)^2}, \qquad \frac{h_n^{**}}{h_n^*}=\alpha^{n+1} \qquad \qquad \Diamond$
\end{enumerate}
\end{conjecture}
We note in particular that this generalizes the well-known result
on the Hankel transforms of $C(n)$ and $C(n+1)$. We can in fact
easily modify the proof of the fact that the Hankel transform of
$C(n)$ is the all $1$'s sequence given in \cite{Rad1} to yield
\begin{proposition} The Hankel transform $h_n^*$ of the sequence
$C(n)\alpha^n$ is given by $h_n^*=\alpha^{n(n+1)}$.
\end{proposition}
\begin{proof} The coefficient of $x^{i+j+1}$ in $(1-\alpha
x)^2(1+\alpha x)^{2i+2j}$ is given by
$$\left\{\binom{2i+2j}{i+j+1}-2\binom{2i+2j}{i+j}+\binom{2i+2j}{i+j-1}\right\}\alpha^{i+j+1}=-2C(i+j)\alpha^{i+j+1}.$$
On the other hand, the coefficient of $x^k$ in $(1-\alpha
x)(1+\alpha x)^{2i}$ is equal to
$$\left\{\binom{2i}{k}-\binom{2i}{k-1}\right\}\alpha^k=\binom{2i}{k}\frac{2i-2k+1}{2i-k+1}\alpha^k.$$
Proceeding as in \cite{Rad1}, we obtain that
$$C(i+j)\alpha^{i+j}=\sum_{k=0}^{\min(i,j)}T_{i,k}T_{j,k}$$ where
$$T_{n,k}=\frac{\binom{2n}{n+k}(2k+1)}{n+k+1}\alpha^{n}.$$
Now $H_n=T_nU_n$ where $U_n$ is the transpose of $T_n$, where $T_n$
is the $(n+1) \times (n+1)$ matrix with general term $T_{i,k}$
\begin{displaymath}\mathbf{T_n}=\left(\begin{array}{ccccccc} 1 & 0 & 0
& 0 & 0 & 0 & \ldots \\\alpha & \alpha & 0 & 0 & 0 & 0 & \ldots \\
2\alpha^2 & 3\alpha^2 & \alpha^2 & 0 & 0 & 0 & \ldots \\ 5\alpha^3 & 9\alpha^3 & 5\alpha^3 & \alpha^3 & 0 & 0 & \ldots \\
14\alpha^4 & 28\alpha^4 & 20\alpha^4 & 7\alpha^4 & \alpha^4 & 0 &
\ldots
\\42\alpha^5 & 90\alpha^5 & 75\alpha^5 & 35\alpha^5 & 9\alpha^5 & \alpha^5 &\ldots\\ \vdots & \vdots & \vdots & \vdots & \vdots
& \vdots & \ddots\end{array}\right)\end{displaymath} Hence $h_n^*$
is the square of the product of the diagonal elements, that is
$$h_n^*=(\alpha^{\binom{n+1}{2}})^2=\alpha^{n(n+1)}.$$
\end{proof}
\end{example}
\hrule
\noindent 2000 {\it Mathematics Subject Classification}: Primary
11B83; Secondary 11C20, 11Y55
\noindent \emph{Keywords:} Hankel transform,
Catalan numbers, series reversion.
\end{document} |
\begin{document}
\title[Harmonic maps of finite uniton type ]{\bf{Harmonic maps of finite uniton type into inner symmetric spaces}}
\author{Josef F. Dorfmeister, Peng Wang}
\maketitle
\begin{abstract}
In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ ``of finite uniton type", from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type. {This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces.} To be more concrete, we prove that every harmonic map of finite uniton type from any Riemann surface into any {compact or non-compact} inner symmetric space has a normalized potential taking values in some nilpotent Lie subalgebra, as well as a normalized frame with initial condition identity. This provides a straightforward way to construct all such harmonic maps. We also illustrate the above results exclusively by Willmore surfaces, since this problem is motivated by the study of Willmore two--spheres in spheres.
\end{abstract}
{\bf Keywords:} harmonic maps of finite uniton type; non-compact inner symmetric spaces; normalized potential; Willmore surfaces.\\
MSC(2010): 58E20; 53C43; 53A30; 53C35
\tableofcontents
\section{Introduction}
Harmonic maps from Riemann surfaces into symmetric spaces arise naturally in geometry and mathematical physics and hence became important objects in several mathematical fields, including the study of minimal surfaces, CMC surfaces, Willmore surfaces and related integrable systems. For harmonic maps into compact symmetric spaces or compact Lie groups, one of the most foundational and important papers is the {description} of all harmonic two spheres into $U(n)$ by Uhlenbeck \cite{Uh}. It was shown that harmonic two-spheres satisfy a very restrictive condition. Uhlenbeck coined the expression `` finite uniton number'' for this property. Since the uniton number is an integer, one also obtains this way a subdivision of harmonic maps into $U(n)$.
Uhlenbeck's work was generalized in a very elegant way to harmonic two spheres in all compact semi-simple Lie groups {and into all compact inner symmetric spaces} by Burstall and Guest in \cite{BuGu}. Using Morse theory for loop groups in the spirit of Segal's work, they showed that {harmonic maps of finite uniton number} in compact Lie groups can be related to {meromorphic } maps into (finite dimensional) nilpotent Lie algebras. They also provided a concrete method to find all such nilpotent Lie algebras. Finally, via the Cartan embedding, harmonic maps into compact inner symmetric spaces are considered as harmonic maps into Lie groups satisfying some algebraic ``twisting'' conditions. Therefore the theory of Burstall and Guest provides a description of {harmonic maps of finite uniton number} into compact Lie groups and compact inner symmetric spaces, in a way which is not only theoretically satisfying, but can also be implemented well for concrete computations \cite{Gu2002}.
{Let us take a somewhat closer look at the paper \cite{BuGu}. The title and the introduction of loc.cit. only refer to harmonic maps $\mathcal F: S^2 \rightarrow \mathbb{S}$, where $\mathbb{S}$ is a compact inner symmetric space. A large part of the body of the paper, however, deals with harmonic maps $\mathcal F: M \rightarrow \mathbb{S}$, where $\mathbb{S}$ is as above and $M$ is an arbitrary, connected, compact or non-compact, Riemann surface satisfying the following conditions:
\begin{itemize}
\item There exists an extended solution $\Phi(z, \bar z, \lambda): M \rightarrow \mathbb{S}$, $\lambda \in S^1$,
such that
\[ \hbox{$\Phi$ has a finite uniton number $k \geq 0$.}\]
\end{itemize}
Such harmonic maps are said to be of finite uniton number $k$ (See Section 1 of \cite{BuGu} for more details).}
{In the present paper we use the DPW method to investigate harmonic maps into non-compact inner symmetric spaces. Therefore we consider extended frames, not extended solutions.} These frames are defined on the universal cover $\tilde{M}$ of $M$.
{ (One can include, with some caveat, the case $M = S^2$, see Section 3 of \cite{DoWa12} and
$(2)$ of Remark \ref{sphere}).}
{The two conditions above defining finite uniton number harmonic maps translate into the following two properties of extended frames} {(see Proposition \ref{typeequivnumber} and Proposition \ref{prop-fut}) :}
\begin{theorem}
{Let $\mathcal F: M\rightarrow \mathbb{S}$ be a harmonic map from a connected Riemann surface $M$ into a compact, inner symmetric space.
Then $\mathcal F$ has a finite uniton number $($$0 \leq k$ for some integer $k$$)$ if and only if}
\begin{enumerate}
\item { There exists an extended frame $F : \tilde{M} \rightarrow \Lambda G_\sigma$
for $\mathcal F$ which has trivial monodromy relative to the action of $\pi_1(M)$ on $\tilde{M}$.}
\item {There exists some frame $F : \tilde{M} \rightarrow \Lambda G_\sigma$ for $\mathcal F$, whose Fourier expansion relative to $\lambda$ is a Laurent polynomial.}
\end{enumerate}
Here $ \Lambda G_\sigma$ denotes the twisted loop group associated to $\mathbb{S}$.
\end{theorem}
Note that property $(1)$ for some extended frame is equivalent to property $(1)$ for all extended frames. Harmonic maps $\mathcal F$, as well as the corresponding extended frames, are said to be of {\bf finite uniton type} if the properties $(1)$ and $(2)$ are satisfied.
We will apply the notion of ``finite uniton type" analogously, if $\mathbb{S}$ is a non-compact symmetric space (see Definition \ref{def-uni}). A discussion of the properties $(1)$ and $(2)$ will be given in Section 3. Both properties together (i.e. the case of finite uniton type harmonic maps) will be investigated in Section 4. Moreover, in
Proposition \ref{typeequivnumber} we will show that for a compact Riemann surface $M$ harmonic maps of finite uniton type are in a bijective relation with finite uniton number harmonic maps in the sense of Uhlenbeck \cite{Uh} (see also \cite{BuGu}).
{While in the literature primarily harmonic maps $\mathcal F: M\rightarrow \mathbb{S}$ were considered, where $\mathbb{S}$ is a compact symmetric space, in the theory of Willmore surfaces in $S^n$, and many other surface classes, one has to deal with ``Gauss type maps" which are harmonic maps into non-compact symmetric spaces. }
It is the general goal of this paper to generalize results of \cite{BuGu} to harmonic maps of finite uniton type into a non-compact inner symmetric space. In particular, we want to describe simple potentials (in the sense of {the DPW method}) which generate such surfaces. For a compact inner symmetric target space this task has been carried out fairly explicitly in \cite{BuGu}, see subsections \ref{f.u.alaBuGu} and \ref{BuGu<->DPW} below for a description in our notation.
For the case of a non-compact symmetric target space no such description is known yet. Thus one looks for an approach which permits to apply the work of \cite{BuGu} in such a way that one can also find simple potentials for the case of a non-compact inner symmetric target space of a harmonic map.
In \cite{DoWa13}, when dealing with harmonic maps into $SO^+(1,n+3)/SO(1,3)\times SO(n)$, the authors found a simple way to relate harmonic maps into a non-compact inner symmetric space $G/K$ to harmonic maps into the {dual compact} inner symmetric space $U/(U\cap K^{\mathbb{C}})$ dual to $G/K$.
These two harmonic maps have a simple, but very important relationship: they share the {same} meromorphic extended framing and the normalized potential (see Theorem 1.1 of \cite{DoWa13} and Theorem \ref{thm-noncompact} in this paper). Here the normalized extended framing and the normalized potential are meromorphic data related to a harmonic map in terms of the language of the DPW method \cite{DPW}, which is a generalized Weierstrass type representation for harmonic maps into symmetric spaces.
Interpreting the work of Burstall and Guest, one will see that for harmonic maps of finite uniton type into a compact symmetric space, their work considers normalized potentials which take values in some (fixed) nilpotent Lie subalgebra (of the originally given finite dimensional complex Lie algebra) and their extended meromorphic frames take values in the loop group of the corresponding unipotent Lie subgroup. Therefore, for each fixed value of the loop parameter these meromorphic extended frames
take values in the finite dimensional unipotent Lie group mentioned above
(see Theorem \ref{thm-finite-uniton2} {in this paper}, or Theorem 1.11 \cite{Gu2002}).
It thus turns out that by combining the dualization procedure of \cite{DoWa13} with the {grouping by the uniton number} of finite uniton number harmonic maps of \cite{BuGu}, one is able to characterize all harmonic maps of finite uniton type into non-compact inner symmetric spaces by characterizing all the normalized extended frames and the normalized potentials of harmonic maps of finite uniton type into compact inner symmetric spaces, which, according to the theory of Burstall and Guest, can be
{described precisely.} Simply speaking, the case of a harmonic map into a non-compact inner symmetric space $G/K$ comes exactly from the case of a harmonic map into the compact dual inner symmetric space $U/(U\cap K^{\mathbb{C}})$ by choosing the same normalized potential for both harmonic maps, but using the two different real forms $G$ and $U$ of $G^\mathbb{C}$ for the loop group construction of the corresponding harmonic maps {(see Theorem \ref{thm-finite-uniton-n-com})}.
From a technical point of view it is important to observe (as pointed out above already) that in \cite{BuGu} the construction of harmonic maps uses ``extended solutions", while the loop group method uses extended frames. It is therefore a priori difficult to relate these two construction schemes to each other. {For the convenience of the reader and to fix notation we start a comparison of these two methods by recalling the relationship
between the extended solutions and the extended frames associated to a harmonic map into a symmetric space. Then we introduce the main results of Burstall and Guest on harmonic maps of finite uniton type \cite{BuGu}, as well as a description of their work in terms of {\em normalized potentials}, some of which has appeared in \cite{BuGu} and \cite{Gu2002}. Applying the duality theorem \cite{DoWa13}, we obtain the Burstall-Guest theory for the cases of non-compact inner symmetric spaces.}
{Both theories occurring in this paper consider group valued (actually ``matrix valued") function systems satisfying certain (partial) differential equations in the space variable with dependence on some ``loop parameter".
The ``extended solution approach" only fixes the solutions for two values of the loop parameter, while the ``extended frame approach = DPW method" fixes values of these frames for each loop parameter at a fixed basepoint in the domain of definition. As a simple consequence, the DPW method works with unique solutions and essentially bijective relations between potentials and harmonic maps. For simplicity we frequently say : the frame $F$ has initial condition $F(z_0, \bar z_0,\lambda)$, when we should spell out more explicitly : the initial condition of the frame $F(z, \bar z,\lambda)$ at the base point $z_0$ is $F(z_0, \bar z_0,\lambda)$.}
{Finally, here comes the problem of initial conditions in the study of harmonic maps of finite uniton type: This turns out be crucial in the cases of non-compact inner symmetric spaces, although it is not a big problem in the compact cases. It comes from the fact that the Iwasawa decomposition for compact loop groups is global while the Iwasawa decomposition for non-compact ones is local (See for example Theorem \ref{thm-decomposition}). Also note that the freedom of initial conditions also is equivalent to the freedom of special dressing actions. In this sense, a fixed initial condition will simplify the classification of harmonic maps of finite uniton type further, which makes it more simple to derive geometric properties of harmonic maps via normalized potentials. A standard example is the description of minimal surfaces in $\mathbb{R}^n$ via potentials in \cite{Wang-2}.} {In Theorem \ref{thm-finite-uniton-in} we can show that the initial condition of the meromorphic extended frame can be set without loss of generality to be identity.}
Our main motivation for the study of such harmonic maps is to provide the background for a detailed study of a wide variety of different types of Willmore surfaces in $S^n$, surfaces of compact or non-compact type. As an application of this paper a rough classification of Willmore two-spheres (whose conformal Gauss maps take values in the non-compact symmetric space $SO^+(1,n+3)/SO(1,3)\times SO(n)$) has been worked out in \cite{Wang-1}. For the convenience of the reader we include the main result of \cite{Wang-1} by presenting its coarse classification of Willmore two-spheres in $S^{2n}$, in terms of the normalized potentials of their conformal Gauss maps. Moreover, we also present a new Willmore two-sphere constructed by using \cite{DoWa11,DoWa12} and the results of this paper. This example also responds to an open problem posed by Ejiri in 1988 \cite{Ejiri1988}.
This paper is organized as follows. In Section 2 we review the basic results of the loop group theory for harmonic maps.
In Section 3, we provide a detailed description of harmonic maps of finite uniton type.
Several equivalent definitions are given for such maps.
Moreover, we also discuss briefly the monodromy and dressing actions for harmonic maps of finite uniton type. {In Section 4 we first introduce the main results of Burstall and Guest on harmonic maps of finite uniton type \cite{BuGu}, as well as a description of their work in terms of {\em normalized potentials}, most of which have appeared in \cite{BuGu} and \cite{Gu2002}. Then we prove that for harmonic maps of finite uniton numbers into compact inner symmetric spaces, the initial condition of the extended frame and the extended meromorphic frame can be identity at some chosen base point in $M$.
Finally in Section 5, we first apply the above result to get the same description of harmonic maps of finite uniton type into non-compact inner symmetric spaces. As an illustration, an outline of applications to the study of Willmore surfaces is listed.} \\
\section{Review of basic loop group theory}
For any inner involution $\sigma$ of a semi-simple {real} Lie group $G$ the center of $G$ is contained in the connected component of the fixed point set of $\sigma$.
To begin with, we first recall some notation. Let $G$ be a connected real semi-simple Lie group, compact or non-compact, represented as a matrix Lie group. Let $G/K$ be an inner symmetric space with the involution
$\sigma: G\rightarrow G$ such
that $G^{\sigma}\supset K\supset(G^{\sigma})^0$, where ``0" denotes ``identity component".
{\em For the purposes of this paper the actual choice of $K$ will be of little importance. The reader may thus simply assume that $K = \hat{K} = G^\sigma$ holds.}
In particular, we can assume without loss of generality that $G$ has trivial center.
We will keep this assumption throughout this paper, except where we state the opposite.
Note that {(the tangent bundle of) } $G/K$ carries a left-invariant non-degenerate symmetric bilinear form.
Let $\mathfrak{g}$ and $\mathfrak{k}$ denote the
Lie algebras of $G$ and $K$ respectively. The involution $\sigma$ induces
a decomposition of $\mathfrak{g}$ into eigenspaces, the (generalized) Cartan decomposition
\[\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p},\hspace{5mm} \hbox{ with }\ [\mathfrak{k},\mathfrak{k}]\subset\mathfrak{k},
~~~ [\mathfrak{k},\mathfrak{p}]\subset\mathfrak{p}, ~~~
[\mathfrak{p},\mathfrak{p}]\subset\mathfrak{k}.\]
Let $\pi:G\rightarrow G/K$ denote the projection of $G$ onto $G/K$.
Now let $\mathfrak{g^{\mathbb{C}}}$ be the complexification of $\mathfrak{g}$ and $G^{\mathbb{C}}$ the connected complex (matrix) Lie group with Lie algebra $\mathfrak{g^{\mathbb{C}}}$.
Let $\tau$ denote the complex anti-holomorphic involution
$g \rightarrow \bar{g}$, of $G^{\mathbb{C}}$. Then $G=Fix^{\tau}(G^{\mathbb{C}})^0$.
The inner involution $\sigma: G \rightarrow G $ commutes with the complex conjugation $\tau$ and extends to
the complexified Lie group $G^\mathbb{C}$, $\sigma: G^{\mathbb{C}}\rightarrow G^{\mathbb{C}}$. Let $K^{\mathbb{C}}\subset \hbox{Fix}^{\sigma}(G^{\mathbb{C}})$ denote the smallest complex subgroup of $G^{\mathbb{C}}$ containing $K$. Then the Lie algebra of $K^{\mathbb{C}}$ is
$\mathfrak{k^{\mathbb{C}}}$.
Occasionally we will also use another complex anti-linear involution, $\theta$, which commutes with $\sigma$ and $\tau$ and has as fixed point set
$ Fix^{\theta}(G^{\mathbb{C}})$, a maximal compact subgroup of
$G^{\mathbb{C}}$. For more details on the basic setting we refer to \cite{DoWa13}.
\begin{remark}
In this paper we only consider inner symmetric spaces. However, several of our results
also hold for arbitrary symmetric spaces. To keep the presentation of the paper
as simple as possible we will not consider { the case of outer symmetric spaces in any detail in this paper. }
\end{remark}
\subsection{Harmonic maps into symmetric spaces}
Let $G/K$ be {an inner symmetric} space as above and let $\mathcal{F}:M\rightarrow G/K$ be a harmonic map {where $M$ is a connected Riemann surface.}
{\bf In this paper we will always assume that $\mathcal{F}$ is ``full"} {in the following sense. We will mention the assumption "full" only where it seems to be particularly important.}
{
\begin{definition} \label{deffull}
A harmonic map $\mathcal{F}:M\rightarrow G/K$ is called ``full" if only $g = e$ fixes every element
of $\mathcal{F}(M)$.
That is, if there exists $g\in G$ such that $g \mathcal{F}(p)=\mathcal{F}(p)$ for all
$p\in M$, then $g=e$.
\end{definition}}
{Let $\tilde{\pi} : \tilde{M} \rightarrow M$ be the universal cover of $M$ and $z_0 \in \tilde{M}$ satisfying $\tilde{\pi}(z_0) = p_0$.
Then $\mathcal{F}$ has a natural lift $\tilde{\mathcal{F}}: \tilde{M} \rightarrow G/K$ satisfying
$\tilde{\mathcal{F}} = \mathcal{F} \circ \tilde{\pi}$ and obviously $\tilde{\mathcal{F}}(z_0,\bar z_0) = eK$. Moreover, there exists a frame $F: \tilde{M} \rightarrow G$ such that $\tilde{\mathcal{F}}=\pi \circ F$ and $F(z_0,\bar z_0) = e.$}
Let $\alpha$ denote the Maurer-Cartan form of $F$. Then $\alpha$ satisfies the Maurer-Cartan equation
and we have
\begin{equation*}F^{-1}\mathrm{d} F= \alpha, \hspace{5mm} \mathrm{d} \alpha+\frac{1}{2}[\alpha\wedge\alpha]=0.
\end{equation*}
Decomposing $\alpha$ with respect to $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ we obtain
\[\alpha=\alpha_{ \mathfrak{k} } +\alpha_{ \mathfrak{p} }, \
\alpha_{\mathfrak{k }}\in \Gamma(\mathfrak{k}\otimes T^*M),\
\alpha_{ \mathfrak{p }}\in \Gamma(\mathfrak{p}\otimes T^*M).\] Moreover, considering the complexification $TM^{\mathbb{C}}=T'M\oplus T''M$, we decompose $\alpha_{\mathfrak{p}}$ further into the $(1,0)-$part $\alpha_{\mathfrak{p}}'$ and the $(0,1)-$part $\alpha_{\mathfrak{p}}''$. Set \begin{equation} \label{alphalambda}
\alpha_{\lambda}=\lambda^{-1}\alpha_{\mathfrak{p}}'+\alpha_{\mathfrak{k}}+\lambda\alpha_{\mathfrak{p}}'', \hspace{5mm} \lambda\in S^1.
\end{equation}
\begin{lemma} $($\cite{DPW}$)$ The map $\mathcal{F}:M\rightarrow G/K$ is harmonic if and only if
\begin{equation}\label{integr}\mathrm{d}
\alpha_{\lambda}+\frac{1}{2}[\alpha_{\lambda}\wedge\alpha_{\lambda}]=0,\ \ \hbox{for all}\ \lambda \in S^1.
\end{equation}
\end{lemma}
\begin{definition}\label{def-1} Let $\mathcal{F}:M\rightarrow G/K$ be harmonic
and {let $F: \tilde{M} \rightarrow G$ be a frame satisfying $\mathcal{F}=\pi \circ F$
and $F(z_0,\bar z_0) = e$, stated as above.}
Define $\alpha_{\lambda}$ as {in \eqref{alphalambda} and consider
on $\tilde{M}$ a solution $F(z,\bar z, \lambda), \lambda \in \mathbb{C}^*$, to the equation }
\begin{equation}\label{eq-F-int}
{\mathrm{d} F(z,\bar z, \lambda)= F(z,\bar z, \lambda)\alpha_{\lambda}, \hspace{2mm} F(z_0,\bar z_0,\lambda) = e.}
\end{equation}
{As a consequence, the choice of initial condition stated determines $F(*,*,\lambda)$ uniquely by $F$. Moreover,
we then also obtain $F(z,\bar z, \lambda)|_{\lambda = 1} = F(z,\bar z)$
for all $z \in \tilde M$.
Any such solution will be called an {\em extended frame} for the harmonic map $\mathcal{F}$.}
{Finally we would like to point out that $\mathcal F_{\lambda}:=F(z,\bar z, \lambda)\mod K$ gives a family of harmonic maps with $\mathcal F_{\lambda}|_{\lambda=1}=\mathcal F$. It will be called the {\bf associated family of $\mathcal F$}.}
\end{definition}
\begin{remark}
{In this paper, we will usually assume for a harmonic map the conventions introduced above.
In particular, we will assume (without further saying) the existence of a base point $z_0$ and an extended frame such that {$F(z_0,\bar z_0, \lambda) = e$} holds. However, in Theorem 4.14 below we will encounter a case, where, a priori, the initial condition may be necessarily much more involved. {Fortunately}, in Section 4.5 it will be shown that ``it suffices to consider the cases where the initial condition in Theorem \ref{thm-finite-uniton2} is the identity matrix".}
\end{remark}
\subsection{Loop groups and decomposition theorems}
For the construction of harmonic maps we will always employ the loop group method. In this context we consider the twisted loop groups of $G$ and $G^{\mathbb{C}}$
and some of their frequently occurring subgroups:
\begin{equation*}
\begin{array}{llll}
\Lambda G^{\mathbb{C}}_{\sigma} ~&=\{\gamma:S^1\rightarrow G^{\mathbb{C}}~|~ ,\
\sigma \gamma(\lambda)=\gamma(-\lambda),\lambda\in S^1 \},\\[1mm]
\Lambda G_{\sigma} ~&=\{\gamma\in \Lambda G^{\mathbb{C}}_{\sigma}
|~ \gamma(\lambda)\in G, \hbox{for all}\ \lambda\in S^1 \},\\[1mm]
\Omega G_{\sigma} ~&=\{\gamma\in \Lambda G_{\sigma}|~ \gamma(1)=e \},\\[1mm]
\Lambda^{-} G^{\mathbb{C}}_{\sigma} ~&=
\{\gamma\in \Lambda G^{\mathbb{C}}_{\sigma}~
|~ \gamma \hbox{ extends holomorphically to } {D_\infty \}},\\[1mm]
\Lambda_{*}^{-} G^{\mathbb{C}}_{\sigma} ~&=\{\gamma\in \Lambda G^{\mathbb{C}}_{\sigma}~
|~ \gamma \hbox{ extends holomorphically to }D_{\infty},\ \gamma(\infty)=e \},\\[1mm]
\Lambda^{+} G^{\mathbb{C}}_{\sigma} ~&=\{\gamma\in \Lambda G^{\mathbb{C}}_{\sigma}~
|~ \gamma \hbox{ extends holomorphically to }D_{0} \},\\[1mm]
\Lambda_{S}^{+} G^{\mathbb{C}}_{\sigma} ~&=\{\gamma\in
\Lambda^+ G^{\mathbb{C}}_{\sigma}~|~ \gamma(0)\in S \},\\[1mm]
\end{array}\end{equation*}
where $D_0=\{z\in \mathbb{C}| \ |z|<1\}$, { $D_\infty=\{z\in \mathbb{C}| \ |z|>1\} \cup\{\infty\}$ }and $S$ is some subgroup of $K^\mathbb{C}$.
If the group $S$ is chosen to be $S = (K^\mathbb{C})^0$, then we {write $\Lambda_{\mathcal{C}}^{+} G^{\mathbb{C}}_{\sigma} $.}
If the group $S$ is chosen to be $S = \{e\}$, then we write {$\Lambda_{\star}^{+} G^{\mathbb{C}}_{\sigma} $.}
{In some cases subgroups $S$, different from the above, are chosen, like in cases where there exists a Borel subgroup. In such cases one can derive a unique decomposition of any loop group element. In other cases, like in \cite{DoWa12}, one can only derive unique decompositions
for elements in some open subset of the given loop group:
\begin{remark}
If $G = SO^+(1,n+3)$ and $K = SO^+(1,3)\times SO(n)$, then there exists a closed, connected solvable subgroup $S \subseteq (K^\mathbb{C})^0$ such that
the multiplication $\Lambda G_{\sigma}^0 \times \Lambda^{+}_S G^{\mathbb{C}}_{\sigma}\rightarrow
{(\Lambda G^{\mathbb{C}}_{\sigma})^0}$ is a real analytic diffeomorphism onto the open subset
$ \Lambda G_{\sigma}^0 \cdot \Lambda^{+}_S G^{\mathbb{C}}_{\sigma} \subset(\Lambda G^{\mathbb{C}}_{\sigma})^0$.
\end{remark}
}
We frequently use the following decomposition theorems
(see \cite{Ke1}, \cite{DPW}, \cite{PS}, \cite{DoWa12}).
\begin{theorem} \label{thm-decomposition}\
\begin{enumerate}
\item {\em (Iwasawa decomposition)}
\begin{enumerate}
\item
$ (\Lambda G^{\mathbb{C}})_{\sigma} ^0=
\bigcup_{\delta \in \Xi }( \Lambda G)_{\sigma}^0\cdot \delta\cdot
\Lambda^{+}_\mathcal{C} G^{\mathbb{C}}_{\sigma},$
where $\Xi $ denotes {a (discrete) }set of representatives for the double-coset decomposition;
\item The multiplication $\Lambda G_{\sigma}^0 \times \Lambda_\mathcal{C}^{+} G^{\mathbb{C}}_{\sigma}\rightarrow
(\Lambda G^{\mathbb{C}}_{\sigma})^0$ is a real analytic map onto the connected open subset
$ \Lambda G_{\sigma}^0 \cdot \Lambda_\mathcal{C}^{+} G^{\mathbb{C}}_{\sigma} = {\mathcal{I}_e} \subset \Lambda G^{\mathbb{C}}_{\sigma}$.
{Here $\mathcal{I}_e$ denotes the (connected) open Iwasawa cell containing the identity element.}
\end{enumerate}
\item {\em (Birkhoff decomposition)}
\begin{enumerate}
\item
{$(\Lambda {G}^\mathbb{C} )^0= \bigcup _{\omega \in \mathcal{W}} \Lambda^{-}_{\mathcal{C}} {G}^{\mathbb{C}}_{\sigma} \cdot \omega \cdot \Lambda^{+}_{\mathcal{C}} {G}^{\mathbb{C}}_{\sigma}$
where $\mathcal{W}$ denotes a (discrete) set of representatives for the double coset
decomposition}
\item The multiplication $\Lambda_{*}^{-} {G}^{\mathbb{C}}_{\sigma}\times
\Lambda^{+}_\mathcal{C} {G}^{\mathbb{C}}_{\sigma}\rightarrow
\Lambda {G}^{\mathbb{C}}_{\sigma}$ is an analytic diffeomorphism onto the
open and dense subset $\Lambda_{*}^{-} {G}^{\mathbb{C}}_{\sigma}\cdot
\Lambda^{+}_\mathcal{C} {G}^{\mathbb{C}}_{\sigma}$ {\em ( big Birkhoff cell )}.
\end{enumerate}
\end{enumerate}
\end{theorem}
\begin{remark} \ \label{S2}
\begin{enumerate}
\item {We would like to recall that for inner symmetric spaces the twisted loop algebras
are isomorphic to the untwisted ones.
For the algebraic case see, e.g. \cite{Kac}, chapter 8, and our case follows by completion in the topology used in this paper.}
\item {The middle terms of the Birkhoff decomposition, item $(2)(a)$ above, form, in the untwisted case, the Weyl group
of the corresponding untwisted loop algebra (see e.g. \cite{PS}) and therefore form a discrete subset of the loop group.}
\item {The middle terms in item $(1)(a)$ above can be determined quite precisely by using \cite{Ke1}.
Roughly speaking, they correspond to factors in a natural product decomposition of the Weyl group elements of the Birkhoff decomposition and thus they form a discrete subset of the loop group as well. For the present paper we will not need any special information about these factors.}
\item {It is well known that in the Birkhoff decomposition only one of the double cosets is an open subset of the loop group under consideration. Therefore the name "open cell" seems to be appropriate.
In the case of the Iwasawa decomposition, in general, several open double cosets can occur
(see, e.g., \cite{Ke1} for an explicit example and also \cite{D:open cells}).
But the open cell $\mathcal{I}_e$ containing the identity element plays naturally a special role.
Therefore it gets a name.}
\end{enumerate}
\end{remark}
Loops which have a finite Fourier expansion are called {\it algebraic loops} and
denoted by the subscript $``alg"$, like
$\Lambda_{alg} G_{\sigma},\ \Lambda_{alg} G^{\mathbb{C}}_{\sigma},\
\Omega_{alg} G_{\sigma} $ as in \cite{BuGu}, \cite{Gu2002}. And we define
\begin{equation}\label{eq-alg-loop}\Omega^k_{alg} G_{\sigma}:
=\left\{\gamma\in
\Omega_{alg} G_{\sigma}|
Ad(\gamma)=\sum_{|j|\leq k}\lambda^jT_j \right\}\subset \Omega_{alg} G_{\sigma} .\end{equation}
\subsection{ The DPW method and its potentials}
With the loop group decompositions as stated above, we obtain a
construction scheme of harmonic maps from a surface into $G/K$.
\begin{theorem}\label{thm-DPW}\cite{DPW}, \cite{DoWa12}, \cite{Wu}.
Let $\mathbb{D}$ be a contractible open subset of $\mathbb{C}$ and $z_0 \in \mathbb{D}$ a base point.
Let $\mathcal{F}: \mathbb{D} \rightarrow G/K$ be a harmonic map with $\mathcal{F}(z_0)=eK.$
The associated family $\mathcal{F}_{\lambda}$ (See Definition \ref{def-1}) of $\mathcal F$ can be lifted to a map
$F:\mathbb{D} \rightarrow \Lambda G_{\sigma}$, the extended frame of $\mathcal{F}$, and we can assume without loss of generality that $F(z_0,\bar z_0, \lambda)= e$ holds.
Under this assumption,
\begin{enumerate}
\item
The map $F$ takes only values in
{$\mathcal{I}_e\subset \Lambda G^{\mathbb{C}}_{\sigma}$, i.e. in the open Iwasawa cell containing the identity element.}
\item There exists a discrete subset $\mathbb{D}_0\subset \mathbb{D}$ such that on $\mathbb{D}\setminus \mathbb{D}_0$
we have the decomposition
\[F(z,\bar{z},\lambda)=F_-(z,\lambda) F_+(z,\bar{z},\lambda),\]
where \[F_-(z,\lambda)\in\Lambda_{*}^{-} G^{\mathbb{C}}_{\sigma}
\hspace{2mm} \mbox{and} \hspace{2mm} F_+(z,\bar{z},\lambda)\in \Lambda^{+}_{\mathcal{C}} G^{\mathbb{C}}_{\sigma}.\]
and $F_-(z,\lambda)$ is meromorphic in $z \in \mathbb{D}$ and satisfies
$F_-(z_0,\lambda) = e$.
Moreover,
\[\eta= F_-(z,\lambda)^{-1} \mathrm{d} F_-(z,\lambda)\]
{is a $\lambda^{-1}\cdot\mathfrak{p}^{\mathbb{C}} \textendash \hbox{valued}$ meromorphic $(1,0) \textendash$form} with poles at points of $\mathbb{D}_0$ only.
\item Spelling out the converse procedure in detail we obtain:
Let $\eta$ be a { $\lambda^{-1}\cdot\mathfrak{p}^{\mathbb{C}} \textendash \hbox{valued}$ meromorphic $(1,0) \textendash$form} such that the solution
to the ODE
\begin{equation}
F_-(z,\lambda)^{-1} \mathrm{d} F_-(z,\lambda)=\eta, \hspace{5mm} F_-(z_0,\lambda)=e,
\end{equation}
is meromorphic on $\mathbb{D}$, with $\mathbb{D}_0$ as set of possible poles.
Then on $\mathbb{D}_{\mathcal{I}}
= \{ z \in \mathbb{D}\setminus {\mathbb{D}_0}\ |\ {F_-(z,\lambda) \in \mathcal{I}_e}\}$
we
define $\tilde{F}(z,\lambda)$ by the Iwasawa decomposition
\begin {equation}\label{Iwa}
F_-(z,\lambda)=\tilde{F}(z,\bar{z},\lambda) \tilde{F}_+(z,\bar{z},\lambda)^{-1}.
\end{equation}
This way one obtains an extended frame
\[\tilde{F}(z,\bar{z},\lambda)=F_-(z,\lambda) \tilde{F}_+(z,\bar{z},\lambda)\]
of some harmonic map from $ \mathbb{D}_{\mathcal{I}} $ to $G/K$ satisfying
$\tilde{F}(z_0,\bar{z}_0,\lambda)= e$.
\item Any harmonic map $\mathcal{F}: \mathbb{D}\rightarrow G/K$ can be derived from a
{$\lambda^{-1}\cdot\mathfrak{p}^{\mathbb{C}} \textendash \hbox{valued}$ meromorphic
$(1,0) \textendash$form} $\eta$ on $\mathbb{D}$.
Moreover, the two constructions outlined above are inverse to each other (on appropriate domains of definition).
\end{enumerate}
\end{theorem}
\begin{remark}\ \label{sphere}
\begin{enumerate}
\item {A typical application of the theorem above arises as follows: one considers a harmonic map $\mathcal{F}: M \rightarrow G/K$, where $M$ is any Riemann surface and $G/K$ any inner semi-simple symmetric space and considers the natural lift $\tilde{\mathcal{F}}: \tilde{M} \rightarrow G/K$. Then, if $M$ is non-compact or compact of positive genus, then $\tilde{M}$ is contractible and one can apply the theorem above to $\tilde{\mathcal{F}}$.}
\item {So the question is: what happens if $M = S^2$? This case has been discussed in detail in Section 3.2 of \cite{DoWa12}. Basically, the theorem above still holds, if one admits some
singular points. More precisely, if $\mathcal{F}: S^2 \rightarrow G/K$ is harmonic, then {(see e.g. loc.cit. Theorem 3.11)} after removing at most two {(different, but otherwise arbitrary)} points $\{p_1,p_2\}$ from $S^2$ one can find an extended frame
$F^\prime : S^2 \setminus{ \{p_1,p_2\}} \rightarrow \Lambda G_\sigma$ for
$\mathcal{F}^\prime = \mathcal{F}| S^2 \setminus{ \{p_1,p_2\}} : S^2 \setminus{ \{p_1,p_2\}} \rightarrow G/K$. Moreover, $F^\prime_-$, formed by Birkhoff decomposing $F^\prime$, extends meromorphically to $S^2$ and the normalized potential formed with $F^\prime_-$ extends meromorphically to $S^2.$
The converse{, the construction of a harmonic map defined on $S^2$ from a normalized potential,} can be carried out {as usual} if one admits at most two singularities {in the extended frame associated with the original normalized potential. For more details we refer to loc.cit.}
Of course, if one wants to obtain a harmonic map defined {and smooth} on all of $S^2$, then additional conditions at the {poles of the original normalized potential} need to be imposed.
We will use this result at several places below. }
\item
{The restriction above to factorizations on $\mathbb{D}_{\mathcal{I}}$ implies that on this set we have globally an Iwasawa decomposition of the form $(2.6)$ with $F_{\pm}$ globally smooth. This implies, of course, the smoothness of the associated harmonic map. At isolated points of $\mathbb{D}_{\mathcal{I}}$ in $\mathbb{D}$ the frames generally exhibit singular behaviour.
In some cases, however, the corresponding harmonic maps are nevertheless non-singular.
When considering, e.g., Willmore surfaces, singularities in the frame may or may not induce singularities in the associated harmonic map \cite{DoWa12}, \cite{Wang-3}.}
{For example, singularities can occur in the extended frame, while both the associated harmonic conformal Gauss map as well as the corresponding Willmore surface stays smooth at the singularities \cite{Wang-3}. See also Example \ref{example} and \cite{Wang-3} for detailed discussions. Therefore, when discussing concrete examples of global immersions one needs to determine separately for all singularities of the frame, whether the final surface has a singularity, i.e. a branch point, or whether it is smooth and an immersion there. We refer to \cite{Wang-3} for examples of Willmore surfaces with branch points.}
\end{enumerate}
\end{remark}
\begin{definition}\cite{DPW},\ \cite{Wu}.
{With the conventions as above, let $\mathcal{F}: M \rightarrow G/K$ be a
harmonic map with basepoint $p_0$, $\tilde{\pi}: \tilde{M} \rightarrow M$ the universal cover of $M$
and $\tilde{\mathcal{F}}: \tilde{M} \rightarrow G/K$
the natural lift of $\mathcal{F}$ with basepoint $z_0,$ where $\tilde{\pi}(z_0) = p_0$.
Then the
$\lambda^{-1}\cdot \mathfrak{p}^{\mathbb{C}} \textendash \hbox{valued}$
meromorphic $(1,0)\textendash$form $\eta$ defined in $(2)$ of the last theorem for
$\tilde{\mathcal{F}}$ is called the {\em normalized potential} for the harmonic
map $\mathcal{F}$ with the point $z_0$ as the reference point. And $F_-(z,\lambda)$ given above is called the {corresponding} meromorphic extended frame.}
\end{definition}
The normalized potential is uniquely determined, {since the extended frames are normalized to $e$ at some fixed base point on $\tilde{M}$.}
The normalized potential is meromorphic {in $z \in \tilde{M}$.}
In many applications it is much more convenient to use potentials which have a Fourier expansion containing more than one power of $\lambda$.
And when permitting many (maybe infinitely many) powers of $\lambda$, one can
{obtain holomorphic coefficients:}
\begin{theorem}\cite{DPW}, \cite{DoWa12}.\label{thm-CC}
Let $\mathbb{D}$ be a contractible open subset of $\mathbb{C}$.
Let $F(z,\bar{z},\lambda)$ be the frame of some harmonic map
into $G/K$. Then there exists {some real-analytic $V_+: \mathbb{D} \rightarrow \Lambda^{+} G^{\mathbb{C}}_{\sigma} $ such that $C(z,\lambda) =
F(z, \bar z, \lambda) V_+ (z, \bar z,\lambda) $} is holomorphic in $z\in\mathbb{D}$ and in $\lambda \in \mathbb{C}^*$.
Then the Maurer-Cartan form $\eta = C^{-1} \mathrm{d} C$ of $C$ is a holomorphic
$(1,0) \textendash$form on $\mathbb{D}$ and it is easy to verify that $\lambda \eta$ is holomorphic for $\lambda \in \mathbb{C}$.
Conversely, Let $\eta\in\Lambda\mathfrak{g}^{\mathbb{C}}_{\sigma}$ be a holomorphic
$(1,0)\textendash$form such that $\lambda \eta$ is holomorphic in $\lambda$ for $\lambda \in \mathbb{C}$, then by the same process {as} given in Theorem \ref{thm-DPW} we obtain a harmonic map $\mathcal{F}: \mathbb{D} \rightarrow G/K$.
\end{theorem}
{\begin{definition}\label{rm-C}
The matrix function $C(z,\lambda)$ associated with the holomorphic $(1,0) \textendash$form $\eta$ as in Theorem \ref{thm-CC} will be called a {\em holomorphic extended frame} for the harmonic map $\mathcal{F}$.
\end{definition}}
\subsection{Symmetries and monodromy}
{It is natural to investigate harmonic maps with symmetries.}
Since harmonic maps frequently occur as ``Gauss maps" of some surfaces,
{the
investigation of harmonic maps with symmetries also has implications for surface theory.
}
\begin{definition}
{Let $\mathcal{F}: M \rightarrow G/K$ be a harmonic map. Then a pair, $(\gamma,R)$, is called a symmetry of $\mathcal{F}$, if
$\gamma$ is an automorphism of $M$ and R is an automorphism of $G/K$ satisfying
$$\mathcal{F}(\gamma.p) = R.\mathcal{F}(p)$$
for all $p \in M$.}
\end{definition}
{We would like to point out that an intuitive notion of "symmetry" for $\mathcal{F}$ would be an automorphism $R$ of $G/K$ such that $ R \mathcal{F}(M) = \mathcal{F}(M)$. In some cases one can prove that this intuitive definition implies the actual definition given just above.}
\begin{lemma} \label{frametransform}
{ Let $\mathcal{F}: M \rightarrow G/K$ be a harmonic map {and} $(\gamma,R)$ a symmetry
of $\mathcal{F}.$
Let $\tilde{M}$ denote the universal cover of $M$
and $\tilde{\mathcal{F}}: \tilde{M} \rightarrow G/K, \tilde{\mathcal{F}} = \mathcal{F} \circ \pi$, its natural lift. Then:
\begin{enumerate}
\item $\tilde{\mathcal{F}}$ satisfies
\[\tilde{\mathcal{F}} (\gamma.z) = R.\tilde{\mathcal{F}}(z).\]
\item
For any frame $F : \tilde{M} \rightarrow G$ of $\mathcal{F}$ one obtains
\begin{equation} \label{symmetry-nolambda}
{\gamma^*F(z,\bar z) = RF(z,\bar z)k(z,\bar z),}
\end{equation}
where $k(z,\bar z)$ is a function from $\tilde{M}$ into $K$.
\item For the extended frame {$F(z,\bar z,\lambda) : \tilde{M} \rightarrow \Lambda G_\sigma$
of $\mathcal{F}$} there exists some map
$\rho_\gamma: \mathbb{C}^* \rightarrow \Lambda G_\sigma$ such that
\begin{equation} \label{symmetry}
\gamma^*F(z,\bar z,\lambda) = \rho_\gamma (\lambda) F(z,\bar z,\lambda) {k(z, \bar z)},
\end{equation}
where $k$ is the $\lambda \textendash$independent function from $\tilde{M}$ into $K$ occurring in the previous equation.
Moreover, $\rho_{\gamma}(\lambda)|_{ \lambda= 1} = R$ holds.
\end{enumerate}}
\end{lemma}
{Note, since $\mathcal{F}$ is full, for each symmetry $(\gamma,R)$ the automorphism $R$ of $G/K$
is uniquely determined by $\gamma$. We therefore write $\rho_\gamma (\lambda) = \rho(\gamma,\lambda)$ and ignore $R$ in this notation. Also note that $\rho_\gamma$ actually is defined and holomorphic for all $\lambda \in \mathbb{C}^*$.}
\begin{proof}
{The first two equations follow immediately from the definitions. In view of (\ref{alphalambda})
the equality of Maurer-Cartan forms of (\ref{symmetry-nolambda}) implies the equality of the Maurer-Cartan forms of (\ref{symmetry}). Therefore only the last statement needs to be proven. But evaluating (\ref{symmetry}) at the base point $z_0$ of $\tilde{\mathcal{F}}$ yields
{$F(\gamma.z_0,\overline{\gamma.z_0},\lambda) = \rho_\gamma (\lambda) k (z_0,\bar z_0).$} Hence we obtain
$\rho_\gamma: \mathbb{C}^* \rightarrow \Lambda G_\sigma,$ and putting $\lambda = 1$ we infer
{$F(\gamma.z_0,\overline{\gamma.z_0}) = \rho_\gamma k(z_0,\bar z_0).$} On the other hand, from (\ref{symmetry-nolambda})
we obtain {$F(\gamma.z_0,\overline{\gamma.z_0}) = R k(z_0,\bar z_0)$}, whence $R =\rho_{\gamma}(\lambda)|_{ \lambda= 1} $.}
\end{proof}
\begin{definition}
{With the notation above, the matrix $\rho_\gamma (\lambda), \lambda \in S^1,$ ( for all $\lambda\in\mathbb{C}^* $ in fact) occurring in (\ref{symmetry}) is called the
monodromy (loop) matrix of $\gamma$ for $\mathcal{F}.$}
\end{definition}
The following result has been proven in Theorem 4.8 of {\cite{Do-Wa-sym}.}
\begin{theorem}
Let $M$ be a Riemann surface which is either non-compact or compact of positive genus.
\begin{enumerate}
\item
Let $\mathcal{F}:M \rightarrow G/K$ be a harmonic map and
{$\tilde{\mathcal{F}}: \tilde{M} \rightarrow G/K$ its natural lift to the universal cover $\tilde{M}$ of $M$.} Then there exists a normalized potential and a holomorphic potential for $\mathcal{F}$, namely the corresponding {potential} for $\tilde{\mathcal{F}}$.
\item Conversely, starting from some potential producing a harmonic map
$\tilde{\mathcal{F}}$ from $\tilde{M}$ to $G/K$, one obtains a harmonic map $\mathcal{F}$ on $M$ if and only if
\begin{enumerate}
\item The monodromy matrices $\chi(g, \lambda)$ associated with
$g \in \pi_1 (M)$, considered as automorphisms of $\tilde{M}$, are elements of $(\Lambda G_{\sigma})^0$.
\item There exists some $\lambda_0 \in S^1$ such that {$\chi(g, \lambda)|_ { \lambda= \lambda_0} =e$, i.e.
\begin{equation*}
\begin{split} F(g.z,\overline{g.z},\lambda)|_{ \lambda= \lambda_0}&\equiv
\chi(g, \lambda)|_{ \lambda= \lambda_0} F(z, \bar{z}, \lambda)|_{ \lambda= \lambda_0}\ \mod\ K\\
&\equiv F(z, \bar{z}, \lambda)|_{ \lambda= \lambda_0}\ \mod\ K
\end{split}
\end{equation*}}
for all $g \in \pi_1 (M)$.
\end{enumerate}
\end{enumerate}
\end{theorem}
{We also need the existence of normalized potentials for harmonic maps from a $2-$sphere
to an inner symmetric space, compact or non-compact. An important difference to the previously discussed cases is that the extended frames will not be smooth globally on $S^2$, but will be smooth (actually real analytic) on
$S^2 \setminus {\hbox{\{two points\}}}.$ See Remark \ref{sphere}.
The details can be found in $(2)$ of Remark \ref{sphere}
or Theorem 3.11 of \cite{DoWa12} which we recall for the convenience of the reader:}
\begin{theorem}\label{normalized-potential-sphere} (Theorem 3.11 of \cite{DoWa12} )
{Every harmonic map from $S^2$ to any Riemannian or pseudo-Riemannian
symmetric space $G/K$ admits an extended frame with at most two singularities and it admits a global meromorphic extended frame. In particular, every {spacelike conformal } harmonic map from $S^2$ to any Riemannian or pseudo-Riemannian symmetric space $G/K$ can be obtained from some meromorphic normalized potential.}
\end{theorem}
{While we generally restrict in this paper to inner symmetric spaces, the last and the following theorem are stated more generally because of their importance. As stated before, the case of outer symmetric spaces will not be considered in any detail in this paper.}
{ If $M$ has a non-trivial fundamental group, then the invariant potentials are of particular interest. The first part of the theorem below (the case of a non-compact $M$) has been proven first in \cite{Do-Ha2} for the case of harmonic maps into $S^2$. The case of harmonic maps from a compact Riemann surface $M$
into the (inner) symmetric space $G/K=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ was treated in
Section 6 of \cite{Do-Wa-sym}. }
{For the case of compact $M$ no proof for a general inner symmetric space is known.
For the case of non-compact $M$ the published proofs are not very clear nor explicit.
We therefore include a new proof using work of Bungart \cite{Bungart} and R\"ohrl \cite{Roehrl}.}
\begin{theorem} \label{thm-monodr}
Let $M$ be a Riemann surface.
\begin{enumerate}
\item If $M$ is non-compact, then every harmonic map from $M$ to any symmetric space {$G/K$} can be generated from some holomorphic potential on $M$, i.e. it can be generated from some holomorphic potential on the universal cover $\tilde{M}$ of $M$ which is invariant under the fundamental group of $M$.
\item If $M = S^2$, then every harmonic map from $S^2$ to any symmetric space can be generated from some meromorphic potential on $S^2$.
\item If $M$ is compact, but different from $S^2$, then every harmonic map from $M$ to
{the inner symmetric space $G/K = SO^+(1,n+3)/SO(1,3)\times SO(n)$} can be generated by some meromorphic potential defined on $M$, i.e. from a meromorphic potential defined on the (contractible) universal cover $\tilde{M}$ of $M$ which is invariant under the fundamental group of $M$.
\end{enumerate}
\end{theorem}
\begin{proof}
{For item $(2)$ see $(2)$ of Remark \ref{sphere} and for item $(3)$ see Section 6 of \cite{Do-Wa-sym}. So let us assume now that $M$ is non-compact, $G/K$ any symmetric space
and $\mathcal{F} : M \rightarrow G/K$ a harmonic map. Let $F(z,\bar z,\lambda)$ denote an extended frame
for $\mathcal{F}.$
Then from Lemma 4.11 in \cite{DPW} we obtain a $($real analytic$)$ matrix function
$\tilde V_+: \tilde{ M} \rightarrow \Lambda^+ G^\mathbb{C}_\sigma$ such that the matrix
$C$ defined by
\begin{equation}
C(z,\lambda) := F(z,\bar z,\lambda) \tilde{V}_+(z,\bar z, \lambda)
\end{equation}
is holomorphic in $z \in \tilde{M}$ and $\lambda \in \mathbb{C}^*$.
Moreover, $F(z,\bar z,\lambda)$ satisfies (\ref{symmetry}).
As a consequence, $C$ inherits
from $F(z,\bar z,\lambda)$ the transformation behaviour
\begin{equation}
C(\tau.z, \lambda) = M(\tau, \lambda) C(z,\lambda) W_+(\tau, z, \lambda),
\end{equation}
where $\tau \in \pi_1(M)$ and $W_+: \tilde{ M} \rightarrow \Lambda^+ {G^\mathbb{C}}_\sigma$
is holomorphic in $z$ and $\lambda \in \mathbb{C}^*$. It is straightforward to verify the ``cocycle condition"
\begin{equation}
W_+(\tau \mu,z,\lambda) =
W_+(\tau, \mu.z,\lambda) W_+(\mu,z,\lambda) \hbox{ for all $\tau, \mu \in \pi (M ).$}
\end{equation}
Our goal is to split the cocycle $W_+(\tau,z,\lambda)$ in $\Lambda^+ G^\mathbb{C}_\sigma$. For this we begin by following the first few lines of the proof of Theorem 3 of \cite{Roehrl}.The paper refers to complex Lie groups,
which, in our case we consider the complex Banach Lie group $H = \Lambda^+ G^\mathbb{C}_\sigma$.}
If $H$ is a complex Banach Lie group, then we denote by
${(H_\omega})^\mathcal{C}$ the sheaf of continuous sections
from open subsets of $M$ to $H$.
Similarly, by ${(H_\omega})^\mathcal{H}$ we denote the sheaf of holomorphic sections
from open subsets of $M$ to $H$.
{First we prove:
\begin{itemize}
\item[(a)] Let $M$ be a non-compact Riemann surface and $H^\mathbb{C}$ a complex Banach Lie group
then $H^1(M, {(H_\omega)}^\mathcal{C})$ = 0.
\end{itemize}
}
For the convenience of the reader we translate the first 10 or so
lines of this proof:
For $\xi\in H^1(M,H)$ we need to prove that in the principal bundle associated
with $\xi$ there exists a continuous section. Since the principal bundle can contain, for dimension reasons, at most two-dimensional obstructions, it suffices for the existence of a continuous section the verification that the two-dimensional obstruction vanishes. But this obstruction is an element of $H^2(M, \pi_1(H))$. Moreover, for a non-compact Riemann surface $M$ it is known that $H_2(M,\mathbb{Z})$ vanishes, whence by the universal coefficient theorem also $H^2(M, \pi_1(H))$ vanishes. This proves claim \textrm{(a)}.
From this we derive the complex Banach group version of
\cite[Theorem 3]{Roehrl}:
\begin{itemize}
\item[(b)]
Let $M$ be a non-compact Riemann surface and $H$ a complex Banach Lie group
then $H^1(M, {(H_\omega)}^\mathcal{H}) = 0.$
\end{itemize}
{But \cite[Theorem 8.1]{Bungart} implies}
\[
H^1(X, {(G^\mathbb{C}_\omega)}^\mathcal{C}) \cong H^1(X, {(G^\mathbb{C}_\omega)}^\mathcal{H}),
\]
and claim $(b)$ follows.
To finish the proof of the splitting theorem for the cocycle $W_+(\tau, z, \lambda),$ we can now use
\cite[Exercise 31.1]{Forster}.
For a detailed proof one can follow the proof of
\cite[Theorem 31.2]{Forster}, but with
$\Psi_i(z)=W_+(\eta_i(z)^{-1},z,
\lambda)^{-1}$ where we refer for notation to loc.cit.
We now know $ W_+(\tau, z, \lambda) = P_+(z,\lambda) P_+(\tau.z,\lambda)^{-1} $ and consider
$\hat{C} = C P_+$.
A simple computation shows
$ \hat{C}(\tau.z,\lambda) = M(\tau, \lambda) C(z, \lambda)$
for all $\tau \in \pi_1(M)$ and all
$\lambda \in \mathbb{C}^*$.
{As a consequence, the differential one-form $ \eta = \hat{C}^{-1} \mathrm{d}\hat{C}$
is invariant under $\pi_1(M)$.
This finishes the proof of the theorem.}
\end{proof}
{ The differential $1-$form $ \eta = \hat{C}^{-1} \mathrm{d}\hat{C}$ as just above we will call
an {\rm invariant} holomorphic potential for the harmonic map $\mathcal{F}$.}
\section{Harmonic maps of finite uniton type}
This section aims at interpreting all harmonic maps of finite uniton type in terms of loop group language. For this purpose, we focus first on simply-connected Riemann surfaces. Then we apply the results to Riemann surfaces with {non-trivial} topology, where monodromy appears naturally. We also provide the relations between harmonic maps of finite uniton type into non-compact symmetric spaces and their dual harmonic maps into compact symmetric spaces, which is vital for later applications. We end this section by some discussions on dressing actions on harmonic maps of finite uniton type.
\subsection{Finite uniton type harmonic maps defined on simply-connected Riemann surfaces} \label{f.u.1-connM}
In this subsection we denote by $\tilde{M}$ a simply-connected Riemann surface, i.e., $S^2$, $\mathbb{E}=\{z\in\mathbb{C}\ | \ |z|<1 \}$, or $\mathbb{C}$. Let $G/K$ denote an inner symmetric space and $\mathcal{F}:\tilde{M}\rightarrow G/K$ a harmonic map. We would like to recall our conventions: {Let $\mathcal{F}:\tilde{M} \rightarrow G/K$ be a harmonic map defined on a simply-connected Riemann surface $\tilde{M}$. Then we assume {at least tacitly} that a basepoint $z_0\in \tilde {M}$ is chosen and we also assume that any extended frame $F$, the corresponding normalized extended frame $F_-$ and {any} holomorphic extended frame $C$ {associated with an extended frame $F$} all attain the $e$ at $z_0$.}
{The case of $M = S^2$ has been addressed in item {$(2)$} of Remark \ref{sphere}, including the reference given there.}
\begin{definition} (Finite uniton {type -- simply-connected} Riemann surface ) Let $\tilde{M}$ be a simply-connected Riemann surface. A harmonic map $\mathcal{F}:\tilde{M}\rightarrow G/K$ {is said} to be of finite uniton type if some extended frame $F$ of $\mathcal{F},$ satisfying $F(z_0,\bar z_0, \lambda)=e$ for some base point $z_0\in \tilde{M},$ is a Laurent polynomial in $\lambda$.
\end{definition}
{Note that the condition
{$F(z_0,\bar z_0,\lambda) = e$ is not a restriction. Assume some $\hat{F}$
satisfies all conditions of the definition above, except $\hat{F}(z_0,\bar z_0,\lambda) = e$,
then $F(z,\bar z,\lambda) = \hat{F}(z_0,\bar z_0,\lambda)^{-1} \hat{F}(z,\bar z,\lambda)$} satisfies all conditions.
We would also like to point out that for any extended frame $F$ of some harmonic map
which is a Laurent polynomial in $\lambda$, both factors in the unique meromorphic Birkhoff decomposition {$F(z,\bar z,\lambda) = F_-(z,\lambda) F_+(z,\bar z,\lambda)$}, i.e. assuming
$F_-(z,\lambda) = e + \mathcal{O}(\lambda^{-1})$, also are Laurent polynomial.}
\begin{proposition}\label{prop-fut} Let $\mathcal{F}:\tilde{M} \rightarrow G/K$ be a harmonic map defined on a {contractible} Riemann surface $\tilde{M}$.
{ Let $z_0\in \tilde {M}$ be a base point. Then the following statements are equivalent:}
\begin{enumerate}
\item $\mathcal{F}$ is of finite uniton type.
\item There exists an extended frame {$F(z,\bar z,\lambda)$} of $\mathcal{F}$ which is a Laurent polynomial in $\lambda$.
\item The normalized extended frame {$\hat{F}_-(z,\lambda)$ of any extended frame
$\hat{F}(z,\bar z,\lambda)$ of $\mathcal{F}$} is a Laurent polynomial in $\lambda$.
\item Every holomorphic {extended frame $\hat{C}(z,\lambda)$} associated with any extended frame
{$\hat{F}(z,\bar z,\lambda)$} of $\mathcal{F}$ only contains finitely many negative powers of $\lambda$.
\item There exists a holomorphic extended { frame $C^\sharp(z,\lambda)$ } which { only contains} finitely many negative powers of $\lambda$.
\end{enumerate}
For the case $\tilde{M} = S^2,$ the above {equivalences remain} true, if one replaces in the last two statements the word ``holomorphic" by ``meromorphic" and {also admits for all
frames at most two singularities.}
\end{proposition}
\begin{proof}
{For a contractible domain, the definition of "finite uniton type" for a harmonic map can be
rephrased by $(1) \Leftrightarrow (2)$. Hence we only need} to show that $(2), (3), (4)$ and $(5)$ are equivalent.
(2) $\mathbb{R}ightarrow $ (3): {Let $F(z,\bar z,\lambda)$ and $\hat{F}(z,\bar z,\lambda)$ be as in $(2)$ and $(3)$ respectively. Then
$\hat{F} (z,\bar z,\lambda)= F (z,\bar z,\lambda)k(z,\bar z,\lambda)$. Inserting the unique Birkhoff decompositions $F(z,\bar z,\lambda) = F_-(z,\lambda)F_+(z,\bar z,\lambda)$ and $\hat{F}(z,\bar z,\lambda) = \hat{F}_- (z,\lambda)\hat{F}_+(z,\bar z,\lambda)$ we infer $F_-(z,\lambda) = \hat{F}_-(z,\lambda)$ and the claim follows.}
(3) $\mathbb{R}ightarrow$ (4): {By Theorem \ref{thm-CC},} { $ \hat{C}(z,\lambda) = \hat{F} (z,\bar z,\lambda)\hat{V}_+(z,\bar z,\lambda)$, where
$\hat{V}_+ $ is actually real-analytic.
Inserting the Birkhoff decomposition $\hat{F}(z,\bar z,\lambda) = \hat{F}_-(z,\lambda) \hat{F}_+(z,\bar z,\lambda)$, we obtain
$\hat{C} (z,\lambda)= \hat{F}_-(z,\lambda) \hat{F}_+(z,\bar z,\lambda) V_+(z,\bar z,\lambda) $ and the claim follows.}
(4) $\mathbb{R}ightarrow$ (2): {Let $\hat{C}(z,\lambda)$ be any holomorphic extended frame for $\mathcal{F}$. Consider the Iwasawa decomposition $F(z,\bar z,\lambda)=\hat{C}(z,\lambda) \tilde{C}_+(z,\bar z,\lambda)$ near $z_0$, where $\tilde{C}_+\in \Lambda^{+} G^{\mathbb{C}}_{\sigma}$
and where $F(z,\bar z,\lambda)$ and $\tilde{C}_+(z,\bar z,\lambda)$ attain the $e$ at $z_0$.
Since $\hat{C}(z,\lambda) $ only contains finitely many negative powers of $\lambda$ by assumption, also $F(z,\bar z,\lambda)$ contains only finitely many negative powers of $\lambda$. Since $F(z,\bar z,\lambda)$ is real, it is a Laurent polynomial.}
(4) $ \Leftrightarrow$ (5): Note that for any two holomorphic extended frames $C_1(z,\lambda)$ and $C_2(z,\lambda)$ there exists some $W_+(z,\lambda)\in \Lambda^{+}G^{\mathbb{C}}_{\sigma}$ such that $C_1(z,\lambda)=C_2(z,\lambda)W_+(z,\lambda)$ holds.
For the case $\tilde{M}=S^2$, {using the results just proven} for $M_1=S^2\setminus\{\infty\}$ and $M_2=S^2\setminus\{0\}$ respectively, one will obtain {the last claim of the proposition. For more details see
$(2)$ of Remark \ref{sphere} and Section 3.2 of \cite{DoWa12}.}
\end{proof}
We wonder, under what conditions the
based (at $z_0$) normalized extended frame {$F_-(z,\lambda)$} will be a Laurent polynomial.
Let $\eta$ denote the normalized potential { $F_-(z,\lambda)^{-1} \mathrm{d} F_-(z,\lambda) = \eta$.
Then $F_-(z,\lambda), F_-(z,\lambda)|_{z=z_0} = e$}, can be obtained from $\eta$ by an application of the {standard Picard iteration of the theory of ordinary differential equations. Since $\eta$ is a multiple of
$\lambda^{-1},$} it is easy to see that each step of the Picard iteration
{decreases the occurring power of $\lambda$ by $-1$.} So $F_-(z,\lambda)$ is a Laurent polynomial if and only if the Picard iteration stops after finitely many steps. { See for example, Section 1 of \cite{Gu2002}}.
The most natural reason for the Picard iteration to stop is that the normalized potential $\eta$ takes values in some nilpotent Lie algebra (Note: If $\eta(z)$ is only nilpotent for every $z\in \tilde M$, it does not follow, in general, that the Picard iteration will stop.).
The following result is a slight generalization of Appendix B of \cite{BuGu}, (1.1) of \cite{Gu2002}. We would like to point out that in particular $G$ does not need to be compact.
\begin{proposition} Let $\mathfrak{n}\subset\mathfrak{g}^{\mathbb{C}}$ be a nilpotent subalgebra and assume that $\eta$ is a ( holomorphic or meromorphic ) potential of some harmonic map such that $\eta(z)\in\mathfrak{n}$ for all
$z\in \tilde M \setminus \lbrace$ poles of $\eta \rbrace $, and $\eta$ only contains finitely many positive powers of $\lambda$. Let $\mathcal{F}:\tilde{M}_0\rightarrow G/K$ be the harmonic map associated with $\eta$ on an open subset $\tilde{M}_0\subset\tilde{M}$. Then $\mathcal{F}$ is of finite uniton type.
\end{proposition}
\begin{proof}
When $\eta(z)\in\mathfrak{n}$ for all $z\in \tilde M$, then it is easy to see that the Picard iteration producing the solution
{$\mathrm{d} C(z,\lambda)= C(z,\lambda) \eta$,} $C(z_0,\lambda)=e$, stops after finitely many steps.
Since $\eta$ is a Laurent polynomial in $\lambda$, {so is the solution} $C$. By Proposition \ref{prop-fut}, $\mathcal{F}$ is of finite uniton type.
\end{proof}
\begin{remark}
\
\begin{enumerate}
\item
The conformal Gauss maps of many Willmore surfaces (see \cite{DoWa11}, \cite{Wang-1} and \cite{Wang-3}) can be constructed as discussed in the proposition above.
In these examples the group $G$ is non-compact.
\item For harmonic maps into arbitrary Lie groups $G$ with a bi--invariant non--degenerate metric, one can also produce harmonic maps of finite uniton type following the above procedure by considering $G$ as the symmetric space
$(G \times G)/G$. Harmonic $2-$spheres in $U(4)$
provide standard examples for the proposition (see Section 5 of \cite{BuGu} or Appendix B of \cite{Gu2002}). In this case the group $G = U(4)$ is compact.
\end{enumerate}
\end{remark}
For the construction of examples of Willmore spheres it is very important to know that harmonic maps defined on $M=S^2$ are of finite uniton type.
First of all { we reiterate that the loop group approach in the sense of this paper, {i.e. the DPW method,} using extended frames and not extended solutions,
also applies to harmonic maps from $S^2$ to any compact or non-compact
inner symmetric space ( see Remark \ref{sphere} above). }
{Using the same approach one can retrieve the result of Uhlenbeck Theorem \cite{Uh},
Segal \cite{Segal}, Burstall-Guest \cite{BuGu} for compact $G/K$
and generalize it to the case of non-compact inner symmetric spaces $G/K$.}
\begin{theorem} \label{S2isfu}(Theorem 3.6, of \cite{DoWa13}) For every compact or non-compact {inner} symmetric space $G/K$ every harmonic map $\mathcal{F}: S^2 \rightarrow G/K$ is
of finite uniton type.
\end{theorem}
\subsection{ {Harmonic maps from Riemann surfaces to inner symmetric spaces which have a trivial monodromy representation}} \label{trivmonorep}
In the last subsection we considered exclusively simply-connected Riemann surfaces $M$.
Obviously then, the monodromy representation of any harmonic map $\mathcal{F}:M \rightarrow G/K$
is trivial. But there also exist non-simply-connected Riemann surfaces which admit harmonic maps
$\mathcal{F}:M \rightarrow G/K$ of {``finite uniton type"} and thus have, by definition, a trivial monodromy representation. {Next we introduce the notion of ``finite uniton type".}
\subsubsection{ {Definition of ``finite uniton type"}}
In \cite{BuGu}, Burstall and Guest generalize the work of Uhlenbeck on harmonic maps
$\mathcal{F}:S^2 \rightarrow G$, $G=U(n)$, to harmonic maps of ``finite uniton number" from a general Riemann surface into compact Lie groups $G$ as well as compact inner symmetric spaces $G/K$.
In our {definition below} we separate out two properties/parts contained implicitly in the definition of \cite{BuGu}, just above loc.cit., Theorem 1.2, and we also admit non-compact Lie groups and inner symmetric spaces. The first property is that one considers maps which actually are defined on $M$, whence do not have any monodromy. The second property is that one can assume w.l.g. that the extended solution used in \cite{BuGu} is a Laurent polynomial in $\lambda$. These two properties define two classes of harmonic maps, the intersection of which gives the harmonic maps of finite uniton number. It would be interesting to investigate these two classes separately.
\begin{definition}\label{def-uni} (Finite uniton type -- arbitrary Riemann surface) Let $M$ be a Riemann surface {and let $G/K$ {be} an inner symmetric space}. A harmonic map
$\mathcal{F}:M\rightarrow G/K$ is said to be of finite uniton type if there exists some extended frame $F$ of $\mathcal{F}$, defined on the universal cover
$\tilde{M}$ of {$M,$ satisfying $F(z_0,\bar{z}_0,\lambda)=e$}
and having the following two properties:
\begin{enumerate}[$(U1)$]
\item
For all $\lambda \in S^1,$ the extended frame $F(z,\bar{z},\lambda): \tilde{M} \rightarrow (\Lambda {G _{\sigma})} $ has trivial monodromy and {, in view of the last equation of
Lemma \ref{frametransform}, therefore descends to a well defined map on $M$, i.e.
$F(z,\bar{z},\lambda): M \rightarrow (\Lambda G_{\sigma})/K$,}
up to two singularities in the case of $M = S^2$.
\item $F(z,\bar{z},\lambda)$ is a Laurent polynomial in $\lambda$.
\end{enumerate}
We will say ``$\mathcal{F}$ has a trivial monodromy representation" if (U1) is satisfied.
{We would also like to point out that we will use, by abuse of notation, the same notation for the frame $F$ with values in
$\Lambda G _{\sigma}$ and its projection with values in $ \Lambda G_{\sigma}/K$.}
\end{definition}
\begin{remark}\
\begin{enumerate}
\item Note that in Section 4 below we discuss the relations between the DPW theory (using extended frames) and the Burstall-Guest theory (using extended solutions). It turns out that {the Definition \ref{def-uni} for
the notion of a finite uniton type harmonic map is equivalent to the notion of a minimal (finite) uniton number
harmonic map defined on page 546 of \cite{BuGu}. ( Also see Proposition \ref{typeequivnumber}.)}
\item
We mainly want to use objects occurring in our approach (like extended frames) and also want to make it as clear as possible to what extent the two properties which make up the notion of
{finite uniton type and minimal (finite) uniton number respectively contribute to statements and proofs.}
\item Condition $(U1)$ is a very strong condition. Of course, in the case of a non-compact simply connected Riemann surface $M$ {and {of} $S^2$ respectively,} it is always satisfied. {We would like to point out that although the assumption $(U1)$ was not stated in \cite{BuGu} explicitly, it was used implicitly by using exclusively extended solutions defined on all of $M$, like in \cite{BuGu}, Theorem 4.5, where $M$ is an
{arbitrary, compact or non-compact,} Riemann surface.}
\end{enumerate}
\end{remark}
\subsubsection{ {Harmonic maps with trivial monodromy} {representation}}
{In this subsubsection we consider all harmonic maps satisfying property $(U1)$. Recall, that we assume that all harmonic maps in this paper are considered to be full,
{see Definition \ref{deffull}.} In the following proposition we will use, as in the definition just above, by abuse of notation the same notation for the maps $C $ and $F_-$
and their projections respectively to some image quotient space.}
\begin{proposition} \label{prop-frame}
Let {$M \neq S^2$} be any Riemann surface and let $\mathcal{F}: M\rightarrow G/K$ be a harmonic map which is full in $G/K$. Then the following statements are equivalent
\begin{enumerate}
\item The monodromy matrices $\chi(g,\lambda)$, $g\in\pi_1(M)$, $\lambda\in S^1$, satisfy $\chi(g,\lambda)=e,\ \hbox{ for all } g\in\pi_1(M),\ \lambda\in S^1$.
\item Any extended frame $F$ of $\mathcal{F}$ is defined
on $M$ ``modulo $K$'', i.e. $F$ descends from a map defined on $\tilde{M}$ with values in
${\Lambda G}_{\sigma}$ to
$F(z, \bar{z},\lambda ): M \rightarrow ({\Lambda G}_{\sigma}) / K$.
\item Any holomorphic extended frame $C$ of $\mathcal{F}$ is defined on $M$
``modulo $\Lambda^+G^{\mathbb{C}}_{\sigma}$'' , i.e. $C$ descends from a map defined on $\tilde{M}$ with values in $\Lambda G^{\mathbb{C}}_{\sigma}$ to
$C(z,\lambda):M\rightarrow \Lambda G^{\mathbb{C}}_{\sigma}/\Lambda^+ G^{\mathbb{C}}_{\sigma}$.
\item The normalized extended frame $F_-$ relative to any base point is defined on $M$, i.e. $F_-$ descends from a map defined on $\tilde{M}$ with values in $\Lambda^- G^{\mathbb{C}}_{\sigma}$ to $F_-(z,\lambda):M\rightarrow \Lambda^- G^{\mathbb{C}}_{\sigma}$.
\end{enumerate}
{For $M = S^2$ the four statements just above still hold in view of {(2) of Remark \ref{sphere} of Section 2,} if one admits up to two singularities {in the extended frames.}}
\end{proposition}
\begin{proof}
{In view of Theorem 3.6 the case $M = S^2$ is trivially satisfied. Hence let us assume $M$ is any Riemann surface different from $S^2$.}
By general theory we have for every $g\in\pi_1(M)$ and all $\lambda\in S^1$
on the universal cover $\tilde{M}$ the equations:
\[\begin{split}
F(g.z,\overline{g.z},\lambda)&= \chi(g,\lambda) F(z,\bar{z},\lambda) K(g,z,\bar{z}),\\
C(g.z,\lambda)&=\chi(g,\lambda) C(z,\lambda) W_+(g,z,\bar{z},\lambda),\\
F_-(g.z,\lambda)&=\chi(g,\lambda) F_-(z,\lambda) L_+(g,z,\bar{z},\lambda),\\
\end{split}\]
for some maps
$K(g,z,\bar{z}): \tilde{M} \rightarrow K $,
$ W_+ (g,z,\bar{z},\lambda),
\ L_+(g,z,\bar{z},\lambda): \tilde{M}\rightarrow \Lambda^+ G^{\mathbb{C}}_{\sigma}$, and $\chi(g,\lambda)\in\Lambda G_{\sigma}$. Therefore (1) implies (2), (3) and (4).
``(2) $\mathbb{R}ightarrow$ (1)": From the assumption we obtain $F(g.z,\overline{g.z},\lambda)= F(z,\bar{z},\lambda)\tilde{K}(g,z,\bar{z})$. This shows \[\chi(g,\lambda)F(z,\bar{z},\lambda)K(g,z,\bar{z})=F(z,\bar{z},\lambda)\tilde{K}(g, z,\bar{z}).\] Since $\mathcal{F}\equiv F\mod K$, this equation implies
\[\mathcal{F}(z,\bar{z},\lambda)=\chi(g,\lambda)\mathcal{F}(z,\bar{z},\lambda), \hbox{ for all } z\in M,\ \lambda\in S^1.\]
When $\mathcal{F}$ is full, $\chi(g,\lambda)=e$ follows.
``(4) $\mathbb{R}ightarrow$ (2)": If $F_-$ is defined on $M$, then $F_-(g.z,\lambda)=F_-(z,\lambda)$ for all $g\in \pi_1(M)$, $\lambda\in S^1.$ But then $F=F_-F_+$ satisfies
\[\begin{split}
F(g.z,\overline{g.z},\lambda)&=F_- (g.z,\lambda) F_+(g.z,\overline{g.z},\lambda)\\
&=F_- (z,\lambda) F_+(g.z,\overline{g.z},\lambda)\\
&=F(z,\bar{z},\lambda)F_+(z,\bar{z},\lambda)^{-1}F_+(g.z,\overline{g.z},\lambda).\\
\end{split}\]
The reality of $F(g.z,\overline{g.z},\lambda)$ and $F(z,\bar{z},\lambda)$ yields $F_+(z,\bar{z},\lambda)^{-1}F_+(g.z,\overline{g.z},\lambda)=K(g, z,\bar{z})$. Hence (2) follows.
``(3) $\mathbb{R}ightarrow$ (4)": Assume $C$ is a holomorphic extended frame defined on $M$. Then $C(g.z,\lambda)=C(z,\lambda)B_+(g,z,\bar{z},\lambda)$
by assumption and $F_-=CS_+$ implies
\[F_-(g.z,\lambda)=C(z,\lambda)B_+(g,z,\bar{z},\lambda)S_+(g.z,\overline{g.z},\lambda)=F_-(z,\lambda) T_+(g,z,\bar{z},\lambda).\]
But since $F_-=e+O(\lambda^{-1})$, this Birkhoff decomposition is unique and $T_+(g,z,\bar{z},\lambda)=e$ follows.
\end{proof}
\begin{corollary}
{ Let $M $ be any Riemann surface and $G/K$ any inner symmetric space and let $\mathcal{F}: M\rightarrow G/K$ be any harmonic map in $G/K$. Then the normalized extended frame $F_-$, is fixed under the action of $\pi_1(M)$.
In particular, we have $F_- (g.z, \lambda) = F_- (z, \lambda)$ for all $z \in \tilde{M}$ and $g \in \pi_1(M)$.
As a consequence, the normalized potential {$\eta_- = F_-(z,\lambda)^{-1} \mathrm{d} F_-(z,\lambda)$,} is fixed under the action of
each $g \in \pi_1(M)$.
}
\end{corollary}
If we assume $M$ to be non-compact then we obtain the following stronger result:
\begin{theorem}\label{thm-invariant-potential} The following statements are equivalent for harmonic maps
$\mathcal{F}:M\rightarrow G/K$ into the
inner symmetric space $G/K$, compact or non-compact, where $M$ is any non-compact Riemann surface:
\begin{enumerate}
\item $\mathcal{F}_{\lambda}: \tilde{M} \rightarrow G/K$ has trivial monodromy for all $\lambda \in S^1$.
\item There exists some extended frame $F$ which satisfies
\[F(g.z,\overline{g.z},\lambda)=F(z,\bar{z},\lambda) k(g, z, \bar z)~~ \hbox{ for all }~~\lambda\in S^1,\ g\in \pi_1(M)
\hbox{ and some } k(g, z, \bar z) \in K.\]
\item There exists a holomorphic extended frame $C$ for the harmonic map $\mathcal{F}$ which satisfies
\[C(g.z,\lambda)=C(z,\lambda)\ \hbox{ for all }\ \lambda\in S^1,\ g\in\pi_1(M).\]
\item The integrated normalized potential $F_-$ of the harmonic map $ \mathcal{F}:M\rightarrow G/K$ satisfies
\[F_-(g.z,\lambda)=F_-(z,\lambda)\ \hbox{ for all }\ \lambda\in S^1,\ g\in\pi_1(M).\]
\end{enumerate}
\end{theorem}
\begin{proof}
From Proposition \ref{prop-frame} we know that $(1)$ and $(4)$ are equivalent.
{
By the proof of part $(1)$ of Theorem \ref{thm-monodr} we know that in general there exists some holomorphic extended frame satisfying $C(g.z,\lambda)=M(g,\lambda)C(z,\lambda)$ for all $g \in \pi_1(M).$
Therefore, by using such a $C(z,\lambda)$ in $(3)$ of
Proposition \ref{prop-frame} we observe that the monodromy is trivial if and only if
$C(z,\lambda)$ is invariant under $\pi_1(M).$ Finally, $(2)$ is equivalent to $(2)$ of
Proposition \ref{prop-frame}.}
\end{proof}
\begin{remark}
It is not known (except for some examples) for which $M$ and which $G/K$ the factor $k$ in $(2)$ can be removed.
\end{remark}
{ For Willmore surfaces we consider the inner symmetric space
$SO^+(1,n+3)/SO^+(1,3)\times SO(n)$. In this case we have in view of Theorem \ref{thm-monodr}:
\begin{theorem}
If the inner symmetric space is $SO^+(1,n+3)/SO^+(1,3)\times SO(n),$ then the
statements of Theorem \ref{thm-invariant-potential} hold for any Riemann surface, compact or non-compact.
\end{theorem}
}
\subsubsection{Dressing and trivial monodromy}
Dressing is a very useful operation which permits to construct new harmonic maps from a given one.
Before applying this to finite uniton harmonic maps we
recall the definition of ``dressing" (e.g. \cite{DPW}, see also \cite{Gu-Oh}, \cite{BP}, \cite{Do-Ha3}, \cite{TU1}).
Let $\mathcal{F}:\tilde{M}\rightarrow G/K,$ a harmonic map, and $F$ an extended frame satisfying
$F(z_0,\bar{z}_0,\lambda) = e$ for some base point $z_0$. Let $h_+\in\Lambda^+G^{\mathbb{C}}_{\sigma}$ and consider the Iwasawa splitting (on an open subset $\tilde{M}'$ of $\tilde{M}$ containing $z_0$)
\begin{equation}
\label{eq-dress1} h_+F=\hat{F}\hat{W}_+.
\end{equation}
Then $\hat{F}(z, \bar z,\lambda)$ defines a {family of harmonic maps on $\tilde M'$}
\begin{equation}
\label{eq-dress2}
{\hat{\mathcal{F}}_\lambda : = h_+ \sharp {\mathcal{F}_\lambda} : \tilde{M}' \rightarrow G/K,\
\hat{\mathcal{F}}_{\lambda} ( z, \bar z) :=\hat{F} (z, \bar z,\lambda) \mod K . }
\end{equation}
\begin{lemma}
Using the notation introduced just above, {$\hat{F}(z, \bar z,\lambda)$} is an extended frame of
$\hat{\mathcal{F}}(z, \bar z,\lambda) = { (h_+\sharp\mathcal{F})(z, \bar z, \lambda)}$ and satisfies w.l.g. {$\hat{F}(z_0, \bar z_0,\lambda) = e$}.
If $M(g,\lambda)$ denotes the monodromy representation of $F(z, \bar z,\lambda)$, $g \in \pi_1(M)$, then the
monodromy representation of $\hat{F}$ is given by
\begin{equation}
\hat{M}(g,\lambda) = h_+ M(g,\lambda) h_+^{-1}.
\end{equation}
\end{lemma}
\begin{corollary} \label{Cor1}
If $\mathcal{F}_\lambda$ descends for some $\lambda_0 \in S^1$ to a harmonic map $\mathcal{F}_{\lambda_0}:M\rightarrow G/K$, i.e. if {$M(g,\lambda_0) = e$} for all $g \in \pi_1(M)$,
then the corresponding dressed harmonic map $h_+\sharp\mathcal{F}_{\lambda_0}$ will descend to
$M' = \tilde{M}' \mod \pi_1(M)$.
\end{corollary}
A particularly interesting feature of harmonic maps with trivial monodromy representation
is that {the property of trivial monodromy representation is preserved under dressing action}.
\begin{theorem}\label{thm-dress} If $\mathcal{F}:\tilde{M}\rightarrow G/K$ is a harmonic map with
trivial monodromy representation, then the associated family $\mathcal{F}_{\lambda}$ consists of harmonic maps with
trivial monodromy and all dressed harmonic maps $h_+\sharp\mathcal{F}_{\lambda}$, $h_+\in\Lambda^+G^{\mathbb{C}}_{\sigma}$,
have trivial monodromy representations.
\end{theorem}
\subsection{ {Relating harmonic maps into a non-compact inner symmetric
space $G/K$ to harmonic maps into the compact dual inner symmetric space of $G/K$}}
{In \cite{BuGu}, Burstall and Guest have given for compact inner symmetric spaces $G/K$ an explicit
description of those normalized potentials which produce finite uniton type harmonic maps.
To make their work applicable to the non-compact case, we established in \cite{DoWa13} a duality theorem between harmonic maps into non-compact inner symmetric spaces and their compact dual. The most important feature of this result is that the corresponding harmonic maps share the same normalized potential (See the next theorem).}
{We will show briefly how this duality relation works and what it implies for the properties $(U1)$
and $(U2)$}.
{Let $G/K$ be a non-compact inner symmetric space, defined by
$\sigma$, and $\tilde{G}$ the (connected) simply-connected cover of $G$.
Then $G/K = \tilde{G}/\tilde{K}$ for some closed subgroup $\tilde{K}$ of $\tilde{G}.$ }
{Let $\tilde{U},$ be a (connected, simply-connected and semi-simple) maximal compact Lie subgroup
of $\tilde{G}^{\mathbb{C}}$, the complexification of $\tilde{G}.$} We {can also assume w.l.g. that $\tilde{G}^{\mathbb{C}}$ is a complex matrix Lie group with Lie subgroup $G$.
Moreover, we can assume w.l.g. that $\tilde{U}$ is invariant under the $\mathbb{C}-$linear extension of $\sigma$ to $G^\mathbb{C}$. Finally, by abuse of notation, for $G = SL(2,\mathbb{R})$ we also denote by $\tilde{G}$ and $\tilde{K}$ the natural image of $\tilde{G}$ and $\tilde{K}$ in $G^\mathbb{C}$. See, e.g. \cite{Hoch}.}
We consider the compact dual $\tilde{U}/(\tilde{U}\cap \tilde{K}^{\mathbb{C}})$ of $G/K$ which
clearly also is defined by $\sigma$ (see \cite{DoWa13} for more details).
Moreover, $ (\tilde{U}\cap \tilde{K}^{\mathbb{C}})^\mathbb{C} = \tilde{K}^{\mathbb{C}}$ holds
(see {Theorem 1.1 of \cite{DoWa13}}). From this we infer:
\begin{equation}\label{eq-loop-com-n-com}
\Lambda \tilde{G}^{\mathbb{C}}_{\sigma}=\Lambda \tilde{U}^{\mathbb{C}}_{ {\sigma}},\ \Lambda_*^- \tilde{G}^{\mathbb{C}}_{\sigma}=\Lambda_*^- \tilde{U}^{\mathbb{C}}_{ {\sigma}},\ \Lambda^+\tilde{ G}^{\mathbb{C}}_{\sigma}=\Lambda ^+ \tilde{U}^{\mathbb{C}}_{ {\sigma}}.
\end{equation}
As a consequence, for any extended framing $F(z,\bar{z},\lambda)$ of $\mathcal{F}: M \to G/K$, the decomposition
{\[F(z,\bar{z},\lambda)=F_{\tilde{G,-}} (z,\lambda) F_{\tilde{G},+}(z,\bar{z},\lambda)=F_{\tilde{U},-}(z,\lambda) F_{\tilde{U},+}(z,\bar{z},\lambda)\]
shows
\begin{equation}\label{eq-norm-framing}
\begin{split}F_{\tilde{G},-} (z,\lambda)&=F_{\tilde{U},-}(z,\lambda),\\
\eta&=\lambda^{-1}\eta_{-1}\mathrm{d}z=
(F_{\tilde{G},-})(z,\lambda)^{-1}\mathrm{d}F_{\tilde{G},-}(z,\lambda)=(F_{\tilde{U},-})^{-1}(z,\lambda)\mathrm{d}F_{\tilde{U},-}(z,\lambda).
\end{split}
\end{equation}}
\begin{theorem} \label{thm-noncompact}$($\cite{DoWa13}$)$
Let
$G/K = \tilde{G} / \tilde{K} $ be a non-compact inner symmetric space with $ \tilde{G} $ simply-connected.
Let $\tilde{U} / (\tilde{U} \cap \tilde{K}^{\mathbb{C}})$ denote the dual compact symmetric space. Then the space
$ \tilde{U} / ( \tilde{U} \cap \tilde{K}^{\mathbb{C}} )$ is inner and $\Lambda \tilde{G}_{\sigma}^{\mathbb{C}} =
\Lambda \tilde{U}_{\sigma}^{\mathbb{C}} $ holds.
Let $\mathcal{F}:\tilde{M} \rightarrow G/K = \tilde{G} / \tilde{K} $ be a harmonic map, where $ \tilde{M} $
is a simply-connected Riemann surface. Let {$F(z,\bar{z},\lambda)$ denote an extended frame
of $ \mathcal{F} $. For $F(z,\bar{z},\lambda)$ define
$ F_ {\tilde{U}} $ via the Iwasawa decomposition $F(z,\bar{z},\lambda)=F_{\tilde{U}}(z,\bar{z},\lambda) S_+(z,\bar{z},\lambda)$, with
$F_{\tilde{U}}(z,\bar{z},\lambda)\in \Lambda \tilde{U}_{\sigma}$, $S_+ (z,\bar{z},\lambda)\in \Lambda^+ \tilde{U}_{\sigma}^{\mathbb{C}}$.
Then \[\hbox{$\mathcal{F}_{\tilde{U}} :\tilde{M} \rightarrow \tilde{U} / \tilde{U} \cap \tilde{K}^{\mathbb{C}}$,
$\mathcal{F}_{\tilde{U}} \equiv F_{\tilde{U}}(z,\bar{z},\lambda) \mod \tilde{U} \cap \tilde{K}^{\mathbb{C}}$}\]} is a harmonic map {for each fixed $\lambda\in S^1$}.
Moreover, the harmonic maps $\mathcal{F}$ and $\mathcal{F}_{\tilde{U}}$ have the same normalized potential.
\end{theorem}
As a consequence of the theorem above we obtain (see also Theorem 3.6 of \cite{DoWa13})
\begin{corollary}\label{cor-finite}
Let {$F(z,\bar{z},\lambda)$ and $F_{\tilde{U}}(z,\bar{z},\lambda)$} be the extended frames defined as above. Then
\begin{enumerate}
\item
$F(z,\bar{z},\lambda)$ satisfies $(U1)$ if and only if $F_{\tilde{U}}(z,\bar{z},\lambda)$ satisfies $(U1)$;
\item
$F(z,\bar{z},\lambda)$ satisfies $(U2)$ if and only if $F_{\tilde{U}}(z,\bar{z},\lambda)$ satisfies $(U2)$.
\end{enumerate}
Therefore $\mathcal{F}$ is of finite uniton type if and only if $\mathcal{F}_{\tilde{U}}$ is of finite uniton type.
\end{corollary}
\begin{proof} Let {$F(z,\bar{z},\lambda): \tilde{M}\rightarrow \tilde{G} / \tilde{K}$ denote an extended frame of $\mathcal{F}$.
Then we have $F(z,\bar{z},\lambda)=F_{\tilde{U}}(z,\bar{z},\lambda)S_+(z,\bar{z},\lambda)$} as stated in Theorem \ref{thm-noncompact}. We need to verify
the properties (U1) and (U2) in Definition \ref{def-uni}.
(U2): If {$F(z,\bar{z},\lambda)$ is a Laurent polynomial in $\lambda$, then $F_{\tilde{U}}(z,\bar{z},\lambda)$ contains only finitely many
negative powers of $\lambda$ since $F_{\tilde U}(z,\bar{z},\lambda)=F(z,\bar{z},\lambda)(S_+(z,\bar{z},\lambda))^{-1}$. Since $F_{\tilde U}(z,\bar{z},\lambda)$ satisfies also a reality
condition, $F_{\tilde U}(z,\bar{z},\lambda)$ is a Laurent polynomial. The converse statement follows by interchanging $F(z,\bar{z},\lambda)$ and $F_{\tilde{U}}(z,\bar{z},\lambda)$.}
(U1): We distinguish two cases: The case of $M=S^2$ is trivial, but one needs to recall from Section 3.2 of \cite{DoWa12} that in this case extended frames have some singular points in general.
If $M$ is different from $S^2$, then we obtain
\[F(g.z,\overline{g.z},\lambda)=\chi(g,\lambda)F(z,\bar{z},\lambda) k(g,z,\bar{z}) \ \hbox{ for all }\ g\in\pi_1(M)\]
with $k(g,z,\bar{z})\in K$ and $\chi\in \Lambda {\tilde{G}_{\sigma}}$. Assume now that $F$ satisfies (U1).
Then, without loss of generality $\chi(g,\lambda)=e$ for all $g\in\pi_1(M)$.
Inserting $F=F_{\tilde{U}}S_+$ we obtain
\[F_{\tilde{U}}(g.z,\overline{g.z},\lambda)=F_{\tilde{U}}(z,\bar{z},\lambda)u(g, z,\bar{z}), u\in {\tilde{U}}\cap {\tilde{K}}^{\mathbb{C}},\]
where $u(g,z,\bar{z}) = S_+ (z, \bar{z},\lambda) k(g,z,\bar{z}) S_+ (g.z, \overline{g.z},\lambda)^{-1}.$
{Hence both $F_{\tilde{U}}$ and $\mathcal{F}_{\tilde U}$ satisfies $(U1)$.}
Assume now that $F_{\tilde{U}}$ satisfies $(U1).$
Hence $F_{\tilde{U}}(g.z,\overline{g.z},\lambda)=F_{\tilde{U}}(z,\bar{z},\lambda){u(g,z,\bar{z})}$
for all $g\in\pi_1(M)$
and with $u \in \Lambda^+U^\mathbb{C}_\sigma = \Lambda G^\mathbb{C}_\sigma$. Substituting $F_{\tilde{U}} = F (S_+)^{-1}$ from above we obtain
\[
F(g.z,\overline{g.z},\lambda)=F(z,\bar{z},\lambda) L_+(g,z,\bar{z},\lambda),
\]
where $L_+(g,z,\bar{z},\lambda) = S_+(z,\bar{z},\lambda)^{-1} u(g,z,\bar{z}) S_+(g.z,\overline{g.z},\lambda)$. But then $L_+(g,z,\bar{z},\lambda) = k(g,z,\bar{z}) \in K$ and $F$ satisfies $(U1)$.
Hence $F$ and then also $\mathcal{F}$ satisfy $(U1)$.
\end{proof}
Thus finding the normalized potentials of the finite uniton type harmonic maps $\mathcal{F}$ into the
non-compact inner symmetric space $G/K$ means finding the normalized potentials for the corresponding
finite uniton type harmonic maps $\mathcal{F}_{\tilde{U}}$ into the compact dual ${\tilde{U}}/({\tilde{U}}\cap {\tilde{K}}^{\mathbb{C}})$.
The latter task has been greatly simplified by \cite{BuGu}.\\
\subsection{Remarks on monodromy, dressing and some application to Willmore surfaces}
Above we have defined finite uniton harmonic maps by the conditions (U1) and (U2) of Definition \ref{def-uni}.
The condition (U1) is always satisfied if $M$ is simply connected (including $S^2$ as it was remarked
in $(2)$ of Remark \ref{sphere} that for the case of $M = S^2$, the usual loop group approach
applies, modulo a minor precaution).
Corollary \ref{cor-finite} yields immediately
\begin{proposition}\label{prop-trivial-monod} With the assumptions and the notations of Theorem \ref{thm-noncompact}
the following statements are equivalent:
\begin{enumerate}
\item $ \mathcal{F}$ has trivial monodromy {representation,}
\item $\mathcal{F}_{\tilde{U}}$ has trivial monodromy {representation.}
\end{enumerate}
\end{proposition}
\begin{corollary}\label{cor-finite-dress}
If $\mathcal{F}:M\rightarrow G/K$ is a harmonic map of finite uniton type and $h_+\in\Lambda^+G^{\mathbb{C}}_{\sigma}$.
Then the dressed harmonic map $h_+\sharp\mathcal{F}_{\lambda}$ is of finite uniton type on $M$.
\end{corollary}
\begin{proof} Condition (U1) for $h_+\sharp\mathcal{F}_{\lambda}$ follows from Theorem \ref{thm-dress}.
It is easy to verify that
$h_+F=\hat{F}\hat{W}_+$ implies that $\hat{F}$ only contains finitely many negative powers of $\lambda$ if $F$ does.
But $\hat{F}$ is real and the claim follows. Also note that if $ {F(z_0,\bar{z}_0,\lambda) = e}$ then one can assume
without loss of generality that also $\hat{F}(z_0,\bar{z}_0,\lambda) = {e}$ holds.
\end{proof}
\section{ {The work of Burstall and Guest} and the DPW formalism}
In this section, we will recall the work of Burstall and Guest on harmonic maps of finite uniton {number} needed in this paper and then translate it into the language of { the DPW method} \cite{DPW}. For more details on Burstall and Guest's work, we refer to \cite{BuGu} and \cite{Gu2002}. Note that, in their work, $G$ is assumed to be a connected, compact, semi-simple real Lie group with trivial center, {as it is in this section, }
and $\mathfrak{g}$ denotes its Lie algebra and $G^{\mathbb{C}}$ its complexification.
{We can assume w.l.g. that $G^\mathbb{C}$ is a semisimple simply-connected matrix Lie group and $G$ a subgroup of $G^\mathbb{C}$ \cite{Hoch}.} {Note also that \cite{BuGu} only considers inner symmetric spaces, as does this paper.}
\subsection{Review of extended solutions}
In this subsection, we compare/unify the notation used in \cite{Uh}, \cite{BuGu} and \cite{DPW}.
For a harmonic map $\mathbb{F}:\mathbb{D}\rightarrow G$, in \cite{Uh}, \cite{BuGu} ``extended solutions" are considered, while in \cite{DPW} always ``extended frames'' are used. In this subsection we will explain the relation between these methods. Here we include primarily the details which we will need to use.
For the convenience of the reader and to fix notation we start with a simple remark.
\subsubsection{Inner symmetric spaces and the modified Cartan embedding}
Consider the inner compact symmetric space $G/\hat{K}$ with inner involution $\sigma$, given by $\sigma(g) = h g h^{-1}$ and with $\hat{K} = Fix^\sigma (G)$. Note that $h^2 \in Center (G) = \{e\}. $ Then with $R_h(g)= gh$ we consider
the map
\[
\begin{tikzcd}[column sep=6mm,row sep=4mm]
G/\hat{K} \ar{r}{\mathfrak{C}} & G \ar{r}{R_h} & G \\
g \ar{r}{} & g \sigma(g)^{-1} = ghg^{-1} h^{-1} \ar{r}{} & \mathfrak{C}_h:=g h g^{-1}.
\end{tikzcd}
\]
In this way $G/\hat K$ is an isometric, totally geodesic submanifold of $G$ \cite{BuGu}, and ${\mathfrak{C}_h}$ will be called the ``modified Cartan embedding". Note that for outer symmetric spaces the above Cartan embedding does not apply directly.
\subsubsection{Extended frames for harmonic maps $\mathcal{F} : M \rightarrow G/\hat{K}$ and modified harmonic maps
$\mathfrak{C}_h\circ \mathcal{F}$}
Using the notation introduced above, consider a harmonic map $\mathcal{F}: M \rightarrow G/\hat{K}$.
By $F : \tilde{M} \rightarrow \Lambda G_{\sigma}$ we denote the extended frame of $\mathcal{F}$ which is normalized to $F(z_0, \bar{z}_0, \lambda) = e$ at some base point $z = z_0$ for all $\lambda \in S^1$.
The extended frame of a harmonic map $\mathcal{F}$ actually is for each fixed $\lambda$ the frame of the corresponding immersion $\mathcal{F}_\lambda$ of the
associated family of $\mathcal{F}$. Obviously, the twisting condition in our case means
\[{\sigma(\gamma)(\lambda) = h \gamma(-\lambda)h^{-1} = \gamma(\lambda)}\hbox{
for all
$\gamma \in \Lambda G_{\sigma}$.}\]
Next we consider the composition of the family of harmonic maps $\mathcal{F}_\lambda$ with the
modified Cartan embedding $\mathfrak{C}_h$. In our setting, since
$\mathcal{F}_\lambda =F( z, \bar{z}, \lambda)\mod \hat{K}$, this yields the
$\lambda-$dependent harmonic map $\mathfrak{F}^h_\lambda$ given by
\begin{equation} \label{harmonicrelation}
\mathfrak{F}^h_{\lambda}=
F( z, \bar{z}, \lambda) h F( z, \bar{z}, \lambda))^{-1}.
\end{equation}
Note that $\mathfrak{F}^h_\lambda $ is a $\lambda-$dependent harmonic map satisfying
$(\mathfrak{F}^h_\lambda )^2=e, $ where the square denotes the product in the group $G$.
Moreover, we also have $\mathfrak{F}^h_\lambda (z_0, \bar{z}_0,\lambda) = h.$ Harmonic maps into $G$ satisfying these two properties will be called ``modified harmonic maps".
\begin{theorem}
We retain the notation and the assumptions made just above. In particular, $z_0$ is a fixed basepoint in the Riemann surface $M$ Then
there is a bijection between harmonic maps $\mathcal{F} : M \rightarrow G/\hat{K}$ satisfying
$\mathcal{F}(z_0,\bar{z}_0,\lambda) = e\hat{K}$ and modified harmonic maps, i.e. harmonic maps
$\mathfrak{F}^h_{\lambda}: M \rightarrow G_h = \{ ghg^{-1}; g \in G \}$ satisfying
$(\mathfrak{F}^h_{\lambda})^2 = e$ and $\mathfrak{F}^h_{\lambda}(z_0, \bar{z}_0, \lambda) = h$ for all $\lambda \in \mathbb{C}^*$.
This relation is given by composition with the modified Cartan embedding (and its inverse respectively).
\end{theorem}
\begin{proof}
We have shown ``$\Longrightarrow$" above. Assume now we have a harmonic map
$\mathbb{F}: M \rightarrow {{G}_h}$ satisfying
$\mathbb{F}^2 = e$ and $\mathbb{F}(z_0, \bar{z}_0, \lambda) = h$ for all $\lambda \in \mathbb{C}^*$.
Since $\mathfrak{C}_h$ is an isometric diffeomorphism onto its image, we consider
(for all $\lambda \in \mathbb{C}^*$) the $\lambda$-dependent harmonic map
$\mathcal{F}_\lambda = (\mathfrak{C}_h)^{-1} \circ \mathbb{F}_\lambda : M \rightarrow G/\hat{K}$.
Clearly, then we have $\mathcal{F}(z_0, \bar z_0, \lambda) = e\hat{K}$. Moreover, by what was shown above, we now infer
$\mathbb{F}(z, \bar z, \lambda) = \mathfrak{F}^h_{\lambda} (z, \bar z, \lambda) =
F( z, \bar{z}, \lambda) h F( z, \bar{z}, \lambda))^{-1}$, where $F$ denotes the extended frame of $\mathcal{F}$.
Since $\mathcal{F} (z_0, \bar{z}_0, \lambda) = e$,
$h = F( z_0 \bar{z}_0, \lambda) h F( z_0, \bar{z}_0, \lambda))^{-1}= \mathfrak{F}^h_{\lambda} ( z_0, \bar{z}_0, \lambda)$ holds.
\end{proof}
\subsubsection{Extended solutions for harmonic maps into Lie groups}
We have shown, among other things, in the subsubsections above that it is essentially sufficient
for our purposes to consider harmonic maps into Lie groups $G$.
In this subsubsection we consider harmonic maps into Lie groups following the approach of
\cite{Uh} and \cite{BuGu}.
We start by relating the different loop parameters
used in \cite{Uh} , \cite{BuGu} and \cite{DPW} respectively to each other.
To begin with, we recall the definition of {\em extended solutions}
following Uhlenbeck \cite{Uh},\cite{BuGu}. Let $\mathbb{D} \subset \mathbb{C}$ be a
simply-connected domain and $\mathbb{F}: \mathbb{D}\ \rightarrow G$ a harmonic map.
Set
\[\mathbb{A}=\frac{1}{2} \mathbb{F}^{-1}\mathrm{d} \mathbb{F} =\mathbb{A}^{(1,0)}+\mathbb{A}^{(0,1)}.\]
Consider {for $\tilde{\lambda} \in \mathbb{C}^*$} the equations
\begin{equation}\label{eq-Uh1}
\left\{
\begin{split}
\partial_z \Phi \mathrm{d}z&=(1-\tilde{\lambda}^{-1})\Phi\mathbb{A}^{(1,0)},\\
\partial_{\bar{z}} \Phi \mathrm{d} \bar{z}&=(1-\tilde{\lambda})\Phi\mathbb{A}^{(0,1)}.\\
\end{split}
\right.
\end{equation}
with $\Phi: \mathbb{D} \rightarrow \Omega G$, where the corresponding loop parameter
is denoted here by $\tilde{\lambda}.$
Then, by Theorem 2.2 of \cite{Uh} (Theorem 1.1 of \cite{BuGu}),
there exists a solution $\Phi(z,\bar{z},\tilde{\lambda})$ to the above
equations such that
\begin{equation}
\Phi(z,\bar{z},\tilde{\lambda}=1)=e,~\ \mbox{and}
~ \Phi(z,\bar{z},\tilde{\lambda}=-1)= \mathbb{F} (z, \bar z)
\end{equation}
hold. This solution is unique up to multiplication by some
$\gamma \in \Omega G = \{ g \in \Lambda G^{\mathbb{C}}_\sigma |, g(\lambda = 1) = I \}$ satisfying $\gamma (-1) = e$. Such solutions $\Phi$ are said to be {\bf extended solutions}.
If we also have $\mathbb{F}(z_0)=e$ , we can also choose $\Phi(z_0,\bar{z}_0, \tilde\lambda)=e$.
Although the assumption $\mathbb{F}(z_0)=e$ was
used in \cite{Uh}, we will, as in \cite{BuGu}, not assume this, since it is not satisfied
in a large part of this section. The following statement is straightforward.
\begin{lemma}\label{lemma-es} \cite{BuGu} Let $\Phi(z,\bar{z}, \tilde{\lambda})$ be
{\em an
extended solution} of the harmonic map $\mathbb{F}:\mathbb{D}\rightarrow G$. Let $\gamma\in \Omega G$.
Then $\gamma(\lambda)\Phi(z,\bar{z},\tilde{\lambda})$ is an extended solution of the
harmonic map $\gamma(-1)\mathbb{F}(z,\bar{z})$.
\end{lemma}
\begin{remark}
Next we show how the ``DPW approach" without the basepoint assumption \cite{DPW} naturally leads to
Uhlenbeck's extended solutions \cite{Uh}.
We follow Section 9 of \cite{Do-Es} and consider the Lie group $G$ as the outer symmetric space
$G = (G \times G)/ \mathbb{D}elta,$
where the defining symmetry $\tilde{\sigma}$ is given by
$$\tilde{\sigma} (a,b) = ( b,a)$$ and we have $$\mathbb{D}elta = \{ (a,b) \in G \times G; a=b \}.$$
For the purposes of our loop group method it is necessary to consider the $G \times G-$loop group
$\Lambda(G \times G)$
twisted by $\tilde{\sigma}$. We thus consider the automorphism of the loop group
$\Lambda(G \times G) = \Lambda G \times \Lambda G$ given by
$$\hat{\tilde{\sigma}} ((a,b)) (\lambda) = \tilde{\sigma}( a(\lambda), b(\lambda)) =
(b(\lambda), a(\lambda)).$$
It is straightforward to verify that the twisted loop group $\Lambda(G \times G)_{\tilde{\sigma}}$
is given by
$$\Lambda(G \times G)_{\tilde{\sigma}} = \{ (g(-\lambda), g(\lambda)) ; g(\lambda) \in \Lambda G \}
\cong \Lambda G. $$
Let's consider now a harmonic map $\mathbb{F}: M \rightarrow G.$
Then the map
$\mathfrak{F}:M \rightarrow G \times G$, given by
$\mathfrak{F}(z, \bar{z}) = \left(\mathbb F(z,\bar{z}),e\right)$, is a global frame of $\mathbb{F}$.
Following \cite{DPW} one needs to decompose the Maurer-Cartan form
$\mathfrak{A} = \mathfrak{F}^{-1} \mathrm{d} \mathfrak{F}$ of $\mathfrak{F}$ into the eigenspaces of
$\tilde\sigma$ and to introduce the loop parameter $\lambda$.
One obtains (see \cite{Do-Es}, formula $(68)$):
\begin{equation}
\mathfrak{A}_{\lambda} = \left( \left(1+\lambda^{-1}\right) \mathbb A^{(1,0)} + \left(1+\lambda\right)\mathbb A^{(0,1)} ,\ \left(1-\lambda^{-1}\right) \mathbb A^{(1,0)} + \left(1-\lambda\right) \mathbb A^{(0,1)} \right).
\end{equation}
\end{remark}
\begin{theorem}
Let $G$ be a connected, compact or non-compact, semi-simple real Lie group with trivial center.
Let $\mathbb{F}:\mathbb{D}\rightarrow G$ be a harmonic map. Then, when representing $G$ as the
symmetric space $G = (G \times G)/\mathbb{D}elta,$
any extended frame $\mathfrak{F} : \mathbb{D} \rightarrow \Lambda(G \times G)_{\tilde\sigma}$
of $\mathbb{F}$ satisfying
$\mathfrak{F}(z, \bar{z}, \lambda = 1) = \left(\mathbb F(z,\bar{z}),e\right)$
is given by a pair of functions,
\[\mathfrak{F}(z, \bar{z},\lambda) = ( \Phi (z, \bar{z},-\lambda), \Phi (z, \bar{z},\lambda)),\]
where the matrix function $ \Phi (z, \bar{z},\lambda)$ is an extended solution for $\mathbb{F}$ in the sense of Uhlenbeck \cite{Uh} as introduced above.
\end{theorem}
\begin{proof}
Since the two components of $\mathfrak{A} $ only differ by a minus sign in $\lambda$, any solution to
the equation $\mathfrak{A}_\lambda = \mathfrak{F}(z, \bar z, \lambda)^{-1} \mathrm{d} \mathfrak{F}(z, \bar z, \lambda)$
is of the form $ \mathfrak{F}(z, \bar z, \lambda) =
(B(-\lambda) \Psi(z, \bar z, - \lambda) , B(\lambda) \Psi(z, \bar z, \lambda))$,
where $\Psi$ solves the equations \eqref{eq-Uh1}. Moreover, we can assume w.l.g. that $\Psi$ satisfies the two conditions for extended solutions stated above for $\lambda = \pm1$.
Now
$\mathfrak{F}(z, \bar{z}, \lambda = 1) = \left(\mathbb F(z,\bar{z}),e\right)$ implies
{$B(1) = B(1) \Psi (z, \bar{z},\lambda= 1) =e$ and $B( -1)\Psi(z, \bar z, -1) =
B(-1) \mathbb{F}(z, \bar z) = \mathbb{F}(z, \bar z),$
whence $B(-1) = e$ follows.
Therefore, $\Phi (z, \bar z, \lambda) = B(\lambda) \Psi(z, \bar z, \lambda)$}
yields the claim.
\end{proof}
Note that in this theorem no normalization is required. Moreover, the loop parameter used in \cite{Uh} is the same as the one used in \cite{DPW}. However, the matrix functions $B(\lambda)$ and $\Phi (z, \bar z, \lambda)$
are not uniquely determined which causes the DPW procedure to yield quite arbitrary potentials, not easily permitting any converse construction procedure.
\subsubsection{Extended solutions and extended frames for harmonic maps into symmetric spaces} \label{414}
Consider as before a harmonic map $\mathcal{F}: M \rightarrow G/\hat{K}$ into a symmetric space
with inner involution $\sigma$, given by $\sigma(g) = h g h^{-1}$ and with $\hat{K} = Fix^\sigma (G)$.
As above we consider the modified harmonic map $\mathbb{F}: \rightarrow G$ given by
$$\mathbb{F}(z, \bar z, \lambda) = \mathfrak{F}^h_{\lambda} (z, \bar z, \lambda) =
F( z, \bar{z}, \lambda) h F( z, \bar{z}, \lambda))^{-1}.$$
For this $\lambda-$family of harmonic maps $\mathbb{F}_\lambda$ we compute
\begin{equation*}
\begin{split}
\mathbb{A}&=\frac{1}{2}\mathbb{F}_{\lambda}^{-1}\mathrm{d}\mathbb{F}_{\lambda} \\
\ &=\frac{1}{2}\left(F(z,\bar{z},\lambda)hF(z,\bar{z},\lambda)^{-1}\right)^{-1}\mathrm{d}\left(F(z,\bar{z},\lambda)hF(z,\bar{z},\lambda)^{-1}\right)\\
\ &=\frac{1}{2}F(z,\bar{z},\lambda) h^{-1}\alpha_{\lambda} h F(z,\bar{z},\lambda)^{-1}-\frac{1}{2}\mathrm{d}F(z,\bar{z},\lambda)F(z,\bar{z},\lambda)^{-1}\\
\ &=\frac{1}{2}F(z,\bar{z},\lambda)\left(\alpha_{-\lambda}-\alpha_{\lambda}\right)F(z,\bar{z},\lambda)^{-1}\\
\ &=-F(z,\bar{z},\lambda)\left(\lambda^{-1}\alpha_{\mathfrak{p}}'+\lambda\alpha_{\mathfrak{p}}''\right)F(z,\bar{z},\lambda)^{-1}.\\
\end{split}
\end{equation*}
Following Uhlenbeck's approach we need to introduce a new ``loop parameter" $\tilde{\lambda}$ now and consider Uhlenbeck's differential equation \eqref{eq-Uh1} for
$\Phi(z,\bar{z},\lambda,\tilde{\lambda})$ with conditions for $\tilde{\lambda} = \pm 1$:
\begin{equation}\label{eq-Uh2}
\left\{
\begin{split}
\partial_z \Phi \mathrm{d}z&=-\Phi(1-\tilde{\lambda}^{-1})\lambda^{-1}F(z,\bar{z},\lambda) \alpha_{\mathfrak{p}}' F(z,\bar{z},\lambda)^{-1}\\
\partial_{\bar{z}} \Phi\mathrm{d}\bar z&=-\Phi(1-\tilde{\lambda})\lambda F(z,\bar{z},\lambda) \alpha_{\mathfrak{p}}'' F(z,\bar{z},\lambda)^{-1}\\
\end{split}
\right.\end{equation}
From (\ref{eq-Uh2}) it is natural to consider the Maurer-Cartan form $\widetilde{\mathbb{A}} $ of $\Phi (z,\bar{z},\lambda,\tilde{\lambda})F(z,\bar{z},\lambda)$. One obtains:
\begin{equation}
\begin{split}
\widetilde{\mathbb{A}}&=(\Phi F)^{-1} \mathrm{d} ( \Phi F)\\
&=F^{-1}\mathrm{d} F+F^{-1}(\Phi^{-1}\mathrm{d} \Phi) F\\
&=\alpha_{\lambda}-F^{-1}\Phi^{-1}\left(\Phi(1-\tilde{\lambda}^{-1})\lambda^{-1}F \alpha_{\mathfrak{p}}' F ^{-1}+\Phi(1-\tilde{\lambda})\lambda F \alpha_{\mathfrak{p}}'' F ^{-1}\right)F\\
&=\alpha_{\lambda}- \left( (1-\tilde{\lambda}^{-1})\lambda^{-1} \alpha_{\mathfrak{p}}' +(1-\tilde{\lambda})\lambda \alpha_{\mathfrak{p}}''\right)\\
&=\lambda^{-1} \alpha_{\mathfrak{p}}'+\alpha_{\mathfrak k}+\lambda \alpha_{\mathfrak{p}}''- \left( (1-\tilde{\lambda}^{-1})\lambda^{-1} \alpha_{\mathfrak{p}}' +(1-\tilde{\lambda})\lambda \alpha_{\mathfrak{p}}''\right)\\
&=\tilde{\lambda}^{-1}\lambda^{-1} \alpha_{\mathfrak{p}}'+\alpha_{\mathfrak k}+\tilde{\lambda}\lambda \alpha_{\mathfrak{p}}'' \\
&=\alpha_{\tilde{\lambda}\lambda}. \\
\end{split}
\end{equation}
From this we derive immediately the relation
\begin{equation}\label{eq-lawson}
F(z,\bar{z}, \lambda\tilde{\lambda})=A(\lambda, \tilde{\lambda}) \Phi (z,\bar{z},\lambda,\tilde{\lambda})F(z,\bar{z}, \lambda).
\end{equation}
Substituting here $z = z_0$ we derive, in view of the normalization of $F$ at $z = z_0$:
\begin{equation}\label{eq-A}
A(\lambda, \tilde{\lambda}) = \Phi (z_0 ,{\bar{z}}_0,\lambda,\tilde{\lambda})^{-1}.
\end{equation}
In particular, setting $\lambda=1$ in \eqref{eq-lawson} we obtain
\begin{equation}
F(z,\bar{z}, \tilde{\lambda})=A(1, \tilde{\lambda}) \Phi (z,\bar{z},1,\tilde{\lambda})F(z,\bar{z},1).
\end{equation}
Setting $\tilde\lambda=-1$ in \eqref{eq-lawson} we obtain (by using the twisting condition for $F$):
\begin{equation}
A(\lambda, -1)=F(z,\bar{z},-\lambda)\left( \Phi (z,\bar{z},\lambda,-1)F(z,\bar{z},\lambda)\right)^{-1}=\textcolor{violet}{h}.
\end{equation}
Hence
\[\Phi (z, \bar{z},1,-1)={A(1,-1)^{-1} } F(z,\bar{z}, -1)F(z,\bar{z}, 1)^{-1}=F(z,\bar{z}, 1)hF(z,\bar{z}, 1)^{-1}=\mathbb{F}(z,\bar z,1).\]
In summary we obtain (by setting $\lambda = 1$ and replacing $\tilde{\lambda}$ by $\lambda$):
\begin{corollary}\label{cor-Phi-F}
The extended solution $\Phi$, and the $\sigma-$twisted extended frame $F$ satisfy
\begin{equation}\label{eq-lawson2}
\Phi (z, \bar{z},1,\lambda) = A(1, \lambda)^{-1}F(z,\bar{z}, \lambda)F(z,\bar{z}, 1)^{-1}.
\end{equation}
In particular, $\Phi (z, \bar{z},1,\lambda)$ is contained in the based loop group $\Omega G$. Moreover,
for $\lambda = -1$ we obtain the harmonic map $ \mathbb{F}(z,\bar z,1) = \mathbb{F}(z,\bar z) $.
\end{corollary}
\subsection{Finite {uniton type \`{a}}
la Burstall-Guest for harmonic maps into compact Lie groups}
Let us recall that in Definition \ref{def-uni} we have given the definition of harmonic maps of finite uniton type into Lie groups. Now we want to define the notion of a ``finite uniton number''.
{It has been introduced by Uhlenbeck \cite{Uh} for $U(n)$ and by Burstall-Guest
\cite{BuGu} for a general compact real Lie group $G$.}
\begin{definition} \label{def-f.u.}
Let $\mathbb{F}:M \rightarrow G$ be a harmonic map into a real Lie group $G$.
Assume there exists a global extended
solution $\Phi(z,
\bar{z},\tilde\lambda):M\rightarrow \Lambda G^{\mathbb{C}}$ (i.e., $\mathbb{F}$ has trivial monodromy).
We say that $\mathbb{F}$ has {\it finite uniton number $k$} if
(see \eqref{eq-alg-loop} for the definition of $ \Omega^k_{alg} G $)
\begin{equation}\Phi(M)\subset \Omega^k_{alg} G ,\
\hbox{ and } \Phi(M)\nsubseteq \Omega^{k-1}_{alg} G .\end{equation}
In this case we write $r(\Phi)=k$ and the minimal uniton number of $\mathbb{F}$ is defined
as \[r(\mathbb{F}):=min\{r(\gamma Ad(\Phi))| \gamma\in \Omega_{alg} Ad G \}.\]
\end{definition}
{
The notion of finite uniton number harmonic maps is related to extended solutions. In this paper we usually use extended framings of harmonic maps and read off the notion of ``finite uniton type" from these extended frames. It is important to this paper that these two notions describe the same class of harmonic maps.}
\begin{proposition} \label{typeequivnumber}
$\mathcal F$ is a harmonic map of finite uniton type in $G/K$ if and only if
{$\mathbb{F} = \mathfrak{C} \circ \mathcal{F} \cdot h = \mathfrak{C}_h \circ \mathcal{F}$ given in section 4.1.1, where $\mathfrak{C}_h$ is the modified Cartan }embedding of $G/K$ into $G$, is a harmonic map of finite uniton number.
\end{proposition}
\begin{proof}
$``\mathbb{R}ightarrow"$ By definition, $F(z,\bar z, \lambda\tilde\lambda)$ is a Laurent polynomial in $\lambda \tilde{\lambda}$ and hence also in $\tilde\lambda$. As a consequence of this and
\eqref{eq-lawson} we have that \[A(\lambda,\tilde\lambda)\Phi(z,\bar z, \lambda,\tilde\lambda)=F(z,\bar{z}, \lambda\tilde\lambda)F(z,\bar{z}, \lambda)^{-1}\]
is also a Laurent polynomial in $\tilde\lambda$. Hence, by the above definition
$\mathbb{F}(z,\bar z,\lambda)$ is of finite uniton number.
$``\Leftarrow"$ By definition, there exists some $\Phi(z,\bar z, \lambda,\tilde\lambda)$ being a Laurent polynomial in $\tilde\lambda$ and hence by \eqref{eq-A} $A(\lambda,\tilde\lambda)=\Phi (z_* ,{\bar{z}}_*,\lambda,\tilde{\lambda})^{-1}$ is also a Laurent polynomial in $\tilde\lambda$. As a consequence, by
\eqref{eq-lawson} we have that $F(z,\bar{z}, \lambda\tilde\lambda)$ is also a Laurent polynomial in $\tilde\lambda$ and hence also in $\lambda$, i.e., $\mathcal F$ is of finite uniton type.
Consequently, $\mathbb{F}(z,\bar z,\lambda)$ is of finite uniton number.
\end{proof}
\begin{remark}\
\begin{enumerate}
\item One of the main goals of this paper is a characterization of the normalized potentials
of all finite uniton type harmonic maps into $G/K$.
The potential for such a harmonic map is the same as the potential
for the induced harmonic map into $G/\hat{K}$,
where $\hat{K} = G^{\sigma}$, since the different Cartan maps have the same images $\mathfrak{C}(gK)=\mathfrak{C}(g\hat{K})$ for all $g\in G$. We will therefore always assume in this section that
$\hat{K} = G^{\sigma}$. In this case the Cartan map $\mathfrak{C}$ actually is an embedding.
\item {If $M$ is simply connected, then a global extended solution
$\Phi(z,\bar{z},\lambda):M\rightarrow \Omega G $ always exists, including the case $M=S^2$ (see Theorem 2.2 of \cite{Uh}, also see \cite{Segal}, Theorem 1.1 of \cite{BuGu})}.
This is in contrast to the case of extended
frames, in which case we have explained above that on $S^2$ the extended frame needs to have (two)
singularities due to the topology of $S^2$. In general, the extended solution may not exist globally on $M$ if $M$ is not simply connected.
\end{enumerate}
\end{remark}
Now let us turn to the Burstall-Guest theory for harmonic maps into Lie groups of {finite uniton number}.
Let $\mathrm{T}\subset G $ be a maximal {torus of $G$} with $\mathfrak{t}$ the Lie algebra of $\mathrm{T}$.
We can identify all the homomorphisms from $S^1$ to $\mathrm{T}$ with the integer lattice $\mathcal{I}:=(2\pi)^{-1}\exp^{-1}(e)\cap\mathfrak{t}$ in $\mathfrak{t}$ via the map
\begin{equation}
\begin{array}{llllll}
\mathcal{I}=(2\pi)^{-1}\exp^{-1}(e)\cap\mathfrak{t}&\longrightarrow \hbox{\{homomorpisms from $S^1$ to $\mathrm{T}$\}}, \\
\ \ \ \ \ \ \ \ \ \ \ \ \xi \ \ \ &\longmapsto \gamma_{\xi},\\
\end{array}
\end{equation}
where $\gamma_{\xi}:S^1\rightarrow T$ is defined by
\begin{equation}\gamma_{\xi}(\lambda):=\exp(t\xi),\ \ \hbox{ for all }\ \ \lambda=e^{it}\in S^1.
\end{equation}
Let {$\mathcal {C}_0$} be a fundamental Weyl chamber of $\mathfrak{t}$. Set $\mathcal{I}'=\mathcal {C}_0\cap \mathcal{I}$. Then $\mathcal{I}'$ parameterizes the
conjugacy classes of homomorphisms $S^1\rightarrow G$.
Let $\mathbb{D}elta$ be the set of roots of $\mathfrak{g}^{\mathbb{C}}$. We have the root space decomposition
$\mathfrak{g}^{\mathbb{C}}=\mathfrak{t}^{\mathbb{C}}\oplus (\underset{\theta\in\mathbb{D}elta}{\oplus}\mathfrak{g}_{\theta} ).$ Decompose $\mathbb{D}elta$ as $\mathbb{D}elta=\mathbb{D}elta^-\cup\mathbb{D}elta^+$ according to {$\mathcal {C}_0$}.
Let $\theta_1,\cdots,\theta_l\in \mathbb{D}elta^{+}$ be the simple roots. We denote by $\xi_1,\cdots,\xi_l\in \mathfrak{t}$ the basis of $\mathfrak{t}$ which is dual to
$\theta_1,\cdots,\theta_l$ in the sense that $\theta_j(\xi_k)=\sqrt{-1}\delta_{jk}$.
\begin{definition} $($ {p.555 of \cite{BuGu}}$)$
An element $\xi$ in $\mathcal{I}'\backslash\{0\}$ is called a {\em canonical} element, if $\xi=\xi_{j_1}+\cdots+\xi_{j_k}$ with $\xi_{j_1},\cdots,\xi_{j_k}\in \{\xi_1,\cdots,\xi_l\}$ pairwise different. In other words, for every simple root $\theta_j$, we have that $\theta_{j}(\xi)$ only attains the values $0$ or $\sqrt{-1}$.\end{definition}
For $\theta\in\mathbb{D}elta$ and $X\in\mathfrak{g}_{\theta}$ we obtain
$$ad\xi X=\theta(\xi)X\ \ \hbox{ and }\ \theta(\xi)\in\sqrt{-1}\mathbb{Z}.$$
Let $\mathfrak{g}^{\xi}_{j}$ be the $\sqrt{-1}\cdot j-\hbox{eigenspace}$ of $\hbox{ad}\xi$. Then
\begin{equation} \label{defgjxi}
\mathfrak{g}^{\xi}_j=\underset{\theta(\xi)=\sqrt{-1} j}{\oplus}\mathfrak{g}_{\theta}, \hbox{ and } \mathfrak{g}^{\mathbb{C}}=\underset{j}{\oplus}\ \mathfrak{g}^{\xi}_j.
\end{equation}
We define the {\em height} of $\xi$ as the non-negative integer \begin{equation}r(\xi)=\hbox{ max}\{j|\ \mathfrak{g}^{\xi}_j\neq0\ \}.\end{equation}
\begin{lemma}\label{lemma-xi} (Lemma 3.4 of \cite{BuGu}) Let $\xi=\sum_{i=1}^kn_{j_i}\xi_{j_i}\in \mathcal{I}'$ with $n_{j_i}>0$. Set $\xi_{can}=\sum_{i=1}^k\xi_{j_i}$. Then we have
\begin{equation}
\mathfrak{g}^{\xi}_0=\mathfrak{g}^{\xi_{can}}_0,\ \ \ ~~ \sum_{0\leq j\leq r(\xi)-1} \mathfrak{g}^{\xi}_{j+1}=\sum_{0\leq j\leq r(\xi_{can})-1} \mathfrak{g}^{\xi_{can}}_{j+1}.
\end{equation}
\end{lemma}
Set
\begin{equation}\left\{
\begin{array}{llllll}
&\mathfrak{f}^{\xi}_j&:= &\underset{k\leq j}{\oplus}\mathfrak{g}^{\xi}_k,\\
&(\mathfrak{f}^{\xi}_j)^{\perp}&:=&\underset{j< k \leq r(\xi)}{\oplus} \mathfrak{g}^{\xi}_k, \\
&\mathfrak{u}^0_{\xi}&:=&\underset{0\leq j <r(\xi)} {\oplus}\lambda^{j}(\mathfrak{f}^{\xi}_j)^{\perp}\in \Lambda^+\mathfrak{g}^\mathbb{C}_{\sigma}.\\
\end{array}\right.
\end{equation}
{Now we can state (in our notation) some of the main results of \cite{BuGu}.}
\begin{theorem}\label{thm-finite-uniton0} (Theorem 1.2, Theorem 4.5, and {p.560} of \cite{BuGu})
Assume $G$ is connected, compact, and semisimple with trivial center.
\begin{enumerate}
\item Let $\Phi:M\rightarrow \Omega^k_{alg}G$ be an extended solution of finite uniton number. Then there exists some canonical $\xi\in \mathcal{I}'$, some $\gamma\in \Omega_{alg}G$, and some discrete subset $D'\subset M$, such that on $M\setminus D'$, {the following Iwasawa decomposition of $\exp C \cdot \gamma_{\xi}$ holds:}
\begin{equation}\label{eq-finite}
{\gamma\Phi=\exp C \cdot \gamma_{\xi} \cdot (\Phi^+_{\xi})^{-1},}
\end{equation}
where $C:M\rightarrow \mathfrak{u}^0_{\xi}$ is a (vector-valued) {\em meromorphic {function}} with poles in $D'$
and {$\Phi^+_{\xi}: M \backslash D' \rightarrow \Lambda^+G^\mathbb{C}_{\sigma}$. Moreover, the Maurer-Cartan form of $\exp C$ is given by}
\begin{equation}\label{eq-C}
{(\exp C)^{-1}\mathrm{d}(\exp C)=\sum_{0\leq j\leq r(\xi)-1}\lambda^j A_j'\mathrm{d} z,}
\end{equation}
{with $A_j':M\rightarrow \mathfrak{g}^{\xi}_{j+1}$ being meromorphic functions with poles contained in $D'$ for each $j$.}
\item
Conversely, let $\xi\in \mathcal{I}'$ be a canonical element and $C:M \rightarrow \mathfrak{u}^0_{\xi}$ a meromorphic function satisfying \eqref{eq-C}. Let $D'\subset M$ be the set of poles of $C$,
{and $\Phi=(\exp C \cdot \gamma_{\xi}) \cdot (\Phi^+_{\xi})^{-1} $ be an Iwasawa decomposition of $ \exp C \cdot \gamma_{\xi}$. Then $\Phi$ is an extended solution of finite uniton number on $M$.}
\end{enumerate}
\end{theorem}
\begin{remark}\
\begin{enumerate}
\item
{Let $z_0\in M\setminus D'$ be some point. Then $C_0=C(z_0)\in \mathfrak{u}^0_{\xi}$
and $\Phi(z_0,\bar{z}_0,\lambda)\in\Omega^k_{alg}G$ gives some initial condition.}
\item By Lemma \ref{lemma-es}, since $\Phi(z,\bar{z},\lambda)$ is an extended solution of a harmonic map
$\Phi(z,\bar{z},-1)$, then $\gamma\Phi(z,\bar{z},\lambda)$ is an extended solution of the harmonic map
$\gamma(-1)\Phi(z,\bar{z},-1)$, which is congruent to the harmonic map $\Phi(z,\bar{z},-1)$ up to the group action of $G$ from the left. {So for the harmonic map $\gamma(-1)\Phi(z,\bar{z},-1)$ we have an extended solution for which without loss of generality we can always assume $\gamma=e$ in $(1)$ of the theorem above.}
\item
On page 560 of \cite{BuGu}, the elements $A'_j$ are defined so that they take values in $\mathfrak{f}^{\xi}_{j+1}=\sum_{k\leq j+1}\mathfrak{g}^{\xi}_{k}$ which is due to the harmonicity of $\mathbb{F}$. In view of the restriction on $C$ to take values in $\mathfrak{u}_{\xi}^0$, the computations on page 561 of \cite{BuGu} for $(\exp C)^{-1}(\exp C)_z$ imply that
$A'_j$ takes values in $\sum_{k\geq j+1}\mathfrak{g}^{\xi}_{k}$. Therefore $A'_j \in \mathfrak{g}^{\xi}_{j+1}$ as stated above.
\end{enumerate}
\end{remark}
\subsection{Finite uniton {type \`{a} la} Burstall-Guest for harmonic maps into symmetric spaces} \label{f.u.alaBuGu}
{Consider a harmonic map $\mathcal{F}: M \rightarrow G/K$ from $M$ into the inner symmetric space
$G/\hat K$. As stated in \cite{BuGu}, the inner symmetric space $G/\hat{K}$, can be embedded into $G$ via the modified Cartan map $\mathfrak{C}_h$ such that $\mathfrak{C}_h(G/\hat{K})$ is a connected component of
$\sqrt{e}$,\ where $\sqrt{e}:=\{g\in G| g^2=e \}.$}
Let $$ {\mathbb{T}}:\Omega G\rightarrow \Omega G,\ \ {\mathbb{T}}(\gamma)(\lambda)=\gamma(-\lambda)\gamma(-1)^{-1}$$
be an involution of $\Omega G$ (see Section 2.2) with the fixed point set
\[(\Omega G)_{ {\mathbb{T}}}=\{\gamma\in\Omega G| {\mathbb{T}}(\gamma)=\gamma\}.\]
{ We give a version of Proposition 5.2 of \cite{BuGu} adapted to our approach.}
\begin{proposition}\label{eq-check-xi} { Let $\mathcal F$ be a harmonic map of finite uniton number with an extended frame $F(z,\bar z, \lambda)$. Let $\Phi(z,\bar z,1,\lambda)=A(1, {\lambda})^{-1} F(z,\bar{z}, \lambda)F(z,\bar{z}, 1)^{-1}$ be an extended solution for the harmonic map $\mathbb{F}=\mathfrak{C}_h\circ\mathcal F$ as described in
(\ref{eq-lawson2}) of Corollary \ref{cor-Phi-F}.} Then there exists some $\tilde\xi \in \mathfrak{t}$ satisfying
$\exp( \pi \tilde{\xi}) = h$, where $\mathfrak{t}$ denotes the Lie algebra of {the maximal torus
$\mathrm{T}$ in $G$.}
Setting $\gamma_{\tilde\xi} (\lambda) = \exp (t \tilde\xi)$ for $\lambda = e^{it}$ and $\tilde{\gamma} (\lambda) = \gamma_{\tilde\xi} (\lambda) A(1, \lambda)$ we obtain
$\tilde{\gamma}(\lambda)\in\Omega G$ , $\tilde{\gamma}(-1)=e$ and
\begin{equation}\label{eq-TP}
{\mathbb{T}}(\tilde{\gamma}(\lambda)\Phi(z,\bar z,1,\lambda))=\tilde{\gamma}(\lambda)\Phi(z,\bar z,1,\lambda).
\end{equation}
Moreover, $\tilde{\gamma}(\lambda)\Phi$ is also an extended solution for
$\mathbb F$.
Furthermore, if $\Phi(z,\bar z,1,\lambda)$ takes {values} in $\Omega _{alg}G$, then we also have $\tilde{\gamma}(\lambda)\in \Omega _{alg}G$ and $\tilde{\gamma}(\lambda)\Phi(z,\bar z,1,\lambda)$ takes {values} in $(\Omega _{alg}G)_{ {\mathbb{T}}}$.
\end{proposition}
\begin{proof} The proof follows closely the one of \cite{BuGu}. First we note that $h$ is contained in the maximal torus. This implies the existence of $\tilde\xi$.
Next we verify \eqref{eq-TP}. Since \[\gamma_{\tilde\xi}(-\lambda)=\exp((\pi+t)\tilde \xi)=
\gamma_{\tilde\xi}(\lambda)h\ \hbox{ and }\ F(z,\bar{z}, -\lambda)h=hF(z,\bar{z}, \lambda),\] we have
\[\begin{split}
{\mathbb{T}}(\tilde{\gamma}(\lambda)\Phi(z,\bar z,1,\lambda))
&= {\mathbb{T}}\left(\gamma_{\tilde\xi}(\lambda)F(z,\bar{z}, \lambda)F(z,\bar{z}, 1)^{-1}\right)\\
&=\gamma_{\tilde\xi}(-\lambda)F(z,\bar{z}, -\lambda)F(z,\bar{z}, 1)^{-1}F(z,\bar{z}, 1)F(z,\bar{z}, -1)^{-1}\gamma_{\tilde\xi}(-1)\\
&=\gamma_{\check\xi}(\lambda)hF(z,\bar{z}, -\lambda)F(z,\bar{z}, -1)^{-1}h\\
&=\gamma_{\check\xi}(\lambda)F(z,\bar{z}, \lambda)hF(z,\bar{z}, 1)^{-1}\\
&=\tilde{\gamma}(\lambda)\Phi(z,\bar z,1,\lambda).
\end{split}\]
\end{proof}
{By Proposition \ref{eq-check-xi} we can assume without loss of generality that $\Phi$ has the form \begin{equation}\label{eq-Phi-F}
\Phi=\gamma_{\tilde\xi}(\lambda)F(z,\bar{z}, \lambda)F(z,\bar{z}, 1)^{-1}.\end{equation}}
With this $\Phi$ we obtain:
\begin{theorem}\label{thm-finite-uniton1}(Proposition 5.3, Theorem 5.4 {and p.567 of } \cite{BuGu})
Assume that $G$ is connected, compact, and semisimple with trivial center.
\begin{enumerate}
\item Let $\Phi:M\rightarrow (\Omega^k_{alg}G)_\mathbb{T}$ be an extended solution for some harmonic map
{ $\mathcal F$ of finite uniton number related by \eqref{eq-Phi-F}.} Then there exists some canonical element $\xi$ in $\mathcal{I}'$, some $\gamma \in \Omega_{alg}G$ {satisfying $\gamma(-1) = e,$ }and some discrete subset $D'$ of $M$ such that on $M\setminus D'$,
\begin{equation}
{ \gamma\Phi =\exp C \cdot \gamma_{\xi} \cdot (\Phi^+_{\xi})^{-1},}
\end{equation}
where {$C:M\rightarrow (\mathfrak{u}^0_{\xi})_\mathbb{T}$ is a {\em meromorphic map} on $M$ with poles in $D'$} and $$(\mathfrak{u}^0_{\xi})_\mathbb{T}=\bigoplus_{0\leq 2j <r(\xi)}\lambda^{2j}(\mathfrak{f}^{\xi}_{2j})^{\perp}.$$
Moreover, $\xi$ satisfies \begin{equation}G/\hat{K} \cong\{g(\exp \pi \xi)g^{-1}| g\in G\}.
\end{equation}
{Note that both $\Phi$ and $\gamma \Phi$ are extended solutions of $\mathcal F$.}
\item
Conversely, let $\xi\in \mathcal{I}'$ be a canonical element.
{Assume that $C:M \rightarrow (\mathfrak{u}^0_{\xi})_\mathbb{T}$ is a meromorphic function} such that
{\begin{equation}
(\exp C)^{-1}\mathrm{d}(\exp C)=\sum_{0\leq 2j\leq r(\xi)-1}\lambda^j A_{2j}'\mathrm{d} z,\
\end{equation}
with $A_{2j}':M \rightarrow \mathfrak{g}^{\xi}_{2j+1}$ being meromorprhic,
$\ 0\leq 2j\leq r(\xi)-1$.}
Let $D'\subset M$ be the set of poles of $C$, then
{$\Phi_{M\setminus D'}=\gamma_{\xi}^{-1} \cdot \exp C \cdot \gamma_{\xi} \cdot (\Phi^+_{\xi})^{-1} $} is an extended solution { for some harmonic map of finite uniton number $F:M\setminus D' \rightarrow G/\hat{K}\cong\{g(\exp \pi \xi)g^{-1}| g\in G\}\subset G$.}
\end{enumerate}\end{theorem}
\begin{remark} A generalization of Burstall and Guest's theory for outer compact symmetric spaces has been published in \cite{Esch-Ma-Qu}.
\end{remark}
\subsection{The Burstall-Guest theory in relation to standard DPW theory} \label{BuGu<->DPW}
Using the above theorems, we can derive the normalized potential of harmonic maps of finite uniton type, showing that they are meromorphic 1-forms taking values in a fixed nilpotent Lie algebra.
{Note that these results have been explained in Appendix B of \cite{BuGu} and Theorem 1.11 of \cite{Gu2002}. Here we rewrite them in terms of our language as well as a proof for later applications and for the convenience of readers. We also would like to point out that the work of Burstall and Guest does not consider initial conditions, e.g. for extended frames. So in general the value of an extended frame will
be different from $e$ at some fixed basepoint $z_0$. We will show below in Theorem 4.20 below that also the standard DPW theory with prescribed initial condition at some fixed base point produces all harmonic maps of finite uniton number as well. All statements in the theorems below can be derived from the work of Burstall and Guest. But the presentation uses substantially the DPW method. Therefore, for the convenience of the reader, we include a proof.}
\begin{theorem} \label{thm-finite-uniton2}We retain the {the notation and the assumptions of Theorem \ref{thm-finite-uniton0} and Theorem \ref{thm-finite-uniton1} as needed.}
\begin{enumerate}
\item
Let $\mathbb{F}:M\rightarrow G$ be a harmonic map of finite uniton number {with extended solution $\Phi(z,\bar z,\lambda)$ as stated in Theorem \ref{thm-finite-uniton0}. Then ${\Phi_-}:=\gamma_{\xi}^{-1}\cdot \exp C \cdot \gamma_{\xi}=\gamma_{\xi}^{-1}\cdot\Phi\cdot\Phi_+$} \textcolor{violet}{has} a Maurer-Cartan form
{\begin{equation}\label{eq-eta} {\mathbb A_-}:= {\Phi_-}^{-1}\mathrm{d} {\Phi_-}=\lambda^{-1}\sum_{0\leq j\leq r(\xi)-1} A_j'\mathrm{d} z, \end{equation}
where each $A_j':M\rightarrow \mathfrak{g}^{\xi}_{j+1}$ is a meromorphic function {on $M$} with poles in $D'$.}
Moreover, at some base point $z_0\in M\setminus D'$ we have
\begin{equation}\label{eq-initial1}
{\Phi_-}(z_0)= {\Phi_{-0}}:= {\gamma_{\xi}^{-1} \cdot \exp C(z_0) \cdot \gamma_{\xi}} \in \Lambda^- G^{\mathbb{C}},\ C(z_0)\in\mathfrak{u}^0_{\xi}.
\end{equation}
Conversely, given $ {\mathbb A_-}$ which takes values in $\lambda^{-1}\cdot\sum_{0\leq j\leq r(\xi)-1}\mathfrak{g}^{\xi}_{j+1}$ and an initial condition of $ {\Phi_-}$ of the form \eqref{eq-initial1}, {assume that on $ M$,
there exists a global meromorphic solution $ {\Phi_-}$ (with poles in $\mathbb{D}'$)} \textcolor{violet}{to}
\begin{equation}\label{eq-A-1} {\Phi_-}^{-1}\mathrm{d} {\Phi_-}= {\mathbb A_-}, \ ~~~ {\Phi_-}(z_0)= {\Phi_{-0}}= {\gamma_{\xi}^{-1} \cdot \exp C_0 \cdot \gamma_{\xi}\ } \in \Lambda^- G^{\mathbb{C}}\hbox{ with } C_0\in\mathfrak{u}^0_{\xi}.\end{equation}
{The Iwasawa decomposition of $\gamma_{\xi}\Phi_-$ gives a harmonic map $\mathbb{F}$ of finite uniton number in $G$.}
{The above two procedures are inverse to each other when the initial conditions match.}
\item Let $\mathcal{F}:M\rightarrow G/\hat{K}$ be a harmonic map of finite uniton number
{with extended frame $F(z,\bar z,\lambda)$ based at $z_0 \in M$ with initial value $e$.} Embed $G/\hat{K}$ into $G$ as totally geodesic submanifold via the { modified} Cartan embedding. Then there exists some canonical $\xi\in \mathcal{I}'$,
some discrete subset $D'\subset M$, such that
\begin{equation}
G/\hat{K}\cong\{g(\exp \pi \xi)g^{-1}| g\in G\},
\end{equation}
and that $ {F_-= \gamma_{\xi}^{-1} \cdot \exp C \cdot \gamma_{\xi}}$ is a meromorphic extended frame of $\mathcal{F}$ with the normalized potential having the form
{\begin{equation}\eta=F_-^{-1}\mathrm{d}F_-=\lambda^{-1}\sum_{0\leq 2j\leq r(\xi)-1} A_{2j}'\mathrm{d} z, \end{equation}
where $A_{2j}':M\rightarrow \mathfrak{g}^{\xi}_{2j+1}$ is a meromorphic function on $M$ with poles in $D'$ for each $j$.} And at the base point $z_0$
\begin{equation}\label{eq-initial2}F_-(z_0)=F_{-0}:= {\gamma_{\xi}^{-1} \cdot \exp C(z_0) \cdot \gamma_{\xi} } \ in\ \Lambda^- G^{\mathbb{C}},\ C(z_0)\in(\mathfrak{u}^0_{\xi})_T.
\end{equation}
Conversely, given a meromorphic normalized potential $\eta$ which takes values in $\lambda^{-1}\cdot\sum_{0\leq 2j\leq r(\xi)-1}\mathfrak{g}^{\xi}_{2j+1}$ and an initial condition
{$F_{-0}$ of $F_-$ of } the form \eqref{eq-initial2},
{assume that on $M$ there exists a global solution $F_-$ (with poles in $\mathbb{D}'$)} {to}
\begin{equation}\label{eq-initial3}F_-^{-1}\mathrm{d}F_-=\eta, \ ~~~F_-(z_0)=F_{-0}. \end{equation}
{The Iwasawa decomposition of $F_-(z,\lambda)$ gives the extended frame of a harmonic} map of finite uniton number into $ \{g(\exp \pi \xi)g^{-1}| g\in G\}\cong G/\hat{K} $.
{The above two procedures are inverse to each other when the initial conditions match.}
\end{enumerate}
\end{theorem}
{Note that part (1) of Theorem \ref{thm-finite-uniton2} is essentially Proposition B1 of Appendix B of \cite{BuGu}.}
\begin{proof}
(1) First we note that by page 557 in \cite{BuGu},
\[ \gamma_{\xi}^{-1}X\gamma_{\xi}=\lambda^{-j-1}X,~ \hbox{ for any element } ~X\in\mathfrak{g}^{\xi}_{j+1}.\]
Together with the definition of $C$ { in Theorem \ref{thm-finite-uniton0},} we have
{ ${\gamma_{\xi}}^{-1} \cdot \exp C \cdot \gamma_{\xi} \in\Lambda^-G^{\mathbb{C}}$.}
Now consider the Maurer-Cartan form of {$\gamma_{\xi}^{-1} \cdot \exp C \cdot \gamma_{\xi}$.} We have
\begin{equation*}\begin{split}( {\gamma_{\xi}^{-1} \cdot \exp C \cdot \gamma_{\xi})^{-1}\cdot }\mathrm{d}( {\gamma_{\xi}^{-1} \cdot \exp C \cdot \gamma_{\xi}})&=( {\gamma_{\xi}^{-1} \cdot
\exp C \cdot \gamma_{\xi}})^{-1}( {\gamma_{\xi}^{-1} \cdot \exp C \cdot \gamma_{\xi}})_z\mathrm{d}z \\
&=
\gamma_{\xi}^{-1}\left( \exp C^{-1}(\exp C)_z\right) \gamma_{\xi}\mathrm{d}z,\end{split}\end{equation*}
since $\gamma_{\xi}$ is independent of $z$.
By \eqref{eq-C} in Theorem \ref{thm-finite-uniton0},
\begin{equation*}\label{eq-exp-C} (\exp C)^{-1}(\exp C)_z=\sum_{j=0}^{r(\xi)-1}\lambda^j A'_j \ \ \hbox{ with }\ \ \ A'_j:M \rightarrow \mathfrak{g}^{\xi}_{j+1},\ 0\leq j\leq r(\xi)-1.\end{equation*}
So
\begin{equation*}( {\gamma_{\xi}^{-1} \cdot \exp C \cdot \gamma_{\xi}})^{-1}
( {\gamma_{\xi}^{-1}\cdot \exp C \cdot \gamma_{\xi}})_z=\gamma_{\xi}^{-1} \left( \sum_{1\leq j\leq r(\xi)-1}\lambda^j A'_j\right) \gamma_{\xi} =\sum_{1\leq j\leq r(\xi)-1} \lambda^j(\gamma_{\xi}^{-1} A'_j \gamma_{\xi}).
\end{equation*}
By page 557 in \cite{BuGu}, for any element $X\in\mathfrak{g}^{\xi}_{j+1}$, $ \gamma_{\xi}^{-1}X\gamma_{\xi}=\lambda^{-j-1}X$. Since $ A'_j $ takes values in $\mathfrak{g}^{\xi}_{j+1}$, we have
\[\gamma_{\xi}^{-1} A'_j \gamma_{\xi}=\lambda^{-j-1}A'_j.\]
As a consequence,
\[( {\gamma_{\xi}^{-1}\cdot \exp C \cdot \gamma_{\xi}})^{-1}( {\gamma_{\xi}^{-1} \cdot
\exp C \cdot \gamma_{\xi}})_z=\sum_{1\leq j\leq r(\xi)-1} \lambda^j(\gamma_{\xi}^{-1} A'_j \gamma_{\xi})= \lambda^{-1}\sum_{1\leq j\leq r(\xi)-1} A'_j .\]
Conversely, let $\Phi_-$ be a solution to \eqref{eq-A-1} as assumed. Then $\Phi_-$ takes values in
$\Omega_{alg}G$, since $\lambda\mathbb A_-(\frac{\partial}{\partial z})$ takes values in a nilpotent Lie algebra.
{Letting $ \gamma_{\xi}\Phi_- =\Phi (\Phi_+)^{-1}$ } be the Iwasawa decomposition of $\gamma_{\xi}\Phi_-$. It hence produces a harmonic map of finite uniton number.
(2) First one needs to restrict the above results to the case $A'_{2j+1}=0$ for all $j$ by Theorem \ref{thm-finite-uniton1}. By $(1)$ of Theorem \ref{thm-finite-uniton1} we can choose the extended solution associated to the harmonic map $\mathcal{F}$ such that
$\Phi=\gamma_{\tilde\xi}(\lambda)F(z,\bar{z}, \lambda)h F(z,\bar{z}, 1)^{-1}$ holds. We see that in this case the Iwasawa decomposition of $\Phi$ yields
$ \Phi_-=\Phi\Phi_+$, whence
{ \[\Phi_-=\Phi\Phi_+=\gamma_{\tilde\xi}(\lambda)F(z,\bar{z}, \lambda)hF(z,\bar{z}, 1)^{-1}\Phi_+,\]
where the second equality follows from (\ref{eq-Phi-F}).}
{Consider $\tilde F(z,\bar z,\lambda):=\gamma_{\xi}(\lambda)^{-1}\gamma_{\tilde\xi}(\lambda)F(z,\bar{z}, \lambda)$. It is also an extended frame of the harmonic map $\mathcal F(z,\bar z,\lambda=1)$, but at the basepoint $z_0$ it might have an initial condition different from the initial condition of $F(z,\bar z, \lambda)$: $\tilde F(z_0,\bar z_0,\lambda)=\gamma_{\xi}(\lambda)^{-1}\gamma_{\tilde\xi}(\lambda)F(z_0,\bar{z}_0, \lambda)$.} Also note that $\tilde F(z,\bar z,1)=F(z,\bar{z}, 1)$. We can w.l.g. replace $F(z,\bar z,\lambda)$ by $\tilde F(z,\bar z,\lambda)$, since we do not assume any special initial conditions. Then we have
\[\Phi_-=F(z,\bar{z}, \lambda)h F(z,\bar{z}, 1)^{-1}\Phi_+=F_-(z,\lambda)F_+(z,\bar{z}, \lambda)h F(z,\bar{z}, 1)^{-1}\Phi_+.\]
So $F_-(z,\lambda)=\Phi_-$. By (1) of this Theorem, we finish the proof of (2).
\end{proof}
\begin{remark}\
\begin{enumerate}
\item Note that $\mathbb{A}_-$ in \eqref{eq-initial1} is not the usual normalized potential, since the harmonic map is mapped into the Lie group $G$ instead of $G/K$ and we do not have the initial condition $e$. Also by the discussion in Section 4.1.1 and for convenience we still call it ``normalized potential". For more relations between $\mathbb{A}_-$ and the real normalized potential (embedding $G$ to construct a harmonic map into $G\times G/G$), we also refer the readers to \cite{Do-Es}.
\item The initial condition {$\Phi_{-0}$ and $F_{-0}$ respectively} of Theorem \ref{thm-finite-uniton2} can be removed by using dressing (see Theorem 1.11 of \cite{Gu2002}). For instance, assume that
$\hat{F}_-^{-1}\mathrm{d}\hat{F}_-=\eta, \ {\hat{F}_-(z_0,\lambda)}=e$. Then $F_-=F_{-0}\hat{F}_-$. By Iwasawa splitting we have
\[F_-=FF_+,\ \ \hat{F}_-=\hat{F}\hat{F}_+.\ \]
Assume that $F_{-0}=\gamma_0\gamma_+$ with $\gamma_0\in\Lambda G,$ $\gamma_+\in\Lambda^+ G^{\mathbb{C}}$. Therefore we have
\[FF_+=\gamma_0\gamma_+\hat{F}\hat{F}_+.\]
As a consequence, we obtain
\begin{equation*} \gamma_+\hat{F}=\gamma_0^{-1}FF_+\hat{F}_+^{-1}.\end{equation*}
Hence
\begin{equation}\gamma_+\sharp\hat{F}=\gamma_0^{-1}F.\end{equation}
So up to a rigid motion $\gamma_0^{-1}$, $F$ is the dressing of $\hat{F}$ by $\gamma_+$ (compare with Corollary \ref{cor-finite-dress}).
\end{enumerate}
\end{remark}
\subsection{On the initial conditions of normalized meromorphic frames: compact case}
In the last theorem we have given a precise relation between certain ``meromorphic potentials" and harmonic maps of finite uniton type. For this one needs very specific initial conditions and { very specific dressing matrices} respectively.
{ In the case of the last theorem the initial conditions occurring are very complicated and very difficult to produce. So it is important and useful to show that all harmonic maps of finite uniton type into compact inner symmetric spaces can be determined by the data as given, but with initial condition $e$, which is the goal of this subsection}.
To this end, we need some preparations.
First, for an arbitrary element {$\xi \in \mathcal{I}'$} we have the following decompositions
{(see e.g. \eqref{defgjxi}):}
\begin{equation}\label{eq-pq-decom-g}
{ \mathfrak{g}^{\mathbb{C}}=\sum_j \mathfrak{g}^{\xi}_j=\left(\sum_{j\geq0} \mathfrak{g}^{\xi}_j\right)\oplus\left(\sum_{j<0} \mathfrak{g}^{\xi}_j\right)=\mathfrak{pr}\oplus\mathfrak{q}.}
\end{equation}
On the Lie group level, let $\mathds{PR}$ be the {complex connected} Lie subgroup of $G^{\mathbb{C}}$ with Lie algebra
$\mathfrak{pr}$ and let $\mathds{Q}$ denote the {complex connected} Lie subgroup of $G^{\mathbb{C}}$ with Lie algebra $\mathfrak{q}$.
Let $\mathfrak{W}$ denote the Weyl group of $G$. Then we have the decomposition
\begin{equation}\label{eq-pq-decom-G}
G^{\mathbb{C}}=\bigcup_{\omega\in \mathfrak{W_{\xi}}}\mathds{PR}\cdot\omega\cdot\mathds{Q}.
\end{equation}
where $\mathfrak{W_{\xi}}$ is {some quotient of $\mathfrak{W}$.} This follows from Corollary 3.2.3 of \cite{Do-Gr-Sz}.
\begin{theorem} \label{thm-PR-Q-decomposition}
{The set $\mathds{PR} \cdot \mathds{Q} \subset G^{\mathbb{C}}$ obtained by pointwise multiplication is open in $G^\mathbb{C}$ and the pointwise multiplication map $ \mathds{PR}\times\mathds{Q} \longrightarrow \mathds{PR}\cdot\mathds{Q}$ is biholomorphic.}
\end{theorem}
\begin{proof} By Theorem 2.4.1 (a) of \cite{Do-Gr-Sz}, $ \mathds{PR}\cdot\mathds{Q}$ is open in $G^{\mathbb{C}}$. By Theorem 2.4.1 (b) of \cite{Do-Gr-Sz},
$ \mathds{PR}\times\mathds{Q}\longrightarrow \mathds{PR}\cdot\mathds{Q}$ is a holomorphic diffeomorphism.
\end{proof}
We also need the following { lemma:}
\begin{lemma} \label{lemma-} {Let $\xi \in \mathcal{I}'$ so that $h=\exp(\pi\xi)$. Then we obtain}
\begin{equation}\label{eq-pr-p}
\mathfrak{pr}\cap \mathfrak{p}^{\mathbb{C}}= \sum_{j\geq0}g^{\xi}_{2j+1}.
\end{equation}
\end{lemma}
\begin{proof}
{Recall that $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$
with $\mathfrak{k}=Lie(K)$, and $\mathfrak{g}^{\mathbb{C}}=\mathfrak{k}^{\mathbb{C}}\oplus\mathfrak{p}^{\mathbb{C}}$.}
We have that $\mathfrak{k}=\{~X\in\mathfrak{g}~|~hX=Xh\ \},~~\mathfrak{k}^{\mathbb{C}}=\{~X\in\mathfrak{g}^{\mathbb{C}}~|~hX=Xh\ \}.$
For any element $X$ in $g^{\xi}_{2j}$,
\[hXh^{-1}=\exp( \pi\xi)\cdot X\cdot\exp(-\pi\xi)=\exp(\pi \cdot ad\xi) X=e^{2 j\pi \sqrt{-1} }X=X.\]
Similarly, for any element $X$ in $g^{\xi}_{2j+1}$,
\[hXh^{-1}=\exp( \pi\xi)\cdot X\cdot\exp(-\pi\xi)=\exp( \pi\cdot ad\xi) X=e^{(2j+1)\pi \sqrt{-1}}X=-X.\]
\end{proof}
{Next we will apply Theorem \ref{thm-finite-uniton2} and Theorem \ref{thm-PR-Q-decomposition} to show that for a harmonic map $\mathcal F$ an extended frame with initial condition $e$, Theorem \ref{thm-finite-uniton2} still holds.}
\begin{theorem} \label{thm-finite-uniton-in}
Let G be a connected, semisimple, compact Lie group with trivial center. Let $\mathcal{F}:M\rightarrow G/\hat{K}$ be a harmonic map of
finite uniton type into the compact inner symmetric space $G/\hat{K}$. {Let $z_0 \in M$ and $F(z, \bar z, \lambda)$ an extended frame of $\mathcal{F}$ satisfying
$F(z_0, \bar z_0, \lambda) = e$ for all $\lambda \in S^1$. Then there exists some canonical $\xi_{can}\in \mathcal{I}'$,
some discrete subset $D'\subset M$, such that
$\mathfrak{C}_h(G/\hat{K})=\{ghg^{-1}| g\in G\}$ with $h=\exp (\pi \xi_{can})$,
and the normalized potential of $\mathcal{F}$ has the form
\begin{equation}\eta=F_-^{-1}\mathrm{d}F_-=\lambda^{-1}\sum_{0\leq 2j\leq r(\xi_{can})-1} A_{2j}'\mathrm{d}z, \end{equation}}
{where $A_{2j}':M\rightarrow \mathfrak{g}^{\xi_{can}}_{2j+1}$ is a meromorphic function with poles in $D'$ for each $j$}. In particular, the normalized potential of $ \mathcal{F}$ is contained in a nilpotent {Lie algebra
and it is invariant} under the fundamental group of $M$.
\end{theorem}
\begin{proof} {
By Proposition \ref{eq-check-xi}, there exists some $\tilde\xi\in\mathcal I'$ such that
$\Phi=\gamma_{\tilde\xi} F(z,\bar{z}, \lambda)F(z,\bar{z}, 1)^{-1}$ is an extended solution of $\mathbb F(z,\bar z,1)=F(z, \bar z, 1)hF(z, \bar z, 1)^{-1}$ which is ${\mathbb{T}}-$invariant and satisfies $\exp(\pi\tilde\xi)=h$. Then by Proposition 4.1 of \cite{BuGu}, there exists some $\hat\xi\in\mathcal I'$ which may not be canonical, and some discrete subset $D'\subset M$ such that on $M\backslash D'$ we have
\[\gamma^{-1}_{\hat\xi}\Phi=\gamma^{-1}_{\hat\xi} \cdot \exp C \cdot \gamma_{\hat\xi}\cdot W_+(z,\bar{z},\lambda)=\hat{F}_- (z,\lambda) W_+(z,\bar{z},\lambda), \text{where } \hat{F}_-(z,\lambda)= \gamma^{-1}_{\hat\xi}
\cdot \exp C \cdot \gamma_{\hat\xi},\]
with $C:M\rightarrow(\mathfrak{u}^0_{\hat\xi})_{\mathbb{T}}$ being meromorphic with poles in $D'$ and
{$W_+: M \rightarrow \Lambda^{+}G^{\mathbb{C}}$.} Note that {$\gamma_{\hat\xi}\in(\Omega_{alg}G)_{\mathbb{T}}$}
since both $\Phi$ and $\gamma_{\tilde\xi} F(z,\bar{z}, \lambda)F(z,\bar{z}, 1)^{-1}$ are contained in $(\Omega_{alg}G)_{\mathbb{T}}$. In particular, $\exp(\pi\hat\xi)=h$ and both of $\gamma_{\tilde\xi}^{-1}$ and $\gamma_{\hat\xi}$ take values in $\mathds{PR}$, {where we define $\mathds{PR}$ by $\hat\xi$}. }
{Let \[\hat{F}(z,\bar{z},\lambda)=\gamma_{\hat\xi}^{-1}\gamma_{\tilde\xi}F(z,\bar{z},\lambda).\]
Since all of $\gamma_{\xi}$, $\gamma_{\tilde\xi}$ and $F(z,\bar{z},\lambda)$ are $\sigma-$twisted, we see that $\hat{F}(z,\lambda)$ is $\sigma-$twisted and hence is an extended frame of $\mathcal F$ with different initial condition {$\hat{F}(z_0,\bar{z}_0,\lambda)=\gamma_{\hat\xi}(\lambda)^{-1}\gamma_{\tilde\xi}(\lambda)$.}
Moreover, since $ \hat{F}_-(z,\lambda)= {\gamma^{-1}_{\hat\xi} \cdot \exp C \cdot \gamma_{\hat\xi}}$ takes values in $\Lambda^-G^{\mathbb C}_{\sigma}$, we have the Birkhoff decomposition of $\hat F(z,\bar{z},\lambda)$:
\[\hat{F}=\gamma^{-1}_{\xi}\Phi(z,\bar{z},\lambda) F(z,\bar{z},1)=\hat{F}_-(z,\lambda) \hat{ V}_+(z,\bar{z},\lambda) \ \hbox{ with }\ \hat{ V}_+(z,\bar{z},\lambda)=W_+(z,\bar{z},\lambda)F(z,\bar{z},1).\]
Now near $z_0$, we also have the Birkhoff decomposition of $F(z,\bar{z},\lambda)$:
\[F(z,\bar{z},\lambda)=F_-(z,\lambda)V_+
=\gamma_{\tilde\xi}^{-1}\gamma_{\hat\xi}\hat{F}(z,\bar{z},\lambda)
=\gamma_{\tilde\xi}^{-1}\gamma_{\hat\xi}\hat{F}_-(z,\lambda)\hat{ V}_+(z,\bar{z},\lambda).\]
Decompose $\hat{ V}_+(z,\bar{z},\lambda)$ according to \eqref{eq-pq-decom-G}:
\[\hat{ V}_+(z,\bar{z},\lambda)=\mathbf{R}\omega \mathbf{Q}~~ \hbox{ with }~ \mathbf{R}\in\mathds{PR} \hbox{ and } \mathbf{Q}\in\mathds{Q}.\]
By Theorem \ref{thm-PR-Q-decomposition}, since $\hat V_+$ is holomorphic in $\lambda$ for all $\lambda\in\mathbb C$, $\mathbf{R}$ and $\mathbf{Q}$ are also holomorphic in $\lambda$ for all $\lambda\in\mathbb C$, i.e., $\mathbf{R}, \mathbf{Q}\in\Lambda^+G^{\mathbb{C}}$.}
{Since $F(z,\bar{z},\lambda)\rightarrow e$ as $z\rightarrow z_0$, we obtain \[\gamma_{\tilde\xi}^{-1}\gamma_{\hat\xi}\hat{F}_-(z,\lambda) \mathbf{R}\omega \mathbf{Q}\rightarrow e, \hbox{ if $z\rightarrow z_0$}.\] Note that all of
$\gamma_{\tilde\xi}^{-1}$, $\gamma_{\hat\xi}$ and $\hat{F}_-(z,\lambda)$ take values in $\mathds{PR}$. Since $\mathds{PR}\cdot\mathds{Q}$ is open and $e\in\mathds{PR}\cdot\mathds{Q}$, $\omega=e$ near $z_0$. As a consequence, when $z\rightarrow z_0$, $\gamma_{\tilde\xi}^{-1}\gamma_{\hat\xi}\hat{F}_-(z,\lambda) \mathbf{R}\rightarrow e$ and $\mathbf{Q}\rightarrow e$. Moreover, considering $F_-$, we obtain
\[F_-=(\gamma_{\tilde\xi}^{-1}\gamma_{\hat\xi}\hat{F}_-(z,\lambda) \mathbf{R})_-.\]
Since $\mathbf{R}\in\mathds {PR}$ and all of
$\gamma_{\tilde\xi}^{-1}$, $\gamma_{\hat\xi}$ and $\hat{F}_-(z,\lambda)$ also take values in $\mathds{PR}$, we see that $F_-$ also takes values in $\mathds{PR}$.
Consider \[\eta_-=F_-^{-1}\mathrm{d}F_-=\lambda^{-1}\eta_{-1}\mathrm{d}z.\] Since $F_-$ takes values in $\mathds{PR}$, $\eta_{-1}$ takes values in $\mathfrak{pr}$. On the other hand, $\eta_{-1}$ also takes values in $\mathfrak{p}^{\mathbb{C}}$. In a sum, by \eqref{eq-pr-p}, $\eta_{-1}$ takes values in \[\mathfrak{pr}\cap\mathfrak{p}^{\mathbb{C}}=\sum_{0\leq 2j\leq r(\hat\xi)-1} \mathfrak{g}^{\hat\xi}_{2j+1}.\] Let $\xi_{can}$ be the canonical element derived from $\hat\xi$ as in Lemma \ref{lemma-xi}. By Lemma \ref{lemma-xi} we have
\[\sum_{0\leq j\leq r(\hat\xi)-1} \mathfrak{g}^{\hat\xi}_{j+1}=\sum_{0\leq j\leq r(\xi_{can})-1} \mathfrak{g}^{\xi_{can}}_{j+1}.\]
From the proof of Theorem 5.4 of \cite{BuGu}, we also obtain
\[ \sum_{0\leq 2j\leq r(\hat\xi)-1} \mathfrak{g}^{\hat\xi}_{2j+1}=\sum_{0\leq 2j\leq r(\xi_{can})-1} \mathfrak{g}^{\xi_{can}}_{2j+1}.\]
So $\eta_{-1}$ takes values in $\sum_{0\leq 2j\leq r(\xi_{can})-1} \mathfrak{g}^{\xi_{can}}_{2j+1}$, which is contained in the same nilpotential Lie subalgebra as $\hat\eta_{-1}=\hat F_{-}^{-1}(\hat F_{-})_{z}$. Since $F(z_0,\bar{z}_0,\lambda)=e$, we have $F_{-}(z_0,\lambda)=e$ as well. }
\end{proof}
\section{Harmonic maps of finite uniton type into non-compact inner symmetric spaces}
{In the last section we have shown how one can relate the construction schemes of Burstall-Guest and DPW respectively of harmonic maps into compact inner symmetric spaces into each other. In this section, we will show that the DPW interpretation of the theory of Burstall and Guest \cite{BuGu} also holds for harmonic maps of finite uniton type into non-compact inner symmetric spaces. We will also review briefly the application of this theory to the coarse} classification of Willmore surfaces \cite{Wang-2}.
\subsection{ {The case of non-compact inner symmetric spaces}}
{We will apply the results of Section 3.3. Let $G/K$ is a non-compact inner symmetric space} and $U$ a maximal compact subgroup of the complexification $G^\mathbb{C}$ of $G$ which is left invariant by the natural real form involution of $G$ and the extension of $\sigma$ to $G^\mathbb{C}$. Combining Theorem \ref{thm-noncompact}, Theorem \ref{thm-finite-uniton2}, {Theorem \ref{thm-finite-uniton-in}} and Corollary \ref{cor-finite}, we obtain
\begin{theorem}\label{thm-finite-uniton-n-com}
Let $\mathcal{F}: \tilde{M}\rightarrow G/K$ be a harmonic map of finite uniton type, and
${\mathcal{F}_ {U}:} \tilde{M} \rightarrow U/(U\cap K^{\mathbb{C}})$ the compact dual harmonic map of $\mathcal{F}$, with base point $z_0\in\tilde{M}$ {such that $\mathcal{F}_{z=z_0}=eK$ and $(\mathcal{F}_{U})_{z=z_0}=e(U\cap K^{\mathbb{C}})$.} {Then the normalized potential $\eta_-$ and the normalized extended framing $F_-$ derived from Theorem \ref{thm-finite-uniton-in} for
$\mathcal{F}_U$, provides also a normalized potential $\eta_-$ and a normalized extended framing $F_-$ of $\mathcal{F}$, with initial condition $F_-(z_0,\lambda)=e$.} {In particular, all harmonic maps of finite uniton type $\mathcal{F}: \tilde{M}\rightarrow G/K$ can be obtained in this way.}
\end{theorem}
{\begin{remark}\
\begin{enumerate}
\item By Theorem \ref{thm-finite-uniton-n-com}, on the (nilpotent) normalized potential level, one can classify all harmonic maps $\mathcal{F}: \tilde{M}\rightarrow G/K$ of finite uniton type by classifying all harmonic maps of finite uniton type from $\tilde{M}$ to $U/(U\cap K^{\mathbb{C}})$.
\item Note that the DPW method yields a 1-1 relation between normalized potentials
and harmonic maps if one chooses a base point and an initial condition $e$ for all relevant extended frames.
\end{enumerate}
\end{remark}}
\subsection{Nilpotent normalized potentials of Willmore surfaces of finite uniton type}
We will end this paper with a {brief view} of a coarse classification and a construction of new Willmore surfaces of finite uniton type, in the spirit of Section 4. To be concrete, we give a coarse classification of Willmore two-spheres in $S^{n+2}$ by classifying all the possible nilpotent Lie sub-algebras related to the corresponding harmonic conformal Gauss maps, see Theorem 3.1, Theorem 3.3 of \cite{Wang-1}. This classification indicates that Willmore two spheres may inherit more complicated and new geometric structures. Moreover, by concrete computations of Iwasawa decompositions, we construct new Willmore two-spheres (Section 3.1 and Section 3.2 of \cite{Wang-3}).
To state the coarse classification and constructions, we first recall that a Willmore surface $y:M\rightarrow S^{n+2}$ is globally related to a harmonic conformal Gauss map $\mathcal{F}:M\rightarrow SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ satisfying some isotropy condition (\cite{Bryant1984}, \cite{Ejiri1988}, \cite{DoWa11}, \cite{Wang-1}). Here
$SO^+(1,n+3)=SO(1,n+3)_0$ is the connected subgroup of
\begin{equation*}SO(1,n+3):=\{A\in Mat(n+4,\mathbb{C})\ |\ A^tI_{1,n+3}A=I_{1,n+3}, \det A=1,\ A=\bar{A}
\},
\end{equation*}
and the subgroup $K=SO^+(1,3)\times SO(n)\subset SO^+(1,n+3)$ is defined by the involution
\begin{equation}\begin{array}{ll}
\sigma: SO^+(1,n+3)&\rightarrow SO^+(1,n+3)\\
\ \ \ \ \ \ \ A &\mapsto DAD^{-1}.
\end{array}\end{equation}
with $D=diag\{-I_4,I_n\}.$ Moreover
\begin{equation*} \mathfrak{so}(1,n+3):=\{A\in Mat(n+4,\mathbb{R})\ |\ A^tI_{1,n+3}+I_{1,n+3}A\}.
\end{equation*}
Let $\mathfrak{k}$ be the Lie algebra of $SO^+(1,3)\times SO(n)\subset SO^+(1,n+3)$ and $\mathfrak{so}(1,n+3)=\mathfrak{k}\oplus\mathfrak{p}$. Then
\[\mathfrak{k}=\left\{\left(
\begin{array}{cc}
A_1 & 0 \\
0 & A_2 \\
\end{array}
\right)| A_1\in Mat(4,\mathbb{R}),\ A_2\in Mat(n,\mathbb{R}),\ A_1^tI_{1,3}+I_{1,3}A_1=0,\ A_2^t+A_2=0
\right\}\]
and
\[\mathfrak{p}=\left\{\left(
\begin{array}{cc}
0 & B_1 \\
-B_1^tI_{1,3} &0 \\
\end{array}
\right)| B_1\in Mat(4\times n,\mathbb{R})
\right\}.\]
{It is straightforward to see that the compact dual of $SO^+(1,n+3)$ and $SO^+(1,3)\times SO(n)$ are $SO(n+4)$ and $SO(4)\times SO(n)$ respectively.}
Following \cite{DoWa11} that we call a conformally harmonic map $\mathcal{F}:M\rightarrow SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ a strongly conformally \footnote{ {Note that in \cite{BW} the notion of ``strongly conformal" has been used in another sense.}} harmonic map if the $\mathfrak{p}^{\mathbb{C}}-$part of its Maurer-Cartan form,
\[\left(
\begin{array}{cc}
0 & {B}_1 \\
-{B}^{t}_1I_{1,3} & 0 \\
\end{array}
\right)\mathrm{d}z,\]
satisfies the isotropy condition
\begin{equation}\label{eq-isotropic}
B_1^tI_{1,3}B_1=0.
\end{equation}
Now assume $n+4=2m$ so that $SO^+(2m)/SO^+(1,3)\times SO(2m)$ is an inner symmetric space.
\begin{theorem}(Theorem 3.1 of \cite{Wang-1})
Let $\mathcal{F}:M\rightarrow SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ be a finite-uniton type strongly conformally harmonic map, with $n+4=2m$. Let
$$\eta=\lambda^{-1}\left(
\begin{array}{cc}
0 & \hat{B}_1 \\
-\hat{B}^{t}_1I_{1,3} & 0 \\
\end{array}
\right)\mathrm{d}z $$
be the normalized potential of $\mathcal{F}$, with $$\hat{B}_1=(\hat{B}_{11},\cdots,\hat{B}_{1,m-2}),\ \hat{B}_{1j}=(\mathrm{v}_{j },\hat{ \mathrm{v}}_{j })\in Mat(4\times 2,\mathbb{C}).$$
Then up to some conjugation by a constant matrix, every $\hat{B}_{j1}$ of $\hat{B}_1$ has one of the two forms:
\begin{equation}(i) \ \mathrm{v}_{j}= \left(
\begin{array}{ccccccc}
h_{1j} \\
h_{1j} \\
h_{3j} \\
ih_{3j} \\
\end{array}
\right),\ \hat{ \mathrm{v}}_{j } =
\left(
\begin{array}{ccccccc}
\hat{h}_{1j} \\
\hat{h}_{1j} \\
\hat{h}_{3j} \\
i\hat{h}_{3j} \\
\end{array}
\right);\
(ii) \ \mathrm{v}_{j}= \left(
\begin{array}{ccccccc}
h_{1j} \\
h_{2j} \\
h_{3j} \\
h_{4j} \\
\end{array}
\right),\ \ \hat{ \mathrm{v}}_{j } = i \mathrm{v}_{j}= \left(
\begin{array}{ccccccc}
ih_{1j} \\
ih_{2j} \\
ih_{3j} \\
ih_{4j} \\
\end{array}
\right).
\end{equation}
And all of $\{\mathrm{v}_j,\ \hat{ \mathrm{v}}_{j }\}$ satisfy the equations
$$\mathrm{v}_j^tI_{1,3}\mathrm{v}_l=\mathrm{v}_j^tI_{1,3}\hat{\mathrm{v}}_l=\hat{\mathrm{v}}_j^tI_{1,3}\hat{\mathrm{v}}_l=0, \ j,l=1,\cdots,m-2.$$
In other words, there are $m-1$ types of normalized potentials with $\hat{B}_{1}$ satisfying $\hat{B}_{1 }^tI_{1,3}\hat{B}_{1 }=0$, namely those being of one of the following $m-1$ forms (up to some conjugation):
$(1)$ (all pairs are of type (i))
\begin{equation}
\hat{B}_{1}= \left(
\begin{array}{ccccccc}
h_{11} & \hat{h}_{11} & h_{12} & \hat{h}_{12} &\cdots & h_{1,m-2}& \hat{h}_{1,m-2} \\
h_{11} & \hat{h}_{11} & h_{12}& \hat{h}_{12}&\cdots & h_{1,m-2} & \hat{h}_{1,m-2}\\
h_{31}& \hat{h}_{31} & h_{32}& \hat{h}_{32} &\cdots & h_{3,m-2}& \hat{h}_{3,m-2} \\
ih_{31}& i\hat{h}_{31} & ih_{32}& i\hat{h}_{32}&\cdots & ih_{3,m-2}& i\hat{h}_{3,m-2} \\
\end{array}
\right);\end{equation}
$(2)$ (the first pair is of type (ii), all others are of type (i))
\begin{equation}
\hat{B}_{1}= \left(
\begin{array}{ccccccc}
h_{11} & i {h}_{11} & h_{12} & \hat{h}_{12} &\cdots & h_{1,m-2}& \hat{h}_{1,m-2} \\
h_{21} & i {h}_{21} & h_{12}& \hat{h}_{12}&\cdots & h_{1,m-2} & \hat{h}_{1,m-2}\\
h_{31}& i {h}_{31} & h_{32}& \hat{h}_{32} &\cdots & h_{3,m-2}& \hat{h}_{3,m-2} \\
h_{41}& i {h}_{41} & ih_{32}& i\hat{h}_{32}&\cdots & ih_{3,m-2}& i\hat{h}_{3,m-2} \\
\end{array}
\right);\end{equation}
Introducing consecutively more pairs of type (ii), one finally arrives at
$(m-1)$ (all pairs are of type (ii))
\begin{equation}
\hat{B}_{1}= \left(
\begin{array}{ccccccc}
h_{11} & i {h}_{11} & h_{12} & i {h}_{12} &\cdots & h_{1,m-2}& i{h}_{1,m-2} \\
h_{21} & i {h}_{21} & h_{22}& i {h}_{22}&\cdots & h_{2,m-2} & i{h}_{2,m-2}\\
h_{31}& i {h}_{31} & h_{32} & i {h}_{32} &\cdots & h_{3,m-2}& i{h}_{3,m-2} \\
h_{41}& i {h}_{41} & h_{42} & i {h}_{42}&\cdots & h_{4,m-2}& ih_{4,m-2} \\
\end{array}
\right).\end{equation}
{Here $r(f)\leq 2$ for Case $(1)$ and Case $(m-1)$. For other cases, $r(f)\leq 3,4,5,6$ or $8$, depending on the structure of the potential.}
\end{theorem}
{This theorem follows from the classification of nilpotent Lie sub-algebras related to the symmetric space $SO^+(1,n+3)/SO^+(1,3)\times SO(n)$, together with a restriction of the isotropy condition \eqref{eq-isotropic} on potentials (to derive Willmore surfaces). See \cite{Wang-1} for more details.}
\begin{example}(\cite{DoWa12}, \cite{Wang-3})\label{example}
Let \[\eta=\lambda^{-1}\left(
\begin{array}{cc}
0 & \hat{B}_1 \\
-\hat{B}_1^tI_{1,3} & 0 \\
\end{array}
\right)\mathrm{d}z,\ ~ \hbox{ with } ~\ \hat{B}_1=\frac{1}{2}\left(
\begin{array}{cccc}
2iz& -2z & -i & 1 \\
-2iz& 2z & -i & 1 \\
-2 & -2i & -z & -iz \\
2i & -2 & -iz & z \\
\end{array}
\right).\]
{Each extended frame $F(z,\bar z,\lambda)$ derived from this potential has singularities, while the corresponding harmonic maps and the corresponding Willmore two-spheres are globally well-defined.}
The associated family of Willmore two-spheres $x_{\lambda}$, $\lambda\in S^1$, corresponding to $\eta$, is
\begin{equation}\label{example1}
\begin{split}x_{\lambda}&=\frac{1}{ \left(1+r^2+\frac{5r^4}{4}+\frac{4r^6}{9}+\frac{r^8}{36}\right)}
\left(
\begin{array}{c}
\left(1-r^2-\frac{3r^4}{4}+\frac{4r^6}{9}-\frac{r^8}{36}\right) \\
-i\left(z- \bar{z})(1+\frac{r^6}{9})\right) \\
\left(z+\bar{z})(1+\frac{r^6}{9})\right) \\
-i\left((\lambda^{-1}z^2-\lambda \bar{z}^2)(1-\frac{r^4}{12})\right) \\
\left((\lambda^{-1}z^2+\lambda \bar{z}^2)(1-\frac{r^4}{12})\right) \\
-i\frac{r^2}{2}(\lambda^{-1}z-\lambda \bar{z})(1+\frac{4r^2}{3}) \\
\frac{r^2}{2} (\lambda^{-1}z+\lambda \bar{z})(1+\frac{4r^2}{3}) \\
\end{array}
\right)\\
\end{split}
\end{equation}
$x_{\lambda}:S^2\rightarrow S^6$ is a Willmore immersion in $S^6$, which is non S-Willmore, full, and totally isotropic. In particular, $x_\lambda$ does not have any branch points. The uniton number of $x_{\lambda}$ is $2$ and therefore its conformal Gauss map is $S^1-$invariant by Corollary 5.6 of \cite{BuGu}.
\end{example}
{ \bf Acknowledgements}\ \
This work was started when {PW} visited the Department of Mathematics of TU M\"{u}nchen, and the Department of Mathematics of Tuebingen University. The paper was continued and finished during several mutual visits of the authors.
They would like to express their sincere gratitude for both the hospitality and the financial support. {PW} is thankful to Professor Changping Wang and Xiang Ma for their suggestions and encouragement. {PW} was partly supported by the Project 11971107 of NSFC. PW is also thankful to the ERASMUS MUNDUS TANDEM Project for the financial supports to visit the TU M\"{u}nchen.
{\footnotesize
\defReferences{References}
}
{\small\
Josef F. Dorfmeister
Fakult\" at f\" ur Mathematik, TU-M\" unchen,
Boltzmann str.3, D-85747, Garching, Germany
{\em E-mail address}: [email protected]\\
Peng Wang
School of Mathematics and Statistics, FJKLMAA,
Fujian Normal University, Qishan Campus,
Fuzhou 350117, P. R. China
{\em E-mail address}: {[email protected]}
\end{document} |
\betaegin{document}
\title{Fourier transform for D-algebras}
\betaigskip
This paper is devoted to the construction of an analogue of the Fourier
transform for a certain
class of non-commutative algebras. The model example which initiated this
study is the equivalence
between derived categories of ${\cal D}$-modules on an abelian variety and
${\cal O}$-modules on the universal
extension of the dual abelian variety by a vector space (see [L], [R2]).
The natural framework for
a generalization of this equivalence is provided by the language of
$D$-algebras developed by
A.~Beilinson and J.~Bernstein in [BB]. We consider a subclass of
$D$-algebras we call special $D$-algebras.
We show that whenever
one has an equivalence of categories of ${\cal O}$-modules on two varieties $X$
and $Y$, it gives rise to a correspondence between special
$D$-algebras on $X$ and $Y$ such
that the corresponding derived categories of modules are equivalent. When
$X$ is an abelian
variety, $Y$ is the dual abelian variety, according to Mukai [M] the
categories of ${\cal O}$-modules on
$X$ and $Y$ are equivalent, so our construction gives in particular the
Fourier transform between
modules over rings of twisted differential operators ({\it tdo}, for short,
see section
\ref{gen-sec} for a definition) with non-degenerate first Chern class on
$X$ and $Y$.
We also deal with the microlocal version of the Fourier transform. The
microlocalization
of a special filtered $D$-algebra on $X$ is an NC-scheme in the sense of
Kapranov (see [K]) over
$X$, i.e. a ringed space whose structure ring is complete with respect to
the topology defined by
commutator filtration (see section \ref{et-sec}). We show that in our
situation the derived
categories of coherent sheaves on microlocalizations are also equivalent.
In the case of rings of
twisted differential operators on the dual abelian varieties $X$ and $Y$
one can think about the
corresponding microlocalized algebras as deformation quantizations of the
cotangent spaces $T^*X$
and $T^*Y$. The projections $p_X:T^*X\rightarrow T_0^*X$ and $p_Y:T^*Y\rightarrow T_0^*Y$
can be considered as
completely integrable systems with dual fibers (a choice of a
non-degenerate tdo on $X$ induces an
identification of the bases $T_0^*X$ and $T_0^*Y$ of these systems). We
conjecture that our
construction generalizes to other dual completely integrable systems. This
hope is based on the
following observation: the relative version of our transform gives a
Fourier transform for modules
over relative tdo's on dual families of abelian varieties, while
deformation quantizations are
usually subalgebras in these tdo's.
An example of dual completely integrable systems appears in the geometric
Langlands program (see [BD]). Namely, one may consider Hitchin systems for
Langlands dual
groups. The analogue of the Fourier transform in this situation should lead
to the equivalence
between modules over a microlocalized tdo on moduli spaces of principal
bundles for Langlands
dual groups.
An important aspect of our work is that in an appropriate sense, the
microlocalized Fourier
transform is in fact \'etale local. Given a special filtered $D$-algebra
$\cal A$ on $X$, let $\cal A_{ml}$ be the corresponding microlocalized
scheme. Then $\cal A_{ml}$
is a non-commutative thickening of the product of $X$ with a scheme $Z$.
Denoting by $\Phi(\cal A)$ the corresponding $D$-algebra on $Y$, $\Phi(\cal
A)_{ml}$
is a thickening of $Y\times Z$. We then prove that the microlocalized Fourier
transform is \'etale local in $Z$. It should be noted that $\cal A_{ml}$
and $\Phi(\cal A)_{ml}$ are not schemes over $Z$, so one does not have a
straightforward base-change
argument. Rather,
we develop in section
\ref{et-sec} the non-commutative version of the theory of \'etale morphisms
in the framework of
Kapranov's NC-schemes, and establish the equivalence by a version of the
topological invariance of
\'etale morphisms.
Our work is motivated in part by Krichever's construction of solutions of
the KP-hierarchy, [Kr]. (See section \ref{geom-subsec}.)
Let $W$ be a smooth variety
of dimension $r$, embedded in its Albanese variety, $X$. Let
$D\subset W$ be an ample hypersurface and
$V\subset H^0(D,{\cal O}_D(D))$ an $r$-dimensional basepoint free subspace. Let
$\phi:D\rightarrow\Bbb P(V^*)$ denote the corresponding morphism and let $U\subset D$ be
an open subset such that $\phi|_U$ is \'etale. Let $Y=Pic^0(W)$. Then
$V$ maps to the space of vector fields on $Y$, and hence one has a
subalgebra of the differential
operators on $Y$, consisting of those operators which differentiate only in
the ``$V$" directions.
Denote this algebra by $\Phi(\cal A_V)$. That is, $\Phi(\cal A_V)$ is dual to a
$D$-algebra $\cal A_V$ on $X$. Then the microlocalizations of these
$D$-algebras are
thickenings of $Y\times\Bbb P(V^*)$ and $X\times\Bbb P(V^*)$ respectively. In
particular,
one has the
\'etale localizations $\cal A_{ml,U}$ and $\Phi(\cal A)_{ml,U}$ supported
on $X\times U$ and
$Y\times U$ respectively. Let $U_{\infty}$ denote the formal neighborhood
of $U$ in $W$. Then
the diagonal embedding $U\rightarrow X\times U$ extends to an embedding
${\cal D}elta_{\infty}:U_{\infty}\rightarrow
\cal A_{ml,U}$. On the other hand, $U\times Y$ sits in both $X\times Y$ and
$\Phi(\cal A)_{ml,U}$. Denote by $\cal L_{\infty}$ the Fourier transform of
${\cal D}elta_{\infty *}(\cal O_{U_{\infty}})$. We prove that $\cal L_{\infty}$
is a locally free
rank-one left $\cal O$ module on $\Phi(\cal A)_{ml,U}$, whose restriction
to $U\times Y$ is the
restriction of the Poincar\'e line bundle. Thus $\cal L_{\infty}$ is a
deformation
of the Poincar\'e line bundle. Furthermore, for any positive integer $k$,
the Fourier
transform of ${\cal D}elta_{\infty *}(\cal O_{U_{\infty}}(kU))$ is $\cal
L_{\infty}(k(U\times Y))$.
The point is that the ring $H^0(W,\cal O(*D))$ acts by $\cal
A_{ml,U}$-endomorphisms on
\betareak
${\cal D}elta_{\infty *}(\cal O_{U_{\infty}}(*U))$. Functoriality of the Fourier
transform then
gives us a representation
\betaegin{equation}\label{representation}
H^0(W,\cal O(*D))\rightarrow End_{\Phi(\cal A)_{ml,U}}(\cal
L_{\infty}(*(U\times Y)))\ .\end{equation}
When $W$ is a curve and $D$ is a point, this representation reduces to the
Burchnall-Chaundy [BC] representation of $H^0(W,\cal O(*D))$ by
differential operators. We intend to study
the representation (\ref{representation}) further in a future work. In
particular, the problem of
characterizing the image of this representation is quite interesting, and
should lead to
generalizations of the KP-hierarchy.
\betaigskip
\betaegin{notation}
Fix a scheme $S$. Given an $S$-scheme $U$, denote by $\pi^U_S$ the structural
morphism. By ``associative $S$-algebra on $U$" we mean a sheaf of
associative rings $\cal A$ on
$U$ equipped with a morphism of sheaves of rings from ${\pi^U_S}^{-1}(\cal
O_S)$ to the center of $\cal A$. We abbreviate
``$\otimes_{{\pi^U_S}^{-1}(\cal O_S)}$" by ``$\ots$". For a scheme $U$
we denote by
$\cal D^b(U)$ the bounded derived category of quasicoherent sheaves on $U$.
Throughout the
paper, $X$ and $Y$ are flat, separated $S$-schemes. \end{notation}
\section{$D$-algebras and Lie algebroids}\label{gen-sec} \subsetsection{}
Let us recall some definitions from [BB]. A {\it differential} $\cal
O_X$-{\it bimodule} $M$ is a quasicoherent sheaf on $X \times_S X$ supported on
the diagonal $X \subsetset X \times_S X$. One can consider the category of
differential $\cal O_X$-bimodules as a subcategory in the category of all
sheaves of $\cal O_X$-bimodules on $X$. A $D$-{\it algebra} on $X$ is a
sheaf of flat, associative $S$-algebras $\cal A$ on $X$ equipped with a
morphism of $S$-algebras $i: \cal O_X \rightarrow \cal A$ such that $\cal
A$ is a differential $\cal O_X$ - bimodule. This means that $\cal A$ has an
increasing filtration $0=\cal A_{-1} \subsetset \cal A_0 \subsetset \cal A_1
\subsetset \cdots$ such that $\cal A =
\cup \cal A_n$ and $ ad(f) (\cal A_k) \subsetset \cal A _{k-1}$ for any $k \ge
0$ and $f \in \cal O_X$ where $ ad(m): = rm-mr$. We denote by $b(\cal A)$
the quasi-coherent sheaf on $X\times_S X$ (supported on the diagonal)
corresponding to $\cal A$.
Also we denote
by $ \cal M ( \cal A)$ the category of sheaves of left $\cal A$-modules on
$X$ which
are quasicoherent as $\cal O_X$-modules.
\subsetsection{}\label{circle} Let us describe some basic operations with
$D$-algebras and modules over them. Let
$\cal A_X$ and $\cal A_Y$ be $D$-algebras over $X$ and $Y$ respectively.
One defines a $D$-algebra
$\cal A_X\betaoxs
\cal A_Y$
on $X\times_S Y$ by gluing $D$-algebras over products of affine opens $U\times_S V$
corresponding to $\cal A_X(U)\ots \cal A_Y(V)$. A module $M\in\cal M(\cal
A_X\betaoxs \cal A_Y)$ is the same as a quasicoherent $\cal O_{X\times_S Y}$-module
together with commuting actions of $p_X^{-1}(\cal A_X)$ and $p_Y^{-1}(\cal
A_Y)$
which are compatible with the $\cal O_{X\times_S Y}$-module structure (where
$p_X$ and $p_Y$ are projections from $X\times_S Y$ to $X$ and $Y$) . In
particular, we have the natural structure of $D$-algebra on
$p_X^*\cal A_X\simeq \cal A_X\betaoxs\cal O_Y$ and $p_Y^*\cal A_Y\simeq \cal
O_X\betaoxs\cal A_Y$ and natural embeddings of $D$-algebras $p_X^*\cal
A_X\hookrightarrow \cal A_X\betaoxs \cal A_Y$, $p_Y^*\cal A_Y\hookrightarrow
\cal A_X \betaoxs\cal A_Y$.
For a pair of modules $M_X\in\cal M(\cal A_X)$ and $M_Y\in\cal M(\cal A_Y)$
there is a natural structure of $\cal A_X\betaoxs\cal A_Y$-module on $M_X\betaoxs
M_Y$.
Now assume that we have $D$-algebras $\cal A_X$, $\cal A_Y$, and $\cal A_Z$
on $X$, $Y$ and $Z$ respectively. Then we can define an operation
$$\circ_{\cal A_Y}:\cal D^-(\cal M(\cal A_X\betaoxs\cal A_Y^{op}))\times \cal
D^-(\cal M(\cal A_Y\betaoxs\cal A_Z))\rightarrow \cal D^-(\cal M(\cal
A_X\betaoxs\cal A_Z)).$$ The definition is the globalization of the operation
of tensor product of bimodules.
Namely, for a pair of objects $M\in {\cal D}^-(\cal M(\cal A_X\betaoxs \cal
A_Y^{op}))$ and $N\in {\cal D}^-(\cal M(\cal A_Y\betaoxs \cal A_Z))$ we can form the
external tensor product $M\betaoxs N\in
{\cal D}^-(\cal M(\cal A_{XYZ}))$ where $\cal A_{XYZ}= \cal A_X\betaoxs \cal
A_Y^{op}\betaoxs\cal A_Y\betaoxs\cal A_Z$ is a $D$-algebra on $X\times_S Y\times_S Y\times_S Z$.
Note that there is a natural structure of left $\cal A_Y\betaoxs\cal
A_Y^{op}$-module on
$b(\cal A_Y)$ given by the multiplication in $\cal A_Y$. Hence, we can
consider the tensor product $$(\cal A_X\betaoxs
b(\cal A_Y)\betaoxs\cal A_Z) \overlineerset{\Bbb L}{\otimes}_ {\cal A_{XYZ}}
(M{\overset{\Bbb L}{\boxtimes}}_S} \def\ots{\otimes_S N)$$
as an object in the category
${\cal D}^-(p_{XZ}^{-1}(\cal A_X\betaoxs\cal A_Z))$ where $p_{XZ}^{-1}$ denotes a
sheaf-theoretical inverse image.
Finally, we set
$$M\circ_{\cal A_Y}N=Rp_{XZ*}((\cal A_X\betaoxs b(\cal A_Y)\betaoxs\cal A_Z)
\overlineerset{\Bbb L}{\otimes}_
{\cal A_{XYZ}} (M{\overset{\Bbb L}{\boxtimes}}_S} \def\ots{\otimes_S N))
.$$
There is also the following equivalent definition: $$M\circ_{\cal A_Y}N=
Rp_{XZ*}((M\betaoxs\cal A_Z)\overlineerset{\Bbb L}{\otimes}_ {{\cal
A_X}^{op}\betaoxs\cal A_Y\betaoxs\cal A_Z}({\cal A_X}^{op}\betaoxs N))\ .$$
Specializing to the case that $Z=S$ and $\cal A_Z=\cal O_S$, we see that
every $\cal A_X\betaoxs\cal A_Y^{op}$-module $F$ on $X\times_S Y$ defines a functor
$G\mapsto F\circ_{\cal A_Y} G$
from $\cal D^-\cal M(\cal A_Y)$ to $\cal D^-\cal M(\cal A_X)$.
\betaegin{Prop}
\betaegin{enumerate}
\item The operation $\circ$ is associative in the natural sense.
\vskip 5pt
\item One has $M\circ b(\cal A_Y)\simeq M$ and $b(\cal A_Y)\circ N\simeq N$
canonically, for
\betareak\hbox{$M\in \cal D^-\cal M(\cal
A_X\betaoxs\cal A_Y^{op})$} and $N\in \cal D^-\cal M(\cal A_Y\betaoxs\cal A_Z)$.
\vskip 5pt
\item If $M$ is a differential $\cal O_X$-bimodule and $N$ is an $\cal
O_{X\times_S Y}$-module,
then
\betareak \hbox{$
b(M)\circ_{\cal O_X}N=p_X^{-1}(M)\otimes_{p_X^{-1}(\cal O_X)}N$.}
\end{enumerate}
\end{Prop}
It follows that if $\cal A$ is a $D$-algebra on $X$, the structural morphism
on $\cal A$ may be
viewed as a morphism
$b(\cal A)\circ_{\cal O_X} b(\cal A)\rightarrow b(\cal A)$, and if $M$ is a
left $\cal A$-module,
the action of $\cal A$ on $M$ is given by a morphism $b(\cal A)\circ_{\cal
O_X} M\rightarrow M$.
Moreover, an
$\cal A_X\betaoxs\cal A_Y^{op}$-module structure on an $\cal O_{X\times_S
Y}$-module $M$ is the same
as a pair of morphisms $b(\cal A_X)\circ_{\cal O_X} M\rightarrow M$ and
$M\circ_{\cal O_Y}b(\cal
A_Y)\rightarrow M$ making
$M$ a (left $b(\cal A_X)$)-(right $b(\cal A_Y)$)-module with respect to
$\circ$, such that the
two module structures commute.
\subsetsection{}
Recall that a {\it Lie algebroid} $L$ on $X$ is a (quasicoherent) $\cal
O_X$-module equipped with a morphism of $\cal O_X$-modules $\sigma: L
\rightarrow \cal T (:= \text{Der}_S \cal O_X = $\text{relative tangent
sheaf of $X$}) and an $S$-linear Lie bracket
$[\cdot,\cdot] : L\ots L \rightarrow L$ such that $\sigma$ is a
homomorphism of Lie algebras and the following identity is satisfied:
$$
[\ell_1, f \ell_2] = f\cdot [\ell_1, \ell_2] + \sigma (\ell_1) (f) \ell_2
$$
where $\ell_1, \ell_2 \in L, f \in \cal O_X$. To every Lie algebroid $L$
one can associate a $D$-algebra $\cal U (L)$ called the {\it universal
enveloping algebra} of $L$. By definition $\cal U(L)$ is a sheaf of
algebras equipped with the morphisms of sheaves $i: \cal O_X \rightarrow
\cal U(L) , i_L: L \rightarrow \cal U(L)$, such that $\cal U(L)$ is
generated, as an algebra, by the images of these morphisms and the only
relations are:
\betaegin{enumerate}
\renewcommand{(\arabic{enumi})} \end{Lem}{(\roman{enumi})} \item $i$ is a morphism of
algebras;
\item $i_L$ is a morphism of Lie algebras; \item $i_L(f \ell)= i(f) i_L
(\ell),\ [ i_L (\ell) , i(f)] = i (\sigma (\ell) (f))$, where $f \in \cal
O_X, \ell \in L$. \renewcommand{(\arabic{enumi})} \end{Lem}{(\alpharabic{enumi})}
\end{enumerate}
\subsetsection{}
Let $L$ be a Lie algebroid on $X$. A central extension of $L$ by $\cal O_X$
is a Lie algebroid $\tilde{L}$ on $X$ equipped with an embedding of $\cal
O_X$ - modules $c:\cal O_X \hookrightarrow \tilde{L}$ such that $[c(1),
\tilde{\ell}] = 0$ for every $\tilde{\ell} \in \tilde{L}$ ( in particular,
$c(\cal O_X)$ is an ideal in $\tilde{L}$), and an isomorphism of Lie
algebroids $\tilde{L}/c(\cal O_X) \simeq L$. For such a central extension
we denote by $\cal U^{\circ}(\tilde{L})$ the quotient of $\cal
U(\tilde{L})$ modulo the ideal generated by the central element $i(1) -
i_{\tilde{L}} (c(1))$.
\betaegin{Lem}\label{locallyfree}
Let $L$ be a locally free $\cal O_X$-module of finite rank. Then there is a
bijective correspondence between isomorphism classes of the following data:
\betaegin{enumerate}
\renewcommand{(\arabic{enumi})} \end{Lem}{(\roman{enumi})} \item a structure of a Lie
algebroid on $L$ and a central extension $\tilde{L}$ of $L$ by $\cal O_X$.
\item a $D$-algebra $\cal A$ equipped with an increasing algebra filtration
$\cal O_X = \cal A_0 \subsetset \cal A_1 \subsetset \cal A_2 \subsetset \dots$ such
that $\cup \cal A_n = \cal A$ and an isomorphism of the associated graded
algebra $\operatorname{gr} A$ with the symmetric algebra $S^\betaullet L$. \end{enumerate}
\renewcommand{(\arabic{enumi})} \end{Lem}{(\alpharabic{enumi})} \end{Lem}
The correspondence between (i) and (ii) maps a central extension $
\tilde{L}$ to $\cal U^\circ (\tilde{L})$.
\subsetsection{}
Assume that $X$ is smooth over $S$. Then one can take $L= \cal T$ with its
natural Lie algebroid structure. The corresponding central extensions
$\tilde{\cal T}$ of $\cal T$ by $\cal O$ are called {\it Picard algebroids}
and the associated $D$-algebras are called {\it algebras of twisted
differential operators}; or simply {\it tdo}'s. If $\cal D$ is a tdo, $\cal
D_{-1} = 0 = \cal D_{0} \subsetset \cal D_1 \subsetset \cal D_2 \subsetset
\dots$ is its maximal $D$-filtration, i.e. $$
{\cal D}_i = \{d \in {\cal D} | ad(f) d \in {\cal D}_{i-1}\ ,\ f\in \cal O_X \},$$ then $\operatorname{gr}{\cal D}
\simeq S^\betaullet \cal T$.
\betaegin{Lem}\label{isomorphism}
For a locally free $\cal O_X$-module of finite rank $E$ one has a canonical
isomorphism
$$
\operatorname{Ext}^1_{\cal O_{X \times_S X}}({\cal D}elta_* E, {\cal D}elta_*\cal O_X) \simeq \operatorname{Hom}_{\cal
O_X} (E, \cal T) \oplus \operatorname{Ext}^1_{\cal O_X} (E,\cal O_X), $$
where $X\overlineerset {\cal D}elta\rightarrow X\times_S X$ is the diagonal embedding. \end{Lem}
\betaegin{pf}
Since ${\cal D}elta_* E \simeq p^*_1 E \otimes_{\cal O_{X\times_S X}} (\cal O_{X\times_S
X}/J)$,
where $J$ is the ideal sheaf of the diagonal, we have an exact sequence
$$
0 \rightarrow \operatorname{Hom} (p^*_1 E \otimes_{\cal O_{X\times_S X}} J,{\cal D}elta_*\cal O_X)
\rightarrow \operatorname{Ext}^1 ({\cal D}elta_*E, {\cal D}elta_*\cal O_X) \rightarrow \operatorname{Ext}^1 (p^*_1
E,{\cal D}elta_*\cal O_X)
$$
Note that the first and last terms are isomorphic to $\operatorname{Hom}(E, \cal T)$ and
$\operatorname{Ext}^1(E,\cal O_X)$ respectively. It remains to note that there is a
canonical splitting ${\cal D}elta_*: \operatorname{Ext}^1(E,\cal O_X) \rightarrow
\operatorname{Ext}^1({\cal D}elta_*
E, {\cal D}elta_* \cal O)$.
\end{pf}
Note that the projection $\operatorname{Ext}^1({\cal D}elta_*E, {\cal D}elta_*\cal O_X) \rightarrow
\operatorname{Hom}(E, \cal T)$ can be described as follows. Given an extension
\[
\betaegin{CD}
0 @>>> {\cal D}elta_* \cal O_X @>>> \tilde{E} @>>> {\cal D}elta_* E @>>> 0 \end{CD}
\]
the action of $J/J^2$ on $\tilde{E}$ induces the morphism $J/J^2 \otimes
\tilde{E} \rightarrow {\cal D}elta_* \cal O$, which factors through $J/J^2
\otimes {\cal D}elta_* E$, since $J$ annihilates ${\cal D}elta_* \cal O$. Hence we get
a morphism ${\cal D}elta_* E \rightarrow {\cal D}elta_* \cal T$.
Now if $\cal A$ is a $D$-algebra,
equipped with a filtration $\cal
A_{\betaullet}$ such that
$\operatorname{gr}\cal A \simeq S^{\betaullet}(E)$, then we consider the corresponding
extension of $\cal O_X$-bimodules \[
\betaegin{CD}
0 @>>> \cal O_X = \cal A_0 @>>> \cal A_1 @>>> E=\cal A_1/\cal A_0@>>> 0
\end{CD}
\]
as an element in $\operatorname{Ext}^1_{\cal O_{X \times X}}({\cal D}elta_* E, {\cal D}elta_* \cal
O_X)$. By definition, $\cal A$ is a tdo if the projection of this element
to $\operatorname{Hom}_{\cal O_X}(E, \cal T)$ is a map $E \rightarrow \cal T$ which is an
isomorphism.
\section{Equivalences of categories of modules over $D$-algebras}
\subsetsection{} Let
$P$ be
an object in ${\cal D}^b(X\times_S Y)$, $Q$ be an object in ${\cal D}^b(Y\times_S X)$ such that
$$P\circ_{\cal O_Y} Q\simeq{\cal D}elta_*\cal O_Y,\\ Q\circ_{\cal O_X}
P\simeq{\cal D}elta_*\cal O_X.$$ where ${\cal D}elta$ denotes the diagonal
embedding. In this case the functors
$$\Phi_P:M\mapsto P\circ M,\\
\Phi_Q:N\mapsto Q\circ N$$ establish an equivalence of categories ${\cal D}^-(X)$
and ${\cal D}^-(Y)$.
For example, we have these data in the following situation: $X$ is an
abelian $S$-scheme,
$Y=\hat{X}$ is the dual abelian $S$-scheme, $P=\cal P$ is the normalized
Poincar\'e line bundle on $X \times_S \hat{X}$, $Q=\sigma^*\cal
P^{-1}\omega_{X}^{-1}[-g]$ where $\sigma:\hat{X}\times_S X\rightarrow X\times_S \hat{X}$ is
the permutation of factors, $g = \dim X$.
\subsetsection{} Let us call a quasicoherent sheaf $K$ on $X \times_S X$ {\it
special} if there is a
filtration $0 = K_{-1}\subsetset K_0 \subsetset K_1 \subsetset \dots$ of $K$ and a
sequence of sheaves
of flat, quasicoherent $\cal O_S$-modules $F_i$ such that $\cup K_i = K$
and $K_i/K_{i-1} \simeq {\cal D}elta_* {\pi_S^X}^*(F_i)$ for every $i \ge 0$. We
denote by $\cal S_X$ the exact category of special sheaves on $X \times_S X$.
The following properties are easily verified. \betaegin{Lem}\label{sheaves on
S} Let $F\in {\cal D}^-(\cal M(\cal O_S))$. \betaegin{enumerate}
\item Let $G\in D^-(\cal M(\cal O_{X\times_S Y}))$. Then
${\cal D}elta_{X*}{\pi_S^X}^*(F)\circ_{\cal O_X}G= {\pi_S^{X\times_S
Y}}^*(F)\overlineerset{\Bbb L}{\otimes}_{\cal O_{X\times_S Y}}G$. \item $P
\circ_{\cal O_Y} {\pi_S^Y}^*(F) \circ_{\cal O_Y} Q={\pi_S^X}^*(F)$.
\end{enumerate}
\end{Lem}
The following proposition then follows.
\betaegin{Prop}
\betaegin{enumerate}
\item
For every $K \in \cal S_Y$, the functor $\cal M(\cal O_{Y\times
Z})\rightarrow \cal M(\cal O_{Y\times Z})$, $M\mapsto K\circ_{\cal O_Y} M$,
is exact. \vskip 5pt
\item
For every
pair of special sheaves $K, K' \in \cal S_Y$, $ K{\circ}_{\cal O_Y} K'$ is
special.
\end{enumerate}
\end{Prop}
\betaegin{Prop} The functor $\Phi: K \mapsto P \circ_{\cal O_Y} K \circ_{\cal
O_Y} Q$
defines an equivalence of categories $\Phi: \cal S_Y \rightarrow \cal S_X$.
\end{Prop}
\betaegin{pf}
>From lemma \ref{sheaves on S}, together with the fact that the
operation $\circ$ commutes with inductive limits, we have $\Phi(K) \in \cal
S_X$ for every $K \in
\cal S_Y$. It remains to notice that there is an inverse functor to $\Phi$
given by $$
\Phi^{-1}(K')=Q {\circ}_{\cal O_X} K' {\circ}_{\cal O_X} P $$
where $K' \in \cal S_X$.
\end{pf}
\betaegin{Prop}\label{homomorphism}
For $K, K' \in \cal S_Y, M \in \cal D^b(Y)$ one has a canonical isomorphism
of $\cal O_X$-bimodules $$
\Phi(K \circ_{\cal O_{Y}} K') \simeq \Phi K \circ_{\cal O_X} \Phi K', $$
and a canonical isomorphism in $\cal D^b(X)$ $$ \Phi (K \circ_{\cal O_Y} M)
\simeq \Phi K \circ_{\cal O_X} \Phi M. $$
\end{Prop}
\betaegin{Def}A $\it special$ $D$-$\it algebra$ on $X$ is a $D$-algebra $\cal
A$ such that
the sheaf $b(\cal A)$ on
$X\times_S X$ is special.\end{Def}
It follows from the above proposition that for any special $D$-algebra $
\cal A$ on $Y$
there exists a
canonical $D$-algebra $\Phi \cal A$ on $X$ such that $$b(\Phi\cal A)\simeq
\Phi(b(\cal A)).$$
Namely one just has to apply
$\Phi$ to structural morphisms $b(\cal A) \circ_{\cal O_{Y}} b(\cal A)
\rightarrow b(\cal A)$
and ${\cal D}elta_* \cal O_{Y} \rightarrow b(\cal A)$. Futhermore, we now prove
that the derived
categories of modules over $\cal A$ and $\Phi \cal A$ are equivalent.
\betaegin{Thm}\label{main1} Assume that $P$ and $Q$ are quasi-coherent sheaves
up to a shift (i.e. they have only one cohomology). Then for every special
$D$-algebra $\cal A$ on $Y$ there is a canonical exact equivalence $\Phi:
\cal D^- \cal M(\cal A) \rightarrow \cal D^- \cal M(\Phi \cal A)$ such that
the following diagram of functors is commutative: \end{Thm} \[
\betaegin{CD}
\cal D^b \cal M(\cal A) @>\Phi>> \cal D^b \cal M(\Phi \cal A)\\ @VVV@VVV\\
\cal D^b (Y) @>\Phi>> \cal D^b (X)
\end{CD}
\]
where the vertical arrows are the forgetting functors. \betaegin{pf} Let us
consider the following
object in ${\cal D}^b(X\times_S Y)$: $$ \cal B = P {\circ}_{\cal O_{Y}} b(\cal A) = P
\otimes_{\cal O_{X\times_S Y}}
p_{Y}^*\cal A\ . $$ Note that $\cal B$ is actually concentrated in one
degree so we can consider
it as a quasicoherent sheaf on
$X\times_S Y$ (perhaps shifted).
We claim that there is a canonical $\Phi \cal A\betaoxtimes \cal
A^{op}$-module structure on $\cal B$. Indeed, it suffices to construct
commuting actions $b(\Phi \cal A)\circ\cal B\rightarrow\cal B$ and $\cal
B\circ b(\cal A)\rightarrow\cal B$
compatible with $\cal O_{X\times_S Y}$-module structure. The right action of
$\cal A$ is obvious
while the left action of $\Phi \cal A$ on $\cal B$ is given by the
following map:
$$
\Phi b(\cal A) \circ_{\cal O_{X}} \cal B = P \circ_{\cal O_{Y}} b(\cal A)
{\circ}_{\cal O_{Y}} Q {\circ}_{\cal O_X} P \circ_{\cal O_{Y}} b(\cal A)
\rightarrow P \circ_{\cal O_{Y}} b(\cal A) \circ_{\cal O_{Y}} b(\cal A)
\rightarrow P \circ_{\cal O_{Y}} b(\cal A) = \cal B\ , $$ where the
last arrow is induced by multiplication in $\cal A$. It is clear that this
is a morphism of right
$p_{Y}^{-1}\cal A$ modules. On the other hand, there is a natural
isomorphism of sheaves on $X\times_S
Y$ $$ b(\Phi \cal A)\circ_{\cal O_{X}} P\simeq P\circ_{\cal O_{Y}} b(\cal
A)\circ_{\cal O_{Y}} Q
\circ_{\cal O_X} P\simeq P\circ_{\cal O_{Y}} b(\cal A)\simeq\cal B\ . $$
One can easily check that
the above left action of $p_X^{-1}\Phi \cal A$ on $\cal B$ is compatible
with the natural left
$p_X^{-1}\Phi \cal A$-module structure on $b(\Phi \cal A)\circ_{\cal O_{X}}
P$ via this
isomorphism. Thus, $\cal B$ is an object of ${\cal D}^b(\cal M(\Phi\cal
A\betaoxtimes\cal A^{op}))$
(concentrated in one degree).
So we can define the functor
$$
\Phi: {\cal D}^b\cal M(\cal A) \rightarrow {\cal D}^b
\cal M(\Phi \cal A) : M \mapsto \cal B{\circ}_{\cal A} M $$
Similarly,
we define an $\cal A\betaoxtimes\Phi \cal A^{op}$-module (perhaps shifted) on
$Y\times_S X$:
$$
\cal B' = b(\cal A) \circ_{\cal O_Y} Q \simeq \cal Q \circ_{\cal O_X}
b(\Phi \cal A)
$$
and the functor
$$
\Phi^\prime: {\cal D}^b \cal M(\Phi \cal A) \rightarrow {\cal D}^b \cal M(\cal A) : N
\mapsto B^\prime
{\circ}_{\Phi \cal A} N.
$$
One has an isomorphism in the derived
category of right $\cal O_Y\betaoxtimes\cal A$-modules on $Y\times_S Y$ $$ \cal
B^\prime {\circ}_{\Phi \cal A} \cal B \simeq ( Q {\circ}_{\cal O_X} b(\Phi
\cal A)){\circ}_{\Phi \cal A} \cal B \simeq Q {\circ}_{\cal O_X} \cal B
\simeq Q {\circ}_{\cal O_X} P \circ_{\cal O_{Y}} b(\cal A) \simeq b(\cal A).
$$
Similarly, there is an isomorphism in the derived category of left $\cal
A\betaoxtimes\cal O_Y$-modules
$$
\cal B^\prime{\circ}_{\Phi\cal A} \cal B \simeq \cal B^\prime{\circ}_{\Phi
\cal A} (b(\Phi \cal A) {\circ}_{\cal O_X} P) \simeq \cal
B^\prime\circ_{\cal O_X}
P \simeq b(\cal A)\circ_{\cal O_{Y}}
Q{\circ}_{\cal O_X} P \simeq b(\cal A).
$$
Moreover, both these isomorphisms
coincide with the following isomorphism of ${\cal O}_{Y\times_S Y}$-modules
$$\cal B'\circ_{\Phi\cal
A} \cal B\simeq Q\circ_{\cal O_X} b(\Phi\cal A)\circ_{\Phi\cal A}
b(\Phi\cal A)\circ_{\cal O_X}
P\simeq Q\circ_{\cal O_X} b(\Phi\cal A)\circ_{\cal O_X} P \simeq b(\cal
A).$$ It follows that
$\cal B^\prime{\circ}_{\Phi\cal A} \cal B \simeq b(\cal A)$ in the derived
category of $\cal
A\betaoxtimes\cal A^{op}$-modules,
Similarly,
$\cal B{\circ}_{\cal A}\cal B'\simeq b(\Phi\cal A)$. It follows that the
compositions $\Phi \Phi': {\cal D}^b \cal M(\Phi \cal A) \rightarrow {\cal D}^b \cal
M(\Phi \cal A)$ and $\Phi' \Phi: {\cal D}^b \cal M(\cal A) \rightarrow \cal D^b
\cal M(\cal A)$ are identity functors.
The composition of $\Phi$ with the forgetting functor can be easily
computed: $$
\cal B {\circ}_{\cal A} M
\simeq ( P {\circ}_{\cal O_Y} b(\cal A)) \circ_{\cal A} M \simeq
P{\circ}_{\cal O_Y} M\ .
$$
Hence, forgetting $\Phi \cal A$-module structure, we just get the transform
with kernel $P$.
\end{pf}
\betaegin{Rems}
1. In the situation of the theorem if we have another special $D$-algebra
$\cal A^{\prime}$ and a homomorphism of $D$-algebras $\cal A\rightarrow
\cal A^{\prime}$ then we have the corresponding induction and restriction
functors $M\mapsto \cal A^{\prime}\otimes_{\cal A} M$ and $N\mapsto N$
between categories of $\cal A$-modules and $\cal A^{\prime}$-modules. It is
easy to check that the corresponding derived functors commute with our
functors $\Phi$ constructed for $\cal A$ and $\cal A^{\prime}$.
\noindent
2. Let $A$ be an abelian variety, $\hat{A}$ be the dual abelian variety.
Then as was shown in [L] and [R2] the Fourier-Mukai equivalence ${\cal
D}^b(A)\simeq{\cal D}^b(\hat{A})$ extends to an equivalence of the derived
categories of $\cal D$-modules on $A$ and ${\cal O}$-modules on the universal
extension of $\hat{A}$ by a vector space. The latter category is equivalent
to the category of modules over the commutative sheaf of algebras $\cal A$
on $\hat{A}$ which is constructed as follows.
Let
$$0\rightarrow \cal O\rightarrow\cal E\rightarrow H^1(\hat{A},\cal O)\otimes\cal O\rightarrow 0$$ be the
universal extension. Then
$\cal A=\operatorname{Sym}(\cal E)/(1_{\cal E}-1)$ where $1_{\cal E}$ is the image of
$1\in\cal O$ in $\cal E$. It is easy to see that $\cal A$ is the dual
special $D$-algebra to the algebra of differential operators on $A$, so our
theorem implies the mentioned equivalence of categories.
\noindent
3. In the case of abelian varieties one can generalize the notion of
special $D$-algebra as follows.
Instead of considering special sheaves on $X\times X$ one can consider
quasi-coherent sheaves on
$X\times X$ admitting filtration with quotients of the form $(\operatorname{id}, t_x)_*L$
where $(\operatorname{id},
t_x):X\rightarrow X\times X$ is the graph of the translation by some point
$x\in X$, $L$ is a line
bundle algebraically equivalent to zero on $X$. Let us call such sheaves
quasi-special. It is easy
to see that quasi-special sheaves are flat over $X$ with respect to both
projections $p_1$ and
$p_2$, so the operation $\circ$ is exact on them. We can define a
quasi-special algebra as a
quasi-special sheaf $K$ on $X\times X$ together with the associative
multiplication $K\circ
K\rightarrow K$ admitting a unit ${\cal D}elta_*\cal O_X\rightarrow K$. Then
there is a Fourier duality
for quasi-special algebras and equivalence of the corresponding derived
categories. The proof of
the above theorem works literally in this situation. Note that modules over
quasi-special algebras
form much broader class of categories than those over special $D$-algebras.
Among these categories
we can find some categories of modules over 1-motives and our Fourier
duality coincides with the
one defined by G. Laumon in [L]. For example, a homomorphism $\phi:\Bbb
Z\rightarrow X$ defines a
quasi-special algebra on $X$ which is a sum of structural sheaves of graphs
of translations by
$\phi(n)$, $n\in\Bbb Z$. The corresponding category of modules is the
category of $\Bbb
Z$-equivariant $\cal O_X$-modules. The Fourier dual algebra corresponds to
the affine group over
$\hat{X}$ which is an extension of $\hat{X}$ by the multiplicative group.
\end{Rems}
\subsetsection{}
Let $L$ be a Lie algebroid on $Y$ such that $L \simeq \cal O^d_{Y}$ as an $
\cal O_{Y}$ - module. Then for any central extension $\tilde{L}$ of $L$ by
$\cal O_{Y}$, the $D$-algebra \ \ $\cal U^\circ (\tilde{L})$ is special.
Futhermore, one has $\Phi \cal U^\circ (\tilde{L}) \simeq \cal
U^\circ(\tilde{L^\prime})$ for some central extension $ \tilde{L^\prime}$
of a Lie algebroid $L^\prime$ on $X$ such that $L^\prime \simeq \cal O^d_X$
as an $\cal O_X$-module. Indeed, this follows essentially from Lemma
\ref{locallyfree}. One just has to notice that if a $D$-algebra $\cal A$ on
$Y$ has an algebra filtration $\cal A_\betaullet$ with $\operatorname{gr} \cal A_\betaullet
\simeq S^\betaullet (\cal O^d_{Y})$, then $\Phi \cal A$ has an algebra
filtration $\cal F \cal A_\betaullet$ with $\operatorname{gr} \Phi \cal A_\betaullet \simeq
S^\betaullet(\cal O^d_X)$. Note that if $L$ is a successive extension of
trivial bundles then the $D$-algebra $\cal U^\circ(\tilde{L})$ is still
special, but $\Phi \cal U^\circ(\tilde{L})$ is not necessarily of the form
$\cal U^\circ (\tilde{L'})$.
\subsetsection{}
Assume now that $X$ is an abelian variety. Let $\cal P$ be a Picard
algebroid on $\hat X, \cal D= \cal U^\circ(\cal P)$ be the corresponding
$tdo$. Then $\cal P/ \cal O_{\hat X} \simeq \cal T_{\hat X} \simeq \hat
{\frak g} \otimes_k \cal O _{\hat X}$ is a trivial $\cal O_{\hat
X}$-module, hence $\Phi \cal D \simeq \cal U^{\circ}(\tilde{L^\prime})$ for
some Lie algebroid ${L^\prime}$ on $X$ and its central extension
$\tilde{L^\prime}$ by $\cal O_X$.
\betaegin{Prop}
Let $\cal D$ be a tdo on $\hat X, \cal P$ be the corresponding Picard
algebroid. Then $\Phi\cal D$ is a tdo on $X$ if and only if the map
$\hat{\frak g} \rightarrow H^1 (\hat X, \cal O)$, induced by the extension
of $\cal O_{\hat X}$-modules $$
0 \rightarrow \cal O_{\hat X}\rightarrow \cal P \rightarrow \hat{\frak g}
\otimes_k \cal O_{\hat X}\rightarrow 0,
$$
is an isomorphism.
\end{Prop}
\betaegin{pf}
Let $\cal D_\betaullet$ be the canonical filtration of $\cal D$. Then $\Phi
\cal D$ is a tdo if only if the class of the extension of $\cal
O_X$-bimodules $$
0 \rightarrow \cal O_X \simeq \Phi \cal D_0 \rightarrow \Phi \cal D_1
\rightarrow
\Phi(\cal D_1/ \cal D_0) \simeq \hat{\frak g} \otimes \cal O_X \rightarrow
0 $$ induces an isomorphism $\hat{\frak g} \otimes \cal O_{\hat X}
\rightarrow \cal T_X$. Thus, it is
sufficient to check that the components of the canonical decomposition $$
\operatorname{Ext}^1_{\cal O_{X \times X}} ({\cal D}elta_* \cal O_X, {\cal D}elta_*, \cal O_X) \simeq
H^0(X, \cal T)\oplus H^1(X, \cal O_X)
$$
introduced in Lemma \ref{isomorphism}, get interchanged by the
Fourier-Mukai transform, if
we take into
account the natural isomorphisms
$$
H^0(X, \cal T) \simeq \frak g \simeq H^1(\hat X, \cal O), $$ $$
H^1 (X,\cal O_X) \simeq \hat{\frak g} \simeq H^0 (\hat X,\cal T). $$ We
leave this to the reader as a pleasant exercise on Fourier-Mukai transform.
\end{pf}
\subsetsection{}
Let us describe in more details the data describing a Lie algebroid $L$ on
an abelian
variety $X$ such that $L \simeq V \otimes_k \cal O_X$ as $\cal O_X$-module,
where $V$ is a finite-dimensional $k$-vector space, and a central extension
$\tilde {L}$ of $L$ by $\cal O_X$. First of all, $V=H^0(X,L)$ has a
structure of
Lie algebra, and the structural morphism $L \rightarrow \cal T$ is given by
some $k$-linear map $\betaeta : V
\rightarrow \frak g = H^0 (X,\cal T)$ which is a homomorphism of Lie
algebras (where $\frak g$ is an
abelian Lie algebra). The central extension $\tilde {L}$ is described (up
to an isomorphism) by a class $\widetilde{\alphalpha}$ in the first
hypercohomology space space $\Bbb
H^1(X,L^*
\rightarrow \wedge^2 L^* \rightarrow \wedge^3 L^* \rightarrow \dots)$ of
the truncated Koszul complex of $L$. In particular, we have the
corresponding class $\alphalpha
\in H^1(X, L^*)$, which is just the class of the extension of $\cal
O_X$-modules
$$
0 \rightarrow \cal O_X \rightarrow \tilde{L} \rightarrow L \rightarrow 0\ .
$$ We can consider $\alphalpha$ as a linear map $V \rightarrow H^1(X, \cal O_X)
= \hat{\frak g}$.
The maps $\alphalpha$ and $\betaeta$ get interchanged by the Fourier transform, up
to a sign.
By definition the $D$-algebra associated with $\widetilde{L}$ is a tdo if
and only if $\betaeta:V\rightarrow\frak g$ is an isomorphism. If an addition
$\alphalpha:V\rightarrow\hat{\frak g}$ is an isomorphism then the dual
$D$-algebra is also a tdo. Thus, we have a bijection between tdo's with
non-degenerate first Chern class on $X$ and $\hat{X}$ such that the
corresponding derived categories of modules are equivalent. According to
[BB] isomorphism classes of tdo on $X$ are classified by $\Bbb
H^2(X,{\cal O}mega^{\ge 1})$ which is an extension of $H^1(X,{\cal O}mega^1)\simeq
\operatorname{Hom}(\frak g,\hat{\frak g})$ by
$H^0(X,{\cal O}mega^2)=\wedge^2{\frak g}^*$.
Let $U_X\subset \Bbb H^2(X,{\cal O}mega^{\ge 1})$ be the subset of elements with
non-degenerate projection to $H^1(X,{\cal O}mega^1)$. The duality gives an
isomorphism between $U_X$ and $U_{\hat{X}}$. It is easy to see that under
this isomorphism multiplication by $\lambda\in k^*$ on $U_X$ corresponds to
multiplication by $\lambda^{-1}$ on $U_{\hat{X}}$.
On the other hand, let $\cal A$ be a tdo with trivial $c_1$. In other
words, $\cal A$ corresponds to some global 2-form $\omega$ on $X$. Modules
over $\cal A$ are ${\cal O}$-modules equipped with a connection having the scalar
curvature $\omega$. Let $\cal B$ be the dual $D$-algebra on $\hat{X}$ and
let $\widetilde{L}\rightarrow L=H^0(X,T_X)\otimes{\cal O}_{\hat{X}}$ be the corresponding
central extension
of Lie algebroids. We claim that $L$ is just an ${\cal O}_{\hat{X}}$-linear
commutative Lie algebra while the central extension $\widetilde{L}$ is given by
the class
$(e,\omega)\in H^1(L^*)\oplus H^0(\wedge^2 L^*)$, where $e$ is the
canonical element in $H^1(L^*)\simeq H^1(\hat{X},{\cal O})\otimes
H^1(\hat{X},{\cal O})^*$. Indeed, as an ${\cal O}_{\hat{X}}$-module $\widetilde{L}$ is a
universal extension of $H^1(\hat{X},{\cal O})\otimes{\cal O}$ by ${\cal O}$. Hence, the Lie
bracket defines a morphism of ${\cal O}$-modules $\wedge^2 L\rightarrow\widetilde{L}$. Since
$H^0(\widetilde{L})=H^0({\cal O})$ it follows that $[\widetilde{L},\widetilde{L}]\subsetset
{\cal O}\subsetset\widetilde{L}$. It is easy to see that the Lie bracket is just given by
$\omega:\wedge^2 L\rightarrow{\cal O}$.
Recall that the Neron-Severi group of $X$ is identified with
$\operatorname{Hom}^{\operatorname{sym}}(X,\hat{X})\otimes\Bbb Q$ where $\operatorname{Hom}^{\operatorname{sym}}(X,\hat{X})$ is the
group of symmetric homomorphisms $X\rightarrow\hat{X}$. Namely, to a line bundle
$L$ therecorrespondsa symmetrichomomorphism
$\phi_L:X\rightarrow\hat{X}$ sending a point $x$ to $t_x^*L\otimes L^{-1}$ where
$t_x:X\rightarrow X$ is the translation by $x$. One has the natural homomorphism
$c_1:NS(X)\rightarrow \Bbb H^2(X,{\cal O}mega^{\ge 1})$ sending a line bundle $L$ to the
class of the ring $D_L$ of differential operators on $L$. For $\mu\in
NS(X)$ we denote by $D_{\mu}$ the corresponding tdo. If $\mu\in NS(X)$ is a
non-degenerate class so that $c_1(\mu)\in U_X$ then the Fourier dual tdo to
$D_{\mu}$ is
\betaegin{equation}\label{detisom}
\Phi(D_{\mu})=D_{-\mu^{-1}}
\end{equation}
Indeed, it suffices to check this when $\mu$ is a class of a line bundle
$L$, in which case it follows easily from the isomorphism
$$\phi_L^*\det\Phi(L)\simeq L^{-\operatorname{rk}\Phi(L)}$$ and the fact that the dual
tdo to $D_L$ acts on $\Phi(L)$.
Let $E$ be a coherent sheaf which is a module over some tdo on $X$ (then
$E$ is automatically locally free). Following [BB] we say in this case that
there is an integrable projective connection on $E$.
\betaegin{Prop}
Let $E$ be a vector bundle on $X$ equipped with an integrable projective
connection. Assume that $\det E$ is a
non-degenerate line bundle.
Then $H^i\Phi(E)$ are vector bundles with canonical integrable projective
connections, and the following equality
holds:
$$\phi_{\det E}^*c_1(\Phi(E))
=-\chi(X,E)\cdot\operatorname{rk} E\cdot c_1(E).$$
\end{Prop}
\betaegin{pf} The first statement follows immediately from the fact that
$\Phi(E)$ is quasi-isomorphic to a complex of modules over the tdo on
$\hat{X}$ dual to $D_{(\det E)^{\frac{1}{r}}}$ where $r=\operatorname{rk} E$. On the
other hand, this tdo acting on $\Phi(E)$ is isomorphic $D_{(\det
\Phi(E))^{\frac{1}{r'}}}$ where $r'=\operatorname{rk}\Phi(E)=\chi(X,E)$. Considering
classes of these dual tdo's and using the isomorphism (\ref{detisom})
applied to $\mu=\frac{1}{r}\phi_{\det E}$ we get the above formula.
\end{pf}
\subsetsection{} The following two natural questions arise: 1) whether for
every $\mu\in NS(X)$ there exists a vector bundle $E$ on $X$ which is a
module over $D_{\mu}$, 2) what vector bundles on an abelian variety admit
integrable projective connections. To answer these questions we use the
following construction.
Let $\pi:X_1\rightarrow X_2$ be an isogeny of abelian varieties and $E$ be a vector
bundle with an integrable projective connection on $X_1$. Then there is a
canonical integrable projective connection on $\pi_*E$. Indeed, the
simplest way to see this is to use Fourier duality. If $E$ is a module over
some tdo $D_{\lambda}$ on $X_1$ then $\Phi(E)$ is a module over the dual
$D$-algebra $\Phi(D_{\lambda})$ on $\hat{X}_1$. Now we use the formula
$$\pi_*E\simeq \Phi^{-1}\hat{\pi}^*(\Phi(E)),$$ where $\Phi^{-1}$ is the
inverse Fourier transform on $X_2$, hence $\pi_*E$ is a module over
$\Phi^{-1}\hat{\pi}^*\Phi(D_{\lambda})$ which is a tdo on $X_2$.
In particular, the push-forwards of line bundles under isogenies have
canonical integrable projective connections. Also it is clear that if $E$
is a vector
bundle with an integrable projective connection and $F$ is a flat vector
bundle then $E\otimes F$ has a natural integrable projective connection.
Now we can answer the above questions.
\betaegin{Thm} For every $\mu\in NS(X)$ there exists a vector bundle $E$ which
is a module over $D_{\mu}$.
\end{Thm}
\betaegin{pf} We can write $\mu=[L]/n$ where $n>0$ is an integer, $[L]$ is a
class of a line bundle $L$ on $X$. Let $[n]_A:A\rightarrow A$ be an endomorphism of
multiplication by $n$. Then $[n]_A^*(\mu)\in NS(X)$ is represented by a
line bundle $L'$. Now we claim that the push-forward $[n]_{A,*}L'$ has a
structure of a module over $D_{\mu}$. Indeed, it suffices to check that
$$c_1([n]_{A,*}L')/\deg([n]_A)=\mu.$$
Let $\operatorname{Nm}_n: NS(X)\rightarrow NS(X)$ be the norm homomorphism corresponding to the
isogeny $[n]_A$. Then the LHS of the above equality is
$\operatorname{Nm}_n([L'])/\deg([n]_A)$. Hence, the pull-back of the LHS by $[n]_A$ is
equal to $[L']=[n]_A^*(\mu)$ which implies our claim. \end{pf}
\betaegin{Thm} Let $E$ be an indecomosable vector bundle with an integrable
projective connection on an abelian variety $X$. Then there exists an
isogeny of abelian varieties $\pi:X'\rightarrow X$, a line bundle $L$ on $X'$ and a
flat bundle $F$ on $X$ such that $E\simeq \pi_*L\otimes F$. \end{Thm}
\betaegin{pf}
The main idea is to analyze the sheaf of algebras $A=End(E)$. Namely, $A$
has a flat connection such that the multiplication is covariantly constant.
In other words, it corresponds to a representation of the fundamental group
$\pi_1(X)$ in automorphisms of the matrix algebra. Since all such
automorphisms are inner we get a homomorphism $$\rho:\pi_1(X)\rightarrow PGL(E_0)$$
where $E_0$ is a fiber of $E$ at zero. Now the central extension
$SL(E_0)\rightarrow PGL(E_0)$ induces a central extension of $\pi_1(X)=\Bbb Z^{2g}$ by
the group of roots of unity of order $rk E$. This central extension splits
on some subgroup of finite index $H\subset\pi_1(X)$. In other words, the
restriction of $\rho$ to $H$ lifts to a homomorphism $\rho_H:H\rightarrow GL(E_0)$.
Let $\pi:\widetilde{X}\rightarrow X$ be an isogeny corresponding to $H$, so that $\widetilde{X}$
is an abelian variety with $\pi_1(\widetilde{X})=H$. Then $\rho_H$ defines a flat
bundle $\widetilde{F}$ on $\widetilde{X}$ such that $$\pi^*A\simeq End(\widetilde{F})$$ as
algebras with connections. It follows that $\pi^*E\simeq L\otimes\widetilde{F}$
for some line bundle $L$ on $\widetilde{X}$. Thus, $L$ is a direct summand of
$\pi_*(L\otimes\widetilde{F})$.
Note that there exists a flat bundle $F$ on $X$ such that $\widetilde{F}\simeq
\pi^*F$ (again the simplest way to see this is to use the Fourier duality).
Hence, $L$ is a direct summand of $\pi_*L\otimes F$. It remains to check
that all indecomposable summands of the latter bundle have the same form.
This follows from the following lemma.
\end{pf}
\betaegin{Lem} Let $\pi:X_1\rightarrow X_2$ be an isogeny of abelian varieties, $L$ be
a line bundle on $X_1$, $F$ be an indecomposable flat bundle on $X_1$.
Assume that $\pi_*(L\otimes F)$ is decomposable. Then there exists a
non-trivial factorization of $\pi$ into a composition
$$X_1\stackrel{\pi'}{\rightarrow}X'_1\rightarrow X_2$$
such that
$L\simeq (\pi')^*L'$ for some line bundle $L'$ on $X'_1$. \end{Lem}
\betaegin{pf} By adjunction and projection formula we have
$$\operatorname{End}(\pi_*(L\otimes F))\simeq \operatorname{Hom}(\pi^*\pi_*(L\otimes F), L\otimes
F)\simeq\oplus_{x\in K}\operatorname{Hom}(t_x^*L\otimes F,L\otimes F)$$ where $K\subsetset
X_1$ is the kernel of $\pi$. If $t_x^*L\simeq L$ for some $x\in K$, $x\neq
0$ then $L$ descends to a line bundle on the quotient of $X_1$ by the
subgroup generated by $x$. Otherwise, we get $\operatorname{End}(\pi_*(L\otimes
F))\simeq\operatorname{End}(F)$, hence, $\pi_*(L\otimes F)$ is indecomposable. \end{pf}
\section{Noncommutative \'Etale morphisms} \label{et-sec}
This section provides the setting we will need to discuss microlocalization.
\subsetsection{}
\betaegin{Def} (cf. [K]) A ring homomorphism $A'\morph{\phi}A$ is called a central
extension if $\phi$ is surjective, $ker(\phi)$ is a central ideal and
$ker(\phi)^2=0$.
\end{Def}
\betaegin{Def}\label{formally etale} Let $\text{\bf Rings}$ denote the category of
asssociative rings.
Let
$\cal C\subsetset\text{\bf Rings}$ be a full subcategory. A morphism $R\morph{\alpha}S$ in
$\cal C$
is formally \'etale if, for every
commutative diagram $\alphalpha, \betaeta, \gamma, \delta$ \vskip 2pt
\betaegin{equation}\label{etale morphism}
\betaegin{array}{ccc} R & \lrar{\delta} & A' \\ \ldar{\alphalpha}
&\lbrurar{\epsilon} & \ldar{\gamma} \\ S & \lrar{\betaeta} & A \end{array}
\end{equation}
in $\cal C$, with $\gamma$ a central extension, there exists a unique morphism
\dsp{S\morph{\epsilon}A'} such that diagram (\ref{etale morphism})
commutes. \end{Def}
\betaegin{Ex}\label{standard ring example} Let $R$ be a ring and let
$a_0,a_1,....,a_n\in R$.
Let
$S$ be the
$R$-algebra generated by elements $z,u$ subject to the relations
\betaegin{equation}\sum a_iz^i=0\ ,\ u\sum i a_iz^{i-1}=1\ ,\ \sum i
a_iz^{i-1}u=1\
.\end{equation} Then the natural map \dsp{R\morph{\alpha}S} is formally \'etale
in $\text{\bf Rings}$.\end{Ex}
\betaegin{pf} Consider a commutative diagram \vskip 2pt
\betaegin{equation}
\betaegin{array}{ccc} R & \lrar{\delta} & A' \\ \ldar{\alphalpha} && \ldar{\gamma}
\\ S & \lrar{\betaeta} & A \end{array}
\end{equation}
as in definition \ref{formally etale}. Let $I=ker(\gamma)$. Choose
$x,y\in A'$ such that $\gamma(x)=\beta(z)$, $\gamma(y)=\beta(u)$. It must be
shown that there
are unique elements $p,q\in I$ such that \betaegin{equation}
\betaegin{matrix}
\sum \delta(a_i)(x+p)^i&=0\ ,\\
\ (y+q)\sum i \delta(a_i)(x+p)^{i-1}&=1\ ,\\ \ \sum i
\delta(a_i)(x+p)^{i-1}(y+q)&=1\ .\end{matrix}\end{equation} Since $I^2=0$,
the equations are uniquely solved by setting \betaegin{equation}
p=-y\sum \delta(a_i)x^i\ \ ,\ \ q=y(1-\sum i\delta(a_i)(x+p)^{i-1} y)\
.\end{equation}
\end{pf}
\subsetsection{}
Let us recall some definitions from [K]. For any associative algebra $R$, the
NC-filtration on
$R$ is the decreasing filtration $\{F^d R\}_{d\ge 0}$ defined by setting
$$F^d R=\sum_{i_1+\ldots+i_m=d} R\cdot R_{i_1} \cdot R \cdot \ldots \cdot
R\cdot R_{i_m}
\cdot R$$ where $R_0=R$, $R_{i+1}=[R,R_i]$ are the terms of the lower
central series for $R$
considered as a Lie algebra (we use a different indexing from Kapranov's).
This filtration
is compatible with multiplication and the associated graded algebra is
commutative.
The category ${\cal N}_d$ is the category of associative algebras $R$ with
$F^{d+1}R=0$. For
example,
${\cal N}_0$ is the category of commutative algebras. For every $d$ there is a
pair of adjoint
functors
$r_d:{\cal N}_d\rightarrow{\cal N}_{d-1}$ and $i_d:{\cal N}_{d-1}\rightarrow{\cal N}_d$, where $i_d$ is the
natural inclusion,
$r_d(R)=R/F_dR$. Note that if $R\in{\cal N}_d$ then $F^dR\subset R$ is a central
ideal with zero
square. Thus, $R$ is a central extension of $r_d(R)$. Indeed, ${\cal N}_d$ is
the category of rings $A$ which are obtained as the composition of $d$
central extensions,
$$
A\rightarrow A_1\rightarrow A_2\rightarrow ...\rightarrow A_d$$
with $A_d$ commutative.
\betaegin{Lem}\label{der} Let $R\rightarrow S$ be a formally \'etale morphism in
${\cal N}_d$, $M$ be an
$S^{ab}$-module. Then the natural map $\operatorname{Der}(S,M)\rightarrow\operatorname{Der}(R,M)$ is a
bijection. \end{Lem}
\betaegin{pf} Given a central $S$-bimodule $M$ we can define a trivial central
extension of $S$
by
$M$: $\widetilde{S}=S\oplus M$. Then derivations from $S$ to $M$ are in bijective
correspondence
with splittings of the projection $\widetilde{S}\rightarrow S$. Hence, the assertion. \end{pf}
\betaegin{Prop}\label{sasha's nice observation} Let \dsp{R\morph{\alphalpha}S} be
a formally \'etale
morphism in
${\cal N}_d$. Let
\betaegin{equation}
\betaegin{array}{ccc} R & \lrar{\delta} & A' \\ \ldar{\alphalpha} & &
\ldar{\gamma} \\ S & \lrar{\betaeta} & A \end{array}
\end{equation} be a commutative diagram in $\text{\bf Rings}$, such that $\betaeta$ is
surjective and $\gamma$ is a central extension. Then $A'\in{\cal N}_d$.\end{Prop}
\betaegin{pf}
Note that apriori we know from this diagram that $A\in{\cal N}_{d}$, hence,
$A'\in{\cal N}_{d+1}$. We want to prove that $F^{d+1}A'=0$. Since $F^{d+2}A'=0$
it suffices to
prove that for every sequence of positive numbers $i_1,\ldots,i_m$ such
that $i_1+\ldots+i_m=d+1$ one has
$$A'_{i_1}\cdot A'_{i_2}\cdot
\ldots\cdot A'_{i_m}=0.$$ We use descending induction in $m$. Assume that
this is true for
$m+1$. Then we can define a map
\betaegin{align*} &D:S^{m+d+1}\rightarrow I: \\ &(s_1,\ldots,s_{m+d+1})\mapsto
[a'_1,[\ldots,[a'_{i_1},a'_{i_1+1}]\ldots]]\cdot
[a'_{i_1+2},[\ldots,[a'_{i_1+i_2+1},a'_{i_1+i_2+2}]\ldots]]]\cdot \ldots
\end{align*} where $a'_i\in A'$ are such that $\gamma(a'_i)=\betaeta(s_i)$.
This map is
well-defined since
$a'_i$ are well-defined modulo $I$ which is a central ideal. Now the
induction assumption
implies that $D$ is a derivation in every argument. Hence, applying Lemma
\ref{der} we
conclude that $D=0$.
\end{pf}
\betaegin{Thm}\label{enlarging the category} Let \dsp{R\morph{\alphalpha}S} be a
formally \'etale
morphism in
${\cal N}_d$. Then \dsp{R\morph{\alphalpha}S}is a formally \'etale morphism in
$\text{\bf Rings}$.\end{Thm}
\betaegin{pf} This follows easily from propostion \ref{sasha's nice
observation}.\end{pf}
Let ${\cal N}_{\infty}$ denote the category of rings that are complete with
respect to the
NC-filtration.
\betaegin{Thm} Let \dsp{R\morph{\alphalpha}S} be formally \'etale in
${\cal N}_{\infty}$, with
$R\in{\cal N}_d$. Then $S\in{\cal N}_d$.\end{Thm}
\betaegin{pf} The natural morphism \dsp{R\rightarrow r_{d+i}(S)} is formally \'etale
in ${\cal N}_{d+i}$ for all $i\ge 0$. Proposition \ref{sasha's nice observation}
applied to the
diagram
\betaegin{equation}
\betaegin{array}{ccc} R & \lrar{} & r_{d+i+1}(S) \\ \ldar{} & & \ldar{} \\
r_{d+i}(S) & \lrar{=} & r_{d+i}(S) \end{array}
\end{equation}
shows that $F^{d+i}(S)\subsetset F^{d+i+1}(S)$ for all $i\ge 1$. Hence the
assertion.\end{pf}
\betaegin{Ex}\label{standard example in Nd} Let $R$ be a ring belonging
${\cal N}_d$ for some $d$.
Let
\dsp{R\morph{\alpha}S} be as in example \ref{standard ring example}. Let $\hat
S$ denote
the completion of $S$ with respect to the NC-filtration. Then $\hat S$
belongs to ${\cal N}_d$
and the
natural morphism $R\rightarrow \hat{S}$ is formally \'etale (in $\text{\bf Rings}$). As in
the commutative case, we call such a morphism {\em standard}.\end{Ex}
\subsetsection{}
The category $\operatorname{NCS}^d$ of NC-schemes of degree $d$ (in Kapranov's
terminology ``NC-nilpotent of
degree $d$") is constructed in the same way as the commutative category of
schemes using
${\cal N}_d$ instead of ${\cal N}_0$ as coordinate rings of affine schemes. For a
scheme $X\in \operatorname{NCS}^d$
we denote by
$\operatorname{NCS}^d_{/X}$ the category of NC-schemes of degree $d$ over $X$. The
morphisms in
$\operatorname{NCS}^d_{\/X}$ are denoted by $\operatorname{Hom}_X(\cdot,\cdot)$.
As in affine case we have natural adjoint functors $r_d:\operatorname{NCS}^d\rightarrow
\operatorname{NCS}^{d-1}$ and
$i_d:\operatorname{NCS}^{d-1}\rightarrow \operatorname{NCS}^d$. In particular, we have the abelianization
functor $\operatorname{NCS}^d\rightarrow
\operatorname{NCS}^0:X\mapsto X^{ab}$ given by the composition $r_1r_2\ldots r_d$.
A morphism $Z\rightarrow\widetilde{Z}$ of NC-schemes of degree $d$ is called a nilpotent
thickening if it
induces an isomorphism of underlying topological spaces and
${\cal O}_{\widetilde{Z}}\rightarrow{\cal O}_Z$ is a
surjection with nilpotent kernel.
\betaegin{Def} A morphism $Y\rightarrow X$ in $\operatorname{NCS}^d$ is called formally smooth
(resp. formally
unramified) if for every nilpotent thickening $Z\subset \widetilde{Z}$ in
$\operatorname{NCS}^d_{/X}$ the map
$\operatorname{Hom}_X(\widetilde{Z},Y)\rightarrow\operatorname{Hom}_X(Z,Y)$ is surjective (resp. injective). A
morphism is called
formally
\'etale if it is both formally smooth and formally unramified. A morphism
is called \'etale
if it is \'etale and the corresponding morphism of commutative schemes
$Y^{ab}\rightarrow X^{ab}$ is
locally of finite type. \end{Def}
\betaegin{Prop}\label{gen} Let $P$ be a property of being formally smooth
(resp. formally
unramified, resp. formally \'etale). \newline a) Let $f:Y\rightarrow X$ be a
morphism in $\operatorname{NCS}^d$
with property $P$. Then the same property holds for $r_d(f)$. \newline b)
If $f:Y\rightarrow X$ and
$g:Z\rightarrow Y$ are morphisms in $\operatorname{NCS}^d$ having property $P$ then $f\circ g$
also has this
property. \newline c) If $f:Y\rightarrow X$ is formally unramified morphism in
$\operatorname{NCS}^d$, $g:Z\rightarrow Y$
is a morphism in $\operatorname{NCS}^d$ such that $f\circ g$ has property $P$ then $g$
also has this
property. \newline d) An open morphism $U\rightarrow X$ is \'etale. \newline e) A
morphism $f:Y\rightarrow X$ is \'etale if and only if there exists an open
covering
$X=\cup X_i$ and for every $i$ an open covering $Y_{ij}$ of $f^{-1}(X_i)$
such that all the
induced morphisms
$Y_{ij}\rightarrow X_i$ are \'etale.
\end{Prop}
The proof is straightforward.
Theorem \ref{enlarging the category} has the following global version.
\betaegin{Thm} Let $f$ be a formally \'etale morphism in $\operatorname{NCS}^{d-1}$. Then
$i_d(f)$ is a
formally
\'etale morphism in $\operatorname{NCS}^d$. \end{Thm}
Now we observe that the topological invariance of \'etale morphisms remains
valid in the
present context.
\betaegin{Thm} For any $X\in \operatorname{NCS}^d$ the canonical functor $Y\mapsto Y^{ab}$
from the category
of
\'etale $X$-schemes to that of \'etale $X^{ab}$-schemes is an equivalence.
\end{Thm}
\betaegin{pf} First we claim that the functor in question is fully faithful.
Indeed, let $Y_1,
Y_2\in\operatorname{NCS}^d$ be \'etale $X$-schemes. Then since $Y_1^{ab}\rightarrow Y_1$ is a
nilpotent thickening
and
$Y_2\rightarrow X$ is \'etale the natural map
$$\operatorname{Hom}_X(Y_1,Y_2)\rightarrow\operatorname{Hom}_X(Y_1^{ab},Y_2)\simeq
\operatorname{Hom}_{X^{ab}}(Y_1^{ab},Y_2^{ab})$$ is an isomorphism as required.
To prove the surjectivity of the functor it suffices to do it locally. Thus
we may assume
that the morphism $Y^{ab}\rightarrow X^{ab}$ is a standard \'etale extension of
commutative rings
$R^{ab}\rightarrow S^{ab}$ where $S^{ab}=(R^{ab}[z_0]/(f_0(z_0)))_{f'_0(z_0)}$,
$f_0$ is a unital polynomial. But such a morphism lifts to a standard
\'etale morphism in
${\cal N}_d$, as in example \ref{standard example in Nd}. \end{pf}
\betaegin{Cor}
A morphism $f:Y\rightarrow X$ in $\operatorname{NCS}^d$ is \'etale if and only if there exists an
open
covering
$Y=\cup Y_i$ such that all the
induced morphisms
$Y_{i}\rightarrow f(Y_i)$ are standard \'etale morphisms. \end{Cor}
\section{Microlocalization}
\subsetsection{}
Let us return to the setting of Theorem (\ref{main1}). We assume that the
$D$-algebra $\cal A$ is equipped with an increasing algebra filtration $\cal
A_{\betaullet}$ (so $\cal A_i \cal A_j\subsetset \cal A_{i+j}$) such that $\cal
A_{-1}=0$, $\cal
A_0=\cal O_Y$, the associated graded algebra $\operatorname{gr}(\cal A)$ is commutative
and is generated by
$\operatorname{gr}(\cal A)_1$ over $\cal O_Y$. In particular, the left and right actions
of $\cal O_Y$ on
$\operatorname{gr}(\cal A)_i$ are the same.
We will call such a filtration {\it special} if there exists a sheaf of
flat, commutative graded
$\cal O_S$-algebras $C$, generated over $\cal O_S$ by $C_1$, and an
isomorphism of
graded algebras
$\operatorname{gr}(\cal A)\simeq {\pi_S^Y}^*( C)$. By lemma \ref{sheaves on S}, such an
isomorphism induces
an isomorphism
$\operatorname{gr}(\Phi\cal A)\simeq {\pi_S^X}^*( C)$.
Given a $D$-algebra $\cal A$ with a special filtration $\cal A_{\betaullet}$
we can form the
corresponding sequence of graded algebras $\operatorname{gr}_{(n)}(\cal A)$ for $n\ge 0$
by setting $$\operatorname{gr}_{(n)}(\cal A)=\oplus_{i=0}^{\infty} \cal A_i/\cal
A_{i-n-1}.$$ In particular,
$\operatorname{gr}_{(0)}(\cal A)=\operatorname{gr}(\cal A)$ is commutative while for $n\ge 1$ there is
a central element $t$ in
$\operatorname{gr}_{(n)}(\cal A)_1=\cal A_1$ (corresponding to $1\in \cal A_0\subsetset \cal
A_1$) such that $t^{n+1}=0$ and $\operatorname{gr}_{(n)}(\cal A)/(t)=\operatorname{gr}(\cal A)$.
These algebras form a projective system via the natural projections
$\operatorname{gr}_{(n+1)}(\cal A)\rightarrow \operatorname{gr}_{(n)}(\cal A)$.
Consider for each $n$ the NC-scheme
$\Bbb P_{n}(\cal A)=\operatorname{Proj}(\operatorname{gr}_{(n)}(\cal A))$ corresponding to $\operatorname{gr}_{(n)}(\cal
A)$ via the noncommutative analogue of $\operatorname{Proj}$ construction. We denote by
$\cal
D^-(\Bbb P_{n}(\cal A))$ the (bounded from above) derived category of left
quasi-coherent sheaves
$\Bbb P_{n}(\cal A)$. Similar to the commutative case there is a natural
localization functor
$M\mapsto\widetilde{M}$ from the category of graded $\operatorname{gr}_{(n)}(\cal
A)$-modules to the category of
quasi-coherent sheaves on $\Bbb P_{n}(\cal A)$.
If $M$ is a left $\cal A$-module (quasi-coherent over ${\cal O}_Y$)
equipped with an increasing module filtration $M_{\betaullet}$ then for every
$n>0$ we can form the
corresponding graded $\operatorname{gr}_{(n)}(\cal A)$-module $\oplus M_i/M_{i-n}$, hence
the corresponding
quasi-coherent sheaf $$\operatorname{ml}_{n}(M)=(\oplus_i M_i/M_{i-n})^{\widetilde{}}.$$
The above NC-schemes are connected by a sequence of closed embeddings
$i_n:\Bbb P_n(\cal A)\hookrightarrow \Bbb P_{n+1}(\cal A)$ and the quasi-coherent
sheaves
$\operatorname{ml}_n(M)$ satisfy $\operatorname{ml}_n(M)=\operatorname{ml}_{n+1}(M)|_{\Bbb P_n(\cal A)}$. In other
words, the system
$(\operatorname{ml}_n(M))$ corresponds to a quasi-coherent sheaf on the formal NC-scheme
$\Bbb P_{\infty}(\cal
A)=\operatorname{inj.}\lim\Bbb P_n(\cal A)$.
Our aim now is to establish an equivalence of derived categories of sheaves
on $\Bbb P_n(\cal A)$ and $\Bbb P_n(\Phi \cal A)$. We will prove something
stronger -- namely
that such an equivalence exists \'etale locally on $Proj(C)$.
With begin a Zariski local version. Clearly, $\circ$ commutes with flat
base change on $S$, so
we may assume $S$ is affine.
The isomorphisms
$\operatorname{gr}(\cal A)\simeq {\pi_S^Y}^*(\cal C)$ and $\operatorname{gr}(\Phi\cal A)\simeq
{\pi_S^X}^*(\cal C)$ give us isomorphisms $\Bbb P_1(\cal A)=Y\times_S\operatorname{Proj}(C)$
and $\Bbb P_1(\Phi\cal A)= X\times_S\operatorname{Proj}(C)$.
Let $f$ be a section of $C_1$.
It defines a Zariski open subset $D_f\subsetset\operatorname{Proj}(C)$ which is the spectrum of
$C_{(f)}$, the degree zero part in the localization of $C$ by $f$. Hence,
we have the
corresponding open subset
$Y\times_S D_f\subset\Bbb P_0(\cal A)$. Since $\Bbb P_n$ has the same underlying
topological space as $\Bbb P_0$
we have the corresponding open subscheme $\Bbb P_n(\cal A)_f\subsetset\Bbb P_n(\cal
A)$ for every $n\ge 0$.
We claim that we can identify $\Bbb P_n(\cal A)_f$ with the spectrum of some
sheaf of algebras over
$Y$. Namely, we have the surjection
$\cal A_1\rightarrow\operatorname{gr}(\cal A)_1$ and locally we can lift $f$ to a section
$\widetilde{f}\in\cal A_1$.
Consider the graded ${\cal O}_Y$-algebra
$$\operatorname{gr}_{(n)}(\cal A)_{f}:=\operatorname{gr}_{(n)}(\cal A)_{\widetilde{f}}.$$ It is easy to see
that this algebra
doesn't depend on a choice of the lifting element $\widetilde{f}$ (since two
liftings differ by a
nilpotent), hence this algebra is defined globally over $Y$. Now let
$\operatorname{gr}_{(n)}(\cal A)_{(f)}$ be
the degree zero component in $\operatorname{gr}_{(n)}(\cal A)_f$. Then
$\operatorname{Spec}(\operatorname{gr}_{(n)}(\cal A)_{(f)})$ is an
open subscheme in $\Bbb P_n(\cal A)$ with the underlying open subset $Y\times_S
D_f$, hence
$\operatorname{Spec}(\operatorname{gr}_{(n)}(\cal A)_{(f)})=\Bbb P_n(\cal A)_f$.
For a graded $\operatorname{gr}_{(n)}(\cal A)$-module $M$ we have the corresponding
quasi-coherent
sheaf $\widetilde{M}$ on $\Bbb P_n(\cal A)$. The restriction of $\widetilde{M}$ to
$\Bbb P_n(\cal A)_f$ is the sheaf
associated with $\operatorname{gr}_{(n)}(\cal A)_{(f)}$-module $M_{(f)}$, the degree zero
part in the
localization of $M$ with respect to some local lifting of $f$.
Let as called a graded sheaf {\it graded special} if every of its graded
components is a special sheaf.
\betaegin{Lem}\label{lemloc} For every element $f\in C_1$ the algebra
$\operatorname{gr}_{(n)}(\cal A)_f$ is a graded special $D$-algebra on $Y$. There is a
canonical
isomorphism of graded algebras \betaegin{equation}\label{Philoc}
\Phi(\operatorname{gr}_{(n)}(\cal A)_f)\simeq\operatorname{gr}_{(n)}(\Phi\cal A)_f. \end{equation}
\end{Lem}
\betaegin{pf}
First of all the Lemma is obvious for $n=0$: in this case $\operatorname{gr}(\cal
A)_f\simeq {\pi_S^Y}^*
(C_f)$ and
$$\Phi(\operatorname{gr}(\cal A)_f)\simeq \operatorname{gr}(\Phi\cal A)_f\simeq {\pi_S^X}^* (C_f).$$
Now for
$n>0$ consider the filtration on $\operatorname{gr}_{(n)}(\cal A)$ by two-sided principal
ideals $I^k=(t^k)$
where $t\in\operatorname{gr}_{(n)}(\cal A)_1$ is the central element corresponding to
$1\in\cal A_1$. Then
$I^0=\cal A$, $I^n=0$, and $I^k/I^{k+1}\simeq\operatorname{gr}(\cal A)(-k)$ as
$\operatorname{gr}_{(n)}(\cal A)$-module for
$0\le k<n$. Localizing this filtration we get a filtration by two
sided-ideals $I^k_f$ in
$\operatorname{gr}_{(n)}(\cal A)_f$ with associated graded quotients $\operatorname{gr}(\cal A)_f(-k)$.
Thus, $\operatorname{gr}_{(n)}(\cal
A)_f$ is graded special.
To construct an isomorphism (\ref{Philoc}) we notice that since
$\Phi(\operatorname{gr}_{(n)}(\cal A)_f)$ is a nilpotent extension of $\Phi(\operatorname{gr}(\cal
A)_f)\simeq
\operatorname{gr}(\Phi\cal A)_f$, any local lifting of $f$ is invertible in
$\Phi(\operatorname{gr}_{(n)}(\cal A)_f)$.
Therefore, by universal property we get a homomorphism $$\operatorname{gr}_{(n)}(\Phi\cal
A)_f\rightarrow
\Phi(\operatorname{gr}_{(n)}(\cal A)_f).$$
Using the above filtration on $\operatorname{gr}_{(n)}$ one immediately checks that
this is an isomorphism.
\end{pf}
\betaegin{Thm}\label{main2}
Assume that $H^0(X,\cal O_X)=H^0(Y,{\cal O}_Y)=\Bbb C$. Then for every $n\ge 0$
there is a canonical equivalence of categories $$\Phi_{(n)}:\cal
D^-(\Bbb P_{n}(\cal A))\rightarrow \cal D^-(\Bbb P_{n}(\Phi \cal A))$$
commuting with functors $i_{n*}$ and $i_n^*$. Moreover, assume that $M$ is
a left $\cal A$-module with an increasing module filtration such that
for some integer $d$ and for $i$ sufficiently large $\Phi(M_i/M_{i-1})$ is
concentrated in degree
$d$ (as an object of $\cal D(X)$). Then
$$\operatorname{ml}_n(\Phi(M))\simeq\Phi_{(n)}(\operatorname{ml}_n(M)).$$ \end{Thm} \betaegin{pf}
The proof of this theorem is similar to the proof of Theorem \ref{main1}.
First we note that the definition of the operation $\circ$ from
\ref{circle} works for
non-commutative schemes as well (the only difference is that now whenever
we need to take the
opposite $D$-algebra we have to pass to the opposite scheme as well). Now
we just want to
construct some quasi-coherent sheaves (perhaps shifted) on $\Bbb P_n(\cal
A)\times\Bbb P_n(\Phi\cal
A^{op})$ and on $\Bbb P_n(\Phi\cal A)\times\Bbb P_n(\cal A^{op})$ such that both
their
$\circ$-composition are equal to the
structure sheaves of diagonals (note that although there is no embedding of
a noncommutative scheme $Z$ into $Z\times Z^{op}$ we still can define an
analogue
of the structure sheaf of the diagonal $\delta_Z$ which is a quasi-coherent
sheaf on $Z\times
Z^{op}$). The definition of these sheaves is the following. First we
observe that
$\Bbb P_n(\Phi\cal A)\times \Bbb P_n(\cal A^{op})=\operatorname{Proj}(\cal A_{XY})$ where $\cal
A_{XY}$ is the following graded algebra on $X\times Y$: $$\cal
A_{XY}=\oplus_i \operatorname{gr}_{(n)}(\Phi\cal A)_i\betaoxtimes \operatorname{gr}_{(n)}(\cal A^{op})_i.$$
Next we remark that the sheaf
$$\cal B=P\circ_{\cal O_Y} b(\cal A)=b(\Phi\cal A)\circ_{\cal O_X} P$$
introduced
in the proof of Theorem \ref{main1} has a natural filtration $$\cal
B_i=P\circ_{\cal O_Y} b(\cal A)_i= b(\Phi\cal A)_i\circ_{\cal O_X} P.$$ It
follows that the sheaf
$\oplus_i\cal B_{2i}/\cal B_{2i-n}$ has a natural structure of graded $\cal
A_{XY}$-module, so we can set $$\cal B_{ml}=(\oplus_i \cal B_{2i}/\cal
B_{2i-n})^{\widetilde{}}$$ which is a quasi-coherent sheaf on
$\Bbb P_n(\Phi\cal A)\times \Bbb P_n(\cal
A^{op})$. Similarly, one can define the quasi-coherent sheaf $\cal B'_{ml}$
on $\Bbb P_n(\cal
A)\times\Bbb P_n(\Phi\cal A^{op})$ starting with the sheaf $\cal
B'=Q\circ_{\cal O_X} b(\Phi\cal A)$.
It remains to compute
$\cal B_{ml}\circ_{\cal O_{\Bbb P_n(\cal A)}}\cal B'_{ml}$ and $\cal
B'_{ml}\circ_{\cal O_{\Bbb P_n(\Phi\cal A)}}\cal B_{ml}$. The idea is the
following: we cover
$\operatorname{Proj}(C)$ by affine subsets $D_f$, where $f$ runs through $C_1$. For every
$f\in C_1$ we'll construct a canonical isomorphism between restrictions of
$\cal B_{ml}\circ_{\cal O_{\Bbb P_n(\cal A)}}\cal B'_{ml}$ and the structure
sheaf of diagonal in
$\Bbb P_n(\Phi\cal A)\times \Bbb P_n(\Phi\cal A^{op})$ to the open subscheme
$\Bbb P_n(\Phi\cal A)_f\times\Bbb P_n(\Phi\cal A^{op})_f$. These isomorphisms
will be compatible on intersections, so they will glue into a global
isomorphism.
The following notation will be useful: for a sheaf $\cal F$ on one of our
schemes and an element $f\in C_1$ we denote by $\cal F_f$ the restriction
of $\cal F$ to the open subscheme defined by $f$.
Under identification of the underlying topological space of $\Bbb P_n(\Phi\cal
A)\times\Bbb P_n(\cal A^{op})$ with $X\times\operatorname{Proj}(C)\times Y\times\operatorname{Proj}(C)$
the support of $\cal B_{ml}$ is the diagonal $X\times Y\times\operatorname{Proj}(C)$.
Using this fact it is fairly easy to see that $$(\cal B_{ml}\circ_{\cal
O_{\Bbb P_n(\cal A)}}\cal B'_{ml})_f= \cal B_{ml,f}\circ_{\cal O_{\Bbb P_n(\cal
A)_f}}\cal B'_{ml,f}$$ It remains to compute the $\circ$-composition in the
RHS. This is easier than the original problem because the sheaf $\cal
B_{ml,f}$ (resp. $\cal B'_{ml,f}$) live on affine schemes over $X\times Y$
(resp. $Y\times X$). Namely, $$\Bbb P_n(\Phi\cal A)_f\times\Bbb P_n(\cal
A^{op})_f= \operatorname{Spec}(\operatorname{gr}_{(n)}(\Phi\cal A)_{(f)}\betaoxtimes\operatorname{gr}_{(n)}(\cal
A^{op})_{(f)}).$$
According to Lemma \ref{lemloc} we have dual $D$-algebras $\operatorname{gr}_{(n)}(\cal
A)_{(f)}$ on $Y$
and $\operatorname{gr}_{(n)}(\Phi\cal A)_{(f)}$ on $X$. Hence,
we can apply Theorem \ref{main1} to these $D$-algebras. Let us denote by
$\cal B_{(f)}$ the
$\operatorname{gr}_{(n)}(\Phi\cal A)_{(f)}\betaoxtimes\operatorname{gr}_{(n)}(\cal A^{op})_{(f)}$-module
constructed in the proof
of the cited theorem (where it is called $\cal B$). We claim that there is
a canonical isomorphism
of the $\cal B_{ml,f}$ with the sheaf on $\operatorname{Spec}(\operatorname{gr}_{(n)}(\Phi\cal
A)_{(f)}\betaoxtimes\operatorname{gr}_{(n)}(\cal
A^{op})_{(f)})$ obtained by localization of $\cal B_{(f)}$. This claim
(together with an easy check
of the compatibility of isomorphisms on intersections) would allow to
finish the proof by referring
to Theorem \ref{main1}. It remains to construct an isomorphism between the two
$\operatorname{gr}_{(n)}(\Phi\cal A)_{(f)}\betaoxtimes\operatorname{gr}_{(n)}(\cal A^{op})_{(f)}$-modules:
$\cal B_{(f)}$ and
$(\oplus_i\cal B_{2i}/\cal B_{2i-n})_{(f\otimes f)}$ (the localization of
the latter module is
clearly $\cal B_{ml,f}$). Recall that by definition
$\cal B_{(f)}=P\circ_{\cal O_Y}\operatorname{gr}_{(n)}\cal A_{(f)}$. Also, it is clear
that $(\oplus_i\cal B_{2i}/\cal B_{2i-n})_{(f\otimes f)}$ is isomorphic to
the degree zero part in the localization of $$\operatorname{gr}_{(n)}(\cal
B)=\oplus_i\cal B_i/\cal B_{i-n}= P\circ_{\cal O_Y}\operatorname{gr}_{(n)}(\cal A)$$ by
$\widetilde{f}\otimes 1$ and $1\otimes \widetilde{f'}$, where $\widetilde{f}$ is a local lifting
of $f$ to $\Phi\cal A_1$, $\widetilde{f'}$ is a local lifting of $f$ to $\cal
A_1$. Thus, it suffices to construct a graded isomorphism between
$P\circ_{\cal O_Y}\operatorname{gr}_{(n)}(\cal A)_f$ and $\operatorname{gr}_{(n)}(\cal
B)_{\widetilde{f}\otimes 1,1\otimes \widetilde{f'}}$. According to Lemma \ref{lemloc} we
have
$$P\circ_{\cal O_Y}\operatorname{gr}_{(n)}(\cal A)_f\simeq \operatorname{gr}_{(n)}(\Phi\cal
A)_f\circ_{\cal O_X} P$$ so the assertion follows. \end{pf}
Note that we have
canonical invertible ${\cal O}_{\Bbb P_n}$-bimodules on $\Bbb P_n(\cal A)$:
$${\cal O}_{\Bbb P_n}(m)=\operatorname{ml}_n(\operatorname{gr}_{(n)}(\cal A)(m))$$ where $M\mapsto M(m)$ denotes
the shift of grading. In particular, we have the automorphism $$M\mapsto
M(1)={\cal O}(1)\otimes_{{\cal O}} M$$
of the category ${\cal D}^-(\Bbb P_n(\cal A))$. It is easy to see that the above
equivalence respects these automorphisms.
\subsetsection{} One can generalize Theorem \ref{main1} to the case of
NC-schemes of finite degree. Namely, there is a natural notion of support
of a quasi-coherent sheaf on such a scheme (just the support of the
corresponding sheaf on the reduced commutative scheme), hence, the
definition of $D$-algebra makes sense. Now the proof of Theorem \ref{main1}
works almost literally in this case. Moreover, it seems plausible for the
NC-schemes $\Bbb P_n(\cal A)$ one can consider slightly more general
$D$-algebras than special ones. Namely, instead of requiring the existence
of filtration with graded factors isomorphic to ${\cal O}$ it suffices to require
the existence of filtration with factors ${\cal O}^{ab}$, plus one should require
$D$-algebra to be flat as left and right ${\cal O}$-module.
\subsetsection{\'Etale local version of the equivalence}
Let $U\rightarrow Z$ be an \'etale morphism of $S$-schemes. Then we have the
corresponding \'etale morphism $Y\times U\rightarrow\Bbb P_1(\cal A)$. By topological
invariance of \'etale
category for every $n\ge 1$ this morphism extends to an \'etale morphism of
NC-schemes
$$j:\Bbb P_n(\cal A)_U\rightarrow\Bbb P_n(\cal A).$$
Similarly we have an NC-scheme $\Bbb P_n(\Phi\cal A)_U$, and an \'etale
morphism $j:\Bbb P_n(\Phi\cal A)_U\rightarrow\Bbb P_n(\cal A)$.
\betaegin{Thm} In the above situation the categories $\cal D^-(\Bbb P_n(\cal
A)_U)$ and
$\cal D^-(\Bbb P_n(\Phi\cal A)_U)$ are canonically equivalent. \end{Thm}
\betaegin{pf} Recall that in the proof of Theorem \ref{main2} we have
constructed a quasi-coherent sheaf (up to shift) $\cal B_{ml}$ on
$\Bbb P_n(\Phi\cal A)\times \Bbb P_n(\cal A^{op})$ supported on the diagonal
${\cal D}elta_Z:X\times Y\times Z\hookrightarrow X\times Z\times Y\times Z$. Moreover, the
restriction of $B_{ml}$ to $\Bbb P_1(\Phi\cal A)\times\Bbb P_n(\cal A^{op})$ is
actually obtained from the sheaf $P$ on $X\times Y$ via first pulling back
to $X\times Y\times Z$ and then pushing forward by ${\cal D}elta_Z$. We have the
following diagram of \'etale morphisms of NC-schemes: \betaegin{equation}
\betaegin{array}{ccc}
\Bbb P_n(\Phi\cal A)_U\times \Bbb P_n(\cal A^{op})_U &\lrar{\operatorname{id}\times j}&
\Bbb P_n(\Phi\cal A)_U\times \Bbb P_n(\cal A^{op})\\ \ldar{j\times\operatorname{id}} & &
\ldar{j\times\operatorname{id}}\\ \Bbb P_n(\Phi\cal A)\times \Bbb P_n(\cal A^{op})_U
&\lrar{\operatorname{id}\times j} &\Bbb P_n(\Phi\cal A)\times \Bbb P_n(\cal A^{op}) \end{array}
\end{equation}
Now we claim that there exists a quasi-coherent sheaf $\cal B_{ml,U}$ on
$\Bbb P_n(\Phi\cal A)_U\times \Bbb P_n(\cal A^{op})_U$ supported on the diagonal
${\cal D}elta_U:X\times Y\times U\hookrightarrow X\times U\times Y\times U$ such that
\betaegin{equation}\label{iso1}
(\operatorname{id}\times j)_*\cal B_{ml,U}\simeq (j\times\operatorname{id})^*\cal B_{ml} \end{equation}
\betaegin{equation}\label{iso2}
(j\times\operatorname{id})_*\cal B_{ml,U}\simeq (\operatorname{id}\times j)^*\cal B_{ml}.
\end{equation} and such that the restriction of $\cal B_{ml,U}$ to
$\Bbb P_1(\Phi\cal A)_U\times\Bbb P_1(\cal A)$ is isomorphic to
${\cal D}elta_{U,*}(p_{XY}^*P)$. Indeed, consider the quasi-coherent sheaf
$(j\times j)^*\cal B_{ml}$ on $\Bbb P_n(\Phi\cal A)_U\times \Bbb P_n(\cal
A^{op})_U$. It is is supported on $(j\times j)^{-1}(X\times Y\times Z)$
where $X\times Y\times Z$ is the relative diagonal in $X\times Y\times
Z\times Z$. Now since $j$ is \'etale, the relative diagonal $X\times
Y\times U$ is a connected component in $(j\times j)^{-1}(X\times Y \times
Z)$. Now we just set $\cal B_{ml,U}$ to be the direct summand of $(j\times
j)^*\cal B_{ml}$ concentrated on this component, i.e. $$\cal
B_{ml,U}=(j\times j)^*\cal B_{ml}|_{X\times Y\times U}.$$ The above
properties of $\cal B_{ml,U}$ are clear from this definition.
Similarly, we construct the sheaf $\cal B'_{ml,U}$ on $\Bbb P_n(\cal A)\times
\Bbb P_n(\Phi\cal A^{op})$.
It remains to compute the relevant $\circ$-products. This is easily done
using isomorphisms (\ref{iso1}), (\ref{iso2}). Namely, one should start by
computing
$(j\times\operatorname{id})_*(\cal B_{ml,U}\circ_{\Bbb P_n(\cal A)_U}\cal B'_{ml,U})$ on
$\Bbb P_n(\Phi\cal A)\times\Bbb P_n(\Phi\cal A^{op})_U$. We have \betaegin{align*}
&(j\times\operatorname{id})_*(\cal B_{ml,U}\circ_{\Bbb P_n(\cal A)_U}\cal B'_{ml,U})\simeq
((j\times\operatorname{id})_*\cal B_{ml,U})\circ_{\Bbb P_n(\cal A)_U}\cal B'_{ml,U}\simeq
((\operatorname{id}\times j)^*\cal B_{ml})\circ_{\Bbb P_n(\cal A)_U}\cal B'_{ml,U}\simeq\\
&\cal B_{ml}\circ_{\Bbb P_n(\cal A)}((j\times\operatorname{id})_*\cal B'_{ml,U})\simeq \cal
B_{ml}\circ_{\Bbb P_n(\cal A)}((\operatorname{id}\times j)^*\cal B'_{ml})\simeq (\operatorname{id}\times
j)^*(\cal B_{ml}\circ_{\Bbb P_n(\cal A)}\cal B'_{ml})\simeq\\ &(\operatorname{id}\times
j)^*(\delta_{\Bbb P_n(\Phi\cal A)})\simeq (j\times\operatorname{id})_*
(\delta_{\Bbb P_n(\Phi\cal A)_U} )
\end{align*}
Now the situation looks locally as follows: we have an \'etale extension of
NC-algebras $A\rightarrow A_1$, an $A_1\otimes A_1^{op}$-module $M$, and an
isomorphism of $A\otimes A_1^{op}$-modules $M\simeq A_1$. Furthermore, we
have a 2-sided ideal $I\subset A$ such that $A/I$ is commutative and $IM=MI$,
and we know that the induced isomorphism $M/IM\simeq A_1/IA_1$ is an
isomorphism of $A_1/IA_1\otimes A_1/IA_1$-modules (notice that $A_1/IA_1$
is commutative). We claim that in such a situation the above isomoprhism
commutes with the left action of $A_1$. Indeed, the left action of $A_1$ on
$M$ induces a homomorphism $\phi:A_1\rightarrow A_1$ such that $\phi|_A=\operatorname{id}$ and
$\phi\mod IA_1$ is the identity on $A_1/IA_1$. Now the formal \'etaleness
implies that $\phi=\operatorname{id}$.
Thus, we conclude that
$\cal B_{ml,U}\circ_{\Bbb P_n(\cal A)_U}\cal B'_{ml,U}\simeq
\delta_{\Bbb P_n(\Phi\cal A)_U}$ as required. \end{pf}
\subsetsection{} The sheaf of rings ${\cal O}_{\Bbb P_n}$ on $\Bbb P_n$ can be naturally
enlarged as follows.
The central element $t\in\operatorname{gr}_{(n)}(\cal A)_1$ induces a sequence of
embeddings of ${\cal O}_{\Bbb P_n}$-bimodules
$${\cal O}_{\Bbb P_n}\rightarrow{\cal O}_{\Bbb P_n}(1)\rightarrow{\cal O}_{\Bbb P_n}(2)\rightarrow\ldots$$ Now using the
natural morphisms
$${\cal O}_{\Bbb P_n}(m)\otimes_{{\cal O}_{\Bbb P_n}}{\cal O}_{\Bbb P_n}(l)\rightarrow {\cal O}_{\Bbb P_n}(m+l)$$
we can define the ring structure on
the direct limit
$$\widetilde{{\cal O}}_{\Bbb P_n(\cal A)}=
\operatorname{inj.}\lim({\cal O}\rightarrow{\cal O}(1)\rightarrow{\cal O}(2)\rightarrow\ldots)$$ For example,
if $Y$ is smooth and
$\cal A=D_Y$ is the sheaf of differential operators on $Y$ then
$\widetilde{{\cal O}}_{\Bbb P_{\infty}}$ is the sheaf of (formal) pseudo-differential
operators (the underlying topological space of $\Bbb P_{\infty}$ is the
projectivized cotangent bundle of $Y$). The subsheaf ${\cal O}_{\Bbb P_{\infty}}$
consists of (formal) pseudo-differential operators of negative order.
Now let $\Bbb P_1(\cal A)=Y\times Z$ and $U\rightarrow Z$ be an \'etale morphism. Then
one can define invertible ${\cal O}_{\Bbb P_n(\cal A)_U}$-bimodules ${\cal O}_{\Bbb P_n(\cal
A)_U}(m)$ as follows.
We can regard ${\cal O}_{\Bbb P_n(\cal A)}$ as a sheaf on $\Bbb P_n(\cal A)\times
\Bbb P_n(\cal A^{op})$ supported on the diagonal. Let $V$ be a thickening of
the diagonal in $\Bbb P_n(\cal A)\times\Bbb P_n(\cal A^{op})$ on which
${\cal O}_{\Bbb P_n(\cal A)}(m)$ lives. Then there is a canonical \'etale morphism
$V_U\rightarrow V$ and an embedding $V_U\rightarrow \Bbb P_n(\cal A)_U\times\Bbb P_n(\cal
A^{op})_U$. Now by definition ${\cal O}_{\Bbb P_n(\cal A)_U}(m)$ is obtained from
${\cal O}_{\Bbb P_n(\cal A)}(m)$ by first pulling back to $V_U$ and then pushing
forward to $\Bbb P_n(\cal A)_U\times\Bbb P_n(\cal A^{op})_U$. It is easy to see
that we still have morphisms of bimodules ${\cal O}(m)\rightarrow{\cal O}(m+1)$ and
${\cal O}(n)\otimes{\cal O}(m)\rightarrow{\cal O}(n+m)$ so we can define the algebra
$\widetilde{{\cal O}}_{\Bbb P_n(\cal A)_U}$.
\betaegin{Thm} In the preceding two theorems one can replace the categories of
${\cal O}$-modules by the categories of $\widetilde{{\cal O}}$-modules.
\end{Thm}
The proof is an application of the analogue of Theorem \ref{main1} for
$D$-modules on NC-schemes.
\section{Noncommutative deformation of the Poincar\'e line
bundle}\label{geom-subsec}
Consider
the following data:
\newline
\noindent
$W$ is a smooth projective variety over $\Bbb C$ of dimension $r$,\newline
\noindent
$D\subset W$ is a reduced irreducible effective divisor,\newline \noindent
$V\subset H^0(D,{\cal O}_D(D))$ is an $r$-dimensional subspace, such that the
corresponding rational morphism $\phi:D\rightarrow\Bbb P(V^*)$ is generically finite,
\newline \noindent
$U\subset D$ is an open subset such that $\phi|_U$ is \'etale.
From the exact sequence
$$0\rightarrow {\cal O}_W\rightarrow{\cal O}_W(D)\rightarrow{\cal O}_D(D)\rightarrow 0$$
we get a boundary homomorphism
$$V\rightarrow H^0(D,{\cal O}_D(D))\rightarrow H^1(W,{\cal O}_W).$$
Now let $X$ be the Albanese variety of $W$, $a:W\rightarrow X$ be the Abel-Jacobi map
(associated with some point of $W$). Then we have the canonical isomorphism
$$H^1(X,{\cal O}_X)\widetilde{\rightarrow}
H^1(W,{\cal O}_W),$$ in particular, we get a homomorphism
$V\rightarrow H^1(W,{\cal O}_W)$. Let
$$0\rightarrow{\cal O}_X\rightarrow{\cal E} \rightarrow H^1(X,{\cal O}_X)\otimes_{\Bbb C}{\cal O}_X\rightarrow 0$$ be the
universal extension.
Taking the pull-back of this extension under the map $V\rightarrow H^1(W,{\cal O}_W)$ we
obtain an extension
$$0\rightarrow{\cal O}_X\rightarrow{\cal E}_V\rightarrow V\otimes_{\Bbb C}{\cal O}_X\rightarrow 0.$$ Now we define a
commutative sheaf of
algebras on $X$ as follows
$$\cal A_V=\operatorname{Sym} ({\cal E}_V)/(1_{\cal E}-1)$$ where $1_{\cal E}$ is the
image of
$1\in{\cal O}_X\rightarrow{\cal E}_V$. Note that $\cal A_V$ is equipped with the
filtration satisfying the
conditions of the previous section. Also by construction we have a
canonical morphism of sheaves
$${\cal E}_V\rightarrow a_*{\cal O}_W(D)$$ which induce the homomorphism of
${\cal O}_X$-algebras $${\cal A}_V\rightarrow
a_*{\cal O}_W(*D)$$ compatible with natural filtrations, where ${\cal O}_W(*D)=
\operatorname{inj.}\lim
{\cal O}_W(nD)$.
If the map $V\rightarrow H^1(V,{\cal O}_V)$ is an embedding then the dual $D$-algebra
$\Phi{\cal A}_V$
is the algebra of differential operators ``in directions $V$", where we
consider $V$ as a subspace
in $H^1(X,{\cal O}_X)\simeq H^0(\hat{X},T_{\hat{X}})$.
By Theorem \ref{main1} we get a functor from the derived category of ${\cal
A}_V$-modules to
the derived category of $\Phi{\cal A}_V$-modules. We can restrict this
functor to the category of
${\cal O}_W(*D)$-modules. For example, if $D$ is ample the Fourier transform of
${\cal O}_W(*D)$ is a coherent
$\Phi{\cal A}_V$-module. In the case when $W$ is a curve, $D$ is a point,
and $a(D)=0\in X$ one can
show that the latter $\Phi{\cal A}_V$-module is free of rank 1 at general
point. However, in
general $\cal F({\cal O}_W(*D))$ is not free as $\Phi\cal A_V$-module even at
general point unless
$a(D)=0$. To get a module which is free of rank 1 at general point we have
to pass to
microlocalization and use \'etale localization "in vector fields direction"
as described below.
Let $D_{(n)}\subset W$ be the closed subscheme corresponding to the divisor
$nD$. Then
$\operatorname{Proj}(\operatorname{gr}_{(n)}({\cal O}_W(*D)))\simeq D_{(n)}$, hence by functoriality we have
a morphism
$$a_n:D_{(n)}\rightarrow\Bbb P_n(\cal A_V)$$
and an isomorphism
$$a_{n,*}{\cal O}_{D_{(n)}}\simeq\operatorname{ml}_n({\cal O}_W(*D))$$ where ${\cal O}_W(*D)$ is considered
as a
$\cal A_V$-module.
Let us start with the case $n=1$. Note that
$$a_1:D\rightarrow \Bbb P_1(\cal A_V)\simeq X\times \Bbb P(V^*)$$
is the natural map induces by $a$ and by
$\phi$. Hence, applying Fourier-Mukai transform to $a_{1,*}{\cal O}_D$ over a
general point
of $\Bbb P(V^*)$
one gets a free module of rank equal to the degree of $\phi$. To get a free
module of rank 1
at
general point we use the \'etale base change $U\rightarrow\Bbb P(V^*)$. Namely, we
replace $\Bbb P_1(\cal
A_V)=X\times\Bbb P(V^*)$ by
$\Bbb P_1(\cal A_V)_U=X\times U$ and $a_1$ by the morphism
$$a^U_1:U\rightarrow\Bbb P_1(\cal A_V)_U=X\times U:u\mapsto (a(u),u).$$
Then the following lemma is clear.
\betaegin{Lem} The relative
Fourier transform of $a^U_{1,*}{\cal O}_U$ is the line bundle $(id\times
a|_U)^*({\cal P})$ on
$\hat{X}\times U$, where $\cal P$ is the Poincar\'e line bundle. \end{Lem}
Now let $\Bbb P_n(\cal A_V)_U$ be the \'etale scheme over $\Bbb P_n(\cal A_V)$
which is a thickening of $X\times U$. Let also $U_{(n)}$ be the open subset
of $D_{(n)}$ which is a nilpotent thickening of $U$.
Then we have a commutative diagram
\betaegin{equation}
\betaegin{array}{ccc}
U & \lrar{} & \Bbb P_n(\cal A_V)_U \\
\ldar{} & & \ldar{} \\
U_{(n)} & \lrar{} & \Bbb P_n(\cal A_V)
\end{array}
\end{equation}
where the top horizontal arrow is the composition of $a_1^U$ and the closed
embedding
$\Bbb P_1(\cal A_V)_U\rightarrow\Bbb P_n(\cal A_V)_U$, the bottom horizontal arrow is the
restriction of $a_n$. Since the right vertical morphism is \'etale this
diagram gives rise to a morphism
$$a^U_n: U_{(n)}\rightarrow\Bbb P_n(\cal A_V)_U$$
filling the diagonal in the above commutative square. It is easy to check
that $a_n^U|_{U_{(n-1)}}=a_{n-1}^U$. Now we define the sequence of coherent
sheaves on $\Bbb P_n(\Phi\cal A_V)_U$ by setting
$$\cal L_n=\Phi_{(n)}(a^U_{n,*}{\cal O}_{U_{(n)}}).$$ These sheaves satisfy
$\cal L_{n+1}|_{\Bbb P_n}\simeq \cal L_n$, hence we can consider the
projective limit $\cal L_{\infty}$ of $\cal L_n$ which is a coherent sheaf
on the formal NC-scheme
$\Bbb P_{\infty}(\Phi\cal A_V)_U$.
\betaegin{Prop} The ${\cal O}_{\Bbb P_{\infty}(\Phi\cal A_V)_U}$-module $\cal
L_{\infty}$ is locally free of rank 1. \end{Prop}
\betaegin{pf} One has an exact sequence
$$0\rightarrow a_{n-1,*}^U\cal O_{U_{(n-1)}}(-1)\rightarrow a_{n,*}^U\cal O_{U_{(n)}}\rightarrow
a_{1,*}^U\cal O_U\rightarrow 0.$$ Applying the functor $\Phi_{(n)}$ and passing to
the limit we obtain an exact sequence
$$0\rightarrow \cal L_{\infty}(-1) \stackrel{t}{\rightarrow} \cal L_{\infty}\rightarrow \cal L_1\rightarrow 0$$
where $t$ is the canonical central element in $\cal O(1)$. It remains to
use the following simple algebraic fact. Let $A$ be a noetherian ring,
$t\in A$ be a non zero divisor such that $At=tA$ and
$A=\operatorname{proj.}\lim A/t^nA$.
Let $M$ be a finitely generated left $A$-module such that $t$ is not a
divisor of zero in $M$ and $M/tM$ is a free $A/tA$-module
of rank 1. Then $M$ is a free $A$-module of rank 1. \end{pf}
Notice that in the case when $W=C$ is a curve $D=P$ is a point, $U=D$ the
module $\cal L_{\infty}$ was used in [R1] to construct Krichever's solution
to the KP hierarchy. In this case $\cal L_{\infty}$ is a locally free
module of rank-1 over the microlocalization of
the subalgebra ${\cal O}[\xi]$ in the ring of the differential operators on the
Jacobian $J(C)$ generated by the vector field $\xi$ which comes from the
boundary homomorphism $H^0(O_P(P))\rightarrow H^1({\cal O}_C)$. The key point is that
$\cal L_{\infty}$ has also an action of completion of ${\cal O}_C(*P)$ at $P$
(which is isomorphic to the ring of Laurent series) commuting with the
action of pseudo-differential operators in $\xi$.
\betaigskip
\noindent
{\betaf References}
\betaigskip
\noindent
[BB] A.~Beilinson, J.~Bernstein, {\it A proof of
Jantzen conjectures}. I.~M.~Gelfand Seminar, 1--50, Adv. Soviet Math., 16,
Part 1,
Amer. Math. Soc., Providence, RI, 1993.
\noindent
[BD] A.~Beilinson, V.~Drinfeld, {\it Quantization of
Hitchin's fibration and Langlands' program}. Algebraic and geometric
methods in mathematical physics (Kaciveli, 1993), 3--7, Math. Phys. Stud.,
19, Kluwer Acad. Publ., Dordrecht, 1996.
\noindent
[BC] J.~Burchnall, T.~Chaundy, {\it Commutative ordinary differential
operators},
Proc. London Math. Soc., 21 (1923), 420--440;
{\it Commutative ordinary differential
operators II}, Proc. Royal Soc.
London (A), 134 (1932), 471--485.
\noindent
[Kr] I.M.~Krichever,
{\it Algebro-geometric construction of
the Zaharov-Shabat equations and their periodic solutions}.
Soviet Math. Dokl. 17 (1976), 394--397;
{\it Integration of nonlinear equations by the
methods of nonlinear geometry}, Funk. Anal. i Pril, 11 (1977), 15-- 31.
\noindent
[L] G.~Laumon, {\it Transformation de Fourier generalisee}, preprint
alg-geom 9603004.
\noindent
[K] M.~Kapranov, {\it Noncommutative geometry based on commutator
expansions}, preprint math.AG/9802041.
\noindent
[M] S.~Mukai, {\it Duality between $D(X)$ and $D(\hat{X})$ with its
application to Picard sheaves}.
Nagoya Math. J. 81 (1981), 153--175.
\noindent
[R1] M.~Rothstein, {\it Connections on the Total Picard Sheaf and the KP
Hierarchy}, Acta Applicandae Math. 42 (1996), 297--308.
\noindent
[R2] M.~Rothstein, {\it Sheaves with connection on abelian varieties}, Duke
Math. Journal 84 (1996), 565--598.
{\sc Department of Mathematics, Harvard University, Cambridge,
MA 02138
Department of Mathematics, University of Georgia, Athens, GA 30602}
{\it E-mail addresses:} apolish@@math.harvard.edu, rothstei@@math.uga.edu
\end{document} |
\begin{document}
\title{Quantum Algorithms for Estimating Physical Quantities using Block-Encodings}
\author{Patrick Rall}
\affiliation{Quantum Information Center, University of Texas at Austin}
\date{\today}
\begin{abstract}
We present quantum algorithms for the estimation of $n$-time correlation functions, the local and non-local density of states, and dynamical linear response functions. These algorithms are all based on block-encodings - a versatile technique for the manipulation of arbitrary non-unitary matrices on a quantum computer. We describe how to `sketch' these quantities via the kernel polynomial method which is a standard strategy in numerical condensed matter physics. These algorithms use amplitude estimation to obtain a quadratic speedup in the accuracy over previous results, can capture any observables and Hamiltonians presented as linear combinations of Pauli matrices, and are modular enough to leverage future advances in Hamiltonian simulation and state preparation.\end{abstract}
\maketitle
\section{Introduction}
A central goal of quantum algorithms is to aid in the study of large quantum systems. It is well established, for example, that quantum computers can simulate the dynamics of most Hamiltonians of interest \cite{1906.07115}. Hamiltonian simulation algorithms, sometimes combined with the quantum Fourier transform, have led to quantum algorithms for some physical quantities, including correlation functions \cite{1401.2430} and dynamical linear response functions \cite{1804.01505}. Both of these examples are crucial for the understanding of phenomena in condensed matter physics like electron and neutron scattering \cite{west, sears}, conductivity and magnetization \cite{diventra}.
Recent work in Hamiltonian simulation has yielded algorithms with exponential improvements in accuracy \cite{1511.02306} over Trotterization and guarantee linear scaling with the simulation time \cite{1906.07115}. The strategies employed by these works can be neatly encompassed in terms of `block-encodings' - a tool that allows quantum computers to represent non-unitary matrices. These block-encodings can be built using linear combinations of unitaries (LCUs) \cite{1501.01715, 1511.02306} and manipulated using quantum singular value transformation \cite{1806.01838}. In addition to providing new and better algorithms, block-encodings provide an intuitive and powerful framework for performing linear algebra on a quantum computer.
In this work we use block-encodings along with amplitude amplification \cite{0005055, 1908.10846, 1912.05559} to construct quantum algorithms for some physical quantities: $n$-time correlation functions, the local and non-local density of states, and dynamical linear response functions. These algorithms are more versatile than previous works \cite{1401.2430, 1804.01505} in that they can compute more general versions of the functions with greater accuracy.
The local and non-local density of states and linear response functions are all functions of the energy $f(E)$. We are usually interested in obtaining the general shape of $f(E)$ over a range of energies, i.e. in obtaining a `sketch' of $f(E)$. We show how to perform two sketching strategies from modern classical numerical condensed matter physics \cite{0504627, 1101.5895, 1811.07387}. First, we show how to compute integrals of $f(E)$ over a range of energies: $\int_{E_A}^{E_B} f(E) dE$. Second, we show how to compute the moments of a Chebyshev expansion of $f(E)$: briefly assuming $|E| \leq 1$ for ease of explanation, if $T_n(E)$ is the $n$'th Chebyshev polynomial of the first kind, then we show how to compute constants $c_n$ such that
\begin{align}
f(E) \approx \frac{1}{\pi\sqrt{1-E^2}} \cdot \sum_{n=0}^{N} c_n T_n(E).
\end{align}
This procedure is known as the kernel polynomial method \cite{0504627} and is intuitively similar to sketching a function by computing the first few coefficients in its Fourier series. Very recent work \cite{2004.04889} shows how similar methods can also perform point-estimates of the density of states by approximating a delta function with a polynomial close to a narrow Gaussian.
Algorithms that compute physical quantities often face barriers from complexity theory, since computing expectations of observables on ground states of Hamiltonians is $\mathsf{QMA}$-complete \cite{0406180}. This remains true even when severe restrictions are placed on Hamiltonians \cite{1212.6312}. For this reason we employ strategies that sidestep these barriers. For correlation functions, we do not provide algorithms for preparing ground states or other states of interest, since the best algorithms for their preparation must use properties of the particular Hamiltonian in question. Evaluating the density of states at particular energies is \#$\mathsf{P}$-complete \cite{1010.3060}, but sketching the density of states via integrals and Chebyshev expansions is in $\mathsf{BQP}$.
The structure of our paper is as follows. In section~II. we review block-encoding techniques. In section~III. we employ these techniques to study $n$-time correlation functions. If we have a set of observables ${O_i}$ and times $\{t_i\}$ we compute expectations of the form
\begin{align}\left\langle O_1(t_1) O_2(t_2) ... \right\rangle\end{align}
employing the Heisenberg picture. In section~IV. we outline quantum singular value transformation and tools for computing Chebyshev moments and integrals over energy intervals. In section~V. we employ these techniques to compute the density of states and the local density of states. If $H$ has eigenvalues $\{E_i\}$ and dimension $D$ then the density of states is:
\begin{align} \rho(E) = \frac{1}{D} \sum_i \delta(E_i - E).\label{eqn:dos}\end{align}
Furthermore, say $H$ is a Hamiltonian describing a particle with some set of positions $\{ \vec r \}$ and position eigenstates $\{\ket{\vec r}\}$. If the eigenvectors of $H$ are $\{\ket{\psi_i}\}$, then the local density of states is:
\begin{align} \rho_{\vec r}(E) = \sum_i \delta(E_i - E) |\braket{\psi_i | \vec r}|^2 \label{eqn:ldos}\end{align}
Finally in section~VI. we show how to sketch linear response functions of the form
\begin{align} A(E) = \left\langle B \delta(E - H + E_0)C \right\rangle \label{eqn:lres} \end{align}
where $E_0$ is the ground state energy of $H$ and $B,C$ are some observables. In the appendix we show how to construct optimal polynomial approximations to the window function, which we require to compute integrals of $\rho(E), \rho_{\vec r}(E)$ and $A(E)$.
\section{Block-Encoding Techniques}
Block encodings allow quantum computers to perform manipulations with non-unitary matrices. If $A$ is \emph{any} matrix with $|A| \leq 1$ where $|A|$ is the largest singular value, then a block-encoding is a unitary $U_A$ such that $A$ occupies the top left corner of $U_A$:
\begin{align}U_A = \begin{bmatrix} A & \hspace{1mm}\cdot\hspace{1mm} \\ \cdot & \cdot \end{bmatrix}\end{align}
Below we give a more formal definition involving an explicit Hilbert space $\mathcal{H}$ for $A$ and an ancillary Hilbert space $\mathbb{C}^k$ for postselection\footnote{In the general case when $A$ is a rectangular matrix that maps $\mathcal{H}\to\mathcal{H}'$ then the input ancilla space $\mathbb{C}^k$ and output ancilla space $\mathbb{C}^l$ must be chosen so that $\mathcal{H}\otimes \mathbb{C}^k$ and $\mathcal{H}'\otimes \mathbb{C}^l$ have the same dimension. For this paper we assume that $A$ is square so we can pick $l=k$.}. We also give a notion of accuracy and a notion of scaling to allow for $|A| > 1$. The number of qubits needed to realize these spaces is bounded by the circuit complexity of $U_A$. We denote the computational basis for ancillary Hilbert spaces $\mathbb{C}^k$ by $\{\ket{0}_k, \ket{1}_k, \ldots\}$.
\begin{definition} \label{def:block} Say $A$ is a matrix on $\mathcal{H}$ with $|A| \leq \alpha$. A unitary $U_A$ on $\mathbb{C}^k \otimes \mathcal{H} $ is an $\varepsilon$-accurate $\alpha$-scaled $Q$-block-encoding of $A$ if $U_A$ is implementable using $Q$ elementary gates and for some $k$ we have
\begin{align}|A/\alpha - (\bra{0}_k\otimes I)U_A(\ket{0}_k\otimes I)| \leq \varepsilon. \end{align}
If `$\varepsilon$-accurate' is omitted then 0-accurate (exact) is implied, and if `$\alpha$-scaled' is omitted then 1-scaled is implied.
\end{definition}
In our work we will only be interested in block-encodings of products of observables, so $A$ will be square and often Hermitian. The Pauli matrices are a basis for Hermitian matrices, but since they are also unitary they have trivial ($U_P = P$) block-encodings. A key property of block-encodings is that a quantum computer can easily prepare products and linear combinations of them.
\begin{lemma} \label{lemma:lcu} Say the matrices $\{A_i\}$ each have $\alpha_i$-scaled $Q_i$-block-encodings. Then:
\begin{enumerate}
\item the product $\prod_i A_i$ has a $\left(\prod_i \alpha_i\right)$-scaled $O\left(\sum_i Q_i \right)$-block-encoding, and
\item for any $\beta_i \in \mathbb{C}$ the linear combination $\sum_i \beta_i A_i$ has a $\left(\sum_i \alpha_i |\beta_i|\right)$-scaled $O\left(\sum_i Q_i\right)$-block-encoding.
\end{enumerate}
\end{lemma}
\begin{proof} A complete construction and analysis of these circuits is given in \cite{1806.01838}, although the core techniques were put forth earlier \cite{1501.01715, 1511.02306}. The construction of block-encodings of products is rather trivial, and we give a brief sketch of the proof that a linear combination of Pauli matrices $O = \sum_{i=1}^k\beta_i P_i$ has a $O\left(\sum_i\beta_i\right)$-scaled $O(k)$-block-encoding $U_O$:
\begin{align}
V_\beta \ket{0}_k &:= \frac{1}{\sqrt{\sum_i |\beta_i|}}\sum_{i=1}^k \sqrt{|\beta_i|} \ket{i}_k \\
V_{P} &:= \sum_{i=1}^k \ket{i}_k\bra{i}_k \otimes \frac{\beta_i}{|\beta_i|} P_i \\
U_{O} &:= (V_\beta^\dagger \otimes I) V_P (V_\beta \otimes I)
\end{align}
The gate complexity is dominated by $V_P$ with complexity $O(k)$. Generalizing to non-trivial block-encodings involves swapping $P_i$ with $U_{A_i}$ and dealing with the control registers.
\end{proof}
Lemma~\ref{lemma:lcu} has the crucial consequence that the vast majority of Hamiltonians in physics have efficient block-encodings, since they can be written as linear combinations of not too many Pauli matrices. In these cases we have $k,\alpha \in O(\text{poly}(n))$ where $n$ is the number of qubits required to encode $\mathcal{H}$.
The algorithms in this work construct block encodings of a desired $A$ and estimate $\text{Tr}(A \rho)$ for some given $\rho$. To do so we assume that there is a unitary that prepares a purification of $\rho$, which is any pure state such that $\rho$ can be obtained by tracing out some ancillary space $\mathbb{C}^l$.
\begin{definition} \label{def:prep} Let $\rho$ be a density operator on $\mathcal{H}$ and let $\ket{\textbf{0}}$ be some easy-to-prepare state in $\mathcal{H}$. A unitary $U_\rho$ on $\mathcal{H}\otimes \mathbb{C}^l$ for some $l$ is an $R$-preparation-unitary of $\rho$ if we have
\begin{align} \rho = \text{Tr}_{\mathbb{C}^l} \left( \ket{\rho}\bra{\rho} \right), \end{align}
where $\ket{\rho} = U_\rho(\ket{\mathbf{0}}\ket{0}_l)$ and $U_\rho$ is implementable using $R$ elementary gates.
\end{definition}
Often we are interested in correlation functions and linear response with respect to ground states or thermal states of some Hamiltonian. Depending on the situation performing state preparation can be an extremely difficult computational task, and the identification of specific practical situations where state preparation is easy is an area of active research \cite{1609.07877}. We consider the problem of state preparation itself out of scope for this work, but aim to present our algorithms in an abstract manner to maximize their versatility and permit the leveraging of future results. We do point out the existence of the following generic tool for constructing thermal states.
\begin{lemma} \label{lemma:thermal} Let $H$ be a Hamiltonian on a $D$-dimensional Hilbert space with an $\alpha$-scaled $Q$-block-encoding. Then for any $\beta \geq 0$ there exists an $R$-preparation unitary for a state $\varepsilon$-close in trace distance to the thermal state $e^{-\beta H} / Z$ where $Z = \text{Tr}(e^{-\beta H})$ and:
\begin{align} R \in O\left( Q\alpha \cdot \sqrt{\frac{D\beta}{Z}} \log\left(\sqrt{\frac{D}{Z}}\frac{1}{\varepsilon}\right)\right) \end{align}
\end{lemma}
\begin{proof} This is the main result of \cite{1603.02940}, combined with the newer Hamiltonian simulation results of \cite{1610.06546, 1606.02685} with corrections from \cite{1806.01838}. Briefly, the strategy is to construct a block-encoding of $e^{-\beta H /2}$ from $e^{iHt}$ using the Hubbard-Stratonovich transformation, and multiply it onto a purification of the maximally mixed state using a strategy called robust oblivious amplitude amplification.
\end{proof}
We now show how to use amplitude estimation to estimate the expectation of block encoded observables.
\begin{lemma} \label{lemma:observ} If $A$ is Hermitian and has an $\alpha$-scaled $Q$-block-encoding and $\rho$ has an $R$-preparation-unitary, then for every $\varepsilon,\delta > 0$ there exists an algorithm that produces an estimate $\xi$ of $\text{Tr}(\rho A)$ such that
\begin{align} |\xi - \text{Tr}(\rho A) | \leq \varepsilon \end{align}
with probability at least $(1-\delta)$. The algorithm has circuit complexity $O\left((R+Q) \cdot \frac{\alpha}{\varepsilon}\log \frac{1}{\delta}\right)$.
\end{lemma}
\begin{proof} The algorithm is as follows:
\patbox{
\textbf{Algorithm: Observable Estimation} \\[2mm]
Let $\bar A = (I + A/\alpha)/2$, and let $U_{\bar A}$ be its 1-scaled $O(Q)$-block-encoding which exists by Lemma~\ref{lemma:lcu}. Let $U_{\bar A}$ have control register dimension $k$ as in Definition~\ref{def:block}, and let $l$ and $\ket{\mathbf{0}}$ be as in Definition~\ref{def:prep}. Let:
\begin{align}
\ket{\rho} &:= U_\rho \ket{\mathbf{0}} \ket{0}_l\\
\ket{\Psi} &:= (U_{\bar A} \otimes I) \ket{0}_k \ket{\rho}\\
\Pi &:= \ket{0}_k\bra{0}_k \otimes \ket{\rho}\bra{\rho}
\end{align}
Perform amplitude estimation to obtain an estimate $\xi_0$ of $|\Pi\ket{\Psi}|$ to precision $\varepsilon/(2\alpha)$ with probability at least $(1-\delta)$. Return $\xi := (2\xi_0 + 1)\alpha$.
}
For details on how to perform amplitude estimation we refer to recent results \cite{1908.10846, 1912.05559} that avoid using the quantum Fourier transform, which was required by the traditional method \cite{0005055} from 2002. These results establish that $|\Pi\ket{\Psi}|$ can be estimated to additive error $\varepsilon$ and probability at least $(1-\delta)$ using $O\left(\frac{1}{\varepsilon} \log \frac{1}{\delta}\right)$ applications of a Grover operator:
\begin{align}G := -( I - 2\Pi )(I - 2\ket{\Psi}\bra{\Psi})\end{align}
This operator requires four uses of $U_\rho$ and two uses of $U_{\bar A}$, so it has circuit complexity $O(R+Q)$. This completes the runtime analysis.
Amplitude estimation estimates:
\begin{align}
|\Pi\ket{\Psi}| &= | \bra{0}_k\bra{\rho} (U_{\bar A} \otimes I)\ket{0}_k\ket{\rho} |\\
&= |\bra{\rho} (\bar A \otimes I) \ket{\rho} | \\
&= | \text{Tr}( \ket{\rho}\bra{\rho} (\bar A \otimes I))| \\
&= | \text{Tr}\left(\text{Tr}_{\mathbb{C}^l}(\ket{\rho}\bra{\rho})\bar A\right) | = |\text{Tr}(\rho \bar A)|
\end{align}
Since $\bar A$ has $|\bar A| \leq 1$ its eigenvalues lie in the range $[-1,1]$, so $\bar A$ is positive semi-definite. Therefore $\xi_0$ approximates $|\text{Tr}(\rho \bar A)| = \text{Tr}(\rho \bar A) = (1 + \text{Tr}(\rho A)/\alpha)/2$ to error $\varepsilon/(2\alpha)$, so $\xi$ approximates $\text{Tr}(\rho A)$ to error $\varepsilon$ as desired.
\end{proof}
In addition to providing a simple framework for manipulating observables on a quantum computer, block-encodings are often the starting point for modern Hamiltonian simulation algorithms \cite{1501.01715, 1906.07115}. Once a block-encoding of a Hamiltonian $H$ is constructed, we can apply functions to its eigenvalues using quantum singular value transformation discussed in section IV.
\section{Correlation Functions}
In this section we show how to estimate $n$-time correlation functions, improving on an algorithm presented in \cite{1401.2430}. This algorithm does not require any new technical tools. We include it primarily to illustrate how simple it is to construct algorithms for complex quantities via block-encodings. We also show how to estimate non-Hermitian block-encoded observables, a tool we will require later in section VI. Consider a system evolving under a time-independent Hamiltonian $H$. If $O_i$ is some Hermitian operator then in the Heisenberg picture:
\begin{align} O_i(t_i) := e^{iHt_i} O_i e^{-iHt_i}\end{align}
To prepare block-encodings of observables in the Heisenberg picture we leverage a modern result in Hamiltonian simulation for time-independent Hamiltonians. For simplicity we focus on time-independent Hamiltonians but there also exist block-encodings for time evolution under time-dependent Hamiltonians \cite{1906.07115, 1805.00582, 1805.00675}.
\begin{lemma} \label{lemma:hamsim} Let $H$ be a Hamiltonian on a $D$-dimensional Hilbert space with an $\alpha$-scaled $Q$-block-encoding. Then for any $t,\varepsilon > 0$ there exists an $\varepsilon$-accurate $T(t,\varepsilon)$-block-encoding of $e^{i H t}$ where:
\begin{align} T(t,\varepsilon) \in O\left( Q\alpha|t| + \frac{Q\log(1/\varepsilon)}{\log(e + \log(1/\varepsilon) / (\alpha |t| ))} \right)\label{eqn:Tcomplex}\end{align}
\end{lemma}
\begin{proof} This result originated in \cite{1606.02685, 1610.06546}, but it is cleanly re-stated with minor corrections as Corollary~60 of \cite{1806.01838}.
\end{proof}
Using this result we can state and analyze the estimation algorithm.
\begin{theorem} Let:
\begin{itemize}
\item $H$ be a Hamiltonian with an $\alpha$-scaled $Q$-block-encoding,
\item $O_1,...,O_n$ be some observables with $\beta_i$-scaled $R_i$-block-encodings,
\item $t_1,...,t_n$ be some times,
\item and $\rho$ be a state with an $S$-preparation unitary.
\end{itemize}
Then for every $\varepsilon,\delta > 0$ there exists an algorithm that produces estimate an estimate $\xi\in \mathbb{C}$ of $\text{Tr}\left( \rho \prod_i O_i(t_i) \right)$ to additive precision $\varepsilon$ in the real and imaginary parts with probability at least $(1-\delta)$. It has circuit complexity $O\left((S+W) \cdot \frac{\gamma}{\varepsilon} \log\frac{1}{\delta}\right)$ where $\gamma= \prod_i \beta_i$ and
\begin{align} W \in& O\left( \sum_{j=1}^n R_j + \sum_{j=0}^n T\left(\tau_j, \frac{\varepsilon}{2(n+1)^2} \right) \right)\label{eqn:Wcomplex}\\
\subset &O\left( \sum_{j=1}^n R_j + Q \alpha \sum_{j=0}^n |\tau_j| + Q n^2 \log\left(\frac{n}{\varepsilon}\right)\right) \label{eqn:roundedWcomplex}\end{align}
where $T(t,\varepsilon)$ is defined in Lemma~\ref{lemma:hamsim} and $\tau_j = t_{j+1} - t_{j}$, padding the list of times with $t_0 = t_{n+1} = 0$.
\end{theorem}
\begin{proof}
The algorithm is as follows:
~
\patbox{
\textbf{Algorithm: $n$-time correlation functions} \\[2mm]
Making use of $e^{-iHt_j}e^{iHt_{j+1}} = e^{iH(t_{j+1}- t_{j})} = e^{iH\tau_j} $, we rewrite the product of observables as follows:
\begin{align}
\prod_{j=1}^n O_j(t_j) &= e^{iHt_1}O_1e^{iH(t_2-t_1)} ... O_n e^{-iHt_n}\\
&= e^{iH\tau_0} \prod_{j=1}^n O_j e^{iH\tau_{j}}
\end{align}
Invoking Lemma~\ref{lemma:hamsim} we obtain $\frac{\varepsilon}{2(n+1)^2}$-accurate block-encodings of $e^{iH\tau_j}$, and we multiply them together with the block-encodings of $O_i$ using Lemma~\ref{lemma:lcu}. We obtain a $W$-block-encoding $U_\Gamma$ of an operator $\Gamma$ that approximates $\prod_i O_i(t_i)$.
Observe that $U_\Gamma^\dagger$ is a block-encoding of $\Gamma^\dagger$. This allows us to use Lemma~\ref{lemma:lcu} to construct $\gamma$-scaled $W$-block-encodings of the Hermitian and anti-Hermitian parts of $\Gamma$, as below. Then we invoke Lemma~\ref{lemma:observ} with target accuracy $\varepsilon/2$ for each of the below to obtain $\varepsilon$-accurate estimates of the real and imaginary parts of $\text{Tr}\left( \rho \prod_i O_i(t_i) \right)$.
\begin{align}
\Re\left(\xi\right) :=& \text{ estimate of } \text{Tr}\left( \rho \cdot \frac{\Gamma + \Gamma^\dagger}{2} \right) \label{eqn:herm}\\
\Im\left(\xi\right) :=& \text{ estimate of } \text{Tr}\left( \rho \cdot \frac{\Gamma - \Gamma^\dagger}{2i} \right) \label{eqn:antiherm}
\end{align}
}
Since the block-encodings of $e^{iH\delta t_j}$ are 1-scaled, the only contribution to $\gamma$ are the scalings of the $O_i$, so $\gamma= \prod_i \beta_i$. The runtime is dominated by the complexity $W$ of the block-encoding for $\Gamma$, which by Lemma~\ref{lemma:lcu} is clearly given by (\ref{eqn:Wcomplex}). To obtain (\ref{eqn:roundedWcomplex}) we loosely bound $1/\log(e + \log(1/\varepsilon)/(\alpha|t|)) \leq 1$ in (\ref{eqn:Tcomplex}). This looseness overestimates the runtime in situations where $n$ is very large but the $\tau_j$ are very small.
It remains to show that $\Gamma$ is $\varepsilon/2$-close in spectral norm to $\prod_i O_i(t_i)$, given that the block-encodings of $e^{iH\tau_j}$ are $\frac{\varepsilon}{2(n+1)^2}$-accurate. From there the $\varepsilon/2$-closeness of the Hermitian and anti-Hermitian parts, and the $\varepsilon$-accuracy of the final estimates follow. In general, Lemma~54 of \cite{1806.01838} gives an argument that if $|A - U| \leq \varepsilon_0$ and $|B - V| \leq \varepsilon_1$ then
\begin{align}|AB - UV| \leq \varepsilon_0 + \varepsilon_1 + 2\sqrt{\varepsilon_0\varepsilon_1} .\end{align}
Iterating this bound for a product of $\prod_{i=0}^n U_i$ where $|U_i - A_i| \leq \varepsilon_0$ we obtain by solving a recurrence relation:
\begin{align}\left|\prod_{i=0}^n U_i - \prod_{i=0}^n A_i\right| \leq (n+1)^2 \varepsilon_0 .\end{align}
Plugging in $\varepsilon_0 := \frac{\varepsilon}{2(n+1)^2} $ gives the desired upper bound of $\varepsilon/2$.
\end{proof}
This algorithm improves over \cite{1401.2430} in several ways. First, \cite{1401.2430} restricts to Pauli observables since they are unitary. Here $O_i$ do not have to be unitary. Secondly, since we are using amplitude estimation to obtain $\xi$ we obtain a quadratic speedup in the accuracy dependence. Finally, \cite{1401.2430} restricts to Hamiltonians where exact Hamiltonian simulation can be achieved using circuit identities. Of course, for situations where these restrictions apply and the accuracy speedup can be sacrificed, their construction yields significantly smaller circuits which may be more amenable to near-term quantum computers.
\section{Integrals and Chebyshev Moments of Functions of the Energy}
In this section we introduce some tools we will require for our quantum algorithms for computing the density of states and linear response functions.
Say a Hermitian matrix $A$ has an eigenvalue-eigenvector decomposition $A = \sum_i \lambda_i \ket{\phi_i}\bra{\phi_i}$. Given a block-encoding of $A$, quantum singular value transformation allows us to construct block-encodings of $p(A) = \sum_i p(\lambda_i) \ket{\phi_i}\bra{\phi_i}$, for polynomials $p(x)$. This requires $p(x)$ to be appropriately bounded, and the complexity of the encoding scales linearly in the degree of the polynomial. This method can also be generalized to non-Hermitian $A$ with some caveats. Singular value transformation is an extremely powerful result, and is a culmination of a long line of research in quantum algorithms, presented in its full generality in \cite{1806.01838}.
\begin{lemma} \label{lemma:svt} Let $A$ have a $Q$-block-encoding, and let $p(x)$ be a degree-$d$ polynomial satisfying $|p(x)| \leq 1$ for $x \in [-1, 1]$. Then for every $\delta > 0$ there exists a $\frac{1}{2}$-scaled $\delta$-accurate $O(Qd)$-block-encoding of $p(A)$. A description of the circuit can be computed in time $\text{poly}\left(d,\log \frac{1}{\delta}\right)$.
\end{lemma}
\begin{proof} This strategy originated in \cite{1606.02685, 1610.06546} and is developed in \cite{1806.01838} where it is formalized as Theorem~56. Calculating the circuit demands careful consideration of numerical precision. Recent work \cite{2003.02831} describes an elegant strategy for dealing with this issue.
\end{proof}
The expressions for density of states (\ref{eqn:dos},\ref{eqn:ldos}) and linear response (\ref{eqn:lres}) are both functions of the energy $f(E)$ roughly of the form:
\begin{align} f(E) := \sum_{i} \delta(E - E_i) \bra{\psi_i}A\ket{\psi_i}\label{eqn:fdef} \end{align}
where $\{E_i\}$ and $\{\ket{\psi_i}\}$ are the eigenvalues and eigenvectors of the Hamiltonian and $A$ is some Hermitian matrix. Rather than computing point-estimates of $f(E)$ we will be interested in computing integrals of $f(E)$ over a range $[a,b]$ as well as the moments of a Chebyshev expansion of $f(E)$. To obtain the scaling requirements of Lemma~\ref{lemma:svt} we observe that an $\alpha$-scaled block-encoding of a Hamiltonian $H$ guarantees that $|H/\alpha| \leq 1$. Rescaling $\bar a = a/\alpha$ and $\bar b = b/\alpha$, we construct a polynomial $w(x)$ that allows us to approximate integrals over the range $[\bar a, \bar b]$:
\begin{theorem} \label{thm:windowfunc}For every $\eta > 0$ and any $\bar a, \bar b$ with $-1 < \bar a < \bar b < 1$ there there exists a polynomial $w(x)$ such that for all $f(\alpha x)$ bounded by $f_\text{max}$ (defined below in (\ref{eq:boundedby})):
\begin{align}\left|\int_{-1}^{1} f(\alpha x) w(x) dx - \int_{\bar a}^{\bar b} f(\alpha x) dx \right| \leq \eta \end{align}
The polynomial has degree $d \in O( \frac{f_\text{max}}{\eta}\ln\frac{f_\text{max}}{\eta } ) $ and satisfies the requirement $|w(x)| \leq 1$ of Lemma~\ref{lemma:svt}.
\end{theorem}
\begin{proof} There exist several strategies for constructing approximating polynomials for window and step functions, which we could adapt for our purposes via shifting and scaling \cite{dolph, 1409.3305, 1707.05391, 0604324, 1907.11748}. We adapt an elegant approach that relies on standard strategies in approximation theory discussed in \cite{0902.3757} leveraging amplifying polynomials and Jackson's theorem \cite{rivlin} which constructs a polynomial that accomplishes our requirements directly. We postpone the argument to Appendix~\ref{section:window}.
\end{proof}
Our accuracy analysis requires a bound on $f(\alpha x)$, which is a bit subtle to define since $f(\alpha x)$ is a sum of many delta functions. However, we only ever perform integrals of $f(\alpha x)$. Therefore when we say `$f(\alpha x)$ is bounded by $f_\text{max}$' we mean that for all $\bar c<\bar d$:
\begin{align}
\int_{\bar c}^{\bar d} f(\alpha x) dx \leq f_\text{max} \cdot (\bar d-\bar c) \label{eq:boundedby}
\end{align}
The polynomial $w(x)$ immediately yields a strategy for computing integrals since the value can be expressed as a trace inner product.
\begin{align}
&\int_{a}^b f(E) dE = \int_{\bar a}^{\bar b} f(\alpha x)\cdot \alpha dx\label{eqn:windowderiv1}\\
&\approx \alpha \int_{-1}^{1} f(\alpha x) w(x) dx \\
&= \alpha \int_{-1}^{1}\sum_{i} \delta(\alpha x - E_i) \bra{\psi_i}A\ket{\psi_i} w(x) dx \\
&= \text{Tr}\left( A \sum_{i} \int_{-1}^{1} \delta(x - E_i/\alpha) w(x) dx\ket{\psi_i}\bra{\psi_i} \right)\label{eqn:deltastep}\\
&=\text{Tr}\left( A \sum_{i} w(E_i/\alpha) \ket{\psi_i}\bra{\psi_i} \right)\\
&= \text{Tr}\left(A w(H/\alpha)\right)\label{eqn:windowderiv5}
\end{align}
In step (\ref{eqn:deltastep}) we used the identity $\delta(\alpha x) = \delta(x)/\alpha$. This final expression can then be estimated using Lemma~\ref{lemma:observ}.
Next we briefly outline our strategy for sketching $f(E)$ using the kernel polynomial method \cite{0504627}. A sketch $f^\text{KPM}(E)$ is a linear combination of Chebyshev polynomials of the first kind $T_n(x)$ weighted by coefficients $\mu^f_n g_n$. The $\mu^f_n$ are the Chebychev moments of $f(E)$ and the $g_n$ are $f(E)$-independent smoothing coefficients (see for example the proof of Jackson's theorem in \cite{rivlin}). Since Chebyshev expansions are performed on the domain $[-1,1]$ we calculate moments of $f(\alpha x)$ for $x \in[-1,1]$.
\begin{align}
\mu^f_n &:= \int_{-1}^1 T_n(x) f(\alpha x) dx\\
f^\text{KPM}(\alpha x) &:= \frac{1}{\pi\sqrt{1 - x^2}} \left(g_0 \mu^f_0 + 2 \sum_{n=0}^N \mu^f_ng_n T_n(x)\right)
\end{align}
For this work we concern ourselves only with estimation of $\mu^f_n$ and defer to \cite{0504627, 1811.07387} for details on how to construct $f^\text{KPM}(E)$. A similar derivation to (\ref{eqn:windowderiv1}-\ref{eqn:windowderiv5}) yields the identity:
\begin{align}
\mu^f_n &:= \int_{-1}^1 T_n(x) f(\alpha x) dx = \text{Tr}\left( A T_n(H/\alpha) \label{eqn:mudef}\right)
\end{align}
Conveniently, quantum singular value transformation is particularly simple for Chebyshev polynomials.
\begin{lemma}\label{lemma:cheby} Let $A$ have a $Q$-block-encoding. Then for every $n$ there exists an $O(nQ)$-block-encoding of $T_n(A).$
\end{lemma}
\begin{proof} This is Lemma~9 of \cite{1806.01838}.\end{proof}
Now we have all the technical tools to state the main algorithms.
\section{Density of States}
In this section we show how to sketch the density of states (DOS):
\begin{align} \rho(E) = \frac{1}{D} \sum_i \delta(E_i - E).\end{align}
This is easily rewritten in the form in (\ref{eqn:fdef}) by choosing $A = I/D$. Following (\ref{eqn:windowderiv1}-\ref{eqn:windowderiv5}) and (\ref{eqn:mudef}) we obtain:
\begin{align}
\int_{a}^b \rho(E) dE &\approx \text{Tr}\left( \frac{I}{D} w(H/\alpha) \right)\\
\mu^{\rho}_n &= \text{Tr}\left( \frac{I}{D} T_n(H/\alpha) \right)
\end{align}
This argument makes use of of Theorem~\ref{thm:windowfunc} which requires a bound on $\rho(E)$. Observe that in the sense of (\ref{eq:boundedby}), $\rho(\alpha x)$ is bounded by any upper bound on the dimension of the largest eigenspace of $H$ which we call $\rho_\text{max}$.
These quantities can be estimated by leveraging the fact that $I/D$ has an $O(\log(D))$-preparation unitary.
\begin{theorem} Let $H$ have an $\alpha$-scaled $Q$-block-encoding and take any $\varepsilon, \delta > 0$. Then:
\begin{enumerate}
\item For any $a,b$ such that $-\alpha < a < b < \alpha$ there exists a quantum algorithm that produces an estimate $\xi$ of $\int_{a}^b \rho(E) dE$ with circuit complexity
\begin{align}O\left(\left( Q\cdot \frac{\rho_\text{max}}{\varepsilon}\log \frac{\rho_\text{max}}{\varepsilon} + \log D\right) \cdot \frac{1}{\varepsilon} \log\frac{1}{\delta} \right) \label{eqn:densintcomplexity}\end{align}
and $O(\text{poly}(\rho_\text{max}/\varepsilon))$ classical pre-processing, where $\rho_\text{max}$ is some upper bound on the dimension of the largest eigenspace of $H$.
\item For any $n$ there exists a quantum algorithm that produces an estimate $\zeta$ of $\mu^\rho_n$ with circuit complexity
\begin{align}O\left(\left(Q\cdot n+ \log D \right) \cdot \frac{1}{\varepsilon} \log\frac{1}{\delta} \right).\label{eqn:denschebycomplexity}\end{align}
\end{enumerate}
The estimates $\xi$ and $\zeta$ have error $\varepsilon$ with probability at least $(1-\delta)$.\label{thm:dosalg}
\end{theorem}
\begin{proof} Observe that a preparation unitary for $I/D$ simply prepares a Bell state on $\mathcal{H}\otimes \mathcal{H}$, call it $\ket{\text{Bell}(\mathcal{H})}$. If $\mathcal{H}$ is encoded as some subspace of a $n$-qubit system where $n = \lceil \log_2(D)\rceil$ then $\ket{\text{Bell}(\mathcal{H})}$ can be obtained from $\ket{\text{Bell}(\mathbb{C}^{2^n})}$ via amplitude amplification. This procedure can be made exact via the following standard trick involving an ancilla qubit. Observe that \begin{align}\beta := \braket{\text{Bell}(\mathbb{C}^{2^n})|\text{Bell}(\mathcal{H})} = \sqrt{D / 2^n}\end{align} is known exactly. If $U$ satisfies
\begin{align}U\ket{0^{2n}} &= \ket{\text{Bell}(\mathbb{C}^{2^n})}\\ &= \beta \ket{\text{Bell}(\mathcal{H})} + \sqrt{1-\beta^2}\ket{\phi_\perp} \end{align}
for some $\ket{\phi_\perp} \perp \ket{\text{Bell}(\mathbb{C}^{2^n})}$ then define $U'$ such that:
\begin{align}U'\ket{0^{2n+1}} &= \gamma U\ket{0^{2n}}\ket{0} + \sqrt{1-\gamma^2}\ket{0^{2n}}\ket{1}\\
&= \gamma\beta \ket{\text{Bell}(\mathcal{H})}\ket{0} + \sqrt{1-(\gamma\beta)^2}\ket{\psi_\perp} \end{align}
for some $\ket{\phi_\perp} \perp \ket{\text{Bell}(\mathbb{C}^{2^n})}\ket{0}$ where $\gamma$ is the largest number $\leq 1$ such that \begin{align}\sin( (2k+1) \arcsin(\gamma\beta)) = 1\end{align} has a solution where $k$ is a positive integer. Then, if $\theta = \arcsin(\gamma\beta)$ and $\Pi_\mathcal{H}$ is a projection onto the $\mathcal{H}\otimes \text{span}(\ket{0}\bra{0})$ subspace of $\mathbb{C}^{2n+1}$, then we can define a Grover operator $G$ that exactly prepares $\ket{\text{Bell}(\mathcal{H})}$.
\begin{align}
G = U'(I - &2\ket{0^{2n+1}}\bra{0^{2n+1}})(U')^\dagger(I - 2\Pi_\mathcal{H})\\
G^k \ket{\text{Bell}(\mathbb{C}^{2^n})} &= \sin((2k+1) \theta))\ket{\text{Bell}(\mathcal{H})}\ket{0} \nonumber\\ &+ \cos((2k+1) \theta)\ket{\psi_\perp}\\
&= \ket{\text{Bell}(\mathcal{H})}\ket{0}
\end{align}
Since $2^n < 2D$ we have $\beta \in \Omega(1)$ so $k \in O(1)$, so the circuit complexity is dominated by $U$, which can be constructed using $n$ Hadamard gates and $n$ CNOT gates. Thus the state $I/D$ on a Hilbert space $\mathcal{H}$ encoded in $\mathbb{C}^n$ has an $O(\log(D))$-preparation-unitary. \\
The algorithm for estimating integrals is as follows:
\patbox{
\textbf{Algorithm: Integral of the Density of States}
\begin{enumerate}
\item Use Theorem~\ref{thm:windowfunc} to construct the polynomial $w(x)$ with $\eta := \frac{\varepsilon}{3}$.
\item Use Lemma~\ref{lemma:svt} to construct an $\frac{\varepsilon}{3}$-accurate $\frac{1}{2}$-scaled block-encoding of $w(H/\alpha)$. Say that this is an exact $\frac{1}{2}$-scaled block-encoding of $\tilde w(H/\alpha)$.
\item Use Lemma~\ref{lemma:observ} to produce an $\frac{\varepsilon}{3}$-accurate estimate $\xi$ of $\text{Tr}\left(\frac{I}{D} \cdot \tilde w(H/\alpha) \right)$ with probability at least $(1-\delta)$.
\end{enumerate}
}
By the triangle inequality the total error is at most $\varepsilon$.
The polynomial $w(x)$ has degree:
\begin{align}
d \in O\left( \frac{\rho_\text{max}}{\varepsilon} \log \frac{\rho_\text{max}}{\varepsilon} \right)
\end{align}
The approximate block-encoding of $\tilde w(H/\alpha)$ has circuit complexity $O(dQ)$ and the preparation unitary for $I/D$ has circuit complexity $\log D$. Combining these with the number of samples required by Lemma~\ref{lemma:observ} gives the overall complexity (\ref{eqn:densintcomplexity}).
The algorithm for Chebyshev Moments is significantly simpler:
\patbox{
\textbf{Algorithm: Chebyshev Moments of Density of States}
\begin{enumerate}
\item Use Lemma~\ref{lemma:cheby} to construct a block-encoding of $T_n(H/\alpha)$.
\item Use Lemma~\ref{lemma:observ} to produce an $\varepsilon$-accurate estimate $\zeta$ of $\text{Tr}\left(\frac{I}{D} \cdot T_n(H/\alpha)\right)$ with probability at least $(1-\delta)$.
\end{enumerate}
}
Since the block-encoding and state preparation are exact, the error stems entirely from the estimation procedure in Lemma~\ref{lemma:observ}. The circuit complexity from Lemma~\ref{lemma:cheby} is $O(nQ)$, so the overall complexity (\ref{eqn:denschebycomplexity}) also follows from Lemma~\ref{lemma:observ}.
\end{proof}
Estimation of integrals of $\rho(E)$ benefit from knowledge of an upper bound $\rho_\text{max}$. Indeed even in pathological cases where $H \propto I$ we have $\rho_\text{max} = 1$, so the circuit complexity can never suffer from high densities of state. We argue that in practical situations prior information on $H$ can be used to bound $\rho_\text{max}$, thereby improving the complexity. For example, the DOS of quantum many body systems with local interactions is often close to a Gaussian due to the central limit theorem. In particular, \cite{0406100} discusses the DOS of a nearest-neighbor Hamiltonian acting on a spin chain. From their work on the transverse-field Ising model with $n$ sites we can derive:
$$\rho_\text{max} = \frac{C}{D}\binom{n}{n/2} \approx C\pi\sqrt{\frac{2}{n}} $$
for some constant $C$ (see the discussion surrounding equation 30 in \cite{0406100}). Here $\rho_\text{max}$ decreases with the number of sites.
Furthermore, exact degeneracy in a Hamiltonian is connected to the Hamiltonian's symmetries \cite{1608.02600}. If there exists a degenerate subspace of dimension $D\rho_\text{max}$ then any unitary transformations on that subspace must preserve the Hamiltonian. Thus, prior knowledge of the symmetries could be used to obtain a bound on $\rho_\text{max}$. However, if only a subset of the symmetries is known then this only leads to a lower bound on the dimension of the largest eigenspace, which is not useful here.
Of course, the efficiency of the algorithm relies on the $1/D$ factor in our definition of $\rho(E)$. If we were interested in the actual number of states within an interval, the circuit complexity would scale with $D$ (for fixed $\varepsilon$). This is to be expected since the number of states in the ground space of a Hamiltonian is \#$\mathsf{P}$-hard to compute exactly and $\mathsf{NP}$-hard to estimate to within relative error \cite{1010.3060}.
Next we consider the local density of states. Say we are working with a Hamiltonian describing a single particle in real space or some space with a notion of locality so that for every position $\vec r$ there is a state $\ket{\psi(\vec r)}$ denoting the state with the particle at $\vec r$. Then local density of states (LDOS) at $\vec r$ is given by \cite{0504627, diventra, 1309.5730}:
\begin{align} \rho_{\vec r}(E) = \sum_i \delta(E_i - E) |\braket{\psi_i | \vec r}|^2 \end{align}
The algorithms for sketching the LDOS are a simple modification of the algorithms for DOS: instead of preparing a maximally mixed state we simply prepare $\ket{\psi(\vec r)}$. Indeed if $\ket{\psi(\vec r)}$ has an $O(R)$-preparation unitary, the new circuit complexities are the same as those in Theorem~\ref{thm:dosalg} but with $\log D$ replaced with $R$.
If $H$ is a lattice Hamiltonian, e.g. a Fermi-Hubbard model, then the states $\ket{\psi(\vec r)}$ are trivial to prepare since the Jordan-Wigner transformation that maps $H$ to qubits preserves locality. For Hamiltonians describing a particle in real-space, the cost of preparing $\ket{\psi(\vec r)}$ depends on the particular choice of basis functions, e.g. Hartree-Fock, used to encode $H$ on the quantum computer.
Similarly to the DOS, estimation of LDOS can benefit from bounds on $\rho_\text{max}$ and it remains true that even for pathological Hamiltonians like $H \propto I$ we have $\rho_\text{max} \leq 1$. However, it no longer makes sense to bound $\rho_\text{max}$ via a central limit theorem since there is only one particle involved.
\section{Linear Response}
In this section we show how to sketch correlation functions of the form:
\begin{align} A(E - E_0) = \left\langle B \delta(E - H)C \right\rangle \label{eqn:lres} \end{align}
We shift the function by the ground state energy $E_0$ since we consider estimation of the ground state energy out of scope. This work improves on an quantum algorithm by \cite{1804.01505} and is useful to compare to a classical algorithm based on matrix product states \cite{1101.5895} that also uses the kernel polynomial method.
Following a similar argument to (\ref{eqn:windowderiv1}-\ref{eqn:windowderiv5}) and (\ref{eqn:mudef}), we connect the desired quantities to expectations of observables that can be represented by block-encodings:
\begin{align}
\int_a^b A(E-E_0) dE &\approx \left\langle B w(H/\alpha) C \right\rangle\\
\mu^A_n &= \left\langle B T_n(H/\alpha) C \right\rangle
\end{align}
This naturally yields quantum algorithms quite similar to those presented in Theorem~\ref{thm:dosalg}, just with some constants changed.
\begin{theorem} Let:
\begin{itemize}
\item $H$ have an $\alpha$-scaled $Q$-block-encoding,
\item $\rho$ have an $R$-preparation-unitary,
\item $B$ have $\beta$-scaled $S_B$-block-encoding and $C$ have $\gamma$-scaled $S_C$-block-encoding.
\end{itemize}
\hspace{1cm}
Then for any $\varepsilon,\delta > 0$:
\begin{enumerate}
\item For any $a,b$ such that $-\alpha < a < b < \alpha$ there exists a quantum algorithm that produces an estimate $\xi$ of $\int_{a}^b A(E) dE$ with circuit complexity
\begin{align}O\left( \left( Q d + S_B + S_C + R\right) \cdot \frac{\beta\gamma}{\varepsilon} \log\frac{1}{\delta} \right) \label{eqn:respintcomplexity}\end{align}
and $O(\text{poly}(d))$ classical pre-processing, where $\rho_\text{max}$ a bound on the dimension of the largest eigenspace and
\begin{align}d =O\left( \frac{\rho_\text{max} \beta\gamma}{\varepsilon}\log\frac{\rho_\text{max} \beta\gamma}{\varepsilon}\right).\end{align}
\item For any $n$ there exists a quantum algorithm that produces an estimate $\zeta$ of $\mu^A_n$ with circuit complexity
\begin{align}O\left((Qn + S_B + S_C + R)\cdot \frac{\beta\gamma}{\varepsilon} \right).\label{eqn:respchebycomplexity}\end{align}
\end{enumerate}
The estimates $\xi$ and $\zeta$ have error $\varepsilon$ with probability at least $(1-\delta)$ in their real and imaginary parts.\label{thm:respalg}
\end{theorem}
\begin{proof} The algorithm for computing integrals is as follows:
\patbox{
\textbf{Algorithm: Integrals of Linear Response Functions}
\begin{enumerate}
\item Use Theorem~\ref{thm:windowfunc} to construct the polynomial $w(x)$ with $\eta := \frac{\varepsilon}{3}$.
\item Use Lemma~\ref{lemma:svt} to construct an $\frac{\varepsilon}{3}$-accurate $\frac{1}{2}$-scaled block-encoding of $w(H/\alpha)$, and say it is an exact $\frac{1}{2}$-scaled block-encoding of $ \tilde w(H/\alpha)$.
\item Use Lemma~\ref{lemma:lcu} to construct a $\frac{1}{2}\beta\gamma$-scaled block-encoding of $\Xi := B \tilde w(H/\alpha) C$.
\item Use Lemma~\ref{lemma:observ} to produce an $\frac{\varepsilon}{3}$-accurate estimates of the real and imaginary parts of $\xi$ with probability at least $(1-\delta)$, corresponding to the Hermitian and anti-Hermitian parts of $\Xi$ as in (\ref{eqn:herm},\ref{eqn:antiherm}).
\end{enumerate}
}
The accuracy and complexity analysis is almost identical to that in Theorem~\ref{thm:dosalg}, except for the fact that since $|B| \leq \beta$ and $|C| \leq \gamma$ we observe that $A(\alpha x) $ is bounded by $\rho_\text{max}\beta\gamma$ when invoking Theorem~\ref{thm:windowfunc}. The algorithm for Chebyshev moments is as follows:
\patbox{
\textbf{Algorithm: Chebyshev Moments of Linear Response Functions}
\begin{enumerate}
\item Use Lemma~\ref{lemma:cheby} to construct a block-encoding of $T_n(H/\alpha)$.
\item Use Lemma~\ref{lemma:lcu} to construct a $\beta\gamma$-scaled block-encoding of $Z := B T_n(H/\alpha)C$.
\item Use Lemma~\ref{lemma:observ} to produce an $\varepsilon$-accurate estimates of the real and imaginary parts of $\zeta$ with probability at least $(1-\delta)$, corresponding to the Hermitian and anti-Hermitian parts of $Z$ as in (\ref{eqn:herm},\ref{eqn:antiherm}).
\end{enumerate}
}
\end{proof}
This technique is significantly more versatile than that of \cite{1804.01505}, which only treats the case when $B = C$ and when $\rho = \ket{\psi_0}\bra{\psi_0}$. Their algorithm runs Hamiltonian simulation under $B$ for a short amount of time to approximately prepare the state $B\ket{\psi_0}$, which is an additional source of error. Furthermore their work also does not capitalize on accuracy improvements from amplitude estimation.
The classical strategy \cite{1101.5895} relies on Matrix Product State (MPS) representations of states $\ket{t_n} = T_n(H/\alpha)C\ket{\psi_0}$. When accurate and efficient MPS representations of $\ket{t_n}$ exist (and $\ket{\psi_0}$ can be efficiently obtained - an assumption we also make), then quantum strategies are not needed. Indeed for many physical systems ground states obey area laws (see e.g. \cite{1905.11337}), which lends MPS strategies their power. Quantum strategies will still be useful for ground states with large amounts of entanglement where efficient classical representations do not exist.
\section{Conclusion}
We have demonstrated that block-encodings provide a powerful framework for the matrix arithmetic on a quantum computer. This modern and versatile toolkit for quantum algorithms encompasses fundamental strategies such as amplitude amplification and estimation, and novel results in active areas like Hamiltonian simulation can be immediately leveraged due to its modularity. Furthermore, once all the necessary tools are assembled, algorithms based on block-encodings are trivial to analyze. We believe that block-encodings are the state-of-the-art technique for estimating physical quantities on a quantum computer. This claim should be further tested by attempting to quantize other numerical strategies in condensed matter physics.
\appendix
\section{Constructing a Polynomial Approximation of the Window Function\label{section:window}}
In this section we prove Theorem~\ref{thm:windowfunc} by following a construction in \cite{0902.3757}. We make use of an important theorem in approximation theory:
\begin{theorem*} (Jackson's Theorem \cite{rivlin}.) For any continuous function $g(x)$ on the interval $[-1,1]$ there exists a polynomial $J(x)$ of degree at most $n$ so that for all $x \in [-1,1]$:
\begin{align} |J(x) - g(x)| \leq 6\omega_g(1/n), \end{align}
where $\omega_g(\delta)$ is the modulus of continuity of $g(x)$:
\begin{align}\omega_g(\delta) := \text{sup}\big\{& \text{ } |g(x) - g(y)|\nonumber\\ &\text{ for } x,y \in [-1,1] \text{ with } |x-y| \leq \delta \big\}. \end{align}
\end{theorem*}
Below we prove Theorem~\ref{thm:windowfunc} with $\eta$ rescaled to $\eta f_\text{max}$. If Jackson's theorem were to be used to construct the desired polynomial approximation directly then the degree would scale with $O(\eta^{-2})$. By introducing an amplifying polynomial we improve this to $O( \frac{1}{\eta}\ln\frac{1}{\eta} )$.
\begin{theorem*} (Theorem~\ref{thm:windowfunc} restated.) For every $\eta > 0$ and any $\bar a, \bar b$ with $-1 < \bar a < \bar b < 1$ there there exists a polynomial $w(x)$ such that for all $f(x)$ bounded by $f_\text{max}$:
\begin{align}\left|\int_{-1}^{1} f(x) w(x) dx - \int_{\bar a}^{\bar b} f(x) dx \right| \leq \eta f_\text{max} \end{align}
The polynomial has degree $d \in O( \frac{1}{\eta}\ln\frac{1}{\eta} ) $ and $w(x)/2$ satisfies the requirements of Lemma~\ref{lemma:svt}.
\end{theorem*}
\begin{proof}
Let $\kappa := \eta/4$. We begin by applying Jackson's theorem to a function $g(x)$ sketched in FIG.~1~a). $g(x) = 1$ in the region $[\bar a, \bar b]$ and $g(x) = -1$ outside of $[\bar a - \kappa ,\bar b + \kappa]$ and interpolates linearly between the gaps. We have $\omega_g(\delta) = \delta/\kappa$, so if we choose $n := 24/\kappa$ we obtain:
\begin{align} |J(x) - g(x)| \leq 6\omega_g(1/n) = \frac{6}{\kappa n} = \frac{1}{4} \end{align}
$J(x)$ is sketched in FIG.~1~b), and is guaranteed to stay inside the shaded region. Next we define the amplifying polynomial $A_k(x)$:
\begin{align} A_k(x) := \sum_{j \geq k/2} \binom{k}{j} \left(\frac{1+x}{2}\right)^j \left(\frac{1-x}{2}\right)^{k-j} \end{align}
Let $X$ be a random variable distributed as the sum of $k$ i.i.d. Bernoulli random variables, each with expectation $\frac{1+x}{2}$, and observe that $A_k(x) = \text{Pr}[X \geq k/2]$. Then it follows from the Chernoff bound that $A_k(x)$ stays inside the shaded region of FIG.~1~c) where $\tau := e^{-k/6}$. Pick $k := \left\lceil 6\ln \frac{4}{\eta} \right\rceil$ so that $\tau \leq \eta/4$.
Finally, we use $A_k(x)$ to amplify the error of $J(x)$.
\begin{align}
w(x) := A_k\left( \frac{4}{5} J(x) \right)
\end{align}
This polynomial $w(x)$ is inside the shaded region of FIG.~1~d) and has degree:
\begin{align}d := n \cdot k \in O\left( \frac{1}{\eta}\ln\frac{1}{\eta} \right)\end{align}
Now we bound the error, which is intuitive from FIG.~1~d). In the region inside $[\bar a, \bar b]$ and outside $[\bar a - \kappa , \bar b + \kappa]$ we have an error at most $\tau$ and inside the interpolation regions we have error at most $1$. The regions have length $2-2\kappa$ and $2\kappa$ respectively, so:
\begin{align}&\left|\int_{-1}^{1} f(x) w(x) dx - \int_{\bar a}^{\bar b} f(x) dx \right|\cdot \frac{1}{f_\text{max}} \nonumber\\
&\leq \tau(2-2\kappa) + 2\kappa \leq 2\tau + 2\kappa \leq \frac{\eta}{2} + \frac{\eta}{2} = \eta
\end{align}
Here we implicitly use (\ref{eq:boundedby}). Note that a more careful choice of the division of error between regions may improve $d$ by a constant factor.
\end{proof}
\onecolumngrid
\begin{figure}
\caption{Functions involved in the proof of Theorem~\ref{thm:windowfunc}
\end{figure}
\end{document} |
\begin{document}
\title[On geometric properties of Morrey spaces]{On geometric properties of Morrey spaces}
\author[H.~Gunawan]{Hendra Gunawan}
\address{Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Bandung 40132, Indonesia}
\email{[email protected]}
\author[D.I.~Hakim]{Denny I. Hakim}
\address{Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Bandung 40132, Indonesia}
\email{[email protected]}
\author[A.S.~Putri]{Arini S. Putri}
\address{Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Bandung 40132, Indonesia}
\email{[email protected]}
\subjclass[2010]{46B20}
\keywords{Morrey spaces, uniformly non-$\ell^1_n$-ness, $n$-th James constant, $n$-th Von Neumann-Jordan constant}
\begin{abstract}
In this article, we show constructively that Morrey spaces are not uniformly non-$\ell^1_n$
for any $n\ge 2$. This result is sharper than those previously obtained in \cite{GKSS, MG},
which show that Morrey spaces are not uniformly non-square and also not uniformly non-octahedral.
We also discuss the $n$-th James constant $C_{{\rm J}}^{(n)}(X)$ and the $n$-th Von
Neumann-Jordan constant $C_{{\rm NJ}}^{(n)}(X)$ for a Banach space $X$, and obtain
that both constants for any Morrey space $\mathcal{M}^p_q(\mathbb{R}^d)$ with
$1\le p<q<\infty$ are equal to $n$.
\end{abstract}
\maketitle
\section{Introduction}
For $1\leq p\leq q<\infty$, the {\em Morrey space} $\mathcal{M}^p_q=\mathcal{M}^p_q(\mathbb{R}^d)$
is the set of all measurable functions $f$ such that
\begin{equation*}
\|f\|_{\mathcal{M}^p_q} := \sup_{{a\in \mathbb{R}^d},{R>0}}|B(a,R)|^{\frac{1}{q}-\frac{1}{p}}
\biggl(\int\limits_{B(a,R)} |f(y)|^p dy\biggr)^{\frac{1}{p}}<\infty,
\end{equation*}
where $|B(a,R)|$ denotes the Lebesgue measure of the open ball $B(a,R)$ in $\mathbb{R}^d$, with center $a$ and
radius $R$. Morrey spaces are Banach spaces (see, e.g., \cite{sawano}). For $p=q$, the space $\mathcal{M}^q_q$ is
identical with the space $L^q=L^q(\mathbb{R}^d)$, the space of $q$-th power integrable functions on $\mathbb{R}^d$.
In \cite{GKSS}, three geometric constants have been computed for Morrey spaces. The first two constants, namely
{\em Von Neumann--Jordan constant} and {\em James constant}, are closely related to the notion of uniformly
non-squareness of (the unit ball of) a Banach space \cite{james, jimenez, kato}. For a Banach space $(X,\|\cdot\|_X)$
in general, the constants are defined by
\begin{equation*}
C_{\rm NJ}(X) := \sup\biggl\{\frac{\|x+y\|_X^2+\|x-y\|_X^2}{2(\|x\|_X^2+\|y\|_X^2)} : x,y \in X\setminus \{0\} \biggr\},
\end{equation*}
and
\begin{equation*}
C_{\rm J}(X) := \sup\bigl\{\min\{\|x+y\|_X,\|x-y\|_X\} : x,y\in S_X \bigr\},
\end{equation*}
respectively. Here $S_X:=\{x\in X : \|x\|_X=1\}$ denotes the unit sphere in $X$. A few basic facts about these constants are:
\begin{itemize}
\item $1\leq C_{\rm NJ}(X)\leq 2$ and $C_{\rm NJ}(X)=1$ if and only if $X$ is a Hilbert space \cite{jordan1935inner}.
\item $\sqrt{2}\leq C_{\rm J}(X)\leq 2$ and $C_{\rm J}(X)=\sqrt{2}$ if $X$ is a Hilbert space \cite{gao1990geometry}.
\end{itemize}
Note also that, for $1\le p\le \infty$, we have $C_{\rm NJ}(L^p)=\max\{2^{\frac{2}{p}-1},2^{1-\frac{2}{p}}\}$
and $C_{\rm J}(L^p)=\max\{2^{\frac{1}{p}},2^{1-\frac{1}{p}}\}$ \cite{clarkson, gao1990geometry}.
As for Morrey spaces, we have the following results.
\begin{theorem}{\rm \cite{GKSS}}\label{Morrey}
For $1\le p<q<\infty$, we have $C_{\rm NJ}(\mathcal{M}^p_q)=C_{\rm J}(\mathcal{M}^p_q)=2$.
\end{theorem}
This theorem tells us that Morrey spaces are not {\em uniformly non-square} (since a Banach space is uniformly non-square
if and only if $C_{\rm J}(X)<2$ or, equivalently, $C_{\rm NJ}(X)<2$). In \cite{MG}, we are also told that Morrey spaces
are not {\em uniformly non-octahedral}, that is, there does not exist a $\delta>0$ such that
\[
\min \|f\pm g\pm h\|_{\mathcal{M}^p_q} \le 3(1-\delta)
\]
for every $f,g,h\in \mathcal{M}^p_q$ with $\|f\|_{\mathcal{M}^p_q}=\|g\|_{\mathcal{M}^p_q}=\|h\|_{\mathcal{M}^p_q}=1$.
Here the minimum is taken over all choices of signs in the expression $f\pm g\pm h$. (A Banach space $(X,\|\cdot\|_X)$
is uniformly non-octahedral if there exists a $\delta>0$ such that
\[
\min \|x\pm y\pm z\|_X \le 3(1-\delta)
\]
for every $x,y,z\in S_X$.) Precisely, we have the following theorem.
\begin{theorem}{\rm \cite{MG}}\label{Octahedral}
Let $1\leq p< q<\infty$. Then, for every $\delta>0$, there exist $f,g,h\in \mathcal{M}^p_q$ (depending
on $\delta$) with $\|f\|_{\mathcal{M}^p_q}=\|g\|_{\mathcal{M}^p_q}=\|h\|_{\mathcal{M}^p_q}=1$ such that
\[
\|f\pm g\pm h\|_{\mathcal{M}^p_q}> 3(1-\delta)
\]
for all choices of signs.
\end{theorem}
In this article, we shall show constructively that Morrey spaces are not {\em uniformly non}-$\ell^1_n$. The result is not only more
general than the previous ones, but also sharper than knowing that Morrey spaces are neither uniformly non-square nor
uniformly non-octahedral (for if $X$ is not uniformly non-$\ell^1_n$ for $n\ge3$, then $X$ is not uniformly non-$\ell^1_{n-1}$).
In addition, given a Banach space $X$, we shall discuss the $n$-th Von Neumann-Jordan constant $C_{{\rm NJ}}^{(n)}(X)$ and
the $n$-th James constant $C_{{\rm J}}^{(n)}(X)$ for $n\ge2$. These two constants were studied in \cite{KTH} and \cite{MNPZ},
respectively. We show that for any Morrey space $\mathcal{M}^p_q$ with $1\le p<q<\infty$ both constants are equal to $n$.
We also indicate that Morrey spaces are not {\em uniformly $n$-convex} for $n\ge2$.
\section{$\mathcal{M}^p_q$ is not uniformly non-$\ell^1_n$}
Before we present our main theorems, we prove several lemmas. We assume that the readers know well how to compute
the integral of a radial function over a ball centered at 0 using polar coordinates. Unless otherwise stated,
we assume that $1\le p<q<\infty$.
\begin{lemma}\label{le:140320-1}
Let $f(x):=|x|^{-d/q}$. Then $f\in \mathcal{M}^p_q$ with
\[
\|f\|_{\mathcal{M}^p_q}=\Bigl(\frac{\omega_{d-1}}{d}\Bigr)^{\frac{1}{q}} \Bigl(\frac{q}{q-p}\Bigr)^{\frac{1}{p}},
\]
where $\omega_{d-1}$ denotes the `area' of the unit sphere in $\mathbb{R}^d$.
\end{lemma}
\begin{proof}
For each $r>0$, one may compute that
\[
|B(0,r)|^{1/q-1/p}\Bigl(\int_{B(0,r)} |x|^{-dp/q}dx\Bigr)^{1/p}=\Bigl(\frac{\omega_{d-1}}{d}\Bigr)^{1/q}
\Bigl(\frac{q}{q-p}\Bigr)^{1/p},
\]
which is independent of $r$. Since the integral of $f$ over $B(a,r)$ will be less than that over $B(0,r)$
for every $a\in\mathbb{R}^d$, we conclude that $\|f\|_{\mathcal{M}^p_q}=
\bigl(\frac{\omega_{d-1}}{d}\bigr)^{\frac{1}{q}} \bigl(\frac{q}{q-p}\bigr)^{\frac{1}{p}}$, as claimed.
\end{proof}
\begin{lemma}\label{le:140320-2}
Let $f(x):=|x|^{-d/q}$ and $R>1$. Then, for any $c_1,c_2>0$, we have
\begin{align}\label{eq:140320-3}
|B_{c_1R}|^{1/q-1/p}\Bigl(\int_{\{x:c_1<|x|<c_1R\}} |f(x)|^p dx\Bigr)^{1/p}=|B_{c_2R}|^{1/q-1/p}
\Bigl(\int_{\{x:c_2<|x|<c_2R\}}|f(x)|^p dx\Bigr)^{1/p},
\end{align}
where $B_{c_1R}$ and $B_{c_2R}$ are balls centered at $0$ with radii $c_1R$ and $c_2R$.
\end{lemma}
\begin{proof}
It suffices to prove that \eqref{eq:140320-3} holds for arbitrary $c_1>0$ and $c_2=1$. But this is
immediate by the change of variable $x=c_1x'$.
\end{proof}
As a consequence of the above lemma, we have the following corollary, which is an important key to
our main theorems.
\begin{cor}\label{cor:140320-1}
Let $f(x):=|x|^{-d/q}$. For $\varepsilon\in(0,1)$ and $k\in\mathbb{N}$, put $f_{\varepsilon,k}:=
f\chi_{\{x:\varepsilon^{k+1}<|x|<\varepsilon^{k}\}}$. Then $f_{\varepsilon,k}\in \mathcal{M}^p_q$
with
\begin{align}\label{eq:140320-2}
\|f_{\varepsilon,k}\|_{\mathcal{M}^p_q}\ge (1-\varepsilon^{d-dp/q})^{1/p}\|f\|_{\mathcal{M}^p_q}.
\end{align}
\end{cor}
\begin{proof}
In view of Lemma \ref{le:140320-2}, it suffices to prove that
\[
\|f_{\varepsilon,0}\|_{\mathcal{M}^p_q}\ge (1-\varepsilon^{d-dp/q})^{1/p}\|f\|_{\mathcal{M}^p_q}.
\]
We observe that
\begin{align*}
\|f_{\varepsilon,0}\|_{\mathcal{M}^p_q}
\ge |B(0,1)|^{\frac{1}{q}-\frac1p}\Bigl(\int_{\{x:\varepsilon<|x|<1\}} |f(x)|^pdx\Bigr)^{1/p}=
\Bigl(\frac{\omega_{d-1}}{d}\Bigr)^{\frac{1}{q}} \Bigl(\frac{q}{q-p}\Bigr)^{\frac{1}{p}}(1-\varepsilon^{d-dp/q})^{1/p}.
\end{align*}
Hence, by Lemma \ref{le:140320-1}, the desired inequality follows.
\end{proof}
We are now ready to state our main results. Our first theorem is the following.
\begin{theorem}\label{unl1n}
For $1\leq p< q<\infty$, the Morrey space $\mathcal{M}^p_q$ is not uniformly non-$\ell^1_n$ for any $n\ge 2$,
that is, for every $\delta\in (0,1)$, there exist $f_1,f_2\dots, f_n\in \mathcal{M}^p_q$
(depending on $\delta$) with $\|f_i\|_{\mathcal{M}^p_q}=1$ for $i=1,2,\dots,n$, such that
\[
\|f_1\pm f_2\pm \cdots \pm f_n\|_{\mathcal{M}^p_q}> n(1-\delta)
\]
for all choices of signs.
\end{theorem}
\begin{proof}
To understand the idea of the proof, let us first ilustrate how the proof goes for $n=3$. Given $\delta \in (0,1)$,
choose $\displaystyle \varepsilon \in \bigl(0, (1-(1-\delta)^p)^{\frac{q}{dq-dp}}\bigr).$
For $f(x):=|x|^{-d/q}$ and $k\in\mathbb{N}$, put $f_{\varepsilon,k}:=f\chi_{\{x:\varepsilon^{k+1}<|x|<\varepsilon^{k}\}}$.
Now write
\begin{align*}
f_1&:=(+1,+1,+1,+1):=f_{\varepsilon,3}+f_{\varepsilon,2}+f_{\varepsilon,1}+f_{\varepsilon,0},\\
f_2&:=(+1,+1,-1,-1):=f_{\varepsilon,3}+f_{\varepsilon,2}-f_{\varepsilon,1}-f_{\varepsilon,0},\\
f_3&:=(+1,-1,+1,-1):=f_{\varepsilon,3}-f_{\varepsilon,2}+f_{\varepsilon,1}-f_{\varepsilon,0}.
\end{align*}
Observe that $\|f_i\|_{\mathcal{M}^p_q}=\|f\chi_{\{x:\varepsilon^4<|x|<1\}}\|_{\mathcal{M}^p_q}$ for $i=1,2,3$, and that
\begin{align*}
3f_{\varepsilon,3} &\le |f_1+f_2+f_3| \le 3f\chi_{\{x:\varepsilon^4<|x|<1\}},\\
3f_{\varepsilon,2} &\le |f_1+f_2-f_3| \le 3f\chi_{\{x:\varepsilon^4<|x|<1\}},\\
3f_{\varepsilon,1} &\le |f_1-f_2+f_3| \le 3f\chi_{\{x:\varepsilon^4<|x|<1\}},\\
3f_{\varepsilon,0} &\le |f_1-f_2-f_3| \le 3f\chi_{\{x:\varepsilon^4<|x|<1\}}.
\end{align*}
By virtue of Corollary \ref{cor:140320-1}, we have
\[
3(1-\varepsilon^{d-dp/q})^{1/p}\|f\|_{\mathcal{M}^p_q} \le \|f_1\pm f_2 \pm f_3\|_{\mathcal{M}^p_q} \le
3\|f\chi_{\{x:\varepsilon^4<|x|<1\}}\|_{\mathcal{M}^p_q}
\]
for all choices of signs. For $i=1,2,3$, define $ \displaystyle F_i:=\frac{f_i}{\|f_i\|_{\mathcal{M}^p_q}}$.
Then, $\|F_i\|_{\mathcal{M}^p_q}=1$ for $i=1,2,3$, and
\[
\|F_1\pm F_2 \pm F_3\|_{\mathcal{M}^p_q}
=\frac{\|f_1\pm f_2 \pm f_3\|_{\mathcal{M}^p_q}}{\|f\chi_{\{x:\varepsilon^4<|x|<1\}}\|_{\mathcal{M}^p_q}}
\ge
\frac{3(1-\varepsilon^{d-dp/q})^{1/p}\|f\|_{\mathcal{M}^p_q}}{\|f\|_{\mathcal{M}^p_q}}
>3(1-\delta).
\]
This proves that $\mathcal{M}^p_q$ is not uniformly non-$\ell^1_3$.
In order to reveal the pattern, we shall now present the proof for $n=4$. With similar notations as above, we write
\begin{align*}
f_1&:=(+1,+1,+1,+1,+1,+1,+1,+1),\\
f_2&:=(+1,+1,+1,+1,-1,-1,-1,-1),\\
f_3&:=(+1,+1,-1,-1,+1,+1,-1,-1),\\
f_4&:=(+1,-1,+1,-1,+1,-1,+1,-1),
\end{align*}
where the $i$-th term corresponds to the sign of $f_{\varepsilon,{8-i}}$ for $i=1,2, \ldots,8$.
Observe that $\|f_i\|_{\mathcal{M}^p_q}=\|f\chi_{\{x:\varepsilon^8<|x|<1\}}\|_{\mathcal{M}^p_q}$ for $i=1,\dots,4$, and that
\begin{align*}
4f_{\varepsilon,7} &\le |f_1+f_2+f_3+f_4| \le 4f\chi_{\{x:\varepsilon^8<|x|<1\}},\\
4f_{\varepsilon,6} &\le |f_1+f_2+f_3-f_4| \le 4f\chi_{\{x:\varepsilon^8<|x|<1\}},\\
4f_{\varepsilon,5} &\le |f_1+f_2-f_3+f_4| \le 4f\chi_{\{x:\varepsilon^8<|x|<1\}},\\
4f_{\varepsilon,4} &\le |f_1+f_2-f_3-f_4| \le 4f\chi_{\{x:\varepsilon^8<|x|<1\}},\\
4f_{\varepsilon,3} &\le |f_1-f_2+f_3+f_4| \le 4f\chi_{\{x:\varepsilon^8<|x|<1\}},\\
4f_{\varepsilon,2} &\le |f_1-f_2+f_3-f_4| \le 4f\chi_{\{x:\varepsilon^8<|x|<1\}},\\
4f_{\varepsilon,1} &\le |f_1-f_2-f_3+f_4| \le 4f\chi_{\{x:\varepsilon^8<|x|<1\}},\\
4f_{\varepsilon,0} &\le |f_1-f_2-f_3-f_4| \le 4f\chi_{\{x:\varepsilon^8<|x|<1\}}.
\end{align*}
Taking the Morrey norms, we get
\begin{align}
4(1-\varepsilon^{d-dp/q})^{1/p}\|f\|_{\mathcal{M}^p_q} \le \|f_1\pm f_2 \pm f_3 \pm f_4\|_{\mathcal{M}^p_q} \le
4\|f\chi_{\{x:\varepsilon^8<|x|<1\}}\|_{\mathcal{M}^p_q}
\end{align}
for all choices of signs. Taking $F_i:=\frac{f_i}{\|f_i\|_{\mathcal{M}^p_q}}$, we obtain
$\|F_i\|_{\mathcal{M}^p_q}=1$ for $i=1,\dots,4$. By our choice of $\varepsilon$ and the
fact that $\|f_i\|_{\mathcal{M}^p_q}=\|f\chi_{\{x:\varepsilon^8<|x|<1\}}\|_{\mathcal{M}^p_q}$
for $i=1,\dots,4$, we get
\[
\|F_1\pm F_2 \pm F_3 \pm F_4\|_{\mathcal{M}^p_q} > 4(1-\delta).
\]
Hence $\mathcal{M}^p_q$ is not uniformly non-$\ell^1_4$. Continuing the pattern, we see that
$\mathcal{M}^p_q$ is not uniformly non-$\ell^1_n$ for $n\ge2$.
\end{proof}
\section{$n$-th Von Neumann-Jordan Constant and $n$-th James Constant}
In this section, let $n\ge2$. For a Banach space $(X,\|\cdot\|_X)$, the $n$-th Von Neumann-Jordan constant
$C_{{\rm NJ}}^{(n)}(X)$ \cite{KTH} and the $n$-th James constant $C_{{\rm J}}^{(n)}(X)$ \cite{MNPZ} are defined by
\[
C_{{\rm NJ}}^{(n)}(X):=\sup \left\{\frac{\sum_{\pm} \|x_1\pm\cdots \pm x_n\|_X^2}{2^{n-1}\sum_{i=1}^n \|x_i\|_X^2}\,:\,x_i\not=0,
\ i=1,\dots,n\right\},
\]
and
\[
C_{{\rm J}}^{(n)}(X):=\sup \bigl\{\min \{\|x_1\pm \cdots \pm x_n\|_X:{\rm all~possible~choices~of~signs}\}:
x_i\in S_X,\ i=1,\dots,n\bigr\},
\]
respectively. In the definition of $C_{{\rm NJ}}^{(n)}(X)$, the sum $\sum_{\pm}$ is taken over all possible choices of signs.
We state some results about the two constants. The last one is specific for Morrey spaces.
\begin{theorem}\label{NJ-constants}\cite{KTH}
For a Banach space $(X,\|\cdot\|_X)$ in general, we have $1\le C_{{\rm NJ}}^{(n)}(X) \le n$.
In particular, $C_{{\rm NJ}}^{(n)}(X)=1$ if and only if $X$ is a Hilbert space.
\end{theorem}
\begin{theorem}\label{J-constant}\cite{MNPZ}
For a Banach space $(X,\|\cdot\|_X)$ in general, we have $1\le C_{{\rm J}}^{(n)}(X) \le n$.
If dim$(X)=\infty$, then $\sqrt{n}\le C_{{\rm J}}^{(n)}(X) \le n$.
For a Hilbert space $(X,\langle\cdot,\cdot\rangle_X)$, we have $C_{{\rm J}}^{(n)}(X)=\sqrt{n}$.
\end{theorem}
\begin{theorem}\label{J-NJ-ineq}
For a Banach space $(X,\|\cdot\|_X)$ in general, we have
\[
[C_{{\rm J}}^{(n)}(X)]^2 \le nC_{{\rm NJ}}^{(n)}(X).
\]
\end{theorem}
\begin{proof}
For every $x_i\in S_X,\ i=1,\dots,n$, let $m:=\min \{\|x_1\pm \cdots \pm x_n\|_X:{\rm all~possible~choices~of~signs}\}$.
Then, clearly $m\le \Bigl( \prod_\pm \|x_1\pm \cdots \pm x_n\|_X\Bigr)^{1/n}$, where the product is taken over all
possible choices of signs. Next, by the GM-QM inequality and the last inequality, we have
\begin{align*}
m
&\le \left(\frac{\sum_{\pm} \|x_1\pm\cdots \pm x_n\|^2_X}{2^{n-1}}\right)^{1/2}\\
&=\left(n\cdot \frac{\sum_{\pm} \|x_1\pm\cdots \pm x_n\|^2_X}{2^{n-1}\sum_{i=1}^n \|x_i\|_X^2}\right)^{1/2}\\
&\le \bigl(nC_{{\rm NJ}(n)(X)}\bigr)^{1/2}.
\end{align*}
Taking the supremum over all $x_i\in S_X,\ i=1,\dots,n$, the desired inequality follows.
\end{proof}
\begin{theorem}
For $1\le p<q<\infty$, we have $C_{{\rm J}}^{(n)}(\mathcal{M}^p_q)=C_{{\rm NJ}}^{(n)}(\mathcal{M}^p_q)=n$.
\end{theorem}
\begin{proof}
It follows immediately from Theorem \ref{unl1n} that
$C_{{\rm J}}^{(n)}(\mathcal{M}^p_q)=n$. Combining this fact and Theorem \ref{J-NJ-ineq},
we get $C_{{\rm NJ}}^{(n)}(\mathcal{M}^p_q)\ge n$.
On the other hand, by Theorem \ref{NJ-constants}, we have
$C_{{\rm NJ}}^{(n)}(\mathcal{M}^p_q)\le n$. Thus, $C_{{\rm NJ}}^{(n)}(\mathcal{M}^p_q)=n$.
\end{proof}
\section{Concluding Remarks}
Before we end our paper, let us consider a Banach space $(X,\|\cdot\|_X)$ which is {\it uniformly $n$-convex},
that is, for every $\varepsilon\in(0,n)$ there exists $\delta\in(0,1)$ such that for every $x_1,\dots,x_n\in S_X$
with $\|x_1\pm\cdots\pm x_n\|_X$ $>\varepsilon$ for all choices of signs except for $\|x_1+\cdots+x_n\|_X$, we have
$\|x_1+\cdots+x_n\|_X\le n(1-\delta)$. This condition is stronger than the uniformly non-$\ell^1_n$ condition, as
we state in the following theorem.
\begin{theorem}
If $X$ is uniformly $n$-convex, then $X$ is uniformly non-$\ell^1_n$.
\end{theorem}
\begin{proof}
Assuming that $X$ is uniformly $n$-convex, we have to find a $\delta>0$ such that for every $x_1,\dots,x_n\in S_X$
with $\|x_1\pm\cdots\pm x_n\|_X>n(1-\delta)$ for all choices of signs except for $\|x_1+\cdots+x_n\|_X$, we have
$\|x_1+\cdots+x_n\|_X\le n(1-\delta)$. To do so, just take an $\varepsilon\in(0,n)$ and choose a corresponding
$\delta\in(0,1)$ such that for every $x_1,\dots,x_n\in S_X$ with $\|x_1\pm\cdots\pm x_n\|_X>\varepsilon$ for all
choices of signs except for $\|x_1+\cdots+x_n\|_X$, we have $\|x_1+\cdots+x_n\|_X\le n(1-\delta)$. Observe that if
$n(1-\delta)\ge\varepsilon$, then we are done. Otherwise, we choose $\delta_0\in(0,\delta)$ such that $n(1-\delta_0)
\ge\varepsilon$. This $\delta_0$ satisfies the uniformly non-$\ell^1_n$ condition.
\end{proof}
As a consequence of the above theorem and the fact that, for $1\le p<q<\infty$, the Morrey space $\mathcal{M}^p_q$
is not uniformly non-$\ell^1_n$, we conclude that $\mathcal{M}^p_q$ is not uniformly $n$-convex.
\noindent{\bf Acknowledgements}. This work is supported by P3MI-ITB 2020 Program. We thank
H.~Batkunde, Ifronika, and N.K.~Tumalun for useful discussions regarding the $n$-th James
constant and $n$-th Von Neumann-Jordan constant for Banach spaces.
\end{document} |
\begin{document}
\title{Noncommutative Henselizations}
\begin{abstract}
\noindent
In this paper, the familiar notion of a Henselian pair is extended to the noncommutative case. Furthermore, the problem of Henselizations is studied in the noncommutative context, and it is shown that every (not necessarily commutative) pair which is Hausdorff with respect to a certain topology has a left (and right) Henselization. \\
\textit{Keywords:}
Noncommutative Henselian pair,
Noncommutative Henselization
\end{abstract}
\begin{section}{Introduction}
The notion of a Henselian ring, introduced by Azumaya in \cite{azumaya1951maximally}, is well-known in the commutative case. This concept has been extended to the noncommutative case, see \cite{aryapoor2009non}. However, the theory of noncommutative Henselian rings is not as well-studied as its commutative counterpart, and many problems concerning noncommutative Henselian rings are still open. One such problem, discussed in \cite{aryapoor2009non}, is the problem of noncommutative Henselizations. In the commutative case, it is known that every local ring has a Henselization, see \cite{nagata1962local}.
The aim of this article is to investigate the notion of Henselization in the noncommutative context. One of our results is that every noncommutative local ring satisfying a kind of ``commutativity'' condition has a Henselization, see Subsection \ref{localringHensel}.
In Section \ref{section2}, we present the preliminaries. In particular, the notion of a noncommutative Henselian ring, introduced in \cite{aryapoor2009non}, is generalized in two different ways. First, we introduce the notions of left Henselian rings and right Henselian rings, which is more natural in the noncommutative context. Second, this concept is generalized to pairs as done by Lafon in the commutative case, see \cite{lafon1963anneaux}. The final section is devoted to the notions of left Henselizations and right Henselizations. It is shown that every perfect pair has a left (and right) Henselization which is unique up to unique isomorphism.
\end{section}
\begin{section}{Henselian pairs} \label{section2}
In this section, we present the preliminaries, and in particular, the notions of left Heneslian rings and right Henselian rings. In this paper, we shall assume that all rings have a unit, and all ring homomorphisms are unit-preserving.
\begin{subsection}{The category of pairs}
The category of pairs has been introduced in the commutative setting, see \cite{lafon1963anneaux}. One can enlarge this category to include noncommutative rings. More precisely, the objects of the category $\mathcal{P}$ of pairs are pairs $(A,I)$ where $A$ is a (unitary but not necessarily commutative) ring and $I$ is an ideal of $A$. A morphism $\phi:(A,I)\to (B,J)$ of pairs is a ring homomorphism $\phi:A\to B$ satisfying $\phi^{-1}(J)=I$. It is straightforward to verify that any directed system in $\mathcal{P}$ has a direct limit. Likewise, any inverse system in $\mathcal{P}$ has an inverse limit.
For a pair $(A,I)$, we denote the image of $a\in A$ in $A/I$ by $\bar{a}$. Given a morphism $\phi:(A,I)\to (B,J)$ of pairs, we have a canonical homomorphism $\bar{\phi}:A/I\to B/J$ induced by $\phi$.
The proof of the following lemma is straightforward and left to the reader.
\begin{lemma}\label{morphism}
Let $\phi:(A,I)\to (C,K)$ and $\psi:(B,J)\to (C,K)$ be morphisms of pairs. If there is a ring homomorphism $\alpha:A\to B$ such that $\phi=\psi\circ\alpha$ as ring homomorphisms, then $\alpha:(A,I)\to (B,J)$ defines a morphism of pairs.
\end{lemma}
\end{subsection}
\begin{subsection}{Coprime polynomials}
For a ring $A$, the ring of polynomials $F(x)=\sum_{i=0}^na_ix^i$ over $A$, where the indeterminate $x$ commutes with all $a\in A$, is denoted by $A[x]$. For a subset $S\subset A$, the set of all polynomials $\sum_{i=0}^na_ix^i\in A[x]$, where $a_0,...,a_n\in S$, is denoted by $S[x]$.
Two polynomials $F_1(x),F_2(x)\in A[x]$ are called \textit{left coprime} if
$$A[x]F_1(x)+A[x]F_2(x)=A[x].$$
The notion of right coprime polynomials is defined in the obvious way. It is easy to see that polynomials $F_1(x),F_2(x)\in A[x]$ are left coprime if and only if there are polynomials $G_1(x),G_2(x)\in A[x]$ such that $$G_1(x)F_1(x)+G_2(x)F_2(x)=1.$$
For a pair $(A,I)$, we denote the canonical homomorphism $A[x]\to (A/I)[x]$ by $F(x)=\sum_i a_i x^i\mapsto \bar{F}(x)=\sum_i \bar{a}_i x^i$.
\end{subsection}
\begin{subsection}{Euclidean algorithm}
Let $A$ be a ring. The following version of Euclidean algorithm holds in $A[x]$.
\begin{lemma}\label{EuclidAlg}
Let $I$ be a right ideal of $A$. For every monic polynomial $F(x)\in A[x]$ and every polynomial $G(x)\in I[x]$, there exist polynomials $Q(x),R(x)\in I[x]$ such that
$$G(x)=Q(x)F(x)+R(x) \; \text{and}\; \deg(R(x))<\deg(F(x)).$$
\end{lemma}
\begin{proof}
We use induction on $\deg(G(x))$. If $\deg(G(x))<\deg(F(x))$, we set $Q(x)=0$ and $R(x)=G(x)$, so we are done. Let $G(x)=\sum_{i=0}^mb_ix^i$ and $$F(x)=a_0+a_1x+\cdots+a_{n-1}x^{n-1}+x^n,$$
where $m\geq n$. The polynomial $G_1(x)=G(x)-b_m x^{m-n}F(x)$ belongs to $I$ because $b_m\in I$ and $I$ is a right ideal. Since
$\deg(G_1(x))<\deg(G(x)),$ by induction, there are
polynomials $Q_1(x),R(x)\in I[x]$ such that
$$G_1(x)=Q_1(x)F(x)+R(x) \; \text{and}\; \deg(R(x))<\deg(F(x)).$$
Setting $Q(x)=b_mx^{m-n}+Q_1(x)$, we obtain
$$G(x)=Q(x)F(x)+R(x),$$
where $Q(x),R(x)$ satisfy the desired conditions, and we are done.
\end{proof}
\end{subsection}
\begin{subsection}{Jacobson pairs}
A pair $(A,I)$ is called a \textit{Jacobson pair} if $I\subset rad(A)$ where $rad(A)$ is the Jacobson radical of $A$. We have the following result concerning Jacobson pairs.
\begin{lemma}\label{coprime-reduction}
Let $(A,I)$ be a Jacobson pair. Then, monic polynomials $F_1(x),F_2(x)\in A[x]$ are left (resp. right) coprime if and only if the polynomials $\bar{F}_1(x),\bar{F}_2(x)\in (A/I)[x]$ are left (resp. right) coprime.
\end{lemma}
\begin{proof}
The ``only if'' part is trivial. To prove the other direction, suppose that $\bar{F}_1(x),\bar{F}_2(x)\in (A/I)[x]$ are left coprime. It follows that $$A[x]F_1(x)+A[x]F_2(x)+I[x]=A[x]$$
Considering $$M=\frac{A[x]}{A[x]F_1(x)}$$ as a left $A$-module in the obvious way, we see that
$M=N+IM$ where $N$ is the submodule of $M$ generated by $F_2(x)$.
The module $M$ is a finitely generated $A$-module because $F_1(x)$ is monic. Since $I\subset rad(A)$, Nakayama's lemma
implies that $N=M$, that is,
$$A[x]F_1(x)+A[x]F_2(x)=A[x],$$
and we are done.
\end{proof}
\end{subsection}
\begin{subsection}{Local homomorphisms}
A ring homomorphism $\phi:A\to B$ is called \textit{local} if $\phi$ sends every nonunit in $A$ to a nonunit in $B$. Here, we provide some facts concerning local maps. The proof of the first lemma is easy and left to the reader.
\begin{lemma}\label{localmaps}
Let $\phi:A\to B$ and $\psi:B\to C$ be ring homomorphisms. If $\psi\circ \phi$ is local, then $\phi$ is local too.
\end{lemma}
\begin{lemma}\label{localmapsLocalrings}
Let $\phi:A\to B$ be a local homomorphism. If $B$ is a local ring, then $A$ is a local ring whose maximal ideal is $\phi^{-1}(I)$ where $I$ is the maximal ideal of $B$.
\end{lemma}
\begin{proof}
To show that $A$ is a local ring, we need to prove that if $a+b$ is invertible in $A$ for some $a,b\in A$, then either $a$ or $b$ is invertible in $A$, see Theorem 19.1 in \cite{lam2013first}. If $a+b$ is invertible in $A$ for some $a,b\in A$, then $\phi(a)+\phi(b)$ is invertible in $B$ which implies that $\phi(a)$ or $\phi(b)$ is invertible in $B$, because $B$ is local. It follows that either $a$ or $b$ is invertible in $A$ because $\phi$ is local. Therefore, $A$ is a local ring. Since $\phi$ is a local homomorphism and $B$ is a local ring, we see that $\phi^{-1}(I)$ consists of all nonunits in $A$, that is, $\phi^{-1}(I)$ is the maximal ideal of $A$.
\end{proof}
\end{subsection}
\begin{subsection}{Localizations}\label{localizationsection}
Let $\phi:A\to B$ be a ring homomorphism. In what follows, we introduce a ring $A_\phi$, a ring homomorphism
$\Lambda_\phi: A\to A_\phi$ and a local homomorphism $\Psi_\phi: A_\phi\to B$ such that $\phi=\Psi_\phi \circ \Lambda_\phi$. Let $S_1$ be the set of all elements $s\in A$ such that $\psi_1(s)$ is invertible in $B$. Let $A_1=A_{S_1}$ be the localization of $A$ at $S_1$ and $\lambda_1:A\to A_{1}$ be the canonical homomorphism, consult \cite{cohn_1995}
for the definition and elementary properties of localizations. It follows from the universal property of $A_{S_1}$ that there exists a unique homomorphism $\psi_1:A_{1}\to B$ such that $\phi=\psi_1 \circ \lambda_1$. Let $S_2$ be the set of all elements $s\in A_1$ such that $\phi_1(s)$ is invertible in $B$. Consider the localization $A_2=(A_{1})_{S_2}$ of $A_1$ at $S_2$ and let $\lambda_2:A_1\to A_{2}$ be the canonical homomorphism. It follows from the universal property of $A_{2}$ that there exists a homomorphism $\psi_2:A_{2}\to B$ such that $\psi_1=\psi_2 \circ \lambda_2$. Continuing this process, we obtain a sequence of rings $A_1,A_2,A_3,...$, homomorphisms
$\lambda_{i}:A_{i-1}\to A_{i}$ and $\psi_i:A_i\to B$ such that $\psi_{i-1}=\psi_{i} \circ \lambda_i$ for $i=1,2,...,$ where $\psi_0=\phi$. We set $A_\phi=\varinjlim A_i$. We have a canonical homomorphism $\Lambda_\phi:A\to A_\phi$ and a canonical homomorphism
$\Psi_\phi: A_\phi\to B$. In the following proposition, we provide some facts about this construction, see Section 2 in \cite{aryapoor2010f} for a proof of this proposition and a detailed discussion of the localization ring $A_\phi$.
\begin{proposition}\label{localization}
The ring homomorphism $\Psi_\phi: A_\phi\to B$ is local and satisfies the equality $\phi=\Psi_\phi \circ\Lambda_\phi$.
\end{proposition}
\end{subsection}
\begin{subsection}{Commutativity with respect to a filtration}
Let $(A,I)$ be a pair. A descending sequence $\mathcal{F}$ of ideals
$$I_1\supset I_2 \supset ...$$
of $A$ is called a \textit{filtration} on $(A,I)$ if $I_1=I$.
The pair $(A,I)$ is called \textit{commutative with respect to} $\mathcal{F}$, or $\mathcal{F}$-\textit{commutative} for short, if
$[A,I_n]\subset I_{n+1}$ for all $n=1,2,...$. The notation $[S,T]$, where $S,T\subset A$, stands for the set of all elements of the form $st-ts$ where $s\in S, t\in T$.
\end{subsection}
\begin{subsection}{The commutator filtration}
For every pair $(A,I)$, one can define a sequence $I^{(1)}, I^{(2)},...$ of ideals of $A$ as follows: $I^{(1)}=I$; for $n=1,2,...$, the ideal $I^{(n+1)}$ is the ideal generated by $[A,I^{(n)}]$. It is easy to see that the sequence $I^{(1)}, I^{(2)},...$ is a filtration on $(A,I)$ called the \textit{commutator filtration} of $(A,I)$. Clearly, $(A,I)$ is commutative with respect to its commutator filtration. Note that if $\phi:(A,I)\to (B,J)$ is a morphism of pairs then $\phi(I^{(n)})\subset J^{(n)}$ for all $n=1,2,...$.
\end{subsection}
\begin{subsection}{Topologies defined by filtrations}
Let $(A,I)$ be a pair and $\mathcal{F}: I_1=I, I_2,...$ be a filtration on $(A,I)$.
The $\mathcal{F}$-\textit{topology} on $(A,I)$ is the linear topology on $A$ for which the sets $I,I_2,\dots$ form a fundamental system of neighborhoods of $0$. The pair $(A,I)$ is called \textit{separated with respect to} $\mathcal{F}$, or $\mathcal{F}$\textit{-separated}
for short, if $A$ is Hausdorff with respect to the $\mathcal{F}$-topology. Note that $(A,I)$ is $\mathcal{F}$-separated if and only if $\cap_{n=1}^\infty I_n =\{0\}$.
The pair $(A,I)$ is called \textit{complete with respect to} $\mathcal{F}$, or $\mathcal{F}$\textit{-complete} for short, if it is complete with respect to the $\mathcal{F}$-topology.
\end{subsection}
\begin{subsection}{Unique factorization pairs}
A pair $(A,I)$ is called a \textit{left unique factorization pair}, or \textit{LUFP} for short, if for every
factorization $\bar{F}(x)=f_1(x)f_2(x)$ of a monic polynomial $F(x)\in A[x]$ over $A/I$, where
$f_1(x)f_2(x)\in (A/I)[x]$ are left coprime monic polynomials, there exists at most one factorization $F(x)=F_1(x)F_2(x)$ such that $F_1(x), F_2(x)\in A[x]$ are monic polynomials, and
$\bar{F}_1(x)=f_1(x), \bar{F}_2(x)=f_2(x)$. The notion of a right unique factorization pair (RUFP) is defined in a similar way.
A pair is called a \textit{unique factorization pair}, or \textit{UFP} for short, if it is both an LUFP and an RUFP.
\end{subsection}
\begin{subsection}{Henselian pairs}
A pair $(A,I)$ is called \textit{left Henselian} if $(A,I)$ is a Jacobson pair, and the following version of Hensel's lemma holds in A. For every monic polynomial $F(x)\in A[x]$, if $\bar{F}(x)=f_1(x)f_2(x)$, where $f_1(x),f_2(x)\in (A/I)[x]$ are left coprime monic polynomials, then there exist unique monic polynomials
$F_1(x),F_2(x)\in A[x]$ satisfying $F(x)=F_1(x)F_2(x)$, $\bar{F}_1(x)=f_1(x)$ and $\bar{F}(x)=f_2(x)$. We note that the polynomials $F_1(x)$ and $F_2(x)$ are left coprime, see Lemma \ref{coprime-reduction}. The notion of a right Henselian pair is defined in a similar fashion. A pair which is both left and right Henselian is called \textit{Henselian}. Obviously, every (resp. left or right) Henselian ring is a (resp. left or right) UFP. We note that if $A/I$ is a commutative ring, then $(A,I)$ is left Henselian if and only if it is right Henselian.
\end{subsection}
\begin{subsection}{A class of left Henselian pairs}
The following result generalizes Theorem 2.1 in \cite{aryapoor2009non}.
\begin{theorem}\label{HenselianRings}
Let $(A,I)$ be a pair and $\mathcal{F}: I_1=I,I_2,...$ be a filtration on $(A,I)$ such that $(A,I)$ is
$\mathcal{F}$-commutative, $\mathcal{F}$-separated and $\mathcal{F}$-complete. If $I_n[A,A]\subset I_{n+1}$ and $I_n^2\subset I_{n+1}$ for all $n=1,2,...$, then $(A,I)$ is left Henselian.
\end{theorem}
\begin{proof}
First, we show that $(A,I)$ is a Jacobson pair. Let $a\in I$ be given. The condition $I_n^2\subset I_{n+1}$ for all $n=1,2,...$, implies that
the geometric series $$1-a+a^2-\cdots$$ converges to a unique limit in $A$ because $(A,I)$ is
$\mathcal{F}$-separated and $\mathcal{F}$-complete. The limit of this series is the inverse of $1+a$. Therefore, every element in $1+I$ is invertible, which implies that $I\subset rad(A)$, that is, $(A,I)$ is a Jacobson pair.
To show that $(A,I)$ is left Henselian,
let $F(x)\in A[x]$ be a monic polynomial such that $\bar{F}(x)=f_1(x)f_2(x)$ where $f_1(x),f_2(x)\in (A/I)[x]$
are left coprime monic polynomials. We need to show that we can lift this factorization to $A[x]$ in a unique way.
First, we show that there are sequences of monic polynomials $$ F_{1,1}(x), F_{1,2}(x),..., F_{1,i}(x),... $$ and
$$ F_{2,1}(x), F_{2,2}(x),..., F_{2,i}(x),... $$ in $A[x]$ such that
$$\bar{ F}_{1,i}(x)=f_1(x),\; \bar{ F}_{2,i}(x)=f_2(x),$$
$$F_{1,i+1}(x)-F_{1,i}(x) \in I_{i}[x],\; F_{2,i+1}(x)-F_{2,i}(x) \in I_{i}[x],$$
$$F(x)-F_{1,i}(x)F_{2,i}(x)\in I_{i}[x],$$ for all $i\geq 1$. To construct these sequences, we use induction on $i$.
Since the canonical map $A\to A/I$ is onto, we can find monic polynomials $$F_{1,1}(x), F_{2,1}(x)\in A[x]$$ such that $\bar{F}_{1,1}(x)=f_1(x), \bar{ F}_{2,1}(x)=f_2(x)$. Clearly, $F_{1,1}(x)$ and $F_{2,1}(x)$ satisfy the desired conditions.
Having found $F_{1,i}(x)$ and $F_{2,i}(x)$, we find $F_{1,i+1}(x)$ and $F_{2,i+1}(x)$ as follows. Set
$$G(x)=F(x)-F_{1,i}(x)F_{2,i}(x).$$
I claim that there are polynomials $R_1(x), R_2(x)\in I_i[x]$ such that
$$\deg(R_1(x))<\deg(F_{2,i}(x)), \deg(R_2(x))<\deg(F_{1,i}(x)),$$ and
$$R_2(x) F_{1,i}(x)+R_1(x) F_{2,i}(x)-G(x)\in I_{i+1}[x]$$
By Lemma \ref{coprime-reduction}, there exit polynomials $H_1(x), H_2(x)\in A[x]$ such
$$H_1(x)F_{1,i}(x)+H_2(x)F_{2,i}(x)=1.$$
It follows that
$$G_1(x)F_{1,i}(x)+G_2(x)F_{2,i}(x)=G(x),$$
where both $G_1(x)=G(x)H_1(x)$ and $G_2(x)=G(x)H_2(x)$ belong to $I_i[x]$.
By Lemma \ref{EuclidAlg}, there are polynomials $Q(x),R_1(x)\in I_i[x]$ such that
$$G_1(x)=Q(x)F_{2,i}(x)+R_1(x) \; \text{and}\; \deg(R_1(x))<\deg(F_{2,i}(x)).$$
It follows that
$$G(x)=R_1(x)F_{1,i}(x)+(G_2(x)+Q(x)F_{1,i}(x))F_{2,i}(x)+$$
$$Q(x)(F_{2,i}(x)F_{1,i}(x)-F_{1,i}(x)F_{2,i}(x))$$
Using the condition $I_i[A,A]\subset I_{i+1}$, we see that
$$ R_1(x)F_{1,i}(x)+(G_2(x)+Q(x)F_{1,i}(x))F_{2,i}(x)-G(x)\in I_{i+1}[x].$$
Let
$G_2(x)+Q(x)F_{1,i}(x)=\sum_{i=0}^mb_ix^i.$
Assume that $m\geq \deg(F_{1,i}(x))$.
Since $F_{2,i}(x)$ is monic, and
$$\deg(R_1(x)F_{1,i}(x))< \deg(F_{1,i}(x))+\deg(F_{2,i}(x)),$$
$$\deg(G(x))<\deg(F_{1,i}(x))+\deg(F_{2,i}(x)),$$
the relation
$$ R_1(x)F_{1,i}(x)+(G_2(x)+Q(x)F_{1,i}(x))F_{2,i}(x)-G(x)\in I_{i+1}[x]$$
implies that $b_m\in I_{i+1}$. Therefore, we have
$$ R_1(x)F_{1,i}(x)+(G_2(x)-b_mx^m)F_{2,i}(x)-G(x)\in I_{i+1}[x].$$
Using induction, we conclude that
$$ R_1(x)F_{1,i}(x)+R_2(x)F_{2,i}(x)-G(x)\in I_{i+1}[x],$$
where $R_2(x)=\sum_{i<\deg(F_1(x))}b_ix^i$, proving the claim. We set
$$F_{1,i+1}(x)=F_{1,i}(x)+R_2(x),$$
$$F_{2,i+1}(x)=F_{2,i}(x)+R_1(x).$$
Clearly, we have
$$\bar{ F}_{1,i+1}(x)=f_1(x),\; \bar{ F}_{2,i+1}(x)=f_2(x)$$
$$F_{1,i+1}(x)-F_{1,i}(x) \in I_{i}[x],\; F_{2,i+1}(x)-F_{2,i}(x) \in I_{i}[x]$$
Moreover, we have
$$F(x)-F_{1,i+1}(x)F_{2,i+1}(x)=$$
$$(F(x)-F_{1,i}(x)F_{2,i}(x))-F_{1,i}(x)R_1(x)-R_2(x)F_{2,i}(x)-R_2(x)R_1(x)=$$
$$(G(x)-R_1(x)F_{1,i}(x)-R_2(x)F_{2,i}(x))+$$
$$(R_1(x)F_{1,i}(x)-F_{1,i}(x)R_1(x))-R_2(x)R_1(x).$$
Since $(A,I)$ is $\mathcal{F}$-commutative and $I_i^2\subset I_{i+1}$, we deduce that
$$F(x)-F_{1,i+1}(x)F_{2,i+1}(x)\in I_{i+1}[x].$$
Having constructed the desired sequences, we proceed as follows. Since $A$ is $\mathcal{F}$-complete,
the limits $$F_1(x)=\lim_{i\to\infty} F_{1,i}(x)\, \text{and}\, F_2(x)=\lim_{i\to\infty} F_{2,i}(x)$$ exist. Clearly, we have $\bar{ F}_{1}(x)=f_1(x)$ and $ \bar{ F}_{2}(x)=f_2(x)$. Since $A$ is $F$-Hausdorff, we have $F(x)=F_1(x)F_2(x)$ and $F_1,F_2$ are monic polynomials. Since $(A,I)$ is $\mathcal{F}$-commutative, it is easy to see that $I^{(n)}\subset I_n$ for all $n$. It follows that $(A,I)$ is also separated with respect to its commutator filtration. Therefore,
by Proposition \ref{uniquenesslifting}, this factorization is unique, and we are done.
\end{proof}
\end{subsection}
\end{section}
\begin{section}{Noncommutative Henselizations}
The notion of the Henselization of a pair has been introduced in the commutative case, see \cite{lafon1963anneaux}.
It turns out that one can develop a similar theory in the noncommutative case. However, we focus our attention on a special subcategory $\mathcal{P}_0$ of the category $\mathcal{P}$ of pairs and prove that every object in this subcategory has a left (and a right) Henselization in $\mathcal{P}_0$, see Theorem \ref{NHenselCom}. Our treatment of Henselization is somewhat similar to the one given in \cite{greco1969henselization}.
\begin{subsection}{The category of perfect pairs}
A pair $(A,I)$ is called \textit{perfect} if $(A,I)$ is separated with respect to its commutator filtration, that is, $
\cap_{n=1}^\infty I^{(n)}=\{0\}$. The full subcategory of the category $\mathcal{P}$ consisting of perfect pairs is denoted by $\mathcal{P}_0$. For any pair $(A,I)$, it is easy to see that the pair
$$\digamma{(A,I)}=(\frac{A}{\cap_{n=1}^\infty I^{(n)}},\frac{I}{\cap_{n=1}^\infty I^{(n)}})$$
is a perfect pair. Furthermore, the assignment $(A,I)\mapsto \digamma{(A,I)}$ gives rise to a functor $\digamma:\mathcal{P}\to \mathcal{P}_0$. One can easily check that $\digamma$ is a left adjoint of the inclusion function $\iota:\mathcal{P}_0\to \mathcal{P}$.
\end{subsection}
\begin{subsection}{Perfect Jacobson pairs}
The reason for restricting our attention to $\mathcal{P}_0$ is in the following result.
\begin{proposition}\label{uniquenesslifting}
Every perfect Jacobson pair is a UFP.
\end{proposition}
\begin{proof}
Let $(A,I)$ be a perfect Jacobson pair. We only show that $(A,I)$ is an LUFP. The proof that A is an RUFP is similar.
Assume, on the contrary, that there are different factorizations $$F(x)=F_1(x)F_2(x)=G_1(x)G_2(x)$$
of a monic polynomial $F(x)\in A[x]$ such that $f_1(x)=\bar{F}_1(x)=\bar{G}_1(x)$ and
$f_2(x)=\bar{F}_2(x)=\bar{G}_2(x)$ are left coprime monic polynomials. Without loss of generality, we may assume
$F_2(x)\neq G_2(x)$. The facts that $F_2(x)$ and $G_2(x)$ are monic polynomials, and $\bar{F}_2(x)=\bar{G}_2(x)$, imply that there exists a polynomial $N(x)\in I[x]$ such that $$F_2(x)=G_2(x)+N(x)\; \text{and} \; \deg(N(x))<\deg(G_2(x)).$$
Since $\cap_{n=1}^\infty I^{(n)}=\{0\}$ and $N(x)\neq 0$, there exists $d\geq 1$ such that
$$N(x)\in I^{(d)}[x]\; \text{but} \; N(x)\notin I^{(d+1)}[x].$$
Since $\bar{F}_1(x)$ and $\bar{G}_2(x)$ are left coprime, it follows from Lemma \ref{coprime-reduction}
that there are polynomials $H_1(x),H_2(x)\in A[x]$ such that
$$H_1(x)F_1(x)+H_2(x)G_2(x)=1.$$
We can write
$$F_2(x)=H_1(x)F_1(x)F_2(x)+H_2(x)G_2(x)F_2(x)=$$
$$(H_1(x)G_1(x)+H_2(x)F_2(x))G_2(x)+H_2(x)(G_2(x)N(x)-N(x)G_2(x)).$$
\newline
Setting $K(x)=H_1(x)G_1(x)+H_2(x)F_2(x)$, we see that $F_2(x)=K(x)G_2(x)$ as elements of $(A/I^{(d+1)})[x]$, because $(A,I)$ is commutative with respect to its commutator filtration. Since $F_2(x)$ and $G_2(x)$ are monic polynomials of the same degree, we conclude that $F_2(x)=G_2(x)$ as elements of $(A/I^{(d+1)})[x]$, that is, $N(x)=F_2(x)-G_2(x)\in I^{(d+1)}[x],$ a contradiction .
\end{proof}
\end{subsection}
\begin{subsection}{Factorizations of polynomials over perfect pairs}
In this part, we prove the following result.
\begin{proposition}\label{factorizationN}
Let $(A,I)$ be a perfect pair and $F(x)\in A[x]$ be a monic polynomial. Suppose that $\bar{F}(x)$ has a factorization $\bar{F}(x)=f_1(x)f_2(x)$ over $A/I$ where $f_1(x),f_2(x)\in (A/I)[x]$ are left coprime monic polynomials. Then, there exists a perfect Jacobson pair
$(A\langle F;f_1,f_2\rangle,I\langle F;f_1,f_2\rangle)$ and a morphism
$$\Phi_{\langle F;f_1,f_2\rangle}:(A,I)\to (A\langle F;f_1,f_2\rangle,I\langle F;f_1,f_2\rangle)$$ of pairs having the following universal property. For every morphism
$$\phi:(A,I)\to (B,K)$$ of pairs, where $(B,K)$ is a perfect Jacobson pair, if $\phi(F(x))=G_1(x)G_2(x)$ for some monic polynomials
$G_1(x), G_2(x)\in B[x]$ such that
$$\bar{G}_1(x)=\bar{\phi}(f_1(x)), \bar{G}_2(x)=\bar{\phi}(f_2(x)),$$
then there exists a unique morphism
$$\psi:(A\langle F;f_1,f_2\rangle,I\langle F;f_1,f_2\rangle)\to (B,K)$$ of pairs such that $\phi=\psi\circ \Phi_{\langle F;f_1,f_2\rangle}$.
\end{proposition}
\begin{proof}
First, we give a construction of the pair $(A\langle F;f_1,f_2\rangle,I\langle F;f_1,f_2\rangle)$ after which we prove its universal property.
Let $$F(x)=a_0+a_1x+\cdots+a_{d-1}x^{d-1}+x^d,$$
$$f_1(x)=b_0+b_1x+\cdots+b_{d_1-1}x^{d_1-1}+x^{d_1},$$
$$f_2(x)=c_0+c_1x+\cdots+c_{d_2-1}x^{d_2-1}+x^{d_2}.$$
We consider the free $A$-ring
$A\langle y_0,...,y_{d_1-1},z_0,...,z_{d_2-1}\rangle$ generated by the noncommutating variables
$y_0,...,y_{d_1-1},z_0,...,z_{d_2-1}$. We have
$$(y_0+y_{1}x+\cdots+y_{d_1-1}x^{d_1-1}+x^{d_1})(z_0+z_{1}x+\cdots+z_{d_2-1}x^{d_2-1}+x^{d_2})=$$
$$g_0+g_1x+\cdots+
g_{d-1}x^{d-1}+x^d$$
where $g_0,...,g_{d-1}\in A\langle y_0,...,y_{d_1-1},z_0,...,z_{d_2-1}\rangle$.
Consider the ring homomorphism
$$\alpha:A\langle y_0,...,y_{d_1-1},z_0,...,z_{d_2-1}\rangle\to \frac{A}{I}$$ defined by
$$\alpha(y_0)=b_0,...,\alpha(y_{d_1-1})=b_{d_1-1}, \alpha(z_0)=c_0,...,\alpha(z_{d_2-1})=c_{d_2-1},$$
$$\alpha(a)=\bar{a}\, \;\text{where}\; a\in A.$$
The ideal $\langle g_0-a_0,...,g_{d-1}-a_{d-1}\rangle$ generated by $g_0-a_0,...,g_{d-1}-a_{d-1}$ is contained in the kernel of $\alpha$
because $\bar{F}(x)=f_1(x)f_2(x)$. Therefore, $\alpha$ gives rise to a ring homomorphism
$$\beta:\frac{A\langle y_0,...,y_{d_1-1},z_0,...,z_{d_2-1}\rangle}{\langle g_0-a_0,...,g_{d-1}-a_{d-1}\rangle}\to \frac{A}{I}$$
Consider the following localization ring (see subsection \ref{localizationsection})
$$R= \Big{(}
\frac{A\langle y_0,...,y_{d_1-1},z_0,...,z_{d_2-1}\rangle}{\langle g_0-a_0,...,g_{d-1}-a_{d-1}\rangle}\Big{)}_\beta.$$
We have canonical ring homomorphisms $\gamma:R\to \frac{A}{I}$
and $\eta:A\to R$
which satisfy $\gamma(\eta(a))=\bar{a}$ for every $a\in A$, see Proposition \ref{localization}. We set $J=\ker(\gamma)$.
By Proposition \ref{localization}, $\gamma$ is local from which it follows that $J\subset rad (A\langle F;f_1,f_2\rangle)$, that is, $(R,J)$ is a Jacobson pair.
Since the quotient homomorphism $A\to A/I$ is a morphism $(A,I)\to (A/I,0)$ of pairs, we can use Lemma \ref{morphism} to deduce that
$\eta:(A,I)\to (R,J)$ is a morphism of pairs.
Finally, we set
$$(A\langle F;f_1,f_2\rangle,I\langle F;f_1,f_2\rangle)=\digamma{(R,J)}$$
The canonical morphism $\eta:(A,I)\to (R,J)$ composed with the quotient morphism $(R,I)\to \digamma{(R,J)}$ gives a morphism
$$\Phi_{\langle F;f_1,f_2\rangle}:(A,I)\to (A\langle F;f_1,f_2\rangle,I\langle F;f_1,f_2\rangle)$$
of pairs.
To prove the universal property of $\Phi_{\langle F;f_1,f_2\rangle}$, let $\phi:(A,I)\to (B,K)$ be a morphism of pairs where $(B,K)$ is a perfect Jacobson pair. Suppose that $\phi(F)(x)=G_1(x)G_2(x)$ where $G_1(x), G_2(x)\in B[x]$ are monic polynomials and
$$\bar{G}_1=\bar{\phi}(f_1), \bar{G}_2=\bar{\phi}(f_2).$$
Let $$G_1(x)=e_0+e_1x+\cdots+e_{d_1-1}x^{d_1-1}+x^{d_1},$$
$$G_2(x)=f_0+f_1x+\cdots+f_{d_2-1}x^{d_2-1}+x^{d_2}.$$
One can easily check that the assignments
$$y_0\mapsto e_0,...,y_{d_1-1}\mapsto e_{d_1-1}, z_0\mapsto f_0,...,z_{d_2-1}\mapsto f_{d_2-1}$$
yield a ring homomorphism
$$\psi_1:\frac{A\langle y_0,...,y_{d_1-1},z_0,...,z_{d_2-1}\rangle}{\langle g_0-a_0,...,g_{d-1}-a_{d-1}\rangle}\to B$$
which extends the ring homomorphism $\phi$.
Furthermore, since $K\subset rad(B)$ and $(B,K)$ is perfect, one can extend $\psi_1$ to a ring homomorphism
$$\psi:A\langle F;f_1,f_2\rangle\to B$$
satisfying $\phi=\psi\circ \Phi_{\langle F;f_1,f_2\rangle}$ as ring homomorphisms.
Since $\bar{G}_1(x)=\bar{\phi}(f_1(x)), \bar{G}_2=\bar{\phi}(f_2(x))$, the following diagram is commutative
$$\begin{array}{ccc}
A\langle F;f_1,f_2\rangle & \xrightarrow{\psi} & B \\
\downarrow{} & & \downarrow \\
A/I & \xrightarrow{\bar{\phi}} & B/K
\end{array}$$
Using the commutativity of this diagram and Lemma \ref{morphism}, we conclude that
$$\psi:(A\langle F;f_1,f_2\rangle, I\langle F;f_1,f_2\rangle)\to (B,K)$$
is, in fact, a morphism of pairs.
The uniqueness of $\psi$ follows from the fact that $(B,K)$ is a UFP by Proposition \ref{uniquenesslifting}.
\end{proof}
\end{subsection}
\begin{subsection}{LF-extensions}
Let $(A,I)$ be a perfect pair. A morphism $\phi:(A,I)\to (B,J)$ of pairs is called a \textit{simple left factorization extension} (or simple \textit{LF-extension} for short) of $(A,I)$ if $\phi=\Phi_{\langle F;f_1,f_2\rangle }$ for some polynomials $F(x)\in A[x]$, $f_1(x),f_2(x)\in
(A/I)[x]$ satisfying the conditions in Proposition \ref{factorizationN}. An \textit{LF-extension} of $(A,I)$ is a morphism $\phi:(A,I)\to (B,J)$ of pairs which is obtained by a finite sequence of simple LF-extensions, that is, there are simple LF-extensions
$$\phi_i:(A_i,I_i)\to (A_{i+1},I_{i+1}), \, \text{where}\; i=1,...,d,$$ such that $(A_1,I_1)=(A,I)$, $(A_{d+1},I_{d+1})=(B,J)$ and $\phi=\phi_{d}\circ\phi_{d-1}\circ\cdots\circ\phi_1.$
Obviously, the collection of all LF-extensions of a perfect pair $(A,I)$ is a set which we denote by $LFext(A,I)$. Given morphisms
$$\phi_1:(A,I)\to (B_1,J_1),\; \text{and}\; \phi_2:(A,I)\to (B_2,J_2)$$ in $LFext(A,I)$, we write $\phi_1\leq \phi_2$ if there exists a morphism
$$\psi: (B_1,J_1)\to (B_2,J_2)$$ of pairs such that $\phi_2=\psi\circ\phi_1$. Clearly, the relation $\leq$ defines a partial order on $LFext(A)$. Furthermore, we have the following result.
\begin{lemma}\label{directedN}
Let $(A,I)$ be a perfect pair. (i) For all $\phi_1:(A,I)\to (B_1,J_1)$ and $\phi_2:(A,I)\to (B_2,J_2)$ in $LFext(A)$, there exists at most one morphism $\psi: (B_1,J_1)\to (B_2,J_2)$ of pairs such that $\phi_2=\psi\circ \phi_1$. (ii) The partial order $\leq$ on $LFext(A)$ is directed.
\end{lemma}
\begin{proof}
Part (i) can be proved using induction and the fact that $(B_2,J_2)$ is a UFP. To prove (ii), we first assume that
$\phi_1:(A,I)\to (B_1,J_1)$ is a simple LF-extension. Therefore, $\phi=\Phi_{\langle F;f_1,f_2\rangle }$ for some polynomials $F(x)\in A[x]$, $f_1(x),f_2(x)\in (A/I)[x]$ satisfying the conditions in Proposition \ref{factorizationN}. Let $G_1(x)=\phi_2(F(x))$,
$g_1(x)=\bar{\phi}_2(F(x))$ and $g_2(x)=\bar{\phi}_2(F(x))$. It is easy to see that the polynomials $G_1(x),g_1(x),g_2(x)$ satisfy the conditions in Proposition \ref{factorizationN}, giving rise to a morphism
$$\Phi_{\langle G;g_1,g_2\rangle}:(B_2,J_2)\to (B\langle G;g_1,g_2\rangle,I\langle G;g_1,g_2\rangle)$$
of pairs. Clearly
$$\Phi_{\langle G;g_1,g_2\rangle}\circ\phi_2:(A,I)\to (B\langle G;g_1,g_2\rangle,I\langle G;g_1,g_2\rangle)$$
is an LF-extension. By the universal property of $\phi=\Phi_{\langle F;f_1,f_2\rangle }$, we see that there exists a morphism
$$\psi:(B_1,J_1)\to (B\langle G;g_1,g_2\rangle,I\langle G;g_1,g_2\rangle)$$
such that $\Phi_{\langle G;g_1,g_2\rangle}\circ\phi_2=\psi\circ\phi_1$. It follows that
$$\phi_1\leq \Phi_{\langle G;g_1,g_2\rangle}\circ\phi_2\;\text{and} \; \phi_2\leq \Phi_{\langle G;g_1,g_2\rangle}\circ\phi_2.$$
The general case is proved by induction.
\end{proof}
\end{subsection}
\begin{subsection}{Left Henselizations}
In this subsection, we prove the following result concerning the concept of Henselization.
\begin{theorem}\label{NHenselCom}
Let $(A,I)$ be a perfect pair. Then, there exists a left Henselian
pair $(A^{lh},I^{lh})$ and a morphism
$\phi^{lh}:(A,I)\to (A^{lh},I^{lh})$ of pairs having the following universal property. For every morphism $\phi:(A,I)\to (B,J)$ of pairs from $(A,I)$ to a left Henselian prefect pair $(B,J)$, there exists a unique morphism
$\psi:(A^{lh},I^{lh})\to (B,J)$ of pairs such that $\phi=\psi \circ\phi^{lh}$.
\end{theorem}
\begin{proof}
By Lemma \ref{directedN}, the direct limit $(A^{lh},I^{lh})$ of elements in $LFext(A)$ exists. Moreover, it is a perfect pair.
We also have a canonical morphism
$$\phi^{lh}:(A,I)\to (A^{lh},I^{lh})$$ of pairs. The universal property of $\phi^{h}$ follows from Proposition
\ref{factorizationN} and properties of direct limits. So, it remains to show that $(A^{lh},I^{lh})$ is left Henselian. Since
$(A^{lh},I^{lh})$ is a direct limit of Jacobson pairs, it is a Jacobson pair. Let a monic polynomial $F(x)\in A^{lh}[x]$ be given such that $\bar{F}(x)=f_1(x)f_2(x)$ for some left coprime
monic polynomials $f_1(x),f_2(x)\in (A^{lh}/I^{lh})[x]$. Since $F(x)$ has only finitely many (nonzero) coefficients, there exists an LF-extension $\phi:(A,I)\to (B,J)$ such that $F(X)\in B[X]$. By Proposition \ref{factorizationN}, the polynomial $F(X)$ has a factorization $F(x)=F_1(x)F_2(x)$ over $B(F;f_1,f_2)$, hence over $A^{lh}$, such that $F_1,F_2\in A^{lh}[X]$ are monic, and $\bar{F}_1(x)=f_1(x)$, $\bar{F}(x)=f_2(x)$. Since $(A^{lh},I^{lh})$ is a perfect Jacobson ring, it is a UFP, see Proposition \ref{uniquenesslifting}. Therefore, the factorization $F(x)=F_1(x)F_2(x)$ is unique. It follows that $(A^{lh},I^{lh})$ is a left Henselian pair, and we are done.
\end{proof}
The pair $(A^{lh},I^{lh})$ is called the \textit{left Henselization} of $(A,I)$. It is easy to see that the pair $(A^{lh},I^{lh})$ is unique up to unique isomorphism. Similarly, one can show that
every perfect pair $(A,I)$ has a right Henselizaiton $$\phi^{rh}:(A,I)\to (A^{rh},I^{rh})$$ satisfying the corresponding universal property.
We note that if $A/I$ is, in addition, a commutative ring, then the right Henselization of $(A,I)$ is also the left Henselization of $(A,I)$, and vice versa.
\end{subsection}
\begin{subsection}{Commutative Henselizations}
A pair $(A,I)$ is called \text{commutative} if $A$ is a commutative ring. It is known that every commutative pair has a Henselization in the category $\mathcal{P}_c$ of commutative pairs, see \cite{lafon1963anneaux}. The Henselization of a commutative pair $(A,I)$ in $\mathcal{P}_c$ is referred to as the commutative Henselization of $(A,I)$, and is denoted by $(A^{ch},I^{ch})$.
Obviously, the category $\mathcal{P}_c$ is a full subcategory of $\mathcal{P}_0$.
The following result determines commutative Henselizations in terms of left Henselizations.
\begin{proposition}\label{NHenselCom}
Let $(A,I)$ be a commutative pair. Then, the ideal $J$ generated by $[A^{lh},A^{lh}]$ is contained in $I^{lh}$. Moreover, the morphism
$$q\circ \phi^{lh}:(A,I)\to (\frac{A}{J}^{lh},\frac{I}{J}^{lh}),$$
where $q:(A^{lh},I^{lh})\to (A^{lh}/J,I^{lh}/J)$ is the quotient morphism, is the commutative Henselization of $(A,I)$.
\end{proposition}
\begin{proof}
Let $(A^{ch},I^{ch})$ be the commutative Henselization of $(A,I)$ and
$$\phi^{ch}:(A,I)\to (A^{ch},I^{ch}),$$ be the corresponding morphism of pairs. Since $(A^{ch},I^{ch})$ is left Henselian, there exists a unique morphism
$$\phi:(A^{lh},I^{lh})\to (A^{ch},I^{ch}),$$
of pairs such that $\phi^{ch}=\phi\circ\phi^{lh}$. The fact that $(A^{ch},I^{ch})$ is commutative implies that
the ideal $J$ generated by $[A^{lh},A^{lh}]$ is contained in the ideal
$$\ker(\phi)=\phi^{-1}(0)\subset \phi^{-1}(I^{ch})=I^{lh}.$$
Moreover, there exists a unique morphism
$$\psi:(\frac{A}{J}^{lh},\frac{I}{J}^{lh})\to (A^{ch},I^{ch})$$ such that $\phi=\psi\circ q$. Using the universal properties of $\phi^{lh}$ and $\phi^{ch}$, one can verify that $\psi$ is an isomorphism and the morphism
$$q\circ \phi^{lh}:(A,I)\to (\frac{A}{J}^{lh},\frac{I}{J}^{lh})$$
is the commutative Henselization of $(A,I)$.
\end{proof}
\end{subsection}
\begin{subsection}{Henselizations of local rings}\label{localringHensel}
We conclude this article with a discussion of Henselizations of local rings. A ring $A$ is called \textit{local} if the set of all nonunits in $A$ form an ideal. Every local ring $A$ has a unique maximal ideal $I$. A pair $(A,I)$ is called a local pair if $A$ is a local ring and $I$ is its maximal ideal. We note that any local pair is a Jacobson pair.
\begin{proposition}\label{NHensellocal}
The left (right) Henselization of any perfect local pair is a local pair.
\end{proposition}
\begin{proof}
Since the direct limit of local pairs is a local pair, it is enough to show that any simple LF-extension of a perfect local pair is a local pair. Let
$$\Phi_{\langle F;f_1,f_2\rangle}:(A,I)\to (A\langle F;f_1,f_2\rangle,I\langle F;f_1,f_2\rangle)$$
be a simple LF-extension where $(A,I)$ is a local ring. Referring to the notations used in Proposition \ref{factorizationN}, one can see that the ring homomorphism $\gamma:R\to A/I$ is a local homomorphism. Since $A/I$ is a local ring, Lemma \ref{localmapsLocalrings}
implies that $R$ is a local ring whose maximal ideal is $J=\gamma^{-1}(0)$. Using the relation
$$(A\langle F;f_1,f_2\rangle,I\langle F;f_1,f_2\rangle)=\digamma{(R,J)},$$
we conclude that $(A\langle F;f_1,f_2\rangle,I\langle F;f_1,f_2\rangle)$ is a local pair, and we are done.
\end{proof}
\end{subsection}
\end{section}
\end{document} |
\begin{document}
\title{Tree Compression with Top Trees Revisited}
\author{Lorenz Hübschle-Schneider \\ \texttt{\href{mailto:[email protected]}{[email protected]}}\\Institute of Theoretical Informatics\\Karlsruhe Institute of Technology\\Germany \and Rajeev Raman \\ \texttt{\href{mailto:[email protected]}{[email protected]}} \\Department of Computer Science\\University of Leicester\\United Kingdom}
\date{}
\maketitle
\setcounter{footnote}{0}
\begin{abstract}
We revisit tree compression with top trees
(Bille et al.~\cite{TopTrees2013}),
and present several improvements to the compressor and
its analysis. By significantly reducing the amount of information stored and
guiding the compression step using a RePair-inspired heuristic, we obtain a fast
compressor achieving good compression ratios, addressing
an open problem posed by~\cite{TopTrees2013}.
We show how, with relatively small overhead, the
compressed file can be converted into an in-memory representation that
supports basic navigation operations in worst-case logarithmic time without
decompression. We also show a much improved worst-case bound on the size
of the output of top-tree compression (answering an open question
posed in a talk on this algorithm by Weimann in 2012).
\end{abstract}
\section{Introduction}\label{s:intro}
Labelled trees are one of the most frequently used nonlinear data structures in
computer science, appearing in the form of suffix trees, XML files, tries, and
dictionaries, to name but a few prominent examples. These trees are
frequently very large, prompting a need for compression for on-disk storage.
Ideally, one would like specialized tree compressors to certainly get
much better compression ratios than general-purpose compressors such as
\texttt{bzip2} or \texttt{gzip}, but also for the compression to
be fast; as Ferragina et al. note~\cite[p4:25]{FerraginaSuccinct2009}.
\footnote{Their remark is about XML tree compressors but applies to
general ones as well.}
In fact, it is also frequently necessary to hold such trees in main memory and
perform complex navigations to query or mine them. However, common in-memory
representations use pointer data structures that have significant
overhead---e.g. for XML files,
standard DOM\footnote{\emph{Document Object Model}, a common interface for
interacting with XML documents} representations are typically~8-16~times larger than
the (already large) XML file~\cite{SiXMLWhitePaper,SpaceEffDOM2007}. To
process such large trees, it is essential to
have compressed in-memory representations that
\emph{directly} support rapid navigation and queries, without partial or
full decompression.
Before we describe previous work, and compare it with
ours, we give some definitions.
A \emph{labelled tree} is an ordered, rooted tree whose nodes have labels from
an alphabet $\Sigma$ of size $\card{\Sigma}=\sigma$. We consider the following
kinds of redundancy in the tree structure. \emph{Subtree repeats} are repeated
occurrences of \emph{rooted subtrees}, i.e.\ a node and all of its descendants,
identical in structure and labels. \emph{Tree pattern repeats} or
\emph{internal repeats} are repeated occurrences of \emph{tree patterns}, i.e.\
connected subgraphs of the tree, identical in structure as well as labels.
\subsection{Previous Work}\label{s:intro:prev}
Nearly all existing compression methods for labelled trees follow one of three
major approaches: \emph{transform-based compressors} that transform the tree's
structure, e.g. into its minimal DAG, \emph{grammar-based compressors} that
compute a tree grammar, and--although not compression--\emph{succinct
representations} of the tree.
\paragraph*{Transform-Based Compressors.}
We can replace subtree repeats by edges to a single shared instance of the
subtree and obtain a smaller Directed Acyclic Graph (DAG) representing the tree.
The smallest of these, called the \textit{minimal DAG}, is unique and can be computed in linear
time~\cite{MinDag1980}. Navigation and path queries can be supported in
logarithmic time~\cite{PQXML2003,RandomAccessGrammars2011}. While its size can
be exponentially smaller than the tree, no compression is achieved in the worst
case (a chain of nodes with the same label is its own minimal DAG, even though
it is highly repetitive). Since DAG minimization only compresses repeated
subtrees, it misses many internal repeats, and is thus insufficient in many
cases.
Bille et al.\ introduced tree compression with top
trees~\cite{TopTrees2013}, which this paper builds upon.
Their method exploits both repeated subtrees and tree structure repeats, and
can compress exponentially better than DAG minimization.
They give a $\log_\sigma^{0.19}{n}$
worst-case compression ratio for a tree of size $n$ labelled from an alphabet of
size $\sigma$ for their algorithm. They show that navigation and a
number of other operations are supported in $O(\log n)$ time directly
on the compressed representation.
However, they do not give any practical evaluation, and indeed state as an
open question whether top-tree compression has practical value.
\paragraph*{Tree Grammars.}
A popular approach to exploit the redundancy of tree patterns is to represent the tree using a
formal grammar that generates the input tree, generalizing grammar compression
from strings to trees~\cite{BPLEX2004, TreeCompression2004, GrammarComprExp2006,
GrammarCompression2005, TreeRePair2013, ApproxSmallest2013}.
These can be exponentially smaller than the minimal
DAG~\cite{GrammarComprExp2006}. Since it is NP-Hard to compute the smallest
grammar~\cite{SmallestGrammar2005}, efficient heuristics are required.
One very simple yet efficient heuristic method is RePair~\cite{RePair2000}. A
string compressor, it can be applied to a parentheses bitstring representation
of the tree. The output grammars produced by RePair can support a variety
of navigational operations and random access, in time logarithmic in the input
tree size, after additional processing~\cite{RandomAccessGrammars2011}. These
methods, however, appear to require significant engineering effort
before their practicality can be assessed.
TreeRePair~\cite{TreeRePair2013} is a generalization of RePair
from strings to trees. It achieves the best grammar compression ratios currently
known. However, navigating TreeRePair's grammars in sublinear time with respect
to their depth, which can be linear in their size~\cite{TopTrees2013}, is an
open problem. For relatively small documents (where the output of
TreeRePair fits in cache), the navigation
speed for simple tree traversals
is about 5 times slower than succinct representations~\cite{TreeRePair2013}.
Several other popular grammar compressors exist for trees. Among them,
BPLEX~\cite{BPLEX2004,GrammarCompression2005} is probably best-known,
but is much slower than TreeRePair.
The~\textsf{TtoG} algorithm is the first to achieve a good theoretical
approximation ratio~\cite{ApproxSmallest2013}, but has not been evaluated in
practice.
\paragraph*{Succinct Representations.} Another approach is to represent the
tree using near-optimal space without applying compression methods to its
structure, a technique called \textit{succinct data structures}. Unlabelled
trees can be represented using~$2n+\oh{n}$ bits~\cite{JacobsonSuccinct1989} and
support queries in constant time~\cite{MunroSuccinct2001}.
There are a few $n \log{\sigma} + \Oh{n}$ bit-representations for labelled
trees, most notably that by Ferragina et al.~\cite{FerraginaSuccinct2009},
which also yields a compressor, XBZip. While XBZip has good performance
on XML files \emph{in their entirety}, including text, attributes etc.,
evidence suggests that it does not beat TreeRePair on pure labelled trees.
As the authors admit, it is also slow.
\begin{figure}
\caption{Five kinds of cluster merges in top trees. Solid nodes are boundary
nodes, hollow ones are boundary nodes that become internal. Source of this
graphic and more details:~\cite[Section 2.1]{TopTrees2013}
\label{fig:mergetypes:a}
\label{fig:mergetypes:b}
\label{fig:mergetypes:c}
\label{fig:mergetypes:d}
\label{fig:mergetypes:e}
\label{fig:mergetypes}
\end{figure}
\subsection{Our Results}\label{s:intro:results}
Our primary aim in this paper is to address the question
of Bille et al.~\cite{TopTrees2013} regarding the
practicality of the top tree approach, but we make some
theoretical contributions as well. We first give some
terminology and notation.
A \emph{top tree}~\cite{TopTrees2005} is a hierarchical decomposition of a
tree into \emph{clusters}, which represent subgraphs of the original tree.
Leaf clusters correspond to single edges, and inner clusters represent the union
of the subgraphs represented by their two children. Clusters are formed in one
of five ways, called \emph{merge types}, shown in Figure~\ref{fig:mergetypes}. A
cluster can have one or two \textit{boundary nodes}, a top- and optionally a
bottom boundary node, where other clusters can be attached by merging. A top
tree's minimal DAG is referred to as a \textit{top DAG}. For further details on
the fundamentals of tree compression with top trees, refer to~\cite{TopTrees2013}. Throughout this paper, let~$T$ be any ordered, labelled tree with~$n_{\Tree}$
nodes, and let~$\Sigma$ denote the label alphabet
with~$\sigma := \card{\Sigma}$. Let~$\mathcal{T}$ be the top tree and~$\mathcal{TD}$ the top DAG
corresponding to~$T$, and~$n_{\Tree}TD$ the total size (nodes plus edges) of~$\mathcal{TD}$. We assume a standard word RAM model with
logarithmic word size, and measure space
complexity in terms of the number of words used.
Then:
\begin{theorem}\label{thm:dagsize}
The size of the top DAG is $n_{\Tree}TD =
\Oh{\frac{n_{\Tree}}{\log_\sigma{n_{\Tree}}} \cdot \log\log_\sigma{n_{\Tree}}}$.
\end{theorem}
\noindent This is only a factor of $\Oh{\log \log _\sigma{n_{\Tree}}}$
away from the information-theoretic lower bound, and greatly
improves the bound of $\Oh{n/\log_\sigma^{0.19}{n}}$ obtained by
Bille et al. and answers an open question posed in a talk by
Weimann.
Next, we show that if only
basic navigation is to be performed, the amount of information that
needs to be stored can be greatly reduced, relative
to the original representation~\cite{TopTrees2013},
without affecting the asymptotic running time.
\begin{theorem}\label{thm:nav}
We can support navigation with the operations \textsf{Is
Leaf}, \textsf{Is Last Child}, \textsf{First Child}, \textsf{Next Sibling}, and
\textsf{Parent} in $\Oh{\log{n_{\Tree}}}$ time, full decompression in time
$\Oh{n_{\Tree}}$
on a representation of size $\Oh{n_{\Tree}TD}$ storing only the top
DAG's structure, the merge types of inner nodes (an integer from $[1..5]$),
and leaves' labels.
\end{theorem}
We believe this approach will have
low overhead and fast running times in practice for in-memory
navigation without decompression, and sketch how one would approach
an implementation.
Furthermore, we introduce the notion of \textit{combiners} that determine the
order in which clusters are merged during top tree construction.
Combiners aim to improve the compressibility of the top tree, resulting in a
smaller top DAG. We present one such combiner that
applies the basic idea of RePair~\cite{RePair2000} to top tree compression,
prioritizing merges that produce subtree repeats in the top tree, in
Section~\ref{s:combiner}. We give a relatively naive encoding of the
top tree, primarily using Huffman codes, and evaluate its compression
performance. Although the output of the modified top tree compressor
is up to 50\,\% larger than the state-of-the-art TreeRePair, it is about
six times faster. We believe that the compression gap can be narrowed
while maintaining the speed gap.
\section{Top Trees Revisited}\label{s:contrib}
\subsection{DAG Design Decisions}\label{s:contrib:dag}
The original top tree compression paper~\cite{TopTrees2013} did not try to
minimize the amount of information that actually needs to be stored. Instead,
the focus was on implementing a wide variety of navigation operations in
logarithmic time while maintaining~$\Oh{n_{\Tree}TD}$ space \textit{asymptotically}.
Here, we reduce the amount of additional information stored about the clusters
to obtain good compression ratios.
Instead of storing the labels of both endpoints of a leaf cluster's
corresponding edge, we store only the child's label, not the parent's.
In addition to reducing
storage requirements, this reduces the top tree's alphabet size from~$\sigma^2+5$
to~$\sigma+5$, as each cluster has either one label or a merge type. This
increases the likelihood of identical subtrees in the top tree, improving
compression. Note that this change implies that there is exactly one leaf
cluster in the top DAG for each distinct label in the input. To code the root,
we perform a merge of type~(a) (see Section~\ref{s:intro:results} and
Figure~\ref{fig:mergetypes}) between a dummy edge leading to the root and the
last remaining edge after all other merges have completed.
With these modifications, we reduce the amount of information stored with the
clusters to the bare minimum required for decompression, i.e.\ leaf clusters'
labels and inner clusters' merge types.
Lastly, we speed up compression by directly constructing the top DAG
during the merge process. We initialize it with all distinct leaves, and
maintain a mapping from cluster IDs to node IDs in~$\mathcal{TD}$, as well as a hash
map mapping DAG nodes to their node IDs. When two edges are merged into a new
cluster, we look up its children in the DAG and only need to add a new node
to~$\mathcal{TD}$ if this is its first occurrence. Otherwise, we simply update the
cluster-to-node mapping.
\subsection{Navigation}\label{s:contrib:nav}
We now explain how to navigate the top DAG with our reduced information set. We
support full decompression in time $\Oh{n_{\Tree}}$, as well as operations to move
around the tree in time proportional to the height of the top DAG,
i.e.~$\Oh{\log{n_{\Tree}}}$. These are: determining whether the current
node is a leaf or its parent's last child, and moving to its first child, next
sibling, and parent. Accessing a node's label is possible in constant time given
its node number in the top DAG.
\par
\begin{proof}[ (Theorem~\ref{thm:nav})]
As a node in a DAG can be the child of any number of other nodes, it does not
have a unique parent. Thus, to allow us to move back to a node's parent in the DAG, we need to maintain
a stack of parent cluster IDs along with a bit to indicate whether we descended into
the left or right child. We refer to this as the \textit{DAG stack}, and update
it whenever we move around in~$\mathcal{TD}$ with the operations below. Similarly,
we also maintain a \textit{tree stack} containing the DAG stack of each ancestor
of the current node in the (original) tree.
\paragraph*{Decompression:} We traverse the top DAG in pre-order, undoing the merge
operations to reconstruct the tree. We begin with~$n_{\Tree}$ isolated nodes,
and then add back the edges and labels as we traverse the top DAG. As this
requires constant time per cluster and edge, we can decompress the top DAG
in~$\Oh{n_{\Tree}}$~time.
\paragraph*{Label Access:} Since only leaf clusters store labels, and these are
coded as the very first clusters in the top DAG (cf. Section~\ref{s:contrib:
encoding}), their node indices come before all other nodes'. Therefore,
a leaf's label index~$i$ is its node number in the top DAG. We access the label
array in the position following the~$(i-1)$th null byte, which we can find with
a~$\mathsf{Select}_{0}(i-1)$ operation, and decode the label string until we
reach another null byte or the end.
\paragraph*{Is Leaf:} A node is a leaf iff it is no cluster's top boundary node.
Moving up through the DAG stack, if we reach a cluster of
type~(a) or~(b) from the \textit{left} child, the node is not a leaf (the
left child of such a cluster is the \textit{upper} one in
Figure~\ref{fig:mergetypes}). If, at any point before encountering such a
cluster, we exhaust the DAG stack or reach a cluster of type~(b) or~(c) from the right, type~(d) from the
left, or type~(e) from either side, the node is a leaf.
This can again be seen in Figure~\ref{fig:mergetypes}.
\paragraph*{Is Last Child:} We move up the DAG stack until we reach a cluster of
type~(c),~(d), or~(e) from its left child. Upon encountering a cluster of
type~(a) or~(b) from the right, or emptying the DAG stack completely, we abort
as the upward search lead us to the node's parent or exhausted the tree, respectively.
\paragraph*{First Child and Next Sibling:} First, we check whether the node is a
leaf (\textsf{First Child}) or its parent's last child (\textsf{Next Sibling}),
and abort if it is. \textsf{First Child} then pushes a copy of the DAG stack onto the tree
stack. Next, we re-use the upward search performed by the previous check,
removing the elements visited by the search from the DAG stack, up until the
cluster with which the search ended. We descend into its right child and keep
following the left child until we reach a leaf.
\paragraph*{Parent:} Since \textsf{First Child} pushes the DAG stack
onto the tree stack, we simply reset the DAG stack to the tree stack's top
element, which is removed.
\end{proof}
We note here that the tree stack could, in theory, grow to a size of~$\Oh{n_{\Tree}
\log{n_{\Tree}}}$, as the tree can have linear height and the logarithmically sized
DAG stack is pushed onto it in each \textsf{First Child} operation. However, we
argue that due to the low depth of common labelled trees, especially XML files,
this stack will remain small in practice. Even when pessimistically assuming a
\emph{very} large tree with a height of~80 nodes, with a top tree of height~50,
the tree stack will comfortably fit into~32\,kB when using~64-bit node IDs. Our
preliminary experiments confirm this.
To improve the worst-case tree stack size in theory, we can instead keep a log
of movements in the top DAG, which is limited in size to the distance travelled
therein. We expect this to be significantly less than~$\Oh{n_{\Tree} \log{n_{\Tree}}}$
in expectation.
\subsection{Worst-Case Top DAG size}
\label{contrib:bounds}
Bille et al. show that a tree's top tree has at most~$\Oh{n_{\Tree} / \log_{\sigma}
^{0.19}{n_{\Tree}}}$ distinct clusters~\cite{TopTrees2013}. This bound, however, is
an artifact of the proof. By modifying the definition of a
\textit{small cluster} in the compression analysis and carefully exploiting the
properties of top trees, we are able to show a new, tighter, bound, which
directly translates to an improvement on the worst-case compression ratio.
Before we can prove Theorem~\ref{thm:dagsize}, we need to show the following
essential lemmata. Let~$\SubS{v}$ be the size of~$v$'s subtree,
and~$p(v)$ denote its parent.
\begin{lemma}\label{lem:logheight}
Let~$T$ be any ordered labelled tree of size~$n_{\Tree}$, and let~$\mathcal{T}$ be
its top tree. For any node~$v$ of~$\mathcal{T}$, the height of its subtree is
at most~$\lfloor\log_{8/7}{\SubS{v}}\rfloor$.
\end{lemma}
\begin{proof}
Consider the incremental construction process of a top tree~$\mathcal{T}$. During
the merge process, the algorithm builds up a tree by joining clusters into
larger clusters. We start with a forest of~$n_{\Tree}+1$ nodes, each representing an
edge of~$T$. Every merge operation joins two clusters and thus reduces the number
of connected components in $\mathcal{T}$ by one. These connected components are
subtrees of the final top tree. We can thus think of them as the top trees for
tree patterns of the input tree.
Note that a subtree of the top tree is not the top tree of a rooted subtree for
two reasons. For one, it might represent some, but not all, siblings of a node.
This is due to horizontal merges operating on pairs of edges to consecutive
siblings. Thus, a cluster could, for example, represent a node and the subtrees
of the first two of its five children. Secondly, if the cluster has a bottom
boundary node~$vC$ (drawn as a filled node at the bottom in Figure~\ref{fig:mergetypes}),
the subtree of~$T$ that is rooted at~$vC$ is \textit{not} contained in the
cluster. Thus, the cluster does not correspond to a subtree of~$T$, but
rather a tree pattern, i.e.\ a connected subgraph.
Therefore, a subtree of a top tree is the top tree of a tree pattern of~$T$,
and the same bounds apply to its height. As each iteration of merges in the top
tree construction reduces the number of strongly connected components by a factor of~$c \geq
8/7$~\cite{TopTrees2013}, there are at most~$\lceil\log_{8/7}{n_{\Tree}}\rceil$
iterations, each of which increases the height of the top tree by exactly 1.
Being a full binary tree with~$n_{\Tree}+\nobreak1$ leaves, the top tree has~$2 n_{\Tree}$
edges. Thus, the height of any top tree of size~$n$ is bounded
by~$\lceil\log_{8/7}{\frac{n}{2}}\rceil <\nobreak \lfloor \log_{8/7}{n} \rfloor \approx\nobreak 5.2\log{n}$.
By the above, this also applies to subtrees of top trees. \end{proof}
\begin{lemma}\label{lem:smallclusters}
Let~$T$ be any ordered labelled tree of size~$n_{\Tree}$, let~$\mathcal{T}$ be its
top tree, and~$t$ be an integer. Then~$\mathcal{T}$ contains at
most~$\Oh{(n_{\Tree} / t) \cdot \log{t}}$ nodes~$v$ so
that~$\SubS{v} \leq t$ and~$\SubS{p(v)} > t$.
\end{lemma}
\begin{proof}
We will call any node~$v$ of the top tree a \textit{light} node
iff~$\SubS{v} \leq t$, otherwise we refer to it as \textit{heavy}.
With this terminology, we are looking to bound the number of light nodes whose
parent is heavy.
As~$\mathcal{T}$ is a full binary tree, there are four cases to distinguish. We are
not interested in the children of light nodes, nor are we interested in heavy
nodes with two heavy children. This leaves us with two interesting cases:
\begin{enumerate}
\item A heavy node~$vA$ with two light children~$vB$
and~$vC$. Then,~$\SubS{vB} + \SubS{vC} \geq t$.
Thus, there are at most~$2n_{\Tree}/t = \Oh{n_{\Tree} / t}$ light nodes
with a heavy parent and a light sibling.
\item A heavy node with one light and one heavy child. We will consider
this case in the remainder of the proof.
\end{enumerate}
Consider any heavy node~$v$. We say that~$v$ is in
\textit{class~$i$} iff~$\SubS{v} \in \left[2^i,2^{i+1}\!-1\right]$. Observe
that only classes~$i \geq \lfloor\log_2{t}\rfloor$ can contain heavy nodes,
and that the highest non-empty class is~$\lfloor\log_2{n_{\Tree}}\rfloor$. Let a
\textit{top class~$i$ node} be a node in class~$i$ whose parent is in class~$j >
i$, and a \textit{bottom class~$i$ node} one for which both children are in
classes lower than~$i$. We now make two propositions:
\begin{description}
\item[Proposition 1] A node~$vA$ of class~$i$ can have at most one
child in class~$i$.
\\\emph{Proof:} Assume both children~$vB, vC$ of $vA$ are in
class~$i$. Then, the subtree size of~$vA$ is~$\SubS{vA} = 1 +
\SubS{vB} + \SubS{vC} \geq 1 + 2^i + 2^i > 2^{i+1}$, and thus by
definition~$vA$ is not in class~$i$.
\item[Proposition 2] Let~$vB$ be a top class~$i$ node. There are at
most~$\Oh{i}$ light nodes in the subtree of~$vB$ that are children of
class~$i$ nodes.
\\\textit{Proof:} By Proposition 1, there exists exactly one path of
class~$i$ nodes in the subtree of~$vB$. This path begins at~$vB$
and ends at the bottom class~$i$ node of the subtree of~$vB$, which we
refer to as~$vC$. There are no other
class~$i$ nodes in the subtree of~$vB$. Being a full binary tree, the
height of~$vC$'s subtree fulfills~$\Height{vC} \geq\nobreak
\log_2{\SubS{vC}} \geq\nobreak \log_2{2^i} = i$. We now use
Lemma~\ref{lem:logheight} to obtain an upper bound on~$\Height{vB}$ of $\Height{vB} \leq
\lfloor\log_{8/7}{\SubS{vB}}\rfloor \leq \frac{i+1}{\log_2{8/7}} \approx 5.2 \cdot
(i+1)$. Thus, the path from the top class~$i$ node to the bottom class~$i$
node has a length of~$l \leq \Height{vB} - \Height{vC} =
\Oh{i}$. Each node on the path can have at most one light child by Proposition~1. Thus, there
are at most~$\Oh{i}$ light nodes that are children of class~$i$ nodes in the
subtree of a top class~$i$ node.
\end{description}
Combining Proposition 2 with the observation that the number of top class~$i$
nodes is clearly at most~$n_{\Tree} / 2^i$, we obtain a bound on the number of
class~$i$ nodes with one heavy and one light child of~$n_{\Tree} / 2^i \cdot
\Oh{i}$. We then sum over all classes containing heavy nodes to obtain the total
number of heavy nodes with one light child,
\begin{equation*}
\sum_{i=\lfloor\log_2{t}\rfloor}^{\lfloor\log_2{n_{\Tree}}\rfloor}{\frac{
n_{\Tree}}{2^i}\cdot\Oh{i}} = \Oh{\frac{n_{\Tree}}{t} \cdot \log{t}}
\end{equation*}
Thus, there are at most~$\Oh{n_{\Tree} / t \cdot \log{t}} + 2n_{\Tree} /
t = \Oh{n_{\Tree} / t \cdot \log{t}}$ light nodes whose parent is
heavy. This concludes the proof.
\end{proof}
\begin{proof}[ (Theorem~\ref{thm:dagsize})]
We define a \textit{small cluster} as one whose subtree contains at
most~$2^j+1$ nodes and set~$j=\log_2{(0.5 \log_{4\sigma}{n_{\Tree}})}$. We call a
small cluster \textit{maximal} if its parent's subtree exceeds the size limit of
a small cluster. A cluster that is not small is called a \textit{large} cluster.
Note that this is a special case of our distinction between light and heavy
nodes in the proof of Lemma~\ref{lem:smallclusters}.
As each of the~$n_{\Tree}$ leaves of the top tree is contained in exactly one
maximal small cluster, and the top tree is a full binary tree, there is exactly one
large cluster less than there are maximal small clusters. Thus, it suffices to
show that there are at most~$\Oh{(n_{\Tree} \cdot \log{\log_\sigma{n_{\Tree}}) /
\log_\sigma{n_{\Tree}}}}$ maximal small clusters, and that the total number of
distinct small clusters does not exceed said bound.
Recall that each inner node of the top tree is labelled with one of the five merge types, and
that each leaf stores the label of its edge's child node, as described in
Section~\ref{s:contrib:dag}. Therefore, the top tree is labelled with an
alphabet of size~$\sigma+5 = \Oh{\sigma}$.
To bound the total number of distinct small clusters, we consider the number of
distinct labelled trees of size at most~$x$, which is~$\Ohsmall{(4\sigma)^{x+1}}$, and
can be rewritten as~$\Ohsmall{\sigma^2\hbox{\rlap{$\sqcap$}$\sqcup$}rt{n_{\Tree}}}$ by setting~$x =\nobreak 2^j+1$~\cite{TopTrees2013}.
If~$\sigma<n_{\Tree}^{1/8}$, this further reduces to~$\Ohsmall{n_{\Tree}^{3/4}}$. Otherwise, the
theorem holds trivially as $\log_\sigma{n_{\Tree}}=\Oh{1}$.
We now bound the number of maximal small clusters with
Lemma~\ref{lem:smallclusters} by choosing the threshold $t =\nobreak 2^j+\nobreak1 =\nobreak 0.5
\log_{4\sigma}{n_{\Tree}} + 1 = \Oh{\log_{\sigma}{n_{\Tree}}}$. As a maximal small cluster
is a light node whose parent is heavy, we can use Lemma~\ref{lem:smallclusters} to
bound the number of maximal small clusters by~$\Oh{(n_{\Tree} \cdot \log{t})
/ t} = \Oh{(n_{\Tree} \cdot \log{\log_\sigma{n_{\Tree}}}) / \log_\sigma{n_{\Tree}}}$.
This concludes the proof.
\end{proof}
\subsection{Encoding}\label{s:contrib:encoding}
\newcommand{core\xspace}{core\xspace}
In the top DAG, we need to be able to access a cluster's left and right child,
as well as its merge type for inner clusters or the child node's label for
the edge that it refers to for leaf clusters.
To realize this interface, we decompose the top DAG into a binary
\textit{core\xspace} tree and a pointer array. The core\xspace tree is defined by removing all
incoming edges from each node, except for the one coming from the node with
lowest pre-order number. All other occurrences are replaced by a dummy leaf node
storing the pre-order number of the referenced node. Leaves in the top DAG are
assigned new numbers as label pointers, which are smaller than the IDs of all inner nodes. All references to
leaves, including the dummy nodes, are coded in an array of \textit{pointers},
ordered by the pre-order number of the originating node. Similarly, the inner
nodes' merge types are stored in an array in pre-order. Lastly, the core\xspace tree itself
can be encoded using two bits per inner node, indicating whether the left and
right children are inner nodes in the core\xspace tree.
Using this representation, all that is required for efficient navigation is an
entropy coder providing constant-time random access to node pointers and merge
types, and a data structure providing \textsf{rank} and \textsf{select} for the
core\xspace tree and label strings. All of these building blocks can be treated as black
boxes, and are well-studied and readily available, e.g.~\cite{Succinter2008} and
the excellent SDSL~\cite{SDSL2014} library.
\paragraph{Simple Encoding} To obtain file size results with reasonable effort,
we now describe a very simple encoding that does not lend itself to navigation
as easily. We compress the core\xspace tree bitstring and merge types using blocked
Huffman coding. The pointer array and null byte-separated concatenated label
string are encoded using a Huffman code. The Huffman trees are coded like the
core\xspace tree above. The symbols are encoded using a fixed length and
concatenated. Lastly, we store the sizes of the four Huffman code segments as a
file header.
\section{Heuristic Combiners}\label{s:combiner}
As described in the original paper~\cite{TopTrees2013}, the construction of the
top tree \textit{exposes} internal repetitions. However, it does not attempt to
maximize the size or number of identical subtrees in the top tree, i.e.\ its
compressibility. Instead, the merge process sweeps through the tree linearly
from left to right and bottom to top. This is a straight-forward cluster
combining strategy that fulfills all the requirements for constructing a top
tree, but does not attempt to maximize compression performance. We therefore
replace the standard combining strategy with heuristic methods that try to increase
compressibility of the top tree. Here, we present one such combiner that applies
the basic idea of RePair to the horizontal merge step of top tree compression.
(In preliminary experiments, it proved detrimental to apply the heuristic to
vertical merges, and we limit ourselves to the horizontal merge step, but note
that this is not a general restriction on combiners.)
We hash all clusters in the top tree as they are created. The hash value
combines the cluster's label, merge type, and the hashes of its left and right
children if these exist. As the edges in the auxiliary tree correspond to
clusters in the top tree during its construction, we assign the cluster's hashes
to the corresponding edges. Defining a digram as two edges whose clusters can
be merged with one of the five merge types from Figure~\ref{fig:mergetypes}, we
can apply the idea of RePair, identifying the edges by their hash values. In
descending order of digram frequency, we merge all non-overlapping occurrences,
updating the remaining edges' hash values to those of the newly created
clusters.
Since this procedure does not necessarily merge a constant fraction of the edges
in each iteration, we may need to additionally apply the normal horizontal merge
algorithm if too few edges were merged by the heuristic. The constant upon which
this decision is based thus becomes a tuning parameter. Note that we need to
ensure that every edge is merged at most once per iteration.
\section{Evaluation}\label{s:eval}
\begin{table}[t]
\caption{XML corpus used for our experiments. File sizes are given for stripped
documents, i.e.\ after removing whitespace and tags' attributes and contents.}\label{tbl:corpus}
\footnotesize
\centering
\begin{minipage}{.497\linewidth}
\begin{tabular}{l rrr}
\toprule
\textbf{File name} & \textbf{size (MB)} & \textbf{~~~\#\,nodes} & \textbf{height}\\\midrule
1998statistics & 0.60 & 28\,306 & 6 \\
dblp & 338.87 & 20\,925\,865 & 6 \\
enwiki-latest-p & 229.78 & 14\,018\,880 & 5 \\
factor12 & 359.36 & 20\,047\,329 & 12 \\
factor4 & 119.88 & 6\,688\,651 & 12 \\
factor4.8 & 143.80 & 8\,023\,477 & 12 \\
factor7 & 209.68 & 11\,697\,881 & 12 \\
JST-gene.chr1 & 5.79 & 173\,529 & 7 \\
\bottomrule
\end{tabular}
\end{minipage}
\begin{minipage}{.497\linewidth}
\begin{tabular}{l rrr}
\toprule
\textbf{File name} & \!\textbf{size (MB)} & \textbf{~~~\#\,nodes} & \textbf{height}\\\midrule
JST-snp.chr1 & 27.31 & 803\,596 & 8 \\
nasa & 8.43 & 476\,646 & 8 \\
NCBI-gene.chr1 & 35.30 & 1\,065\,787 & 7 \\
proteins & 365.12 & 21\,305\,818 & 7 \\
SwissProt & 45.25 & 2\,977\,031 & 5 \\
treebank-e & 25.92 & 2\,437\,666 & 36 \\
uwm & 1.30 & 66\,729 & 5 \\
wiki & 42.29 & 2\,679\,553 & 5 \\
\bottomrule
\end{tabular}
\end{minipage}
\end{table}
We now present an experimental evaluation of top tree compression. In this
section, we demonstrate its qualities as a fast and efficient compressor,
compare it against other compressors, and show the effectiveness of our
RePair-inspired combiner.
\paragraph*{Experimental Setup}
All algorithms were implemented in C++11 and compiled with the GNU C++ compiler
\texttt{g++} in version 4.9.2 using optimization level~\texttt{fast} and
profile-guided optimizations. The experiments were conducted on a commodity PC
with an Intel Core i7-4790T CPU and 16\,GB of DDR3 RAM, running Debian Linux
from the \texttt{sid} channel. We used \texttt{gzip 1.6-4} and \texttt{bzip2
1.0.6-7} from the standard package repositories. Default compression settings
were used for all compressors, except the \texttt{-9} flag for gzip. All input
and output files were located in main memory using a \texttt{tmpfs} RAM disk to
eliminate I/O delays.
\paragraph*{XML corpus}
We evaluated the compressor and our heuristic improvement on a corpus of common
XML files~\cite{UWXML,JSNP,WikiXML,Delpratt2009}, listed in
Table~\ref{tbl:corpus}. In our experiments, we give file sizes for our simple
encoding, which represent pessimistic results that can serve as an upper bound
of what to expect from a more optimized encoding. We give these file sizes to
demonstrate that even a simple encoding yields good results with regard to
file size, speed, and ease of navigation (see Section~\ref{s:contrib:nav}).
\begin{figure}
\caption{Comparison of compression ratios, measured by comparing file sizes
against a succinct encoding of the input file (higher is better)}
\label{fig:vs_succ}
\end{figure}
\begin{figure}
\caption{Comparison of output file sizes produced by top tree compression with
and without the RePair combiner, measured against TreeRePair file sizes (lower
is better)}
\label{fig:vs_trp}
\end{figure}
\paragraph*{Results}
We use a minimum merge ratio of $c=1.26$ for the horizontal merge step using our
RePair-inspired heuristic combiner in all our experiments. This is the result of an
extensive evaluation which showed that values $c \in [1.2, 1.27]$ work
very well on a broad range of XML documents. We observed that values close to 1
can improve compression by up to 10\,\% on some files, while causing a
deterioration by a similar proportion on others. Thus, while better choices
of~$c$ exist for individual files, we chose a fixed value for all files to
provide a fair comparison, similar to the choice of 4 as the maximum rank of the
grammar in TreeRePair~\cite{TreeRePair2013}.
We use a parenthesis bitstring encoding of the input tree as a baseline to
measure compression ratios. The unique label strings are concatenated, separated
by null bytes. Indices into this array are stored as fixed-length numbers
of~$\lceil \log_2{\#\text{labels}} \rceil$ bits.
TreeRePair\footnote{\url{https://code.google.com/p/treerepair}\label{fn:trp}},
which has been carefully optimized to produce very small output files, serves us
as a benchmark. We are, however, reluctant to compare tree compression with top
trees to TreeRePair directly, as our methods have not been optimized to the same
degree.
In Figure~\ref{fig:vs_succ} we give a compression ratios relative to the
succinct encoding. We evaluated our implementation of top tree compression using
the combining strategy from~\cite{TopTrees2013} as well as our RePair-inspired
combiner. We also give the file sizes achieved by TreeRePair and those of RePair
on a parentheses bitstring representation of the input tree and the concatenated
nullbyte-separated label string (note that no deduplication is performed here,
as this is up to the compressor). We represent RePair's grammar production rules
as a sequence of integers with an implicit left-hand side and encode this
representation using a Huffman code. Figure~\ref{fig:vs_succ} shows that top
tree compression consistently outperforms RePair already, but does not achieve
the same level of compression as TreeRePair at this stage. We can also clearly
see the impact of our RePair-inspired heuristic combiner, which improves
compression on nearly all files in our corpus and is studied in more detail in
the next paragraph. Table~\ref{tbl:size} gives the exact numbers for the output
file sizes, supplementing them with results for general-purpose compressors.
\def2.93{2.93}
\def2.93Rel{1.93}
\def2.93Percent{193}
\deffactor12{factor12}
\def1.52{1.52}
\def1.52Rel{0.52}
\def1.52Percent{52}
\defNCBI-gene.chr1{NCBI-gene.chr1}
\def1.64{1.64}
\def1.47{1.47}
\def1.39{1.39}
\def1.39{1.39}
\def10.9{10.9}
\def5.0{5.0}
\paragraph*{RePair Combiner.}
\begin{absolutelynopagebreak}
Figure~\ref{fig:vs_trp} compares the two versions of top tree compression, using
TreeRePair as a benchmark. The RePair combiner's effect is clearly visible,
reducing the maximum disparity in compression relative to TreeRePair from a
file~2.93\xspace times the size (\texttt{
factor12}) to one that is~1.52Percent\,\%
larger (\texttt{NCBI-gene.chr1}). This constitutes nearly a four-fold
decrease in overhead (from 2.93Rel\xspace to 1.52Rel).
On average, files are~1.39\xspace times the size of TreeRePair's, down
from a factor of~1.64\xspace before. On
our corpus, using the heuristic combiner reduced file sizes
by~10.9\,\% on average, with the median being
a~5.0\,\% improvement compared to classical top tree
compression. Reduced compression performance was observed on few files only,
particularly smaller ones, while larger files tended to fare better.
\end{absolutelynopagebreak}
\def10.5{10.5}
\def9.3{9.3}
\def6.2{6.2}
\def5.5{5.5}
\def2.0{2.0}
\def1.6{1.6}
\def3.3{3.3}
\def2.6{2.6}
\def19.6{19.6}
\def21.2{21.2}
\def1.6Percent{62}
\def15.4{15.4}
\def11.9{11.9}
\def9.7{9.7}
\def7.3{7.3}
\def1.8{1.8}
\def1.6{1.6}
\paragraph*{Speed.}
Using our RePair-inspired combiner increases the running time of the top tree
creation stage, doubling it on average. Our implementation of classical top tree
compression was~10.5\xspace times faster than TreeRePair on
average over the corpus from Table~\ref{tbl:corpus}, and
still~6.2\xspace times faster when using our RePair combiner.
Detailed running time measurements are given in Table~\ref{tbl:runtime}. In
particular, classical top tree compression takes only twice as long as
\texttt{gzip -9} on average, and~3.3\xspace times when using our
RePair combiner (TreeRePair:~21.2). In contrast,
\texttt{bzip2} is~15.4\xspace times \emph{slower} than top
tree compression on average, and~9.7\xspace times when using
our RePair combiner. This strikingly demonstrates the method's qualities as a
fast compressor.
\paragraph*{Performance on Random Trees.}
\begin{figure}
\caption{Compression ratio in terms of the average number of edges, divided by
information-theoretical compression ratio bound~$\log_\sigma{n_{\Tree}
\label{fig:exp:rand}
\end{figure}
We examine random trees to show that tree compression with top trees is a very
versatile method, and that it does not rely on typical characteristics of XML
files. By definition, random trees do not compress well. Thus, we can use them
to approximate worst-case behaviour. We generate trees uniformly at random
using a method developed by Atkinson and Sack~\cite{RandomTrees1992}---note
that the method is not limited to binary trees. For the generated trees, we compare the average
number of edges~$n_{\Tree}TD$ in the Top DAG to the information-theoretic lower
bound of~$\Omega(n_{\Tree}/\log_\sigma{n_{\Tree}})$ for a tree of size~$n_{\Tree}$. The
results of this are shown in Figure~\ref{fig:exp:rand} for~$\sigma=2$. We can
see that apart from some oscillation, the values are in a very small range
between~$0.0889$ and~$0.0904$ and do not show an overall tendency to grow or
shrink, except for the amplitude of oscillation. This suggests that tree
compression with top trees performs asymptotically optimal on random trees.
The oscillation or zig-zag behaviour exhibited in Figure~\ref{fig:exp:rand}
poses a riddle. The period duration doubles with each quarter of a period,
exhibiting exponential growth, while the wave's amplitude appears to grow by a
constant amount per quarter period. We do not have a definitive explanation
for the causes of this behaviour. However, we can speculate about possible
contributing factors. For one, consider the subtrees that could be shareable
in the top tree. Their height, and therefore number, grows logarithmically
with the height of the top tree, which in turn grows logarithmically with
respect to the input tree's size. Thus, the number of potentially shareable
subtrees grows proportional to~$\log{\log{n_{\Tree}}}$. As the potential for DAG
compression grows with the number of shareable subtrees, we would expect a
sawtooth-like pattern in the compression ratios, spiking whenever the
shareable subtree height increases.
This could contribute to the exponential growth in period duration. Further
investigation beyond the scope of this paper would be required to account for
the smoothness and amplitude of the curve.
\section{Conclusions}\label{s:concl}
We have demonstrated that tree compression with top trees is viable, and
suggested several enhancements to improve the degree of compression achieved.
Using the notion of combiners, we demonstrated that significant improvements can
be obtained by carefully choosing the order in which clusters are merged during
top tree creation. We showed that the worst-case compression ratio is
within a~$\log\log_\sigma{n_{\Tree}}$ factor of the information-theoretical bound,
and experiments with random trees suggest that actual behaviour is asymptotically
optimal.
Further, we gave efficient methods to navigate the compressed representation,
and described how the top DAG can be encoded to support efficient navigation
without prior decompression.
We thus conclude that tree compression with top trees is a very promising
compressor for labelled trees, and has several key advantages over other
compressors that make it worth pursuing. It is our belief that its great
flexibility, efficient navigation, high speed, simplicity, and provable bounds
should not be discarded easily. While further careful optimizations are required
to close the compression ratio gap, tree compression with top trees is already a
good and fast compressor with many advantages.
\paragraph*{Future Work} We expect that significant potential for improvement
lies in more sophisticated combiners. The requirements for combiners give us a
lot of space to devise better merging algorithms. Combiners might also be used
to improve locality in the top tree in addition to compression performance,
leading to better navigation performance. Moreover, additional compression
improvements should be achievable with carefully engineered output
representations. Since the vast majority of total running time is currently
spent on the construction of the top DAG, using more advanced encodings may
improve compression without losing speed. One starting point to replace our
relatively naïve representation could be a decomposition of the top DAG into two
spanning trees~\cite{ESP2013}.
\begin{table}[b!]
\caption{Running times in seconds, median over ten iterations}\label{tbl:runtime}
\footnotesize
\centerline{
\begin{tabular}{l rrrrrrr}
\toprule
\textbf{File name}~~ & ~~\textbf{TopTrees} & ~~\textbf{TT+RePair} & ~~\textbf{TreeRePair}
& ~~\textbf{RePair} & ~~\textbf{gzip -9} & ~~\textbf{bzip2} \\\midrule
1998statistics & 0.00 & 0.01 & 0.05 & 0.04 & 0.00 & 0.13 \\
dblp & 6.00 & 11.21 & 45.72 & 39.57 & 2.46 & 74.77 \\
enwiki-latest-p & 3.92 & 7.14 & 32.98 & 28.33 & 1.29 & 49.12 \\
factor12 & 7.16 & 11.54 & 109.47 & 54.19 & 4.48 & 81.86 \\
factor4.8 & 2.82 & 4.70 & 46.22 & 21.61 & 1.79 & 33.09 \\
factor4 & 2.40 & 3.92 & 39.47 & 17.75 & 1.49 & 28.21 \\
factor7 & 4.21 & 6.84 & 67.83 & 31.55 & 2.61 & 48.60 \\
JST-gene.chr1 & 0.04 & 0.06 & 0.38 & 0.54 & 0.03 & 1.27 \\
JST-snp.chr1 & 0.24 & 0.38 & 2.12 & 3.40 & 0.17 & 6.33 \\
nasa & 0.15 & 0.23 & 0.94 & 0.85 & 0.06 & 1.86 \\
NCBI-gene.chr1 & 0.31 & 0.51 & 2.25 & 3.33 & 0.20 & 7.97 \\
proteins & 6.92 & 11.88 & 50.17 & 53.27 & 2.41 & 81.92 \\
SwissProt & 1.13 & 2.13 & 12.35 & 5.74 & 0.50 & 11.15 \\
treebank-e & 1.35 & 1.99 & 12.70 & 4.00 & 2.80 & 3.62 \\
uwm & 0.01 & 0.02 & 0.11 & 0.09 & 0.00 & 0.28 \\
wiki & 0.78 & 1.17 & 5.59 & 4.28 & 0.22 & 9.21 \\
\bottomrule
\end{tabular}
}
\end{table}
\begin{table}[b!]
\caption{Compressed file sizes in Bytes}\label{tbl:size}
\footnotesize
\centerline{
\begin{tabular}{l rrrrrrrr}
\toprule
\textbf{File name}~~ & ~~\textbf{Succinct} & ~~\textbf{TopTrees} & ~~\textbf{TT+RePair} & ~~\textbf{TreeRePair}
& ~~~~\textbf{RePair} & ~~~~\textbf{gzip -9} & ~~~~~~\textbf{bzip2}\\\midrule
1998statistics & 18\,426 & 788 & 851 & 692 & 1\,327 & 4\,080 & 1\,301 \\
dblp & 13\,740\,160 & 1\,486\,208 & 1\,416\,538 & 1\,093\,533 & 2\,037\,878 & 2\,476\,347 & 1\,116\,311 \\
enwiki-latest-p & 7\,901\,904 & 516\,638 & 525\,532 & 379\,410 & 866\,161 & 1\,490\,278 & 544\,606 \\
factor12 & 16\,402\,888 & 2\,069\,437 & 948\,167 & 705\,740 & 3\,092\,194 & 6\,342\,947 & 2\,913\,894 \\
factor4.8 & 6\,565\,499 & 1\,070\,045 & 784\,519 & 548\,853 & 1\,587\,043 & 2\,542\,773 & 1\,168\,654 \\
factor4 & 5\,473\,158 & 937\,660 & 704\,105 & 490\,945 & 1\,429\,872 & 2\,119\,269 & 973\,463 \\
factor7 & 9\,571\,503 & 1\,421\,376 & 855\,063 & 625\,094 & 2\,248\,370 & 3\,702\,132 & 1\,700\,043 \\
JST-gene.chr1 & 96\,159 & 5\,332 & 5\,523 & 3\,672 & 8\,273 & 33\,316 & 7\,027 \\
JST-snp.chr1 & 547\,594 & 29\,084 & 27\,039 & 20\,654 & 50\,347 & 194\,862 & 49\,857 \\
nasa & 341\,161 & 42\,077 & 39\,883 & 29\,310 & 60\,394 & 83\,231 & 34\,404 \\
NCBI-gene.chr1 & 721\,803 & 17\,880 & 17\,418 & 11\,459 & 29\,912 & 199\,308 & 47\,901 \\
proteins & 17\,315\,832 & 905\,613 & 860\,366 & 614\,892 & 1\,537\,249 & 3\,214\,663 & 1\,141\,697 \\
SwissProt & 2\,343\,730 & 598\,960 & 574\,466 & 395\,417 & 699\,757 & 829\,119 & 398\,197 \\
treebank-e & 2\,396\,061 & 1\,173\,463 & 1\,170\,304 & 830\,324 & 1\,537\,334 & 1\,858\,722 & 1\,032\,303 \\
uwm & 37\,491 & 2\,177 & 2\,070 & 1\,366 & 3\,101 & 7\,539 & 2\,102 \\
wiki & 1\,242\,418 & 110\,686 & 102\,371 & 75\,090 & 171\,075 & 247\,898 & 93\,858 \\
\bottomrule
\end{tabular}
}
\end{table}
\end{document} |
\begin{document}
\title{ extbf {Uniformly polynomially stable approximations for a class of second order evolution equations}
\abstract{}
In this paper we study time semi-discrete approximations of a class of polynomially stable infinite dimensional systems modeling the damped vibrations. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly polynomially stable with respect to the discretization parameter.\\\\\
\textbf{Key words and phrases}: Polynomial stabilization, observability inequality, discretization, viscosity term.\\
\textbf{2010 MSC}: 93D15, 93B07, 49M25.
\section{Introduction}
Let $X$ and $Y$ be real Hilbert spaces ( $Y$ will be identified to its dual space) with norms denoted respectively by $\Vert .\Vert_X$ and $\Vert .\Vert_Y$.\\
Let $\mathcal{A} : D(\mathcal{A})\to X$ be a self-adjoint positive operator with $\mathcal{A}^{-1}$ compact in $X$ and let $\mathcal{B}\in\mathfrak{L}(Y,X)$.
We consider the system described by
\begin{equation}\label{1}
\left \{
\begin{array}{lcr}
\ddot{w}(t)+\mathcal{A}w(t)+\mathcal{B}\mathcal{B}^{*}\dot{w}(t)=0,\;\;t\geq 0\;\;\; \\
w(0)=w_0,\; \dot{w}(0)=w_1.
\end{array}
\right.
\end{equation}
Most of the linear equations modeling the damped vibrations of elastic structures can be written in the form (\ref{1}).\\
We define the energy of solutions at instant $t$ by
\begin{equation}\label{2}
E(t)=\frac{1}{2}\left\{\Vert \dot{w}(t)\Vert_X^2+\Vert \mathcal{A}^{\frac{1}{2}}w(t)\Vert_X^2\right\},
\end{equation}
which satisfies
\begin{equation}\label{3}
\frac{dE}{dt}(t)=-\Vert \mathcal{B}^*\dot{w}(t)\Vert_Y^2,\;\;\;\;\; \forall t\geq 0.
\end{equation}
It is well known that the natural well-posedness space for (\ref{1}) is \\
$H=V\times X$ where $V=D(\mathcal{A}^{\frac{1}{2}})$ and $\Vert x\Vert_V=\Vert \mathcal{A}^{\frac{1}{2}}x\Vert_X, \forall x\in V$.\\
The existence and uniqueness of finite energy solutions of (\ref{1}) can be obtained by standard semigroup methods.\\
We consider the undamped system associated to (\ref{1}):
\begin{equation}\label{4}
\left \{
\begin{array}{lcr}
\ddot{\partialhi}(t)+\mathcal{A}\partialhi(t)=0,\;\;t\geq 0\;\;\; \\
\partialhi(0)=w_0,\; \dot{\partialhi}(0)=w_1.
\end{array}
\right.
\end{equation}
We assume that system (\ref{4}) satisfy a "weakened" observability inequality (that is the case when the damping operator is effective on a subdomain
where the Geometric Control Condition is not fulfilled [4]), that is there exist positive constants $T, C>0$ and $\beta>-\frac{1}{2}$ such that for all $(w_0, w_1)\in D(\mathcal{A})\times V$ we have
\begin{equation}\label{5}
\int_{0}^{T} \Vert \mathcal{B}^{*}\partialhi'(t)\Vert_Y^2 dt \geq C \Vert (w_0, w_1)\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2,
\end{equation}
where
\begin{equation*}
\left \{
\begin{array}{lcr}
X_{\beta} = D(\mathcal{A}^{\beta}), \;\; \beta \geq 0,\\
X_{-\beta}= (D({\mathcal{A}^{\beta}}))',\;\;\; \beta \geq 0.
\end{array}
\right.
\end{equation*}
The dual space is obtained by means of the inner product in $X$.\\
Then system (\ref{1}) is polynomially stable \cite{2}, that is there exist a constant $C_1>0$ such that for all $t>0$ and for all $(w_0, w_1)\in D(\mathcal{A})\times V$ we have
$$ E(t)\leq \frac{C_1}{t^{\frac{1}{2\beta+1}}}\Vert (w_0, w_1)\Vert_{D(\mathcal{A})\times V}^2.$$
Our goal is to get, as consequence of (\ref{5}), polynomial stability results for time-discrete systems.
If we introduce $z(t):= \left( \begin{array}{c}
w(t) \\
\dot{w}(t)
\end{array} \right),$ \;\;\;\; $y(t)=\left( \begin{array}{c}
\partialhi(t) \\
\dot{\partialhi}(t)
\end{array} \right),$\\
then $z$ satisfies
$$\dot{z}(t)= \left( \begin{array}{c}
\dot{w}(t) \\
-\mathcal{A}w(t)-\mathcal{B}\mathcal{B}^*\dot{w}(t)
\end{array} \right).$$
Consequently the problem (\ref{1}) may be rewritten as the first order evolution equation
\begin{equation}\label{6}
\left \{
\begin{array}{lcr}
\dot{z}(t) =A z(t)-B B^* z(t),\\
z(0)= z_0=(w_0, w_1),
\end{array}
\right.
\end{equation}
where $A:D(A)\to H , A=\left(\begin{array}{c}
0\;\;\;\;\;\;\;\;\;\;I\\
-\mathcal{A}\;\;\;\;\;\;0
\end{array} \right),$\;\;\;$B=\left( \begin{array}{c}
0 \\
\mathcal{B}
\end{array} \right),$\\
$B^*=(0,\mathcal{B}^*)$ ,\;\; $D(A)= D(\mathcal{A})\times V$ and $H=V\times X$.\\
With this notation, (\ref{5}) becomes
\begin{equation}\label{8}
\int_{0}^{T} \Vert B^*\; y(t)\Vert_Y^2 dt \geq C \Vert z_0 \Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2.
\end{equation}
\\\\\\
In recent years an important literature was devoted to the space/time semi-discrete
approximations of a class of exponentially stable infinite dimensional
systems. Let us also mention the recent work [1], where polynomial stability was discussed for space discrete schemes of (\ref{1}). It has been proved that exponential/polynomial
stability may be lost under numerical discretization as the mesh size tends
to zero due to the existence of high-frequency spurious solutions. \\
Several remedies have been proposed and
analyzed to overcome this difficulties. Let us quote the Tychonoff regularization [11, 22, 21, 23, 8, 1 ],
a bi-grid algorithm [9, 19], a mixed finite element method [10, 3, 5, 6, 18], or filtering the high
frequencies [13, 16, 25, 7, 24].
As in [1, 8, 21, 22, 23] our goal is to damp the spurious high frequency modes by introducing a numerical
viscosity in the approximation schemes. Though our paper is inspired from \cite{8}, it differs from that paper on the following points:\\
i)\; We analyze the polynomial decay of the discrete schemes when the continuous problem has such a decay.\\
ii)\; For the proof of the discrete observability inequality, we will use a method based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. But, for the low frequency, contrary to \cite{8} where a Hautus-type test is required, we use a spectral approach and a discrete Ingham type inequalities ( when the spectrum of the spatial operator $A$ associated
with the undamped problem satisfies such a gap condition). \\
Note however that we cannot apply these methods when the damped operator $B$ is not bounded, as in [2], where
the wave equation is damped by a feedback law on the boundary. Dealing with unbounded damping operators $B$ needs
further work.\\
Despite all the existing literature, this article seems to be the first one to provide a systematic way of transferring
polynomial decay properties from the continuous to the time-discrete setting.\\
The paper is organized as follows. In section 2, we prove a uniform "weakened" observability after the addition of numerical viscosity term by using, as we said, a decoupling argument and a spectral approach. Section 3 is devoted to prove the main result of this paper. We illustrate our results by presenting different examples in Section 4. Finally, some further comments and open problems are collected in section 5.\\
In the following, we will write $E\sim F$ instead of $c_1 E \leq F \leq c_2 E$ for brevity, where $c_1, c_2>0$ are constants.\\
\section{observability of time-discrete systems}
In this section, we assume that system (\ref{6}) is polynomially stable and $B^*\in\mathfrak{L}(H,Y)$, i.e. there exists a constant $K_B$ such that
$$\Vert B^*z\Vert_Y\leq K_B\Vert z\Vert_H,\;\;\;\forall\;z\in H.$$
We start considering the following time-discretization scheme for the continuous system (\ref{1}) or equivalent for the system (\ref{6}).
For any $\Delta t>0$, we denote by $z^k$ the approximation of the solution $z$ of system (\ref{6}) at time $t_k=k\Delta t$, for $k\in\mathbb{N}$, and we consider time discretization of system (\ref{6}):
\begin{equation}\label{9}
\left \{
\begin{array}{lcr}
\frac{\tilde{z}^{k+1}-z^k}{\Delta t} = A\left(\frac{z^k+\tilde{z}^{k+1}}{2}\right)-BB^*\left(\frac{z^k+\tilde{z}^{k+1}}{2}\right),\;k\in \mathbb{N}, & \\\\
\frac{z^{k+1}-\tilde{z}^{k+1}}{\Delta t}={(\Delta t)^2}{A^2}{z^{k+1}} ,\;k\in\mathbb{N},\\\\
z^0=z_0.
\end{array}
\right.
\end{equation}
The numerical viscosity term $(\Delta t)^2A^2$ in (\ref{9}) is introduced in order to damp the high frequency modes.\\
We can define the discrete energy by:
\begin{equation}\label{10}
E^k=\frac{1}{2}\Vert z^k\Vert_H^2,\;\;k\geq 0.
\end{equation}
The energy satisfies (\cite{8}):
\begin{equation*}
E^{k+1}+(\Delta t)^3\left\Vert Az^{k+1}\right\Vert_H^2+\frac{(\Delta t)^6}{2}\left\Vert {A^2}{z}^{k+1}\right\Vert_H^2+ \Delta t \left\Vert B^*\left(\frac{z^k+\tilde{z}^{k+1}}{2}\right)\right\Vert_Y^2 = E^k.
\end{equation*}
Summing from $j=0$ to $l=[T/\Delta t]$, it follows then that :\\
$E^0-E^{l+1}=\Delta t\displaystyle{\sum_{j=0}^{l}}(\Delta t)^2\left\Vert Az^{j+1}\right\Vert_H^2+\frac{\Delta t}{2}\displaystyle{\sum_{j=0}^{l}}(\Delta t)^5 \left\Vert {A^2}{z^{j+1}}\right\Vert_H^2$
\begin{equation}\label{11}
\hspace{4cm} +\;\Delta t\displaystyle{\sum_{j=0}^{l}}\left\Vert B^*\left(\frac{z^j+\tilde{z}^{j+1}}{2}\right)\right\Vert_Y^2.
\end{equation}
Note that this numerical scheme is based on the decomposition of the operator $A-BB^*+(\Delta t)^2A^2$ into its
conservative and dissipative parts, that we treat differently. Indeed, the midpoint scheme is appropriate for conservative
systems since it preserves the norm conservation property. This is not the case for dissipative systems, since midpoint
schemes do not preserve the dissipative properties of high frequency solutions. Therefore, we rather use an implicit
Euler scheme, which efficiently preserves these dissipative properties.\\\\
The convergence of the solutions of (\ref{9}) towards those of the original system (\ref{6}) when $\Delta t \to 0$ holds in a suitable
topology (\cite{8}).\\
As the continuous level, we will prove that the uniform polynomial decay of system (\ref{6}) is a consequence of the following "weakened" observability inequality for every solution of the following time-discrete system:
\begin{equation}\label{12}
\left \{
\begin{array}{lcr}
\frac{\tilde{u}^{k+1}-u^k}{\Delta t} = A\left(\frac{u^k+\tilde{u}^{k+1}}{2}\right),\;k\in \mathbb{N}, & \\\\
\frac{u^{k+1}-\tilde{u}^{k+1}}{\Delta t}={(\Delta t)^2}{A^2}{u^{k+1}},\;k\in\mathbb{N},\\\\
u^0=u_0.
\end{array}
\right.
\end{equation}
We want to show that there exist positive constants $T$, $c$ and $\beta>-\frac{1}{2}$ such that, for any $\Delta t>0$ every solution $u^k$ of (\ref{12}) satisfies:\\
$ c \Vert u^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2\leq \Delta t \displaystyle{\sum_{k\Delta t\in [0,T]}}\left\Vert B^*\left(\frac{\tilde{u}^{k+1}+u^k}{2}\right)\right\Vert_Y^2$\\
$\hspace{4cm} +\;\Delta t \displaystyle{\sum_{k\Delta t\in [0,T]}}(\Delta t)^2\left\Vert Au^{k+1}\right\Vert_H^2$\\
\begin{equation}\label{13}
\hspace{5cm}+\;\Delta t\displaystyle{\sum_{k\Delta t\in [0,T]}}(\Delta t)^5\left\Vert {A^2}{u}^{k+1}\right\Vert_H^2.
\end{equation}
Here and in the sequel $c$ denotes a generic positive constant that may vary from line to line but is independent of $\Delta t$.\\
In this section, we show how to obtain the observability inequality (\ref{13}). Before giving spectral conditions to obtain polynomial decay, we need to introduce some notations.\\\\
Since $A$ is a skew-adjoint operator with compact resolvent, its spectrum is discrete and $\sigma(A)=\{i\mu_j:\;j\in\mathbb{Z^*}\}$ where $(\mu_j)_{j\in\mathbb{Z^*}}$ is a sequence of real numbers such that $\vert \mu_j\vert\to\infty$ when $j\to\infty$. Set
$(\varphi_j)_{j\in\mathbb{Z^*}}$ an orthonormal basis of eigenvectors of $A$ associated to the eigenvalues $(i\mu_j)_{j\in\mathbb{Z^*}}$, that is\\
$A\varphi_j=i\mu_j\varphi_j$, with $\mu_j=
\left \{
\begin{array}{lcr}
\sqrt{\eta_j}\;\;\text{if}\;j\in\mathbb{N}^*,\\
-\sqrt{\eta_{-j}}\;\;\text{if}\;(-j)\in\mathbb{N}^*,
\end{array}
\right.
$ \\
and \;\;$\varphi_j=\frac{1}{\sqrt{2}}\left(\begin{array}{l}
\frac{1}{i\mu_j}\partialhi_j\\
\partialhi_j
\end{array}\right)$\;\;\;$\forall\; j\in\mathbb{Z}^*$,\;\;\;\; ( we define\\
$\partialhi_{-j}=\partialhi_j,\;\forall j\in\mathbb{N}^*$)\\
where $\eta_j$ and $\partialhi_j$ are the eigenvalues and the corresponding eigenvectors of $\mathcal{A}$.\\
Moreover, define
$$C_s = span \{\varphi_j:\;\text{the corresponding}\;i\mu_j\;\text{satisfies}\;\vert\mu_j\vert\leq s\}.$$
\\
Now, we recall some results about Discrete Ingham type inequalities.\\
\begin{teo}(\cite{14}) Assume that there exist a positive number $\gamma$ satisfying
\begin{equation}\label{14}
\vert\omega_{k}-\omega_n\vert \geq \gamma \;\; \text{for all}\; k\neq n,
\end{equation}
where $(\omega_k)_{k\in\mathbb{Z}}$ is a family of real numbers.\\
Given $0<\sigma\leq \partiali/\gamma$ arbitrarily, fix an integer $J$ such that $J\sigma > \partiali/\gamma$. Then there exist two positive constants $c_1$ and $c_2$, depending only on $\gamma$ and $J\sigma$, such that, for every $t\in\mathbb{R}$, we have
\begin{equation}\label{15}
\sigma \displaystyle{\sum_{j=-J}^{J}}\left\vert\sum_{k\in\mathbb{Z}} x_k e^{i\omega_k(t+ j\sigma)}\right\vert^2\sim \sum_{k\in\mathbb{Z}}\vert x_k\vert^2,
\end{equation}
with complex coefficients $x_k$ satisfying the condition
\begin{equation}\label{16}
x_k=0 \;\;\text{whenever}\;\; \vert w_k\vert \geq \frac{\partiali}{\sigma}-\frac{\gamma}{2},
\end{equation}
\end{teo}
\begin{teo}(\cite{15}) Assume that there exist a positive number $\gamma_1$ satisfying
\begin{equation}\label{17}
\omega_{k+2}-\omega_k\geq 2\gamma_1 \;\; \forall\; k,
\end{equation}
where $(\omega_k)_{k\in\mathbb{Z}}$ is a family of real numbers.\\
Given $0<\sigma_1\leq \partiali/\gamma_1$ arbitrarily, fix an integer $J$ such that $J\sigma_1 > \partiali/\gamma_1$. Then there exist two positive constants $c_3$ and $c_4$, depending only on $\gamma_1$ and $J\sigma_1$,
\begin{equation}\label{18}
c_3 Q(x)\leq \sigma_1 \displaystyle{\sum_{j=-J}^{J}}\left\vert\sum_{k\in\mathbb{Z}} x_k e^{i\omega_k j\sigma_1}\right\vert^2\leq c_4 Q(x),
\end{equation}
with complex coefficients $x_k$ satisfying the condition
\begin{equation}\label{19}
x_k=0 \;\;\text{whenever}\;\; \vert w_k\vert \geq \frac{\partiali}{\sigma_1}-\frac{\gamma_1}{2},
\end{equation}
and where
$$ Q(x)=\sum_{k\in \mathbb{A}_1}\vert x_k\vert^2+\sum_{k\in \mathbb{A}_2}\vert x_k + x_{k+1}\vert^2 + (\omega_{k+1}-\omega_k)^2(\vert x_k\vert^2+\vert x_{k+1}\vert^2).$$
\end{teo}
The above equivalence means that ( see \cite{20} for more details)
$$ \sigma_1 \displaystyle{\sum_{j=-J}^{J}}\left\vert\sum_{k\in\mathbb{Z}} x_k e^{i\omega_k j\sigma}\right\vert^2\geq c_3\sum_{l=1}^{2}\sum_{k\in \mathbb{A}_l} \Vert B_{k}^{-1}C_{k}\Vert_2^2,$$
where $\Vert .\Vert_2$ means the Euclidean norm of the vector, for $k\in \mathbb{A}_l$ the vector $C_{k}$ and the $l\times l$ matrix $B_{k} $ are given by\\
$$C_{k}=x_k,\; B_{k}^{-1}=1\;\;\text{if}\;\; l=1,$$
and
$$C_{k}=\left( \begin{array}{c}
x_k \\
x_{k+1}
\end{array} \right),\;B_{k}^{-1}=\left( \begin{array}{c}
1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; 1\\
0 \;\;\;\;\;\;\;\;\;\;\;\omega_{k+1}-\omega_k\end{array} \right)\;\;\;\text{if}\;\; l=2.$$
Now, as in \cite{20}, let $U$ be a separable Hilbert space (in the sequel, $U$ will be $Y$ ). For a vector $d=\left( \begin{array}{c}
d_1\\
.\\
.\\
.\\
d_m\end{array} \right)\in U^m$, we set $\Vert .\Vert_{U,2}$ the norm in $U^m$ defined by
$$\Vert d\Vert_{U,2}^2=\sum_{j=l}^{m}\Vert d_j\Vert_U^2.$$
Let $\delta\in(0,\delta_0)$ where $\delta_0=\min(\partiali-\frac{\Delta t \gamma}{2}, \partiali-\frac{\Delta t \gamma_1}{2})$.\\
Then we obtain the discrete inequality of Ingham's type in $U$ :\\\\\\
\begin{pro}
If $\mu_n$ satisfy (\ref{14}), then for all sequence $(a_n)_n$ in $U$, the function
$$x^k=\sum_{\vert \mu_n\vert\leq\delta/\Delta t} a_n e^{ik\Delta t \mu_n}$$
satisfies the estimates
$$\Delta t \sum_{k\Delta t\in[0,T]}\Vert x^k\Vert_U^2\sim\sum_{\vert \mu_n\vert\leq\delta/\Delta t}\Vert a_n\Vert_U^2,$$
for $T>T_0=\frac{2\partiali}{\gamma}$.
\end{pro}
\begin{proof}
Since $U$ is a separable Hilbert space, there exists a Hilbert basis $(\partialsi_j)_{j\geq 1}$ of $U$. Therefore, $a_n\in U$ can be written as
$$a_n=\sum_{j=1}^{+\infty}a_n^j\partialsi_j.$$
We truncate $a_n$ as follows: for $K\in\mathbb{N}^*$, let $a_n^{(K)}=\displaystyle{\sum_{j=1}^{K}} a_n^j\partialsi_j$ \\
and set $x_K^k=\displaystyle{\sum_{j=1}^{K}}\left(\displaystyle{\sum_{\vert \mu_n\vert\leq\delta/\Delta t}} a_n^j e^{ik\Delta t \mu_n}\right)\partialsi_j.$\\
Since $(\partialsi_j)_{j\geq 1}$ is a Hilbert basis, we have by Parseval's theorem
$$\Vert x_K^k\Vert_U^2=\displaystyle{\sum_{j=1}^{K}}\left\vert\;\;\displaystyle{\sum_{\vert \mu_n\vert\leq\delta/\Delta t}}a_n^j e^{ik\Delta t \mu_n}\right\vert^2.$$
Thus, by applying discrete Ingham type inequality, we have
$$\Delta t \displaystyle{\sum_{k\Delta t\in[0,T]}}\Vert x_K^k\Vert_U^2=\Delta t \displaystyle{\sum_{j=1}^{K}}\;\;\displaystyle{\sum_{k\Delta t\in[0,T]}}\left\vert\displaystyle{\sum_{\vert \mu_n\vert\leq\delta/\Delta t}}a_n^j e^{ik\Delta t \mu_n}\right\vert^2.$$\\
$$\sim\displaystyle{\sum_{j=1}^{K}} \;\;\displaystyle{\sum_{\vert \mu_n\vert\leq\delta/\Delta t}}(a_n^j)^2.$$
$$\sim\displaystyle{\sum_{\vert \mu_n\vert\leq\delta/\Delta t}}\;\;\displaystyle{\sum_{j=1}^{K}}(a_n^j)^2.$$
Therefore
$$\Delta t \displaystyle{\sum_{k\Delta t\in[0,T]}}\Vert x_K^k\Vert_U^2\sim\displaystyle{\sum_{\vert \mu_n\vert\leq\delta/\Delta t}}\Vert a_n^{(K)}\Vert_U^2.$$
Since $x_K^k\to x^k$ and $a_n^{(K)}\to a_n$ when $K\to +\infty$, we obtain the result.
\end{proof}
\begin{cor}
With the same hypothesis of Theorem 2.2, for all sequence $(a_n)_{n\in\mathbb{Z^*}}$ in $U$, the function
$$ f^k=\sum_{n\in\mathbb{Z^*}} a_n e^{i\omega_n k\Delta t},$$
satisfy, for $T>T_1=\frac{2\partiali}{\gamma_1}$, the inequality
$$\Delta t \displaystyle{\sum_{k\Delta t\in[0,T]}}\Vert f^k \Vert_U^2\geq c \sum_{l=1}^{2}\sum_{n\in\mathbb{A}_l} \Vert B_{n}^{-1}C_{n}\Vert_{U,2}^2,$$
with $ a_n=0 \;\;\text{whenever}\;\; \vert w_n\vert \geq \frac{\partiali}{\Delta t}-\frac{\gamma_1}{2},$ and $c>0$.
\end{cor}
\begin{pro}
Assume that $Y$ is separable. Let $y^k$ the solution of the following system
\begin{equation}\label{20}
\left \{
\begin{array}{lcr}
\frac{y^{k+1}-y^{k}}{\Delta t}=A\left(\frac{y^{k+1}+y^{k}}{2}\right),\;\; k\in\mathbb{N},\\
y^{0}=z_{0}=(w_0, w_1).
\end{array}
\right.
\end{equation}
\begin{enumerate}
\item
Assume that $\mu_j$ satisfy (\ref{14}) and for all $y^0 \in V\times X$ we have
\begin{equation}\label{21}
\exists\; \theta>0,\;\forall\;j\geq 1,\;\Vert B^*\varphi_j\Vert_Y\geq \frac{\theta}{\mu_j^{2\beta+1}}
\end{equation}
for some constant $\theta>0$ and for a fixed real number $\beta> -\frac{1}{2}$.\\
Then, there exist a time $T > T_0$ and a constant $C > 0$ such that
\begin{equation}\label{22}
C\Vert y^{0}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2 \leq \Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T \right]}
}\left\Vert B^*\left(\frac{y^k+y^{k+1}}{2}\right)\right\Vert_Y^2,\; \forall\; y^0 \in C_{\delta/ \Delta t}.
\end{equation}
\item
Assume that $\mu_n$ verify (\ref{17}) and, for all $y^0 \in V\times X$,
\begin{equation}\label{23}
\exists\; \theta>0,\;\forall\;l=1,2,\;\;\forall\;n\in \mathbb{A}_l,\;\forall\;\xi\in\mathbb{R}^2,\;\Vert B_n^{-1}\partialhi_n\xi\Vert_{Y,2}\geq \frac{\theta}{\mu_n^{2\beta+1}}\Vert \xi\Vert_2,
\end{equation}
\\
then there exist a time $T > T_1$ and a constant $C > 0$ such that (\ref{22}) holds true.\\
\end{enumerate}
\end{pro}
\begin{rem}
In the last proposition, we have chosen $\delta/ \Delta t$ the filtering parameter. Indeed, this scale is linked with the paper \cite{8}. The question of optimality of this choice, in our case, remains open.
\end{rem}
\begin{proof}
We first show that (\ref{21})$\Longrightarrow$ (\ref{22}).\\
Simple formal calculations give
$$y^{k+1}=(I+\frac{\Delta t}{2}A)^{-1}(I-\frac{\Delta t }{2}A)y^k$$
$\hspace{3.8cm}=e^{i\alpha_j\Delta t}y^k,$\\
where $e^{i\alpha_j\Delta t}=\frac{1+\frac{\Delta t}{2}i\mu_j}{1-\frac{\Delta t}{2}i\mu_j}.$\\
Writing
$$y^0=\displaystyle{\sum_{\vert \mu_j\vert\leq\delta/\Delta t}}c_j \varphi_j=\left(\begin{array}{c} w_0\\
w_1\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{c} \displaystyle{\sum_{\vert \mu_j\vert\leq\delta/\Delta t}}\frac{1}{i\mu_j}c_j\partialhi_j\\
\displaystyle{\sum_{\vert \mu_j\vert\leq\delta/\Delta t}}c_j\partialhi_j\end{array}\right),$$
We have $\Vert w_0\Vert_{X_{-\beta}}^2=\Vert w_1\Vert_{X_{-\beta-\frac{1}{2}}}^2\sim\frac{1}{2}\displaystyle{\sum_{\vert \mu_j\vert\leq\delta/\Delta t}} c_j^2\eta_j^{-2\beta-1}.$\\
The solution $y^k$ is given by
$$y^k=\displaystyle{\sum_{\vert \mu_j\vert\leq\delta/\Delta t} }c_j e^{i\alpha_j k\Delta t}\varphi_j,$$
where
$$ \alpha_j=\frac{2}{\Delta t}\arctan(\frac{\mu_j\Delta t}{2}).$$
Consequently
$$B^*(\frac{y^{k+1}+y^k}{2})=\displaystyle{\sum_{\vert \mu_j\vert\leq\delta/\Delta t}}c_j\cos (\frac{\alpha_j\Delta t}{2})e^{i\alpha_j (k+\frac{1}{2})\Delta t} B^*\varphi_j .$$
It is easy to check that $\vert\alpha_k-\alpha_n\vert\geq \gamma\partialrime=\frac{\gamma}{2}$ (for all $k\neq n$) for $\Delta t $ sufficiently small, and
$$\cos^2(\frac{\alpha_j\Delta t}{2})=\cos^2(\arctan(\frac{\mu_j\Delta t}{2}))=\frac{1}{1+\frac{(\mu_j\Delta t)^2}{4}}\geq \frac{1}{1+\frac{\delta^2}{4}}.$$
Now, using Ingham's inequality in $Y$, for $T> T_0$, we get\\
$$\Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T \right]}
}\left\Vert B^*\left(\frac{y^k+y^{k+1}}{2}\right)\right\Vert_Y^2 \geq C_1 \displaystyle\sum_{\vert \mu_j\vert\leq\delta/\Delta t} c_j^2 \Vert \mathcal{B}^*\partialhi_j\Vert_Y^2.$$\\
By (\ref{21}), we get
$$\Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T \right]}
}\left\Vert B^*\left(\frac{y^k+y^{k+1}}{2}\right)\right\Vert_Y^2 \geq C_1 \displaystyle{\sum_{\vert \mu_j\vert\leq\delta/\Delta t}} c_j^2 \mu_j^{-2(2\beta+1)}$$\\
$$\hspace{4cm}=C_1\left\Vert y^{0}\right\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2.$$
The proof of (\ref{23})$\Longrightarrow$ (\ref{22}) is similar to the first one but now we use the other discrete Ingham type inequality presented in Corollary 2.1. \end{proof}
Applying Proposition 2.2, for any $\delta >0$ defined as above, choosing a time $T^*>T_2=\max (T_0,T_1)$ there exists a positive constant $C=C_{T^*,\delta}$ such that the inequality (\ref{22}) holds for any solution $y^k$ with $y^0\in C_{\delta/\Delta t}$ and where the gap condition (\ref{14}) or (\ref{17}) is satisfied. In the sequel, we fix $T^*=2T_2$.
\begin{lem}
If $\mu_j$ verify (\ref{14}) or (\ref{17}) , then there exists a constant $c > 0$ such that (\ref{13}) holds with $T=T^*$ for all solutions $u^k$
of (\ref{12}) uniformly with respect to $\Delta t$.
\end{lem}
\begin{proof}
The proof can be done similarly as the one of Lemma 5.2.4. in \cite{8}, we decompose the solution $u^k$ of (\ref{12}) into its low and high frequency parts. To be more precise, we consider
$$u_l^k=p_{\delta/ \Delta t}u^k,\;\;\;u_h^k=(I-p_{\delta/ \Delta t})u^k,$$\\
where $\delta>0$ is the positive number that we have been chosen above, and $p_{\delta/ \Delta t}u$ is the orthogonal projection on $C_{\delta/ \Delta t}$.\\
Note both $u_l^k$ and $u_h^k$ are solutions of (\ref{12}).\\
In addition, $u_h^k$ lies in the space $ C_{\delta/ \Delta t}^\bot$, in which we have:
\begin{equation}\label{24}
\left \{
\begin{array}{lcr}
\Delta t \Vert Ay\Vert_H\geq \delta \Vert y\Vert_H,\;\;\forall\;y\in C_{\delta/ \Delta t}^\bot,\;\; \text{and also}\\\\
\Delta t \Vert Ay\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}\geq \delta \Vert y\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}},\;\;\; \forall\;y\in C_{\delta/ \Delta t}^\bot.
\end{array}
\right.
\end{equation}
\\\\\\
{\bf The low frequencies.} First we compare $u_l^k$ with $y_l^k$ solution of (\ref{20})\\
with initial data $y_l^k(0)= u_l^k(0)$. Set $w_l^k=u_l^k-y_l^k$. From (\ref{22}), which is\\
valid for solutions of (\ref{20}) with initial data in $C_{\delta/ \Delta t}$, we get\\
$$C\Vert u_l^{0}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2 \leq 2\Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T^* \right]}
}\left\Vert B^*\left(\frac{u_l^k+\tilde{u}_l^{k+1}}{2}\right)\right\Vert_Y^2$$
\begin{equation}\label{25}
\hspace{4.1cm} + 2\Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T^* \right]}
}\left\Vert B^*\left(\frac{w_l^k+\tilde{w}_l^{k+1}}{2}\right)\right\Vert_Y^2.
\end{equation}
Now, we write the equation satisfied by $w_l^k$, which can be deduced from (\ref{12}) and (\ref{20}):
\begin{equation}\label{26}
\left \{
\begin{array}{lcr}
\frac{\tilde{w}_l^{k+1}-w_l^k}{\Delta t} = A\left(\frac{w_l^k+\tilde{w}_l^{k+1}}{2}\right),\;k\in \mathbb{N}, & \\\\
\frac{w_l^{k+1}-\tilde{w}_l^{k+1}}{\Delta t}={(\Delta t)^2}{A^2}{u_l^{k+1}},\;k\in\mathbb{N},\\\\
w_l^0=0.
\end{array}
\right.
\end{equation}
the energy estimates for $w_l^k$ give:
\begin{equation}\label{27}
\left \{
\begin{array}{lcr}
\Vert \tilde{w}_l^{k+1}\Vert_H^2=\Vert w_l^k\Vert_H^2,\\\\
\Vert w_l^{k+1}\Vert_H^2=\Vert \tilde{w}_l^{k+1}\Vert_H^2-2(\Delta t)^3 \left\langle Au_l^{k+1}, A\left(\frac{w_l^{k+1}+\tilde{w}_l^{k+1}}{2}\right)\right\rangle_H.
\end{array}
\right.
\end{equation}
Note that $w_l^k$ and $\tilde{w}_l^{k+1}$ belong to $C_{\delta/ \Delta t}$ for all $k\in\mathbb{N}$, since $u_l^k$ and $y_l^k$ both belong to $C_{\delta/ \Delta t}$. Therefore, the energy estimates for $w_l^k$ lead, for $k\in\mathbb{N}$, to
$$\Vert w_l^k\Vert_H^2=-2\Delta t \sum_{j=1}^{k} (\Delta t)^2 \left\langle Au_l^j, A\left(\frac{w_l^j+\tilde{w}_l^{j+1}}{2}\right)\right\rangle_H$$
$$\hspace{2cm} \leq \Delta t \sum_{j=1}^{k} (\Delta t)^2\Vert Au_l^j\Vert_H^2+\delta^2 \Delta t\sum_{j=1}^{k}\left\Vert\frac{w_l^j+\tilde{w}_l^{j+1}}{2}\right\Vert_H^2$$
$$ \hspace{1cm} \leq \Delta t \sum_{j=1}^{k} (\Delta t)^2\Vert Au_l^j\Vert_H^2+\delta^2 \Delta t\sum_{j=1}^{k}\left\Vert w_l^j\right\Vert_H^2,$$
where we used the first line of (\ref{27}).\\
Gr$\ddot{o}$nwall's Lemma applies and allows to deduced from (\ref{25}) and the fact that the operator $B$ is bounded, the existence of a positive\\
constant ( that may change from line to line) independent of $\Delta t$ such that\\
$$c\Vert u_l^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2\leq \Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T^* \right]}
}\left\Vert B^*\left(\frac{u_l^k+\tilde{u}_l^{k+1}}{2}\right)\right\Vert_Y^2$$
$$\hspace{2.7cm} +\Delta t \displaystyle { \sum_{k \Delta t \in \left]0,T^* \right]}
} (\Delta t)^2\Vert Au_l^k\Vert_H^2.$$\\
Besides,
$$\Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T^* \right]}
}\left\Vert B^*\left(\frac{u_l^k+\tilde{u}_l^{k+1}}{2}\right)\right\Vert_Y^2 \leq 2\Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T^* \right]}
}\left\Vert B^*\left(\frac{u^k+\tilde{u}^{k+1}}{2}\right)\right\Vert_Y^2$$
$$ \hspace{5.4cm} +2\Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T^* \right]}
}\left\Vert B^*\left(\frac{u_h^k+\tilde{u}_h^{k+1}}{2}\right)\right\Vert_Y^2$$\\
and, since $u_h^k$ and $\tilde{u}_h^{k+1}$ belong to $C_{\delta/ \Delta t}^\bot$ for all $k$, we get from (\ref{24}) that\\
$\Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T^* \right]}
}\left\Vert B^*\left(\frac{u_h^k+\tilde{u}_h^{k+1}}{2}\right)\right\Vert_Y^2\leq K_B^2\Delta t \displaystyle {\sum_{k \Delta t \in \left[0,T^* \right]}
}\left\Vert \frac{u_h^k+\tilde{u}_h^{k+1}}{2}\right\Vert_H^2$
$$ \hspace{3cm} \leq K_B^2\Delta t \displaystyle {\sum_{k \Delta t \in \left[0,T^* \right]}}\Vert u_h^k\Vert_H^2$$
$$ \hspace{5cm}\leq \frac{K_B^2}{\delta^2}\Delta t \displaystyle {\sum_{k \Delta t \in \left]0,T^* \right]}}\Vert Au_h^k\Vert_H^2 + K_B^2\Delta t \Vert u_h^0\Vert_H^2,$$
since, from the first line of (\ref{12}),
$$\Vert \tilde{u}_h^{k+1}\Vert_H^2=\Vert u_h^k\Vert_H^2,\;\;\forall\;k\in\mathbb{N}.$$
It follows that there exists $c>0$ independent of $\Delta t$ such that \\
$c\Vert u_l^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2\leq \Delta t \displaystyle { \sum_{k \Delta t \in \left[0,T^* \right]}
}\left\Vert B^*\left(\frac{u^k+\tilde{u}^{k+1}}{2}\right)\right\Vert_Y^2$\\
\begin{equation}\label{28}
\hspace{3.8cm} +\Delta t \displaystyle { \sum_{k \Delta t \in \left]0,T^* \right]}
} (\Delta t)^2\Vert Au_l^k\Vert_H^2+ \Delta t \Vert u_h^0\Vert_H^2.
\end{equation}
\\
{\bf The high frequencies.} We now discuss the decay properties of solutions of (\ref{12}) with initial data $u_h^0\in C_{\delta/ \Delta t}^\bot.$ It is easy to check that for all $k\in\mathbb{N}$, $u_h^k\in C_{\delta/ \Delta t}^\bot$. But, simple calculations give:\\
$\Vert (I-(\Delta t)^3A^2)u_h^{k+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2$\\
$\hspace{1.2cm} =\Vert u_h^{k+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2 +2(\Delta t)^3\Vert Au_h^{k+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2$\\
$\hspace{4cm} +(\Delta t)^6\Vert A^2u_h^{k+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2$\\
\begin{equation}\label{29}
= \Vert \tilde{u}_h^{k+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2=\Vert u_h^k\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2,\;\; k\in\mathbb{N}.
\end{equation}
Due to (\ref{24}), we get:
$$ (1+2(\Delta t)\delta^2)\Vert u_h^{k+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2\leq \Vert u_h^k\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2.$$\\
We deduce that
$$ \Vert u_h^{k+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2\leq \frac{1}{1+2(\Delta t)\delta^2}\Vert u_h^k\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2,\;\; k\in\mathbb{N},$$\\
which implies
\begin{equation}\label{30}
\Vert u_h^k\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2\leq\left(\frac{1}{1+2(\Delta t)\delta^2}\right)^k\Vert u_h^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2,\;\;k\in\mathbb{N}.
\end{equation}
Taking $k^*=[T^*/\Delta t]$, we get a constant $\tau<1$ independent of $\Delta t>0$ such that
$$\Vert u_h^{k^*}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2\leq \tau \Vert u_h^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2.$$
From (\ref{29}), we have that, for $k\in\mathbb{N}$,\\
$$\Vert u_h^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2=\Vert u_h^k\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2 +2\Delta t\sum_{j=0}^{k-1}(\Delta t)^2\Vert Au_h^{j+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2$$
$$ \hspace{4cm}+\Delta t \sum_{j=0}^{k-1}(\Delta t)^5\Vert A^2u_h^{j+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2,$$
taking $k=k^*$, we deduce the existence of a positive constant $c_1$, which depends only on $T^*$ and $\delta$ such that \\
$$c_1 \Vert u_h^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2 \leq \Delta t\sum_{j=0}^{k^*-1}(\Delta t)^2\Vert Au_h^{j+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2$$
$$\hspace{4.7cm} +\Delta t \sum_{j=0}^{k^*-1}(\Delta t)^5\Vert A^2u_h^{j+1}\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2.$$\\
Using the fact that $ H\subset X_{-\beta}\times X_{-\beta-\frac{1}{2}}$, with continuous embedding, we deduce the existence of a positive constant $c_2$, which depends only on $T^*$ and $\delta$ such that
\begin{equation}\label{31}
c_2 \Vert u_h^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2 \leq \Delta t\sum_{j=0}^{k^*-1}(\Delta t)^2\Vert Au_h^{j+1}\Vert_H^2 +\Delta t \sum_{j=0}^{k^*-1}(\Delta t)^5\Vert A^2u_h^{j+1}\Vert_H^2,
\end{equation}
holds uniformly with respect to $\Delta t>0$ for any solution of (\ref{12}) with initial data $u^0\in C_{\delta/ \Delta t}^\bot.$\\
Combining (\ref{28}) and (\ref{31}) yields Lemma 2.1, since $u_h$ and $u_l$ lie in orthogonal spaces with respect to the scalar product $\langle ., .\rangle_{X_{-\beta}}\times X_{-\beta-\frac{1}{2}}$ and $\langle A., A.\rangle_H$.
\end{proof}
\begin{rem}
The assumptions (\ref{14}) and (\ref{21}) or (\ref{17}) and (\ref{23}) hold true if we have (\ref{8}) ( see \cite{20}).
\end{rem}
\section{Polynomial stability via a "weakened" observability inequality and main result}
The main result of this paper reads as follows:\\
\begin{teo}
Assume that there exist positive constants $T$ and $c$ and $\beta>-\frac{1}{2}$ such that for all initial data $z^0\in D(A)=D(\mathcal{A})\times V$, we have (\ref{13}).\\
Then there exists $M>0$ such that
\begin{equation}\label{32}
E^k\leq\frac{M}{(1+t_k)^{\frac{1}{1+2\beta}}}\Vert z^0\Vert_{D(A)}^2, \;\;\;\forall k\geq 0,
\end{equation}
holds uniformly with respect to $0<\Delta t<1$,\; with $t_k=k\Delta t.$
\end{teo}
For the proof of this theorem, we need a technical lemma ( see Lemma 4.4 in \cite{2}).
\begin{lem}
Let $(\mathcal{E}^k)$ be a sequence of positive real numbers satisfying
$$\mathcal{E}_{k+1}\leq \mathcal{E}_{k}-C \mathcal{E}_{k+1}^{2+\alpha}, \;\;\;\forall k\geq 0,$$
where $C>0$ and $\alpha>-1$ are constants. Then there exists a positive constant M (depending only on $C$ and $\alpha$) such that
$$\mathcal{E}_k\leq\frac{M}{(k +1)^{\frac{1}{\alpha+1}}},\;\forall k\geq 0.$$
\end{lem}
\begin{proof}
We decompose the solution $z^k$ of (\ref{9}) as $z^k=w^k+u^k$ with $z^0=u^0$\\
where $u^k$ is the solution of (\ref{12}) and $w^k$ is the solution of\\
\begin{equation}\label{33}
\left \{
\begin{array}{lcr}
\frac{\tilde{w}^{k+1}-w^k}{\Delta t} = A\left(\frac{w^k+\tilde{w}^{k+1}}{2}\right)-BB^*\left(\frac{z^k+\tilde{z}^{k+1}}{2}\right),\;k\in \mathbb{N}, & \\\\
\frac{w^{k+1}-\tilde{w}^{k+1}}{\Delta t}={(\Delta t)^2}{A^2}{w^{k+1}},\;k\in\mathbb{N},\\\\
w^0=0.
\end{array}
\right.
\end{equation}\\\
\\\\\\\\\\\\\\\\\\
Applying Lemma 2.1 to $u^k = z^k - w^k$, we get: \\
$ c \Vert z^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2\leq 2\Bigg(\Delta t \displaystyle{\sum_{k\Delta t\in [0,T^*]}}\left\Vert B^*\left(\frac{z^k+\tilde{z}^{k+1}}{2}\right)\right\Vert_Y^2$\\
$+ \Delta t \displaystyle{\sum_{k\Delta t\in [0,T^*]}}(\Delta t)^2\left\Vert Az^{k+1}\right\Vert_H^2 +\Delta t\displaystyle{\sum_{k\Delta t\in [0,T^*]}}(\Delta t)^5\left\Vert {A^2}{z}^{k+1}\right\Vert_H^2 \Bigg )$\\
$ + 2\Bigg( \Delta t \displaystyle{\sum_{k\Delta t\in [0,T^*]}}\left\Vert B^* \left(\frac{w^k+\tilde{w}^{k+1}}{2}\right)\right\Vert_Y^2+ \Delta t \displaystyle{\sum_{k\Delta t\in [0,T^*]}}(\Delta t)^2\left\Vert Aw^{k+1}\right\Vert_H^2$\\
\begin{equation}\label{26}
+\Delta t\displaystyle{\sum_{k\Delta t\in [0,T^*]}}(\Delta t)^5\left\Vert {A^2}{w}^{k+1}\right\Vert_H^2 \Bigg).
\end{equation}
Now we follow the same approach as in the proof of Theorem 1.1 in \cite{8}, there exists a constant $G>0$ (independent of $\Delta t$) such that\\
$$\Delta t \displaystyle{\sum_{k\Delta t\in [0,T^*]}}\left\Vert B^* \left(\frac{w^k+\tilde{w}^{k+1}}{2}\right)\right\Vert_Y^2 + \Delta t \displaystyle{\sum_{k\Delta t\in [0,T^*]}}(\Delta t)^2\left\Vert Aw^{k+1}\right\Vert_H^2 $$\\
$\hspace{5cm}+ \Delta t\displaystyle{\sum_{k\Delta t\in [0,T^*]}}(\Delta t)^5\left\Vert {A^2}{w}^{k+1}\right\Vert_H^2$\\
$$\hspace{2cm}\leq G\Delta t \displaystyle{\sum_{j\Delta t\in [0,T^*]}}\left\Vert B^* \left(\frac{z^j+\tilde{z}^{j+1}}{2}\right)\right\Vert_Y^2.$$
Combining this inequality and (\ref{26})), we get the existence of a constant $c$ such that\\
$$ c \Vert z^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2\leq \Delta t \displaystyle{\sum_{k\Delta t\in [0,T^*]}}\left\Vert B^* \left(\frac{z^k+\tilde{z}^{k+1}}{2}\right)\right\Vert_Y^2$$
$$ \hspace{4cm}+ \Delta t \displaystyle{\sum_{k\Delta t\in [0,T^*]}}(\Delta t)^2\left\Vert Az^{k+1}\right\Vert_H^2$$
$$ \hspace{4cm}+\Delta t\displaystyle{\sum_{k\Delta t\in [0,T^*]}}(\Delta t)^5\left\Vert {A^2}{z}^{k+1}\right\Vert_H^2.$$
Combining this inequality and (\ref{11}), it follows that:
$$E^{l+1}\leq E^0- c \Vert z^0\Vert_{X_{-\beta}\times X_{-\beta-\frac{1}{2}}}^2.$$
By using a simple interpolation inequality ( see Proposition 2.3 in \cite{17}) and the fact that the function $E^k$ is nonincreasing, we obtain the existence of a constant $C>0$ such that
\begin{equation}\label{35}
E^{l+1}\leq E^0- C\frac{\{E^{l+1}\}^{2(1+ \beta)}}{\Vert z^0\Vert_{D(A)}^{2(1+2 \beta)}}.
\end{equation}
Estimate (\ref{35}) remains valid in successive intervals $[k(l+1), (k + 1)(l+1)]$, so, we have
$$E^{(k+1)(l+1)}\leq E^{k(l+1)}- C\frac{\{E^{(k+1)(l+1)}\}^{2(1+ \beta)}}{\Vert z^0\Vert_{D(A)}^{2(1+2 \beta)}}.$$
If we adopt the notation
$$\mathcal{H}^k=\frac{E^{k(l+1)}}{\Vert z^0\Vert_{D(A)}^2},$$
the last inequality gives
$$\mathcal{H}^{k+1}\leq\mathcal{H}^k- C(\mathcal{H}^{k+1})^{2(1+\beta)},\;\;\forall k\geq 0.$$
By using Lemma 3.1, we obtain the existence of a constant $M>0$ such that
$$\mathcal{H}^k\leq \frac{M}{(1+k)^{\frac{1}{1+2\beta}}},\;\forall k\geq 0,$$
and consequently, for all $\Delta t <1$, we have
$$E^{k(l+1)}\leq \frac{M}{(1+t_k)^{\frac{1}{1+2\beta}}}\Vert z^0\Vert_{D(A)}^2,\;\forall k\geq 0,$$
which obviously implies (\ref{32}).
\end{proof}
\section{Applications}
\subsection{Two coupled wave equations}
We consider the following system
\begin{equation*}
\left\{
\begin{array}{lcr}
u_{tt}(x,t)-u_{xx}(x,t)+\alpha y(x,t)=0,\;\;0<x<1,\;t>0,\\\\
y_{tt}(x,t)-y_{xx}(x,t)+\alpha u(x,t)+\gamma y_t(x,t)=0,\;\;0<x<1,\;t>0,\\\\
u(0,t)=u(1,t)=y(0,t)=y(1,t)=0,\;\;t>0,\\\\
u(x,0)=u_0(x),\;u_t(x,0)=u_1(x),\; y(x,0)=y_0(x),\;y_t(x,0)=y_1(x),\;\;0<x<1.\end{array}
\right.
\end{equation*}
with $\gamma >0$ and $\alpha>0$ small enough. Take $H=L^2(0,1)^4$, the operator $B$ defined by
$$B=\left(\begin{array}{l}
0\\
0\\
0\\
\sqrt{\gamma}\end{array}\right),\;\;\;(B^*=(0,0,0,\sqrt{\gamma}))$$
which is a bounded operator from $Y=L^2(0,1)$ into $H$ and the operator $A$ as follows
$$A=\left( \begin{array}{c}
0\;\;\;\;\;\;\;0\;\;\;\;\;\;\;\;I\;\;\;\;\;\;\;0 \\
0\;\;\;\;\;\;\;0\;\;\;\;\;\;\;\;0\;\;\;\;\;\;\;I\\
\partialartial_{xx}\;\;\;\;-\alpha\;\;\;\;0\;\;\;\;\;\;0\\
-\alpha\;\;\;\;\;\;\partialartial_{xx}\;\;\;\;\;0\;\;\;\;\;\;0
\end{array} \right),$$\\\\
with $D(A)=(H_0^1(0,1)\cap H^2(0,1))^2\times L^2(0,1)^2$.\\
If $\alpha$ is small enough, namely if $\alpha <\partiali^2$, this operator $A$ is a skew-adjoint\\
operator in $H$, then the above system is equivalent to system (\ref{6}) where $Z=\left(\begin{array}{l}
u\\
y\\
u_t\\
y_t\end{array}\right)$.
We use the same method in \cite{1}, we show that the eigenvalues of $A$ are
$$sp(A)=\{i\mu_{+,k}\}\cup\{i\mu_{-,k}\}\;\;k\in\mathbb{Z^*},$$
with $\mu_{+,k}=\sqrt{\alpha+k^2\partiali^2}$,\; $\mu_{-,k}=\sqrt{-\alpha+k^2\partiali^2}$ and $\mu_{+,-k}=-\mu_{+,k}$\;\; $\mu_{-,k}=-\mu_{-,k},\;\forall k\in\mathbb{N^*}.$\\
The corresponding eigenvectors are, respectively, given by
$$w_{+,k}=\left(\begin{array}{l}
\frac{1}{i\mu_{+,k}}\sin(k\partiali x)\\
\frac{1}{i\mu_{+,k}}\sin(k\partiali x)\\
\sin(k\partiali x)\\
\sin(k\partiali x)\end{array}\right),\;\;w_{-,k}=\left(\begin{array}{l}
\frac{1}{i\mu_{-,k}}\sin(k\partiali x)\\
-\frac{1}{i\mu_{-,k}}\sin(k\partiali x)\\
\sin(k\partiali x)\\
-\sin(k\partiali x)\end{array}\right)\;\;k\in\mathbb{Z^*},$$
with $w_{+,-k}=w_{+,k}$ and $w_{-,-k}=w_{-,k}$,\;\;$\forall\;k\in\mathbb{N^*}$.\\
(\ref{17}) is satisfied and (\ref{23}) holds with $\beta=0$ (see \cite{1} for more details), thus the above system is weakly observable \cite{20}, and consequently polynomially stable. Now, according to Theorem 3.1 we have
\begin{pro}
The solutions of
\begin{equation*}
\left \{
\begin{array}{lcr}
\frac{\tilde{Z}^{k+1}-Z^k}{\Delta t} = A\left(\frac{Z^k+\tilde{Z}^{k+1}}{2}\right)-BB^*\left(\frac{Z^k+\tilde{Z}^{k+1}}{2}\right),\;k\in \mathbb{N}, & \\\\
\frac{Z^{k+1}-\tilde{Z}^{k+1}}{\Delta t}={(\Delta t)^2}{A^2}{Z^{k+1}} ,\;k\in\mathbb{N}, \\\\
Z^0=(u_0,y_0,u_1,y_1).
\end{array}
\right.
\end{equation*}
are polynomially uniformly decaying in the sense of (\ref{32}) with $\beta=0$.
\end{pro}
\subsection{Two boundary coupled wave equations}
We consider the following system
\begin{equation*}
\left\{
\begin{array}{lcr}
u_{tt}(x,t)-u_{xx}(x,t)=0,\;\;0<x<1,\;t>0,\\\\
y_{tt}(x,t)-y_{xx}(x,t)+\gamma y_t(x,t)=0,\;\;\;0<x<1,t>0,\\\\
u(0,t)=y(0,t)=0,\;\;\;t>0,\\\\
y_x(1,t)=\alpha u(1,t),\;\; t>0,\\\\
u(x,0)=u_0(x),\;u_t(x,0)=u_1(x),\; y(x,0)=y_0(x),\;y_t(x,0)=y_1(x),\;\;\;\;0<x<1.\end{array}
\right.
\end{equation*}
when $\alpha,\beta\in\mathbb{R}$ with $\gamma>0$ and $\alpha>0$ small enough. Hence it is written\\
in the form (\ref{6}) with the following choices: Take $H=L^2(0,1)^4$, the \\
operator $B$ as follows:
$$B=\left(\begin{array}{l}
0\\
0\\
0\\
\sqrt{\gamma}\end{array}\right),\;\;\;(B^*=(0,0,0,\sqrt{\gamma}))$$
which is a bounded operator from $Y=L^2(0,1)$ into $H$ and the operator\\
$A$ defined by
$$D(A)=\{(u,y,u_t,y_t)\in (V\cap H^2(0,1))^2\times L^2(0,1)^2: y_x(1)=\alpha u(1); u_x(1)=\alpha y(1)\}$$
when $V=\{v\in H^1(0,1); v(0)=0\}$ and
$$AZ=\left( \begin{array}{l}
u_t \\
y_t\\
u_{xx}\\
y_{xx}
\end{array} \right),$$
when $Z=\left(\begin{array}{l}
u\\
y\\
u_t\\
y_t\end{array}\right)$.
If $\alpha$ is small enough, namely if $\alpha <1$, this operator $A$ is skew-adjoint in $H$.\\
As in \cite{1}, the eigenvalues of $A$ are
$$sp(A)=\{i\mu_{+,k}\}\cup\{i\mu_{-,k}\}\;\;k\in\mathbb{Z^*},$$
with $\mu_{+,k}=\frac{\partiali}{2}+k\partiali +\epsilon_{+,k}$,\; $\mu_{-,k}=\frac{\partiali}{2}+k\partiali -\epsilon_{-,k}$ and $\mu_{+,-k}=-\mu_{+,k}$\;\; $\mu_{-,k}=-\mu_{-,k},\;\forall k\in\mathbb{N^*},$\\
where $\epsilon_{+,k}=\arctan(\frac{\alpha}{\mu_{+,k}})$ and $\epsilon_{-,k}=\arctan(\frac{\alpha}{\mu_{-,k}}).$\\
The corresponding eigenvectors are, respectively, given by
$$w_{+,k}=\left(\begin{array}{l}
-\frac{1}{i\mu_{+,k}}b_{+,k}\sin(\mu_{+,k}.)\\
\frac{1}{i\mu_{+,k}}b_{+,k}\sin(\mu_{+,k}.)\\
-b_{+,k}\sin(\mu_{+,k}.)\\
b_{+,k}\sin(\mu_{+,k}.)
\end{array}\right),\;\;w_{-,k}=\left(\begin{array}{l}
\frac{1}{i\mu_{-,k}}b_{-,k}\sin(\mu_{-,k}.)\\
\frac{1}{i\mu_{-,k}}b_{-,k}\sin(\mu_{-,k}.)\\
b_{-,k}\sin(\mu_{-,k}.)\\
b_{-,k}\sin(\mu_{-,k}.)\end{array}\right)
\;\;k\in\mathbb{Z^*},$$
with $w_{+,-k}=w_{+,k}$ and $w_{-,-k}=w_{-,k}$,\;\;$\forall\;k\in\mathbb{N^*}$,\\
and where $b_{+,k}$ and $b_{-,k}$ are chosen to normalize the eigenvectors.\\\\\\\\
(\ref{17}) is satisfied and (\ref{23}) holds with $\beta=0$ (see \cite{1} for more details), thus the above system is weakly observable \cite{20}, and consequently polynomially stable. Now, applying Theorem 1.3 we get
\begin{pro}
The solutions of
\begin{equation*}
\left \{
\begin{array}{lcr}
\frac{\tilde{Z}^{k+1}-Z^k}{\Delta t} = A\left(\frac{Z^k+\tilde{Z}^{k+1}}{2}\right)-BB^*\left(\frac{Z^k+\tilde{Z}^{k+1}}{2}\right),\;k\in \mathbb{N}, & \\\\
\frac{Z^{k+1}-\tilde{Z}^{k+1}}{\Delta t}={(\Delta t)^2}{A^2}{Z^{k+1}} ,\;k\in\mathbb{N}, \\\\
Z^0=(u_0,y_0,u_1,y_1).
\end{array}
\right.
\end{equation*}
are polynomially uniformly decaying in the sense of (\ref{32}) with $\beta=0$.
\end{pro}
\section{Further comments}
\begin{enumerate}
\item
As we mentioned in the introduction, our methods and results require the assumption that the damping operator
$B$ is bounded. We use the fact that the polynomial
decay of the energy is a consequence of the observability properties of the conservative system. That is the case, even in the continuous setting. However, in several relevant applications when the feedback law is unbounded \cite{2}, our method does not apply.
\item
Another drawback of our method is that it is restrictive for a class of operators, that is the spectrum of the operator $A$ associated
with the undamped problem satisfies such a gap condition. This is due to the method we employ, which is based on a discrete Ingham type inequalities. One could ask if we have some results about polynomial stability for the time semi-discrete scheme when the following generalized gap condition $w_{k+N}-w_k\geq N\gamma,\;k\in\mathbb{N}$, is satisfied for $N\geq 3$. To our knowledge, we don't have a discrete Ingham type inequalities when the last gap condition is verified, and this issue is widely open.\\
In our context, it would be also relevant to ask if our methods allow to deal with stabilization properties of fully discrete approximation scheme with numerical viscosity or under a suitable CFL type condition on the time and space discretization parameters as in the exponential case \cite{8}.
\item
Other question arise when discretizing in time semilinear wave equations. For instance, in \cite{12}, under suitable properties of the nonlinearity it is proved that the polynomial decay property of solutions holds. It would be interesting to analyze
whether the same polynomial decay property holds, uniformly with respect to the time-step, for the numerical schemes
analyzed in this article in this semilinear setting.
\end{enumerate}
\end{document} |
\begin{document}
\parbox{13 cm}
{
\begin{flushleft}
\vspace* {1.2 cm}
{\Large\bf
{Revealing non-classical behaviours in the oscillatory motion of a trapped ion
}
}\\
\vskip 1truecm
{\large\bf
{
B.Militello$^1$, A.Messina$^1$, A.Napoli$^1$
}
}\\
\vskip 5truemm
{
$^1$)
INFM, MIUR
and Dipartimento di Scienze Fisiche ed Astronomiche \\Via Archirafi 36, 90123 Palermo (ITALY)
\\e-mail: [email protected]
}
\end{flushleft}
}
\vskip 0.5truecm
{\bf Abstract:\\}
{
\noindent
The possibility of revealing non-classical behaviours in the dynamics of a trapped ion via measurements of the mean value of suitable operators is reported.
In particular we focus on the manifestation known as \lq\lq Parity Effect\rq\rq which may be observed \emph{directly measuring} the expectation value of an appropriate correlation operator.
The experimental feasibility of our proposal is discussed.
}
\vskip 0.1 cm
\noindent
PACS: 03.65.Ta; 32.80.Pj; 42.50.Ct; 42.50.Hz
\section{Introduction}
Trapped ions furnish a physical scenario wherein many interesting applications may be realized and quantum mechanical manifestations observed.
In a Paul trap, an ion is confined in such a way that its center of mass is substantially nothing but a quantized tridimensional harmonic oscillator, the three quadratic well frequencies being determined by the ion properties and trapping field parameters\cite{nist,Toschek}.
Subjecting the trapped ion to suitable laser field configurations, it is possible to generate a wide variety of vibronic couplings involving both the center of mass degrees of freedom and the internal atomic state variables\cite{nist,Vogel-Rass,Vogel-Modelli}.
Under the action of such interactions the ion, prepared in appropriate dynamical configurations, exhibits non-classical features. We will focus our attention on a particular quantum mechanical effect known as \lq\lq Parity Effect\rq\rq, brougth to light in cavity QED\cite{Parity-Anna} and then found in trapped ions systems too\cite{Parity-Sabry}.
The detection of this effect may be performed using, for example, methods already proposed for reconstructing the Wigner distribution\cite{Davidovich}. In this way, one con observe all dynamical manifestations of the system and in particular the Parity Effect. Nevertheless this effect also reveals itself in a simpler way, for example in the temporal evolution of the mean value of an appropriate correlation operator relative to two components of the center of mass vibrations. Such a correlation exhibits very different behaviours depending on the parity of the initial number of vibrational quanta. Hence, in some sense, the \lq\lq observation\rq\rq of just the expectation value of a suitable correlation operator is enough to clearly reveal the occurrence of the effect we are interested in.
In this paper we present a method to perform a \emph{direct measurement} of the mean value of a correlation operator appropriate to reveal the Parity Effect. We also show that our procedure is applicable to other vibrational variables.
As direct measurement we mean the possibility of obtaining just the expectation value of an observable without reconstructing the eigenvalue probability distribution at all.
In the next section we will describe in detail the occurrence of the Parity Effect and in the subsequent section we will present our approach to vibrational variables expectation value measurements focusing on the correlation operator appropriate for Parity effect detection. Finally, in the last section, we will give some conclusive remarks and the generalization of the presented procedure.
\section{Non-Classical behaviours: Parity Effect}
Consider a two level ion confined in a Paul trap, so that its free dynamics is governed by the Hamiltonian
\begin{equation} \label{unperturbed}
\hat{H}_0=\hbar\sum_{j=x,y,z}\omega_{j}\hat{a}_j^{\dag}\hat{a}_j
+\hbar\omega_{A} \hat{\sigma}_{z}
\end{equation}
Here $\omega_j$ ($j=x,y,z$) are the frequencies of the trap, $\hat{a}_j(\hat{a}_j^{\dag})$ with $j=x,y,z$ are the annihilation (creation) operators of the centre of mass oscillatory motion. The axial symmetry of the trap implies $\omega_{x}=\omega_{y}$.
Moreover $\omega_A$ is the atomic transition frequency and $\hat{\sigma}_z$ the $z$ component of the pseudospin operator $\stackrel{\rightarrow}{\sigma}\equiv(\hat{\sigma}_x,\hat{\sigma}_y,\hat{\sigma}_z)$ associated to the internal ionic dynamics. Here $\hat{\sigma}_x=\hat{\sigma}_+ +\hat{\sigma}_-$, $\hat{\sigma}_y=i(\hat{\sigma}_+ -\hat{\sigma}_-)$, $\hat{\sigma}_+=\KetBra{+}{-}$,
$\hat{\sigma}_-=\KetBra{-}{+}$, $\hat{\sigma}_z=\KetBra{+}{+}-\KetBra{-}{-}$,
and $\Ket{\pm}$ are the two effective atomic levels.
Prepare the ion in a vibrational $SU(2)$ coherent state
\begin{equation} \label{SU2-state}
\Ket{\psi_{in}}=\sum_{k=0}^{N}\frac{1}{2^{\frac{N}{2}}}\left(\frac{N!}{(N-k)!k!}\right)^{\frac{1}{2}}\Ket{n_x=k, n_y=N-k, n_z=0}\Ket{-}
\end{equation}
This is a state with a well defined bosonic excitation numberm, $N$, in the $xy$ plane, which is nothing but a Fock state along the bisector of the first $xy$-quadrant.
Subject now the particle to a suitable laser configuration responsible for the \emph{two-mode and two-phonon Jaynes-Cummings model}\cite{Parity-Sabry}
\begin{equation} \label{JC-Model}
\hat{H}_{int}=\hbar g \left(\hat{a}_x\hat{a}_y\hat{\sigma}_+ + h.c.\right)
\end{equation}
The dynamics of the trapped ion under such conditions has been studied in detail in ref\cite{Parity-Sabry} where has been brought to light that, at certain instant of time, the system discriminate between the two cases corresponding to \lq\lq $N$ even\rq\rq or \lq\lq $N$ odd\rq\rq. Such a discrimination reveals itself, for example, in the behaviour of the mean value of an operator which correlates the two motions along $x$ and $y$, that is
\begin{equation} \label{C_xy}
\hat{C}_{xy}=\hat{a}_x^{\dag}\hat{a}_y+\hat{a}_x\hat{a}_y^{\dag}
\end{equation}
or its square, $\hat{C}_{xy}^2$.
The mean value of $\hat{C}_{xy}$ is evaluated in the state of the ion after an electronic \emph{conditional measurement} has been performed. More precisely, we consider the unitary time evolution of the system under the action of the hamiltonian (\ref{JC-Model}) and then induce a collapse of the electronic state (observing it). If the ion is found in the atomic ground state, we consider the mean value $\MeanValue{\hat{C}_{xy}}$ in the collapsed wave function.
In fig.1 are shown the graphics of $\MeanValue{\hat{C}_{xy}}$ evaluated at time $t$ for two different values of $N$: $N=20$ and $N=21$.
It is well visible that there exists an instant of time at which the mean value of the correlation operator $\hat{C}_{xy}$ assume the value $(-1)^N N$, discriminating in a \emph{mesoscopic} way (a difference almost equal to $2N=40$ in the mean value) between two just \emph{microscopically} distinguishable initial conditions (different just for $1$ vibrational quantum).
\begin{figure}
\caption{\footnotesize
Temporal evolution of $\langle\hat{C}
\end{figure}
\section{Direct measurement of $\MeanValue{\hat{C}_{xy}}$}
In order to directly measure the mean value of the correlation operator $\hat{C}_{xy}$ one can use the procedure outlined in ref\cite{My-Work}. This requires the implementation of a suitable vibronic coupling
\begin{equation} \label{interaction}
\hat{H}^{(I)}_I=\hbar\gamma\hat{C}_{xy}\hat{\sigma}_{x}
\end{equation}
which induces atomic transitions whose speed is determined by the mean value of the motional operator $\hat{C}_{xy}$. Hence, monitoring the electronic state after the action of hamiltonian (\ref{interaction}), it is possible to obtain information about the mean value of $\hat{C}_{xy}$.
The vibronic coupling (\ref{interaction}) may be realized just considering two lasers directed along the axis $x'$ and $y'$ which are $x$ and $y$ axis $\frac{\pi}{4}$-rotated about $z$ respectively. These two lasers must have the same intensities, be $\pi$ out of phases and both tuned to the electronic transition frequency, $\omega_A$.
The evaluation of the effective hamiltonian in the Lamb-Dicke limit and in the Rotating Wave Approximation just leads to the hamiltonian in eq.(\ref{interaction}).
Let us consider now the dynamics induced by this interaction.
If the ion is described by the state
\begin{equation} \label{init-cond-z}
\Ket{\psi(0)}=\Ket{\psi_{vibr}}\Ket{-}
\end{equation}
one may \emph{rotate} the electronic state (via a \emph{non dispersive} interaction due to a Raman laser tuned to $\omega_A$, that is via a hamiltonian of the form $\hat{H}_{I}\propto \delta \hat{\sigma}_x$) in order to obtain
\begin{equation} \label{init-cond-y}
\Ket{\overline{\psi}(0)}=\Ket{\psi_{vibr}}\Ket{-}_y
\end{equation}
where $\Ket{-}_y$ is an eigenstate of the operator $\hat{\sigma}_y$. Applying now the interaction in eq.(\ref{interaction}) and evaluating the mean value of $\hat{\sigma}_z$ one obtains
\begin{eqnarray}\label{Mean-Sigma-L}
\MeanValue{\hat{\sigma}_z(t)}
=
\MatrixEl{\overline{\psi}(0)}
{e^{i\frac{\hat{H}^{(I)}_I}{\hbar}t}
\hat{\sigma}_z
e^{-i\frac{\hat{H}^{(I)}_I}{\hbar}t}}
{\overline{\psi}(0)}=
-\MatrixEl{\psi_{vibr}}{\sin(2\gamma t \hat{C}_{xy})}{\psi_{vibr}}
\end{eqnarray}
Suppose now that the mean value of the operator $\sin(2\gamma t \hat{C}_{xy})$ may be linearized, meaning that the following approximation is valid
\begin{equation} \label{linearization}
\MeanValue{\sin(2\gamma t \hat{C}_{xy})} \simeq \MeanValue{2\gamma t \hat{C}_{xy}}
\end{equation}
Under such a hypothesis the mean value of $\hat{\sigma}_z$, evaluated after the action of the hamiltonian in eq.(\ref{interaction}), is proportional to the mean value of $\hat{C}_{xy}$ before such a vibronic interaction.
Observe now that in typical experiments performed with trapped ions, the number of bosonic excitations doesn't exceed 30. Moreover, since the correlation operator $\hat{C}_{xy}$ is canonically equivalent to the operator $\hat{a}_y^{\dag}\hat{a}_y-\hat{a}_x^{\dag}\hat{a}_x$ (as well visible transforming it considering a $\frac{\pi}{4}$-rotation about z), we can assume that no eigenstate of $\hat{C}_{xy}$ corresponding to an eigenvalue $c>c_{max}=30$ is involved in the series expansion of $\Ket{\psi_{vibr}}$.
This is a central assumption of our protocol that legitimates the linearization of the sinusoidal operator.
Indeed, in this situation we can choose the interaction time relative to the hamiltonian in eq.(\ref{interaction}) in such a way that $\sin(2\gamma t c)\simeq 2\gamma t c$ for $|c|<c_{max}$ and this easily leads to the approximation in eq.(\ref{linearization}).
The last step of our analysis consists in estimating the necessary duration, $t$, of the interaction due to hamiltonian (\ref{interaction}), and in verifying its consistency with decoherence and typical experimental times.
To this end consider now characteristic values for the coupling strength, for instance $\gamma\sim 10 KHz$\cite{exp-values-nist,exp-values-Inn}, and $0.4$ as the bound of sinus linearizability zone (meaning that we assume $\sin x\simeq x$ for $|x|<0.4$). From these assumptions we deduce an interaction time of the order of magnitude of $1\mu s$, which is compatible both with decoherence (in the sense that coherences are maintained during the vibronic interaction) \cite{exp-values-nist}
and \lq\lq laser switch-on/switch-off\rq\rq typical times.
\section{Conclusive remarks}
Summarizing we have recalled the Parity Effect occurring in the dynamics of a trapped ion under some particular conditions, stressing the fact that such a behaviour may be revealed considering the mean value of a suitable correlation operator $\hat{C}_{xy}$.
Then we have presented a proposal for measuring the mean value of a suitable correlation operator \emph{without reconstructing its eigenvalue probability distribution function at all}.
The possibility of implementing the vibronic coupling (\ref{interaction}) and the values of the quantities involved in the protocol above stated (strength coupling, interaction time, maximum value of the eigenvalues of $\hat{C}_{xy}$) both imply a good degree of experimental feasibility.
We conclude this paper stressing that the mean value measurement protocol here presented is also applicable to a wide variety of motional observable like for example, energy, position, momentum, angular momentum\cite{My-Work}. Indeed, our procedure substantially rests upon the implementation of the hamiltonian (\ref{interaction}) \emph{mutatis mutandis} and in the individuation of a \emph{cut-off} for the eigenvalues of the observable under scrutiny. This last assumption is valid for vibrational energy (we have already recalled the maximum value of 30 excitations), obviously for position (by definition of trapped ion) and even for angular momentum (which, as the correlation operator, may be unitarily transformed into the difference of two phononic number operators).
\end{document} |
\begin{document}
\title{Being, Becoming and the Undivided Universe:
A Dialogue between Relational Blockworld and the Implicate Order
Concerning the Unification of Relativity and Quantum Theory}
\institute{Michael Silberstein \at Department of Philosophy \\ Elizabethtown College \\
Elizabethtown, PA 17022 \\ Tel. 717-361-1253\\
\email{[email protected]} \and W.M. Stuckey \at Department of
Physics
\\ Elizabethtown College \\ Elizabethtown, PA 17022 \\ Tel.
717-361-1436 \\
\email{[email protected]}
\and Timothy McDevitt \at Department of Mathematical Sciences \\
Elizabethtown College \\ Elizabethtown, PA 17022 \\ Tel.
717-361-1337\\
\email{[email protected]}}\maketitle
\begin{abstract}
In this paper two different approaches to unification will be
compared, Relational Blockworld (RBW) and Hiley's implicate order.
Both approaches are monistic in that they attempt to derive matter
and spacetime geometry `at once' in an interdependent and background
independent fashion from something underneath both quantum theory
and relativity. Hiley's monism resides in the implicate order via
Clifford algebras and is based on process as fundamental while RBW's
monism resides in spacetimematter via path integrals over graphs
whereby space, time and matter are co-constructed per a global
constraint equation. RBW's monism therefore resides in being
(relational blockworld) while that of Hiley's resides in becoming
(elementary processes). Regarding the derivation of quantum theory
and relativity, the promises and pitfalls of both approaches will be
elaborated. Finally, special attention will be paid as to how
Hiley's process account might avoid the blockworld implications of
relativity and the frozen time problem of canonical quantum gravity.
\end{abstract}
\thispagestyle{empty}
\section{Introduction}
\begin{quotation}Listening not to me but to the Logos it is wise to agree that
all things are one.
-Heraclitus\end{quotation}
\begin{quotation}There remains, then, but one word by which to express the
[true] road: Is. And on this road there are many signs that What Is
has no beginning and never will be destroyed: it is whole, still,
and without end. It neither was nor will be, it simply is-now,
altogether, one, continuous ...
-Parmenides\end{quotation}
\label{section1}
\subsection{Modeling Fundamental Reality and Ultimate Explanation: A Schism in Physics}
There has been a very long standing debate in Western philosophy and
physics regarding the following three pairs of choices about how
best to model the universe: 1) the fundamentality of being versus
becoming, 2) monism versus atomism and 3) algebra versus geometry
broadly construed; more generally, which of the myriad formalisms
will be most unifying.
Regarding 1, from very early on Western thinkers have generally
assumed that everything can be explained. Perhaps the cosmological
argument for the existence of God is the classic example of such
thinking. In that argument Leibniz appeals to a version of the
principle of sufficient reason (PSR) which states\cite{melamed} ``no
fact can be real or existing and no statement true without a
sufficient reason for its being so and not otherwise.'' Leibniz uses
the principle to argue that the sufficient reason for the ``series
of things comprehended in the universe of creatures must exist
outside this series of contingencies and is found in a necessary
being that we call God''\cite{melamed}. While physics dispensed with
appeals to God at some point, it did not jettison PSR, merely
replacing God with fundamental dynamical laws, e.g., as anticipated
for a Theory of Everything (TOE), and initial conditions (the big
bang or some condition leading to it). In keeping with everyday
experience a very early assumption of Western physics--reaching its
apotheosis with Newtonian mechanics--is that the fundamental
phenomena in need of explanation are \emph{motion} and \emph{change
in time}, so explanation will involve dynamical laws most
essentially.
In the quest to unify all of physics, it is the combination of PSR
plus the dynamical perspective writ large (call it dynamism) that
has in great part motivated the particular kind of unification being
sought, i.e., the search for a TOE, quantum gravity (QG) and the
like. Therefore, almost all attempts to unify relativity and quantum
theory opt for becoming (dynamism) as fundamental in some form or
another. Such theories may deviate from the norm by employing
radical new fundamental dynamical entities (branes, loops, ordered
sets, etc.), but the game is always dynamical, broadly construed
(vibrating branes, geometrodynamics, sequential growth process,
etc).
However, it is also important to note that from fairly early on in
Western physics there have also been adynamical explanations that
focused on the role of the future in explaining the past as well as
the reverse, such as integral (as opposed to differential) calculus
and various least action principles of the sort Richard Feynman
generalized to produce the path integral approach to quantum
mechanics. And of course there are the various adynamical
constraints in physics such as conservation laws and the symmetries
underlying them that constrain if not determine the various
equations of motion. But nonetheless, dynamism is still the reigning
assumption in physics.
Dynamism then encompasses three claims: A) the world, just as
appearances and the experience of time suggest, evolves or changes
in time in some objective fashion, B) the best explanation for A
will be be some dynamical law that ``governs'' the evolution of the
system in question, and C) the fundamental entities in a TOE will
themselves be dynamical entities evolving in some space however
abstract, e.g., Hilbert space. In spite of the presumption of
dynamism, those who want fundamental explanation in physics to be
dynamical and those who want a world that evolves in time in some
objective fashion, face well-known problems concerning: 1) the
possible blockworld implications of relativity (both special and
general) and 2) canonical QG, the quantization of a generally
covariant classical theory leading to ``frozen time.'' As for
whether relativity (both special and general) implies a blockworld,
there is much debate\cite{savitt}. Regarding special relativity
(SR), many of us have argued\cite{peterson} that given certain
widely held innocuous assumptions and the Minkoswski formulation,
special relativity does indeed imply a blockworld. In the words of
Geroch\cite{geroch}:
\begin{quotation}There is no dynamics within space-time itself:
nothing ever moves therein; nothing happens; nothing changes.
In particular, one does not think of particles as moving through
space-time, or as following along their world-lines. Rather, particles
are just in space-time, once and for all, and the world-line represents,
all at once, the complete life history of the particle.\end{quotation}
In addition there is the problem of time in canonical general
relativity (GR). That is, in a particular Hamiltonian formulation of
GR the reparametrization of spacetime is a gauge symmetry.
Therefore, all genuinely physical magnitudes are constants of
motion, i.e., they don't change over time. In short, change is
merely a redundancy of the representation.
Finally, the problem of frozen time in canonical QG (unification of
GR and quantum field theory) is that if the canonical variables of
the theory to be quantized transform as scalars under time
reparametrizations, which is true in practice because they have a
simple geometrical meaning, then ``the Hamiltonian is (weakly) zero
for a generally covariant system''\cite{henneaux}. The result upon
canonical quantization is the famous Wheeler-DeWitt equation, void
of time evolution. While it is too strong to say a generally
covariant theory must have H = 0, there is no well-developed theory
of quantum gravity that has avoided it to date\cite{kiefer1}. It is
supremely ironic that the dynamism and unificationism historically
driving physics led us directly to blockworld and frozen time.
Two basic reactions to this tension between blockworld and frozen
time on the one hand and dynamism on the other are to either embrace
the former and show that at least the appearances of dynamism, if
not the substance, can be maintained with resources intrinsic to
relativity or the particular QG scheme in
question\cite{savitt}\cite{kiefer2}, or reject the former whether
conceptually or formally and attempt to construct a fundamental
theory that has something definitively dynamical at bottom. The idea
is to somehow make time or change fundamental in some way, as
opposed to merely emergent as in the case of string theory or an
illusion as in the case of Wheeler-Dewitt. Smolin, for example,
suggests a radically ``neo-Heraclitean'' solution wherein change and
becoming are fundamental in that axiomatic dynamical laws, the
values of constants that figure in those laws and configuration
space itself evolve in time or meta-time\cite{smolin}. Though he
does not necessarily frame it this way, Smolin is advocating for
something like a fundamentally Whiteheadian process conception of
reality, a process-based physics where change or flux itself is
fundamental. In doing so, Smolin joins Bohm and Hiley who have been
advocating such an approach for many decades\cite{bohm1}.
However, what isn't clear is if Smolin appreciates what a radical
departure a process-conception of reality is from atomism wherein
reality has some fundamental dynamical building blocks (atoms,
particles, waves, strings, loops, etc.) from which everything else
is constructed, determined or realized. This brings us to choice
point number 2, atomism versus monism. Despite all the tension that
quantum theory has created for atomism as originally conceived, most
physicists still assume there is something fundamentally entity-like
at bottom, however strange it may be by classical lights. But on the
process view, potentia, activity, flux or change itself is
fundamental, not entities/things changing in time such as particles
or strings. In this monistic physics (what Bohm and Hiley call
``undivided wholeness''), all talk of such dynamical entities would
emerge from, and be derived from, the more fundamental flux together
with, and inseparably from, spacetime in a background independent
fashion (the formal question remains of course as to how this move
would resolve for example the problem of frozen time). Thus Bohm and
Hiley are constructing a monistic model wherein ``the whole is prior
to its parts, and thus views the cosmos as fundamental, with
metaphysical explanation dangling downward from the
One''\cite{schaffer}. However, the motivations for a process-based
physics are not exclusively physical, but are also driven by the
desire to have fundamental concepts of physical time correspond with
time and change as experienced such that time as experienced isn't
merely a subjective psychological feature of humans with no clear
physical correspondence. Following Price\cite{price}, the key
elements to time as experienced are: \emph{objectively dynamical}
(flow or flux-like), \emph{present moment} objectively
distinguished, and \emph{objective direction}.
This brings us to choice point number 3, algebra versus geometry
broadly construed. There is a dizzying array of formalisms at work
in physics. In quantum mechanics alone we have matrix mechanics,
Schr\"{o}dinger dynamics, Clifford algebras, and path integrals, to
name a few, and in quantum field theory (QFT) we have canonical
quantization, covariant quantization, path integral method,
Becchi-Rouet-Stora-Tyupin (BRST) approach, Batalin-Vilkovisky (BV)
quantization, and Stochastic quantization\cite{kaku1}. When we get
to QG and unification the list is even longer and more
diverse\cite{kiefer3}. Throughout history there have always been
differences of opinion, some pragmatic and some principled, about
which formalism(s) best models fundamental physical reality. Indeed,
one of the striking things about the state of unification is the
heterogeneity of formal approaches and the lack of consensus despite
the juggernaut of string theory and its progeny. Hiley for example,
likes to say that in his program, geometry (spacetime) is derived
from algebra (process), rather than the other way
around\cite{hiley1}. Other approaches, such as ours, proceed along
something closer to the opposite direction. Hiley enumerates several
advantages to using orthogonal Clifford algebras in quantum
mechanics: 1) they provide a mathematical hierarchy of nested
algebras in which to naturally embed the Dirac, Pauli and
Schr\"{o}dinger particles, 2) the approach is fully algebraic, which
allows a more general approach to quantum phenomena, 3) because it
is an algebraic theory, it provides a natural mathematical setting
for the Heisenberg `matrix' mechanics, 4) because it is
representation free, it avoids the use of multiple indices on
spinors , and 5) it removes the \emph{ad hoc} features of the
earlier attempts to extend the Bohm approach to spin and
relativity\cite{hiley2}. But, what is interesting from the
perspective of foundations of physics is that while there is no
necessary connection between a formalism and a particular model or
metaphysical interpretation, we see that theorists sometimes pick a
formalism based in part on their prior metaphysical biases and
background beliefs about the nature of reality, in addition to other
physical and formal considerations pertaining to unification such as
those Hiley gives above. For example, one of the main reasons Hiley
adopts an algebraic approach at bottom is that he thinks algebra can
better model process whereas the geometrization of time in
relativity leads exactly to blockworld, a conception of reality he
rejects as too static. Indeed, at least on the surface it is hard to
imagine a cosmology less comforting to a process conception of
reality than blockworld or H = 0. At any rate, what should now be
clear is that each of our three choice points has implications for
the others.
\subsection{Prelude: RBW versus the Implicate Order}
In this paper two different approaches to unification will be
compared, the Relational Blockworld (RBW) emphasizes being over
becoming formally and conceptually, while the Implicate Order of
Hiley emphasizes the converse. RBW has something closer to geometry
at bottom (discrete graphical structure) while Hiley has Clifford
algebras as fundamental. Each of these programs was originally
spawned by two diametrically opposed solutions to foundational
issues in non-relativistic quantum mechanics (NRQM) and QFT, rather
than starting life as models of
QG\cite{stuckey1}\cite{silberstein}\cite{bohm2}. As we will see,
while both are cast in the monistic spirit, Hiley's monism resides
in Bohm's implicate order and is based on process while RBW's monism
resides in ``spacetimematter,'' whereby space, time and matter are
co-constructed per a global constraint equation; RBW's monism
therefore resides in being while that of Hiley resides in becoming.
Both these programs have proposed new formalisms for quantum physics
and are in the process of extrapolating their approaches to
unification and quantum
gravity\cite{hiley1}\cite{stuckey2}\cite{hiley3}\cite{hiley4}\cite{hiley5}.
The Implicate Order of Hiley extends Bohmian mechanics to the
relativistic regime and unites spacetime geometry and material
processes, as he doesn't want things happening in a background
spacetime but wants to ``start from something more primitive from
which both geometry and material process unfold
together''\cite{hiley6}. That which he considers ``more primitive''
is elementary process. Hiley calls the fundamental process/potentia
the ``holomovement'' and it has two intertwined aspects, the
``implicate order'' (characterized algebraically) and all the
physics derived from it, such as spacetime geometry, the ``explicate
(or manifest) order.'' The holomovement is thus the whole ground
form of existence, which contains orders that are both implicate and
explicate, wherein the latter expresses aspects of the former. Hiley
reduces the Clifford algebra $C_{4,1}$ to $C_{1,3}$ whence he
derives the vector space of M4 by mapping the Dirac gamma matrices
to the orthonormal vectors spanning $V_{1,3}$ of M4. He then defines
Bohm momentum and energy densities in the Dirac equation in analogy
with his earlier work with Bohm\cite{hiley7}. From the perspective
of the implicate order, rather than point particles being evolved in
time aided by instantaneous updating by the quantum potential or
pilot wave, the fundamental evolution is one of processes that give
rise to explicate structures (``moments'' or ``durons'') extended in
space and time. In short, particles and pilot waves are not
fundamental but are at best emergent from the implicate order (see
section \ref{section4}). The irony is not lost on Hiley that the
Bohm and Hiley work on interpreting NRQM has done more than perhaps
any other interpretation to bolster a particle ontology and a
``mechanical'' conception of the quantum modeled on an analogy with
classical mechanics\cite{goldstein}. Indeed, as we will see in
section \ref{section2}, much of Hiley's later work is trying to get
out from under such a pseudo-classical model and emphasize the
undivided wholeness instead.
However, in order for Hiley to finish his program, presumably, he
will need to accommodate any Lagrangian, not just that of the Dirac
equation. For example, he will need to compute cross sections for
the various collision experiments of high energy physics. If he
proceeds along the lines of ``current algebra''\cite{kaku2}, as
suggested by his approach to
date\cite{hiley1}\cite{hiley4}\cite{hiley5}, perhaps he could
produce a Bohmian explanation for why the commutators between some
currents in the Standard Model do not close, producing the so-called
Schwinger terms. But, even if he were able to find an algebra of
process for the Standard Model that provided Bohm momenta and
energies for all the particles, he would still have only ``a first
approximation to the true theory of subatomic
particles''\cite{kaku3}, since the Standard Model is plagued with
twenty-some-odd free parameters. He would be in the same boat as
everyone else, needing to account for the free parameters of the
Standard Model and include gravity (see section \ref{section3}). The
point is that Hiley would have to join the ranks of theorists who
are still looking for a `super-algebra' whence the Lagrangian
unifying the Standard Model and gravity.
As with Hiley's implicate order, our account of quantum physics,
which we call the Relational Blockworld (RBW), is based on a form of
monism, i.e., the unity of space, time and matter at the most
fundamental level. We call this fundamental unity
``spacetimematter'' and use it to recover dynamical or process-like
classical physics only statistically. Thus, we do not attempt to
derive geometry from algebra but in a sense, the other way round
(see section \ref{section2}). In order to appreciate how GR
``emerges'' on our view, it is important to understand that, unlike
Hiley's account, our approach is fundamentally adynamical and
acausal, again, in contrast also to other fundamental theories
attempting to quantize gravity (M-theory, loop quantum gravity,
causets, etc.). According to RBW, as we will explain in detail in
section \ref{section2}, quantum physics is the continuous
approximation of a more fundamental, discrete graph theory whereby
the transition amplitude Z is not viewed as a sum over all paths in
configuration space, but is a measure of the symmetry of the
difference matrix and source vector of the discrete graphical action
for a 4D process (Figure \ref{fig1}a). We have proposed that the
source vector and difference matrix of the discrete action in the
path integral be constructed from boundary operators on the graph so
as to satisfy an adynamical constraint equation we call the
``self-consistency criterion'' (SCC), (see section \ref{section2}
for details). While itself adynamical, the SCC guarantees the graph
will produce divergence-free classical dynamics in the appropriate
statistical limit (Figure \ref{fig2}a), and provides an acausal
global constraint that results in a self-consistent co-construction
of space, time and matter that is \emph{de facto} background
independent. Thus, in RBW one has an acausal, adynamical unity of
``spacetimematter'' at the fundamental level that results
statistically in the causal, dynamical ``spacetime + matter'' of
classical physics. This graphical amalgam of spacetimematter is the
basis for all quantum phenomena as viewed in a classical context
(Figure \ref{fig2}b), that is, we represent this unity of
spacetimematter with 4D graphs constructed per the SCC, and a
Wick-rotated Z provides a partition function for the distribution of
graphical relations responsible statistically for a particular
classical process (Figures \ref{fig1} and \ref{fig2}).
Thus, RBW provides a wave-function-epistemic account of quantum
mechanics with a time-symmetric explanation of interference via
acausal global constraints\cite{silberstein}. Quantum physics is
simply providing a distribution function for graphical relations
responsible for the experimental equipment and process from
initiation to termination. So, while according to some such as
Bohmian mechanics, EPR-correlations and the like evidence
superluminal information exchange (quantum non-locality), and
according to others such correlations represent non-separable
quantum states (quantum non-separability), per RBW these phenomena
are actually evidence of the deeper graphical unity of
spacetimematter responsible for the experimental set up and process,
to include outcomes\cite{stuckey1}\cite{silberstein}. RBW is
therefore integral calculus thinking writ
large\cite{stuckey1}\cite{stuckey2}.
As regards the ``emergence'' or derivation of GR from RBW (see
section \ref{section3}), since we recover classical physics in terms
of the ``average spacetime geometry'' over the graphical unity of
spacetimematter, our discrete average/classical result is a modified
Regge calculus \footnote{Interestingly, in direct correspondence,
Hiley noted that he and Bohm had considered Regge calculus, but
found it emphasized the `structure' too much and lost the notion of
`process'. By turning to the notion of an `algebra', Hiley found he
could keep the structure aspect, but emphasize more the process.}.
Ordinary Regge calculus is a discrete approximation to GR where the
discrete counterpart to Einstein's equations is obtained from the
least action principal on a 4D graph\cite{misner1}. This generates a
rule for constructing a discrete approximation to the spacetime
manifold of GR using small, contiguous 4D graphical `tetrahedra'
called ``simplices.'' The smaller the legs of the simplices, the
better one may approximate a differentiable manifold via contiguous
simplices. Our proposed modification of Regge calculus (and,
therefore, GR) requires all simplex legs contain non-zero
stress-energy contributions (per spacetimematter), so our simplices
can be both large and non-contiguous. Consequently, per RBW, GR is
seen as a continuous approximation to a modified Regge calculus
wherein the simplices can be large and non-contiguous.
Clearly, Hiley's Implicate Order and RBW differ formally (algebraic
vs path integral) and conceptually (process-oriented vs adynamical).
The monistic character of Hiley's process-oriented approach is
housed in the implicate order, i.e., the Clifford algebra. That
which we observe (the explicate order) is a projection from the
implicate order. Thus, the implicate order accounts for EPR
correlations, which appear to require quantum non-locality (as in
Bohmian mechanics) and/or non-separability in the explicate order of
spacetime. The monistic character of RBW is housed in
spacetimematter which underwrites the spacetime + matter classical
world of our observations. Thus spacetimematter accounts for EPR
correlations, which appear to require quantum non-locality and/or
non-separability in the spacetime + matter of our classical
perspective\cite{stuckey1}\cite{silberstein}. Therefore, both
approaches want to explain such observed quantum phenomena from a
more fundamental theory underneath quantum theory itself, though
these are quite opposing fundamental theories. More specifically,
both approaches want to derive GR and quantum theory from something
more fundamental in a background independent fashion such that the
explanation for quantum entanglement and EPR correlations, rather
than creating tensions with spacetime and relativity, requires
neither non-locality nor non-separability in spacetime. Rather, such
quantum effects (their phenomenology) are explained at the more
fundamental level whether graphical or algebraic. In section
\ref{section2} we provide a brief overview of Hiley's implicate
order (details are already published elsewhere) and a technical
overview of RBW. In sections \ref{section3} and \ref{section4} we
explore their respective prospects for providing progress in the
quest for unification and quantum gravity, and discuss their
perspectives on dynamism.
\section{Quantum Field Theory: Implicate Order Versus RBW}
\label{section2}
\subsection{Hiley's Implicate Order}
Hiley has issued the following challenge\cite{hiley6}:
\begin{quotation}Since the advent of general relativity in which matter and
geometry codetermine each other, there is a growing realisation that
starting from an \emph{a priori} given manifold in which we allow
material processes to unfold is, at best, limited. Can we start from
something more primitive from which both geometry and material
process unfold together? The challenge is to find a formalism that
would allow this to happen.\end{quotation} Hiley then refers to
Bohm's early attempt\cite{hiley6}:
\begin{quotation}David Bohm introduced the notion of a discrete structural
process in which he takes as basic, not matter or fields in
space-time, but a notion of `structure process' from which the
geometry of space-time and its relationship to matter emerge
together providing a way that could underpin general relativity and
quantum theory.\end{quotation}
While Hiley's view may seem radical to some, he is not alone in
appreciating what quantum theory and GR have wrought and what their
unification may require\cite{rovelli}:\begin{quotation}General
relativity (GR) altered the classical understanding of the concepts
of space and time in a way which...is far from being fully
understood yet. QM challenged the classical account of matter and
causality, to a degree which is still the subject of controversies.
After the discovery of GR we are no longer sure of what is spacetime
and after the discovery of QM we are no longer sure of what matter
is. \emph{The very distinction between space-time and matter is
likely to be ill-founded}....I think it is fair to say that today we
do not have a consistent picture of the physical world. [italics
added]\end{quotation}
With regard to QFT, Hiley's own response to his challenge employs
``Clifford algebras taken over the reals'' to provide ``a coherent
mathematical setting for the Bohm formalism.'' In particular, he is
concerned with finding the Bohm momentum and energy in a
relativistic theory, i.e., the Dirac theory, since a common
criticism of Bohm's view is that it cannot be applied in the
relativistic domain. Early attempts by Bohm at making his approach
relativistically invariant focused on the conserved Dirac current
$J^{\mu}=\langle\bar{\Psi}|\gamma^{\mu}|\Psi\rangle$ which results
from global gauge invariance $\psi\rightarrow e^{i\theta}\psi$.
Hiley finds another conserved current associated with the Dirac
particle, the energy-momentum density current
$2iT^{\mu0}=\psi^\dag(\partial^{\mu}\psi)-(\partial^{\mu}\psi^\dag)\psi$
which results from invariance under spacetime translations. Hiley
argues that this energy-momentum density current is the relativistic
counterpart to Bohm energy and momentum for the Schr\"{o}dinger
particle, $E_B=-\partial_tS$ and $\vec{p}_B=\nabla S$. This differs
from the standard treatment of the Dirac particle whereby the
energy-momentum current is only integrated for global conservation
of energy and momentum. In standard field theory, the Dirac current
is stressed, since it couples to the gauge field. Hiley's view leads
to a curious split of the Dirac particle into a `Bohm' part and a
`gauge' part. The split is unique to the relativistic regime, as
there is no such split for the Schr\"{o}dinger or Pauli particles.
So, what does this relativistic dual nature suggest?
Hiley speculates it is indicative of a composite or extended nature
of the Dirac particle. While this idea would apply to baryons, as
they are understood as extended and composed of quarks, it would not
appear relevant to leptons, which are understood as point-like and
fundamental. And what, for example, would we expect for a Bohmian
explanation of the twin-slit experiment using Dirac particles? Would
the resulting interference pattern be explained by trajectories for
the energy-momentum density current in analogy with the Bohmian
Schr\"{o}dinger particle? If so, how would the change in this
interference pattern in the Aharanov-Bohm experiment be explained?
Since it is the Dirac current that couples to the gauge field and it
is the gauge field that is responsible for the Aharanov-Bohm shift
in the interference pattern, we would expect the Bohmian
trajectories to adhere in some respect to the Dirac current. We
suspect that this is indicative of an underlying problem, i.e.,
trying to understand relativistic quantum phenomena in the context
of a particular Lorentz frame, as is done by generating his minimal
left ideal with the idempotent $\epsilon_1=(1+\gamma^0)/2$. We don't
see any problem with his suggested correspondence between his Dirac
energy-momentum density current and its non-relativistic, non-spin
limit of the Bohm energy and momentum for the Schr\"{o}dinger
particle, i.e., $\rho E_B = T^{00}$ and $\rho P_B = T^{k0}$.
However, the fact that it is the energy-momentum density current
that makes this correspondence, rather than the Dirac current,
suggests to us a breakdown in the Bohmian view (quantum potential
defined per a particular Lorentz frame), as would be expected when
going to the relativistic regime.
Regardless of whether or not the notion of Bohmian trajectories can
be preserved in the relativistic regime, Hiley's implicate order
does offer a process-based approach to quantum physics via ``a
hierarchy of Clifford algebras which fit naturally the physical
sequence: Twistors $\rightarrow$ relativistic particle with spin
$\rightarrow$ non-relativistic particle with spin $\rightarrow$
non-relativistic particle without spin''\cite{hiley8}. And this
approach does unite spacetime geometry and material process via the
primitive notion of process algebra. What is unique about the shadow
manifolds (explicate order) that are projected from his Clifford
algebras is that they lead to an equivalence class of Lorentz
observers, rather than a single Minkowski spacetime manifold (M4).
Any particular Lorentz frame serves as the base space for a Clifford
bundle. Assuming this base space is a flat Riemannian manifold M,
Hiley constructs a derivative D from space-like derivatives on M and
the generators of his Clifford bundle. Thus defined, D is a
connection on M and the momentum operator of quantum mechanics
(Schr\"{o}dinger, Pauli, Dirac equations). He then uses this D to
construct a Hamiltonian whence ``the two dynamical equations that
form the basis of the Bohm approach to quantum mechanics - a
Louville type conservation of probability equation and a quantum
Hamilton-Jacobi equation''\cite{hiley9}. While it may seem like a
weakness that he produces shadow manifolds rather than M4, we see
this as a potential advantage in dealing with the problems of
blockworld and ``frozen time,'' as we will discuss in section
\ref{section4}. For now, we simply point out the obvious challenge,
i.e., he must find a connection with curvature for the tangent space
bundle to the base space manifold so as to recover GR. He speculates
this might be done by analyzing phase information in the exchange of
light signals, since ``the Moyal algebra for relating phase
information can be obtained from a deformed Poisson algebra, which
is obtained via the hidden Heisenberg algebra''\cite{hiley10}. As he
has not begun this project, we can offer only limited speculation on
such an attempt in section \ref{section3}.
Our more general concern is about Hiley's motivation for wanting to
obtain a complete relativistic version of the Bohm model for the
Dirac particle, given that he clearly rejects the fundamentality of
particles and pilot (guide) waves, they are emergent at best.
Consider the following passages from Hiley:
\begin{quotation}We strive to find the elementary objects, the quarks,
the strings, the loops and the M-branes from which we try to
reconstruct the world. Surely we are starting from the wrong
premise. Parker-Rhodes (1981) must be right, so too is Lou Kauffman
(1982)! We should start with the whole and then make distinctions.
Within these distinctions we can make finer distinctions and so
on\cite{hiley11}.\end{quotation}
\begin{quotation}In this paper we want to draw specific attention to a sixth
advantage, namely, that it allows us to apply Clifford algebras to
the Bohm approach outlined in Bohm and Hiley. In fact it provides,
for the first time, an elegant, unified approach to the Bohm model
of the Schr\"{o}dinger, Pauli, and Dirac particles, in which we no
longer have to appeal to any analogy to classical mechanics to
motivate the approach as was done by Bohm in his original
paper\cite{hiley12}.\end{quotation}
When Hiley speaks of analogies to classical mechanics, not only is
he jettisoning point particles as fundamental but also the wave
function and apparently the guide wave:
\begin{quotation}In our
approach, the information normally encoded in the wave function is
already contained within the algebra itself, namely, in the elements
of its minimal left ideals\cite{hiley13}.\end{quotation}
\begin{quotation}Thus we see that at no stage is it necessary to
appeal to classical mechanics and therefore there is no need to
identify the classical action with the phase to motivate the
so-called `guidance' equation $\vec{p}=\nabla S$ as was done in
Bohm's original work\cite{callaghan}.\end{quotation}
\begin{quotation}Then it is not diffcult to show that this again reduces, in the non-relativistic
limit, to the Bohm momentum found in the Pauli case and reduces
further, if the spin is suppressed, to the well-known
Schr\"{o}dinger expression $P_B = \nabla S$. This condition is
sometimes known as the guidance condition, but here we have no
`waves', only process, so this phrase is inappropriate in this
context\cite{hiley14}.\end{quotation}
\begin{quotation} Thus by choosing $\alpha = \frac{1}{2}$ we see that our $\rho P_j$ is simply the momentum
density. Furthermore it also means that $\vec{P} = \vec{p_B}$, the
Bohm momentum. Because this can be written in the form $\vec{p_B} =
\nabla S$. Some authors call this the `guidance' condition, but here
it is simply a bilinear invariant and any notion of `guidance' is
meaningless\cite{hiley15}.\end{quotation}
It seems to us that there has always been a tension in Bohm and
Hiley's ``undivided wholeness'' and the pseudo-classical Bohmian
mechanics conceived as a modal interpretation of NRQM with particles
communicating instantaneously with one another, especially in a
relativistic setting. Why spend so much energy trying to recover a
relativistic Bohmian version of the Dirac particle complete with
particle trajectories when such particles and the guidance wave are
at best emergent, and the wave function is merely epistemic? In the
earlier work it was thought that the Dirac current would provide a
means of calculating particle trajectories\cite{hiley15a}. In Hiley
and Callaghan's recent work they show that the Dirac current is in
fact different from the Bohm energy-momentum current, leaving them
with two different sets of trajectories\cite{hiley16}; again, all of
which raising the question whether Bohmian trajectories can be
recovered in the relativistic case after all. But even if such
trajectories can be recovered, what's the point of trying to
establish that the Bohmian model is relativistically invariant when
Hiley rejects the fundamentality of, if not realism about, that very
model? If it's the monism \emph{a la} process that matters most to
Hiley, then recovering ordinary quantum mechanics or QFT from the
algebraic base is sufficient, nothing is added by recovering a
relativistically Bohmian mechanics as the latter is just a competing
interpretation of quantum mechanics, one that only makes sense to
pursue if you take seriously point particles and pilot waves, which
apparently Hiley does not. Furthermore, it isn't enough to render
Bohmian mechanics Lorentz invariant, it must also be explained how
the non-locality in that model can be squared with the relativity of
simultaneity. Presumably this problem would get solved by Hiley at
the level of the implicate order as a kind of conspiracy theory, but
again, then why bother with recovering Bohmian trajectories and the
like? In the next section we will see that these problems don't
arise for RBW because that model makes a much cleaner break from the
ontology of particles and wave functions even at the level of
ordinary quantum mechanics in spacetime.
\subsection{RBW and Spacetimematter}
We believe the real issue is the fact that QFT involves the
quantization of a classical field\cite{wallace1} when one would
rather expect QFT to originate independently of classical field
theory, the former typically understood as fundamental to the
latter. Herein we propose a new, fundamental origin for QFT.
Specifically, we follow the possibility articulated by
Wallace\cite{wallace2} that, ``QFTs as a whole are to be regarded
only as approximate descriptions of some as-yet-unknown deeper
theory,'' which he calls ``theory X,'' and we propose a new discrete
path integral formalism over graphs for ``theory X'' underlying QFT.
Accordingly, sources $\tilde{J}J$ , space and time are self-consistently
co-constructed per a graphical self-consistency criterion (SCC)
based on the boundary of a boundary principle\cite{misner2} on the
graph ($\partial_1\cdot
\partial_2 = 0$)\footnote{In a graphical representation of QFT, part of $\tilde{J}J$
represents field disturbances emanating from a source location
(Source) and the other part represents field disturbances incident
on a source location (sink).}. We call this amalgam
``spacetimematter.'' The SCC constrains the difference matrix and
source vector in Z, which then provides the probability for finding
a particular source-to-source relationship in a quantum experiment,
i.e., experiments which probe individual source-to-source relations
(modeled by individual graphical links) as evidenced by discrete
outcomes, such as detector clicks. Since, in QFT, all elements of an
experiment, e.g., beam splitters, mirrors, and detectors, are
represented by interacting sources, we confine ourselves to the
discussion of such controlled circumstances where the empirical
results evidence individual graphical links\footnote{Hereafter, all
reference to ``experiments'' will be to ``quantum experiments.''}.
In this approach, the SCC ensures the source vector is
divergence-free and resides in the row space of the difference
matrix, so the difference matrix will necessarily have a nontrivial
eigenvector with eigenvalue zero, a formal characterization of gauge
invariance. Thus, our proposed approach to theory X provides an
underlying origin for QFT, accounts naturally for gauge invariance,
i.e., via a graphical self-consistency criterion, and excludes
factors of infinity associated with gauge groups of infinite volume,
since the transition amplitude $Z$ is restricted to the row space of
the difference matrix and source vector.
While the formalism we propose for theory X is only suggestive, the
computations are daunting, as will be evident when we present the
rather involved graphical analysis underlying the Gaussian
two-source amplitude which, by contrast, is a trivial problem in its
QFT continuum approximation. However, this approach is not intended
to replace or augment QFT computations. Rather, our proposed theory
X is fundamental to QFT and constitutes a new program for physics,
much as quantum physics relates to classical physics. Therefore, the
motivation for our theory X is, at this point, conceptual and while
there are many conceptual arguments to be made for our
approach\cite{stuckey1}\cite{silberstein}, we restrict ourselves
here to the origins of gauge invariance and QFT.
\subsubsection{The Discrete Path Integral Formalism} We understand
the reader may not be familiar with the path integral formalism, as
Healey puts it\cite{healey}, ``While many contemporary physics texts
present the path-integral quantization of gauge field theories, and
the mathematics of this technique have been intensively studied, I
know of no sustained critical discussions of its conceptual
foundations.'' Therefore, we begin with an overview and
interpretation of the path integral formalism, showing explicitly
how we intend to use ``its conceptual foundations.'' We employ the
discrete path integral formalism because it embodies a 4Dism that
allows us to model spacetimematter. For example, the path integral
approach is based on the fact that\cite{feynman} ``the [S]ource will
emit and the detector receive,'' i.e., the path integral formalism
deals with Sources and sinks as a unity while invoking a description
of the experimental process from initiation to termination. By
assuming the discrete path integral is fundamental to the
(conventional) continuum path integral, we have a graphical basis
for the co-construction of time, space and quantum sources via a
self-consistency criterion (SCC). We will then show how the
graphical amalgam of spacetimematter underlies QFT.
\subsubsection{Path Integral in Quantum Physics}
In the conventional path integral formalism as used by Zee\cite{zee1} for non-relativistic quantum mechanics (NRQM)
one starts with the amplitude for the propagation from the initial point in configuration space $q_I$ to
the final point in configuration space $q_F$ in time $T$ via the unitary operator $e^{-iHT}$,
i.e., $\displaystyle \left \langle q_F \left | e^{-iHT} \right | q_I \right \rangle$. Breaking the
time $T$ into $N$ pieces $\delta t$ and inserting the identity between each pair of operators $e^{-iH\delta t}$ via
the complete set $\int dq | q \rangle \langle q | =1$ we have
\begin{multline*}
\left \langle q_F \left | e^{-iHT} \right | q_I \right \rangle = \left [ \prod_{j=1}^{N-1} \int dq_j \right ]
\left \langle q_F \left | e^{-iH\delta t} \right | q_{N-1} \right \rangle
\left \langle q_{N-1} \left | e^{-iH\delta t} \right | q_{N-2} \right \rangle
\ldots \\
\left \langle q_2 \left | e^{-iH\delta t} \right | q_1 \right \rangle
\left \langle q_1 \left | e^{-iH\delta t} \right | q_I \right \rangle.
\end{multline*}
\begin{equation*}
\left \langle q_2 \left | e^{-iH\delta t} \right | q_1 \right \rangle
\left \langle q_1 \left | e^{-iH\delta t} \right | q_I \right \rangle.
\end{equation*}
With $H=\hat{p}^2/2m + V(\hat{q})$ and $\delta t \rightarrow 0$ one
can then show that the amplitude is given by
\begin{equation}
\left \langle q_F \left | e^{-iHT} \right | q_I \right \rangle =
\int Dq(t) \exp \left [ i \int_0^T dt L(\dot{q},q) \right ],
\label{eqn1}
\end{equation}
where $L(\dot{q},q) = m \dot{q}^2/2-V(q)$ . If $q$ is the spatial
coordinate on a detector transverse to the line joining Source and
detector, then $\displaystyle \prod_{j=1}^{N-1}$ can be thought of
as $N-1$ ``intermediate'' detector surfaces interposed between the
Source and the final (real) detector, and $\int dq_j$ can be thought
of all possible detection sites on the $j^{\mbox{th}}$ intermediate
detector surface. In the continuum limit, these become $\int Dq(t)$
which is therefore viewed as a ``sum over all possible paths'' from
the Source to a particular point on the (real) detector, thus the
term ``path integral formalism'' for conventional NRQM is often
understood as a sum over ``all paths through space.''
To obtain the path integral approach to QFT one associates $q$ with
the oscillator displacement at a {\em particular point} in space
($V(q) = kq^2/2$). In QFT, one takes the limit $\delta x \rightarrow
0$ so that space is filled with oscillators and the resulting
spatial continuity is accounted for mathematically via $q_i(t)
\rightarrow q(t,x)$, which is denoted $\phi(t,x)$ and called a
``field.'' The QFT transition amplitude $Z$ then looks like
\begin{equation}
Z = \int D\phi \exp \left [ i \int d^4 x L( \dot{\phi}, \phi ) \right ]
\label{eqn2}
\end{equation}
where $L(\dot{\phi},\phi) = (d\phi)^2/2 - V(\phi)$ . Impulses $J$
are located in the field to account for particle creation and
annihilation; these $J$ are called ``sources'' in QFT and we have
$L(\dot{\phi},\phi) = (d\phi)^2/2 - V(\phi) + J(t,x) \phi(t,x)$,
which can be rewritten as $L(\dot{\phi},\phi) = \phi D \phi/2 +
J(t,x) \phi(t,x)$, where $D$ is a differential operator. In its
discrete form (typically, but not necessarily, a hypercubic
spacetime lattice), $D \rightarrow \vec{\vec{K}}$ (a difference matrix),
$J(t,x)\rightarrow \tilde{J}J$ (each component of which is associated with
a point on the spacetime lattice) and $\phi \rightarrow \tilde{Q}Q$ (each
component of which is associated with a point on the spacetime
lattice). Again, part of $\tilde{J}J$ represents field disturbances
emanating from a source location (Source) and the other part
represents field disturbances incident on a source location (sink)
in the conventional view of path integral QFT and, in particle
physics, these field disturbances are the particles. We will keep
the partition of $\tilde{J}J$ into Sources and sinks in our theory X, but
there will be no vacuum lattice structure between the discrete set
of sources. The discrete counterpart to (\ref{eqn2}) is
then\cite{zee2}
\begin{equation}
Z = \int \ldots \int dQ_1 \ldots dQ_N \exp \left[ \frac {i}{2} \tilde{Q}Q
\cdot \vec{\vec{K}} \cdot\tilde{Q}Q + i \tilde{J}J \cdot\tilde{Q}Q \right ]. \label{eqn3}
\end{equation}
In conventional quantum physics, NRQM is understood as $(0+1)-$dimensional QFT.
\subsubsection{Our Interpretation of the Path Integral in Quantum
Physics}
We agree that NRQM is to be understood as $(0+1)-$dimensional QFT, but point out
this is at conceptual odds with our derivation of (\ref{eqn1}) when $\int Dq(t)$
represented a sum over all paths in space, i.e., when $q$ was understood as a
location in space (specifically, a location along a detector surface). If NRQM
is $(0+1)-$dimensional QFT, then $q$ is a field displacement at a single location
in space. In that case, $\int Dq(t)$ must represent a sum over all field values at a particular
point on the detector, not a sum over all paths through space from the Source to a particular
point on the detector (sink). So, how {\em do} we relate a point on the detector (sink) to the Source?
In answering this question, we now explain a formal difference
between conventional path integral NRQM and our proposed approach:
our links only connect and construct discrete sources $\tilde{J}J$, there
are no source-to-spacetime links (there is no vacuum lattice
structure, only spacetimematter). Instead of $\delta x \rightarrow
0$, as in QFT, we assume $\delta x$ is measureable for (such) NRQM
phenomenon. More specifically, we propose starting with (\ref{eqn3})
whence (roughly) NRQM obtains in the limit $\delta t \rightarrow 0$,
as in deriving (\ref{eqn1}), and QFT obtains in the additional limit
$\delta x \rightarrow 0$, as in deriving (\ref{eqn2}). The QFT limit
is well understood as it is the basis for lattice gauge theory and
regularization techniques, so one might argue that we are simply
{\em clarifying} the NRQM limit where the path integral formalism is
not widely employed. However, again, we are proposing a discrete
starting point for theory X, as in (\ref{eqn3}). Of course, that
discrete spacetime is fundamental while ``the usual continuum theory
is very likely only an approximation''\cite{Feinberg} is not new.
\subsubsection{Discrete Path Integral is Fundamental}
The version of theory X we propose is a discrete path integral over
graphs, so (\ref{eqn3}) {is not a discrete approximation of
(\ref{eqn1}) \& (\ref{eqn2})}, but rather {\em (\ref{eqn1}) \&
(\ref{eqn2}) are continuous approximations of (\ref{eqn3})}. In the
arena of quantum gravity it is not unusual to find discrete
theories\cite{loll} that are in some way underneath spacetime theory
and theories of ``matter'' such as QFT, e.g., causal dynamical
triangulations\cite{ambjorn}, quantum graphity\cite{konopka} and
causets\cite{sorkin1}. While these approaches are interesting and
promising, the approach taken here for theory X will look more like
Regge calculus quantum gravity (see Bahr \& Dittrich \cite{bahr} and
references therein for recent work along these lines) modified to
contain no vacuum lattice structure.
Placing a discrete path integral at bottom introduces conceptual and
analytical deviations from the conventional, continuum path integral
approach. Conceptually, (\ref{eqn1}) of NRQM represents a sum
over all field values at a particular point on the detector, while
(\ref{eqn3}) of theory X is a mathematical machine that measures
the ``symmetry'' (strength of stationary points) contained in the
core of the discrete action
\begin{equation}
\frac 12 \vec{\vec{K}} + \tilde{J}J
\label{eqn4}
\end{equation}
This core or {\em actional} yields the discrete action after
operating on a particular vector $\tilde{Q}Q$ (field). The actional
represents a {\em fundamental/topological, 4D description of the
experiment} and $Z$ is a measure of its symmetry\footnote{In its
Euclidean form, which is the form we will use, $Z$ is a partition
function.}. For this reason we prefer to call $Z$ the symmetry
amplitude of the 4D experimental configuration. Analytically,
because we are {\em starting} with a discrete formalism, we are in
position to mathematically explicate trans-temporal identity,
whereas this process is unarticulated elsewhere in physics. As we
will now see, this leads to our proposed self-consistency criterion
(SCC) underlying $Z$.
\subsubsection{Self-Consistency Criterion} Our use of a
self-consistency criterion is not without precedent, as we already
have an ideal example in Einstein's equations of GR. Momentum, force
and energy all depend on spatiotemporal measurements (tacit or
explicit), so the stress-energy tensor cannot be constructed without
tacit or explicit knowledge of the spacetime metric (technically,
the stress-energy tensor can be written as the functional derivative
of the matter-energy Lagrangian with respect to the metric). But, if
one wants a ``dynamic spacetime'' in the parlance of GR, the
spacetime metric must depend on the matter-energy distribution in
spacetime. GR solves this dilemma by demanding the stress-energy
tensor be ``consistent'' with the spacetime metric per Einstein's
equations. For example, concerning the stress-energy tensor, Hamber
and Williams write\cite{hamber}, ``In general its covariant
divergence is not zero, but consistency of the Einstein field
equations demands $\nabla^{\alpha} T_{\alpha \beta} = 0$ .'' This
self-consistency hinges on divergence-free sources, which finds a
mathematical underpinning in $\partial
\partial = 0$. So, Einstein's equations of GR are a mathematical
articulation of the boundary of a boundary principle at the
classical level, i.e., they constitute a self-consistency criterion
at the classical level, as are quantum and classical
electromagnetism\cite{misner3}\cite{wise}. We will provide an
explanation for this fact later, but essentially the graphical SCC
of our theory X gives rise to continuum counterparts in QFT and
classical field theory.
In order to illustrate the discrete mathematical co-constuction of
space, time and sources $\tilde{J}J$, we will use graph theory \emph{a la}
Wise\cite{wise} and find that $\partial_1\cdot \partial_1^T$, where
$\partial_1$ is a boundary operator in the spacetime chain complex
of our graph satisfying $\partial_1\cdot \partial_2 = 0$ , has
precisely the same form as the difference matrix in the discrete
action for coupled harmonic oscillators. Therefore, we are led to
speculate that $\vec{\vec{K}} \propto\partial_1\cdot \partial_1^T$. Defining
the source vector $\tilde{J}J$ relationally via $\tilde{J}J \propto
\partial_1\cdot \vec{e}$ then gives tautologically per $\partial_1\cdot
\partial_2 = 0$ both a divergence-free $\tilde{J}J$ and $\vec{\vec{K}}\cdot \vec{v}
\propto \tilde{J}J$, where $\vec{e}$ is the vector of links and $\vec{v}$ is the
vector of vertices. $\vec{\vec{K}}\cdot \vec{v} \propto \tilde{J}J$ is our SCC
following from $\partial_1\cdot\partial_2 = 0$, and it defines what
is meant by a self-consistent co-construction of space, time and
divergence-free sources $\tilde{J}J$, thereby constraining $\vec{\vec{K}}$ and
$\tilde{J}J$ in $Z$. Thus, our SCC provides a basis for the discrete
action and supports our view that (\ref{eqn3}) is fundamental to
(\ref{eqn1}) \& (\ref{eqn2}), rather than the converse.
Conceptually, that is the basis of our discrete, graphical path
integral approach to theory X. We now provide the details.
\subsubsection{The General Approach}
Again, in theory X, the symmetry amplitude $Z$ contains a discrete
action constructed per a self-consistency criterion (SCC) for space,
time and divergence-free sources $\tilde{J}J$. As introduced above and
argued later below, we will codify the SCC using $\vec{\vec{K}}$ and $\tilde{J}J$;
these elements are germane to the transition amplitude $Z$ in the
Central Identity of Quantum Field Theory\cite{zee3},
\begin{equation}
Z = \int D \vec{\phi} \exp \left [ - \frac 12 \vec{\phi} \cdot \vec{\vec{K}} \cdot \vec{\phi} - V(\vec{\phi}) + \tilde{J}J \cdot \vec{\phi} \right ] \\
= \exp \left [ -V \left ( \frac {\delta}{\delta J} \right ) \right ] \exp \left [\frac 12 \tilde{J}J \cdot \vec{\vec{K}}^{-1} \cdot \tilde{J}J \right ].
\label{eqn5}
\end{equation}
While the field is a mere integration variable used to produce $Z$,
it must reappear at the level of classical field theory. To see how
the field makes it appearance per theory X, consider (\ref{eqn5}) for the simple Gaussian theory ($V(\phi) = 0$). On a
graph with $N$ vertices, (\ref{eqn5}) is
\begin{equation}
Z = \int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} dQ_1
\ldots dQ_N \exp \left [-\frac 12 \tilde{Q}Q \cdot \vec{\vec{K}} \cdot \tilde{Q}Q + \tilde{J}J
\cdot \tilde{Q}Q \right ] \label{eqn6}
\end{equation}
with a solution of
\begin{equation}
Z = \left ( \frac {(2\pi)^N}{\det \vec{\vec{K}}} \right )^{1/2} \exp \left [\frac 12 \tilde{J}J \cdot \vec{\vec{K}}^{-1} \cdot \tilde{J}J \right ].
\label{eqn7}
\end{equation}
It is easiest to work in an eigenbasis of $\vec{\vec{K}}$ and (as will argue
later) we restrict the path integral to the row space of $\vec{\vec{K}}$,
this gives
\begin{equation}
Z = \int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} d\tilde{Q}_1
\ldots d\tilde{Q}_{N-1} \exp \left [\sum_{j=1}^{N-1} \left (-\frac 12
\tilde{Q}_j^2 a_j + \tilde{J}_j \tilde{Q}_j \right ) \right ] \label{eqn8}
\end{equation}
where $\tilde{Q}_j$ are the coordinates associated with the eigenbasis of
$\vec{\vec{K}}$ and $\tilde{Q}_N$ is associated with eigenvalue zero, $a_j$ is the
eigenvalue of $\vec{\vec{K}}$ corresponding to $\tilde{Q}_j$, and $\tilde{J}_j$ are the
components of $\tilde{J}J$ in the eigenbasis of $\vec{\vec{K}}$. The solution of
(\ref{eqn8}) is
\begin{equation}
Z = \left ( \frac {(2\pi)^{N-1}}{\prod_{j=1}^{N-1} a_j} \right )^{1/2} \prod_{j=1}^{N-1} \exp \left ( \frac {\tilde{J}_j^2}{2a_j} \right ).
\label{eqn9}
\end{equation}
On our view, the experiment is described fundamentally by $\vec{\vec{K}}$ and
$\tilde{J}J$ on our topological graph. Again, per (\ref{eqn9}), there is
no field $\tilde{Q}$ appearing in $Z$ at this level, i.e., $\tilde{Q}$ is only
an integration variable. $\tilde{Q}$ makes its first appearance as
something more than an integration variable when we produce
probabilities from $Z$. That is, since we are working with a
Euclidean path integral, $Z$ is a partition function and the
probability of measuring $\tilde{Q}_k=\tilde{Q}_0$ is found by computing the
fraction of $Z$ which contains $\tilde{Q}_0$ at the $k^{\mbox{th}}$
vertex\cite{lisi}. We have
\begin{equation}
P \left ( \tilde{Q}_k = \tilde{Q}_0 \right ) = \frac {Z \left ( \tilde{Q}_k = \tilde{Q}_0
\right )}{Z} = \sqrt{\frac {a_k}{2\pi}} \exp \left ( - \frac 12
\tilde{Q}_0^2 a_k + \tilde{J}_k \tilde{Q}_0 - \frac {\tilde{J}_k^2}{2a_k} \right )
\label{eqn10}
\end{equation}
as the part of theory X approximated in the continuum by QFT. The
most probable value of $\tilde{Q}_0$ at the $k^{\mbox{th}}$ vertex is then
given by
\begin{equation}
\delta P \left ( \tilde{Q}_k = \tilde{Q}_0 \right ) = 0 \Longrightarrow \delta
\left ( - \frac 12 \tilde{Q}_0^2 a_k + \tilde{J}_k \tilde{Q}_0 - \frac {\tilde{J}_k^2}{2a_k}
\right ) = 0 \Longrightarrow a_k \tilde{Q}_0 = \tilde{J}_k. \label{eqn11}
\end{equation}
That is, $\vec{\vec{K}} \cdot \tilde{Q}Q_0 = \tilde{J}J$ is the part of theory X that
obtains statistically and is approximated in the continuum by
classical field theory. We note that the manner by which $\vec{\vec{K}} \cdot
\tilde{Q}Q_0 = \tilde{J}J$ follows from $P(\tilde{Q}_k = \tilde{Q}_0) = Z(\tilde{Q}_k = \tilde{Q}_0)/Z$
parallels the manner by which classical field theory follows from
QFT via the stationary phase method\cite{zee4}. Thus, one may obtain
classical field theory by the continuum limit of $\vec{\vec{K}} \cdot \tilde{Q}Q_0
= \tilde{J}J$ in theory X (theory X $\rightarrow$ classical field theory),
or by first obtaining QFT via the continuum limit of $P(\tilde{Q}_k =
\tilde{Q}_0) = Z(\tilde{Q}_k = \tilde{Q}_0)/Z$ in theory X and then by using the
stationary phase method on QFT (theory X $\rightarrow$ QFT
$\rightarrow$ classical field theory). In either case, QFT is not
quantized classical field theory in our approach. In summary:
\begin{enumerate}
\item $Z$ is a partition function for an experiment described topologically by $\vec{\vec{K}}/2+ \tilde{J}J$ (Figure \ref{fig1}a).
\item $P(\tilde{Q}_k = \tilde{Q}_0) = Z(\tilde{Q}_k = \tilde{Q}_0)/Z$ gives us the probability for a particular geometric outcome in that experiment (Figures \ref{fig1}b and \ref{fig2}b).
\item $\vec{\vec{K}}\cdot \tilde{Q}Q_0 = \tilde{J}J$ gives us the most probable values of the experimental outcomes which are then averaged to produce the geometry for the experimental procedure at the classical level (Figure \ref{fig2}a).
\item $P(\tilde{Q}_k = \tilde{Q}_0) = Z(\tilde{Q}_k = \tilde{Q}_0)/Z$ and $\vec{\vec{K}}\cdot \tilde{Q}Q_0 = \tilde{J}J$ are the parts of theory X approximated in the continuum by QFT and classical field theory, respectively.
\end{enumerate}
\subsubsection{The Two-Source Euclidean Symmetry Amplitude/Partition
Function}
Typically, one identifies fundamentally interesting physics with
symmetries of the action in the Central Identity of Quantum Field
Theory, but we have theory X fundamental to QFT, so our method of
choosing fundamentally interesting physics must reside in the
topological graph of theory X. Thus, we seek a constraint of $\vec{\vec{K}}$
and $\tilde{J}J$ in our graphical symmetry amplitude $Z$ and this will be
in the form of a self-consistency criterion (SCC). In order to
motivate our general method, we will first consider a simple graph
with six vertices, seven links and two plaquettes for our
$(1+1)-$dimensional spacetime model (Figure \ref{fig3}). Our goal
with this simple model is to seek relevant structure that might be
used to infer an SCC. We begin by constructing the boundary
operators over our graph.
The boundary of $\mbox{\boldmath $p$}_1$ is $\mbox{\boldmath $e$}_4 + \mbox{\boldmath $e$}_5 - \mbox{\boldmath $e$}_2 - \mbox{\boldmath $e$}_1$,
which also provides an orientation. The boundary of $\mbox{\boldmath $e$}_1$ is
$\mbox{\boldmath $v$}_2 - \mbox{\boldmath $v$}_1$, which likewise provides an orientation. Using
these conventions for the orientations of links and plaquettes we
have the following boundary operator for $C_2 \rightarrow C_1$,
i.e., space of plaquettes mapped to space of links in the spacetime
chain complex:
\begin{equation}
\partial_2 = \left [ \begin{array}{rr}
-1 & 0 \\
-1 & 1 \\
0 & -1 \\
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & -1 \end{array} \right ]
\label{eqn12}
\end{equation}
Notice the first column is simply the links for the boundary of
$\mbox{\boldmath $p$}_1$ and the second column is simply the links for the boundary
of $\mbox{\boldmath $p$}_2$. We have the following boundary operator for $C_1
\rightarrow C_0$, i.e., space of links mapped to space of vertices
in the spacetime chain complex:
\begin{equation}
\partial_1 = \left [ \begin{array}{rrrrrrr}
-1 & 0 & 0 & -1 & 0 & 0 & 0 \\
1 & -1 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 1 & -1 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 \end{array} \right ]
\label{eqn13}
\end{equation}
which completes the spacetime chain complex, $C_0 \leftarrow C_1
\leftarrow C_2$. Notice the columns are simply the vertices for the
boundaries of the edges. These boundary operators satisfy
$\partial_1\cdot\partial_2 = 0$, i.e., the boundary of a boundary
principle.
The potential for coupled oscillators can be written
\begin{equation}
V(q_1,q_2) = \sum_{a,b} \frac 12 k_{ab} q_a q_b = \frac 12 k q_1^2 + \frac 12 k q_2^2 + k_{12} q_1 q_2
\label{eqn14}
\end{equation}
where $k_{11} = k_{22} = k>0$ and $k_{12} = k_{21}<0$ per the
classical analogue (Figure \ref{fig4}) with $k = k_1 + k_3 = k_2 +
k_3$ and $k_{12} = -k_3$ to recover the form in (\ref{eqn14}).
The Lagrangian is then
\begin{equation}
L = \frac 12 m \dot{q}_1^2 + \frac 12 m \dot{q}_2^2 - \frac 12
kq_1^2 - \frac 12 k q_2^2 - k_{12} q_1q_2 \label{eqn15}
\end{equation}
so our NRQM Euclidean symmetry amplitude is
\begin{equation}
Z = \int Dq(t) \exp \left [ - \int_0^T dt \left ( \frac 12 m
\dot{q}_1^2 + \frac 12 m \dot{q}_2^2 + V(q_1, q_2) - J_1 q_1 - J_2
q_2 \right )\right ] \label{eqn16}
\end{equation}
after Wick rotation. This gives
\begin{equation}
\vec{\vec{K}} = \left [ \begin{array}{rrrrrr}
\left ( \frac m{\Delta t} + k \Delta t \right ) & -\frac m{\Delta t} & 0 & k_{12} \Delta t & 0 & 0 \\
-\frac m{\Delta t} & \left ( \frac {2m}{\Delta t} + k \Delta t \right ) & -\frac m{\Delta t} & 0 & k_{12} \Delta t & 0 \\
0 & -\frac m{\Delta t} & \left ( \frac m{\Delta t} + k \Delta t \right ) & 0 & 0 & k_{12} \Delta t \\
k_{12} \Delta t & 0 & 0 & \left ( \frac m{\Delta t} + k \Delta t \right ) & -\frac m{\Delta t} & 0 \\
0 & k_{12} \Delta t & 0 & -\frac m{\Delta t} & \left ( \frac {2m}{\Delta t} + k \Delta t \right ) & -\frac m{\Delta t} \\
0 & 0 & k_{12} \Delta t & 0 & -\frac m{\Delta t} & \left ( \frac m{\Delta t} + k \Delta t \right ) \end{array} \right ]
\label{eqn17}
\end{equation}
on our graph. Thus, we borrow (loosely) from Wise\cite{wise} and
suggest $\vec{\vec{K}} \propto \partial_1\cdot\partial_1^T$ since
\begin{equation}
\partial_1\cdot\partial_1^T = \left [ \begin{array}{rrrrrr}
2 & -1 & 0 & -1 & 0 & 0 \\
-1 & 3 & -1 & 0 & -1 & 0 \\
0 & - 1& 2 & 0 & 0 & -1 \\
-1 & 0 & 0 & 2 & -1 & 0 \\
0 & -1 & 0 & -1 & 3 & -1 \\
0 & 0 & -1 & 0 & -1 & 2 \end{array} \right ]
\label{eqn18}
\end{equation}
produces precisely the same form as (\ref{eqn17}) and quantum theory
is known to be ``rooted in this harmonic paradigm''\cite{zee5}. [In
fact, these matrices will continue to have the same form as one
increases the number of vertices in Figure \ref{fig3}.] Now we
construct a suitable candidate for $\tilde{J}J$, relate it to $\vec{\vec{K}}$ and
infer our SCC.
Recall that $\tilde{J}J$ has a component associated with each vertex so
here it has components, $J_n$, $n = 1, 2, \ldots, 6$; $J_n$ for $n =
1, 2, 3$ represents one source and $J_n$ for $n = 4, 5, 6$
represents the second source. We propose $\tilde{J}J \propto
\partial_1\cdot\vec{e}$, where $e_i$ are the links of our graph, since
\begin{equation}
\partial_1\cdot\vec{e} =
\left [ \begin{array}{rrrrrrr}
-1 & 0 & 0 & -1 & 0 & 0 & 0 \\
1 & -1 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 1 & -1 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 \end{array} \right ]
\left [ \begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e_4 \\ e_5 \\ e_6 \\ e_7 \end{array} \right ]
= \left [\begin{array}{c} -e_1-e_4 \\ e_1 - e_2-e_3 \\ e_3 - e_7 \\ e_4 - e_5 \\ e_2 + e_5 - e_6 \\ e_6 + e_7 \end{array}\right ]
\label{eqn19}
\end{equation}
automatically makes $\tilde{J}J$ divergence-free, i.e., $\displaystyle
\sum_i J_i = 0$, and relationally defined. Such a relationship on
discrete spacetime lattices is not new. For example, Sorkin showed
that charge conservation follows from gauge invariance for the
electromagnetic field on a simplicial net\cite{sorkin2}.
With these definitions of $\vec{\vec{K}}$ and $\tilde{J}J$ we have, \emph{ipso
facto}, $\vec{\vec{K}}\cdot \vec{v} \propto \tilde{J}J$ as the basis of our SCC since
\begin{equation}
\partial_1\cdot\partial_1^T \cdot \vec{v} = \left [ \begin{array}{rrrrrr}
2 & -1 & 0 & -1 & 0 & 0 \\
-1 & 3 & -1 & 0 & -1 & 0 \\
0 & - 1& 2 & 0 & 0 & -1 \\
-1 & 0 & 0 & 2 & -1 & 0 \\
0 & -1 & 0 & -1 & 3 & -1 \\
0 & 0 & -1 & 0 & -1 & 2 \end{array} \right ] \left [
\begin{array}{c} v_1 \\ v_2 \\ v_3 \\ v_4 \\ v_5 \\ v_6 \end{array}
\right ] = \left [\begin{array}{c} -e_1-e_4 \\ e_1 - e_2-e_3 \\ e_3 - e_7 \\
e_4 - e_5\\ e_2 + e_5 - e_6 \\ e_6 + e_7 \end{array} \right ] =
\partial_1\cdot\vec{e} \label{eqn20}
\end{equation}
where we have used $e_1 = v_2 - v_1$ (etc.) to obtain the last
column. You can see that the boundary of a boundary principle
underwrites (\ref{eqn20}) by the definition of ``boundary'' and from
the fact that the links are directed and connect one vertex to
another, i.e., they do not start or end `off the graph'. Likewise,
this fact and our definition of $\tilde{J}J$ imply $\displaystyle \sum_i
J_i = 0$, which is our graphical equivalent of a divergence-free,
relationally defined source (every link leaving one vertex goes into
another vertex). Thus, the SCC $\vec{\vec{K}}\cdot \vec{v} \propto \tilde{J}J$ and
divergence-free sources $\displaystyle \sum_i J_i = 0$ obtain
tautologically via the boundary of a boundary principle. The SCC
also guarantees that $\tilde{J}J$ resides in the row space of $\vec{\vec{K}}$ so,
as will be shown, we can avoid having to ``throw away infinities''
associated with gauge groups of infinite volume as in Faddeev-Popov
gauge fixing. $\vec{\vec{K}}$ has at least one eigenvector with zero
eigenvalue which is responsible for gauge invariance, so {\em the
self-consistent co-construction of space, time and divergence-free
sources entails gauge invariance.}
Moving now to $N$ dimensions, the Wick rotated version of (\ref{eqn3}) is (\ref{eqn6})
and the solution is (\ref{eqn7}). Using $\tilde{J}J = \alpha
\partial_1\cdot\vec{e}$ and $\vec{\vec{K}} = \beta \partial_1\cdot
\partial_1^T$ ($\alpha, \beta \in \mathbb{R}$) with the SCC gives
$\vec{\vec{K}}\cdot \vec{v} = (\beta/\alpha) \tilde{J}J$, so that $\vec{v} =
(\beta/\alpha) \vec{\vec{K}}^{-1}\cdot \tilde{J}J$. However, $\vec{\vec{K}}^{-1}$ does not
exist because $\vec{\vec{K}}$ has a nontrivial null space, therefore the row
space of $\vec{\vec{K}}$ is an $(N-1)-$dimensional subspace of the
$N-$dimensional vector space\footnote{This assumes the number of
degenerate eigenvalues always equals the dimensionality of the
subspace spanned by their eigenvectors.}. The eigenvector with
eigenvalue of zero, i.e., normal to this hyperplane, is $\left[
\begin{array}{ccccc} 1 & 1 & 1 & \ldots & 1 \end{array} \right ]^T$,
which follows from the SCC as shown supra. Since $\tilde{J}J$ resides in
the row space of $\vec{\vec{K}}$ and, on our view, $Z$ is a functional of
$\vec{\vec{K}}$ and $\tilde{J}J$ which produces a partition function for the
various $\vec{\vec{K}}/2+\tilde{J}J$ associated with different 4D experimental
configurations, we restrict the path integral of (\ref{eqn6}) to the
row space of $\vec{\vec{K}}$. Thus, our approach revises (\ref{eqn7}) to give
(\ref{eqn9}).
Since this is linear, we do not expect to recover GR in this manner.
Instead, we expect to make correspondence with GR via a modification
to Regge calculus, a form of lattice gravity.
\section{Recovering General Relativity: RBW Versus Hiley's Implicate Order}
\label{section3}
The modeling of ``undivided wholeness'' (monism) in each formalism
leads to the same problem for both approaches when dealing with GR,
i.e., how to relate/connect different M4 frames. This is simply to
say the essence of gravity in GR is spacetime curvature, i.e., the
relative acceleration of `neighboring' geodesics, whereas the other
forces are modeled via deviation from geodetic motion in a flat
spacetime. Consider, for example, the phenomenon of gravitational
lensing that produces an Einstein ring image of a distant quasar by
an intervening galaxy. The explanation per GR is that empty
spacetime around the worldtube of the intervening galaxy is curved
so that null geodesics near its worldtube are deformed or `bent'
thereby `lensing' the photons as they proceed from the quasar around
the galaxy to Earth. We note that the principle explanatory
mechanism, i.e., spacetime curvature, doesn't have anything to do
with the stress-energy tensor of the quasar, or of the photons
passing through that region of space, or of Earth. Yet, the monistic
view doesn't allow for a separation of this sort - if we're relating
the quasar, galaxy, photons, and Earth, then the stress-energy
tensor for all these objects must be produced together with the
geometry of spacetime from a single `entity'.
For Hiley, this `entity' will be a process-based algebra of the
implicate order. Specifically, he speculates, a deformed Poisson
algebra obtained via the hidden Heisenberg algebra gives the Moyal
algebra for relating phase information for our electromagnetic
interactions. If he proceeds with a current algebra approach (again,
as inferred by his approach to Schr\"{o}dinger, Pauli and Dirac
particles), presumably, he will have to promote the spacetime metric
to a field so that it will have its own particle and current. Then,
he will have to produce commutation relations between the
electromagnetic current and the gravitational current to describe
the possible outcomes at interaction vertices. The problem is, of
course, there are no spacetime locations for the interaction
vertices, since one result of the calculation itself must be the
spacetime geometry. Of course, if this algebra produces dual
currents as with the Dirac particle, one is again left with the
problem of figuring out which currents correspond to actual detector
outcomes. But, suppose he takes the hint from his Dirac result and
gives up on the idea of ``Bohmian trajectories,'' as he has with the
``Bohmian guidance equation,''
\cite{callaghan}\cite{hiley14}\cite{hiley15} and proceeds with a
canonical quantization. Since his shadow manifolds are particular
Lorentz frames rather than the full M4 for the Dirac equation, the
logical counterpart to his approach (if it exists) for GR would be a
particular foliation of the curved spacetime manifold. That is, a
shadow manifold would be a particular path through all possible
three geometries and matter fields in the solution space of H = 0.
For RBW, the single `entity' responsible for its monism is
spacetimematter and we note immediately that for us the GR
explanation of the Einstein ring in the above example must be
corrected since there is no ``empty spacetime.'' Thus, per RBW, GR
is only an approximation to the `correct' theory of gravity. Of
course this is not new, the same can be said of Newtonian gravity
given GR and Newtonian mechanics given special relativity. The
questions are, what is the `correct' theory of gravity and in what
sense is it approximated by GR? Since our underlying approach is
graphical, we start with the graphical version of GR, called Regge
calculus, and propose modifications thereto.
In Regge calculus, the spacetime manifold is replaced by a lattice
geometry where each cell is Minkowskian (flat). Typically, this
lattice spacetime is viewed as an approximation to the continuous
spacetime manifold, but the opposite could be true and that is what
we will advocate. The lattice reproduces a curved manifold as the
cells (typically 4D `tetrahedra' called ``simplices'') become
smaller (Figure \ref{fig5}). Curvature is represented by ``deficit
angles'' (Figure \ref{fig5}) about any plane orthogonal to a
``hinge'' (triangular side to a tetrahedron, which is a side of a
simplex). A hinge is two dimensions less than the lattice dimension,
so in 2D a hinge is a zero-dimensional point (Figure \ref{fig5}).
The Hilbert action for a vacuum lattice is $I_R = \frac{1}{8
\pi}\displaystyle \sum_{\sigma_i\in L}\varepsilon_{i} A_{i}$ where
$\sigma_i$ is a triangular hinge in the lattice \emph{L}, $A_i$ is
the area of $\sigma_i$ and $\varepsilon_i$ is the deficit angle
associated with $\sigma_i$. The counterpart to Einstein's equations
is then obtained by demanding $\frac {\delta
I_R}{\delta\ell_{j}^{2}}=0$ where $\ell_{j}^{2}$ is the squared
length of the $j^{th}$ lattice edge, i.e., the metric. To obtain
equations in the presence of matter-energy, one simply adds the
matter-energy action $I_{M}$ to $I_R$ and carries out the variation
as before to obtain $\frac {\delta I_R}{\delta\ell_{j}^{2}}= -\frac
{\delta I_{M}}{\delta\ell_{j}^{2}}$. One finds the stress-energy
tensor is associated with lattice edges, just as the metric, and
Regge's equations are to be satisfied for any particular choice of
the two tensors on the lattice. Thus, Regge's equations are, like
Einstein's equations, a self-consistency criterion for the
stress-energy tensor and metric.
It seems to us that the most glaring deviation from GR phenomena
posed by directly connected sources per theory X would be found in
the exchange of photons on cosmological scales. Therefore, using
Regge calculus, we constructed a Regge differential equation for the
time evolution of the scale factor $a(t)$ in the Einstein-de Sitter
cosmology model (EdS) and proposed two modifications to the Regge
calculus approach: (1) we allowed the graphical links on spatial
hypersurfaces to be large, as when the interacting sources reside in
different galaxies, and (2) we assumed luminosity distance $D_L$ is
related to graphical proper distance $D_p$ by the equation $D_L =
(1+z)\sqrt{\overrightarrow{D_p}\cdot \overrightarrow{D_p}}$, where
the inner product can differ from its usual trivial
form\cite{stuckey3}. There are two reasons we made this second
assumption. First, in our view, space, time and sources are
co-constructed, yet $D_p$ is found without taking into account EM
sources responsible for $D_L$. That is to say, in Regge EdS (as in
EdS) we assume that pressureless dust dominates the stress-energy
tensor and is exclusively responsible for the graphical notion of
spatial distance $D_p$. However, even though the EM contribution to
the stress-energy tensor is negligible, EM sources are being used to
measure the spatial distance $D_L$. Second, in our view, there are
no ``photon paths being stretched by expanding space,'' so we cannot
simply assume $D_L = (1+z)D_p$ as in EdS. The specific form of
$\vec{\vec{K}}\cdot \tilde{Q}Q_0 = \tilde{J}J$ that we used to find the inner product for
$D_L$ was borrowed from linearized gravity in the harmonic gauge,
i.e., $\partial^2 h_{\alpha\beta} = -16 \pi G (T_{\alpha\beta} -
\frac{1}{2} \eta_{\alpha\beta} T)$. That is, $D_L = (1+z)\sqrt{1 +
h_{11}}D_p$ and we use $\vec{\vec{K}}\cdot \tilde{Q}Q_0 = \tilde{J}J$ to find $h_{11}$.
We emphasize that $h_{\alpha\beta}$ here corrects the graphical
inner product $\eta_{\alpha\beta}$ in the inter-nodal region between
the worldlines of photon emitter and receiver, where
$\eta_{\alpha\beta}$ is obtained via a matter-only stress-energy
tensor. Since the EM sources are negligible in the matter-dominated
solution and we're only considering a classical deviation from a
classical background, we have $\partial^2 h_{\alpha\beta} = 0$ to be
solved for $h_{11}$. Obviously, $h_{11} = 0$ is the solution that
gives the trivial relationship, but allowing $h_{11}$ to be a
function of $D_p$ allows for the possibility that $D_L$ and $D_p$
are not trivially related. We have $h_{11} = AD_p + B$ where $A$ and
$B$ are constants and, if the inner product is to reduce to
$\eta_{\alpha\beta}$ for small $D_p$, we have $B = 0$. Presumably,
$A$ should follow from the corresponding theory of quantum gravity,
so an experimental determination of its value provides a guide to
quantum gravity per our view of classical gravity. As we will show,
our best fit to the Union2 Compilation data gives $A^{-1}$ = 8.38
Gcy, so the correction to $\eta_{11}$ is negligible except at
cosmological distances, as expected.
The modified Regge calculus model (MORC), EdS and the concordance
model $\Lambda$CDM (EdS plus a cosmological constant to account for
dark energy) were compared using the data from the Union2
Compilation, i.e., distance moduli and redshifts for type Ia
supernovae\cite{amanullah} (see Figures \ref{fig8} and \ref{fig9}).
We found that a best fit line through $\displaystyle
\log{\left(\frac{D_L}{\mbox{Gpc}}\right)}$ versus $\log{z}$ gives a
correlation of 0.9955 and a sum of squares error (SSE) of 1.95. By
comparison, the best fit $\Lambda$CDM gives SSE = 1.79 using $H_o$ =
69.2 km/s/Mpc, $\Omega_{M}$ = 0.29 and $\Omega_{\Lambda}$ = 0.71.
The parameters for $\Lambda$CDM yielding the most robust fit to
``the Wilkinson Microwave Anisotropy Probe data with the latest
distance measurements from the Baryon Acoustic Oscillations in the
distribution of galaxies and the Hubble constant
measurement\cite{komatsu}'' are $H_o$ = 70.3 km/s/Mpc, $\Omega_{M}$
= 0.27 and $\Omega_{\Lambda}$ = 0.73, which are consistent with the
parameters we find for its Union2 Compilation fit. The best fit EdS
gives SSE = 2.68 using $H_o$ = 60.9 km/s/Mpc. The best fit MORC
gives SSE = 1.77 and $H_o$ = 73.9 km/s/Mpc using $R = A^{-1}$ = 8.38
Gcy and $m = 1.71\times 10^{52}$ kg, where $R$ is the coordinate
distance between nodes, $A^{-1}$ is the scaling factor from our
non-trival inner product explained above, and $m$ is the mass
associated with nodes\footnote{Strictly speaking, the stress-energy
tensor is associated with graphical links, not nodes. Our
association of mass with nodes is merely conceptual.}. A current
(2011) ``best estimate'' for the Hubble constant is $H_o$ = (73.8
$\pm$ 2.4) km/s/Mpc \cite{riess2}. Thus, MORC improves EdS as much
as $\Lambda$CDM in accounting for distance moduli and redshifts for
type Ia supernovae even though the MORC universe contains no dark
energy is therefore always decelerating.
This is but one test of the RBW approach and MORC must pass more
stringent tests in the context of the Schwarzschild solution where
GR is well confirmed. However, MORC's empirical success in dealing
with dark energy gives us reason to believe this formal approach to
classical gravity may provide creative new techniques for solving
other long-standing problems, e.g., quantum gravity, unification,
and dark matter. In particular, if MORC passes empirical muster in
the context of the Schwarzschild solution, then information such as
$A^{-1}$ might provide guidance to a theory of quantum gravity
underlying a graphical classical theory of gravity.
\section{The Problems of Time: RBW versus the Implicate
Order on Being and Becoming} \label{section4}
\subsection{The Implicate Order}
It is obvious that a process conception of fundamental reality does
not sit well with blockworld or frozen time. In the case of
blockworld there is no unique `now' successively coming into
existence. There are an indenumerably infinite number of time-like
foliations of M4, each representing a unique global `now' at various
values of its foliating time, and a particular spatial hypersurface
in foliation A (a `now' for observer A) contains events on many
different spatial hypersurfaces in foliation B (different `nows' for
observer B). That events which are simultaneous for observer A are
not simultaneous for observer B is called the ``relativity of
simultaneity'' and negates an objective passage of time. That is to
say, there is no objective (frame independent) distinction in
spacetime between past, present and future events respectively and
therefore no objective distinction to be had about the occurrence or
non-occurrence of events. In the words of Costa de
Beauregard\cite{beauregard}:
\begin{quotation}This is why first
Minkowski, then Einstein, Weyl, Fantappi\`{e}, Feynman, and many
others have imagined space-time and its material contents as spread
out in four dimensions. For those authors, of whom I am one ...
relativity is a theory in which everything is ``written'' and where
change is only relative to the perceptual mode of living
beings.\end{quotation}
And we have seen that the canonical or gauge interpretation of GR
leads to an even ``blockier'' world than SR! As Earman puts
it\cite{earman}:
\begin{quotation}Taken at face value, the gauge interpretation of
GTR implies a truly frozen universe: not just the `block universe'
that philosophers endlessly carp about -- that is, a universe
stripped of A-series change or shifting `nowness' -- but a universe
stripped of its B-Series change in that no genuine physical
magnitude (= gauge invariant quantity) changes its value with
time.\end{quotation}
As for the problem of frozen time in canonical QG, as we said, the
dynamics of the theory are given by a Hamiltonian operator
$\hat{H}$, which is defined on a space of spin network states via
the equation $\hat{H}|\Psi\rangle=0$, i.e., the Wheeler-DeWitt
equation mentioned earlier. It is hard to see how to avoid the
problem of frozen time in canonical QG because, unlike the standard
Schrödinger equation
$\hat{H}|\Psi\rangle=i\hbar\frac{\partial|\Psi\rangle}{\partial t}$,
the RHS of the Wheeler-DeWitt equation disappears. Because time is
part of the physical system being quantized, there is no external
time with respect to which the dynamics could unfold, only the
analogous gauge symmetries are there.
Therefore, in order to preserve his process model of reality, at the
end of the day Hiley must end up with a fundamental physical theory
that avoids the blockworld of relativity and the frozen time of
canonical QG. We can only speculate as to exactly how Hiley will
address these concerns or even exactly how his program will recover
GR, therefore the reader should consider our suggestions tentative.
Let's discuss SR first and extrapolate from there.
Any argument from SR to blockworld requires, as a premise, realism
about the geometric properties of M4. As we indicated earlier, we
think Hiley might be in a position to reject such realism because in
his scheme each shadow manifold constitutes a particular Lorentz
frame. Every Lorentz observer will construct his own space and time.
These space-times can exist together, but we cannot ascribe a
sharply defined `time' as to when they all exist together. Therefore
every frame is its own coordinate origin of its own explicate
manifold. Think of the implicate order as the head of an octopus and
the various explicate shadow manifolds (e.g., individual
perspectives or proper times) as the many tentacles produced from
the implicate order by the holomovement. Every event is described by
an infinity of times and spatial locations even though there is only
one event that all Lorentz observers are observing, as related
formally by the Lorentz group. The shadow manifolds are not
connected directly and thus there is no M4 as conceived by
Minkowski-there isn't one spacetime. As for the problem of time in
canonical GR and in canonical QG, again, assuming he gives up on
Bohmian trajectories and guide waves and uses canonical quantization
per his yet-to-be-determined process algebra for gravity and all
other forces, then his shadow manifolds correspond to particular
paths through all possible three geometries and matter fields in the
solution space of H = 0. Thus, Hiley avoids ``frozen time'' in GR
and QG exactly like he avoids it in SR -- by giving up on the idea
of a unique explicate order \emph{a la} M4, leaving the unification
of perspective to the implicate order as dictated by the
holomovement. Whether or not such a view is Hiley's considered view
and whether or not it is any better off than solipsism, we do not
know. Hiley is clear however that blockworld defined as the reality
of all events past, present and future is inconsistent with his
process ontology. This means he either rejects realism about M4 at
its root or provides a physically and formally acceptable preferred
foliation in addition to the structure of M4.
Given that Hiley rejects blockworld it would be reasonable to assume
that he embraces some form of presentism (only the present is real).
However in his theory of moments\cite{hiley17} he clearly rejects
presentism. According to Hiley's theory of moments, the holomovement
gives rise to ``moments/durons'' (which involves information from
the past and the future). Of moments he goes on to say that: ``For a
process with a given energy cannot be described as unfolding at an
instant except in some approximation''\cite{hiley18}. As we
understand it, the idea is that the holomovement can explicate
either a small region or a large region of spacetime (to include the
future) `at once', though never the entire universe. The extent of
the explicate domain (how much of the future exists) depends on the
properties of the holomovement in each particular case and the
process is apparently stochastic. In Hiley's model therefore, just
as the past can effect what unfolds in the future, so the future can
influence what unfolds in the present and what unfolded in the past.
Hiley is clear that what happens in the future cannot be made to
rewrite the past, but that the future possibilities can influence
the unfolding of the present. What is less clear is whether these
moments pass in and out of existence or always stay in existence
once explicated. All this suggests that each individual shadow
manifold is constantly changing in its own time (evolving `now')
such that the past is consistent with the present and the future is
understood probabilistically. Again, the solipsistic view of
individual shadow manifolds connected via the implicate order per
the holomovement avoids the blockworld implication of M4. One could
imagine other hybrid models of blockworld and presentism (or at
least becoming) such as entire blockworld universes winking
discretely in and out existence, each one different in some way from
the last. How to formalize models such as these, whether in an
algebraic program or some other, is unclear to us. What is clear to
us at the end of the day, merely advocating for fundamental physics
based on process isn't enough to secure every feature of dynamism.
Whether or not quantum theory and relativity can be unified in such
a way as to uphold all of dynamism is a formal question that has yet
to be resolved.
\subsection{RBW}
Of course we happily accept the implication of relativity theory
that it is a block universe and we are not bothered by the problem
of frozen time in canonical QG because we reject dynamism at its
foundation. For those wedded to dynamism these results are puzzling
embarrassments that require some sort of compatibilist response or a
completely new process-based ontology and formalism. In RBW we start
at bottom with an adynamical global constraint, a self-consistency
criterion (SCC) that allows us to construct discrete spacetimematter
graphs from which all the other effective theories and their
concomitant phenomena emerge. According to RBW, what quantum theory
and relativity theory are both trying to tell us is that every facet
of dynamism is false. If we succeed in our program of unification,
we will have shown that nothing in physics itself demands dynamism,
rather it was just a historical contingency based in the fact that
all physics must start with experience. Perhaps RBW offers a fourth
possibility regarding the nature of time, i.e., time as part of a
fundamental (pregeometric) regime wherein the notions of space, time
and matter are co-defined and co-determining. Technically, time,
space and matter as stand-alone concepts are not fundamental,
emergent or illusions in RBW. We note that it is only from a God's
eye Point of view (the view from nowhere and nowhen) that time and
change are an illusion and in a fundamentally relational model such
as ours there are no perspectives ``external to the universe.'' The
conceptual foundation of our dynamical reality isn't a so-called
``initial singularity,'' but the adynamical SCC upon which all
dynamic theories reside. The SCC characterizing spacetimematter at
the bottom of RBW is not a dynamical law or initial condition, but
it is responsible for the discrete action. Therefore, if
higher-level physical theories are truly recovered from the discrete
action, then there is nothing left to explain at bottom, regardless
what phenomena one counts as initial/boundary conditions versus
laws. The point of all this is that in RBW there will be no quantum
cosmology as is currently conceived. We also note that the universe
comes with many physically significant modes of temporal passage and
change such as proper time, cosmic time, etc. Certainly these
constitute objective notions of becoming (objectively dynamical
flow) even if they are mere patterns in a block universe. Therefore,
RBW does not negate change and becoming, it merely internalizes and
relativizes them.
Of course, all this falls short of getting every facet of time as
experienced into fundamental physics. There is no objectively
distinguished present moment, and there is no objectively dynamical
becoming in the sense of bringing events into existence that never
existed before from a God's eye point of view. However, perhaps the
standard wisdom that time as experienced is either a physical
feature of reality or merely a psychological feature of conscious
beings is a false dichotomy. Perhaps what all this suggests is that
conscious temporal experience is fundamental as well, so instead of
spacetimematter at bottom we have the super-monistic
spacetimematterexperience at bottom. This is shear speculation of
course, it would require working out a new formal model and much
else conceptually. We can say however that the alternatives are not
very appetizing if we take the frozen block universe seriously. The
image of consciousness crawling along the worldtube of individuals
illuminating the present and moving it toward the future is an
unhelpful and non-explanatory kind of dualism which simply exempts
conscious experience from the rules of the block
universe\cite{weyl}. The other alternative, that conscious
experience emerges from or is realized in neuro-dynamical activity,
is problematic in a block universe in which everything, past,
present and future is just there `at once' (including conscious
experiences throughout the block) and brains are just worldtubes
like everything else. One might find correlations between brain
states and the experience of the objective specialness of the `now'
and the experience of objectively dynamical becoming, but it cannot
be said that brain dynamics produce or bring into being conscious
states (themselves worldtubes). In such a universe brain processes
are not metaphysically or causally more fundamental than conscious
processes. Again, the idea of spacetimematterexperience is
half-baked, but if we take it seriously, perhaps it moves RBW closer
to Hiley's process conception of reality since process (objectively
distinguished present and objectively dynamical) is the nature of
ordinary conscious experience and the experience of time partially
motivates the process model.
\begin{figure}
\caption{(a) Topological Graph - This spacetimematter graph depicts
four sources, i.e., the columns of squares. The graph's actional
$\vec{\vec{K}
\label{fig1}
\end{figure}
\begin{figure}
\caption{(a) Classical Physics - Classical Objects result when the
most probable field values $\tilde{Q}
\label{fig2}
\end{figure}
\begin{figure}
\caption{Graph with six vertices, seven links $e_i$ and two
plaquettes $p_i$.}
\label{fig3}
\end{figure}
\begin{figure}
\caption{Coupled harmonic oscillators.}
\label{fig4}
\end{figure}
\begin{figure}
\caption{Reproduced from Misner, C.W., Thorne, K.S., Wheeler, J.A.:
Gravitation. W.H. Freeman, San Francisco (1973), p. 1168. Permission
pending.}
\label{fig5}
\end{figure}
\begin{figure}
\caption{Plot of transformed Union2 data along with the best fits for linear
regression (thin black), EdS (dashed), $\Lambda$CDM (gray), and MORC (dotted).}
\label{fig8}
\end{figure}
\begin{figure}
\caption{Plot of Union2 data along with the best fits for EdS (dashed),
$\Lambda$CDM (gray), and MORC (dotted). The MORC curve is terminated at $z$ = 1.4 in this figure
so that the $\Lambda$CDM curve is visible underneath.}
\label{fig9}
\end{figure}
\end{document} |
\begin{document}
\title{Essential Trigonometry Without Geometry}
\author{
John Gresham\\
Bryant Wyatt\\
Jesse Crawford\\
Tarleton State University}
\date{\today}
\begin{titlepage}
\maketitle
\thispagestyle{empty}
\setcounter{page}{0}
\end{titlepage}
\begin{abstract}
The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics demands a more rigorous approach. Our purpose here is to revisit elementary trigonometry from an entirely analytic perspective. We will give a comprehensive treatment of the sine and cosine functions and will show how to derive the familiar theorems of trigonometry without reference to geometric definitions or constructions.
\end{abstract}
\subsection*{Introduction}
As we approach trigonometry from an analytic perspective, our understanding deepens and old theorems become new again. For this study, we will assume a familiarity with calculus, differential equations, and real analysis.
\subsection*{Definitions and Basic Properties}
We begin by considering the solution of the second-order homogeneous linear
differential equation
\[
f\,^{\prime \prime }\left( x\right) +f\left( x\right) =0\text{ with }
f\left( 0\right) =0\text{ and }f\,^{\prime }\left( 0\right) =1.
\]
By the Existence and Uniqueness Theorem we know that a unique solution
exists [Nagle, Saff, and Snider, p. 171]. If this solution has a power series representation around the ordinary point $x=0$, it must have the form
\[
f\left( x\right) =\sum_{n=0}^{\infty }c_{n}x^{n}
\]
Note that $f\left( 0\right) =c_{0}=0$ and $f\,^{\prime }\left( 0\right)
=c_{1}=1$. We also have
\begin{align*}
f\,^{\prime \prime }\left( x\right) &=\sum_{n=2}^{\infty }\left( n\right)
\left( n-1\right) c_{n}x^{n-2} \\
&=\sum_{n=0}^{\infty }\left( n+2\right) \left( n+1\right) c_{n+2}x^{n}
\end{align*}
Then
\[
\sum_{n=0}^{\infty }\left( n+2\right) \left( n+1\right)
c_{n+2}x^{n}+\sum_{n=0}^{\infty }c_{n}x^{n}=\sum_{n=0}^{\infty }\left(
\left( n+2\right) \left( n+1\right) c_{n+2}+c_{n}\right) x^{n}=0
\]
Since this power series is $0$ for all $x$, we get the general recursion relation
\[
\left( n+2\right) \left( n+1\right) c_{n+2}+c_{n}=0
\]
so that
\[
c_{n+2}=-\frac{c_{n}}{\left( n+2\right) \left( n+1\right) }.
\]
Because $c_{0}=0$, we have for all even indices $2n$
\[
c_{2n}=0
\]
Let us now examine the coefficients with odd indices $2n+1$.
\begin{align*}
c_{1} &=1\text{ \ \ initial condition} \\
c_{3} &=-\frac{1}{3\cdot 2}=-\frac{1}{3!} \\
c_{5} &=-\genfrac{}{}{1pt}{0}{-\frac{1}{3!}}{5\cdot 4}=\frac{1}{5!} \\
c_{7} &=-\genfrac{}{}{1pt}{0}{\frac{1}{5!}}{7\cdot 6}=-\frac{1}{7!}
\end{align*}
and in general,
\[
c_{2n+1}=\left( -1\right) ^{n}\frac{1}{\left( 2n+1\right) !}
\]
The power series about $x=0$ must have the form
\[
\sum_{n=0}^{\infty }\left( -1\right) ^{n}\dfrac{x^{2n+1}}{\left( 2n+1\right)
!}
\]
Using the Ratio Test, it is easy to show that this series converges for all
real $x$.
The function represented by this power series is the unique solution of the
differential equation
\[
f\,^{\prime \prime }\left( x\right) +f\left( x\right) =0\text{ with }
f\left( 0\right) =0\text{ and }f\,^{\prime }\left( 0\right) =1.
\]
We call this function the \textbf{sine }function\textbf{, }denoted $
\sin x$, or $\sin \left( x\right) $.\textbf{
}
\begin{defn}[Sine Function]
\[
\sin x=\sum_{n=0}^{\infty }\left( -1\right) ^{n}\dfrac{x^{2n+1}}{\left(
2n+1\right) !}
\]
\end{defn}
We define the \textbf{cosine} to be the derivative of the sine function.
\begin{defn}[\textit{Cosine Function}]
\[
\cos x=\dfrac{d}{dx}\sum_{n=0}^{\infty }\left( -1\right) ^{n}\dfrac{x^{2n+1}
}{\left( 2n+1\right) !}=\sum_{n=0}^{\infty }\left( -1\right) ^{n}\dfrac{
x^{2n}}{\left( 2n\right) !}
\]
\end{defn}
The following are elementary consequences of the
definitions.
\begin{enumerate}
\item $\sin 0=0$
\item $\cos 0=1$
\item The function $\sin x$ is odd because all exponents in its power series
are odd.
\item The function $\cos x$ is even because all exponents in its power
series are even.
\item The functions $\sin x$ and $\cos x$ are both continuous since they are
differentiable.
\item The derivatives of $\sin x$ are cyclic with order four.
\end{enumerate}
\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$f\left( x\right) $ & $f\,^{\prime }\left( x\right) $ & $f\,^{\prime \prime
}\left( x\right) $ & $f\,^{\prime \prime \prime }\left( x\right) $ & $
f\,^{\prime \prime \prime \prime }\left( x\right) $ \\ \hline
$\sin x$ & $\cos x$ & $-\sin x$ & $-\cos x$ & $\sin x$ \\ \hline
\end{tabular}
\]
\subsection*{Key Theorems}
This section presents the Pythagorean and Sine Sum identities which, along with the smallest positive critical value of $\sin x$, enable the development of several important identities and analytic results in elementary trigonometry.
First, we prove the Pythagorean Identity.
\begin{thm}[Pythagorean Identity]
For all $x$,
\[
\sin ^{2}x+\cos ^{2}x=1
\]
\end{thm}
\begin{proof}[Proof:\nopunct] Consider the derivative of the left side.
\begin{align*}
\dfrac{d}{dx}\left( \sin ^{2}x+\cos ^{2}x\right) &=2\sin x\cos x+2\cos x\left( -\sin x\right)\\&=0
\end{align*}
Since the derivative is $0$, $\sin ^{2}x+\cos ^{2}x$ is a constant.
Because $\sin 0=0$, and $\cos 0=1$, this
constant must be $1$.
\end{proof}
Next, we consider the identity for the sine of the sum of $x$ and $y$. The
proof in most elementary trigonometry texts involves a geometric
construction with triangles or the unit circle. In our geometry-free
approach, we will use only power series.
\begin{thm}[Sine Sum Identity]For all $x$, $y$,
\[
\sin \left( x+y\right) =\sin x\cos y+\cos x\sin y
\]
\end{thm}
\begin{proof}[Proof:\nopunct]
Consider the series expansion
\[
\sin \left( x+y\right) =\sum_{n=0}^{\infty }\left( -1\right) ^{n}\dfrac{
\left( x+y\right) ^{2n+1}}{\left( 2n+1\right) !}
\]
Now examine the general $n^{th}$ term $a_{n}$ of this series using the Binomial
Theorem:
\begin{align*}
a_{n}
&=\left( -1\right) ^{n}\dfrac{\left( x+y\right) ^{2n+1}}{\left( 2n+1\right) !}\\
&=\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}\left( x+y\right)
^{2n+1} \\
&=\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}\sum_{i=0}^{2n+1}
\dbinom{2n+1}{i}x^{2n+1-i}y^{i} \\
&=\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}\sum_{i=0}^{2n+1}
\dfrac{\left( 2n+1\right) !}{i!\left( 2n+1-i\right) !}x^{2n+1-i}y^{i} \\
&=\left( -1\right) ^{n}\sum_{i=0}^{2n+1}\dfrac{1}{i!\left( 2n+1-i\right) !}
x^{2n+1-i}y^{i}
\end{align*}
This last sum has $2n+2$ terms. We will re-write it as two sums
each having $n+1$ terms.
\begin{align*}
\left( -1\right) ^{n}\sum_{i=0}^{2n+1}\dfrac{1}{i!\left( 2n+1-i\right) !}
x^{2n+1-i}y^{i} &=\left( -1\right) ^{n}\underbrace{\sum_{i=0}^{n}\dfrac{
x^{2i+1}y^{2n-2i}}{\left( 2i+1\right) !\left( 2n-2i\right) !}}+\left(
-1\right) ^{n}\underbrace{\sum_{i=0}^{n}\dfrac{x^{2n-2i}y^{2i+1}}{\left(
2n-2i\right) !\left( 2i+1\right) !}} \\
&\text{\ \ \ \ \ \ \ \ increasing odd powers of }x\text{ \ \ \ \ \ \ \
decreasing\ even powers of }x \\
&=\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}\sum_{i=0}^{n}\dfrac{
\left( 2n+1\right) !x^{2i+1}y^{2n-2i}}{\left( 2i+1\right) !\left(
2n-2i\right) !}+\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}
\sum_{i=0}^{n}\dfrac{\left( 2n+1\right) !x^{2n-2i}y^{2i+1}}{\left(
2n-2i\right) !\left( 2i+1\right) !} \\
&=\underbrace{\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}
\sum_{i=0}^{n}\dbinom{2n+1}{2i+1}x^{2i+1}y^{2n-2i}}+\underbrace{\dfrac{
\left( -1\right) ^{n}}{\left( 2n+1\right) !}\sum_{i=0}^{n}\dbinom{2n+1}{2i+1}
x^{2n-2i}y^{2i+1}} \\
&\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ [2]}
\end{align*}
This last line represents the $n^{th}$ term of the expansion of $
\sin \left( x+y\right) $. We now turn our attention to the right side
\[
\sin x\cos y+\cos x\sin y
\]
and consider the series expansion of the term $\sin x\cos y$.
Since the series for $\sin x$ and for $\cos x$ both converge absolutely, we
can write $\sin x\cos y$ as the Cauchy product of the two series
\[
\sin x\cos y=\sum_{n=0}^{\infty }c_{n}
\]
where
\[
c_{n}=\sum_{i=0}^{n}a_{i}b_{n-i\,}\text{, \ }n=0,1,2,3,...
\]
and the $a_{i}$, $b_{n-i}$ terms come from the series for $\sin x$ and $\cos x$, respectively [Rudin, p. 63ff]. Let us examine the general term $c_{n}$ of
this Cauchy product.
\begin{align*}
c_{n} &=\sum_{i=0}^{n}\left( -1\right) ^{i}\dfrac{x^{2i+1}}{\left(
2i+1\right) !}\cdot \left( -1\right) ^{n-i}\dfrac{y^{2n-2i}}{\left(
2n-2i\right) !} \\
&=\sum_{i=0}^{n}\left( -1\right) ^{n}\dfrac{x^{2i+1}y^{2n-2i}}{\left(
2i+1\right) !\left( 2n-2i\right) !} \\
&=\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}\sum_{i=0}^{n}\dfrac{
\left( 2n+1\right) !}{\left( 2i+1\right) !\left( 2n-2i\right) !}
x^{2i+1}y^{2n-2i} \\
&=\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}\sum_{i=0}^{n}\dbinom{
2n+1}{2n-2i}x^{2i+1}y^{2n-2i}
\end{align*}
Then the term $c_{n}$ is the odd powers of $x$ in part [1] of the
general binomial expansion above.
By switching $x$ with $y$ in the previous equation, we get the general term $d_{n}$ for the Cauchy product of the series for $\sin y$ and $\cos x$.
\begin{align*}
d_{n} &=\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}
\sum_{i=0}^{n}\dbinom{2n+1}{2n-2i}y^{2i+1}x^{2n-2i} \\
&=\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}\sum_{i=0}^{n}\dbinom{
2n+1}{2n-2i}x^{2n-2i}y^{2i+1} \\
&=\dfrac{\left( -1\right) ^{n}}{\left( 2n+1\right) !}\sum_{i=0}^{n}\dbinom{
2n+1}{2i+1}x^{2n-2i}y^{2i+1}
\end{align*}
This matches the even powers of $x$ in part [2] of the general binomial
expansion.
Therefore
\[a_{n}=c_{n}+d_{n}\]
and
\[
\sin \left( x+y\right) =\sin x\cos y+\cos x\sin y.
\]
\end{proof}
We now turn our attention to a special value, the smallest positive critical value of $\sin x$, a number we will call $Q$.
\begin{thm}[Critical Value]
There exists a smallest positive critical value of $\sin x$, that is, a smallest positive zero of $\cos x$.
\end{thm}
\begin{proof}
We have already seen that $\cos 0=1$. Now observe that
\[
\cos 2=1-\frac{2^{2}}{2!}+\frac{2^{4}}{4!}-\frac{2^{6}}{6!}+\cdots
\]
We now write
\begin{align*}
\cos 2 &=\left( 1-\frac{2^{2}}{2!}+\frac{2^{4}}{4!}-\frac{2^{6}}{6!}\right)
+R_{3} \\
&=\left( -\frac{19}{45}\right) +R_{3} \\
&\leq -\frac{19}{45}+\left\vert R_{3}\right\vert
\end{align*}
The Remainder Theorem for alternating series tells us that
\begin{align*}
\left\vert R_{3}\right\vert &\leq a_{4}=\frac{2^{8}}{8!}\text{ \ and so} \\
\cos 2 &\leq -\frac{19}{45}+\frac{2}{315}=-\frac{131}{315}
\end{align*}
Since $\cos 0>0$ and $\cos 2<0$, by the Intermediate Value Theorem,
there is at least one real number $c\in \left( 0,2\right) $ with $\cos c=0$.
The nonempty set $\left\{ x|\cos x=0\right\} $ is the inverse image of the
closed point set $\left\{ 0\right\}$ under the continuous function $\cos x$.
Therefore the set $\left\{ x|\cos x=0\right\} $ is closed. It follows that the set
\[
\left\{ x|\cos x=0\right\} \cap \left[ 0,2\right]
\]
is nonempty, closed, bounded, and is therefore compact [Willard, p. 120]. It
must contain its least element which we shall call, temporarily, $Q$.
\end{proof}
\textbf{Definition of $Q$}
\[
Q=\min \left( \left\{ x|\cos x=0\right\} \cap \left[ 0,2\right] \right)
\]
\subsection*{Consequences of the Key Theorems}
The Pythagorean Identity leads directly to the following corollary.
\begin{cor}For all $x$,
\[
\left\vert \sin x\right\vert \leq 1\text{ \ and \ }\left\vert \cos
x\right\vert \leq 1.
\]
\end{cor}
\begin{proof}[Proof:\nopunct]
If $\left\vert \sin x\right\vert > 1$, then $\cos^{2} x < 0$ and $\cos x$ is not a real number. Similarly, if $\left\vert \cos x\right\vert > 1$, then $\sin x$ is not a real number. In this study, we are restricting our work to real numbers.
\end{proof}
The next two corollaries follow from the Pythagorean Identity and the special properties of $Q$.
\pagebreak
\begin{cor} $\sin Q=1$ and $\sin x$ has an absolute maximum value of $1$ at $
x=Q$.
\end{cor}
\begin{proof}[Proof:\nopunct] Since $\cos 0=1$ and $\cos x$ is an even function, for $x\in ( -Q,Q)$, we have $\cos x>0$. Therefore $\sin x$ is strictly increasing on $(-Q,Q)$. Since $0<Q$ we have $0=\sin 0<\sin Q$. From the Pythagorean Identity we know that
\[
\sin ^{2}Q+\cos ^{2}Q=1
\]
Since $\cos Q=0$, it must be the case that $\sin Q=1$. We have already
observed that
\[
\left\vert \sin x\right\vert \leq 1
\]
and therefore $1$ is an absolute maximum of $\sin x$.
\end{proof}
\begin{cor} The range of $\sin x$ is $\left[ -1,1\right] $.
\end{cor}
\begin{proof}[Proof:\nopunct] Because $\sin x$ is an odd function we have $\sin \left(
-Q\right) =-\sin Q=-1$ is an absolute minimum. The range $\left[ -1,1
\right] $ follows from the continuity of $\sin x$ and the Intermediate Value
Theorem.\end{proof}
Later will will see that the range of cosine is also $[-1,1]$.
Our next two corollaries follow from the Sine Sum Theorem.
\begin{cor} $\sin \left( x-y\right) =\sin x\cos y-\cos x\sin y$
\end{cor}
\begin{proof}[Proof:\nopunct]
Because $\sin x$ is an odd function and $\cos x$ is even, we have the following:
\begin{align*}
\sin \left( x-y\right) &=\sin \left( x+\left( -y\right) \right) \\
&=\sin x\cos \left( -y\right) +\cos x\sin \left( -y\right) \\
&=\sin x\cos y-\cos x\sin y
\end{align*}
\end{proof}
\begin{cor}$
\sin 2x=2\sin x\cos x$
\end{cor}
\begin{proof}[Proof:\nopunct]
\begin{align*}
\sin 2x &=\sin \left( x+x\right) \\
&=\sin x\cos x+\cos x\sin x \\
&=2\sin x\cos x
\end{align*}
\end{proof}
We now consider the cofunction rules that follow from the Sine Sum Identity and the properties of $Q$. We will use these later to show that the sine and cosine functions are periodic.
\begin{cor}[Cofunction Rule] $\sin \left( Q-x\right)
=\cos x$
\end{cor}
\begin{proof}[Proof:\nopunct]
\begin{align*}
\sin \left( Q-x\right) &=\sin Q\cos x-\cos Q\sin x \\
&=1\cdot \cos x-0\cdot \sin x \\
&=\cos x
\end{align*}
\end{proof}
\begin{cor}[Cofunction Rule]$\cos \left( Q-x\right) =\sin x$
\end{cor}
\begin{proof}[Proof:\nopunct]
\begin{align*}
\cos \left( Q-x\right) &=\sin \left( Q-\left( Q-x\right) \right) \\
&=\sin x
\end{align*}
\end{proof}
In the following corollaries we complete the sum, difference, and double angle rules.
\begin{cor} \ $\cos \left( x+y\right) =\cos x\cos y-\sin x\sin y$
\end{cor}
\begin{proof}[Proof:\nopunct]
\begin{align*}
\cos \left( x+y\right) &=\sin \left( Q-\left( x+y\right) \right) \\
&=\sin \left( \left( Q-x\right) -y\right) \\
&=\sin \left( Q-x\right) \cos y-\cos \left( Q-x\right) \sin y \\
&=\cos x\cos y-\sin x\sin y
\end{align*}
\end{proof}
The following corollaries now follow.
\begin{cor} \ $\cos \left( x-y\right) =\cos x\cos y+\sin x\sin y$
\end{cor}
\begin{proof}[Proof:\nopunct]
\begin{align*}
\cos \left( x-y\right) &=\cos(x+(-y)) \\
&=\cos x \cos(-y)- \sin x \sin(-y) \\
&=\cos x \cos y + \sin x \sin y
\end{align*}
\end{proof}
\begin{cor} \ $\cos 2x =2\cos^{2} x-1$
\end{cor}
\begin{proof}[Proof:\nopunct]
\begin{align*}
\cos 2x &=\cos \left( x+x\right) \\
&=\cos x\cos x-\sin x\sin x \\
&=\cos^{2}x-\sin^{2}x\\
&=\cos^{2}x-(1-\cos^{2}x)\\
&=2\cos^{2} x-1
\end{align*}
\end{proof}
We have seen that the three key theorems have led to the familiar difference formulas as well as double angle formulas. From these follow the other identities such as half-angle and product-to-sum rules. In particular, we will later need the
identity
\[
\cos ^{2}x=\frac{1}{2}+\frac{1}{2}\cos 2x
\]
\subsection*{Periodicity}
We will need the sine and cosine function values of $4Q$ to show periodicity. Here is a sequence of steps to arrive at this point.
\begin{enumerate}
\item $\sin 2Q=2\sin Q\cos Q=2(1)(0)=0$
\item $\cos 2Q=\sin \left( Q-2Q\right) =\sin \left( -Q\right) =-\sin Q=-1$.\\
From this it follows that the range of $\cos x$ is $[-1,1].$
\item $\sin 3Q=\sin \left( Q+2Q\right) =\sin Q\cos 2Q+\cos Q\sin 2Q=-1$
\item $\cos 3Q=\sin \left( Q-3Q\right) =\sin \left( -2Q\right) =-\sin 2Q=0$
\item $\sin 4Q=2\sin 2Q\cos 2Q=0$
\item $\cos 4Q=\sin \left( Q-4Q\right) =\sin \left( -3Q\right)=-\sin\left(3Q\right)=-(-1) =1
$
\end{enumerate}
We now have the machinery needed to prove the periodicity of $\sin x$ and $
\cos x$.
\begin{defn}
A function $f\left( x\right) $ is \textit{periodic}
if there is a positive number $p$ such that
\[
f\left( x+p\right) =f\left( x\right)
\]
for all $x$. If there is a \textsl{smallest} positive number $p$ for which
this holds, then $p$ is called the \textit{period of }$f$.
\end{defn}
\begin{thm}[Periodicity of Sine] The sine function is periodic and its period is $4Q$.
\end{thm}
\begin{proof}[Proof:\nopunct] We first show that sine is periodic.
\begin{align*}
\sin \left( x+4Q\right) &=\sin x\cos 4Q+\cos x\sin 4Q \\
&=\sin x\left( 1\right) +\cos x\left( 0\right) \\
&=\sin x
\end{align*}
This shows that $\sin x$ is periodic, but does not show that the
period is $4Q$. To show that $4Q$ is the period, assume, to the
contrary, that there exists a number $R$ such that $0<4R<4Q$ and for all $x$,
\[
\sin \left( x+4R\right) =\sin x
\]
Observe that $0<R<Q$. For $x\in \left( 0,Q\right) $ we have $\cos x>0$ because
$\cos 0=1$ and $Q$ is the smallest value with $\cos Q=0$. We also have $
\sin x>0$ since $\sin 0=0$ and $\sin $ is increasing on $\left( 0,Q\right)$.
Now examine $\sin Q$:
\begin{align*}
\sin Q &=\sin \left( Q+4R\right) \\
&=\sin Q\cos 4R+\cos Q\sin 4R \\
&=\cos 4R \\
&=\cos 2\left( 2R\right) \\
&=2\cos ^{2}\left( 2R\right) -1
\end{align*}
Because $\sin Q=1$,
\begin{align*}
1 &=2\cos ^{2}\left( 2R\right) -1 \\
1 &=\cos ^{2}\left( 2R\right) \\
\cos 2R &=1\text{ or }\cos 2R=-1
\end{align*}
We now have two cases:\\
\pagebreak
\ Case I: \ $\cos 2R=1.$
\ \ \ \ \ \ \ \ \ \ \ \ \ \ Then by the double angle identity,
\begin{align*}
2\cos ^{2}R-1 &=1 \\
\cos ^{2}R &=1
\end{align*}
If $\cos^{2} R=1$, then by the Pythagorean Identity, $\sin R=0$, a contradiction
to the fact that $\sin R>0$.
\ Case II: \ $\cos 2R=-1$.
\qquad Then
\begin{align*}
2\cos ^{2}R-1 &=-1 \\
\cos R &=0
\end{align*}
This last statement contradicts the choice of $Q$ as the smallest positive
number in $\left[ 0,2\right] $ with $\cos Q=0$. Therefore such a number $R$ does not exist, and the period of $\sin $ is $4Q$.
\end{proof}
\begin{cor}[Periodicity of Cosine]The cosine function is periodic with period $4Q$.
\end{cor}
\begin{proof}[Proof:\nopunct] We can write $\cos x$ as \[\cos x=-\sin(x-Q)\]
Because horizontal translations and vertical rotations about the x-axis do not change the period of a function, $\cos x$ is periodic with period $4Q$.
\end{proof}
\subsection*{Connection to Geometry}
With this result we now show the connection between the analytic and
geometric approaches to trigonometry.
\begin{thm}[Connection with $\pi$] \[ \int_{0}^{1}\sqrt{1-x^{2}}\, dx=\frac{Q}{2} \]
\end{thm}
\begin{proof}[Proof:\nopunct] Use the substitution
\[
x=\sin \theta
\]
with the values
\begin{tabular}{c|c}
$x$ & $\theta $ \\ \hline
$0$ & $0$ \\
$1$ & $Q$ \\
\end{tabular}
\ so that the integral becomes
\begin{align*}
\int_{0}^{Q}\sqrt{1-\sin ^{2}\theta }\cos \theta \,d\theta
&=\int_{0}^{Q}\cos ^{2}\theta \,d\theta \\
&=\int_{0}^{Q}\frac{1}{2}\left( 1+\cos 2\theta \right) \,d\theta \\
&=\frac{1}{2}\left[ \theta +\frac{1}{2}\sin 2\theta \right] _{0}^{Q} \\
&=\frac{1}{2}\left[ \left( Q+\frac{1}{2}\sin 2Q\right) -\left( 0+\frac{1}{2}
\sin \left( 2\cdot 0\right) \right) \right] \\
&=\frac{1}{2}Q
\end{align*}
\end{proof}
The integral $\int_{0}^{1}\sqrt{1-x^{2}}\,dx$ represents the
quarter-circle area enclosed by the unit circle, the nonnegative $x$-axis,
and the nonnegative $y$-axis, and so we are led to the conclusion that \[Q=\pi/2\].
Using what we have previously developed about multiples of $Q$, we have a table restating the values for sine and cosine in terms of $\pi $ instead of $Q$.
\begin{center}
\begin{tabular}{c|c|c|r|r|c}
$x$ & $0$ & $\pi /2$ & $\pi $ & $3\pi /2$ & $2\pi $ \\ \hline
$\sin x$ & $0$ & $1$ & $0$ & $-1$ & $0$ \\
$\cos x$ & $1$ & $0$ & $-1$ & $0$ & $1$ \\
\end{tabular}
\end{center}
From this follows the usual information about the graphs of the sine and
cosine: intervals for positive/negative values, intervals for
increasing/decreasing, local (and absolute) maximums/minimums.
Without geometry, we can find the values of sine and cosine of $\dfrac{\pi }{
4}$, $\dfrac{\pi }{6}$, $\dfrac{\pi }{3}$ using only the sum and difference
identities. We include the development of these values in Appendix A. In Appendix B we present the mathematics that connects the sine and cosine functions, defined here as power series, to the trig functions defined using the unit circle.
\subsection*{Pythagorean Identity Revisited}
We conclude this study with the observation that the \textbf{converse} of the
Pythagorean Identity also holds.
\begin{thm} If $f:\mathbb{R} \to \mathbb{R}$ is analytic, $f\,^{\prime
}\left( 0\right) =1$, $\ f\left( 0\right) =0$, and $f$ satisfies the
Pythagorean Identity
\[
\left( f\left( x\right) \right) ^{2}+\left( f\,^{\prime }\left( x\right)
\right) ^{2}=1
\]
for all $x$, then $f$ $\left( x\right) \equiv \sin x$.
\end{thm}
\begin{proof}[Proof:\nopunct] Differentiation of both sides gives
\[
2\left( f\left( x\right) \right) f\,^{\prime }\left( x\right) +2f\,^{\prime
}\left( x\right) f\,^{\prime \prime }\left( x\right) =0
\]
so that
\[
2f\,^{\prime }\left( x\right) \left( f\left( x\right) +f\,^{\prime \prime
}\left( x\right) \right) =0
\]
Since $f'(0)=1$, and $f$ is analytic, $f'$ is positive on some open interval containing $0$. Therefore, \textbf{on this interval},
\[
f\left( x\right) +f\,^{\prime \prime }\left( x\right) =0,
\]
and $f(x) = \sin(x)$. Moreover, if two analytic functions agree on an open interval, then they agree on $\mathbb{R}$.
\end{proof}
\subsection*{Summary and Conclusions}
We have developed the theorems and identities of basic trigonometry using the definition of the sine function as the solution, expressed as a power series, of a certain second order linear homogeneous differential equation. The key theorems in this study are the Pythagorean Identity, the Sine Sum Identity, and the special value $Q$, which turned out to be $\pi/2$. From these the other identities follow. The interested reader is referred to Landau, chapter 16, in which the sine and cosine functions are developed from a power series definition. In a brief note, Appendix III in Hardy uses the definition of the inverse tangent function as an integral to lead to the definitions of sine, cosine, and their sum laws.
In a future study we plan to consider a generalization of the sine and cosine functions, and show that versions of the Key Theorems still hold in these settings.
\pagebreak
\subsection*{REFERENCES}
\noindent
Hardy, G. H. \ 1921. \textit{A Course of Pure Mathematics, Third Edition}. Mineola, New York:
Dover\\
\indent
Publications, Inc., 2018. An unabridged republication of the work originally published by\\
\indent
the Cambridge University Press, Cambridge.\\
\noindent
Landau, Edmund. 1965. \textit{Differential and Integral Calculus}. Providence, Rhode Island: AMS\\
\indent
Chelsea Publishing. Reprinted by the American Mathematical Society, 2010, from a translation\\
\indent
of Edmund Landau's 1934 \textit{Einf\"{u}hring in die Differentialrechnung und Integralrechnung}.\\
\noindent
Nagle, R. Kent, Edward B. Saff, and Arthur David Snider. 2008. \textit{Fundamentals of
Differential Equations}.\\
\indent
Boston: Pearson Addison Wesley.\\
\noindent
Rudin, Walter. 1964. \textit{Principles of Mathematical Analysis}. New York: McGraw-Hill Book Company.\\
\noindent
Willard, Stephen. 1970. \textit{General Topology}. Reading, Massachusetts: Addison-Wesley Publishing Company.
\pagebreak
\subsection*{Appendix A: Trig Functions of Special Angles}
First we consider $\dfrac{\pi }{4}$.
\[
\sin \left( \dfrac{\pi }{4}\right) =\cos \left( \dfrac{\pi }{2}-\dfrac{\pi }{
4}\right) =\cos \left( \dfrac{\pi }{4}\right)
\]
Since
\[
1=\sin ^{2}\left( \dfrac{\pi }{4}\right) +\cos ^{2}\left( \dfrac{\pi }{4}
\right) =2\sin ^{2}\left( \dfrac{\pi }{4}\right)
\]
we obtain
\[
\sin \left( \dfrac{\pi }{4}\right) =\dfrac{\sqrt{2}}{2}=\cos \left( \dfrac{
\pi }{4}\right)
\]
To find values for $\sin \dfrac{\pi }{6}$, we need the triple-angle identity
\[
\sin \left( 3\theta \right) =3\sin \theta -4\sin ^{3}\theta
\]
This follows from expanding $\sin(2\theta+\theta)$. We can now write
\begin{align*}
1 &=\sin \dfrac{\pi }{2} \\
&=\sin \left( 3\cdot \dfrac{\pi }{6}\right) \\
&=3\sin \dfrac{\pi }{6}-4\sin ^{3}\dfrac{\pi }{6}
\end{align*}
We solve this cubic equation in $\sin \dfrac{\pi }{6}$ to obtain a double
solution $\dfrac{1}{2}$ and single solution $-1$. Because $\sin \dfrac{\pi
}{6}>0$,
\begin{align*}
\sin \dfrac{\pi }{6} &=\dfrac{1}{2}\text{, and} \\
\cos \dfrac{\pi }{6} &=\dfrac{\sqrt{3}}{2}
\end{align*}
Here's a summary table:
\[
\begin{tabular}{c|c|c|c|c|c}
$x$ & $0$ & $\pi /6$ & $\pi /4$ & $\pi /3$ & $\pi /2$ \\ \hline
$\sin x$ & $0$ & $1/2$ & $\sqrt{2}/2$ & $\sqrt{3}/2$ & $1$ \\
$\cos x$ & $1$ & $\sqrt{3}/2$ & $\sqrt{2}/2$ & $1/2$ & $0$ \\
\end{tabular}
\]
\subsection*{Appendix B: Connection to Unit Circle Trigonometry}
\begin{lem}
If $f$ is continuous on $\left[ a,b\right] $ and strictly
increasing on $\left( a,b\right) $, then $f$ is strictly increasing on $
\left[ a,b\right] $.
\end{lem}
\begin{proof}[Proof:\nopunct] We first show that for any $x$ in the interval $\left(
a,b\right) $, we must have $f\left( a\right) <f\left( x\right) $. Assume, to the
contrary, that there exists $c$, $\ a<c<b$, such that $f\left( a\right) \geq
f\left( c\right) $.
Case I: \ $f\left( a\right) >f\left( c\right) $. \ \ Let
\[
\varepsilon =\frac{f\left( c\right) -f\left( a\right) }{2}
\]
Then by right-hand continuity of $f$ at $a$, there exists $\delta $, $
0<\delta <c-a$, such that if $a<x<a+\delta $ then
\[
f\left( a\right) -\varepsilon <f\left( x\right) <f\left( a\right)
+\varepsilon
\]
Then
\begin{align*}
f\left( x\right) &>f\left( a\right) -\frac{f\left( c\right) -f\left(
a\right) }{2} \\
&=\frac{f\left( a\right) +f\left( c\right) }{2} \\
&>\frac{f\left( c\right) +f\left( c\right) }{2} \\
&=f\left( c\right)
\end{align*}
Since $x$, $c$ are both in $\left( a,b\right) $ \ and $x<a+\delta <a+\left(
c-a\right) =c$, we must have $f\left( x\right) <f\left( c\right) $, a
contradiction. Therefore it cannot be the case that $f\left( a\right) \geq
f\left( c\right) $ for some $c\in \left( a,b\right) $.
Case II: \ $f\left( a\right) =f\left( c\right) $. Then consider $\frac{a+c}{2}\in \left( a,c\right) \subset \left( a,b\right) $. Since $f$ is strictly
increasing on $\left( a,b\right) $, it follows that
\[
f\left( \frac{a+c}{2}\right) <f\left( c\right) =f\left( a\right)
\]
and we have the situation of Case I.
A similar argument shows that if $a<x<b$, then $f\left( x\right) <f\left(
b\right) $.
\end{proof}
\begin{thm}
The function $\sin x$ is strictly increasing on $\left[
-\frac{\pi }{2},\frac{\pi }{2}\right] $.
\end{thm}
\begin{proof}[Proof:\nopunct]
By our development and definition of $\frac{\pi }{2}$
as least in $\left[ 0,2\right] $ with $\cos \left( \frac{\pi }{2}\right) =0$
, $\cos x$ is positive on $\left[ 0,\frac{\pi }{2}\right) .$ Since it is an
even function, it is positive on $\left( -\frac{\pi }{2},\frac{\pi }{2}
\right) $ and therefore $\sin x$ is increasing on $\left( -\frac{\pi }{2},
\frac{\pi }{2}\right) $. The function $\sin x$ is differentiable and
therefore continuous at all $x$, and by the lemma must be increasing on $
\left[ -\frac{\pi }{2},\frac{\pi }{2}\right] $.
\end{proof}
We see, then, that the sine function restricted to $\left[ -\frac{\pi }{2},
\frac{\pi }{2}\right] $ is a bijection onto $\left[ -1,1\right] $.
\begin{defn}[Inverse Sine Function]\ The \textit{inverse sine function of $x$}, written here as $\arcsin x$, is defined:
\[
\arcsin x:\left[ -1,1\right] \rightarrow \left[ -\frac{\pi }{2},\frac{\pi }{2
}\right]
\]
is the inverse of the sine function restricted to the domain $\left[ -\frac{\pi }{2},
\frac{\pi }{2}\right] $.
\end{defn}
We now consider the derivative of $\arcsin x$, $-1<x<1$.
\begin{thm}
If $-1<x<1$ then $\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{
1-x^{2}}}$.
\end{thm}
\begin{proof}[Proof:\nopunct]
If $y=\arcsin x$, $-1<x<1$, then $-\frac{\pi }{2}<y<
\frac{\pi }{2}$ and
\[
\sin y=x
\]
Using implicit differentiation,
\[
\cos y\frac{dy}{dx}=1
\]
and
\[
\frac{dy}{dx}=\frac{1}{\cos y}
\]
What is $\cos y=\cos (\arcsin x)$? By the Pythagorean Identity,
\begin{align*}
\cos ^{2}y+\sin ^{2}y &=1 \\
\cos ^{2}y+x^{2} &=1
\end{align*}
and so $\cos y=\sqrt{1-x^{2}}$ or $\cos y=-\sqrt{1-x^{2}}$. Since $-\frac{
\pi }{2}<y<\frac{\pi }{2}$, we have $\cos y>0$. Thus $\cos y=\sqrt{1-x^{2}}$ and
\[
\frac{dy}{dx}=\frac{1}{\sqrt{1-x^{2}}}
\]
\end{proof}
\begin{cor}
If $-1<x<1$, then
\[
\arcsin x=\int_{0}^{x}\frac{1}{\sqrt{1-t^{2}}}dt
\]
\end{cor}
\begin{proof}[Proof:\nopunct]
This follows from the Fundamental Theorem of Calculus.
\end{proof}
Does this equation hold for $x=1$ and for $x=-1$?
\begin{thm}
\[
\int_{0}^{1}\frac{1}{\sqrt{1-t^{2}}}dt=\frac{\pi }{2}
\]
\end{thm}
\begin{proof}[Proof:\nopunct] Let $g\left( x\right) =\displaystyle\int_{0}^{x}\dfrac{dt}{\sqrt{
1-t^{2}}}$, $0\leq x\leq 1$.
What is the value of $g\left( 1\right) $, an improper integral? First, we
know that $g\left( x\right) =\arcsin x$ for $x\in \lbrack 0,1)$ and that
\[
\sin \dfrac{\pi }{4}=\cos \left( \dfrac{\pi }{2}-\dfrac{\pi }{4}\right)
=\cos \left( \dfrac{\pi }{4}\right)
\]
Since $\sin ^{2}x+\cos ^{2}x=1$, we have
\[
\sin \dfrac{\pi }{4}=\cos \dfrac{\pi }{4}=\frac{\sqrt{2}}{2}
\]
Therefore
\[
g\left( \tfrac{\sqrt{2}}{2}\right) =\dfrac{\pi }{4}
\]
We are now ready to find $g\left( 1\right) $.
\begin{align*}
g\left( 1\right) &=\lim_{b\rightarrow 1^{-}}\int_{0}^{b}\frac{dt}{\sqrt{
1-t^{2}}} \\
&=\int_{0}^{\sqrt{2}/2}\frac{dt}{\sqrt{1-t^{2}}}+\lim_{b\rightarrow
1^{-}}\int_{\sqrt{2}/2}^{b}\frac{dt}{\sqrt{1-t^{2}}} \\
&=g\left( \tfrac{\sqrt{2}}{2}\right) +\lim_{b\rightarrow 1^{-}}\int_{\sqrt{
2}/2}^{b}\frac{dt}{\sqrt{1-t^{2}}} \\
&=\dfrac{\pi }{4}+\lim_{b\rightarrow 1^{-}}\int_{\sqrt{2}/2}^{b}\frac{dt}{
\sqrt{1-t^{2}}}
\end{align*}
With the substitution $u=\sqrt{1-t^{2}}$ we obtain for this last integral
\begin{align*}
\lim_{b\rightarrow 1^{-}}\int_{\sqrt{2}/2}^{b}\frac{dt}{\sqrt{1-t^{2}}}
&=\lim_{b\rightarrow 1^{-}}\int_{\sqrt{2}/2}^{\sqrt{1-b^{2}}}\dfrac{1}{u}
\cdot \dfrac{-u}{\sqrt{1-u^{2}}}du \\
&=-\lim_{b\rightarrow 1^{-}}\int_{\sqrt{2}/2}^{\sqrt{1-b^{2}}}\dfrac{1}{
\sqrt{1-u^{2}}}du \\
&=-\int_{\sqrt{2}/2}^{0}\dfrac{1}{\sqrt{1-u^{2}}}du \\
&=\int_{0}^{\sqrt{2}/2}\frac{du}{\sqrt{1-u^{2}}} \\
&=g\left( \tfrac{\sqrt{2}}{2}\right) \\
&=\frac{\pi }{4}
\end{align*}
so that
\[
g\left( 1\right) =g\left( \tfrac{\sqrt{2}}{2}\right) +g\left( \tfrac{\sqrt{2}
}{2}\right) =\dfrac{\pi }{2}
\]
\end{proof}
\begin{cor} $\displaystyle\int_{0}^{-1}\frac{1}{\sqrt{1-t^{2}}}dt=-\frac{\pi }{2}$
\end{cor}
\begin{proof}[Proof:\nopunct]
Since $\frac{1}{\sqrt{1-t^{2}}}$ is an even function,
for $0\leq a<1$, we have
\[
\int_{-a}^{0}\frac{1}{\sqrt{1-t^{2}}}dt=\int_{0}^{a}\frac{1}{\sqrt{1-t^{2}}}
dt
\]
and
\begin{align*}
\int_{0}^{-1}\frac{1}{\sqrt{1-t^{2}}}dt &=-\int_{-1}^{0}\frac{1}{\sqrt{
1-t^{2}}}dt \\
&=-\lim_{a\rightarrow 1^{-}}\int_{-a}^{0}\frac{1}{\sqrt{1-t^{2}}}dt \\
&=-\lim_{a\rightarrow 1^{-}}\int_{0}^{a}\frac{1}{\sqrt{1-t^{2}}}dt \\
&=-\frac{\pi }{2}
\end{align*}
\end{proof}
We now complete the connection to unit circle trigonometry.
\begin{thm}If $-1\leq a\leq b\leq 1$ then the arc length of the graph
of $y=\sqrt{1-x^{2}}$ from $x=a$ to $x=b$ is \[\arcsin b-\arcsin a.\]
\end{thm}
\begin{proof}[Proof:\nopunct] First, note that
\[
\frac{d}{dx}\sqrt{1-x^{2}}=-\frac{x}{\sqrt{1-x^{2}}}
\]
Using the arc length formula and the previous result,
\begin{align*}
\int\nolimits_{a}^{b}\sqrt{1+\left( \frac{dy}{dx}\right) ^{2}}
&=\int_{a}^{b}\sqrt{1+\left( -\frac{x}{\sqrt{1-x^{2}}}\right) ^{2}}\enspace dx \\
&=\int_{a}^{b}\sqrt{1+\frac{x^{2}}{1-x^{2}}}\enspace dx \\
&=\int_{a}^{b}\sqrt{\frac{1}{1-x^{2}}}\enspace dx \\
&=\int_{a}^{b}\frac{1}{\sqrt{1-x^{2}}}\enspace dx \\
&=\arcsin b-\arcsin a
\end{align*}
\end{proof}
In the particular case that $b=1$, we have that the arc length $s$ along
the upper unit circle from $x=a$ to $x=1$ is
\[
s=\frac{\pi }{2}-\arcsin a
\]
Then
\begin{align*}
\cos s &=\cos \left( \frac{\pi }{2}-\arcsin a\right) \\
&=\sin \left( \arcsin a\right) \\
&=a
\end{align*}
and
\begin{align*}
\sin s &=\sin \left( \frac{\pi }{2}-\arcsin a\right) \\
&=\cos \left( \arcsin a\right) \\
&=\sqrt{1-a^{2}}
\end{align*}
This shows that a point $P\left( a,\sqrt{1-a^{2}}\right) $ on the upper unit
circle with $-1\leq a\leq 1$ has coordinates $P\left( \cos s,\sin s\right)$
where $s$ is the arc length along the upper unit circle from the point $P$ to $
A\left( 1,0\right) $. This arc length is the definition of the radian
measure of angle $AOP$ where $O=\left( 0,0\right) $.
The connection from geometry-free trigonometry to unit circle trigonometry is complete.
\end{document} |
\begin{document}
\title{Pruning has a disparate impact on model accuracy}
\begin{abstract}
Network pruning is a widely-used compression technique that is able to significantly scale down overparameterized models with minimal loss of accuracy. This paper shows that pruning may create or exacerbate disparate impacts. The paper sheds light on the factors to cause such disparities, suggesting differences in gradient norms and distance to decision boundary across groups to be responsible for this critical issue. It analyzes these factors in detail, providing both theoretical and empirical support, and proposes a simple, yet effective, solution that mitigates the disparate impacts caused by pruning.
\end{abstract}
\section{Introduction}
\label{sec:intro}
As deep learning models evolve and become more powerful, they also
become larger and more costly to store and execute. The trend hinders
their deployment in resource-constrained platforms, such as embedded
systems or edge devices, which require efficient models in time and
space. To address this challenge, studies have developed a variety
of techniques to prune the relatively insignificant or insensitive
parameters from a neural network while ensuring competitive
accuracy \cite{aghli2021combining,
baykal2019sipping,
blalock2020state,
Renda2020Comparing,
Han2015NIPS,sehwag2019compact,
zhang2018systematic}.
When a model needs to be developed to
fit given and certain requirements in size and resource consumption,
a pruned model which is derived from a large, rigorously-trained, and
(often) over-parameterized model, is regarded as a de-facto standard.
That is because it performs incomparably better than a same-size dense
model which is trained from scratch, when the same amount of
effort and resources are invested.
In spite of its strengths, pruning has been showed to induce or
exacerbate disparate effects in the accuracy of the resulting
reduced models \cite{Hooker2020CharacterisingBI, Hooker2020WhatDC}.
Intuitively, the removal of model weights affects the process
in which the network separates different classes, which can have
contrasting consequences for different groups of individuals.
This paper further shows that the accuracy of the pruned models tends
to increase (decrease) more in classes that had already
high (low) accuracy in the original model, leading to a
``the rich get richer'' and ``the poor get poorer'' effect.
This \emph{Matthew} effect is illustrated in Figure \ref{fig:motivation}.
The figure shows the accuracy of a facial recognition task on
different demographic groups for several pruning rates (indicating
the percentage of parameters removed from the original models).
Notice how the accuracy of the majority group (White) tends to
increase while that of the minority groups tends to decrease as the
pruning ratio increases.
\begin{figure*}
\caption{Accuracy of each demographic group in the UTK-Face dataset using Resnet18 \cite{he2016deep}
\label{fig:motivation}
\end{figure*}
Following these observations, we shed light on the factors to
cause such disparities. The theoretical findings suggest the presence
of two key factors responsible for why accuracy disparities arise
in pruned models:
{\bf(1)} disparity in \emph{gradient norms} across groups, and {\bf
(2)} disparity in \emph{Hessian matrices} associated with the loss function computed using a group's data.
Informally, the former carries information about the groups' local optimality,
while the latter relates to model separability.
We analyze these factors in detail, providing both theoretical and
empirical support on a variety of settings, networks, and datasets.
By recognizing these factors, we also develop a simple yet
effective training technique that largely mitigates the disparate impacts caused by pruning. The method is based on an alteration of the loss
function to include components that penalize disparity of the average
gradient norms and distance to decision boundary across groups.
These findings are significant: {\em Pruning is a key enabler for
neural network models in embedded systems with deployments in
security cameras and sensors for autonomous devices for applications
where fairness is an essential need.
Without careful consideration of the fairness impact of these
techniques, the resulting models can have profound effects on our
society and economy}.
\subsubsection*{Related work}
Fairness and network pruning have been long studied in isolation.
The reader is referred to the related papers and surveys on fairness
\citep{
barocas2017fairness,caton2020fairness,dwork2012fairness,NIPS2016_9d268236,mehrabi2021survey} and
pruning \citep{
aghli2021combining,baykal2019sipping,blalock2020state,
Renda2020Comparing,Han2015NIPS,sehwag2019compact,Wiebke2022,zhang2018systematic}
for a review on these areas.
The recent interest in assessing societal values of machine learning models
has seen an increase of studies at the intersection of different properties
of a learning model and their effects on fairness. For example, \citet{xu2021robust}
studies the setting of adversarial robustness and show that adversarial training
introduces unfair outcomes in term of accuracy parity \cite{zhao2019inherent}.
\citet{zhu2021rich} show that semisupervised settings can introduce unfair outcomes
in the resulting accuracy of the learned models.
Finally, several authors have also shown that private training can have
unintended disparate impacts to the resulting models' outputs \cite{NEURIPS2019_eugene,fioretto:arxiv22b,
Fioretto:NeurIPS21b,Uniyal2021DPSGDVP,Zhu:AAAI2021}
and downstream decisions \cite{pujol:20,Tran:IJCAI21}.
Network compression has also been shown to have a profound impact towards
the model fairness. For example, several works observed empirically that
network compression may amplify unfairness in different learning tasks~\cite{paganini2020prune,
Hooker2020WhatDC, Hooker2020CharacterisingBI, Joseph2020GoingBC}.
Most of the focus has been on vision tasks and in identifying the
set of \emph{Pruning Identified Exemplars} (PIEs), the samples that are
impacted most under the compression scheme and conclude that PIEs belongs
to low frequency groups (those observed at the tail of the data distribution).
\citet{Blakeney2021SimonSE} further investigate how bias could be evaluated
and mitigated in pruned neural networks using knowledge distillation while
\citet{hosseinilearning} observed empirically that knowledge distillation
processes may produce unfair student models.
The impact of network compression towards fairness has also been assessed
in natural language tasks. For example, \citet{Du2021WhatDC} and
\citet{xu-etal-2021-beyond} empirically measure the robustness of
compressed large language models, while \citet{Ahia2021TheLD} look
into how compression schemes affects data-limited regimes. Finally,
\citet{Xu2022CanMC} investigate ways to improve fairness in generative
language models by compressing them.
We also note that, concurrently to this work, \citet{good:22} studied
the relative distortion in recall for various classes. They show that pruning
has a Matthews effect on the recall for various classes and
propose an algorithm to attenuate such an effect.
This paper builds on this body of work and their important empirical
observations and provides a step towards a deeper theoretical understanding
of the fairness issues arising as a result of pruning. It derives conditions
and studies the causes of unfairness in the context of pruning as well
as it introduces mitigating guidelines.
\section{Problem settings and goals}
The paper considers datasets $D$ consisting of $n$ datapoints
$(\bm{x}_i, a_i, y_i)$, with $i \in [n]$, drawn i.i.d.~from an unknown
distribution $\Pi$. Therein, $\bm{x}_i \in \cX$ is a feature vector,
$a_i \in \cA$ with $\cA = [m]$ (for some finite $m$) is a demographic group
attribute, and $y_i \in \cY$ is a class label. For example, consider the
case of a face recognition task. The training example feature $\bm{x}_i$ may
describe a headshot of an individual, the protected attribute $a_i$
may describe the individual's gender or ethnicity, and $y_i$ represents
the identity of the individual.
The goal is to learn a predictor $f_{\bm{\theta}} : \cX \to \cY$, where ${\bm{\theta}}$
is a $k$-dimensional real-valued vector of parameters that minimizes
the empirical risk function:
\begin{align}
\label{eq:erm}
\optimal{{\bm{\theta}}} = \argmin_{\bm{\theta}} J(\bm{{\bm{\theta}}}; D) =
\frac{1}{n} \sum_{i=1}^n \ell(f_{\bm{\theta}}(\bm{x}_i), y_i),
\end{align}
where $\ell: \cY \times \cY \to \RR_+$ is a non-negative \emph{loss function}
that measures the model quality.
We focus on analyzing properties arising when extracting a small model $f_{\bar{{\bm{\theta}}}}$ with $\bar{{\bm{\theta}}} \subset\, \optimal{{\bm{\theta}}}$ of size $|\bar{{\bm{\theta}}}| = \bar{k} \ll k$.
Model $f_{\bar{{\bm{\theta}}}}$ is constructed by pruning the least important
values or filters from vector $\optimal{{\bm{\theta}}}$ (i.e., those with smaller
values in magnitude) according to a prescribed criterion, such as an $\ell_p$
norm \cite{NIPS1988_07e1cd7d, Han2015NIPS}.
The paper focuses on understanding the fairness impacts (as defined next)
arising when pruning general classifiers, such as neural networks.
\paragraph{Fairness}
The fairness analysis focuses on the notion of \emph{excessive loss},
defined as the difference between the original and the pruned risk
functions over some group $a \in \cA$:
\begin{equation}
\label{eq:2}
R(a) = J(\bar{{\bm{\theta}}}; D_a) - J(\optimal{{\bm{\theta}}}; D_a),
\end{equation}
where $D_a$ denotes the subset of the dataset $D$ containing samples
($\bm{x}_i, a_i, y_i$) whose group membership $a_i = a$.
Intuitively, the excessive loss represents the change in loss (and thus, in
accuracy) that a given group experiences as a result of pruning.
Fairness is measured in terms of the maximal \emph{excessive loss difference},
also referred to as \emph{fairness violation}:
\begin{equation}
\label{eq:3}
\xi(D) = \max_{a, a' \in \cA} |R(a) - R(a')|,
\end{equation}
defining the largest excessive loss difference across all protected groups.
(Pure) fairness is achieved when $\xi(D) = 0$, and thus
a fair pruning method aims at minimizing the excessive loss difference.
The goal of this paper is to shed light on why fairness issues arise
(i.e., $ R(a) > 0)$ as a result of pruning, why some groups suffer more than others (i.e., $R(a) > R(a'))$, and what mitigation measures could be taken
to minimize unfairness due to pruning.
We use the following notation: variables are denoted by calligraph
symbols, vectors or matrices by bold symbols, and sets by uppercase symbols.
Finally, $\| \cdot \|$ denotes the Euclidean norm and we use
$f_{{\bm{\theta}}}(\bm{x})$ to refer to the model' \emph{soft} outputs.
All proofs are reported in Appendix~\ref{sec:missing_proofs}.
\iffalse
To encourage accuracy parity fairness of the classifier, we can add the following equality constraints to the objective function in Equation \ref{eq:erm}.
\begin{equation}
J(\mathcal{D}_a, {\bm{\theta}}) = J(\mathcal{D}, {\bm{\theta}}) \, \forall a \in \cA
\end{equation}
The above fairness constraint requires the average classification loss per group $a$ is similar with that's at population level. To learn the fair classifier ${\bm{\theta}}_f$ under such fairness constraint, we can employ the Dual Lagrangian framework \cite{fioretto2020lagrangian}. First, the Lagrangian loss of the fair problem can be written as follows:
\begin{equation}
\cL(\theta, \lambda, \cD) = J(\cD, \theta) + \sum_{a \in \cA} \lambda_a | J(\cD, \theta) - J(\cD, \theta),
\end{equation}
where $\lambda_a \geq 0, \forall a \in \cA$ are the multipliers. The training process consists of primal and dual update steps.
In the primal update steps, we are updating the classifiers' parameters $\theta$ given fixed multipliers $\lambda$.
\begin{equation}
\theta^*(\lambda) = \mbox{argmin}_{\theta} \cL(\theta, \lambda, \cD)
\end{equation}
In the dual update steps, we are updating the multipliers $\lambda$ given fixed model parameters $\theta$.
\begin{equation}
\lambda^*(\theta) = \mbox{argmax}_{\lambda} \cL(\theta, \lambda, \cD)
\end{equation}
Both the primal and dual update steps can follow the traditional SGD training scheme \cite{fioretto2020lagrangian}.
\paragraph{Network pruning} In the scope of this paper, we focus on single shot network pruning.[CITE] This network pruning scheme consists of the following three steps:
\begin{enumerate}
\item \textbf{Standard training} In the first step, we perform standard training as in Equation \ref{eq:erm} to obtain optimal parameters $\theta^*$.
\item \textbf{Pruning} In the second step, we prune/remove the least important parameters/filters based on some criteria (e.g., $l_1$/$l_2$ norm based) from $\theta_*$ to obtain $\bar{\theta}$.
\item \textbf{Fine-tuning} In the last step, we fine-tune the pruned parameter $\bar{\theta}$ to obtain parameter $\tilde{\theta}$
\end{enumerate}
We are interested in quantifying the disparate impact on accuracy of using the $\tilde{\theta}$ compared to the before pruning parameter $\theta_*$ to different classes.
\fi
\section{Fairness analysis in pruning: Roadmap}
\label{sec:pruning_impact}
To gain insights on how pruning may introduce unfairness, we start with providing a useful upper bound for a group's excessive loss. Its goal is to isolate key aspects of model pruning that are responsible for the observed unfairness. The following discussion assumes the loss function $\ell(\cdot)$ to be at least twice differentiable, which is the case for common ML loss functions, such as mean squared error or cross entropy loss.
\begin{theorem}
\label{thm:taylor}
The \emph{excessive loss} of a group $a \in \cA$ is upper bounded by\footnote{
With a slight abuse of notation, the results refer to $\bar{{\bm{\theta}}}$ as the homonymous vector which is extended with $k-\bar{k}$ zeros.
}:
\begin{align}
R(a) \leq
\left\| \bm{g}_a^{\ell} \right\|
\times \left\| \bar{{\bm{\theta}}} - \optimal{{\bm{\theta}}}\right\|
+
\frac{1}{2} \lambda \left( \bm{H}_{a}^{\ell} \right)
\times
\left\| \bar{{\bm{\theta}}} - \optimal{{\bm{\theta}}}\right\|^2
+
\cO\left( \left\|\bar{{\bm{\theta}}} - \optimal{{\bm{\theta}}} \right\|^3 \right),
\label{eq:thm1}
\end{align}
where
$\bm{g}_a^\ell = \nabla_{{\bm{\theta}}} J( \optimal{{\bm{\theta}}}; D_{a})$ is the vector of gradients associated with the loss function $\ell$ evaluated at $\optimal{{\bm{\theta}}}$ and computed using group data $D_a$,
$\bm{H}_{a}^{\ell} =
\nabla^2_{{\bm{\theta}}} J(\optimal{{\bm{\theta}}}; D_{a})$ is the Hessian matrix of the loss function $\ell$, at the optimal parameters vector $\optimal{{\bm{\theta}}}$, computed using the group data $D_a$ (henceforth simply referred to as \emph{group hessian}), and
$\lambda(\Sigma)$ is the maximum eigenvalue of a matrix $\Sigma$.
\end{theorem}
The bound above follows from a second order Taylor expansion of the loss function, Cauchy-Schwarz inequality, and properties of the Rayleigh quotient.
Notice that, in addition to the difference in the original and pruned parameters vectors, two key terms appear in Equation \eqref{eq:thm1}:
{\bf (1)} The norms of the gradients $\bm{g}_a^\ell$ and {\bf (2)} the maximum eigenvalue of the Hessian matrix $\bm{H}_a^\ell$ for a group $a$. Informally, the former is associated with the groups' local optimality while the latter relates to the ability of the model to separate the groups data.
As we will show next these components represent the main sources of unfairness due to model pruning.
The following is an important corollary of Theorem \ref{thm:taylor}. It shows that the larger the pruning, the larger will be the excessive loss for a given group.
\begin{corollary}
\label{cor:1}
Let $\bar{k}$ and $\bar{k}'$ be the size of parameter vectors $\bar{{\bm{\theta}}}$ and $\bar{{\bm{\theta}}}'$, respectively, resulting from pruning model $f_{\optimal{{\bm{\theta}}}}$, where $\bar{k} < \bar{k}'$ (i.e., the former model prunes more weight than the latter one). Then, for any group $a \in \cA$,
\begin{equation}
\tilde{R}(a, \bar{{\bm{\theta}}}) \geq \tilde{R}(a, \bar{{\bm{\theta}}}'),
\end{equation}
where $\tilde{R}(a, \bm{\omega})$ is the excessive loss upper bound computed using pruned model parameters $\bm{\omega}$ (Eq.~\eqref{eq:thm1}).
\end{corollary}
The corollary above indicates that the excess risk for a group increases as the pruning regime increase. Building on this result, the paper illustrates next why unfairness can become more significant as the pruning regime increases.
The next sections analyze the effect of gradient norms and the Hessian to unfairness in the pruned models. The theoretical claims are supported and complemented by analytical results. These results use the UTKFace dataset \cite{zhifei2017cvpr}
for a vision task whose goal is to classify ethnicity. The experiments use a ResNet-18 architecture and the pruning counterparts remove the $P\%$ parameters with the smallest absolute values for various $P$. All reported metrics are normalized and an average of 10 repetitions.
While the theoretical analysis focuses on the notion of disparate impacts under the lens of excessive loss, the empirical results report differences in accuracy of the resulting models. The empirical results thus reflect the setting commonly adopted when measuring accuracy parity \cite{zhao2019inherent}.
We report a glimpse of the empirical results, with the purpose of supporting the theoretical claims, and extended experiments, as well as additional descriptions of the datasets and settings, are reported in Appendix \ref{sec:additional_results}.
\input{gradientCpnt2}
\section{Why disparity in groups' Hessians causes unfairness?}
\label{sec:hessian_analysis}
Having examined the properties of the groups gradients and their
relation to unfairness in pruning, this section turns on analyzing
how the
Hessian associated with the loss function for a group is linked to the unfairness observed during pruning. In more detail, it connects the groups' Hessian to the distance to the decision boundary for the samples in that group and their resulting model errors (Theorem \ref{thm:hessian_norm_bound}), it illustrates a strong positive correlation between groups' Hessian and gradient norms, and links these concepts with the excessive loss (Theorem \ref{thm:taylor}) to show that unfairness in model pruning is controlled by the difference in maximum eigenvalues of the Hessians among groups.
\paragraph{Group Hessians and accuracy.}
The section first shows that groups presenting large Hessian values may suffer larger disparate impacts due to pruning, when compared with groups that have smaller Hessians. It does so by connecting the maximum eigenvalues of the groups Hessians with their distance to decision boundary and the group accuracy. The following result sheds light on these observations. It restricts its attention to models trained under binary cross entropy losses, for clarity of explanation, although an extension to a multi-class case is directly attainable.
\begin{theorem}
\label{thm:hessian_norm_bound}
Let $f_\theta$ be a binary classifier trained using a binary cross entropy loss. For any group $a \in \cA$, the maximum eigenvalue of the group Hessian $\lambda(\bm{H}_a^{\ell})$ can be upper bounded by:
\begin{equation}
\lambda(\bm{H}_a^{\ell}) \leq \frac{1}{|D_a|}
\sum_{(\bm{x}, y)\in D_a}
\underbrace{
\left( {f}_{\optimal{\bm{\theta}}}(\bm{x}) \right)
\left( 1 - {f}_{\optimal{\bm{\theta}}}(\bm{x}) \right)}_{\textit{Closeness to decision boundary}}
\times
\left\| \nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm {x}) \right\|^2
+
\underbrace{\left| f_{\optimal{\bm{\theta}}}(\bm{x}) - y \right|}_{\textit{Error}}
\times
\lambda\left( \nabla^2_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right).
\label{eq:hessian_norm_bound}
\end{equation}
\end{theorem}
The proof relies on derivations of the Hessian associated with model loss function and Weyl inequality.
In other words, Theorem \ref{thm:hessian_norm_bound} highlights a direct connection between the maximum eigenvalue of the group Hessian and {\bf (1)} the closeness to the decision boundary of the group samples, and {\bf (2)} the accuracy of the group. The distance to the decision boundary is derived from \cite{cohen2019certified}. Intuitively this term is maximized when the classifier is highly uncertain about the prediction: ${f}_{\optimal{\bm{\theta}}}(\bm{x}) \to 0.5$, and minimized when
it is highly certain ${f}_{\optimal{\bm{\theta}}}(\bm{x}) \to 0$ or $1$, as showed in the following proposition.
\begin{proposition}
\label{prop:dist_bndry}
Consider a binary classifier $f_{{\bm{\theta}}}(\bm{x}) $. For a given sample $\bm{x} \in D $, the term ${f}_{\optimal{\bm{\theta}}}(\bm{x}) (1 - {f}_{\optimal{\bm{\theta}}}(\bm{x}) )$ is maximized when ${f}_{\optimal{\bm{\theta}}}(\bm{x}) = 0.5$ and minimized when ${f}_{\optimal{\bm{\theta}}}(\bm{x}) \in \{0,1\} $.
\end{proposition}
\begin{wrapfigure}[10]{r}{143pt}
\centering
\includegraphics[width=\linewidth]{figures/H_and_acc.pdf}
\caption{\small Group Hessians, distance to decision boundary, and~accuracy.}
\label{fig:H_vs_acc}
\end{wrapfigure}
Observe that a group consisting of samples that are far from the decision boundary will have smaller Hessians and, thus, be less subject to a drop in accuracy due to model pruning. These results can be appreciated in Figure \ref{fig:H_vs_acc}. Notice the inverse relationship between maximum eigenvalues of the groups' Hessians and their average distance to the decision boundary. The same relation also holds for accuracy: the higher the Hessians maximum eigenvalues, the smaller the accuracy.
This is intuitive as samples which are close to the decision boundary
will be more prone to errors due to small changes in the model due
to pruning, when compared with samples lying far from the decision boundary.
\paragraph{Correlation between group Hessians and gradient norms.}
This section observes a positive correlation between maximum eigenvalues of the Hessian of a group and their gradient norms. This relation can be appreciated in Figure \ref{fig:H_vs_grad}.
While mainly empirical, this observation is important as it illustrates that both the Hessian $\lambda(\bm{H}_a^\ell)$ and the gradient $\| \bm{g}_a^\ell \|$ terms appearing in the upper bound of the excessive loss $R(a)$ reported in Theorem \ref{thm:taylor} are in agreement. This relation was observed in all our experiments and settings.
Such observation allows us to infer that it is the combined
effect of gradient norms and group Hessians that is responsible for
the excessive loss of a group and, in turn, for the exacerbation of
unfairness in the pruned models.
\begin{wrapfigure}[11]{r}{130pt}
\centering
\includegraphics[width=\linewidth]{figures/H_and_grads.pdf}
\caption{\small Group Hessians and~gradient norms.}
\label{fig:H_vs_grad}
\end{wrapfigure}
\paragraph{The role of the group Hessian in pruning.}
Having highlighted the connection between Hessian for a group with the resulting accuracy of the model on such a group, this section provides theoretical intuitions on the role of the Hessians in the disparate group losses during pruning.
In Theorem \ref{thm:taylor}, notice that the excessive loss is controlled by term $\|\bm{H}_a^\ell \| \times \| \bar{\bm{\theta}} - \optimal{\bm{\theta}} \|^2$.
As also noted in the previous section, the term $\| \bar{\bm{\theta}} - \optimal{\bm{\theta}} \|$
regulates the impact of pruning on the excessive loss as the difference between the pruned and non-pruned parameter vectors directly depends on the pruning rate. Similar to the observation for gradient norms, with a fixed pruning rate, groups with different Hessians will have a disparate effect on the resulting term. In particular, groups with small Hessians eigenvalues (those generally distant from the decision boundary and highly accurate) will be less sensitive to the effects of the pruning rate. Conversely, groups with large Hessians eigenvalues will be affected by the pruning rate to a greater extent, \emph{typically} resulting in larger excessive losses.
These observations can further be appreciated empirically in Figures~\ref{fig:global_pruning_accuracy_avg} (for accuracy) and \ref{fig:nonnormal_hessian} (for maximum group Hessian eigenvalues) on the UTKFace datasets for a variety of pruning rates.
\section{Mitigation solution and evaluation}
\label{sec:mitigation}
The previous sections highlighted the presence of two key factors playing a role in the observed model accuracy disparities due to pruning: the difference in gradient norms, and the difference in Hessians losses across groups. This section first shows how to leverage these findings to provide a simple, yet effective solution to reduce the disparate impacts of pruning. Then, the section illustrates the benefits of this mitigating solution on a variety of tasks, datasets, and network architectures.
\subsection{Mitigation solution}
To achieve fairness, the aforementioned findings suggest to equalize the disparity associated with gradient norms $\| \bm{g}_a^\ell \|$ and Hessians
$\lambda(\bm{H}_a^\ell)$ across different groups $a \in \cA$. For this goal, we adopt a constrained empirical risk minimization approach:
\begin{align}
\label{eq:LD_slow}
\minimize{\bm{\theta}}\;\; J(\bm{\theta}; D)
\quad
\text{such that:}\;\; \| \bm{g}^\ell_a\| = \|\bm{g}^\ell\|, \;\;
\lambda(\bm{H}_a^\ell) = \lambda(\bm{H}^\ell) \;\; \forall a \in \cA,
\end{align}
where $\bm{g}^\ell = \nabla_{\bm{\theta}} J({\bm{\theta}}; D)$ and $\bm{H}^\ell = \nabla^2_{\bm{\theta}} J({\bm{\theta}}; D)$ refer to the gradients and Hessian associated with loss function $\ell$, respectively, and are computed using the whole dataset $D$. The approach \eqref{eq:LD_slow} is a common strategy adopted in fair learning tasks, and
the paper uses the Lagrangian Dual method of \citet{fioretto2020lagrangian} which exploits Lagrangian duality to extend the loss function with trainable and weighted regularization terms that encapsulate constraints violations (see Appendix \ref{sec:additional_results} for additional details).
A shortcoming of this approach is, however, that requires computing the gradient norms and Hessian matrices of the group losses in each and every training iteration, rendering the process computationally unviable, especially for deep, overparametrized networks.
To overcome this computational burden, we will use two observations made earlier in the paper. First, recall the strong relation between gradient norms for a group and their associated losses. This aspect was noted in Proposition \ref{thm:acc_vs_grad}. That is, when the losses across the groups are similar, the gradient norms across such groups will also tend to be similar. Next, Theorem \ref{thm:hessian_norm_bound} noted a positive correlation between model errors (and thus loss values) for a group and its associated Hessian eigenvalues. Thus, when the losses across the groups are similar, the group Hessians will also tend to be similar. This intuition is also complemented by the strong correlation between group Hessians and gradient norms reported in Section \ref{sec:hessian_analysis}.
Based on the above observations, we propose a simpler version of the constrained minimizer \eqref{eq:LD_slow} defined as
\begin{align}
\label{eq:LD_fast}
\minimize{\bm{\theta}}\;\; J(\bm{\theta}; D)
\quad
\text{such that:}\;\; J(\bm{\theta}; D_a) = J(\bm{\theta}; D)
\;\; \forall a \in \cA,
\end{align}
that substitutes the gradient norms and max eigenvalues of group Hessians equality constraints with proxy terms capturing the group $J({\bm{\theta}}; D_a)$ and population $J({\bm{\theta}}; D)$ losses.
\begin{wrapfigure}[17]{r}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/fair_models_grad_hessian_UTK-ethnicity.pdf}
\caption{Effects of fairness constraints in balancing not only group accuracy (left) but also gradient norms (middle) and group average distance to the decision boundary (right).}
\label{fig:example_mitigation}
\end{wrapfigure}
The impact of such proxy terms in the fairness-constrained problem above can be appreciated, empirically, in Figure \ref{fig:example_mitigation}. The plots, that use the UTK-Face dataset, with Ethnicity as protected group, show an original unfair model (top) and a fair counterpart obtained through problem \eqref{eq:LD_fast} (bottom).
Both top and bottom sub-figures use an unpruned model. The top sub-figure shows the performance of an original unpruned model trained by minimizing the empirical risk function while the bottom one shows the effect of solving Problem \eqref{eq:LD_fast}, i.e., it constrains the empirical risk function with the various group loss terms.
Notice how enforcing balance in the group losses also helps reducing and balancing the gradient norms and group's average distance to the decision boundary. As a consequence, the resulting model fairness is dramatically enhanced (bottom-left subplot).
\subsection{Assessment of the mitigation solution}\label{sec:AssmtMitigation}
\paragraph{Datasets, models, and settings.}
This section analyzes the results obtained using the proposed mitigation solution with ResNet50 and VGG19 on the UTKFace dataset \cite{zhifei2017cvpr}, CIFAR-10~\cite{cifar10}, and SVHN~\cite{Netzer2011ReadingDI} for various protected attributes. The experiments compare the following four models:
$\bullet$ {\sl No Mitigation}: it refers to the standard pruning approach which uses no fairness mitigation strategy.\\
$\bullet$ {\sl Fair Bf Pruning}: it applies the fairness mitigation process (Problem \eqref{eq:LD_fast}) exclusively to the original large network, thus \emph{before} pruning.\\
$\bullet$ {\sl Fair Aft Pruning}: it applies the mitigation exclusively to the pruned network, thus \emph{after} pruning. \\
$\bullet$ {\sl Fair Both}: it applies the mitigation both to the original large network and to the pruned network.
The experiments report the overall accuracy of resulting models as well as their fairness violations, defined here as the difference between the maximal and minimal group accuracy. The reported metrics are the average of $10$ repetitions. Additional details on datasets, architectures, and hyper-parameters adopted, as well as additional and extended results are reported in Appendix \ref{sec:additional_results}.
\begin{figure*}
\caption{Accuracy and Fairness violations attained by all models on ResNet50, UTK-Face dataset with {\sl ethnicity}
\label{fig:mitigation_pruning-1}
\end{figure*}
\begin{figure*}
\caption{Accuracy and Fairness violations attained by all models on VGG-19, CIFAR-10 dataset (left) and SVHN (right) with 10 class labels also used as group attribute.}
\label{fig:mitigation_pruning-2}
\end{figure*}
\paragraph{Effects on accuracy.}
The section first focuses on analyzing the effects of accuracy drop due to applying the proposed mitigation solution for fair pruning. Figure \ref{fig:mitigation_pruning-1} compares the four models on the UTK-Face dataset using a ResNet50 architecture. The left subplots use {\sl ethnicity} as protected group and class label, with $|\cY| = 5$, while the right subplots use {\sl age} as protected group and class label, with $|\cY| = 9$. Notice that, as expected, all compared models present some accuracy deterioration as the pruning rate increases. However, notably, the deterioration of the models that apply the fair mitigation steps are comparable to (or even improved) those of the "{\sl No mitigation}" model, which applies standard pruning.
A similar trend can be seen in Figure \ref{fig:mitigation_pruning-2} that reports results on CIFAR (left) and SVHN (right). Both use the ten class labels as protected attributes. These results clearly illustrate the ability of the mitigating solution to preserve highly accurate models.
A comparison of the ``full'' (Equation \ref{eq:LD_slow}) and ``relaxed'' (Equation \ref{eq:LD_fast}) versions of the proposed mitigation solutions is
provided in Table~\ref{tab:full_relaxed_results}. We note that while the "full" version leads to fairer results, the reduction in the various groups accuracy is often insubstantial. We also note that the running time of the "full" version is largely (over an order magnitude) longer than the relaxed counterpart. This is due to the calculation of gradient norms and the Hessian terms associated with each group.
\begin{table}[t]
\centering
\resizebox{0.85\columnwidth}{!}{
\begin{tabular}{@{}rr@{\hspace{10pt}}ll@{}}
\toprule
{\bf Dataset} & {\bf version} & {\bf Class-wise accuracy} & {\bf Overall accuracy} \\
\midrule
\multirow{2}{*}{\bf UTK-age bins} & {\bf full} & 0.856, 0.128, 0.145, 0.319, 0.331, 0.342, 0.181, 0.334, 0.512 & 0.395 \\
& {\bf relaxed} & 0.810, 0.096, 0.141, 0.284, 0.385, 0.324, 0.227, 0.257, 0.533 & 0.390 \\[0.25em]
\multirow{2}{*}{\bf UTK-gender} & {\bf full} & 0.830, 0.876 & 0.857\\
& {\bf relaxed} & 0.868, 0.845 & 0.852\\[0.25em]
\multirow{2}{*}{\bf SVHN} & {\bf full} & 0.864, 0.911, 0.869, 0.819, 0.887, 0.784, 0.840, 0.877, 0.805, 0.856 & 0.857\\
& {\bf relaxed} & 0.824, 0.910, 0.775, 0.726, 0.827, 0.752, 0.747, 0.789, 0.713, 0.755 & 0.795\\[0.25em]
\multirow{2}{*}{\bf MNIST} & {\bf full} & 0.998, 0.996, 0.993, 0.998, 0.994, 0.991, 0.991, 0.993, 0.992, 0.985 & 0.993\\
& {\bf relaxed} & 0.994, 0.988, 0.989, 0.986, 0.987, 0.979, 0.981, 0.988, 0.969, 0.994 & 0.986\\
\bottomrule
\end{tabular}
}
\caption{Full (Equation \ref{eq:LD_slow}) vs relaxed (Equation \ref{eq:LD_fast}) versions of the proposed mitigation solutions.}
\label{tab:full_relaxed_results}
\end{table}
\paragraph{Effects on fairness.}
The section next illustrates the ability of the proposed solution to achieve fair pruned models.
Table \ref{tab:protected_attr_results} illustrates the results for the UTKFaces dataset with ethnicity as class labels and age as
protected attributes for a CNN with two convolutional layers and three linear layers and prune amounts: 30\%, 50\%, 70\%, and 90\%.
Notice how Fair Aft pruning and Fair both achieve relatively lesser fairness violations compared to the No mitigation and
the Fair bf Pruning methods in most cases.
\begin{table}[t]
\centering
\resizebox{0.6\textwidth}{!}{
\begin{tabular}{@{}r@{\hspace{10pt}} l@{\hspace{5pt}}l@{\hspace{5pt}}l@{\hspace{5pt}}l @{\hspace{10pt}} l@{\hspace{5pt}}l@{\hspace{5pt}}l@{\hspace{5pt}}l@{}}
\toprule
{\bf Methods} & \multicolumn{4}{c}{\bf Overall accuracy} & \multicolumn{4}{c}{\bf Fairness violations} \\
& 30\% & 50\% & 70\% & 90 \% & 30\% & 50\% & 70\% & 90\% \\
\midrule
{\bf No mitigation} & 0.546 & 0.545 & 0.529 & 0.559 & 0.179 & 0.186 & 0.152 & 0.134 \\
{\bf Fair bf Pruning} & 0.539 & 0.557 & 0.529 & 0.540 & 0.189 & 0.190 & 0.174 & 0.238 \\
{\bf Fair Aft Pruning} & 0.538 & 0.532 & 0.497 & 0.472 & 0.172 & 0.161 & 0.163 & 0.05 \\
{\bf Fair both} & 0.525 & 0.541 & 0.508 & 0.484 & 0.170 & 0.144 & 0.156 & 0.073 \\
\bottomrule
\end{tabular}
}
\caption{Accuracy and fairness violations for the UTKFaces dataset with \emph{ethnicity} as class labels and \emph{age} as protected attributes and prune amounts of 30\%, 50\%, 70\%, and 90\%.}
\label{tab:protected_attr_results}
\end{table}
Next, the second and fourth subplots presented in Figures \ref{fig:mitigation_pruning-1} and \ref{fig:mitigation_pruning-2} illustrate the fairness violations obtained by the four models analyzed on different datasets and settings.
We make the following observations: First, all the plots exhibit a consistent trend in that the mitigation solution produces models which improve the fairness of the baseline, "{\sl No mitigation}" model.
Observe that, as already illustrated in Figure \ref{fig:example_mitigation}, the fair models tend to equalize the gradient norms and group Hessians components (and thus the distance to the decision boundary across groups). Thus, the resulting pruned models also attain better fairness, when compared to their standard counterparts.
Next, notice that "{\sl Fair Aft Pruning}" often achieves better fairness violations than "{\sl Fair Bf Pruning}", especially at high pruning regimes. This is because the former has the advantage to apply the mitigation solution directly to the pruned model to ensure that the resulting model has low differences in gradient norms and group Hessians. The presentation also illustrates the application of the mitigation strategies both before and after pruning ({\sl Fair Both}) which shows once again the significance of applying the mitigation solution over the pruned network.
Finally, it is notable that "{\sl Fair Aft Pruning}" achieves good reductions in fairness violation. Indeed, pre-trained large (non-pruned) fair models may not be available and the ability to retrain these large models prior to pruning may be hindered by their size and complexity.
\section{Discussion and limitations}
\label{sec:limitations}
This section discusses three key messages found in this study. First, we notice that pruning affecting model separability and distance to the decision boundary is related to concepts also explored in robust machine learning \cite{https://doi.org/10.48550/arxiv.1412.6572,papernot2016transferability}. Not surprisingly, some recent literature in network pruning has empirically observed that pruning may have a negative impact on adversarial robustness \cite{guo2019sparse}. These observations raise questions about the connection between pruning, robustness, and fairness, which we believe is an important direction to further investigate.
Next, although the solution proposed in Problem \eqref{eq:LD_fast} allows it to be adopted in large models, the size of modern ML models (together with the amount of hyperparameters searches) may hinder retraining such original massive models from incorporating fairness constraints. Notably, however, the proposed mitigation solution can be used as a post-processing step to be applied during the pruning operation directly. The previous section shows that the proposed method delivers desirable performance in terms of both accuracy and fairness.
Finally, we notice that the results analyzed in this paper pertain to losses that are twice differentiable. Lifting such an assumption will be an interesting and challenging future research avenue.
\paragraph{Ethical considerations.}
The analyses and solutions reported in this paper should not be intended as an endorsement for using the developed techniques to aid facial recognition systems. We hope this work creates further awareness of the unfairness caused by pruning mechanisms in ML systems in the context of models that could be deployed in energy-efficient devices, such as smart cameras or access control systems, etc.
\section{Conclusion}
This work observed that pruning, while effective in compressing large models with minimal loss of accuracy, can result in substantial disparate accuracy impacts. The paper examined the factors causing such disparities both theoretically and empirically showing that: {\bf (1)} disparity in gradient norms across groups and {\bf (2)} disparity in Hessian matrices associated with the loss functions computed using a groups' data are two key factors responsible for such disparities to arise.
By recognizing these factors, the paper also developed a simple yet effective retraining technique that largely mitigates the disparate impacts caused by pruning.
As reduced versions of large, overparameterized models become increasingly adopted in embedded systems to facilitate autonomous decisions, we believe that this work makes an important step toward {\em understanding} and {\em mitigating} the sources of disparate impacts observed in compressed learning models and hope it will spark awareness in this important area.
\section*{Acknowledgments}
This research is partly funded by NSF grants SaTC-1945541, SaTC-2133169, and CAREER-2143706.
F.~Fioretto is also supported by a Google Research Scholar Award and an Amazon Research Award.
The views and conclusions of this work are those of the authors only.
\section*{NeurIPS 2022 Paper Checklist}
\newcommand{\ans}[1]{{\color{purple}{{\em #1}}}}
\begin{enumerate}
\item For all authors
\begin{enumerate}[label=\alph*)]
\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
\ans{Yes. The paper contributions are stated in the abstract and listed in the Introduction.}
\item (b) Have you read the ethics review guidelines and ensured that your paper conforms to them?
\ans{Yes.}
\item Did you discuss any potential negative societal impacts of your work?
\ans{This work sheds light on the reasons behind the observed disparate impacts and fairness violations through pruning. Pruning is a widely-used compression technique for large-scale models which are deployed in settings with less resources. Thus, the insights generated by this work may have a positive societal impact.}
\item Did you describe the limitations of your work?
\ans{Yes. Please, see section \ref{sec:limitations}.}
\end{enumerate}
\item If you are including theoretical results...
\begin{enumerate}
\item Did you state the full set of assumptions of all theoretical results?
\ans{Yes. The assumptions were stated in or before each Theorem and also reported in the Appendix.}
\item Did you include complete proofs of all theoretical results?
\ans{Yes. While the main paper only contains proof sketches or intuitions, all complete proofs are reported in Appendix \ref{sec:missing_proofs}.}
\end{enumerate}
\item If you ran experiments...
\begin{enumerate}
\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
\ans{Yes. Code, datasets and experiments were submitted in
the supplemental material. We also provide a link in Appendix \ref{sec:additional_results}.}
\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
\ans{Yes. See Appendix \ref{sec:additional_results}.}
\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
\ans{The main evaluation metric adopted in this work is the excessive loss (see Equations \eqref{eq:2} and \eqref{eq:3})
which implicitly captures the randomness of the private mechanisms. Providing error bars would be misleading.}
\item Did you include the amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
\ans{Yes. See Appendix \ref{sec:additional_results}}
\end{enumerate}
\item If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
\begin{enumerate}
\item If your work uses existing assets, did you cite the creators?
\ans{Yes. See References section.}
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\ans{Yes, when available.}
\item Did you include any new assets either in the supplemental material or as a URL?
\ans{No new asset was required to perform this research.}
\item Did you discuss whether and how consent was obtained from people whose data you're using/curating?
\ans{Yes. The paper uses public datasets.}
\item Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
\ans{No. The adopted data is composed of standard benchmarks that have been used extensively in the ML literature and we believe the above does not apply.}
\end{enumerate}
\item If you used crowdsourcing or conducted research with human subjects...
\ans{No. This research did not use crowdsourcing.}
\end{enumerate}
\appendix
\setcounter{theorem}{0}
\setcounter{lemma}{0}
\setcounter{proposition}{0}
\setcounter{property}{0}
\section{Missing Proofs}
\label{sec:missing_proofs}
\begin{theorem}
\label{thm:taylor}
The \emph{excessive loss} of a group $a \in \cA$ is upper bounded by\footnote{
With a slight abuse of notation, the results refer to $\bar{{\bm{\theta}}}$ as the homonymous vector which is extended with $k-\bar{k}$ zeros.
}:
\begin{align}
R(a) \leq
\left\| \bm{g}^{\ell}_a \right\|
\times \left\| \bar{{\bm{\theta}}} - \optimal{\bm{\theta}}\right\|
+
\frac{1}{2} \lambda \left( \bm{H}_{a}^{\ell} \right)
\times
\left\| \bar{{\bm{\theta}}} - \optimal{\bm{\theta}}\right\|^2
+
\cO\left( \left\|\bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right\|^3 \right),
\label{eq:thm1}
\end{align}
where
$\bm{g}^{\ell}_a = \nabla_{\optimal{\bm{\theta}}} J( \optimal{\bm{\theta}}; D_{a})$ is the vector of gradients associated with the loss function $\ell$ evaluated at $\optimal{\bm{\theta}}$ and computed using group data $D_a$,
$\bm{H}_{a}^{\ell} =
\nabla^2_{\optimal{\bm{\theta}}} J(\optimal{\bm{\theta}}; D_{a})$ is the Hessian matrix of the loss function $\ell$, at the optimal parameters vector $\optimal{\bm{\theta}}$, computed using the group data $D_a$ (henceforth simply referred to as \emph{group hessian}), and
$\lambda(\Sigma)$ is the maximum eigenvalue of a matrix $\Sigma$.
\end{theorem}
\begin{proof}
Using a second order Taylor expansion around $\optimal{\bm{\theta}}$, the excessive loss $R(a)$ for a group $a \in \cA$ can be stated as:
\begin{align*}
R(a)
&= J(\bar{{\bm{\theta}}}; D_{a}) - J(\optimal{\bm{\theta}}; D_{a}) \\
& = \left[ J\left(\optimal{\bm{\theta}}; D_{a}\right)
+ \left(\bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right)^\top \,
\nabla_{\theta} J\left(\optimal{\bm{\theta}}; D_{a} \right)
+ \frac{1}{2} \left(\bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right)^\top\,
\bm{H}_{a}^\ell \left(\bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right)
+ \cO \left( \left\| \optimal{\bm{\theta}} - \bar{{\bm{\theta}}} \right\|^3 \right)
\right]
- J\left(\optimal{\bm{\theta}}; D_{a}\right) \\
&= \left(\bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right)^\top\, \bm{g}^{\ell}_a
+ \frac{1}{2} \left(\bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right)^\top \,
\bm{H}_a^\ell \left(\bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right)
+ \cO \left( \|\optimal{\bm{\theta}} - \bar{{\bm{\theta}}}\|^3 \right)
\end{align*}
The above, follows from the loss $\ell(\cdot)$ being at least twice differentiable, by assumption.
By Cauchy-Schwarz inequality, it follows that
\[
\left( \bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right)^\top
\bm{g}^{\ell}_a \leq
\left\| \bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right\| \times
\left\|\bm{g}^{\ell}_a \right\|.
\]
In addition, due to the property of Rayleigh quotient we have:
\[
\frac{1}{2} \left(\bar{{\bm{\theta}}} - \optimal{\bm{\theta}}\right)^\top
\bm{H}_{a}^\ell \left(\bar{{\bm{\theta}}} - \optimal{\bm{\theta}}\right)
\leq
\frac{1}{2} \lambda \left(\bm{H}_a^\ell \right) \times
\left\| \bar{{\bm{\theta}}} - \optimal{\bm{\theta}} \right\|^2.
\]
The upper bound for the excessive loss $R(a)$ is thus obtained by combining these two inequalities.
\end{proof}
\begin{proposition}
\label{thm:grad_imbalance}
Consider two groups $a$ and $b$ in $\cA$ with $|D_a| \geq |D_b|$. Then
\(
\left\| \bm{g}_a^\ell \right\| \leq \left\| \bm{g}_b^\ell \right\|.
\)
\end{proposition}
\begin{proof}
By the assumption that the model converges to a local minima, it
follows that:
\begin{align*}
\nabla_{\theta} \cL(\optimal{\bm{\theta}}; D)
&= \sum_{a \in \cA} \frac{|D_a|}{|D|} \nabla_{\theta}
J(\optimal{\bm{\theta}}; D_a) \\
&= \frac{|D_a|}{|D|} \bm{g}^{\ell}_a + \frac{|D_b|}{|D|}
\bm{g}^{\ell}_b
= \bm{0}
\label{eq:grad_decomp}
\end{align*}
Thus, $\bm{g}^{\ell}_a = -\frac{|D_b|}{|D_a|} \bm{g}_b$. Hence $\| \bm
{g}^{\ell}_a \| = \frac{|D_b|}{|D_a|} \| \bm{g}^\ell_b \| \leq \| \bm{g}^\ell_b\| $, because
$|D_a| \geq |D_b| $.
\end{proof}
\begin{proposition}
\label{thm:acc_vs_grad}
For a given group $a \in \cA$, gradient norms can be upper bounded as:
\[
\|\bm{g}_a^\ell \| \in
\cO\left(
\sum_{(\bm{x}, y) \in D_a}
\underbrace{\|f_{\optimal{\bm{\theta}}}(\bm{x}) - y \|}_{\textit{Accuracy}}
\times
\left\| \nabla_{\optimal{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right\|
\right).
\]
\end{proposition}
The above proposition is presented in the context of cross entropy loss or mean squared error loss functions. These two cases are reviewed as follows
\paragraph{Cross Entropy Loss.}
Consider a classification task with cross entropy loss:
$\ell(f_{\optimal{\bm{\theta}}}(\bm{x}), y) = - \sum_{z \in \cY} f^z_{\optimal{\bm{\theta}}}
(\bm{x}) \bm{y}^z$, where $f_{\optimal{\bm{\theta}}}^z(\bm{x})$ represents the $z$-th element of the output associated with the soft-max
layer of model $f_{\optimal{\bm{\theta}}}$, and $\bm{y}$ is a one-hot encoding of the true label $y$, with $\bm{y}^z$ representing its $z$-th element, then,
\begin{align*}
\| \bm{g}_a\|
= \left\| \nabla_{{\bm{\theta}}} J(\optimal{\bm{\theta}}; D_a,) \right\|
&= \left\| \nicefrac{1}{|D_a|} \sum_{(\bm{x},y) \in D_{a}}
\nabla_{f} \ell(f_{\optimal{\bm{\theta}}}(\bm{x}), y) \times
\nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right\|\\
&= \left\| \nicefrac{1}{|D_a|} \sum_{(\bm{x}, y) \in D_a}
(f_{\optimal{\bm{\theta}}}(\bm{x}) - \bm{y}) \times
\nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right\|\\
&\leq \nicefrac{1}{|D_a|}\sum_{(\bm{x}, y) \in D_a}
\left\| f_{\optimal{\bm{\theta}}}(\bm{x}) - \bm{y} \right\| \times
\left\| \nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right\|,
\end{align*}
where the third equality is due to that the gradient of the cross entropy loss reduces to $f_{\optimal{\bm{\theta}}}(\bm{x}) - \bm{y}$.
\paragraph{Mean Squared Error.}
Next, consider a regression task with mean squared error loss
$\ell(f_{\optimal{\bm{\theta}}}(\bm{x}), y) = ( f_{\optimal{\bm{\theta}}}(\bm{x}) - y)^2$. Using the same notation as that made above, if follows:
\begin{align*}
\| \bm{g}_a\|
= \left\| \nabla_{{\bm{\theta}}} J(\optimal{\bm{\theta}}; D_a,) \right\|
&= \left\| \nicefrac{1}{|D_a|} \sum_{(\bm{x},y) \in D_{a}}
\nabla_{f} \ell(f_{\optimal{\bm{\theta}}}(\bm{x}), y) \times
\nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right\|\\
&= \left\| \nicefrac{2}{|D_a|} \sum_{(\bm{x}, y) \in D_a}
(f_{\optimal{\bm{\theta}}}(\bm{x}) - \bm{y}) \times
\nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right\|\\
&\leq \nicefrac{2}{|D_a|}\sum_{(\bm{x}, y) \in D_a}
\left\| f_{\optimal{\bm{\theta}}}(\bm{x}) - \bm{y} \right\| \times
\left\| \nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right\|,
\end{align*}
where the third equality is due to that the gradient of the mean squared error loss w.r.t.~$f_{\optimal{\bm{\theta}}}(\cdot)$ reduces to $2( f_{\optimal{\bm{\theta}}}(\bm{x}) - \bm{y})$.
\begin{theorem}
\label{thm:hessian_norm_bound}
Let $f_\theta$ be a binary classifier trained using a binary cross entropy loss. For any group $a \in \cA$, the maximum eigenvalue of the group Hessian $\lambda(\bm{H}_a^{\ell})$ can be upper bounded by:
\begin{equation}
\lambda(\bm{H}_a^{\ell}) \leq \frac{1}{|D_a|}
\sum_{(\bm{x}, y)\in D_a}
\underbrace{
\left( {f}_{\optimal{\bm{\theta}}}(\bm{x}) \right)
\left( 1 - {f}_{\optimal{\bm{\theta}}}(\bm{x}) \right)}_{\textit{Distance to decision boundary}}
\times
\left\| \nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm {x}) \right\|^2
+
\underbrace{\left| f_{\optimal{\bm{\theta}}}(\bm{x}) - y \right|}_{\textit{Accuracy}}
\times
\lambda\left( \nabla^2_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right).
\label{eq:hessian_norm_bound}
\end{equation}
\end{theorem}
\begin{proof}
First notice that an upper bound for the Hessian loss computed on a group $a \in \cA$ can be derived as:
\begin{align}
\lambda(\bm{H}_a^{\ell})
&= \lambda\left(
\frac{1}{|D_a|} \sum_{(\bm{x}, y) \in D_a} \bm{H}_{\bm{x}}^{\ell} \right)
\leq
\frac{1}{|D_a|} \sum_{(\bm{x}, y) \in D_a}
\lambda\left( \bm{H}_{\bm{x}}^{\ell} \right)
\label{eq:hessian_decompose}
\end{align}
where $\bm{H}_{\bm{x}}^{\ell}$ represents the Hessian loss associated with a sample $\bm{x} \in D_a$ from group $a$.
The above follows Weily's inequality which states that for any two symmetric matrices $A$ and $B$, $\lambda(A + B) \leq \lambda(A) + \lambda(B)$.
Next, we will derive an upper bound on the Hessian loss associated to a sample $\bm{x}$. First, based on the chain rule a closed form expression for the Hessian loss associated to a sample $\bm{x}$ can be written as follows:
\begin{align}
\bm{H}^{\ell}_{\bm{x}} &=
\nabla^2_{f} \ell\left(f_{\optimal{\bm{\theta}}}(\bm{x}), y\right)
\left[ \nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x})
\left( \nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right)^\top \right]
+
\nabla_{f} \ell\left(f_{\optimal{\bm{\theta}}}(\bm{x}), y\right)
\nabla^2_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}).
\label{eq:hessian_sample}
\end{align}
The next follows from that
\begin{align*}
\nabla_{f} \ell\left( f_{\optimal{\bm{\theta}}}(\bm{x}), y\right)
&= (f_{\optimal{\bm{\theta}}}(\bm{x}) - y), \\
\nabla^2_{f} \ell\left(f_{\optimal{\bm{\theta}}}(\bm{x}), y\right)
&= f_{\optimal{\bm{\theta}}}(\bm{x}) \left(1 - f_{\optimal{\bm{\theta}}}(\bm{x})\right).
\end{align*}
Applying the Weily inequality again on the R.H.S.~of Equation \ref{eq:hessian_sample}, we obtain:
\begin{align}
\lambda(\bm{H}_{\bm{x}}^{\ell}) & \notag
\leq f_{\optimal{\bm{\theta}}}(\bm{x})
\left(1 - f_{\optimal{\bm{\theta}}}(\bm{x})\right) \times
\left\| \nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right\|^2
+ \lambda\left( f_{\optimal{\bm{\theta}}}(\bm{x}) - y\right) \times
\nabla^2_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x})\\
& \leq
f_{\optimal{\bm{\theta}}}(\bm{x})
\left(1 - f_{\optimal{\bm{\theta}}}(\bm{x})\right) \times
\left\| \nabla_{{\bm{\theta}}} f_{\optimal{\bm{\theta}}}(\bm{x}) \right\|^2
+
\left| f_{\optimal{\bm{\theta}}}(\bm{x}) - y \right|
\lambda \left( \nabla^2_{{\bm{\theta}}}
f_{\optimal{\bm{\theta}}}(\bm{x}) \right)
\label{eq:hessian_bound}
\end{align}
The statement of Theorem \ref{thm:hessian_norm_bound} is obtained combining Equations \ref{eq:hessian_bound} with \ref{eq:hessian_decompose}.
\end{proof}
\begin{proposition}
\label{prop:dist_bndry}
Consider a binary classifier $f_{{\bm{\theta}}}(\bm{x}) $. For a given sample $\bm{x} \in D $, the term ${f}_{\optimal{\bm{\theta}}}(\bm{x}) (1 - {f}_{\optimal{\bm{\theta}}}(\bm{x}) )$ is maximized when ${f}_{\optimal{\bm{\theta}}}(\bm{x}) = 0.5$ and minimized when ${f}_{\optimal{\bm{\theta}}}(\bm{x}) \in \{0,1\} $.
\end{proposition}
\begin{proof}
First, notice that $f_{\optimal{\bm{\theta}}}(\bm{x}) \in [0,1]$, as it represents the soft prediction (that returned by the last layer of the network), thus ${f}_{\optimal{\bm{\theta}}}(\bm{x}) \geq f^2_{\optimal{\bm{\theta}}}(\bm{x})$. It follows that:
\begin{equation}
f_{\optimal{\bm{\theta}}}(\bm{x}) \left( 1 - f_{\optimal{\bm{\theta}}}(\bm{x}) \right)
= f_{\optimal{\bm{\theta}}}(\bm{x}) - f^2_{\optimal{\bm{\theta}}}(\bm{x}) \geq 0.
\end{equation}
In the above, it is easy to observe that the equality holds when either $f_{\optimal{\bm{\theta}}}(\bm{x}) = 0$ or $f_{\optimal{\bm{\theta}}}(\bm{x}) = 1$.
Next, by the Jensen inequality, it follows that:
\begin{equation}
f_{\optimal{\bm{\theta}}}(\bm{x}) \left( 1 - f_{\optimal{\bm{\theta}}}(\bm{x}) \right)
\leq \frac{\left( f_{\optimal{\bm{\theta}}}(\bm{x})+1
- f_{\optimal{\bm{\theta}}}(\bm{x}) \right)^2}{4}
= \frac{1}{4}.
\end{equation}
The above holds when $f_{\optimal{\bm{\theta}}}(\bm{x}) = 1 - f_{\optimal{\bm{\theta}}}(\bm{x})$, in other words, when $f_{\optimal{\bm{\theta}}}(\bm{x}) = 0.5$.
Notice that, in the case of binary classifier, this refers to the case when the sample $\bm{x}$ lies on the decision boundary.
\end{proof}
\iffalse
\section{Additional Theoretical Results}
\label{sec:additional_theory}
We provide additional theoretical results regarding convergence in gradient norm for majority/minority group under non-convex settings here. The results from Proposition \ref{thm:grad_imbalance} relies on the assumption that under standard training such as SGD the model converges to a local minima.
\paragraph{Settings} To begin with, we consider the standard stochastic gradient descent (SGD)
training, in which the model is started at ${\bm{\theta}}^0$ and then is
updated by following: ${\bm{\theta}}^t = {\bm{\theta}}^{t-1} - \eta \bm
{g}^t_B$, where $g^t_B = \frac{1}{|B|}\sum_{\bm{x} \in B} \nabla_
{{\bm{\theta}}} \ell(f_{{\bm{\theta}}^t}(\bm{x}), y) $ is the gradient of loss
defined on a random mini-batches $B \subseteq D$ and $\eta$ is the learning rate. Note that since the samples
in the mini-batch $B$ are i.i.d. then it follows that $\mathbf{E}[ \bm
{g}^t_B] = \bm{g}^t$ the gradient defined on overall dataset $D$ at
$t$-iteration. Denote $g^t_{i} =\nabla_{\theta} J(D_i, \theta^t) $
is the gradient of loss over group $i \in \{a, b\}$ at $t$-iteration
with parameter ${\bm{\theta}}^t$.
Our convergence analysis is based on the following assumption:
\begin{enumerate}
\item The variance of gradient $g^t_{B}$ defined on random
mini-batch $B$ is bounded, i.e $\mathbf{E} [\|\bm{g}^t_B - \bm
{g}^t\|^2 ] \leq \sigma^2$.
\item $\max_{\theta} \frac{\|g_i \|}{\|g\|} = L_i$ for $i \in
\{a, b\}$. In other words, for any parameter $\theta$ the
maximum difference between gradient norm of a group $i$ and the
gradient norm on dataset $D$ is bounded by $L_i$.
\item We also ssume that the loss function over group $i \in \{a, b\} $ $J(D_i, {\bm{\theta}})$ is
$\beta_i$-smooth.
\item Finally we assume that the learning rate is small enough, $\eta \leq \min_i \frac{1}{\beta_i}$
\end{enumerate}
\begin{theorem} Given the above assumptions, then we have the
following results regarding the convergence property of gradient
norm of each group $i \in \{a, b\}$.
\begin{equation}
\frac{1}{T}\sum^{T}_{t=0} \mathbf{E}[ \|g^t_{i}\| ] \leq \frac{1}
{1-m} \big( \frac{J(D_i, {\bm{\theta}}^0) }{T \eta } + \eta \beta_i^2 \sigma^2 \big)
\end{equation}
\end{theorem}
\begin{proof}
First, it is known that for any $\beta$ smooth function $f(x)$ then the the following hold: $f(y) \leq ( f(x) \leq \nabla_{x} f
(x), y-x) + \frac{\beta}{2}\|y-x\|^2 $. Since the loss function over group $i$ is $\beta_i$-smooth, hence the loss function at iteration $t$ of group $i$ can be upper bounded by:
\begin{equation} J(D_i, \theta^t) \leq J(D_i, \theta^{t-1}) - (g^
{t-1}_{i})^T (\theta^t- \theta^{t-1}) + \frac{\beta}
{2} \| \theta^t - \theta^{t-1}\|.
\end{equation}
Because $\theta^t = \theta^{t-1} - \eta g^{t-1}_{B}$, and $\mathbf{E}[g^{t-1}_
{B}] = g^{t-1}$. Replace the quantity $\theta^t - \theta^{t-1} = \eta
g^t_B$ and taking expectation on both sides w.r.t randomness of $B$,
we obtain:
\begin{equation} J(D_i, \theta^t) \leq J(D_i, \theta^{t-1}) - \eta(g^
{t-1}_{i})^T g^t + \frac{\beta}{2} \eta^2\big( \| \bm{g}^t\|^2 + \mbox
{Var}[\|g^{t-1}_{B}\|] \big)
\end{equation}
Due to our assumption that the variance of gradient on on mini-batches
is bounded by $\sigma^2$. Hence, it follows that:
\begin{equation} J(D_i, \theta^t) \leq J(D_i, \theta^{t-1}) - \eta(g^
{t-1}_{i})^T g^{t-1} + \frac{\beta}{2} \eta^2 \| \bm{g}^{t-1}\|^2 + \frac{\beta}{2} \eta^2 \sigma^2
\label{eq:19}
\end{equation}
Because of the assumption that $\eta \leq \frac{1}{\beta_i}$ hence $\frac{\beta}{2}\eta \|\bm{g}^{t-1}\|^2 \leq \eta \|\bm{g}^{t-1}\|^2 $, hence:
\begin{equation} J(D_i, \theta^{t-1}) \leq J(D_i, \theta^{t-1}) - \eta \big( \| \bm{g}^{t-1}\|^2 - 2 (\bm{g}_i^{t-1})^T \bm{g}^{t-1} \big) + \frac{\beta}{2} \eta^2 \sigma^2
\label{eq:20}
\end{equation}
\end{proof}
\fi
\section{Dataset and Experimental Settings}
\label{sec:additional_results}
\subsection{Datasets}
The paper uses the following datasets to validate the findings discussed in the main paper:
\begin{itemize}
\item {\bf UTK-Face}~\cite{zhifei2017cvpr}.
A large-scale face dataset with a long age span (range from 0 to 116 years old). The dataset consists of over 20,000 face images with annotations of age, gender, and ethnicity. The images cover large variations in pose, facial expression, illumination, occlusion, resolution, etc.
The experiments adopt the following attributes for classification (e.g., $\cY$) and as protected group ($\cA$): {\sl ethnicity}, {\sl age bins}, {\sl gender}.
\item {\bf CIFAR-10}~\cite{cifar10}.
This dataset consists of 60,000 32$\times$32 RGB images in 10 classes, with 6,000 images per class. The 10 different classes represent airplanes, cars, birds, cats, deer, dogs, frogs, horses, ships, and trucks.
\item {\bf SVHN}~\cite{Netzer2011ReadingDI}
Street View House Numbers (SVHN) is a digit classification dataset that contains 600,000 32$\times$32 RGB images of printed digits (from 0 to 9) cropped from pictures of house number plates.
\end{itemize}
\subsection{Architectures, Hyper-parameters, and Settings}
The study adopts the following architectures to validate the results of the main paper:
\begin{itemize}
\item {\bf ResNet18}: An 18-layer architecture, with 8 residual blocks. Each residual block consists of two convolution layers. The model has $\sim 11$ million trainable parameters.
\item {\bf ResNet50} This model contains 48 convolution layers, 1 MaxPool layer and a AvgPool layer. ResNet50 has $\sim 25$ million trainable parameters.
\item {\bf VGG-19} This model consists of 19 layers (16 convolution layers, 3 fully connected layers, 5 MaxPool layers and 1 SoftMax layer). The model has $\sim 143$ million parameters.
\end{itemize}
Hyperparameters for each of the above models was performed over a grid search (for different learning rates = $[0.0001, 0.001, 0.01, 0.1, 0.5, 0.05, 0.005, 0.0005]$) over a cluster of NVIDIA RTX A6000 with the above networks using the UTKFace dataset.
The models with the highest accuracy were chosen and employed for the assessment of the mitigation solution in Sec.~\ref{sec:AssmtMitigation}. The running time required for all sets of experiments which include mitigation solutions was about \textasciitilde{3} \emph{days}.
The training uses the SGD optimizer with a momentum of $0.9$ and weight\_decay of $1e^{-4}$.
Finally, the Lagrangian step size adopted in the Lagrangian dual learning framework \cite{fioretto2020lagrangian} is set to $0.001$.
All the models developed were implemented using Pytorch 3.0. The training was performed using NVidia Tesla P100-PCIE-16GB GPUs and 2GHz Intel Cores. The model was run for 100 epochs for the CIFAR-10 and SVHN and 40 epochs for UTK-Face dataset.
Each reported experiment is an average of 10 repetitions.
In all experiments, the protected group set coincides with the target label set: i.e., $\cA = \cY$.
\section{Additional Experimental Results}
\label{sec:additional_results}
\subsection{Impact of pruning on fairness}
This section shows and affirms the impact of pruning towards accuracy disparity through VGG-19 network. The same training procedures as employed with ResNet18 in Fig~\ref{fig:motivation} were followed. Each demographic group's accuracy is shown before and after pruning on the UTK-Face dataset in two cases: when \emph{ethnicity} is a group attribute as in Figure \ref{fig:utk_race_vgg_motivation}, and when \emph{gender} is a group attribute as in Figure \ref{fig:utk_gender_vgg_motivation}. A consistent message is that under a higher pruning rate, the accuracies are more imbalanced across different groups, emphasizing the negative impact of pruning on fairness.
\begin{figure*}
\caption{Accuracy of each demographic group in the UTK-Face dataset with ethnicity (5 classes) as group attribute using VGG19 over increasing pruning rates.}
\label{fig:utk_race_vgg_motivation}
\end{figure*}
\begin{figure*}
\caption{Accuracy of each demographic group in the UTK-Face dataset with gender (2 classes) as group attribute using VGG19 over increasing pruning rates.}
\label{fig:utk_gender_vgg_motivation}
\end{figure*}
\subsection{Correlation of gradient/hessian norm and average distance to the decision boundary}
This subsection elaborates the impact of gradient norms and group Hessians towards the fairness issues shown in Figures \ref{fig:utk_race_vgg_motivation} and \ref{fig:utk_gender_vgg_motivation}. In Section \ref{sec:grad_analysis}, it has been shown that the group with a larger gradient norm before pruning will be penalized more than the groups with a smaller gradient norm. Figures \ref{fig:utk_race_vgg_grad} and \ref{fig:utk_gender_vgg_grad} show the gradient norm of each demographic group for UTK-Face dataset under two choices of protected attributes for VGG 19 networks. The results indicate that a group penalized less will have a smaller gradient norm with respect to those of the other groups.
In addition, Section \ref{sec:hessian_analysis} supports that Hessian norm is another factor. More precisely, the groups with a larger Hessian norm will be penalized more (drop much more in accuracy) than groups with a smaller Hessian norm. Evidence is provided for the claim on VGG19 in Figures \ref{fig:utk_gender_vgg_grad} and \ref{fig:utk_race_vgg_grad}. These results on VGG19 again confirm the theoretical findings.
Finally, in Section \ref{sec:hessian_analysis}, a positive correlation between gradient norms and Hessian groups is shown in Theorem \ref{thm:hessian_norm_bound}, and a negative correlation between Hessian groups and distance to the decision boundary is shown in Proposition \ref{prop:dist_bndry}. These important results again are supported by the results in Figures \ref{fig:utk_gender_vgg_grad} and \ref{fig:utk_race_vgg_grad}.
\begin{figure*}
\caption{Gradient/Hessian norm and average distance to the decision boundary of each demographic group in the UTK-Face dataset with gender (2 classes) as group attribute using VGG19 with no pruning.}
\label{fig:utk_gender_vgg_grad}
\end{figure*}
\begin{figure*}
\caption{Gradient/Hessian norm and average distance to the decision boundary of each demographic group in the UTK-Face dataset with ethnicity (5 classes) as group attribute using VGG19 with no pruning.}
\label{fig:utk_race_vgg_grad}
\end{figure*}
\subsection{Impact of group sizes to gradient norm}
This section presents additional empirical results to support Theorem \ref{thm:grad_imbalance}, stating that the group with more samples will tend to have a smaller gradient norm. In these experiments, run on a ResNet50 network, one group is chosen and upsampled 1$\times$, 5$\times$, 10$\times$, and 20$\times$ times. Note that by increasingly upsampling it, the group becomes the majority group in that dataset. A group with \emph{more samples} is expected to end up with a \emph{smaller gradient norm} when the training convergences.
\paragraph{UTK-Face with gender}
Since the UTK-Face is balanced with regard to gender (Female/Male), the number of samples in Female, and Male groups is upsampled in turn. Figure \ref{fig:utk_gender_group_sizes_impact} reports the respective gradient norms at the last training iteration when upsampling Females (left) and Males (right.) Note how the Male group, initially with no upsampling, has a larger gradient norm than the Female group (right sub-plot). However, if the number of Male samples is increased enough, its gradient norm becomes smaller than that of the Female group.
\begin{figure*}
\caption{Impact of group sizes to the gradient norm per group in UTK-Face datase where groups are Male and Female.}
\label{fig:utk_gender_group_sizes_impact}
\end{figure*}
\paragraph{UTK-Face with age bins}
Similar experiments are performed with UTK-Face on nine age bin groups. Three age bins are randomly chosen, 0, 2, 4, and the number of samples for each group is upsampled in turn. The gradient norms of nine age bin groups are shown in Figure \ref{fig:utk_age_bins_group_sizes}, where the upsampled groups are highlighted with dotted thick lines. The results echo that if a group's number of samples is increased enough, its gradient norm at convergence will be smaller than the other 8 age bin groups.
\begin{figure*}
\caption{Impact of group sizes to the gradient norm per group in UTK-Face dataset where groups are nine age bins. The group with dotted thick line is a \emph{majority group}
\label{fig:utk_age_bins_group_sizes}
\end{figure*}
\end{document} |
\begin{document}
\title{On $1-$point densities for Arratia flows with drift}
\author[A.A.~Dorogovtsev]{Andrey A. Dorogovtsev}
\author[M.B.~Vovchanskyi]{Mykola B. Vovchanskyi}
\address{Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska str. 3, 01601, Kiev, Ukraine}
\maketitle
\begin{abstract}
We show that if drift coefficients of Arratia flows converge in $L_1(\mathbb R)$ or $L_{\infty}(\mathbb R)$ then the 1-point densities associated with these flows converge to the density for the flow with the limit drift.
Keywords: Brownian web, Arratia flow, point process, point density.
2020 Mathematics Subject Classification: Primary 60H10; Secondary 60G55, 35C10
\mathrm{e}nd{abstract}
The Arratia flow, a continual system of coalescing Wiener processes that are independent until they meet, was introduced independently as a limit of coalescing random walks in \cite{Ar79Coalescing}, as a system of reflecting Wiener processes in \cite{SouBaWer00Reflection} and as a limit of stochastic homeomorphic flows in \cite{Do04OneBrownian}. If interpreted as a collection of particles started at $0,$ it is a part of the Brownian web \cite{FonIsoNewRa04Brownian}. At the same time, one can construct the Arratia flow with drift using flows of kernels defined in \cite{JanRai04Flows} (see \cite[{\S 6}]{Ria18Duality} for a short explanation) or directly via martingale problems by adapting the method used in \cite{Ha84Coalescing} to build coalescing stochastic flows with more general dependence between particles (see \cite[Chapter 7]{Do07MeasureEng} for this approach).
Following \cite[Chapter 7]{Do07MeasureEng}, we consider a modification of the Arratia flow that introduces drift affecting the motion of a particle within the flow. So by an Arratia flow $X^a\mathrm{e}quiv\{X^a(u,t)\mid u\in{\mathbb R}, t\in{\mathbb R}_+\}$ with bounded measurable drift $a$ we understand a collection of random variables such that
\begin{enumerate}
\item for every $u$ the process $X^a(u,\cdot)$ is an It\^o process with diffusion coefficient $1$ and drift $a;$
\item for all $t\ge 0$ the mapping $X^a(\cdot,t)$ is monotonically increasing;
\item for any $u_1, u_2$
the joint quadratic covariation of the martingale parts of $X^a(u_1,\cdot)$ and $X^a(u_2,\cdot)$ equals $(t-\inf\{s\mid X^a(u_1,s)=X^a(u_2,s) \})_+,$
with $\inf \mathrm{e}mptyset$ being equal $\infty$ by definition.
\mathrm{e}nd{enumerate}
\begin{theorem}[{\cite[Lemma 7.3.1]{Do07MeasureEng}}]
For any measurable and locally bounded function $a$ the flow $X^a$ exists and is unique in the terms of finite-dimensional distributions.
\mathrm{e}nd{theorem}
\begin{remk}
\label{remark:existence}
\cite[Lemma 7.3.1]{Do07MeasureEng} is stated for Lipschitz continuous drift, but, being based on the classical existence and uniqueness theorems for the martingale problem, the proof holds verbatim for any measurable locally bounded drift.
\mathrm{e}nd{remk}
Since the set $\{X^a(u,t)\mid u\in {\mathbb R}\}$ is known to be locally finite for all $t>0$ \cite[Chapter 7]{Do07MeasureEng}, one defines for any $t>0$ the point process $\{X^a(u, t) \mid u \in {\mathbb R}\}.$
Studying of such a process can be performed using point densities (see: \cite{MunRajTriZab06Multi, TriZab11Pfaffian} for a definition and a representation in terms of Pfaffians in the case of zero drift, respectively; \cite{DorVov20ApproximationsEng, DoVov20Representations} for representations in the case of non-trivial drift; \cite{GliFom18Limit,Fo16Distribution} for applications to the study of the above-mentioned point process). In accordance with \cite[Appendix B]{MunRajTriZab06Multi},
the following definition is adopted in the present paper: the 1-point density at time $t$ of the point process associated with $X^a$ is given via
\begin{equation}
\label{eq:density.defin}
p^{a,1}_t(u) = \lim_{\delta\to 0+} \delta^{-1} \Prob \big( X^a({\mathbb R}, t) \cap [u;u+\delta] \not= \mathrm{e}mptyset \big),
\mathrm{e}nd{equation}
where the limit exists a.e.
The well-known relation between PDEs and SDEs is used to show that $p^{a_n,1}_t\to p^{a,1}_t, n\to \infty, a.e.$ whenever $a_n\to a, n\to\infty,$ in $L_1({\mathbb R})$ or $L_\infty({\mathbb R})$ under additional natural assumptions, which is the main result of the article.
It should be noted that one may approach the problem stated utilizing the representation of $p_t^{a,1}$ given in \cite[Theorem 2.1]{DoVov20Representations}. However, the method used in the present paper allows us to avoid a discussion of properties (e.g. uniform integrability) of certain stochastic exponentials involving Brownian bridges. An additional benefit is that the term equal to $p^{0,1}_t$ is separated from those containing drift in the resulting expression. Thus there is a possibility to obtain local estimates of the density in the terms of drift, in particular, in the studies of point concentration when external force modeled with non-zero drift is present. Moreover, following the route proposed in \cite[\S 4, Appendixes A, B]{MunRajTriZab06Multi} one can estimate $n-$dimensional densities in the terms of $p^{a,1}_t$ given estimates for the corresponding transitional density.
\begin{prop}
\label{prop:density.existence}
The density $p^{a,1}_t$ exists.
\mathrm{e}nd{prop}
\proof
Following \cite[Appendix B]{MunRajTriZab06Multi} one considers the measure $\nu^{a,1}$ on ${\mathcal B}({\mathbb R})$ given via
\[
\nu_t^{a,1}(A) = \E |X^a({\mathbb R}, t) \cap A|, \quad A \in {\mathcal B}({\mathbb R}),
\]
where given $B\in {\mathcal B}({\mathbb R})$ $X^a(B, t) = \{X^a(u,t)\mid u\in B\}$ by definition and $|B|$ equals the number of distinct points in $B.$
Then
\begin{equation}
\label{eq:measure.as.limit}
\E |X^a({\mathbb R}, t) \cap A| = \lim_{U\to\infty} \E |X^a([-U;U], t) \cap A|.
\mathrm{e}nd{equation}
The Girsanov theorem for the Arratia flow \cite[Theorem 7.3.1]{Do07MeasureEng} states that given $T>0$ and a bounded Lipshitz continuous function $a$ the distribution of $X^a$ as a random element in the Skorokhod space $D({\mathbb R}, C([0; T]))$ is absolutely continuous w.r.t the distribution of $X^0.$ To extend this result to the case of $a\in L_\infty({\mathbb R}),$ one proceeds as follows. The result of \cite[{Lemma 7.3.2}]{Do07MeasureEng} states that for any $t>0$ there exists $C_t > 0$ such that
\begin{equation}
\label{eq:prob.not.meeting.estimate}
\Prob\left\{ X^a(u, t) \not= X^a(v, t) \right\} \le C_t (v-u), \quad u, v\in{\mathbb R}, u<v.
\mathrm{e}nd{equation}
and is valid for $a\in L_\infty({\mathbb R}).$ Using this estimate one extends the proof of \cite[Theorem 2]{Do05Remarks}, so the flow $X^a$ admits a version in $D({\mathbb R}, C([0; T])).$ Then it is left to notice that the proof of \cite[Theorem 7.3.1]{Do07MeasureEng} uses only the standard Girsanov theorem and the estimate \cite[{Lemma 7.3.2}]{Do07MeasureEng}. Therefore, the Girsanov theorem for the Arratia flow holds for $a\in L_\infty({\mathbb R}),$ too.
Since $\nu_t^{0,1}$ is absolutely continuous w.r.t. the Lebesgue measure~\cite[Appendix B]{MunRajTriZab06Multi}, for any $A\in{\mathcal B}({\mathbb R}) $ with $\mathrm{Leb}(A)=0$ the assumption
\[
|X^0([-U;U], t) \cap A| = 0 \quad \mathrm{a.s.}, \quad U \ge 0,
\]
implies
\[
\E |X^a([-U;U], t) \cap A | = 0, \quad U \ge 0.
\]
Hence by \mathrm{e}qref{eq:measure.as.limit} there exists the density $q_t^{a,1}$ such that
\[
\int_A q_t^{a,1}(x) dx = \nu_t^{a,1}(A) = \E \sum_{v\in X^a({\mathbb R}, t)} 1\!\!\,{\rm I}_A(v), \quad A\in{\mathcal B}({\mathbb R}).
\]
Analogously one proves the existence of $n-$dimensional densities $q_t^{a,n}$ such that
\[
\E\sum_{
\begin{subarray}{c}v_1, \ldots, v_{n}\in X^a({\mathbb R}, t),\\
v_1, \ldots, v_{n} \ \mbox{\small are distinct}
\mathrm{e}nd{subarray}
}
1\!\!\,{\rm I}_A((v_1, \ldots, v_{n}))=\int_{A}q_t^{a,n}(x)dx = \nu_t^{a,n}(A), \quad A \in {\mathcal B}({\mathbb R}^n).
\]
By the Lebesgue differentiation theorem, for almost all $x=(x_1, \ldots, x_n)\in{\mathbb R}^n,$
\begin{equation*}
\label{eq:p.leb.diff}
q_t^{a,n}(x) = \lim_{\delta\to 0+} \delta^{-n} \nu_t^{a,n} \left( \mathop{\times}_{j=\overline{1,n}} [x_k;x_k+\delta] \right).
\mathrm{e}nd{equation*}
In particular, for almost all $x$
\[
q_t^{a,1}(x) = \lim_{\delta\to 0+} \delta^{-1} \E |X^a({\mathbb R}, t) \cap [x;x+\delta]|.
\]
In order to obtain \mathrm{e}qref{eq:density.defin}, one still needs to replace $|X^a({\mathbb R}, t) \cap [x;x+\delta]|$ with $1\!\!\,{\rm I}[X^a({\mathbb R}, t) \cap [x;x+\delta] \not= \mathrm{e}mptyset]$ in the last expression. The argument presented in \cite[Appendix B]{MunRajTriZab06Multi} for $a=0$ is applicable here.
If $a$ is additionally assumed to be Lipschitz continuous it is shown in \cite[Corollary 6.1]{Ria18Duality} that the dual flow \cite[Section 2.2]{TriZab11Pfaffian} to the flow $X^a$ is $X^{-a}.$ The same proof can be extended to the case of a just bounded function $a.$ Thus, due to the analog of \mathrm{e}qref{eq:prob.not.meeting.estimate} for $-a$ we have, for $u,v\in{\mathbb R}, u < v,$
\begin{align*}
\sup_{x\in{\mathbb R}} \Prob\left\{ X^a(x,t) \in [u;v] \right\} &\le \Prob\left\{ X^a({\mathbb R},t) \cap [u;v] \not = \mathrm{e}mptyset \right\} \nonumber \\
&= \Prob\left\{ X^{-a}(u,t) \not = X^{-a}(v,t) \right\} \nonumber \\
&\le C_t(v-u).
\mathrm{e}nd{align*}
Using this estimate and repeating the arguments in \cite[\S 4]{MunRajTriZab06Multi}, we get
\begin{align*}
\label{eq:induction.est.5}
&\Prob\left\{ X^a({\mathbb R}, t) \cap [u_k,v_k] \not = \mathrm{e}mptyset, k =\overline{1,n}\right\} \le C_t^n \prod_{k=\overline{1,n}} (v_k - u_k), \notag \\
& \qquad u_1 < v_1 < u_2 < \ldots u_n < v_n, \notag \\
& \qquad n \in {\mathbb N}.
\mathrm{e}nd{align*}
Exactly as in \cite[Appendix B]{MunRajTriZab06Multi} we then conclude
\begin{equation*}
\nu_t^{a,n} (A) \le C_t^n \, \mathrm{Leb}(A), \quad A\in {\mathcal B}({\mathbb R}^n), n\in{\mathbb N}.
\mathrm{e}nd{equation*}
Thus
\begin{align*}
\lim_{\delta \to 0+} & \delta^{-1} \E \left( |X^a({\mathbb R}, t) \cap [x;x+\delta]| - 1\!\!\,{\rm I}\big[ X^a({\mathbb R}, t) \cap [x;x+\delta] \not= \mathrm{e}mptyset \big] \right) \\
& \quad \le \lim_{\delta \to 0+} \delta^{-1} \E |X^a({\mathbb R}, t) \cap [x;x+\delta]| \big( |X^a({\mathbb R}, t) \cap [x;x+\delta]| - 1\big) \\
& \quad = \lim_{\delta \to 0+} \delta^{-1} \nu_t^{a,2} \big([x;x+\delta] \times [x;x+\delta]\big) \\
& \quad \le C_t^2 \lim_{\delta \to 0+} \delta \\
& \quad = 0,
\mathrm{e}nd{align*}
which concludes the proof. \qed
\begin{remk}
If $a$ is additionally Lipschitz continuous one can prove that $p_t^{a,n}$ has a continuous version, using the aforementioned representations from \cite{DoVov20Representations} or the analog of \cite[Theorem 3.1]{DorVov20ApproximationsEng}. The same conclusion for $a\in L_\infty({\mathbb R})$ follows from Theorem \ref{th:density.prop.repr}.
\mathrm{e}nd{remk}
Let $D_2 = \{u\in{\mathbb R}^2\mid u_1 < u_2\},$ and put for $a\in L_\infty({\mathbb R})$
\[
\nabla^a_x = \sum_{k=1}^2 a(x_k) \partial_{x_k}, \quad x=(x_1,x_2) \in {\mathbb R}^2.
\]
In what follows, we construct a series $W^a \in C(\overline D_2 \times (0;\infty)) \cap C(D_2 \times [0;\infty)) \cap C^{1,0}(D_2 \times [0;\infty))$ such that $W^a$ is a distribution solution \cite[p. 31]{Fol95Introduction} in ${\mathcal D}^\prime(D_2\times (0;t)),$ the space of (Schwartz) distributions over infinitely differentiable compactly supported in $D_2\times (0;t)$ test functions, to
\begin{align}
\label{eq:main.pde}
\partial_s W^a & = \frac{1}{2} \Delta W^a - \nabla^a_x W^a.
\mathrm{e}nd{align}
Additionally, as a function, $W^a$ satisfies
\begin{align}
\label{eq:main.pde.part2}
W^a(x,0) &= 1, \quad x \in D_2,\nonumber \\
W^a(x,s) &= 0, \quad x\in \partial D_2, \ s > 0.
\mathrm{e}nd{align}
The density $p^{a,1}_t$ admits a representation as a derivative of $W^{a}.$ As a result, in order to prove the convergence of the 1-point densities it is sufficient to consider he convergence of the corresponding series.
It is supposed $a\in L_\infty({\mathbb R})$ throughout the paper.
Let $x=(x_1,x_2)\in D_2$ and let $\xi^a_x=(\xi^a_{x_1},\xi^a_{x_2})$ be the unique weak solution of the Cauchy problem \cite[Corollary 3.11]{KaShre91Brownian}
\begin{align}
\label{eq:xi^a}
d\xi^a_{x_k}(t)& = - a(\xi^a_{x_k}(t))dt + dw_k(t), \nonumber \\
\xi^a_{x_k}(0)& = x_k, \quad k=1, 2,
\mathrm{e}nd{align}
where $w_1, w_2$ are independent standard Wiener processes started at $0.$ Define
\begin{equation}
\label{eq:theta^a}
\theta^a_x = \inf\{s \mid \xi^a_x \in \partial D_2\}.
\mathrm{e}nd{equation}
\begin{prop}
\label{lem:density.via.prob}
Let $a\in L_\infty({\mathbb R}), t>0.$ The density $p^{a,1}_t$ admits the representation
\[
p^{a,1}_t(u) = \lim_{\delta\to 0+} \delta^{-1} \Prob\left(\theta^{a}_{(u,u+\delta)} > t\right),
\]
whenever the limit exits.
\mathrm{e}nd{prop}
\proof As in the proof of Proposition \ref{prop:density.existence}, one gets, using the notion of the dual flow,
\begin{align*}
\Prob \big( X^a({\mathbb R}, t) \cap [u;u+\delta] \not= \mathrm{e}mptyset \big) = & \Prob \big( X^{-a}(u+\delta, t) > X^{-a}(u, t)\big) \\
= & \Prob\left(\theta^{a}_{(u,u+\delta)} > t\right)
\mathrm{e}nd{align*}
for all $\delta>0.$
\qed
\begin{exam}
Using the previous result one immediately gets $p^{0,1}_t(u) = \frac{1}{\sqrt{\pi t}}.$ Let $a_c(x) = cx, c\not= 0.$ The existence of the Arratia flow $X^{a_c}$ follows from Remark \ref{remark:existence}. Moreover, using the arguments from \cite[proof of Lemma 2.2]{DorVov20ApproximationsEng}, it is possible to establish the analog of Proposition \ref{lem:density.via.prob}. Solving the SDE for $\xi^{a_c}_{u_2}-\xi^{a_c}_{u_1}$ explicitly and using the arguments from \cite[proof of Lemma 2.2]{DorVov20ApproximationsEng} one gets
\[
p^{a_c,1}_t(u) = \sqrt{\frac{2}{\pi}} \frac{|c|^{1/2}}{\psi(t,c)}, \quad u\in{\mathbb R},
\]
where
\begin{align*}
\psi(t,c) =& \begin{cases}
\left(\mathrm{e}^{2tc}-1\right)^{1/2}, & c >0,\ t >0,\\
\left(1-\mathrm{e}^{2tc}\right)^{1/2}, & c <0,\ t >0.
\mathrm{e}nd{cases}
\mathrm{e}nd{align*}
Thus $p^{0,1}_t = \lim_{c\to 0} p^{a_c,1}_t$ $a.e.$ in ${\mathbb R}.$
\mathrm{e}nd{exam}
Note that due to the domain $D_2$ being unbounded the problem \mathrm{e}qref{eq:main.pde}--\mathrm{e}qref{eq:main.pde.part2} does not admit, in general, a unique solution. However, as usually, one expects the unique bounded solution $W^a$ to have a probabilistic representation, that is, to satisfy $W^a(x, s)=\Prob\left(\theta^{a}_{x} > s\right).$ At the same time, in the case of $a$ smooth enough the Duhamel principle \cite[\S 2.3.1.c]{Evans10Partial} states
\[
W^a(x,s) = \int_{D_2} dy_0 \ g_{s}(x,y_0) + \int_0^s dr \int_{D_2} dy \ g_{s-r}(x,y) \left( - \nabla^a_y W^a(y,r)\right),
\]
where
\[
g_r(x, y) = \frac{1}{2\pi r}\left( \mathrm{e}^{-\frac{\|x-y\|^2}{2r}} - \mathrm{e}^{-\frac{\|x-y^\ast\|^2}{2r}}\right),
\]
with $y^\ast = (y_2,y_1),$ is
the transition density of the 2-dimensional Wiener process killed when it reaches $\partial D_2.$ Using this formal relation as a starting point we define, for any $a\in L_\infty({\mathbb R}),$
\begin{align*}
W^a(x,s)=&\sum_{n\ge 0} W^a_{n}(x,s),\\
W^a_{0}(x,s) =& \int_{D_2} dy_0 \ g_{s}(x,y_0), \\
W^a_{n}(x,s)=& (-1)^n \int_{\Delta_n(s)} dr_1 \ldots dr_n \int_{D_2^{n+1}} dy_0 \ldots dy_n \ g_{s-r_n}(x,y_n) \prod_{j={1}}^n \nabla^a_{y_j} g_{r_j-r_{j-1}}(y_j,y_{j-1}), \\
& n\ge 1,
\mathrm{e}nd{align*}
where hereinafter $\Delta_k(r) = \{u\in{\mathbb R}^k\mid 0 \le u_1 \le \ldots \le u_k \le r\}, r> 0, k \ge 1,$ so that
\[
W^a_{n}(x,s) = - \int_0^s dr_n \int_{D_2} dy_n \ g_{s-r_n}(x,y_n) \nabla^a_{y_n} W^a_{n-1}(y_n, r_n), \quad n \ge 1.
\]
Note that always
\begin{align*}
W^a_{0}((x_1,x_2),s) &= \Prob\left(\theta^0_{(x_1,x_2)} > s\right) = \Prob\left( \sup_{r \le s} w(r) < \frac{x_2-x_1}{\sqrt{2}} \right) \\
&= \sqrt\frac{2}{\pi} \int_0^{\frac{x_2-x_1}{\sqrt{2s}}} da \ \mathrm{e}^{-\frac{a^2}{2}},
\mathrm{e}nd{align*}
where $w$ is a standard Wiener process, and
\begin{align*}
\partial_{x_2}W^a_0((u,u),s) = - \partial_{x_1}W^a_0((u,u),s) = \frac{1}{\sqrt{\pi s}}, \quad s > 0.
\mathrm{e}nd{align*}
The next lemma gathers technical results from calculus that are needed afterwards. In what follows $\overline D_2$ is the closure of $D_2,$ the derivatives $\partial_{x_k}, k=1,2$ on $\partial D_2$ are understood as one-sided derivatives and, for $y=(y_1,y_2),$
\[
G_a(y) = \min\Big\{ \sum_{j=1}^2 |a(y_{j})|, \|a\|_{L_\infty({\mathbb R})} \Big\}.
\]
\begin{lem}
\label{lem:calculus}
\phantom{a} \\
\begin{enumerate}
\item $\mathrm{e}xists K, \gamma > 0 \ \forall a\in L_\infty({\mathbb R}) \ \forall r > 0$
\begin{align*}
&\left| \partial_{y_{1,k}} g_r(y_1,y_2) \right| \le K r^{-3/2} \mathrm{e}^{-\gamma \frac{\|y_1-y_2\|^2}{r}} \ \ in \ \overline D_2, \quad k = 1,2, \ \\
&\left| \nabla^a_{y_1} g_r(y_1,y_2) \right| \le K G_a(y_1) r^{-3/2} \mathrm{e}^{-\gamma \frac{\|y_1-y_2\|^2}{r}} \ \ a.e. \ in \ \overline D_2,
\mathrm{e}nd{align*}
where $y_1 =(y_{1,1}, y_{1,2});$
\item $\forall \alpha > 0 \ \forall y\in {\mathbb R}^2$
\[
\int_{D_2} dy_0 \ \mathrm{e}^{-\alpha\|y_0 -y\|^2} \le \frac{\pi}{\alpha};
\]
\item put $r_0 = 0;$ then $\forall s > 0 \ \forall n \ge 1$
\begin{align*}
&\int_{\Delta_n(s)}dr_1 \ldots dr_n \ (s-r_n)^{-1/2} \prod_{j=1}^n (r_j-r_{j-1})^{-1/2}= \frac{\pi^{\frac{n+1}{2}} s^{\frac{n-1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)}, \\
&\int_{\Delta_n(s)}dr_1 \ldots dr_n \ \prod_{j=1}^n (r_j-r_{j-1})^{-1/2}= \frac{2\pi^{\frac{n}{2}} s^{\frac{n}{2}}}{n\Gamma\left(\frac{n}{2}\right)};
\mathrm{e}nd{align*}
where here and hereinafter $\Gamma$ is the gamma-function;
\item
for any $n\ge 1$ the derivatives $\partial_{x_k} W^a_n, k=1,2,$ exist on $\overline D_2\times (0;+\infty),$ and $W^a_n, \partial_{x_k} W^a_n \in C(\overline D_2\times (0;+\infty));$ additionally,
\begin{align*}
\sup_{x\in \overline D_2}\left| W^a_n(x,s)\right| &\le C_1 \frac{C_2^{n} \|a\|_{L_\infty({\mathbb R})}^{n} s^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}, \\
\sup_{x\in \overline D_2}\left| \partial_{x_k} W^a_n(x,s)\right| &\le C_1 \frac{C_2^{n} \|a\|_{L_\infty({\mathbb R})}^{n} s^{\frac{n-1}{2}}}{\Gamma\left(\frac{n}{2}\right)}, \quad s > 0, k=1,2, n\ge 1,
\mathrm{e}nd{align*}
for some absolute positive constants $C_1, C_2$ independent of $s, x, n$ and $a;$
\item for any $t>0$ the series for $W^a$ converges uniformly on $\overline D_2\times[0;t];$ the derivatives $\partial_{x_k} W^a, k=1,2,$ exist and are continuous on $\overline D_2\times (0;+\infty).$
\mathrm{e}nd{enumerate}
\mathrm{e}nd{lem}
\proof
Denote the coordinates of the point $y_2$ by $y_{2,1}, y_{2,2},$ and put $y_2^\ast = (y_{2,2}, y_{2,1}).$ Note that, for $k=1,2,$
\begin{align*}
|\partial_{y_{1,k}} g_r(y_1,y_2) | &\le \frac{1}{2\pi r^2} \left( \mathrm{e}^{-\frac{\|y_1-y_2\|^2}{2r}} |y_{1,k} - y_{2,k}| + \mathrm{e}^{-\frac{\|y_1-y_2^\ast\|^2}{2r}} |y_{1,k} - y_{2,k}^\ast| \right)\\
& \le \frac{1}{2\pi r^2} \left( \mathrm{e}^{-\frac{\|y_1-y_2\|^2}{2r}} \|y_{1} - y_{2}\| + \mathrm{e}^{-\frac{\|y_1-y_2^\ast\|^2}{2r}} \|y_{1} - y_{2}^\ast\| \right) \\
& \le \frac{C}{r^{3/2}} \left( \mathrm{e}^{-\frac{\|y_1-y_2\|^2}{4r}} + \mathrm{e}^{-\frac{\|y_1-y_2^\ast\|^2}{4r}} \right),
\mathrm{e}nd{align*}
for some absolute constant $C.$
Since $\mathrm{e}^{-\frac{\|y_1-y_2^\ast\|^2}{4r}} \le \mathrm{e}^{-\frac{\|y_1-y_2\|^2}{4r}}$ for $y_1,y_2 \in \overline D_2$ this yields Item (1).
For Item (2), write
\begin{align*}
\int_{D_2} dy_0 \ \mathrm{e}^{-\alpha\|y_0 -y\|^2} \le 2\pi \int_0^\infty d\rho \ \rho \mathrm{e}^{-\alpha\rho^2} = \frac{\pi}{\alpha}.
\mathrm{e}nd{align*}
Item (3) is a direct consequence of the integral representation given in \cite[\S 4.3]{Carl77Special} after the change of variables $r_1 = s u_1 , r_2 = s(u_1 + u_2), \ldots, r_n = s(u_1 + \ldots + u_n )$.\footnote{The authors are grateful to a colleague for providing the reference.}
Fix $n$ and put $r_{n+1}=s, y_{n+1}=x.$ Combining Items (1) and (2) we get
\begin{align*}
\left| W^a_n(x,s)\right| &\le K^n \|a\|_{L_\infty({\mathbb R})}^{n} \int_{\Delta_n(s)} dr_1 \ldots dr_n \int_{D_2^{n+1}} dy_0 \ldots dy_n \ g_{s-r_n}(x,y_n) \times \\
& \qquad \qquad \qquad \times \prod_{j=1}^{n} \frac{\mathrm{e}^{-\gamma \frac{\|y_j-y_{j-1}\|^2}{r_j-r_{j-1}}}}{(r_j-r_{j-1})^{3/2}} \\
&\le K^n \|a\|_{L_\infty({\mathbb R})}^{n} \left(\frac{\pi}{\gamma}\right)^{n} \int_{\Delta_n(s)} dr_1 \ldots dr_n \ \prod_{j=1}^{n} (r_j-r_{j-1})^{-1/2}, \\
\left|\partial_{x_k} W^a_n(x,s)\right| &\le K^{n+1} \|a\|_{L_\infty({\mathbb R})}^{n} \int_{\Delta_n(s)} dr_1 \ldots dr_n \int_{D_2^{n+1}} dy_0 \ldots dy_n \ \prod_{j=1}^{n+1} \frac{\mathrm{e}^{-\gamma \frac{\|y_j-y_{j-1}\|^2}{r_j-r_{j-1}}}}{(r_j-r_{j-1})^{3/2}} \\
&\le K^{n+1} \|a\|_{L_\infty({\mathbb R})}^{n} \left(\frac{\pi}{\gamma}\right)^{n+1} \int_{\Delta_n(s)} dr_1 \ldots dr_n \ \prod_{j=1}^{n+1} (r_j-r_{j-1})^{-1/2},
\mathrm{e}nd{align*}
so the rest follows by calculus.
\qed
In \cite{PaWin08First} a general version of the problem \mathrm{e}qref{eq:main.pde}--\mathrm{e}qref{eq:main.pde.part2} in a bounded domain is studied in terms of weak (variational) solutions\footnote{The statement of \cite[Theorem 2.7]{PaWin08First} contains a misprint: $Q(t,x)$ should be equal $\Prob(\tau_A > t).$} so the smoothness of the solution follows by standard results. Since it is easy to see that neither $\partial_{x_k x_k} W^a_n$ nor $\partial_{s} W^a_n$ exists for $n\ge 1$ this is not a viable approach for us, and we use the hypoellipticity of an operator $\frac{1}{2} \Delta - \nabla^a_x$ for $a\in C^{\infty}({\mathbb R})\cap L_\infty({\mathbb R})$ instead to check that the distribution solution $W^a$ is sufficiently smooth.
The proof of the following lemma is trivial.
\begin{lem}
\label{lem:approx.a}
Let $M\in {\mathcal B}({{\mathbb R}^m})$ for some $m\in{\mathbb N},$ and $k\in L_1(M).$ Suppose functions $f_n\colon M \mapsto {\mathbb R}, n\ge 0,$ are such that
\[
\sup_{n\ge 0} \|f_n\|_{L_\infty(M)} < \infty.
\]
Suppose that one of the following conditions holds:
\begin{enumerate}
\item $f_n\in L_1(M), n\ge 0,$ and $f_n\to f_0, n\to \infty,$ in $L_1(M);$
\item
$f_n\to f_0, n\to \infty,$ in $L_\infty(M).$
\mathrm{e}nd{enumerate}
Then
\[
\int_M dy\ f_n(y) k(y)\to \int_M dy\ f_0(y) k(y), \quad n\to\infty.
\]
\mathrm{e}nd{lem}
\begin{prop}
\label{prop:distr.solution.W_n}
For all $n\ge 1$ in the sense of Schwartz distributions
\[
\partial_s W^a_n = \frac{1}{2} \Delta W^a_n- \nabla^a_x W^a_{n-1}
\]
in $D_2 \times (0;\infty) .$
\mathrm{e}nd{prop}
\proof Suppose that $t >0$ is fixed throughout the proof. We have, for $n\ge 1,$
\[
W^a_n(x,s) = \int_0^s \int_{D_2} dr dy\ g_{s-r}(x, y) f_n(r, y), \quad x \in D_2, s \in (0;t),
\]
where
\[
f_n(r, y) = - \nabla^a_{y} W^a_{n-1}(y, r).
\]
By Item 4 of Lemma \ref{lem:calculus}
\begin{equation}
\label{eq:bound.f_n}
\sup_{n\ge 1} \mathrm{e}sssup_{r\in(0;t), y\in D_2} |f_n(r,y)| \le F r^{-\frac{1}{2}},
\mathrm{e}nd{equation}
for an absolute constant $F.$
Consider for $h>0$
\begin{align}
\label{eq:defn.H_k}
W^a_n(x, s+h) - W^a_n(x, s)&= \int_0^s \int_{D_2} dr dy\ \left( g_{s+h-r}(x,y) - g_{s-r}(x,y)\right) f_n(r,y) \notag \\
& \quad + \int_s^{s+h} \int_{D_2} dr dy\ g_{s+h-r}(x,y) f_n(r,y) \notag \\
=& H_1(h,x,s) + H_2(h,x,s).
\mathrm{e}nd{align}
Given a compactly supported in $D_2$ function $v\in C^{\infty}(D_2)$ we have, by the Fubini theorem, properties of the Gaussian density and the Green formula,
\begin{align*}
\label{eq:10}
h^{-1}\int_{D_2} dx\ & v(x) H_1(h,x, s) = \int_{D_2} dx\ v(x) h^{-1} \int_0^s \int_{D_2} dr dy \int_{s-r}^{s+h-r} d\tau\ \partial_\tau g_\tau(x,y)f_n(r,y) \notag \\
&= \int_0^s \int_{D_2} dr dy \int_{s-r}^{s+h-r} d\tau\ h^{-1} f_n(r,y) \int_{D_2} dx\ \partial_\tau g_\tau(x,y) v(x) \notag \\
&= \int_0^s \int_{D_2} dr dy \int_{s-r}^{s+h-r} d\tau\ h^{-1} f_n(r,y) \int_{D_2} dx\ \frac{1}{2}\Delta_{x} g_\tau(x,y) v(x) \notag \\
&= -\frac{1}{2} \int_0^s \int_{D_2} dr dy \int_{s-r}^{s+h-r} d\tau\ h^{-1} f_n(r,y) \int_{D_2} dx\ \nabla_{x} g_\tau(x,y) \cdot \nabla_x v(x) \notag \\
&= -\frac{1}{2} \int_{D_2} \int_0^s \int_{D_2} dx dr dy \ \nabla_x v(x) \cdot \left( h^{-1} \int_{r}^{r+h} d\tau\ f_n(s-r,y) \nabla_{x} g_\tau(x,y) \right),
\mathrm{e}nd{align*}
where $\cdot$ stands for the inner product in ${\mathbb R}^2.$ For all $r$
\begin{equation*}
\label{eq:11}
h^{-1} \int_{r}^{r+h} d\tau\ \partial_{x_k} g_\tau(x,y) \to \partial_{x_k} g_{r}(x,y), \quad k =1,2.
\mathrm{e}nd{equation*}
Thus to prove
\begin{align}
\label{eq:13}
h^{-1}\int_{D_2} dx\ v(x) H_1(h,x, s) &\to -\frac{1}{2}\int_{D_2}\int_0^s \int_{D_2} dx dr dy\ \nabla_x v(x) \cdot f_n(r,y) \nabla_{x} g_{s-r}(x,y) \notag \\
&= -\frac{1}{2}\int_{D_2}dx \ \nabla_x v(x) \cdot \nabla_x W^a_n(x,s), \quad h\to 0+,
\mathrm{e}nd{align}
by the means of dominated convergence theorem it is sufficient to show that both families
\begin{align*}
&\left\{ (x,r,y) \mapsto h^{-1} \partial_{x_k} v(x) f_n(s-r,y) \int_{r}^{r+h} d\tau\ \partial_{x_k} g_\tau(x,y) \mid h\in (0;1) \right\}, \\ &\qquad \qquad k= 1,2,
\mathrm{e}nd{align*}
are uniformly integrable on $\mathop{\rm supp} v \times (0;t) \times D_2.$ For that, note that by Lemma \ref{lem:calculus} and \mathrm{e}qref{eq:bound.f_n} for almost all $(r,y)$
\begin{align*}
\left| h^{-1} f_n(s-r,y) \int_{r}^{r+h} d\tau\ \partial_{x_k} g_\tau(x,y) \right| &\le KF h^{-1} (s-r)^{-\frac{1}{2}}\int_{r}^{r+h} d\tau\ \tau^{-\frac{3}{2}} \mathrm{e}^{-\gamma\frac{\|x-y\|^2}{\tau}},
\mathrm{e}nd{align*}
where, for some positive $C,$
\begin{align*}
h^{-1} \int_{r}^{r+h} d\tau\ \tau^{-\frac{3}{2}} \mathrm{e}^{-\gamma\frac{\|x-y\|^2}{\tau}} & \le
\begin{cases}
r^{-\frac{3}{2}} \mathrm{e}^{-\gamma\frac{\|x-y\|^2}{r}}, & \frac{2}{3} < \frac{r}{\gamma \|x-y\|^2}, \\
(r+h)^{-\frac{3}{2}} \mathrm{e}^{-\gamma\frac{\|x-y\|^2}{r+h}}, & \frac{2}{3} > \frac{r+h}{\gamma \|x-y\|^2}, \\
\frac{C}{\|x-y\|^3}, & \frac{2}{3}\in \left[\frac{r}{\gamma \|x-y\|^2};\frac{r+h}{\gamma \|x-y\|^2}\right], \\
\mathrm{e}nd{cases}
\mathrm{e}nd{align*}
so that, for some $C_1, C_2 > 0,$ all $\alpha\in(1;\frac{4}{3})$ and $\delta\in(0;1),$ we have, due to Item 2 of Lemma \ref{lem:calculus}, for almost all $r$
\begin{align*}
\label{eq:un.integr.est}
& \int_{D_2} dy \left| f_n(s-r,y) h^{-1} \int_{r}^{r+h} d\tau\ \partial_{x_k} g_\tau(x,y) \right|^\alpha \notag \\
& \qquad \le \frac{C_1}{(s-r)^{\frac{\alpha}{2}}} \int_{D_2} dy \Bigg\{ r^{-\frac{3\alpha}{2}} \mathrm{e}^{-\frac{\alpha\gamma \|x-y\|^2}{r}} + (r+h)^{-\frac{3\alpha}{2}} \mathrm{e}^{-\frac{\alpha\gamma \|x-y\|^2}{r+h}} \notag \\
& \qquad \qquad + \frac{1}{\|x-y\|^{3\alpha}} 1\!\!\,{\rm I}\left[\frac{2}{3}\in \left[\frac{r}{\gamma \|x-y\|^2};\frac{r+h}{\gamma \|x-y\|^2}\right]\right] \Bigg\} \notag \\
& \qquad \le \frac{C_1}{(s-r)^{\frac{\alpha}{2}}} \Bigg\{ \frac{\pi}{\alpha\gamma} \left( r^{1-\frac{3\alpha}{2}} + (r+h)^{1-\frac{3\alpha}{2}} \right) + 2\pi \int_0^\infty d\rho \ \rho^{1-3\alpha} 1\!\!\,{\rm I}\left[ \frac{r}{\rho^2} \le \frac{2\gamma}{3} \le \frac{r+h}{\rho^2} \right] \Bigg\} \notag \\
& \qquad \le \frac{C_1 \left(\frac{\pi}{\alpha\gamma} + 2\pi\right)}{(s-r)^{\frac{\alpha}{2}}} \Bigg\{ \left( r^{1-\frac{3\alpha}{2}} + (r+h)^{1-\frac{3\alpha}{2}} \right) + \int_{\sqrt{ \frac{3r}{2\gamma}}}^{\sqrt{\frac{3(s+1)}{2\gamma}}} d\rho \ \rho^{1-3\alpha} \Bigg\} \notag \\
& \qquad \le \frac{C_1 \left(\frac{\pi}{\alpha\gamma} + 2\pi\right)}{(s-r)^{\frac{\alpha}{2}}} \Bigg\{ \left( r^{1-\frac{3\alpha}{2}} + (r+h)^{1-\frac{3\alpha}{2}} \right) + \left(\frac{3r}{2\gamma}\right)^{1-\frac{3\alpha+\delta}{2}} \int_0^{\sqrt{\frac{3(s+1)}{2\gamma}}} d\rho \ \rho^{-1+\delta} \Bigg\} \notag \\
& \qquad \le \frac{C_2}{(s-r)^{\frac{\alpha}{2}}} \Bigg\{ \left( r^{1-\frac{3\alpha}{2}} + (r+h)^{1-\frac{3\alpha}{2}} \right) + r^{1-\frac{3\alpha+\delta}{2}} (s+1)^{\frac{\delta}{2}} \Bigg\}.
\mathrm{e}nd{align*}
Choosing $\delta$ in such a way that $1-\frac{3\alpha+\delta}{2} > -1$ gives, for some $C_3 >0,$
\begin{align*}
&\sup_{x\in D_2} \int_0^s \int_{D_2} dr dy \left| h^{-1} f_n(s-r,y) \int_{r}^{r+h} d\tau\ \partial_{x_k} g_\tau(x,y) \right|^\alpha \notag \\
& \qquad \le C_2 \Bigg\{ 2 \int_{0}^{s}dr \ \frac{r^{1-\frac{3\alpha}{2}}}{(s-r)^{\frac{\alpha}{2}}} + (s+1)^{\frac{\delta}{2}} \int_0^s dr \ \frac{r^{1-\frac{3\alpha+\delta}{2}}}{(s-r)^{\frac{\alpha}{2}}} \Bigg\} \notag \\
& \qquad \le C_3 \int_0^s dr \ \frac{r^{1-\frac{3\alpha+\delta}{2}}}{(s-r)^{\frac{\alpha}{2}}} \notag \\
& \qquad < +\infty,
\mathrm{e}nd{align*}
which yields \mathrm{e}qref{eq:13}.
For $H_2,$ we have, in terms of the process $\xi^a_x$ and the stopping time $\theta^a_x$ defined in \mathrm{e}qref{eq:xi^a}-\mathrm{e}qref{eq:theta^a}
\begin{align*}
h^{-1} H_2(h,x, s) = h^{-1} \int_s^{s+h} dr\ k_{x,s}(h,r),
\mathrm{e}nd{align*}
where
\[
k_{x,s}(h,r) = \E f_n\left(r, \xi^a_x(s+h-r)\right) 1\!\!\,{\rm I}\left[ \theta^a_x > s+h-r \right].
\]
Since $a$ is bounded, the Novikov condition holds on $(0;t)$ so by the Girsanov theorem
\label{eq:17}
\begin{align*}
k_{x,s}(h,r) = \E f_n\left(r, \xi^0_x(s+h-r)\right) 1\!\!\,{\rm I}\left[ \theta^0_x > s+h-r \right] {\mathcal E}_t^a,
\mathrm{e}nd{align*}
where ${\mathcal E}_t^a$ is the corresponding stochastic exponential. The family of random variables
\[
\left\{ f_n\left(r, \xi^0_x(s+h-r)\right) 1\!\!\,{\rm I}\left[ \theta^0_x > s+h-r \right] {\mathcal E}_t^a \mid (r,h) \in A\right\},
\]
where $A = \{(u_1,u_2)\mid u_1\in [s;s+u_2], u_2\in [0;1]\},$ is uniformly integrable, so since the distribution of $\theta_x^0$ has no atoms the function $k_{x,s}$ is uniformly continuous on $A$ for any $(x,s).$ Fix $\mathrm{e}psilon > 0.$ Then there exists $h_0$ such that
\[
h^{-1} \int_s^{s+h} dr\ |k_{x,s}(h_0,r) -k_{x,s}(h,r)| \le \mathrm{e}psilon
\]
whenever $h\le h_0.$ By the Lebesgue differentiation theorem for fixed $h_0$
\[
h^{-1} \int_s^{s+h} dr\ k_{x,s}(h_0,r) \to k_{x, s}(h_0,s), \quad h\to0+,
\]
where
\[
k_{x, s}(h_0,s) \to k_{x, s}(0,s) = f_n(s,x), \quad h_0\to 0+.
\]
Therefore
\begin{equation*}
\lim_{h\to 0+} h^{-1} H_2(h,x, s) = \lim_{h_0\to 0+} k_{x,s}(h_0,s) = f_n(s,x) = -\nabla^a_x W^a_{n-1}(x,s).
\mathrm{e}nd{equation*}
Consider the analog of \mathrm{e}qref{eq:defn.H_k} for a negative increment: for $h>0$
\begin{align*}
W^a_n(x, s-h) - W^a_n(x, s)&= \int_0^{s-h} \int_{D_2} dr dy\ \left( g_{s-h-r}(x,y) - g_{s-r}(x,y)\right) f_n(r,y) \\
& \quad + \int_{s-h}^{s} \int_{D_2} dr dy\ g_{s-h-r}(x,y) f_n(r,y) \notag \\
=& \widetilde H_1(h,x,s) + \widetilde H_2(h,x,s),
\mathrm{e}nd{align*}
and the reasoning for $\widetilde H_1$ and $\widetilde H_2$ is the same as for their counterparts $H_1$ and $H_2,$ which concludes the proof.
\qed
\begin{theorem}
\label{th:W^a.prob.repr}
Let $a\in L_\infty({\mathbb R}).$ For all $s>0$ and $x\in D_2$
\[
W^a(x,s) = \Prob\left( \theta^a_x > s\right)
\]
where $\theta^a_x$ is defined in \mathrm{e}qref{eq:theta^a}.
\mathrm{e}nd{theorem}
\proof Assume $a\in C^\infty({\mathbb R})$ additionally. It follows from Proposition \ref{prop:distr.solution.W_n} and the completeness of the space of Schwartz distributions that in the sense of Schwartz distributions
\[
\partial_s W^a = \frac{1}{2} \Delta W^a- \nabla^a_x W^a
\]
in $(0;\infty)\times D_2.$ Since the operator $\frac{1}{2} \Delta - \nabla^a_x$ is hypoelliptic, $W^a\in C^\infty(D_2 \times (0;\infty))$ \cite[Theorem 3.4.1]{Stroock2008Partial}. By Item 4 of Lemma \ref{lem:calculus}, $W^a\in C(\overline D_2 \times (0;\infty)) \cap C(D_2 \times [0;\infty)).$ Since
\[\
g_r((u,u), y) = 0, \quad r\ge 0, u\in{\mathbb R}, y\in{\mathbb R}^2,
\] $W^a$ satisfies the Dirichlet boundary condition as a function. To show
\begin{equation}
\label{eq:prob.repr}
W^a(x,s) = \Prob\left( \theta^a_x > s\right),
\mathrm{e}nd{equation}
one proceeds as follows.
Let
\[
K_n = \left\{ x\in D_2 \mid \|x\| \le n, \inf_{y\in\partial D_2} \|x -y\| \ge \frac{1}{n} \right\}, \quad n\ge 1,
\]
and define, for fixed $T>0,$
\begin{align*}
\theta^{n}_{t,x; T} &= T \wedge \inf\left\{s \ge t \mid \xi^a_{t,x}(s) \not\in K_n \right\}, \\
\theta_{t,x} &= \inf\left\{s \ge t \mid \xi^a_{t,x}(s) \not\in D_2 \right\}, \quad t > 0, x \in D_2,
\mathrm{e}nd{align*}
where, analogously to \mathrm{e}qref{eq:xi^a}, for a standard Wiener process $w$
\begin{align*}
d\xi^a_{t,x}(s)& = - a(\xi^a_{t,x}(s))ds + dw(s), \quad s \ge t, \nonumber \\
\xi^a_{t,x}(t)& = x.
\mathrm{e}nd{align*}
The standard reasoning via the It\'o formula and the exhaustion method (e.g. \cite[\S 2.2]{Freid85Functional} or \cite[Ch.8, \S 5]{GikhSko77IntroEng}) gives, for any fixed $h>0, t \in (0; T),$
\begin{equation}
\label{eq:repr.}
\E W^a\left(h+T-\theta^n_{t,x; T}, \xi^a_{t,x}(\theta^{n}_{t,x; T})\right) = W^a(h+T-t,x), \quad n\ge 1, x\in D_2.
\mathrm{e}nd{equation}
Since $W^a\in C(D_2 \times [0;\infty))$
\begin{equation*}
\label{eq:repr.rhs}
W^a(h+T-t,x) \to W^a(T-t, x), \quad h\to 0+.
\mathrm{e}nd{equation*}
We have
\begin{align*}
\label{eq:repr.lhs}
\E W^a\left(h+T-\theta^n_{t,x; T}, \xi^a_{t,x}(\theta^{n}_{t,x; T})\right) = \E W^a\left(h + T - t - \theta^n_{0,x; T-t}, \xi^a_{0,x}(\theta^n_{0,x; T-t})\right).
\mathrm{e}nd{align*}
The process $\xi^a_{0,x}$ is bounded on $[0;T-t]$ with probability $1.$ On the set $\{\theta_{0,x} \le T-t\}$
\[
W^a\left(h + T - t - \theta^n_{0,x; T-t}, \xi^a_{0,x}(\theta^n_{0,x; T-t})\right) \to W^a\left(h + T - t - \theta_{0,x}, \xi^a_{0,x}(\theta_{0,x})\right) = 0, \quad n\to\infty,
\]
since $W^a\in C(\overline D_2 \times (0;\infty)).$ On the set $\{\theta_{0,x} > T-t\}$
\[
W^a\left(h + T - t - \theta^n_{0,x; T-t}, \xi^a_{0,x}(\theta^n_{0,x; T-t})\right) \to W^a\left(h, \xi^a_{0,x}(T-t)\right), \quad n\to\infty,
\]
and, since $\xi^a_{0,x}(T-t)\in D_2$ now and $W^a\in C(D_2 \times [0;\infty)),$
\[
W^a\left(h, \xi^a_{0,x}(T-t)\right) \to W^a(0,x)=1, \quad h\to 0+.
\]
Returning to \mathrm{e}qref{eq:repr.}, passing to limit w.r.t $n$ and $h$ and introducing $s= T-t > 0$, we get
\[
W^a(x, s) = \E 1\!\!\,{\rm I}\left[\theta_{0,x} > s \right],
\]
which coincides with \mathrm{e}qref{eq:prob.repr}.
Consider the case $a\in L_{\infty}.$ Let $\mathrm{e}ta$ be a mollifier and put $a_n = a \ast n\mathrm{e}ta(\frac{\cdot}{n}), n\ge 1.$ Additionally put $a_0 = a.$ Then $a_n\in C^\infty({\mathbb R}),$ $|a_n| \le |a|, n\ge 1,$ and $a_n\to a, n\to\infty,$ in $L_1({\mathbb R}).$ Consider $s\in (0;t)$ for a fixed $t.$ Then the Novikov condition
\[
\sup_{n\ge 0} \E \mathrm{e}xp\Big\{ \frac{1}{2} \int_0^t ds \sum_{j=1}^2 a_n^2 \left(\xi^a_{x_j}(s)\right)\Big\} < \infty,
\]
holds, where $\xi^a_x = \xi^a_{0, x}.$ Thus, by the Girsanov theorem,
\begin{align*}
\Prob\left(\theta^{a_n}_x > s\right) =& \E \varkappa{\mathcal E}_n,
\mathrm{e}nd{align*}
where
\begin{align*}
\varkappa =& 1\!\!\,{\rm I}\left[\theta^0_x > s\right], \\
{\mathcal E}_n =& \mathrm{e}xp\Big\{\sum_{j=1}^2 \int_0^t a_n \left(\xi^0_{x_j}(r)\right) d\xi^0_{x_j}(r) - \frac{1}{2}\sum_{j=1}^2 \int_0^t dr\ a_n^2 \left(\xi^0_{x_j}(r)\right) \Big\}, \\
& n\ge 0.
\mathrm{e}nd{align*}
Since $a\in L_\infty({\mathbb R})$ the Cauchy inequality implies that the sequence $\{\varkappa{\mathcal E}_n\}_{n\ge 1}$ is uniformly integrable. Consequently, in order to prove
\begin{equation}
\label{eq:add.1}
\Prob\left(\theta^{a_n}_x > s\right) = \E \varkappa{\mathcal E}_n \to \E\varkappa{\mathcal E}_0 = \Prob\left(\theta^{a}_x > s\right), \quad n\to\infty,
\mathrm{e}nd{equation}
it is sufficient to check that ${\mathcal E}_n\to {\mathcal E}_0$ in probability. For that, consider, for fixed $j,$
\begin{align}
\label{eq:add}
\E &\left|\int_0^t dr\ a_n^2 \left(\xi^0_{x_j}(r)\right) - \int_0^t dr\ a_0^2 \left(\xi^0_{x_j}(r)\right) \right|\le \int_0^t dr \E \left| a_n^2 \left(\xi^0_{x_j}(r)\right) -a_0^2 \left(\xi^0_{x_j}(r)\right) \right|.
\mathrm{e}nd{align}
Since for every $s$ the sequence
\[
\Big\{ u\mapsto \frac{\left| a^2_n(u) -a^2_0(u) \right|\mathrm{e}^{-\frac{\|u-x_j\|^2}{2s}}}{s} \mid n\ge 1 \Big\}
\]
is uniformly integrable w.r.t. the Lebesgue measure in ${\mathbb R}$ the dominated convergence theorem implies
\[
\E \left| a_n^2 \left(\xi^0_{x_j}(r)\right) -a_0^2 \left(\xi^0_{x_j}(r)\right)\right| \to 0, \quad n\to\infty,
\]
for all $s\in(0;t).$ Using the dominated convergence theorem one more time in \mathrm{e}qref{eq:add} we obtain
\[
\int_0^t dr\ a_n^2 \left(\xi^0_{x_j}(r)\right) \to \int_0^t dr\ a_0^2 \left(\xi^0_{x_j}(r)\right), \quad n\to\infty,
\]
in probability. Analogously, one proves
\[
\int_0^t a_n \left(\xi^0_{x_j}(r)\right) d\xi^0_{x_j}(r) \to \int_0^t a_0 \left(\xi^0_{x_j}(r)\right) d\xi^0_{x_j}(r), \quad n\to\infty,
\]
in probability, which finishes the proof of \mathrm{e}qref{eq:add.1}.
Thus
\[
W^{a_n}(x,s) = \Prob\left( \theta^{a_n}_x > s\right) \to \Prob\left( \theta^{a}_x > s\right), \quad n\to\infty.
\]
Therefore, it is left to show
\begin{equation}
\label{eq:add.4}
W^{a_n}(x,s)\to W^{a}(x,s), \quad n\to\infty.
\mathrm{e}nd{equation}
Note that by Item 4 of Lemma \ref{lem:calculus} for all $s\in(0;t)$ and $x\in D_2$
\begin{equation}
\label{eq:add.2}
\sup_{n\ge 1} \left| \sum_{m\ge M} W^{a_n}_m(x,s) -\sum_{m\ge M} W^{a}_m(x,s)\right| \to 0, \quad M \to\infty.
\mathrm{e}nd{equation}
For all $m\ge 1$ by Lemma \ref{lem:calculus}
\begin{align*}
&\Big| \prod_{j={1}}^m \nabla^a_{y_j} g_{r_j-r_{j-1}}(y_j,y_{j-1}) - \prod_{j={1}}^m \nabla^{a_n}_{y_j} g_{r_j-r_{j-1}}(y_j,y_{j-1})\Big| \\
& \qquad \le \sum_{k=1}^m \Big| \prod_{j={1}}^{k-1} \nabla^{a_n}_{y_j} g_{r_j-r_{j-1}}(y_j,y_{j-1})\ \nabla^{a_n-a}_{y_k} g_{r_k-r_{k-1}}(y_k,y_{k-1}) \prod_{j={k+1}}^m \nabla^{a}_{y_j} g_{r_j-r_{j-1}}(y_j,y_{j-1})\Big| \\
&\qquad \le K^m \|a\|_{L_\infty({\mathbb R})}^{m-1} \prod_{j={1}}^{m} \frac{\mathrm{e}^{-\gamma \frac{\|y_1-y_2\|^2}{r_j-r_{j-1}}}}{(r_j-r_{j-1})^{3/2}} \sum_{k=1}^m \sum_{j=1}^2 |a_n(y_{k,j})-a(y_{k,j})| ,
\mathrm{e}nd{align*}
where the product over the empty set is equal to $1$ by definition. Therefore, due to \mathrm{e}qref{eq:add.2} and the definition of $W^a$ it is sufficient to show that
\[
\int_{\Delta_m(s)} dr_1 \ldots dr_m \int_{D_2^{m+1}} dy_0 \ldots dy_m\ \prod_{j={1}}^{m} \frac{\mathrm{e}^{-\gamma \frac{\|y_1-y_2\|^2}{r_j-r_{j-1}}}}{(r_j-r_{j-1})^{3/2}} |a_n(y_{k,j})-a(y_{k,j})| \to 0, \quad n\to\infty,
\]
for fixed $m$ and $j.$ Setting $f_n(y_{k,j}) = |a_n(y_{k,j})-a(y_{k,j})|$ in Lemma \ref{lem:approx.a} and applying Items 2 and 3 of Lemma \ref{lem:calculus} finishes the proof. We omit the details.\qed
\begin{theorem}
\label{th:density.prop.repr}
Let $a\in L_\infty({\mathbb R}).$ Then for any $t>0$ the density $p^{a,1}_t$ has a continuous version
\[
p^{a,1}_t(x) = \partial_{x_2} W^a((x,x),t) = \sum_{n\ge 0} \partial_{x_2} W^a_n((x,x),t).
\]
\mathrm{e}nd{theorem}
\proof Combining Proposition \ref{lem:density.via.prob}, Theorem \ref{th:W^a.prob.repr} and the fact that $W^a((x,x), t) = 0$ we get
\begin{align*}
p^{a,1}_t(x) &= \lim_{\delta\to 0+} \delta^{-1} \Prob\left(\theta^{a}_{(x,x+\delta)} > t\right) \notag \\
&= \lim_{\delta\to 0+} \delta^{-1} \left(W^a((x,x+\delta),t) - W^a((x,x), t)\right),
\mathrm{e}nd{align*}
so the conclusion follows by Lemma \ref{lem:calculus} and calculus.
\qed
The next result is stated for a continuous version of the $1-$dimensional density.
\begin{theorem}
\label{th:density.conver}
Assume $a_n\in L_\infty({\mathbb R}), n\ge 0; \sup_{n\ge 0} \|a_n\|_{L_\infty({\mathbb R})}< \infty.$ Let one of the following conditions hold:
\begin{enumerate}
\item $a_n\to a_0, n\to\infty,$ in $L_\infty({\mathbb R});$
\item $a_n\in L_1({\mathbb R}), n\ge 0$, and $a_n\to a_0, n\to\infty,$ in $L_1({\mathbb R}).$
\mathrm{e}nd{enumerate}
Then for any $t>0$ and all $x\in{\mathbb R}$
\[
p^{a_n,1}_t(x)\to p^{a_0,1}_t(x), \quad n\to\infty.
\]
\mathrm{e}nd{theorem}
\proof By Theorem \ref{th:density.prop.repr}, the conclusion of the theorem is equivalent to
\[
\sum_{k\ge 0} \partial_{x_2} W^{a_n}_k((x,x),t) \to \sum_{k\ge 0} \partial_{x_2} W^{a_0}_k((x,x),t), \quad n\to\infty.
\]
Since $\sup_{n\ge 0} \|a_n\|_{L_\infty({\mathbb R})}< \infty,$ one applies the same reasoning as in the proof of \mathrm{e}qref{eq:add.4}, utilizing the uniform estimates for $\partial_{x_k}W^{a_n}(x,s)$ of Item 4 in Lemma \ref{lem:calculus}, and uses both statements of Lemma \ref{lem:approx.a} subsequently. We omit the details. \qed
\end{document} |
\begin{document}
\title[Volume invariant and maximal representation]
{Volume invariant and maximal representations of discrete subgroups of Lie groups}
\author{Sungwoon Kim}
\address{School of Mathematics,
KIAS, Hoegiro 85, Dongdaemun-gu, Seoul, 130-722, Republic of Korea}
\email{[email protected]}
\author{Inkang Kim}
\address{School of Mathematics
KIAS, Hoegiro 85, Dongdaemun-gu, Seoul, 130-722, Republic of Korea}
\email{[email protected]}
\footnotetext[1]{2000 {\sl{Mathematics Subject Classification.}}
22E46, 57R20, 53C35}
\footnotetext[2]{{\sl{Key words and phrases.}}
volume invariant, lattice, representation variety, semisimple Lie group, Toledo invariant, maximal representation.}
\footnotetext[3]{The second
author gratefully acknowledges the partial support of KRF grant
(0409-20060066).}
\begin{abstract}
Let $\Gamma$ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors.
We introduce a volume invariant for representations of $\Gamma$ into
$G$, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman.
Then, we show that the maximality of this volume invariant exactly characterizes discrete, faithful representations of $\Gamma$ into $G$.
\end{abstract}
\maketitle
\section{Introduction}
A volume invariant is defined to characterize discrete, faithful representations of a discrete group $\Gamma$
into a connected semisimple Lie group $G$. For a uniform lattice $\Gamma$, Goldman \cite{Go92} introduced a volume invariant $\upsilon(\rho)$ of a representation $\rho \colon\thinspace \Gamma \rightarrow G$ as follows:
Let $X$ be the associated symmetric space of dimension $n$ and $M=\Gamma\backslash X$.
To every representation $\rho \colon\thinspace \Gamma \rightarrow G$, a bundle $E_\rho$ over $M$ with fibre $X$ and structure group $G$ is associated.
One can obtain a closed $n$--form $\omega_\rho$ on $E_\rho$ by
spreading the $G$--invariant volume form $\omega$ on $X$ over the fibres of $E_\rho$. Then, the volume invariant $\upsilon(\rho)$ of $\rho$ is defined by
$$ \upsilon (\rho) = \int_M f^*\omega_\rho,$$
where $f$ is a section of $E_\rho$.
The definition of the volume invariant $\upsilon(\rho)$ is independent of the choice of a section since $X$ is contractible. It can be easily seen that the volume invariant $\upsilon(\rho)$ satisfies an inequality
\begin{eqnarray}\label{eqn:1.1} |\upsilon(\rho)| \leq \mathrm{Vol}(M),\end{eqnarray} which recovers the Milnor-Wood inequality
for $G=\mathrm{PSL}_2(\mathbb{R})$. Note that the volume invariant $\upsilon(\rho)$ is available only for representations of uniform lattices.
Goldman \cite{Go92} conjectured the following and gave a positive answer for all connected semisimple Lie groups except for $\mathrm{SU}(n,1), \mathrm{Sp}(n,1), \mathrm{F}_4^{-20}$.
\begin{conj}\label{con:1.1}
Equality holds in (\ref{eqn:1.1}) if and only if $\rho$ is a discrete, faithful representation
of $\Gamma$ into $G$.
\end{conj}
Numerical invariants such as the volume invariant have been used to study a representation variety $\mathrm{Hom}(\Gamma,G)$ consisting of homomorphisms $\rho \colon\thinspace \Gamma \rightarrow G$.
For example, Goldman \cite{Go88} characterized $(4g-3)$--connected components of the representation variety $\mathrm{Hom}(\pi_1(S),\mathrm{PSL}_2 (\mathbb{R}))$ for a closed surface $S$ of genus $g$ via the Toledo invariant.
Moreover, he verified that the connected component of $\mathrm{Hom}(\pi_1(S),\mathrm{PSL}_2 (\mathbb{R}))$ with maximal Toledo invariant is exactly the embedding of the Teichm\"{u}ller space of $S$ into $\mathrm{Hom}(\pi_1(S),\mathrm{PSL}_2 \mathbb{R})$ \cite{Go80}.
Burger, Iozzi and Wienhard \cite{BIW10} generalize the theories of a closed surface representation variety in $\mathrm{PSL}_2 (\mathbb{R})$ to other Lie groups such as split simple Lie groups and Lie groups of Hermitian type.
In comparison with uniform lattices, numerical invariants for representations of nonuniform lattices have been rarely defined. The main reason for this is that the fundamental class of open manifolds vanishes in the top dimensional singular homology. Recently, Burger, Iozzi and Wienhard \cite{BIW11} define
the Toledo invariant for representations of a compact surface with boundary by using its relative
fundamental class. Then, they show that this Toledo invariant exactly detects hyperbolic structures on the surface.
The aim of this paper is to introduce a new invariant for representations of arbitrary lattices $\Gamma$ in $G$ which detects discrete, faithful representations in the representation variety $\mathrm{Hom}(\Gamma,G)$. One advantage of the new invariant is that it provides a tool for studying the representation varieties of nonuniform lattices in semisimple Lie groups. In addition, we explore the relation between the new invariant and $\upsilon(\rho)$. Then, we give a proof of Conjecture \ref{con:1.1}.
Let $\Gamma$ be a lattice in $G$. Every representation $\rho \colon\thinspace \Gamma \rightarrow G$ induces canonical pullback maps $\rho^*_b \colon\thinspace H^\bullet_{c,b}(G,\mathbb{R})\rightarrow H^\bullet_b(\Gamma,\mathbb{R})$ in continuous bounded cohomology.
Let $c \colon\thinspace H^\bullet_{c,b}(G,\mathbb{R})\rightarrow H^\bullet_c(G,\mathbb{R})$ be the comparison map induced
from the inclusion of the continuous bounded cochain complex of $G$ into the continuous cochain complex of $G$.
The Van Est isomorphism gives an isomorphism $H^n_c(G,\mathbb{R})\colon\thinspaceng \mathbb{R}\cdot \omega$,
where $\omega$ is the $G$--invariant volume form on the associated symmetric space $X$.
Then, we define a new invariant $\mathrm{Vol}(\rho)$ by
$$\mathrm{Vol}(\rho) = \inf \{ |\langle \rho^*_b(\omega_b),\alpha \rightarrowngle| \text{ }|\text{ }c(\omega_b)=\omega \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \},
$$
where $[M]^{\ell^1}_\mathrm{Lip}$ is the set of all $\ell^1$--homology classes in $H^{\ell^1}_n(M,\mathbb{R})$ that are represented by at least one locally finite fundamental cycle with finite Lipschitz constant.
Note that $\rho^*_b (\omega_b)$ is regarded as a bounded
cohomology class in $H^n_b(M,\mathbb{R})$ by the canonical isomorphism between $H^n_b(\Gamma,\mathbb{R})$ and $H^n_b(M,\mathbb{R})$. Thus, $\rho^*_b(\omega_b)$ can be evaluated on $\ell^1$--homology classes in $H^{\ell^1}_n(M,\mathbb{R})$ and hence, the definition of $\mathrm{Vol}(\rho)$ makes sense.
For more details on the definition and properties of the volume invariant $\mathrm{Vol}(\rho)$, see Section \ref{sec:3}.
An essential ingredient in defining the volume invariant $\mathrm{Vol}(\rho)$ is the geometric simplicial volume of $M$, introduced by Gromov \cite{Gr82}.
Indeed, Gromov defined two kinds of simplicial volumes for open Riemannian manifolds. One is defined as the $\ell^1$--seminorm of the locally finite fundamental class of $M$. This is a topological invariant. The other is defined by the infimum over all $\ell^1$--norms of locally finite fundamental cycles of $M$ with finite Lipschitz constant.
The latter is called the geometric simplicial volume of $M$ because the Riemannian structure on $M$ is involved in its definition. Note that this is not a topological invariant anymore.
One can notice that the volume invariant $\mathrm{Vol}(\rho)$ can be defined via
locally finite fundamental cycles of $M$ instead of locally finite fundamental cycles with finite Lipschitz constant.
However, it turns out that if the volume invariant $\mathrm{Vol}(\rho)$ is defined via locally finite fundamental cycles, then this invariant does not always detect discrete, faithful representations. For further discussion of this, see Section \ref{sec:3.1}.
\begin{thm}\label{thm:1.2}
Let $\Gamma$ be an irreducible lattice in a connected semisimple Lie group $G$ with trivial center and
no compact factors. Let $\rho \colon\thinspace \Gamma \rightarrow G$ be a representation. Then, the volume invariant $\mathrm{Vol}(\rho)$ satisfies an inequality
$$ \mathrm{Vol}(\rho) \leq \mathrm{Vol}(M),$$
where $X$ is the associated symmetric space and $M=\Gamma\backslash X$.
Moreover, equality holds if and only if
$\rho$ is a discrete, faithful representation.
\end{thm}
Theorem \ref{thm:1.2} implies that the volume invariant $\mathrm{Vol}(\rho)$ exactly characterizes discrete, faithful representations
in the representation variety $\mathrm{Hom}(\Gamma,G)$. In particular, when $\Gamma$ is a uniform lattice, we verify that
\begin{eqnarray}\label{eqn:1.2} \mathrm{Vol}(\rho)=|\upsilon(\rho)|. \end{eqnarray}
From the view of Equation (\ref{eqn:1.2}), the volume invariant $\mathrm{Vol}(\rho)$ can be regarded as an invariant for representations of arbitrary lattices extending the volume invariant $\upsilon(\rho)$ only for representations of uniform lattices. Note that Theorem \ref{thm:1.2} covers the remaining cases $\mathrm{SU}(n,1), \mathrm{Sp}(n,1), \mathrm{F}_4^{-20}$ that Goldman's proof in \cite{Go92} did not cover. In fact, one can easily notice that Conjecture \ref{con:1.1} is able to be proved by using the Besson-Courtois-Gallot technique in \cite{BCG99}.
In a similar way, we define a volume invariant $\mathrm{Vol}(\rho)$ for representations $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ of lattices $\Gamma$ in $\mathrm{SO}(n,1)$. A representation $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ is said to be a \emph{totally geodesic representation} if there is a totally geodesic $\mathbb{H}^n \subset \mathbb{H}^m$ so that the image of the representation lies in the subgroup $G \subset \mathrm{SO}(m,1)$ that preserves this $\mathbb{H}^n$ and that the $\rho$--equivariant map $F \colon\thinspace \mathbb{H}^n \rightarrow \mathbb{H}^m$ is a totally geodesic isometric embedding. Then, we show that this volume invariant characterizes totally geodesic representations.
\begin{thm}
Let $\Gamma$ be a lattice in $\mathrm{SO}(n,1)$ and $M=\Gamma \backslash \mathbb{H}^n$. The volume invariant $\mathrm{Vol}(\rho)$ of a representation $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ for $m\geq n \geq 3$ satisfies an inequality $$\mathrm{Vol}(\rho) \leq \mathrm{Vol}(M).$$
Moreover, equality holds if and only if $\rho$ is a totally geodesic representation.
\end{thm}
Finally using bounded cohomology theory and volume invariant,
we can formulate the local rigidity phenomena of complex hyperbolic uniform lattices. Specially we prove that
\begin{thm}Let $\Gamma\subset \mathrm{SU}(n,1)$ be a uniform lattice and $\rho:\Gamma\rightarrow \mathrm{SU}(m,1)$, $m\geq n \geq 2$ a representation. Then it is
a maximal volume representation if and only if it is a totally geodesic representation. For the natural inclusion
$\Gamma\subset \mathrm{SU}
(n,1)\subset \mathrm{SU}(m,1)\subset \mathrm{Sp}(m,1)$, it is locally rigid, in the sense that the nearby representations stabilize a copy of ${\mathbb H}^n_{\mathbb C}$ inside ${\mathbb H}^m_{\mathbb H}$.
\end{thm}
This paper is organized as follows: We review the simplicial volume, $\ell^1$--homology and continuous (bounded) cohomology in order to define the new invariant $\mathrm{Vol}(\rho)$ in Section \ref{sec:2}. We describe the basic properties of the volume invariant $\mathrm{Vol}(\rho)$ in Section \ref{sec:3}. Then, we devote ourselves to proving Theorem \ref{thm:1.2} for the case that $G$ is a semisimple Lie group of higher rank in Section \ref{sec:4}, $G$ is a simple Lie group of rank $1$ except for $\text{SO}(2,1)$ in Section \ref{sec:5} and $G$ is $\mathrm{SO}(2,1)$ in Section \ref{sec:6}.
We deal with a volume invariant for representations $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ of lattices $\Gamma$ in $\mathrm{SO}(n,1)$ in Section \ref{sec:7}. Lastly, we reformulate the rigidity phenomenon of uniform lattices of $\mathrm{SU}(n,1)$ in $\mathrm{SU}(m,1)$ or $\mathrm{Sp}(m,1)$ via the volume invariant in Section \ref{sec:8}.
\section{Preliminaries}\label{sec:2}
\subsection{Simplicial volume}
Let $M$ be an $n$--dimensional manifold. The simplicial $\ell^1$--norm $\| \cdot \|_1$ on the singular chain complex $C_\bullet(M,\mathbb{R})$ is defined by
the $\ell^1$--norm with respect to the basis given by all singular simplices.
The simplicial $\ell^1$--norm induces a $\ell^1$--seminorm on $H_\bullet(M,\mathbb{R})$ as follows:
$$\| \alpha \|_1 =\inf \| c \|_1$$
where $c$ runs over all singular cycles representing $\alpha \in H_\bullet(M,\mathbb{R})$.
For an oriented, connected, closed $n$--manifold $M$, the simplicial volume $\| M \|$ of $M$ is defined as the $\ell^1$--seminorm of the fundamental class $[M]$ in $H_n(M,\mathbb{R})$.
If $M$ is an oriented, connected, open $n$--manifold, then $M$ has a fundamental class $[M]$ in the locally finite homology $H^\mathrm{lf}_n(M,\mathbb{R})$.
The locally finite homology of $M$ is defined as the homology of the locally finite chain complex $C_\bullet^\mathrm{lf}(M,\mathbb{R})$.
More precisely, let $S_k(M)$ be the set of singular $k$--simplices of $M$ and $S^\mathrm{lf}_k(M)$ denote the set of all locally finite subsets of $S_k(M)$, that is, if $A \in S^\mathrm{lf}_k(M)$, any compact subset of $M$ intersects the image of only finitely many elements of $A$. Then, the locally finite chain complex $C_\bullet^\mathrm{lf}(M,\mathbb{R})$ is defined by
$$C_\bullet^\mathrm{lf}(M,\mathbb{R})= \left\{\sum_{\sigma \in A} a_\sigma \sigma \ \bigg| \ A \in S^\mathrm{lf}_\bullet(X) \text{ and }a_\sigma \in \mathbb{R} \right\}.$$
A $\ell^1$--seminorm on $H^\mathrm{lf}_\bullet(M,\mathbb{R})$ is
induced from the simplicial $\ell^1$--norm on the locally finite chain complex $C^\mathrm{lf}_\bullet(M,\mathbb{R})$ with respect to the basis given by all singular simplices. The simplicial volume $\| M \|$ of $M$ is defined as the $\ell^1$--seminorm of the locally finite fundamental class $[M]$ of $M$.
In addition, Gromov introduces the geometric simplicial volume of oriented, connected, open Riemannian manifolds. Fixing a metric on the standard $k$--simplex $\Delta^k$ by the Euclidean metric, the Lipschitz constant $\mathrm{Lip}(\sigma)$ of a singular simplex $\sigma \colon\thinspace \Delta^k \rightarrow M$ is defined. Subsequently, for a locally finite chain $c \in C^\mathrm{lf}_\bullet(M,\mathbb{R})$, define the Lipschitz constant $\mathrm{Lip}(c)$ of $c$ by the supremum over all Lipschitz constants of the simplices occurring in $c$.
The subcomplex $C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$ of $C^\mathrm{lf}_\bullet(M,\mathbb{R})$ consisting of all chains with finite Lipschitz constant
induces the homology with Lipschitz locally finite support, denoted by $H^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$. Indeed, $H^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$ is isomorphic to $H^\mathrm{lf}_\bullet(M,\mathbb{R})$ \cite[Theorem 3.3]{LS09}. Hence, it has a distinguished generator $[M]_\mathrm{Lip}$ in $H^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$ corresponding to the locally finite fundamental class $[M]$ in $H^\mathrm{lf}_\bullet(M,\mathbb{R})$. The geometric simplicial volume of $M$ is defined as the $\ell^1$--seminorm of $[M]_\mathrm{Lip}$, denoted by $\| M \|_\mathrm{Lip}$.
Gromov \cite{Gr82} proves the proportionality principle for the geometric simplicial volume as follows.
\begin{thm}[Gromov]\label{thm:2.1}
Let $M$ be a closed Riemannian manifold and $N$ be a complete Riemannian manifold of
finite volume. If the universal covers of $M$ and $N$ are isometric, then
$$\frac{\|M\|_\mathrm{Lip}}{\mathrm{Vol}(M)}=\frac{\|N\|_\mathrm{Lip}}{\mathrm{Vol}(N)}.$$
\end{thm}
The simplicial volume of a smooth manifold gives a lower bound of its minimal volume. Hence,
the question was naturally raised as to which manifolds have nonzero simplicial volumes. Gromov \cite{Gr82} and Thurston \cite{Th78} first
show that the simplicial volume of complete Riemannian manifolds of finite volume with pinched negative sectional curvature is nonzero.
Moreover, it is shown that closed locally symmetric spaces of noncompact type have positive simplicial volumes \cite{LS06}.
In contrast, the simplicial volume of open, complete locally symmetric spaces of noncompact type with finite volume may vanish. For instance,
the simplicial volume of locally symmetric spaces of noncompact type with $\mathbb{Q}$--rank at least $3$ vanishes \cite{LS09}. On the other hand, it turns out that the simplicial volume of $\mathbb{Q}$--rank $1$ locally symmetric spaces covered by a product of $\mathbb{R}$--rank $1$ symmetric spaces is positive \cite{KK} and moreover, it is equal to their geometric simplicial volume \cite{BKK} for amenable boundary group cases. The $\mathbb{Q}$--rank $2$ cases remain open.
\subsection{$\ell^1$--homology}
Let $M$ be an oriented, connected $n$--manifold.
The $\ell^1$--chain complex of $M$ is the $\ell^1$--completion $C_\bullet^{\ell^1}(M,\mathbb{R})$ of
the normed chain complex $C_\bullet(M,\mathbb{R})$ with respect to the simplicial $\ell^1$--norm $\| \cdot \|_1$. Then, the $\ell^1$--homology $H^{\ell^1}_\bullet(M,\mathbb{R})$ of $M$ is defined as the homology of $\ell^1$--chain complex of $M$,
$$H^{\ell^1}_\bullet(M,\mathbb{R}) = H_\bullet ( C_\bullet^{\ell^1}(M,\mathbb{R})).$$
The natural inclusion $C_\bullet(M,\mathbb{R}) \hookrightarrow C_\bullet^{\ell^1}(M,\mathbb{R})$ induces a comparison map $H_\bullet(M,\mathbb{R}) \rightarrow H_\bullet^{\ell^1}(M,\mathbb{R})$. Note that this map is an isometric inclusion because $C_\bullet(M,\mathbb{R})$ is a dense subcomplex of $C_\bullet^{\ell^1}(M,\mathbb{R})$ \cite[Proposition 2.4]{Lo08}.
Similarly, inclusions $C_\bullet(M,\mathbb{R}) \subset C_\bullet^\mathrm{lf}(M,\mathbb{R}) \cap C_\bullet^{\ell^1}(M,\mathbb{R}) \subset C_\bullet^{\ell^1}(M,\mathbb{R})$ imply that the middle complex is dense in $C_\bullet^{\ell^1}(M,\mathbb{R})$. Hence, the induced map $H_\bullet (C_\bullet^\mathrm{lf}(M,\mathbb{R}) \cap C_\bullet^{\ell^1}(M,\mathbb{R})) \rightarrow H^{\ell^1}_\bullet(M,\mathbb{R})$ is an isometric inclusion.
From this point of view, the simplicial volume of $M$ can be computed in terms of the $\ell^1$--homology of $M$ as follows:
$$ \| M \| = \inf \{ \| \alpha \|_1 \text{ }|\text{ }\alpha \in [M]^{\ell^1} \subset H^{\ell^1}_n(M,\mathbb{R}) \},$$
where $[M]^{\ell^1}$ is the set of
all $\ell^1$--homology classes that are represented by at least one locally finite fundamental cycle.
In a similar way, the geometric simplicial volume of $M$ is computed by
$$ \| M \|_\mathrm{Lip} = \inf \{ \| \alpha \|_1 \text{ }|\text{ }\alpha \in [M]^{\ell^1}_\mathrm{Lip} \subset H^{\ell^1}_n(M,\mathbb{R}) \},$$
where $[M]^{\ell^1}_\mathrm{Lip}$ is the set of
all $\ell^1$--homology classes that are represented by at least one locally finite fundamental cycle with finite Lipschitz constant. We refer the reader to \cite[Section 6]{Lo08} for more detailed explanations.
\subsection{Continuous bounded cohomology}
Let $G$ be a topological group. Consider the continuous cocomplex $C^\bullet_c(G,\mathbb{R})$ with the homogeneous coboundary operator, where $$C^k_c(G,\mathbb{R})=\{ f \colon\thinspace G^{k+1} \rightarrow \mathbb{R}\text{ }|\text{ }f \text{ is continuous} \}.$$
The action of $G$ on $C^k_c(G,\mathbb{R})$ is given by
$$(g\cdot f)(g_0,\ldots,g_k)=f(g^{-1}g_0,\ldots,g^{-1}g_k).$$
The continuous cohomology $H^\bullet_c(G,\mathbb{R})$ of $G$ with trivial coefficients is defined as the cohomology of the $G$--invariant continuous cocomplex $C^\bullet_c(G,\mathbb{R})^G$.
For a cochain $f\colon\thinspace G^{k+1} \rightarrow \mathbb{R}$, define its sup norm by
$$\|f\|_\infty = \sup \{ |f(g_0,\ldots,g_k)|\text{ }|\text{ } (g_0,\ldots,g_k)\in G^{k+1}\}.$$
The sup norm turns $C^\bullet_c(G,\mathbb{R})$ into normed real vector spaces. The continuous bounded cohomology $H^\bullet_{c,b}(G,\mathbb{R})$ of $G$ is defined as the cohomology of the subcocomplex $C^\bullet_{c,b}(G,\mathbb{R})^G$ of
$G$--invariant continuous bounded cochains in $C^\bullet_c(G,\mathbb{R})^G$.
The inclusion of $C^\bullet_{c,b}(G,\mathbb{R})^G \subset C^\bullet_c(G,\mathbb{R})^G$ induces a comparison map $c \colon\thinspace H^\bullet_{c,b}(G,\mathbb{R}) \rightarrow H^\bullet_c(G,\mathbb{R})$. The sup norm induces seminorms on both $H^\bullet_c(G,\mathbb{R})$ and $H^\bullet_{c,b}(G,\mathbb{R})$, denoted by $\| \cdot \|_\infty$. Note that for $\beta \in H^k_c(G,\mathbb{R})$,
$$ \|\beta \|_\infty = \inf \{ \| \beta_b \|_\infty \text{ }|\text{ } \beta_b \in H^k_{c,b}(G,\mathbb{R}) \text{ and } c(\beta_b)=\beta \}.$$
For a connected semisimple Lie group $G$ with trivial center and no compact factors, the continuous cohomology $H^\bullet_c(G,\mathbb{R})$ is isomorphic to the set of $G$--invariant differential forms on the associated symmetric space $X$ according to the Van Est isomorphism. In particular,
the continuous cohomology of $G$ in the top degree is generated by the $G$--invariant volume form $\omega$ on $X$.
Let $\Gamma_0$ be a uniform lattice in $G$ and $M=\Gamma_0 \backslash X$.
Bucher-Karlsson \cite{Bu08} reformulates a proof of Gromov's proportionality principle in the language of continuous bounded cohomology and moreover, shows that $$\frac{\| M \|}{\mathrm{Vol}(M)}=\frac{1}{\| \omega \|_\infty}.$$
It is easy to see that $\|M\|_\mathrm{Lip}=\|M\|$ because $M$ is closed. Let $\Gamma$ be an arbitrary lattice in $G$ and $N=\Gamma \backslash X$. It follows from Gromov's proportionality principle that
\begin{eqnarray}\label{eqn:2.1}
\frac{\| N \|_\mathrm{Lip}}{\mathrm{Vol}(N)}=\frac{\| M \|_\mathrm{Lip}}{\mathrm{Vol}(M)}=\frac{\| M \|}{\mathrm{Vol}(M)}=\frac{1}{\| \omega \|_\infty}.
\end{eqnarray}
Note that the proportionality principle fails in general for the ordinary simplicial volume.
\section{Volume invariant}\label{sec:3}
In this section, we define a new invariant $\mathrm{Vol}(\rho)$ and explore its properties.
Throughout the paper, $G$ denotes a connected semisimple Lie group with trivial center and no compact factors,
and $\Gamma$ denotes a lattice in $G$. As usual, $X$ denotes the associated symmetric $n$--space and $M$ denotes the locally symmetric space $\Gamma\backslash X$. The symbol $\omega$ denotes the $G$--invariant volume form on $X$.
\subsection{Volume invariant}\label{sec:3.1}
Let $\rho \colon\thinspace \Gamma \rightarrow G$ be a representation. Then, $\rho$ induces canonical pullback map
$\rho^*_c \colon\thinspace H^\bullet_c(G,\mathbb{R}) \rightarrow H^\bullet(\Gamma,\mathbb{R})$ in continuous cohomology. This canonical pullback map is realized on the level of cocomplex as follows:
For a continuous map $f \colon\thinspace G^{k+1}\rightarrow \mathbb{R}$, define a map $\rho^*(f) \colon\thinspace \Gamma^{k+1} \rightarrow \mathbb{R}$ by
$$\rho^*(f)(\gamma_0,\ldots,\gamma_k)=f(\rho(\gamma_0),\ldots,\rho(\gamma_k)),$$
for $(\gamma_0,\ldots,\gamma_k) \in \Gamma^{k+1}$.
This defines a chain map $\rho^* \colon\thinspace C^\bullet_c(G,\mathbb{R}) \rightarrow C^\bullet(\Gamma,\mathbb{R})$.
Moreover, $\rho^*$ maps $G$--invariant cochains to $\Gamma$--invariant cochains and hence, it induces a homomorphism $\rho^*_c \colon\thinspace H^\bullet_c(G,\mathbb{R}) \rightarrow H^\bullet(\Gamma,\mathbb{R})$ in continuous cohomology. In the same manner, $\rho$ induces a homomorphism $\rho^*_b \colon\thinspace H^\bullet_{c,b}(G,\mathbb{R}) \rightarrow H^\bullet_b(\Gamma,\mathbb{R})$ in continuous bounded cohomology.
For a connected semisimple Lie group $G$ with trivial center and no compact factors, it is well known that the $G$--invariant volume form $\omega \in H^n_c(G,\mathbb{R})$ is bounded. In other words, there exists a continuous bounded cohomology class $\omega_b \in H^n_{c,b}(G,\mathbb{R})$ such that $c(\omega_b)=\omega$ for the comparison map $c \colon\thinspace H^n_{c,b}(G,\mathbb{R}) \rightarrow H^n_c(G,\mathbb{R})$.
By pulling back $\omega_b$ by $\rho$, we obtain a bounded cohomology class $\rho^*_b(\omega_b)\in H^n_b(\Gamma,\mathbb{R})$. Subsequently, we identify the bounded cohomology class $\rho^*_b(\omega_b)$ in $H^n_b(\Gamma,\mathbb{R})$ with a bounded cohomology class in $H^n_b(M,\mathbb{R})$ via the canonical isomorphism between $H^\bullet_b(\Gamma,\mathbb{R})$ and $H^\bullet_b(M,\mathbb{R})$ \cite{Gr82}.
Then, the bounded cohomology class $\rho^*_b(\omega_b)$ can be evaluated on $\ell^1$--homology classes in $H^{\ell^1}_n(M,\mathbb{R})$ by the Kronecker products
$$ \langle\cdot ,\cdot \rightarrowngle \colon\thinspace H^\bullet_b(M,\mathbb{R}) \otimes H^{\ell^1}_\bullet(M,\mathbb{R}) \rightarrow \mathbb{R}.$$
Now, we define a \emph{volume invariant $\mathrm{Vol}(\rho)$ of $\rho$} by
$$\mathrm{Vol}(\rho) = \inf \{ |\langle \rho^*_b(\omega_b),\alpha \rightarrowngle| \text{ }|\text{ }c(\omega_b)=\omega \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \}.$$
It is easy to see that the volume invariant $\mathrm{Vol}(\rho)$ is finite since $\omega$ is bounded and the geometric simplicial volume of $M$ is strictly positive. Furthermore, a upper bound on the volume invariant $\mathrm{Vol}(\rho)$ can be obtained from its definition immediately as follows.
\begin{prop}\label{pro:3.1}
Let $\rho \colon\thinspace \Gamma \rightarrow G$ be a representation. Then, the volume invariant $\mathrm{Vol}(\rho)$ of $\rho$ satisfies an inequality $$ \mathrm{Vol}(\rho) \leq \mathrm{Vol}(M).$$
\end{prop}
\begin{proof}
For a continuous cohomology class $\beta \in H^n_c(G,\mathbb{R})$,
$$ \| \beta \|_\infty = \inf \{ \| \beta_b \|_\infty \ | \ c(\beta_b)=\beta \},$$
where $c \colon\thinspace H^n_{c,b}(G,\mathbb{R}) \rightarrow H^n_c(G,\mathbb{R})$ is the comparison map.
From the definition of the volume invariant $\mathrm{Vol}(\rho)$, we have
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
\mathrm{Vol}(\rho) &=& \inf \{ |\langle \rho^*_b(\omega_b),\alpha \rightarrowngle| \ | \ c(\omega_b)=\omega \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \} \\
&\leq& \inf \{ \| \rho^*_b(\omega_b) \|_\infty \cdot \| \alpha \|_1 \ | \ c(\omega_b)=\omega \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \} \\
&\leq& \inf \{ \|\omega_b\|_\infty \ | \ c(\omega_b)=\omega \} \cdot \inf \{ \|\alpha \|_1 \ | \ \alpha\in [M]^{\ell^1}_\mathrm{Lip} \} \\
&=& \|\omega\|_\infty \cdot \| M \|_\mathrm{Lip} \\
&=& \mathrm{Vol}(M).
\end{eqnarray*}}
The last equation comes from Equation (\ref{eqn:2.1}).
\end{proof}
\begin{rem}
If we define the volume invariant $\mathrm{Vol}(\rho)$ via $[M]^{\ell^1}$ instead of $[M]^{\ell^1}_\mathrm{Lip}$, we obtain the following inequality in a similar way as above
$$\mathrm{Vol}(\rho) \leq \|\omega\|_\infty \cdot \| M \|.$$
If $\Gamma$ is a lattice of $\mathbb{Q}$--rank at least $3$, it is known that $\| M \|=0$ \cite{LS09}. This implies that $\mathrm{Vol}(\rho)=0$ for all representations $\rho \colon\thinspace \Gamma \rightarrow G$.
Then, this volume invariant cannot detect discrete, faithful representations. This is the reason why we use the notion of the geometric simplicial volume of $M$ to define the volume invariant $\mathrm{Vol}(\rho)$ instead of the ordinary simplicial volume of $M$.
\end{rem}
\subsection{Volume invariant and $\rho$--equivariant map}
Goldman \cite{Go92} defined the volume invariant $\upsilon(\rho)$ by using a section $s \colon\thinspace M \rightarrow E_\rho$. Indeed, a section $s \colon\thinspace M \rightarrow E_\rho$ corresponds to a $\rho$--equivariant map $s \colon\thinspace X \rightarrow X$.
In a similar way, the volume invariant $\mathrm{Vol}(\rho)$ can be reformulated in terms of $\rho$--equivariant map. In this section, we devote ourselves to explaining this and verifying $\mathrm{Vol}(\rho)=|\upsilon(\rho)|$ for representations $\rho \colon\thinspace \Gamma \rightarrow G$ of uniform lattices $\Gamma$.
First, we describe another useful cocomplexes for both continuous and continuous bounded cohomology of $G$. For a nonnegative integer $k$, define
$$C^k_c(X,\mathbb{R}) = \{ f \colon\thinspace X^{k+1} \rightarrow \mathbb{R} \ | \ f \text{ is continuous} \}.$$
Consider the sup norm $\| \cdot \|_\infty$ on $C^k_c(X,\mathbb{R})$ defined by
$$\| f \|_\infty = \sup \{ |f(x_0,\ldots,x_k)| \ | \ (x_0,\ldots,x_k)\in X^{k+1} \}.$$
Let $C^k_{c,b}(X,\mathbb{R})$ be the subspace consisting of continuous bounded $k$--cochains.
Then, $C^\bullet_c(X,\mathbb{R})$ with the homogeneous coboundary operator becomes a cochain complex. Moreover, the homogeneous coboundary operator on $C^\bullet_c(X,\mathbb{R})$ restricts to $C^\bullet_{c,b}(X,\mathbb{R})$.
The $G$--action on $C^\bullet_c(X,\mathbb{R})$ is defined analogously to the one on $C^\bullet_c(G,\mathbb{R})$.
It is a standard fact that the continuous cohomology $H^\bullet_c(G,\mathbb{R})$ of $G$ is isometrically isomorphic to the cohomology of the cocomplex $C^\bullet_c(X,\mathbb{R})^G$. For a proof, see \cite[Chapter 3]{Gu80}. The continuous bounded cohomology $H^\bullet_{c,b}(G,\mathbb{R})$ of $G$ is isometrically isomorphic to the cohomology of the subcocomplex $C^\bullet_{c,b}(X,\mathbb{R})^G$ of $C^\bullet_c(X,\mathbb{R})^G$.
The comparison map $c \colon\thinspace H^\bullet_{c,b}(G,\mathbb{R})\rightarrow H^\bullet_c(G,\mathbb{R})$ is induced by the natural inclusion $C^\bullet_{c,b}(X,\mathbb{R})^G \subset C^\bullet_c(X,\mathbb{R})^G$.
Furthermore, both $H^\bullet(\Gamma,\mathbb{R})$ and $H^\bullet_b(\Gamma,\mathbb{R})$ are isometrically isomorphic to the cohomologies of cocomplexes $C^\bullet_c(X,\mathbb{R})^\Gamma$ and $C^\bullet_{c,b}(X,\mathbb{R})^\Gamma$ respectively. See \cite[Corollary 7.4.10]{Mo01} for a detailed proof.
We describe here an explicit map on the level of cocomplex which induces an isometric isomorphism between $H^\bullet(C^\bullet_c(X,\mathbb{R})^G)$ and $H^\bullet_c(G,\mathbb{R})$. Let us fix a base point $o\in X$.
Define a map $\phi_o \colon\thinspace C^k_c(X,\mathbb{R}) \rightarrow C^k_c(G,\mathbb{R})$ by
$$\phi_o(f)(g_0,\ldots,g_k)=f(g_0\cdot o,\ldots, g_k\cdot o).$$
The map $\phi_o$ is a $G$-morphism between two cocomplexes and restricts to the subcocomplexes of continuous bounded cochains. Then, $\phi_o$ induces an isometric isomorphism $\phi^G_c \colon\thinspace H^\bullet(C^\bullet_c(X,\mathbb{R})^G) \rightarrow H^\bullet_c(G,\mathbb{R})$ in continuous cohomology.
Note that $\phi^G_c$ is independent of the choice of the base point $o\in X$ even though $\phi_o$ depends on $o\in X$. Hence, we denote the induced map in continuous cohomology by $\phi^G_c$ without the subscript ``o". In a similar way, the map $\phi_o$ induces isometric isomorphisms, $\phi^G_b \colon\thinspace H^\bullet(C^\bullet_{c,b}(X,\mathbb{R})^G) \rightarrow H^\bullet_{c,b}(G,\mathbb{R})$ and
$\phi^\Gamma_b \colon\thinspace H^\bullet(C^\bullet_{c,b}(X,\mathbb{R})^\Gamma) \rightarrow H^\bullet_b(\Gamma,\mathbb{R})$.
Let $s \colon\thinspace X \rightarrow X$ be a $\rho$--equivariant continuous map for a representation $\rho \colon\thinspace \Gamma \rightarrow G$. Then, $s$ induces a map $s^* \colon\thinspace C^k_c(X,\mathbb{R}) \rightarrow C^k_c(X,\mathbb{R})$ defined by $$s^*(f)(x_0,\ldots,x_k)=f(s(x_0),\ldots,s(x_k)),$$
for a cochain $f$ in $C^k_c(X,\mathbb{R})$.
Due to the $\rho$--equivariance and continuity of $s \colon\thinspace X\rightarrow X$, it follows that $s^*$ maps $G$--invariant continuous (bounded) cochains to $\Gamma$--invariant continuous (bounded) cochains. Hence, $s^*$ induces homomorphisms $s^*_c \colon\thinspace H^\bullet( C^\bullet_c (X,\mathbb{R})^G) \rightarrow H^\bullet( C^\bullet_c (X,\mathbb{R})^\Gamma)$ in continuous cohomology and $s^*_b \colon\thinspace H^\bullet( C^\bullet_{c,b} (X,\mathbb{R})^G) \rightarrow H^\bullet( C^\bullet_{c,b} (X,\mathbb{R})^\Gamma)$ in continuous bounded cohomology. Now, consider the following diagram:
$$ \xymatrixcolsep{4pc}\xymatrix{
C^\bullet_c(X,\mathbb{R})^G \ar[r]^-{\phi_o} &
C^\bullet_c(G,\mathbb{R})^G \\
C^\bullet_{c,b}(X,\mathbb{R})^G \ar[r]^-{\phi_o} \ar[d]_-{s^*} \ar[u]^-{i} &
C^\bullet_{c,b}(G,\mathbb{R})^G \ar[d]^-{\rho^*} \ar[u]_-{i} \\
C^\bullet_{c,b}(X,\mathbb{R})^\Gamma \ar[r]^-{\phi_o} &
C^\bullet_b(\Gamma,\mathbb{R})^\Gamma.
}$$
In this diagram, it is clear that the upper diagram commutes.
On the other hand, the lower diagram does not commute. However, one can notice that it commutes in cohomology as follows.
Let $f\in C^k_{c,b}(X,\mathbb{R})^G$ be a $G$--invariant continuous bounded cocycle. Define $b \in C^{k-1}_b(\Gamma,\mathbb{R})$ by
$$b(\gamma_0,\ldots,\gamma_{k-1})=\sum_{i=0}^{k-1} (-1)^i f(\rho(\gamma_0) \cdot o, \ldots, \rho(\gamma_i) \cdot o, \rho(\gamma_i) \cdot s(o),\ldots,\rho(\gamma_{k-1})\cdot s(o)).$$
Then, $b$ is a $\Gamma$--invariant bounded cochain since $f$ is a $G$--invariant continuous bounded cocycle.
Also, it is a straightforward computation that
$$(\rho^* \circ \phi_o - \phi_o \circ s^*)(f)(\gamma_0,\ldots,\gamma_k) = \delta b (\gamma_0,\ldots,\gamma_k).$$ This implies that the lower diagram commutes in the cohomology level and hence, we have the following commutative diagram:
$$ \xymatrixcolsep{4pc}\xymatrix{
H^\bullet(C^\bullet_c(X,\mathbb{R})^G) \ar[r]^-{\phi^G_c}_-{\colon\thinspaceng} &
H^\bullet_c(G,\mathbb{R}) \\
H^\bullet(C^\bullet_{c,b}(X,\mathbb{R})^G) \ar[r]^-{\phi^G_b}_-{\colon\thinspaceng} \ar[d]_-{s^*_b} \ar[u]^-{c} &
H^\bullet_{c,b}(G,\mathbb{R}) \ar[d]^-{\rho^*_b} \ar[u]_-{c} \\
H^\bullet(C^\bullet_{c,b}(X,\mathbb{R})^\Gamma) \ar[r]^-{\phi^\Gamma_b}_-{\colon\thinspaceng} &
H^\bullet_b(\Gamma,\mathbb{R})
}$$
Each cohomology class in $H^\bullet_c(G,{\mathbb R})$, $H^\bullet_{c,b}(G,{\mathbb R})$ and $H^\bullet_b(\Gamma,{\mathbb R})$ is canonically identified with a cohomology class in $H^\bullet(C^\bullet_c(X,\mathbb{R})^G)$, $H^\bullet(C^\bullet_{c,b}(X,\mathbb{R})^G)$ and $H^\bullet(C^\bullet_{c,b}(X,\mathbb{R})^\Gamma)$ via the isomorphisms induced by $\phi_o$ respectively.
Let $\omega_b$ be a continuous bounded cohomology class in $H^n_{c,b}(G,{\mathbb R})$ representing the $G$--invariant volume form $\omega \in H^n_c(G,{\mathbb R})$. We use the same notations $\omega$ and $\omega_b$ for the cohomology class in $H^\bullet(C^\bullet_c(X,\mathbb{R})^G)$ and $H^\bullet(C^\bullet_{c,b}(X,\mathbb{R})^G)$ identified with $\omega \in H^n_c(G,\mathbb{R})$ and $\omega_b \in H^n_{c,b}(G,\mathbb{R})$ via $\phi^G_c$ and $\phi^G_b$, respectively.
Noting that the cohomologies $H^\bullet(C^\bullet_{c,b}(X,\mathbb{R})^\Gamma)$ and $H^\bullet_b(\Gamma,\mathbb{R})$ are canonically identified with the bounded cohomology $H^\bullet_b(M,\mathbb{R})$, one can conclude that $s^*_b(\omega_b) = \rho^*_b(\omega_b)$ in $H^n_b(M,{\mathbb R})$ via the canonical isomorphisms. Hence,
$$\{ s^*_b(\omega_b) \in H^n_b(M,{\mathbb R}) \ | \ c(\omega_b) =\omega \} =\{ \rho^*_b(\omega_b) \in H^n_b(M,{\mathbb R}) \ | \ c(\omega_b) =\omega \}.$$
Therefore, the volume invariant $\mathrm{Vol}(\rho)$ can be reformulated in terms of $\rho$--equivariant map as follows:
$$\mathrm{Vol}(\rho)= \inf \{|\langle s^*_b(\omega_b),\alpha \rightarrowngle| \ | \ c(\omega_b)=\omega \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \}.$$
Note that the above reformulation of the volume invariant $\mathrm{Vol}(\rho)$ is independent of the choice of $\rho$--equivariant map $s \colon\thinspace X \rightarrow X$
as observed.
To define the volume invariant $\upsilon(\rho)$, Goldman \cite{Go92} uses a smooth section of the associated bundle. The reformulation of the volume invariant $\mathrm{Vol}(\rho)$ in terms of $\rho$--equivariant map makes it possible to verify the relation between two invariants $\upsilon(\rho)$ and $\mathrm{Vol}(\rho)$.
\begin{lemma}\label{lem:3.3}
Let $\Gamma$ be a uniform lattice in $G$ and $\rho \colon\thinspace \Gamma \rightarrow G$ be a representation. Then,
$$\mathrm{Vol}(\rho) = |\upsilon (\rho)| = \left| \int_M s^* \omega \right|,$$
where $s \colon\thinspace M \rightarrow E_\rho$ is a smooth section of the associated bundle $E_\rho$.
\end{lemma}
\begin{proof}
A section $s \colon\thinspace M \rightarrow E_\rho$ corresponds to a $\rho$--equivariant map $X\rightarrow X$, denoted by $s \colon\thinspace X \rightarrow X$. Since $M=\Gamma\backslash X$ is a closed manifold, the set $[M]^{\ell^1}_\mathrm{Lip}$ contains exactly one element, namely, the class $i_*[M]$, where
$[M]$ is the fundamental class of $M$, and
$i_* \colon\thinspace H_n(M,\mathbb{R}) \rightarrow H^{\ell^1}_n(M,\mathbb{R})$ is the map induced by the inclusion
$C_\bullet(M,\mathbb{R}) \subset C_\bullet^{\ell^1}(M,\mathbb{R})$.
Hence, the volume invariant $\mathrm{Vol}(\rho)$ is computed by
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
\mathrm{Vol}(\rho) &=& \inf \{ |\langle s^*_b(\omega_b),\alpha \rightarrowngle | \ | \ c(\omega_b)=\omega \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \} \\
&=& \inf \{ |\langle s^*_b(\omega_b),i_*[M] \rightarrowngle | \ | \ c(\omega_b)=\omega \}.
\end{eqnarray*}}
Considering the following commutative diagram,
$$ \xymatrixcolsep{4pc}\xymatrix{
H^n_{c,b}(G,\mathbb{R}) \ar[r]^-{c} \ar[d]_-{s^*_b} &
H^n_c(G,\mathbb{R}) \ar[d]^-{s^*_c} \\
H^n_b(\Gamma,\mathbb{R}) \ar[r]^-{c} &
H^n(\Gamma,\mathbb{R})
}$$
we have $c(s^*_b(\omega_b))=s^*_c(c(\omega_b))=s^*_c\omega$. Note that $s^*_c\omega$ is represented by a $\Gamma$--invariant cocycle $s^*f$ where $f \colon\thinspace X^{n+1} \rightarrow \mathbb{R}$ is the $G$--invariant cocycle representing $\omega$, which is defined by
$$ f(x_0,\ldots,x_n) = \int_{[x_0,\ldots,x_n]} \omega.$$
Also, one can consider another $\Gamma$--invariant cocycle $h \colon\thinspace X^{n+1} \rightarrow \mathbb{R}$ defined by
$$h(x_0,\ldots,x_n) = \int_{[x_0,\ldots,x_n]} s^*\omega.$$
Here, $s^*\omega$ is the pull-back of the $G$--invariant volume form $\omega$ by $s \colon\thinspace X \rightarrow X$.
It is easy to see that $h$ also represents the continuous cohomology class $s^*_c \omega$ because the geodesic straightening map is chain homotopic to the identity.
Let $c$ be a fundamental cycle representing $[M]$.
Since $h$ represents the cohomology class $s^*_c\omega$ in $H^n(\Gamma,\mathbb{R}) \colon\thinspaceng H^n(M,\mathbb{R})$, we have
$$ \langle s^*_b(\omega_b), i_*[M] \rightarrowngle =\langle s^*_c \omega, [M] \rightarrowngle = \langle h, c \rightarrowngle=\int_M s^*\omega$$
for any $\omega_b \in c^{-1}(\omega)$. The last equation follows from the de Rham theorem. This completes the proof.
\end{proof}
Goldman proves that $\upsilon (\rho)$ exactly characterizes discrete, faithful representations of $\Gamma$ into $G$ for the case that $G$ is either a connected semisimple Lie group of higher rank or $\text{SO}(n,1)$. This implies that $\mathrm{Vol}(\rho)$ does so by Lemma \ref{lem:3.3}.
\section{Semisimple Lie groups of higher rank}\label{sec:4}
In this section, we prove Theorem \ref{thm:1.2} for the case that $G$ is a semisimple Lie group of higher rank.
Recall the restriction maps $$res_c \colon\thinspace H^\bullet_c(G,\mathbb{R}) \rightarrow H^\bullet(\Gamma,\mathbb{R}) \text{ and } res_b \colon\thinspace H^\bullet_{c,b}(G,\mathbb{R}) \rightarrow H^\bullet_b(\Gamma,\mathbb{R}),$$ induced from the inclusions $C^\bullet_c(X,\mathbb{R})^G \subset C^\bullet_c(X,\mathbb{R})^\Gamma$ and $C^\bullet_{c,b}(X,\mathbb{R})^G \subset C^\bullet_{c,b}(X,\mathbb{R})^\Gamma$ respectively.
Note that $res_b$ is an isometric embedding because $\Gamma$ is a lattice in $G$.
We first observe that $$\langle res_b(\omega_b), \alpha \rightarrowngle = \mathrm{Vol}(M)$$ for all $\omega_b \in c^{-1}(\omega)$ and all $\alpha \in [M]^{\ell^1}_\mathrm{Lip}$. To verify this, we need to prove the existence of the geodesic straightening map on the locally finite chain complex with finite Lipschitz constant.
The geodesic straightening map on the singular chain complex of a nonpositively curved manifold is introduced by Thurston \cite[Section 6.1]{Th78}.
Let $X$ be a simply connected, complete Riemannian manifold with nonpositive sectional curvature.
A geodesic simplex is defined inductively as follows: Let $x_0,\ldots,x_k \in X$. First, the geodesic $0$--simplex $[x_0]$ is the point $x_0 \in X$ and the geodesic $1$--simplex $[x_0,x_1]$ is the unique geodesic from $x_1$ to $x_0$. In general, the geodesic $k$--simplex $[x_0,\ldots,x_k]$ is the geodesic cone over $[x_0,\ldots,x_{k-1}]$ with the top point $x_k$.
Let $M$ be a connected, complete Riemannian manifold with nonpositive sectional curvature. Then, \emph{the geodesic straightening map} $str \colon\thinspace C_\bullet (M,\mathbb{R}) \rightarrow C_\bullet (M,\mathbb{R})$ is defined by
$$ str(\sigma) = \pi_M \circ [\tilde{\sigma}(e_0),\ldots,\tilde{\sigma}(e_k)],$$
for a singular $k$--simplex $\sigma \colon\thinspace \Delta^k \rightarrow M$ where $\pi_M \colon\thinspace \widetilde{M} \rightarrow M$ is the universal covering map, $e_0,\ldots,e_k$ are the vertices of the standard $k$--simplex $\Delta^k$, and $\tilde{\sigma}$ is a lift of $\sigma$ to the universal cover $\widetilde{M}$.
\begin{prop}\label{pro:4.1}
Let $M$ be a connected, complete, locally symmetric space of noncompact type. Then, geodesic straightening map is well-defined on $C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$ and moreover, it is chain homotopic to the identity.
\end{prop}
\begin{proof}
Let $A \in S^\mathrm{lf,Lip}_k(M)$ for a nonnegative integer $k$. This means that any compact subset of $M$ intersects the image of only finitely many elements of $A$ and there exists a constant $C_A>0$ such that $\mathrm{Lip}(\sigma)<C_A$ for all $\sigma\in A$.
Let $str \colon\thinspace C_\bullet(M,\mathbb{R}) \rightarrow C_\bullet(M,\mathbb{R})$ denote the geodesic straightening map. Define $str(A)$ by
$$str(A)= \{ str(\sigma) \text{ }|\text{ } \sigma \in A \}.$$
To show that geodesic straightening map is well defined on $C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$, it is sufficient to show that $str(A) \in S^\mathrm{lf,Lip}_k(M)$.
We first claim that $str(A)$ has finite Lipschitz constant.
Let $\mathrm{Diam}(\sigma)$ denote the diameter of $\sigma(\Delta^k)$ for a singular simplex $\sigma \colon\thinspace \Delta^k \rightarrow M$. For all $\sigma \in A$,
$$\mathrm{Diam}(\sigma) \leq C_A \cdot \mathrm{Diam}(\Delta^k)$$
since $\sigma \colon\thinspace \Delta^k \rightarrow M$ has Lipschitz constant $C_A$.
Hence, each $\sigma \in A$ is contained in a closed ball of diameter $D_A= C_A \cdot \mathrm{Diam}(\Delta^k)$ in $M$. Because every closed ball in $X$ is geodesically convex,
both $\sigma$ and $str(\sigma)$ are contained in the same closed ball of diameter $D_A$ for every $\sigma \in A$. This implies that $\mathrm{Diam}(str(\sigma))<D_A$ for all $\sigma \in A$.
For every $D>0$ and $k\in \mathbb{N}$, there is $L>0$ such that every geodesic $k$--simplex $\tau$ of diameter less than $D$ satisfies $\| T_x\tau \|<L$ for every $x \in \Delta^k$ \cite[Proposition 2.4]{LS09}. Hence,
there exists $L_A>0$ such that $\mathrm{Lip}(str(\sigma)) < L_A $ for all $\sigma \in A$, that is, $str(A)$ has finite Lipschitz constant $L_A$.
Next, to verify that $str(A)$ has locally finite support, we need to show that every compact subset of $M$ intersects the image of only finitely many elements of $str(A)$. Let $K$ be a compact subset of $M$ and $\mathcal{N}_{D_A}(K)$ be the $D_A$--neighborhood of $K$.
Suppose $\overline{\mathcal{N}_{D_A}(K)}\cap \sigma= \emptyset$ for some $\sigma \in A$.
As observed above, both $\sigma$ and $str(\sigma)$ are contained in a closed ball $B_\sigma$ of diameter $D_A$.
It is obvious that $B_\sigma \cap (M-\overline{\mathcal{N}_{D_A}(K)}) \neq \emptyset$ because of $\sigma \subset B_\sigma$.
Then, $B_\sigma$ can never touch $K$, which implies $str(\sigma) \cap K = \emptyset$.
Thus, $K$ can intersect the image of $str(\sigma)$ only for $\sigma \in A$ with $\overline{\mathcal{N}_{D_A}(K)}\cap \sigma \neq \emptyset$.
There exist finitely many such elements of $A$ since $\overline{\mathcal{N}_{D_A}(K)}$ is the compact subset of $M$, and $A$ has locally finite support. Finally, we can conclude that $str(A)$ is a locally finite subset of $S_k(M)$ with finite Lipschitz constant, that is, $str(A) \in S^\mathrm{lf,Lip}_k(M)$.
From the above observation, we have a well-defined map $$str^\mathrm{lf} \colon\thinspace C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})\rightarrow C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$$ extending the geodesic straightening map $str \colon\thinspace C_\bullet (M,\mathbb{R}) \rightarrow C_\bullet (M,\mathbb{R})$.
It is obvious that $str^\mathrm{lf}$ is a chain map.
Now, to construct a chain homotopy from $str^\mathrm{lf}$ to the identity, recall
the chain homotopy $H_\bullet \colon\thinspace C_\bullet(M,\mathbb{R}) \rightarrow C_{\bullet+1}(M,\mathbb{R})$ from the geodesic straightening map $str$ to the identity.
Let $G_\sigma \colon\thinspace \Delta^k \times [0,1] \rightarrow M$ be the canonical straight line homotopy from $\sigma$ to $str(\sigma)$ for a singular $k$--simplex $\sigma$ in $M$. Let $\{e_0,\ldots,e_k \}$ denote the set of vertices in $\Delta^k$ for each $k$. The chain homotopy $H_k \colon\thinspace C_k(M,\mathbb{R})\rightarrow C_{k+1}(M,\mathbb{R})$ is defined by
$$H_k(\sigma) = \sum_{i=0}^k (-1)^i G_\sigma \circ \eta_i,$$
where $\eta_i \colon\thinspace \Delta^{k+1} \rightarrow \Delta^k \times [0,1]$ is the affine map that maps $e_0,\ldots,e_{k+1}$ to $(e_0,0),\ldots,(e_i,0),(e_i,1),\ldots,(e_k,1)$ for $i=0,\ldots,k$.
Let $c=\sum_{\sigma \in A} a_\sigma \sigma$ be a $k$--chain in $C_k^\mathrm{lf,Lip}(M,\mathbb{R})$ for $A\in S_k^\mathrm{lf,Lip}(M)$. Then, as we observed previously, $\mathrm{Lip}(\sigma) < C_A$ and $\mathrm{Lip}(str(\sigma)) < L_A$ for all $\sigma \in A$.
Moreover, the canonical line homotopy $G_\sigma$ from $\sigma$ to $str(\sigma)$ has finite Lipschitz constant that depends only on $C_A$, $L_A$ by \cite[Proposition 2.1]{LS09}. Noting that the Lipschitz constant of $\eta_i$ is also uniformly bounded from above for all $i=0,\ldots,k$, it follows that the Lipschitz constant of $H_k(\sigma)$ is uniformly bounded from above by a constant depending only on $C_A$, $L_A$ for all $\sigma \in A$. This means that the Lipschitz constant $\mathrm{Lip}(H_k(c))$ of $H_k(c)$ is finite.
To see that $H_k(c)$ has locally finite support, note that if $\sigma$ is contained in a closed ball, then the images of both $str(\sigma)$ and $H_k(\sigma)$ are contained in the same closed ball because every closed ball in $X$ is geodesically convex. As in the proof that $str(A)$ has locally finite support, any compact subset $K$ of $M$ can intersect the image of singular $(k+1)$--simplices occurring in $H_k(\sigma)$ only for $\sigma \in A$ with $\overline{\mathcal{N}_{D_A}(K)}\cap \sigma \neq \emptyset$. The set of such elements of $A$ are finite due to $A\in S^\mathrm{lf,Lip}_k(M)$. Moreover, since $H_k(\sigma)$ is a finite sum of $(k+1)$--simplices, $K$ intersects the image of finitely many $(k+1)$--simplices occurring in $H_k(c)$. This implies that $H_k(c)$ has locally finite support.
Now, we have a well-defined map,
$$H_\bullet^\mathrm{lf} \colon\thinspace C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R}) \rightarrow C^\mathrm{lf,Lip}_{\bullet+1}(M,\mathbb{R}).$$
Since $H_\bullet^\mathrm{lf} \colon\thinspace C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R}) \rightarrow C^\mathrm{lf,Lip}_{\bullet+1}(M,\mathbb{R})$ is the map extending the chain homotopy $H_\bullet \colon\thinspace C_\bullet (M,\mathbb{R}) \rightarrow C_{\bullet +1}(M,\mathbb{R})$ between the geodesic straightening map $str$ and the identity, it clearly satisfies $$\partial \circ H_k^\mathrm{lf} + H_{k-1}^\mathrm{lf} \circ \partial =str^\mathrm{lf} -id.$$
Hence, $H^\mathrm{lf}_\bullet$ is a chain homotopy from $str^\mathrm{lf}$ to the identity.
Therefore, we can conclude that $str^\mathrm{lf} \colon\thinspace C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R}) \rightarrow C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$ is a chain homotopic to the identity.
\end{proof}
The existence of the geodesic straightening map on $C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$ allows us to get a straight cycle from an arbitrary cycle without changing its homology class. By using the straightening map $str^\mathrm{lf} \colon\thinspace C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R}) \rightarrow C^\mathrm{lf,Lip}_\bullet(M,\mathbb{R})$, we can prove the following Lemma.
\begin{lemma}\label{lem:4.2}
Let $G$ be a connected semisimple Lie group with trivial center and no compact factors. Let $\Gamma$ be a lattice in $G$. Then,
$$\langle res_b(\omega_b) , \alpha \rightarrowngle = \mathrm{Vol}(M)$$
for all $\omega_b \in c^{-1}(\omega)$ and all $\alpha \in [M]^{\ell^1}_\mathrm{Lip}$.
\end{lemma}
\begin{proof}
On the continuous cochain complex $C^\bullet_c(X,\mathbb{R})$, the $G$--invariant volume form $\omega$ is represented by a cocycle $f \colon\thinspace X^{n+1} \rightarrow \mathbb{R}$ defined by
$$f(x_0,\ldots,x_n)=\int_{[x_0,\ldots,x_n]} \omega.$$
Let $f_b \colon\thinspace X^{n+1} \rightarrow \mathbb{R}$ be a cocycle representing $\omega_b \in c^{-1}(\omega)$. Both $f$ and $f_b$ represent the same cohomology class $\omega$ in the continuous cohomology of $G$. Hence, there exists a $G$--invariant continuous cochain $b$ in $C^{n-1}_c(X,\mathbb{R})^G$ such that $$f_b =f +\delta b.$$
Let $c=\sum_{i=1}^\infty a_i \sigma_i$ be a locally finite fundamental $\ell^1$--cycle with finite Lipschitz constant representing $\alpha$.
Due to Proposition \ref{pro:4.1}, $str^\mathrm{lf}(c)$ is also a locally finite fundamental $\ell^1$--cycle with finite Lipschitz constant and represents $\alpha$. By \cite[Proposition 4.4]{LS09}, we have
$$ \langle f, str^\mathrm{lf}(c) \rightarrowngle = \mathrm{Vol}(M).$$
Now, we claim that $\langle \delta b, str^\mathrm{lf}(c) \rightarrowngle =0$. Let $\sigma_i^j$ denote the $j$-th face of $\sigma$ for $j=0,\ldots,n$. Then, $\partial \sigma_i = \sum_{j=0}^n (-1)^j \cdot \sigma_i^j$ and
\begin{eqnarray}\label{eqn:4.1}
\langle \delta b, str^\mathrm{lf}(c) \rightarrowngle = \sum_{i=1}^\infty \sum_{j=0}^n (-1)^j a_i \cdot \langle b, str(\sigma_i^j) \rightarrowngle.
\end{eqnarray}
Since the Lipschitz constant of $str^\mathrm{lf}(c)$ is finite, there exists $R>0$ such that
each $\sigma_i$ is contained in a closed ball with radius $R$ for all $i \in \mathbb{N}$. Fix a closed ball $B$ with radius $R$ in $X$. Then, there exists $g_i \in G$ for each $\sigma_i$ such that $g_i\cdot str(\sigma_i) \subset B$ since $G$ acts transitively on $X$. Due to the $G$--invariance of $b$, we have $\langle b, str(\sigma_i^j) \rightarrowngle = \langle b, g_i \cdot str(\sigma_i^j) \rightarrowngle$ for all $i\in \mathbb{N}$ and $j=0,\ldots,n$. This implies that $$\langle b, str(\sigma_i^j) \rightarrowngle = b(x_0, \ldots,x_{n-1})$$ for some $(x_0,\ldots,x_{n-1})\in B^n$.
Since $b$ is continuous and $B$ is the compact subset of $X$, there exists a upper bound $C>0$ on $\langle b, str(\sigma_i^j) \rightarrowngle $ for all $i\in \mathbb{N}$ and $j=0,\ldots,n$. Furthermore, $c$ is a $\ell^1$--cycle and hence,
$$\sum_{i=1}^\infty \sum_{j=0}^n | (-1)^j a_i \cdot \langle b, str(\sigma_i^j) \rightarrowngle | < n C \cdot \sum_{i=1}^\infty |a_i| < \infty.$$
In other words, the series in Equation (\ref{eqn:4.1}) absolutely converges. Thus, all rearrangements of the series in Equation (\ref{eqn:4.1}) converge to the same value. From the cycle condition of $str^\mathrm{lf}(c)$,
there exists a permutation $\tau$ of $\mathbb{N}\times \{0,\ldots,n\}$ such that
$$ \sum_{i=1}^\infty \sum_{j=0}^n (-1)^{\tau(j)} a_{\tau(i)} \cdot str(\sigma_{\tau(i)}^{\tau(j)})=0.$$
Under this permutation $\tau$, we can conclude that $\langle \delta b, str^\mathrm{lf}(c) \rightarrowngle = 0$.
Finally, we have
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
\langle res_b(\omega_b), \alpha \rightarrowngle &=& \langle f+\delta b, str^\mathrm{lf}(c) \rightarrowngle \\ &=&
\langle f, str^\mathrm{lf}(c) \rightarrowngle + \langle \delta b, str^\mathrm{lf}(c) \rightarrowngle \\
&=& \mathrm{Vol}(M)
\end{eqnarray*}}
The second equation is available since all series in the equation absolutely converge.
\end{proof}
\begin{defi}
A representation $\rho \colon\thinspace \Gamma \rightarrow G$ is \emph{maximal} if
$$\mathrm{Vol}(\rho)=\mathrm{Vol}(M).$$
\end{defi}
For reader's convenience, we recall Margulis's normal subgroup theorem \cite{Margulis}.
\begin{thm}Let $G$ be a connected semisimple Lie group with finite center with ${\mathbb R}$-rank $\geq 2$, and let $\Gamma\subset G$ be an irreducible lattice. If $N\subset \Gamma$ is a normal subgroup of $\Gamma$, then either $N$ lies in the center of $G$ or the quotient $\Gamma/N$ is finite.
\end{thm}
\begin{thm}\label{thm:4.4}
Let $G$ be a connected semisimple Lie group of higher rank with trivial center and no compact factors.
Let $\Gamma$ be an irreducible lattice in $G$. Then, a representation $\rho \colon\thinspace \Gamma \rightarrow G$ is maximal if and only if $\rho$ is a discrete, faithful representation.
\end{thm}
\begin{proof}
First, suppose that $\rho$ is discrete and faithful.
Margulis Superrigidity Theorem implies that $\rho$ extends to an automorphism $\tilde{\rho} \colon\thinspace G \rightarrow G$. Then, a representation $\rho \colon\thinspace \Gamma \rightarrow G$ is written as a composition $\rho = \tilde{\rho} \circ i$ where $i \colon\thinspace \Gamma \rightarrow G$ is the natural inclusion of $\Gamma$ into $G$.
The canonical pullback map $\rho^*_b \colon\thinspace H^\bullet_{c,b}(G,\mathbb{R}) \rightarrow H^\bullet_b(\Gamma,\mathbb{R})$ in continuous bounded cohomology is realized as a composition $\rho^*_b=res_b \circ \tilde{\rho}^*_b$,
$$ \xymatrixcolsep{2pc}\xymatrix{
H^\bullet_{c,b}(G,\mathbb{R}) \ar[r]^{\tilde{\rho}^*_b} &
H^\bullet_{c,b}(G,\mathbb{R}) \ar[r]^{res_b} &
H^\bullet_b(\Gamma,\mathbb{R}).
}$$
Since $\tilde{\rho}$ is an automorphism of $G$, it induces an automorphism of the continuous (bounded) cohomology of $G$. In particular, it is easy to see that $\tilde{\rho}^*_c(\omega) = \pm \omega$ in $H^n_c(G,\mathbb{R})$.
Considering the commutative diagram
$$ \xymatrixcolsep{4pc}\xymatrix{
H^n_{c,b}(G,\mathbb{R}) \ar[r]^-{c} \ar[d]_-{\tilde{\rho}^*_b} &
H^n_c(G,\mathbb{R}) \ar[d]^-{\tilde{\rho}^*_c} \\
H^n_{c,b}(G,\mathbb{R}) \ar[r]^-{c} &
H^n_c(G,\mathbb{R})
}$$
the automorphism $\tilde{\rho}^*_b \colon\thinspace H^n_{c,b}(G,\mathbb{R}) \rightarrow H^n_{c,b}(G,\mathbb{R})$ permutes the set of $c^{-1}(\omega)$ up to sign. Hence,
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
\mathrm{Vol}(\rho) &=& \inf \{ |\langle \rho^*_b (\omega_b),\alpha \rightarrowngle| \ | \ c(\omega_b)=\omega \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \} \\
&=& \inf \{|\langle res_b( \tilde{\rho}^*_b (\omega_b)),\alpha \rightarrowngle | \ | \ c(\omega_b)=\omega \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \} \\
&=& \inf \{ |\langle res_b( \omega_b),\alpha \rightarrowngle | \ | \ c(\omega_b)=\omega \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \}.
\end{eqnarray*}}
According to Lemma \ref{lem:4.2}, $\langle res_b(\omega_b),\alpha \rightarrowngle =\mathrm{Vol}(M)$ for all $\omega_b \in c^{-1}(\omega)$ and all $\alpha \in [M]^{\ell^1}_\mathrm{Lip}$. Therefore, $\mathrm{Vol}(\rho)= \mathrm{Vol}(M)$.
Conversely, suppose that $\rho \colon\thinspace \Gamma \rightarrow G$ is not a discrete, faithful representation. If $\rho$ has nontrivial kernel, then $\rho(\Gamma)$ is a finite group by the Margulis's normal subgroup theorem.
If $\rho$ is a nondiscrete, faithful representation, then $\rho(\Gamma)$ is precompact by the Margulis superrigidity theorem. In either case, $\rho(\Gamma)$ is an amenable subgroup of $G$.
Regarding $\rho$ as a composition $\rho = i \circ \rho$,
$$ \xymatrixcolsep{2pc}\xymatrix{
\Gamma \ar[r]^-{\rho} &
\rho(\Gamma) \ar[r]^-{i} &
G
}$$
one can realize $\rho^*_b \colon\thinspace H^\bullet_{c,b}(G,\mathbb{R})\rightarrow H^\bullet_b(\Gamma,\mathbb{R})$ as a composition $\rho^*_b \circ res_b$,
$$ \xymatrixcolsep{2pc}\xymatrix{
H^\bullet_{c,b}(G,\mathbb{R}) \ar[r]^-{res_b} &
H^\bullet_{c,b}(\rho(\Gamma),\mathbb{R}) \ar[r]^-{\rho^*_b} &
H^\bullet_b(\Gamma,\mathbb{R}).
}$$
The continuous bounded cohomology $H^\bullet_{c,b}(\rho(\Gamma),\mathbb{R})$ is trivial because
$\rho(\Gamma)$ is amenable. This implies that $\rho^*_b(\omega_b) = \rho^*_b(res_b(\omega_b))=0$ for all $\omega_b \in c^{-1}(\omega)$. Hence, $\mathrm{Vol}(\rho)=0$.
This completes the proof of this theorem.
\end{proof}
\section{Simplie Lie groups of rank $1$}\label{sec:5}
In this section, we give a proof of Theorem \ref{thm:1.2} for the case that $G$ is a simple Lie group of rank $1$ except for $\text{SO}(2,1)$. The Besson-Courtois-Gallot technique is a central ingredient here.
\begin{defi}
Let $F \colon\thinspace X \rightarrow Y$ be a smooth map between Riemannian manifolds $X$ and $Y$. The $p$--Jacobian $\mathrm{Jac}_pF$ of $F$ is defined by $$\mathrm{Jac}_pF(x) =\sup \| d_xF(u_1) \wedge \cdots \wedge d_xF(u_p) \|,$$ where $\{u_1,\ldots,u_p\}$ varies on the set of orthonormal $p$--frames at $x\in X$.
\end{defi}
Let $X$ and $Y$ be complete, simply connected, Riemannian manifolds.
Suppose that the sectional curvature $K_Y$ on $Y$ satisfies $K_Y \leq -1$.
Let $\Gamma$ and $\Gamma'$ be discrete subgroups of $\text{Isom}(X)$ and $\text{Isom}(Y)$ respectively. For any representation $\rho \colon\thinspace \Gamma \rightarrow \Gamma'=\rho(\Gamma)$,
Besson, Courtois and Gallot show that for all $\epsilon >0$, $p\geq 3$, there exists a $\rho$--equivariant map $F_\epsilon \colon\thinspace X \rightarrow Y$ such that
$$\mathrm{Jac}_p F_\epsilon (x) \leq \left( \frac{\delta(\Gamma)}{p-1}(1+\epsilon) \right)^p,$$
for all $x \in X$ where $\delta(\Gamma)$ is the critical exponent of $\Gamma$.
Furthermore, they show that if $X$ has strictly negative sectional curvature, $\Gamma$ and $\Gamma'$ are convex cocompact and $\rho$ is injective,
then there exists the natural map $F \colon\thinspace X\rightarrow Y$ in \cite[Theorem 1.10]{BCG99}.
Note that one can make use of Besson-Courtois-Gallot's method
if there exists a $\rho$--equivariant measurable map from the visual boundary $\partial X$ of $X$ to $\partial Y$ for a representation $\rho :\Gamma \rightarrow \mathrm{Isom}(Y)$.
\begin{prop}\label{pro:5.2}
Let $G$ and $H$ be connected simple Lie groups of rank $1$ with trivial center and no compact factors.
Let $X$ and $Y$ be the symmetric spaces associated with $G$ and $H$ respectively.
Assume that the symmetric metrics on $X$ and $Y$ are normalized so that their curvatures lie between $-4$ and $-1$.
Let $\Gamma$ be a lattice in $G$ and $\rho \colon\thinspace \Gamma \rightarrow H$ be a representation whose image is nonelementary. Then, there exists a map $F \colon\thinspace X\rightarrow Y$ such that
\begin{itemize}
\item[(1)] $F$ is smooth.
\item[(2)] $F$ is $\rho$--equivariant.
\item[(3)] For all $k\geq 3$, $\mathrm{Jac}_kF(x) \leq (\delta(\Gamma)/(k-1))^k$.
\item[(4)] If $\mathrm{dim}(X) \geq \mathrm{dim}(Y) \geq 3$, then $\mathrm{Jac}_nF(x) \leq (\delta(\Gamma)/(n+d-2))^n$ where $d$ is the real dimension of the field or the ring under consideration for $G$.
Moreover, equality holds for some $x\in X$ if and only if $D_xF$ is a homothety from $T_xX$ to $T_{F(x)}Y$.
\end{itemize}
\end{prop}
\begin{proof}
By the assumption of the sectional curvatures on $X$ and $Y$, the associated symmetric spaces $X$ and $Y$ are $\mathrm{CAT}(-1)$--spaces. Since any lattice in $G$ is a discrete divergence subgroup of $G$, it follows from \cite[Theorem 0.2]{BM96} that there exists the unique $\rho$--equivariant measurable map $\varphi \colon\thinspace \partial X \rightarrow \partial Y$ and it takes almost all its values in the limit set of $\rho(\Gamma)$.
Let $\{ \nu_x \}_{x\in X}$ denote the family of Patterson-Sullivan measures on $\partial X$ for $\Gamma$. Let $\mu_x $ be the pushforward of $\nu_x$ by $\varphi$, that is, $\mu_x = \varphi_* \nu_x$. It can be easily seen that $\{\mu_x \}_{x\in X}$ is $\rho$--equivariant and moreover, the measures $\mu_x$ and $\mu_y$ are in the same measure class for all $x,y \in X$.
We claim that the barycenter of $\mu_x$ is well defined for all $x\in X$. Recall that if $\mu_x$ is not concentrated on two points, then the barycenter of $\mu_x$ is well defined. Assume that $\mu_x$ is concentrated on two points. Let $p$ be one of them. Then, $\mu_x$ must have positive weights on each $\rho(\Gamma)$--orbit of $p$ because $\mu_x$ and $\mu_{\gamma x}=\rho(\gamma)_* \mu_x$ are in the same measure class for all $\gamma \in \Gamma$. However, the set of $\rho(\Gamma)$--orbits of $p$ contains more than two points because $\rho(\Gamma)$ is nonelementary. This contradicts the assumption that $\mu_x$ is concentrated on only two points. Therefore, the claim holds.
As Besson, Courtois and Gallot construct the natural map in \cite{BCG99}, define a map $F \colon\thinspace X\rightarrow Y$ by the composition $bar \circ \varphi_* \circ \mu$ of maps
$$ \xymatrixcolsep{2pc}\xymatrix{
X \ar[r]^-{\mu} &
\mathcal{M}^+ (\partial X) \ar[r]^-{\varphi_*} &
\mathcal{M}^+ (\partial Y) \ar[r]^-{bar} &
Y
}$$
where $\mathcal{M}^+ (\partial X)$ denotes the set of positive Borel measures on $\partial X$.
Then, this map $F$ is a $\rho$--equivariant.
Furthermore, the properties $(1) \sim (4)$ of the natural map $F \colon\thinspace X \rightarrow Y$ can be proved by the same argument as in \cite[Section 2]{BCG99}.
\end{proof}
The map $F \colon\thinspace X\rightarrow Y$ as above is called the \emph{natural map} for a representation $\rho \colon\thinspace\Gamma \rightarrow H$.
\begin{thm}\label{thm:5.3}
Let $G$ be a connected simple Lie group of rank $1$ with trivial center and no compact factors, except for $\mathrm{SO}(2,1)$.
Let $\Gamma$ be a lattice in $G$. Then, a representation $\rho \colon\thinspace \Gamma \rightarrow G$ is maximal if and only if $\rho$ is a discrete, faithful representation.
\end{thm}
\begin{proof}
Suppose that $\rho \colon\thinspace \Gamma \rightarrow G$ is a discrete, faithful representation.
Let $X$ be the associated symmetric space of dimension $n$ and $M=\Gamma\backslash X$.
Then, $\rho$ extends to an automorphism $\tilde{\rho} \colon\thinspace G \rightarrow G$ due to the Mostow's rigidity theorem. In a similar argument as in the proof of Theorem \ref{thm:4.4}, we have $\mathrm{Vol}(\rho)=\mathrm{Vol}(M).$
Conversely, we now suppose that $\mathrm{Vol}(\rho)=\mathrm{Vol}(M)$. If $\rho(\Gamma)$ is elementary, then $\rho^*_b(\omega_b)=0$ for all $\omega_b \in c^{-1}(\omega)$ and thus, $\mathrm{Vol}(\rho)=0$. Hence, we can assume that $\rho(\Gamma)$ is nonelementary. Assume that the sectional curvature on $X$ lies between $-4$ and $-1$.
Then, there exists the natural map $F \colon\thinspace X \rightarrow X$ according to Proposition \ref{pro:5.2}.
Because of the critical exponent $\delta(\Gamma)=n+d-2$ for any lattice $\Gamma$ in $G$
where $d$ is the real dimension of the field or the ring under consideration for $G$, we have $$\mathrm{Jac}_nF(x) \leq 1.$$
Define a continuous function $f \colon\thinspace X^{n+1} \rightarrow \mathbb{R}$ by
$$f(x_0,\ldots,x_n) = \int_{[x_0,\ldots,x_n]}\omega.$$
It can be easily seen that $f \colon\thinspace X^{n+1} \rightarrow \mathbb{R}$ is a $G$--invariant continuous bounded cocycle representing the $G$--invariant volume form $\omega \in H^n_c(G,\mathbb{R})$ on $X$.
Hence, $f$ determines a continuous bounded cohomology class $\omega_b \in c^{-1}(\omega)$.
Recall that the $\Gamma$--invariant bounded cocycle $F^* f \colon\thinspace X^{n+1} \rightarrow \mathbb{R}$ is defined by $$F^* f(x_0,\ldots,x_n)=f(F(x_0),\ldots,F(x_n))=\int_{[F(x_0),\ldots,F(x_n)]}\omega.$$
Considering the pullback $F^*\omega$ of the $G$--invariant volume form $\omega$ on $X$ by the natural map $F$, one can define another $\Gamma$--invariant continuous bounded cocycle $h \colon\thinspace X^{n+1}\rightarrow \mathbb{R}$ by $$h(x_0,\ldots,x_n)=\int_{[x_0,\ldots,x_n]}F^*\omega.$$
The change of variables formula implies
$$h(x_0,\ldots,x_n)=\int_{[x_0,\ldots,x_n]}F^*\omega=\int_{F([x_0,\ldots,x_n])} \omega.$$
It is clear that $[F(x_0),\ldots,F(x_n)]= str(F([x_0,\ldots,x_n]))$. From the canonical straight line homotopy $H_\bullet \colon\thinspace C_\bullet(X,\mathbb{R})\rightarrow C_{\bullet+1}(X,\mathbb{R})$ between the geodesic straightening map and the identity, we have
$$ [F(x_0),\ldots,F(x_n)]-F([x_0,\ldots,x_n])=(\partial \circ H_n + H_{n-1} \circ \partial) (F([x_0,\ldots,x_n])).$$
It is a straightforward computation that $ h - F^* f = \delta \eta$ where
$$\eta(x_0,\ldots,x_{n-1})=\int_{H_{n-1}\circ F([x_0,\ldots,x_{n-1}])} \omega .$$
\begin{lemma} If $G$ is not $\mathrm{SO}(3,1)$, $\eta$ is a $\Gamma$-invariant continuous bounded cochain, which implies that $h$ and $F^* f$ represent the same bounded cohomology class $F^*_b(\omega_b)$ in $H^n_b(\Gamma,\mathbb{R})$.
\end{lemma}
\begin{proof}
In the case that $G$ is not $\mathrm{SO}(3,1)$, the associated symmetric space $X$ has dimension at least $4$.
Then, the property ($3$) in Proposition \ref{pro:5.2} shows $$\mathrm{Jac}_{n-1}F(x) \leq \left( \frac{n+d-2}{n-2} \right)^{n-1},$$ for all $x \in X$. Hence, the volume of $F([x_0,\ldots,x_{n-1}])$ has a uniform upper bound. The volume of the straight line homotopy between $F([x_0,\ldots,x_{n-1}])$ and $[F(x_0),\ldots,F(x_{n-1})]$ is uniformly bounded from above
since the volumes of both $F([x_0,\ldots,x_{n-1}])$ and $[F(x_0),\ldots,F(x_{n-1})]$ are uniformly bounded from above and the sectional curvature on $X$ is bounded from above by $-1$. More precisely, one can approximate the straight line homotopy by the union of small cones $C_i$ whose bases are on $[F(x_0),\ldots,F(x_{n-1})]$ and whose apexes are on $F([x_0,\ldots,x_{n-1}])$, and small cones $C_j$ whose bases are on $F([x_0,\ldots,x_{n-1}])$ and whose apexes are on $[F(x_0),\ldots,F(x_{n-1})]$. On the other hand, it can be shown, see for example \cite{Gr82} (page 19), that
$$\mathrm{Vol}(Cone)\leq (n-1)^{-1}\mathrm{Vol}(Base).$$ This shows that the volume of the straight line homotopy is bounded uniformly by the sum of volumes of
$F([x_0,\ldots,x_{n-1}])$ and $[F(x_0),\ldots,F(x_{n-1})]$.
Thus, $\eta$ is a $\Gamma$-invariant continuous bounded cochain, which implies that $h$ and $F^* f$ represent the same bounded cohomology class $F^*_b(\omega_b)$ in $H^n_b(\Gamma,\mathbb{R})$.
\end{proof}
Let $\alpha \in [M]^{\ell^1}_\text{Lip}$ and $c$ be a locally finite fundamental $\ell^1$--cycle with finite Lipschitz constant representing $\alpha$.
We now assume that $G$ is not $\mathrm{SO}(3,1)$. Maximality condition $\mathrm{Vol}(\rho)=\mathrm{Vol}(M)$ gives us an inequality
\begin{eqnarray}\label{naturalvol}
|\langle F^*_b (\omega_b), \alpha \rightarrowngle | = | \langle F^* f, c \rightarrowngle | = |\langle h, c \rightarrowngle| = \left| \int_M F^*\omega \right|\geq \mathrm{Vol}(M).
\end{eqnarray}
Since $\text{Jac}_nF(x)\leq 1$ almost everywhere, inequality (\ref{naturalvol}) actually implies that
$$ \left| \int_M F^*\omega \right| = \mathrm{Vol}(M),$$
and hence, $\text{Jac}_nF(x)=1$ everywhere. Then, it follows from the property $(4)$ of the natural map in Proposition \ref{pro:5.2} that $F$ is an isometry.
Therefore, $\rho :\Gamma \rightarrow G$ is a discrete, faithful representation.
The theorem for the case $G=\mathrm{SO}(3,1)$ can be covered by the result of Bucher, Burger and Iozzi \cite{BBI}. In their paper \cite{BBI}, an invariant for representations of lattices in $\mathrm{SO}(n,1)$ is defined in the same manner as the invaraint for representations of lattices in $\mathrm{SO}(2,1)$ in \cite{BIW10}. Moreover, they show that the invariant detects discrete, faithful representations for $n \geq 3$. In fact, it is easy to see that the absolute value of the invariant for representations $\rho$ of hyperbolic lattices is equal to the volume invariant $\mathrm{Vol}(\rho)$. This follows from the same argument in the proof of Proposition \ref{prop:6.2}. Hence, the theorem holds for the case $G=\mathrm{SO}(3,1)$. We finally complete the proof.
\end{proof}
From Lemma \ref{lem:3.3}, it is easy to see that Theorem \ref{thm:5.3} covers the remaining cases $\mathrm{SU}(n,1), \mathrm{Sp}(n,1), \mathrm{F}_4^{-20}$ that Goldman's proof in \cite{Go92} did not cover. Hence, we complete the proof of Conjecture \ref{con:1.1}.
\section{$\text{SO}(2,1)$}\label{sec:6}
In this section, we deal with $\text{PU}(1,1)$ instead of $\text{SO}(2,1)$ for convenience.
Let $\Gamma$ be a lattice in $\text{PU}(1,1)$ and $\rho \colon\thinspace \Gamma \rightarrow \text{PU}(1,1)$ be a representation.
The unit ball $\mathbb{D}$ in the complex plane $\mathbb{C}$ is the associated symmetric space and $S=\Gamma\backslash \mathbb{D}$ is a surface of finite topological type with negative Euler number.
If $\Gamma$ is a uniform lattice, then the volume invariant $\mathrm{Vol}(\rho)$ is equal to
$|\upsilon(\rho)|$ as we see this in Lemma \ref{lem:3.3}. Hence, Theorem \ref{thm:1.2} for uniform lattices in $\text{PU}(1,1)$ follows from Goldman's proof. We refer the reader to \cite{Go81} for a detailed proof of this.
From now on, we assume that $\Gamma$ is a nonuniform lattice in $\text{PU}(1,1)$. In this case, Burger, Iozzi and Wienhard define the Toledo invariant as follows.
Let $\Sigma$ be a connected, oriented, compact surface with boundary $\partial \Sigma$ whose interior is homeomorphic to $S$. Let $\rho \colon\thinspace \pi_1(\Sigma) \rightarrow \text{PU}(1,1)$ be a representation.
The second continuous cohomology $H^2_c(\text{PU}(1,1),\mathbb{R})$ of $\text{PU}(1,1)$ is generated by the K\"{a}hler form $\kappa$ on $\mathbb{D}$. There is the unique continuous bounded K\"{a}hler class $\kappa_b \in H^2_{c,b}(\text{PU}(1,1),\mathbb{R})$ since the comparison map $c \colon\thinspace H^\bullet_{c,b}(\mathrm{PU}(1,1),\mathbb{R}) \rightarrow H^\bullet_c(\mathrm{PU}(1,1),\mathbb{R})$ is an isomorphism in degree $2$. By pulling back the bounded K\"{a}hler class $\kappa_b$ via $\rho$, one can obtain a bounded cohomology class $$\rho^*_b(\kappa_b) \in H^2_b(\pi_1(\Sigma),\mathbb{R}) \colon\thinspaceng H^2_b(\Sigma,\mathbb{R}).$$
The canonical map $C^\bullet_b(\Sigma, \partial \Sigma, \mathbb{R}) \rightarrow C^\bullet_b(\Sigma,\mathbb{R})$ induces an isomorphism $j \colon\thinspace H^2_b(\Sigma,\partial \Sigma,\mathbb{R}) \rightarrow H^2_b(\Sigma,\mathbb{R})$ in bounded cohomology. The Toledo invariant $\mathrm{T}(\Sigma, \rho)$ of $\rho$ is defined by
$$\text{T}(\Sigma, \rho)= \langle j^{-1}(\rho^*_b(\kappa_b)),[\Sigma,\partial \Sigma]\rightarrowngle,$$
where $j^{-1}(\rho^*_b(\kappa_b))$ is considered as an ordinary relative cohomology class and $[\Sigma,\partial \Sigma]$ is the relative fundamental class. Burger, Iozzi and Wienhard obtain a kind of the Milnor inequality
$$ | \mathrm{T}(\Sigma,\rho) | \leq \chi(\Sigma),$$
where $\chi(\Sigma)$ is the Euler number of $\Sigma$. Moreover, they generalize Goldman's characterization of maximal representations for closed surfaces to the cases of surfaces with boundary.
\begin{thm}[Burger, Iozzi and Wienhard]
Let $\Sigma$ be a connected oriented surface with negative Euler number. A representation $\rho \colon\thinspace \pi_1(\Sigma) \rightarrow \mathrm{PU}(1,1)$ is maximal if and only if it is the holonomy representation of a complete hyperbolic metric on the interior of $\Sigma$.
\end{thm}
In fact, a similar argument holds for a representation of $\pi_1(\Sigma)$ into a Lie group of Hermitian type.
We refer the reader to \cite{BIW10} for more details.
\begin{prop}\label{prop:6.2}
Let $\Gamma$ be a nonuniform lattice in $\mathrm{PU}(1,1)$. Then
$$\mathrm{Vol}(\rho)=2\pi |\mathrm{T}(\Sigma,\rho) |.$$
\end{prop}
\begin{proof}
Let $S=\Gamma\backslash \mathbb{D}$ and $\Sigma$ be the compact surface with boundary whose interior is homeomorphic to $S$. We think of $S$ as the interior of $\Sigma$.
Let $\omega$ be the $\text{PU}(1,1)$--invariant volume form on $\mathbb{D}$. Then, $\omega= 2\pi \kappa$ for the K\"{a}hler form $\kappa$ on $\mathbb{D}$. Hence,
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
\mathrm{Vol}(\rho)&=& \inf \{ |\langle \rho^*_b (\omega_b), \alpha \rightarrowngle| \ | \ c(\omega_b)=\omega \text{ and }\alpha \in [S]^{\ell^1}_\mathrm{Lip} \} \\
&=& 2\pi \cdot \inf \{| \langle \rho^*_b (\kappa_b), \alpha \rightarrowngle | \ | \ \alpha \in [S]^{\ell^1}_\mathrm{Lip} \}.
\end{eqnarray*}}
We claim that $\langle \rho^*_b (\kappa_b), \alpha \rightarrowngle = \text{T}(\Sigma,\rho)$ for all $\alpha \in [S]^{\ell^1}_\mathrm{Lip}$. Consider a collar neighborhood of $\partial \Sigma$ in $\Sigma$ that is homeomorphic to $\partial \Sigma \times [0,1)$. Let $K$ be the complement of the collar neighborhood of $\partial \Sigma$. Note that $K$ is a compact subsurface with boundary that is a deformation retract of $\Sigma$. Consider the following commutative diagram,
$$ \xymatrixcolsep{2pc}\xymatrix{
C^\bullet_b(S,\mathbb{R}) &
C^\bullet_b(\Sigma,\mathbb{R}) \ar[l]_-{i_1} &
C^\bullet_b(\Sigma,\partial \Sigma,\mathbb{R}) \ar[l]_-{j} \\
C^\bullet_b(S,S-K,\mathbb{R})\ar[u]^-{p_1} & C^\bullet_b(\Sigma, \Sigma-K, \mathbb{R}) \ar[l]_-{i_2} \ar[u]^-{p_2} \ar[ru]_-{p_3} &
}$$
where every map in the above diagram is the map induced from the canonical inclusion.
Every map in the diagram induces an isomorphism in bounded cohomology in degree $2$.
Thus, there exists a cocycle $z\in C^2_b(\Sigma,\Sigma-K,\mathbb{R})$ such that
$p_2(z)$ represents $\rho_b^*(\kappa_b)$ in $H^2_b(\Sigma,\mathbb{R})$ and
$i_1 (p_2(z))$ represents $\rho_b^*(\kappa_b)$ in $H^2_b(S,\mathbb{R})$ and
$p_3(z)$ represents $j^{-1}(\rho_b^*(\kappa_b))$ in $H^2_b(\Sigma,\partial \Sigma, \mathbb{R})$.
Here, we use the same notation $\rho^*_b(\kappa_b)$ for the bounded cohomology classes in $H^2_b(\Sigma,\mathbb{R})$ and $H^2_b(S,\mathbb{R})$ identified with $\rho^*_b(\kappa_b) \in H^2_b(\Gamma,\mathbb{R})$ via the canonical isomorphisms $H^2_b(\Sigma,\mathbb{R}) \colon\thinspaceng H^2_b(\Gamma,\mathbb{R})$ and $H^2_b(S,\mathbb{R}) \colon\thinspaceng H^2_b(\Gamma,\mathbb{R})$ respectively.
Let $c=\sum_{i=1}^\infty a_i \sigma_i$ be a locally finite fundamental $\ell^1$--cycle with finite Lipschitz constant representing $\alpha \in [S]^{\ell^1}_\mathrm{Lip}$. Then, we have
$$\langle \rho^*_b(\kappa_b),\alpha \rightarrowngle = \langle i_1 (p_2(z)), c \rightarrowngle = \langle z, c|_K \rightarrowngle,$$
where $c|_K=\sum_{\mathrm{im}\sigma_i \cap K \neq \emptyset} a_i \sigma_i$. It is a standard fact that $c|_K$ represents the relative fundamental class $[S,S-K]$ in $H_2(S,S-K,\mathbb{R})$.
Since the fundamental cycle representing $[S,S-K]$ is also a representative of the fundamental class $[\Sigma,\Sigma-K]$ by the canonical inclusion, $c|_K$ represents the fundamental class $[\Sigma,\Sigma -K]$ in $H_2(\Sigma,\Sigma-K,\mathbb{R})$. Let $[z]$ denote the cohomology class in $H^2(\Sigma,\Sigma-K,\mathbb{R})$ determined by $z$. From the viewpoint of the Kronecker product $\langle \cdot, \cdot \rightarrowngle \colon\thinspace H^2(\Sigma, \Sigma-K, {\mathbb R}) \otimes H_2(\Sigma,\Sigma-K,{\mathbb R}) \rightarrow {\mathbb R}$, we have
$$\langle z, c|_K \rightarrowngle = \langle [z], [\Sigma, \Sigma-K] \rightarrowngle.$$
Let $d \in C_2(\Sigma,\partial \Sigma)$ be a cycle representing the fundamental cycle $[\Sigma,\partial \Sigma]$ in $H_2(\Sigma,\partial \Sigma,\mathbb{R})$. Since $p_3(z)$ represents $j^{-1}( \rho^*_b(\kappa_b))$,
$$ \langle j^{-1}(\rho_b^*(\kappa_b)), [\Sigma,\partial \Sigma] \rightarrowngle = \langle p_3(z), d \rightarrowngle = \langle z, d|_K \rightarrowngle.$$
For any relative fundamental cycle $d$ in $C_2(\Sigma,\partial \Sigma,\mathbb{R})$, $d|_K$ represents the fundamental class $[\Sigma, \Sigma-K]$ in $H_2(\Sigma,\Sigma-K,\mathbb{R})$.
Hence, $$\langle z,d|_K \rightarrowngle = \langle [z], [\Sigma,\Sigma-K] \rightarrowngle.$$
Therefore, we can finally conclude that
$$\langle \rho^*_b(\kappa_b),\alpha \rightarrowngle = \langle j^{-1}(\rho^*_b(\kappa_b)), [\Sigma,\partial \Sigma] \rightarrowngle = \langle [z], [\Sigma,\Sigma-K] \rightarrowngle,$$
which implies this proposition.
\end{proof}
The equation $\mathrm{Vol}(\rho)=2 \pi |\text{T}(\rho)|$ implies that the structure theorem for maximal representations of compact surfaces into $\text{PU}(1,1)$ with respect to the Toledo invariant $\text{T}(\rho)$ holds for the volume invariant $\mathrm{Vol}(\rho)$.
\begin{thm}\label{thm:6.3}
Let $\Gamma$ be a lattice in $\mathrm{PU}(1,1)$. Then, a representation $\rho \colon\thinspace \Gamma \rightarrow G$ is maximal if and only if $\rho$ is a discrete, faithful representation.
\end{thm}
Theorem \ref{thm:1.2} follows from Proposition \ref{pro:3.1}, Theorem \ref{thm:4.4}, \ref{thm:5.3} and \ref{thm:6.3}.
\begin{thm}\label{thm:6.4}
Let $\Gamma$ be an irreducible lattice in a connected semisimple Lie group $G$ with trivial center and
no compact factors. Let $\rho \colon\thinspace \Gamma \rightarrow G$ be a representation. Then, the volume invariant $\mathrm{Vol}(\rho)$ satisfies an inequality
$$ \mathrm{Vol}(\rho) \leq \mathrm{Vol}(M),$$
where $X$ is the associated symmetric space and $M=\Gamma\backslash X$.
Moreover, equality holds if and only if
$\rho$ is a discrete, faithful representation.
\end{thm}
\section{Representations of lattices in $\mathrm{SO}(n,1)$ into $\mathrm{SO}(m,1)$}\label{sec:7}
In this section, we introduce a volume invariant $\mathrm{Vol}(\rho)$ for representations $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ of lattices $\Gamma$ in $\mathrm{SO}(n,1)$ for $m\geq n$.
Let $\mathbb{H}^k$ denote the hyperbolic $k$--space for each $k\in \mathbb{N}$. Define a map $f_n^m \colon\thinspace (\mathbb{H}^m)^{n+1} \rightarrow \mathbb{R}$ by
$$f_n^m(x_0,\ldots,x_n)= \mathrm{Vol}_n^m([x_0,\ldots,x_n]),$$
where $\mathrm{Vol}_n^m([x_0,\ldots,x_n])$ is the $n$--dimensional volume of the geodesic $n$--simplex $[x_0,\ldots,x_n]$ in $\mathbb{H}^m$. Clearly, $f_n^m$ is a $\mathrm{SO}(m,1)$--invariant continuous (bounded) cochain in $C^n_c(\mathbb{H}^m,\mathbb{R})$.
Observing that the geodesic $n$--simplex $[x_0,\ldots,x_n]$ is contained in a copy of $\mathbb{H}^n$ in $\mathbb{H}^m$, it is easy to see that $f_n^m$ is a continuous (bounded) cocycle and moreover,
$$\| \omega_n^m \|_\infty = v_n$$ where $\omega_n^m \in H^n_c(\mathrm{SO}(m,1),\mathbb{R})$ is the continuous cohomology class determined by the cocycle $f_n^m$ and $v_n$ is the volume of a regular ideal geodesic simplex in $\mathbb{H}^n$.
According to the Van Est isomorphism, the continuous cohomology class $\omega_n^m$ corresponds to a $\mathrm{SO}(m,1)$--invariant, differential $n$--form $\omega_n^m$ on $\mathbb{H}^m$.
The restriction of the differential form $\omega_n^m$ to any totally geodesic $\mathbb{H}^n$ in $\mathbb{H}^m$ is the Riemannian volume form on the totally geodesic $\mathbb{H}^n$ in $\mathbb{H}^m$.
Let $\Gamma$ be a lattice in $\mathrm{SO}(n,1)$ and $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ be a representation for $m\geq n$.
Let $c \colon\thinspace H^*_{c,b}(\mathrm{SO}(m,1),\mathbb{R}) \rightarrow H^*_c(\mathrm{SO}(m,1),\mathbb{R})$ be the comparison map and $M=\Gamma \backslash \mathbb{H}^n$.
Then, we define a volume invariant $\mathrm{Vol}(\rho)$ of $\rho$ by
$$ \mathrm{Vol}(\rho) = \inf \{ | \langle \rho^*_b (\omega_{n,b}^m), \alpha \rightarrowngle | \ | \ c(\omega_{n,b}^m)=\omega_n^m \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip}\}.$$
It satisfies an inequality
$$ \mathrm{Vol}(\rho) \leq \| \omega_n^m \|_\infty \cdot \|M\|_\mathrm{Lip} = v_n \cdot \frac{\mathrm{Vol}(M)}{v_n}=\mathrm{Vol}(M).$$
Recall that a representation $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ is said to be a \emph{totally geodesic representation} if there is a totally geodesic $\mathbb{H}^n \subset \mathbb{H}^m$ so that the image of the representation lies in the subgroup $G \subset \mathrm{SO}(m,1)$ that preserves this $\mathbb{H}^n$
and that the $\rho$--equivariant map $F \colon\thinspace \mathbb{H}^n \rightarrow \mathbb{H}^m$ is a totally geodesic isometric embedding.
Note that the subgroup $G$ of $\mathrm{SO}(m,1)$ is of the form $H\times K$ where $H$ is isomorphic to $\mathrm{SO}(n,1)$ and $K$ is isomorphic to the compact group $\mathrm{SO}(m-n)$.
A totally geodesic representation $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ splits into $\rho = \rho_1 \times \rho_2$ where $\rho_1$ is conjugate to $\Gamma$ by the Mostow rigidity theorem.
\begin{thm}
Let $\Gamma$ be a lattice in $\mathrm{SO}(n,1)$ and $M=\Gamma \backslash \mathbb{H}^n$. The volume invariant $\mathrm{Vol}(\rho)$ of a representation $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ for $m\geq n \geq 3$ satisfies an inequality $$\mathrm{Vol}(\rho) \leq \mathrm{Vol}(M).$$
Moreover, equality holds if and only if $\rho$ is a totally geodesic representation.
\end{thm}
\begin{proof}
We only need to show the second statement. In fact, a proof of the theorem is given by Bucher, Burger and Iozzi in \cite{BBI}. We give here an independent proof of the theorem for $m \geq n >3$. Suppose that a representation $\rho \colon\thinspace \Gamma \rightarrow \mathrm{SO}(m,1)$ is a totally geodesic representation.
Then, there exists a $\rho$--equivariant totally geodesic isometric embedding $F \colon\thinspace \mathbb{H}^n \rightarrow \mathbb{H}^m$.
The $\rho$--equivariant map $F$ induces homomorphisms $F^*_c \colon\thinspace H^\bullet_c(\mathrm{SO}(m,1),\mathbb{R})\rightarrow H^\bullet (\Gamma,\mathbb{R})$ and
$F^*_b \colon\thinspace H^\bullet_{c,b}(\mathrm{SO}(m,1),\mathbb{R})\rightarrow H^\bullet_b(\Gamma,\mathbb{R})$. The volume invariant $\mathrm{Vol}(\rho)$ of $\rho$ can be computed by
$$\mathrm{Vol}(\rho) = \inf \{ | \langle F^*_b (\omega_{n,b}^m), \alpha \rightarrowngle | \ | \ c(\omega_{n,b}^m)=\omega_n^m \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip}\}.$$
Since $F$ is an isometric embedding, we have
{\setlength\arraycolsep{2pt}
\begin{eqnarray*}
F^* f^m_n(y_0,\ldots,y_n) &=& f^m_n(F(y_0),\ldots,F(y_n)) \\
&=& \mathrm{Vol}_n^m([F(y_0),\ldots,F(y_n)]) \\
&=& \mathrm{sign}(F) \cdot \mathrm{Vol}^n_n([y_0,\ldots,y_n]),
\end{eqnarray*}}
where $\mathrm{sign}(F)=1$ if $F$ is orientation-preserving and $\mathrm{sign}(F)=-1$ if $F$ is orientation-reversing.
This implies that $F^*_c(\omega^m_n)=\mathrm{sign}(F) \cdot res_c(\omega_n)$ where $\omega_n$ is the $\mathrm{SO}(n,1)$--invariant volume form on $\mathbb{H}^n$. Hence, it immediately follows that $F^*_b(\omega^m_{n,b})= \mathrm{sign}(F) \cdot res_b(\omega_{n,b})$ for some $\omega_{n,b} \in c^{-1}(\omega_n)$.
By Lemma \ref{lem:4.2}, we have
$$ \langle F^*_b(\omega^m_{n,b}), \alpha \rightarrowngle = \langle \mathrm{sign}(F) \cdot res_b(\omega_{n,b}), \alpha \rightarrowngle = \mathrm{sign}(F) \cdot \mathrm{Vol}(M),$$
for all $\alpha \in [M]^{\ell^1}_\mathrm{Lip}$. Thus, we can conclude that $$ \mathrm{Vol}(\rho) = \inf \{ | \langle F^*_b (\omega_{n,b}^m), \alpha \rightarrowngle | \ | \ c(\omega_{n,b}^m)=\omega_n^m \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip}\} = \mathrm{Vol}(M).$$
Conversely, we suppose that $\mathrm{Vol}(\rho)=\mathrm{Vol}(M)$.
Recall that the natural map $F \colon\thinspace \mathbb{H}^n \rightarrow \mathbb{H}^m$ satisfies:
\begin{itemize}
\item $F$ is smooth.
\item $F$ is $\rho$--equivariant.
\item For all $k\geq 3$, $\text{Jac}_kF(x) \leq (\delta(\Gamma)/(k-1))^k$.
\item If $\| D_xF(u_1)\wedge \cdots \wedge D_xF(u_k)\|= (\delta(\Gamma)/(k-1))^k$ for an orthonormal $k$-frame $u_1,\ldots,u_k$ at $x \in \mathbb{H}^n$, then the restriction of $D_xF$ to the subspace generated by $u_1,\ldots,u_k$ is a homothety.
\end{itemize}
Because of $\delta(\Gamma)=n-1$ for a lattice $\Gamma$ in $\mathrm{SO}(n,1)$, $\text{Jac}_nF(x)\leq 1$.
By an argument similar to the one used in the proof of Theorem \ref{thm:5.3}, we can conclude that
$$\left| \int_M F^*\omega_n^m \right| = \mathrm{Vol}(M).$$
Hence, $\mathrm{Jac}_n F(x) =1$ almost everywhere after possibly reversing the orientation of $X$.
Then, $F$ is a global isometry of $\mathbb{H}^n$. For a detailed proof about this, we refer to \cite{FK06}.
Therefore, $\rho$ is a totally geodesic representation.
\end{proof}
\section{Toledo invariant of complex hyperbolic representations}\label{sec:8}
In this section we consider only uniform lattices $\Gamma\subset \mathrm{SU}(n,1),\ n\geq 2$.
\subsection{On complex hyperbolic space}
Let $\Gamma\subset \mathrm{SU}(n,1)$ be a uniform lattice that $M=\Gamma\backslash {\mathbb H}^n_{\mathbb C}$ and $\rho \colon\thinspace \Gamma\rightarrow G=\mathrm{SU}(m,1),\ m\geq n$ be a representation.
Let $\omega$ be a K\"ahler form on ${\mathbb H}^m_{\mathbb C}$. Then $\frac{1}{n!}\omega^n$ will be a $\mathrm{SU}(m,1)$-invariant form.
Then it defines an element $\omega_c\in H^{2n}_{c}(G,{\mathbb R})$ via Van-Est isomorphism. Denote $\omega_b\in H^{2n}_{b,c}(G,{\mathbb R})$ a bounded class such that $c(\omega_b)=\omega_c$ under the comparison map. Define the volume of the representation $\rho$ by
$$ \mathrm{Vol}(\rho) = \inf \{ |\langle \rho^*_b(\omega_b),\alpha \rightarrowngle| \ | \ c(\omega_b)=\omega_c \text{ and } \alpha \in [M]^{\ell^1}_\mathrm{Lip} \}.$$
Then, it satisfies the usual inequality
$$\mathrm{Vol}(\rho) \leq \|\omega_c\|_\infty \cdot \| M \|_\mathrm{Lip}.$$
But, since $\frac{1}{n!}\omega^n$ is the volume form on ${\mathbb H}^n_{\mathbb C}$, $\mathrm{Vol}(\rho) \leq \mathrm{Vol}(M)$.
Suppose $\mathrm{Vol}(\rho)=\mathrm{Vol}(M)$. If $\rho$ is not reductive, the image will be contained in a parabolic group, and the volume will be zero. Hence assume that $\rho$ is reductive.
Let $F \colon\thinspace {\mathbb H}^n_{\mathbb C}\rightarrow {\mathbb H}^m_{\mathbb C}$ be a $\rho$--equivariant smooth harmonic map. Then
some class $\rho^*_b(\omega_b)$ is represented by $F^*(\frac{1}{n!}\omega^n)$ and the pairing satisfies
$$|\langle \rho^*_b(\omega_b),\alpha \rightarrowngle|=\left| \int_M F^* \left( \frac{1}{n!}\omega^n \right) \right| \geq\mathrm{Vol}(M).$$ This implies that the rank of $dF$ at some point $x\in {\mathbb H}^n_{\mathbb C}$ is maximal. By Siu's argument \cite{Siu}, $F$ is holomorphic. It is shown
in \cite{BCG99} that $\mathrm{Jac}_{2n}F\leq 1$ for holomorphic map $F$.
Consequently
$$\left| \int_M F^* \left( \frac{1}{n!}\omega^n \right) \right|
= \mathrm{Vol}(M)$$ and $F$ is an isometric embedding.
Hence we obtain using the same proof of section \ref{sec:7} and the above argument
\begin{thm}Let $\Gamma\subset \mathrm{SU}(n,1)$ be a uniform lattice and $\rho \colon\thinspace \Gamma\rightarrow \mathrm{SU}(m,1)$, $ m\geq n$ be a representation. Then $\rho \colon\thinspace \Gamma\rightarrow \mathrm{SU}(m,1)$ is a maximal volume representation if and only if $\rho$ is a totally geodesic representation.
\end{thm}
This is a reformulation of Corlette's result in \cite{Corlette} in terms of the bounded cohomology theory.
See also \cite{KM} and \cite{BI} for defferent formulations.
Note that this theorem implies both Goldman-Millson and Corlette's results.
\begin{cor}\label{cor:8.2} Let $\Gamma\subset \mathrm{SU}(n,1)\subset \mathrm{SU}(m,1)$ be a uniform lattice. Then it is locally rigid up to compact group.
\end{cor}
\begin{proof}Suppose $\rho_t \colon\thinspace \Gamma\rightarrow \mathrm{SU}(m,1)$ is an one-parameter family of representations near $\rho_0=i$, the canonical inclusion.
Note that $\mathrm{Vol}(\rho_t)=\left| \int_M f_t^* \left( \frac{1}{n!}\omega^n \right) \right| $ by Lemma \ref{lem:3.3} where $f_t$ is a $\rho_t$--equivariant map $f_t \colon\thinspace {\mathbb H}^n_{\mathbb C}\rightarrow {\mathbb H}^m_{\mathbb C}$.
Note that $f_0^*(\frac{1}{n!}\omega^n)\in H^{2n}(M,{\mathbb Z})$ is a Chern class and hence $[f_t^*(\frac{1}{n!}\omega^n)]=[f_0^*(\frac{1}{n!}\omega^n)]\in H^{2n}(M,{\mathbb Z})$. This implies that $\mathrm{Vol}(\rho_t)=\mathrm{Vol}(\rho_0)$. Since $\rho_0$ is a maximal volume representation, $\rho_t$ is also maximal, hence
they are all conjugate each other up to compact group.
\end{proof}
\subsection{On quaternionic hyperbolic space}
We can also formulate the rigidity phenomenon of uniform lattices of $\mathrm{SU}(n,1)$ in $\mathrm{Sp}(m,1)$ as follows.
A homogeneous space $D=\mathrm{Sp}(m,1)/\mathrm{Sp}(m)\times\mathrm{U}(1)$ over ${\mathbb H}^m_{\mathbb H}$ with fiber ${\mathbb C} P^1$ is called a twistor space. The vertical bundle $\cal V$ tangent to the fibers is a smooth subbundle of $TD$ and a unique $\mathrm{Sp}(m,1)$-invariant complement to this vertical subbundle is called the holomorphic
horizontal subbundle $\cal H$. This twistor space $D$ posseses a pseudo-K\"ahler metric $g$ which is negative definite on $\cal V$ and positive definite on $\cal H$, whose associated (1,1) form $\omega_D$ is closed.
The quaternionic hyperbolic space ${\mathbb H}^m_{\mathbb H}$ has one dimensional space of $\mathrm{Sp}(m,1)$--invariant four forms. We pick a four form $\alpha$ so that its restriction to a totally geodesic complex hyperbolic subspace ${\mathbb H}^m_{\mathbb C}$ is $\omega^2_{{\mathbb H}^m_{\mathbb C}}$ where $\omega_{{\mathbb H}^m_{\mathbb C}}$ is a K\"ahler form on complex hyperbolic space. The relation between $\alpha$ and $\omega_D$ is as follows, see \cite[Lemma 1]{DG}.
$$\pi^*\alpha=\omega_D^2 + d\beta$$ for some $\mathrm{Sp}(m,1)$--invariant 3--form $\beta$. Note that $\omega_D^n\in H^{2n}_{c}(\mathrm{Sp}(m,1),{\mathbb R})$.
One can define the volume of a representation $\rho \colon\thinspace \Gamma\rightarrow \mathrm{Sp}(m,1)$ by$$ \mathrm{Vol}(\rho) = \inf \{ |\langle \rho^*_b(\omega_b),\beta \rightarrowngle| \ | \ c(\omega_b)=\omega^n_D \text{ and } \beta \in [M]^{\ell^1}_\mathrm{Lip} \}.$$
Then $\mathrm{Vol}(\rho)\leq \| \omega^n_D \|_\infty \cdot \| M \|_{\mathrm{Lip}}$.
Suppose $\mathrm{Vol}(\rho)=\mathrm{Vol}(M)$. Note that such a value is realized when $\rho$ is a totally geodesic embedding since the restriction of $\omega_D$ to ${\mathbb H}^n_{\mathbb C}$ is a K\"ahler form. Let $f\colon\thinspace {\mathbb H}^n_{\mathbb C}\rightarrow {\mathbb H}^m_{\mathbb H}$ be a $\rho$--equivariant harmonic map and let
$F$ be a lift of $f$ to the twister space $D$ so that $\pi\circ F=f$.
If $f^*\alpha=0$, then $F^*\omega_D^2=-dF^*\beta$ on $M$ and thus, we have $$\int_M F^*\omega^n_D =-\int_M F^*(\omega^{n-2}_D\wedge d\beta)=-\int_M dF^*(\omega^{n-2}_D\wedge \beta)=0.$$
Hence we may assume that $f^*\alpha\neq 0$, then the rank of $f$ is at least four at some point. By \cite{CD}, one can choose $F$ to be a holomorphic horizontal lift.
Then $F^* \omega_D^n$ represents some class
$\rho^*_b(\omega_b)$ with $c(\omega_b)=\omega^n_D$, and
$$\mathrm{Vol}(\rho)=\mathrm{Vol}(M)\leq \int_M F^*\omega_D^n. $$ Since $F$ is holomorphic, $F^*\omega_D\leq \omega_M$, hence
$$\int_M F^*\omega_D^n\leq \int_M \omega_M^n=\mathrm{Vol}(M).$$
This forces $F$ to be totally geodesic embedding.
Hence the local rigidity of
$\Gamma\subset \mathrm{SU}(n,1)\subset \mathrm{Sp}(m,1)$ also follows as in Corollary \ref{cor:8.2}, which is part of a result in \cite{KKP}.
\end{document} |
\begin{document}
\title{Value sets of bivariate Chebyshev maps over finite fields}
\author{\"{O}mer K\"{u}\c{c}\"{u}ksakall{\i}}
\address{Middle East Technical University, Mathematics Department, 06800 Ankara,
Turkey.}
\email{[email protected]}
\date{\today}
\begin{abstract}
We determine the cardinality of the value sets of bivariate Chebyshev maps over
finite fields. We achieve this using the dynamical properties of these maps
and the algebraic expressions of their fixed points in terms of roots of unity.
\end{abstract}
\subjclass[2010]{11T06}
\keywords{Chebyshev map; value set}
\maketitle
\section*{Introduction}
The Chebyshev polynomials show remarkable properties and they have applications
in many areas of mathematics. There is a generalization of these polynomials to
several variables introduced by Lidl and Wells \cite{lidlwells}. It is a well
known fact that the Dickson polynomial, a normalization of one variable
Chebyshev polynomial, induces a permutation on $\mathbf F_q$ if and only if
$\gcd(k,q^s-1)=1$ for $s=1,2$. It is in perfect analogy with one variable case
that the $n$ variable Chebyshev map is a bijection of $\mathbf F_q^n$ if and only if
$\gcd(k,q^s-1)=1$ for $s=1,2,\ldots,n+1$.
Let $f:\mathbf F_q^n\rightarrow\mathbf F_q^n$ be a polynomial map in $n$ variables defined
over $\mathbf F_q$. Denote its value set by $V(f,\mathbf F_q^n)=\{f(c):c\in\mathbf F_q^n\}$. Clearly
$f$ is a bijection of $\mathbf F_q^n$ if and only if its value set has cardinality
$q^n$. If $f$ is not a bijection, then it is natural to ask how far it is away
from being a bijection. There are several results in the literature which give
bounds on the cardinality of the value set. We refer to the work of Mullen, Wan
and Wang \cite{mullen} for a nice introduction to this problem which include
several references and historical remarks.
For an arbitrary polynomial map, there is no easy formula giving the cardinality
of the value set. However Chou, Gomez-Calderon and Mullen \cite{chou} achieve
in finding such a formula for the Dickson polynomials. In our previous work
\cite{kucuksakalli}, we gave a shorter proof of this formula by using a singular
cubic curve and generalized those computations to the elliptic case. More
precisely we have found the cardinality of the value sets of Latt\`{e}s maps,
which are induced by isogenies of elliptic curves, over finite fields.
In this paper we study the Chebyshev maps with two variables. The bivariate
Chebyshev map $\mathcal{T}_k$ is given by the formula
\[ \mathcal{T}_k(x,y) = (g_k(x,y),g_k(y,x)) \]
where $g_k(x,y)$ is the generalized Chebyshev polynomial defined by Lidl
and Wells \cite{lidlwells}. We have $g_{-1}(x,y)=y, g_{0}(x,y)=3 $ and
$g_{1}(x,y)=x$ and these polynomials satisfy the recurrence relation
\[ g_k(x,y) = xg_{k-1}(x,y) - yg_{k-2}(x,y) + g_{k-3}(x,y). \]
The recurrence relation work in both ways and $g_{k}(x,y)$ is defined for all
integers $k\in\mathbf Z$. Note that $g_k$ has integral coefficients and one can
consider the map induced on finite fields. The main result of this paper is
Theorem~\ref{main} which provides a formula for the cardinality of the
value set
\[V(\mathcal{T}_k,\mathbf F_q^2)=\{\mathcal{T}_k(x,y):(x,y)\in\mathbf F_q^2\}.\]
We achieve in finding such a formula by using the dynamical properties of
bivariate Chebyshev maps over complex numbers which are studied by Uchimura
\cite{uchimura}. Uchimura shows that the set of points with bounded orbits is a
certain closed domain $S$ in $\mathbf C^2$. This set is enclosed by Steiner's
hypocycloid, see Figure~\ref{fig:domains}. Moreover he shows that the
number of periodic points of order $n$ is equal to $|k|^{2n}$ if $|k|\geq2$.
Another tool for our computations is the nice expression of periodic points of
$\mathcal{T}_k$. Algebraically each periodic point is a triple sum of roots of unity,
a characterization due Koornwinder \cite{koornwinder}. We combine these facts
together with the identity $\mathcal{T}_q(x,y) \equiv (x^q,y^q) \mathfrak{p}mod{p}$.It follows
that $q^2$ fixed points of $\mathcal{T}_q$ reduce
to distinct elements in $\mathbf F_q^2$ modulo a certain prime ideal of a number field.
This is the idea we have used in order to compute the size of the value sets for
Latt\`{e}s maps \cite{kucuksakalli}. After characterizing the elements in
$\mathbf F_q^2$ in a compatible fashion under the action of $\mathcal{T}_q$, determining the
cardinality of $V(\mathcal{T}_k,\mathbf F_q^2)$ reduces to a combinatoric argument.
The organization of the paper is as follows: In the first section we
give an alternative computation of the value set of Dickson polynomials in
order to give the idea of our computations in the bivariate case. In the second
section, we review some known facts about the dynamics of bivariate Chebyshev
maps and classify the points which have bounded orbits. In the
third section we focus on the periodic and preperiodic points of $\mathcal{T}_k$
over complex numbers and their algebraic expressions. In the last section, we
find the cardinality of the value set of bivariate Chebyshev maps. We finish
our paper by giving an example.
\section{Single variable case}
In this section, we will consider the Dickson polynomials, a normalization of
one variable Chebyshev polynomials, and give an alternative computation of the
cardinality of their value sets. This alternative computation will be a summary
of the ideas that will be used in the rest of the paper.
The family of Chebyshev polynomials (of the first kind) are defined by the
recurrence relation $T_{k+1}(x) = 2xT_k(x) - T_{k-1}(x)$ where $T_0(x)=1$ and
$T_1(x)=x$. One can normalize the Chebyshev polynomials by the relation
$D_k(x)=2T(\frac{x}{2})$ to obtain the Dickson polynomials (of the first
kind). This is done in order to cancel the repeating factors of two and
consider the arithmetic over fields of characteristic two. The Dickson
polynomials satisfy a similar recurrence relation $D_{k+1}(x) = xD_{k}(x) -
D_{k-1}(x)$ where $D_0(x)=2$ and $D_1(x)=x$.
The first few Chebyshev and Dickson polynomials are:
\begin{align*}
T_0(x)&=1, & D_0(x) &=2,\\
T_1(x)&=x, & D_1(x) &=x,\\
T_2(x)&=2x^2-1, & D_2(x) &=x^2-2,\\
T_3(x)&=4x^3-3x, & D_3(x) &=x^3-3x,\\
T_4(x)&=8x^4-8x^2+1, & D_4(x) &=x^4-4x^2+2,\\
T_5(x)&=16x^5-20x^3+5x, &D_5(x) &=x^5 - 5x^3 + 5x.
\end{align*}
The dynamics of Dickson polynomials $D_k(x)$ over complex numbers is
well-understood. We refer to Silverman \cite{sil-dyn} for a nice summary of
these results. In contrast with our terminology, Silverman uses the term
Chebyshev polynomials for the family $D_k$.
In order to emphasize the analogy between the one variable case and the two
variable case, let us set
\[\alpha(\sigma)=e^{2\mathfrak{p}i i \sigma}+e^{-2\mathfrak{p}i i \sigma}, \hspace{20pt}
\sigma\in\mathbf R.\]
Let $k$ be a fixed integer with $k\geq 2$. It is a well-known fact that the
Julia set of $D_k(x)$ is the closed interval $J=[-2,2]$ in $\mathbf C$. Observe that
the Julia set can be given as $J=\{ \alpha(\sigma) : \sigma\in \mathbf R \}$. The
(forward) orbit of $x$ under $D_k$ is the set $\mathcal{O}(x)= \{D_k^n(x):n\geq0\}$ by
definition. Observe that the interval $[-2,2]$ can also be obtained as the set
of complex numbers $x$ whose orbits $\mathcal{O}(x)$ are bounded sets. Note that any
preperiodic or periodic point must be in the set $[-2,2]$ since their orbits
have finitely many elements.
The value sets of Dickson polynomials are first computed by Chou,
Gomez-Calderon and Mullen \cite{chou}. In \cite{kucuksakalli}, we gave an
alternative computation of their result by using a singular cubic curve. Now we
will give another approach which is a summary of ideas that will be used in the
rest of the paper.
The map $\alpha:\mathbf R/\mathbf Z \rightarrow [-2,2]$ is a two to one covering with two
exceptional points. Namely the points $-2=\alpha(1/2)$ and $2=\alpha(0)$. The
fixed points of $D_k$ satisfy the relation $D_k(x)=x$ by definition. Moreover
$x=\alpha(\sigma)$ for some $\sigma\in\mathbf R$ and we have $D_k(\alpha(\sigma)) =
\alpha(k\sigma)$. For real numbers $\sigma$ and $\tilde{\sigma}$, we have
$\alpha(\sigma) = \alpha(\tilde{\sigma})$ if and only if $\{\sigma,1-\sigma\}$
and $\{\tilde{\sigma},1-\tilde{\sigma}\}$ are equal as subsets of $\mathbf R/\mathbf Z$. It
follows from these observations that any fixed point of $D_k$ is of the form
$x=\alpha(r)$ for some rational number $r$. Moreover if $r$ is written in its
lowest terms, its denominator must be relatively prime to $k$. Using this
characterization, we can write
\[\mathbf Fix(D_k,\mathbf C)=\left\{\alpha\left(\frac{a}{k-1}\right):a\in\mathbf Z\right\} \cup
\left\{\alpha\left(\frac{a}{k+1}\right):a\in\mathbf Z\right\}.\]
Now let us count the elements in $\mathbf Fix(D_k,\mathbf C)$. Both sets in the above union
contain the element $2=\alpha(0)$. If $k$ is odd, then $-2$ is in their
intersection as well. Other than these two elements, the above sets are disjoint
since $\gcd(k-1,k+1)|2$. It follows easily that there are $k$ distinct elements
in $\mathbf Fix(D_k,\mathbf C)$ by the inclusion and exclusion principle.
Let $\mathbf F_q$ be a finite field of characteristic $p$. Consider the number field $K
= \mathbf Q( \mathbf Fix(D_q,\mathbf C) )$ which is obtained by adjoining the fixed points of $D_q$
to the rational numbers. Let $\mathfrak{p}$ be a prime ideal of $K$ lying over $p$.
We have $D_q(x)\equiv x^q \mathfrak{p}mod{p}$ and therefore the fixed points of $D_q$
reduced modulo $\mathfrak{p}$ are the solutions of $x^q-x=0$. Thus each element of
$\mathbf F_q$ is obtained by reducing a fixed point modulo $\mathfrak{p}$. Since there are
$q$ fixed points of $D_q$, we conclude that there is a one-to-one correspondence
\[\mathbf Fix(D_q,\mathbf C) \longleftrightarrow \mathbf F_q \]
which is obtained by the reduction modulo $\mathfrak{p}$.
From this point on finding a formula for the size of the value set is
straightforward. One can use the representations $\alpha(a/(q\mathfrak{p}m1))$ of
elements
in $\mathbf Fix(D_q,\mathbf C)$ in order to analyze the value set of $D_k$. Applying the
inclusion and exclusion principle, we find that
\[|V(D_k,\mathbf F_q)|=\frac{q-1}{2\gcd(k,q-1)}+\frac{q+1}{2\gcd(k,q+1)}+\eta(k,q).\]
Here $\eta(k,q)$ is a function which takes the values $0$ or $1/2$. More
precisely, $\eta(k,q)=0$ if and only if $\gcd(k,q-1) \equiv \gcd(k,q+1)
\mathfrak{p}mod{2}$.
\section{Bivariate Chebyshev maps}
There is a generalization of Chebyshev maps to higher dimensions introduced by
Lidl and Wells \cite{lidlwells}. For any integer $n$, the polynomial
equation $z^2-nz+1=0$ has roots $y$ and $1/y$ in the complex numbers. If $y^k$
and $1/y^k$ are the roots of $z^2-n'z+1=0$ then $n'$ is also an integer, and
it is a well known fact that $D_k(n)=n'$ where $D_k$ is the Dickson
polynomial (of the first kind).
Lidl and Wells generalize this construction by considering a polynomial
equation of degree $n+1$ with integral coefficients and with roots $t_1,t_2,
\ldots, t_{n+1}$. Then they consider another polynomial equation with
roots $t_1^k,t_2^k \ldots, t_{n+1}^k$. It turns out that there
is a system of polynomials with integral coefficients which give the
coefficients of the latter equation in terms of the coefficients of the former
equation. They are called the generalized Chebyshev polynomials. For details
see \cite{lidlwells}, or \cite{lidlnied}.
Now we focus on the bivariate case. Suppose that $x=t_1+t_2+t_3$ and $y=t_1t_2 +
t_1t_3 + t_2t_3$ with $t_1t_2t_3=a$. We assume that $a=1$ for simplicity.
According to the construction of Lidl and Wells, there exists a bivariate
polynomial $g_k(x,y)$ with integer coefficients which maps $(x,y)$ to $t_1^k
+ t_2^k + t_3^k$. Moreover it turns out that $g_k(y,x)$ is equal to $t_1^kt_2^k
+ t_1^kt_3^k + t_2^kt_3^k$. It is easy to see that $g_{-1}(x,y)=y, g_0(x,y)=3$
and $g_1(x,y)=x$. Further this family satisfies the recurrence relation
\[ g_k(x,y) = xg_{k-1}(x,y) - yg_{k-2}(x,y) + g_{k-3}(x,y). \]
The first few bivariate Chebyshev polynomials are:
\begin{align*}
g_0(x,y)&=3,\\
g_1(x,y)&=x,\\
g_2(x,y)&=x^2-2y,\\
g_3(x,y)&=x^3-3xy+3,\\
g_4(x,y)&=x^4- 4x^2y + 2y^2+4x,\\
g_5(x,y)&=x^5 - 5x^3y + 5xy^2 + 5x^2 - 5y.
\end{align*}
As we have seen from the first section, the dynamical properties of Dickson
polynomials play an important role in the analysis of the map induced over
finite fields. Thus we start with reviewing some known facts about the
bivariate Chebyshev map which is defined by
\[ \mathcal{T}_k(x,y) = (g_k(x,y),g_k(y,x)). \]
Dynamical properties of $\mathcal{T}_k$ on complex numbers are studied by Uchimura
\cite{uchimura}. Uchimura shows that the map $\mathcal{T}_k$ admits an invariant
plane $\{x=\bar{y}\}\subseteq\mathbf{C}^2$. The restriction of $\mathcal{T}_k$ to
this plane is the polynomial considered by Koornwinder \cite{koornwinder}. With
Koornwinder's notation, we have $\mathcal{T}_k(x,\bar{x}) = P_{(k,0)}^{-1/2}
(x,\bar{x})$. The key property we get from Koornwinder's work is the nice action
of Chebyshev maps on certain elements. Define
\[ \alpha(\sigma,\tau)=e^{2\mathfrak{p}i i\sigma}+e^{2\mathfrak{p}i i\tau}+e^{2\mathfrak{p}i i
(-\sigma-\tau)}, \hspace{20pt} \sigma,\tau\in\mathbf R.\]
We have
\[ \mathcal{T}_k\left(\alpha(\sigma,\tau), \overline{\alpha(\sigma,\tau)} \right)=
(\alpha(k\sigma,k\tau),\overline{\alpha(k\sigma,k\tau)}). \]
Let $k$ be a fixed integer with $|k|\geq2$. Uchimura shows that any point in
$\mathbf C^2$, whose orbit under $\mathcal{T}_k$ is a bounded set, must be in $\{(x,
\bar{x}): x\in S\}$ where
\[ S=\{\alpha(\sigma,\tau):\sigma,\tau\in\mathbf R\}. \]
Note that any periodic or preperiodic point must have a bounded orbit and
therefore it must be in $\{(x, \bar{x}): x\in S\}$ too. If we write $x=u+vi$,
then the set $S$ is a closed domain enclosed by Steiner's hypocycloid
\[ (u^2+v^2+9)^2+8(-u^3+3uv^2)-108=0. \]
The Steiner's hypocycloid is a simple closed curve which can also be
parametrized by $\alpha(\sigma,\sigma)$ with $0 \leq \sigma \leq 1$.
\begin{figure}
\caption{The domain $S$.}
\label{fig:domains}
\end{figure}
There is a symmetry under multiplication by a third root of unity because we
have $\alpha(\sigma,\tau)\zeta_3 = \alpha(\sigma+1/3,\tau+1/3)$.
Periodic points will play an important role in our computations. A periodic
point must have a bounded orbit thus its coordinates are of the form
$\alpha(\sigma,\tau)$ for some $\sigma,\tau\in\mathbf R$. The following lemma is the
key to count the elements in the value sets of bivariate Chebyshev maps over
finite fields.
\begin{lemma}\label{equal}
The complex numbers $\alpha(\sigma,\tau)$ and $\alpha(\tilde{\sigma} ,
\tilde{\tau})$ are equal if and only if $\{\sigma, \tau, -(\sigma+\tau)\}$ and
$\{\tilde{\sigma}, \tilde{\tau}, - (\tilde{\sigma} + \tilde{\tau})\}$ are equal
as subsets of $\mathbf R/\mathbf Z$.
\end{lemma}
\begin{proof}
To understand the representations of elements in $S$ in terms of
$\alpha(\sigma,\tau)$, a useful idea is to consider the tangent lines to
the hypocycloid
\[C=\{\alpha(\sigma,\sigma):\sigma\in [0,1]\}.\]
Define $\ell_\sigma$ to be the line passing through the point
$\alpha(\sigma,\sigma)$ with slope $-\tan(\mathfrak{p}i\sigma)$. If $\sigma\in 1/2+\mathbf Z$,
then set $\ell_\sigma$ as the vertical line $\mathfrak{Re}(z)=-1$. Note that the
lines
$\ell_\sigma$ are distinct for $\sigma\in [0,1)$.
The subset $\{\sigma, \tau, -(\sigma+\tau)\}$ of $\mathbf R/\mathbf Z$ has one element if
$\alpha(\sigma,\tau)$ is one of the three corner points of $C$. It has two
elements if $\alpha(\sigma,\tau)$ is on $C$ but not a corner point. The lemma
is trivially true in these cases.
We assume that $\alpha(\sigma,\tau)$ is an interior point of $S$. In this case
the subset $\{\sigma, \tau, -(\sigma+\tau)\}$ of $\mathbf R/\mathbf Z$ has three elements
and the lines $\ell_\sigma$, $\ell_\tau$ and $\ell_{-(\sigma+\tau)}$ are
pairwise distinct. Observe that each interior point of $S$ is realized
precisely
three times by the lines $\ell_\sigma$ as $\sigma$ varies on the interval
$[0,1)$. We will show that the lines $\ell_\sigma$, $\ell_\tau$ and
$\ell_{-(\sigma+\tau)}$ intersect at $\alpha(\sigma,\tau)$. This geometric
result will finish the proof because it gives a one-to-one correspondence
between the interior points of $S$ and the subsets of $\mathbf R/\mathbf Z$ with three
elements.
Consider $L_\sigma=\{ (\sigma,t):t\in\mathbf R \}$, a vertical line in $\mathbf R^2$. We
claim that $\alpha$, regarded as a map from $\mathbf R^2$ to $\mathbf C$, maps $L_\sigma$ to
the line segment $\ell_\sigma \cap S$. It follows that $\alpha(\sigma,\tau)$
lies on $\ell_\sigma$. By symmetry $\alpha(\sigma,\tau)$ lies on $\ell_\tau$,
too. Therefore the lines $\ell_\sigma$ and $\ell_\tau$ intersects at
$\alpha(\sigma,\tau)$. It is clear that $\ell_{-(\sigma+\tau)}$ passes through
the same point because $\alpha(\sigma,\tau)=\alpha(\sigma,-(\sigma+\tau))$. To
justify the claim $\alpha(L_\sigma) = \ell_\sigma \cap S$, we start with
\[ \alpha(L_\sigma) = \{ \zeta^{2 \mathfrak{p}i i\sigma}+\zeta^{2\mathfrak{p}i it}+\zeta^{-2\mathfrak{p}i
i(\sigma+t)}:t\in \mathbf R \}.\]
The parametric curve $\alpha(L_\sigma)$ in $\mathbf C$ has the following components:
\begin{align*}
f(t)=\mathfrak{Re}(\alpha(L_\sigma)) &= \cos(2\mathfrak{p}i \sigma)+\cos(2\mathfrak{p}i t) +
\cos(2\mathfrak{p}i(\sigma+t))\\
g(t)=\mathfrak{Im}(\alpha(L_\sigma)) &= \sin(2\mathfrak{p}i \sigma)+\sin(2\mathfrak{p}i t) -
\sin(2\mathfrak{p}i(\sigma+t)).
\end{align*}
The slope of the tangent line to the curve $\alpha(L_\sigma)$ at any point
$\alpha(\sigma,t)$ is given by $m=g'(t)/f'(t)$ provided that $f'(t)\neq
0$. We have
\begin{align*}
f'(t)/(2\mathfrak{p}i)&=-\sin(2\mathfrak{p}i t)-\sin(2\mathfrak{p}i(\sigma+t)=-2\sin(\mathfrak{p}i(\sigma
+ 2t))\cos(\mathfrak{p}i \sigma)\\
g'(t)/(2\mathfrak{p}i)&=\cos(2\mathfrak{p}i t)-\cos(2\mathfrak{p}i(\sigma+t))=2\sin(\mathfrak{p}i(\sigma +
2t))\sin(\mathfrak{p}i \sigma).
\end{align*}
Here, the second equalities follow from the sum to product formulas for the
trigonometric functions. Thus, $m=-\tan(\mathfrak{p}i \sigma)$. This computation shows
that $L_\sigma$ is mapped under $\alpha$ to a line segment with slope
$-\tan(\mathfrak{p}i
\sigma)$. If $\sigma\in 1/2+\mathbf Z$, then it is mapped to the vertical line
$\mathfrak{Re}(z)=-1$. Moreover, $\alpha(L_\sigma)$ is a line segment with end
points lying on the hypocycloid $C$. To see this, note that the functions
$f(t)$
and $g(t)$ have common critical values if and only if $\sin( \mathfrak{p}i(\sigma+2t)
)=0$. This is possible only at $t=1/2-\sigma/2$ and $t=1-\sigma/2$ for those
$t\in[0,1)$. The points corresponding to these two $t$ values are on the
hypocycloid $C$.
\end{proof}
As a final observation note that the restricted map $\alpha:[0,1)\times[0,1)
\rightarrow S$ is a six to one map unless $\alpha(\sigma,\tau)$ is on the
boundary $C=\{\alpha(\sigma,\sigma):\sigma\in \mathbf R\}$. A point on $C$ which is
not a corner point can be represented in three different ways and the corner
points $\alpha(0,0),\alpha(1/3,1/3)$ and $\alpha(2/3,2/3)$ can be represented
uniquely. We finish this section with an illustration of the lines $\ell_\sigma$
within the proof of Lemma\ref{equal}.
\begin{example}
The point $-1+\sqrt{-3}\in S$ can be represented by any of the
following six expressions: $\alpha(1/6,1/3), \alpha(1/3,1/6), \alpha(1/6,1/2),
\alpha(1/2,1/6), \alpha(1/2,1/3)$ and $\alpha(1/3,1/2)$. Note that the lines
$\ell_{1/6}, \ell_{1/3}$ and $\ell_{1/2}$ intersect at the same point, namely
$-1+\sqrt{-3}$. See Figure~\ref{fig:tangency}.
\end{example}
\begin{figure}
\caption{There lines meeting at $-1+\sqrt{-3}
\label{fig:tangency}
\end{figure}
\section{Periodic and preperiodic points}
The family of Dickson polynomials $D_k(x)$ has very explicit dynamical
properties. For example a point with bounded orbit must lie in the interval
$[-2,2]$ in $\mathbf C$. Moreover a point $x$ is preperiodic if and only if
$x=2\cos(2\mathfrak{p}i r)$ for some rational number $r$.
We want to classify all periodic and preperiodic points of $\mathcal{T}_k$. The
cases $k=-1$ $k=0$ and $k=1$ are trivial so we assume that $|k| \geq 2$. All
points with bounded orbit lie in $\{(x,\bar{x}):x\in S\}$ and their coordinates
are of the form $\alpha(\sigma,\tau)$ for some real numbers $\sigma$ and
$\tau$. Periodic and preperiodic points have bounded orbits since there are
finitely many elements in their orbits. Thus their coordinates are given by
$\alpha(\sigma,\tau)$. Moreover we have the following
\begin{lemma}
Let $k$ be a fixed integer with $|k|\geq 2$. A point $(x,y)\in\mathbf C^2$ is a
preperiodic point of $\mathcal{T}_k$ if and only if there exist rational
numbers $r,s\in\mathbf Q$ such that $x=\alpha(r,s)$ and $y=\bar{x}=\alpha(-r,-s)$.
Moreover if $r$ and $s$ are written in their lowest terms then
$(\alpha(r,s),\alpha(-r,-s))$ is a periodic point of $\mathcal{T}_k$ if and only if
the denominators of $r$ and $s$ are both relatively prime to $k$.
\end{lemma}
\begin{proof}
Suppose that $x=\alpha(r,s)$ where $r$ and $s$ are rational numbers. Then it is
easy to see that $(x,\bar{x})$ is a preperiodic point of $\mathcal{T}_k$. For the
converse, let $\alpha(\sigma,\tau)$ be a preperiodic point under $\mathcal{T}_k$ with
$|k| \geq 2$. It follows that
$$\alpha(k^n\sigma,k^n\tau) = \alpha(k^l\sigma,k^l\tau)$$
for some positive integers $n\leq l$. This is possible if and only if
\[\{k^n\sigma, k^n\tau, -k^n(\sigma+\tau)\} = \{k^l\sigma, k^l\tau,
-k^l(\sigma+\tau) \}\]
as subsets of $\mathbf R/\mathbf Z$ by Lemma~\ref{equal}. There are six possibilities. We
will prove only one case. The proofs for the other cases are similar. Suppose
that we have
\begin{align*}
k^n\sigma & \equiv k^l\tau \mathfrak{p}mod{\mathbf Z}, \\
k^n\tau & \equiv -k^l(\sigma+\tau) \mathfrak{p}mod{\mathbf Z}.
\end{align*}
We omit the third equation since it can be obtained from these two. Starting
with the former equation and then using the latter equation, we obtain
\[k^n\sigma \equiv k^l\tau \equiv k^{l-n}k^n\tau \equiv
k^{l-n}[-k^l(\sigma+\tau)] \mathfrak{p}mod{\mathbf Z}.\]
Now we plug in the first equation again and get
\[k^n\sigma \equiv k^{l-n}[-k^l\sigma-k^n\sigma] \mathfrak{p}mod{\mathbf Z}. \]
Therefore
\[ k^n\sigma+k^{2l-n}\sigma+k^l\sigma\equiv 0 \mathfrak{p}mod{\mathbf Z}. \]
From this congruence, we see that $\sigma$ is a rational number with
denominator $k^{2l-n}+k^l+k^n$. Since $k^n\sigma \equiv k^l\tau \mathfrak{p}mod{\mathbf Z}$, the
number $\tau$ must be rational too.
Now suppose that $(x,y)$ is a periodic point under $\mathcal{T}_k$. Then there exist
rational numbers $r=a/b$ and $s=c/d$ for some integers $a,b,c$ and $d$ such that
$x=\alpha(r,s)$ and $y=\alpha(-r,-s)$. Suppose that $r$ and $s$ are written in
their lowest terms, i.e. $\gcd(a,b)=1$ and $\gcd(c,d)=1$. Without loss of
generality we can assume that $(x,y)$ is fixed by $\mathcal{T}_k$. The general result
will follow from the identity $\mathcal{T}_k\circ \mathcal{T}_m = \mathcal{T}_{km}$ which is
valid on $\{ (x,\bar{x}):x\in S \}$. It follows by Lemma~\ref{equal} that the
set $\{r,s,-(r+s)\}$ modulo $\mathbf Z$ is permuted under multiplication by $k$. Thus
$r\equiv k^6r \mathfrak{p}mod{\mathbf Z}$ and therefore $r(k^6-1)\equiv 0 \mathfrak{p}mod{\mathbf Z}$. From this
we conclude that the denominator of $r$ is relatively prime to $k$ since it is
a
divisor of $k^6-1$. The same result holds for $s$ as well. To see the converse
let $f$ be order of $k$ modulo the least common multiple of denominators of $r$
and $s$. Then $\mathcal{T}_k^f$ fixes the point $(x,y)$ and therefore it is an
$f$-periodic point of $\mathcal{T}_k$.
\end{proof}
Now we want to describe the set of points in $\mathbf C$ which are fixed under
$\mathcal{T}_k$. Consider the following sets for $|k|\geq 2$:
\begin{align*}
\mathcal{A}_k & =\left\{\alpha\left( \frac{d}{k-1}, \frac{e}{k-1}
\right):d,e\in\mathbf Z \right\},\\
\mathcal{B}_k & =\left\{\alpha\left( \frac{d}{k^2-1}, \frac{dk}{k^2-1}
\right):d\in\mathbf Z \right\},\\
\mathcal{C}_k & =\left\{\alpha\left( \frac{d}{k^2+k+1} , \frac{dk}{k^2+k+1}
\right):d\in\mathbf Z \right\}.
\end{align*}
It is obvious that $\mathbf Fix(\mathcal{T}_k,\mathbf C^2) \supseteq \{ (x, \bar{x}):
x\in \mathcal{A}(k) \cup \mathcal{B}(k)
\cup \mathcal{C}(k)\}$. The converse inclusion is also true.
\begin{theorem}\label{fixed}
Let $k$ be a fixed integer with $|k|\geq 2$. Then
\[ \mathbf Fix(\mathcal{T}_k,\mathbf C^2) = \{ (x, \bar{x}): x\in
\mathcal{A}_k\cup\mathcal{B}_k\cup\mathcal{C}_k \}. \]
\end{theorem}
\begin{proof}
It is enough to show that the union $\mathcal{A}_k\cup\mathcal{B}_k\cup\mathcal{C}_k$ has $k^2$
elements. In order to do this we will apply the inclusion and exclusion
principle.
We start with counting the elements in $\mathcal{A}_k$. The set $\mathcal{A}_k$ have
elements of the form $\alpha(d/(k-1) , e/(k-1))$. It is enough to consider $0
\leq d,e \leq k-2$ because of the periodicity. There are $(k-1)^2$ such pairs of
$(d,e)$. These pairs do not result in distinct elements because there are some
repetitions. A fixed point will be represented six times among these pairs
unless $d=e$. An element of the form $\alpha(d/(k-1), d/(k-1))$ will be
represented three times unless $3d\equiv0\mathfrak{p}mod{k-1}$. Moreover three corner
points $\alpha(0,0),\alpha(1/3,1/3)$ and $\alpha(2/3,2/3)$ are in $\mathcal{A}_k$ if
$3|k-1$. If $3 \nmid k-1$, then only $\alpha(0,0)$ is in $\mathcal{A}_k$ among the
corner points. Thus
\[ |\mathcal{A}_k| = \frac{(k-1)^2+3(k-1)+2\gcd(k-1,3)}{6}. \]
The set $\mathcal{B}_k$ have elements of the form $\alpha(d/(k^2-1) , dk/(k^2-1))$.
It is enough to consider $0 \leq d \leq k^2-2$ because of the periodicity. A
fixed point will be represented two times among these values unless $d$ is a
multiple of $k+1$. In that case the representation will be unique. Therefore
\[ |\mathcal{B}_k| = \frac{(k^2-1)+(k-1)}{2} \]
The set $\mathcal{C}_k$ consists of elements of the form $\alpha(d/(k^2+k+1) ,
dk/(k^2+k+1))$ with $0 \leq d \leq k^2+k$. Every element is represented three
times in this case unless $k-1$ is divisible by $3$. Thus
\[ |\mathcal{C}_k| = \frac{(k^2+k+1)+2\gcd(k-1,3)}{3} \]
Now we consider the intersections. We start with $\mathcal{A}_k\cap
\mathcal{B}_k$. An element of the form $\alpha(d/(k^2-1) , dk/(k^2-1))$ is in
$\mathcal{A}_k$ if and only if $d$ is a multiple of $k+1$. There are $k-1$ such
integers among $\{0,1,2,\ldots,k^2-2\}$, each of which is represented uniquely.
As a result $|\mathcal{A}_k\cap\mathcal{B}_k|=k-1$. The set $\mathcal{A}_k\cap\mathcal{C}_k$ may
only have elements $\alpha(0,0), \alpha(1/3,1/3)$ or $\alpha(2/3,2/3)$ since
$\gcd(k-1,k^2+k+1)$ divides $3$. Thus $|\mathcal{A}_k\cap\mathcal{C}_k|=\gcd(k-1,3)$.
Similarly $|\mathcal{B}_k \cap \mathcal{C}_k| = \gcd(k-1,3)$, and $|\mathcal{A}_k \cap
\mathcal{B}_k
\cap \mathcal{C}_k| = \gcd(k-1,3)$. Now it is trivial to verify that $|\mathcal{A}_k \cup
\mathcal{B}_k\cup\mathcal{C}_k|=k^2$ by applying the inclusion and exclusion principle.
\end{proof}
\section{Value sets over finite fields}
It is a well known fact that the Dickson polynomial $D_k(x)$ induces a
permutation on $\mathbf F_q$ if and only if $\gcd(k,q^s-1)=$ for $s=1,2$. It is in
perfect analogy with one variable case that the $n$ variable Chebyshev map is a
bijection of $\mathbf F_q^n$ if and only if $\gcd(k,q^s-1)=1$ for $s=1,2,\ldots,n+1$
\cite{lidlwells}. In this section we compute the cardinality of
$V(\mathcal{T}_k,\mathbf F_q^2)$. As a corollary, we recover the result of Lidl and
Wells in the case $n=2$.
The coefficients of the Dickson polynomials $D_k(x)$ can be computed using the
following formula:
\[ D_k(x) = \sum_{i=0}^{\lfloor k/2\rfloor} \frac{k(-1)^i}{k-i}\binom{k-i}{i}
x^{k-2i}.\]
Let $q$ be a power of a prime $p$. It is easily verified using this formula
that $D_q(x)\equiv x^q \mathfrak{p}mod{p}$. Lidl and Wells provide a similar formula for
the bivariate Chebyshev polynomials \cite[p.~110]{lidlwells}. We have
\[ g_k(x,y) = \sum_{i=0}^{\lfloor k/2\rfloor} \sum_{j=0}^{\lfloor k/3\rfloor}
\frac{k(-1)^i}{k-i-2j}\binom{k-i-2j}{i+j} \binom{i+j}{i}x^{k-2i-3j}y^i \]
where only those terms occur for which $k\geq 2i+3j$. Recall that
$\mathcal{T}_k(x,y)
= (g_k(x,y),g_k(y,x))$. It is clear by this formula that
\[\mathcal{T}_q(x,y) \equiv (x^q,y^q) \mathfrak{p}mod{p}.\]
This congruence enables us to observe that the elements in $\mathbf F_q^2$ can be
obtained by reducing the elements of $\mathbf Fix(\mathcal{T}_q,\mathbf C^2)$ modulo a certain
prime ideal. Because there are precisely $q^2$ fixed points of $\mathcal{T}_q$, we
obtain the following lemma.
\begin{lemma}
Let $\mathbf F_q$ be a finite field of characteristic $p$. Consider the number field
$K=\mathbf Q(\mathbf Fix(\mathcal{T}_q,\mathbf C^2))$ which is obtained by adjoining the coordinates of
fixed points of $\mathcal{T}_q$ to the rational numbers. Let $\mathfrak{P}$ be a prime
ideal of $K$ lying over $p$. Then there exists a one-to-one correspondence
\[ \mathbf Fix(\mathcal{T}_q,\mathbf C^2) \longleftrightarrow \mathbf F_q^2 \]
which is given by the reduction modulo $\mathfrak{P}$.
\end{lemma}
After characterizing the elements in $\mathbf F_q^2$ in a compatible fashion under the
action of $\mathcal{T}_q$, determining the cardinality of $V(\mathcal{T}_k,\mathbf F_q^2)$
reduces
to a combinatoric argument. This is the idea we have used in order to compute
the size of the value sets for Latt\`{e}s maps \cite{kucuksakalli}.
\begin{theorem}\label{main}
Let $k$ be a nonzero integer and let $\mathbf F_q$ be a finite field of characteristic
$p$. Set
\[a=\frac{q-1}{\gcd(k,q-1)},\ \ b=\frac{q^2-1}{\gcd(k,q^2-1)}\ \ \textnormal{
and }\ \ c=\frac{q^2+q+1}{\gcd(k,q^2+q+1)}.\]
Then the cardinality of the value set is
\[|V(\mathcal{T}_k,\mathbf F_q^2)|=\frac{a^2}{6}+\frac{b}{2}+\frac{c}{3}+\eta(k,q)\]
where $\eta(k,q)$ is given by
\[\begin{array}{c|c|c}
\eta(k,q) & 3\nmid k \textnormal{ or } 3\nmid a & 3|k \textnormal{ and } 3|a\\
\hline
2 \nmid k \textnormal{ or } 2\nmid b & 0 & 2/3\\ \hline
2|k \textnormal{ and } 2| b & a/2 & a/2+2/3
\end{array}\]
In particular if $\gcd(k,6)=1$, then $\eta(k,q)=0$.
\end{theorem}
\begin{proof} We have $\mathbf Fix(\mathcal{T}_q, \mathbf C^2) = \{ (x, \bar{x}): x\in
\mathcal{A}_q\cup\mathcal{B}_q\cup\mathcal{C}_q \}$ and there is a one-to-one correspondence
between $\mathbf Fix(\mathcal{T}_q,\mathbf C^2)$ and $\mathbf F_q^2$. There will be three types of
elements $\mathcal{T}_k(x,y)$ in the value set $V(\mathcal{T}_k, \mathbf F_q^2)$ depending on $x$
being in $\bar{\mathcal{A}}_q, \bar{\mathcal{B}}_q$ and $\bar{\mathcal{C}}_q$. We will refer to
these elements as Type-I, Type-II and Type-III, respectively. The proof of the
case $k=1$ is similar to the proof of Theorem~\ref{fixed}. Other cases require a
more detailed investigation. For each type we give the form of $x$ and the
number of elements in that type by the following table:
\begin{center}
\begin{tabular}{c|c|c}
Type-I & $\alpha\left(\frac{d}{a}, \frac{e}{a}\right)$ &
$\frac{a^2+3a+2\gcd(a,3)}{6}$\\ \hline
Type-II & $\alpha\left(\frac{d}{b}, \frac{dq}{b}\right)$ &
$\frac{b+\gcd(b,q-1)}{2}$\\ \hline
Type-III & $\alpha\left(\frac{d}{c}, \frac{dq}{c}\right)$ &
$\frac{c+2\gcd(c,3)}{3}$ \\
\end{tabular}
\end{center}
The number of elements which fit into different types are given by the following
table:
\begin{center}
\begin{tabular}{c|c|c|c}
Type-I\&II & Type-I\&III & Type-II\&III & Type-I\&II\&III\\ \hline
$\gcd(a,b)$ & $\gcd(a,c)$ & $\gcd(b,c)$ & $\gcd(a,b,c)$
\end{tabular}
\end{center}
Applying the inclusion and exclusion principle we see that the cardinality
of the value set $V(\mathcal{T}_k,\mathbf F_q^2)$ can be written as
\begin{align*}
|V(\mathcal{T}_k,\mathbf F_q^2)|=& \left(\frac{a^2}{6}+\frac{b}{2}+\frac{c}{3}\right) +
\left(\frac{a}{2}+\frac{\gcd(b,q-1)}{2}-\gcd(a,b) \right) \\
& +\left( \frac{\gcd(a,3)+2\gcd(c,3)}{3}- \gcd(a,c)-\gcd(b,c)+\gcd(a,b,c)
\right).
\end{align*}
The second term in the above sum is $0$ unless $2|b$ and $2|k$. To see this
note that if $2|b$ and $2|k$, then $\frac{\gcd(b,q-1)}{2}=\frac{a}{2}$ and
$\gcd(a,b)=\frac{a}{2}$. If $2\nmid b$ or $2\nmid k$ then $\gcd(a,b)$
becomes $a$. A case by case investigation shows that the third term is $0$
unless $3|k$ and $3 |a$. If $3|k$ and $3|a$, then each greatest common divisor
appearing in the third term is equal to $1$ except $\gcd(a,3)=3$. Thus the third
term of the sum becomes $2/3$.
\end{proof}
We recover the result of Lidl and Wells for bivariate Chebyshev maps by
Theorem~\ref{main}. More precisely we have a sufficient and necessary condition
for bivariate Chebyshev maps for being a permutation of $\mathbf F_q^2$.
\begin{corollary}\label{maincor}
The bivariate Chebyshev map $\mathcal{T}_k(x,y)$ induces a permutation of $\mathbf F_q^2$
if and only $\gcd(k,q^s-1)=1$ for $s=1,2,3$.
\end{corollary}
We finish our paper by giving an example to illustrate the invariants
introduced in Theorem~\ref{main}.
\begin{example}
Let $k=6^i$ with $i=0,1,2,\ldots$ and consider the bivariate Chebyshev map
$\mathcal{T}_{6^i}$ on $\mathbf F_{73}^2$. We have
$\mathcal{T}_6(x,y)=(g_6(x,y),g_6(y,x))$ where
\[g_6(x,y)=x^6 - 6yx^4 + 9y^2x^2 + 6x^3 - 2y^3 - 12yx + 3.\]
The maps $\mathcal{T}_{6^i}$ are not bijections of $\mathbf F_{73}^2$ for $i=1, 2, 3,
\ldots$ since $6$ is not relatively prime to $73-1,73^2-1$ and $73^3-1$. We find
the cardinality of the value set by using Theorem~\ref{main} and obtain the
following table.
\[\begin{array}{c|c|c|c|c|c|c|c}
k & 6^0 & 6^1 & 6^2 & 6^3 & 6^4 & 6^5 & \ldots\\ \hline
a & 72 & 12 & 2 & 1 & 1 & 1 & \ldots\\ \hline
b & 5328 & 888 & 148 & 74 & 37 & 37 & \ldots\\ \hline
c & 5403 & 1801 & 1801 & 1801 & 1801 & 1801 & \ldots\\ \hline
\eta(k,73) & 0 & 12/2+2/3 & 2/2 & 1/2 & 0 & 0 & \ldots\\ \hline
|V(\mathcal{T}_k,\mathbf F_{73}^2)| & 5329 & 1075 & 676 & 638 & 619 & 619 & \ldots
\end{array}\]
Note that the size of the value set of $\mathcal{T}_{6^i}$ will be $619$ for $i \geq
4$ since $a,b$ and $c$ are relatively prime to $6$ from that point on.
\end{example}
{\small
\defReferences{References}
\newcommand{\etalchar}[1]{$^{#1}$}
\end{document} |
\begin{document}
\title[time crystalline order]{Stability of the discrete time-crystalline order in spin-optomechanical and open cavity QED systems}
\author{Zhengda Hu}
\affiliation{Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA}
\affiliation{School of Science, Jiangnan University, Wuxi 214122, China}
\author{Xingyu Gao}
\affiliation{Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA}
\affiliation{School of Science, Jiangnan University, Wuxi 214122, China}
\author{Tongcang Li}
\email{[email protected]}
\affiliation{Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA}
\affiliation{Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA}
\affiliation
{Birck Nanotechnology Center, Purdue University, West Lafayette,
IN 47907, USA}
\affiliation
{Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA}
\date{\today}
\begin{abstract}
Discrete time crystals (DTC) have been demonstrated experimentally in several different quantum systems in the past few years. Spin couplings and cavity losses have been shown to play crucial roles for realizing DTC order in open many-body systems out of equilibrium. Recently, it has been proposed that eternal and transient DTC can be present with an open Floquet setup in the thermodynamic limit and in the deep quantum regime with few qubits, respectively.
In this work, we consider the effects of spin damping and spin dephasing on the DTC order in spin-optomechanical and open cavity systems in which the spins can be all-to-all coupled.
In the thermodynamic limit, it is shown that the existence of dephasing can destroy the coherence of the system and finally lead the system to its trivial steady state.
Without dephasing, eternal DTC is displayed in the weak damping regime, which may be destroyed by increasing the all-to-all spin coupling or the spin damping.
By contrast, the all-to-all coupling is constructive to the DTC in the moderate damping regime. We also focus on a model which can be experimentally realized by a suspended hexagonal boron nitride (hBN) membrane with a few spin color centers under microwave drive and Floquet magnetic field. Signatures of transient DTC behavior are demonstrated in both weak and moderate dissipation regimes without spin dephasing.
Relevant experimental parameters are also discussed for realizing transient DTC order in such an hBN optomechanical system.
\end{abstract}
\maketitle
\section{Introduction}
In recent years, periodically driven (Floquet) quantum many-body systems have attracted considerable attention since they are crucial for understanding new non-equilibrium Floquet many-body localization (MBL)~\cite{Abanin2019} phase and may have potential applications in quantum metrology ~\cite{Lyu2020}.
One example of a non-equilibrium Floquet-MBL phase is the discrete time-crystalline (DTC) order~\cite{Sacha2015,Else2016,Khemani2016},
which is different from a continuous time crystal~\cite{Wilczek2012,Li2012,Huang2018,Huang2020} and is characterized by the breaking of discrete time-translation symmetry (TTS)~\cite{Sacha2020}. The DTC order has been realized experimentally in several quantum systems in the past few years~\cite{Choi2017,Zhang2017,Randall2021,Kyprianidis2021}. Under driving with a period $T$, the system can exhibit stroboscopic response with a period $nT$ and it is expected to be robust against imperfection of the driving~\cite{von Keyserlingk2016,Yao2017}.
Recently, the DTCs in open Floquet systems have been \mbox{reported~\cite{Lazarides2017,Else2017,Gong2018,Zhu2019,KL2021,Lazarides2020,Riera-Campeny2020}.}
Since any realistic systems will be unavoidably coupled to its surroundings and the influences of baths can be either negative or positive, the mechanisms of stabilizing DTC in dissipative systems will be important to explore.
\begin{figure}
\caption{Sketches of the setups for realizing the DTC order: ({\bf a}
\label{fig:fig1}
\end{figure}
Meanwhile, recent development of optomechanical systems~\cite{Fabre1994,Mancini1994,RMP2014,Yin2015,Xu2021casimir} has facilitated breakthroughs of quantum technologies such as ground state cooling~\cite{Chan2011,Liu2013}, optical sensing~\cite{Xiong2017,Liu2017,Krause2012,Ahn2020, LiBB2021}, and quantum information processing~\cite{Stannigel2010,Stannigel2012}.
With nanoscale cavity optomechanical devices, the coupling between light and motion of mechanical resonators can be flexibly modulated with controllable loss~\cite{Karg2020}, which may even reach ultrastrong coupling regime~\cite{Frisk Kockum2019}.
A natural choice of mechanical modes is to use membranes of two-dimensional materials due to their excellent mechanical properties~\cite{Akinwande2017}.
Recently, hexagonal boron nitride (hBN) has drawn great interest and served as a promising platform for exploring both quantum and nanophotonic effects~\cite{Tran2016,Cadiz2018,Liu2019,Xia2014,Klusek2010}. hBN has a very wide bandgap and outstanding chemical and thermal stability beyond that of graphene. As a type of van der Waals materials, hBN can be integrated with plasmonic, nanophotonic, and potentially more complex structures~\cite{Tran2017,Caldwell2019,Gao2020,Wu2021,Xu2021}. The hBN membranes have low mass, small out-of-plane stiffness, high elasticity modulus and strong tensile strength, which make them a promising candidate for high-Q mechanical resonators and high-sensitivity sensors~\cite{Kim2018,Shandilya2019}.
A spin-mechanical system based on color centers in a suspended hBN mechanical resonator has been proposed~\cite{Abdi2017,Abdi2019},
which can even simulate the Rabi model in the ultrastrong coupling regime. Very recently, optically addressable spin defects were observed in hBN~\cite{Gottscholl2020,Chejanovsky2021,Gao2021}. As the DTC order has been found in $N$ atoms in a lossy cavity ~\cite{Gong2018,Zhu2019,KL2021}, it is interesting to explore the DTC in such spin-optomechanical systems with incoherent noise (spin damping or~dephasing).
In this work, we consider the DTC behaviors in an open Floquet system as $N$ qubits in a (mechanical) cavity via switching on and off of the spin-cavity coupling. In the thermodynamic limit, it describes a cavity QED model with a large ensemble of trapped spins while, in the deep quantum regime (with few qubits), it characterizes an optomechanical model as a suspended hBN monolayer membrane with a few spin defects under a microwave drive and a Floquet magnetic field (Figure \ref{fig:fig1}). We discuss stroboscopic dynamics in both regimes and explore whether stroboscopic oscillations are stable to spin damping and spin dephasing as well as the effect of all-to-all spin coupling.
\section{Perfect DTC in the Thermodynamic Limit}
We consider an open system as $N$ qubits with all-to-all interactions in a (mechanical) cavity (Figure \ref{fig:fig1}). The all-to-all coupling can be mediated by a photon in an optical cavity~\cite{Gong2018} or a phonon in a mechanical oscillator~\cite{Abdi2017,Abdi2019,LiB2020}. The Hamiltonian is given by~\cite{Gong2018,Zhu2019,Abdi2017,Abdi2019,Morrison2008,Russomanno2017}
\begin{equation}\label{H}
\hat{H}(h,\lambda)=\omega_{0}\sum_{i}\hat{s}_{i}^{z}+\omega\hat{a}^{\dag}
\hat{a}+\frac{2h}{N}\sum_{i<j}\hat{s}_{i}^{z}\hat{s}_{j}^{z}+\frac{2\lambda
}{\sqrt{N}}(\hat{a}+\hat{a}^{\dag})\sum_{i}\hat{s}_{i}^{x},
\end{equation}
where $\hat{a}$ ($\hat{a}^{\dag}$) is the annihilation (creation) operator of the photon field with optical frequency $\omega$,
$\hat{s}_{i}^{\mu}$ ($\mu=x,y,z$) is the spin-$\frac{1}{2}$ angular momentum operator along the $\mu$ axis for the $i$-th qubit
of transition frequency $\omega_{0}$, and $h$ ($\lambda$) is related to the spin-spin (spin-cavity) coupling strength.
For convenience, a more compact version can be derived as
\begin{equation}
\hat{H}(h,\lambda)=\omega_{0}\hat{J}_{z}+\omega\hat{a}^{\dag}\hat{a}+\frac
{h}{N}\hat{J}_{z}^{2}+\frac{2\lambda}{\sqrt{N}}(\hat{a}+\hat{a}^{\dag})\hat
{J}_{x},
\end{equation}
by introducing the collective angular moment operator $\hat{J}_{\mu}=\sum_{i}\hat{s}_{i}^{\mu}$ and neglecting a constant term.
We consider a general decoherent model by including both the spin and cavity losses.
Then, the dynamics of the system can be described by the master equation (setting $\hbar=1$)
\begin{equation}\label{mseq}
\frac{\mathrm{d}\hat{\rho}}{\mathrm{d}t}=-\mathrm{i}[\hat{H},\hat{\rho
}]+\gamma D[\hat{a}]\hat{\rho}+\frac{\Gamma}{N}D[\hat{J}_{-}]\hat{\rho}
+\frac{\tilde{\Gamma}}{N}D[2\hat{J}_{z}]\hat{\rho},
\end{equation}
where $\hat{J}_{-}=\hat{J}_{x}-\mathrm{i}\hat{J}_{y}$ is the collective lowering operator and
$D[\hat{o}]\hat{\rho}=\hat{o}\hat{\rho}\hat{o}^{\dag}-(\hat{o}^{\dag}\hat{o}\hat{\rho}+\hat{\rho}\hat{o}^{\dag}\hat{o})/2$. Here, $\gamma=\omega/Q$
is the cavity damping rate with $Q$ the quality factor.
In addition, $\Gamma$ and $\tilde{\Gamma}$ are the spin relaxation and dephasing rate, respectively. Previous works mainly focused on the DTC in cavity QED systems with merely the cavity loss or the nearest-neighbor (short-range) spin coupling~\cite{Gong2018,Zhu2019,KL2021}. They have neither discussed stabilizing DTC in dissipative systems with all-to-all coupling nor considered the effects of spin damping and spin dephasing.
\begin{figure*}
\caption{Stroboscopic dynamics (\textbf{top}
\label{fig:fig2}
\end{figure*}
First, we would like to consider the robustness of DTC behavior in the thermodynamic limit $N\rightarrow\infty$.
By performing the mean-field approximation and factorizing the means of operator product, we obtain a closed set of semiclassical equations as
\begin{align}\label{mfeqns}
\dot{j}_{x} & =-\omega_{0}j_{y}-hj_{y}j_{z}+\frac{\Gamma}{2}j_{x}
j_{z}-\tilde{\Gamma}j_{x},\nonumber\\
\dot{j}_{y} & =\omega_{0}j_{x}-2\lambda\sqrt{2\omega}xj_{z}+hj_{x}
j_{z}+\frac{\Gamma}{2}j_{y}j_{z}-\tilde{\Gamma}j_{y},\nonumber\\
\dot{j}_{z} & =2\lambda\sqrt{2\omega}xj_{y}+\frac{\Gamma}{2}(j_{z}^{2}-1),\nonumber\\
\dot{x} & =p-\frac{\gamma}{2}x,\nonumber\\
\dot{p} & =-\omega^{2}x-\frac{\gamma}{2}p-2\lambda\sqrt{2\omega}j_{x},
\end{align}
where $j_{\mu}= \langle \hat{J}_{\mu} \rangle /j$ with $j=N/2$ and ${\sum_{\mu}}j_{\mu}^{2}=1$, $x=\langle \hat{a}+\hat{a}^{\dag}\rangle/\sqrt{2N\omega}$, and $p=\mathrm{i}\langle \hat{a}^{\dag}-\hat{a}\rangle /\sqrt{2N/\omega}$.
The set of Equation~(\ref{mfeqns}) is a generalization of that in Reference~\cite{Gong2018} which is a special case as $h=0$ here.
The introduction of spin-spin coupling $h$ breaks the original stable attractors $(j_x, j_y, j_z)_\mathrm{st}=(\pm \sqrt{1-\mu^2},0,-\mu)/2$ and $(x,p)_\mathrm{st}=\mp[\lambda\sqrt{2\omega(1-\mu^2)}/(\omega^2+\gamma^{2}/4)](1,\gamma/2)$,
with $\mu=(\lambda_\mathrm{c}/\lambda)^2$ and the critical spin-cavity coupling strength $\lambda_\mathrm{c}=\sqrt{(\omega_0/\omega)(\omega^2+\gamma^2/4)}/2$.
We would also like to focus on the steady-state solutions as Reference~\cite{Gong2018}, which is instead numerically found out due to the more complexity considered.
It is clear that there exist trivial steady-state solutions as $x=p=j_{x}=j_{y}=0$ and $j_{z}=\pm1$.
Besides, as long as the dephasing exists ($\tilde{\Gamma}\neq0$), the steady-state solutions will fall into be trivial.
This can be understood as that the existence of dephasing will finally destroy the coherence (non-diagonal terms of density matrix)
and leads to the final state as either $\left\vert +N/2\right\rangle$ or $\left\vert -N/2\right\rangle$ when the $\mathbb{Z}_2$ symmetry is broken at $\lambda>\lambda_\mathrm{c}$. Here, $\left\vert \pm N/2\right\rangle $ are the eigenstates of $\hat{J}_{z}$ with
$\hat{J}_{z}$ $\left\vert \pm N/2\right\rangle =\pm N/2\left\vert \pm N/2\right\rangle$.
Therefore, we set $\tilde{\Gamma}=0$ in the following discuss, unless specifically mentioned.
Besides, we assume the spins are initially in the eigenstate $\left\vert \rightarrow\rightarrow\cdots\rightarrow\rightarrow
\right\rangle $ with $j_{x}|_{t=0}=1$, $j_{y}|_{t=0}=0$, and $j_{z}|_{t=0}=0$ and the cavity mode is initially in a coherent state $\left\vert
\alpha\right\rangle $ with $x|_{t=0}=p|_{t=0}=0$. If we consider the symmetry-broken regime $\lambda>\lambda_\mathrm{c}$, it is clear that the final state will fall into either one of the two nontrivial stable states. To observe a DTC order, we perform the Floquet driving protocol similar to Reference~\cite{Gong2018}: the spin-cavity coupling $\lambda$ is artificially switched off in the second-half period, i.e., $\lambda=0$ for $(n+1/2)T \leq t<(n+1)T$ with $n=0,1,2,\ldots$. From an alternative viewpoint, the Floquet driving is that we let the spins periodically driven by a leaky cavity in every first-half period $nT \leq t<(n+1/2)T$. We introduce the imperfection parameter $\varepsilon$ via a detuning between $\omega$ and $\omega_0$ as $\omega=(1-\varepsilon)\omega_\mathrm{T}$ and $\omega_0=(1+\varepsilon)\omega_\mathrm{T}$ with $\omega_\mathrm{T}=2\pi/T$. In the perfect case ($\varepsilon=0$), it is not difficult to check that the unitary dynamics during the second-half period contributes a parity operator $P=\mathrm{e}^{-\mathrm{i}\pi(a^{\dag}a+J_z)}$ which flips the stable state to the other one. If certain observables of the spins (say $j_\mu$) or the cavity mode (say $x,p$) exhibit period doubling oscillations which are robust against imperfection driving $\varepsilon$, then a DTC order may be identified. We also consider nonunitary imperfections due to decoherence of the system. To observe the long-time behavior, we numerically solve a Floquet--Lindblad master equation (setting $\lambda$ in Equation~(\ref{mfeqns}) be periodically time-dependent as characterized above) up to $500$ periods $T$ by means of the Runge-Kutta method. We shall remark that we have also tried more periods such as $5000$ periods as in Reference~\cite{Gong2018} but there is no qualitative difference. For convenience, we set $\omega_\mathrm{T}=1$ and $\lambda=1$ to illustrate the perfect DTC in the $\lambda>\lambda_\mathrm{c}$ regime.
\begin{figure*}
\caption{Stroboscopic dynamics
(\textbf{top}
\label{fig:fig3}
\end{figure*}
In Figures~\ref{fig:fig2} and~\ref{fig:fig3}, we plot the stroboscopic dynamics of the scaled angular momentum vector $\vec{j}=(j_x,j_y,j_z)$ as well as their stroboscopic trajectories on the Bloch sphere for the perfect driving ($\varepsilon=0$) and imperfect driving ($\varepsilon\neq 0$) cases, respectively.
By comparing the first row a-d where there is no spin-spin coupling with $h=0$, we clearly observe different stroboscopic dynamics in different dissipation regimes. First, the DTC order is well preserved by the existence of weak spin damping $\Gamma$ as shown in Figure~\ref{fig:fig2}a and robust again imperfection $\varepsilon$ as shown in Figure~\ref{fig:fig3}a. As the spin damping rate $\Gamma$ increases, the DTC dynamics becomes irregular with the trajectory of $\vec{j}$ scattered on the Bloch sphere (Figures~\ref{fig:fig2}b and ~\ref{fig:fig3}b). However, the dynamics will become more regularly with the area of stroboscopic trajectories reduced if the cavity loss rate $\gamma$ increases (Figures~\ref{fig:fig2}c and ~\ref{fig:fig3}c). The eternal stroboscopic oscillations will occur again with the trajectories almost collapse into the two stable points for $\gamma\gg \Gamma$ (Figure~\ref{fig:fig2}d), which is robust against imperfection $\varepsilon$ (Figure~\ref{fig:fig3}d) so as to identify the DTC order.
Besides, by comparing the second row (e-h) where there is spin-spin coupling $h\neq 0$, different stroboscopic dynamics from that of $h=0$ is also demonstrated in different dissipation regimes. From Figures~\ref{fig:fig2}e--h (perfect $\varepsilon=0$ case) with growing all-to-all coupling $h$, we observe that DTC oscillations is gradually destroyed and the system finally falls into one of the trivial stable states with $j_x=j_y=0$ and $j_z=-1$ (Figure~\ref{fig:fig2}h).
By contrast, in the imperfect case ($\varepsilon\neq 0$) as shown in Figures~\ref{fig:fig3}e--h, we surprisingly find that the DTC order may be rebuilt by appropriate $h$ in the moderate damping regime, by comparing Figure~\ref{fig:fig3}g with Figure~\ref{fig:fig2}g.
\section{Transient DTC Behavior in the Deep Quantum Regime}
We proceed to focus on the few-atom cases [$N\sim O(1)$], which corresponds to the hBN optomechanical system as displayed in Figure~\ref{fig:fig1}b.
It is expected that a DTC behavior may still survive in few atom cases, the so-called deep quantum regime~\cite{Gong2018}.
In this regime, we do not perform semiclassical approximation so that all the quantumness of the system is well maintained.
The interplay among spin-spin coupling, spin-cavity coupling and dissipations may give rise to more subtle behaviors for transiently long DTC
in this deep quantum regime. By transiently long we mean that the DTC lasts much longer than the decay time $\gamma^{-1}$. The initial state is chosen to be $\left\vert \Rightarrow \right\rangle $ $\otimes$ $\left\vert \alpha\right\rangle $,
where $\left\vert\Rightarrow\right\rangle \equiv\otimes_{j=1}^{N}\left\vert \rightarrow\right\rangle $ is the eigenstate of $\hat{J}_{x}$ with the eigenvalue $N/2$ and $\left\vert \alpha\right\rangle $ is a coherent state with $\hat{a}\left\vert \alpha\right\rangle =\alpha\left\vert \alpha\right\rangle $.
The Floquet--Lindblad dynamics extended from Equations~(\ref{H}) and (\ref{mseq}) is directly solved under a truncation of $16$ photons for $\alpha=0.01$.
\begin{figure*}
\caption{Stroboscopic
dissipative dynamics of the scaled angular momenta of $j_x$ (red solid curve), $j_y$ (green dashed curve), and $j_z$ (blue dotted curve) in the two-qubit $N=2$ case. The inset shows quadratures $x$ (purple solid) and $p$ (black dashed) behaviors. We consider weak dissipation in (\textbf{a}
\label{fig:fig4}
\end{figure*}
\begin{figure*}
\caption{Stroboscopic
dissipative dynamics of the scaled angular momenta of $j_x$ (red solid curve), $j_y$ (green dashed curve), and $j_z$ (blue dotted curve) for the
three-qubit $N=3$ case. The parameter setups are the same as those in Figure~\ref{fig:fig5}
\label{fig:fig5}
\end{figure*}
Figure \ref{fig:fig4}a shows the stroboscopic dynamics of the scaled angular momenta $j_{\mu}$ and quadratures $x$, $p$ (inset)
in the strong coupling regime ($\lambda=1$)
and weak dissipation regime ($\gamma=\Gamma=0.05$) for the two-qubit case ($N=2$).
We clearly observe that $j_{x}$ and $x$ exhibit stroboscopic oscillations with doubling period $2T$ after $t\sim5T$,
which persists even at $t\sim50T$ and thus is much longer than the decay time (here $\gamma^{-1}=\Gamma^{-1}\sim3T$). In this sense, a transient DTC order is established in the deep quantum regime before reaching the stationary state.
In Figure~\ref{fig:fig4}b, we plot the stroboscopic dynamics in the moderate dissipation regime ($\gamma=\Gamma=0.3$). In this case, the decay time
can be estimated as $\gamma^{-1}=\Gamma^{-1}\sim 0.5 T$ so that the stroboscopic dynamics occur immediately and lasts over $10 T$,
which still maintains a transient DTC order.
Moreover, if the spin dephasing $\tilde{\Gamma}\approx 2 \Gamma$ as predicted in Reference~\cite{Abdi2017} is additionally considered, as shown in Figures~\ref{fig:fig4}c,d, we find that the oscillation time is merely comparable to the decay time and thus no transient DTC order exists. Besides, we observe similar phenomena if more spins are involved such as the case of $N=3$ shown in Figure~\ref{fig:fig5}. One effect of increasing the spin number $N$ is that the transient oscillations evolve into an eternal one as predicted at the thermodynamic limit $N\rightarrow \infty$ in Figure~\ref{fig:fig2}. Another effect of increasing $N$ may be that the stroboscopic oscillations is more robust against the spin dephasing as comparing the oscillation dynamics of quadrature $x$ (purple solid) in Figures~\ref{fig:fig4}c and ~\ref{fig:fig5}c.
Before ending, we would like to discuss the setup of experimental parameters for realizing TDC order in the optomechanical system of hBN monolayer membrane. According to Referecne~\cite{Abdi2017}, a maximum magnetic field gradient $270$ $\mathrm{G}/\mathrm{nm}$ may be reached such that the spin-cavity coupling $\lambda$ may become comparable or even larger than the oscillator frequency $\omega$. In this work, we consider $\lambda=\omega_{\mathrm{T}}$, $\omega=(1-\varepsilon)\omega_{\mathrm{T}}$ and $\omega_0=(1+\varepsilon)\omega_{\mathrm{T}}$ with $\varepsilon\leq 10 \%$, and the cavity loss rate $\gamma\leq 1.5 \omega_{\mathrm{T}}$, which indicates $\lambda_\mathrm{c}\leq 0.65 \omega_{\mathrm{T}}$.
To insure the occurrence of transient DTC dynamics, we need operate in the regime of $\lambda >\lambda_\mathrm{c}$. Then, the minimal spin-cavity coupling is to achieve $\lambda >0.65 \omega_{\mathrm{T}}$, which is realizable in a suspended circular hBN membrane with radius $R\sim 1$ $ \mu \mathrm{m}$. Another important aspect is to control the dephasing rate $\tilde{\Gamma}$ which is detrimental to the DTC order. According to Reference~\cite{Abdi2017}, the spin dephasing mainly stems from optical polarization $\tilde{\Gamma}_\mathrm{o}$ and membrane vibrations $\tilde{\Gamma}_\mathrm{v}$, which is proportional to the vibration frequency $\omega$. Therefore, to suppress the dephasing rate, it is suggested to reduce the cavity frequency $\omega$, which also corresponds to enhance the membrane radius $R$. Last, but not least, the cavity loss promotes spin cooling and localization, which is crucial to the emergence of DTC. However, as can be indicated by comparing Figure ~\ref{fig:fig4}b with Figure ~\ref{fig:fig4}a (or Figure ~\ref{fig:fig5}b with Figure ~\ref{fig:fig5}a), a too strong cavity loss $\gamma$ (corresponding to extremely low $Q$) may overdamp the system dynamics and destroy the DTC order. Besides, a stronger $\gamma$ leads to a higher critical spin-cavity coupling $\lambda_\mathrm{c}$ such that stronger spin-cavity coupling $\lambda$ is needed, which imposes a challenge to its experimental realization. For cavity loss rate $\gamma=0.05 \omega_{\mathrm{T}}$ as considered in Figures~\ref{fig:fig4}c and ~\ref{fig:fig5}c, the quality factor $Q$ is about $20$, which provides a balance between spin cooling and loss to make experimental realization more feasible ~\cite{Shandilya2019}. Overall, negligible spin dephasing, weak spin damping and appropriate cavity loss are suggested in realizing transient DTC order in such an optomechanical system.
\section{Conclusions}\label{sec:sec5}
In summary, we have investigated DTC order in a Floquet open system composed of $N$ qubits trapped in a (mechanical) cavity. The influences of all-to-all spin interactions, spin damping, spin dephasing as well as cavity loss are explored both in the thermodynamic limit and in the deep quantum regime. It is shown that the existence of dephasing will destroy the coherence of the system and finally leads the system to its trivial steady state. Without dephasing and all-to-all spin coupling, different stroboscopic dynamics in different dissipation regimes is demonstrated. First, with weak spin damping and weak all-to-all coupling, eternal DTC oscillations are observed and robust against imperfection. As the spin damping rate increases, the stroboscopic dynamics evolves irregularly accompanied by the trajectory of the scaled angular momentum vector scattered on the Bloch sphere. However, with enhancement of cavity loss, the dynamics will become more regularly and the eternal eternal DTC order will reemerge at strong cavity loss.
Besides, by growing the all-to-all coupling, we demonstrate that stroboscopic oscillations are gradually destroyed in the weak damping regime. It is interesting to show that the DTC order may be rebuilt by appropriate all-to-all coupling in the moderate damping regime.
We also focus on the few-atom cases, the so-called deep quantum regime, the model of which describes a suspended hBN monolayer membrane with a few spin defects under a microwave drive and a Floquet magnetic field. A transient DTC lasting much longer than the decay time can be found in both weak and moderate dissipation regimes when there is no spin dephasing. Nonetheless, the existence of dephasing will destroy transient oscillations and leads the system fast to a trivial steady state, which is consistent with the results obtained by semiclassical approximation in the thermodynamic limit. We also find that stroboscopic oscillations may be more robust against the spin dephasing by increasing the spin number. Finally, the parameters in the experimental aspect are briefly discussed and how to realizing transient DTC order in such an hBN optomechanical system is suggested.
T.L. acknowledges the support from NSF (Grant No. PHY-2110591). Z.H. acknowledges the support from the Fundamental Research Funds for the Central Universities (Grant No. JUSRP21935) and the China Scholarship Council (CSC).
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\end{thebibliography}
\end{document} |
\begin{document}
\begin{abstract}
We propose a two-component mixture of a noninformative (diffuse) and an informative prior distribution, weighted through the data in such a way to prefer the first component if a prior-data conflict arises. The data-driven approach for computing the mixture weights makes this class data-dependent. Although rarely used with any theoretical motivation, data-dependent priors are often used for different reasons, and their use has been a lot debated over the last decades. However, our approach is justified in terms of Bayesian inference as an approximation of a hierarchical model and as a conditioning on a data statistic. This class of priors turns out to provide less information than an informative prior, perhaps it represents a suitable option for not dominating the inference in presence of small samples. First evidences from simulation studies show that this class could also be a good proposal for reducing mean squared errors.
\textit{Keywords}: Informative prior, Prior-data conflict, Data-dependent prior, Mixture prior, Small sample size, Hierarchical approximation, Mean squared error.
\end{abstract}
\section{Introduction}
Prior elicitation is the core of every Bayesian analysis and the prior should represent the belief of the statistician before observing the data. But for several reasons in the last decades many attempts for including data information in the elicitation process have been proposed. Roughly speaking, the resulting data-dependent prior is just a prior that depends on the data and suffers from two main criticisms: data are used twice and the calculus of the Bayes' theorem may not be performed directly.
Despite this evident contravention of the Bayesian philosophy, many statisticians dealt with the double use of the data in Bayesian inference, and many others use data-dependent priors for complex models. However, as invoked by \cite{wasserman2000asymptotic} almost twenty years ago, a theoretical justification for these distributions is missing and the need for data-dependent priors may become more common as the complexity for applied problems increases. Apparently, the call for the data-dependent Bayesians did not remain silent in these last years. As far as we can tell from reviewing the literature, we may recognize at least three frameworks for justifying the data-dependent approach within the Bayesian inference: the approximation of a hierarchical model through the estimation of some hyperparameters \citep{gelmandatadependent}; the definition of an adjusted data-dependent paradigm allowing for the Bayes' Theorem computation \citep{darnieder2011bayesian}; and the definition of a data-dependent prior as a measurable function from the data space $\mathcal{Y}^{m}$ to the set of priors $\mathcal{P}$ \citep{wasserman2000asymptotic}. In this paper we propose a class of data-dependent prior distributions that may be theoretically justified under all these frameworks. Moreover, the methodology presented in this paper turns out to be interpreted also in terms of a penalized likelihood framework \citep{cole2013maximum} for regression models, where the penalty term is the kernel of a prior distribution and the weight of such penalization is not fixed in advance ---as it happens for instance through cross-validation or empirical Bayes techniques.
Why proposing a new data-dependent prior formulation? We acknowledge at least two reasons. From a Bayesian point of view, we want to investigate the information's extent of a prior distribution, and our proposal follows the words of \cite{gelmanpriors}, when he says that we need a compromise between the information carried by a ``\textit{wildly unrealistic in most settings prior informative distribution and a noninformative prior, feasible only in settings where data happen to be strongly informative about all parameters}''. And from a broader statistical point of view, we are interested in the global quality of the model and on the assumptions we propose, and we believe our prior might be a good solution in case of model/prior misspecification.
According to the first argument, we are aware that the use of informative priors ---or, at least, weakly informative priors \citep{gelman2008weakly}--- is strongly encouraged by subjectivist Bayesians, especially when a prior information for a specific application is actually available. However, even if the model is simple, when the sample size is `small' it is not trivial to elicit an informative prior that does not dominate the inference. Using an informative prior distribution elicited from historical data ---as it is usual in medical studies, for instance--- could result in a mismatch between the prior and the observed data, the so called \textit{prior-data conflict} \citep{evans2006checking, mutsvari2016addressing}. Thus, it emerges clearly that measuring the information contained in a prior distribution is not referred only as a mathematical exercise, but turns out to be helpful in terms of inference and prediction purposes. For instance, \cite{morita2008determining} developed the so called prior effective
sample size (ESS), an index which measures the amount of
information contained in a proposed prior distribution $\pi$ for
the parameter $\theta$, computed with respect to a posterior
$q_{m}(\theta|y)$ resulting from a baseline prior $\pi_{b}$, with $\pi_{b}$ less informative than $\pi$. When fitting a Bayesian model to a dataset consisting of 10 observations, an effective sample size of 1 is reasonable, whereas a value of 20 implies that the prior, rather than the data, dominates the inference: with a few data, there is the risk of being `too much informative'.
Motivated by these considerations, our method uses data for dealing directly with the priors construction. Given a pair of distributions consisting of an informative and a diffuse prior, our procedure measures the distance between the data at
hand and an additional set of data generated under the informative prior until the resulting posteriors may be considered approximately equal. The corresponding value of such a distance ---bounded in the interval $[0,1]$--- is plugged into a two-components mixture of the prior distributions considered above. The greater is this value, the farther are the data (simulated and real) from the informative prior,
and consequently the stronger is the influence of the diffuse
prior in our specification.
We prove that the so obtained class of mixture data-dependent priors ---hereafter MDD priors--- satisfies some nice properties. Among these, the distributions of this class always have a closed form in conjugate models and preserve the conjugacy. Under mild conditions, they yield a lower effective sample size than that provided by the informative prior ---substantially they provide less information. Moreover, evidences from simulation studies in the supplementary material accompanying this paper show that they also yield lower mean squared errors in presence of both model or prior misspecification.
It is worth noting that the use of mixture priors ---possibly with one relative precise component and the other more vague--- is not a novelty in Bayesian statistic. They have been introduced for making the inference robust in terms of a Bayesian perspective \citep{berger1986robust}, and developed for assessing any prior-data conflict \citep{schmidli2014robust,mutsvari2016addressing}. A mixture specification turns out to be useful also in Bayesian variable selection: a `spike and slab' prior \citep{miller2002subset} with fixed hyperparameters is assigned to the regression coefficients in the stochastic search variable selection approach ---see \cite{o2009review} for an overview on variable selection methods.
The paper is organized as follows. Section~\ref{sec:datadep} reviews the existing data-dependent approaches and presents in a few details the frameworks proposed by \cite{darnieder2011bayesian} and \cite{gelmandatadependent}; moreover, this section puts also in evidence the connection between the double use of the data and the penalized likelihood methods under a Bayesian perspective. In Section~\ref{sec:mixture} we introduce the MDD density class and describe the resampling algorithms required for building these priors. After introducing the notion of effective sample size, in Section~\ref{sec:theor} we focus on some theoretical results for the MDD priors; still, in this section we put in evidence the distribution-constant behaviour of the Hellinger distance in some special cases, if used as a data statistic. The information of the proposed class of priors is discussed in two examples for non standard models in Section~\ref{sec:case}: an exponential model with a Jeffreys prior and a logistic regression for determining the greatest amount of tolerable dose in phase I trial. Section~\ref{sec:concl} concludes.
\section{Using data twice in Bayesian inference}
\label{sec:datadep}
The commonly used expression `using data twice' in some Bayesian procedures does not mean nothing really precise, actually. However, it is not of interest for us taking an overview on all those tools which make use of the data twice for checking the fit of the model ---posterior predictive checkings, posterior Bayes fators, etc.--- or reviewing the empirical Bayes methods \citep{carlin2000bayes}. In this section we focus on those priors' procedures which explicitly consider data in the elicitation process.
As widely known, using data or the data mechanism process in the priors' elicitation is not properly Bayesian and suffers from two main criticisms: using data twice and not allowing for the direct computation of the Bayes' Theorem. However, some authors have attempted to circumvent these criticisms. In what follows, we take a brief overview on some existing data-dependent approaches. Firstly, we present the theoretical framework proposed by \cite{darnieder2011bayesian}, who formalized the so called Adjusted Data-dependent Bayesian paradigm, a new approach which introduces an adjustment in order to obtain a proper Bayesian inference starting from a data-dependent prior. Then, we present and formalize the considerations presented by \cite{gelmandatadependent}, who proposed to approximate a hierarchical model by using a data-dependent prior. We refer at \cite{wasserman2000asymptotic} for the formulation of data-dependent priors that yield proper posteriors for finite mixture-models.
Finally, we draw a parallel between data-dependent priors and the penalized likelihood methods commonly used in Bayesian variable selection. Although this paper does not explicitly take in consideration regression models, it is of future interest for us to implement our procedure also for regression purposes, and we consider this subsection as a grounding motivation for future work.
\subsection{Darnieder's approach}
\label{sec:darnieder}
Let $\bm{y}$ denote the sample of the data at hand, $\bm{\theta}$ the vector of parameters and $T( \bm{y})$ a statistic computed on the data. Let $\pi(\bm{\theta}| T( \bm{y}))$ denote a data-dependent prior whose dependence through the data is expressed by the statistic $T( \bm{y})$. \cite{darnieder2011bayesian} espresses the joint probability density of $(\bm{\theta}, \bm{y}, T( \bm{y}))$ as:
\begin{align*}
p( \bm{\theta}, \bm{y}, T( \bm{y}))= & p( T( \bm{y})| \bm{\theta}, \bm{y})p( \bm{\theta}| \bm{y}) m( \bm{y}) \\
= & f(\bm{y}|\bm{\theta}, T( \bm{y}) )\pi(\bm{\theta}| T( \bm{y}))m( T( \bm{y}))
\end{align*}
where $m(\bm{y})$ is the marginal (or integrated) likelihood. By isolating the posterior distribution on the left side, we obtain
\begin{equation}
p( \bm{\theta}| \bm{y})= \frac{f(\bm{y}|\bm{\theta}, T( \bm{y}) )\pi(\bm{\theta}| T( \bm{y}))m( T( \bm{y}))}{p( T( \bm{y})| \bm{\theta}, \bm{y})m( \bm{y})}
\end{equation}
Now, we observe that given $\bm{y}$, $T( \bm{y})| \bm{\theta}, \bm{y} $ is not random, and that the ratio $m( T( \bm{y}))/m( \bm{y})$ depends only on the observed data. Hence, we may write the above expression as
\begin{equation}
p( \bm{\theta}| \bm{y}) \propto f(\bm{y}|\bm{\theta}, T( \bm{y}) )\pi(\bm{\theta}| T( \bm{y})).
\label{eq:naive_posterior}
\end{equation}
As stated by \cite{darnieder2011bayesian}, the posterior in~\eqref{eq:naive_posterior} is obtained through a \textit{naive} approach. The equation is suggesting that using a data-dependent prior requires that also the likelihood of the model should be conditioned on the statistic $T( \bm{y})$. This formula is mathematically appealing, but the update of $\pi(\bm{\theta}| T( \bm{y}))$ is often not straightforward. Hence, after some simple algebra, the posterior may be expressed as
\begin{equation}
p( \bm{\theta}| \bm{y}) \propto \frac{f(\bm{y}|\bm{\theta} )\pi(\bm{\theta}| T( \bm{y}))}{ g( T( \bm{y})| \bm{\theta}) }=f(\bm{y}|\bm{\theta} ) \frac{\pi(\bm{\theta}| T( \bm{y}))}{g( T( \bm{y})| \bm{\theta})}
\label{eq:adjusted_posterior}
\end{equation}
where the ratio $\pi(\bm{\theta}| T( \bm{y})/g( T( \bm{y})| \bm{\theta})$ is the actual data-dependent prior, updated with the usual unconditioned likelihood $f(\bm{y}|\bm{\theta})$. \cite{darnieder2011bayesian} defines the posterior in \eqref{eq:adjusted_posterior} as an \textit{adjusted} posterior, obtained through an adjusted procedure. He also shows a relationship between a genuine Bayesian approach and the data-dependent Bayesian approach, putting in evidence the following identity:
\begin{equation}
1= \frac{p( \bm{\theta}| \bm{y})m(\bm{y})}{f(\bm{y}|\bm{\theta} )\pi(\bm{\theta})}= \frac{\pi(\bm{\theta}| T( \bm{y}))m( T( \bm{y}))}{g( T( \bm{y})| \bm{\theta}) \pi(\bm{\theta})}
\label{eq:bayesian_comparison}
\end{equation}
By dividing this expression by the genuine prior $\pi(\bm{\theta})$, we can state the following proportionality, the so called data-dependent Bayesian Principle:
\begin{equation}
\frac{p( \bm{\theta}| \bm{y})}{f(\bm{y}|\bm{\theta} )}\propto \frac{\pi(\bm{\theta}| T( \bm{y}))}{g( T( \bm{y})| \bm{\theta})}
\label{eq:bayesian_principle}
\end{equation}
which formally coincides with \eqref{eq:adjusted_posterior}, but suggests something even stronger. In fact, this expression highlights that the principle is satisfied whether a genuine prior $\pi(\bm{\theta})$ exists or not. With the adjusted procedure we provide a posterior distribution which is directly implied by Bayes' Theorem, whatever is the choice for $\pi(\bm{\theta})$.
A natural question concerns the choice of the statistic $T(\bm{y})$. There are no particular guidelines for choosing $T(\bm{y})$, but \cite{darnieder2011bayesian} lists some theorems that are useful for this aim. For example, it is trivial to show that if $T(\bm{y})$ is sufficient for $\bm{y}$, then the data-dependent prior $\pi(\bm{\theta}| T( \bm{y}))$ coincides with the genuine posterior $p( \bm{\theta}| \bm{y})$. And the following theorem in case of a distribution-constant statistic $T(\bm{y})$ will be useful later.
\begin{thm}
Suppose $T(\bm{y})$ is distribution-constant for $\bm{\theta}$, then the naive expression~\eqref{eq:naive_posterior} and the adjusted expression~\eqref{eq:adjusted_posterior} coincide. Furthermore, the data-dependent prior $\pi(\bm{\theta}| T( \bm{y}))$ coincides with the genuine prior $\pi(\bm{\theta})$.
\label{eq:thm_1}
\end{thm}
For a quick proof see the Appendix. As suggested by \cite{darnieder2011bayesian}, it is hard to imagine a beneficial conditioning on a distribution-constant statistic, unless for those priors which depend only on the data sample size. However, in Section~\ref{sec:theor} we will use this result for showing that, within some particular cases, our data-dependent prior procedure only depends on the sample size of our dataset and yields some good properties in terms of global information, frequentist coverage and mean squared errors.
\subsection{Gelman's approach}
\label{sec:gelman}
\cite{gelmandatadependent} draws an appealing framework considering the data-dependent priors as an approximation of a hierarchical model. He moves from a concrete example of regression models with standardized predictors: rescaling a bunch of predictors based on the data and then putting informative priors on their coefficients means eliciting a prior that depends on the data. He doesn't go in depth with mathematical notation, but we consider challenging to formalize this setup.
As usual in hierarchical models \citep{gelman2014bayesian}, let $\bm{y}$ represent the data-vector, $\bm{\theta}$ denote the generic vector of parameters and $\bm{\phi}$ the vector of hyperparameters. The likelihood of the model is $p(\bm{y}|\bm{\theta})$. The joint prior distribution for $(\bm{\theta}, \bm{\phi}) $ is
$$p(\bm{\theta}, \bm{\phi})= p(\bm{\phi})p(\bm{\theta}|\bm{\phi}), $$
and the joint posterior distribution is
\begin{equation}
p(\bm{\theta}, \bm{\phi}|\bm{y}) \propto p(\bm{\theta}, \bm{\phi}) p(\bm{y}| \bm{\theta}, \bm{\phi})=p(\bm{y}| \bm{\theta})p(\bm{\theta}|\bm{\phi})p(\bm{\phi}),
\label{eq:hierarchical_model}
\end{equation}
with the further assumption that the hyperparameter $\bm{\phi}$ affects $\bm{y}$ only through $\bm{\theta}$. In a full Bayesian model, $\bm{\phi}$ is not known and is assigned a prior distribution $p(\bm{\phi})$; however, in some circumstances it may be possible to consider $\bm{\phi}$ as known, or estimate it. As in the Gelman's example, if this hyperparameter, say a \textit{population} parameter, is estimated from the data, then we denote this estimate with ${\bm{\phi}}(\bm{y})$ and the population distribution $p(\bm{\theta}|\phi)$ reduces to $p(\bm{\theta} | {\phi}(\bm{y}) )$, which actually is a data-dependent prior according to \cite{darnieder2011bayesian}.
If we replace $\bm{\phi}$ with an estimate, $\bm{\theta}$ still preserves the dependence from ${\bm{\phi}}(y)$, but the joint posterior distribution in~\eqref{eq:hierarchical_model} reduces to the following approximate hierarchical joint posterior,
\begin{equation}
p(\bm{\theta}, \bm{\phi}(\bm{y})|\bm{y}) \propto p(\bm{\theta}| \bm{\phi}(\bm{y}), \bm{y}) p( \bm{\phi}(\bm{y})| \bm{y}) \propto p(\bm{\theta}| \bm{\phi}(\bm{y}), \bm{y}),
\label{eq:approx_joint}
\end{equation}
where $p(\bm{\theta}| \bm{\phi}(\bm{y}), \bm{y})$ may be interpreted as the marginal approximate posterior for $\bm{\theta}$ ---analogous to the pseudo-posterior distribution in empirical Bayes methods \citep{petrone2014empirical}, where $\bm{\phi}(\bm{y})$ is usually obtained through marginal maximum likelihood estimation. We may derive an explicit form for this quantity by applying the Bayes' Theorem and the assumption $p(\bm{y}| \bm{\theta}, \bm{\phi}(\bm{y}))=p(\bm{y}| \bm{\theta})$:
\begin{equation}
p(\bm{\theta}| \bm{\phi}(\bm{y}), \bm{y}) \propto p(\bm{y}| \bm{\theta}, \bm{\phi}(\bm{y}))p(\bm{\theta}, \bm{\phi}(\bm{y}) ) \propto p(\bm{y}| \bm{\theta})p(\bm{\theta}|\bm{\phi}(\bm{y})).
\label{eq:appr_hierarchical_model}
\end{equation}
The comparison between this latter expression and~\eqref{eq:hierarchical_model},~\eqref{eq:approx_joint} highlights the relationship existing between a full Bayesian hierarchical model and an approximate hierarchical model, where $\bm{\phi}(\bm{y})$ naturally acts in place of $\bm{\phi}$ and Bayes' Theorem is guaranteed by the product between the usual likelihood and the data-dependent prior $p(\bm{\theta}|\bm{\phi}(\bm{y}))$. The framework above has the merit of interpreting a data-dependent prior as an approximation of a further level of hierarchy within hierarchical models, through the use of a data-statistic $\phi(\bm{y})$ as a plug-in estimate for the hyperparameter $\phi$; moreover, it proposes the definition of a pseudo-posterior $p(\bm{\theta}| \phi(\bm{y}), \bm{y})$.
\subsection{Penalized likelihood}
\label{sec:penalized}
In the penalized likelihood approaches for regression models ---Lasso \citep{tibshirani1996regression}, Ridge regression, Bridge regression--- it is usual to penalize some coefficients by inducing a certain amount of shrinkage in order to (i) overcome problems in the
stability of parameter estimates due to a relatively flat likelihood and (ii) reduce the global mean squared error. A penalized log-likelihood with quadratic penalization is
\begin{equation}
\log L(\bm{\beta}, \bm{y}) -\frac{r}{2}(\bm{\beta}-\bm{g})^{2},
\label{eq:penalized}
\end{equation}
where $\bm{\beta}=(\beta_{1},...,\beta_{J)}$ is the vector of regression parameters, $\bm{g}=(g_{1},...,g_{J})$ is a vector of values which should be good guesses for the vector parameter $\bm{\beta}$, and $(\bm{\beta}-\bm{g})^{2}= \sum_{j=1}^{J}(\beta_{j}-g_{j})^{2}$ is the quadratic penalty. The formula above may be easily interpreted in terms of a Bayesian perspective. In fact, if $\beta_{j} \sim \mathcal{N}(g_{j}, 1/r)$, then~\eqref{eq:penalized} represents a log-likelihood penalized by the log-density of the prior distribution for $\beta_{j}$, where $r$ is the precision (the inverse of the prior variance) and is usually called the \textit{tuning} parameter. Thus, the quadratic log-likelihood penalization reduces to eliciting independent normal priors on the parameters with prior mean $g_{j}$ and prior variance $1/r$. The ordinary Lasso of Tibshirani can be interpreted as a Bayesian Lasso \citep{park2008bayesian}, i.e. as a Bayesian posterior mode estimate when regression parameters have Laplace independent priors. And more generally Bridge regression is a direct generalization for Lasso and Ridge regression, where the penalty is $(\bm{\beta}-\bm{g})^{q}$ for some $q \ge 0$ ($q=1$ corresponds to the ordinary Lasso, $q=2$ to the Ridge regression). Many approaches for estimating the tuning parameter $r$ have been proposed: cross-validation, general cross-validation, empirical Bayes methods through marginal maximum likelihood estimation. But only assigning a diffuse hyperprior is purely Bayesian. Using data for estimating the tuning parameter makes in fact the Bayesian penalized log-likelihood approach affected by the data process and, more precisely, the prior on $\beta$ affected by the data. In Section~\ref{sec:theor} we put in evidence that our methodology allows for a hierarchical approximation and may be also justified in terms of log-likelihood penalization.
\section{Mixture Data-dependent priors}
\label{sec:mixture}
Let $ \bm{y}_{m}= (y_{1},...,y_{m})$ be a data vector from a given sampling distribution $f(\bm{y}_{m}|\theta)$, with $\theta \in
\mathbb{R}$. Let $\pi_{b}(\theta) $ denote a diffuse prior
distribution for $\theta$ ---hereafter called \textit{baseline} prior--- and suppose that, from a preliminary knowledge about the problem (for instance historical information), we are somehow able to assign
a more informative prior distribution $\pi(\theta)$.
When data consist of a relatively small number of observations, the choice between these two priors' options is
not trivial, since the support and the shape of the posterior are sensitive to the choice of the prior distribution.
Thus, the information contained in the prior could turn out to be dominant when the dataset is small. This is one of the reasons for combining our previous information about the problem with our data at hand ----precisely, with an augmented version of it, as will be clarified later--- and proposing a data-dependent approach for eliciting a particular class of mixture prior distributions. We may then introduce the mixture data-dependent (MDD) prior $\varphi(\theta)$ with mixture weight $\psi_{m^{*}}$
\begin{equation}
\varphi(\theta)= \psi_{m^{*}}\pi_{b}(\theta)+(1-\psi_{m^{*}})\pi(\theta),
\label{eq:mixture:prior}
\end{equation}
belonging to the corresponding MDD class $$\Phi= \{ \varphi : \varphi(\theta)= \psi_{m^{*}}\pi_{b}(\theta)+(1-\psi_{m^{*}})\pi(\theta) ,\ \theta \in \Theta,\ 1 \ge\psi_{m^{*}}\ge 0 \}.$$ The MDD prior~\eqref{eq:mixture:prior} may then be viewed as a compromise between an informative prior and a noninformative one, with weights $\psi_{m^{*}}, \ 1-\psi_{m^{*}}$ obtained through a data augmentation with global length $m^{*}$. Note that mixture priors designed for overcoming the prior-data conflict and for robustness purposes have been already proposed by \cite{mutsvari2016addressing} and \cite{schmidli2014robust}: however, the authors do not propose any procedure for computing/assigning the mixture weights, and this is a crucial point for us, as explained in the next section.
\subsection{The resampling algorithms for the mixture weigths}
\label{sec:alg}
\begin{figure}
\caption{Normal-Normal model, resampling-algorithm 1. (\textit{Top}
\label{normal_mixture}
\end{figure}
Assume to have observed the
data vector $\bm{y}_{m}$, which represents our data at hand. Let simulate $\theta^{*} \sim \pi
(\theta)$ and define a modified version of the sampling distribution $f$ as $f(\bm{y}_{m}|\theta^{*})$. Assuming that $\theta_{0}$ is the true value of the parameter $\theta$ which generates our data at hand, we compute the Hellinger distance $\mathcal{H}$ ---closely related to the Bhattacharyya distance \citep{bhattacharyya1946measure}--- between our data generating process $f(\bm{y}_{m}| \theta_{0})$ and $f(\bm{y}_{m}|\theta^{*})$, defined as:
\begin{equation}
\Psi_{m} \equiv \mathcal{H}( f(\bm{y}_{m}|\theta_{0}), f(\bm{y}_{m}|\theta^{*}))=\frac{1}{\sqrt{2}} \left[ \int |\sqrt{f}-\sqrt{f^{*}}|^{2}d\bm{y_{m}} \right]^{\frac{1}{2}}
\label{eq:Hellinger:dist}
\end{equation}
where $f^{*}$ is an abbreviate notation for $f(\bm{y}_{m}|\theta^{*})$. For any couple of density functions $g, h$, the Hellinger distance satisfies the property $0 \le \mathcal{H}(g,h) \le 1$. It is worth noting that in~\eqref{eq:Hellinger:dist} we are treating $\theta_{0}$ as known, but in most of the statistical applications it is unknown and we need to estimate it. Among the others, one possibility could be that of using the maximum likelihood (ML) estimate $\hat{\theta}_{0}$, obtained equating at zero the log-derivative of the sampling distribution. Let $\psi_{m}$ denote the observed value of the Hellinger distance~\eqref{eq:Hellinger:dist}, bounded between 0 and 1. In an analogous way, let $\omega_{m}$ be the observed value of the Hellinger distance
\begin{equation}
\Omega_{m} \equiv \mathcal{H}(q_{m}(\theta|\bm{y}_{m}), \pi_{m}(\theta|\bm{y}_{m}))
\label{eq:Hellinger:dist:K}
\end{equation}
between the baseline posterior $q_{m}(\theta|\bm{y}_{m})$ and the informative posterior $\pi_{m}(\theta|\bm{y}_{m})$. The key-point of our procedure is that of sequentially generating $\varkappa$ new values $\bm{y}_{\varkappa}=(y_{m+1},...,y_{m+\varkappa})$, and re-computing the distances~\eqref{eq:Hellinger:dist},~\eqref{eq:Hellinger:dist:K} for each new draw, until a certain condition of similarity between the posterior distributions $\pi$ and $q$ is satisfied. Precisely, the stop condition is expressed by
\begin{equation}
\varkappa={\mbox{inf}}\ \{k \in \mathbb{N}\ | \Omega_{m+k} < \epsilon, \ \epsilon>0 \}
\label{eq:pess}
\end{equation}
for a fixed tolerance $\epsilon$. Thus, the so obtained $\psi_{m^{*}}$ is the observed value of $\Psi_{m^{*}}$, in correspondence of the dimension $m^{*}=m+\varkappa$ of the augmented dataset. This posterior similarity may be seen as an approximate matching between the proposed posterior distributions.
Note that the idea of matching the posterior uncertainty carried by two different posteriors doesn't represent a novelty, and a procedure based on the average posterior uncertainty is proposed by \cite{reimherr2014being}. The use of Hellinger distance is appropriate for some nice theoretical properties, as will be clarified in Section~\ref{sec:theor}.
As mentioned above, a crucial point is the generation of the additional data. Given the specific problem at hand, there is not a unique way for achieving this task. We propose two possible procedures, respectively named resampling-algorithm 1 and resampling-algorithm 2: for a deep illustration of these methods see the Appendix. For illustration purposes only, Figure~\ref{normal_mixture} displays a graphical example for the mixture prior and posterior (blue lines) obtained through resampling-algorithm 1 for a simple Normal-Normal model. However, in both the procedures as $\psi_{m^{*}}$ approximates 1 (maximal distance), the mixture prior~
\eqref{eq:mixture:prior} approximates the baseline prior
distribution $\pi_{b}(\theta)$; conversely, as $\psi_{m^{*}}$
approximates 0 (minimal distance), the mixture prior approximates the informative prior
$\pi(\theta)$. In this formulation, the data dependence is expressed
by the presence in~\eqref{eq:mixture:prior} of the observed
Hellinger distance $\psi_{m^{*}}$ between the actual and the further
data at the $m^{*}$-th iteration. However, one could simply use the
current set of data without the need of generating additional data.
In such a case, Equation~\eqref{eq:mixture:prior} will be the same,
but the weight $\psi_{m^{*}}$ may be
computed as the observed value of the Hellinger distance between the
informative prior $\pi(\theta)$ and the informative posterior
$\pi_{m}(\theta|y_{m})$. Along the rest of the paper, we will refer to this
formulation as the \textit{natural} MDD prior. Whereas MDD prior-res1 and MDD prior-res2 will denote respectively the MDD priors obtained with the resampling-algorithm 1 and 2.
\section{Theoretical results}
\label{sec:theor}
In this section we present some theoretical results for the MDD class presented in Section~\ref{sec:mixture} within the univariate conjugate models. Precisely, we introduce here the notion of effective sample size proposed by \cite{morita2008determining}, showing that the information of the MDD prior is always lower than the information of any informative prior. Moreover, we frame the MDD prior class in the theoretical approaches of \cite{darnieder2011bayesian} and \cite{gelmandatadependent}, summarized in Section~\ref{sec:datadep}. According to the first reference, we review the notion of distribution-constant statistics and we put in evidence that in some special cases ---e.g. the Normal-Normal model, but generally all the statistical models for which the Fisher information doesn't depend on the parameter--- the Hellinger distance is a distribution-constant statistic. This property implies that in these special models our proposed methodology substantially reduces to choosing a genuine prior.
\begin{table}
\caption{\label{tab:01} $\theta \in \mathbb{R}$, $c\ge 1$. Suppose $\bm{y}_{m}=(y_{1},...,y_{m}) \sim f(\bm{y}_{m}|\theta)$.
Prior $\pi(\theta)$, baseline prior $\pi_{b}(\theta)$, MDD prior $\varphi(\theta)$,
likelihood $f(\bm{y}_{m}|\theta) $, baseline posterior $q_{m}(\theta| \bm{y}_{m}) $
and MDD posterior $\varphi_{m}(\theta| \bm{y}_{m}) $ for the univariate conjugate models: Normal-Normal (NN), Gamma-Poisson (GP), Gamma-Exponential (GExp) and Beta-Binomial (BB). Following \cite{gelman2014bayesian},
we denote $\mathcal{N}(\mu, \sigma^2)$, $\mathcal{G}\mbox{a}(\alpha, \beta)$, $\mathcal{B}\mbox{e}(\alpha, \beta)$, $\mathcal{B}\mbox{in}(n, \theta)$, $\mathcal{P}\mbox{ois}(\theta)$ and $\mathcal{E}\mbox{xp}(\theta)$ for the normal, gamma, beta, binomial, Poisson and exponential distributions.
For the Normal-Normal model let $\bar{\mu}(\tau^{2})= (\frac{\mu}{\tau^{2}}+\frac{m}{\sigma^{2}}\bar{y})/ (\frac{1}{\tau^{2}}+\frac{m}{\sigma^{2} })$ denote the posterior mean in function of the prior variance $\tau^{2}$, and $\bar{\tau}^{2}(\tau^{2})=(\frac{1}{\tau^{2}}+\frac{m}{\sigma^{2} })^{-1}$ the posterior variance in function of the prior variance $\tau^{2}$.}
\begin{small}
\begin{tabular}{|lll|}
\multicolumn{3}{c}{}\\
\hline
& \textit{NN} & \textit{GP} \\
\hline\\
\small $\pi_{b}(\theta)$ & $ \mathcal{N}(\mu, c\tau^{2})$ & $\mathcal{G}\mbox{a}( \frac{\alpha}{c}, \frac{\beta}{c})$ \\
\small $\pi(\theta) $ & $ \mathcal{N}(\mu, \tau^{2})$ &$ \mathcal{G}\mbox{a}(\alpha, \beta)$ \\
\small $\varphi(\theta)$ & $ \psi_{m^{*}} \mathcal{N}(\mu, c\tau^{2})+$& $\psi_{m^{*}}\mathcal{G}\mbox{a}(\frac{\alpha}{c}, \frac{\beta}{c})+$ \\
& $ (1-\psi_{m^{*}})\mathcal{N}(\mu, \tau^{2})$ & $(1-\psi_{m^{*}})\mathcal{G}\mbox{a}(\alpha, \beta)$ \\
\small $f(\bm{y}_{m}|\theta)$ & $\mathcal{N}(\theta, \sigma^{2})$ & $\mathcal{P}\mbox{ois}(\theta)$\\
\small $q_{m}(\theta| \bm{y}_{m})$ &$ \mathcal{N}( \bar{\mu}(c \tau^{2}),\bar{\tau}^{2}(c\tau^{2}) )$ & $\mathcal{G}\mbox{a}(\frac{\alpha}{c}+\sum y_{i}, \frac{\beta}{c}+m)$ \\
\small $\varphi_{m}(\theta| \bm{y}_{m})$ & $\psi_{m^{*}} \mathcal{N}( \bar{\mu}(c\tau^{2}),\bar{\tau}^{2}(c\tau^{2}) )+$ & $\psi_{m^{*}}\mathcal{G}\mbox{a}(\frac{\alpha}{c}+\sum y_{i}, \frac{\beta}{c}+m)+$ \\
& $ (1-\psi_{m^{*}})\mathcal{N}( \bar{\mu}(\tau^{2}),\bar{\tau}^{2}(\tau^{2}) )$ & $(1-\psi_{m^{*}})\mathcal{G}\mbox{a}(\alpha+\sum y_{i}, \beta+m)$ \\
\hline
\multicolumn{3}{c}{}\\
\hline
& \textit{GExp} & \textit{BB} \\
\hline\\
\small $\pi_{b}(\theta)$ & $\mathcal{G}\mbox{a}(\frac{\alpha}{c}, \frac{\beta}{c})$ & $\mathcal{B}\mbox{e}(\frac{\alpha}{c}, \frac{\beta}{c})$\\
\small $\pi(\theta) $ & $\mathcal{G}\mbox{a}(\alpha, \beta)$ & $\mathcal{B}\mbox{e}(\alpha, \beta)$\\
\small $\varphi(\theta)$ & $ \psi_{m^{*}}\mathcal{G}\mbox{a}(\frac{\alpha}{c}, \frac{\beta}{c})+$ & $\psi_{m^{*}}\mathcal{B}\mbox{e}(\frac{\alpha}{c}, \frac{\beta}{c})+$\\
& $(1-\psi_{m^{*}})\mathcal{G}\mbox{a}(\alpha, \beta)$ & $(1-\psi_{m^{*}})\mathcal{B}\mbox{e}(\alpha, \beta)$\\
\small $f(\bm{y}_{m}|\theta)$ & $\mathcal{E}\mbox{xp}(\theta)$& $\mathcal{B}\mbox{in}(m, \theta)$\\
\small $q_{m}(\theta| \bm{y}_{m})$ & $\mathcal{G}\mbox{a}(\frac{\alpha}{c}+m, \frac{\beta}{c}+m \bar{y})$ &$\mathcal{B}\mbox{e}(\frac{\alpha}{c}+m \bar{y}, \frac{\beta}{c}+m-m \bar{y})$\\
\small $\varphi_{m}(\theta| \bm{y}_{m})$ & $\psi_{m^{*}}\mathcal{G}\mbox{a}(\frac{\alpha}{c}+m, \frac{\beta}{c}+m \bar{y})+$ & $\psi_{m^{*}}\mathcal{B}\mbox{e}(\frac{\alpha}{c}+m \bar{y}, \frac{\beta}{c}+(m-m \bar{y}) )+$\\
& $(1-\psi_{m^{*}})\mathcal{G}\mbox{a}(\alpha+m, \beta+m \bar{y})$ & $(1-\psi_{m^{*}})\mathcal{B}\mbox{e}(\alpha+m \bar{y}, \beta+m-m \bar{y})$\\
\hline
\end{tabular}
\end{small}
\end{table}
Before proceeding, we introduce here a general vector notation that turns out to be helpful in the following sections. Without loss of generality, let $\bm{\theta}$, $\bm{\theta} \in \mathbb{R}^{d}$, denote the parameters' vector, with $d \ge1$. Let the symbols $\pi_{b}(\bm{\theta})$, $\pi(\bm{\theta})$ denote as before respectively a baseline prior and an informative prior for $\bm{\theta}$. Let $m$ denote the generic sample size and $f( \bm{y}_{m}|\bm{\theta})$ the likelihood for our sample $\bm{y}_{m}=(y_{1},...,y_{m})$. Finally, let $q_{m}(\bm{\theta}|\bm{y}_{m})$ denote the baseline posterior for our parameter $\bm{\theta}$. In Section~\ref{sec:mixture} we used the symbols $m$ for the initial sample size, $\varkappa$ for the sample size of the generated sample of data and, consequently, $m^{*}=m+\varkappa$ for the global dimension of the data vector, comprising both the data at hand and those generated via resampling-algorithm 1 or 2. The MDD prior presented in this section obviously relies on $\psi_{m^{*}}$ and on a preliminary generation of $\varkappa$ values with one of the resampling algorithms introduced in~\ref{sec:alg}. The further technical assumptions are
\begin{align}
\begin{split}
&E_{\pi_{b}}(\bm{\theta})=E_{\pi}(\bm{\theta})\\
& \mbox{Corr}_{\pi}(\theta_{i}, \theta_{j})= \mbox{Corr}_{\pi_{b}}(\theta_{i}, \theta_{j}), \ i \ne j \\
& \mbox{Var}_{\pi_{b}}(\theta_{j}) >> \mbox{Var}_{\pi}(\theta_{j}), \ j=1,...,d.
\end{split}
\label{eq:assumption}
\end{align}
\begin{table}
\caption{\label{tab:02} \small $\theta \in \mathbb{R}, \ c\ge 1, \ m$ is the generic sample size. Negative second derivatives of the log densities and
effective sample sizes for the baseline prior $\pi_{b}(\theta)$, the informative prior $\pi(\theta)$ and the MDD prior
$\varphi(\theta)$, for the univariate conjugate models. Let $\bar{\theta}=E_{\pi}(\theta)$ denote the plug-in estimate. See Table~\ref{tab:01} for the priors' specification.}
\begin{small}
\begin{tabular}{|lllll|}
\hline
& \textit{NN} &\textit{GP} &\textit{GExp}& \textit{BB} \\
\hline
$D_{\pi_{b}}(\theta)$ & $1/c \tau^{2}$ & $\frac{(\alpha/c-1)}{\bar{\theta}^{2}}$ & $\frac{(\alpha/c-1)}{\bar{\theta}^{2}}$ & $(\frac{\alpha}{c}-1)\frac{1}{\bar{\theta}^{2}}+(\frac{\beta}{c}-1)\frac{1}{(1-\bar{\theta})^{2}}$ \\
$D_{\pi} (\theta)$ & $1/ \tau^{2}$ & $(\alpha-1)\bar{\theta}^{-2}$ & $(\alpha-1)\bar{\theta}^{-2}$& $\frac{(\alpha-1)}{\bar{\theta}^{2}}+\frac{(\beta-1)}{(1-\bar{\theta})^{2}}$ \\
$D_{q}(m,\theta, \bm{y}_{m})$ &$m/\sigma^{2}$ & $\frac{(\alpha/c+\sum y_{i}-1)}{\bar{\theta}^{2}}$ & $\frac{(\alpha/c+m-1)}{\bar{\theta}^{2}}$ & $\frac{(\frac{\alpha}{c}+\sum_{i}y_{i}-1)}{\bar{\theta}^{2}}+ \frac{(\frac{\beta}{c}+m-\sum_{i}y_{i}-1)}{(1-\bar{\theta})^{2}}$ \\
$ESS(\pi_{b}(\theta))$ & $\sigma^{2}/c\tau^{2}$ & 0 & 0 & 0\\
$ESS(\pi(\theta))$ & $\sigma^{2}/\tau^{2}$ & $\frac{\alpha-\alpha/c}{\bar{y}}$ & $\alpha-\alpha/c$ & $\alpha+\beta$ \\
\hline
\end{tabular}
\end{small}
\end{table}
\subsection{Effective sample size (ESS)}
\label{sec:ess}
The idea of measuring and quantifying the amount of information contained in a prior distribution is of a great theoretical appeal. Nevertheless, it has been not yet studied by many authors and many technical difficulties arise, including the impossibility of encompassing in a unique philosophical and mathematical framework the task of assessing the impact of a prior distribution: several distance measures and many definitions of prior sample size may be in fact adopted. In what follows we will refer to the work of \cite{morita2008determining}, who defined the prior effective sample size (ESS) of $\pi(\bm{\theta})$, with respect to the likelihood $f(\bm{y}_{m}|\bm{\theta})$ as that integer $m$ which minimizes the distance between $\pi(\bm{\theta})$ and the baseline posterior $q_{m}(\bm{\theta}|\bm{y}_{m})$. To define this distance, they used the second derivatives of the log densities (the observed informations)
\begin{equation}
D_{\pi,j}(\bm{\theta})= -\frac{\partial^{2} \log(\pi(\bm{\theta}))}{\partial \theta^{2}_{j}}, \ \ D_{q,j}(m, \bm{\theta}, \bm{y}_{m})=-\frac{\partial^{2} \log(q_{m}(\bm{\theta}|\bm{y}_{m}))}{\partial\theta^{2}_{j}}, \ j=1,...,d.
\label{eq:morita:derivative}
\end{equation}
In what follows, we will sometimes use the simplified notations $\pi, q_{m}$ in place of $\pi(\bm{\theta}), q_{m}(\bm{\theta}|\bm{y}_{m})$ and $D_{\pi,j}, D_{q_{m},j}$ in place of $D_{\pi,j}(\bm{\theta}), D_{q,j}(m, \bm{\theta}, \bm{y}_{m})$.
Let $D_{\pi,+}=\sum_{j=1}^{d} D_{\pi,j}$ and $D_{q_{m},+}=\sum_{j=1}^{d} \int D_{q_{m},j}f(\bm{y}_{m})d\bm{y}_{m}$ denote the global information for the prior $\pi$ and the posterior $q_{m}$, respectively. The distance between the prior and the posterior for the sample size $m$ is then defined as
\begin{equation}
\delta(m, \bar{\bm{\theta}}, \pi, q_{m})=| D_{\pi,+}(\bar{\bm{\theta}})-D_{q_{m},+}(\bar{\bm{\theta}})|,
\label{eq:morita_distance}
\end{equation}
evaluated in $\bar{\bm{\theta}}=E_{\pi}(\bm{\theta})$, the prior informative mean. The ESS for $\pi$ is defined as
\begin{equation}
ESS(\pi(\bm{\theta}))=\underset{m \in \mathbb{N}}{\mbox{Argmin}} \{ \delta(m, \bar{\bm{\theta}}, \pi, q_{m}) \}.
\label{eq:morita:ess}
\end{equation}
When $d=1$, we will simply write $D_{\pi}, D_{q_{m}}$, suppressing the subscript `+'. Table~\ref{tab:01} shows an example of the priors and the posteriors for four univariate conjugate models: Normal-Normal, Gamma-Poisson, Gamma-Exponential and Beta-Binomial. Note that, under the assumptions in~\eqref{eq:assumption}, the baseline prior mean corresponds to the informative prior mean, and the hyperparameter $c$ is a large constant chosen to inflate the baseline variance.
Table~\ref{tab:02} reports the distances and the effective sample sizes for these univariate conjugate models. Similarly to the general expression in~\eqref{eq:morita_distance}, the distance between the MDD prior $\varphi(\theta)$ and the baseline posterior $q_{m}(\theta|\bm{y}_{m})$ evaluated in $\bar{\theta}=E_{\pi}(\theta)$ is defined as
\begin{equation}
\delta(m, \bar{\theta}, \varphi, q_{m})=| D_{\varphi}(\bar{\theta})-D_{q_{m}}(\bar{\theta})|,
\label{eq:egidi_distance}
\end{equation}
where $ D_{\varphi}$ has not in general a closed form and it is computed through an $\mathsf{R}$ routine. The effective sample size $ESS(\varphi(\theta))$ is computed for the MDD prior analogously as in~\eqref{eq:morita:ess}. For the univariate conjugate models the following theorem holds.
\begin{thm}
Given $\theta \in \mathbb{R}$, the likelihood $f(\bm{y}_{m}|\theta)$, an informative prior $\pi(\theta)$, a baseline prior $\pi_{b}(\theta)$, the baseline posterior $q_{m}(\theta|\bm{y}_{m})$ and the MDD prior $\varphi(\theta)$ defined in~\eqref{eq:mixture:prior}, assume to be in a conjugate case and that the technical conditions in \eqref{eq:assumption}
hold.
Then
\begin{equation}
ESS(\varphi(\theta)) \le ESS(\pi(\theta))
\label{eq:thesis}
\end{equation}
\label{eq:thm_2}
\end{thm}
Formula~\eqref{eq:thesis} provides an upper bound for the effective sample size of the MDD prior class, and yields an intuitive result. Although an analytic form of the ESS for this class of priors is not available, the interpretation is that whatever are the observed weights and the priors $\pi_{b}, \pi$ used in the formulation, the information contained in the MDD prior is never greater than the information contained in $\pi$. From a practical point of view, this prior distribution provides a lower information than that contained in the prior $\pi$, and is then more likely to not dominate the likelihood.
\subsection{Distribution-constant statistics}
In this section we frame the MDD priors approach within the general theoretical framework for the data-dependent priors proposed by \cite{darnieder2011bayesian} ---and summarized in Section~\ref{sec:datadep}--- and we draw an appealing theoretical comparison between the MDD priors and the Bayesian approach, under certain technical conditions.
As alluded in Section~\ref{sec:datadep}, one of the key-points of the Darnieder's approach concerns the choice of the statistic $T(\bm{y})$ on which conditioning the prior distribution. As widely explained in Section~\ref{sec:mixture}, the MDD prior depends on the data only through the Hellinger distance defined in~\eqref{eq:Hellinger:dist}. For illustration purposes only and without loss of generality ---the theorems listed below preserve their validity in a multidimensional case--- let consider $\theta$ as a scalar parameter, $\theta \in \mathbb{R}$, and put $r(\theta, \theta + \triangle) \equiv \mathcal{H}( f(\bm{y}_{m}|\theta), f(\bm{y}_{m}|\theta+\triangle))$, where the parameters' difference $\triangle$ is not a parameter, but just an observed quantity which may be computed for each $m$, as $\triangle=\theta^{*}-\theta^{(0)}$ (see Section~\ref{sec:alg}). Let $I_{m}(\theta;f)=mI(\theta) $ denote the Fisher information for the parametric family $\{f(\bm{y}_{m}; \theta): \theta \in \Theta \}$ in case of independent observations. \cite{borovkovmathematical} state the following theorem.
\begin{thm}
If the function $\sqrt{f(\bm{y}_{m}|\theta)}$ is differentiable with respect to $\bm{\theta}$, and $I_{m}(\theta;f) $ is continuous, than there exists the limit:
\begin{equation}
\lim_{\triangle \rightarrow 0} \frac{r(\triangle)}{\triangle^{2}}=I_{m}(\theta;f)
\end{equation}
\label{eq:thm_4}
\end{thm}
This Theorem provides a limiting behaviour for the Hellinger distance, as the difference $\triangle$ approximates zero. Furthermore, he also provides some uniform bounds for $r(\triangle)/\triangle^{2}$:
\begin{thm}
If the parameters set $\Theta$ is compact, $f(\bm{y}_{m}| \theta) \ne f(\bm{y}_{m}| \theta+\triangle)$ whenever $\triangle >0$ and if $0 < I(\theta)\le h < \infty$ for a given constant $h$, then there exists a constants $g>0$ such that the following relation holds:
\begin{equation}
g < \frac{r(\triangle)}{\triangle^{2}}< h
\label{eq:unoform_bounds}
\end{equation}
\label{eq:thm_5}
\end{thm}
Theorem~\eqref{eq:thm_5} is stating that, for every choice of $\theta$, $ r(\triangle)$ is bounded between $g \triangle^{2}$ and $h \triangle^{2}$. Hence, denoting with $ \{ \theta^{(m)} \}$ a generic parameter sequence depending on the sample size $m$ and with $r^{(m)}(\triangle)$ the corresponding Hellinger distance, we may state the following corollary:
\begin{corl}
As $m \rightarrow \infty$, the distribution of $r^{(m)}(\triangle)$ doesn't depend on the parameter $\theta$ but only on the parameters' difference $\triangle$.
\end{corl}
In our framework, the dependence on the data for the MDD class is expressed by the observed Hellinger distance $\psi_{m^{*}}$; thus, we naturally set $T(\bm{y}_{m^{*}})= r^{(m^{*})}(\triangle)$. If $I_{m^{*}}(\theta;f)$ doesn't depend on the parameter $\theta^{(m^{*})}$ ---this happens for instance for the Normal, LogNormal, Cauchy and Logistic distributions--- then, as $m^{*} \rightarrow \infty$, the distribution of $T(\bm{y}_{m^{*}})$ doesn't depend on $\theta$, but only on the parameters' difference $\triangle$: in other words, $T(\bm{y}_{m^{*}})$ is distribution-constant and Theorem~\ref{eq:thm_1} in Section~\ref{sec:darnieder} holds. We may summarize these results and state the following theorem.
\begin{thm}
Given a parametric family of continuous distributions $\{f(\bm{y}_{m^{*}}| \theta), \theta \in \Theta \}$, if the Fisher information $I_{m^{*}}(\theta;f)$ doesn't depend on $\theta$, then the Hellinger distance $r^{(m^{*})}(\triangle)$ doesn't depend on $\theta^{(m^{*})}$ but only on the difference $\triangle$. This means that the statistic $T(\bm{y}_{m^{*}})= r^{(m^{*})}(\triangle) $ is distribution-constant and the MDD prior \eqref{eq:mixture:prior} $\pi(\theta | T(\bm{y}_{m^{*}}))$ reduces to the genuine prior $\pi(\theta)$.
\end{thm}
It is straightforward to show that, in this particular case, the MDD prior still depends on the data, but exhibits their dependence on the data only through conditioning on the sample size $m$, plus an augmented sample size $\varkappa$. And, as \cite{darnieder2011bayesian} suggests, there is no need of doing any adjustment, since the sample size $m$ is intrinsic in the likelihood and does not convey any information about $\theta$. However, preliminary simulation in the supplementary material show that conditioning on such a statistic yields some advantages in terms of frequentist coverage and mean squared errors, especially when the genuine prior distribution is not well posed.
By concluding, we found some special cases that, due to the presence of distribution-constant statistics, may be reduced to a genuine Bayesian approach even conditioning the prior on a data statistic.
\subsection{Approximation of a hierarchical model}
As suggested by \cite{gelmandatadependent}, data-dependent priors may sometimes be interpreted as an approximation of a hierarchical model, and in Sect.~\ref{sec:gelman} we provide a brief formalization of this intuition. Using again the Normal-Normal model as a toy example, let consider the following hierarchical model:
\begin{equation}
y_{ij} \sim \mathcal{N}(\theta_{j[i]}, \sigma^{2}), \ i=1\ldots m, \ j=1,\ldots,J
\label{eq:hierarchical_model1}
\end{equation}
\begin{equation}
\theta_j \sim \mathcal{N}(0, \tau^{2}_{j})
\label{eq:hierarchical_model2}
\end{equation}
\begin{equation}
\tau^{2}=\begin{cases}
\zeta^{2} \ \ \mbox{with } p \\
c\zeta^{2} \ \mbox{with } 1-p
\end{cases}
\label{eq:hierarchical_model3}
\end{equation}
where the nested index $j[i]$ codes as usual in the hierarchical models \citep{gelman2006data} the group membership for the statistical unit $i$; the group-level parameter $\theta_{j}$ is assigned a normal prior distribution; the prior variance $\tau^2$ may assume two different values with probabilities $p$ and $1-p$; $c,\zeta^2$ are for simplicity fixed hyperparameters. If we fit this model according to the Bayesian paradigm, we should also assign a prior distribution to the probability $p$, for instance $p\sim \mathcal{B}\mbox{e}(a,b)$, depending on some hyperparameters $a,b$. The MDD prior for $\theta$, $\theta \sim p\mathcal{N}(0, c\zeta^2)+(1-p)\mathcal{N}(0, \zeta^2)$, is another way for expressing equations~\eqref{eq:hierarchical_model2},~\eqref{eq:hierarchical_model3}. We may then argue that the MDD class is a natural approximation of the model above, with the parameter $p$ that is not assigned a prior but estimated from the data through the resampling algorithms in Section~\ref{sec:alg}.
For illustration purposes only, Figure~\ref{hierarchical_comp} displays a comparison, obtained through simulation using RStan \citep{rstan}, the R \citep{rcore} interface to
the Stan C++ library \citep{stan}, between the mean squared errors obtained from the hierarchical model in~\eqref{eq:hierarchical_model1},~\eqref{eq:hierarchical_model2},~\eqref{eq:hierarchical_model3}, the MDD prior-res1, and the MDD prior-res2, with $c=100, \ \zeta^2=1,\ \sigma^2=5,\ m=5$. The MDD priors show lower MSEs as the true value $\theta_{0}$ moves away from zero, the prior mean.
\begin{figure}
\caption{Comparison between the MSE of the hierarchical model (dashed black line) and of the MDD prior-res1, MDD-prior-res2, with weights $\psi_{m^{*}
\label{hierarchical_comp}
\end{figure}
\subsection{Model for the tuning parameter}
As mentioned in Sect.~\ref{sec:penalized}, the relationship of the penalized likelihood to Bayesian theory is explained by the penalty through the kernel of the prior log-density. However, the estimation of the penalty weight remains open. \cite{hastie2002elements} suggest to use cross-validation, whereas \cite{efron2012large} propose empirical Bayes methods. Otherwise, \cite{cole2013maximum} set different values and examine the results for these different inputs. The MDD prior specification may be seen as a natural alternative for estimating the tuning parameter in the penalized likelihood approach. For illustration purposes only, let consider the regression model
$$y_{i}=\beta_{0}+\sum_{j=1}^{J}\beta_{j}x_{ij} +\epsilon_{i},$$
where $\epsilon_{i}\sim \mathcal{N}(0, \sigma^{2})$. And consider now the penalized log-likelihood with quadratic penalty for this model
\begin{equation}
\l(\bm{\beta}; \bm{y}) -\frac{1}{2\tau^{2}}\bm{\beta}^{2},
\label{eq:penalized_hierarchical}
\end{equation}
where $\beta_{j} \sim \mathcal{N}(0, \tau^{2})$ according to the Bayesian interpretation of the Ridge regression. The penalty weight/tuning parameter is $r=1/\tau^{2}$, the inverse of the prior variance. Instead of estimating directly this factor, specifying a MDD prior for $\beta_{j}$ is an automatic tool for introducing an auxiliary level for the variance, as in~\eqref{eq:hierarchical_model3}, and estimating the proportion $p$ through the resampling algorithms in Sect.~\ref{sec:alg}:
\begin{align}
\l(\bm{\beta}; \bm{y})& -\frac{1}{2\tau^{2}}\bm{\beta}^{2}\\
\tau^{2}=&\begin{cases}
\zeta^{2} \ \ \mbox{with } p \\
c\zeta^{2} \ \mbox{with } 1-p.
\end{cases}
\label{eq:penalized_hierarchical2}
\end{align}
Although we use the Normal-Normal model, this approach allows flexibility also for other types of prior distributions \citep{wood2017generalized}.
The penalized methods ---Lasso, Ridge regression, etc.--- are designed for reducing the mean squared errors, and the MDD class of priors, together with the resampling algorithms, represents a built-in method for addressing the same objective. Further work should be developed in order to implement the MDD priors for regression models and within the Bayesian variable selection framework.
\section{Examples to Some Nonstandard Models}
\label{sec:case}
In the previous sections we dealt with a pair of priors $\pi$ and $\pi_{b}$ belonging to the same family of distributions, under the technical condition in~\eqref{eq:assumption}. This is the same choice adopted by \cite{morita2008determining} and allows for inflating the noninformative variance by a factor $c$ and falling into the conjugate models. However, one may be interested in exploring other prior choices for $\pi_{b}$, possibly automatic priors, and attempting to measure the information carried by the MDD prior~\eqref{eq:mixture:prior}, by taking unchanged the informative prior $\pi$. In this section we explore this possibility and we focus on the corresponding amount of priors' information through a toy example and through a real case from a phase I trial study.
\subsection{Jeffreys prior for an exponential model}
\label{sec:Jeffreys_exp}
Let $\bm{y}_{m}=(y_{1},..., y_{m}) \underset{iid}{\sim} \mathcal{E}xp(\theta)$, with $\pi(\theta)= \mathcal{G}a(\alpha,\beta)$. The likelihood is then
\begin{equation}
L_{m}(\theta; \bm{y}_{m})=\prod_{i=1}^{m} f(y_{i})= \theta^{m} \exp(-\theta \sum_{i} y_{i}).
\label{eq:exponential_lik}
\end{equation}
We introduce the Fisher information for the exponential model computed for a single observation:
$$I_{\theta}=E \left[- \frac{d^{2} \log f(y; \theta)}{d \theta^{2}} \right]=$$
$$= -E \left[ \frac{d^{2}}{d \theta^{2}} \left[ \log(\theta)-\theta y\right] \right]=-E \left[\frac{d}{d \theta}[1/{\theta}-y ] \right]=E \left[ \frac{1}{\theta^{2}} \right]= \frac{1}{\theta^{2}}.$$
Let $\pi_{b}(\theta)=j(\theta)$, where $j(\theta)= I^{1/2}_{\theta}$ is the Jeffreys prior.
For the exponential model, the Jeffreys prior for $\theta$ is
\begin{equation}
j(\theta)=I^{1/2}_{\theta}=1/\theta.
\label{eq:Jeffreys_exp}
\end{equation}
Now we compute the Jeffreys posterior $q_{m}(\theta|y_{1},...,y_{m})= j_{m}(\theta|y_{1},...,y_{m} )$:
\begin{equation}
j_{m}( \theta| \bm{y}_{m}) \propto j(\theta) L_{m}(\theta; \bm{y}_{m})
=\theta^{-1}\prod_{i=1}^{m}\theta\exp \{-\theta y_{i} \}
=\theta^{m-1} \exp \{ -\theta \sum_{i=1}^{m} y_{i} \}
\label{eq:Jeffreys_post}
\end{equation}
We immediately realize that this is the kernel of a Gamma distribution, $\mathcal{G}a(m, \sum_{i} y_{i})$
$$ j_{m}( \theta |\bm{y}_{m} )= \frac{(\sum_{i}y_{i})^{m}}{\Gamma(m)} \theta^{m-1} \exp \{ -\theta \sum_{i=1}^{m} y_{i} \}. $$
We compute the negative second log derivative of $j_{m}(\theta |\bm{y}_{m})$ and we find the familiar result for a Gamma distribution
\begin{equation}
D_{j_{m}}=-\frac{d^{2}}{d \theta^{2}}\left[j_{m}( \theta| \bm{y}_{m}) \right]= \frac{m-1}{\theta^{2}}
\label{eq:Morita_Jeffreys}
\end{equation}
Finally, by using the plug-in estimate $\bar{\theta}=\alpha/\beta$, we may compute: 1) the distance~\eqref{eq:morita_distance} between the informative prior $\pi$ and the Jeffreys posterior $j_{m}$; 2) the distance between the Jeffreys prior $j$ and the Jeffreys posterior $j_{m}$; 3) the distance~\eqref{eq:egidi_distance} between the MDD prior $\varphi$ and the Jeffreys posterior $j_{m}$. Fig.~\ref{Jeffreys_esp} shows these distances according to three different values for the Hellinger distance, where the informative prior is set to $\pi(\theta)= \mathcal{G}a(4,8)$. The distance for $\varphi$ is always bounded between the distances of $j$ and $\pi$: hence, the ESS ---the value which minimizes these quantities--- for $\varphi$ is bounded between the effective sample sizes respectively for $j$ and $\pi$. As is intuitive, as the mixture weight increases, $ESS(\varphi(\theta))$ approximates $ESS(j(\theta))$.
\begin{figure}
\caption{Exponential model with Jeffreys prior: on $y-$axis the distances $\delta(m, \bar{\theta}
\label{Jeffreys_esp}
\end{figure}
\subsection{Logistic regression for phase I trial}
\cite{thall2003practical} proposed a logistic regression to determine the greatest amount of tolerable dose in a phase I trial. In this section we follow the approach of \cite{morita2008determining}, who used the same example for studying the properties of the effective sample size for different values of the hyperparameters.
The level of dose which each patient may receive is one among 100, 200, 300, 400, 500, 600 mg/m$^{2}$, denoted by $x_{1},\ldots,x_{6}$. These values are then standardized on the log scale and denoted with $X_{1},...,X_{6}$. The response variable is $y_{i}=1$ if patient $i$ suffers toxicity, $y_{i}=0$ if not. They assume the following logistic model:
\begin{equation}
P(y_{i}=1) \equiv \pi(X_{i}, \bm{\theta})=logit^{-1}(\mu+\beta X_{i}) , \ i=1,...,m
\label{eq:logit}
\end{equation}
where $logit^{-1}(x)=e^{x}/(1+e^{x})$. Unlike the conjugate models considered in Section~\ref{sec:ess}, here the dimension of the parameters' space is $d=2$, $\bm{\theta}=(\mu, \beta)$, where $\mu$ is the intercept of the linear predictor and $\beta$ is the coefficient associated to the different levels of the doses. In order to compute the effective sample size, we need the extension to the multivariate case outlined by \cite{morita2008determining}. The likelihood for a sample of $m$ patients $\bm{y}_{m}=(y_{1},...,y_{m})$ is
\begin{equation}
f(\bm{y}_{m}| X, \bm{\theta})=\prod_{i=1}^{m}\pi(X_{i}, \theta)^{y_{i}}(1-\pi(X_{i}, \theta))^{1-y_{i}}
\label{eq:likelihood_logistic}
\end{equation}
\cite{thall2003practical} elicited two independent informative priors for $\mu$ and $\beta$ based on preliminary sensitivity analysis:
\begin{align}
\begin{split}
\mu &\sim \pi(\mu)=\mathcal{N}(\tilde{\mu}_{\mu}, \tilde{\sigma}^{2}_{\mu})= \mathcal{N}(-0.11313, 2^{2})\\
\beta & \sim \pi(\beta)= \mathcal{N}(\tilde{\mu}_{\beta}, \tilde{\sigma}^{2}_{\beta})= \mathcal{N}(2.3980, 2^{2}).
\end{split}
\label{eq:phaseI_thall_priors}
\end{align}
Hence, the baseline posterior is $q_{m}(\bm{\theta}| \bm{y}) = \mathcal{N}(\tilde{\mu}_{\mu}, c\tilde{\sigma}^{2}_{\mu})\mathcal{N}(\tilde{\mu}_{\beta}, c\tilde{\sigma}^{2}_{\beta})$, where the hyperparameter $c$ is fixed at 10000. We follow the steps of the algorithm formulated by \cite{morita2008determining} for determining (i) the effective sample size of each subvector and (ii) the global effective sample size of the parameter vector $\bm{\theta}$ as those values which respectively minimize the distances $\delta_{1}(m_{\mu},\bar{\bm{\theta}}, \pi_{\mu}, q_{m_{\mu}}), \delta_{2}(m_{\beta},\bar{\bm{\theta}}, \pi_{\beta}, q_{m_{\beta}})$ and $ \delta(m,\bar{\bm{\theta}}, \pi, q_{m})$, by using the plug-in vector $\bar{\bm{\theta}}=(\tilde{\mu}_{\mu}, \tilde{\mu}_{\beta})$. See the Appendix for a deep illustration of the algorithm. In this way, we compute the effective sample size of each parameter's subvector and then the global effective sample size of the logistic model. Given the two priors $\pi_{\mu}, \pi_{\beta}$ in~\eqref{eq:phaseI_thall_priors}, we will denote the first two quantities with $ ESS(\pi(\mu)), ESS(\pi(\beta))$, and the third one simply with $ESS$. Table~\ref{tab:03} in the Appendix reports these effective sample sizes, obtained replicating the experiment of \cite{morita2008determining} and evaluated with respect to different values of the priors variances $\sigma^{2}_{\mu}, \sigma^{2}_{\beta}$. As intuitive, the information contained in the prior distributions decreases as the variances increase. In any case, the parameter $\beta$, associated to
the effect of the doses, yields a greater knowledge than the parameter $\mu$, which represents the average response.
We repeat the same steps above adopting our mixture data-dependent prior by specifying for the vector parameter $\bm{\theta}$ the priors
\begin{align}
\begin{split}
\mu \sim & \ \varphi(\mu)= \psi \mathcal{N}(\tilde{\mu}_{\mu}, c\tilde{\sigma}^{2}_{\mu})+ (1-\psi)\mathcal{N}(\tilde{\mu}_{\mu}, \tilde{\sigma}^{2}_{\mu})\\
\beta \sim & \ \varphi(\beta)= \psi \mathcal{N}(\tilde{\beta}_{\beta}, c\tilde{\sigma}^{2}_{\beta})+ (1-\psi)\mathcal{N}(\tilde{\mu}_{\beta}, \tilde{\sigma}^{2}_{\beta})
\end{split}
\label{eq:phaseI_priors}
\end{align}
where the hyperparameter $c$ is fixed at 10000 as before and $\psi$ is the mixture weight. Being in absence of actual data at hand, here we do not adopt the algorithms of Section~\ref{sec:alg} for computing the observed value $\psi_{m^{*}}$ of the Hellinger distance $\Psi_{m^{*}}$: thus, for illustration purposes only, we drop the subscript $m^{*}$ and we consider three different values for $\psi$, $\psi= \{0.2,0.5,0.8 \}$. Then, we compare the so obtained results with those obtained with the above mentioned prior distributions. As may be noticed from Table~\ref{tab:04}, as $\psi$ increases the effective sample sizes for the MDD priors~\eqref{eq:phaseI_priors} slightly decrease, as expected. However, the values obtained under these mixture priors are quite close to those obtained under the above priors $\pi(\mu),\ \pi(\beta)$ originally chosen by \cite{thall2003practical}. It would be worth assessing how much varies the information of the mixture priors $\varphi$ by choosing other baseline priors instead of flat normal distributions. Let us consider two improper priors, $\pi_{b}(\mu) \propto 1, \ \pi_{b}(\beta) \propto 1$. The resulting mixture priors $\varphi(\mu), \ \varphi(\beta)$ are then defined as
\begin{align}
\begin{split}
\mu \sim & \ \varphi(\mu)= \psi + (1-\psi)\mathcal{N}(\tilde{\mu}_{\mu}, \tilde{\sigma}^{2}_{\mu})\\
\beta \sim & \ \varphi(\beta)= \psi + (1-\psi)\mathcal{N}(\tilde{\mu}_{\beta}, \tilde{\sigma}^{2}_{\beta}).
\end{split}
\label{eq:phaseI_priors2}
\end{align}
Table~\ref{tab:05} in the Appendix reports the effective sample sizes for the priors in~\eqref{eq:phaseI_priors2}. In this case, there is an evident decrease of the information associated to the mixture priors $\varphi$: as $\psi$ increases and the improper priors are then preferred, the effective sample size rapidly decreases. This is intuitive, since the improper priors which appear in~\eqref{eq:phaseI_priors2} provide less information than two flat normal priors in~\eqref{eq:phaseI_priors}. The example suggests that even inflating the noninformative variances by a great factor $c$ doesn't affect in a sensible way the amount of information contained in the mixture prior. We may conclude that the best way for reducing an extra amount of information is combining an informative prior with an improper or ---when possible--- with a Jeffreys prior as in Section~\ref{sec:Jeffreys_exp}.
\section{Concluding remarks}
\label{sec:concl}
In this paper a new class of data-dependent prior distributions is proposed. This class consists of a two-component mixture of a baseline (flat) prior $\pi_{b}$ and an informative prior $\pi$, weighted through resampling methods in such a way to prefer $\pi_{b}$ if the additional set of data generated under $\pi$ appears to be far from the data at hand. This prior turns out to be a good proposal for avoiding prior-data conflict in presence of small sample size and first evidences from simulation studies suggest good performances for reducing the mean squared errors.
Using the notion of effective sample size within conjugate models, we proved that the MDD prior class always provides a lower information than an informative prior.
Furthermore, different solutions for eliciting the baseline prior $\pi_{b}$ are explored: flat prior belonging to the same family of $\pi$, Jeffreys prior, improper prior. As is just partially intuitive, different strategies for the noninformative prior yield different extents of information for the MDD prior.
Further work should be done in many directions. We should in fact explore more complex models, whose a brief sketch is only outlined in this paper. Performing a proper sensitivity test for the selected priors $\pi_{b}, \pi$ is also a task of future interest. Finally, we strongly believe that extending the proposed methodology for regression models in terms of Bayesian variable selection is one crucial point in future research.
\section*{Appendix}
\subsection*{Resampling algorithms}
\begin{small}
According to the resampling-algorithm 1, we directly generate a sample $\bm{y}_{\varkappa}=(y_{m+1},...,y_{m^{*}}) $ from $f(\bm{y}_{m}| \theta^{*})$. At each step $k, \ k=1, \ldots, \varkappa$, we compute the Hellinger distances
\begin{align}
\begin{split}
\Psi_{m+k} \equiv & \mathcal{H}( f(\bm{y}_{m}|\theta_{0}), \bm{y} _{m+k} ) \\
\Omega_{m+k} \equiv &\mathcal{H}(q_{m+k}(\theta|\bm{y}_{m+k}), \pi_{m+k}(\theta|\bm{y}_{m+k})),
\end{split}
\label{eq:Hellinger_m1}
\end{align}
where the first equation in~\eqref{eq:Hellinger_m1} is the Hellinger distance between an absolute continuous distribution $f(\bm{y}_{m}|\theta_{0})$ and a numerical sample of length $m+k$ \footnote{We used the R function {\tt{HellingerDist}} of the {\tt{distrEx}} package \citep{kohl2007distrex}.}; whereas in the second equation $ q_{m+k}(\theta|\bm{y}_{m+k})$ denotes the baseline posterior computed in correspondence of the sample size $m+k$.
According to the resampling-algorithm 2, we generate the values $\bm{y}_{\varkappa}=(y_{m+1},...,y_{m^{*}}) $ from the sampling distribution $f(\bm{y}_{m}| \theta_{0})$ and at each step we compute the Hellinger distances
\begin{align}
\begin{split}
\Psi_{m+k} \equiv & \mathcal{H}( f(\bm{y}_{m}|\hat{\theta}^{(k)}_{0}), f(\bm{y}_{m}| \theta^{*}) )\\
\Omega_{m+k} \equiv &\mathcal{H}(q_{m+k}(\theta|\bm{y}_{m+k}), \pi_{m+k}(\theta|\bm{y}_{m+k})),
\end{split}
\label{eq:Hellinger_m2}
\end{align}
where the Equation~\eqref{eq:Hellinger_m2} is the Hellinger distance between two absolute continuous distributions, and $\hat{\theta}^{(k)}_{0}$ the ML estimate for $\theta_{0}$ at step $k$, based on $y_{1},\ldots,y_{m},$ $y_{m+1},\ldots,y_{m+k}$.
Resampling-algorithm 1 implies a data generation from the informative prior and compares these further data with those at hand: in some sense, this method is actually checking whether the informative prior is close to the data generating process. Perhaps, this procedure is oriented to assess the \textit{prior misspecification}. While in the resampling-algorithm 2, the data are generated according to the true model $f$: this second algorithm assesses the \textit{model misspecification}.
\framebox[\textwidth][l]{\parbox{\textwidth}{\textbf{Resampling-algorithm 1}:
\\
Given $y_{1},...,y_{m} \sim f(\bm{y}_{m}| \theta)$, generate $\theta^{*} \sim \pi
(\theta)$.
Fix the tolerance $\epsilon$.
Given $\Psi_{m} \equiv \mathcal{H}( f(\bm{y}_{m}|\theta_{0}), f(\bm{y}_{m}|\theta^{*}))$ and $\Omega_{m} \equiv \mathcal{H}(q_{m}(\theta|\bm{y}_{m}), \pi(\theta|\bm{y}_{m}))$ compute the observed\\ values $\psi_{m}$, $\omega_m$. If the true value $\theta_{0}$ is unknown, provide an estimate for it.
Set $k=1$.
$\ \ \ \Diamond$ generate $y_{m+k}$ from $f(\bm{y}_{m}|\theta^{*})$. Given \begin{align*}
\Psi_{m+k} &\equiv \mathcal{H}( f(\bm{y}_{m}|\theta_{0}), \bm{y} _{m+k} )\\
\Omega_{m+k} &\equiv \mathcal{H}(q_{m+k}(\theta|\bm{y}_{m+k}), \pi_{m+k}(\theta|\bm{y}_{m+k}))
\end{align*}
$\ \ \ \Diamond\Diamond$ Compute the observed values $\psi_{m+k}, \omega_{m+k}$.
$\ \ \ $ \textbf{while} $\{\omega_{m+k} > \epsilon\}$ set $k=k+1$ and go back to $\Diamond$.
Save $\psi_{m+\varkappa}$, $\omega_{m+\varkappa}$ and the new sample size $m^{*}=m+\varkappa$.
Set the prior~\eqref{eq:mixture:prior} with $\psi_{m^{*}}$.
}}
\framebox[\textwidth][l]{\parbox{\textwidth}{\textbf{Resampling-algorithm 2}:
\\
Given $y_{1},...,y_{m} \sim f(\bm{y}_{m}| \theta)$, generate $\theta^{*} \sim \pi
(\theta)$.
Fix the tolerance $\epsilon$.
Given $\Psi_{m} \equiv \mathcal{H}( f(\bm{y}_{m}|\theta_{0}), f(\bm{y}_{m}|\theta^{*}))$, $\Omega_{m} \equiv \mathcal{H}(q_{m}(\theta|\bm{y}_{m}), \pi(\theta|\bm{y}_{m}))$ compute the observed values $\psi_{m}$, $\omega_m$. If the true value $\theta_{0}$ is unknown, provide an estimate for it.
Set $k=1$.
$\ \ \ \triangle$ generate $y_{m+k}$ from $f(\bm{y}_{m}|\hat{\theta}^{(k)}_{0})$. Given \begin{align*}
\Psi_{m+k} &\equiv \mathcal{H}( f(\bm{y}_{m}|\hat{\theta}^{(k)}_{0}), f(\bm{y}_{m}| \theta^{*}) )\\
\Omega_{m+k} &\equiv \mathcal{H}(q_{m+k}(\theta|\bm{y}_{m+k}), \pi_{m+k}(\theta|\bm{y}_{m+k}))
\end{align*}
$\ \ \ $ with $\hat{\theta}^{(k)}_{0}$ the ML estimate for $\theta_{0}$ at step $k$.
$\ \ \ \triangle\triangle$ Compute the observed values $\psi_{m+k}, \omega_{m+k}$.
$\ \ \ $ \textbf{while} $ \{\omega_{m+k} > \epsilon \}$ set $k=k+1$ and go back to $\triangle$.
Save $\psi_{m+\varkappa}$, $\omega_{m+\varkappa}$ and the new sample size $m^{*}=m+\varkappa$.
Set the prior~\eqref{eq:mixture:prior} with $\psi_{m^{*}}$.
}}
\end{small}
\subsection*{Proof of Theorem \ref{eq:thm_1}}
Due to distribution-constant definition, $g( T( \bm{y})| \bm{\theta})= g( T( \bm{y}))$ and then $$p( \bm{\theta}| \bm{y}) \propto f(\bm{y}|\bm{\theta} )\pi(\bm{\theta}| T( \bm{y}))/ g( T( \bm{y})| \bm{\theta}) \propto f(\bm{y}|\bm{\theta} )\pi(\bm{\theta}| T( \bm{y})).$$
Furthermore, $ \pi(\bm{\theta}| T( \bm{y})) \propto g( T( \bm{y})| \bm{\theta}) \pi(\bm{\theta}) \propto \pi(\bm{\theta}). \ \Box$
\subsection*{Proof of Theorem \ref{eq:thm_2}}
\begin{small}
\textit{Proof.} For simplicity of notation we denote with $\alpha$ the baseline prior $\pi_{b}(\theta)$, with $\gamma$ the informative prior $\pi(\theta)$ and with $\beta$ the mixture prior $\varphi(\theta)= \psi_{m^{*}}\pi_{b}(\theta)+(1-\psi_{m^{*}})\pi(\theta)$. Furthermore, we abbreviate the weight $\psi_{m^{*}}$ as $\psi$. Unless otherwise stated, the dependence of the quantities introduced in Section \ref{sec:theor} on the parameter $\theta \in \mathbb{R}$ is here implicit. We compute the negative second log-derivative for the mixture prior \eqref{eq:mixture:prior} in general terms as
\begin{align}
D_{\varphi}= &-\frac{d^{2} \log \{\varphi(\theta)\}}{d \theta^{2}}= -\frac{d^{2} \log \{\psi\pi_{b}(\theta)+(1-\psi)\pi(\theta) \}}{d \theta^{2}} =\\
& = -\frac{d}{ d\theta} \left[ \frac{\psi\alpha^{'}+(1-
\psi)\gamma^{'}}{\psi\alpha +(1-\psi)\gamma } \right]=\\
& =\frac{(\psi\alpha^{'}+(1-
\psi)\gamma^{'})^2- (\psi\alpha^{''}+(1-
\psi)\gamma^{''}) (\psi\alpha +(1-\psi)\gamma )}{(\psi\alpha +(1-\psi)\gamma )^2}
\label{eq:D}
\end{align}
After some simple expansions we can rewrite \eqref{eq:D} and apply some minorations:
$$ D_{\varphi}= \frac{ \psi^2 [ (\alpha^{'})^2- \alpha^{''}\alpha ]
+(1- \psi)^2 (\gamma^{'})^2+2\psi(1- \psi) \gamma^{'}\alpha^{'}}{(\psi\alpha +(1-\psi)\gamma )^2} -$$
$$-\frac{\psi(1- \psi)\alpha^{''}\gamma+ \psi(1- \psi)\alpha\gamma^{''}+(1- \psi)^2 \gamma
\gamma^{''} }{(\psi\alpha +(1-\psi)\gamma )^2} \le $$
$$ \le \left[\frac{(\alpha^{'})^2- \alpha^{''}\alpha}{\alpha^2} \right]+ \frac{(1- \psi)^2 (\gamma^{'})^2-(1- \psi)^2 \gamma
\gamma^{''}}{(1-\psi)^2\gamma^2}+$$
$$+\frac{2\psi(1- \psi) \gamma^{'}\alpha^{'}-\psi(1- \psi)\alpha^{''}\gamma- \psi(1- \psi)\alpha\gamma^{''}}{\psi^2\alpha^2}=$$
\begin{equation}
=D_{\alpha}+ K_{1}
\label{diseq:D1}
\end{equation}
where $ K_{1}$ collects all the terms which do not enter in $D_{\alpha}$. Analogously, we can find another minoration:
$$ D_{\varphi} \le \left[\frac{(\gamma^{'})^2- \gamma^{''}\gamma}{\gamma^2} \right]+ \frac{\psi^2 (\alpha^{'})^2-\psi^2 \alpha
\alpha^{''}+2\psi(1- \psi) \gamma^{'}\alpha^{'}}{\psi^2\alpha^2}-$$
$$-\frac{\psi(1- \psi)\alpha^{''}\gamma+ \psi(1- \psi)\alpha\gamma^{''}}{\psi^2\alpha^2} =$$
\begin{equation}
=D_{\gamma+ K_{2}}
\label{diseq:D2}
\end{equation}
From \eqref{diseq:D1} and \eqref{diseq:D2} it stems that
$$K_{1}-K_{2}= \left[\frac{(\gamma^{'})^2- \gamma^{''}\gamma}{\gamma^2} \right]- \left[\frac{(\alpha^{'})^2- \alpha^{''}\alpha}{\alpha^2} \right]= D_{\gamma}-D_{\alpha}$$
with $D_{\gamma}-D_{\alpha}>0$ for assumption (see Table~ \ref{tab:01}). In what follows we abbreviate $D_{\varphi}$ as $D$. Hence we have found the following conditions
\begin{equation}
\begin{cases}
\mathbf{A} \ D \le D_{\alpha}+ K_{1}\\
\mathbf{B} \ D \le D_{\gamma}+ K_{2}\\
\end{cases}
\label{diseq:system}
\end{equation}
Condition $\mathbf{B}$ implies $D \le D_{\gamma}+ K_{2}+(K_{1}-K_{2})=D_{\gamma}+K_{1}$ and yields the further condition
$$\mathbf{C} \ D \le D_{\gamma}+K_{1}$$
Thus, we may collect the three conditions already found
\begin{equation}
\begin{cases}
\mathbf{A} \ D \le D_{\alpha}+ K_{1}\\
\mathbf{B} \ D \le D_{\gamma}+ K_{2}\\
\mathbf{C} \ D \le D_{\gamma}+K_{1}\\
\end{cases}
\label{diseq:system2}
\end{equation}
Now we may distinguish three separate cases which satisfy the condition $K_{1}-K_{2}>0$:\\
(a) $ K_{1},\ K_{2}>0$
We use conditions $\mathbf{B}, \mathbf{C}$
\begin{equation}
\begin{cases}
\mathbf{B} \ D \le D_{\gamma}+ K_{2}\\
\mathbf{C} \ D \le D_{\gamma}+K_{1}\\
\end{cases} \rightarrow
\begin{cases}
2D \le 2D_{\gamma}+ 2K_{2}\\
D \le D_{\gamma}+2K_{1}\\
\end{cases} \rightarrow
\begin{cases}
D \le D_{\gamma}+2(K_{2}-K_{1})=D_{\gamma}\\
-\\
\end{cases}
\label{diseq:system3}
\end{equation}
and we conclude that $D \le D_{\gamma}$.\\
(b) $ K_{1}>0,\ K_{2}<0$
By applying condition $\mathbf{B}$ , it follows $D \le D_{\gamma}$.\\
(c) $ K_{1}<0,\ K_{2}<0$
By applying condition $\mathbf{B}$ or $\mathbf{C}$ , it follows $D \le D_{\gamma}$.\\
We have proved that for any possible sign of $ K_{1}, \ K_{2}$, $D \le D_{\gamma}$. By definition of effective sample size from \cite{morita2008determining} we know that
\begin{align*}
ESS(\varphi(\theta))&=\underset{m \in \mathbb{N}}{\mbox{Argmin}}\{\delta(m, \bar{\theta}, \varphi, q_{m}) \}=\\
&=\underset{m \in \mathbb{N}}{\mbox{Argmin}} \{ | D
-D_{q_{m}}(\bar{\theta})| \} \\
\end{align*}
evaluated in the plug-in estimate $\bar{\theta}=E_{\pi}[\theta]$. From Table~\ref{tab:01} we also know that the observed information of the baseline posterior $D_{q_{m}}$ is a linear function of the sample size $m$ and is increasing: $$\frac{d D_{q_{m}} }{dm}>0, \ \forall m \in \mathbb{N}$$ Thus we may conclude that from $D \le D_{\pi}$ it follows:
\begin{align*}
ESS(\varphi(\theta))&= \underset{m \in \mathbb{N}}{\mbox{Argmin}} \{ | D
_{\varphi}(\bar{\theta})
-D_{q_{m}(\theta|y)}(\bar{\theta})| \} \le \\
&\le \underset{m \in \mathbb{N}}{\mbox{Argmin}} \{ | D
-D_{q_{m}}(\bar{\theta})| \} = ESS(\pi(\theta))\ . \ \ \Box
\end{align*}
\end{small}
\subsection*{Logistic regression for phase I trial}
\begin{small}
\textit{Algorithm for computing the ESS} \citep{morita2008determining}
\begin{itemize}
\item According to the definitions in \eqref{eq:morita:derivative}, we compute the following quantities:\\ $D_{\pi,1}=(\tilde{\sigma}^{2}_{\mu})^{-1},\ D_{\pi,2}=(\tilde{\sigma}^{2}_{\beta})^{-1}$.
\item We need to compute $D_{q,1}(m,\bm{\theta}, X_{m}, \bm{y}_{m})=\sum_{i=1}^{m} \pi(X_{i}, \theta)\{ 1-\pi(X_{i}, \theta) \}$,\\ $ D_{q,2}(m,\bm{\theta}, X_{m}, \bm{y}_{m})=\sum_{i=1}^{m}X^{2}_{i} \pi(X_{i}, \theta)\{ 1-\pi(X_{i}, \theta) \} $.
\item It turns out that $\int D_{q_{m},j}f(\bm{y}_{m})d\bm{y}_{m}$ ---where $f(\bm{y}_{m})$ is the likelihood \eqref{eq:likelihood_logistic} evaluated in correspondence of fixed values for $\bm{\theta}$ and $\bm{X}$--- cannot be computed analytically and need to be computed through Monte Carlo simulation. Before of proceeding, let us notice that $D_{q,1}(m,\bm{\theta}, X_{m}, \bm{y}_{m})$ and $D_{q,2}(m,\bm{\theta}, X_{m}, \bm{y}_{m})$ depend on $X_{m}$ but not on $\bm{y}_{m}$, and this simplifies the simulation procedure. We may replace them respectively with the new notations $D_{q,1}(m,\bm{\theta}, X_{m})$ and $D_{q,2}(m,\bm{\theta}, X_{m})$.
\item Assuming a uniform distribution for the doses, we draw $X^{(t)}_{1},...,X^{(t)}_{6}$ independently from $\{X_{1},...,X_{6} \}$ with probability 1/6 each, for $t=1,...,100000$.
\item Use the Monte Carlo average $T^{-1} \sum_{t=1}^{T}D_{q,j}(m,\bm{\theta}, X_{m})$ in place of $\int D_{q_{m},j}f(\bm{y}_{m})d\bm{y}_{m}$, for $ j=1,2$.
\item Compute $\delta_{1}(m_{\mu},\bar{\bm{\theta}}, \pi_{\mu}, q_{m_{\mu}})$, $ \delta_{2}(m_{\beta},\bar{\bm{\theta}}, \pi_{\beta}, q_{m_{\beta}})$ and $ \delta(m,\bar{\bm{\theta}}, \pi, q_{m})$.
\item $ ESS(\pi(\mu)), ESS(\pi(\beta))$ and $ESS$ are the interpolated values of the sample sizes $m_{\mu}, m_{\beta}, m$ minimizing $\delta_{1}, \delta_{2}$ and $\delta$ respectively.
\end{itemize}
\begin{table}
\caption{\label{tab:03}Effective sample sizes $ESS(\pi(\mu)), ESS(\pi(\beta))$ for the tolerable dose in a phase I trial.}
\begin{tabular}{|llll|}
\hline
& \tiny{$ ESS$} & \tiny{$
ESS(\pi(\mu))$}& \tiny{$ESS(\pi(\beta))$}\\
\hline
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=0.5^{2}$ & 37.00 & 22.73 & 98.11 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=1^{2}$ & 10.00 & 5.75 & 25.56 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=2^{2}$ & 3.00 & 1.37 & 6.53 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=3^{2}$ & 2.00 & 1.03 & 3.06 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=5^{2}$ & 1.00 & 1.00 & 1.38 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{\label{tab:04} \small Effective sample sizes $ESS(\varphi(\mu)), \ ESS(\varphi(\beta))$ for the MDD priors $\varphi(\mu)=\psi \mathcal{N}(\tilde{\mu}_{\mu}, c\tilde{\sigma}^{2}_{\mu})+ (1-\psi)\mathcal{N}(\tilde{\mu}_{\mu}, \tilde{\sigma}^{2}_{\mu}), \ \varphi(\beta)=\psi \mathcal{N}(\tilde{\mu}_{\beta}, c\tilde{\sigma}^{2}_{\beta})+ (1-\psi)\mathcal{N}(\tilde{\mu}_{\beta}, \tilde{\sigma}^{2}_{\beta})$ according to different values of the mixture weight $\psi$.}
\begin{small}
\begin{tabular}{|l|lll|lll|lll|}
\hline
& \multicolumn{3}{|c|}{$\psi=0.2$}& \multicolumn{3}{|c|}{$\psi=0.5$}&\multicolumn{3}{|c|}{$\psi=0.8$}\\
& \tiny{$
ESS$} & \tiny{$ ESS(\varphi(\mu))$}&
\tiny{$ESS(\varphi(\beta))$} & \tiny{$ ESS
$} & \tiny{$ ESS(\varphi(\mu))$}& \tiny{$ESS(\varphi(\beta))$}&
\tiny{$ ESS$} & \tiny{$ ESS(\varphi(\mu))
$}& \tiny{$ESS(\varphi(\beta))$} \\
\hline
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=0.5^{2}$ & 37.00 & 22.70 & 98.06 & 37.00 & 22.62 & 97.90 & 37.00 & 22.30 & 97.18 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=1^{2}$ & 10.00 & 5.73 & 25.50 & 10.00 & 5.69 & 25.31 & 9.00 & 5.52 & 24.58 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=2^{2}$ & 3.00 & 1.37 & 6.49 & 3.00 & 1.37 & 6.42 & 3.00 & 1.31 & 6.06 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=3^{2}$ & 2.00 & 1.03 & 3.03 & 2.00 & 1.03 & 3.01 & 2.00 & 1.03 & 2.68 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=5^{2}$ & 1.00 & 1.00 & 1.38 & 1.00 & 1.00 & 1.37 & 1.00 & 1.00 & 1.26 \\
\hline
\end{tabular}
\end{small}
\end{table}
\begin{table}
\caption{\label{tab:05} \small Effective sample sizes $ESS(\varphi(\mu)), \ ESS(\varphi(\beta))$ for the MDD priors $\varphi(\mu)=\psi+(1-\psi)\pi_{\mu}, \ \varphi(\beta)=\psi+(1-\psi)\pi_{\beta}$ according to different values of the mixture weight $\psi$.}
\begin{small}
\begin{tabular}{|l|lll|lll|lll|}
\hline
& \multicolumn{3}{|c|}{$\psi=0.2$}& \multicolumn{3}{|c|}{$\psi=0.5$}&\multicolumn{3}{|c|}{$\psi=0.8$}\\
& \tiny{$
ESS$} & \tiny{$ ESS(\varphi(\mu))$}&
\tiny{$ESS( \varphi(\beta))$} & \tiny{$ ESS
$} & \tiny{$ ESS(\varphi(\mu))$}& \tiny{$ESS(\varphi(\beta))$}&
\tiny{$ ESS$} & \tiny{$ ESS(\varphi(\mu))
$}& \tiny{$ESS(\varphi(\beta))$} \\
\hline
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=0.5^{2}$ & 32.00 & 19.71 & 87.65 & 23.00 & 14.03 & 62.43 & 11.00 & 6.55 & 29.06 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=1^{2}$ & 6.00 & 3.58 & 15.78 & 3.00 & 1.68 & 7.42 & 1.00 & 1.03 & 2.48 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=2^{2}$ & 1.00 & 1.00 & 1.99 & 1.00 & 1.00 & 1.14 & 1.00 & 1.00 & 1.03 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=3^{2}$ & 1.00 & 1.00 & 1.10 & 1.00 & 1.00 & 1.03 & 1.00 & 1.00 & 1.03 \\
$\sigma^{2}_{\mu}=\sigma^{2}_{\beta}=3^{2}$ & 1.00 & 1.00 & 1.03 & 1.00 & 1.00 & 1.03 & 1.00 & 1.00 & 1.03 \\
\hline
\end{tabular}
\end{small}
\end{table}
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\end{document} |
\begin{document}
\mathfrak{v}space*{1em}
\begin{center}
{\LARGE\bf
The complexity of intersecting subproducts with subgroups in Cartesian powers
}
\mathfrak{v}space{1em}
{\Large\bf
P.\ Spelier
}
\mathfrak{v}space{2em}
\end{center}
\begin{abstract}
Given a finite abelian group $G$ and a natural number $t$, there are two natural substructures of the Cartesian power $G^t$; namely, $S^t$ where $S$ is a subset of $G$, and $x+H$ a coset of a subgroup $H$ of $G^t$. A natural question is whether two such different structures have non-empty intersection. This turns out to be an NP-complete problem. If we fix $G$ and $S$, then the problem is in $\mathcal{P}$ if $S$ is a coset in $G$ or if $S$ is empty, and NP-complete otherwise; if we restrict to intersecting powers of $S$ with subgroups, the problem is in $\mathcal{P}$ if $\bigcap_{n\in\mathfrak{m}athbb{Z} \mathfrak{m}id nS \subset S} nS$ is a coset or empty, and NP-complete otherwise. These theorems have applications in the article \citep{artorders}, where they are used as a stepping stone between a purely combinatorial and a purely algebraic problem.
\end{abstract}
\section{Introduction}
\label{section:intro}
In this article, we take a look at intersecting cosets of submodules of Cartesian powers of modules, with powers of subsets of those modules. Although our final classification theorems hold only for abelian groups, i.e. $\mathfrak{m}athbb{Z}$-modules, some of our lemmas in between also hold for more general modules.
We now give our definitions and theorems.
\begin{defn}
\label{defn:pgs}
Let $R$ be a commutative ring that is finitely generated as a $\mathfrak{m}athbb{Z}$-module, let $G$ be a finite $R$-module and $S$ a subset of $G$. Then define the problem $P_{G,S}^R$ as follows. With input $t \in \mathfrak{m}athbb{Z}_{\geq 0}, x_* \in G^t$, the $t$-th Cartesian power of $G$, and $H$ a submodule of $G^t$ given by a list of generators, decide whether $(x_* + H) \cap S^t$ is non-empty. Write $P_{G,S}$ for $P_{G,S}^\mathfrak{m}athbb{Z}$.
\end{defn}
\begin{defn}
\label{defn:pigs}
Let $R$ be a commutative ring that is finitely generated as a $\mathfrak{m}athbb{Z}$-module, let $G$ be a finite $R$-module and $S$ a subset of $G$. Then define the problem $\mathcal{P}i_{G,S}^R$ as the subproblem of $P_{G,S}^R$ where $x_* = 0$. I.e., with input $t \in \mathfrak{m}athbb{Z}_{\geq 0}$ and $H$ a submodule of $G^t$ given by a list of generators, decide whether $H \cap S^t$ is non-empty. Write $\mathcal{P}i_{G,S}$ for $\mathcal{P}i_{G,S}^\mathfrak{m}athbb{Z}$.
\end{defn}
\begin{rem}
If $R$ is not finitely generated as a $\mathfrak{m}athbb{Z}$-module, we can replace it by its image in $\End(G)$.
\end{rem}
\begin{rem}
Note that $R,G,S$ are \mathfrak{t}extit{not} part of the input of the problem. In particular, computations inside $G$ can be done in $O(1)$.
\end{rem}
Note these problems are certainly in $\mathfrak{m}athbb{N}P$, as one can easily give an $R$-linear combination of the generators (and add $x_*$ if necessary), and check that it lies in $S^t$. These problems are further studied in Section \mathfrak{r}ef{section:group}. For $R = \mathfrak{m}athbb{Z}$, an $R$-module is just an abelian group; we prove two theorems that completely classify the problems $P_{G,S}$ and $\mathcal{P}i_{G,S}$, in the sense that for each problem we either have a polynomial time algorithm or a proof of NP-completeness.
\begin{defn}
With $G$ an abelian group and $S \subset G$, we call $S$ a \emph{coset} if there is some $x \in G$ such that $S - x$ is a subgroup of $G$.
\end{defn}
\begin{thm}
\label{thm:pgs}
If $S$ is empty or a coset, then we have $P_{G,S} \in \mathcal{P}$. In all other cases, $P_{G,S}$ is NP-complete.
\end{thm}
\begin{thm}
\label{thm:pigs}
If $S$ is empty or $\mathfrak{t}heta(S) := \bigcap_{a \in \mathfrak{m}athbb{Z} \mathfrak{m}id aS \subset S} aS$ is a coset, then we have $\mathcal{P}i_{G,S} \in \mathcal{P}$. In all other cases, $\mathcal{P}i_{G,S}$ is NP-complete.
\end{thm}
\begin{rem}
\label{rem:psi}
Note that if $0 \in S$, then $\mathfrak{t}heta(S) = \{0\}$; additionally, if $G$ is a group with order a prime power and $S$ does not contain 0, then $\mathfrak{t}heta(S) = S$, as will be proven in Lemma~\mathfrak{r}ef{lem:psipowerp}.
\end{rem}
This paper is based on the author's thesis \cite{spelier2018}.
\section{Group-theoretic \mathfrak{t}exorpdfstring{$\mathfrak{m}athbb{N}P$}{NP}-complete problems}
\label{section:group}
In this section we will completely classify the group-theoretic problems $P_{G,S}$ and $\mathcal{P}i_{G,S}$. Some of the lemmas we use to prove Theorems \mathfrak{r}ef{thm:pgs} and \mathfrak{r}ef{thm:pigs} we give for general $P_{G,S}^R$ (resp. $\mathcal{P}i_{G,S}^R$) and some only for $P_{G,S}$ (resp. $\mathcal{P}i_{G,S}$). Throughout this section, let $R$ be a commutative ring, finitely generated as a $\mathfrak{m}athbb{Z}$-module. All abelian groups and $R$-modules we consider in this section are finite. We will use definitions and notations for $\mathcal{P}$ and $\mathfrak{m}athbb{N}P$ as given in Appendix \mathfrak{r}ef{section:pnp}.
\subsection{Proof of Theorem~\mathfrak{r}ef{thm:pgs}}
\label{subs:pgs}
We first introduce four general lemmas that will help with the proof of Theorem~\mathfrak{r}ef{thm:pgs}. As defined in the appendix, for two problems $P,Q$ we use the notation $P \leq Q$ if there is a reduction from $P$ to $Q$ and we write $P \approx Q$ if $P \leq Q$ and $Q \leq P$.
\begin{lem}
\label{lem:transinvar}
We have $P_{G,S}^R \approx P_{G,S+g}^R$ for all $g \in G$ (i.e., the problem is translation invariant).
\end{lem}
\begin{proof}
For the reduction $P_{G,S}^R \leq P_{G,S+g}^R$, we send an instance $(t,x_*,H)$ to $(t,x_* + (g,\ldots,g),H)$. By symmetry, we also have $P_{G,S+g}^R \leq P_{G,S}^R$; by the definition of $\approx$, we are done.
\end{proof}
\begin{lem}
\label{lem:restrict}
If $G'$ is a submodule of $G$, then we have $P_{G',G' \cap S}^R \leq P_{G,S}^R$.
\end{lem}
\begin{proof}
Given an instance $(t,x_*,H)$ of the first $P_{G',G' \cap S}^R$, we see it is also an instance of $P_{G,S}^R$, and as $H \cap S^t \subset G'^t \cap S^t = (G' \cap S)^t$, we see it is a yes-instance of the first problem exactly if it is a yes-instance of the second one.
\end{proof}
\begin{lem}
\label{lem:divideout}
Let $G'$ be a submodule of $G$ and $S'$ a subset of $G$, and define $S = S' + G'$. Then we have $P_{G/G',S'}^R \approx P_{G,S}^R$.
\end{lem}
\begin{proof}
For the reduction $P_{G/G',S'}^R \leq P_{G,S}^R$ we send an instance $(t,x_*,H)$ to $(t,x_*,H + G'^t)$; this works exactly because of the property $S = S' + G'$. For the reduction $P_{G,S}^R \leq P_{G/G',S'}^R$, we pass everything through the map $G \mathfrak{r}ightarrow G/G'$.
\end{proof}
For the last lemma, we first introduce a definition.
\begin{defn}
Let $G$ be an $R$-module. A \emph{transformation} on $G$ is a map $\mathfrak{v}arphi : G \mathfrak{t}o G$ of the form $x \mathfrak{m}apsto c(x) + g$ with $c$ an $R$-linear endomorphism of $G$ and $g \in G$. For $S \subset G$, we write $S_{\mathfrak{v}arphi}$ for $S \cap \mathfrak{v}arphi^{-1}(S)$.
\end{defn}
\begin{rem}
Since $R$ is commutative, multiplication by $r \in R$ is an $R$-linear endomorphism of $G$.
\end{rem}
\begin{lem}
\label{lem:transformation}
Let $G$ be an $R$-module, $S$ a subset of $G$ and $\mathfrak{v}arphi$ a transformation on $G$. Then $P_{G,S_{\mathfrak{v}arphi}}^R \leq P_{G,S}^R$.
\end{lem}
\begin{proof}
Let $c$ be an $R$-linear endomorphism of $G$, and $g \in G$ such that $\mathfrak{v}arphi$ is given by $x\mathfrak{m}apsto c(x) + g$. Let $(t,x_*,H)$ be an instance of $P_{G,S_{\mathfrak{v}arphi}}$. Define $\Gamma = G^{t} \mathfrak{t}imes G^t$ with $\mathfrak{m}athfrak{p}i_1,\mathfrak{m}athfrak{p}i_2$ the two projections, let $x_*' = (x_*,c(x_*) + g) \in \Gamma$ and $H' = \{(h,c(h)) \mathfrak{m}id h \in H\} \subset \Gamma$. Note that $H$ is naturally isomorphic to $H'$ by $f: h \mathfrak{m}apsto (h,c(h))$, as $R$ is commutative. We then see that for $h \in H$ we have that $x_*' + f(h) \in S^{2t}$ if and only if $\mathfrak{m}athfrak{p}i_1(x_*' + f(h)),\mathfrak{m}athfrak{p}i_2(x_*' + f(h)) \in S^{t}$ if and only if $x_* + h \in S$ and $c(x_*+h)+g = \mathfrak{v}arphi(x_* +h) \in S$, which is equivalent to $x_* + h \in S_{\mathfrak{v}arphi}$. This shows that $(2t,x_*',H')$ is a yes-instance of $P_{G,S}^R$ if and only if $(t,x_*,H)$ is a yes-instance of $P_{G,S_{\mathfrak{v}arphi}}^R$.
\end{proof}
We will now prove the easy part of Theorem~\mathfrak{r}ef{thm:pgs} with the following lemma.
\begin{lem}
\label{lem:pgseasy}
If $S \subset G$ is empty or a coset of some subgroup of $G$, then $P_{G,S}^R \in \mathcal{P}$.
\end{lem}
\begin{proof}
As a submodule is in particular a subgroup, we have the inequality $P_{G,S}^R \leq P_{G,S}$, so it suffices to prove the lemma assuming that $R = \mathfrak{m}athbb{Z}$. If $S$ is empty, then the problem is easy --- the answer is always no for $t > 0$ and yes for $t = 0$. If $S$ is of the form $a + G'$ with $a \in G$ and $G'$ a submodule of $G$, then by Lemma~\mathfrak{r}ef{lem:divideout} the problem is equivalent to $P_{G/G',\{a\}}$. To solve $P_{G/G',\{a\}}$ in polynomial time, we only need to decide whether the single element $(a,\ldots,a)-x_*$ is in $H$: this is simply checking whether a linear system of equations over $\mathfrak{m}athbb{Z}$ has a solution, which can be done in polynomial time as proven in \mathfrak{t}extsection14 of \citep{lenstra2008lattices}. Here we use that $R$ is finitely generated as a $\mathfrak{m}athbb{Z}$-module. Hence we indeed find that $P_{G/G',\{a\}}$ admits a polynomial time algorithm.
\end{proof}
We will prove the NP-complete part of Theorem~\mathfrak{r}ef{thm:pgs} by induction on $|S|$. There are two base cases: $|S| = 2$ for any group $G$, and $|S| = |G| - 1$ for $G = C_2^2$ where $C_2$ is the cyclic group of order $2$. Both cases follow from a reduction from $n$-colorability.
\begin{defn}
Let $n \in \mathfrak{m}athbb{Z}_{\geq 1}$ be given. Then define the problem $n$-colorability as follows. Given as input a graph $(V,E)$, decide whether there exists a mapping $V \mathfrak{t}o \{1,\dots,n\}$ such that adjacent vertices have different images.
\end{defn}
\begin{lem}
Let $G$ be an $R$-module of cardinality at least $3$, and $S$ a subset of cardinality $|G|-1$. Then $P_{G,S}^R$ is NP-complete.
\end{lem}
\begin{proof}
By translating, we can assume $S = G \setminus \{0\}$. We will reduce from $|G|$-colorability.
Let $(V,E)$ be an instance of $|G|$-colorability. Note that $G^V = \{(g_v)_{v\in V} \mathfrak{m}id g_v \in G\}$ can be thought of as all ways of assigning elements of $G$ to the vertices. Let $f$ be the homomorphism from $G^V$ to $G^E$ defined by $(g_v)_{v\in V} \mathfrak{m}apsto (g_u-g_v)_{(u,v) \in E}$, and note that an assignment in $G^V$ is a $|G|$-coloring if and only if it is sent to an element of $S^E$. Then we can take $H$ to be the submodule of $G^E$ generated by the images of $R$-generators of $G^V$, of which we need at most $|G||V|$. This is a valid reduction as $(V,E)$ will be $|G|$-colorable if and only if $H \cap S^E \not=\mathfrak{v}arnothing$; we can take $x_*$ to be zero.
As we have $|G| \geq 3$, the $|G|$-colorability problem is NP-complete \citep{garey} hence $P_{G,S}$ is NP-complete.
\end{proof}
\begin{lem}
\label{lem:s2}
Let $G$ be an $R$-module and $S$ a subset of cardinality $2$ which is not a coset of a subgroup. Then $P_{G,S}^R$ is NP-complete.
\end{lem}
\begin{proof}
Write $S = \{s,s+d\}$. Because of Lemma~\mathfrak{r}ef{lem:transinvar} we can take $s = 0$. Since $S$ is of cardinality $2$ and not a subgroup, we have $-d,2d \not\in S$. By Lemma~\mathfrak{r}ef{lem:restrict} we are allowed to take $G = Rd$, and by renaming we can take $d = 1$, the module $G$ some finite quotient of $R$ and $S = \{0,1\}$ with $-1,2 \not\in S$.
We reduce from $3$-colorability. Let $C$ be our set of three colors. Given a graph $(V,E)$, we will construct a subgroup $H \subset \Gamma := G^{V \mathfrak{t}imes C} \mathfrak{t}imes G^{V} \mathfrak{t}imes G^V \mathfrak{t}imes G^{E\mathfrak{t}imes C}$ and $x_* \in \Gamma$ such that $H + x_*$ has an element in $T := S^{V \mathfrak{t}imes C} \mathfrak{t}imes S^{V} \mathfrak{t}imes S^V \mathfrak{t}imes S^{E\mathfrak{t}imes C} $ exactly if $(V,E)$ is 3-colorable. Let $\mathfrak{m}athfrak{p}i_1,\mathfrak{m}athfrak{p}i_2,\mathfrak{m}athfrak{p}i_3,\mathfrak{m}athfrak{p}i_4$ denote the four projections from $\Gamma$ on the four factors $G^{V \mathfrak{t}imes C},G^{V} ,G^V ,G^{E\mathfrak{t}imes C}$. We take $H$ to be the image of $G^{V \mathfrak{t}imes C}$ under the map
\begin{align*}
\mathfrak{v}arphi: G^{V \mathfrak{t}imes C} &\mathfrak{t}o \Gamma \\
f &\mathfrak{m}apsto (f,\sigma(f),\sigma(f),\mathfrak{t}au(f)) \\
\end{align*}
where we define
\begin{align*}
\sigma(f)(v) &= \sum_{c \in C} f(v,c)\\
\mathfrak{t}au(f)(e,c) &= \sum_{v \in e} f(v,c).
\end{align*} Furthermore, we choose $x_* = (0,0,-1,0)$. We see $\mathfrak{v}arphi$ gives an isomorphism $G^{V \mathfrak{t}imes C} \mathfrak{t}o H$, with inverse $\mathfrak{m}athfrak{p}i_1$. Note that $\mathfrak{m}athfrak{p}i_1(\mathfrak{v}arphi(f) + x_*)$ needs to be in $\{0,1\}^{V \mathfrak{t}imes C}$ for $\mathfrak{v}arphi(f) + x_*$ to be in $T$, and $\mathfrak{m}athfrak{p}i_1(\mathfrak{v}arphi(f) + x_*)\in \{0,1\}^{V \mathfrak{t}imes C}$ happens if and only if $f$ itself is in $\{0,1\}^{V \mathfrak{t}imes C}$.
To prove this is truly a reduction, we interpret $\{0,1\}^{V \mathfrak{t}imes C}$ as assignments of subsets of $C$ to the vertices $V$, using the bijection between $\mathfrak{m}athbb{F}un(V,\{0,1\}^C)$ and $\mathfrak{m}athbb{F}un(V\mathfrak{t}imes C,\{0,1\}) = \{0,1\}^{V \mathfrak{t}imes C}$. A $3$-coloring of $(V,E)$ can then be equivalently redefined as such an assignment $f \in \{0,1\}^{V \mathfrak{t}imes C}$ with the property that for every vertex $v \in V$ we have $\sigma(f)(v) =1$, i.e., each vertex gets a single color, and that for every $c \in C,\{i,j\} \in E$ we have $f(i)(c),f(j)(c)$ not both $1$. It suffices to show that these colorings map under $\mathfrak{v}arphi$ exactly to those $h \in \mathfrak{v}arphi(\{0,1\}^{V \mathfrak{t}imes C})$ with $h + x_* \in T$.
Let $f \in \{0,1\}^{V \mathfrak{t}imes C}$ be such a coloring with subsets of $C$, and let $h = \mathfrak{v}arphi(f)$ be the corresponding element of $H$. Then note that $\mathfrak{m}athfrak{p}i_2(h + x_*) \in \{0,1\}^{V \mathfrak{t}imes C'}$ if and only if $\sigma(f)(v)$ is either $0$ or $1$. Also, $\mathfrak{m}athfrak{p}i_3(h+x_*)(v) = \sigma(f)(v) - 1$, which is $-1 \not\in S$ if $\sigma(f)(v) = 0$. Hence $(\mathfrak{m}athfrak{p}i_2(h+x_*),\mathfrak{m}athfrak{p}i_3(h+x_*)) \in \{0,1\}^{V} \mathfrak{t}imes \{0,1\}^{V}$ if and only if for every vertex $v \in V$ we have $\sigma(f)(v) = 1$.
Finally, note that $\mathfrak{m}athfrak{p}i_4(h + x_*)(e,c)$ is in $\{0,1\}$ exactly if the two endpoints of $e$ do not both have color $c$. This completes the proof that the elements of $\{0,1\}^{V \mathfrak{t}imes C}$ that are $3$-colorings correspond to $h \in H$ with $h + x_* \in T$, and hence the reduction is completed.
\end{proof}
\subsubsection{Induction step for a special kind of group}
\label{subsubs:special}
First, we will do the induction step for a special family of finite groups: $G = \langle a,b \mathfrak{r}angle$ with $a\not= b$ and $S$ containing $0,a,b$, but not $a+b$.
\begin{lem}
\label{lem:pigsspecial}
Let $G$ be a finite abelian group generated by two distinct elements $a, b$, and $S$ a subset of $G$ containing $0,a,b$ but not $a + b$. Assume Theorem~\mathfrak{r}ef{thm:pgs} holds for all $P_{G',S'}$ with $|G'| + |S'| < |G| + |S|$. Then $P_{G,S}$ is NP-complete.
\end{lem}
\begin{proof}
In this proof, we will heavily use Lemma~\mathfrak{r}ef{lem:transformation}. We restrict to bijective transformations of the form $x \mathfrak{m}apsto cx + g$ with $c = \mathfrak{m}athfrak{p}m 1$. If $\mathfrak{v}arphi$ is a transformation on $G$ and $P_{G,S_{\mathfrak{v}arphi}}$ is NP-complete, so is $P_{G,S}$. For the NP-completeness of the former, we only need $2 \leq |S_{\mathfrak{v}arphi}| < |S|$ and $S_{\mathfrak{v}arphi}$ not a coset, and then we are done by the induction hypothesis. We can also interpret this in another way: if two of the three conditions on $S_{\mathfrak{v}arphi}$ hold, then either we are done immediately, or the third one does not hold, which gives us more information about $S$. If $\mathfrak{v}arphi$ is bijective, then $|S_{\mathfrak{v}arphi}| \leq |S|$ with equality if and only if $\mathfrak{v}arphi$ induces a permutation of $S$.
We will now prove the following claim: let $A = \{g \in G \mathfrak{m}id g,g+a,g+b \in S, g+a+b \not\in S\}$. We already know that $0 \in A$. We will prove that if $g \in A$, then $\mathcal{P}i_{G,S}$ is NP-complete or $g - 2a \in A$. For the proof, we can by Lemma~\mathfrak{r}ef{lem:transinvar} assume that $g = 0$.
We do a case distinction, based on whether $a-b$ is in $S$ or not. First, we assume it is. Let $\mathfrak{v}arphi_1: x\mathfrak{m}apsto a+b-x$, and note that $a,b\in S_{\mathfrak{v}arphi_1},0\not\in S_{\mathfrak{v}arphi_1}$, so either we are done or $S_\mathfrak{m}athfrak{p}hi \ni a,b$ is a coset, implying that $a + \langle b-a \mathfrak{r}angle \subset S$, which we now assume. Now let $\mathfrak{v}arphi_2$ be the transformation $x \mathfrak{m}apsto a-b + x$. Note $0,a,b\in S_{\mathfrak{v}arphi_2}$, so $2 \leq |S|$ and $S_{\mathfrak{v}arphi_2}$ is not a coset. This now tells us that either we are done, or $S = S_{\mathfrak{v}arphi_2}$, meaning we can write $S = \Sigma + \langle b-a \mathfrak{r}angle$ for some set $\Sigma$. Writing $\Gamma = G/\langle b-a \mathfrak{r}angle$ we see $\Sigma$ is not a coset in $\Gamma$. Furthermore by Lemma~\mathfrak{r}ef{lem:divideout} we know $P_{G,S} \approx P_{\Gamma,\Sigma}$. Since $b-a \not= 0$, we have that $|\Sigma| + |\Gamma|$ is strictly smaller than $|S| + |G|$, which means that by the induction hypothesis we know $P_{\Gamma,\Sigma}$ to be NP-complete. Hence $P_{G,S} \in \mathfrak{m}athbb{N}PC$ as we wanted to show.
In the remaining case, we have $a-b\not\in S$ and similarly we can assume that $b-a \not\in S$ holds as well. Looking at $x \mathfrak{m}apsto b-x$ or $x \mathfrak{m}apsto a -x$ we see we can assume $\langle a \mathfrak{r}angle, \langle b \mathfrak{r}angle \subset S$. Now we look at $\mathfrak{v}arphi_3: x\mathfrak{m}apsto x-a-b$. If $-a-b\in S$, all conditions are met and we are done. So assume $-a-b\not\in S$. Finally taking $\mathfrak{v}arphi_4: x\mathfrak{m}apsto a-b+x$, we can see that we must have $-a + \langle a-b \mathfrak{r}angle \subset S$. We now have have $0-2a,a-2a,b-2a\in S, a+b-2a\not\in S$, hence $-2a \in A$. This proves the claim.
Now $A$ is closed under $g \mathfrak{m}apsto g - 2a$ and by symmetry also under $g \mathfrak{m}apsto g- 2b$. As $a,b$ are of finite order, we find $2G \subset A$ and hence $S = \{ka + \ell b \mathfrak{m}id k,\ell\in\mathfrak{m}athbb{Z}, k\ell \equiv 0 \bmod 2\}$ and $G \setminus S = a+b+\langle 2a, 2b\mathfrak{r}angle$, meaning $A = 2G$. Dividing out by $A$ and using Lemma~\mathfrak{r}ef{lem:divideout} we see we $P_{G,S}$ is equivalent to $P_{G/A,S'}$ where $S' = \{0,a,b\}$. Note that $|G/A| = 4$; we know $0,a+b$ are different in $G/A$ as $x \in \langle 2a, 2b\mathfrak{r}angle$ implies $x \in S$, and then $a$ is non-zero as we have that $0+b \in S$ but $a + b \not\in S$, hence $a \not \in \langle 2a, 2b\mathfrak{r}angle$. So $G/A = C_2^2$ and $|S'| = |G/A|-1$. We have already proven this to be NP-complete, so we are done.
\end{proof}
\subsubsection{Induction step for general case}
Finally, we will prove Theorem~\mathfrak{r}ef{thm:pgs} in the general case, by reducing to the case in Lemma~\mathfrak{r}ef{lem:pigsspecial}. For this, we first prove the following little lemma.
\begin{lem}
\label{lem:sab}
Let $G$ be an abelian group, and $S$ a subset of $G$. If $S$ has at least three elements, and the following statement holds
\[
\forall s,a,b: \left(s,s+a,s+b \in S \wedge a\not=b\mathfrak{r}ight) \Rightarrow s+a+b\in S,
\]
then $S$ is a coset.
\end{lem}
\begin{proof}
Since the statement is translation invariant, assume $0 \in S$; we will prove that $S$ is a subgroup. Let $\{0,x,y\}$ be a subset of $S$ of size three, i.e. $x,y$ non-zero and different. As per the property for $s =0, a = x, b = y$, we already have $x+y \in S$, it suffices to prove that $x+x,-x \in S$. As $x+y\in S$, we can apply the property with $(x+y,-x,-x-y)$ to see $-x \in S$ and with $(y,x,-y+x)$ to get $2x \in S$, concluding the proof.
\end{proof}
Now for any instance $P_{G,S}$ with $S$ not a coset and with at least three elements, we can by contraposition of Lemma~\mathfrak{r}ef{lem:sab} find $s,a,b$ with $s,s+a,s+b\in S$ and $a\not=b$ and $s+a+b\not\in S$; by translating, we can assume $s = 0$. Then, we set $G' = \langle a,b \mathfrak{r}angle$ and $S' = S \cap G'$. By Lemma~\mathfrak{r}ef{lem:pigsspecial}, we know $P_{G',S'}$ is NP-complete, and then by Lemma~\mathfrak{r}ef{lem:restrict} we find $P_{G,S}$ is NP-complete, as we wanted to show. This concludes the proof of Theorem~\mathfrak{r}ef{thm:pgs}.
\subsection{Proof of Theorem~\mathfrak{r}ef{thm:pigs}}
\label{subs:pigs}
We will show that Theorem~\mathfrak{r}ef{thm:pigs} follows immediately from Theorem~\mathfrak{r}ef{thm:pgs} and a lemma that relates the two problems. That lemma holds in the generality of $R$-modules, so we first will give a general definition of a useful function $\mathfrak{t}heta$.
\begin{defn}
Let $G$ be an $R$-module, and $S$ a subset of $G$. Then we write
\[
\mathfrak{t}heta(S) = \bigcap_{r \in R \mathfrak{m}id rS \subset S} rS.
\]
\end{defn}
\begin{lem}
\label{lem:pgspigs}
With $G$ a finite $R$-module, $S \subset G$ we have the following equivalence of problems
\[
\mathcal{P}i_{G,S}^R = \mathcal{P}i_{G,\mathfrak{t}heta(S)}^R \approx \mathcal{P}i_{R\mathfrak{t}heta(S),\mathfrak{t}heta(S)}^R \approx P_{R\mathfrak{t}heta(S),\mathfrak{t}heta(S)}^R
\]
where $R\mathfrak{t}heta(S)$ means the $R$-module generated by $\mathfrak{t}heta(S)$.
\end{lem}
\begin{proof}
We have to prove three equivalences, where the first is an equality. As defined in the appendix, two problems are the same if they have the same set of instances and the same set of yes-instances.
For the first one, note that if $(t,H)$ is a yes-instance of $\mathcal{P}i_{G,S}^R$ with certificate $h \in H \cap S^{t}$, then
\[
\left(\mathfrak{m}athfrak{p}rod_{r \in \im(R \mathfrak{t}o \End(G)) : rS \subset S} r\mathfrak{r}ight) h
\]
is in $\mathfrak{t}heta(S)^t$ as $R$ is commutative, hence $(t,H)$ is a yes-instance of $\mathcal{P}i_{G,\mathfrak{t}heta(S)}^R$. The other way around, if $(t,H)$ is a yes-instance of $\mathcal{P}i_{G,\mathfrak{t}heta(S)}^R$, then it is a yes-instance of $\mathcal{P}i_{G,S}^R$ since $\mathfrak{t}heta(S)$ is a subset of $S$, proving the first equality.
For the second one, write $S' = \mathfrak{t}heta(S), G' = RS'$ and note that $\mathcal{P}i_{G,S'} \leq \mathcal{P}i_{G',S'}$ by taking any instance $H$ of the first problem and intersecting it with $G'^t$ using the kernel algorithm from \mathfrak{t}extsection14 of \citep{lenstra2008lattices}, since $H \cap S'^t = (H \cap G'^t) \cap S'^t$. And by Lemma~\mathfrak{r}ef{lem:restrict}, the inequality $\mathcal{P}i_{G',S'} \leq \mathcal{P}i_{G,S'}$ holds as well.
The real work happens in the third equivalence. Note $\mathcal{P}i_{G',S'}^R \leq P_{G',S'}^R$ by taking $x_* = 0$. To show $P_{G',S'}^R \leq \mathcal{P}i_{G',S'}^R$, let $(t,x_*,H)$ be an instance of $P_{G',S'}^R$. Let $n = |S'|$, and enumerate $S' = \{s_1,\dots,s_n\}$. We will construct an instance $(t',H')$ of the first problem; we will take $t' = t + n$ and $H' = H \mathfrak{t}imes \{0\}^t + R\cdot y_*$ where $y_*$ is defined as $(x_*,s_1,\dots,s_n) $.
We need to check that if $(t,x_*,H)$ is a yes-instance, so is $(t',H')$ and vice versa. The first implication is trivial; if $h \in H$ has $x_* + h \in S^t$, then $h' = y_* + (h,0)$ is an element of $H'$, and lies in $S'$ on every coordinate, hence we see $h' \in H' \cap S'^{t'}$. For the other implication, let $h' = ay_* + (h,0)$ be an element of $H' \cap S'^{t'}$. Looking at the last $|S'|$ coordinates, we see $aS' \subset S'$. But as $\mathfrak{t}heta(S') = \mathfrak{t}heta^2(S) = \mathfrak{t}heta(S) = S'$, we must have $aS' = S'$. Since $a$ induces a bijection on $S'$ and $S'$ generates $G'$ we see $a$ is a unit in $\End(G')$, using the commutativity of $R$. Then some power of $a$ is its inverse in $\End(G')$. Hence we can multiply $h'$ with $a^{-1} \in \End(G')$, and since $a^{-1}S' = S'$ we see $y_* + a^{-1}(h,0) \in S'^{t'}$. Restricting to the first $t$ places, we see $x_* + h \in S'^{t}$, hence $(x_* + H) \cap S'^t \not= \mathfrak{v}arnothing$ as we wanted to show.
\end{proof}
\begin{proof}[Proof of Theorem~\mathfrak{r}ef{thm:pigs}]
Theorem~\mathfrak{r}ef{thm:pigs} now follows trivially from Theorem~\mathfrak{r}ef{thm:pgs} and Lemma~\mathfrak{r}ef{lem:pgspigs}.
\end{proof}
As promised in the introduction, we will also prove a short lemma about $\mathfrak{t}heta$ which generalises the case that $G$ has prime power cardinality.
\begin{lem}
\label{lem:psipowerp}
Let $G$ be an $R$-module such that $A := \im(R \mathfrak{t}o \End(G))$ is local. Let $S$ be a subset of $G$. Then $\mathfrak{t}heta(S)$ equals $\{0\}$ if $0 \in S$ and $S$ otherwise.
\end{lem}
\begin{proof}
Obviously, if $0 \in S$ then for every integer $a$ we have $0 \in aS$ so $\{0\} \subset \mathfrak{t}heta(S)$, and $0S \subset S$ hence $\mathfrak{t}heta(S) \subset \{0\}$, proving the first part. For the second part, in a local finite ring the powers of the maximal ideal must stabilise, which by Nakayama's lemma mean they must become zero. So every element of $A$ is either invertible or nilpotent. If $r \in A$ is nilpotent, and $0 \not\in S$, then $rS \not\subset S$; otherwise, we would have $r^k S \subset S$ for every $k \in \mathfrak{m}athbb{Z}_{> 0}$, contradicting with the nilpotency of $r$ and $0 \not \in S$. That means that if $rS \subset S$ then on $G$, we have that $r$ induces an automorphism, and $rS \subset S$ then implies by cardinality that $rS = S$. Hence in this case $\mathfrak{t}heta(S) = S$, as we set out to prove.
\end{proof}
\begin{rem}
Some important examples of when the conditions are satisfied, are the case where $R$ itself is local, and the case where $R = \mathfrak{m}athbb{Z}$ and $G$ has prime power cardinality.
\end{rem}
\appendix
\section{\mathfrak{t}exorpdfstring{$\mathcal{P}$}{P} and \mathfrak{t}exorpdfstring{$\mathfrak{m}athbb{N}P$}{NP}}
\label{section:pnp}
In this appendix we will briefly set up a formal, mathematical framework for the complexity classes $\mathcal{P},\mathfrak{m}athbb{N}P,\mathfrak{m}athbb{N}PC$. For more information, we refer to a standard work like \cite{garey}, but reader beware; there are some differences between the definitions in \cite{garey} and in this appendix.
\begin{defn}
\label{defn:prob}
A \emph{decision problem}, briefly a problem, is defined to be a pair of sets $(I,Y)$ with $Y \subset I \subset \mathfrak{m}athbb{Z}_{> 0}$, where we call $I$ the set of \emph{instances} of the problem, and $Y$ the set of \emph{yes-instances}. A problem is called \emph{trivial} if $Y = I$ or $Y = \mathfrak{v}arnothing$.
\end{defn}
\begin{rem}
If the input is not an integer, we simply encode the input as natural numbers in an algorithmically nice way. In the present article we will simply gloss over this.
\end{rem}
We will not define algorithms formally; we treat them as a black box.
We can now say when an algorithm solves a problem in polynomial time.
\begin{defn}
An algorithm \emph{solves a problem $(I,Y)$ in polynomial time} if it runs in polynomial time in the length of the input $i \in I$, and ends in an accepting state if and only if $i \in Y$, and otherwise in a rejecting state; informally, outputting ``yes'' and ``no'' respectively. Contrary to \cite{garey}, we do not put any restrictions on what the algorithm does given input in $\mathfrak{m}athbb{Z}_{>0} \setminus I$.
\end{defn}
Now we can define the class of problems $\mathcal{P}$.
\begin{defn}
\[
\mathcal{P} = \{\mathfrak{t}ext{problems for which there exists a polynomial time algorithm}\}.
\]
\end{defn}
We will now give a definition of $\mathfrak{m}athbb{N}P$.
\begin{defn}
\label{defn:np}
We say a problem $(I,Y)$ is in $\mathfrak{m}athbb{N}P$ if there exists a set theoretic function $c: Y \mathfrak{t}o \mathfrak{m}athbb{Z}_{> 0}$ with $\log c(y)$ polynomially bounded in $\log y$ and a function $t: I \mathfrak{t}imes \mathfrak{m}athbb{Z}_{> 0} \mathfrak{t}o \{\mathfrak{t}ext{yes},\mathfrak{t}ext{ no}\}$ that can be calculated in polynomial time in the length of the input such that $t|_{(I\setminus Y) \mathfrak{t}imes \mathfrak{m}athbb{Z}_{> 0}}$ is the constant ``no'' function and $y\mathfrak{m}apsto t(y,c(y))$ is the constant ``yes'' function. This $c(y)$ is also called a \emph{certificate} for $y$.
\end{defn}
Next we will define the notion of polynomial time reductions.
\begin{defn}
Let $\mathcal{P}i_1 = (I_1,Y_1),\mathcal{P}i_2 = (I_2,Y_2)$ be two problems. We say $\mathcal{P}i_1$ is \emph{reducible} to $\mathcal{P}i_2$, notation $\mathcal{P}i_1 \leq \mathcal{P}i_2$, if there exists a function $f: I_1 \mathfrak{t}o I_2$, called a \emph{reduction}, that can be calculated by an algorithm in polynomial time, such that $Y_1 = f^{-1}(Y_2)$.
\end{defn}
We can now define the following equivalence relation.
\begin{defn}
\label{defn:approx}
Let $\mathcal{P}i_1, \mathcal{P}i_2$ be two problems. We say $\mathcal{P}i_1$ is \emph{equivalent} to $\mathcal{P}i_2$, notation $\mathcal{P}i_1 \approx \mathcal{P}i_2$, if $\mathcal{P}i_1 \leq \mathcal{P}i_2$ and $\mathcal{P}i_2 \leq \mathcal{P}i_1$.
\end{defn}
Note that the non-trivial problems in $\mathcal{P}$ form one of the equivalence classes of this relation.
With the notation we have developed, we can define NP-complete problems.
\begin{defn}
\[
\mathfrak{m}athbb{N}PC = \{\mathcal{P}i \in \mathfrak{m}athbb{N}P \mathfrak{m}id \forall R \in \mathfrak{m}athbb{N}P : R \leq \mathcal{P}i\}.
\]
\end{defn}
\addcontentsline{toc}{section}{References}
\end{document} |
\begin{document}
\title{Schoenberg Representations and Gramian Matrices of Mat\'ern Functions}
\begin{itemize}
\item[{}] {\bf Abstract.} We represent Mat\'ern functions in terms of Schoenberg's integrals
which ensure the positive definiteness and prove the systems of translates
of Mat\'ern functions form Riesz sequences in $L^2(\mathbb{R}^n)$ or Sobolev spaces.
Our approach is based on a new class of integral transforms that
generalize Fourier transforms for radial functions.
We also consider inverse multi-quadrics and obtain similar results.
\end{itemize}
{\mathbb{S}mall
\begin{itemize}
\item[{}]{\bf Keywords.} Bessel function, Fourier transform, Gramian matrix, Hankel-Schoenberg transform, inverse multi-quadrics, Mat\'ern function,
positive definite, Riesz sequence, Schoenberg matrix, Sobolev space.
\item[{}] 2010 Mathematics Subject Classification: 33C10, 41A05, 42B10, 60E10.
\end{itemize}}
\mathbb{S}ection{Introduction}
In many areas of Mathematics, the functions of type
\begin{equation}
M_\alpha(z) = K_\alpha(z) z^\alpha\qquad(\alpha\in\mathbb{R},\,z>0)
\end{equation}
arise frequently, referred to as the Mat\'ern functions, where
$K_\alpha(z)$ stands for the modified Bessel function of the second kind
of order $\alpha$.
Intimately connected is the family of functions of type
\begin{equation}
\phi_\beta(r) = (1+r^2)^{-\beta}\qquad(\beta>0, \,r\ge 0)
\end{equation}
whose radial extensions to the Euclidean spaces are referred to as the inverse multi-quadrics
in the theory of interpolations or spatial statistics.
In a fixed Euclidean space, both class of functions, if radially extended with suitably rearranged $\,\alpha, \beta,\,$
provide essential ingredients of Sobolev spaces.
In their pioneering work \cite{AK}, N. Aronszajn and K. T. Smith introduced the Sobolev space
$H^\alpha(\mathbb{R}^n),\,\alpha>0,\,$ as the space of Bessel potentials, that is, the convolutions
$\,(G_{\alpha/2}\ast u)(\mathbf{x}),\,u\in L^2(\mathbb{R}^n),\,$ where $G_{\alpha/2}$ denotes the radial
extension of a special kind of Mat\'ern functions defined as follows.
\begin{definition} For a positive integer $n$ and $\,\alpha>0,$
\begin{equation}\label{G1}
G_\alpha(z) = \frac{1}{2^{\alpha-1 + \frac n2}\,\pi^{\frac n2}\,
\Gamma(\alpha)}\,K_{\alpha-\frac n2}(z) z^{\alpha - \frac n2}\qquad(z>0).
\end{equation}
For its radial extension to the Euclidean space $\mathbb{R}^n$, we write
$$G_\alpha(\mathbf{x}) = G_\alpha(|\mathbf{x}|), \quad |\mathbf{x}| = \mathbb{S}qrt{\mathbf{x}\cdot\mathbf{x}}\qquad(\mathbf{x}\in\mathbb{R}^n).$$
\end{definition}
A characteristic feature of the kernel $G_\alpha$ is the Fourier transform
\begin{align*}
\widehat{G_\alpha}(\xi) =\int_{\mathbb{R}^n} e^{-i \xi\cdot\mathbf{x}}\,G_\alpha(\mathbf{x}) d\mathbf{x}= \left( 1+|\xi|^2\right)^{-\alpha}\,,
\end{align*}
which, together with the intrinsic properties of $K_{\alpha-n/2}$, enables the authors to obtain a comprehensive list of functional properties.
Let us state only a few of their list which are relevant to the present work (see also \cite{C}).
\begin{itemize}
\item[(a)] The Sobolev space $H^\alpha(\mathbb{R}^n)$ is identified with
\begin{equation*}
H^\alpha(\mathbb{R}^n) = \left\{ u\in L^2(\mathbb{R}^n) : \int_{\mathbb{R}^n} \left( 1+|\xi|^2\right)^{\alpha}|\widehat{u}(\xi)|^2 d\xi <\infty
\right\}.
\end{equation*}
In particular, $\,G_\beta\in H^\alpha(\mathbb{R}^n)\,$ if and only if $\,\beta>(2\alpha +n)/4.$
\item[(b)] $\,\left(G_\alpha\ast G_\beta\right)(\mathbf{x}) = G_{\alpha +\beta}(\mathbf{x})\,$ for $\,\alpha>0,\,\beta>0.$
\item[(c)] In the case $\,\alpha>n/2,\,$ $G_{\alpha}$ is positive definite on $\mathbb{R}^n$.
The symmetric kernel $G_{\alpha}(\mathbf{x} - \mathbf{y})$
is in fact a reproducing kernel for the Hilbert space $H^\alpha(\mathbb{R}^n)$ under the inner product
$$\big(u, v\big)_{H^\alpha(\mathbb{R}^n)} =
(2\pi)^{-n}\int_{\mathbb{R}^n} \widehat{u}(\xi)\,\overline{\widehat{v}(\xi)}\,(1+|\xi|^2)^\alpha\,d\xi.$$
\end{itemize}
Our primary purpose in the present work is to obtain a set of invariants for both classes
of functions, that is, those properties valid in any Euclidean space, related with
the positive definiteness and Fourier transforms.
We recall that a univariate function $\phi$ defined on the interval $[0, \infty)$
is said to be {\it positive semi-definite on $\mathbb{R}^n$}
if it satisfies
\begin{align}\label{G3}
\mathbb{S}um_{j=1}^N\mathbb{S}um_{k=1}^N \,\phi\left(\left|\mathbf{x}_j - \mathbf{x}_k\right|\right)\alpha_j \overline{\alpha_k}\,\ge\, 0
\end{align}
for any choice of $\,\alpha_1, \cdots, \alpha_N\in\mathbb{C}\,$ and
distinct points $\,\mathbf{x}_1, \cdots, \mathbf{x}_N\in\mathbb{R}^n,\,$ where $N$ is arbitrary. If equality in \eqref{G3} holds only if
$\,\alpha_1=\cdots=\alpha_N=0,\,$ then it is said to be {\it positive definite on $\mathbb{R}^n$}.
A univariate function which is positive semi-definite or positive definite on every $\mathbb{R}^n$
takes the following specific form:
\begin{itemize}
\item[{}]{\bf Criterion I} (I. J. Schoenberg \cite{Sc2}).
{\it A continuous function $\phi$ on $[0, \infty)$ is
positive semi-definite on every $\mathbb{R}^n$ if and only if
\begin{equation}\label{G4}
\phi(r) = \int_0^\infty e^{-r^2 t}\,d\nu(t)
\end{equation}
for a finite positive Borel measure $\nu$ on $[0, \infty).$
Moreover, if $\nu$ is not concentrated at zero,
then $\phi$ is positive definite on every $\mathbb{R}^n$. }
\end{itemize}
Due to the representation formula
\begin{equation}\label{G5}
\phi_\beta(r) = \frac{1}{\Gamma(\beta)}\int_0^\infty e^{-r^2 t}\,e^{-t} t^{\beta -1} dt\qquad(\beta>0),
\end{equation}
it is well known that each $\phi_\beta$ is positive definite on every $\mathbb{R}^n$ (see e.g. \cite{We}).
Our preliminary observation is the following.
\mathbb{S}mallskip
\begin{theorem}\label{theorem1.1}
For $\,\alpha>0,\,$ we have
\begin{equation*}
\frac{2^{1-\alpha}}{\Gamma(\alpha)}\,K_\alpha(z) z^\alpha = \int_0^\infty e^{-z^2 t}\,f_\alpha(t) dt\qquad(z\ge 0),
\end{equation*}
where $f_\alpha$ denotes the probability density defined by
$$ f_\alpha(t) = \frac{1}{2^{2\alpha}\Gamma(\alpha)}\,\exp\left(-\frac{1}{4t}\right) t^{-\alpha-1}.$$
As a consequence, $M_\alpha$ is positive definite on every $\mathbb{R}^n$.
\end{theorem}
\mathbb{S}mallskip
In order to find direct relationships between the functions $M_\alpha$ and $\phi_\beta$, without recourse to
their Euclidean extensions, we shall introduce a new class of integral transforms that incorporates Fourier transforms
for radial measures and Hankel transforms in certain sense.
\begin{definition}\label{def1}
For $\,\lambda>-1,\,$ let $J_\lambda$ denote the Bessel function of the first kind of order $\lambda$ and
define $\,\Omega_\lambda : \mathbb{R}\to \mathbb{R}\,$ by
\begin{align*}
\Omega_\lambda(t) &= \Gamma(\lambda+1)\left(\frac t2\right)^{-\lambda} J_\lambda(t)\\
&=\Gamma(\lambda+1)\mathbb{S}um_{k=0}^\infty\frac{(-1)^k}{k!\,\Gamma(\lambda +k +1)}\,\left(\frac t 2\right)^{2k}.
\end{align*}
\end{definition}
In the special case $\,\lambda = (n-2)/2\,,$ with $n$ a positive integer, $\Omega_\lambda$
arises on consideration of the Fourier transforms for radial functions on $\mathbb{R}^n$. To be specific,
if $F$ is integrable with $\,F(\mathbf{x})= f(|\mathbf{x}|)\,$ for some univariate function $f$ on $[0, \infty)$,
then it is well known (see e.g. \cite{Stw}) that
\begin{align}\label{G6}
\widehat{F}(\xi) &= (2\pi)^{n/2} |\xi|^{-\frac{n-2}{2}}\int_0^\infty J_{\frac{n-2}{2}}
(|\xi|t) f(t) t^{n/2}dt\nonumber\\
&=\frac{2\pi^{n/2}}{\Gamma(n/2)} \int_0^\infty \Omega_{\frac{n-2}{2}}(|\xi|t) f(t) t^{n-1} dt.
\end{align}
More extensively, I. J. Schoenberg noticed that the Fourier transform of any radial measure on $\mathbb{R}^n$
is also representable in the above form and set up the following characterization (see also H. Wendland \cite{We}).
\begin{itemize}
\item[{}]{\bf Criterion II} (I. J. Schoenberg \cite{Sc1}, \cite{Sc2}).
{\it A continuous function $\phi$ on $[0, \infty)$ is
positive semi-definite on $\mathbb{R}^n$ if and only if
\begin{equation}\label{G7}
\phi(r) = \int_0^\infty \Omega_{\frac{n-2}{2}}(rt) d\nu(t)
\end{equation}
for a finite positive Borel measure $\nu$ on $[0, \infty)$.
Moreover, in the case when $\,d\nu(t) = f(t) t^{n-1} dt\,$ with continuous $f$,
$\phi$ is positive definite on $\mathbb{R}^n$ if and only if $\phi$ is nonnegative and non-vanishing.
}
\end{itemize}
Our generalization of Schoenberg's integrals or Fourier transforms for radial measures
takes the following form.
\begin{definition}
The Hankel-Schoenberg transform of order $\,\lambda>-1\,$
of a finite positive Borel measure $\nu$ on $[0, \infty)$ is defined by
\begin{equation*}
\phi(r) = \int_0^\infty\Omega_\lambda(rt)\,d\nu(t)\qquad(0\le r<\infty).
\end{equation*}
\end{definition}
For those Borel measures on $[0, \infty)$ which are absolutely
continuous with respect to Lebesgue measure, it is simple to express the Hankel-Schoenberg transforms
in terms of the classical Hankel transforms for which analogues of
the Fourier inversion theorem and Parseval's relations are available.
Our evaluations will be of the form
\begin{align}
\left(1+r^{2}\right)^{-\alpha-\lambda -1} = c(\alpha, \lambda)\int_{0}^{\infty}\Omega_{\lambda}(rt)\big[K_{\alpha}(t)t^{\alpha}\big]t^{2\lambda+1}dt
\end{align}
for $\,\alpha+\lambda+1>0\,$ with an explicit positive constant $c(\alpha, \lambda)$.
By inversions and order-changing transforms, we shall obtain
a number of representation formulas for the Mat\'ern functions $M_\alpha$ in terms of $\phi_\beta$'s and
vice versa, which suits to Schoenberg's criterion and makes it possible to find
the Fourier transforms of their radial extensions to any Euclidean space.
In accordance with the notation of \cite{GMO}, we introduce
\begin{definition}
For a univariate function $\phi$ on $[0, \infty)$ and a set of distinct
points $\,X= \{\mathbf{x}_j\}_{j\in\mathbb{N}}\mathbb{S}ubset\mathbb{R}^n,\,$
the Schoenberg matrix is defined to be
\begin{equation}
\mathbf{S}_X (\phi) = \Big[ \phi\big(\left|\mathbf{x}_j - \mathbf{x}_k\right|\big)\Big]_{j, \,k\in\mathbb{N}}.
\end{equation}
\end{definition}
The notion of Schoenberg matrix comes up instantly with an attempt to
construct an interpolating functional that matches the values of any function at each point of $X$.
To state briefly, if $\mathbf{S}_X (\phi)$ defines a bounded invertible operator on the space
$\ell^2(\mathbb{N})$, then it is possible to construct a Lagrange-type radial basis sequence
$\,\left\{u_j^*\right\}_{j\in\mathbb{N}}\,$ by setting
$$u_j^*(\mathbf{x}) = \mathbb{S}um_{k=1}^\infty c_{j, k}\,\phi(|\mathbf{x}- \mathbf{x}_k|),\quad j=1,2, \cdots, $$
and solving the infinite system $\,u_j^*(\mathbf{x}_k) = \delta_{j, k},\,$ which has a unique solution
$\,\mathbf{c}_j = (c_{j, 1}, c_{j, 2},\cdots)\in\ell^2(\mathbb{N})\,$ for each $j$. The functional
$$A_X(f)(\mathbf{x}) = \mathbb{S}um_{j=1}^\infty f(\mathbf{x}_j)\,u_j^*(\mathbf{x}),$$
definable on any class of functions, obviously interpolates $f$ at $X.$
The Schoenberg matrices arise under various guises in other fields of Mathematics.
In Functional Analysis, for example, it is common that $\mathbf{S}_X (\phi)$ coincides with
the Gramian matrix of a sequence obtained by translating another function $\psi$ by $X$
in an appropriate Hilbert space $H$, that is,
$$\mathbf{S}_X (\phi) = \Big[\big(\psi(|\cdot - \mathbf{x}_j|),\,\psi(|\cdot - \mathbf{x}_k|)\big)_H\Big]_{j, k\in\mathbb{N}}.$$
In such a circumstance, $\,\left\{\psi(|\mathbf{x}- \mathbf{x}_j|)\right\}_{j\in\mathbb{N}}\,$
is a Riesz sequence in $H$ if and only if $\mathbf{S}_X (\phi)$ defines a bounded invertible operator on $\ell^2(\mathbb{N})$.
Our secondary purpose is to study the Schoenberg or Gramian matrices associated with
the Mat\'ern functions $M_\alpha$ as well as the functions $\phi_\beta$ with our focuses on their
boundedness and invertibility on $\ell^2(\mathbb{N})$.
Our approaches are substantially based on the recent developments
\cite{GMO}, \cite{MS} of L. Golinskii {\it et al.} in which a list of
criteria for the boundedness and invertibility are established from several perspectives.
To illustrate, the authors devoted considerable portions of their work in studying
the $L^2$-based Gramian matrices associated to the Mat\'ern functions $M_\alpha$
and obtained their boundedness and invertibility on $\ell^2(\mathbb{N})$ in the range $\,-n/4<\alpha\le 0.$
As we shall present below, we shall improve their results by extending the range to
$\,\alpha>-n/4\,$ and the boundedness and invertibility results to the aforementioned
Sobolev space-based Gramian matrices. In applications, we shall prove that the system of type
$\,\big\{M_\alpha(|\mathbf{x}-\mathbf{x}_j|)\big\}_{j\in\mathbb{N}},\,$
where $\,(\mathbf{x}_j)\,$ is an arbitrary set of distinct points of $\mathbb{R}^n$, is a Riesz sequence
in $L^2(\mathbb{R}^n)$ or the Sobolev space of certain specified order.
In the same manner, the system of type $\,\big\{\phi_\beta(|\mathbf{x}-\mathbf{x}_j|)\big\}_{j\in\mathbb{N}}\,$ will be shown to be
a Riesz sequence in the Hilbert space of functions on $\mathbb{R}^n$ for which $\,\phi_\beta(|\mathbf{x}-\mathbf{y}|)\,$ is a reproducing
kernel.
\mathbb{S}ection{Bessel functions $K_\alpha$}
In this section we collect some of the basic properties of $K_\alpha$ relevant to
the present work, most of which can be found in \cite{AS}, \cite{E} and \cite{Wa}.
For $\,\alpha\in\mathbb{C},\,$ the modified Bessel function $K_\alpha$ is defined by
\begin{align}
K_\alpha(z) &= \frac{\pi}{2}\left[\frac{\,I_{-\alpha}(z)-I_{\alpha}(z)\,}{\mathbb{S}in\left(\alpha\pi\right)}\right],\quad\text{where}\\
I_{\alpha}(z) &= \mathbb{S}um_{k=0}^{\infty}\frac{1}{k!\,\Gamma\left(k+\alpha+1\right)}\left(\frac{z}{2}\right)^{2k +\alpha}.\label{K1}
\end{align}
In the case when $\alpha$ happens to be an integer, $\,\alpha=n,\,$ this formula should be interpreted as
$\,K_n(z) = \lim_{\alpha\to\, n} K_\alpha(z).\,$ The Bessel functions $\,I_\alpha, \,K_\alpha\,$ form a fundamental
system of solutions to the differential equation
\begin{equation}\label{K2}
z^2\frac{d^2 u}{dz^2} + z \frac{du}{dz} - (z^2 + \alpha^2) u = 0.
\end{equation}
Hereafter, we shall be concerned only with $\,\alpha\in\mathbb{R}\,$ and $\,z>0.$
\begin{itemize}
\item[(K1)] By definition, it is evident $\,K_{-\alpha}(z) = K_\alpha(z).\,$ For each integer $n$,
a series expansion formula for $K_n(z)$ is also available. In particular,
\begin{equation}\label{K5}
K_0(z) = -\log (z/2) I_0(z) + \mathbb{S}um_{k=0}^\infty \frac{\psi(k+1)}{(k!)^2} \left(\frac z2\right)^{2k},
\end{equation}
where $\psi$ denotes the digamma function so that
$$\psi(1) = -\gamma, \quad \psi(k+1) = -\gamma + \mathbb{S}um_{j=1}^k\frac 1j\,,$$
with $\gamma$ being the Euler-Mascheroni constant.
\item[(K2)] For $\,\alpha>-1/2\,$ and $\,z>0,$ Schl\"afli's integrals state
\begin{align}\label{K3}
K_{\alpha}(z) &= \frac{\mathbb{S}qrt{\pi}}{\Gamma(\alpha + 1/2)}\,\left(\frac{z}{2}\right)^{\alpha}
\int_{1}^{\infty} e^{-zt}\left(t^{2}-1\right)^{\alpha-\frac{1}{2}}\,dt\nonumber\\
&=\mathbb{S}qrt{\frac{\pi}{2}\,} \frac{e^{-z} z^\alpha}{\Gamma(\alpha+1/2)}\,
\int_{0}^{\infty} e^{- zt} \left[ t\left( 1 + \frac t2\right)\right]^{\alpha - \frac 12}\,dt
\end{align}
in which the latter follows from the former by suitable substitutions.
Another form of Schl\"afli's integral reads
\begin{equation}\label{K4}
K_\alpha(z) = \frac 12 \int_{-\infty}^\infty e^{-z \cosh t - \alpha t}\, dt,
\end{equation}
which holds for any real $\alpha$ and $\,z>0.$ As a consequence, $K_\alpha(z)$ is
positive on the interval $(0, \infty)$.
\item[(K3)] From the differential equation \eqref{K2}, it follows plainly
$$\frac{d}{dz} \big[K_\alpha(z) z^\alpha\big] = - K_{\alpha -1}(z) z^\alpha.$$
By (K2), hence, the Mat\'ern function $\,M_\alpha(z) = K_\alpha(z) z^\alpha\,$ is positive and strictly decreasing
on the interval $(0, \infty)$.
\item[(K4)] Of great significance is the asymptotic behavior of $K_\alpha$ for $\,\alpha\ge 0.$
\begin{itemize}
\item[(i)] As $\,z\to 0,\,$ the series expansions \eqref{K1} and \eqref{K5} yield\footnote{To be more precise, \eqref{K1} shows
$$K_\alpha(z) = 2^{\alpha-1}\Gamma(\alpha) z^{-\alpha}\big[ 1 + O\left( z^{\alpha_*}\right)\big],$$
where $\,\alpha_* = \min (2\alpha, \,2)\,$ and \eqref{K5} shows
$$K_0(z) = -\log z + \log 2 -\gamma + \big[1-\log(z/2)\big] O\left(z^2\right).$$}
\begin{equation*}
K_\alpha(z) \,\mathbb{S}im\,\left\{\begin{aligned} &{2^{\alpha-1}\Gamma(\alpha) z^{-\alpha}} &{\quad\text{for}\quad \alpha>0},\\
&{-\log z} &{\quad\text{for}\quad \alpha =0}.\end{aligned}\right.
\end{equation*}
\item[(ii)] As $\,z\to\infty,$ a version of Hankel's asymptotic formula states
\begin{align*}
K_{\alpha}(z) = \mathbb{S}qrt{\frac{\pi}{2 z}\,}\,e^{-z}\left[ 1 + \frac{4\alpha^{2}-1}{8z} + O\left(\frac{1}{z^2}\right)\right].
\end{align*}
\end{itemize}
\item[(K5)] In the special case $\,\alpha = n + 1/2\,$ with $n$ an integer, it is simple to express
$K_\alpha$, and hence the Mat\'ern function $M_\alpha$, in closed forms on evaluation of Schl\"afli's integral \eqref{K3}.
To state $M_\alpha$ explicitly,
\begin{align}\label{K6}
M_{n+ \frac 12}(z) &= \mathbb{S}qrt{\frac{\pi}{2}\,}\,e^{-z} z^{n}\mathbb{S}um_{k=0}^n \frac{(n+k)!}{k! (n-k)!}\,(2z)^{-k},\nonumber\\
M_{-n-\frac 12} (z) &= \mathbb{S}qrt{\frac{\pi}{2}\,}\,e^{-z} z^{-n-1}\mathbb{S}um_{k=0}^n \frac{(n+k)!}{k! (n-k)!}\,(2z)^{-k},
\end{align}
where $n$ is a nonnegative integer. A list of positive orders reads
\begin{align}
M_{\frac 12}(z)&= \mathbb{S}qrt{\frac{\pi}{2}\,}\, e^{-z}\,,\nonumber\\
M_{\frac 32}(z)&= \mathbb{S}qrt{\frac{\pi}{2}\,}\,(1+z)e^{-z}\,,\nonumber\\
M_{\frac 52}(z)&= \mathbb{S}qrt{\frac{\pi}{2}\,}\,\left(3 + 3z+ z^{2}\right)e^{-z}
\end{align}
which are of considerable interest in spatial statistics (see \cite{G1}, \cite{G2}).
A list of negative orders reads
\begin{align}
M_{-\frac 12}(z)&= \mathbb{S}qrt{\frac{\pi}{2}\,} \,\frac{e^{-z}}{z}\,,\nonumber\\
M_{-\frac 32}(z)&= \mathbb{S}qrt{\frac{\pi}{2}\,}\,\left(\frac{1}{z^2} + \frac{1}{z^3}\right)e^{-z}\,,\nonumber\\
M_{-\frac 52}(z)&= \mathbb{S}qrt{\frac{\pi}{2}\,}\,\left(\frac{1}{z^3} + \frac{3}{z^4} +\frac{3}{z^5}\right)e^{-z}\,.
\end{align}
\end{itemize}
\mathbb{S}ection{Hankel-Schoenberg transforms}
The purpose of this section is to establish basic properties of the
Hankel-Schoenberg transforms which will be used subsequently.
To begin with, we list the following properties on the kernels $\Omega_\lambda$
which are deducible from those on the Bessel functions $J_\lambda$
(\cite{E}, \cite{Wa}).
\begin{itemize}
\item[(J1)] Each $\Omega_\lambda$ is of class $C^\infty(\mathbb{R})$, even and uniformly bounded by $\,1=\Omega_\lambda(0).\,$
A theorem of Bessel-Lommel states that it is an oscillatory function with an infinity of positive simple zeros.
A modification of Hankel's asymptotic formula for $J_\lambda$ shows that as $\,t\to\infty,$
\begin{equation*}
\Omega_\lambda(t) = \frac{\Gamma(\lambda+1)}{\mathbb{S}qrt{\pi}} \left(\frac t2\right)^{-\lambda -1/2}
\left[\cos\left(t - \frac{(2\lambda+1)\pi}{4}\right) + O\left( t^{-1}\right)\right].
\end{equation*}
\item[(J2)] For $\,\lambda>-1/2,\,$ Poisson's integral reads
\begin{align*}
\Omega_\lambda(t) = \frac{2}{B\left(\lambda + 1/2\,,\,1/2\right)}\,
\int_0^1 \cos(t s)\, (1-s^2)^{\lambda -\frac 12}\,dt,
\end{align*}
where $B$ stands for the Euler beta function defined by
$$B(a, \,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt\qquad(a>0, \,b>0).$$
\item[(J3)] By Liouville's theorem, $\Omega_\lambda$ is expressible
in finite terms by algebraic and trigonometric functions
if and only if $2\lambda$ is an odd integer. Indeed,
the Lommel-type recurrence formula
\begin{align*}
\Omega_\lambda(t) -\Omega_{\lambda-1}(t)
= \frac{t^2}{\,4\lambda(\lambda+1)\,}\,\Omega_{\lambda+2}(t)
\qquad(\lambda>-1)
\end{align*}
may be used to evaluate $\Omega_{n + 1/2}$ for any integer $n$ together with
$$\Omega_{-\frac 12}(t) = \cos t,\quad \Omega_{\frac 12}(t) = \frac{\mathbb{S}in t}{t}\,.$$
\end{itemize}
The Hankel transforms of a function $f$ refer to the integrals
$$\int_0^\infty J_\lambda(rt) f(t) t dt\qquad(\lambda\in\mathbb{C}).$$
It follows by definition that the Hankel-Schoenberg transforms
can be written in terms of the Hankel transforms whenever $\nu$ admits an integrable density $f$,
that is, $\,d\nu(t) = f(t) dt.\,$
The Hankel-Watson inversion theorem (\cite{Wa}) states that if $\,\lambda\ge -1/2\,$
and $\,f(t)\mathbb{S}qrt t\,$ is integrable on $[0, \infty)$, then
\begin{equation*}
\int_0^\infty J_\lambda(rt)\left[\int_0^\infty J_\lambda(ru) f(u)u du\right] rdr
= \frac{\,f(t+0) + f(t-0)\,}{2}
\end{equation*}
at every $\,t>0\,$ such that $f$ is of bounded variation in a neighborhood of $t$.
An obvious modification yields the following inversion formula which may serve as an alternative of
the Fourier inversion theorem for radial functions.
\begin{theorem}\label{inversion} {\rm (Inversion)}
For $\,\lambda\ge -1/2,\,$ assume that
\begin{equation}\label{invc1}
\int_0^\infty |f(t)| t^{-\lambda-1/2}\,dt <\infty.
\end{equation}
Then the following holds for every $\,t>0\,$ at which $f$ is continuous:
\begin{align*}
\left\{\aligned &{\phi(r) = \int_0^\infty \Omega_\lambda(rt) f(t) dt\quad \text{implies}}\\
&{f(t) = \frac{t^{2\lambda+1}}{4^\lambda\left[\Gamma(\lambda+1)\right]^2}\,
\int_0^\infty \Omega_\lambda(rt)\, \phi(r) r^{2\lambda+1}dr.}\endaligned\right.
\end{align*}
\end{theorem}
A version of Parseval's theorem is deducible from its equivalent for the Hankel transforms
in a trivial manner.
\begin{theorem}\label{Parseval}
{\rm (Parseval's relation)}
For $\,\lambda>-1\,,$ let
\begin{align*}
\phi_j(r) = \int_0^\infty\Omega_\lambda(rt) f_j(t) t^{2\lambda +1} dt,\quad j=1, 2.
\end{align*}
If both integrals are absolutely convergent, then
\begin{equation*}
\int_0^\infty f_1(t) f_2(t) t^{2\lambda+1} dt = \frac{1}{4^\lambda\left[\Gamma(\lambda+1)\right]^2}\,
\int_0^\infty \phi_1(r) \phi_2(r) r^{2\lambda+1} dr.
\end{equation*}
\end{theorem}
\begin{lemma}\label{basic}
For $\,\lambda>\rho>-1\,$ and $\,r\ge 0,$
\begin{equation*}
\Omega_\lambda(r) = \frac{2}{B(\rho +1, \,\lambda-\rho)} \int_0^\infty\Omega_\rho(rt)
(1-t^2)_+^{\lambda-\rho-1}t^{2\rho+1} dt.
\footnote{As usual, we write $\,x_+ = \max\,(x, 0)\,$ for $\,x\in\mathbb{R}.\,$}\end{equation*}
\end{lemma}
\begin{proof} If $\nu$ denotes the probability measure
$$d\nu(t) = \frac{2}{B(\rho +1, \,\lambda-\rho)} \,(1-t^2)_+^{\lambda-\rho-1}t^{2\rho+1} dt,$$
then it has finite moments of all orders with
$$\int_0^\infty t^{2k} d\nu(t) = \frac{\Gamma(k+\rho+1)}{\Gamma(\rho+1)}\cdot
\frac{\Gamma(\lambda+1)}{\Gamma(k+\lambda+1)},\quad k=0,1,\cdots.$$
It follows from integrating termwise, readily justified, that
\begin{align*}
\int_0^\infty\Omega_\rho(rt) d\nu(t) &= \Gamma(\rho+1)
\mathbb{S}um_{k=0}^\infty\frac{(-1)^k}{k!\,\Gamma(k+\rho+1)}\left(\frac r2\right)^{2k}
\int_0^\infty t^{2k} d\nu(t)\\
&= \Gamma(\lambda+1)\mathbb{S}um_{k=0}^\infty\frac{(-1)^k}{k!\,\Gamma(k+\lambda +1)}\left(\frac r2\right)^{2k}\\
&=\Omega_\lambda(r).
\end{align*}
\end{proof}
The Hankel-Schoenberg transforms may be regarded as a generalization of the radial Fourier transforms
or Schoenberg's integrals due to the following order-changing interrelations.
\begin{theorem}\label{orderwalk}
Let $\,\lambda>\frac{n-2}{2}\,$ with $n$ a positive integer.
For any finite positive Borel measure $\nu$ on $[0, \infty)$ which is not concentrated at zero,
its Hankel-Schoenberg transform of order $\lambda$ can be represented as
\begin{align*}
& \int_0^\infty \,\Omega_{\lambda}(rt) d\nu(t) =
\int_0^\infty \,\Omega_{\frac{n-2}{2}}(rt)\, W_\lambda(\nu)(t) t^{n-1} dt,\quad\text{where}\\
& W_\lambda(\nu)(t) = \frac{2}{B\left(\frac n2, \,\lambda +1 -\frac n2\right)}
\int_0^\infty\left( 1- \frac{t^2}{s^2}\right)^{\lambda - \frac n2}_+ s^{-n} d\nu(s).
\end{align*}
\mathbb{S}mallskip
\noindent
Moreover, $\,d\mu(t) = W_\lambda(\nu)(t) t^{n-1} dt\,$ defines
a finite positive Borel measure on $[0, \infty)$ with the total mass $\mu\left([0, \infty)\right) =
\nu\left([0, \infty)\right).$
\end{theorem}
\begin{proof}
As a special case of Lemma \ref{basic}, the choice $\,\rho=\frac{n-2}{2}\,$ gives
\begin{equation}\label{O1}
\Omega_\lambda(r) = \frac{2}{B\left(\frac n2, \,\lambda+1-\frac n2\right)} \int_0^\infty\Omega_{\frac{n-2}{2}}(rs)
(1-s^2)_+^{\lambda-\frac n2}s^{n-1} ds,
\end{equation}
whence the result follows by interchanging the order of integrations.
Since
\begin{align*}
\int_0^\infty\left( 1- \frac{t^2}{s^2}\right)^{\lambda - \frac n2}_+ t^{n-1} dt
= \frac{s^n}{2}\int_0^1 (1-u)^{\lambda-\frac n2} u^{\frac n2 -1} du
\end{align*}
for each $\,s>0,$ it is straightforward to find
\begin{align*}
\mu([0, \infty)) &= \int_0^\infty W_\lambda(\nu)(t) t^{n-1} dt\\
&= \frac{2}{B\left(\frac n2, \,\lambda +1 -\frac n2\right)}
\int_0^\infty\int_0^\infty\left( 1- \frac{t^2}{s^2}\right)^{\lambda - \frac n2}_+ s^{-n} d\nu(s) t^{n-1} dt\\
&= \frac{2}{B\left(\frac n2, \,\lambda +1 -\frac n2\right)}
\int_0^\infty \int_0^\infty\left( 1- \frac{t^2}{s^2}\right)^{\lambda - \frac n2}_+ t^{n-1} dt s^{-n} d\nu(s)\\
&=\nu([0, \infty)).
\end{align*}
\end{proof}
\begin{remark}
A positive Borel measure $\nu$ on $[0, \infty)$ is concentrated at zero
if it is a constant multiple of Dirac mass at zero, that is,
$\,\nu = c\,\delta_0\,$ with $\,c>0.$ For such a Borel measure $\nu$,
its Hankel-Schoenberg transform is simply
$$ \int_0^\infty\Omega_{\lambda}(rt) d\nu(t) = c\,\Omega_\lambda(0) = c.$$
\end{remark}
\mathbb{S}ection{Schoenberg representations}
Our aim in this section is to set up Schoenberg's representations for
Mat\'ern functions which ensure their positive definiteness.
\begin{lemma}\label{lemmaS0}
For $\,\alpha\in\mathbb{R}\,$ and $\,z>0,\,$ we have
\begin{equation}\label{S1}
K_\alpha(z) z^\alpha = 2^{-\alpha -1}\int_0^\infty
\exp\left(-z^2 t - \frac {1}{4t}\right) t^{-\alpha -1} dt.
\end{equation}
\end{lemma}
\begin{proof} For any real $\alpha$ and $\,z>0,$ if we make substitution
$\,z e^{-t} = 2s\,$ in the second form of Schl\"afli's integral \eqref{K4}, then
\begin{align*}
K_\alpha(z) &= \frac 12\int_{-\infty}^\infty \exp\left(-z\cosh t-\alpha t\right) dt\\
&= 2^{\alpha-1}z^{-\alpha}\int_0^\infty \exp\left(-s - \frac{z^2}{4s}\right) s^{\alpha -1} ds
\end{align*}
from which \eqref{S1} follows on making another substitution $\,s= 1/4t.$
\end{proof}
In the case $\,\alpha>0,\,$ it follows from the asymptotic behavior of $K_\alpha$
near zero, as stated in (K4), that the Mat\'ern function $M_\alpha$ is well defined
as a continuous function on $[0, \infty)$ with the limiting value $\,M_\alpha(0) = 2^{\alpha -1}\,\Gamma(\alpha).$
For this reason, it will be convenient to consider the following types of Mat\'ern functions
which are frequently used in many fields (see e.g. \cite{G1}).
\begin{definition}
For $\,\alpha>0,\,$ put
\begin{equation}\label{M}
\mathcal{M}_\alpha(z) = \frac{2^{1-\alpha}}{\Gamma(\alpha)}\,K_\alpha(z) z^\alpha\qquad(z\ge 0).
\end{equation}
\end{definition}
We recall that a function $\phi$ is said to be {\it completely
continuous on $[0, \infty)$} if it is continuous on $[0, \infty)$ and satisfies
the condition $\,(-1)^m\phi^{(m)}(z)\ge 0\,$ for all nonnegative integers $m$ and $\,z>0\,$ (see e.g. \cite{We}).
\begin{theorem}\label{corollaryS1}{\rm (Theorem \ref{theorem1.1})}
For $\,\alpha>0,\,$ we have
\begin{equation*}
\mathcal{M}_\alpha(z) = \int_0^\infty e^{-z^2 t} f_\alpha(t) dt\quad\,\,(z\ge0),
\end{equation*}
where $f_\alpha$ is the continuous probability density on $[0, \infty)$ defined by
\begin{equation*}
f_\alpha(t) = \left\{\begin{aligned} &{\frac{1}{4^\alpha\Gamma(\alpha)}\,
\exp\left(-\frac{1}{4t}\right) t^{-\alpha-1}} &{\text{for}\quad t>0},\\
&{\qquad\qquad\,\, 0} &{\text{for}\quad t=0}.\end{aligned}\right.
\end{equation*}
As a consequence, $\mathcal{M}_\alpha$ is continuous and positive definite on every $\mathbb{R}^n$.
In addition, the function $\mathcal{M}_\alpha\left(\mathbb{S}qrt{z}\,\right)$ is also positive definite on every $\mathbb{R}^n$ and
completely continuous on $[0, \infty)$.
\end{theorem}
\begin{proof}
As it is elementary to verify that $f_\alpha$ is a continuous probability density on $[0, \infty)$,
the statements on $\mathcal{M}_\alpha(z)$ are immediate consequences of Lemma \ref{lemmaS0} and Schoenberg's
Criterion I on the positive definiteness.
In the special case $\,\alpha=1/2,\,$ we have
\begin{equation}\label{S2}
e^{-z} = \int_0^\infty e^{-z^2 t} f_{1/2}(t) dt\quad\,\,(z\ge0),
\end{equation}
whence it is straightforward to deduce the integral representations
\begin{align}
\mathcal{M}_\alpha\left(\mathbb{S}qrt z\,\right) &= \int_0^\infty e^{-z t} f_\alpha(t) dt\label{S3}\\
&=\int_0^\infty e^{-z^2 u} g_\alpha(u) du\label{S4},
\end{align}
where $g_\alpha$ stands for the function defined by $\,g_\alpha(0) =0\,$ and
\begin{equation*}
g_\alpha(u) = \frac{u^{-3/2}}{2^{2\alpha+1}\mathbb{S}qrt\pi\,\Gamma(\alpha)}\int_0^\infty
\exp\left(-\frac{1}{4t} -\frac{t^2}{4u}\right) t^{-\alpha} dt
\end{equation*}
for $\,u>0.\,$ As readily verified, $g_\alpha$ is
a continuous probability density on $[0, \infty)$ and hence it follows from \eqref{S4}
and Schoenberg's Criterion I that the function
$\mathcal{M}_\alpha\left(\mathbb{S}qrt{z}\,\right)$ is positive definite on every $\mathbb{R}^n$.
That it is completely continuous on $[0, \infty)$ is a consequence of \eqref{S3}.\footnote{It may be proved either by differentiating
under the integral sign or by applying the well-known theorem of Bernstein-Hausdorff-Widder which states that
a function $f$ is completely
continuous on $[0, \infty)$ if and only if
$$ f(r) = \int_0^\infty e^{-rt} d\mu(t)\qquad(r\ge 0)$$
for some finite positive Borel measure $\mu$ on $[0, \infty)$ (see e.g. \cite{We}).}
\end{proof}
We are now concerned with the second form of Schoenberg's integrals.
For the sake of computational facilitation as well as inversion, it is advantageous to consider
the Hankel-Schoenberg transforms.
As it is conventional, we shall use the notation of Pochhammer and Barnes for the generalized hypergeometric functions
\begin{equation*}
{}_pF_q\left(a_1, \cdots, a_p;\,b_1, \cdots, b_q;\,x\right)
=\mathbb{S}um_{k=0}^\infty\frac{\left(a_1\right)_k\cdots\left(a_p\right)_k}{k!
\left(b_1\right)_k\cdots\left(b_q\right)_k}\,x^k
\end{equation*}
in which the symbol $(a)_k$ for a non-zero real number $a$ stands for
\begin{equation*}
(a)_k = \left\{\begin{aligned} &{a(a+1)\cdots (a+k-1)} &{\text{for}
\quad k\ge 1}, \\
&{\qquad 1} &{\text{for} \quad k = 1}.\end{aligned}\right.
\end{equation*}
The following is easily obtainable from Scl\"afli's integrals.
As it is known, however, we shall omit the proof (see \cite{AS}, \cite{E}, \cite{Wa}).
\begin{lemma}\label{lemmaS1}
For $\,\alpha\in\mathbb{R}\,$ and $\,\beta>|\alpha|,\,$ we have
\begin{equation}\label{S5}
\int_0^\infty K_{\alpha}(t)t^{\beta-1}dt = 2^{\beta-2}
\Gamma\left(\frac{\beta+\alpha}{2}\right)\Gamma\left(\frac{\beta-\alpha}{2}\right).
\end{equation}
\end{lemma}
\begin{lemma}\label{lemmaS2}
Let $\,\alpha\in\mathbb{R}\,$ and $\,\beta>|\alpha|.\,$ For the probability measure
$$d\nu(t) =\frac{1}{2^{\beta-2}
\Gamma\left(\frac{\beta+\alpha}{2}\right)\Gamma\left(\frac{\beta-\alpha}{2}\right)}\,
K_{\alpha}(t)t^{\beta-1}dt,$$
the Hankel-Schoenberg transform of order $\lambda>-1$ is given by
\begin{equation}\label{S6}
\int_{0}^{\infty}\Omega_{\lambda}(rt)d\nu(t)
={}_2F_{1}\left(\frac{\beta-\alpha}{2},\,\frac{\beta+\alpha}{2};\,\lambda+1;\,-r^2\right)\,.
\end{equation}
\end{lemma}
\begin{proof}
A simple modification of \eqref{S5} yields
\begin{align*}
\int_0^\infty t^{2k}d\nu(t)=2^{2k} \left(\frac{\beta+\alpha}{2}\right)_{k}
\left(\frac{\beta-\alpha}{2}\right)_{k},\quad k=0,1,2,\cdots.
\end{align*}
Integrating termwise, we deduce
\begin{align*}
\int_{0}^{\infty}\Omega_{\lambda}(rt)d\nu(t)&=
\mathbb{S}um_{k=0}^\infty \frac{\left(-1\right)^k}{k!\left(\lambda+1\right)_k} \left(\frac{r}{2}\right)^{2k}
\int_0^\infty t^{2k} dt\\
&=\mathbb{S}um_{k=0}^\infty
\frac{\left(\frac{\beta+\alpha}{2}\right)_k\left(\frac{\beta-\alpha}{2}\right)_k}{k!\left(\lambda+1\right)_k}
\left(-r^2\right)^k,
\end{align*}
which is equivalent to the stated formula \eqref{S6}.
\end{proof}
By obvious cancellation effects, the generalized hypergeometric function \eqref{S6}
reduces to the binomial series expansion in the case $\,\beta=\alpha + 2(\lambda+1)\,$
or $\,\beta=-\alpha + 2(\lambda+1).\,$ To be precise, we have the following
general results which include Schoenberg's representations for Mat\'ern functions.
\begin{theorem}\label{theoremS1} Let $\,\lambda>-1\,$ and $\,\alpha+\lambda+1>0.$ For each $\,r\ge 0,$ we have
\begin{align}\label{S7}
(1+r^{2})^{-\alpha-\lambda-1} &=\frac{1}{2^{\alpha+2\lambda}\Gamma(\lambda+1)\Gamma(\alpha+\lambda+1)}
\nonumber\\
&\qquad\times\quad\int_{0}^{\infty}\Omega_{\lambda}(rt) \big[K_{\alpha}(t) t^{\alpha}\big] t^{2\lambda+1} dt.
\end{align}
Moreover, if $\, 2\alpha +\lambda +3/2>0\,$ in addition, then for each $\,z>0,$
\begin{align}\label{S8}
K_\alpha(z) z^\alpha =\frac{2^\alpha\Gamma(\alpha+\lambda+1)}{\Gamma(\lambda+1)}
\int_0^\infty \Omega_\lambda(zt) (1+ t^2)^{-\alpha-\lambda-1} t^{2\lambda+1} dt.
\end{align}
\end{theorem}
\begin{proof}
Formula \eqref{S7} follows from the special case $\,\beta=\alpha+2\lambda+2\,$ of \eqref{S6},
Lemma \ref{lemmaS2}, and Newton's binomial theorem
\begin{align*}
\mathbb{S}um_{k=0}^{\infty}\frac{(\alpha+\lambda+1)_{k}}{k!}\,(-r^2)^{k} =(1+r^2)^{-\alpha-\lambda-1}.
\end{align*}
As the function $\,f(t) = K_{\alpha}(t) t^{\alpha +2\lambda+1} \,$ is continuous on $(0, \infty)$ and
\begin{align*}
\int_0^\infty |f(t)| t^{-\lambda-1/2} dt &= \int_0^\infty K_{\alpha}(t) t^{\alpha +\lambda+1/2} dt\\
&= 2^{\alpha + \lambda -1/2}
\Gamma\left(\alpha + \frac{2\lambda +3}{4}\right)\Gamma\left(\frac{2\lambda+3}{4}\right)<\infty
\end{align*}
by Lemma \ref{lemmaS1}, applicable due to the condition $\,2\alpha +\lambda +3/2>0,\,$
\eqref{S8} follows from inverting \eqref{S7} in accordance with Theorem \ref{inversion}.
\end{proof}
Choosing $\,\alpha, \lambda\,$ suitably or regarding them as variable parameters, one may exploit these formulas
from several perspectives. If we are concerned with the Fourier transforms in a fixed
Euclidean space $\mathbb{R}^n$, for example, the first formula may be applied to yield the following.
\begin{itemize}
\item[(a)] For $\,\alpha>0,\,$ if we recall \eqref{G1}
\begin{equation*}
G_\alpha(z) = \frac{1}{2^{\alpha-1 + \frac n2}\,\pi^{\frac n2}\,
\Gamma(\alpha)}\,K_{\alpha-\frac n2}(z) z^{\alpha - \frac n2},
\end{equation*}
the special case $\,\lambda = (n-2)/2\,$ of \eqref{S7} yields
\begin{equation*}
(1+r^2)^{-\alpha} = \frac{2\pi^{n/2}}{\Gamma\left(n/2\right)} \int_0^\infty\Omega_{\frac{n-2}{2}} (rt) G_\alpha(t) t^{n-1} dt
\end{equation*}
so that we obtain the Fourier transform formula
\begin{equation}\label{S9}
\widehat{G_\alpha}(\xi) = (1 +|\xi|^2)^{-\alpha}.
\end{equation}
\item[(b)] As $\alpha$ varies over $\,\alpha> 0,\,$ \eqref{S9} expresses the inverse multi-quadrics
of any positive order in terms of the Fourier transforms of $G_\alpha(\mathbf{x})$. On the contrary, Hankel-Schoenberg
transform formula \eqref{S7} enables us to obtain such Fourier representations by varying $\lambda$ with a fixed $\alpha$.
To be specific, let us fix $\,\alpha>-n/2\,$ and set
\begin{align}
F_{\alpha, \lambda}(z) &= \frac{1}{2^{\alpha + 2\lambda} \pi^{\frac n2}\Gamma\left(\lambda +1 -\frac n2\right)\Gamma(\alpha + \lambda +1)}
\nonumber\\
&\qquad\qquad\times\quad \int_z^\infty (s^2 - z^2)^{\lambda -\frac n2} \big[K_\alpha(s) s^\alpha\big] s ds
\end{align}
for $\,\lambda>(n-2)/2.$ By Theorem \ref{orderwalk}, we may put \eqref{S7} in the form
\begin{equation*}
(1+r^2)^{-\alpha - \lambda-1} = \frac{2\pi^{n/2}}{\Gamma\left(n/2\right)} \int_0^\infty\Omega_{\frac{n-2}{2}} (rt) F_{\alpha, \lambda}(t) t^{n-1} dt.
\end{equation*}
If we write $\, F_{\alpha, \lambda}(\mathbf{x}) = F_{\alpha, \lambda}(|\mathbf{x}|),\,\mathbf{x}\in\mathbb{R}^n,\,$ then
\begin{equation}\label{S10}
\widehat{ F_{\alpha, \lambda}}(\xi) = (1 +|\xi|^2)^{-\alpha -\lambda-1}.
\end{equation}
As $\lambda$ varies in the range $\,\lambda>(n-2)/2,$ this Fourier transform formula represents
the inverse multi-quadrics of order greater than $\,\alpha + n/2.$
\end{itemize}
A noteworthy feature of Mat\'ern functions is the following invariance
which follows immediately from \eqref{S8} by reformulation.
\begin{corollary}\label{corollaryS1}
If $\,\alpha>0,\,$ then for any $\,\lambda>-1,$
\begin{align}\label{S11}
\mathcal{M}_\alpha(z) = \int_0^{\infty}\Omega_\lambda (zt)\, d\nu_{\alpha, \lambda}(t)
\qquad(z\ge 0),
\end{align}
where $\nu_{\alpha, \lambda}$ denotes the probability measure on $[0, \infty)$ defined by
\begin{align*}
d\nu_{\alpha, \lambda}(t)= \frac{2}{B(\alpha, \,\lambda+1)}(1+t^2)^{-\alpha-\lambda-1}
t^{2\lambda+1} dt.
\end{align*}
\end{corollary}
\begin{remark}
In view of Schoenberg's Criterion II, this integral formula with $\,\lambda = (n-2)/2\,$
provides another proof of the positive definiteness of the Mat\'ern functions.
In particular, the choice of $\,n=1\,$ gives
\begin{align*}
\mathcal{M}_\alpha(z) = \frac{2}{B\left(\alpha,\,1/2\right)}
\int_{0}^{\infty}\frac{\cos (zt)\,dt}{\,\left(1 + t^{2}\right)^{\alpha+ 1/2}\,},
\end{align*}
the formula obtained by Basset, Malmst\'en and Poisson (see \cite{Wa}).
\end{remark}
\mathbb{S}ection{Schoenberg matrices on $\ell^2(\mathbb{N})$}
In this section we shall investigate whether Schoenberg matrices of Mat\'ern
functions or inverse multi-quadrics, in a fixed Euclidean space $\mathbb{R}^n$,
give rise to bounded invertible operators on the Hilbert space
$\ell^{2}(\mathbb{N})$.
As it is common in the theory of scattered data approximations, we shall deal with arbitrary
sets of type $\,X = \left\{ \mathbf{x}_{j}\in \mathbb{R}^n : j\in\mathbb{N}\right\}\,$ satisfying
\begin{equation}\label{M1}
\delta(X) = \inf_{j\neq k} \left|\mathbf{x}_{j}-\mathbf{x}_{k}\right|>0,\quad \dim\left[ {\rm span}(X)\right] = d
\end{equation}
for some $\,1\le d\le n.$
Our analysis will be based on the following.
\begin{proposition}\label{propM} {\rm (\cite{GMO})}
Let $f$ be a nonnegative function defined on $[0, \infty)$.
\begin{itemize}
\item[\rm(i)] Suppose $f$ is monotone decreasing, $\,f(0) =1\,$ and
$\,f(t) t^{d-1}\,$ is integrable on $[0, \infty).$
Then the Schoenberg matrix $\,\mathbf{S}_{X}(f)$ defines a bounded self-adjoint operator on $\ell^{2}(\mathbb{N})$ with
\begin{equation*}
\left\|\mathbf{S}_{X}(f)\right\|\le 1+\frac{ d(5^d-1)}{[\delta(X)]^d}\int_{0}^{\infty}f(t)t^{d-1}dt.
\end{equation*}
Moreover, if $X$ satisfies the additional separation assumption
\begin{equation*}
\delta(X) > \left[d(5^d-1)\int_{0}^{\infty}f(t)t^{d-1}dt\right]^{1/d},
\end{equation*}
then $\,\mathbf{S}_{X}(f)$ defines a bounded invertible operator on $\ell^{2}(\mathbb{N})$.
\item[\rm(ii)] Suppose $\,n\ge 2\,$ and $f$ admits an integral representation
$$f(r) = \int_0^\infty e^{-r^2 t}\, d\nu(t)\quad(r\ge 0)$$
for a finite positive Borel measure $\nu$ such that it is equivalent to Lebesgue measure on $[0, \infty)$
and satisfies the moment condition
$$\int_0^\infty t^{-d/2}\, d\nu(t)<\infty.$$
Then $\,\mathbf{S}_{X}(f)$ defines a bounded invertible operator on $\ell^{2}(\mathbb{N})$.
\end{itemize}
\end{proposition}
\begin{remark}
A positive Borel measure $\nu$ on $[0, \infty)$ is equivalent to Lebesgue measure $|\cdot|$ if both are absolutely
continuous with respect to each other. By the Radon-Nikodym theorem, a necessary and sufficient condition
for $\nu$ to be equivalent to Lebesgue measure is that $\,d\nu(t) = p(t) dt\,$ for a nonnegative density $p$
such that $\,{\rm supp} (p) = [0, \infty)\,$ and
$$\int_I p(t) dt =0 \,\Longleftrightarrow\, |I|=0$$
for any Borel set $\,I\mathbb{S}ubset [0, \infty).$
\end{remark}
As the operator norm bound and the invertibility condition of part (i) are slightly different from
the original ones presented in \cite{GMO}, we shall give a review of their proof for part (i) in the appendix.
Now that Schoenberg's representations are available for Mat\'ern functions of type \eqref{M},
it is a simple matter to prove the following.
\begin{theorem}\label{theoremM1}
Let $X$ be an arbitrary set of points of $\mathbb{R}^n$ satisfying \eqref{M1}. For $\,\alpha>0,\,$
consider the Schoenberg matrix of $\mathcal{M}_\alpha$,
$$\mathbf{S}_X\left(\mathcal{M}_\alpha\right) = \Big[ \mathcal{M}_\alpha\left(\mathbf{x}_j - \mathbf{x}_k\right)\Big]_{j, \,k\in\mathbb{N}}.$$
\begin{itemize}
\item[\rm(i)] $\,\mathbf{S}_X\left(\mathcal{M}_\alpha\right)$ defines a bounded self-adjoint operator on $\ell^{2}(\mathbb{N})$ with
$$\left\| \mathbf{S}_X\left(\mathcal{M}_\alpha\right)\right\| \le 1 + \frac{d\, 2^{d-1} (5^d -1) \Gamma\left(\alpha + \frac d2\right)\Gamma\left(\frac d2\right)}
{\left[\delta(X)\right]^d\,\Gamma(\alpha)}\,.$$
\item[\rm(ii)] For $\,n\ge 2,\,$ $\,\mathbf{S}_X\left(\mathcal{M}_\alpha\right)$ defines a bounded invertible operator on $\ell^{2}(\mathbb{N})$.
In the case $\,n=d=1,\,$ if $X$ satisfies the additional assumption
$$\delta(X) > \frac{4\,\Gamma\left(\alpha + \frac 12\right)\Gamma\left(\frac 12\right)}
{\Gamma(\alpha)}\,,$$
then it defines a bounded invertible operator on $\ell^{2}(\mathbb{N})$.
\end{itemize}
\end{theorem}
\begin{proof}
An application of Lemma \ref{lemmaS1} gives
\begin{align*}
\int_0^\infty \mathcal{M}_\alpha(t) t^{d-1} dt &= \frac{2^{1-\alpha}}{\Gamma(\alpha)}\int_0^\infty K_\alpha(t) t^{\alpha + d-1} dt\\
&= \frac{2^{d-1}\Gamma\left(\alpha + \frac d2\right)\Gamma\left(\frac d2\right)}
{\Gamma(\alpha)}\,.
\end{align*}
Since $\,\mathcal{M}_\alpha(0) = 1\,$ and $\mathcal{M}_\alpha$ is strictly decreasing on the interval $[0, \infty)$ as it is noted in (K3),
the criterion in the first part of Proposition \ref{propM} is applicable and part (i) follows with the stated operator norm bound.
Concerning part (ii), we invoke Corollary \ref{corollaryS1} to represent
$$\mathcal{M}_\alpha(z) = \int_0^\infty e^{-z^2 t} f_\alpha(t) dt \qquad(z\ge 0)$$
in which $f_\alpha$ stands for the probability density
\begin{equation*}
f_\alpha(t) = \left\{\begin{aligned} &{\frac{1}{4^\alpha\Gamma(\alpha)}\,
\exp\left(-\frac{1}{4t}\right) t^{-\alpha-1}} &{\text{for}\quad t>0},\\
&{\qquad\qquad\,\, 0} &{\text{for}\quad t=0}.\end{aligned}\right.
\end{equation*}
Since the measure determined by $\,f_\alpha(t) dt\,$ is obviously equivalent to Lebesgue measure
on $[0, \infty)$ and it is elementary to compute
$$\int_0^\infty t^{-d/2} f_\alpha(t) dt = \frac{2^d\Gamma\left(\alpha + \frac d2\right)}{\Gamma(\alpha)} <\infty,$$
the criterion in the second part of Proposition \ref{propM} implies the invertibility of $\,\mathbf{S}_X\left(\mathcal{M}_\alpha\right)$
in the case $\,n\ge 2.$ The last statement on the invertibility when $\,n=d=1\,$ follows by the first
criterion of Proposition \ref{propM}.
\end{proof}
\begin{theorem}\label{theoremM2}
For $\,\beta> n/2,\,$ put
\begin{equation}
\phi_\beta(r) = (1+ r^2)^{-\beta} \qquad(r\ge 0).
\end{equation}
Let $X$ be an arbitrary set of points of $\mathbb{R}^n$ satisfying \eqref{M1} and
$$\mathbf{S}_X\left(\phi_\beta\right) = \Big[ \phi_\beta\left(\mathbf{x}_j - \mathbf{x}_k\right)\Big]_{j, \,k\in\mathbb{N}}.$$
\begin{itemize}
\item[\rm(i)] $\,\mathbf{S}_X\left(\phi_\beta\right)$ defines a bounded self-adjoint operator on $\ell^{2}(\mathbb{N})$ with
$$\left\| \mathbf{S}_X\left(\phi_\beta\right)\right\| \le 1 + \frac{d(5^d -1) B\left(\beta-\frac d2,\,\frac d2\right)}
{2 \left[\delta(X)\right]^d}\,.$$
\item[\rm(ii)] For $\,n\ge 2,\,$ $\,\mathbf{S}_X\left(\phi_\beta\right)$ defines a bounded invertible operator on $\ell^{2}(\mathbb{N})$.
In the case $\,n=d=1,\,$ if $X$ satisfies the additional assumption
$$\delta(X) > 2 B\left(\beta- \frac 12\,,\, \frac 12\right),$$
then it defines a bounded invertible operator on $\ell^{2}(\mathbb{N})$.
\end{itemize}
\end{theorem}
\begin{proof}
By using the aforementioned integral representation
$$\phi_\beta(r) = \frac{1}{\Gamma(\beta)}\int_0^\infty e^{-r^2 t} e^{-t} t^{\beta-1} dt\qquad(r\ge 0),$$
the proof follows along the same scheme as above.
\end{proof}
\begin{remark} In connection with the problem of interpolating functions at an arbitrary
set of distinct points $X$, it is an immediate consequence of Theorems \ref{theoremM1}, \ref{theoremM2} that
$\,\mathcal{M}_\alpha,\, \phi_\beta,\,$ with $\,\alpha>0,\,\beta>n/2,\,$ could be used
in constructing Lagrange-type radial basis sequences $\,\left\{u_j^*\right\}_{j\in\mathbb{N}},\,$ by the same process
pointed out in the introduction, and the interpolating functional
$$A_X(f)(\mathbf{x}) = \mathbb{S}um_{j=1}^\infty f(\mathbf{x}_j)\,u_j^*(\mathbf{x}).$$
\end{remark}
\mathbb{S}ection{Gramian matrices and Riesz sequences}
Now that Schoenberg matrices of Mat\'ern functions are shown to induce bounded and
invertible operators on $\ell^2(\mathbb{N})$, it is natural to ask if they generate Riesz sequences
or bases in appropriate Hilbert spaces.
We recall that a system $\,\{f_j\}_{j\in\mathbb{N}}\,$ of vectors
in a Hilbert space $H$ is said to be a Riesz sequence if its moment space
is equal to $\ell^2(\mathbb{N})$, that is,
$$\left\{ \mathbf{m}_f = \big\{(f, f_j)_H\big\}_{j\in\mathbb{N}} : f\in H\right\} = \ell^2(\mathbb{N}).$$
If $\,\{f_j\}_{j\in\mathbb{N}}\,$ is complete in addition, it is called a Riesz basis (see \cite{Y}).
A classical theorem of Bari states a necessary and sufficient condition for
the system $\,\{f_j\}_{j\in\mathbb{N}}\,$ to be a Riesz sequence is that
the Gramian matrix
\begin{equation}
{\rm Gram}\Big(\{f_j\}_{j\in\mathbb{N}}\,;\,H\Big) = \big[\left( f_j,\,f_k\right)_H\big]_{j, \,k\in N}
\end{equation}
defines a bounded and invertible operators on $\ell^2(\mathbb{N})$.
As for the sequences constructed from translating Mat\'ern functions by
distinct points, their Gramian matrices in $L^2(\mathbb{R}^n)$ or
Sobolev spaces turn out to be easily identifiable in terms of
Schoenberg matrices.
In order not to entangle with parameters, it is convenient to work with the Bessel
potential kernels of \eqref{G1}
\begin{equation*}
G_\alpha(\mathbf{x}) = \frac{1}{2^{\alpha-1 + \frac n2}\,\pi^{\frac n2}\,
\Gamma(\alpha)}\,K_{\alpha-\frac n2}(|\mathbf{x}|) |\mathbf{x}|^{\alpha - \frac n2}.
\end{equation*}
\mathbb{S}ubsection{Results on $L^2(\mathbb{R}^n)$ space}
Concerning the square integrability, we have the following.
\begin{lemma}\label{lemmaGR1}
For $\,\lambda>-1,\,$ if $\,2\alpha + \lambda +1>0,\,$ then
\begin{equation*}
\int_0^\infty \big[K_\alpha(t) t^\alpha\big]^2 t^{2\lambda +1} dt = \frac{\mathbb{S}qrt{\pi}\,\,\Gamma(\alpha +\lambda +1)\Gamma(2\alpha +\lambda +1)
\Gamma(\lambda+1)}{4\,\Gamma\left(\alpha + \lambda + \frac 32\right)}\,.
\end{equation*}
In particular, if $\, \alpha + n/4>0\,$ with $n$ a positive integer, then
\begin{equation*}
\int_0^\infty \big[K_\alpha(t) t^\alpha\big]^2 t^{n-1} dt = \frac{\mathbb{S}qrt{\pi}\,\,\Gamma\left(\alpha +\frac n2\right)
\Gamma\left(2\alpha +\frac n2\right)\Gamma\left(\frac n2\right)}{4\,\Gamma\left(\alpha + \frac{n+1}{2}\right)}\,.
\end{equation*}
\end{lemma}
\begin{proof}
An application of Parseval's relation, Theorem \ref{Parseval}, for the Hankel-Schoenberg transforms
to formula \eqref{S7} of Theorem \ref{theoremS1} gives
\begin{align*}
\int_0^\infty \big[K_\alpha(t) t^\alpha\big]^2 t^{2\lambda +1} dt = \big[2^{\alpha+\lambda}\,\Gamma(\alpha +\lambda+1)\big]^2
\int_0^\infty \frac{r^{2\lambda+1}\,dr}{(1+ r^2)^{2\alpha+ 2\lambda+ 2}}.
\end{align*}
By making substitution $\, u = 1/(1+r^2),\,$ we compute
\begin{align*}
\int_0^\infty \frac{r^{2\lambda+1}\,dr}{(1+ r^2)^{2\alpha+ 2\lambda+ 2}}
&= \frac 12 \int_0^1 u^{2\alpha + \lambda} (1-u)^\lambda du\\
&= \frac 12\,B(2\alpha +\lambda+1,\,\lambda+1)
\end{align*}
and the stated formula follows on simplifying constants by using Legendre's duplication formula
for the Gamma function. The second stated formula corresponds to a special case of the first one with
$\,\lambda = n/2 -1.$
\end{proof}
\begin{theorem}\label{theoremGR1}
If $\,\alpha>n/4,\,$ then for any $\,\mathbf{x}, \,\mathbf{y}\in\mathbb{R}^n,\,$
\begin{align}\label{GR1}
\big( G_{\alpha}(\cdot-\mathbf{x}),\, G_\alpha(\cdot-\mathbf{y})\big)_{L^2(\mathbb{R}^n)}
= G_{2\alpha} (\mathbf{x}-\mathbf{y}).
\end{align}
As a consequence, for any sequence of distinct points $\,(\mathbf{x}_j)_{j\in\mathbb{N}}\mathbb{S}ubset \mathbb{R}^n,\,$
the Gramian matrix of the system $\,\big\{ G_\alpha(\mathbf{x}- \mathbf{x}_j) \big\}_{j\in\mathbb{N}}\mathbb{S}ubset
L^2(\mathbb{R}^n)\,$ coincides with the Schoenberg matrix of $G_{2\alpha}$, that is,
\begin{align*}
{\rm Gram}\Big(\big\{ G_\alpha(\mathbf{x}- \mathbf{x}_j)\big\}_{j\in\mathbb{N}}\,;\,L^2(\mathbb{R}^n)\Big)
= \Big[G_{2\alpha} \left(\mathbf{x}_j- \mathbf{x}_k\right)\Big]_{j, \,k\in\mathbb{N}}\,.
\end{align*}
\end{theorem}
\begin{proof}
By Lemma \ref{lemmaGR1}, $\,G_\alpha\in L^2(\mathbb{R}^n).\,$ Due to radial symmetry,
\begin{align*}
\big( G_\alpha(\cdot-\mathbf{x}),\, G_\alpha(\cdot-\mathbf{y})\big)_{L^2(\mathbb{R}^n)}
&=\int_{\mathbb{R}^n} G_\alpha(\mathbf{u}-\mathbf{x}) G_\alpha(\mathbf{u} -\mathbf{y}) d\mathbf{u}\\
&= \int_{\mathbb{R}^n} G_\alpha(\mathbf{x} - \mathbf{y} -\mathbf{w}) G_\alpha(\mathbf{w}) d\mathbf{w}\\
&= \left(G_\alpha\ast G_\alpha\right)(\mathbf{x}-\mathbf{y}).
\end{align*}
On the Fourier transform side, formula \eqref{S9} gives
\begin{align*}
\widehat{G_\alpha\ast G_\alpha}(\xi)
= (1+|\xi|^2)^{-2\alpha} = \widehat{G_{2\alpha}}(\xi),
\end{align*}
whence $\,G_\alpha \ast G_\alpha = G_{2\alpha}\,$ and the result follows.
\end{proof}
\begin{remark} This result extends the work of L. Golinskii {\it et al.} \cite{GMO} in which the
authors dealt only with the range $\,n/4<\alpha\le n/2.$
\begin{itemize}
\item[(a)] In the case when both $\,\alpha-n/2\,$ and $\,2\alpha - n/2\,$ are halves of odd integers,
it is possible to write $L^2$ inner products explicitly with the aid of (K5).
As illustrations in $\mathbb{R}^3$, we take $\,\alpha = 1, \,2\,$ to obtain
\begin{align*}
&\qquad\qquad\int_{\mathbb{R}^3} \frac{e^{-|\mathbf{u} -\mathbf{x}| - |\mathbf{u}-\mathbf{y}|}}
{|\mathbf{u}-\mathbf{x}|\,|\mathbf{u}-\mathbf{y}|}\,d\mathbf{u}
= 2\pi\,e^{-|\mathbf{x}-\mathbf{y}|}\,,\\
&\int_{\mathbb{R}^3} e^{-|\mathbf{u} -\mathbf{x}| - |\mathbf{u}-\mathbf{y}|}\,d\mathbf{u}
= \pi\,e^{-|\mathbf{x}-\mathbf{y}|}\left( 1+ |\mathbf{x}-\mathbf{y}| + \frac{|\mathbf{x}-\mathbf{y}|^2}{3}\right)
\end{align*}
for which the first formula is of considerable interest in the spectral
analysis for the Schr\"odinger equations (see \cite{MS}).
\item[(b)] To reformulate \eqref{GR1} in a more direct fashion, put
\begin{equation}\label{GR2}
F_\alpha(\mathbf{x}) = \frac{1}{2^{\alpha +n -1} \pi^{\frac n2}\Gamma\left(\alpha + \frac n2\right)}\,
K_\alpha(|\mathbf{x}|) |\mathbf{x}|^\alpha\,.
\end{equation}
As an alternative of \eqref{GR1}, if $\,\alpha>-n/4,\,$ then
\begin{align}
\big( F_\alpha(\cdot-\mathbf{x}),\, F_\alpha(\cdot-\mathbf{y})\big)_{L^2(\mathbb{R}^n)}
= F_{2\alpha + \frac n2} (\mathbf{x}-\mathbf{y}).
\end{align}
\end{itemize}
\end{remark}
As it is shown in Theorem \ref{theoremM1} that
the Schoenberg matrices of
$$G_{2\alpha}(z) = \frac{\Gamma(2\alpha-n/2)}{(4\pi)^{n/2}\,\Gamma(2\alpha)}\,
\mathcal{M}_{2\alpha -n/2}(z)\qquad(z>0)$$
define bounded and invertible
operators on $\ell^2(\mathbb{N})$ as long as $\,\alpha>n/4,$ we obtain
the following from Bari's theorem and Theorem \ref{theoremGR1}.
\begin{corollary} Let $\,\alpha>n/4\,$ and
$\,X =\left\{\mathbf{x}_j\in\mathbb{R}^n : j\in\mathbb{N}\right\}\,$ be arbitrary with
$$\delta(X) = \inf_{j\ne k}\,\left|\mathbf{x}_j - \mathbf{x}_k\right|\,>0.$$
\begin{itemize}
\item[\rm(i)] If $\,n\ge 2,\,$ then $\,\big\{ G_\alpha(\mathbf{x}- \mathbf{x}_j) \big\}_{j\in\mathbb{N}}\,$
forms a Riesz sequence in $L^2(\mathbb{R}^n).$
\item[\rm(ii)] In the case $\,n=1,\,$ if $X$ is separated with
$$\delta(X)> \frac{4\,\Gamma(2\alpha) \Gamma(1/2)}{\Gamma(2\alpha-1/2)},$$
then $\,\big\{ G_\alpha(x - x_j) \big\}_{j\in\mathbb{N}}\,$
forms a Riesz sequence in $L^2(\mathbb{R}).$
\end{itemize}
\end{corollary}
\mathbb{S}ubsection{Results on Sobolev spaces}
An important feature of the Sobolev space $H^\alpha(\mathbb{R}^n)$ with $\,\alpha>n/2\,$
is that it is a reproducing kernel Hilbert space with the kernel
$G_{\alpha}(\mathbf{x}-\mathbf{y})$ so that it may be viewed as the space of functions of type
$$f(\mathbf{x})= \mathbb{S}um_{j=1}^\infty a_j \,G_{\alpha}(\mathbf{x} - \mathbf{x}_j),$$
where $\,(a_j)\in\ell^2(\mathbb{N})\,$ and $\,(\mathbf{x}_j)\mathbb{S}ubset\mathbb{R}^n\,$ are arbitrary (see \cite{A}).
Thus it is reasonable to expect that the system $\,
\left\{G_{\alpha}(\mathbf{x} - \mathbf{x}_j)\right\}_{j\in\mathbb{N}}\mathbb{S}ubset H^\alpha(\mathbb{R}^n)\,$ may serve as a Riesz sequence
or a Riesz basis in its closed linear span once the translation points $(\mathbf{x}_j)$ were scattered all
over some planes of $\mathbb{R}^n$.
As a matter of fact, the reproducing property implies
\begin{equation}\label{GR3}
\big( G_{\alpha}(\cdot-\mathbf{x}),\, G_{\alpha}(\cdot-\mathbf{y})\big)_{H^\alpha(\mathbb{R}^n)}
= G_{\alpha} (\mathbf{x}-\mathbf{y})
\end{equation}
for all $\,\mathbf{x}, \,\mathbf{y}\in\mathbb{R}^n\,$ and our foregoing analysis yields
\begin{theorem}
Let $\,\alpha>n/2\,$ and $\,X =\left\{\mathbf{x}_j\in\mathbb{R}^n : j\in\mathbb{N}\right\}\,$ be arbitrary with
$$\delta(X) = \inf_{j\ne k}\,\left|\mathbf{x}_j - \mathbf{x}_k\right|>0.$$
\begin{itemize}
\item[\rm(i)] If $\,n\ge 2,\,$ then $\,\big\{ G_{\alpha}(\mathbf{x}- \mathbf{x}_j) \big\}_{j\in\mathbb{N}}\,$
forms a Riesz sequence in $H^\alpha(\mathbb{R}^n).$
\item[\rm(ii)] In the case $\,n=1,\,$ if $X$ is separated with
$$\delta(X)> \frac{4\,\Gamma(\alpha) \Gamma(1/2)}{\Gamma(\alpha- 1/2)},$$
then $\,\big\{ G_{\alpha}(x - x_j) \big\}_{j\in\mathbb{N}}\,$
forms a Riesz sequence in $H^\alpha(\mathbb{R}).$
\end{itemize}
\end{theorem}
Regarding the problem of determining if the sequences of translates by inverse multi-quadrics
give rise to Riesz sequences, we introduce a class of function spaces defined in terms of Fourier transforms as follows.
\begin{definition}
For $\,\alpha>0,$
\begin{align*}
\mathcal{K}_\alpha(\mathbb{R}^n) = \left\{ f\in C(\mathbb{R}^n)\cap L^2(\mathbb{R}^n) :
\int_{\mathbb{R}^n} \frac{\big|\widehat f(\mathbf{\xi})\big|^2 d\mathbf{\xi}}
{ K_{\alpha} (|\xi|) |\xi|^{\alpha}} <\infty\right\}.
\end{align*}
\end{definition}
A theorem of R. Schaback \cite{S} and H. Wendland (\cite{We}, Theorem 10.27) states
if $\,\Phi\in C(\mathbb{R}^n)\cap L^1(\mathbb{R}^n),\,$ real-valued and positive definite,
the Hilbert space of functions on $\mathbb{R}^n$ with the reproducing kernel $\Phi(\mathbf{x}-\mathbf{y})$ coincides with
\begin{align*}
\mathcal{H}(\mathbb{R}^n) = \left\{ f\in C(\mathbb{R}^n)\cap L^2(\mathbb{R}^n) : \int_{\mathbb{R}^d} \frac{\big|\widehat f(\mathbf{\xi})\big|^2 d\mathbf{\xi}}
{\widehat{\Phi}(\mathbf{\xi})} <\infty\right\}
\end{align*}
for which the inner product is defined by
\begin{align*}
\bigl(f,\,g\bigr)_{\mathcal{H}(\mathbb{R}^n)} = (2\pi)^{-n}\int_{\mathbb{R}^n} \frac{\widehat{f}(\mathbf{\xi})
\overline{\,\widehat{g}(\mathbf{\xi})} \,d\mathbf{\xi}}{\widehat{\Phi}(\mathbf{\xi})}.
\end{align*}
As a consequence, it is simple to find that the space $\mathcal{K}_\alpha(\mathbb{R}^n)$ arises as
a reproducing kernel Hilbert space with an appropriate multi-quadrics as its reproducing kernel.
To be precise, we have the following results.
\begin{theorem}
For $\,\beta>n/2,\,$ consider the inverse multi-quadrics
$$\phi_\beta(\mathbf{x}) = (1+|\mathbf{x}|^2)^{-\beta}.$$
\begin{itemize}
\item[\rm(i)] The Hilbert space of functions on $\mathbb{R}^n$ with the reproducing kernel $\phi_\beta$
coincides with $\,\mathcal{K}_{\beta-n/2}(\mathbb{R}^n)\,$ for which the inner product is defined by
\begin{align}\label{GR4}
\bigl(f,\,g\bigr)_{\mathcal{K}_{\beta-n/2}(\mathbb{R}^n)} = (2\pi)^{-2n}\int_{\mathbb{R}^n} \frac{\widehat{f}(\mathbf{\xi})
\overline{\,\widehat{g}(\mathbf{\xi})} \,d\mathbf{\xi}}{G_\beta(\mathbf{\xi})}.
\end{align}
\item[\rm(ii)] Let $\,X =\left\{\mathbf{x}_j\in\mathbb{R}^n : j\in\mathbb{N}\right\}\,$ be arbitrary with
$$\delta(X) = \inf_{j\ne k}\,\left|\mathbf{x}_j - \mathbf{x}_k\right|>0.$$
Then the system $\,\big\{ \phi_\beta(\mathbf{x}- \mathbf{x}_j) \big\}_{j\in\mathbb{N}}\,$
forms a Riesz sequence in $\mathcal{K}_{\beta-n/2}(\mathbb{R}^n)\,$ for any $\,n\ge 2\,$ and for $\,n=1\,$
under the additional assumption
$$\delta(X)> 2 B(\beta-1/2,\,1/2).$$
\end{itemize}
\end{theorem}
\begin{proof}
Obviously, $\phi_\beta$ is continuous, integrable and positive definite.
By an application of the Hankel-Schoenberg transform formula for $\phi_\beta$ as stated in
Corollary \ref{corollaryS1}, we have
$\,\,\widehat{\phi_\beta}(\xi) = (2\pi)^n G_\beta(\xi)\,$
and hence part (i) follows
by the aforementioned theorem of Schaback and Wendland.
By the reproducing property, the Gramian matrix is given by
\begin{align*}
{\rm Gram}\Big(\big\{\phi_\beta(\mathbf{x}- \mathbf{x}_j)\big\}_{j\in\mathbb{N}}\,;\,\mathcal{K}_{\beta-n/2}(\mathbb{R}^n)\Big)
= \Big[\phi_{\beta} \left(\mathbf{x}_j- \mathbf{x}_k\right)\Big]_{j, \,k\in\mathbb{N}}\,.
\end{align*}
and part (ii) follows immediately from Theorem \ref{theoremM2}.
\end{proof}
\begin{remark}
As the Mat\'ern functions of positive order are bounded smooth functions with exponential decays,
it is evident $\,\mathcal{K}_\alpha(\mathbb{R}^n)\mathbb{S}ubset H^\infty(\mathbb{R}^n)\,$ for any $\,\alpha>0.$
In the special case $\,\beta= (n+1)/2,\,$ we note
$$\mathcal{K}_{1/2}(\mathbb{R}^n) = \left\{f\in C(\mathbb{R}^n)\cap L^2(\mathbb{R}^n) :
\int_{\mathbb{R}^n} e^{\,|\xi|} \,\big|\widehat f(\mathbf{\xi})\big|^2\, d\mathbf{\xi} <\infty\right\},
$$
which is the reproducing kernel Hilbert space with the Poisson kernel
$$\phi_{\frac{n+1}{2}}(\mathbf{x}) = (1+|\mathbf{x}|^2)^{-\frac{n+1}{2}}.$$
\end{remark}
\mathbb{S}ection{Appendix: $\ell^2(\mathbb{N})$-Boundedness}
For the sake of completeness, we reproduce the proof of L. Golinskii {\it et al.} \cite{GMO}
for part (i) of Proposition \ref{propM} which states
\begin{itemize}{\it
\item[{}] Suppose that $f$ is a nonnegative monotone decreasing function on $[0, \infty)$ such that
$\,f(0) =1\,$ and the function $\,f(t) t^{d-1}\,$ is integrable on $[0, \infty).$
For any $\,X\mathbb{S}ubset \mathbb{R}^n\,$ satisfying the condition \eqref{M1}, the Schoenberg matrix
$\,\mathbf{S}_{X}(f)$ defines a bounded self-adjoint operator on $\ell^{2}(\mathbb{N})$ with
\begin{equation}\label{A}
\left\|\mathbf{S}_{X}(f)\right\|\le 1+\frac{ d(5^d-1)}{[\delta(X)]^d}\int_{0}^{\infty}f(t)t^{d-1}dt.
\end{equation}}
\end{itemize}
\paragraph{Proof.}
Let us write $\,\delta = \delta(X)\,$ and assume $\,{\rm span}(X)\mathbb{S}imeq\mathbb{R}^{d}\,$ for simplicity.
We fix $j$ and estimate the infinite sum
\begin{align*}
A_j &\equiv \mathbb{S}um_{k=1}^{\infty}f(|\mathbf{x}_k- \mathbf{x}_j|)=1+\mathbb{S}um_{m=1}^{\infty}\mathbb{S}um_{\mathbf{x}_{k}\in X_{m}}f(|\mathbf{x}_k- \mathbf{x}_j|)\,,\quad\text{where}\\
X_{m} &=\big\{\mathbf{x}_{k}\in X : m\delta \le |\mathbf{x}_k- \mathbf{x}_j|< (m+1)\delta\big\}\,.
\end{align*}
In terms of the open balls $\,B(\mathbf{x}_k,\delta/2)\mathbb{S}ubset \mathbb{R}^n\,,$ a geometric inspection reveals
\begin{align*}
\#(X_{m}) &\le \frac{\,\mathrm{vol}\Big(\Big\{\mathbf{y}\in \mathbb{R}^d : \left(m-\frac{1}{2}\right)\delta\le |\mathbf{y} - \mathbf{x}_j|<
\left(m+\frac{3}{2}\right)\delta\Big\}\Big)\,}
{\text{vol}\big(\,B(\mathbf{x}_k,\delta/2)\cap \mathbb{R}^d\,\big)}\\
&=(2m+3)^d-(2m-1)^d\\
&\le (5^d-1)\,m^{d-1},
\end{align*}
which implies
\begin{equation*}
A_j \le 1+\mathbb{S}um_{m=1}^{\infty}(5^d-1)m^{d-1}f(m\delta)\,.
\end{equation*}
As $f$ is monotone decreasing on $[0,\infty)$,
\begin{align*}
\int_{0}^{\infty}f(t\delta)t^{d-1}dt &=\mathbb{S}um_{m=1}^{\infty}\int_{m-1}^{m}f(t\delta)t^{d-1}dt\\
&\ge \mathbb{S}um_{m=1}^{\infty} f(m\delta)\left[\frac{m^{d}-(m-1)^d}{d}\right]\\
&\ge\mathbb{S}um_{m=1}^{\infty} f(m\delta)\frac{m^{d-1}}{d},
\end{align*}
which yields
$$ \mathbb{S}um_{m=1}^{\infty}f(m\delta)m^{d-1}\le \frac{d}{\delta^{d}}\int_{0}^{\infty}f(t)t^{d-1}dt.$$
Inserting this estimate into the above sum, we are led to
\begin{align*}
A_j \le 1+\frac{d(5^d-1)}{\delta^d}\int_{0}^{\infty}f(t)t^{d-1}dt\,.
\end{align*}
Since this estimate is independent of $j$, the result follows by Schur's test.
\begin{remark}
By Schur's test, (\ref{A}) implies
\begin{equation*}
\left\|I-S_X(f)\right\|\le \frac{d(5^d-1)}{[\delta(X)]^d}\int_{0}^{\infty}f(t)t^{d-1}dt
\end{equation*}
and the right side is strictly less than $1$ if
\begin{equation*}
\delta(X)>\left[d(5^d-1)\int_{0}^{\infty}f(t)t^{d-1}dt\right]^{1/d}\,.
\end{equation*}
For such a set $X$, $S_X(f)$ defines a bounded invertible operator on $\ell^{2}(\mathbb{N})$.
\end{remark}
\noindent
{\bf Acknowledgements.} Yong-Kum Cho is supported by National Research Foundation of Korea Grant
funded by the Korean Government (\# 20150301). Hera Yun is supported by the Chung-Ang University
Research Scholarship Grants in 2014.
\noindent
Yong-Kum Cho
\noindent
Department of Mathematics, Chung-Ang University, 84 Heukseok-Ro, Dongjak-Gu, Seoul 156-756, Korea (e-mail: [email protected])
\noindent
Dohie Kim
\noindent
Department of Mathematics, Chung-Ang University, 84 Heukseok-Ro, Dongjak-Gu, Seoul 156-756, Korea (e-mail: [email protected])
\noindent
Kyungwon Park
\noindent
Department of Computer Engineering, Korea Polytechnic University,
237 Sangidaehak-Ro, Siheung-Si 429-793, Korea (e-mail: [email protected])
\noindent
Hera Yun
\noindent
Department of Mathematics, Chung-Ang University,
84 Heukseok-Ro, Dongjak-Gu, Seoul 156-756, Korea (e-mail: [email protected])
\end{document} |
\begin{document}
\title{Global Existence and Asymptotic
Behavior of Solutions to a Chemotaxis-Fluid System on General Bounded Domain}
\author{Jie Jiang\thanks{Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences,
Wuhan 430071, HuBei Province, P.R. China,
\textsl{[email protected]}.}, \ \ Hao Wu\thanks{School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied
Mathematics, Fudan University, Shanghai 200433,\ P.R. China,
\textsl{[email protected]}.}\ \ and \ Songmu Zheng\thanks{School of Mathematical Sciences, Fudan University, Shanghai 200433,\ P.R. China,
\textsl{[email protected]}.}}
\deltaate{\today}
\maketitle
\begin{abstract} In this paper, we investigate an initial-boundary value problem for a chemotaxis-fluid system in a general bounded regular domain $\Omega \subset \mathbb{R}^N$ ($N\in\{2,3\}$), not necessarily being convex. Thanks to the elementary lemma given by Mizoguchi \& Souplet \cite{MizoSoup}, we can derive a new type of entropy-energy estimate, which enables us to prove the following: (1) for $N=2$, there exists a unique global classical solution to the full chemotaxis-Navier-Stokes system, which converges to a constant steady state $(n_\infty, 0,0)$ as $t\to+\infty$, and (2) for $N=3$, the existence of a global weak solution to the simplified chemotaxis-Stokes system. Our results generalize the recent work due to Winkler \cite{WinklerCPDE,WinkARMA}, in which the domain $\Omega$ is essentially assumed to be convex.\\
\noindent {\bf Keywords}: Chemotaxis, Navier-Stokes equation, global existence, general bounded domain.\\
\textbf{AMS Subject Classification}: 35K55, 35Q92, 35Q35, 92C17.
\epsilonnd{abstract}
\section{Introduction}
In this paper, we study the following initial-boundary value problems for a chemotaxis(-Navier)-Stokes system \cite{Lorz2010,PNAS2005}:
\begin{equation}\begin{cases}\langlebel{chemo1}
n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(n\chi(c)\nabla c),\\
c_t+u\cdot\nabla c=\Delta c-nf(c),\\
u_t=\Delta u+\kappa(u\cdot\nabla)u+\nabla p+n\nabla\phi,\\
\nabla\cdot u=0.\epsilonnd{cases}
\epsilonnd{equation}
The chemotaxis-fluid system \epsilonqref{chemo1} was recently proposed in \cite{PNAS2005} to model the motion of swimming bacteria under the effects of diffusion, oxygen-taxis and transport through an incompressible fluid. Here, we assume that $\Omega\subset \mathbb{R}^N$ $(N\in\{2,3\})$ is a bounded domain with smooth boundary ${\partial\Omega}$. The scalar functions $n$ and $c$ denote the concentration of oxygen and bacteria, respectively. The vector $u$ stands for the velocity field of the fluid subject to an incompressible Navier-Stokes type equation with pressure $p$. The scalar function $\phi$ stands for the gravitational potential, while $\chi(c)$ and $f(c)$ are the chemotactic sensitivity and the per-capita oxygen consumption rate that may depend on $c$. $\kappa\in\mathbb{R}$ is a parameter such that if $\kappa\neq 0$, the fluid motion is governed by the Navier-Stokes equation, while $\kappa=0$, the equation for $u$ is simplified to be the Stokes equation. We complement system \epsilonqref{chemo1} with the following Neumann and no-slip boundary conditions
\begin{equation}\langlebel{boundcon}\frac{\partial n}{\partial\nu}=\frac{\partial c}{\partial \nu}=0\quad\text{and}\quad u=0\quad\text{for} \;\; x\in {\partial\Omega},\; t>0\epsilonnd{equation}
with $\nu$ being the unit outward normal to $\partial\Omega$, and the initial conditions
\begin{equation}\langlebel{ini} n(x,0)=n_0(x),\quad c(x,0)=c_0(x),\quad u(x,0)=u_0(x).\epsilonnd{equation}
The chemotaxis-fluid system has recently been extensively studied by many authors. Here, we only mention some contributions in the literature that are mostly related to our initial boundary value problem \epsilonqref{chemo1}--\epsilonqref{ini} in a bounded domain $\Omega\subset\mathbb{R}^N$. We refer the reader to \cite{LL2011,DLM2010,CKL2013,CKLCPDE2013,TZJMAA} for the studies on the Cauchy problem in the whole space $\mathbb{R}^N$, and to \cite{FLM2010,LL2011,TW2013} for the case that the linear diffusion term $\Delta n$ is replaced by a nonlinear porous medium type one $\Delta n^m$. When the domain $\Omega$ is bounded and regular, Lorz \cite{Lorz2010} proved the existence of a local weak solution by Schauder's fixed point theory when $N=3$. If the domain $\Omega$ is further assumed to be {\epsilonm convex}, then using a key observation made in \cite{TWDCDS2012} and some delicate entropy-energy estimates, Winkler \cite{WinklerCPDE} established the existence of a unique global classical solution with large initial data for $\kappa\in\mathbb{R}$ when $N=2$, and the existence of a global weak solution for with $\kappa=0$ when $N=3$. Later, in \cite{WinkARMA}, the same author further proved that the global classical solution obtained in \cite{WinklerCPDE} in 2D will converge to a constant state $(n_\infty, 0,0)$ as times goes to infinity.
The main purpose of this note is to extend the results by Winkler from the convex domain to the general smooth domain.
Throughout this paper, we denote by $L^q(\Omega)$, $W^{k,q}(\Omega)$, $1\leq q\leq\infty$, $k\in\mathbb{N}$ the usual Lebesgue and Sobolev spaces respectively, and as usual, $H^k(\Omega)=W^{k,2}(\Omega)$. $\|\cdot\|_{B}$ denotes the norm in the space $B$; we also use the abbreviation $\|\cdot\|:=\|\cdot\|_{L^2(\Omega)}$.
We introduce the following assumptions on the initial data as in \cite{WinklerCPDE}:
\begin{equation}
\begin{cases}\langlebel{ini1}
n_0\in C^0(\overline{\Omega}), \quad \ n_0>0\ \text{in}\ \overline{\Omega},\\
c_0\in W^{1,q}(\Omega),\ \ \text{for some }\ q>N,\ \ c_0>0\ \text{in}\ \overline{\Omega},\\
u_0\in D(A^{\alpha}),\quad\, \text{for some}\ \ \alpha\in(\frac{N}{4},1),
\epsilonnd{cases}
\epsilonnd{equation}
where $A$ denotes the realization of the Stokes operator in the solenoidal subspace $L^2_{\sigma}(\Omega)$ that is the closure of $\{u\in (C_0^{\infty}(\Omega))^N; \ \nabla \cdot u=0\}$ in $(L^2(\Omega))^N$ (see e.g., \cite{G}). As for the parameter functions, we suppose that
\begin{equation}
\begin{cases}\langlebel{ass1}
\chi\in C^2[0,\infty), \quad \chi>0 \text{ in }[0,\infty),\\
f\in C^2[0,\infty), \quad f(0)=0,\ f>0 \text{ in }(0,\infty),\\
\phi\in C^2(\overline{\Omega}),
\epsilonnd{cases}
\epsilonnd{equation}
and
\begin{equation}
\left(\frac{f}{\chi}\right)'>0,\quad \left(\frac{f}{\chi}\right)''\leq 0,\quad \left(\chi\cdot f\right)'\geq 0,\quad \text{on }[0,\infty).\langlebel{ass4}
\epsilonnd{equation}
Now we state our main result of the paper:
\begin{theorem}\langlebel{MT}
Let $\Omega\in \mathbb{R}^N$ $(N\in\{2,3\})$ be a bounded regular domain. Assume that the assumptions \epsilonqref{ini1}--\epsilonqref{ass4} are satisfied.
\begin{itemize}
\item[(i)] If $N=2$ and $\kappa\in \mathbb{R}$, then problem \epsilonqref{chemo1}--\epsilonqref{ini} admits a unique global classical solution $(n,c,u,p)$ (up to addition of a constant to the pressure $p$) such that for any $T>0$,
\begin{equation}
\begin{cases}
n\in C([0,T]; L^2(\Omega))\cap L^{\infty}(0,T;C^0(\overline{\Omega}))\cap C^{2,1}(\overline{\Omega}\times(0,T)),\\
c\in C([0,T]; L^2(\Omega))\cap L^{\infty}(0,T;W^{1,q}(\Omega))\cap C^{2,1}(\overline{\Omega}\times(0,T)),\\
u\in C([0,T];L^2(\Omega))\cap L^{\infty}(0,T;D(A^{\alpha}))\cap C^{2,1}(\overline{\Omega}\times(0,T)),\\
p\in L^1(0,T;H^1(\Omega)).
\epsilonnd{cases}\nonumber umber
\epsilonnd{equation}
Moreover, the global classical solution satisfies
\begin{equation}
\lim_{t\to +\infty} \|n(\cdot, t)-n_\infty\|_{L^\infty(\Omega)}+\|c(\cdot, t)\|_{L^\infty(\Omega)}+\|u(\cdot, t)\|_{L^\infty}=0,\langlebel{conver}
\epsilonnd{equation}
where $n_\infty=\frac{1}{|\Omega|}\int_\Omega n_0 dx$.
\item[(ii)] If $N=3$ and $\kappa=0$, then there exists at least one global weak solution of \epsilonqref{chemo1}--\epsilonqref{ini} in the sense of \cite[Definition 5.1]{WinklerCPDE}.
\epsilonnd{itemize}
\epsilonnd{theorem}
Before giving the detailed proof, we stress some new features of the present paper. Our initial boundary problem \epsilonqref{chemo1} is proposed on any regular bounded domain $\Omega\subset \mathbb{R}^N$ $(N\in\{2,3\})$, while in \cite{WinklerCPDE} convexity of the domain is essentially used in order to prove the global existence results. The main difficulty comes from an integration term on the boundary that takes the following form:
\begin{equation}
\int_{\partial\Omega}\frac{\chi(c)}{f(c)}\frac{\partial}{\partial\nu}|\nabla c|^2 dS.\nonumber umber
\epsilonnd{equation}
If $\Omega$ is convex, then the above integrand turns out to be non-positive (cf. \cite{PGG1998}) and as a consequence it can be simply neglected like in \cite{WinklerCPDE}. However, if we consider the problem in a general bounded domain, this integrand fails to have a definite sign and has to be estimated in a suitable way. To overcome this difficulty, we make use of the elementary lemma due to Mizoguchi \& Souplet \cite{MizoSoup} together with the trace theorem to control the above boundary integration. Then by delicate estimates, we are able to derive a new type of entropy-energy estimate, whose right-hand side terms turns out to be easier to handle (cf. \epsilonqref{e1} below). This estimate plays a key role in obtaining global existence results for problem \epsilonqref{chemo1}--\epsilonqref{ini} in a similar manner as in \cite{WinklerCPDE}. Moreover, based on it we can further prove that when $N=2$, the global classical solution will converge to a constant steady state $(n_\infty, 0,0)$ as times goes to infinity. Our work removes the restriction on the convexity of $\Omega$ and thus improves the corresponding results by Winkler \cite{WinklerCPDE, WinkARMA}.
The rest of this paper is organized as follows. In Section 2, we handle the above mentioned boundary integration term with the aid of a gradient estimate result on the boundary from Mizoguchi \& Souplet \cite{MizoSoup}. In Section 3, we derive the key entropy-energy estimate by using some delicate calculations and then sketch the proof for Theorem \ref{MT}.
\section{Estimate of Boundary Integration}
To begin with, we first state two basic properties of the solutions to problem \epsilonqref{chemo1}--\epsilonqref{ini} that have already been obtained in \cite{WinklerCPDE,WinkARMA,LL2011}.
\begin{lemma} \langlebel{lla}
We have
\begin{equation}
\int_{\Omega}n(x,t)dx=\int_{\Omega}n_0dx\quad\text{for all }t>0
\epsilonnd{equation}
and
\begin{equation}
t\mapsto\|c(\cdot,t)\|_{L^{\infty}(\Omega)} \quad\text{is non-increasing.}
\epsilonnd{equation}
In particular,
\begin{equation}
\|c(\cdot, t)\|_{L^{\infty}(\Omega)}\leq\|c_0\|_{L^{\infty}(\Omega)}\quad\text{for all }t>0.
\epsilonnd{equation}
\epsilonnd{lemma}
\begin{lemma} The solution satisfies the identity
\begin{equation}
\begin{split} \langlebel{iden}
\frac{d}{dt}&\left\{\int_{\Omega}n\log n dx+\frac{1}{2}\int_{\Omega}|\nabla\psi(c)|^2dx\right\}+\int_{\Omega}\frac{|\nabla n|^2}{n}dx+\int_{\Omega}g(c)|D^2\rho(c)|^2dx\\
&=-\frac{1}{2}\int_{\Omega}\frac{g'(c)}{g^2(c)}|\nabla c|^2(u\cdot \nabla c)dx+\int_{\Omega}\frac{1}{g(c)}\Delta c(u\cdot\nabla c)dx\\
&+\int_{\Omega}F(n)\left(\frac{f(c)g'(c)}{2g^2(c)}-\frac{f'(c)}{g(c)}\right)|\nabla c|^2dx\\
&+\frac{1}{2}\int_{\Omega}\frac{g''(c)}{g^2(c)}|\nabla c|^4dx+\frac{1}{2}\int_{\partial\Omega}\frac{1}{g(c)}\frac{\partial}{\partial\nu}|\nabla c|^2dx
\epsilonnd{split}
\epsilonnd{equation}
for all $t\in(0, T_{max})$, where $D^2\rho$ denotes the Hessian of $\rho$ and we have set
\begin{equation}\langlebel{gpr}
g(s):=\frac{f(s)}{\chi(s)},\ \ \ \psi(s):=\int_1^s\frac{d\sigma}{\sqrt{g(\sigma)}}\ \ \ \ \text{and}\ \ \ \rho(s):=\int_1^s\frac{d\sigma}{g(\sigma)} \ \ \ \text{for }s>0,
\epsilonnd{equation}
and $F\in C^2([0,+\infty))$ is a nonnegative function satisfying $0\leq F'(s)\leq 1$ for all $s\geq 0$.
\epsilonnd{lemma}
Therefore, if the domain $\Omega$ is convex and $\frac{\partial{c}}{\partial \nu}=0$ on $\partial \Omega$, then $\frac{\partial}{\partial \nu}|\nabla c|^2\leq 0$ (cf. \cite{PGG1998}). This implies that the boundary integration term on the right-hand side of \epsilonqref{iden} is non-positive and hence can be simply neglected as in \cite{WinklerCPDE}. However, for general bounded domains, it fails to have a definite sign and thus needs to be estimated in a suitable way.
For this purpose, we introduce a lemma by Mizoguchi \& Souplet \cite[Lemma 4.2]{MizoSoup}, which enables us to deal with the boundary integration in \epsilonqref{iden}.
\begin{lemma} \langlebel{MS} For the bounded domain $\Omega$ and $w\in C^2(\overline{\Omega})$ satisfying $\frac{\partial w}{\partial \nu}=0$ on $\partial\Omega$. We have
\begin{equation} \frac{\partial|\nabla w|^2}{\partial\nu}\leq2\kappa|\nabla w|^2\quad\text{on }\partial\Omega,\epsilonnd{equation}
where $\kappa=\kappa(\Omega)>0$ is an upper bound for the curvatures of $\partial\Omega.$
\epsilonnd{lemma}
Then we can prove the following boundary estimate:
\begin{lemma} \langlebel{BDES}
Suppose that the assumptions of Theorem \ref{MT} hold, and let $g,\psi$ and $\rho$ be defined as in \epsilonqref{gpr}. Then for any $\epsilonpsilon\in (0,1)$, the following estimate holds
\begin{equation}
\begin{split}
&\frac{1}{2}\left|\int_{\partial\Omega}\frac{1}{g(c)}\frac{\partial}{\partial\nu}|\nabla c|^2dS\right| \\
&\ \ \leq \epsilon\int_{\Omega}g(c)|\Delta\rho(c)|^2dx+\epsilon\int_{\Omega}\frac{|g'(c)|^2|\nabla c|^4}{g(c)^3}dx+C_{\epsilon}\|\psi(c)\|^2,
\epsilonnd{split}\langlebel{bde1}
\epsilonnd{equation}
where $C_\epsilon$ is a constant that may depend on $\Omega$, $\kappa$ and $\epsilon$, but not on $c$.
\epsilonnd{lemma}
\begin{proof}
Thanks to Lemma \ref{MS}, we can control the boundary integration term as follows
\begin{equation}
\frac{1}{2}\left|\int_{\partial\Omega}\frac{1}{g(c)}\frac{\partial}{\partial\nu}|\nabla c|^2dS \right|
\leq
\kappa(\Omega)\int_{\partial\Omega}\frac{|\nabla c|^2}{g(c)}dS
=\kappa(\Omega)\int_{\partial\Omega}|\nabla \psi(c)|^2dS.
\langlebel{bde2}
\epsilonnd{equation}
On the other hand, by the trace theorem \cite[Theorem I.9.4]{LM}, it holds
\begin{equation} \int_{\partial\Omega}|\nabla \psi(c)|^2 dS\leq C\|\psi(c)\|_{H^{\frac{3+s}{2}}(\Omega)}^2,
\quad \forall\, s\in (0, \frac12).\langlebel{bde3}
\epsilonnd{equation}
where $C>0$ depends only on $\Omega$. Then we infer from the interpolation inequality (see e.g., \cite[Remark I.9.6]{LM}) that
\begin{equation}
\begin{split}
\|\psi(c)\|_{H^{\frac{3+s}{2}}(\Omega)}^2&\ \ \leq C\|\psi(c)\|_{H^2(\Omega)}^\frac{3+s}{2}\|\psi(c)\|^\frac{1-s}{2}\\
&\ \ \leq C\|\Delta\psi(c)\|^\frac{3+s}{2}\|\psi(c)\|^\frac{1-s}{2}+C\|\psi(c)\|^2
\epsilonnd{split}
\langlebel{inter}
\epsilonnd{equation}
By direct calculations, we have
\begin{equation} \langlebel{id2}
\Delta\psi(c)=\frac{\Delta c}{\sqrt{g(c)}}-\frac{1}{2}\frac{g'(c)|\nabla c|^2}{(g(c))^{\frac{3}{2}}}
\epsilonnd{equation}
and
\begin{equation}
\Delta\rho(c)=\frac{\Delta c}{g(c)}-\frac{g'(c)|\nabla c|^2}{g(c)^2},\nonumber
\epsilonnd{equation}
which yield that
\begin{equation} \sqrt{g(c)}\Delta\rho(c)=\frac{\Delta c}{\sqrt{g(c)}}-\frac{g'(c)|\nabla c|^2}{(g(c))^{\frac{3}{2}}}=\Delta\psi(c)-\frac{1}{2}\frac{g'(c)|\nabla c|^2}{(g(c))^{\frac{3}{2}}}.\epsilonnd{equation}
Then it follows that
\begin{equation}
\langlebel{boun1}
\|\Delta\psi(c)\|^2\leq2\int_{\Omega}g(c)|\Delta\rho(c)|^2dx+\frac{1}{2}\int_{\Omega}\frac{|g'(c)|^2|\nabla c|^4}{g(c)^3}dx.
\epsilonnd{equation}
Using the estimates \epsilonqref{bde2}--\epsilonqref{inter}, \epsilonqref{boun1} and Young's inequality, we can conclude \epsilonqref{bde1}. The proof is complete.
\epsilonnd{proof}
\section{Proof of Theorem \ref{MT}}
Lemma \ref{BDES} enables us to prove the following entropy-energy type estimate for problem \epsilonqref{chemo1}--\epsilonqref{ini}:
\begin{lemma} Suppose that the assumptions of Theorem \ref{MT} hold, and let $g,\psi$ and $\rho$ be defined as in \epsilonqref{gpr}. Then the smooth solution to problem \epsilonqref{chemo1}--\epsilonqref{ini} satisfies
\begin{equation}
\begin{split}
& \frac{d}{dt}\left\{\int_{\Omega}n\log n dx+\frac{1}{2}\int_{\Omega}|\nabla\psi(c)|^2dx \right\}
+\int_{\Omega}\frac{|\nabla n|^2}{n}dx +\frac{1}{2}\int_{\Omega}g(c)|D^2\rho(c)|^2dx \\
&\ \ \leq C\|\nabla u\|^2+C\|\psi(c)\|^2\quad \text{for all}\ t\in(0,T_{max}),
\epsilonnd{split}
\langlebel{e1}
\epsilonnd{equation}
where $C>0$ is independent of $t$, $T_{max}$.
\epsilonnd{lemma}
\begin{proof}
We proceed to estimate the right-hand side of \epsilonqref{iden} term by term. First, applying \cite[Lemma 3.3]{WinklerCPDE}, we obtain that
\begin{equation}
\int_{\Omega}\frac{g'(c)}{g(c)^3}|\nabla c|^4dx \leq(2+\sqrt{N})^2\int_{\Omega}\frac{g(c)}{g'(c)}|D^2\rho(c)|^2dx.
\langlebel{boun2}
\epsilonnd{equation}
Since $g'>0$ on $[0,+\infty)$ and $0\leq c\leq K:=\|c_0\|_{L^{\infty}}$ (see Lemma \ref{lla}), there exist constants
$$
C_1:=\inf_{s\in(0,K)}g'(s)>0,\quad \text{and} \quad C_2=\sup_{s\in(0,K)}g'(s)>0
$$
such that $C_1\leq g'(c)\leq C_2$ in $\Omega\times(0,T_{max})$. Hence, it follows from \epsilonqref{boun2} that
\begin{equation}
\langlebel{boun3}
\int_{\Omega}\frac{|g'(c)|^2}{g(c)^3}|\nabla c|^4dx\leq\frac{C_2}{C_1}(2+\sqrt{N})^2\int_{\Omega}g(c)|D^2\rho(c)|^2dx.
\epsilonnd{equation}
Then for the first two terms on the right-hand side of \epsilonqref{iden}, using \epsilonqref{id2}, after integration by parts, and using the boundary conditions \epsilonqref{boundcon}, we deduce from \epsilonqref{boun3} that
\begin{equation}
\begin{split}
&-\frac{1}{2}\int_{\Omega}\frac{g'(c)}{g(c)^2}|\nabla c|^2(u\cdot \nabla c)dx +\int_{\Omega}\frac{1}{g(c)}\Delta c(u\cdot\nabla c)dx \\
&\ \ =\int_\Omega \frac{1}{\sqrt{g(c)}}\Delta\psi(c)(u\cdot\nabla c) dx\\
&\ \ =\int_{\Omega}\Delta\psi(c)(u\cdot \nabla\psi(c))dx\\
&\ \ =-\int_{\Omega}(\nabla\psi(c)\otimes\nabla\psi(c)): \nabla u dx \\
&\ \ \leq \frac{C_1}{4C_2(2+\sqrt{N})^2}\int_{\Omega}\frac{g'(c)}{g(c)^3}|\nabla c|^4dx +\frac{C_2(2+\sqrt{N})^2}{C_1}\int_{\Omega}\frac{g(c)}{g'(c)}|\nabla u|^2dx\\
&\ \ \leq \frac14\int_{\Omega}g(c)|D^2\rho(c)|^2dx+\frac{C_2(2+\sqrt{N})^2}{C_1}\int_{\Omega}\frac{g(c)}{g'(c)}|\nabla u|^2dx,
\epsilonnd{split}\langlebel{bde2a}
\epsilonnd{equation}
where we have also used the incompressibility of the fluid as well as Young's inequality. Here, the notion $A : B$ denotes $\mathrm{Tr}(AB) = A_{ij}B_{ji}$ for two $N\times N$ matrices $A, B$.
As in \cite{WinklerCPDE}, it follows from the assumption \epsilonqref{ass4} that $g''\leq 0$ in $[0, +\infty)$ and $\frac{fg'}{2g^2}-\frac{f'}{g}=-\frac{(\chi f)'}{2f}\leq 0$ on $[0, +\infty)$. Then the third term on the right-hand side of \epsilonqref{iden} is indeed non-positive and can simply be neglected.
Next, we infer from Lemma \ref{BDES} that the last boundary integration term on the right-hand side of \epsilonqref{iden} can be estimated by using \epsilonqref{bde1}. In view of the pointwise inequality $|\Delta z|^2\leq N|D^2 z|^2$ for $z\in C^2(\overline{\Omega})$, the first term on the right-hand side of \epsilonqref{bde1} can be estimated as follows:
\begin{equation}
\int_{\Omega}g(c)|\Delta\rho(c)|^2dx\leq N \int_{\Omega}g(c)|D^2\rho(c)|^2dx.\langlebel{boun1b}
\epsilonnd{equation}
As a consequence, it follows from \epsilonqref{bde1}, \epsilonqref{boun3} and \epsilonqref{boun1b} that
\begin{equation}
\begin{split}
&\frac{1}{2}\left|\int_{\partial\Omega}\frac{1}{g(c)}\frac{\partial}{\partial\nu}|\nabla c|^2dS\right| \\
&\ \ \leq \epsilon\left(N+\frac{C_2}{C_1}(2+\sqrt{N})^2\right)\int_{\Omega}g(c)|D^2\rho(c)|^2dx+C_{\epsilon}\|\psi(c)\|^2.
\epsilonnd{split}\langlebel{bde1a}
\epsilonnd{equation}
Taking $\epsilon>0$ sufficiently small such that $\epsilon\left(N+\frac{C_2}{C_1}(2+\sqrt{N})^2\right)\leq \frac{1}{4}$ in \epsilonqref{bde1a}, we can deduce from \epsilonqref{iden}, \epsilonqref{bde2a} and \epsilonqref{bde1a} that the inequality \epsilonqref{e1} holds. The proof is complete.
\epsilonnd{proof}
Besides, we can derive the following estimates on the velocity field $u$:
\begin{lemma} Under the assumptions of Theorem \ref{MT}, we have the following estimate
\begin{equation}
\langlebel{bound4}
\|u\|^2+\int_0^T\|\nabla u\|^2dt\leq C\left(\int_0^T\int_{\Omega}\frac{|\nabla n|^2}{n}dx+1\right)^{\frac{N}{6}},\;\;\text{for any}\ 0<T<T_{max},
\epsilonnd{equation}
where the constant $C$ depends on $T$ and $\int_\Omega n_0 dx$.
\epsilonnd{lemma}
\begin{proof}
Multiplying the third equation in \epsilonqref{chemo1} by $u$, integrating over $\Omega$, using the H\"older inequality and the Sobolev embedding theorem, we obtain that
\begin{equation}
\begin{split}
\frac{1}{2}\frac{d}{dt}\int_{\Omega}|u|^2dx +\int_{\Omega}|\nabla u|^2dx
& =\int_{\Omega} n\nabla\phi\cdot u dx \\
& \leq \|u\|_{L^6}\|\nabla \phi\|_{L^\infty}\|n\|_{L^\frac65}\\
& \leq \frac{1}{2}\|\nabla u\|^2+C\|\nabla\phi\|_{L^{\infty}}^2\|n\|_{L^{\frac{6}{5}}}^2.\langlebel{aaa}
\epsilonnd{split}
\epsilonnd{equation}
On the other hand, we infer from \cite[Lemma 4.1]{WinklerCPDE} that
\begin{equation}
\int_0^T\|n\|_{L^{\frac{6}{5}}}^2dt\leq C(T, \int_\Omega n_0 dx)\left(\int_0^T\int_{\Omega}\frac{|\nabla n|^2}{n}dxdt+1\right)^{\frac{N}{6}}.\langlebel{aaaa}
\epsilonnd{equation}
Then integrating \epsilonqref{aaa} with respect to time, we conclude from \epsilonqref{aaaa} that the assertion \epsilonqref{bound4} follows.
\epsilonnd{proof}
As a consequence of the above lemmas, we can obtain the following \epsilonmph{a priori} estimates on solutions to problem \epsilonqref{chemo1}--\epsilonqref{ini}:
\begin{lemma} \langlebel{lemkey} Under the same assumptions of Theorem \ref{MT}, for any $0<T<T_{max}$, we have the following estimates
\begin{equation}\langlebel{bound5a}
\int_{\Omega}n\log n dx+\int_{\Omega}|\nabla\psi(c)|^2dx+\int_{\Omega}|u|^2dx\leq C,
\epsilonnd{equation}
\begin{equation} \int_0^T\int_{\Omega}\frac{|\nabla n|^2}{n}dxdt+\int_0^T\int_{\Omega}g(c)|D^2\rho(c)|^2dxdt+\int_0^T\int_{\Omega}|\nabla u|^2dxdt\leq C,\langlebel{bound5b}
\epsilonnd{equation}
and
\begin{equation}
\langlebel{bound6}
\int_0^T\int_{\Omega}|\nabla c|^4 dxdt \leq C.
\epsilonnd{equation}
\epsilonnd{lemma}
\begin{proof} The estimates \epsilonqref{bound5a}--\epsilonqref{bound5b} easily follow from \epsilonqref{e1} and \epsilonqref{bound4} together with an application of Young's inequality. Then the estimate \epsilonqref{bound6} can be derived in the same way as in \cite[Corollary 4.4, Corollary 5.3]{WinklerCPDE}.
\epsilonnd{proof}
After the previous preparations, we can proceed to prove our main result.
\textbf{Proof of Theorem \ref{MT}}. The proof for the existence of global solutions follows a similar argument like in \cite{WinklerCPDE}. First, since the proof in \cite{WinklerCPDE} for the local existence and uniqueness of classical solutions to problem \epsilonqref{chemo1}--\epsilonqref{ini} does not rely on the fact that $\Omega$ is convex, then for the current case with general domain $\Omega$, we can still apply the same fixed point argument to prove the local wellposeness result. Then by the \epsilonmph{a priori} estimates in Lemma \ref{lemkey}, we are able to prove the existence of global classical solutions for $N=2$ and global weak solutions for $N=3$ in the same way as in \cite[Section 4, Section 5]{WinklerCPDE}. Therefore, the details are omitted here.
Concerning the asymptotic behavior for $N=2$, we notice that in our new entropy-energy inequality \epsilonqref{e1}, the right-hand side terms now become $C\|\nabla u\|^2+C\|\psi(c)\|^2$ instead of $C\int_\Omega |u|^4 dx$ as in \cite[(3.11)]{WinklerCPDE} (see also \cite[(2.15)]{WinkARMA}). In \cite{WinkARMA} this (worse) term $\int_\Omega |u|^4 dx$ is estimated by using the Sobolev embedding theorem and Poincar\'e's inequality such that
$$\|u\|_{L^4(\Omega)}^4\leq C\|\nabla u\|^2\|u\|^2,$$
which is a higher-order nonlinearity than $\|\nabla u\|^2$. Since the current right-hand side of \epsilonqref{e1} is simpler and the second term $\|\psi(c)\|^2$ is uniformly bounded in time due to the definition of $\psi$ (cf. \epsilonqref{gpr}) and Lemma \ref{lla}, one can check that all the arguments in \cite{WinkARMA} can pass through. As a consequence, we can conclude the convergence result \epsilonqref{conver}.
Hence, Theorem \ref{MT} is proved.
\noindent \textbf{Acknowledgments:}
J. Jiang is partially supported by National Natural Science Foundation of China (NNSFC) under the grants No. 11201468 and No. 11275259, H. Wu is partially supported by NNSFC under the grant No. 11371098 and Zhuo Xue program in Fudan University, and S. Zheng is partially supported by NNSFC under the grant No. 11131005.
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\epsilonnd{thebibliography}
\epsilonnd{document} |
{\betaf e}gin{document}
\tauitle{Nash blow-ups of jet schemes}
\alphauthor{Tommaso de Fernex}
\alphaddress[T.\ de Fernex]{
Department of Mathematics\pirotect\linebreak
University of Utah\pirotect\linebreak
155 South 1400 East\pirotect\linebreak
Salt Lake City, UT 48112\pirotect\linebreak
USA
}
\etamail{[email protected]}
\alphauthor{Roi Docampo}
\alphaddress[Roi Docampo]{
Department of Mathematics\pirotect\linebreak
University of Oklahoma\pirotect\linebreak
601 Elm Avenue, Room 423\pirotect\linebreak
Norman, OK 73019\pirotect\linebreak
USA
}
\etamail{[email protected]}
\sigmaubjclass[2010]{Primary 14E18; Secondary 14E04, 14B05.}
\kappaeywords{Jet scheme, Nash blow-up, singularities, Grassmannian, functor of points}
\pirotect\tauhanks@warning{The research of the first author was partially supported by NSF Grant DMS-1700769.}
{\betaf e}gin{abstract}
Given an arbitrary projective birational morphism of varieties, we provide a
natural and explicit way of constructing relative compactifications of the maps
induced on the main components of the jet schemes. In the case the morphism is
the Nash blow-up of a variety, such relative compactifications are shown to be
given by the Nash blow-ups of the main components of the jet schemes.
\etand{abstract}
\muaketitle
\sigmaection{Introduction}
The \etamph{Nash blow-up} of a variety is defined as the universal
projective birational morphism for which the pull-back of the sheaf of
differentials admits a locally free quotient of the same rank.
The name comes from John Nash, who is generally credited for having promoted
the question of whether singularities of algebraic varieties can always be
resolved by finitely many iterations of such blow-ups; before him, the question
had already been considered by Semple \cite{Sem54}. The property is known to
hold for curves of characteristic zero, and to fail in positive characteristics
\cite{Nob75}. A variant of this question, where Nash blow-ups are alternated
with normalizations, has been settled affirmatively for surfaces of
characteristic zero by Spivakovsky \cite{Spi90}, building on \cite{Hir83}.
Higher order Nash blow-ups have been defined and studied by Yasuda
\cite{Yas07}.
The Nash blow-up can be thought as the universal operation separating multiple
limits of tangent spaces, and hence its construction relates to the geometry of
the main component of the first jet scheme of the variety. It is however
unclear a priori how the Nash blow-up of a variety should relate to the Nash
blow-up of such component. Even less obvious is whether there should be a
relationship with the Nash blow-ups of the main components of the higher jet
schemes of the variety.
The following result shows that these Nash blow-ups are not just related, but
in fact they essentially determine each other.
{\betaf e}gin{theorem}
\label{th:jet-Nash-blowup}
Let $X$ be a variety. For every $n$, the main component of the $n$-th jet
scheme of the Nash blow-up of $X$ has an open immersion into the Nash blow-up
of the main component of the $n$-th jet scheme of $X$, and such immersion is
compatible with the respective natural map to the $n$-th jet scheme of $X$.
\etand{theorem}
Denoting by $N(X) \tauo X$ the Nash blow-up of a variety and by $J_n'(X)$ the
main component of the $n$-th jet scheme of $X$, \cref{th:jet-Nash-blowup} can
be rephrased by saying that the Nash blow-up $N(J_n'(X)) \tauo J_n'(X)$ gives a
relative compactification of the map $J_n'(N(X)) \tauo J_n'(X)$ induced on
$n$-jets by the Nash blow-up of $X$.
This implies that the Nash blow-up of a variety $X$ can equivalently be
characterized as the universal projective birational morphism $Y \tauo X$ such
that, for every $n$, the pull-back of $\muathcal{O}m_{J_n'(X)}$ via $J_n'(Y) \tauo J_n'(X)$
has a locally free quotient of the same rank.
The theorem also implies that the Nash blow-up of $J_n'(X)$ induces the Nash
blow-up of $X$ under the natural section (the `zero section') of the projection
$J_n'(X) \tauo X$. It was shown by Ishii \cite{Ish09} that if a variety $X$ is
singular then all of its jet schemes are singular, and
\cref{th:jet-Nash-blowup} implies that, if the ground field is algebraically
closed of characteristic zero, then in fact the main components of the jet
schemes are already singular.
The proof of \cref{th:jet-Nash-blowup} uses the description of the sheaves of
differentials on jet schemes given in \cite{dFD} in combination with
\cref{th:jet-Nash-transform} (stated below), which addresses a related question
in a more general context.
Suppose that $\mu \colon Y \tauo X$ is an arbitrary projective birational morphism
of varieties. By functoriality, $\mu$ induces for every $n$ a morphism on jet
schemes $\mu_n \colon J_n(Y) \tauo J_n(X)$, and hence, by restriction, a
birational morphism $\mu_n' \colon J_n'(Y) \tauo J_n'(X)$ between the main
components of the jet schemes. In general, $\mu_n'$ is not a projective
morphism, and one can ask whether there are natural ways of constructing
relative compactifications of $\mu_n'$. The next theorem provides an answer to
this question.
The morphism $\mu$ can be described as the blow-up of an ideal sheaf $\muathcal{I} \sigmaubset
\muathcal{O}_X$, and a way to approach the question is to look for natural ways of
constructing an ideal sheaf $\muathfrak{a}_n \sigmaubset \muathcal{O}_{J_n'(X)}$ whose blow-up gives a
relative compactification of $\mu_n'$. Doing this directly seems hard: while a
posteriori we will provide an explicit formula for computing the local
generators of such an ideal $\muathfrak{a}_n$ in terms of the generators of $\muathcal{I}$, the
formula will show that the complexity of $\muathfrak{a}_n$ grows fast even in simple
examples, an indication that looking at ideals might not be the best approach.
Instead, we view $\mu$ as the \etamph{Nash transformation} $N(\muathcal{F}) \tauo X$ of a
coherent sheaf $\muathcal{F}$, as defined for instance in \cite{OZ91}. In this language,
the blow-up of an ideal $\muathcal{I} \sigmaubset \muathcal{O}_X$ is the same as the Nash
transformation $N(\muathcal{I}) \tauo X$ of the ideal, and the Nash blow-up of a variety
$X$ is defined to be the Nash transformation $N(\muathcal{O}m_X) \tauo X$ of the sheaf of
differentials of $X$. In general, the Nash transformation of a coherent sheaf
$\muathcal{F}$ of rank $r$ is defined using the Grassmann bundle of locally free
quotients of rank $r$ of $\muathcal{F}$, and is a projective birational morphism.
Conversely, every projective birational morphism $\mu \colon Y \tauo X$ can be
realized as a Nash transformation of some coherent sheaf $\muathcal{F}$ on $X$.
{\betaf e}gin{theorem}
\label{th:jet-Nash-transform}
Let $X$ be a variety over a field $k$, and let $\mu \colon N(\muathcal{F}) \tauo X$ be the
Nash transformation of a coherent sheaf $\muathcal{F}$ on $X$. For every $n$, let
\[
\xiymatrix{
J_n'(X) \tauimes \Deltaelta_n \alphar[d]_{\rho_n'} \alphar[r]^(.65){\gamma_n'} & X \pirotect\linebreak
J_n'(X)
&
}
\]
be the diagram induced by restriction from the universal $n$-jet of $X$; here,
we denote $\Delta_n = {\muathbb S}pec k[t]/(t^{n+1})$. Define
\[
\muathcal{F}_n' := (\rho_n')_*(\gamma_n')^*\muathcal{F}.
\]
Then the induced map $\mu_n' \colon J_n'(N(\muathcal{F})) \tauo J_n'(X)$ factors as
\[
\xiymatrix@C=20pt{
J_n'(N(\muathcal{F})) \alphar@{^(->}[r]^(.55){\iota_n}
& N(\muathcal{F}_n') \alphar[r]^{\nu_n}
& J_n'(X)
}
\]
where $\iota_n$ is an open immersion and $\nu_n$ is the Nash transformation of
$\muathcal{F}_n'$.
\etand{theorem}
If in this theorem we take $\muathcal{F} = \muathcal{I} \sigmaubset \muathcal{O}_X$, an ideal sheaf on $X$, then
$\muathcal{F}_n'$ is not an ideal sheaf. However, the sheaf $\wedge^{(n+1)}\muathcal{F}_n'$,
modulo torsion, is isomorphic to an ideal sheaf $\muathfrak{a}_n$, and $N(\muathcal{F}_n') =
N(\muathfrak{a}_n)$. Our approach enables us to make explicit computations and hence to
provide a formula for the generators of $\muathfrak{a}_n$.
A motivation for \cref{th:jet-Nash-transform} comes from the Nash problem on
families of arcs through the singularities of a variety \cite{Nas95} and, more
specifically, from the problem of lifting wedges \cite{LJ80,Reg06}. In
dimension two, the Nash problem has been settled in characteristic zero in
\cite{FdBPP12} but it remains open in positive characteristics. The algebraic
proof given in \cite{dFD16} may be adaptable to positive characteristics,
provided one can avoid certain wild ramifications that could occur in the
proof. A possible approach is to look for suitable
deformations of wedges, and this requires working with relative
compactifications of the maps $J_n(Y) \tauo J_n(X)$ where $Y \tauo X$ is the
minimal resolution of the surface. \cref{th:jet-Nash-transform} provides a
first step in this direction.
\sigmaubsection{Acknowledgments}
We thank Mircea Musta\c t\u a for pointing out an error in a previous version of the paper
and the referee for a careful reading of the paper and valuable comments and corrections.
\sigmaection{Proofs}
We work over an arbitrary field $k$. For every integer $n \gammae 0$, the
\etamph{$n$-th jet scheme} $J_n(X)$ of a scheme $X$ is the scheme representing
the functor of points defined by
\[
J_n(X)(Z) = X(Z \tauimes \Delta_n)
\]
for any scheme $Z$, where $\Delta_n = {\muathbb S}pec k[t]/(t^{n+1})$. We denote by
\[
\xiymatrix{
J_n(X) \tauimes \Deltaelta_n \alphar[d]_{\rho_n} \alphar[r]^(.65){\gamma_n} & X \pirotect\linebreak
J_n(X)
&
}
\]
the universal $n$-jet of $X$. For generalities about jet schemes, we refer
to \cite{Voj07,EM09}. If $X$ is a variety, then there exists a unique
irreducible component of $J_n(X)$ dominating $X$, and this component has
dimension $(n+1)\partialltaim X$. We shall denote it by $J_n'(X)$ and call it the
\etamph{main component} of $J_n(X)$.
Given a coherent sheaf $\muathcal{F}$ on a scheme $X$, and a positive integer $r$, we
denote by $\Gammar(\muathcal{F},r)$ the \etamph{Grassmann bundle} over $X$ parameterizing
locally free quotients of $\muathcal{F}$ of rank~$r$, where by the term \etamph{quotient}
we mean an equivalence class of surjective maps from the same source where two surjections
are identified whenever they have the same kernel.
This scheme represents the functor of points given by
{\betaf e}gin{align*}
\Gammar(\muathcal{F},r)(Z) &=
\betaig\{ \betaig(Z \xirightarrow{p} X,\, p^*\muathcal{F} \tauwoheadrightarrow {\muathcal Q}\betaig)
\muid \pirotect\linebreak
& \hskip1cm \tauext{${\muathcal Q}$ locally free sheaf on $Z$ of rank $r$}\betaig\}
\etand{align*}
for any scheme $Z$.
Suppose now that $X$ is a variety, and let $\muathcal{F}$ be a coherent sheaf on $X$ of
rank $r$. The \etamph{Nash transformation} of $\muathcal{F}$ is defined to be the
irreducible component of $\Gammar(\muathcal{F},r)$ dominating $X$, and is denoted by
$N(\muathcal{F})$. The natural projection $\Gammar(\muathcal{F},r) \tauo X$ induces the blow-up map
$N(\muathcal{F}) \tauo X$. The \etamph{Nash blow-up} $N(X) \tauo X$ is, by definition, the
Nash transformation of the sheaf of K\"ahler differentials $\muathcal{O}m_X$.
{\betaf e}gin{proof}[Proof of \cref{th:jet-Nash-transform}]
The sheaf $\muathcal{F}_n'$ is the restriction, under the inclusion $J_n'(X) \sigmaubset J_n(X)$, of
the sheaf
\[
\muathcal{F}_n := (\rho_n)_*\gamma_n^*\muathcal{F}.
\]
By construction, $J_n'(N(\muathcal{F}))$ is an irreducible component of the jet scheme
$J_n(\Gammar(\muathcal{F},r))$. Similarly, observing that $\muathcal{F}_n'$ is a sheaf of rank
$(n+1)r$ and keeping in mind that $J_n'(X)$ is an irreducible component of
$J_n(X)$, we see that $N(\muathcal{F}_n')$ is an irreducible component of
$\Gammar(\muathcal{F}_n,(n+1)r)$. We claim that there is a universally injective map
\[
i \colon J_n(\Gammar(\muathcal{F},r)) \hookrightarrow \Gammar(\muathcal{F}_n,(n+1)r),
\]
defined over $X$, which agrees with the natural identification of these schemes
over the open set where $X$ is smooth and $\muathcal{F}$ is locally free,
and restricts to an open immersion from $J_n'(N(X))$ to $N(J_n'(X))$.
Note that the existence of such a map implies the statement of the theorem.
In order to prove this claim, we compare the functors of points of the schemes
$J_n(\Gammar(\muathcal{F},r))$ and $\Gammar(\muathcal{F}_n,(n+1)r)$. For every scheme $Z$, we have
{\betaf e}gin{align*}
J_n(\Gammar(\muathcal{F},r))(Z) &= \Gammar(\muathcal{F},r)(Z \tauimes \Delta_n) \pirotect\linebreak
&= \betaig\{ \betaig(Z \tauimes \Deltaelta_n \xirightarrow{\alphalpha} X,\,
\alphalpha^*\muathcal{F} \tauwoheadrightarrow {\muathcal Q}\betaig)
\muid \pirotect\linebreak
& \hskip1cm \tauext{${\muathcal Q}$ locally free sheaf on $Z \tauimes \Deltaelta_n$ of rank $r$}\betaig\}
\etand{align*}
and
{\betaf e}gin{align*}
\Gammar(\muathcal{F}_n,(n+1)r)(Z)
&= \betaig\{ \betaig(Z \xirightarrow{{\betaf e}ta} J_n(X),\,
{\betaf e}ta^*\muathcal{F}_n \tauwoheadrightarrow {\muathcal R}\betaig)
\muid \pirotect\linebreak
&\hskip1cm \tauext{${\muathcal R}$ locally free sheaf on $Z$ of rank $(n+1)r$}\betaig\}.
\etand{align*}
By the description of $J_n(X)$ via the functor of points, every ${\betaf e}ta \colon Z
\tauo J_n(X)$ corresponds to a unique $\alphalpha \colon Z \tauimes \Deltaelta_n \tauo X$,
and for any such pair of maps there is a commutative diagram
\[
\xiymatrix@C=40pt{
Z \tauimes \Deltaelta_n \alphar[d]_{\pi} \alphar[r]_(.45){\beta\tauimes\id_{\Delta_n}} \alphar@/^20pt/[rr]^\alphalpha
& J_n(X) \tauimes \Deltaelta_n \alphar[d]^{\rho_n} \alphar[r]_(.65){\gamma_n}
& X \pirotect\linebreak
Z \alphar[r]_(.45){\betaf e}ta
& J_n(X)
&
}
\]
where $\pi$ is the projection onto the first factor. Note that taking
push-forward along $\pi$ of a sheaf on $Z \tauimes \Delta_n$ simply means that we are
restricting scalars to $\muathcal{O}_Z$ and forgetting the given $\muathcal{O}_{Z \tauimes
\Delta_n}$-module structure of the sheaf.
By the definition of $\muathcal{F}_n$ and base-change, which holds in this setting
because $\rho_n$ and $\pi$ are affine, we have
\[
\beta^*\muathcal{F}_n = \pi_*\alpha^*\muathcal{F}.
\]
Using the identification $J_n(X)(Z) = X(Z \tauimes \Delta_n)$, the above formula
yields the following alternative description of the functor of points of
$\Gammar(\muathcal{F}_n,(n+1)r)$:
{\betaf e}gin{align*}
\Gammar(\muathcal{F}_n,(n+1)r)(Z)
&= \betaig\{
\betaig(
Z\tauimes\Delta_n \xirightarrow{\alpha} X,\,
\pi_*\alpha^*\muathcal{F} \tauwoheadrightarrow {\muathcal R}
\betaig)
\muid \pirotect\linebreak
& \hskip1cm \tauext{${\muathcal R}$ locally free sheaf on $Z$ of rank $(n+1)r$}\betaig\}.
\etand{align*}
For every locally free sheaf ${\muathcal Q}$ on $Z \tauimes \Deltaelta_n$ of
rank $r$, the push-forward $\pi_*{\muathcal Q}$ is a locally free sheaf on $Z$ of
rank $(n+1)r$. Taking push-forwards via $\pi$ is exact,
and any two quotients of $\alpha^*\muathcal{F}$ are identified (i.e., they define the same kernel in $\alpha^*\muathcal{F}$)
if and only if their push-forwards are identified as quotients of $\pi_*\alpha^*\muathcal{F}$
(i.e., they define the same kernel in $\pi_*\alpha^*\muathcal{F}$).
This means that taking push-forwards via $\pi$ defines a natural injection
\[
J_n(\Gammar(\muathcal{F},r))(Z) \hookrightarrow \Gammar(\muathcal{F}_n,(n+1)r)(Z).
\]
As this holds for every scheme $Z$, we deduce that there is a
naturally defined universally injective morphism
\[
i \colon J_n(\Gammar(\muathcal{F},r)) \hookrightarrow \Gammar(\muathcal{F}_n,(n+1)r).
\]
It is immediate to see that $i$ is defined over $X$ and therefore it agrees with the natural
identification of these schemes over the open set where $X$ is smooth and
$\muathcal{F}$ is locally free. Furthermore,
the restriction of $i$ to $J_n'(N(\muathcal{F}))$ gives a universally injective map
$\iota_n \colon J_n'(N(\muathcal{F})) \tauo N(\muathcal{F}_n')$. To finish the proof, we need to show that $\iota_n$
is a local isomorphism, that is, it induces isomorphisms on all local rings. To
this end, we prove the following property.
{\betaf e}gin{lemma}
\label{th:lifting}
Let $(A,\muathfrak{m})$ be a Noetherian local domain over $k$, and set $U = {\muathbb S}pec A$ and $P =
{\muathbb S}pec A/\muathfrak{m}$. Assume that $f_0$ and $g$ are morphisms as in the diagram
\[
\xiymatrix{
P \alphar[r]^(.33){f_0} \alphar@{^(->}[d] & J_n(\Gammar(\muathcal{F},r)) \alphar@{^(->}[d]^i \pirotect\linebreak
U \alphar[r]_(.25){g} \alphar@{..>}[ru]^(.38){f} & \Gammar(\muathcal{F}_n,(n+1)r)
}
\]
such that the square sub-diagram commutes and the image of $g$ is a dense subset of $N(\muathcal{F}'_n)$.
Then there exists a unique morphism $f$ (marked by the dotted arrow in
the diagram) making the whole diagram commute.
\etand{lemma}
{\betaf e}gin{proof}
Suppose $f_0$ and $g$ are given. Let $\pi_0 \colon P \tauimes\Delta_n \tauo P$ and $\pi
\colon U \tauimes \Delta_n \tauo U$ denote the respective projections to the first
components. By the descriptions of the functors of points, we can write
\[
f_0 = \betaig(P \tauimes \Delta_n \xirightarrow{\alpha_0} X,\, \alpha_0^*\muathcal{F} \sigmaurj \muathcal{Q}\betaig),
\]
where $\muathcal{Q}$ is a locally free $A[t]/(t^{n+1})$-module of rank $r$, and
\[
g = \betaig(U\tauimes\Delta_n \xirightarrow{\alpha} X,\, \pi_*\alpha^*\muathcal{F} \sigmaurj \muathcal{R}\betaig)
\]
where $\muathcal{R}$ is a locally free $A$-module of rank $(n+1)r$. The commutativity of
the square sub-diagram in the statement means that $\alpha_0$ is the restriction of
$\alpha$ and $\muathcal{R} \circtimes_A A/\muathfrak{m} = (\pi_0)_*\muathcal{Q}$. The fact that the image of $g$
is dense in $N(\muathcal{F}'_n)$ implies that $\alpha$ is dominant, and hence
$\pi_*\alpha^*\muathcal{F}$ is a sheaf of rank $(n+1)r$. Since
$\muathcal{R}$ is a locally free quotient of the same rank of $\pi_*\alpha^*\muathcal{F}$, the kernel $\muathcal{K}$
of $\pi_*\alpha^*\muathcal{F} \tauo \muathcal{R}$ is the torsion $A$-submodule of $\pi_*\alpha^*\muathcal{F}$.
Every element of $\sigmaum_{i \gammae 0} t^i\muathcal{K}$, viewed as an $A$-submodule of $\pi_*\alpha^*\muathcal{F}$,
is torsion, and therefore we have $\sigmaum_{i \gammae 0} t^i\muathcal{K} = \muathcal{K}$. This shows that $\muathcal{K}$ is
an $A[t]/(t^{n+1})$-submodule of $\pi_*\alpha^*\muathcal{F}$ and hence $\muathcal{R}$ is an
$A[t]/(t^{n+1})$-module quotient of $\pi_*\alpha^*\muathcal{F}$.
This gives the lift $f$ of $g$ as in the diagram, which is clearly unique
and makes the diagram commute.
\etand{proof}
We apply \cref{th:lifting} to the local rings of $N(\muathcal{F}_n')$ at the
points in the image of $\iota_n$. Using the fact that $i$ is injective on the
functors of points, we deduce that $i$ induces isomorphisms on the local rings.
To see this last implication, let $\muathcal{O}_q$ denote the local ring of
$J_n'(N(\muathcal{F}))$ at a point $q$, and let $\muathcal{O}_p$ denote the local ring of
$N(\muathcal{F}_n')$ at $p = \iota_n(q)$. Let $g \colon {\muathbb S}pec \muathcal{O}_p \tauo
\Gammar(\muathcal{F}_n,(n+1)r)$ and $h \colon {\muathbb S}pec \muathcal{O}_q \tauo J_n(\Gammar(\muathcal{F},r))$ be the natural
maps, and let $j \colon {\muathbb S}pec \muathcal{O}_q \tauo {\muathbb S}pec \muathcal{O}_p$ be the map induced by $\iota_n$.
We have the following diagram:
\[
\xiymatrix{
{\muathbb S}pec\muathcal{O}_q \alphar@{^(->}[r]^(.45)h \alphar[d]^{j} & J_n(\Gammar(\muathcal{F},r)) \alphar[d]^i \pirotect\linebreak
{\muathbb S}pec\muathcal{O}_p \alphar@{^(->}[r]_(.37)g \alphar@{..>}@/^15pt/[u]^s \alphar@{..>}[ur]^(.45)f
& \Gammar(\muathcal{F}_n,(n+1)r).
}
\]
Here, the square sub-diagram is commutative, $f$ exists by \cref{th:lifting}
and hence satisfies
{\betaf e}gin{equation}
\label{eq:if=g}
i \circ f = g,
\etand{equation}
and the universal property of local rings implies that $f$ factors through $h$,
so that we have a morphism $s$, as in the diagram, satisfying
{\betaf e}gin{equation}
\label{eq:hs=f}
h \circ s = f.
\etand{equation}
Using the commutativity of the square sub-diagram and \cref{eq:if=g}, we get
\[
i\circ h = g \circ j = i \circ f \circ j.
\]
Then, using the fact that $i$ is injective at the level of functors of points
and hence is a monomorphism, we deduce that
{\betaf e}gin{equation}
\label{eq:h=fi_q}
h = f \circ j.
\etand{equation}
Now, using \cref{eq:hs=f,eq:h=fi_q}, we get
\[
h = f \circ j = h \circ s \circ j,
\]
and since $h$ is a monomorphism, this implies that $s \circ j$ is the identity
of ${\muathbb S}pec \muathcal{O}_q$. Using \cref{eq:hs=f,eq:h=fi_q} in a different order, we get
\[
f = h\circ s = f \circ j \circ s.
\]
Since $g$ is a monomorphism, it follows by \cref{eq:if=g} that $f$ is a
monomorphism, and this implies that $j \circ s$ is the identity of ${\muathbb S}pec \muathcal{O}_p$.
This proves that $j$ is an isomorphism, which completes the proof of the
theorem.
\etand{proof}
{\betaf e}gin{proof}[Proof of \cref{th:jet-Nash-blowup}]
By \cite[Theorem~B]{dFD}, there is an isomorphism
\[
\muathcal{O}m_{J_n(X)} \cong (\rho_n)_*\gamma_n^*\muathcal{O}m_X,
\]
and this implies that
\[
N(J_n'(X)) = N\betaig((\rho_n)_*\gamma_n^*\muathcal{O}m_X \circtimes_{\muathcal{O}_{J_n(X)}}\muathcal{O}_{J_n'(X)}\betaig)
= N\betaig((\rho_n')_*(\gamma_n')^*\muathcal{O}m_X\betaig),
\]
where $\rho_n'$ and $\gamma_n'$ are the restrictions of $\rho_n$ and $\gamma_n$ to $J_n'(X)
\tauimes \Delta$. Therefore \cref{th:jet-Nash-blowup} reduces to
\cref{th:jet-Nash-transform} with $\muathcal{F} = \muathcal{O}m_X$.
\etand{proof}
{\betaf e}gin{corollary}
\label{th:cor-jet-Nash-blowup}
For any variety $X$, the following properties are equivalent:
{\betaf e}gin{enumerate}
\item
the Nash blow-up $N(J_n'(X)) \tauo J_n'(X)$ is an isomorphism for some $n\gammae 0$;
\item
the Nash blow-up $N(J_n'(X)) \tauo J_n'(X)$ is an isomorphism for every $n\gammae 0$.
\etand{enumerate}
\etand{corollary}
{\betaf e}gin{proof}
By \cref{th:jet-Nash-blowup}, both properties are equivalent to the fact that
the Nash blow-up $N(X) \tauo X$ is an isomorphism.
\etand{proof}
In positive characteristics, there are examples of singular varieties whose
Nash blow-up is an isomorphism (see \cite[Example~1]{Nob75}), and
\cref{th:cor-jet-Nash-blowup} implies that this property, whenever it holds,
propagates through all the jet schemes, and conversely.
By contrast, when the ground field is algebraically closed of characteristic
zero the Nash blow-up is an isomorphism if and only if the variety is smooth
(see \cite[Theorem~2]{Nob75}). It is elementary to show that the jet schemes of
a smooth variety are smooth, and conversely it was proved in \cite{Ish09} that
if $X$ is a singular variety then all its jet schemes $J_n(X)$ are singular.
With the above assumptions on the ground field, we deduce the following
stronger statement from \cref{th:cor-jet-Nash-blowup}.
{\betaf e}gin{corollary}
If $X$ is a singular variety defined over an algebraically closed field of
characteristic zero, then the main component of $J_n'(X)$ of $J_n(X)$ is
singular for every $n$.
\etand{corollary}
\sigmaection{Computational aspects}
After viewing a projective birational morphism $\mu \colon Y \tauo X$ as the Nash
transformation of a coherent sheaf $\muathcal{F}$ on a variety $X$,
\cref{th:jet-Nash-transform} provides a construction of a relative
compactification of the induced map $\mu_n' \colon J_n'(Y) \tauo J_n'(X)$ by
taking the Nash transformation of an explicitly described sheaf $\muathcal{F}_n'$ on
$J_n'(X)$. Such transformation is a projective birational morphism, and
therefore can also be described as the blow-up of an ideal sheaf $\muathfrak{a}_n$ on
$J_n'(X)$. In this section we explain how to compute such ideal.
For simplicity, we assume that $X = {\muathbb S}pec R$ is affine. The following diagram
provides the algebraic counterpart of the restriction to $J_n'(X)$ of the
universal $n$-jet:
\[
\xiymatrix{
R_n'[t]/(t^{n+1})
&
R \alphar[l]_-{(\gammaamma_n')^\sigmaharp}
\pirotect\linebreak
R_n' \alphar[u]_-{(\rhoho_n')^\sigmaharp}
}
\]
Here $R_n'$ is a quotient of $R_n$, the algebra of Hasse--Schmidt differentials
of order at most $n$, $(\rhoho_n')^\sigmaharp$ is the natural inclusion map, and
$(\gammaamma_n')^\sigmaharp$ is induced by the homomorphism
\[
\gamma_n^\sigmaharp \colon R \tauo R_n[t]/(t^{n+1}), \quad
f \muapsto \sigmaum_{i=0}^n D_i(f)\, t^i,
\]
where $(D_0, D_1, \ldots, D_n)$ is the universal Hasse--Schmidt derivation of
order $n$. With this notation, we have $J_n(X) = {\muathbb S}pec R_n$ and $J_n'(X) =
{\muathbb S}pec R_n'$.
If $\tau(\muathcal{F})$ denotes the torsion of $\muathcal{F}$, then the two sheaves $\muathcal{F}_n'$ and
$(\muathcal{F}/\tau(\muathcal{F}))_n'$ have the same torsion free quotient. We can therefore assume
without loss of generality that $\muathcal{F}$ is torsion free. Let $F$ denote the
$R$-module associated to $\muathcal{F}$. If $r$ is the rank of $F$, we can then realize
$F$ as a submodule of $R^r$. Picking a set of generators for $F$ of cardinality
$s$, we obtain a matrix $M \in {\rhom Mat}_{r\tauimes s}(R)$ such that $F = \im M$.
Notice that, to produce an ideal whose blow-up gives $Y \tauo X$, one can take
the ideal generated by the $r \tauimes r$ minors of $M$.
The relative compactification of $\mu_n' \colon J_n'(Y) \tauo J_n'(X)$ constructed
in \cref{th:jet-Nash-transform} is given by the Nash transformation of the
$R_n'$-module
\[
F_n' := (\gammaamma_n')^\sigmaharp(F) \cdot (R_n'[t]/(t^{n+1}))^r,
\]
where the $R_n'$-module structure is defined via $(\rhoho_n')^\sigmaharp$. A
straightforward computation shows that $F_n = \im M_n$ where $M_n \in {\rhom
Mat}_{(n+1)r \tauimes (n+1)s}(R_n')$ is the matrix given in block form by
\[
M_n =
{\betaf e}gin{bmatrix}
D_0(M) & 0 & \cdots & 0 \pirotect\linebreak
D_1(M) & D_0(M) & \cdots & 0 \pirotect\linebreak
\vdots & \vdots & \partialltadots & \vdots \pirotect\linebreak
D_n(M) & D_{n-1}(M) & \cdots & D_0(M) \pirotect\linebreak
\etand{bmatrix}.
\]
Here $D_i(M)$ is the matrix obtained from $M$ by applying $D_i$ to each entry.
By construction, we have the following property.
{\betaf e}gin{proposition}
With the above notation, the morphism $N(F_n') \tauo J_n'(X)$ is the blow-up of
the ideal $\muathfrak{a}_n \sigmaubset R_n'$ generated by the $(n+1)r \tauimes (n+1)r$ minors
of $M_n$.
\etand{proposition}
The next example shows the computation of the first few ideals $\muathfrak{a}_n$ in a
simple case.
{\betaf e}gin{example}
\label{example1}
Consider $X = \muathbb A^2 = {\muathbb S}pec k[x,y]$, and let $Y \tauo X$ be the blow-up of
the maximal ideal $(x,y)$. Taking $F$ to be the maximal ideal, we have
{\betaf e}gin{align*}
M_0 &= {\betaf e}gin{bmatrix} x & y \etand{bmatrix},\pirotect\linebreak
M_1 &= {\betaf e}gin{bmatrix}
x & y & 0 & 0 \pirotect\linebreak
x_1 & y_1 & x & y \pirotect\linebreak
\etand{bmatrix},\pirotect\linebreak
M_2 &= {\betaf e}gin{bmatrix}
x & y & 0 & 0 & 0 & 0 \pirotect\linebreak
x_1 & y_1 & x & y & 0 & 0 \pirotect\linebreak
x_2 & y_2 & x_1 & y_1 & x & y \pirotect\linebreak
\etand{bmatrix}, \pirotect\linebreak
M_3 &= {\betaf e}gin{bmatrix}
x & y & 0 & 0 & 0 & 0 & 0 & 0 \pirotect\linebreak
x_1 & y_1 & x & y & 0 & 0 & 0 & 0 \pirotect\linebreak
x_2 & y_2 & x_1 & y_1 & x & y & 0 & 0 \pirotect\linebreak
x_3 & y_3 & x_2 & y_2 & x_1 & y_1 & x & y \pirotect\linebreak
\etand{bmatrix},
\etand{align*}
where $x_i = D_i(x)$ and $y_i = D_i(y)$. Letting $\muathfrak{a}_n$ be the ideal generated
by the $(n+1)\tauimes(n+1)$ minors of $M_n$, we have
{\betaf e}gin{align*}
\muathfrak{a}_0 &= (x,y),
\pirotect\linebreak
\muathfrak{a}_1 &= \muathfrak{a}_0^2 + (x y_1-y x_1),
\pirotect\linebreak
\muathfrak{a}_2 &= \muathfrak{a}_0\muathfrak{a}_1 + (y_2 x^2-y x_2 x-x_1 y_1 x+y x_1^2,\,
x_2 y^2-x_1 y_1 y-x y_2 y+x y_1^2),
\pirotect\linebreak
\muathfrak{a}_3 &= \muathfrak{a}_0\muathfrak{a}_2 + \muathfrak{a}_1^2 +
\pirotect\linebreak
& \hskip0.5cm
+ (
{\betaf e}gin{aligned}[t]
& y_3 x^3-y x_3 x^2-x_2 y_1 x^2-x_1 y_2 x^2+2 y x_1 x_2 x+x_1^2 y_1 x-y x_1^3
,\pirotect\linebreak
& y_1 y_2 x^2-y y_3 x^2-x_1 y_1^2 x+y^2 x_3 x+y^2 x_1 x_2+y x_1^2 y_1
,\pirotect\linebreak
& x_2 y_1 y^2+x_1 y_2 y^2+x y_3 y^2-x_1 y_1^2 y-2 x y_1 y_2 y+x y_1^3-y^3 x_3
,\pirotect\linebreak
& y_2^2 x^2
-y_1 y_3 x^2
+x_2 y_1^2 x
+y x_3 y_1 x
-2 y x_2 y_2 x
-x_1 y_1 y_2 x \, +\pirotect\linebreak
& \hskip2cm +y x_1 y_3 x
+y^2 x_2^2
-y^2 x_1 x_3
-y x_1 x_2 y_1
+y x_1^2 y_2
).
\etand{aligned}
\etand{align*}
Notice that while the matrices $M_n$ remain simple and have an easily
recognizable structure, the corresponding ideals $\muathfrak{a}_n$ grow in complexity
quite fast.
\etand{example}
{\betaf e}gin{bibdiv}
{\betaf e}gin{biblist}
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}
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{\taut arXiv:1703.07505 [math.AG]}}
}
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\etand{biblist}
\etand{bibdiv}
\etand{document} |
\begin{document}
\centerline{\bf On the $p$-adic Leopoldt transform of a power series}
\centerline{ Bruno Angl\` es\footnote{Universit\'e de Caen,
LMNO CNRS UMR 6139,
BP 5186, 14032 Caen Cedex, France. E-mail: [email protected]} }
${}$\par
Let $p$ be an odd prime number. Let $X$ be the projective limit for the norm maps of the $p$-Sylow subgroups of the ideal class groups of $\mathbb Q(\zeta_{p^{n+1}}),$ $n\geq 0.$ Let $\Delta ={\rm Gal}(\mathbb Q(\zeta_p)/\mathbb Q)$ and let $\theta $ be an even and non-trivial character of $\Delta. $ Then $X$ is a $\mathbb Z_p[[T]]$-module and the characteristic ideal of the isotypic component $X(\omega \theta^{-1})$ is generated by a power series $f(T,\theta)\in \mathbb Z_p[[T]]$ such that (see for example \cite{CS}):
$$\forall n\geq 1,\, n\equiv 0\pmod{p-1},\, f((1+p)^{1-n}-1,\theta)=L(1-n,\theta ),$$
where $L(s,\theta)$ is the usual Dirichlet $L$-series. Therefore, it is natural and interesting to study the properties of the power series $f(T,\theta).$\par
${}$\par
We denote by $\overline{f(T,\theta)}\in \mathbb F_p[[T]]$ the reduction of $f(T,\theta)$ modulo $p.$ Then B. Ferrero and L. Washington have proved (\cite{FW}):
$$\overline{f(T,\theta)}\not =0.$$
Note that, in fact, we have (\cite{ANG}):
$$\overline{f(T,\theta)}\not \in \mathbb F_p[[T^p]].$$
W. Sinnott has proved the following (\cite{SI2}):
$$\overline{f(T,\theta)}\not \in \mathbb F_p(T).$$
But, note that $\forall a\in \mathbb Z_p^*,$ $\mathbb F_p[[T]]=\mathbb F_p[[(1+T)^a -1]].$ Therefore it is natural to introduce the notion of a pseudo-polynomial which is an element $F(T)$ in $\mathbb F_p[[T]]$ such that there exist an integer $r\geq 1,$ $c_1,\cdots c_r\in \mathbb F_p,$ $a_1,\cdots ,a_r\in \mathbb Z_p,$ such that $F(T)=\sum_{i=1}^r c_i (1+T)^{a_i}.$ An element of $\mathbb F_p[[T]]$ will be called a pseudo-rational function if it is the quotient of two pseudo-polynomials. In this paper, we prove that $\overline{f(T,\theta)}$ is not a pseudo-rational function (part 1) of Theorem \ref{Theorem2}). This latter result suggests the following question: is $\overline{f(T,\theta)}$ algebraic over $\mathbb F_p(T)?$ We suspect that this is not the case but we have no evidence for it. Note that, by the result of Ferrero and Washington, we can write:
$$\overline{f(T,\theta)}=T^{\lambda (\theta )}U(T),$$
where $\lambda (\theta) \in \mathbb N $ and $U(T)\in \mathbb F_p[[T]]^*.$ S. Rosenberg has proved that (\cite{ROS}):
$$\lambda (\theta )\leq (4p(p-1))^{\phi (p-1)},$$
where $\phi$ is Euler's totient function. In this paper, we improve Rosenberg's bound (part 2) of Theorem \ref{Theorem2}):
$$\lambda(\theta )<(\frac{p-1}{2})^{\phi (p-1)}.$$
This implies that the lambda invariant of the field $\mathbb Q(\zeta_p)$ is less than $2(\frac{p-1}{2})^{\phi (p-1)+1}$ (see Corollary \ref{Theorem3} for the precise statement for an abelian number field). Note that this bound is certainly far from the truth, because according to a heuristic argument due to Ferrero and Washington (see \cite{LA}) and to Grennberg's conjecture:
$$\lambda (\mathbb Q(\zeta_p))=\sum_{\theta \in \widehat{\Delta},\, \theta \not =1\, {\rm and \, even }}\lambda(\theta )\leq \frac{{\rm Log}(p)}{{\rm Log}({\rm Log}(p)) }.$$
${}$\par
The author is indebted to Warren Sinnott for communicating some of his unpublished works (note that Lemma \ref{Lemma7} is due to Warren Sinnott). The author also thanks Filippo Nuccio for pointing out the work of J. Kraft and L. Washington (\cite{KW}).\par
\section{Notations}\par
${}$\par
Let $p$ be an odd prime number and let $K$ be a finite extension of $\mathbb Q_p.$ Let $O_K$ be the valuation ring of $K$ and let $\pi $ be a prime of $K.$ We set $\mathbb F_q=O_K/\pi O_K,$ it is a finite field having $q$ elements and its characteristic is $p.$ Let $T$ be an indeterminate over $K,$ we set $\Lambda =O_K[[T]].$ Observe that $\Lambda/\pi \Lambda \simeq \mathbb F_q[[T]].$ Let $F(T)\in \Lambda\setminus \{ 0\},$ then we can write in an unique way (\cite{WAS}, Theorem 7.3):
$$F(T)=\pi^{\mu(F)} P(T) U(T),$$
where $U(T)$ in an unit of $\Lambda ,$ $\mu(F)\in \mathbb N,$ $P(T)\in O_K[T]$ is a monic polynomial such that $P(T)\equiv T^{\lambda (F)}\pmod{\pi }$ for some integer $\lambda (F)\in \mathbb N.$ If $F(T)=0,$ we set $\mu(F)=\lambda(F)=\infty .$ An element $F(T)\in \Lambda$ is called a pseudo-polynomial (see also \cite{ROS}, Definition 2) if there exist some integer $r\geq 1,$ $c_1,\cdots , c_r \in O_K,$ $a_1,\cdots ,a_r\in \mathbb Z_p,$ such that:
$$F(T)=\sum_{i=1}^r c_i (1+T)^{a_i}.$$
We denote the ring of pseudo-polynomials in $\Lambda $ by $A.$ Let $\delta \in \mathbb Z/(p-1)\mathbb Z$ and $F(T)\in \Lambda ,$ we set:
$$\gamma_{\delta } (F(T))= \frac{1}{p-1} \sum_{\eta \in \mu_{p-1}} \eta^{\delta } F((1+T)^{\eta }-1).$$
Then $\gamma_{\delta}:\Lambda \rightarrow \Lambda$ is a $O_K$-linear map and:\par
\noindent - for $\delta ,\delta'\in \mathbb Z/(p-1)\mathbb Z,$ $\gamma_{\delta}\gamma_{\delta'}=0$ if $\delta \not = \delta'$ and $\gamma_{\delta}^2=\gamma_{\delta},$\par
\noindent - $\sum_{\delta \in \mathbb Z/(p-1)\mathbb Z} \gamma_{\delta }={\rm Id}_{\Lambda}.$\par
\noindent For $F(T)\in \Lambda,$ we set:
$$D(F(T))=(1+T) \frac{d}{dT} F(T),$$
$$U(F(T))=F(T)-\frac{1}{p}\sum_{\zeta\in \mu_p} F(\zeta (1+T)-1)\, \in \Lambda .$$
Then $D,U: \Lambda \rightarrow \Lambda $ are $O_K$-linear maps. Observe that:\par
\noindent - $U^2=U,$\par
\noindent - $DU=UD,$\par
\noindent- $\forall \delta \in \mathbb Z/(p-1)\mathbb Z,$ $ \gamma_{\delta }U=U\gamma_{\delta},$\par
\noindent - $\forall \delta \in \mathbb Z/(p-1)\mathbb Z,$ $D\gamma_{\delta }=\gamma_{\delta +1}D.$\par
\noindent If $F(T)\in \Lambda,$ we denote its reduction modulo $\pi$ by $\overline{F(T)}\in \mathbb F_q[[T]].$ If $f:\Lambda \rightarrow \Lambda$ is a $O_K$-linear map, we denote its reduction modulo $\pi$ by $\overline{f}: \mathbb F_q[[T]]\rightarrow \mathbb F_q[[T]].$ For all $n\geq 0,$ we set $\omega_n(T)=(1+T)^{p^n}-1.$\par
Let $B$ be a commutative and unitary ring. We denote the set of invertible elements of $B$ by $B^*.$\par
We fix $\kappa $ a topological generator of $1+p\mathbb Z_p.$ Let $x\in \mathbb Z_p$ and let $n\geq 1,$ we denote the unique integer $k\in \{ 0,\cdots , p^n-1\}$ such that $x\equiv k\pmod{p^n}$ by $[x]_n.$ Let $\omega :\mathbb Z_p^*\rightarrow \mu_{p-1}$ be the Teichm\"uller character, i.e. $\forall a\in \mathbb Z_p^*,$ $\omega (a)\equiv a \pmod{p}.$ Let $x,y\in \mathbb Z_p,$ we write:\par
\noindent - $x\sim y$ if there exists $\eta \in \mu_{p-1}$ such that $y=\eta x,$\par
\noindent - $x\equiv y \pmod {\mathbb Q^*}$ if there exists $z\in \mathbb Q^*$ such that $y=zx.$\par
\noindent The function ${\rm Log}_p$ will denote the usual $p$-adic logarithm. $v_p$ will denote the usual $p$-adic valuation on $\mathbb C_p$ such that $v_p(p)=1. $\par
Let $\rho$ be a Dirichlet character of conductor $f_{\rho}.$ Recall that the Bernoulli numbers $B_{n,\rho}$ are defined by the following identity:
$$\sum_{a=1}^{f_{\rho}}\frac{\rho (a) e^{aZ}}{e^{fZ}-1}=\sum_{n\geq 0} \frac{B_{n,\rho }}{n!} Z^{n-1},$$
where $e^Z=\sum_{n\geq 0} Z^n/n!.$ If $\rho =1,$ for $n\geq 2,$ $B_{n,1}$ is the $n$th Bernoulli number.\par
Let $x\in \mathbb R.$ We denote the biggest integer less than or equal to $x$ by $[x].$ The function ${\rm Log}$ will denote the usual logarithm.\par
\section{Preliminaries}
${}$\par
Let $\delta \in \mathbb Z/(p-1)\mathbb Z.$ In this section, we will recall the construction of the $p$-adic Leopoldt transform $\Gamma_{\delta }$ (see \cite{LA}, Theorem 6.2) which is a $O_K$-linear map from $\Lambda $ to $\Lambda .$\par
First, observe that $(\pi^n, \omega_n(T))= \pi^n \Lambda +\omega_n(T)\Lambda ,$ $n\geq 1,$ is a basis of neighbourhood of zero in $\Lambda :$\par
\newtheorem{Lemma1}{Lemma}[section]
\begin{Lemma1} \label{Lemma1}
${}$\par
\noindent 1) $\forall n\geq 1,$ $(\pi , T)^{2n}\subset (\pi^n, T^n)\subset (\pi ,T)^n.$\par
\noindent 2) $\forall n\geq 1,$ $\omega_n (T)\in (p^{[n/2]}, T^{p^{[n/2]+1}}).$\par
\noindent 3) Let $N\geq 1,$ set $n=[{\rm Log}(N)/{\rm Log}(p)].$ We have:
$$T^N\in (p^{[n/2]}, \omega_{[n/2]+1}(T)).$$
\end{Lemma1}
\noindent{\sl Proof} Note that assertion 1) is obvious. Assertion 2) comes from the fact:
$$\forall k\in \{ 1,\cdots, p^n\},\, v_p( \frac{p^n!}{k! (p^n-k)!})=n-v_p(k).$$
To prove assertion 3), it is enough to prove the following:\par
\noindent $\forall n\geq 0,$ there exist $\delta_0^{(n)}(T),\cdots , \delta_n^{(n)}(T) \in \mathbb Z [T]$ such that:
$$T^{p^n}=\sum_{i+j=n} \omega_i(T) p^j \delta_j^{(n)}(T).$$
Let's prove this latter fact by recurrence on $n.$ Note that the result is clear if $n=0.$ Let's assume that it is true for $n$ and let's prove the assertion for $n+1.$ Let $r(T)\in \mathbb Z[T]$ such that:
$$\frac{\omega_{n+1}(T)}{\omega_n(T)} +pr(T)=T^{p^n(p-1)}.$$
Then:
$$T^{p^{n+1}}=T^{p^n} \frac{\omega_{n+1}(T)}{\omega_n(T)} +pr(T) T^{p^n}.$$
Note that there exists $q(T) \in \mathbb Z[T]$ such that:
$$\frac{\omega_{n+1}(T)}{\omega_n(T)}=\omega_n(T)^{p-1}+pq(T).$$
Thus:
$$T^{p^{n+1}}=\omega_{n+1}(T)\delta_0^{(n)}(T) +\, \sum_{i+j=n,\, j\geq 1}( \omega_n(T)^{p-1}+pq(T)) \omega_i(T) p^j \delta_j^{(n)}(T)\, + \sum_{i+j=n} \omega_i(T) p^{j+1} \delta_j^{(n)}(T) r(T).$$
Thus, there exist $\delta_0^{(n+1)}(T), \cdots , \delta_{n+1}^{(n+1)}(T) \in \mathbb Z[T]$ such that:
$$T^{p^{n+1}}=\sum_{i+j=n+1} \omega_i(T)p^j \delta_j^{(n+1)} (T).\, \diamondsuit $$
The following Lemma will be useful in the sequel (for a similar result see \cite{ROS}, Lemma 5):\par
\newtheorem{Lemma2}[Lemma1]{Lemma}
\begin{Lemma2} \label{Lemma2}
Let $F(T)\in A.$ Write $F(T)=\sum_{i=1}^r \beta_i (1+T)^{\alpha_i},$ $\beta_1,\cdots, \beta_r\in O_K,$ $\alpha_1,\cdots ,\alpha_r \in \mathbb Z_p,$ and $\alpha_i\not = \alpha_j$ for $i\not = j.$ Let $N={\rm Max}\{ v_p(\alpha_i-\alpha_j),i\not =j\}.$ Let $n\geq 1$ be an integer. Then:
$$F(T)\equiv 0\pmod{(\pi^n,\omega_{N+1}(T))} \Leftrightarrow \forall i=1,\cdots r, \, \beta_i\equiv 0\pmod{\pi^n}.$$
\end{Lemma2}
\noindent{\sl Proof} We have:
$$F(T)\equiv \sum_{i=1}^r \beta_i (1+T)^{[\alpha_i]_{N+1}}\pmod{\omega_{N+1}(T)}.$$
Therefore $F(T)\equiv 0\pmod{(\pi^n,\omega_{N+1}(T))}$ if and only if we have:
$$\sum_{i=1}^r\beta_i (1+T)^{[\alpha_i]_{N+1}}\equiv 0\pmod{\pi^n}.$$
But for $i\not =j,$ $[\alpha_i]_{N+1}\not = [\alpha_j]_{N+1}.$ Therefore $\sum_{i=1}^r\beta_i (1+T)^{[\alpha_i]_{N+1}}\equiv 0\pmod{\pi^n}$ if and only if:
$$\forall i=1,\cdots r, \, \beta_i\equiv 0\pmod{\pi^n}.\, \diamondsuit $$
Observe that $U,D, \gamma_{\delta}$ are continuous $O_K$-linear maps by Lemma \ref{Lemma1} and the following Lemma:\par
\newtheorem{Lemma3}[Lemma1]{Lemma}
\begin{Lemma3} \label{Lemma3}
Let $F(T)\in \Lambda$ and let $n\geq 0.$\par
\noindent 1) $F(T)\equiv 0\pmod{\omega_n(T)}\Rightarrow \gamma_{\delta}(F(T))\equiv 0\pmod{\omega_n(T)}.$\par
\noindent 2) $F(T)\equiv 0\pmod{\omega_n(T)}\Rightarrow D(F(T))\equiv 0\pmod{(p^n,\omega_n(T))}.$\par
\noindent 3) If $n\geq 1,$ $F(T)\equiv 0\pmod{\omega_n(T)}\Rightarrow U(F(T))\equiv 0\pmod{\omega_n(T)}.$\par
\end{Lemma3}
\noindent{\sl Proof} The assertions 1) and 2) are obvious. It remains to prove 3). Observe that, by \cite{WAS}, Proposition 7.2, we have:
$$\forall G(T)\in \Lambda,\, G(T)\equiv 0\pmod{\omega_n(T)} \Leftrightarrow \forall \zeta \in \mu_{p^n},\, G(\zeta-1)=0.$$
Now, let $F(T)\in \Lambda ,$ $F(T)\equiv 0\pmod{\omega_n(T)}.$ Since the map: $\mu_{p^n}\rightarrow \mu_{p^n},$ $x\mapsto \zeta x,$ is a bijection for all $\zeta\in \mu_{p^n},$ we get:
$$\forall \zeta\in \mu_{p^n},\, U(F)(\zeta-1)=0.$$
Therefore:
$$U(F(T))\equiv 0\pmod{\omega_n(T)}.\, \diamondsuit $$
Let $s\in \mathbb Z_p.$ For $n\geq 0,$ set:
$$k_n(s,\delta)=[s]_{n+1}+\delta_n p^{n+1}\, \in \mathbb N\setminus\{ 0\},$$
where $\delta_n\in \{ 1, \cdots ,p-1\}$ is such that $[s]_{n+1}+\delta_n \equiv \delta \pmod{p-1}.$ Observe that:\par
\noindent - $\forall n\geq 0,$ $k_n(s,\delta)\equiv \delta \pmod{p-1}$ and $k_n(s,\delta)\equiv s\pmod{p^{n+1}},$\par
\noindent - $\forall n\geq 0,$ $k_{n+1}(s,\delta)>k_n(s,\delta),$\par
\noindent - $s={\rm lim}_nk_n(s,\delta).$\par
\noindent In particular:
$$\forall a\in \mathbb Z_p,\, \forall n\geq 0,\, a^{k_{n+1}(s,\delta)}\equiv a^{k_n(s,\delta)}\pmod{p^{n+1}}.$$
Now, let $F(T)\in A.$ Write $F(T)=\sum_{i=1}^r \beta_i (1+T)^{\alpha_i},$ $\beta_1,\cdots ,\beta_r\in O_K,$ $\alpha_1,\cdots ,\alpha_r \in \mathbb Z_p.$ We set:
$$\Gamma_{\delta }(F(T))=\sum_{\alpha_i\in \mathbb Z_p^*} \beta_i \omega^{\delta}(\alpha_i) (1+T)^{\frac{{\rm Log}_p(\alpha_i)}{{\rm Log}_p(\kappa)}}.$$
Thus, we have a surjective $O_K$-linear map: $\Gamma_{\delta }: A\rightarrow A.$ Note that:\par
\newtheorem{Lemma4}[Lemma1]{Lemma}
\begin{Lemma4} \label{Lemma4}
Let $F(T)\in A.$\par
\noindent 1) Let $s\in \mathbb Z_p.$ Then:
$$\forall n\geq 0, \, \Gamma_{\delta} (F)(\kappa^s-1)\equiv D^{k_n(s,\delta)}(F)(0)\pmod{p^{n+1}}.$$
2) Let $n\geq 1.$ Assume that $F(T)\equiv 0\pmod{\omega_n(T)}.$ Then $\Gamma_{\delta }(F(T))\equiv 0\pmod{\omega_{n-1}(T)}.$\par
\end{Lemma4}
\noindent {\sl Proof} For $a\in \mathbb Z_p^*,$ write $a=\omega (a)\, <a>,$ where $<a>\in 1+p\mathbb Z_p.$ Let's write:
$$F(T)=\sum_{i=1}^r\beta_i(1+T)^{\alpha_i},$$
$\beta_1,\cdots ,\beta_r \in O_K,$ $\alpha_1,\cdots ,\alpha_r \in \mathbb Z_p.$ We have:
$$D^{k_n(s,\delta)}(F(T))=\sum_{i=1}^r\beta_i \alpha_i^{k_n(s,\delta)}(1+T)^{\alpha_i}.$$
Thus:
$$D^{k_n(s,\delta)}(F(T))\equiv \sum_{\alpha_i \in \mathbb Z_p^*}\beta_i \omega^{\delta }(\alpha_i)<\alpha_i>^s(1+T)^{\alpha_i}\pmod{p^{n+1}}.$$
But recall that:
$$\Gamma_{\delta}(F)(\kappa^s-1)=\sum_{\alpha_i\in \mathbb Z_p^*}\beta_i \omega^{\delta}(\alpha_i)<\alpha_i>^s.$$
Assertion 1) follows easily. Now, let's suppose that $F(T)\equiv 0\pmod{\omega_n(T)}$ for some $n\geq 1.$ Then:
$$\forall a\in\{ 0,\cdots ,p^n-1\},\, \sum_{\alpha_i\equiv a\pmod{p^n}}\beta_i =0.$$
This implies that:
$$\forall a\in\{ 0,\cdots, p^{n-1}-1\},\, \sum_{\alpha_i\in \mathbb Z_p^*,\, {\rm Log}_p(\alpha_i)/{\rm Log}_p(\kappa)\equiv a \pmod{p^{n-1}}}\omega^{\delta }(\alpha_i) \beta_i =0.$$
But recall that:
$$\Gamma_{\delta }(F(T))=\sum_{\alpha_i\in \mathbb Z_p^*} \beta_i \omega^{\delta}(\alpha_i) (1+T)^{\frac{{\rm Log}_p(\alpha_i)}{{\rm Log}_p(\kappa)}}.$$
Thus $\Gamma_{\delta }(F(T))\equiv 0\pmod{\omega_{n-1}(T)}.\, \diamondsuit $\par
\newtheorem{Proposition1}[Lemma1]{Proposition}
\begin{Proposition1} \label{Proposition1}
Let $F(T)\in \Lambda.$ There exists an unique power series $\Gamma_{\delta}(F(T))\in \Lambda$ such that:
$$\forall s\in \mathbb Z_p\, \forall n\geq 0,\, \Gamma_{\delta} (F)(\kappa^s-1)\equiv D^{k_n(s,\delta)}(F)(0)\pmod{p^{n+1}}.$$
\end{Proposition1}
\noindent{\sl Proof} Let $(F_N(T))_{N\geq 0}$ be a sequence of elements in $A$ such that:
$$\forall N\geq 0,\, F(T)\equiv F_N(T)\pmod{\omega_N(T)}.$$
Fix $N\geq 1.$ Then:
$$\forall m\geq N, F_m(T)\equiv F_N(T)\pmod{\omega_N(T)}.$$
Therefore, by Lemma \ref{Lemma4}, we have:
$$\forall m\geq N, \Gamma_{\delta}(F_m(T))\equiv \Gamma_{\delta}(F_N(T))\pmod{\omega_{N-1}(T)}.$$
This implies that the sequence $(\Gamma_{\delta}(F_N(T)))_{N\geq 1}$ converges in $\Lambda $ to some power series $G(T)\in \Lambda.$ Observe that, since $\Lambda $ is compact, we have:
$$\forall N\geq 1, G(T)\equiv \Gamma_{\delta}(F_N(T))\pmod{\omega_{N-1}(T)}.$$
In particular:
$$\forall N\geq 1, G(\kappa^s-1)\equiv \Gamma_{\delta }(F_N)(\kappa^s-1)\pmod{p^N}.$$
Thus, applying Lemma \ref{Lemma4}, we get:
$$\forall N\geq 1, G(\kappa^s-1)\equiv D^{k_{N-1}(s,\delta )}(F_N)(0)\pmod{p^N}.$$
But:
$$\forall N\geq 1, D^{k_{N-1}(s,\delta )}(F(T))\equiv D^{k_{N-1}(s,\delta )}(F_N(T))\pmod{(p^N,\omega_N(T))}.$$
Therfore:
$$\forall N\geq 1, G(\kappa^s -1)\equiv D^{k_{N-1}(s,\delta )}(F)(0)\pmod{p^N}.$$
Now, set $\Gamma_{\delta }(F(T))=G(T).$ The Proposition follows easily. $\diamondsuit$\par
\section{Some properties of the $p$-adic Leopoldt transform}
${}$\par
We need the following fundamental result:\par
\newtheorem{Proposition2}{Proposition}[section]
\begin{Proposition2} \label{Proposition2}
Let $\delta \in \mathbb Z/(p-1)\mathbb Z$ and let $F(T)\in \Lambda.$ Let $m,n\in \mathbb N\setminus \{ 0\}.$ then:
$$\Gamma_{\delta}(F(T))\equiv 0\pmod{(\pi^n,\omega_{m-1}(T))} \Leftrightarrow \gamma_{-\delta}U(F(T)) \equiv 0\pmod{(\pi^n,\omega_{m}(T))}.$$
\end{Proposition2}
\noindent{\sl Proof} A similar result has been obtained by S. Rosenberg (\cite{ROS}, Lemma 8). We begin by proving that $\Gamma_{\delta}$ is a continuous $O_K$-linear map. By Lemma \ref{Lemma1}, this comes from the following fact:\par
\noindent Let $F(T)\in \Lambda.$ Let $n\geq 1$ and assume that $F(T)\equiv 0 \pmod{\omega_n(T)},$ then $\Gamma_{\delta }(F(T))\equiv 0\pmod{\omega_{n-1}(T)}.$\par
\noindent Indeed, let $(F_N(T))_{N\geq 0}$ be a sequence of elements in $A$ such that:
$$\forall N\geq 0, \, F(T)\equiv F_N(T) \pmod{\omega_{N}(T)}.$$
By the proof of Proposition \ref{Proposition1}:
$$\forall N\geq 1,\, \Gamma_{\delta}(F(T))\equiv \Gamma_{\delta}(F_N(T))\pmod{\omega_{N-1}(T)}.$$
Now, by Lemma \ref{Lemma4}:
$$\Gamma_{\delta}(F_n(T))\equiv 0\pmod{\omega_{n-1}(T)}.$$
The assertion follows.\par
\noindent Now, since $\Gamma_{\delta}, \gamma_{-\delta},U$ are continuous $O_K$-linear maps, it suffices to prove the Proposition in the case where $F(T)\in A.$ Write $F(T)=\sum_{i=1}^r \beta_i (1+T)^{\alpha_i},$ $\beta_1,\cdots ,\beta_r \in O_K,$ $\alpha_1,\cdots ,\alpha_r \in \mathbb Z_p.$ Let $I\subset \{ \alpha_1,\cdots ,\alpha_r\}$ be a set of representatives of the classes of $\alpha_1, \cdots ,\alpha_r$ for the relation $\sim .$ For $x\in I,$ $x\not \equiv 0\pmod{p},$ set:
$$\beta_x=\sum_{\alpha_i\sim x}\beta_i \frac{\alpha_i}{x}.$$
We get:
$$(p-1)\gamma_{-\delta}U(F(T))=\sum_{\eta\in \mu_{p-1}}\sum_{x\in I,\, x\in \mathbb Z_p^*}\eta^{-\delta}\beta_x (1+T)^{\eta x}.$$
Now observe that:
$$\Gamma_{\delta}(F(T)) =\Gamma_{\delta} \gamma_{-\delta }U (F(T))=\sum_{x\in I, \, x\in \mathbb Z_p^*}\beta_x \omega^{\delta}(x) (1+T)^{{\rm Log}_p(x)/{\rm Log}_p(\kappa )}.$$
Therefore $\Gamma_{\delta}(F(T))\equiv 0\pmod{(\pi^n,\omega_{m-1}(T))} $ if and only if:
$$\forall a \in \{ 0,\cdots p^{m-1}-1\},\, \sum_{x\in I,\, x\in \mathbb Z_p^*,\, {\rm Log}_p(x)/{\rm Log}_p(\kappa )\equiv a\pmod{p^{m-1}}}\beta_x \omega^{\delta} (x) \equiv 0\pmod{\pi^n}.$$
Now, observe that for $a\in \{ 0,\cdots , p^{m}-1\},$ there exists at most one $\eta \in \mu_{p-1}$ such that $[\eta x]_m=a,$ and if such a $\eta $ exists it is equal to $\omega (a) \omega^{-1}(x).$ Therefore $\Gamma_{\delta}(F(T))\equiv 0\pmod{(\pi^n,\omega_{m-1}(T))} $ if and only if:
$$\forall a \in \{ 0,\cdots ,p^m-1\},\, \sum _{x\in I,\, x\in \mathbb Z_p^*,\, \exists \eta_x \in \mu_{p-1},[\eta_x x]_m=a} \beta_x \eta_x^{-\delta}\equiv 0\pmod{\pi ^n}.$$
This latter property is equivalent to $ \gamma_{-\delta}U(F(T)) \equiv 0\pmod{(\pi^n,\omega_{m}(T))}.\, \diamondsuit $\par
Now, we can list the basic properties of $\Gamma_{\delta}:$\par
\newtheorem{Proposition3}[Proposition2]{Proposition}
\begin {Proposition3} \label{Proposition3}
Let $\delta \in \mathbb Z/(p-1)\mathbb Z.$\par
\noindent 1) $\Gamma_{\delta}:\Lambda \rightarrow \Lambda$ is a surjective and continuous $O_K$-linear map.\par
\noindent 2) $\forall F(T)\in \Lambda,$ $\Gamma_{\delta}(F(T))=\Gamma_{\delta }\gamma_{-\delta}U(F(T)).$\par
\noindent 3) $\forall a \in \mathbb Z_p^*,$ $\Gamma_{\delta} (F((1+T)^a-1))= \omega^{\delta }(a) (1+T)^{{\rm Log}_p(a)/{\rm Log}_p (\kappa)}\Gamma_{\delta }(F(T)).$\par
\noindent 4) Let $\kappa'$ be another topological generator of $1+p\mathbb Z_p$ and let $\Gamma_{\delta}'$ be the $p$-adic Leopoldt transform associated to $\kappa'$ and $\delta .$ Then:
$$\forall F(T)\in \Lambda ,\, \Gamma_{\delta}'(F(T)) =\Gamma_{\delta}(F)((1+T)^{{\rm Log}_p(\kappa)/{\rm Log}_p(\kappa')}-1).$$
5) Let $F(T)\in \Lambda.$ Then $\mu(\Gamma_{\delta }(F(T)))=\mu (\gamma_{-\delta}U(F(T)))$ and:
$$\forall N\geq 1,\, \lambda (\Gamma_{\delta}(F(T)))\geq p^{N-1} \Leftrightarrow \lambda (\gamma_{-\delta}U(F(T)))\geq p^N.$$
\end{Proposition3}
\noindent{\sl Proof} The assertions 1),2),3),4) come from the fact that $\Gamma_{\delta}, \gamma_{-\delta}, U$ are continuous and that these assertions are true for pseudo-polynomials. The assertion 5) is a direct application of Proposition \ref{Proposition2} . $\diamondsuit$\par
Let's recall the following remarkable result due to W. Sinnott:
\newtheorem{Proposition4}[Proposition2]{Proposition}
\begin{Proposition4} \label{Proposition4}
Let $r_1(T),\cdots ,r_s(T)\in \mathbb F_q(T)\cap \mathbb F_q[[T]].$ Let $c_1,\cdots ,c_s\in \mathbb Z_p\setminus \{ 0\}$ and suppose that:
$$\sum_{i=1}^s r_i((1+T)^{c_i}-1)=0.$$
Then:
$$\forall a\in \mathbb Z_p,\, \sum_{c_i \equiv a\pmod{\mathbb Q^*}}r_i((1+T)^{c_i}-1)\, \in \mathbb F_q.$$
\end{Proposition4}
\noindent{\sl Proof} See \cite{SI2}, Proposition 1. $\diamondsuit$\par
Let's give a first application of this latter result:
\newtheorem{Proposition5}[Proposition2]{Proposition}
\begin{Proposition5} \label{Proposition5}
Let $\delta \in \mathbb Z/(p-1)\mathbb Z$ and let $F(T)\in K(T)\cap \Lambda.$\par
\noindent 1) If $\delta $is odd or if $\delta =0,$ then:
$$\mu (\Gamma_{\delta }(F(T)))=\mu (U(F(T))+(-1)^{\delta }U(F((1+T)^{-1}-1))).$$
2) If $\delta $is even and $\delta \not =0,$ then:
$$\mu (\Gamma_{\delta }(F(T)))=\mu (U(F(T))+U(F((1+T)^{-1}-1))-2U(F)(0)).$$
\end{Proposition5}
\noindent{\sl Proof} The case $\delta =0$ has already been obtained by Sinnott (\cite{SI1}, Theorem 1). We prove 1), the proof of 2) is quite similar. Now, observe that 1) is a consequence of Proposition \ref{Proposition3} and the following fact:\par
\noindent Let $F(T)\in K(T)\cap \Lambda ,$ then $\mu(\gamma_{-\delta }(F(T)))=\mu (F(T)+(-1)^{\delta }F((1+T)^{-1}-1)).$\par
\noindent Let's prove this fact. Let $r(T)\in \Lambda,$ observe that:
$$\gamma_{-\delta }(r(T))= (-1)^{\delta} \gamma_{-\delta}(r((1+T)^{-1}-1)).$$
We can assume that $F(T)+(-1)^{\delta}F((1+T)^{-1}-1)\not =0.$ Write:
$$F(T)+(-1)^{\delta}F((1+T)^{-1}-1)=\pi^{m}G(T),$$
where $m\in \mathbb N,$ and $G(T)\in \Lambda \setminus \pi \Lambda .$ Note that $G(T)\in K(T).$ We must prove that $\gamma_{-\delta}(G(T))\not \equiv 0\pmod{\pi}.$ Suppose that it is not the case, i.e. $\gamma_{-\delta}(G(T)) \equiv 0\pmod{\pi}.$ Then:
$$G(0)\equiv 0\pmod{\pi}.$$
Furthermore, by Proposition \ref{Proposition4}, there exists $c\in O_K$ such that:
$$G(T)+(-1)^{\delta}G((1+T)^{-1}-1)\equiv c\pmod{\pi}.$$
But, we must have $c\equiv 0\pmod{\pi}.$ Observe that:
$$G(T)=(-1)^{\delta}G((1+T)^{-1}-1).$$
Therefore we get $G(T)\equiv 0\pmod{\pi}$ which is a contradiction. $\diamondsuit$\par
\newtheorem{Lemma5}[Proposition2]{Lemma}
\begin{Lemma5} \label{Lemma5}
Let $F(T)\in \mathbb F_q(T)\cap \mathbb F_q[[T]].$ Then $F(T)$ is a pseudo-polynomial if and only if there exists some integer $n\geq 0$ such that $(1+T)^n F(T)\in \mathbb F_q[T].$
\end{Lemma5}
\noindent{\sl Proof} Assume that $F(T)$ is a pseudo-polynomial. We can suppose that $F(T)\not =0.$ Write:
$$F(T)=\sum_{i=1}^r c_i (1+T)^{a_i},$$
where $c_1,\cdots, c_r \in \mathbb F_q^*,$ $a_1,\cdots a_r\in \mathbb Z_p$ and $a_i\not = a_j$ for $i\not =j.$ Since $F(T)\in \mathbb F_q(T)$ there exist $m,n\in \mathbb N\setminus\{0\},$ $m>{\rm Max}\{ v_p(a_i-a_j),\, i\not =j\},$ such that:
$$(T^{q^n}-T)^{q^{m}}F(T)\in \mathbb F_q[T].$$
Thus:
$$\sum_{i=1}^rc_i (1+T)^{a_i+q^{n+m}}-\sum_{i=1}^rc_i (1+T)^{a_i+q^{m}}\, \in \mathbb F_q[T].$$
Observe that:\par
\noindent- $\forall i,j \in \{ 1,\cdots ,r\},$ $a_i+q^{n+m}\not = a_j+q^m,$\par
\noindent - $a_i+q^m=a_j+q^m \Leftrightarrow i=j.$\par
\noindent Thus, by Lemma \ref{Lemma2}, we get:
$$\forall i\in \{ 1,\cdots r\}, \, a_i+q^m \in \mathbb N.$$
Therefore $(1+T)^{q^m}F(T)\in \mathbb F_q[T].$ The Lemma follows. $\diamondsuit$\par
Let's give a second application of Proposition \ref{Proposition4}:
\newtheorem{Proposition6}[Proposition2]{Proposition}
\begin{Proposition6} \label{Proposition6}
Let $\delta \in \mathbb Z/(p-1)\mathbb Z$ and let $F(T)\in \mathbb F_q(T)\cap \mathbb F_q[[T]].$ Suppose that there exist an integer $r\in \{ 0,\cdots, (p-3)/2\},$ $c_1,\cdots c_r \in \mathbb Z_p\setminus\{ 0\},$ $G_1(T),\cdots , G_r(T) \in \mathbb F_q(T)\cap \mathbb F_q[[T]]$ and a pseudo-polynomial $R(T)\in \mathbb F_q[[T]]$ such that:
$$\overline{\gamma_{\delta}}(F(T))=R(T)+\sum_{i=1}^r G_i((1+T)^{c_i}-1).$$
Then, there exists an integer $n\geq 0$ such that:
$$(1+T)^n (F(T)+(-1)^{\delta} F((1+T)^{-1}-1))\, \in \mathbb F_q[T].$$
\end{Proposition6}
\noindent{\sl Proof} Note that if $\eta,\eta'\in \mu_{p-1}:$ $\eta\equiv \eta'\pmod{\mathbb Q^*}\Leftrightarrow \eta=\eta'\, {\rm or}\, \eta =-\eta'.$ Since $r<(p-1)/2,$ by Proposition \ref{Proposition4}, there exists $\eta \in \mu_{p-1}$ such that:
$$\overline{\eta}^{\delta} F((1+T)^{\eta }-1)+\overline{-\eta}^{\delta} F((1+T)^{-\eta }-1)\, {\rm is\, a \, pseudo-polynomial}.$$
Therefore:
$$F(T)+(-1)^{\delta} F((1+T)^{-1}-1)\, {\rm is\, a \, pseudo-polynomial}.$$
It remains to apply Lemma \ref{Lemma5}. $\diamondsuit$\par
Let $F(T)\in \Lambda.$ We say that $F(T)$ is a pseudo-rational function if $F(T)$ is the quotient of two pseudo-polynomials. For example, $\forall a\in \mathbb Z_p,$ $\forall b\in \mathbb Z_p^*,$ $\frac{(1+T)^a -1}{(1+T)^b-1}$ is a pseudo-rational function. We finish this section by giving a generalization of \cite{SI2}, Theorem1:
\newtheorem{Theorem1}[Proposition2]{Theorem}
\begin{Theorem1} \label{Theorem1}
Let $\delta \in \mathbb Z/(p-1)\mathbb Z$ and let $F(T)\in \mathbb F_q(T)\cap \mathbb F_q[[T]].$ Then $\overline{\Gamma_{\delta}}(F(T))$ is a pseudo-rational function if and only if there exists some integer $n\geq 0$ such that:
$$(1+T)^n(\overline{U}(F(T))+(-1)^{\delta}\overline{U}(F((1+T)^{-1}-1)))\, \in \mathbb F_q[T].$$
\end{Theorem1}
\noindent{\sl Proof} Assume that $\overline{\Gamma_{\delta}}(F(T))$ is a pseudo-rational function. Then , by 3) of Proposition \ref{Proposition3} and Proposition \ref{Proposition2}, there exist $c_1,\cdots, c_r \in \mathbb F_q^*,$ $a_1,\cdots ,a_r \in \mathbb Z_p,$ $a_i\not = a_j$ for $i\not =j,$ such that:
$$\overline{\Gamma_{\delta}}\overline{\gamma_{-\delta}}\overline{ U}(\sum_{i=1}^r c_i F((1+T)^{\kappa^{a_i}}-1)))\, {\rm is\, a \, pseudo-polynomial}.$$
This implies, again by Proposition \ref{Proposition2}, that:
$$\overline{\gamma_{-\delta}}\overline{ U}(\sum_{i=1}^r c_i F((1+T)^{\kappa^{a_i}}-1)))\, {\rm is\, a \, pseudo-polynomial}.$$
Set:
$$G(T)=\overline{U}(F(T))+(-1)^{\delta}\overline{U}(F((1+T)^{-1}-1))\, \in \mathbb F_q(T)\cap \mathbb F_q[[T]].$$
Now, by Proposition \ref{Proposition4}, there exist $d_1,\cdots ,d_{\ell}\in \mathbb F_q^*,$ $b_1,\cdots b_{\ell}\in \mathbb Z_p,$ $b_i\not =b_j$ for $i\not =j,$ $\eta_1,\cdots , \eta_{\ell}\in \mu_{p-1},$ with $\forall i,j\in \{1,\cdots ,\ell \},$ $\eta_i\kappa^{b_i}\equiv \eta_j\kappa^{b_j}\pmod{\mathbb Q^*},$ and $\eta_i\kappa^{b_i}\not = \eta_j\kappa^{b_j}$ for $i\not = j,$ such that:
$$\sum_{i=1}^{\ell}d_iG((1+T)^{{\eta_i}\kappa^{b_i}}-1) \, {\rm is\, a \, pseudo-polynomial}.$$
For $i=1,\cdots , \ell,$ write:
$$\eta_i\kappa^{b_i}= \eta_1\kappa^{b_1}x_i,$$
where $x_i\in \mathbb Q^*\cap \mathbb Z_p^*,$ and $x_i\not = x_j$ for $i\not = j.$
Since $G(T)=(-1)^{\delta} G((1+T)^{-1}-1),$ we can assume that $x_1,\cdots x_{\ell}$ are positives. Now, we get:
$$\sum_{i=1}^{\ell}d_iG((1+T)^{x_i}-1) \, {\rm is\, a \, pseudo-polynomial}.$$
Therefore, there exist $N_1,\cdots ,N_{\ell}\in \mathbb N\setminus\{0\},$ $N_i\not =N_j$ for $i\not =j,$ such that:
$$\sum_{i=1}^{\ell}d_iG((1+T)^{N_i}-1) \, {\rm is\, a \, pseudo-polynomial}.$$
Now, by Lemma \ref{Lemma5}, there exists some integer $N\geq 0$ such that:
$$(1+T)^N(\sum_{i=1}^{\ell}d_iG((1+T)^{N_i}-1)) \, \in \mathbb F_q[T].$$
But, since $G(T)\in \mathbb F_q(T)\cap \mathbb F_q[[T]],$ $d_1,\cdots ,d_{\ell}\in \mathbb F_q^*,$ $N_1,\cdots N_{\ell }\in \mathbb N\setminus \{ 0\}$ and $N_i\not =N_J$ for $i\not =j,$ this implies that there exist some integer $n\geq 0$ such that $(1+T)^n G(T)\in \mathbb F_q[T].\, \diamondsuit$\par
\section{Application to Kubota-Leopoldt $p$-adic L-functions}
${}$\par
Let $\theta$ be a Dirichlet character of the first kind, $\theta \not =1$ and $\theta$ even. We denote by $f(T,\theta)$ the Iwasawa power series attached to the $p$-adic L-function $L_p(s,\theta)$ (see \cite{WAS}, Theorem 7.10). Write:
$$\theta =\chi \omega^{\delta +1},$$
where $\chi$ is of conductor $d,$ $d\geq 1$ and $d\not \equiv 0\pmod{p},$ and $\delta \in \mathbb Z/(p-1)\mathbb Z.$ Set $\kappa =1+pd$ and $K=\mathbb Q_p(\chi).$ We set:
$$F_{\chi}(T)=\frac{\sum_{a=1}^d\chi (a)(1+T)^a}{1-(1+T)^d}.$$
Let's give the basic properties of $F_{\chi}(T):$
\newtheorem{Lemma6}{Lemma}[section]
\begin{Lemma6} \label{Lemma6}
${}$\par
\noindent 1) If $d\geq 2,$ $F_{\chi}(T)\in \Lambda.$\par
\noindent 2) If $d=1,$ $\forall \alpha \in \mathbb Z/(p-1)\mathbb Z,$ $\alpha \not =1,$ $\gamma_{\alpha}(F_{\chi }(T))\in \Lambda.$\par
\noindent 3) $U(F_{\chi}(T))=F_{\chi}(T)-\chi(p) F_{\chi}((1+T)^p-1).$\par
\noindent 4) If $d\geq 2,$ $F_{\chi}((1+T)^{-1}-1)=\varepsilon F_{\chi}(T),$ wher $\varepsilon =1$ if $\chi$ is oddd and $\varepsilon=-1$ if $\chi$ is even.\par
\noindent 5) If $d=1,$ $F_{\chi}((1+T)^{-1}-1)=-1-F_{\chi}(T).$\par
\end{Lemma6}
\noindent {\sl Proof} 1), 4) and 5) are obvious.\par
\noindent 2) For $d=1,$ we have:
$$F_{\chi}(T)=-1+\frac{\sum_{a=0}^{p-1}(1+T)^a}{1-(1+T)^p}.$$
Set:
$$G(T)=(1-(1+T)^p)\gamma_{\alpha }(F_{\chi}(T)).$$
Note that:
$$\forall \eta \in \mu_{p-1},\, \frac{1-(1+T)^p}{1-(1+T)^{\eta p}}\equiv \eta^{-1} \pmod{\omega_1(T)}.$$
Therefore:
$$(p-1) G(T) \equiv \sum_{\eta\in \mu_{p-1}}\eta^{\alpha -1}\sum_{a=0}^{p-1}(1+T)^{\eta a}\pmod{\omega_1(T)}.$$
Thus:
$$(p-1) G(T) \equiv \sum_{\eta\in \mu_{p-1}}\eta^{\alpha -1}\sum_{b=0}^{p-1}(1+T)^{b}\pmod{\omega_1(T)}.$$
Since $\alpha \not =1,$ we get:
$$G(T)\equiv 0\pmod{\omega_1(T)}.$$
Therefore $\gamma_{\alpha}(F_{\chi}(T))\in \Lambda.$\par
\noindent 3) For $d=1,$ we have:
$$U(F_{\chi}(T))=\frac{\sum_{a=1}^{p-1}(1+T)^a}{1-(1+T)^p}= F_{\chi} (T) -F_{\chi }((1+T)^p-1).$$
Now, let $d\geq 2.$ Set $q_0=\kappa=1+pd.$ Note that:
$$F_{\chi}(T)=\frac{\sum_{a=1}^{q_0}\chi(a) (1+T)^a}{1-(1+T)^{q_0}}.$$
Therefore:
$$U(F_{\chi}(T))=\frac{\sum_{a=1,\, a\not \equiv 0\pmod{p}}^{q_0}\chi(a) (1+T)^a}{1-(1+T)^{q_0}}.$$
But:
$$F_{\chi}(T)-\chi (p)F_{\chi}((1+T)^p-1)= \frac{\sum_{a=1}^{q_0}\chi(a) (1+T)^a}{1-(1+T)^{q_0}}-\chi (p)\frac{\sum_{a=1}^{d}\chi(a) (1+T)^{pa}}{1-(1+T)^{q_0}}.$$
The Lemma follows easily. $\diamondsuit$\par
\newtheorem{Lemma7}[Lemma6]{Lemma}
\begin{Lemma7} \label{Lemma7}
Assume that $d\geq 2.$ The denominator of $F_{\chi}(T)$ is $\phi_{d}(1+T)$ where $\phi_d(X)$ is the $d$th cyclotomic polynomial and the same is true for $\overline{F_{\chi}(T)}.$\par
\end{Lemma7}
\noindent{\sl Proof} Let $\zeta \in \mu_d.$ If $\zeta$ is not a primite $d$th root of unity, then, by \cite{WAS}, Lemma 4.7, we have:
$$\sum_{a=1}^d \chi (a) \zeta^a=0.$$
If $\zeta $ is a primitive $d$th root of unity, then by \cite{WAS}, Lemma 4.8, we have:
$$\sum_{a=1}^d \chi (a) \zeta ^a \not \equiv 0 \pmod{\widetilde{\pi }},$$
where $\widetilde {\pi }$ is any prime of $K(\mu_d ).$ $\diamondsuit$\par
\newtheorem{Lemma8}[Lemma6]{Lemma}
\begin{Lemma8} \label{Lemma8}
The derivative of $\gamma_{-\delta}(F_{\chi}(T))$ is not a pseudo-polynomial modulo $\pi.$\par
\end{Lemma8}
\noindent{\sl Proof} We first treat the case $d\geq 2.$ By 3) and 4) of Lemma \ref{Lemma6}, Lemma \ref{Lemma7} and Proposition \ref{Proposition6}, $\overline{\gamma_{-\delta}}\overline{U} (\overline{F_{\chi}(T)})$ is not a pseudo-polynomial. But observe that $\overline{U}=\overline{D^{p-1}}. $ Thus $\overline{D}\overline{\gamma_{-\delta}} (\overline{F_{\chi}(T)})$ is not a pseudo-polynomial.\par
\noindent For the case $d=1.$ Set $\widetilde {F_{\chi}(T)}=F_{\chi}(T)-2F_{\chi}((1+T)^2-1)=1-\frac{1}{2+T}.$ Observe that:\par
\noindent - $ \widetilde {F_{\chi}((1+T)^{-1}-1)}=1-\widetilde {F_{\chi}(T)},$\par
\noindent - $U(\widetilde {F_{\chi}(T)})=\widetilde {F_{\chi}(T)}-\widetilde {F_{\chi}((1+T)^p-1)}.$\par
\noindent Therefore, as in the case $d\geq2,$ $ \overline {\gamma_{-\delta}}\overline{U} (\overline{\widetilde{F_{\chi}(T)}})$ is not a pseudo-polynomial. Thus $ \overline{\gamma_{-\delta}}\overline{U} (\overline{F_{\chi}(T)})$ is not a pseudo-polynomial. And one can conclude as in the case $d\geq 2.$ $\diamondsuit$ \par
\newtheorem{Lemma9}[Lemma6]{Lemma}
\begin{Lemma9} \label{Lemma9}
$$\Gamma_{\delta}\gamma_{-\delta}(F_{\chi}(T))=f(\frac{1}{1+T}-1,\theta).$$
\end{Lemma9}
\noindent{\sl Proof} We treat the case $d=1,$ the case $d\geq 2$ is quite similar. Set $T=e^Z-1.$ We get:
$$\gamma_{-\delta}(F_{\chi}(T))=\sum_{n\geq 0,\, n\equiv 1+\delta \pmod{p-1}}\frac{B_n}{n!}Z^{n-1}.$$
Thus, by \cite{WAS}, Theorem 5.11, we get:
$$\forall k\in \mathbb N, k\equiv \delta \pmod{p-1},\, D^k\gamma_{-\delta}U(F_{\chi}) (0)=L_p(-k,\theta).$$
But, by Proposition \ref{Proposition1},we have for $s\in \mathbb Z_p:$
$$\Gamma_{\delta}\gamma_{-\delta}U(F_{\chi})(\kappa^s-1)={\rm lim}_n D^{k_n(s,\delta)}\gamma_{-\delta}U(F_{\chi})(0)=L_p(-s,\theta)=f(\kappa^{-s}-1,\theta).$$
The Lemma follows. $\diamondsuit$\par
We can now state and prove our main result:
\newtheorem{Theorem2}[Lemma6]{Theorem}
\begin{Theorem2} \label{Theorem2}
${}$\par
\noindent 1) $\overline{f(T,\theta )}$ is not a pseudo-rational function.\par
\noindent 2) $\lambda (f(T,\theta ))< (\frac{p-1}{2}\phi(d))^{\phi (p-1)},$ where $\phi$ is Euler's totient function.\par
\end{Theorem2}
\noindent{\sl Proof}${}$\par
\noindent 1) Assume the contrary, i.e. $\overline{f(T,\theta )}$ is a pseudo-rational function. Then $\overline{f(\frac{1}{1+T}-1,\theta )}$ is also a pseudo-rational function. Thus $\overline{\Gamma_{\delta}}\overline {\gamma_{-\delta}}\overline{U} (\overline {F_{\chi}(T)})$ is a pseudo-rational function.\par
\noindent We first treat the case $d\geq 2.$ By Theorem \ref{Theorem1}, there exists an integer $n\geq 0$ such that $(1+T)^n(\overline{U}(\overline{F_{\chi}(T)})+(-1)^{\delta }\overline{U}(\overline{F_{\chi}((1+T)^{-1}-1)}) \in \mathbb F_q[T].$ This is a contradiction by 3) and 4) of Lemma \ref{Lemma6} and Lemma \ref{Lemma7}.\par
\noindent For the case $d=1.$ We work with $\widetilde {F_{\chi}(T)}=F_{\chi}(T)-2F_{\chi}((1+T)^2-1)=1-\frac{1}{2+T}.$ Then , by Proposition \ref{Proposition3}, $\overline{\Gamma_{\delta}}\overline {\gamma_{-\delta}}\overline{U} (\overline {\widetilde{F_{\chi}(T)}})$ is a pseudo-rational function. We get a contradiction as in the case $d\geq 2.$\par
\noindent 2) Our proof is inspired by a method introduced by S. Rosenberg (\cite{ROS}). We first treat the case $d=1.$ Note that we can assume that $\lambda(f(T,\theta ))\geq 1.$ Now, by Lemma \ref{Lemma8}:
$$\mu(\gamma_{-\delta}(F_{\chi}(T)))=0.$$
Futhermore, we have:
$$\gamma_{-\delta}(F_{\chi})(0)\equiv 0\pmod{\pi }.$$
Therefore, by 3) of Lemma \ref{Lemma6}, we get:
$$\lambda(\gamma_{-\delta}U(F_{\chi}(T)))=\lambda (\gamma_{-\delta}(F_{\chi}(T))).$$
Therefore we have to evaluate $\lambda (\gamma_{-\delta}(F_{\chi}(T))).$ Set $F(T)=\frac{-1}{T}.$ Since $\delta$ is odd, we have:
$$\gamma_{-\delta}(F_{\chi}(T))=\gamma_{-\delta}(F(T)).$$
Observe that $F((1+T)^{-1}-1)=1-F(T).$ Let $S\subset \mu_{p-1}$ be a set of representatives of $\mu_{p-1}/\{ 1,-1\}.$ We have:
$$(p-1)\gamma_{-\delta}(F(T))=2\sum_{\eta \in S} \eta^{-\delta}F((1+T)^{\eta}-1)\, -\sum_{\eta \in S}\eta^{-\delta}.$$
Set:
$$G(T)=(\prod_{\eta\in S}((1+T)^{\eta}-1))\gamma_{-\delta}(F(T)).$$
Then:\par
\noindent- $\mu (G(T))=0,$\par
\noindent- $\lambda (G(T))=\frac{p-1}{2}+\lambda (\gamma_{-\delta }(F(T))).$\par
\noindent For $S'\subset S,$ write $t(S')=\sum_{x\in S'}x.$ We can write:
$$G(T)=\sum_{S'\subset S} a_{S'} (1+T)^{t(S')},$$
where $a_{S'}\in O_K.$ Set:
$$N={\rm Max}\{v_p(t(S')-t(S'')),\, S',S''\subset S,\, t(S')\not = t(S'')\}.$$
It is clear that:
$$p^N<(\frac{p-1}{2})^{\phi (p-1)}.$$
But, by Lemma \ref{Lemma2}, we have:
$$\lambda(G(T))<p^{N+1}.$$
Thus, by Propositon \ref{Proposition3}, we get:
$$\lambda(f(T,\theta))=\lambda(f(\frac{1}{1+T}-1,\theta))<p^N<(\frac{p-1}{2})^{\phi (p-1)}.$$
Now, we treat the general case, i.e. $d\geq 2.$ Again we can assume that $\lambda(f(T,\theta ))\geq 1.$ Thus as in the case $d=1,$ we get:
$$\lambda(\gamma_{-\delta}U(F_{\chi}(T)))=\lambda (\gamma_{-\delta}(F_{\chi}(T))).$$
Now, by Lemma \ref{Lemma7}, we can write:
$$F_{\chi}(T)=\frac{\sum_{a=0}^{\phi(d)-1}r_a (1+T)^a}{\phi_d(1+T)},$$
where $r_a\in O_K$ for $a\in \{ 0,\cdots, \phi(d)-1\}.$
Let again $S\subset \mu_{p-1}$ be a set of representatives of $\mu_{p-1}/\{ 1,-1\}.$ By Lemma \ref{Lemma6}, we have:
$$(p-1)\gamma_{-\delta}(F_{\chi}(T))=2\sum_{\eta \in S}\eta^{-\delta}F_{\chi}((1+T)^{\eta }-1).$$
Set:
$$G(T)=(\prod_{\eta \in S}\phi_d((1+T)^{\eta})))\gamma_{-\delta}(F_{\chi}(T)).$$
We have:
$$G(T)=\sum_{a=0}^{\phi(d)-1}\sum_{\eta \in S}\sum_{S'\subset S\setminus \{ \eta \}}\, \, \sum_{\underline{d}=(d_{\eta'})_{\eta'\in S'},\, d_{\eta'}\in \{ 0,\cdots, \phi (d)\}}b_{S',\underline{d}}(1+T)^{a\eta +\sum_{\eta'\in S'}d_{\eta'}\eta'},$$
where $b_{S',\underline{d}}\in O_K.$ Note that again $\mu (G(T))=0$ and that $\lambda (G(T))=\lambda (\gamma_{-\delta}(F_{\chi}(T))).$ Now, for $a,b \in \{ 0,\cdots , \phi (d)-1\},$ $\eta_1,\eta_2\in S,$ $S_1\in S\setminus \{ \eta_1\},$ $S_2\in S \setminus\{ \eta_2\},$ set:
$$V=a\eta_1+\sum_{\eta \in S_1} d_{\eta}\eta \, -b\eta_2-\sum_{\eta \in S_2} d_{\eta }' \eta,$$
where $\forall \eta \in S_1,$ $d_{\eta}Ê\in \{ 0, \cdots , \phi (d)\},$ and $\forall \eta \in S_2,$ $d_{\eta}'Ê\in \{ 0, \cdots , \phi (d)\}.$\par
\noindent If $\eta_1=\eta_2$ then we can write:
$$V=(a-b)\eta_1+\sum_{\eta \in S'} u_{\eta }\eta,$$
where $\mid u_{\eta}\mid \in \{ 0,\cdots ,\phi(d)\}$ and $\mid S'\mid \leq \frac {p-3}{2}.$\par
\noindent If $\eta_1\not =\eta_2,$ we can write:
$$V=a'\eta_1+b'\eta_2+\sum_{\eta\in S'}u_{\eta} \eta,$$
where $\mid a'\mid, \mid b'\mid, \mid u_{\eta }\mid \in\{ 0,\cdots ,\phi (d)\},$ and $\mid S'\mid\leq \frac{p-5}{2}.$
Therefore, if $V\not = 0,$ we get:
$$p^{v_p(V)}<(\frac{p-1}{2}\phi(d))^{\phi(p-1)}.$$
Now, we can conclude as in the case $d=1.$ $\diamondsuit$\par
Let $E$ be a number field and let $E_{\infty}/E$ be the cyclotomic $\mathbb Z_p$-extension of $E.$ For $n\geq 0,$ let $A_n$ be the $p$th Sylow subgroup of the ideal class group of the $n$th layer in $E_{\infty}/E.$Then , by \cite{WAS}, Theorem 13.13, there exist $\mu_p(E)\in \mathbb N, \lambda_p(E)\in \mathbb N$ and $\nu_p(E) \in \mathbb Z,$ such that for all sufficiently large $n:$
$$\mid A_n\mid =p^{\mu_p(E)p^n+\lambda_p(E)n+\nu_p(E)}.$$
Recall that it is conjectured that $\mu_p(E)=0$ and if $E$ is an abelian number field it has been proved by B. Ferrero and L. Washington (\cite{FW}).
\newtheorem{Theorem3}[Lemma6]{Corollary}
\begin{Theorem3} \label{Theorem3}
Let $F$ be an abelian number field of conductor $N.$ Write $N=p^m d,$ where $m\in \mathbb N$ and $d\geq 1,$ $d\not \equiv 0\pmod{p}.$ Then:
$$\lambda_p(F)<2(\frac{p-1}{2}\phi(d))^{\phi(p-1)+1}.$$
\end{Theorem3}
\noindent{\sl Proof} Set, for all $n\geq 0,$ $q_n=p^{n+1}d.$ Then $F\subset \mathbb Q(\mu_{q_m}).$ It is not difficult to see that (see the arguments in the proof of Theorem 7.15 in \cite{WAS}) :
$$\lambda_p(F)\leq \lambda_p(\mathbb Q(\mu_{q_m})).$$
But, note that:
$$\lambda_p(\mathbb Q(\mu_{q_m}))=\lambda_p(\mathbb Q(\mu_{q_0})).$$
Now, by \cite {WAS} Proposition 13.32 and Theorem 7.13:
$$\lambda_p(\mathbb Q(\mu_{q_0}))\leq 2\sum_{\theta \, {\rm even},\, \theta \not =1,\, f_{\theta}\mid q_0}\lambda (f(T,\theta)).$$
It remains to apply Theorem \ref{Theorem2}. $\diamondsuit$\par
Note that the bound of this latter Corollary is certainly far from the truth even in the case $p=3$ (see \cite{KW}).\par
\end{document} |
\begin{document}
\title[One and two level densities]{Type-I contributions to the one and two level densities of quadratic Dirichlet $L$--functions over function fields}
\author{Hung M. Bui, Alexandra Florea and J. P. Keating}
\address{Department of Mathematics, University of Manchester, Manchester M13 9PL, UK}
\email{[email protected]}
\address{Department of Mathematics, Columbia University, New York NY 10027, USA}
\email{[email protected]}
\address{Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK}
\email{[email protected]}
\begin{abstract}
Using the Ratios Conjecture, we write down precise formulas with lower order terms for the one and the two level densities of zeros of quadratic Dirichlet $L$--functions over function fields. We denote the various terms arising as Type-$0$, Type-I and Type-II contributions. When the support of the Fourier transform of the test function is sufficiently restricted, we rigorously compute the Type-$0$ and Type-I terms and confirm that they match the conjectured answer. When the restrictions on the support are relaxed, our results suggest that Type-II contributions become important in the two level density.
\end{abstract}
\allowdisplaybreaks
\maketitle
\section{Introduction}
In this paper we compute the one and the two level densities of zeros of $L$--functions associated to quadratic characters over function fields. We compute certain Type-I contributions (as in the work of Conrey and Keating \cite{ck1, ck2, ck3, ck4, ck5}) and write down explicit conjectural Type-II terms predicted by the Ratios Conjecture \cite{cfz}.
Understanding zeros in families of $L$--functions is a problem of considerable interest which has been much-studied. Katz and Sarnak \cite{katzsarnak, katzsarnak2} conjectured that the behavior of zeros close to the central point in a family of $L$--functions coincides with the distribution of eigenvalues near $1$ of matrices in a certain symmetry group associated to the family. There is an abundance of papers in the literature in which the above mentioned agreement is observed (for example \cite{ILS, hughesrudnick, miller, miller2}).
When computing the $n$--level density of zeros for a particular family of $L$--functions, the Katz and Sarnak conjectures predict the main term in the asymptotic formula. Conrey, Farmer and Zirnbauer \cite{cfz} conjectured formulas for averages of ratios of $L$--functions, and using the Ratios Conjecture, one can write down an explicit formula for the $n$--level density which recovers the Katz-Sarnak main term and further include lower order terms \cite{CS}. In the case of the Riemann zeta-function, the resulting expressions coincide with formulas obtained earlier by Bogomolny and Keating using the Hardy-Littlewood twin-prime conjecture \cite{BK1} (see also \cite{BeKe, BK4, BK5}).
A related problem is that of computing moments in families of $L$--functions. Using analogies with random matrix theory, Keating and Snaith \cite{ks2, ksnaith} conjectured asymptotic formulas with the leading order term for moments in various families. A more refined conjecture, due to Conrey, Farmer, Keating, Rubinstein and Snaith \cite{cfkrs}, and similar in nature to the Ratios Conjecture \cite{cfz}, predicts lower order terms undetected by the random matrix models. More recent work of Conrey and Keating \cite{ck1, ck2,ck3,ck4,ck5} revisits the question of evaluating shifted moments of the Riemann zeta-function from a different perspective, and recovers the lower order terms predicted in \cite{cfkrs}. Conrey and Keating used long Dirichlet polynomials rather than the approximate functional equation, and divide the terms that arise into certain Type-$0$, Type-I and Type-II contributions (depending on the number of swaps in the shifts). This builds on previous work in the case of the $n$-point correlation of the zeros by Bogomolny and Keating \cite{BK2, BK3}, where a similar division was first introduced (see also \cite{ck6, ck7}). Here we use the same ideas to examine asymptotic formulas including lower order terms for the $n$ level density of zeros. Throughout our paper, we use the Conrey and Keating nomenclature for Type-$0$, Type-I and Type-II terms.
For the family of quadratic Dirichlet $L$--functions, \"{O}zl\"{u}k and Snyder \cite{ozluk} computed the one level density of zeros when the support of the Fourier transform of the test function is in $(-2,2)$. The higher densities in this family of $L$--functions were studied by Rubinstein \cite{rubinstein}. For a Schwartz test function $f \in \mathcal{S}(\mathbb{R}^n)$, even in all the variables, Rubinstein computed the $n$--level density when the Fourier transform of $f$ is supported in $\sum_{j=1}^n |u_j|<1$, conditional on the Generalized Riemann Hypothesis. Gao \cite{gao} attempted to double the range in Rubinstein's result. More specifically, he showed that if $f$ is of the form $f(x_1,\ldots,x_n)= \prod_{i=1}^n f_i(x_i)$ and each $\hat{f_i}$ is supported in $|u_i|<s_i$ and $\sum_{i=1}^n s_i<2$, then the $n$--level density of zeros is equal to a complicated combinatorial factor $A(f)$. For $n=2,3$, he showed that $A(f)$ agrees with the Katz and Sarnak conjecture. Recent work of Entin, Roditty-Gershon and Rudnick \cite{entin} showed that indeed the combinatorial factor $A(f)$ obtained by Gao matches the random matrix theory prediction for all $n$. Their novel approach does not involve doing the combinatorics directly, but passing to a function field analog of the problem, taking the limit $q \to \infty$ and using equidistribution results of Katz and Sarnak. An alternative approach was developed in \cite{CS2, MS}.
In the function field setting, Rudnick \cite{R} computed the one level density of zeros for the family of quadratic Dirichlet $L$--functions and showed that there is a transition when the support of the Fourier transform goes beyond $1$. Bui and Florea \cite{bf} obtained infinitely many lower order terms when the support of the Fourier transform is in certain ranges, and further computed the pair correlation of zeros in the family.
In the present paper, we consider the two level density of zeros in the family of quadratic Dirichlet $L$--functions. Let $\mathcal{H}_{2g+1}$ denote the space of monic, square-free polynomials of degree $2g+1$ over $\mathbb{F}_q[x]$. For simplicity, in the definition of the two level density, we take the test function to be equal to $1$. The two level density of zeros is defined to be
\begin{equation}
I_2(N;\alpha,\beta)=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}\sum_{\substack{f_1,f_2\in\mathcal{M}\\d(f_1f_2)\leq N}}\frac{\Lambda(f_1)\Lambda(f_2)\chi_D(f_1f_2)}{|f|_1^{1/2+\alpha}|f|_2^{1/2+\beta}}, \label{i2}
\end{equation}
where $\Lambda(f)$ denotes the von Mangoldt function over function fields, and $\chi_D(f)$ is the quadratic character.
Using the Ratios Conjecture over function fields \cite{AK}, we write down precise formulas for the two level density in terms of Type-$0$, Type-I and Type-II contributions. The Type-I terms kick in when $N \geq 2g$ and Type-II terms appear when $N \geq 4g$. We compute the Type-$0$ and Type-I terms rigorously by estimating sums over primes (i.e.~over monic irreducible polynomials). Our approach in computing the two level density is more direct than the one used by Entin, Roditty-Gershon and Rudnick \cite{entin}, and we do not take $q \to \infty$ (hence we do not use any equidistribution results). The Type-$0$ terms, or the so-called "diagonal", come from prime powers $f_1$ and $f_2$ in \eqref{i2} with the product $f_1 f_2$ being a square. The diagonal terms are relatively straightforward to compute. Evaluating the Type-I terms is more subtle and requires more involved computations. We use the Poisson summation formula for the sum over $D$ (after removing the squarefree condition) and then we compute the contribution from the parameter on the dual side of the Poisson summation formula being a square. We sum up these contributions and then we check that they match the answer conjectured from the Ratios Conjecture.
Type-I terms essentially come from squares on the dual side of the Poisson summation formula over function fields. Our methods do not allow us to identify the Type-II terms which only arise when $N \geq 4g$, but we explicitly write down the conjectured Type-II contribution. This is one of our main goals: to draw attention to the fact that when the methods that have been employed successfully for many years in calculations of the one level density are applied to the two level density they fail to capture all of the terms, underlining the importance of developing methods to compute the Type-II terms in this case.
For the sake of completeness, we also include the computation of the one level density (with a shift) and match the terms we obtain with the Type-$0$ and Type-I contributions.
\subsection{Outline of the paper}
In Section \ref{background} we gather a few useful lemmas we will need. In Section \ref{1rc} we use the Ratios Conjecture to write down formulas for the one level density of zeros with Type-$0$ and Type-I terms (there are no Type-II terms for the one level density). We rigorously compute these terms when $N<4g$ and match them to the conjecture in Section \ref{1compute}. In Section \ref{2rc} we again use the Ratios Conjecture to predict the Type-$0$, Type-I and Type-II contributions for the two level density. The diagonal terms are computed in Section \ref{2diag} and Type-I terms in Section \ref{type1}. In subsection \ref{combine} we combine the various contributions from Sections \ref{type11} and \ref{type12} and show that they agree with the conjecture.
{\bf Acknowledgements.} A. Florea gratefully acknowledges the support of an NSF Postdoctoral Fellowship during part of the research which led to this paper.
J.P. Keating was supported by a Royal Society Wolfson Research Merit Award, EPSRC Programme Grant EP/K034383/1 LMF: $L$-Functions and Modular Forms, and by ERC Advanced Grant 740900 (LogCorRM).
The authors would also like to thank Julio Andrade, Brian Conrey, Chantal David, Steve Gonek and Matilde Lal\'{i}n for many stimulating discussions and useful comments during SQuaRE meetings at AIM.
\section{Lemmas}
\label{background}
Let $q \equiv 1 \pmod 4$ be a prime. We denote the set of monic polynomials over $\mathbb{F}_q[x]$ by $\mathcal{M}$. Let $\mathcal{M}_n$ denote the set of monic polynomials of degree $n$, $\mathcal{H}_n$ the set of monic, squarefree polynomials of degree $n$, and $\mathcal{P}_n$ the monic, irreducible polynomials of degree $n$. The set of monic polynomials of degree less than or equal to $n$ is denoted by $\mathcal{M}_{\leq n}$. For simplicity, we denote the degree of a polynomial $f$ by $d(f)$.
The norm of a polynomial $f$ is defined by $|f|= q^{d(f)}$.
The zeta-function over $\mathbb{F}_q[x]$ is defined by
$$\zeta_q(s) = \sum_{f\in\mathcal{M}} \frac{1}{|f|^s}$$ for $\Re(s)>1$. Since there are $q^n$ monic polynomials of degree $n$, one can easily show that
$$\zeta_q(s) = \frac{1}{1-q^{1-s}},$$ and this provides a meromorphic continuation of $\zeta_q$ with a simple pole at $s=1$. Making the change of variables $u=q^{-s}$, the zeta-function becomes
$$ \mathcal{Z}(u) = \zeta_q(s) = \sum_{f \in\mathcal{M}} u^{d(f)} = \frac{1}{1-qu},$$ which has a simple pole at $u=1/q$. Note that $\mathcal{Z}(u)$ is given by the Euler product
$$ \mathcal{Z}(u) = \prod_P \Big(1-u^{d(P)}\Big)^{-1},$$ for $|u|<1/q$, where the product is over monic, irreducible polynomials in $\mathbb{F}_q[t]$.
The quadratic character over $\mathbb{F}_q[t]$ is defined as follows. For $P$ a monic, irreducible polynomial let
$$ \Big( \frac{f}{P} \Big)=
\begin{cases}
1 & \mbox{ if } P \nmid f, f \text{ is a square modulo }P, \\
-1 & \mbox{ if } P \nmid f, f \text{ is not a square modulo }P, \\
0 & \mbox{ if } P|f.
\end{cases}
$$
We extend the definition of the quadratic residue symbol above to any monic $D \in \mathbb{F}_q[t]$ by multiplicativity, and define the quadratic character $\chi_D$ by
$$\chi_D(f) = \Big( \frac{D}{f} \Big).$$
Since we assumed that $q \equiv 1 \pmod 4$, note that the quadratic reciprocity law takes the following form: if $A$ and $B$ are two monic coprime polynomials, then
$$ \Big( \frac{A}{B} \Big) = \Big( \frac{B}{A} \Big).$$
We define the von Mangoldt function to be
$$\Lambda(f) =
\begin{cases}
d(P) & \mbox{ if } f=cP^k, c \in \mathbb{F}_q^{\times}, \\
0 & \mbox{ otherwise.}
\end{cases}
$$
The following lemma expresses sums over squarefree polynomials in terms of sums over monics.
\begin{lemma}\label{L1}
For $f\in\mathcal{M}$ we have
\[
\sum_{D\in\mathcal{H}_{2g+1}}\chi_D(f)=\sum_{C|f^\infty}\sum_{h\in\mathcal{M}_{2g+1-2d(C)}}\chi_f(h)-q\sum_{C|f^\infty}\sum_{h\in\mathcal{M}_{2g-1-2d(C)}}\chi_f(h),
\]
where the summations over $C$ are over monic polynomials $C$ whose prime factors are among the prime factors of $f$.
\end{lemma}
\begin{proof}
See Lemma $2.2$ in \cite{aflorea}.
\end{proof}
We define the generalized Gauss sum as follows. For $f \in \mathcal{M}$, let
\[
G(V,f):= \sum_{u \pmod f} \chi_f(u)e\Big(\frac{uV}{f}\Big),
\]
where the exponential over function fields was defined in \cite{hayes}. Specifically, for $a \in \mathbb{F}_q((1/t))$,
$$e(a) = e^{2 \pi i a_1/q},$$ where $a= \ldots +a_1/t+ \ldots$.
The following two lemmas are Proposition 3.1 and Lemma 3.2 in \cite{aflorea}.
\begin{lemma}\label{L2}
Let $f\in\mathcal{M}_n$. If $n$ is even then
\[
\sum_{h\in\mathcal{M}_m}\chi_f(h)=\frac{q^m}{|f|}\bigg(G(0,f)+q\sum_{V\in\mathcal{M}_{\leq n-m-2}}G(V,f)-\sum_{V\in\mathcal{M}_{\leq n-m-1}}G(V,f)\bigg),
\]
otherwise
\[
\sum_{h\in\mathcal{M}_m}\chi_f(h)= \frac{q^{m+1/2}} {|f|}\sum_{V\in\mathcal{M}_{n-m-1}}G(V,f).
\]
\end{lemma}
\begin{lemma}\label{L3}
\begin{enumerate}
\item If $(f,h)=1$, then $G(V, fh)= G(V, f) G(V,h)$.
\item Write $V= V_1 P^{\alpha}$ where $P \nmid V_1$.
Then
$$G(V , P^j)=
\begin{cases}
0 & \mbox{if } j \leq \alpha \text{ and } j \text{ odd,} \\
\varphi(P^j) & \mbox{if } j \leq \alpha \text{ and } j \text{ even,} \\
-|P|^{j-1} & \mbox{if } j= \alpha+1 \text{ and } j \text{ even,} \\
\chi_P(V_1) |P|^{j-1/2} & \mbox{if } j = \alpha+1 \text{ and } j \text{ odd, } \\
0 & \mbox{if } j \geq 2+ \alpha .
\end{cases}$$
\end{enumerate}
\end{lemma}
The following lemmas are the equivalent of the Polya-Vinogradov inequality and the Weil bound in function fields.
\begin{lemma}
\label{pv}
We have
$$ \sum_{D \in \mathcal{H}_{2g+1}} \chi_D(P) \ll |P|^{1/2},$$
and for $Q$ a prime polynomial,
$$ \sum_{\substack{D \in \mathcal{H}_{2g+1} \\ (D,Q)=1}} \chi_D(P) \ll \frac{g}{d(Q)}|P|^{1/2} .$$
\end{lemma}
\begin{proof}
See, for example, Lemma $3.5$ and p. $8033$ in \cite{bf}.
\end{proof}
\begin{lemma}[The Weil bound]\label{sumprimes}
For $V\in\mathcal{M}$ not a perfect square we have
$$\sum_{P \in \mathcal{P}_n} \chi_V(P) \ll \frac{d(V)}{n} q^{n/2}.$$
\end{lemma}
\begin{proof}
See equation $2.5$ in \cite{R}.
\end{proof}
\begin{lemma}\label{L4}
For $f\in\mathcal{M}$ we have
\[
\frac{1}{| \mathcal{H}_{2g+1}|}\sum_{D \in \mathcal{H}_{2g+1}} \chi_D(f^{2})=\prod_{P|f}\bigg(1-\frac{1}{|P|+1}\bigg)+O(q^{-2g}).
\]
\end{lemma}
\begin{proof}
See, for example, Lemma $3.7$ in \cite{bf}.
\end{proof}
\section{The one level density - using the Ratios Conjecture}
\label{1rc}
Consider
\begin{equation}\label{maineq}
I_1(N;\alpha)=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}\sum_{f\in \mathcal{M}_{\leq N}}\frac{\Lambda(f)\chi_D(f)}{|f|^{1/2+\alpha}},
\end{equation}
where the shift is assumed to satisfy $|\alpha| \ll 1/g$.
Using an analogue of the Perron formula in the form
\begin{equation}\label{Perron}
\sum_{n\leq N}a(n)=\frac{1}{2\pi i}\oint_{|u|=r}\bigg(\sum_{n=0}^{\infty}a(n)u^{n}\bigg)\frac{du}{u^{N+1}(1-u)}
\end{equation}
we get
\begin{align*}
I_1(N;\alpha)&=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}\frac{1}{2\pi i}\oint_{|u|=r}\sum_{f\in \mathcal{M}}\frac{\Lambda(f)\chi_D(f)u^{d(f)}}{|f|^{1/2+\alpha}}\frac{du}{u^{N+1}(1-u)}\\
&=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}} \frac{1}{2 \pi i} \oint_{|u|=r} \frac{u}{q^{1/2+\alpha}} \frac{ \mathcal{L}'}{\mathcal{L}} \Big(\frac{u}{q^{1/2+\alpha}},\chi_D \Big) \frac{du}{u^{N+1}(1-u)}
\end{align*}
for any $r<q^{-1/2-\varepsilon}$. We enlarge the contour to $|u|=r=q^{-\varepsilon}$. The Ratios Conjecture implies that (see, for example, Theorem $8.1$ in \cite{bf})
\begin{align*}
\frac{1}{|\mathcal{H}_{2g+1}|} &\sum_{D\in\mathcal{H}_{2g+1}}u \frac{\mathcal{L}'}{\mathcal{L}} (u,\chi_D) = u^2 \frac{\mathcal{Z}'}{\mathcal{Z}}(u^2) - \mathcal{B}(u) + (qu^{2})^{g} \mathcal{A}_1(u)\mathcal{Z}\Big(\frac{1}{q^{2}u^{2}}\Big)+O_\varepsilon(q^{-g+\varepsilon g}),
\end{align*}
where \begin{equation}\label{mathcalBr}\mathcal{B}(u) = \sum_P \frac{ d(P) u^{2d(P)}}{(1-u^{2 d(P)})(|P|+1)}\end{equation} and
\begin{align*}
\mathcal{A}_1(u) &= \prod_P \Big(1- \frac{1}{|P|} \Big)^{-1} \Big( 1-\frac{1}{|P|^2u^{2 d(P)} (|P|+1)}-\frac{1}{|P|+1}\Big) \\
&= \prod_P \Big(1-\frac{1}{|P|^2} \Big)^{-1} \Big(1- \frac{1}{|P|^3 u^{2d(P)}} \Big) = \frac{\mathcal{Z}(1/q^{2})}{\mathcal{Z}(1/q^{3}u^{2})}=1+\frac{1-(qu^2)^{-1}}{q-1}.
\end{align*}
Hence, up to an error of size $O_\varepsilon(q^{-g+\varepsilon g})$,
\begin{align}\label{1}
I_1(N;\alpha)&=\frac{1}{2\pi i}\oint_{|u|=r}\frac{du}{u^{N-1}(1-u)(q^{2\alpha}-u^2)}-\frac{1}{2\pi i}\oint_{|u|=r}\frac{\mathcal{B}(u,\alpha)du}{u^{N+1}(1-u)}\\
&\qquad\quad+\frac{q^{-2g\alpha}}{2\pi i}\oint_{|u|=r}\frac{du}{u^{N-2g-1}(1-u)(u^2-q^{2\alpha})}+\frac{q^{-2g\alpha}}{2\pi i(q-1)}\oint_{|u|=r}\frac{du}{u^{N-2g+1}(1-u)},\nonumber
\end{align}
where
\begin{equation*}
\mathcal{B}(u,\alpha)=\mathcal{B}\Big(\frac{u}{q^{1/2+\alpha}}\Big).
\end{equation*}
Enlarging the contours we cross the poles at $u=1$ and $u=\pm q^{\alpha}$ in the first integral, and the only pole at $u=1$ in the second integral. Note that $\mathcal{B}(u,\alpha)$ is absolutely convergent for $|u|<q^{1/2-\varepsilon}$, so in the second integral we shift the contour to $|u|=q^{1/2-\varepsilon}$, obtaining an error term of size $O_\varepsilon(q^{-N/2+\varepsilon N})$. Hence the contribution of the first two terms in \eqref{1} is equal to
\begin{align}\label{type01}
&\frac{q^{-2[N/2]\alpha}-1}{1-q^{2\alpha}}-B(\alpha) +O_\varepsilon(q^{-N/2+\varepsilon N}),
\end{align}
where
\begin{equation}\label{Br}
B(\alpha):=\mathcal{B}(1,\alpha)= \sum_P \frac{ d(P) }{(|P|^{1+2\alpha}-1)(|P|+1)}.
\end{equation}
This should correspond to the diagonal terms.
For the remaining two terms in \eqref{1}, we note that they vanish if $N< 2g$, and if $N\geq 2g$ they contribute
\begin{align}\label{type1}
\frac{q^{-2g\alpha}-q^{-2[N/2]\alpha}}{1-q^{2\alpha}}+\frac{q^{-2g\alpha}}{q-1}.
\end{align}
This should correspond to the Type-I terms. Combining \eqref{type01} and \eqref{type1} we arrive at the following conjecture.
\begin{conjecture}\label{conjecture1level}
We have
\begin{align*}
I_1(N;\alpha)&=\frac{q^{-2[N/2]\alpha}-1}{1-q^{2\alpha}}-B(\alpha)+\mathds{1}_{N\geq 2g}\bigg(\frac{q^{-2g\alpha}-q^{-2[N/2]\alpha}}{1-q^{2\alpha}}+\frac{q^{-2g\alpha}}{q-1}\bigg)\\
&\qquad\qquad+O_\varepsilon(q^{-g+\varepsilon g}) +O_\varepsilon(q^{-N/2+\varepsilon N}).
\end{align*}
\end{conjecture}
\section{The one level density}
\label{1compute}
We assume in this section that $N<4g$.
\subsection{The diagonal}
The diagonal, denoted by $I_1^0(N;\alpha)$, corresponds to the terms $f=P^{2k}$ in \eqref{maineq}, and so in view of Lemma \ref{L4} we have
\[
I_1^0(N;\alpha)=\sum_{1\leq kn\leq [N/2]}\sum_{P\in\mathcal{P}_{n}}\frac{d(P)}{|P|^{k(1+2\alpha)}}-\sum_{1\leq kn\leq [N/2]}\sum_{P\in\mathcal{P}_{n}}\frac{d(P)}{|P|^{k(1+2\alpha)}(|P|+1)}+O_\varepsilon(q^{-2g+\varepsilon g}).
\]
The first term, by the Prime Polynomial Theorem, is equal to
\begin{align*}
\sum_{d(f)\leq [N/2]}\frac{\Lambda(f)}{|f|^{1+2\alpha}}&=\sum_{1\leq n\leq [N/2]}q^{-2n\alpha }=\frac{q^{-2[N/2]\alpha}-1}{1-q^{2\alpha}}.
\end{align*}
For the second term, note that
\begin{align*}
\sum_{1\leq kn\leq [N/2]}\sum_{P\in\mathcal{P}_{n}}\frac{d(P)}{|P|^{k(1+2\alpha)}(|P|+1)}&=B(\alpha)-\sum_{kn> [N/2]}\sum_{P\in\mathcal{P}_{n}}\frac{d(P)}{|P|^{k(1+2\alpha)}(|P|+1)}\\
&=B(\alpha)+O_\varepsilon\big(q^{-N/2+\varepsilon g}\big).
\end{align*}
Hence,
\[
I_1^0(N;\alpha)=\frac{q^{-2[N/2]\alpha}-1}{1-q^{2\alpha}}-B(\alpha)+O_\varepsilon\big(q^{-N/2+\varepsilon g}\big).
\]
Notice that the leading term matches up with \eqref{type01}.
\subsection{Type-I terms}
We now evaluate the off-diagonal terms corresponding to $f=P^{2k+1}$ in \eqref{maineq},
\[
I_1^1(N;\alpha)=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{d(P^{2k+1})\leq N}\frac{d(P)}{|P|^{(2k+1)(1/2+\alpha)}}\sum_{D\in\mathcal{H}_{2g+1}}\chi_D(P).
\]
Combining the Polya-Vinogradov inequality in Lemma \ref{pv} with the Prime Polynomial Theorem, the contribution of the terms with $k\geq 1$ is
\begin{equation*}
\ll q^{-2g}\sum_{n\leq N}\sum_{k\geq1}q^{-(k-1)n}\ll Nq^{-2g},
\end{equation*}
and the contribution of the terms with $d(P)=n$ is
\[
\ll q^{n-2g}.
\] So
\[
I_1^1(N;\alpha)=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{g+1\leq d(P)\leq N}\frac{d(P)}{|P|^{1/2+\alpha}}\sum_{D\in\mathcal{H}_{2g+1}}\chi_D(P)+O(q^{- g}).
\]
From Lemma \ref{L1} we have
\begin{align*}
&\sum_{D\in\mathcal{H}_{2g+1}}\chi_D(P)=\sum_{C|P^\infty}\sum_{h\in\mathcal{M}_{2g+1-2d(C)}}\chi_P(h)-q\sum_{C|P^\infty}\sum_{h\in\mathcal{M}_{2g-1-2d(C)}}\chi_P(h).
\end{align*}
The sums over $h$ are non-zero only if $0\leq 2g\pm1-2d(C)<d(P) $. Since $C|P^\infty$ and $d(P)\geq g+1$, we must have $C=1$ and, consequently, $d(P)\geq 2g$. Thus,
\[
I_1^1(N;\alpha)=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{2g\leq d(P)\leq N}\frac{d(P)}{|P|^{1/2+\alpha}}\bigg(\sum_{h\in\mathcal{M}_{2g+1}}\chi_P(h)-q\sum_{h\in\mathcal{M}_{2g-1}}\chi_P(h)\bigg)+O(q^{- g}).
\]
Consider the terms with $d(P)$ odd. Applying Lemma \ref{L2} and Lemma \ref{L3}, the expression inside the bracket is
\[
\frac{q^{2g+3/2}}{|P|^{1/2}}\sum_{d(V)= d(P)-2g-2}\chi_P(V)-\frac{q^{2g+1/2}}{|P|^{1/2}}\sum_{d(V)= d(P)-2g}\chi_P(V).
\]
Notice that $V$ cannot be a square in the sums, and hence by Lemma \ref{sumprimes}, the contribution of these terms to $I_1^1(N;\alpha)$ is $O(Nq^{N/2-2g})$.
If $d(P)$ is even, then from Lemma \ref{L2} and Lemma \ref{L3} we have
\begin{align*}
\sum_{h\in\mathcal{M}_{2g+1}}\chi_P(h)-q\sum_{h\in\mathcal{M}_{2g-1}}\chi_P(h)=&\frac{q^{2g+1}}{|P|^{1/2}}\bigg(q\sum_{d(V)\leq d(P)-2g-3}\chi_P(V)-\sum_{d(V)\leq d(P)-2g-2}\chi_P(V)\bigg)\\
&\ -\frac{q^{2g}}{|P|^{1/2}}\bigg(q\sum_{d(V)\leq d(P)-2g-1}\chi_P(V)-\sum_{d(V)\leq d(P)-2g}\chi_P(V)\bigg).
\end{align*}
As above, the contribution of the terms $V$ non-square is negligible. For $V=\square$, as $d(V)<d(P)$ we have $\chi_P(V)=1$. Thus, the contribution from $V=\square$ is
\begin{align*}
&\frac{q^{2g+1}}{|P|^{1/2}}\bigg(q\sum_{d(V)\leq d(P)/2-g-2}1-\sum_{d(V)\leq d(P)/2-g-1}1\bigg)-\frac{q^{2g}}{|P|^{1/2}}\bigg(q\sum_{d(V)\leq d(P)/2-g-1}1-\sum_{d(V)\leq d(P)/2-g}1\bigg)\\
&\qquad=\begin{cases}
-\frac{q^{2g}(q-1)}{|P|^{1/2}}& \textrm{if }d(P)\geq 2g+2,\\
q^g & \textrm{if }d(P)=2g.
\end{cases}
\end{align*}
We hence obtain that
\[
I_1^1(N;\alpha)=\mathds{1}_{N\geq 2g}\bigg(-\sum_{g+1\leq n\leq[N/2]}\sum_{P\in\mathcal{P}_{2n}}\frac{d(P)}{|P|^{1+\alpha}}+\frac{q^{-2g\alpha}}{q-1}\bigg)+O(Nq^{N/2-2g})+O(q^{- g}).
\]
Now, in view of the Prime Polynomial Theorem,
\begin{align*}
&\sum_{g+1\leq n\leq[N/2]}\sum_{P\in\mathcal{P}_{2n}}\frac{d(P)}{|P|^{1+\alpha}}=\sum_{g+1\leq n\leq[N/2]}q^{-2n(1+\alpha)}\big(q^{2n}+O(q^n)\big)\\
&\qquad\qquad=\sum_{g+1\leq n\leq[N/2]}q^{-2n\alpha}+O(q^{- g})=-\frac{q^{-2g\alpha}-q^{-2[N/2]\alpha}}{1-q^{2\alpha}}+O(q^{- g}).
\end{align*}
So
\[
I_1^1(N;\alpha)=\mathds{1}_{N\geq 2g}\bigg(\frac{q^{-2g\alpha}-q^{-2[N/2]\alpha}}{1-q^{2\alpha}}+\frac{q^{-2g\alpha}}{q-1}\bigg)+O(Nq^{N/2-2g})+O(q^{- g}).
\]
Notice that the leading term matches up with \eqref{type1}.
\section{The two level density - Using the Ratios Conjecture}
\label{2rc}
\subsection{The Ratios Conjecture}
We would like to study
\[
\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}\frac{L(1/2+\alpha,\chi_D)L(1/2+\beta,\chi_D)}{L(1/2+\gamma,\chi_D)L(1/2+\delta,\chi_D)}
\]
using the recipe in \cite{CS}, where the shifts are assumed to satisfy $|\alpha|, |\beta|, |\gamma|, |\delta| \ll 1/g$.
We use the approximate functional equation for each of the two $L$--functions in the numerator. The contribution coming from the first parts of the approximate functional equations is equal to
\[
\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{f_1,f_2,h_1,h_2}\frac{\mu(h_1)\mu(h_2)}{|f_1|^{1/2+\alpha}|f_2|^{1/2+\beta}|h_1|^{1/2+\gamma}|h_2|^{1/2+\delta}}\sum_{D\in\mathcal{H}_{2g+1}}\chi_D(f_1f_2h_1h_2).
\]
We only keep the terms with $f_1f_2h_1h_2=\square$. The above expression then becomes
\begin{align*}
&\sum_{f_1f_2h_1h_2=\square}\frac{\mu(h_1)\mu(h_2)a(f_1f_2h_1h_2)}{|f_1|^{1/2+\alpha}|f_2|^{1/2+\beta}|h_1|^{1/2+\gamma}|h_2|^{1/2+\delta}},
\end{align*}
where
\[
a(f)=\prod_{P|f}\bigg(1+\frac{1}{|P|}\bigg)^{-1}.
\]
Using multiplicativity, this is equal to
\begin{align*}
&\prod_{P}\sum_{\substack{f_1,f_2,h_1,h_2\\f_1+f_2+h_1+h_2\ \textrm{even}}}\frac{\mu(P^{h_1})\mu(P^{h_2})a(P^{f_1+f_2+h_1+h_2})}{|P|^{(1/2+\alpha)f_1+(1/2+\beta)f_2+(1/2+\gamma)h_1+(1/2+\delta)h_2}}\\
&\qquad\qquad=A(\alpha,\beta,\gamma,\delta)\frac{\zeta_q(1+2\alpha)\zeta_q(1+2\beta)\zeta_q(1+\alpha+\beta)\zeta_q(1+\gamma+\delta)}{\zeta_q(1+\alpha+\gamma)\zeta_q(1+\alpha+\delta)\zeta_q(1+\beta+\gamma)\zeta_q(1+\beta+\delta)},
\end{align*}
where
\begin{align*}
&A(\alpha,\beta,\gamma,\delta)=\prod_P\bigg(1+\frac{1}{|P|}\bigg)^{-1}\bigg(1-\frac{1}{|P|^{1+\alpha+\beta}}\bigg)\bigg(1-\frac{1}{|P|^{1+\gamma+\delta}}\bigg)\\
&\qquad \bigg(1-\frac{1}{|P|^{1+\alpha+\gamma}}\bigg)^{-1}\bigg(1-\frac{1}{|P|^{1+\alpha+\delta}}\bigg)^{-1}\bigg(1-\frac{1}{|P|^{1+\beta+\gamma}}\bigg)^{-1}\bigg(1-\frac{1}{|P|^{1+\beta+\delta}}\bigg)^{-1}\\
&\qquad\qquad\bigg(1+\frac{1}{|P|}+\frac{1}{|P|^{1+\alpha+\beta}}+\frac{1}{|P|^{1+\gamma+\delta}}-\frac{1}{|P|^{1+\alpha+\gamma}}-\frac{1}{|P|^{1+\alpha+\delta}}-\frac{1}{|P|^{1+\beta+\gamma}}-\frac{1}{|P|^{1+\beta+\delta}}\\
&\qquad\qquad\qquad-\frac{1}{|P|^{2+2\alpha}}-\frac{1}{|P|^{2+2\beta}}+\frac{1}{|P|^{2+\alpha+\beta+\gamma+\delta}}+\frac{1}{|P|^{3+2\alpha+2\beta}}\bigg).
\end{align*}
The contributions from the other parts of the approximate functional equations can be determined by using the functional equation
\[
L(\tfrac{1}{2}+\alpha,\chi_D)=q^{-2g\alpha }L(\tfrac{1}{2}-\alpha,\chi_D).
\]Hence we have the following.
\begin{conjecture}
We have
\begin{align*}
&\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}\frac{L(1/2+\alpha,\chi_D)L(1/2+\beta,\chi_D)}{L(1/2+\gamma,\chi_D)L(1/2+\delta,\chi_D)}\\
&\qquad=A(\alpha,\beta,\gamma,\delta)\frac{\zeta_q(1+2\alpha)\zeta_q(1+2\beta)\zeta_q(1+\alpha+\beta)\zeta_q(1+\gamma+\delta)}{\zeta_q(1+\alpha+\gamma)\zeta_q(1+\alpha+\delta)\zeta_q(1+\beta+\gamma)\zeta_q(1+\beta+\delta)}\\
&\qquad\qquad+q^{-2g\alpha }A(-\alpha,\beta,\gamma,\delta)\frac{\zeta_q(1-2\alpha)\zeta_q(1+2\beta)\zeta_q(1-\alpha+\beta)\zeta_q(1+\gamma+\delta)}{\zeta_q(1-\alpha+\gamma)\zeta_q(1-\alpha+\delta)\zeta_q(1+\beta+\gamma)\zeta_q(1+\beta+\delta)}\\
&\qquad\qquad+q^{-2g\beta }A(\alpha,-\beta,\gamma,\delta)\frac{\zeta_q(1+2\alpha)\zeta_q(1-2\beta)\zeta_q(1+\alpha-\beta)\zeta_q(1+\gamma+\delta)}{\zeta_q(1+\alpha+\gamma)\zeta_q(1+\alpha+\delta)\zeta_q(1-\beta+\gamma)\zeta_q(1-\beta+\delta)}\\
&\qquad\qquad+q^{-2g(\alpha+\beta) }A(-\alpha,-\beta,\gamma,\delta)\frac{\zeta_q(1-2\alpha)\zeta_q(1-2\beta)\zeta_q(1-\alpha-\beta)\zeta_q(1+\gamma+\delta)}{\zeta_q(1-\alpha+\gamma)\zeta_q(1-\alpha+\delta)\zeta_q(1-\beta+\gamma)\zeta_q(1-\beta+\delta)}\\
&\qquad\qquad+O_\varepsilon\big(q^{-g+\varepsilon g}\big).
\end{align*}
\end{conjecture}
Notice that for a function $f(u,v)$ analytic at $(u,v)=(r,r)$ and a function $F(s)$ having a simple pole at $s=1$ with residue $r_{F}$, we have
\begin{equation*}\label{trick101}
\frac{\partial}{\partial\alpha}\frac{f(\alpha,\gamma)}{F(1-\alpha+\gamma)}\bigg|_{\alpha=\gamma=r}=-\frac{f(r,r)}{r_{F}}.
\end{equation*}
As $r_{\zeta_q}=1/\log q$, taking derivatives with respect to $\alpha$ and $\beta$, and setting $\gamma=\alpha$, $\delta=\beta$ we obtain
\begin{conjecture}
We have
\begin{align*}
&\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}\frac{L'}{L}(\tfrac12+\alpha,\chi_D)\frac{L'}{L}(\tfrac12+\beta,\chi_D)\\
&\qquad=\frac{\zeta_q'}{\zeta_q}(1+2\alpha)\frac{\zeta_q'}{\zeta_q}(1+2\beta)+\bigg(\frac{\zeta_q'}{\zeta_q}\bigg)'(1+\alpha+\beta)\nonumber\\
&\qquad\qquad+(\log q)B(\alpha)\frac{\zeta_q'}{\zeta_q}(1+2\beta)+(\log q)B(\beta)\frac{\zeta_q'}{\zeta_q}(1+2\alpha)+(\log q)^2C(\alpha,\beta)\\
&\qquad\qquad+q^{-2g\alpha }(\log q)^2A_2(\alpha)T_2(\alpha,\beta)+q^{-2g\beta }(\log q)^2A_2(\beta)T_2(\beta,\alpha)\\
&\qquad\qquad+q^{-2g(\alpha+\beta) }(\log q)^2A(\alpha,\beta)\frac{\zeta_q(1-2\alpha)\zeta_q(1-2\beta)\zeta_q(1-\alpha-\beta)\zeta_q(1+\alpha+\beta)}{\zeta_q(1-\alpha+\beta)\zeta_q(1+\alpha-\beta)}\\
&\qquad\qquad+O_\varepsilon\big(q^{-g+\varepsilon g}\big),
\end{align*}
where
\begin{align*}
A_2(\alpha)&:=A(-\alpha,\beta,\alpha,\beta)\zeta_q(1-2\alpha)=\frac{\zeta_q(2)\zeta_q(1-2\alpha)}{\zeta_q(2-2\alpha)}\\
&=\frac{1}{1-q^{2\alpha}}+\frac{1}{q-1}=\frac{q^{2\alpha}}{1-q^{2\alpha}}+\frac{q}{q-1},
\end{align*}
\begin{align*}
A(\alpha,\beta)&:=A(-\alpha,-\beta,\alpha,\beta)\\
&=\prod_{P}\bigg(1+\frac{1}{|P|}\bigg)^{-1}\bigg(1-\frac{1}{|P|}\bigg)^{-2}\bigg(1-\frac{1}{|P|^{1-\alpha-\beta}}\bigg)\bigg(1-\frac{1}{|P|^{1+\alpha+\beta}}\bigg)\\
&\qquad \bigg(1-\frac{1}{|P|^{1-\alpha+\beta}}\bigg)^{-1}\bigg(1-\frac{1}{|P|^{1+\alpha-\beta}}\bigg)^{-1}\bigg(1-\frac{1}{|P|}+\frac{1}{|P|^{1-\alpha-\beta}}+\frac{1}{|P|^{1+\alpha+\beta}}\\
&\qquad\qquad-\frac{1}{|P|^{1-\alpha+\beta}}-\frac{1}{|P|^{1+\alpha-\beta}}-\frac{1}{|P|^{2-2\alpha}}-\frac{1}{|P|^{2-2\beta}}+\frac{1}{|P|^{2}}+\frac{1}{|P|^{3-2\alpha-2\beta}}\bigg),
\end{align*}
$B(\alpha)$ is defined in \eqref{Br},
\begin{align*}
C(\alpha,\beta)&=B(\alpha)B(\beta)+\sum_{P}\frac{d(P)^2\big(|P|^{2+\alpha+\beta}(|P|+1)(|P|^\alpha-|P|^\beta)^2-(|P|^{1+\alpha+\beta}-1)^3\big)}{(|P|^{1+2\alpha}-1)(|P|^{1+2\beta}-1)(|P|^{1+\alpha+\beta}-1)^2(|P|+1)}\\
&\qquad -\sum_{P}\frac{d(P)^2}{(|P|^{1+2\alpha}-1)(|P|^{1+2\beta}-1)(|P|+1)^2}\\
&=B(\alpha)B(\beta)+\sum_{P}\frac{d(P)^2|P|^{2+\alpha+\beta}(|P|^\alpha-|P|^\beta)^2}{(|P|^{1+2\alpha}-1)(|P|^{1+2\beta}-1)(|P|^{1+\alpha+\beta}-1)^2}\\
&\qquad-\sum_{P}\frac{d(P)^2|P|^{1+\alpha+\beta}}{(|P|^{1+2\alpha}-1)(|P|^{1+2\beta}-1)(|P|+1)}+\sum_{P}\frac{d(P)^2|P|}{(|P|^{1+2\alpha}-1)(|P|^{1+2\beta}-1)(|P|+1)^2}
\end{align*}
and
\begin{align}\label{factno6}
T_2(\alpha,\beta)&=\frac{1}{\log q}\bigg(\frac{\zeta_q'}{\zeta_q}(1+\alpha+\beta)-\frac{\zeta_q'}{\zeta_q}(1-\alpha+\beta)-\frac{\zeta_q'}{\zeta_q}(1+2\beta)-\frac{\partial A_2(-\alpha,b,\alpha,\beta)/\partial b\big|_{b=\beta}}{A_2(-\alpha,\beta,\alpha,\beta)}\bigg)\nonumber\\
&=\sum_{P}\frac{d(P)(|P|^{2(1-\alpha)}-|P|^{1-2\alpha}-|P|^{2-3\alpha+\beta}+|P|^{2-\alpha+\beta})}{(|P|^{2(1-\alpha)}-1)(|P|^{1+2\beta}-1)}.
\end{align}
\end{conjecture}
Equivalently we have
\begin{conjecture}\label{RCF2}
We have
\begin{align*}
&\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}uv\frac{\mathcal{L}'}{\mathcal{L}}(u,\chi_D)\frac{\mathcal{L}'}{\mathcal{L}}(v,\chi_D)\\
&\qquad=u^2v^2\frac{\mathcal{Z}'}{\mathcal{Z}}(u^2)\frac{\mathcal{Z}'}{\mathcal{Z}}(v^2)+u^2v^2\bigg(\frac{\mathcal{Z}'}{\mathcal{Z}}\bigg)'(uv)+\mathcal{B}(u)v^2\frac{\mathcal{Z}'}{\mathcal{Z}}(v^2)+\mathcal{B}(v)u^2\frac{\mathcal{Z}'}{\mathcal{Z}}(u^2)+\mathcal{C}(u,v)\\
&\qquad\qquad+(qu^2)^g\mathcal{A}_2(u)\mathcal{T}_2(u,v)+(qv^2)^g\mathcal{A}_2(v)\mathcal{T}_2(v,u)\\
&\qquad\qquad+(quv)^{2g}\mathcal{A}(u,v)\frac{\mathcal{Z}\big(\frac{1}{q^2u^2}\big)\mathcal{Z}\big(\frac{1}{q^2v^2}\big)\mathcal{Z}\big(\frac{1}{q^2uv}\big)\mathcal{Z}(uv)}{\mathcal{Z}\big(\frac{v}{qu}\big)\mathcal{Z}\big(\frac{u}{qv}\big)}+O_\varepsilon\big(q^{-g+\varepsilon g}\big),
\end{align*}
where
\begin{align*}
\mathcal{A}_2(u)&=\frac{qu^2}{qu^2-1}+\frac{1}{q-1}=\frac{1}{qu^2-1}+\frac{q}{q-1},
\end{align*}
\begin{align*}
\mathcal{A}(u,v)&=\prod_{P}\bigg(1+\frac{1}{|P|}\bigg)^{-1}\bigg(1-\frac{1}{|P|}\bigg)^{-2}\bigg(1-\frac{1}{|P|^{2}(uv)^{d(P)}}\bigg)\Big(1-(uv)^{d(P)}\Big)\\
&\qquad \bigg(1-\frac{v^{d(P)}}{|P|u^{d(P)}}\bigg)^{-1}\bigg(1-\frac{u^{d(P)}}{|P|v^{d(P)}}\bigg)^{-1}\bigg(1-\frac{1}{|P|}+\frac{1}{|P|^{2}(uv)^{d(P)}}+(uv)^{d(P)}\\
&\qquad\qquad-\frac{v^{d(P)}}{|P|u^{d(P)}}-\frac{u^{d(P)}}{|P|v^{d(P)}}-\frac{1}{|P|^{3}u^{2d(P)}}-\frac{1}{|P|^{3}v^{2d(P)}}+\frac{1}{|P|^{2}}+\frac{1}{|P|^{5}(uv)^{2d(P)}}\bigg),
\end{align*}
$\mathcal{B}(u)$ is defined in \eqref{mathcalBr},
\begin{align*}
\mathcal{C}(u,v)&=\mathcal{B}(u)\mathcal{B}(v)+\sum_{P}\frac{d(P)^2(uv)^{d(P)}(u^{d(P)}-v^{d(P)})^2}{(1-u^{2d(P)})(1-v^{2d(P)})(1-(uv)^{d(P)})^2}\\
&\qquad-\sum_{P}\frac{d(P)^2(uv)^{d(P)}}{(1-u^{2d(P)})(1-v^{2d(P)})(|P|+1)}+\sum_{P}\frac{d(P)^2|P|(uv)^{2d(P)}}{(1-u^{2d(P)})(1-v^{2d(P)})(|P|+1)^2}
\end{align*}
and
\begin{align*}
\mathcal{T}_2(u,v)&=\sum_{P}\frac{d(P)(|P|^{3}(uv)^{2d(P)}-|P|^{2}(uv)^{2d(P)}-|P|^{3}u^{3d(P)}v^{d(P)}+|P|^{2}(uv)^{d(P)})}{(|P|^{3}u^{2d(P)}-1)(1-v^{2d(P)})}.
\end{align*}
\end{conjecture}
\subsection{The two level density}
Consider
\begin{equation}\label{2level}
I_2(N;\alpha,\beta)=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}\sum_{\substack{f_1,f_2\in\mathcal{M}\\d(f_1f_2)\leq N}}\frac{\Lambda(f_1)\Lambda(f_2)\chi_D(f_1f_2)}{|f|_1^{1/2+\alpha}|f|_2^{1/2+\beta}}.
\end{equation}
Using the Perron formula \eqref{Perron} this is equal to
\begin{align*}
&\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}\frac{1}{2\pi i}\oint_{|u|=r}\sum_{f_1,f_2\in\mathcal{M}}\frac{\Lambda(f_1)\Lambda(f_2)\chi_D(f_1f_2)u^{d(f_1)+d(f_2)}}{|f|_1^{1/2+\alpha}|f|_2^{1/2+\beta}}\frac{du}{u^{N+1}(1-u)}\\
&\qquad=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}\frac{1}{2\pi i}\oint_{|u|=r} \frac{u^2}{q^{1+\alpha+\beta}} \frac{ \mathcal{L}'}{\mathcal{L}} \Big(\frac{u}{q^{1/2+\alpha}},\chi_D \Big) \frac{ \mathcal{L}'}{\mathcal{L}} \Big(\frac{u}{q^{1/2+\beta}},\chi_D \Big)\frac{du}{u^{N+1}(1-u)}
\end{align*}
for any $r<q^{-1/2-\varepsilon}$. We enlarge to contour to $|u|=r=q^{-\varepsilon}$.
In view of Conjecture \ref{RCF2} we write
\begin{align}
I_2(N;\alpha,\beta)=\frac{1}{2\pi i}\oint_{|u|=r}\sum_{j=1}^{4}R_j(u,\alpha,\beta)\,\frac{du}{u^{N+1}(1-u)}+O_\varepsilon\big(q^{-g+\varepsilon g}\big).
\label{int_density}
\end{align}
The terms coming from the first parts of the approximate functional equations, $R_1(u,\alpha,\beta)$, correspond to the diagonal terms, while the terms coming from only $1$ swap in the approximate functional equations, $R_2(u,\alpha,\beta)$ and $R_3(u,\alpha,\beta)$, correspond to the Type-I terms. Type-II terms are the terms with $2$ swaps, $R_4(u,\alpha,\beta)$.
For the $0$ swap terms we have
\begin{align}\label{type02level}
R_1(u,\alpha,\beta)&=\frac{u^4}{q^{2(1+\alpha+\beta)}}\frac{\mathcal{Z}'}{\mathcal{Z}} \Big(\frac{u^2}{q^{1+2\alpha}}\Big)\frac{\mathcal{Z}'}{\mathcal{Z}}\Big(\frac{u^2}{q^{1+2\beta}}\Big)+\frac{u^4}{q^{2(1+\alpha+\beta)}}\bigg(\frac{\mathcal{Z}'}{\mathcal{Z}}\bigg)'\Big(\frac{u^2}{q^{1+\alpha+\beta}}\Big)\\
&\qquad\qquad+\mathcal{B}(u,\alpha) \frac{u^2}{q^{1+2\beta}}\frac{\mathcal{Z}'}{\mathcal{Z}}\Big(\frac{u^2}{q^{1+2\beta}}\Big)+\mathcal{B}(u,\beta)\frac{u^2}{q^{1+2\alpha}}\frac{\mathcal{Z}'}{\mathcal{Z}}\Big(\frac{u^2}{q^{1+2\alpha}}\Big)+\mathcal{C}(u,\alpha,\beta),\nonumber
\end{align}
where
\begin{align}\label{type02level1}
\mathcal{C}(u,\alpha,\beta)&:=\mathcal{C}\Big(\frac{u}{q^{1/2+\alpha}},\frac{u}{q^{1/2+\beta}}\Big)\nonumber\\
&=\mathcal{B}(u,\alpha)\mathcal{B}(u,\beta)+\sum_{P}\frac{d(P)^2|P|^{2+\alpha+\beta}u^{4d(P)}(|P|^\alpha-|P|^\beta)^2}{(|P|^{1+2\alpha}-u^{2d(P)})(|P|^{1+2\beta}-u^{2d(P)})(|P|^{1+\alpha+\beta}-u^{2d(P)})^2}\nonumber\\
&\qquad\qquad-\sum_{P}\frac{d(P)^2|P|^{1+\alpha+\beta}u^{2d(P)}}{(|P|^{1+2\alpha}-u^{2d(P)})(|P|^{1+2\beta}-u^{2d(P)})(|P|+1)}\\
&\qquad\qquad+\sum_{P}\frac{d(P)^2|P|u^{4d(P)}}{(|P|^{1+2\alpha}-u^{2d(P)})(|P|^{1+2\beta}-u^{2d(P)})(|P|+1)^2}.\nonumber
\end{align}
Concerning the $1$ swap terms we have
\begin{align} \label{r2}
R_2(u,\alpha,\beta)+R_3(u,\alpha,\beta)=q^{-2g\alpha}u^{2g}\mathcal{A}_2(u,\alpha)\mathcal{T}_2(u,\alpha,\beta)+q^{-2g\beta}u^{2g}\mathcal{A}_2(u,\beta)\mathcal{T}_2(u,\beta,\alpha),
\end{align}
where
\begin{align}\label{A22}
\mathcal{A}_2(u,\alpha)&=\mathcal{A}_2\Big(\frac{u}{q^{1/2+\alpha}}\Big)\nonumber\\
&=\frac{u^2}{u^2-q^{2\alpha}}+\frac{1}{q-1}=\frac{q^{2\alpha}}{u^2-q^{2\alpha}}+\frac{q}{q-1}
\end{align}
and
\begin{align*}
\mathcal{T}_2(u,\alpha,\beta)&=\mathcal{T}_2\Big(\frac{u}{q^{1/2+\alpha}},\frac{u}{q^{1/2+\beta}}\Big)\nonumber\\
&=\sum_{P} \frac{u^{2d(P)}(|P|^{2(1-\alpha)}u^{2d(P)}-|P|^{1-2\alpha}u^{2d(P)}-|P|^{2-3\alpha+\beta}u^{2d(P)}+|P|^{2-\alpha+\beta})}{(|P|^{2(1-\alpha)}u^{2d(P)}-1)(|P|^{1+2\beta}-u^{2d(P)})}.
\end{align*}
Note that $1$ swap terms kick in once $N \geq 2g$. In the computation of Type-I terms in section \ref{type1} we also assume that $N <4g$. We write $\mathcal{T}_2(u,\alpha,\beta)$ as a sum of four terms. For the first three terms, we claim that we can truncate the sum over $P$ to those primes $P$ with $d(P)<g$; otherwise the corresponding integrals in equation \eqref{int_density} will be equal to zero. Indeed, in order for the integrals to be non-vanishing, we need $2g+2d(P)=N$. Since $N<4g$ it follows that $d(P)<g$. We write the fourth term in the expression of $\mathcal{T}_2(u,\alpha,\beta)$ as
\begin{align}
\sum_P & \frac{u^{2d(P)} |P|^{2-\alpha+\beta}}{(|P|^{2(1-\alpha)}u^{2d(P)}-1)(|P|^{1+2\beta}-u^{2d(P)})} = \sum_{d(P) <g} \frac{u^{2d(P)} |P|^{2-\alpha+\beta}}{(|P|^{2(1-\alpha)}u^{2d(P)}-1)(|P|^{1+2\beta}-u^{2d(P)})} \label{nonconv} \\
&+ \sum_{d(P) \geq g} \frac{u^{2d(P)} |P|^{2-\alpha+\beta}}{(|P|^{2(1-\alpha)}u^{2d(P)}-1)(|P|^{1+2\beta}-u^{2d(P)})} \nonumber \\
&= \sum_{d(P) <g} \frac{u^{2d(P)} |P|^{2-\alpha+\beta}}{(|P|^{2(1-\alpha)}u^{2d(P)}-1)(|P|^{1+2\beta}-u^{2d(P)})} + \sum_{d(P) \geq g} \frac{1}{|P|^{1+\beta-\alpha}} \nonumber \\
& + \sum_{d(P) \geq g} \frac{|P|^{2(1-\alpha)} u^{4d(P)}+|P|^{1+2\beta}+u^{2d(P)}}{|P|^{1+\beta-\alpha} (|P|^{2(1-\alpha)}u^{2d(P)}-1)(|P|^{1+2\beta}-u^{2d(P)})} \nonumber \\
&= \sum_{d(P) <g} \frac{u^{2d(P)} |P|^{2-\alpha+\beta}}{(|P|^{2(1-\alpha)}u^{2d(P)}-1)(|P|^{1+2\beta}-u^{2d(P)})} + \sum_{d(P) \geq g} \frac{1}{|P|^{1+\beta-\alpha}}+O(q^{-g}). \nonumber
\end{align}
We use the Prime Polynomial Theorem for the sum over $d(P)\geq g$ above and without worrying abut convergence issues since the recipe is a heuristic argument, we replace it by what we get by summing the geometric series.
Then when $N<4g$ we rewrite
\begin{align}
\mathcal{T}_2(u,\alpha,\beta) &= \sum_{d(P)<g} \frac{u^{2d(P)}(|P|^{2(1-\alpha)}u^{2d(P)}-|P|^{1-2\alpha}u^{2d(P)}-|P|^{2-3\alpha+\beta}u^{2d(P)}+|P|^{2-\alpha+\beta})}{(|P|^{2(1-\alpha)}u^{2d(P)}-1)(|P|^{1+2\beta}-u^{2d(P)})} \nonumber \\
&+ q^{g(-\beta+\alpha)} \frac{1}{q^{\alpha-\beta}-1}.
\label{t2}
\end{align}
We remark that although the term in the second line above gives a term involving $q^{-g(\alpha+\beta)}$ in the expression of $R_2(u,\alpha,\beta)$, when we put all the terms together, the contributions of this type will cancel out.
For the $2$ swaps terms we have
\begin{align*}
R_4(u,\alpha,\beta) = q^{-2g(\alpha+\beta)} u^{4g} \mathcal{A} \Big( \frac{u}{q^{1/2+\alpha}}, \frac{u}{q^{1/2+\beta}}\Big) \frac{ \mathcal{Z}( \frac{1}{q^{1-2 \alpha}u^2}) \mathcal{Z}( \frac{1}{q^{1-2 \beta}u^2} ) \mathcal{Z}( \frac{1}{q^{1- \alpha-\beta}u^2} ) \mathcal{Z}( \frac{u^2}{q^{1+\alpha+\beta}} )}{\mathcal{Z} (\frac{1}{q^{1-\alpha+\beta}}) \mathcal{Z}(\frac{1}{q^{1+\alpha-\beta}})}.
\end{align*}
\kommentar{\Hung{I got something slightly different from what you got.}
\acom{I suggest we include the expression for Type-II terms even if we don't actually compute them. Please check that I got this right.
Type-II terms correspond to the integral of $R_4(u,\alpha,\beta)$, where
\begin{align*}
R_4(u,\alpha,\beta) = q^{-2g(\alpha+\beta)} u^{4g} \mathcal{A} \Big( \frac{u}{q^{1/2+\alpha}}, \frac{u}{q^{1/2+\beta}}\Big) \frac{ \mathcal{Z} \Big( \frac{q^{-1+2 \alpha}}{u^2} \Big) \mathcal{Z} \Big( \frac{q^{-1+2 \beta}}{u^2} \Big) \mathcal{Z} \Big( \frac{q^{-1+ \alpha+\beta}}{u^2} \Big) \mathcal{Z} \Big( \frac{u^2}{q^{1+\alpha+\beta}} \Big)}{\mathcal{Z} (q^{-1+\alpha+\beta}) \mathcal{Z}(q^{-1-\alpha-\beta})}.
\end{align*}}
}
\section{The two level density - The diagonal}
\label{2diag}
In this and the following section, we assume that $N<4g$.
The diagonal, denoted by $I_2^0(N;\alpha,\beta)$, comes of the terms with $f_1f_2=\square$ in \eqref{2level}. From Lemma \ref{L4} and the Perron formula \eqref{Perron} we have
\begin{align}\label{600}
I_2^0(N;\alpha,\beta)&=\sum_{\substack{f_1,f_2\in\mathcal{M}\\d(f_1f_2)\leq N\\ f_1f_2=\square}}\frac{\Lambda(f_1)\Lambda(f_2)}{|f|_1^{1/2+\alpha}|f|_2^{1/2+\beta}}\prod_{P|f_1f_2}\bigg(1-\frac{1}{|P|+1}\bigg)+O_\varepsilon(q^{-2g+\varepsilon g})\nonumber\\
&=\frac{1}{2\pi i}\oint_{|u|=r}J_2^0(u,\alpha,\beta)\frac{du}{u^{N+1}(1-u)}+O_\varepsilon(q^{-2g+\varepsilon g})
\end{align}
for any $r<q^{-1/2-\varepsilon}$, where
\[
J_2^0(u,\alpha,\beta)=\sum_{\substack{f_1,f_2\in\mathcal{M}\\f_1f_2=\square}}\frac{\Lambda(f_1)\Lambda(f_2)u^{d(f_1f_2)}}{|f|_1^{1/2+\alpha}|f|_2^{1/2+\beta}}\prod_{P|f_1f_2}\bigg(1-\frac{1}{|P|+1}\bigg).
\]
We write
\[
J_2^0(u,\alpha,\beta)=J_{2}^{0,\textrm{ee}}(u,\alpha,\beta)+J_{2}^{0,\textrm{oo}}(u,\alpha,\beta),
\]
where $J_{2}^{0,\textrm{ee}}(u,\alpha,\beta)$ consists of the terms $f_1=P^{2k}$, $f_2=Q^{2l}$ with $k,l\geq1$, and $J_{2}^{0,\textrm{oo}}(u,\alpha,\beta)$ consists of the terms $f_1=P^{2k+1}$, $f_2=P^{2l+1}$ with $k,l\geq0$.
We have
\begin{align}\label{Jee}
J_{2}^{0,\textrm{ee}}(u,\alpha,\beta)&=\sum_{k,l\geq1}\sum_{P, Q}\frac{d(P)d(Q)u^{2kd(P)+2ld(Q)}}{|P|^{k(1+2\alpha)}|Q|^{l(1+2\beta)}}\bigg(1-\frac{1}{|P|+1}\bigg)\bigg(1-\frac{1}{|Q|+1}\bigg)\nonumber\\
&\qquad\qquad+\sum_{k,l\geq1}\sum_{P}\frac{d(P)^2|P|u^{2(k+l)d(P)}}{|P|^{k(1+2\alpha)+l(1+2\beta)}(|P|+1)^2}\nonumber\\
&=\bigg(\frac{u^2}{q^{1+2\alpha}}\frac{\mathcal{Z}'}{\mathcal{Z}} \Big(\frac{u^2}{q^{1+2\alpha}}\Big)+\mathcal{B}(u,\alpha)\bigg)\bigg(\frac{u^2}{q^{1+2\beta}}\frac{\mathcal{Z}'}{\mathcal{Z}} \Big(\frac{u^2}{q^{1+2\beta}}\Big)+\mathcal{B}(u,\beta)\bigg)\\
&\qquad\qquad+\sum_{P}\frac{d(P)^2|P|u^{4d(P)}}{(|P|^{1+2\alpha}-u^{2d(P)})(|P|^{1+2\beta}-u^{2d(P)})(|P|+1)^2}.\nonumber
\end{align}
On the other hand,
\begin{align*}
J_{2}^{0,\textrm{oo}}(u,\alpha,\beta)&=\sum_{k,l\geq0}\sum_{P}\frac{d(P)^2u^{2(k+l+1)d(P)}}{|P|^{1+\alpha+\beta+k(1+2\alpha)+l(1+2\beta)}}\bigg(1-\frac{1}{|P|+1}\bigg)\\
&=\sum_{P}\frac{d(P)^2|P|^{1+\alpha+\beta}u^{2d(P)}}{(|P|^{1+2\alpha}-u^{2d(P)})(|P|^{1+2\beta}-u^{2d(P)})}\\
&\qquad\qquad-\sum_{P}\frac{d(P)^2|P|^{1+\alpha+\beta}u^{2d(P)}}{(|P|^{1+2\alpha}-u^{2d(P)})(|P|^{1+2\beta}-u^{2d(P)})(|P|+1)}.
\end{align*}
Note that
\[
u^2\bigg(\frac{\mathcal{Z}'}{\mathcal{Z}}\bigg)'(u)=\sum_{P}\frac{d(P)^2u^{d(P)}}{(1-u^{d(P)})^2}.
\]
So
\begin{align}\label{Joo}
&J_{2}^{0,\textrm{oo}}(u,\alpha,\beta)-\frac{u^4}{q^{2(1+\alpha+\beta)}}\bigg(\frac{\mathcal{Z}'}{\mathcal{Z}}\bigg)'\Big(\frac{u^2}{q^{1+\alpha+\beta}}\Big)\nonumber\\
&\qquad\qquad=\sum_{P}\frac{d(P)^2|P|^{2+\alpha+\beta}u^{4d(P)}(|P|^\alpha-|P|^\beta)^2}{(|P|^{1+2\alpha}-u^{2d(P)})(|P|^{1+2\beta}-u^{2d(P)})(|P|^{1+\alpha+\beta}-u^{2d(P)})^2}\\
&\qquad\qquad\qquad\qquad-\sum_{P}\frac{d(P)^2|P|^{1+\alpha+\beta}u^{2d(P)}}{(|P|^{1+2\alpha}-u^{2d(P)})(|P|^{1+2\beta}-u^{2d(P)})(|P|+1)}.\nonumber
\end{align}
We enlarge the contour in \eqref{600} to $|u|=r=q^{-\varepsilon}$. Combining \eqref{Jee} and \eqref{Joo}, and comparing with \eqref{type02level} and \eqref{type02level1} we see that
\[
J_2^0(u,\alpha,\beta)=R_1(u,\alpha,\beta).
\]
\section{The two level density - Type-I terms}
\label{type1}
\subsection{The terms $f_1=P^{2k+1}$, $f_2=Q^{2l}$ with $k\geq0$, $l\geq1$} \label{type11} We denote this contribution by $I_{2}^{\textrm{oe}}(N;\alpha,\beta)$. In this section, we assume $N \geq 2g$. We have
\begin{align*}
I_{2}^{\textrm{oe}}(N;\alpha,\beta)=&\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{\substack{P\ne Q\\d(P^{2k+1}Q^{2l})\leq N}}\frac{d(P)d(Q)}{|P|^{(2k+1)(1/2+\alpha)}|Q|^{l(1+2\beta)}}\sum_{D\in\mathcal{H}_{2g+1}}\chi_D(PQ^2)\\
&\qquad\qquad+\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{d(P^{2k+2l+1})\leq N}\frac{d(P)^2}{|P|^{(2k+1)(1/2+\alpha)+l(1+2\beta)}}\sum_{D\in\mathcal{H}_{2g+1}}\chi_D(P).
\end{align*}
By the Polya-Vinogradov inequality in Lemma \ref{pv}, the second term is $O(N^2q^{-2g})$. We now consider the first term with $P\ne Q$. The same argument also shows the terms with $k\geq1$ are bounded by the same error term, and the contribution of the terms with $d(P)=n$ is $O(Nq^{n-2g})$.
So
\begin{align*}
I_{2}^{\textrm{oe}}(N;\alpha,\beta)=&\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{\substack{d(PQ^{2l})\leq N\\d(P)\geq g+1}}\frac{d(P)d(Q)}{|P|^{1/2+\alpha}|Q|^{l(1+2\beta)}}\sum_{D\in\mathcal{H}_{2g+1}}\chi_D(PQ^2)+O(Nq^{-g}).
\end{align*}
Applying Lemma \ref{L1} and since $d(P) \geq g+1$, we have
\begin{align}\label{P&Q}
\sum_{D \in \mathcal{H}_{2g+1}} \chi_D(PQ^2) &= \sum_{j\geq 0} \bigg(\sum_{h \in \mathcal{M}_{2g+1-2jd(Q)}} \chi_{PQ^2}(h) - q \sum_{h \in \mathcal{M}_{2g-1-2jd(Q)}} \chi_{PQ^2}(h)\bigg).
\end{align}
If $d(P)$ is odd, then using Lemmas \ref{L2} and \ref{L3} it follows that the term in parenthesis is equal to
\begin{align*}
&\frac{q^{2g+3/2}}{|P|^{1/2}|Q|^{2j+2}}\sum_{d(V)= d(P)+(2j+2)d(Q)-2g-2}\chi_P(V)G(V,Q^2)\\
&\qquad\qquad-\frac{q^{2g+1/2}}{|P|^{1/2}|Q|^{2j+2}}\sum_{d(V)= d(P)+(2j+2)d(Q)-2g}\chi_P(V)G(V,Q^2).
\end{align*}
As $V$ cannot be a square in the sums, by Lemma \ref{sumprimes}, the contribution of these terms to $I_{2}^{\textrm{oe}}(N;\alpha,\beta)$ is $O(N^2q^{N/2-2g})$.
Now consider the case $d(P)$ is even. Applying Lemmas \ref{L2} and \ref{L3}, the first sum over $h$ in \eqref{P&Q} is
\begin{align*}
&\frac{q^{2g+1}}{|P||Q|^{2j+2}}\bigg(q\sum_{V\in\mathcal{M}_{\leq d(P)+(2j+2)d(Q)-2g-3}} G(V,P)G(V,Q^2)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad- \sum_{V\in\mathcal{M}_{\leq d(P)+(2j+2)d(Q)-2g-2}}G(V,P)G(V,Q^2) \bigg)\\
&\qquad =-\frac{q^{2g+1}\chi_P(Q)}{|P|^{1/2}|Q|^{2j+1}}\bigg(q\sum_{\substack{V\in\mathcal{M}_{\leq d(P)+(2j+1)d(Q)-2g-3}\\(V,Q)=1}} \chi_P(V)- \sum_{\substack{V\in\mathcal{M}_{\leq d(P)+(2j+1)d(Q)-2g-2}\\(V,Q)=1}}\chi_P(V)\bigg)\\
&\qquad\qquad\qquad+\frac{q^{2g+1}\varphi(Q^2)}{|P|^{1/2}|Q|^{2j+2}}\bigg(q\sum_{V\in\mathcal{M}_{\leq d(P)+2jd(Q)-2g-3}} \chi_P(V)- \sum_{V\in\mathcal{M}_{\leq d(P)+2jd(Q)-2g-2}}\chi_P(V)\bigg).
\end{align*}
As above, the contribution of the first term and that of $V\ne\square$ in the second term to $I_{2}^{\textrm{oe}}(N;\alpha,\beta)$ is bounded by $O(N^2q^{N/2-2g})$. We are thus left with $V=\square$ in the second term above, which is equal to
\begin{align*}
&\frac{q^{2g+1}\varphi(Q^2)}{|P|^{1/2}|Q|^{2j+2}}\bigg(q\sum_{\substack{V\in\mathcal{M}_{\leq d(P)/2+jd(Q)-g-2}}} 1- \sum_{\substack{V\in\mathcal{M}_{\leq d(P)/2+jd(Q)-g-1}}}1\bigg)\\
&\qquad=\begin{cases}
-\frac{q^{2g+1}\varphi(Q^2)}{|P|^{1/2}|Q|^{2j+2}} & \quad\textrm{if } d(P)+2jd(Q)>2g,\\
0 & \quad\textrm{otherwise}.
\end{cases}
\end{align*}
The same argument applies to the second sum over $h$ in \eqref{P&Q}, and hence we obtain
\begin{align*}
&I_{2}^{\textrm{oe}}(N;\alpha,\beta)=-\sum_{\substack{d(PQ^{2l})\leq N\\d(P)\, \textrm{even}\,\geq g+1}}\,\sum_{\substack{d(P)+2jd(Q)> 2g}}\frac{d(P)d(Q)\varphi(Q^2)}{|P|^{1+\alpha}|Q|^{l(1+2\beta)+2j+2}}\\
&\qquad+\frac{1}{q-1}\sum_{\substack{d(PQ^{2l})\leq N\\d(P)\geq g+1}}\,\,\sum_{d(P)+2jd(Q)=2g}\frac{d(P)d(Q)\varphi(Q^2)}{|P|^{1+\alpha}|Q|^{l(1+2\beta)+2j+2}}+O(N^2q^{N/2-2g})+O(Nq^{-g}).
\end{align*}
By the Prime Polynomial Theorem, the condition $d(P)\geq g+1$ can be removed at the cost of an error of size $O(gq^{-g/2})$. The same argument also implies that we can restrict the sum to $jd(Q)<g$. So
\begin{align}
I_{2}^{\textrm{oe}}(N;\alpha,\beta) =\frac{1}{2\pi i}\oint_{|u|=r}J_{2}^{\textrm{oe}}(u,\alpha,\beta)\,\frac{du}{u^{N+1}(1-u)}+O(N^2q^{N/2-2g})+O(gq^{-g/2})
\label{i2}
\end{align}
for any $r<q^{-\varepsilon}$, where
\begin{align*}
J_{2}^{\textrm{oe}}(u,\alpha,\beta)&=-\sum_{l\geq1}\sum_{d(P)\, \textrm{even}}\,\sum_{\substack{d(P)+2jd(Q)>2g\\jd(Q)<g}}\frac{d(P)d(Q)\varphi(Q^2)u^{d(P)+2ld(Q)}}{|P|^{1+\alpha}|Q|^{l(1+2\beta)+2j+2}}\\
&\qquad\qquad+\frac{1}{q-1}\sum_{l\geq1}\,\sum_{\substack{d(P)+2jd(Q)=2g}}\frac{d(P)d(Q)\varphi(Q^2)u^{d(P)+2ld(Q)}}{|P|^{1+\alpha}|Q|^{l(1+2\beta)+2j+2}}\\
&=-\sum_{d(P)\, \textrm{even}}\,\sum_{\substack{d(P)+2jd(Q)>2g\\jd(Q)<g}}\frac{d(P)d(Q)\varphi(Q^2)u^{d(P)+2d(Q)}}{|P|^{1+\alpha}|Q|^{2j+2}(|Q|^{1+2\beta}-u^{2d(Q)})}\\
&\qquad\qquad+\frac{1}{q-1}\,\sum_{\substack{d(P)+2jd(Q)=2g}}\frac{d(P)d(Q)\varphi(Q^2)u^{d(P)+2d(Q)}}{|P|^{1+\alpha}|Q|^{2j+2}(|Q|^{1+2\beta}-u^{2d(Q)})}.
\end{align*}
From the Prime Polynomial Theorem we have
\begin{align*}
\sum_{d(P)\, \textrm{even}\,>2g-2jd(Q)}\frac{d(P)u^{d(P)}}{|P|^{1+\alpha}}&=\sum_{n>g-jd(Q)}\frac{u^{2n}}{q^{2n\alpha}}\big(1+O(q^{-n})\big)\\
&=-q^{-2g\alpha}u^{2g-2jd(Q)}|Q|^{2j\alpha}\frac{u^{2}}{u^2-q^{2\alpha}}+O(q^{-g}|Q|^j).
\end{align*}
\kommentar{\acom{For the error term to have that size I think we need to assume $\alpha>0$.}\Hung{We shall assume $\alpha\ll 1/g$, and I think the above error term is fine in that case.}}
Hence, using \eqref{A22}, we get
\begin{align}
J_{2}^{\textrm{oe}}(u,\alpha,\beta)&=q^{-2g\alpha}u^{2g}\mathcal{A}_2(u,\alpha)\sum_{j\geq0}\sum_{Q\in\mathcal{P}}\frac{d(Q)\varphi(Q^2)u^{-2jd(Q)+2d(Q)}}{|Q|^{2j(1-\alpha)+2}(|Q|^{1+2\beta}-u^{2d(Q)})}+O(gq^{-g})\nonumber\\
&=q^{-2g\alpha}u^{2g}\mathcal{A}_2(u,\alpha)\sum_{Q\in\mathcal{P}}\frac{d(Q)(|Q|^{2(1-\alpha)}-|Q|^{1-2\alpha})u^{4d(Q)}}{(|Q|^{2(1-\alpha)}u^{2d(Q)}-1)(|Q|^{1+2\beta}-u^{2d(Q)})}+O(gq^{-g}),
\end{align}
where in the first line we have removed the condition $jd(Q)<g$ with an admissible error.
Note that we can truncate the sum over $Q$ above to $d(Q)<g$ using a similar argument as in section \ref{2rc}. Indeed, when $d(Q) \geq g$ the corresponding term in integral \eqref{i2} will be equal to zero since there will be no poles inside the contour of integration. Then we rewrite
\begin{equation} \label{factno1}
J_{2}^{\textrm{oe}}(u,\alpha,\beta)= q^{-2g\alpha}u^{2g}\mathcal{A}_2(u,\alpha)\sum_{d(Q)<g}\frac{d(Q)(|Q|^{2(1-\alpha)}-|Q|^{1-2\alpha})u^{4d(Q)}}{(|Q|^{2(1-\alpha)}u^{2d(Q)}-1)(|Q|^{1+2\beta}-u^{2d(Q)})}+O(gq^{-g}).
\end{equation}
\subsection{The terms $f_1=P^{2k+1}$, $f_2=Q^{2l+1}$ with $P\ne Q$ and $k,l\geq0$} \label{type12}
We denote
\begin{align*}
&\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{\substack{P\ne Q\\d(P^{2k+1}Q^{2l+1})\leq N}}\frac{d(P)d(Q)}{|P|^{(2k+1)(1/2+\alpha)}|Q|^{(2l+1)(1/2+\beta)}}\sum_{D\in\mathcal{H}_{2g+1}}\chi_D(PQ)\\
&\qquad\qquad=I_{2,>}^{\textrm{oo}}(N;\alpha,\beta)+I_{2,<}^{\textrm{oo}}(N;\alpha,\beta)+I_{2,=}^{\textrm{oo}}(N;\alpha,\beta),
\end{align*}
corresponding to the terms with $d(P)> d(Q)$, $d(P)<d(Q)$ and $d(P)=d(Q)$, respectively.
Applying Lemma \ref{L1} we have
\begin{align}\label{condition}
\sum_{D \in \mathcal{H}_{2g+1}} \chi_D(PQ) &= \sum_{i,j\geq0} \bigg(\sum_{h \in \mathcal{M}_{2g+1-2id(P)-2jd(Q)}} \chi_{PQ}(h) - q \sum_{h \in \mathcal{M}_{2g-1-2id(P)-2jd(Q)}} \chi_{PQ}(h)\bigg).
\end{align}
As in the previous subsection, the terms with $d(PQ)$ odd shall lead to $V\ne\square$ after applying Lemma \ref{L2}, and their contribution, as before, is bounded by $O(N^2q^{N/2-2g})$. We are left with the terms with $d(PQ)$ even.
From Lemmas \ref{L2} and \ref{L3}, the expression inside the bracket is equal to
\begin{align*}
&\frac{q^{2g+1}}{|P|^{2i+1/2}|Q|^{2j+1/2}}\bigg(q\sum_{V \in \mathcal{M}_{\leq d(PQ)+2id(P)+2jd(Q)-2g-3}} \chi_{PQ}(V)- \sum_{V \in \mathcal{M}_{\leq d(PQ)+2id(P)+2jd(Q)-2g-2}} \chi_{PQ}(V) \bigg)\\
&\ -\frac{q^{2g}}{|P|^{2i+1/2}|Q|^{2j+1/2}}\bigg(q\sum_{V \in \mathcal{M}_{\leq d(PQ)+2id(P)+2jd(Q)-2g-1}} \chi_{PQ}(V)- \sum_{V \in \mathcal{M}_{\leq d(PQ)+2id(P)+2jd(Q)-2g}} \chi_{PQ}(V) \bigg).
\end{align*}
Again the contribution from the terms $V\ne\square$ is negligible and we focus on the term with $V=\square$, which is
\begin{align}\label{500}
&\frac{q^{2g+1}}{|P|^{2i+1/2}|Q|^{2j+1/2}}\bigg(q\sum_{\substack{V \in \mathcal{M}_{\leq d(PQ)/2+id(P)+jd(Q)-g-2}\\(V,PQ)=1}}1- \sum_{\substack{V \in \mathcal{M}_{\leq d(PQ)/2+id(P)+jd(Q)-g-1}\\(V,PQ)=1}} 1\bigg)\\
&\qquad-\frac{q^{2g}}{|P|^{2i+1/2}|Q|^{2j+1/2}}\bigg(q\sum_{\substack{V \in \mathcal{M}_{\leq d(PQ)/2+id(P)+jd(Q)-g-1}\\(V,PQ)=1}}1- \sum_{\substack{V \in \mathcal{M}_{\leq d(PQ)/2+id(P)+jd(Q)-g}\\(V,PQ)=1}} 1 \bigg).\nonumber
\end{align}
First consider $I_{2,>}^{\textrm{oo}}(N;\alpha,\beta)$. The treatment for $I_{2,<}^{\textrm{oo}}(N;\alpha,\beta)$ is similar. From \eqref{condition} we have $id(P)+jd(Q)\leq g$, so
\[
d(V)\leq d(PQ)/2+id(P)+jd(Q)-g\leq d(PQ)/2<d(P),
\] and hence $(V,P)=1$ automatically. Note that
\begin{align*}
&q\sum_{\substack{V \in \mathcal{M}_{\leq d(PQ)/2+id(P)+jd(Q)-g-1}\\(V,Q)=1}}1- \sum_{\substack{V \in \mathcal{M}_{\leq d(PQ)/2+id(P)+jd(Q)-g}\\(V,Q)=1}} 1\\
&\qquad=\bigg(q\sum_{V \in \mathcal{M}_{\leq d(PQ)/2+id(P)+jd(Q)-g-1}}1- \sum_{V \in \mathcal{M}_{\leq d(PQ)/2+id(P)+jd(Q)-g}} 1\bigg)\\
&\qquad\qquad\qquad-\bigg(q\sum_{V \in \mathcal{M}_{\leq (d(P)-d(Q))/2+id(P)+jd(Q)-g-1}}1- \sum_{V \in \mathcal{M}_{\leq (d(P)-d(Q))/2+id(P)+jd(Q)-g}} 1\bigg)\\
&\qquad=\begin{cases}
-1 & \quad\textrm{if } (2i+1)d(P)+(2j-1)d(Q)<2g\leq (2i+1)d(P)+(2j+1)d(Q),\\
0 & \quad\textrm{otherwise}.
\end{cases}
\end{align*}
So
\begin{align*}
\eqref{500}=\begin{cases}
-\frac{q^{2g}(q-1)}{|P|^{2i+1/2}|Q|^{2j+1/2}} & \textrm{if } (2i+1)d(P)+(2j-1)d(Q)<2g< (2i+1)d(P)+(2j+1)d(Q),\\
-\frac{q^{2g+1}}{|P|^{2i+1/2}|Q|^{2j+1/2}} & \textrm{if } (2i+1)d(P)+(2j-1)d(Q)=2g,\\
\frac{q^{2g}}{|P|^{2i+1/2}|Q|^{2j+1/2}} & \textrm{if } (2i+1)d(P)+(2j+1)d(Q)=2g,\\
0 & \textrm{otherwise.}
\end{cases}
\end{align*}
Hence $I_{2,>}^{\textrm{oo}}(N;\alpha,\beta)$ is equal to, up to an error of size $O(N^2q^{N/2-2g})$,
\begin{align*}
&-\sum_{\substack{d(P)>d(Q)\\d(P^{2k+1}Q^{2l+1})\, \textrm{even}\,\leq N}}\,\sum_{\substack{(2i+1)d(P)+(2j-1)d(Q)< 2g\\2g< (2i+1)d(P)+(2j+1)d(Q)}}\frac{d(P)d(Q)}{|P|^{(2k+1)(1/2+\alpha)+2i+1/2}|Q|^{(2l+1)(1/2+\beta)+2j+1/2}}\\
&\quad-\frac{q}{q-1}\sum_{\substack{d(P)>d(Q)\\d(P^{2k+1}Q^{2l+1})\, \textrm{even}\,\leq N}}\,\sum_{(2i+1)d(P)+(2j-1)d(Q)=2g}\frac{d(P)d(Q)}{|P|^{(2k+1)(1/2+\alpha)+2i+1/2}|Q|^{(2l+1)(1/2+\beta)+2j+1/2}}\\
&\quad+\frac{1}{q-1}\sum_{\substack{d(P)>d(Q)\\d(P^{2k+1}Q^{2l+1})\, \textrm{even}\,\leq N}}\,\sum_{(2i+1)d(P)+(2j+1)d(Q)=2g}\frac{d(P)d(Q)}{|P|^{(2k+1)(1/2+\alpha)+2i+1/2}|Q|^{(2l+1)(1/2+\beta)+2j+1/2}}.
\end{align*}
By the Prime Polynomial Theorem, the contribution of the terms with $d(P)=m$ and $d(Q)=n$ to $I_{2,>}^{\textrm{oo}}(N;\alpha,\beta)$ for each $i,j,k,l$ is bounded by
\begin{equation}\label{bound100}
\ll q^{-(k+2i)m-(l+2j)n}.
\end{equation}
Note that $(2i+1)m+(2j+1)n\geq2g$, so this is, in particular, bounded by $O( q^{-2g+(1-k)m+(1-l)n})$. It follows that the contribution of the terms with $m+n\leq g$ is $O(q^{-g})$. For those with $m+n>g$, the condition $m>n$ leads to $m>g/2$, and it follows from \eqref{bound100} that the contribution of such terms with $i+k\geq 1$ is $O(Nq^{-g/2})$. Hence we can restrict to the case $i=k=0$ and get
\begin{align*}
I_{2,>}^{\textrm{oo}}(N;\alpha,\beta)&=-\sum_{\substack{d(P)>d(Q)\\d(PQ^{2l+1})\, \textrm{even}\,\leq N}}\,\sum_{\substack{d(P)+(2j-1)d(Q)< 2g\\2g< d(P)+(2j+1)d(Q)}}\frac{d(P)d(Q)}{|P|^{1+\alpha}|Q|^{(2l+1)(1/2+\beta)+2j+1/2}}\\
&\qquad\qquad-\frac{q}{q-1}\sum_{\substack{d(P)>d(Q)\\d(PQ^{2l+1})\leq N}}\,\sum_{d(P)+(2j-1)d(Q)=2g}\frac{d(P)d(Q)}{|P|^{1+\alpha}|Q|^{(2l+1)(1/2+\beta)+2j+1/2}}\\
&\qquad\qquad+\frac{1}{q-1}\sum_{\substack{d(P)>d(Q)\\d(PQ^{2l+1})\leq N}}\,\sum_{d(P)+(2j+1)d(Q)=2g}\frac{d(P)d(Q)}{|P|^{1+\alpha}|Q|^{(2l+1)(1/2+\beta)+2j+1/2}}\\
&\qquad\qquad+O(N^2q^{N/2-2g})+O(Nq^{-g/2}).
\end{align*}
\kommentar{\acom{In the last two terms we need $d(PQ^{2l+1})$ to be even.}\Hung{$d(P)+(2j\pm1)d(Q)=2g$, and hence $d(PQ^{2l+1})$ even.}}
We shall write
\[
I_{2,>}^{\textrm{oo}}(N;\alpha,\beta)=I_{2,>}^{\textrm{oo}\flat}(N;\alpha,\beta)+I_{2,>}^{\textrm{oo}\dagger}(N;\alpha,\beta)+O(N^2q^{N/2-2g})+O(Nq^{-g/2})
\]
to separate the cases $j+l\geq 1$ and $j=l=0$, respectively. For $I_{2,>}^{\textrm{oo}\flat}(\alpha,\beta)$, by the Perron formula we have
\[
I_{2,>}^{\textrm{oo}\flat}(N;\alpha,\beta)=\frac{1}{2\pi i}\oint_{|u|=r}J_{2,>}^{\textrm{oo}\flat}(u,\alpha,\beta)\,\frac{du}{u^{N+1}(1-u)}
\]
for any $r<q^{-\varepsilon}$, where
\begin{align*}
J_{2,>}^{\textrm{oo}\flat}(u,\alpha,\beta)=&-\sum_{l+j\geq1}\,\sum_{\substack{d(P)>d(Q)\\d(PQ)\, \textrm{even}\\d(P)+(2j-1)d(Q)< 2g\\2g< d(P)+(2j+1)d(Q)}}\frac{d(P)d(Q)u^{d(P)+(2l+1)d(Q)}}{|P|^{1+\alpha}|Q|^{(2l+1)(1/2+\beta)+2j+1/2}}\\
&\qquad\qquad-\frac{q}{q-1}\,\sum_{l+j\geq1}\,\sum_{\substack{d(P)>d(Q)\\d(P)+(2j-1)d(Q)=2g}}\frac{d(P)d(Q)u^{d(P)+(2l+1)d(Q)}}{|P|^{1+\alpha}|Q|^{(2l+1)(1/2+\beta)+2j+1/2}}\\
&\qquad\qquad+\frac{1}{q-1}\,\sum_{l+j\geq1}\,\sum_{\substack{d(P)>d(Q)\\d(P)+(2j+1)d(Q)=2g}}\frac{d(P)d(Q)u^{d(P)+(2l+1)d(Q)}}{|P|^{1+\alpha}|Q|^{(2l+1)(1/2+\beta)+2j+1/2}}.
\end{align*}
Given $Q$, from the Prime Polynomial Theorem we have
\begin{align*}
&\sum_{\substack{dP)>d(Q)\\d(PQ)\, \textrm{even}\\d(P)+(2j-1)d(Q)< 2g\\2g< d(P)+(2j+1)d(Q)}}\frac{d(P)u^{d(PQ)}}{|P|^{1+\alpha}}=\sum_{\max\{d(Q),g-jd(Q)\}<n<g-(j-1)d(Q)}\frac{|Q|^\alpha u^{2n}}{q^{2n\alpha}}\big(1+O(q^{-n}|Q|^{1/2})\big)\\
&=\begin{cases}-q^{-2g\alpha}u^{2g-2jd(Q)}|Q|^{(2j+1)\alpha}\frac{u^{2}}{u^2-q^{2\alpha}}+q^{-2g\alpha}u^{2g-2(j-1)d(Q)}|Q|^{(2j-1)\alpha}\frac{q^{2\alpha}}{u^2-q^{2\alpha}}+O(q^{-g}|Q|^{j+1/2})\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{if }(j+1)d(Q)<g,\\
-u^{2d(Q)}|Q|^{-\alpha}\frac{u^{2}}{u^2-q^{2\alpha}}+q^{-2g\alpha}u^{2g-2(j-1)d(Q)}|Q|^{(2j-1)\alpha}\frac{q^{2\alpha}}{u^2-q^{2\alpha}}+O(|Q|^{-1/2})\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{if }jd(Q)< g\leq (j+1)d(Q).
\end{cases}
\end{align*}
\kommentar{\acom{For the error term I get that we have an extra $|Q|^{2rj}$, and since $r = \alpha-\log_q u$ if $\alpha$ is for example close to $1/4$ then the error term is much bigger.}\Hung{Again, we shall assume $\alpha \ll 1/g$, and I think the above is fine}}
Hence
\begin{align*}
&J_{2,>}^{\textrm{oo}\flat}(u,\alpha,\beta)=\,q^{-2g\alpha}u^{2g}\bigg(\frac{u^{2}}{u^2-q^{2\alpha}}+\frac{1}{q-1}\bigg)\sum_{l+j\geq1}\,\sum_{(j+1)d(Q)<g}\frac{d(Q)u^{2(l-j)d(Q)}}{|Q|^{1-\alpha+\beta+l(1+2\beta)+2j(1-\alpha)}}\\
&\qquad+\frac{u^{2}}{u^2-q^{2\alpha}}\sum_{l+j\geq1}\,\sum_{jd(Q)< g\leq (j+1)d(Q)}\frac{d(Q)u^{2(l+1)d(Q)}}{|Q|^{1+\alpha+\beta+l(1+2\beta)+2j}}\\
&\qquad-q^{-2g\alpha}u^{2g}\bigg(\frac{q^{2\alpha}}{u^2-q^{2\alpha}}+\frac{q}{q-1}\bigg)\sum_{l+j\geq1}\,\sum_{jd(Q)<g}\frac{d(Q)u^{2(l-j+1)d(Q)}}{|Q|^{1+\alpha+\beta+l(1+2\beta)+2j(1-\alpha)}}+O(q^{-g/2}).
\end{align*}
By the Prime Polynomial Theorem again, it is easy to see that the second expression is bounded by $O(q^{-g})$. Also, we can extend the sum over $Q$ in the third expression to all of $Q\in\mathcal{P}$ at the cost of an error of size $O_\varepsilon(q^{-2g+\varepsilon g})$. For the first expression, we write
\begin{align*}
&\sum_{l+j\geq1}\,\sum_{(j+1)d(Q)<g}\frac{d(Q)}{|Q|^{1-\alpha+\beta}}\,x^ly^j = \sum_{d(Q)<g}\frac{d(Q)}{|Q|^{1-\alpha+\beta}}\sum_{\substack{l+j\geq1\\j< g/d(Q)-1}}x^ly^j\\
&\qquad\qquad=\sum_{d(Q)<g}\frac{d(Q)}{|Q|^{1-\alpha+\beta}}\sum_{l+j\geq1}x^ly^j+O_\varepsilon(q^{-g+\varepsilon g})\\
&\qquad\qquad=\sum_{d(Q)<g}\frac{d(Q)}{|Q|^{1-\alpha+\beta}}\,\bigg(\frac{x}{1-x}+\frac{y}{(1-x)(1-y)}\bigg)+O_\varepsilon(q^{-g+\varepsilon g}).
\end{align*}
\kommentar{\acom{I'm not sure that this is correct. For example if $j=1$ and $l=0$ I get $q^{-2g/3} q^{(\alpha-\beta)g/3}$ and again if $\alpha \sim 1/4$ the above can be bigger. Maybe you're assuming everywhere that $\Re(\alpha), \Re(\beta) \ll 1/g$?}\Hung{Same comment as above}}
The identities in \eqref{A22} and an argument similar to that used in the evaluation of $J_{2}^{\textrm{oe}}(u,\alpha,\beta)$ in equation \eqref{factno1} then imply that
\begin{align}\label{factno2}
&J_{2,>}^{\textrm{oo}\flat}(u,\alpha,\beta)=\,q^{-2g\alpha}u^{2g}\mathcal{A}_2(u,\alpha)\bigg(\sum_{d(Q)<g}\frac{d(Q)u^{2d(Q)}}{|Q|^{1-\alpha+\beta}(|Q|^{1+2\beta}-u^{2d(Q)})}\nonumber\\
&\qquad+\sum_{d(Q)<g}\frac{d(Q)|Q|^{\alpha+\beta}}{(|Q|^{2(1-\alpha)}u^{2d(Q)}-1)(|Q|^{1+2\beta}-u^{2d(Q)})}\\
&\qquad-\sum_{d(Q)<g}\frac{d(Q)|Q|^{2-3\alpha+\beta}u^{4d(Q)}}{(|Q|^{2(1-\alpha)}u^{2d(Q)}-1)(|Q|^{1+2\beta}-u^{2d(Q)})}+\sum_{d(Q)<g}\frac{d(Q)u^{2d(Q)}}{|Q|^{1+\alpha+\beta}} \bigg)+O(q^{-g/2}).\nonumber
\end{align}
For $I_{2,>}^{\textrm{oo}\dagger}(N;\alpha,\beta)$, by the Perron formula we have
\[
I_{2,>}^{\textrm{oo}\dagger}(N;\alpha,\beta)=\frac{1}{2\pi i}\oint_{|u|=r}J_{2,>}^{\textrm{oo}\dagger}(u,\alpha,\beta)\,\frac{du}{u^{N+1}(1-u)} \label{idag}
\]
for any $r<q^{-\varepsilon}$, where
\begin{align*}
J_{2,>}^{\textrm{oo}\dagger}(u,\alpha,\beta)&=-\sum_{\substack{d(P)>d(Q)\\d(PQ)\, \textrm{even}}}\,\sum_{\substack{d(P)-d(Q)< 2g\\2g< d(P)+d(Q)}}\frac{d(P)d(Q)u^{d(PQ)}}{|P|^{1+\alpha}|Q|^{1+\beta}}-\frac{q}{q-1}\sum_{d(P)-d(Q)=2g}\frac{d(P)d(Q)u^{d(PQ)}}{|P|^{1+\alpha}|Q|^{1+\beta}}\\
&\qquad\qquad+\frac{1}{q-1}\sum_{d(P)>d(Q)}\,\sum_{d(P)+d(Q)=2g}\frac{d(P)d(Q)u^{d(PQ)}}{|P|^{1+\alpha}|Q|^{1+\beta}}.
\end{align*}
The last two terms can be evaluated using the Prime Polynomial Theorem. Concerning the first term, note that given $Q$,
\begin{align*}
&\sum_{\substack{d(P)>d(Q)\\d(PQ)\, \textrm{even}}}\,\sum_{\substack{d(P)-d(Q)< 2g\\2g< d(P)+d(Q)}}\frac{d(P)u^{d(PQ)}}{|P|^{1+\alpha}}=\sum_{\max\{d(Q),g\}<n<g+d(Q)}\frac{|Q|^\alpha u^{2n}}{q^{2n\alpha}}\big(1+O(q^{-n}|Q|^{1/2})\big)\\
&\qquad\qquad=\begin{cases}
-q^{-2g\alpha}u^{2g}|Q|^{\alpha}\frac{u^{2}}{u^2-q^{2\alpha}}+q^{-2g\alpha}u^{2g+2d(Q)}|Q|^{-\alpha}\frac{q^{2\alpha}}{u^2-q^{2\alpha}}+O(q^{-g/2}) & \textrm{if }d(Q)< g,\\
-|Q|^{-\alpha}u^{2d(Q)}\frac{u^{2}}{u^2-q^{2\alpha}}+q^{-2g\alpha}u^{2g+2d(Q)}|Q|^{-\alpha}\frac{q^{2\alpha}}{u^2-q^{2\alpha}}+O(q^{-g/2}) & \textrm{if }d(Q)\geq g,
\end{cases}
\end{align*}
by writing $d(PQ)=2n$.
\kommentar{\acom{I agree with the answer, but I find the way it is written confusing. I'd write
\begin{align*}
\sum_{\substack{d(P)>d(Q)\\d(PQ)\, \textrm{even}}}\,\sum_{\substack{d(P)-d(Q)< 2g\\2g< d(P)+d(Q)}}\frac{d(P)}{|P|^{1+r}} = \sum_{\substack{\max\{d(Q),2g-d(Q)\}<n<2g+d(Q) \\ n \equiv d(Q) \pmod 2}} \Big(q^{-nr} + O(q^{-n/2-nr}) \Big).
\end{align*}
}\Hung{I try to keep it consistent with the way I wrote before, but either way is fine}}
So
\begin{align*}
&-\sum_{\substack{d(P)>d(Q)\\d(PQ)\, \textrm{even}}}\,\sum_{\substack{d(P)-d(Q)< 2g\\2g< d(P)+d(Q)}}\frac{d(P)d(Q)u^{d(PQ)}}{|P|^{1+\alpha}|Q|^{1+\beta}}=q^{-2g\alpha}u^{2g}\frac{u^{2}}{u^2-q^{2\alpha}}\sum_{d(Q)< g}\frac{d(Q)}{|Q|^{1-\alpha+\beta}}\\
&\qquad\qquad +\frac{u^{2}}{u^2-q^{2\alpha}}\sum_{d(Q)\geq g}\frac{d(Q)u^{2d(Q)}}{|Q|^{1+\alpha+\beta}}-q^{-2g\alpha}u^{2g}\frac{q^{2\alpha}}{u^2-q^{2\alpha}}\sum_{Q\in\mathcal{P}}\frac{d(Q)u^{2d(Q)}}{|Q|^{1+\alpha+\beta}}+O(q^{-g/2}).
\end{align*}
Hence, using \eqref{A22}, we have
\begin{align*}
J_{2,>}^{\textrm{oo}\dagger}(u,\alpha,\beta)&=q^{-2g\alpha}u^{2g}\bigg(\frac{u^{2}}{u^2-q^{2\alpha}}+\frac{1}{q-1}\bigg)\sum_{d(Q)< g}\frac{d(Q)}{|Q|^{1-\alpha+\beta}}+\frac{u^{2}}{u^2-q^{2\alpha}}\sum_{d(Q)\geq g}\frac{d(Q)u^{2d(Q)}}{|Q|^{1+\alpha+\beta}}\nonumber\\
&\qquad\qquad-q^{-2g\alpha}u^{2g}\bigg(\frac{q^{2\alpha}}{u^2-q^{2\alpha}}+\frac{q}{q-1}\bigg)\sum_{Q\in\mathcal{P}}\frac{d(Q)u^{2d(Q)}}{|Q|^{1+\alpha+\beta}}+O(q^{-g/2})\nonumber\\
&=q^{-2g\alpha}u^{2g}\mathcal{A}_2(u,\alpha)\bigg(\sum_{d(Q)< g}\frac{d(Q)}{|Q|^{1-\alpha+\beta}}-\sum_{d(Q)<g}\frac{d(Q)u^{2d(Q)}}{|Q|^{1+\alpha+\beta}}\bigg)\\
&\qquad\qquad+\frac{u^{2}}{u^2-q^{2\alpha}}\sum_{d(Q)\geq g}\frac{d(Q)u^{2d(Q)}}{|Q|^{1+\alpha+\beta}}+O(q^{-g/2}),\nonumber
\end{align*}
where in the second identity we truncated the second sum over $Q$ using a similar argument as before.
For the third term, from the Prime Polynomial Theorem we have
\[
\sum_{d(Q)\geq g}\frac{d(Q)u^{2d(Q)}}{|Q|^{1+\alpha+\beta}}=\sum_{n\geq g}\frac{u^{2n}}{q^{n(\alpha+\beta)}}\big(1+O(q^{-n/2})\big)=-q^{-g(\alpha+\beta)}u^{2g}\frac{q^{\alpha+\beta}}{u^2-q^{\alpha+\beta}}+O(q^{-g/2}).
\]
\kommentar{\acom{Again, we need $\Re(\alpha) \geq 0$ for the error term.}\Hung{I rewrite things a bit differently}}
Thus,
\begin{align}
J_{2,>}^{\textrm{oo}\dagger}(u,\alpha,\beta)&=q^{-2g\alpha}u^{2g}\mathcal{A}_2(u,\alpha)\bigg(\sum_{d(Q)< g}\frac{d(Q)}{|Q|^{1-\alpha+\beta}}-\sum_{d(Q)<g}\frac{d(Q)u^{2d(Q)}}{|Q|^{1+\alpha+\beta}}\bigg)\nonumber \\
&\qquad\qquad-q^{-g(\alpha+\beta)}u^{2g}\frac{q^{\alpha+\beta}u^2}{(u^2-q^{2\alpha})(u^2-q^{\alpha+\beta})}+O(q^{-g/2}). \nonumber \\
\label{factno3}
\end{align}
Combining \eqref{factno2} and \eqref{factno3} we obtain
\begin{align*}
I_{2,>}^{\textrm{oo}}(N;\alpha,\beta)=\frac{1}{2\pi i}\oint_{|u|=r}J_{2,>}^{\textrm{oo}}(u,\alpha,\beta)\,\frac{du}{u^{N+1}(1-u)}+O(N^2q^{N/2-2g})+O(Nq^{-g/2}),
\end{align*}
where
\begin{align}\label{factno4}
J_{2,>}^{\textrm{oo}}(u,\alpha,\beta)&= q^{-2g\alpha}u^{2g}\mathcal{A}_2(u,\alpha)\bigg(\sum_{d(Q)<g}\frac{d(Q)u^{2d(Q)}}{|Q|^{1-\alpha+\beta}(|Q|^{1+2\beta}-u^{2d(Q)})}\nonumber\\
&\qquad\qquad+\sum_{d(q)<g}\frac{d(Q)(|Q|^{\alpha+\beta}-|Q|^{2-3\alpha+\beta}u^{4d(Q)})}{(|Q|^{2(1-\alpha)}u^{2d(Q)}-1)(|Q|^{1+2\beta}-1)}+ \sum_{d(Q)<g} \frac{1}{|Q|^{1-\alpha+\beta}} \bigg) \\
&\qquad\qquad -q^{-g(\alpha+\beta)}u^{2g}\frac{q^{\alpha+\beta}u^2}{(u^2-q^{2\alpha})(u^2-q^{\alpha+\beta})} +O(q^{-g/2}).\nonumber
\end{align}
\kommentar{\begin{align}\label{factno4}
J_{2,>}^{\textrm{oo}}(u,\alpha,\beta)&=-A_2(-r,s,r,s)q^{-2gr}\zeta_q(1-2r)\sum_{Q\in\mathcal{P}}\frac{d(Q)(|Q|^{2-3r+s}-|Q|^{2-r+s})}{(|Q|^{2(1-r)}-1)(|Q|^{1+2s}-1)}\\
&\qquad\qquad-\frac{q^{-g(r+s)}}{1-q^{r-s}}\bigg(\frac{1}{1-q^{r+s}}+\frac{1}{q-1}\bigg).\nonumber
\end{align}
}
Now consider $I_{2,=}^{\textrm{oo}}(N;\alpha,\beta)$. As before we will have $(V,PQ)=1$ automatically in \eqref{500}. So
\begin{align*}
\eqref{500}&=\frac{q^{2g+1}}{|P|^{2i+2j+1}}\bigg(q\sum_{V \in \mathcal{M}_{\leq (i+j+1)d(P)-g-2}}1- \sum_{V \in \mathcal{M}_{\leq (i+j+1)d(P)-g-1}} 1\bigg)\\
&\qquad\qquad-\frac{q^{2g}}{|P|^{2i+2j+1}}\bigg(q\sum_{V \in \mathcal{M}_{\leq (i+j+1)d(P)-g-1}}1- \sum_{V \in \mathcal{M}_{\leq (i+j+1)d(P)-g}} 1 \bigg)\\
&=\begin{cases}
-\frac{q^{2g}(q-1)}{|P|^{2i+2j+1}} & \textrm{if } (i+j+1)d(P)>g,\\
\frac{q^{2g}}{|P|^{2i+2j+1}} & \textrm{if } (i+j+1)d(P)=g,\\
0 & \textrm{otherwise.}
\end{cases}
\end{align*}
Hence
\begin{align*}
&I_{2,=}^{\textrm{oo}}(N;\alpha,\beta)=-\sum_{\substack{P\ne Q\\d(P^{2k+2l+2})\leq N\\d(P)=d(Q)>g/(i+j+1)}}\frac{d(P)^2}{|P|^{(2k+1)(1/2+\alpha)+(2l+1)(1/2+\beta)+2i+2j+1}}\\
&\qquad\qquad+\frac{1}{q-1}\sum_{\substack{P\ne Q\\d(P^{2k+2l+2})\leq N\\d(P)=d(Q)=g/(i+j+1)}}\frac{d(P)^2}{|P|^{(2k+1)(1/2+\alpha)+(2l+1)(1/2+\beta)+2i+2j+1}}+O(N^2 q^{N/2-2g}).
\end{align*}
The same argument as before shows that the contribution of the term with $i+j+k+l\geq 1$ is $O(Nq^{-g})$. For $i=j=k=l=0$, we can ignore the condition $P\ne Q$ at the cost of $O(gq^{-g})$. So using the Perron formula we obtain that
\[
I_{2,=}^{\textrm{oo}}(N;\alpha,\beta)=\frac{1}{2\pi i}\oint_{|u|=r}J_{2,=}^{\textrm{oo}}(u,\alpha,\beta)\,\frac{du}{u^{N+1}(1-u)}+O(N^2 q^{N/2-2g})+O(gq^{-g})
\]
for any $r<q^{-\varepsilon}$, where
\begin{align*}
J_{2,=}^{\textrm{oo}}(u,\alpha,\beta)&=-\sum_{d(P)=d(Q)>g}\frac{d(P)^2u^{2d(P)}}{|P|^{2+\alpha+\beta}}+\frac{1}{q-1}\sum_{d(P)=d(Q)=g}\frac{d(P)^2u^{2d(P)}}{|P|^{2+\alpha+\beta}}.
\end{align*}
From the Prime Polynomial Theorem we get
\begin{align}\label{fact3}
J_{2,=}^{\textrm{oo}}(u,\alpha,\beta)&=-\sum_{n>g} \frac{u^{2n}}{q^{n(\alpha+\beta)}}+\frac{q^{-g(\alpha+\beta)}u^{2g}}{q-1}+O(q^{-g/2}) \nonumber\\
&=q^{-g(\alpha+\beta)}u^{2g}\bigg(\frac{u^2}{u^2-q^{\alpha+\beta}}+\frac{1}{q-1}\bigg)+O(q^{-g/2}).
\end{align}
\subsection{Combining Type-I terms}
\label{combine}
In view of \eqref{r2}, \eqref{t2}, \eqref{factno1} and \eqref{factno4} we obtain
\begin{align*}
I_{2}^{\textrm{oe}}(N;\alpha,\beta)+ I_{2,>}^{\textrm{oo}}(N;\alpha,\beta)&=\frac{1}{2\pi i}\oint_{|u|=r}J_2(u,\alpha,\beta)\,\frac{du}{u^{N+1}(1-u)}+O(N^2q^{N/2-2g})+O(Nq^{-g/2}),
\end{align*}
where
\begin{align*}
J_2(u,\alpha,\beta)&=R_2(u,\alpha,\beta)- \frac{q^{-g(\alpha+\beta)}u^{2g}\mathcal{A}_2(u,\alpha)}{1-q^{\alpha-\beta}}-q^{-g(\alpha+\beta)}u^{2g}\frac{q^{\alpha+\beta}u^2}{(u^2-q^{2\alpha})(u^2-q^{\alpha+\beta})}+ O(q^{-g/2})\\
&=R_2(u,\alpha,\beta)- \frac{q^{-g(\alpha+\beta)}u^{2g}}{(1-q^{\alpha-\beta})(q-1)}-\frac{q^{-g(\alpha+\beta)}u^{2g}}{1-q^{\alpha-\beta}}\frac{u^2}{u^2-q^{\alpha+\beta}}+ O(q^{-g/2}).
\end{align*}
\kommentar{\acom{We get that
\begin{align*}
J_2(u,\alpha,\beta) &= q^{-2\alpha g} u^{2g} \mathcal{A}_2(u,\alpha) \Big(\sum_{d(Q)<g} \frac{d(Q)}{|Q|^{1-\alpha+\beta}} \\
&- \sum_{Q} \frac{d(Q)(|Q|^{3a+b}- |Q|^{-1+3a-b}u^{2d(Q)}-|Q|u^{4d(Q)}+|Q|^2u^{4d}+|Q|^{1+a-b}u^{4d}-|Q|^{2-a+b}u^{4d}}{(u^{2d}-|Q|^{1+2b})(-|Q|^{2a}+|Q|^2 u^{2d})} \Big) \\
& -q^{-g(\alpha+\beta)}u^{2g}\frac{q^{\alpha+\beta}u^2}{(u^2-q^{2\alpha})(u^2-q^{\alpha+\beta})} \\
&= -q^{-2\alpha g} u^{2g} \mathcal{A}_2(u,\alpha) \sum_{Q} \frac{d(Q)(|Q|^{3a+b}- |Q|^{-1+3a-b}u^{2d(Q)}-|Q|u^{4d(Q)}+|Q|^2u^{4d}+|Q|^{1+a-b}u^{4d}-|Q|^{2-a+b}u^{4d}}{(u^{2d}-|Q|^{1+2b})(-|Q|^{2a}+|Q|^2 u^{2d})} \\
&- \frac{q^{-g(\alpha+\beta)}u^{2g} \mathcal{A}_2(u,\alpha)}{1-q^{\alpha-\beta}} - \frac{q^{-2\alpha g} u^{2g} \mathcal{A}_2(u,\alpha)}{1-q^{\beta-\alpha})}-q^{-g(\alpha+\beta)}u^{2g}\frac{q^{\alpha+\beta}u^2}{(u^2-q^{2\alpha})(u^2-q^{\alpha+\beta})}
\end{align*}}}
Similarly,
\begin{align*}
I_{2}^{\textrm{eo}}(N;\alpha,\beta)+ I_{2,<}^{\textrm{oo}}(N;\alpha,\beta)&=\frac{1}{2\pi i}\oint_{|u|=r}J_3(u,\alpha,\beta)\,\frac{du}{u^{N+1}(1-u)}+O(N^2q^{N/2-2g})+O(Nq^{-g/2}),
\end{align*}
where
\[
J_3(u,\alpha,\beta)=R_3(u,\alpha,\beta)- \frac{q^{-g(\alpha+\beta)}u^{2g}}{(1-q^{-\alpha+\beta})(q-1)}-\frac{q^{-g(\alpha+\beta)}u^{2g}}{1-q^{-\alpha+\beta}}\frac{u^2}{u^2-q^{\alpha+\beta}}+ O(q^{-g/2}).
\]
Now note that
\[
\frac{1}{1-q^{\alpha-\beta}}+\frac{1}{1-q^{-\alpha+\beta}}=1,
\]
and hence, by using \eqref{fact3},
\begin{align*}
&I_{2}^{\textrm{oe}}(N;\alpha,\beta)+I_{2}^{\textrm{eo}}(N;\alpha,\beta)+ I_{2}^{\textrm{oo}}(N;\alpha,\beta)\\
&\qquad\qquad=\frac{1}{2\pi i}\oint_{|u|=r}\big(R_2(u,\alpha,\beta)+R_3(u,\alpha,\beta)\big)\,\frac{du}{u^{N+1}(1-u)}+O(N^2q^{N/2-2g})+O(Nq^{-g/2}).
\end{align*}
\end{document} |
\begin{document}
\startdocumentlayoutoptions
\thispagestyle{plain}
\defAbstract{Abstract}
\begin{abstract}
The space of unitary $\ensuremath{C_{0}}$-semigroups on separable infinite-dimensional Hilbert space,
when viewed under the topology of uniform weak operator convergence on compact subsets of $\mathbb{R}NonNeg$,
is known to admit various interesting residual subspaces.
Before treating the contractive case, the problem of
the complete metrisability of this space was raised in
\cite{eisner2010buchStableOpAndSemigroups}.
Utilising Borel complexity computations and automatic continuity results for semigroups,
we obtain a general result, which in particular implies that the
one-/multiparameter contractive $\ensuremath{C_{0}}$-semigroups constitute Polish spaces
and thus positively addresses the open problem.
\end{abstract}
\subjclass[2020]{47D06, 54E35}
\keywords{Semigroups of operators, metrisability, completeness, Polish spaces, Borel complexity.}
\title[On the complete metrisability of spaces of contractive semigroups]{On the complete metrisability of spaces of contractive semigroups}
\author{Raj Dahya}
\email{[email protected]}
\mathop{\textup{ad}}dress{Fakult\"at f\"ur Mathematik und Informatik\newline
Universit\"at Leipzig, Augustusplatz 10, D-04109 Leipzig, Germany}
\maketitle
\setcounternach{section}{1}
\@startsection{section}{1}{\z@}{.7\linespacing\@plus\linespacing}{.5\linespacing}{\formatsection@text}[Introduction]{Introduction}
\label{sec:intro}
\noindent
In
\cite{dahya2022weakproblem}
the space of contractive $\ensuremath{C_{0}}$-semigroups over a separable infinite-dimensional Hilbert space,
and when viewed with the topology of uniform weak operator convergence on compact subsets of $\mathbb{R}NonNeg$,
was shown to constitute a Baire space.
The main application of this is
\cite[Proposition 5.1]{dahya2022weakproblem},
which relies on the approximation result in
\cite[Theorem~2.1]{krol2009}
and shows that residual properties for the unitary case automatically transfer to the contractive case.
In particular, this application renders meaningful the residuality results achieved in
\cite{eisnersereny2009catThmStableSemigroups},
\cite[Corollary~3.2]{krol2009},
and \cite[\S{}III.6 and \S{}IV.3.3]{eisner2010buchStableOpAndSemigroups}.
Note also that residual properties of (contractive) operators on Banach spaces,
as initiated in \cite{eisner2010typicalContraction,eisnermaitrai2010typicalOperators},
have recently been studied in connection with
\emph{hypercyclicity} and
the \emph{Invariant Subspace Problem}
in \cite{grivaux2021localspecLp,grivaux2021invariantsubspaceLp,grivaux2021typicalexamples}.
The continuous case remains to be investigated.
In this paper we improve upon the result in \cite{dahya2022weakproblem}
and show that the space of contractive $\ensuremath{C_{0}}$-semigroups is \highlightTerm{Polish} (\mathop{\textit{id}}est separable, completely metrisable).
In fact, we prove this for spaces of more generally defined semigroups, including multiparameter semigroups.
In particular, our result positively solves a problem raised in
\cite[\S{}III.6.3]{eisner2010buchStableOpAndSemigroups}
(\cf also \cite[Remark~2.2]{Eisner2008kato}).
There it was shown that,
when viewed under the topology of uniform weak operator convergence on compact time intervals,
the space of contractive $\ensuremath{C_{0}}$-semigroups
is not sequentially closed within the larger space of continuous contraction-valued functions.
We shall reinforce this by studying the geometric properties of these spaces in a general setting,
and providing a deeper reason for this failure (see \Cref{cor:broad-class-of-counterexamples-to-Hs-closed:sig:article-str-problem-raj-dahya}).
This renders the complete metrisability problem non-trivial.
The approach in
\cite[Theorem~1.20]{dahya2022weakproblem}
involves studying and transferring properties from the subspace of unitary semigroups,
which is a Baire space.
This method crucially relies on the fact that contractive semigroups
can be weakly approximated by unitary semigroups.
These density results in turn arise from the theory of dilations
(\cf
\cite[Theorem~1]{peller1981estimatesOperatorPolyLp}
and
\cite[Theorem~2.1]{krol2009}
).
By contrast, the approach here bypasses dependency upon dilation.
Instead we directly classify
the space of contractive $\ensuremath{C_{0}}$-semigroups
in terms of its Borel complexity within a larger, completely metrisable space.
This complexity result in turn implies complete metrisability
(see \Cref{thm:main-result:sig:article-str-problem-raj-dahya}).
Our result encompasses a broad class of spaces on which the semigroups are defined.
We provide basic examples in the main text
and broaden this to a larger class in \Cref{app:continuity}.
The generality of the main result may also be of interest to other fields.
Multiparameter semigroups, for example, occur in
structure theorems
(see \exempli \cite{lopushanskyMultiparamFourier}),
the study of diffusion equations in space-time dynamics
(see \exempli \cite{zelik2004}),
the approximation of periodic functions in multiple variables
(see \exempli \cite{terehin1975}),
\etcetera.
\@startsection{section}{1}{\z@}{.7\linespacing\@plus\linespacing}{.5\linespacing}{\formatsection@text}[Definitions of spaces of semigroups]{Definitions of spaces of semigroups}
\label{sec:intro-definitions}
\noindent
Throughout this paper, $\mathcal{H}$ shall denote a fixed separable infinite-dimensional Hilbert space.
Furthermore,
\begin{mathe}[mc]{rcccccl}
\BoundedOps{\mathcal{H}}
&\supseteq &\OpSpaceC{\mathcal{H}}
&\supseteq &\OpSpaceI{\mathcal{H}}
&\supseteq &\OpSpaceU{\mathcal{H}}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
denote (from left to right) the spaces of bounded linear operators, contractions, isometries, and unitaries over $\mathcal{H}$.
These can be endowed with the weak operator topology ($\text{\upshape \scshape wot}$)
or the strong operator topology ($\text{\upshape \scshape sot}$).
Instead of working with semigroups defined on $\mathbb{R}NonNeg$ (continuous time)
or over $\mathbb{N}zero$ (discrete time), we shall more generally work with semigroups parameterised by a topological monoid.
\begin{defn}
\makelabel{defn:semigroup-on-h:sig:article-str-problem-raj-dahya}
Let $(M,\cdot,1)$ be a topological monoid.
A \highlightTerm{semigroup} over $\mathcal{H}$ on $M$
shall mean any operator-valued function, ${T:M\to\BoundedOps{\mathcal{H}}}$,
satisfying ${T(1)=\text{\upshape\bfseries I}}$ and ${T(st)=T(s)T(t)}$ for ${s,t\in M}$.
\end{defn}
In other words, semigroups are just certain kinds of algebraic morphisms.
Observe that the above definition applied to the topological monoid
$(\mathbb{R}NonNeg,+,0)$
yields the usual definition of an operator semigroup.
The continuous contractive semigroups defined on $M$ may be viewed as subspaces
of the function spaces $\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{M}{\OpSpaceC{\mathcal{H}}}$,
where $\OpSpaceC{\mathcal{H}}$
may be endowed with either the $\text{\upshape \scshape wot}$- or $\text{\upshape \scshape sot}$-topology.
We summarise these spaces and their topologies as follows:
\begin{defn}
\makelabel{defn:standard-funct-spaces:sig:article-str-problem-raj-dahya}
Let $M$ be a topological monoid.
Denote via
${\cal{F}^{c}_{s}(M)\mathop{\textup{co}}lonequals\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{M}{(\OpSpaceC{\mathcal{H}},\text{\upshape \scshape sot})}}$
the $\text{\upshape \scshape sot}$-continuous contraction-valued functions
defined on $M$,
and via ${\cal{C}_{s}(M)\mathop{\textup{co}}lonequals\Hom{M}{(\OpSpaceC{\mathcal{H}},\text{\upshape \scshape sot})}}$
the $\text{\upshape \scshape sot}$-continuous contractive semigroups
over $\mathcal{H}$ on $M$.
Denote via $\cal{F}^{c}_{w}(M)$ and $\cal{C}_{w}(M)$ the respective $\text{\upshape \scshape wot}$-continuous counterparts.
\end{defn}
\begin{defn}
\makelabel{defn:loc-wot-sot-convergence:sig:article-str-problem-raj-dahya}
Let $M$ be a topological monoid
and let $X$ be any of the spaces in \Cref{defn:standard-funct-spaces:sig:article-str-problem-raj-dahya}.
Let $\KmpRm{M}$ denote the collection of compact subsets of $M$.
The topologies of
\highlightTerm{uniform \text{\upshape \scshape wot}-convergence on compact subsets of $M$}
(short: $\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}}$),
\respectively of
\highlightTerm{uniform \text{\upshape \scshape sot}-convergence on compact subsets of $M$}
(short: $\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape sot}}$)
are induced via the convergence conditions:
\begin{mathe}[mc]{rcll}
T_{i} \overset{\text{\scriptsize{{{$\mathpzc{k}$}}}-{\upshape \scshape wot}}}{\longrightarrow} T
&:\Longleftrightarrow
&\forall{\xi,\eta\in\mathcal{H}:~}
\forall{K\in\KmpRm{M}:~}
&\sup_{t\in K}|\BRAKET{(T_{i}(t)-T(t))\xi}{\eta}|\underset{i}{\longrightarrow}0\\
T_{i} \overset{\text{\scriptsize{{{$\mathpzc{k}$}}}-{\upshape \scshape sot}}}{\longrightarrow} T
&:\Longleftrightarrow
&\forall{\xi\in\mathcal{H}:~}
\forall{K\in\KmpRm{M}:~}
&\sup_{t\in K}\|(T_{i}(t)-T(t))\xi\|\underset{i}{\longrightarrow}0\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all nets, $(T_{i})_{i}\subseteqX$ and all $T\inX$.
\end{defn}
Working with these definitions, one can readily classify some of these spaces as follows:
\begin{prop}
\makelabel{prop:basic:SpC:basic-polish:sig:article-str-problem-raj-dahya}
Let $M$ be a locally compact Polish monoid.
Then $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$,
$(\cal{F}^{c}_{s}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape sot}})$,
and
$(\cal{C}_{s}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape sot}})$
are Polish spaces.
\end{prop}
For a full proof see
\cite[Propositions~1.16~and~1.18]{dahya2022weakproblem}.
For the reader's convenience, we sketch the arguments here.
\begin{proof}[of \Cref{prop:basic:SpC:basic-polish:sig:article-str-problem-raj-dahya}]
First note that the spaces,
${(\OpSpaceC{\mathcal{H}},\text{\upshape \scshape wot})}$
and
${(\OpSpaceC{\mathcal{H}},\text{\upshape \scshape sot})}$,
are well-known to be Polish
(
see \exempli
\cite[Exercise~3.4~(5) and Exercise~4.9]{kech1994}
).
To prove the first claim, we need to show that
$(\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{M}{Y},\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$
is Polish, where ${Y\mathop{\textup{co}}lonequals(\OpSpaceC{\mathcal{H}},\text{\upshape \scshape wot})}$.
Since $M$ is a locally compact Polish space,
and since for metrisable spaces, separability is equivalent to second countability,
one can readily construct a countable collection of compact subsets,
${\mathop{\textup{co}}mpactcover\subseteq\KmpRm{M}}$,
such that $\{\topInterior{K}\mid K\in\mathop{\textup{co}}mpactcover\}$ covers $M$.
Consider the spaces $\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{K}{Y}$ for $K\in\mathop{\textup{co}}mpactcover$ and endow these with the topology of uniform convergence,
which makes them Polish spaces
(see \exempli
\cite[Theorem~4.19]{kech1994}
or
\cite[Lemma~3.96--7,~3.99]{aliprantis2005}
).
The map
\begin{mathe}[mc]{rcccl}
\Psi &: &\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{M}{Y} &\to &\prod_{K\in\mathop{\textup{co}}mpactcover}\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{K}{Y}\\
&&f &\mapsto &(f\restr{K})_{K\in\mathop{\textup{co}}mpactcover}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
is clearly well-defined.
Since $\{\topInterior{K} \mid K\in\mathop{\textup{co}}mpactcover\}$ covers $M$ and $M$ is locally compact,
the map is also clearly bicontinuous.
The covering property also guarantees that every coherent sequence of continuous functions
$(f_{K})_{K\in\mathop{\textup{co}}mpactcover}\in\prod_{K\in\mathop{\textup{co}}mpactcover}\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{K}{Y}$
corresponds to a unique continuous function,
${f\mathop{\textup{co}}lonequals\bigcup_{K\in\mathop{\textup{co}}mpactcover}f_{K}\in\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{M}{Y}}$,
satisfying $\Psi(f)=(f_{K})_{K\in\mathop{\textup{co}}mpactcover}$.
Thus $\Psi$ is a homeomorphism between $\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{M}{Y}$
and the subspace of coherent sequences of continuous functions.
Since the product of Polish spaces is Polish
(see
\cite[Corollary~3.39]{aliprantis2005}
)
and the subspace of coherent sequences is clearly closed under the product topology,
it follows that $(\@ifnextchar_{\Cts@tief}{\Cts@tief_{}}{M}{Y},\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$ is Polish.
The second claim is obtained in the same manner,
by replacing
$Y$ by $(\OpSpaceC{\mathcal{H}},\text{\upshape \scshape sot})$
and $\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}}$ by $\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape sot}}$
above.
Finally, it is easy to verify that $\cal{C}_{s}(M)$ is a closed subspace within $(\cal{F}^{c}_{s}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape sot}})$,
and thus that $(\cal{C}_{s}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape sot}})$ too is Polish.
\end{proof}
The aim of this paper is to show that $(\cal{C}_{s}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$ is completely metrisable
for some topological monoids $M$, in particular for $M=(\mathbb{R}NonNeg,+,0)$.
We can achieve this for a broad class of monoids, by appealing to the following condition:
\begin{defn}
\makelabel{defn:good-monoids:sig:article-str-problem-raj-dahya}
Call a topological monoid, $M$, \highlightTerm{\usesinglequotes{good}},
if the contractive $\text{\upshape \scshape wot}$-continuous semigroups
over $\mathcal{H}$ on $M$
are automatically $\text{\upshape \scshape sot}$-continuous,
\mathop{\textit{id}}est if $\cal{C}_{w}(M)=\cal{C}_{s}(M)$ holds.
\end{defn}
All discrete monoids (including non-commutative ones) are trivially \usesinglequotes{good}.
By a classical result, $(\mathbb{R}NonNeg,+,0)$ is \usesinglequotes{good}
(\cf
\cite[Theorem~5.8]{Engel1999}
as well as
\cite[Theorem~9.3.1 and Theorem~10.2.1--3]{hillephillips2008}
).
Furthermore, it is easy to see that \usesinglequotes{good} monoids are closed under products:
\begin{prop}
\makelabel{prop:good-monoids-closed-under-products:sig:article-str-problem-raj-dahya}
Let $d\in\mathbb{N}pos$ and $M_{1},M_{2},\ldots,M_{d}$ be \usesinglequotes{good} topological Polish monoids.
Then $\prod_{i=1}^{d}M_{i}$ is \usesinglequotes{good}.
\end{prop}
\begin{proof}
Let ${T:\prod_{i=1}^{d}M_{i}\to\OpSpaceC{\mathcal{H}}}$
be a $\text{\upshape \scshape wot}$-continuous contractive semigroup.
We need to show that $T$ is $\text{\upshape \scshape sot}$-continuous.
For each $k\in\{1,2,\ldots,d\}$
let
${\pi_{k}:\prod_{i=1}^{d}M_{i}\to M_{k}}$
denote the canonical projection,
which is a (continuous) monoid homomorphism,
and
let
${r_{k}:M_{k}\to\prod_{i=1}^{d}M_{i}}$
denote the canonical (continuous) monoid homomorphism
defined by $r_{k}(t)=(1,1,\ldots,t,\ldots,1)$
(the $d$-tuple with $t$ in the $k$-th position and identity elements elsewhere)
for all $t\in M_{k}$.
For each $k\in\{1,2,\ldots,d\}$
observe further that
${T_{k}:M_{k}\to\OpSpaceC{\mathcal{H}}}$
define by ${T_{k} \mathop{\textup{co}}lonequals T\circ r_{k}}$
is a $\text{\upshape \scshape wot}$-continuous homomorphism.
That is, each $T_{k}$ is a $\text{\upshape \scshape wot}$-continuous contractive semigroup over $\mathcal{H}$ on $M_{k}$.
Since each $M_{k}$ is \usesinglequotes{good}, these are $\text{\upshape \scshape sot}$-continuous.
Observe now, that
\begin{mathe}[mc]{rcl}
T(t)
&= &T(\prod_{i=1}^{d}r_{k}(\pi_{k}(t)))\\
&= &T(r_{1}(\pi_{1}(t)))\cdot T(r_{2}(\pi_{2}(t)))\cdot\ldots\cdot T(r_{d}(\pi_{d}(t)))\\
&= &T_{1}(\pi_{1}(t))\cdot T_{2}(\pi_{2}(t))\cdot\ldots\cdot T_{d}(\pi_{d}(t))\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
holds for all $t\in \prod_{i=1}^{d}M_{i}$.
Since the algebraic projections are continuous
and the $T_{k}$ are $\text{\upshape \scshape sot}$-continuous and contractive,
and since multiplication of contractions is $\text{\upshape \scshape sot}$-continuous,
it follows that $T$ is $\text{\upshape \scshape sot}$-continuous.
\end{proof}
Thus we immediately obtain the following examples of \usesinglequotes{good} monoids:
\begin{cor}
\makelabel{cor:multiparameter-auto-cts:sig:article-str-problem-raj-dahya}
For each $d\in\mathbb{N}pos$ the monoid $\mathbb{R}NonNeg^{d}$, viewed under pointwise addition, is \usesinglequotes{good}.
\end{cor}
If one more generally considers monoids which are closed subspaces of locally compact Hausdorff topological groups,
a sufficient topological condition exists,
which guarantees that a monoid is \usesinglequotes{good}
(see
\Cref{app:continuity:defn:conditition-II:sig:article-str-problem-raj-dahya}
and
\Cref{app:continuity:thm:generalised-auto-continuity:sig:article-str-problem-raj-dahya}
).
By
\Cref{
app:continuity:e.g.:extendible-mon:reals:sig:article-str-problem-raj-dahya,
app:continuity:e.g.:extendible-mon:p-adic:sig:article-str-problem-raj-dahya,
app:continuity:e.g.:extendible-mon:discrete:sig:article-str-problem-raj-dahya,
app:continuity:e.g.:extendible-mon:non-comm-non-discrete:sig:article-str-problem-raj-dahya,
}
and
\Cref{app:continuity:prop:monoids-with-cond-closed-under-fprod:sig:article-str-problem-raj-dahya},
the class of monoids satisfying this condition
is closed under finite products
and includes all discrete monoids,
the non-negative reals under addition $(\mathbb{R}NonNeg,+,0)$,
the $p$-adic integers under addition $(\mathbb{Z}_{p},+,0)$ for all $p\in\mathbb{P}$,
and even non-discrete non-commutative monoids including
naturally definable monoids contained within the Heisenberg group of order $2d-3$
for each $d\geq 2$.
\@startsection{section}{1}{\z@}{.7\linespacing\@plus\linespacing}{.5\linespacing}{\formatsection@text}[The $\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}}$-closure of the space of contractive semigroups]{The $\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}}$-closure of the space of contractive semigroups}
\label{sec:convexity}
\noindent
The simplest approach to demonstrate the complete metrisability of
$(\cal{C}_{s}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$
would be to show that this be a closed subspace within the function space
$(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$,
which we already know to be Polish (see \Cref{prop:basic:SpC:basic-polish:sig:article-str-problem-raj-dahya}).
In
\cite[Example~\S{}III.6.10]{eisner2010buchStableOpAndSemigroups}
and
\cite[Example~2.1]{Eisner2008kato}
a construction is provided, which demonstrates that this fails
in particular in the case of one-parameter contractive $\ensuremath{C_{0}}$-semigroups.
In this section we reveal that the deeper reason for this failure
is that the closure of $\cal{C}_{s}(M)$ within $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$ is always convex,
whereas for a broad class of topological monoids, $M$, the subset $\cal{C}_{s}(M)$ is not convex
(see \Cref{cor:broad-class-of-counterexamples-to-Hs-closed:sig:article-str-problem-raj-dahya} below).
Before we proceed, we require a few definitions.
In the following $M$ shall denote an arbitrary topological monoid.
We continue to use $\mathcal{H}$ to denote a separable infinite-dimensional Hilbert space
and $\text{\upshape\bfseries I}_{\mathcal{H}}$ (or simply $\text{\upshape\bfseries I}$) for the identity operator.
\begin{defn}
\makelabel{defn:partition-of-identity-operator:sig:article-str-problem-raj-dahya}
For $n\in\mathbb{N}pos$ and
${u_{1},u_{2},\ldots,u_{n}\in\BoundedOps{\mathcal{H}}}$
we shall call
$(u_{1},u_{2},\ldots,u_{n})$
an \highlightTerm{isometric partition of the identity},
just in case the following axioms hold:
\begin{kompaktenum}{(\bfseries {P}1)}[\rtab][\rtab]
\item\label{ax:partition:1}
$u_{j}^{\ast}u_{i}=\delta_{ij}\cdot\text{\upshape\bfseries I}$ for all $i,j\in\{1,2,\ldots,n\}$.
\item\label{ax:partition:2}
$\sum_{i=1}^{n}u_{i}u_{i}^{\ast}=\text{\upshape\bfseries I}$.
\end{kompaktenum}
\@ifnextchar\bgroup{\nvraum@c}{\vspace*{-\baselineskip}}{1}
\end{defn}
Note that by axiom
(P\ref{ax:partition:1})
the operators in an isometric partition are necessarily isometries.
\begin{rem}
\makelabel{rem:isometric-partitions-of-one-correspond-to-decompositions:sig:article-str-problem-raj-dahya}
Let $n\in\mathbb{N}pos$. If $(u_{1},u_{2},\ldots,u_{n})$ is an isometric partition of $\text{\upshape\bfseries I}$,
then, letting
${\mathcal{H}_{i}\mathop{\textup{co}}lonequals\mathop{\textup{ran}}(u_{i})}$ for $i\in\{1,2,\ldots,n\}$,
it is easy to see that
(P\ref{ax:partition:1}) and (P\ref{ax:partition:2}),
together with the fact that each $u_{i}$ is necessarily an isometry,
imply that the $\mathcal{H}_{i}$ are mutually orthogonal closed subspaces of $\mathcal{H}$
and that $\mathcal{H}=\bigoplus_{i=1}^{n}\mathcal{H}_{i}$.
Conversely, if $\mathcal{H}$ can be decomposed as $\bigoplus_{i=1}^{n}\mathcal{H}_{i}$
where each $\mathcal{H}_{i}\subseteq\mathcal{H}$ is a closed subspace
with $\mathop{\textup{dim}}(\mathcal{H}_{i})=\mathop{\textup{dim}}(\mathcal{H})$ for each $i$,
then letting $u_{i}\in\BoundedOps{\mathcal{H}}$
be any isometries with $\mathop{\textup{ran}}(u_{i})=\mathcal{H}_{i}$
for each $i\in\{1,2,\ldots,n\}$,
one can readily see that
(P\ref{ax:partition:1}) and (P\ref{ax:partition:2})
are satisfied.
Thus, isometric partitions of $\text{\upshape\bfseries I}$
can be constructed from orthogonal decompositions into infinite-dimensional closed subspaces of $\mathcal{H}$
and vice versa.
\end{rem}
Of course, these observations only apply for infinite-dimensional Hilbert spaces.
\begin{defn}
Let $n\in\mathbb{N}pos$,
$(u_{1},u_{2},\ldots,u_{n})$ be an isometric partition of $\text{\upshape\bfseries I}$,
and $T_{1},T_{2},\ldots,T_{n}\in\cal{F}^{c}_{w}(M)$.
Denote via $(\bigoplus_{i=1}^{n}T_{i})_{\quer{u}}$
the operator-valued function ${T:M\to\BoundedOps{\mathcal{H}}}$
given by
\begin{mathe}[mc]{rcl}
T(\cdot) &= &\sum_{i=1}^{n}u_{i} T_{i}(\cdot) u_{i}^{\ast}.\\
\end{mathe}
\@ifnextchar\bgroup{\nvraum@c}{\vspace*{-\baselineskip}}{1}
\end{defn}
\begin{defn}
Let $A\subseteq\cal{F}^{c}_{w}(M)$.
Say that $A$ is \highlightTerm{closed under finite joins}
just in case for all $n\in\mathbb{N}pos$,
all isometric partitions
$\quer{u} \mathop{\textup{co}}lonequals (u_{1},u_{2},\ldots,u_{n})$ of $\text{\upshape\bfseries I}$,
and all $T_{1},T_{2},\ldots,T_{n}\in A$,
it holds that
${(\bigoplus_{i=1}^{n}T_{i})_{\quer{u}}\in A}$.
\end{defn}
The property of being closed under finite joins is a key ingredient
in proving the convexity of the closure of subsets
(see \Cref{lemm:closure-is-convex:sig:article-str-problem-raj-dahya} below).
We first provide some basic observations about which subsets are closed under finite joins.
\begin{prop}
\makelabel{prop:basic-some-subspaces-are-closed-under-finite-stream-addition:sig:article-str-problem-raj-dahya}
Let $A \in \{ \cal{F}^{c}_{w}(M), \cal{F}^{c}_{s}(M), \cal{C}_{w}(M), \cal{C}_{s}(M) \}$.
Then $A$ is closed under finite joins.
\end{prop}
\begin{proof}
First consider the case $A=\cal{F}^{c}_{w}(M)$.
Let $n\in\mathbb{N}pos$,
$\quer{u} \mathop{\textup{co}}lonequals (u_{1},u_{2},\ldots,u_{n})$ be an isometric partition of $\text{\upshape\bfseries I}$,
and $T_{1},T_{2},\ldots,T_{n}\in A$.
We need to show that ${T \mathop{\textup{co}}lonequals (\bigoplus_{i=1}^{n}T_{i})_{\quer{u}}}$ is in $A$.
Applying the properties of the partition yields
\begin{mathe}[mc]{rcl}
\|T(t)\xi\|^{2}
&= &\sum_{i,j=1}^{n}
\BRAKET{u_{i}T_{i}(t)u_{i}^{\ast}\xi}{u_{j}T_{j}(t)u_{j}^{\ast}\xi}\\
&\textoverset{(P\ref{ax:partition:1})}{=}
&\sum_{i=1}^{n}
\BRAKET{u_{i}T_{i}(t)u_{i}^{\ast}\xi}{u_{i}T_{i}(t)u_{i}^{\ast}\xi}\\
&\leq
&\sum_{i=1}^{n}
\|u_{i}T_{i}(t)\|\|u_{i}^{\ast}\xi\|\\
&\leq
&\sum_{i=1}^{n}
\|u_{i}^{\ast}\xi\|\\
&\multispan{2}{\text{since $u_{i}$ is isometric and $T_{i}(t)$ contractive for all $i$}}\\
&= &\BRAKET{\sum_{i=1}^{n}u_{i}u_{i}^{\ast}\xi}{\xi}\\
&\textoverset{(P\ref{ax:partition:2})}{=}
&\BRAKET{\text{\upshape\bfseries I} \xi}{\xi}
= \|\xi\|^{2}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all $\xi\in\mathcal{H}$ and all $t\in M$.
And since by construction, ${T(\cdot)=\sum_{i=1}^{n}u_{i}T_{i}(\cdot)u_{i}^{\ast}}$,
it clearly holds that $T$ is $\text{\upshape \scshape wot}$-continuous.
Thus $T$ is a $\text{\upshape \scshape wot}$-continuous contraction-valued function, \mathop{\textit{id}}est $T\in A$.
Hence $A$ is closed under finite joins.
The case of $A=\cal{F}^{c}_{s}(M)$ is analogous.
Next we consider the case $A=\cal{C}_{w}(M)$.
Let $n\in\mathbb{N}pos$,
$\quer{u} \mathop{\textup{co}}lonequals (u_{1},u_{2},\ldots,u_{n})$ be an isometric partition of $\text{\upshape\bfseries I}$,
and $T_{1},T_{2},\ldots,T_{n}\in A$.
We need to show that ${T \mathop{\textup{co}}lonequals (\bigoplus_{i=1}^{n}T_{i})_{\quer{u}}}$ is in $A$.
Since $A\subseteq\cal{F}^{c}_{w}(M)$ and $\cal{F}^{c}_{w}(M)$ is closed under finite joins,
we already know that $T\in\cal{F}^{c}_{w}(M)$, \mathop{\textit{id}}est that $T$ is contraction-valued and $\text{\upshape \scshape wot}$-continuous.
To show that $T\in A$, it remains to show that $T$ is a semigroup.
Since each of the $T_{i}$ are semigroups, applying the properties of the partition
yields
\begin{mathe}[mc]{rcccccccl}
T(1)
&= &\sum_{i=1}^{n}u_{i}T_{i}(1)u_{i}^{\ast}
&= &\sum_{i=1}^{n}u_{i}\cdot\text{\upshape\bfseries I}\cdot u_{i}^{\ast}
&= &\sum_{i=1}^{n}u_{i}u_{i}^{\ast}
&\textoverset{(P\ref{ax:partition:2})}{=}
&\text{\upshape\bfseries I}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
and
\begin{mathe}[mc]{rcl}
T(s)T(t)
&= &(\sum_{i=1}^{n}u_{i}T_{i}(s)u_{i}^{\ast})(\sum_{j=1}^{n}u_{j}T_{j}(t)u_{j}^{\ast})\\
&= &\sum_{i,j=1}^{n}u_{i}T_{i}(s)u_{i}^{\ast}\cdot u_{j} T_{j}(t)u_{j}^{\ast}\\
&\textoverset{(P\ref{ax:partition:1})}{=}
&\sum_{i=1}^{n}u_{i}T_{i}(s)T_{i}(t)u_{i}^{\ast}\\
&= &\sum_{i=1}^{n}u_{i}T_{i}(st)u_{i}^{\ast}\\
&= &T(st)\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all $s,t\in M$.
Thus $T$ is a $\text{\upshape \scshape wot}$-continuous contractive semigroup, \mathop{\textit{id}}est $T\in A$.
Hence $A$ is closed under finite joins.
The case of $A=\cal{C}_{s}(M)$ is analogous.
\end{proof}
\begin{prop}
\makelabel{prop:closure-is-convex:sig:article-str-problem-raj-dahya}
Let $A\subseteq \cal{F}^{c}_{w}(M)$ and
let $\quer{A}$ be the closure of $A$ within $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$.
If $A$ is closed under finite joins,
then so too is $\quer{A}$.
\end{prop}
\begin{proof}
Let $n\in\mathbb{N}pos$,
$\quer{u} \mathop{\textup{co}}lonequals (u_{1},u_{2},\ldots,u_{n})$ be an isometric partition of $\text{\upshape\bfseries I}$,
and $T_{1},T_{2},\ldots,T_{n}\in\quer{A}$.
We need to show that ${T \mathop{\textup{co}}lonequals (\bigoplus_{j=1}^{n}T_{j})_{\quer{u}}}$ is in $\quer{A}$.
To see this, we may simply fix a net
${((T^{(i)}_{1},T^{(i)}_{2},\ldots,T^{(i)}_{n}))_{i} \subseteq \prod_{j=1}^{n}A}$
such that ${T^{(i)}_{j}\underset{i}{\overset{\text{\scriptsize{{{$\mathpzc{k}$}}}-{\upshape \scshape wot}}}{\longrightarrow}} T_{j}}$
for all $j\in\{1,2,\ldots,n\}$.
Since $A$ is closed under finite joins,
we have
${T^{(i)}\mathop{\textup{co}}lonequals (\bigoplus_{j=1}^{n}T^{(i)}_{j})_{\quer{u}}\in A}$
for all $i$. We also clearly have
\begin{mathe}[mc]{rcccccl}
A \ni T^{(i)}
&= &\sum_{j=1}^{n}u_{j}T^{(i)}_{j}(\cdot)u_{j}^{\ast}
&\underset{i}{\overset{\text{\scriptsize{{{$\mathpzc{k}$}}}-{\upshape \scshape wot}}}{\longrightarrow}}
&\sum_{j=1}^{n}u_{j}T_{j}(\cdot)u_{j}^{\ast}
&= &T(\cdot).\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
Hence $T\in\quer{A}$.
\end{proof}
\begin{lemm}
\makelabel{lemm:closure-is-convex:sig:article-str-problem-raj-dahya}
Let $A\subseteq \cal{F}^{c}_{w}(M)$ be closed under finite joins.
Then the closure, $\quer{A}$, of $A$ within $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$ is convex.
\end{lemm}
\begin{proof}
Since $\mathcal{H}$ is a separable Hilbert space,
it admits a countable orthonormal basis (ONB) ${B\subseteq\mathcal{H}}$,
which we shall fix.
It suffices to show for $S,T\in\quer{A}$ and $\alpha,\beta\in[0,1]$ with ${\alpha+\beta=1}$,
that ${R \mathop{\textup{co}}lonequals \alpha S + \beta T \in \quer{A}}$.
To do this, we need to show that $R$ can be approximated within the $\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}}$-topology
by elements in $\quer{A}$.
In order to achieve this, it suffices to
fix arbitrary
$K\in\KmpRm{M}$,
$F\subseteq B$ finite,
and $\varepsilon>0$,
and show that some $\tilde{R}\in\quer{A}$ exists
satisfying
\begin{mathe}[mc]{lql}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:0:\beweislabel]
\sup_{t\in K}|\BRAKET{(R(t)-\tilde{R}(t))e}{e'}|<\varepsilon,
&\text{for all $e,e'\in F$.}
\end{mathe}
To construct $\tilde{R}$, we first construct an isometric partition, $(w_{0},w_{1})$, of $\text{\upshape\bfseries I}$,
such that $w_{0}$ fixes the vectors in $F$.
This can be achieved as follows:
Since the ONB $B$ is infinite, a partition $\{B_{0},B_{1}\}$ of $B$
exists satisfying $F\subseteq B_{0}$ and $|B_{0}|=|B_{1}|=|B|=\mathop{\textup{dim}}(\mathcal{H})$.
There thus exist bijections ${f_{0}:B\to B_{0}}$ and ${f_{1}:B\to B_{1}}$
and since $B_{0}\supseteq F$, we may assume without loss of generality that $f_{0}\restr{F}=\mathop{\textit{id}}_{F}$.
Using these bijections we obtain (unique) isometries
${w_{0},w_{1}\in\BoundedOps{\mathcal{H}}}$
satisfying ${w_{0}e=f_{0}(e)}$ and ${w_{1}e=f_{1}(e)}$ for all $e\in B$.
In particular,
$\mathop{\textup{ran}}(w_{0})=\quer{\mathop{\textup{lin}}}(B_{0})$
and $\mathop{\textup{ran}}(w_{1})=\quer{\mathop{\textup{lin}}}(B_{1})$.
Now since $\{B_{0},B_{1}\}$ partitions $B$,
we have $\mathcal{H}=\quer{\mathop{\textup{lin}}}(B)=\quer{\mathop{\textup{lin}}}(B_{0})\oplus\quer{\mathop{\textup{lin}}}(B_{1})$.
As per \Cref{rem:isometric-partitions-of-one-correspond-to-decompositions:sig:article-str-problem-raj-dahya}
it follows that
$(w_{0},w_{1})$
satisfies the axioms of an isometric partition of $\text{\upshape\bfseries I}$.
Now set
\begin{mathe}[mc]{rclcrcl}
u_{0} &\mathop{\textup{co}}lonequals &\sqrt{\alpha}w_{0} + \sqrt{\beta}w_{1}
&\text{and}
&u_{1} &\mathop{\textup{co}}lonequals &\sqrt{\beta}w_{0} - \sqrt{\alpha}w_{1}.\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
One can easily derive from the fact that $(w_{0},w_{1})$ is an isometric partition of $\text{\upshape\bfseries I}$,
that $\quer{u} \mathop{\textup{co}}lonequals (u_{0},u_{1})$ also satisfies the axioms of an isometric partition of $\text{\upshape\bfseries I}$.
Moreover, since $w_{0}$ was chosen to fix the vectors in $F$,
applying the properties of the partition $(w_{0},w_{1})$ yields
\begin{mathe}[mc]{rcccccccl}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:sqrt-alpha:\beweislabel]
u_{0}^{\ast}e
&= &u_{0}^{\ast}w_{0}e
&= &\sqrt{\alpha}w_{0}^{\ast}w_{0}e+\sqrt{\beta}w_{1}^{\ast}w_{0}e
&\textoverset{(P\ref{ax:partition:1})}{=}
&\sqrt{\alpha}\text{\upshape\bfseries I} e+\sqrt{\beta}\text{\upshape\bfseries 0} e
&= &\sqrt{\alpha}e\\
u_{1}^{\ast}e
&= &u_{1}^{\ast}w_{0}e
&= &\sqrt{\beta}w_{0}^{\ast}w_{0}e-\sqrt{\alpha}w_{1}^{\ast}w_{0}e
&\textoverset{(P\ref{ax:partition:1})}{=}
&\sqrt{\beta}\text{\upshape\bfseries I} e-\sqrt{\alpha}\text{\upshape\bfseries 0} e
&= &\sqrt{\beta}e\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all $e\in F$.
Finally set $\tilde{R}\mathop{\textup{co}}lonequals(S\bigoplus T)_{\quer{u}}$.
Since $A$ is closed under finite joins,
by \Cref{prop:closure-is-convex:sig:article-str-problem-raj-dahya}
$\quer{A}$ is also closed under finite joins,
and hence the constructed operator-valued function, $\tilde{R}$, lies in $\quer{A}$.
For all $e,e'\in F$ we obtain
\begin{mathe}[mc]{rcl}
\BRAKET{(\tilde{R}(t)-R(t))e}{e'}
&= &\BRAKET{u_{0}S(t)u_{0}^{\ast}e}{e'}
+ \BRAKET{u_{1}T(t)u_{1}^{\ast}e}{e'}
- \BRAKET{R(t)e}{e'}\\
&= &\BRAKET{S(t)u_{0}^{\ast}e}{u_{0}^{\ast}e'}
+ \BRAKET{T(t)u_{1}^{\ast}e}{u_{1}^{\ast}e'}
- \BRAKET{(\alpha S(t) + \beta T(t))e}{e'}\\
&\eqcrefoverset{eq:sqrt-alpha:\beweislabel}{=}
&\BRAKET{S(t)\sqrt{\alpha}e}{\sqrt{\alpha}e'}
+ \BRAKET{T(t)\sqrt{\beta}e}{\sqrt{\beta}e'}
- \BRAKET{(\alpha S(t) + \beta T(t))e}{e'}\\
&= &0\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all $t\in M$.
In particular we have found an $\tilde{R}\in\quer{A}$
which clearly satisfies \eqcref{eq:0:\beweislabel}.
This establishes that the convex hull of $\quer{A}$ is contained in $\quer{A}$.
Thus $\quer{A}$ is convex.
\end{proof}
By \Cref{prop:basic-some-subspaces-are-closed-under-finite-stream-addition:sig:article-str-problem-raj-dahya}
and \Cref{lemm:closure-is-convex:sig:article-str-problem-raj-dahya} we thus immediately obtain the general result:
\begin{cor}
\makelabel{cor:closure-of-Hs-is-convex:sig:article-str-problem-raj-dahya}
For all topological monoids, $M$,
the closure of $\cal{C}_{s}(M)$ within $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$ is convex.
\end{cor}
We now provide a large class of topological monoids, $M$, for which $\cal{C}_{s}(M)$ is not convex.
\begin{defn}
Denote by $\text{\upshape\bfseries I}(\cdot)$
the trivial $\text{\upshape \scshape sot}$-continuous semigroup over $\mathcal{H}$ on $M$,
which is everywhere equal to the identity operator.
Say that $M$ has \highlightTerm{non-trivial unitary semigroups} over $\mathcal{H}$,
just in case there exists a unitary $\text{\upshape \scshape sot}$-continuous semigroup, $U$,
over $\mathcal{H}$ on $M$
with ${U \neq \text{\upshape\bfseries I}(\cdot)}$.
\end{defn}
\begin{defn}
Let $(G,\cdot,{}^{-1},1)$ be a topological group.
Say that $M\subseteq G$ is a \highlightTerm{topological submonoid},
just in case $M$ is endowed with the subspace topology,
contains the neutral element $1$, and is closed under $\cdot$.
\end{defn}
In particular, if $G$ is a topological group and $M\subseteq G$ is a topological submonoid,
then $M$ is itself a topological monoid.
\begin{prop}
\makelabel{prop:gelfand-raikov-monoids:sig:article-str-problem-raj-dahya}
Let $G$ be a locally compact Polish group and suppose that
$M\subseteq G$ is a topological submonoid with $M\neq\{1\}$.
Then $M$ has non-trivial unitary semigroups over $\mathcal{H}$.
\end{prop}
\begin{proof}
Let $t_{0}\in M\setminus\{1\}$.
By the Gelfand-Raikov theorem
(see \exempli \cite[Theorem~6]{yoshi1949}),
there exists a Hilbert space $\mathcal{H}_{0}$
and an irreducible $\text{\upshape \scshape sot}$-continuous unitary representation,
${U_{0}:G\to\OpSpaceU{\mathcal{H}_{0}}}$,
satisfying $U_{0}(t_{0})\neq \text{\upshape\bfseries I}_{\mathcal{H}_{0}}$.
By irreducibility and since $G$ is separable,
it necessarily holds that $\mathop{\textup{dim}}(\mathcal{H}_{0})\leq\aleph_{0}=\mathop{\textup{dim}}(\mathcal{H})$.
If $\mathop{\textup{dim}}(\mathcal{H}_{0})=\mathop{\textup{dim}}(\mathcal{H})$,
then we may assume without loss of generality that $\mathcal{H}_{0}=\mathcal{H}$,
and thus that $U_{0}$ is a representation of $G$ over $\mathcal{H}$.
If $\mathop{\textup{dim}}(\mathcal{H}_{0})$ is finite,
we may assume that $\mathcal{H}_{0}\subset\mathcal{H}$
and view the orthogonal complement $\mathcal{H}_{1}\mathop{\textup{co}}lonequals\mathcal{H}_{0}^{\perp}$
within $\mathcal{H}$.
Replacing $U_{0}$ by
${t\in G\mapsto U_{0}(t)\oplus \text{\upshape\bfseries I}_{\mathcal{H}_{1}}}$
yields an $\text{\upshape \scshape sot}$-continuous unitary representation of $G$ over $\mathcal{H}$
which satisfies $U_{0}(t_{0})\neq\text{\upshape\bfseries I}_{\mathcal{H}}$.
In both cases, restricting $U_{0}$ to $M$
yields a non-trivial $\text{\upshape \scshape sot}$-continuous unitary semigroup over $\mathcal{H}$ on $M$.
Hence $M$ has non-trivial unitary semigroups over $\mathcal{H}$.
\end{proof}
\begin{lemm}
\makelabel{lemm:non-trivial-Hs-is-nonconvex:sig:article-str-problem-raj-dahya}
Suppose that $M$ has non-trivial unitary semigroups over $\mathcal{H}$.
Then $\cal{C}_{s}(M)$ is not a convex subset of $\cal{F}^{c}_{w}(M)$.
\end{lemm}
\begin{proof}
Let ${S\mathop{\textup{co}}lonequals\text{\upshape\bfseries I}(\cdot)}$ be the trivial $\text{\upshape \scshape sot}$-continuous unitary semigroup over $\mathcal{H}$ on $M$.
And by non-triviality we may fix some $\text{\upshape \scshape sot}$-continuous unitary semigroup ${T\in\cal{C}_{s}(M)\setminus\{\text{\upshape\bfseries I}(\cdot)\}}$.
In particular, $T(t_{0})\neq\text{\upshape\bfseries I}$ for some $t_{0}\in M$, which we shall fix.
Choose any $\alpha,\beta\in(0,1)$ with $\alpha+\beta=1$.
It suffices to show that $R \mathop{\textup{co}}lonequals \alpha S + \beta T \notin \cal{C}_{s}(M)$.
Suppose \textit{per contra} that $R \in \cal{C}_{s}(M)$.
Then by the semigroup law we have
\begin{mathe}[mc]{rcl}
\text{\upshape\bfseries 0} &= &R(st)-R(s)R(t)\\
&= &\big(\alpha S(st)+\beta T(st)\big)
-\big(\alpha S(s)+\beta T(s)\big)
\big(\alpha S(t)+\beta T(t)\big)\\
&= &\big(\alpha S(s)S(t)+\beta T(s)T(t)\big)
-\big(\alpha S(s)+\beta T(s)\big)
\big(\alpha S(t)+\beta T(t)\big)\\
&= &\alpha(1-\alpha)\,S(s)S(t)
+\beta(1-\beta)\,T(s)T(t)
-\alpha\beta\,S(s)T(t)
-\beta\alpha\,T(s)S(t)\\
&= &\alpha\beta\,S(s)S(t)
+\beta\alpha\,T(s)T(t)
-\alpha\beta\,S(s)T(t)
-\beta\alpha\,T(s)S(t)\\
&= &\alpha\beta\,(S(s)-T(s))(S(t)-T(t)).\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all $s,t\in M$.
Since $\alpha,\beta \neq 0$ setting $s\mathop{\textup{co}}lonequals t\mathop{\textup{co}}lonequals t_{0}$ in the above yields
\begin{mathe}[mc]{rcl}
(\text{\upshape\bfseries I} - u)^{2} &= &\text{\upshape\bfseries 0},\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
where $u\mathop{\textup{co}}lonequals T(t_{0})\in\OpSpaceU{\mathcal{H}}$.
Since $u$ is unitary, a basic application of the spectral mapping theorem
yields that the spectrum of $u$ is $\{1\}$.
By the Gelfand theorem
(see \cite[Theorem~2.1.10]{murphy1990}),
it follows that $T(t_{0})=u=\text{\upshape\bfseries I}$, which is a contradiction.
\end{proof}
Applying
\Cref{cor:closure-of-Hs-is-convex:sig:article-str-problem-raj-dahya},
\Cref{prop:gelfand-raikov-monoids:sig:article-str-problem-raj-dahya},
and
\Cref{lemm:non-trivial-Hs-is-nonconvex:sig:article-str-problem-raj-dahya}
yields:
\begin{cor}
\makelabel{cor:broad-class-of-counterexamples-to-Hs-closed:sig:article-str-problem-raj-dahya}
Let $G$ be a locally compact Polish group and suppose that
$M\subseteq G$ is a topological submonoid with $M\neq\{1\}$.
Then the closure of $\cal{C}_{s}(M)$ within $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$ is convex,
whilst $\cal{C}_{s}(M)$ itself is not convex.
In particular, $\cal{C}_{s}(M)$ is not a closed subspace within $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$.
\end{cor}
Considering $G=(\mathbb{R}^{d},+,\text{\upshape\bfseries 0})$ with $d\geq 1$
and ${M \mathop{\textup{co}}lonequals \mathbb{R}NonNeg^{d} \subseteq G}$,
the conditions of \Cref{cor:broad-class-of-counterexamples-to-Hs-closed:sig:article-str-problem-raj-dahya} are satisfied.
In particular, the subspace of one-/multiparameter $\ensuremath{C_{0}}$-semigroups
is not closed within the larger space of $\text{\upshape \scshape wot}$-continuous contraction-valued functions.
\@startsection{section}{1}{\z@}{.7\linespacing\@plus\linespacing}{.5\linespacing}{\formatsection@text}[Complete metrisability results]{Complete metrisability results}
\label{sec:results}
\noindent
As indicated in the introduction, we shall demonstrate the complete metrisability of $\cal{C}_{s}(M)$
by directly classifying its Borel complexity within
the larger Polish space, $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$.
By the previous section we know that in a very general setting,
$\cal{C}_{s}(M)$ is not closed within $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$.
Hence we require weaker conditions,
which determine when subspaces are completely metrisable.
For this we rely on the following classical result from descriptive set theory
(see
\cite[Theorem~3.11]{kech1994}
for a proof):
\begin{lemm*}[Alexandroff's lemma]
Let $X$ be a completely metrisable space.
Then $A\subseteqX$ viewed with the relative topology is completely metrisable
if and only if it is a $G_{\delta}$-subset of $X$.
\end{lemm*}
We now present the main result.
\begin{schattierteboxdunn}[
backgroundcolor=leer,
nobreak=true,
]
\begin{thm}
\makelabel{thm:main-result:sig:article-str-problem-raj-dahya}
Let $\mathcal{H}$ denote a separable infinite-dimensional Hilbert space
and $M$ be a locally compact Polish monoid.
If $M$ is \usesinglequotes{good}, then
the space $\cal{C}_{s}(M)$ of contractive $\ensuremath{C_{0}}$-semigroups over $\mathcal{H}$ on $M$
is Polish under the $\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}}$-topology.
\end{thm}
\end{schattierteboxdunn}
\begin{proof}
Since $\{1\}$ is a compact subset of $M$, it is easy to see that
\begin{mathe}[mc]{c}
X \mathop{\textup{co}}lonequals \{T\in\cal{F}^{c}_{w}(M) \mid T(1)=\text{\upshape\bfseries I}\}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
is a closed subset in $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$
and thus $(X,\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$ is Polish
(\cf \Cref{prop:basic:SpC:basic-polish:sig:article-str-problem-raj-dahya}).
By Alexandroff's lemma it thus suffices to prove that $\cal{C}_{s}(M)$ is a $G_{\delta}$-subset of $X$.
To proceed, observe that since $M$ is a locally compact Polish space, it is $\sigma$-compact,
\mathop{\textit{id}}est there exists a countable collection of compact subsets,
${\mathop{\textup{co}}mpactcover\subseteq\KmpRm{M}}$,
such that ${\bigcup_{K\in\mathop{\textup{co}}mpactcover}K=M}$.
\Withoutlog one may assume that $\mathop{\textup{co}}mpactcover$ is closed under finite unions.
Since $\mathcal{H}$ is a separable Hilbert space,
it admits a countable ONB ${B\subseteq\mathcal{H}}$.
For each finite $F\subseteq B$ define
\begin{mathe}[mc]{c}
\pi_{F} \mathop{\textup{co}}lonequals \mathop{\mathrm{Proj}}_{\mathop{\textup{lin}}(F)} \in \BoundedOps{\mathcal{H}},\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
\mathop{\textit{id}}est, the projection onto the closed subspace generated by $F$.
Using these, we construct
\begin{mathe}[mc]{c}
d_{K,F,e,e'}(T) \mathop{\textup{co}}lonequals {\displaystyle\sup_{s,t\in K}}|\BRAKET{(T(s)\pi_{F}T(t)-T(st))e}{e'}|\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for each
${K\in\KmpRm{M}}$,
${F\subseteq B}$ finite,
${e,e'\in\mathcal{H}}$,
and
${T\inX}$,
and
\begin{mathe}[mc]{rcl}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:sets:\beweislabel]
V_{\varepsilon;K,F}(\tilde{T}) &\mathop{\textup{co}}lonequals
&{\displaystyle\bigcap_{e,e'\in F}}
\{
T\inX
\mid
{\displaystyle\sup_{t\in K}}
|\BRAKET{(T(t)-\tilde{T}(t))e}{e'}|
< \varepsilon
\},\\
W_{\varepsilon;K,F,e,e'} &\mathop{\textup{co}}lonequals &\{T\inX \mid d_{K,F,e,e'}(T)<\varepsilon\}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for each
${\varepsilon>0}$,
${K\in\KmpRm{M}}$,
${F\subseteq B}$ finite,
${e,e'\in\mathcal{H}}$,
and ${\tilde{T}\inX}$.
We can now present our strategy for the rest of the proof:
To show that $\cal{C}_{s}(M)$ is a $G_{\delta}$-subset of $X$,
it suffices to show
(I) that the $W$-sets defined in \eqcref{eq:sets:\beweislabel} are open
and
(II) that
\begin{mathe}[tc]{rcccl}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:G-delta-expression:\beweislabel]
\cal{C}_{s}(M) &\subseteq
&\displaystyle\bigcap_{\substack{
\varepsilon\in\mathbb{Q}^{+},~K\in\mathop{\textup{co}}mpactcover,\\
e,e'\in B,\\
F_{0}\subseteq B~\text{finite}
}}\;
\displaystyle\bigcup_{\substack{
F\subseteq B~\text{finite}\\
\text{s.t.}~F\supseteq F_{0}
}}\;
W_{\varepsilon;K,F,e,e'}
&\subseteq
&\cal{C}_{w}(M).\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
If these two statements hold, then by assumption of $M$ being \usesinglequotes{good},
(I) + (II) will yield that $\cal{C}_{s}(M)=\cal{C}_{w}(M)$ are equal to a $G_{\delta}$-subset of $X$,
which will complete the proof.
\null
Towards (I), fix arbitrary
${\varepsilon>0}$,
${K\in\KmpRm{M}}$,
${F\subseteq B}$ finite,
and
${e,e'\in B}$,
and consider an arbitrary element, ${\tilde{T}\in W_{\varepsilon;K,F,e,e'}}$.
We need to show that $\tilde{T}$ is in the interior of $W_{\varepsilon;K,F,e,e'}$.
By continuity of multiplication in the topological monoid, $M$, the set
${K\cdot K=\{st\mid s,t\in K\}}$
is compact.
Setting
${K'\mathop{\textup{co}}lonequals K\cup (K\cdot K)}$ and ${F'\mathop{\textup{co}}lonequals F\cup\{e,e'\}}$,
it suffices to show that
\begin{mathe}[tc]{c}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:openness-of-W:\beweislabel]
V_{\varepsilon';K',F'}(\tilde{T}) \subseteq W_{\varepsilon;K,F,e,e'}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
holds for some ${\varepsilon'>0}$, since clearly $\tilde{T}$ is an element of the left hand side
and by definition of the $\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}}$-topology, the $V$-sets are clearly open.
We determine $\varepsilon'$ as follows.
First note that by virtue of $\tilde{T}$ being in $W_{\varepsilon;K,F,e,e'}$
\begin{mathe}[tc]{c}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:modified-eps:\beweislabel]
r \mathop{\textup{co}}lonequals \varepsilon - d_{K,F,e,e'}(\tilde{T}) > 0\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
holds.
Since the unit disc, ${\cal{D}_{1}=\{z\in\mathbb{C}\mid |z|\leq 1\}}$,
is compact, the map
${(a,b)\in\cal{D}_{1}^{2}\mapsto ab\in\mathbb{C}}$
is uniformly continuous, and hence some ${\varepsilon'>0}$ exists,
such that
\begin{mathe}[tc]{c}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:uniform-cts-func:\beweislabel]
|a'b'-ab| < \frac{r}{4|F|+1}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all ${a,b,a',b'\in\cal{D}_{1}}$
with ${|a-a'|<\varepsilon'}$ and ${|b-b'|<\varepsilon'}$.
We may also assume without loss of generality that $\varepsilon'<\frac{r}{4}$.
With this $\varepsilon'$-value, the left hand side of \eqcref{eq:openness-of-W:\beweislabel} is now determined.
It remains to show that the inclusion holds.
Since the elements in $X$ are all contraction-valued functions
and the ONB, $B$, consists of unit vectors,
it holds that
${\BRAKET{T(t)\xi}{\eta}\in\cal{D}_{1}}$
for all
$T\inX$,
${t\in M}$,
and
${\xi,\eta\in B}$.
Now consider an arbitrary $T$ in the left hand side of \eqcref{eq:openness-of-W:\beweislabel}.
Let $s,t\in K$ be arbitrary.
Then $s,t,st\in K'$,
so that by the choice of $F'$
and by virtue of $T$ being inside $V_{\varepsilon';K',F'}(\tilde{T})$,
we have
\begin{mathe}[mc]{rcl}
|\BRAKET{T(s)e''}{e'}-\BRAKET{\tilde{T}(s)e''}{e'}| &< &\varepsilon',\\
|\BRAKET{T(t)e}{e''}-\BRAKET{\tilde{T}(t)e}{e''}| &< &\varepsilon',\\
|\BRAKET{T(st)e}{e'}-\BRAKET{\tilde{T}(st)e}{e'}| &< &\varepsilon'\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all $e''\in F$.
Since $F$ is an orthonormal collection,
the choice of $\varepsilon'$ and \eqcref{eq:uniform-cts-func:\beweislabel}
yield
\begin{longmathe}[mc]{RCL}
\begin{array}[b]{0l}
|\BRAKET{(T(s)\pi_{F}T(t)-T(st))e}{e'}\\
\;-\BRAKET{(\tilde{T}(s)\pi_{F}\tilde{T}(t)-\tilde{T}(st))e}{e'}|\\
\end{array}
&\leq &{\displaystyle\sum_{e''\in F}}
|
\BRAKET{T(s)e''}{e'}
\BRAKET{T(t)e}{e''}
-
\BRAKET{\tilde{T}(s)e''}{e'}
\BRAKET{\tilde{T}(t)e}{e''}
|\\
&&\quad +\;|\BRAKET{T(st)e}{e'}-\BRAKET{\tilde{T}(st)e}{e'}|\\
&\eqcrefoverset{eq:uniform-cts-func:\beweislabel}{<}
&{\displaystyle\sum_{e''\in F}}\frac{r}{4|F|+1}\;+\;\varepsilon'
<\frac{r|F|}{4|F|+1} + \frac{r}{4}
<\frac{r}{2}\\
\end{longmathe}
\mathop{\textup{co}}ntinueparagraph
for all $s,t\in K$. Thus
\begin{longmathe}[mc]{RCL}
d_{K,F,e,e'}(T)
&= &{\displaystyle\sup_{s,t\in K}}|\BRAKET{(T(s)\pi_{F}T(t)-T(st))e}{e'}|\\
&\leq &{\displaystyle\sup_{s,t\in K}}|\BRAKET{(\tilde{T}(s)\pi_{F}\tilde{T}(t)-\tilde{T}(st))e}{e'}|
+\frac{r}{2}\\
&= &d_{K,F,e,e'}(\tilde{T})+\frac{r}{2}\\
&\eqcrefoverset{eq:modified-eps:\beweislabel}{=}
&\varepsilon-r+\frac{r}{2}
<
\varepsilon,\\
\end{longmathe}
\mathop{\textup{co}}ntinueparagraph
whence ${T\in W_{\varepsilon;K,F,e,e'}}$.
Hence the inclusion in \eqcref{eq:openness-of-W:\beweislabel} holds, as desired.
\null
To prove (II), consider the first inclusion of \eqcref{eq:G-delta-expression:\beweislabel}.
Let ${T\in\cal{C}_{s}(M)}$ be arbitrary.
To show that $T$ is in the $G_{\delta}$-set in the middle of \eqcref{eq:G-delta-expression:\beweislabel},
consider arbitrary fixed
${\varepsilon>0}$,
${K\in\KmpRm{M}}$,
${F_{0}\subseteq B}$ finite,
and
${e,e'\in B}$.
Our goal is to find some finite ${F\subseteq B}$ with ${F\supseteq F_{0}}$,
such that $T \in W_{\varepsilon;K,F,e,e'}$.
To this end,
we rely on the fact that $T$ is a contractive semigroup
and observe that for all finite $F\subseteq B$
the functions
\begin{mathe}[mc]{rcccl}
\tilde{f}_{F} &: &K &\to &\mathbb{R}NonNeg\\
&&t &\mapsto &\|(\text{\upshape\bfseries I}-\pi_{F})T(t)e\|,\\
\\
f_{F} &: &K\times K &\to &\mathbb{R}NonNeg\\
&&(s,t) &\mapsto &|\BRAKET{(T(s)\pi_{F}T(t)-T(st))e}{e'}|\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
satisfy
\begin{mathe}[mc]{rclcccl}
f_{F}(s,t)
&= &|\BRAKET{(T(s)\pi_{F}T(t)-T(s)T(t))e}{e'}|\\
&= &|\BRAKET{(\text{\upshape\bfseries I}-\pi_{F})T(t)e}{T(s)^{\ast}e'}|
&\leq &\|T(s)^{\ast}e'\|\tilde{f}_{F}(t)
&\leq &\tilde{f}_{F}(t)\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all $s,t\in K$.
Furthermore, the $\text{\upshape \scshape sot}$-continuity of $T$ guarantees that $\tilde{f}_{F}$ is continuous.
Now consider the net $(\tilde{f}_{F})_{F}$,
where the indices run over all finite $F\subseteq B$, ordered by inclusion.
Note that the correspondingly indexed net of projections, $(\pi_{F})_{F}$, is monotone,
and, since ${\bigcup_{F\subseteq B~\text{finite}}F=B}$ and $B$ is a basis for $\mathcal{H}$,
it holds that ${\pi_{F}\underset{F}{\longrightarrow}\text{\upshape\bfseries I}}$ weakly (in fact strongly).
Clearly then
\begin{mathe}[mc]{c}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:pt-wise-convergence:\beweislabel]
\tilde{f}_{F} \underset{F}{\longrightarrow} 0\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
pointwise and monotone. Since $K$ is compact and the $\tilde{f}_{F}$ are continuous for all $F$,
by Dini's Theorem (\cf
\cite[Theorem~2.66]{aliprantis2005}
) the monotone pointwise convergence in
\eqcref{eq:pt-wise-convergence:\beweislabel}
is in fact uniform convergence.
Hence, by the definition of the net,
for some finite $F\subseteq B$ with $F\supseteq F_{0}$
\begin{mathe}[mc]{c}
d_{K,F,e,e'}(T)
= \sup_{s,t\in K}f_{F}(s,t)
\leq \sup_{t\in K}\tilde{f}_{F}(t)
< \varepsilon,\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
and thereby ${T\in W_{\varepsilon;K,F,e,e'}}$.
\null
To complete the proof of (II), we treat the second inclusion in \eqcref{eq:G-delta-expression:\beweislabel}.
So let ${T\inX}$ be an arbitrary element in the $G_{\delta}$-set in the middle of \eqcref{eq:G-delta-expression:\beweislabel}.
To show that ${T\in\cal{C}_{w}(M)}$, it is necessary and sufficient
to show that ${T(st)=T(s)T(t)}$ for all ${s,t\in M}$.
So fix arbitrary ${s,t\in M}$.
It suffices to show that
$\BRAKET{(T(s)T(t)-T(st))e}{e'}=0$
for all basis vectors, ${e,e'\in B}$.
So fix arbitrary $e,e'\in B$.
Note that since $\mathop{\textup{co}}mpactcover$ covers $M$ and is closed under finite unions,
there exists some ${K\in\mathop{\textup{co}}mpactcover}$, such that ${s,t\in K}$.
Fix this compact set.
Now consider the net
$(d_{K,F,e,e'}(T))_{F}$,
whose indices run over all finite ${F\subseteq B}$, ordered by inclusion.
Since $T$ is in the set in the middle of \eqcref{eq:G-delta-expression:\beweislabel},
working through the definitions yields
\begin{mathe}[mc]{c}
\forall{\varepsilon\in\mathbb{Q}^{+}:~}
\forall{F_{0}\subseteq B~\text{finite}:~}
\,\exists{F\subseteq B~\text{finite},~\text{s.t.}~F\supseteq F_{0}:~}
\,d_{K,F,e,e'}(T)<\varepsilon,\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
which is clearly equivalent to
\begin{mathe}[mc]{c}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:liminf:\beweislabel]\relax
{\displaystyle\mathop{\ell\mathrm{im}}inf_{F}}\,d_{K,F,e,e'}(T) = 0.\\
\end{mathe}
Now, since $s,t\in K$, it holds that
\begin{mathe}[mc]{c}
|\BRAKET{(T(s)T(t)-T(st))e}{e'}|
\leq
\underbrace{
|\BRAKET{T(s)(\text{\upshape\bfseries I}-\pi_{F})T(t)e}{e'}|
}_{=:d'_{F}}
+\underbrace{
|\BRAKET{(T(s)\pi_{F}T(t)-T(st))e}{e'}|
}_{\leq d_{K,F,e,e'}(T)}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all finite $F\subseteq B$.
Since ${\pi_{F}\underset{F}{\longrightarrow}\text{\upshape\bfseries I}}$ weakly (see above),
we have ${d'_{F}\underset{F}{\longrightarrow}0}$.
Noting \eqcref{eq:liminf:\beweislabel},
taking the limit inferior of the right hand side of the above expression
thus yields ${\BRAKET{(T(s)T(t)-T(st))e}{e'}=0}$.
Since ${e,e'\in B}$ were arbitrarily chosen, and $B$ is a basis for $\mathcal{H}$,
it follows that ${T(st)=T(s)T(t)}$.
This completes the proof.
\end{proof}
\begin{rem}
The proof of \Cref{thm:main-result:sig:article-str-problem-raj-dahya} reveals that in fact claims (I) and (II) hold,
provided the topological monoid $M$ is at least $\sigma$-compact.
And if $M$ is furthermore \usesinglequotes{good}, then these again imply that that $\cal{C}_{s}(M)$ is a $G_{\delta}$-subset in $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$.
The stronger assumption of $M$ being locally compact is only relied upon
to obtain that $(\cal{F}^{c}_{w}(M),\text{{{$\mathpzc{k}$}}}_{\text{\tiny\upshape \scshape wot}})$ is itself completely metrisable
(\cf the proof of \Cref{prop:basic:SpC:basic-polish:sig:article-str-problem-raj-dahya}),
and thus via Alexandroff's lemma
that $G_{\delta}$-subsets of this space are completely metrisable.
\end{rem}
Finally, by \Cref{cor:multiparameter-auto-cts:sig:article-str-problem-raj-dahya} we obtain:
\begin{cor}
The spaces of one-/multiparameter contractive $\ensuremath{C_{0}}$-semigroups on a separable Hilbert space,
viewed under the topology of uniform $\text{\upshape \scshape wot}$-convergence on compact subsets,
are Polish.
\end{cor}
This positively solves the open problem raised in
\cite[\S{}III.6.3]{eisner2010buchStableOpAndSemigroups}.
\setcounternach{section}{1}
\appendix
\@startsection{section}{1}{\z@}{.7\linespacing\@plus\linespacing}{.5\linespacing}{\formatsection@text}[Strong continuity of operator semigroups]{Strong continuity of operator semigroups}
\label{app:continuity}
\noindent
The main result of this paper
is proved for semigroups defined on \usesinglequotes{good} monoids
(see \Cref{defn:good-monoids:sig:article-str-problem-raj-dahya}).
By a well-known result, any $\text{\upshape \scshape wot}$-continuous semigroup
on a Banach space $\mathcal{E}$
over the monoid $(\mathbb{R}NonNeg,+,0)$
is automatically $\text{\upshape \scshape sot}$-continuous,
(\cf \cite[Theorem~5.8]{Engel1999})
and thus $\mathbb{R}NonNeg$ is by our definition a \usesinglequotes{good} monoid.
In this appendix, we provide sufficient conditions for topological monoids
to possess this property, and thus broaden application of the main result.
These conditions are given as follows:
\begin{defn}
\makelabel{app:continuity:defn:conditition-I:sig:article-str-problem-raj-dahya}
A topological monoid, $M$, shall be called \highlightTerm{extendible},
if there exists a locally compact Hausdorff topological group, $G$,
such that $M$ is topologically and algebraically isomorphic to a closed subset of $G$.
\end{defn}
If $M$ is extendible to $G$ via the above definition,
then one can assume \withoutlog that $M \subseteq G$.
\begin{defn}
\makelabel{app:continuity:defn:conditition-II:sig:article-str-problem-raj-dahya}
Let $G$ be a locally compact Hausdorff group.
Call a subset $A \subseteq G$ \highlightTerm{positive in the identity},
if for all neighbourhoods, $U \subseteq G$, of the group identity,
$U \cap A$ has non-empty interior within $G$.
\end{defn}
\begin{e.g.}[The non-negative reals]
\makelabel{app:continuity:e.g.:extendible-mon:reals:sig:article-str-problem-raj-dahya}
Consider ${M \mathop{\textup{co}}lonequals \mathbb{R}NonNeg}$ viewed under addition.
Since ${M\subseteq\mathbb{R}}$ is closed,
we have that $M$ is an extendible locally compact Hausdorff monoid.
For any open neighbourhood, ${U\subseteq\mathbb{R}}$, of the identity,
there exists an ${\varepsilon>0}$, such that ${(-\varepsilon,\varepsilon)\subseteq U}$
and thus ${U\cap M\supseteq(0,\varepsilon)\neq\emptyset}$.
Hence $M$ is positive in the identity.
\end{e.g.}
\begin{e.g.}[The $p$-adic integers]
\makelabel{app:continuity:e.g.:extendible-mon:p-adic:sig:article-str-problem-raj-dahya}
Consider ${M \mathop{\textup{co}}lonequals \mathbb{Z}_{p}}$ with ${p\in\mathbb{P}}$,
viewed under addition and with the topology generated by the $p$-adic norm.
Since ${M\subseteq\mathbb{Q}_{p}}$ is clopen,
it is an extendible locally compact Hausdorff monoid.
Since $M$ is clopen, it is clearly positive in the identity.
\end{e.g.}
\begin{e.g.}[Discrete cases]
\makelabel{app:continuity:e.g.:extendible-mon:discrete:sig:article-str-problem-raj-dahya}
Let $G$ be a discrete group, and let $M\subseteq G$ contain the identity and be closed under group multiplication.
Clearly, $M$ is a locally compact Hausdorff monoid, extendible to $G$ and positive in the identity.
For example one can take the free-group $\mathbb{F}_{2}$ with generators $\{a,b\}$,
and $M$ to be the algebraic closure of $\{1,a,b\}$ under multiplication.
\end{e.g.}
\begin{e.g.}[Non-discrete, non-commutative cases]
\makelabel{app:continuity:e.g.:extendible-mon:non-comm-non-discrete:sig:article-str-problem-raj-dahya}
Let $d\in\mathbb{N}pos$ with $d>1$ and consider the space, $X$, of $\mathbb{R}$-valued $d\times d$ matrices.
Topologised with any matrix norm (equivalently the strong or the weak operator topologies),
this space is homeomorphic to $\mathbb{R}^{d^{2}}$ and thus locally compact Hausdorff.
Since the determinant map ${X\ni T\mapsto \det(T)\in\mathbb{R}}$ is continuous,
the subspace of invertible matrices $\{T\inX\mid \det(T)\neq 0\}$
is open and thus a locally compact Hausdorff topological group.
The subspace, $G$, of upper triangular matrices with positive diagonal entries,
is a closed subgroup and thus locally compact Hausdorff.
Letting
\begin{mathe}[mc]{rcl}
G_{0} &\mathop{\textup{co}}lonequals &\{T\in G\mid \det(T)=1\},\\
G_{+} &\mathop{\textup{co}}lonequals &\{T\in G\mid \det(T)>1\},~\text{and}\\
G_{-} &\mathop{\textup{co}}lonequals &\{T\in G\mid \det(T)<1\},\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
it is easy to see that $M \mathop{\textup{co}}lonequals G_{0}\cup G_{+}$ is a topologically closed subspace containing the identity and is closed under multiplication.
Moreover $M$ is a proper monoid, since the inverses of the elements in $G_{+}$ are clearly in ${G\setminus M}$.
Consider now an open neighbourhood, ${U\subseteq G}$, of the identity.
Since inversion is continuous, $U^{-1}$ is also an open neighbourhood of the identity.
Since, as a locally compact Hausdorff space, $G$ satisfies the Baire category theorem,
and since ${G_{+}\cup G_{-}}$ is clearly dense (and open) in $G$, and thus comeagre,
we clearly have $(U\cap U^{-1})\cap(G_{+}\cup G_{-})\neq\emptyset$.
So either ${U\cap G_{+}\neq\emptyset}$ or else ${U^{-1}\cap G_{-}\neq\emptyset}$,
from which it follows that ${U\cap G_{+}=(U^{-1}\cap G_{-})^{-1}\neq\emptyset}$.
Hence in each case ${U\cap M}$ contains a non-empty open subset, \viz ${U\cap G_{+}}$.
So $M$ is extendible to $G$ and positive in the identity.
Next, consider the subgroup, ${G_{h} \subseteq G}$,
consisting of matrices of the form $T = \text{\upshape\bfseries I} + \tilde{T}$,
where $\tilde{T}$ is a strictly upper triangular matrix
with at most non-zero entries on the top row and right hand column.
That is, $G_{h}$ is the \highlightTerm{continuous Heisenberg group}, $\Heisenberg{2d-3}(\mathbb{R})$, of order ${2d-3}$.
The elements of the Heisenberg group occur in the study of Kirillov's \highlightTerm{orbit method}
(see \cite{kirillov1962})
and have important applications in physics
(see \exempli \cite{kirillov2003}).
Clearly, $G_{h}$ is topologically closed within $G$ and thus locally compact Hausdorff.
Now consider the subspace,
\begin{mathe}[mc]{c}
M_{h} \mathop{\textup{co}}lonequals \{T\in G_{h} \mid \forall{i,j\in\{1,2,\ldots,d\}:~}T_{ij}\geq 0\},\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
of matrices with only non-negative entries.
This is clearly a topologically closed subspace of $G_{h}$ containing the identity and closed under multiplication.
Moreover, for each ${S,T \in M_{h}\setminus\{\text{\upshape\bfseries I}\}}$ we have
\begin{mathe}[mc]{c}
ST = \text{\upshape\bfseries I} + \big((S-\text{\upshape\bfseries I}) + (T-\text{\upshape\bfseries I}) + (S-\text{\upshape\bfseries I})(T-\text{\upshape\bfseries I})\big)
\in M_{h} \setminus \{\text{\upshape\bfseries I}\},\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
which implies that no non-trivial element in $M_{h}$ has its inverse in $M_{h}$, making $M_{h}$ a proper monoid.
Consider now an open neighbourhood, ${U \subseteq G_{h}}$, of the identity.
Since $G_{h}$ is homeomorphic to $\mathbb{R}^{2d-3}$,
there exists some ${\varepsilon>0}$, such that
\begin{mathe}[mc]{c}
U \supseteq \{
T \in G_{h}
\mid
\forall{(i,j)\in\mathcal{I}:~}T_{ij}\in(-\varepsilon,\varepsilon)
\},\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
where $\mathcal{I} \mathop{\textup{co}}lonequals \{(1,2),(1,3),\ldots,(1,d),(2,d),\ldots,(d-1,d)\}$.
Hence
\begin{mathe}[mc]{rcl}
U \cap M_{h} &\supseteq &\{
T \in G_{h}
\mid
\forall{(i,j)\in\mathcal{I}:~}T_{ij}\in(0,\varepsilon)
\} \eqcolon V,\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
where $V$ is clearly a non-empty open subset of $G_{h}$,
since the $2d-3$ entries in the matrices can be freely and independently chosen.
Thus $M_{h}$ is extendible to $G_{h}$ and positive in the identity.
Finally, we may consider the subgroup, ${G_{u} \mathop{\textup{co}}lonequals \mathop{\mathrm{UT}}Unit(d)}$, of upper triangular matrices over $\mathbb{R}$ with unit diagonal.
The elements of $\mathop{\mathrm{UT}}Unit(d)$ have important applications in image analysis
(see \exempli
\cite{kirillov2003}
and
\cite[\S{}5.5.2]{pennec2020}
)
and representations of the group have been studied in
\cite[Chapter~6]{samoilenko1991}.
Setting ${M_{u} \mathop{\textup{co}}lonequals \{T \in G_{u}\mid \forall{i,j\in\{1,2,\ldots,d\}:~}T_{ij}\geq 0\}}$,
one may argue similarly to the case of the continuous Heisenberg group
and obtain that $G_{u}$ is locally compact
and that $M_{u}$ is a proper topological monoid
which is furthermore extendible to $G_{u}$ and positive in the identity.
\end{e.g.}
The following result allows us to generate infinitely many examples from basic ones:
\begin{prop}
\makelabel{app:continuity:prop:monoids-with-cond-closed-under-fprod:sig:article-str-problem-raj-dahya}
Let ${n\in\mathbb{N}pos}$ and let $M_{i}$ be locally compact Hausdorff monoids for ${1\leq i\leq n}$.
Assume for each ${i<n}$ that $M_{i}$ is extendible to a locally compact Hausdorff group $G_{i}$,
and that $M_{i}$ is positive in the identity of $G_{i}$.
Then ${M \mathop{\textup{co}}lonequals \prod_{i=1}^{n}M_{i}}$ is a locally compact Hausdorff monoid
which is extendible to ${G \mathop{\textup{co}}lonequals \prod_{i=1}^{n}G_{i}}$
and positive in the identity.
\end{prop}
\begin{proof}
The extendibility of $M$ to $G$ is clear.
Now consider an arbitrary open neighbourhood, $U$, of the identity in $G$.
For each ${1\leq i\leq n}$, one can find open neighbourhoods, $U_{i}$, of the identity in $G_{i}$,
so that ${U' \mathop{\textup{co}}lonequals \prod_{i=1}^{n}U_{i}\subseteq U}$.
By assumption, $M_{i}\cap U_{i}$ contains a non-empty open set,
${V_{i}\subseteq G_{i}}$ for each ${1\leq i\leq n}$.
Since ${U\supseteq\prod_{i=1}^{n}U_{i}}$,
it follows that
$M\cap U
\supseteq
\prod_{i=1}^{n}(M_{i}\cap U_{i})
\supseteq
\prod_{i=1}^{n}V_{i}\neq\emptyset$.
Thus ${M\cap U}$ has non-empty interior.
Hence $M$ is positive in the identity.
\end{proof}
Our argumentation for the generalised continuity result
builds on
\cite[Theorem~5.8]{Engel1999}.
\begin{schattierteboxdunn}[
backgroundcolor=leer,
nobreak=true,
]
\begin{thm}
\makelabel{app:continuity:thm:generalised-auto-continuity:sig:article-str-problem-raj-dahya}
Let $M$ be a topological monoid
such that
$M$ is extendible to a locally compact Hausdorff group $G$
and such that
$M$ is positive in the identity.
Then for any Banach space, $\mathcal{E}$,
every $\text{\upshape \scshape wot}$-continuous semigroup, ${T:M\to\BoundedOps{\mathcal{E}}}$,
is $\text{\upshape \scshape sot}$-continuous.
In particular, $M$ is \usesinglequotes{good}.
\end{thm}
\end{schattierteboxdunn}
(Note that a semigroup over a Banach space $\mathcal{E}$ on a topological monoid
is defined analogously to \Cref{defn:semigroup-on-h:sig:article-str-problem-raj-dahya}.)
\begin{proof}
First note that the principle of uniform boundedness applied twice to the $\text{\upshape \scshape wot}$-continuous function, $T$,
ensures that $T$ is norm-bounded on all compact subsets of $M$.
Fix now a left-invariant Haar measure, $\lambda$, on $G$ and set
\begin{mathe}[mc]{rcl}
S &\mathop{\textup{co}}lonequals &\{F\subseteq G\mid F~\text{a compact neighbourhood of the identity}\}.\\
\end{mathe}
Consider arbitrary ${F\in S}$ and ${x \in \mathcal{E}}$.
By the closure of $M$ in $G$ as well as positivity in the identity,
${M\cap F}$ is compact and contains a non-empty open subset of $G$.
It follows that ${0<\lambda(M\cap F)<\infty}$.
The $\text{\upshape \scshape wot}$-continuity of $T$, the compactness (and thus measurability) of ${M\cap F}$,
and the norm-boundedness of $T$ on compact subsets
ensure that
\begin{mathe}[mc]{rcl}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:defn-xF:\beweislabel]
\BRAKET{x_{F}}{\altvarphi}
&\mathop{\textup{co}}lonequals &\frac{1}{\lambda(M\cap F)}
\displaystyle\int_{t\in M\cap F}
\BRAKET{T(t)x}{\altvarphi}
~\textup{d}{}t,
\quad\text{for $\altvarphi \in \mathcal{E}^{\prime}$}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
describes a well-defined element $x_{F} \in \mathcal{E}^{\prime\prime}$.
Exactly as in
\cite[Theorem~5.8]{Engel1999},
one may now argue by the $\text{\upshape \scshape wot}$-continuity of $T$ and compactness of ${M\cap F}$
that in fact ${x_{F} \in \mathcal{E}}$ for each ${x \in \mathcal{E}}$ and ${F\in S}$.
Moreover, since $M$ is locally compact,
and $T$ is $\text{\upshape \scshape wot}$-continuous with $T(1)=\text{\upshape\bfseries I}$,
one readily obtains that each $x \in \mathcal{E}$
can be weakly approximated by the net,
$(x_{F})_{F \in S}$,
ordered by inverse inclusion. So
\begin{mathe}[mc]{rcl}
D &\mathop{\textup{co}}lonequals &\{x_{F}\mid x \in \mathcal{E},~F \in S\}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
is weakly dense in $\mathcal{E}$.
Since the weak and strong closures of any convex subset in a Banach space coincide
(\cf \cite[Theorem~5.98]{aliprantis2005}),
it follows that the convex hull, $\mathop{\textup{co}}(D)$, is strongly dense in $\mathcal{E}$.
Now, to prove the $\text{\upshape \scshape sot}$-continuity of $T$,
we need to show that
\begin{mathe}[mc]{rcl}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:map:\beweislabel]
t\in M &\mapsto &T(t)x \in \mathcal{E}\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
is strongly continuous for all $x \in \mathcal{E}$.
Since $M$ is locally compact and $T$ is norm-bounded on compact subsets of $M$,
the set of $x \in \mathcal{E}$ such that \eqcref{eq:map:\beweislabel} is strongly continuous,
is itself a strongly closed convex subset of $\mathcal{E}$.
So, since $\mathop{\textup{co}}(D)$ is strongly dense in $\mathcal{E}$,
it suffices to prove
the strong continuity of \eqcref{eq:map:\beweislabel}
for each ${x\in D}$.
To this end, fix arbitrary ${x \in \mathcal{E}}$, ${F\in S}$ and ${t\in M}$.
We need to show that ${T(t')x_{F}\longrightarrow T(t)x_{F}}$
strongly for ${t'\in M}$ as ${t'\longrightarrow t}$.
First recall, that by basic harmonic analysis,
the canonical \highlightTerm{left-shift},
\begin{mathe}[mc]{rcccl}
L &: &G &\to &\BoundedOps{L^{1}(G)},\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
defined via ${(L_{t}f)(s)=f(t^{-1}s)}$ for ${s,t\in G}$ and $f\in L^{1}(G)$,
is an $\text{\upshape \scshape sot}$-continuous morphism
(\cf
\cite[Proposition~3.5.6 ($\lambda_{1}$--$\lambda_{4}$)]{reiter2000}
).
Now, by compactness, ${f \mathop{\textup{co}}lonequals \text{\textbf{1}}_{M\cap F}\in L^{1}(G)}$
and it is easy to see that
${\|L_{t'}f-L_{t}f\|_{1}=\lambda(t'(M\cap F)\mathbin{\Delta} t(M\cap F))}$
for ${t'\in M}$.
The $\text{\upshape \scshape sot}$-continuity of $L$ thus yields
\begin{mathe}[mc]{rcl}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:1:\beweislabel]
\lambda(t'(M\cap F) \mathbin{\Delta} t(M\cap F)) &\longrightarrow &0\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for ${t'\in M}$ as ${t'\longrightarrow t}$.
Fix now a compact neighbourhood, ${K\subseteq G}$, of $t$.
For ${t'\in M\cap K}$ and ${\altvarphi \in \mathcal{E}^{\prime}}$ one obtains
\begin{mathe}[mc]{rcl}
|\BRAKET{T(t')x_{F}-T(t)x_{F}}{\altvarphi}|
&= &|\BRAKET{x_{F}}{T(t')^{\ast}\altvarphi}-\BRAKET{x_{F}}{T(t)^{\ast}\altvarphi}|\\
&= &\frac{1}{\lambda(M\cap F)}
\cdot\left|
\displaystyle\int_{s\in M\cap F}\textstyle
\BRAKET{T(s)x}{T(t')^{\ast}\altvarphi}~\textup{d}{}s
-
\displaystyle\int_{s\in M\cap F}\textstyle
\BRAKET{T(s)x}{T(t)^{\ast}\altvarphi}~\textup{d}{}s
\right|\\
&\multispan{2}{\text{by construction of $x_{F}$ in \eqcref{eq:defn-xF:\beweislabel}}}\\
&= &\frac{1}{\lambda(M\cap F)}
\cdot\left|
\displaystyle\int_{s\in M\cap F}\textstyle
\BRAKET{T(t's)x}{\altvarphi}~\textup{d}{}s
-
\displaystyle\int_{s\in M\cap F}\textstyle
\BRAKET{T(ts)x}{\altvarphi}~\textup{d}{}s
\right|\\
&\multispan{2}{\text{since $T$ is a semigroup}}\\
&= &\frac{1}{\lambda(M\cap F)}
\cdot\left|
\displaystyle\int_{s\in t'(M\cap F)}\textstyle
\BRAKET{T(s)x}{\altvarphi}~\textup{d}{}s
-
\displaystyle\int_{s\in t(M\cap F)}\textstyle
\BRAKET{T(s)x}{\altvarphi}~\textup{d}{}s
\right|\\
&\multispan{2}{\text{by left-invariance}}\\
&\leq &\frac{1}{\lambda(M\cap F)}
\displaystyle\int_{s\in t'(M\cap F)\mathbin{\Delta} t(M\cap F)}\textstyle
|\BRAKET{T(s)x}{\altvarphi}|~\textup{d}{}s\\
&\leq &\frac{1}{\lambda(M\cap F)}
\cdot\displaystyle\sup_{s\in(M\cap K)(M\cap F)}\textstyle\|T(s)\|
\cdot\|x\|\cdot\|\altvarphi\|
\cdot\lambda(t'(M\cap F)\mathbin{\Delta} t(M\cap F))\\
&\multispan{2}{\text{since $t,t'\in M\cap K$}.}\\
\end{mathe}
Since $K' \mathop{\textup{co}}lonequals (M\cap K)(M\cap F)$ is compact, and $T$ is uniformly bounded on compact sets (see above),
it holds that ${C \mathop{\textup{co}}lonequals \displaystyle\sup_{s\in K'}\|T(s)\|\textstyle<\infty}$.
The above calculation thus yields
\begin{mathe}[mc]{rcl}
\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}[eq:2:\beweislabel]
\|T(t')x_{F}-T(t)x_{F}\|
&= &\displaystyle\sup\textstyle\{
|\BRAKET{T(s)x_{F}-T(t)x_{F}}{\altvarphi}|
\mid
\altvarphi \in \mathcal{E}^{\prime},~\|\altvarphi\|\leq 1
\}\\
&\leq &\frac{1}{\lambda(M\cap F)}
\cdot C
\cdot\|x\|
\cdot\lambda(t'(M\cap F)\mathbin{\Delta} t(M\cap F))\\
\end{mathe}
\mathop{\textup{co}}ntinueparagraph
for all $t'\in M$ sufficiently close to $t$.
By \eqcref{eq:1:\beweislabel}, the right-hand side of \eqcref{eq:2:\beweislabel} converges to $0$
and hence ${T(t')x_{F}\longrightarrow T(t)x_{F}}$
strongly as ${t'\longrightarrow t}$.
This completes the proof.
\end{proof}
\begin{rem}
In the proof of \Cref{app:continuity:thm:generalised-auto-continuity:sig:article-str-problem-raj-dahya},
weak continuity only played a role in obtaining
the boundedness of $T$ on compact sets,
as well as the well-definedness of the elements in $D$.
In \cite[Theorem~9.3.1 and Theorem~10.2.1--3]{hillephillips2008} a proof of the classical continuity result exists under weaker conditions,
\viz weak measurability, provided the semigroups are almost separably valued.
It would be interesting to know whether the above approach can be adapted to these weaker assumptions.
\end{rem}
\@startsection{subsection}{2}{\z@}{\z@}{\z@\hspace{1em}}{\formatsubsection@text}*{Acknowledgement}
\noindent
The author is grateful
to Tanja Eisner for her feedback,
to Konrad Zimmermann for his helpful comments on the results in the appendix,
and
to the referee for their constructive feedback.
\defReferences{References}
\bgroup
\footnotesize
\egroup
\mathop{\textup{ad}}dresseshere
\end{document} |
\begin{document}
\title{On the representation number of a crown graph}
\abstract{A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. It is known that any word-representable graph $G$ is $k$-word-representable for some $k$, that is, there exists a word $w$ representing $G$ such that each letter occurs exactly $k$ times in $w$. The minimum such $k$ is called $G$'s representation number.
A crown graph $H_{n,n}$ is a graph obtained from the complete bipartite graph $K_{n,n}$ by removing a perfect matching. In this paper we show that for $n\geq 5$, $H_{n,n}$'s representation number is $\lceil n/2 \rceil$. This result not only provides a complete solution to the open Problem 7.4.2 in \cite{KL}, but also gives a negative answer to the question raised in Problem 7.2.7 in \cite{KL} on 3-word-representability of bipartite graphs. As a byproduct we obtain a new example of a graph class with a high representation number. \\
\noindent {\bf Keywords:} word-representable graph, crown graph, representation number
}
\section{Introduction}
A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. For example, the cycle graph on 4 vertices labeled by 1, 2, 3 and 4 in clockwise direction can be represented by the word 14213243.
There is a long line of research on word-representable graphs, which is summarised in the recently published book \cite{KL}. The roots of the theory of word-representable graphs are in the study of the celebrated Perkins semigroup \cite{KS} which has played a central role in semigroup theory since 1960, particularly as a source of examples and counterexamples.
It was shown in \cite{KP} that if a graph $G$ is word-representable then it is {\em $k$-word-representable} for some $k$, that is, $G$ can be represented by a $k$-{\em uniform} word $w$, i.~e. a word containing $k$ copies of each letter. In such a context we say that $w$ {\em $k$-represents} $G$. For example, the cycle graph on 4 vertices mentioned above can be 2-represented by the word 14213243. Thus, when discussing word-representability, one need only consider $k$-uniform words. The nice property of such words is that any cyclic shift of a $k$-uniform word represents the same graph \cite{KP}. The minimum $k$ for which a word-representable graph $G$ is $k$-word-representable is called $G$'s {\em representation number}.
The following observation trivially follows from the definitions.
\begin{obs}\label{propY} The class of complete graphs coincides with the class of $1$-word-representable graphs. In particular, the complete graph's representation number is $1$.\end{obs}
\subsection{Representation of crown graphs}
A {\it crown graph} (also known as a {\it cocktail party graph}) $H_{n,n}$ is a graph obtained from the complete bipartite graph $K_{n,n}$ by removing a perfect matching. Formally,
$V(H_{n,n})=\{1,\ldots, n,1',\ldots, n' \}$ and $E(H_{n,n})=\{ ij' \ | i\ne j \}$. First four examples of such graphs are presented in Figure~\ref{crown-pic}.
\begin{figure}
\caption{The crown graph $H_{n,n}
\label{crown-pic}
\end{figure}
Crown graphs are of special importance in the theory of word-representable graphs. More precisely, they appear in the construction of graphs requiring long words representing them \cite{HKP}. Note that these graphs also appear in the theory of partially ordered sets as those defining partial orders that require many linear orders to be represented.
Each crown graph, being a bipartite graph, is a {\em comparability graph} (that is, a transitively orientable graph), and thus it can be represented by a concatenation of permutations \cite{KS}. Moreover, it follows from \cite{HKP}, and also is discussed in Section 7.4 in \cite{KL}, that $H_{n,n}$ can be represented as a concatenation of $n$ permutations but it cannot be represented as a concatenation of a fewer permutations. Thus, the representation number of $H_{n,n}$ is at most $n$. See Table~\ref{ex-k-repr-Hkk} (appearing in \cite{KL}) for the words representing the graphs in Figure~\ref{crown-pic} as concatenation of permutations.
\begin{table}
\begin{center}
\begin{tabular}{c|c}
$n$ & representation of $H_{n,n}$ by concatenation of $n$ permutations\\
\hline
1 & $11'1'1$\\
\hline
2 & $12'21'21'12'$\\
\hline
3 & $123'32'1'132'23'1'231'13'2'$\\
\hline
4 & $1234'43'2'1'1243'34'2'1'1342'24'3'1'2341'14'3'2'$\\
\hline
\end{tabular}
\caption{Representing $H_{n,n}$ as a concatenation of $n$ permutations}\label{ex-k-repr-Hkk}
\end{center}
\end{table}
It was noticed in \cite{K} that, for example, $H_{3,3}$ can be represented using two copies of each letter as $3'32'1'132'23'1'231'1$ (as opposed to three copies used in Table~\ref{ex-k-repr-Hkk} to represent it) if we drop the requirement to represent crown graphs as concatenation of permutations. On the other hand, $H_{4,4}$ is the three-dimensional cube, which is the {\em prism graph} Pr$_4$, so that $H_{4,4}$ is 3-word-representable by Proposition~15 in \cite{KP}, while four copies of each letter are used in Table~\ref{ex-k-repr-Hkk} to represent this graph. Note that $H_{4,4}$ is not 2-word-representable by Theorem~18 in \cite{K}.
These observations led to Problem 7.4.2 on page 172 in \cite{KL} essentially asking to find the representation number of a crown graph $H_{n,n}$. A relevant Problem 7.2.7 on page 169 in \cite{KL} asks whether each bipartite graph is 3-word-representable. When we started to investigate these problems, we established that both $H_{5,5}$ and $H_{6,6}$ are 3-word-representable, which not only suggested that the representation number of a crown graph could be the constant 3, but also that any bipartite graph could be 3-word-representable since crown graphs seem to be the most difficult amoung them to be represented.
In this paper we completely solve the former problem (Problem 7.4.2) and provide the negative answer to the question in the latter problem (Problem 7.2.7) by showing that if $n\ge 5$ then the crown graph $H_{n,n}$, being a bipartite graph, is $\lceil n/2 \rceil$-representable (see Theorem~\ref{thm1}). Thus, crown graphs are another example of a graph class with high representation number. Note that non-bipartite graphs obtained from crown graphs by adding an all-adjacent vertex require roughly twice as long words representing them (see Section 4.2.1 in \cite{KL}).
\subsection{Organization of the paper and some definitions}
The paper is organized as follows. In Section~\ref{sec-lower}, we find a lower bound for the representation number of $H_{n,n}$, while in Section~\ref{sec-upper} we provide a construction of words representing $H_{n,n}$ that match our lower bound. Finally, in Section~\ref{conclusion} we provide some concluding remarks including directions for further research.
We conclude the introduction with a number of technical definitions to be used in the paper.
A {\em factor} of a word is a number of consecutive letters in the word. For example, the set of all factors of the word 1132 of length at most 2 is $\{1,2,3,11,13,32\}$. A {\em subword} of a word is a subsequence of letters in the word. For instance, 56, 5212 and 361 are examples of subwords in 3526162. The subword of a word $w$ {\em induced} by a set $A$ is obtained by removing all elements in $w$ not belonging to $A$. For example, if $A=\{2,4,5\}$ then the subword of 223141565 induced by $A$ is 22455.
For a vertex $v$ in a graph $G$ denote by $N(v)$ the neighbourhood of $v$, i.~e. the set of vertices adjacent to $v$. Clearly, if a graph is bipartite then the neighbourhood of each vertex induces an independent set.
\section{A lower bound for the representation number of $H_{n,n}$}\label{sec-lower}
For a word $w$, let $l(w)$ and $r(w)$ be its first and last letters, respectively.
Let $w$ be a word that $k$-represents a graph $G=(V,E)$. A subset $A\subseteq V$ is {\it splittable} if there is a cyclic shift of the word $w$ such that the subword induced by the set $A$ has the form $P_1\cdots P_k$ where each $P_i$ is a permutation of $A$. For a splittable set $A$, a {\it canonical shift} of $w$, with respect to $A$, is a cyclic shift of $w$ that puts
$l(P_1)$ at the beginning of the word. Note that up to renaming permutations there is a unique canonical shift.
The following proposition gives an example of a splittable set.
\begin{prop}\label{propX} For any vertex $v\in V$ in a word-representable graph $G=(V,E)$, the set $A=N(v)$ is splittable. \end{prop}
\begin{proof} Consider a cyclic shift of a word $w$ $k$-representing $G$ that puts $v$ at the beginning of the word. Then between any two occurrences of $v$ (and after the last one) each letter from $A$ occurs exactly once, i.e.\ the subword induced by $A$ is a concatenation of permutations. Note, however, that this shift is not canonical with respect to $A$. \end{proof}
For a letter $x$, denote by $x_i$ its $i$-th occurrence in a word $w$ (from left to right). We write $x_i<y_j$ if the $i$-th occurrence of $x$ is to the left of the $j$-th occurrence of $y$ in $w$. Clearly, if $A$ is splittable, then for every $a,b\in A$ and for all $i,j$ such that $1\le i<j\le k$, we have $a_i<b_j$.
\begin{lemma}\label{lem1}
Let a word $w$ $k$-represent a graph $G=(V,E)$ and $A\subset V$ be a splittable set. Further, let $a,b\in A$, $x\not\in A$ and $ax,bx\in E$. If in a canonical shift of $w$ $a_1<x_1<b_1$ then
$ab\in E$. \end{lemma}
\begin{proof} Let $a_1<x_1<b_1$. Since $A$ is splittable, $b_i<a_{i+1}$ for each $i$. Since both $a$ and $b$ are adjacent to $x$, we have $a_i<x_i<b_i$ for every $i=1,\ldots, k$. Therefore, $a$ and $b$ alternate in $w$ and must be adjacent in $G$. \end{proof}
\begin{lemma}\label{lem2} If $n\ge 5$ then in any word $w$ $k$-representing $H_{n,n}$ the set $A=\{ 1,\ldots, n\}$ is splittable. \end{lemma}
\begin{proof} By Proposition~\ref{propX}, the set $B:=N(1')=\{2,\ldots, n\}$ is splittable, i.e. there is a cyclic shift of $w$ in which the letters of $B$ form the subword $P'_1\cdots P'_k$, where $P'_i$ is a permutation of $B$. Let a canonical shift of $w$ with respect to $B$ be $P_1I_1\cdots P_kI_k$, where for $i=1,\ldots, k$, the factor $P_i$ begins at $l(P'_i)$ and ends at $r(P'_i)$, and $I_i$s are (possibly empty) factors lying between $r(P'_i)$ and $l(P'_{i+1})$. We begin by proving the following fact. \\
\noindent
{\bf Claim 1.} For every $t\geq 1$ and $i\geq 1$ such that $i+t-1\leq k$, the factor $U=P_iI_i\cdots I_{i+t-2}P_{i+t-1}$ of $w$ contains at most $t$ copies of the letter~1. \\
\noindent
{\bf Proof of Claim 1.} Indeed, suppose not. Using a cyclic shift of $w$ if necessary, without loss of generality we can assume that in a problematic case $i=1$. First consider the case $t=1$. That is, we assume that $P_1$ contains at least two 1s.
Let $a=l(P_1), b= r(P_1)$ and $x\in V\setminus \{ 1,a,b,1',2',\ldots,n'\}$. Recall that $a,b$ belong to the splittable set $B$. Then the letter $x'$ occurs exactly once between any two consecutive occurrences of 1, in particular, between the first two occurrences. Hence we have $a_1<x'_1<b_1$. Since both $a$ and $b$ are adjacent to $x'$, it follows from Lemma~\ref{lem1} that $ab\in E$, contradiction.
Now let $t\ge 2$. Let $a=l(P_1), b= r(P_1), c=r(P_t)$ and $x\in V\setminus \{ 1,a,b,c,1',2',\ldots,n'\}$ (recall that $n\ge 5$), and suppose that there are at least $t+1$ occurrences of 1 between $a$ and $c$. Note that $a\neq b$, but it is possible that $a=c$ or $a=b$. Since $1x'\in E$, there must be at least $t$ occurrences of $x'$ between $a$ and $c$. By Lemma~\ref{lem1}, no $x'$ can appear between $a_1$ and $b_1$. However, $c$ appears exactly once between $a_1$ and $b_1$ (possibly coinciding with one of them) because $P_1$ contains the permutation $P'_1$ over $B$ as a subword. Moreover, there are exactly $t$ occurrences of $c$ in $U$.
Therefore, the subword of $U$ induced by $c$ and $x'$
starts and ends with $c$ and contains at least $t$ copies of $x'$. Clearly, such subword cannot be alternating, which contradicts $cx'\in E$. Claim~1 is proved. \qed \\
It follows from Claim 1 that each $P_iI_i$ contains at most two 1s, since $P_iI_iP_{i+1}$ contains at most two 1s for $1\leq i\leq k-1$. If each of $P_iI_i$ contains exactly one 1 then add 1 to each $P'_i$
to obtain the concatenation of permutations for the set $A$ showing that it is splittable.
Otherwise, some $P_iI_i$ must contain at least two 1s. Without loss of generality, $i=1$
(otherwise, we can apply a cyclic shift and rename the permutations). By Claim 1 applied to $P_1$ and $P_1I_1P_2$, at least one of 1s must be in $I_1$ and $P_2$ contains no 1s. So, add the first occurrence of 1 to $P'_1$ and the second one to $P'_2$. If $I_2$ contains no 1s we apply the same arguments to the word obtained from $w$ by removing the factor $P_1I_1P_2I_2$. Otherwise, again by Claim 1 applied to $P_1I_1P_2I_2P_3$, $I_2$ has one 1, $P_3$ has no 1 and we add this 1 to $P'_3$ and continue in the same way showing that $w$ contains as a subword a concatenation of permutations over $A$, and thus $A$ is splittabe. \end{proof}
\begin{lemma}\label{lem3} Let $n\ge 5$ and $w$ $k$-represents $H_{n,n}$. Also, let $P'_1\cdots P'_k$ be a subword of (a cyclic shift of) $w$ that is a concatenation of permutations over $A=\{ 1,\ldots, n\}$ (existing by Lemma~\ref{lem2}). Then for every $a\in A$ there is $j\in \{1,\ldots, k\}$ such that $a=l(P'_j)$ or $a=r(P'_j)$.\end{lemma}
\begin{proof}
Assume that the letter 1 is never the first or the last letter of any permutation $P'_j$. Consider a canonical shift of $w$ for the set $A$ and define the subwords $P_i$ and $I_i$ for permutations $P'_i$ in the same way as in the proof of Lemma~\ref{lem2}. Since $l(P_1)\ne 1$ and $r(P_1)\ne 1$, no $1'$ can appear between
$l(P_1)$ and $r(P_1)$ by Lemma~\ref{lem1}. This is true for any $P_i$ since we can apply a cyclic shift and rename $P_i$ and $P_1$. Moreover, no $I_i$ can have two or more $1'$s, or no $1'$s at all, because otherwise $1'$ would not be adjacent to the vertices in $\{2,\ldots,n\}$.
But since each $P_j$ for $j=1,\ldots, k$ contains one $1$, the letters 1 and $1'$ alternate in $w$, i.e. the vertices 1 and $1'$ must be adjacent, contradicting the definition of $H_{n,n}$. \end{proof}
\begin{theorem}\label{thm1} For $n\geq 1$, the representation number of $H_{n,n}$ is at least $\lceil n/2 \rceil$.
\end{theorem}
\begin{proof} We consider three cases.
\begin{itemize}
\item The statement is trivial for $n=1,2$ since each graph requires at least one copy of each letter to be represented.
\item None of $H_{n,n}$'s is a complete graph, and thus, by Observation~\ref{propY}, the statement is true for $n=3,4$.
\item Let $n\ge 5$. Since the set $A=\{ 1,\ldots, n\}$ is splittable by Lemma~\ref{lem2}, and each of its $n$ letters must be the first or the last letter of some permutation $P'_j$ for $j=1,\ldots,k$ by Lemma~\ref{lem3}, we have the inequality $2k\ge n$. Since $k$ is an integer, we obtain the bound $k\ge \lceil n/2 \rceil$.
\end{itemize} \end{proof}
\section{An upper bound for the representation number of $H_{n,n}$}\label{sec-upper}
In this section we provide a construction that shows that the bound in Theorem~\ref{thm1} is tight for all $n$ except $n=1,2,4$. We need the following auxiliary fact.
\begin{lemma}\label{lem4} If $n=2k\ge 6$ then for every partition of the set $A=\{ 1,\ldots, 2k\}$ into $k$ pairs $(a_1,b_1),\ldots, (a_k,b_k)$ there exist permutations $P(a_1,b_1), \ldots, P(a_k,b_k)$ such that:
\begin{enumerate}
\item $l(P(a_i,b_i))=a_i, r(P(a_i,b_i))=b_i$ for each $i=1,\ldots, k$, and
\item For every $x,y\in A$ there are $i,j$ such that $x<y$ in $P(a_i,b_i)$ and $y<x$ in $P(a_j,b_j)$.
\end{enumerate}
\end{lemma}
\begin{proof} Let $P$ be an arbitrary permutation over the set $A\setminus \{a_1,a_2, b_1, b_2\}$, $Rev(P)$ be obtained from $P$ by writing it in the reverse order, $P'$ be an arbitrary permutation over the set $A\setminus \{a_1,a_2, a_3,b_1, b_2,b_3\}$ and for each $i=4,\ldots, k$ let $P_i$ be an arbitrary permutation over the set $A\setminus \{a_i, b_i\}$. Define the sought permutations as follows: $P(a_1,b_1)=a_1b_2Pa_2b_1,\ P(a_2,b_2)=a_2b_1Rev(P)a_1b_2,\ P(a_3,b_3)=a_3b_2a_1P'b_1a_2b_3$ and $P(a_i,b_i)=a_iP_ib_i$ for each $i=4,\ldots, k$. It is straightforward to verify that both requirements of the lemma hold for these permutations. \end{proof}
Note that for $n\in \{2,4\}$ Lemma \ref{lem4} is not true.
\begin{theorem}\label{thm2} If $n\ge 5$ then the crown graph $H_{n,n}$ is $\lceil n/2 \rceil$-representable. \end{theorem}
\begin{proof} It is sufficient to prove the theorem only for $n=2k$, $k\geq 3$, because the case of $n=2k-1$ is obtained from the case of $n=2k$ by removing all occurrences of the letters $2k$ and $(2k)'$ from the respective word. First, consider the following $k$-uniform word, where the permutations $P(x,y)$s are defined in Lemma~\ref{lem4} and $P(x',y')$s are obtained from these by adding primes.
$$w'=P(1,2)P(2',3')P(3,4)P(4',5')\cdots P(n-1,n)P(n',1').$$ It follows from the property 2 in Lemma~\ref{lem4} that $w'$ represents the complete bipartite graph $K_{n,n}$. Shift $w'$ cyclicly one position to the left to obtain the word $w''$ where for every even $i$ there is exactly one occurrence of the factor $ii'$ and for every odd $i$ there is exactly one occurrence of the factor $i'i$. Let $w$ be the word obtained from $w''$ by switching $i$ and $i'$ in each of these factors. This operation makes the subword induced by $i$ and $i'$ non-alternating (thus removing the edges $ii'$ in $K_{n,n}$) but does not affect any other alternations in the word. Therefore, $w$ $k$-represents $H_{n,n}$, as desired.\end{proof}
Note that for $n<4$ the graph $H_{n,n}$ is 2-word-representable, which is given by the words $w_1=11'1'1$, $w_2= 12'21'21'12'$ and $w_3=12'3'123'1'231'2'3$, respectively (see pages 172 and 173 in \cite{KL}). As for $n=4$, note that $H_{4,4}$ is the three-dimensional cube, which is the prism graph Pr$_4$. Thus, $H_{4,4}$ is 3-word-representable by Proposition 15 in \cite{KP} and it is not 2-word-representable by Theorem 18 in \cite{K}. An example of 3-representation of $H_{4,4}$ given on page 90 in \cite{KL} is
$$414'343'231'12'24'1'3'44'2'33'11'22'.$$
\section{Concluding remarks}\label{conclusion}
In this paper we found the representation number of any crown graph solving at once two open problems in \cite{KL}. We suspect that crown graphs are (among) the hardest graphs to be represented in the class of bipartite graphs in the sense that they require longest words for representation. It would be interesting to prove or disprove this fact. In either case, one should be able to apply the methods used in this paper, in particular, the notion of a splittable set, to provide a complete classification for the representation number of any bipartite graph.
\end{document} |
\begin{document}
\title{AN ANALOGUE OF THE L\'EVY-CRAM\'ER THEOREM FOR RAYLEIGH DISTRIBUTIONS }
\author{Thu Van Nguyen}
\address{Department of Mathematics;
International University, HCM City;
No.6 Linh Trung ward, Thu Duc District, HCM City;
Email: [email protected]}
\date{June 30, 2009}
\begin{abstract} In the present paper we prove that every k-dimensional Cartesian product of Kingman convolutions can be embedded into a k-dimensional symmetric convolution (k=1, 2, \ldots) and obtain an analogue of the Cram\'er-L\'evy theorem for multi-dimensional Rayleigh distributions.
\end{abstract}\indent
\maketitle{Keywords and phrases: Cartesian products of Kingman convolutions; Rayleigh distributions; radial characteristic functions;
AMS2000 subject classification: 60B07, 60B11, 60B15, 60K99.}
\section{Introduction, Notations and Preliminaries}\label{S:intro}
In probability theory and statistics, the {\bf Rayleigh distribution} is a continuous probability distribution which is widely used to model events that occur in different fields such as medicine, social and natural sciences. A multivariate Rayleigh distribution is the probability distribution of a vector of norms of random Gaussian vectors.
The purpose of this paper, is to introduce and study the fractional indexes multivariate Rayleigh distributions via the Cartesian product of Kingman convolutions and, in particular, to prove an analogue of the L\'evy-Cram\'er theorem for multivariate Rayleigh distributions.
Let $\mathcal P:=\mathcal P(\mathbb R^+)$ denote the set of all probability measures (p.m.'s) on the positive half-line $\mathbb R^+$. Put, for each continuous bounded function f on $\mathbb R^{+}$,
\begin{multline}\label{astKi}
\int_{0}^{\infty}f(x)\mu\ast_{1,\delta}\nu(dx)=\frac{\Gamma(s+1)}{\sqrt{\pi}\Gamma(s+\frac{1}{2})}\\
\int_{0}^{\infty}\int_{0}^{\infty}\int_{-1}^{1}f((x^2+2uxy+y^2)^{1/2})(1-u^2)^{s-1/2}\mu(dx)\nu(dy)du,
\end{multline}
where $\mu\mbox{ and }\nu\in\mathcal P\mbox{ and }\delta=2(s+1)\geq1$ (cf. Kingman \cite{Ki} and
Urbanik \cite{U1}). The convolution algebra $(\mathcal{P},\ast_{1,\delta})$ is
the most important example of Urbanik convolution algebras (cf Urbanik \cite{U1}). In language of the
Urbanik convolution algebras, the {\it characteristic measure}, say $\sigma_s$, of the Kingman convolution
has the Rayleigh density
\begin{equation}\label{Ray}
d\sigma_s(y)= \frac{2{(s+1)^{s+1}}}{\Gamma(s+1)}y^{2s+1}\exp{(-(s+1)y^2)}dy
\end{equation}
with the characteristic exponent $\varkappa=2$ and the kernel
$\Lambda_s$
\begin{equation}\label{eq:Lam}
\Lambda_s(x)= \Gamma(s+1) J_{s}(x)/(1/2x)^{s},
\end{equation}
where $J_s(x)$ denotes the Bessel function of the first kind,
\begin{equation}\label{eq:Bessel}
J_s(x):= \Sigma_{k=0}^{\infty} \frac{(-1)^k
(x/2)^{\nu+2k}}{k!\Gamma(\nu+k+1)}.
\end{equation}
It is known (cf. Kingman \cite{Ki}, Theorem 1), that the kernel $\Lambda_s$ itself is an
ordinary characteristic function (ch.f.) of a symmetric p.m., say $F_s$, defined on the
interval [-1,1]. Thus, if $\theta_s$ denotes a random variable (r.v.) with distribution
$F_s$ then for each $t\in \mathbb R^+$,
\begin{equation}\label{eq:LamThe}
\Lambda_s(t)= E\exp{(it\theta_s)}=\int_{-1}^1\cos{(tx)}dF_s(x).\end{equation}
Suppose that $X$ is a nonnegative r.v. with distribution $\mu\in\mathcal{P}$
and $X$ is independent of $\theta_s$. The {\it radial characteristic function}
(rad.ch.f.) of $\mu$, denoted by $\hat\mu(t),$ is defined by
\begin{equation}\label{ra.ch.f.}
\hat\mu(t) = E\exp{(itX\theta_s)} = \int_0^{\infty}
\Lambda_s(tx)\mu(dx),
\end{equation}
for every $t\in \mathbb R^{+}$.
The characteristic measure of the Kingman convolution $\ast_{1, \delta}$, denoted by $\sigma_s$,
has the Maxwell density function
\begin{equation}\label{Maxwell density}
\frac{d\sigma_s(x)}{dx}=\frac{2(s+1)^{s+1}}{\Gamma(s+1)}x^{2s+1}exp\{-(s+1)x^2\}, \quad(0<x<\infty).
\end{equation}
and the rad.ch.f.
\begin{equation}
\hat\sigma_s(t)=exp\{-t^2/4(s+1)\}.
\end{equation}
Let $\tilde P:=\tilde{\mathcal P}(\mathbb R^+)$ denote the class of symmetric p.m.'s on $\mathbb R^+.$ Putting, for every $G\in \mathcal P$,
\begin{equation*}
F_s(G)=\int_0^{\infty}F_{cs} G(dc),
\end{equation*}
we get a continuous homeomorphism from the Kingman convolution algebra $(\mathcal{P},\ast_{1,\delta})$ onto the ordinary convolution algebra $(\tilde{\mathcal P}, \ast)$ such that
\begin{eqnarray}\label{homeomorphism1}
F_s\{G_1\ast_{1, \delta}G_2\}&=&(F_sG_1)\ast(F_sG_2) \qquad G_1, G_2\in \mathcal P\\
F_s\sigma_s&=&N(0, 2s+1)
\end{eqnarray}
which shows that every Kingman convolution can be embedded into the ordinary convolution $\ast$.
\section{Cartesian product of Kingman convolutions}
Denote by $ \mathbb {R}^{+k}, k=1,2,...$ the k-dimensional nonnegative cone
of $ \mathbb {R}^{k}$ and $\mathcal{P}(\mathbb {\mathbb R}^{+k})$ the class of
all p.m.'s on $\mathbb {\mathbb R}^{+k}$ equipped with the weak convergence. In
the sequel, we will denote the multidimensional vectors and random vectors (r.vec.'s)
and their distributions by bold face letters.
For each point z of any set $A$ let $\delta_z$ denote the Dirac measure (the unit mass) at
the point z. In particular, if $\mathbf x=(x_1, x_2,\cdots,x_k)\in
\mathbb R^{k+}$, then
\begin{equation}\label{proddelta}
\delta_{\mathbf {x}} =
\delta_{x_1}\times\delta_{x_2}\times \ldots\times\delta_{x_k},\quad (k\; times),
\end{equation}
where the sign $"\times"$ denotes the Cartesian product of
measures.
We put, for $\mathbf {x} = (x_1,\cdots, x_k)\mbox{ and }\mathbf {y} =
(y_1,y_2,\cdots, y_k)\in \mathbb R^{+k},$
\begin{equation}\label{convdeltas}\delta_{\mathbf x}\bigcirc_{s, k} \delta_{\mathbf {y}} = \{\delta_{x_1}\circ _s \delta_{y_1}\} \times\{\delta_{x_2} \circ _s\delta_{y_2}\}
\times\cdots\
\times \{\delta_{x_k} \circ_s \delta_{y_k}\},\quad (k\; times),
\end{equation}
here and somewhere below for the sake of simplicity we denote the
Kingman convolution operation $\ast_{1,\delta}, \delta=2(s+1)\ge 1$ simply by $\circ_{s}, s\ge \frac{!}{2}.$
Since convex combinations of p.m.'s of the form
(\ref{proddelta}) are dense in $\mathcal P(\mathbb R^{+k})$ the
relation (\ref{convdeltas}) can be extended to arbitrary p.m.'s $
\mathbf{G}_1 \mbox{ and } \mathbf{G}_2\in\mathcal{P}( \mathbb R^{+k})$.
Namely, we put
\begin{equation}\label{convF}
\mathbf {G}_1 \bigcirc_{s, k} \mathbf {G}_2 = \iint\limits_{ \mathbb R^{+k}}
\delta_{\mathbf {x}} \bigcirc_{s, k} \delta_{\mathbf {y}} {\mathbf G}_1(d\mathbf {x}) {\mathbf G}_2(d\mathbf {y})
\end{equation} which means that for each continuous bounded function $\phi$ defined on $\mathbb R^{+k}$
\begin{equation}\label{convof}
\int\limits_{\mathbb R^{+k}} \phi({\mathbf z}) {\mathbf G}_1 \bigcirc_{s, k} {\mathbf G}_2 (d{\mathbf z})= \iint\limits_{ \mathbb R^{+k}}\big\{\int\limits_{\mathbb R^{+k}} \phi({\mathbf z}) \delta_{{\mathbf x}} \bigcirc_{s, k} \delta_{{\mathbf y}}(d{\mathbf z})\big\}{ \mathbf G}_1(d{\mathbf x}) {\mathbf G}_2(d{\mathbf y}).
\end{equation}
In the sequel, the binary
operation $\bigcirc_{s, k}$ will be called {\it the k-times Cartesian
product of Kingman convolutions} and the pair $(\mathcal P( \mathbb R^{+k}), \bigcirc_{s, k})$ will be called
{\it the k-dimensional Kingman convolution algebra}. It is easy to show that the
binary operation $\bigcirc_{s, k}$ is continuous in the weak topology
which together with (\ref{astKi}) and (\ref{convF}) implies the
following theorem.
\begin{theorem}\label{Theo:Kingmanalgebra} The pair $(\mathcal P{( \mathbb R^{+k})} ,\bigcirc_{s, k})$
is a commutative
topological semigroup with $\delta_{\mathbf 0}$ as the unit element.
Moreover, the operation $\bigcirc_{s, k}$ is distributive w.r.t.
convex combinations of p.m.'s in $\mathcal P( \mathbb R^{+k})$.
\end{theorem}
\ For every ${\mathbf G}\in\mathcal P( \mathbb R^{+k})$ the
k-dimensional rad.ch.f. $\hat{{\mathbf G}}({\mathbf t}), {\mathbf
t}=(t_1, t_2, \cdots t_k)\in \mathbb R^{+k},$ is defined by
\begin{equation}\label{k-ra.ch.f.}
\hat{\mathbf G}(\mathbf {t})=\int\limits_{\mathbb R^{+k}}
\prod_{j=1}^{k}\Lambda_s(t_jx_j){ \mathbf G}(\mathbf {dx}),
\end{equation}
where $\mathbf {x}=(x_1, x_2, \cdots x_k)\in \mathbb R^{+k}.$
Let $\mathbf{\Theta_s} = \{\theta_{s, 1},\theta_{s, 2}, \cdots ,\theta_{s, k}\}$, where $\theta_{s, j}$ are independent r.v.'s with the same distribution
$F_s $.
Next, assume that $ {\mathbf X}=\{X_1, X_2,..., X_k\}$ is a k-dimensional
r.vec. with distribution $\mathbf{G}$ and $\mathbf{X}$ is independent of
$\mathbf{\Theta}_s$. We put
\begin{equation}\label{[Theta,X]}
[{\mathbf\Theta}_s,{\mathbf X}]=\{{\theta_{s,1} X_1, \theta_{s, 2} X_2,...,\theta_{s, k}X_k}\}.
\end{equation}
Then, the following formula is equivalent to (\ref{k-ra.ch.f.}) (cf. \cite{Ng4})
\begin{equation}\label{multiradchf}
\widehat{\mathbf G}({\mathbf t})=Ee^{i<{\mathbf t},[{\mathbf\Theta_s, \mathbf X}]>},\qquad {\mathbf t}\in \mathbb R^{+k}.
\end{equation}
The Reader is referred to Corollary 2.1, Theorems 2.3 \& 2.4 \cite{Ng4} for the principal properties of the above rad.ch.f.
Given $s\ge -1/2$ define a map $F_{s, k}: \mathcal P(\mathbb R^{+k}) \rightarrow \mathcal P(\mathbb R^k)$ by
\begin{equation}\label{k-map}
F_{s, k}({\mathbf G})=\int\limits_{\mathbb R^{+k}} (T_{c_1}F_s)\times(T_{c_2}F_s)\times \ldots\times(T_{c_k}F_s) {\mathbf G}(d{\mathbf c}),
\end{equation}
here and in the sequel, for a distribution $\mathbf G$ of a r.vec. $\mathbf X$ and a real number c we denote by $T_c{\mathbf G}$ the distribution of $c{\mathbf X}$.
Let us denote by $\tilde{ \mathcal P}_{s, k}(\mathbb{R}^{+k})$ the sub-class of $\mathcal P(\mathbb R^k)$ consisted of all p.m.'s defined by the right-hand side of (\ref{k-map}).
By virtue of (\ref{k-ra.ch.f.})-(\ref{k-map}) it is easy to prove the following theorem.
\begin{theorem}\label{symmconvo}
The set $\tilde{ \mathcal P}_{s, k}(\mathbb{R}^{+k})$ is closed w.r.t. the weak convergence and the ordinary convolution $\big.\ast$ and the following equation holds
\begin{equation}\label{Fourier=rad.ch.f.}
\hat{\mathbf G}({\mathbf t})=\mathcal F(F_{s, k}({\mathbf G}))({\mathbf t})\qquad {\mathbf t}\in {\mathbb R^{+k}}
\end{equation}
where $\mathcal F({\mathbf K})$ denotes the ordinary characteristic function (Fourier transform)
of a p.m. ${\mathbf K}$. Therefore, for any ${\mathbf G}_1\mbox{ and } {\mathbf G}_2\in \mathbb R^{+k}$
\begin{equation}\label{convolequality}
F_{s, k}({\mathbf G}_1)\big.\ast F_{s, k}({\mathbf G}_2)=F_{s, k}({\mathbf G}_1\bigcirc_{s, k}{\mathbf G}_2)
\end{equation}
and the map $F_{s, k}$ commutes with convex combinations of distributions and scale changes
$T_c, c>0$. Moreover,
\begin{equation}\label{Gaussian-Rayleigh}
F_{s, k}({\Sigma_{s, k}})=N({\mathbf 0}, 2(s+1){\mathbf I})
\end{equation}
where $\Sigma_{s, k}$ denotes the k-dimensional Rayleigh distribution and $N({\mathbf 0}, 2(s+1){\mathbf I})
$ is the symmetric normal distribution on $\mathbb R^k \mbox{ with variance operator } R= 2(s+1)
{\mathbf I}, {\mathbf I}$ being the identity operator.
Consequently, Every Kingman convolution algebra $\big( \mathcal P(\mathbb R^{+k}), \bigcirc_{s, k}\big)$ is
embedded in the ordinary convolution algebra $\big( \mathcal P_{s, k}(\mathbb{R}^{+k}), \big.\star \big)$ and the map $F_{s, k}$ stands for a homeomorphism.
\end{theorem}
\begin{proof}
First we prove the equation (\ref{Fourier=rad.ch.f.}) by taking the Fourier transform of the right-hand side of (\ref{k-map}). We have, for ${\mathbf t}\in \mathbb R^k,$
\begin{eqnarray}\label{Fourier-r.ch.f}
\mathcal F(F_{s, k}({\mathbf G}))({\mathbf t})&=&\notag
\int\limits_{\mathbb R^k}\Pi_{j=1}^k \cos(t_jx_j)H_{s, k}({\mathbf G})d{\mathbf x}\\
&=&\int\limits_{\mathbb R^k}\int_{\mathbb R^{+k}}
\Pi_{j=1}^k\cos(t_jx_j)(T_{c_j}F_s (d{\mathbf x}){\mathbf G}(d{\mathbf c})\\
&= &\int\limits_{\mathbb R^{+k}} \prod_{j=1}^{k}\Lambda_s(t_jc_j) {\mathbf G}(d{\mathbf c})\notag\\
&=& \hat{\mathbf G}({\mathbf t})\notag
\end{eqnarray}
which implies that the set set $\tilde{ \mathcal P}_{s, k}(\mathbb{R}^{+k})$ is closed w.r.t. the weak convergence and the ordinary convolution $\big.\ast$ and, moreover the equations (\ref{convolequality}) and (\ref{Gaussian-Rayleigh}) hold.
\end{proof}
\begin{definition}\label{k-ID}
A p.m. ${\mathbf F} \in \mathcal P(\mathbb R^{+k})$ is called $\bigcirc_{s, k}-$infinitely divisible
($\bigcirc_{s, k}-$ID), if for every m=1, 2, \ldots there exists $\mathbf F_m\in \mathbf P(\mathbb R^{+k})$ such that
\begin{equation}\label{kID}
{ \mathbf F}={\mathbf F}_m\bigcirc_{s, k} {\mathbf F}_m\bigcirc_{s, k}\ldots \bigcirc_{s, k}{\mathbf F}_m\quad (m\;times).
\end{equation}
\end{definition}
\begin{definition}\label{stability}
$\mathbf F$ is called stable, if for any positive
numbers a and b there exists a positive number c such that
\begin{equation}\label{k-stability}
T_a{\mathbf F}\;{\bigcirc_{s, k}}\;T_b{\mathbf F}=T_c{\mathbf F}
\end{equation}
\end{definition}
By virtue of Theorem \ref{symmconvo} it follows that the following theorem holds.
\begin{theorem}\label{equivdef}
A p.m. $\mathbf G\mbox{ is } \bigcirc_{s, k}-ID$, resp. stable if and only if
$H_{s, k}({\mathbf G})$ is ID, resp. stable, in the usual sense.
\end{theorem}
The following lemma will be used in the representation of $\bigcirc_{s, k}-ID, k\ge 2.$
\begin{lemma}\label{Bessellimittheorem}
(i) For every $t\ge 0$
\begin{equation}\label{Bessellimittheorem 1}
\lim_{x\rightarrow 0}\frac{1-\Lambda_s(tx)}{x^2}=
\lim_{x\rightarrow
0}\frac{1-Ee^{it\theta}}{x^2}=\frac{t^2}{2}.
\end{equation}
(ii) For any ${\mathbf x}=(x_0, x_1,\cdots ,x_k)\mbox{ and }{\mathbf t}=(t_0, t_1, \cdots, t_k)\in\mathbb R^{k+1}, k=1,2, ...$
\begin{equation}\label{bessellimittheorem2}
lim_{\rho\rightarrow 0}\frac{1-\prod_{r=0}^k \Lambda_s
(t_rx_r)}{\rho^2}=\frac{1}{2}\Sigma_{r=0}^k \lambda^2_r( Arg({\mathbf x}))t_r^2,
\end{equation}
where $\rho=||\mathbf x||, Arg({\mathbf x})=\frac{\mathbf x}{||\mathbf x||},\mbox{ and } \lambda_r( Arg({\mathbf x})), r=0,1, ...,k$ are given by
\begin{equation}\label{polarization}
\lambda_r( Arg({\mathbf x}))=
\begin{cases}\cos\phi & r=0,\\
\sin\phi\sin\phi_1\cdots
\sin\phi_{r-1}\cos\phi_{r} &1\le r\le k-2,\\
\sin\phi\sin\phi_1...\sin\phi_{k-2}\cos\psi & r={k-1},\\
\sin\phi\sin\phi_1
...\sin\phi_{k-2}\sin\psi & r=k,
\end{cases}
\end{equation}
where $0\le \psi, \phi, \phi_r\le\pi/2, r=1,2,...,k-2$ are angles
of $\mathbf{x}$ appearing its polar form.
\end{lemma}
The following theorem gives a representation of rad.ch.f.'s of $\bigcirc_{s, k}-$ID distributions
(see \cite{Ng4} ), Theorem 2.6 for the proof).
\begin{theorem}\label{LevyID} A p.m. $\mu\in ID(\bigcirc_{s, k})$ if and only if
there exist a $\sigma$-finite measure M (a L\'evy's measure) on
$ \mathbb R^{+k}$ with the property that $M({\mathbf 0})=0, {\mathbf M}$ is finite outside every neighborhood of ${\mathbf 0}$ and
\begin{equation}\label{integrable w. r. t. weight function}
\int_{\mathbb R^{+k}}\frac {\|{\mathbf x}\|^2} {1+\|{\mathbf x}\|^2}
{\mathbf M}(d{\mathbf x}) < \infty
\end{equation}
and for each ${\mathbf t}=(t_1,...,t_k)\in
\mathbb R^{+k}$
\begin{equation}\label{Levy-Kintchine for k-dim.rad. ch. f.}
-\log{\hat{\mu}({\mathbf t})}=\int_{\mathbb R^{+k}}\{1-\prod_{j=1}^{k}\Lambda_s(t_jx_j)\} \frac
{1+\|{\mathbf x}\|^2} {\|{\mathbf x}\|^2} M({\mathbf {dx}}),
\end{equation}
where, at the origin $\mathbf{0}$, the integrand on the right-hand
side of (\ref{Levy-Kintchine for k-dim.rad. ch. f.}) is assumed to
be
\begin{equation}\label{limiting integrand}
\Sigma_{j=1}^k \lambda^2_j t_j^2 = lim_{\|\mathbf
{x}\|\rightarrow 0 }\{1-\prod_{j=1}^k
\Lambda_s(t_jx_j)\} \frac {1+\|\mathbf x\|^2}
{\|\mathbf {x}\|^2}
\end{equation}
for nonnegative $\lambda_j, j=1, 2,...,k$ given by equations (\ref{polarization})
in Lemma \ref{Bessellimittheorem}.
In particular, if $ M=0, \mbox{ then } \mu $ becomes a Rayleighian distribution with
the rad.ch.f (see definition \ref{Rayleigh})
\begin{equation}\label{kRayleighian rad. ch. f.}
-\log{\hat{\mu}({\mathbf t})}=\frac{1}{2}\sum_{j=1}^k \lambda^2_j
t_j^2,\quad {\mathbf t}\in \mathbb R^{+k},
\end{equation}
for some nonnegative $\lambda_j, j=1,...,k.$
Moreover, the representation (\ref{Levy-Kintchine for k-dim.rad. ch.
f.}) is unique.
\end{theorem}
An immediate consequence of the above theorem is the following:
\begin{corollary}\label{Cor:Pair}
Each distribution $\mu\in ID(\bigcirc_{s, k})$ is uniquely determined by the pair $[\mathbf{M}, \pmb {\lambda}]$, where $\mathbf{M}$ is a Levy's measure
on $\mathbb R^{+k}$ such that $\mathbf{M}(\mathbf{0})=0,$ $\mathbf{M}$ is finite outsite every neighbourhood of $\mathbf{0}$ and the condition (\ref{integrable w. r. t. weight function})
is satisfied and $\pmb{\lambda}:=\{\lambda_1, \lambda_2,\cdots \lambda_k\}\in \mathbb R^{+k}$ is a vector of nonnegative numbers appearing in (\ref{kRayleighian rad. ch. f.}).
Consequently, one can write $\mu\equiv[\mathbf{M}, \pmb {\lambda}].$\\ \indent
In particular, if $\mathbf{M}$ is zero measure then $\mu=[\pmb{\lambda}]$ becomes a Rayleighian p.m. on $\mathbb R^{+k}$ as defined as follows:
\end{corollary}
\begin{definition}\label{Rayleigh}
A k-dimensional distribution, say $\pmb{\mathbf \Sigma}_{s, k}$, is called a {\it Rayleigh distribution}, if \begin{equation}\label{k-dimension Rayleigh}
\pmb{\mathbf \Sigma}_{s, k}=\sigma_s\times\sigma_s\times\cdots\times\sigma_s \quad
(k\;times).
\end{equation}
Further, a distribution ${\mathbf F}\in \mathcal P(\mathbb R^{+k})$ is called a {\it Rayleighian distribution} if there exist nonnegative numbers $\lambda_r,
r=1,2 \cdots k $ such that
\begin{equation}\label{k-dimensional rayleighian}
{ \mathbf F}=\{T_{\lambda_1}\sigma_s\}\times \{T_{\lambda_2}\sigma_s\}
\times\ldots \times\{T_{\lambda_k}\sigma_s\}.
\end{equation}
\end{definition}
\indent
It is evident that every Rayleigh distribution is a Rayleighian distribution. Moreover, every Rayleighian distribution is $\bigcirc_{s, k}-$ID. By virtue of (\ref{Maxwell density} ) and (\ref{k-dimension Rayleigh}) it follows that the k-dimensional Rayleigh density is given by
\begin{equation}\label{density k-dimension Rayleigh}
g({\mathbf x})=\Pi_{j=1}^k\frac{2^k(s+1)^{k(s+1)}}{\Gamma^k(s+1)}x_j^{2s+1}exp\{-(s+1)||{\mathbf x}||^2\},
\end{equation}
where ${\mathbf x}=(x_1, x_2,\ldots, x_k)\in \mathbb R^{+k}$ and the corresponding rad.ch.f. is given by
\begin{equation}
\hat\Sigma_{s, k}({\mathbf t})=Exp(-|{\mathbf t}|^2/4(s+1)),\quad {\mathbf t}\in \mathbb R^{+k}.
\end{equation}
Finally, the rad.ch.f. of a Rayleighian distribution $\mathbf F\mbox{ on } \mathbb R^{+k}$ is given by
\begin{equation}\label{rad.ch.rayleighian}
\hat{\mathbf F}({\mathbf t})=Exp(-\frac{1}{2}\sum_{j=1}^k\lambda_j^2t_j^2)
\end{equation}
where $\lambda_j, j=1, 2, \ldots, k$ are some nonnegative numbers.
\section{An analogue of the L\'evy-Cram\'er Theorem in multi-dimensional Kingman convolution algebras}
We say that a distribution ${\mathbf F \mbox{ on } \mathbb R^k}$ has dimension m, $1\le m \le k$,
if m is the dimension of the smallest hyper-plane which contains the support of $\mathbf F.$
The following theorem can be regarded as a version of the L\'evy-Cram\'er Theorem
for multi-dimensional Kingman convolution.The case k=1 was proved by Urbanik (\cite{U2}).
\begin{theorem}\label{Levy-Cramer}
Suppose that $\mathbf G_i \in \mathcal P(\mathbb R^{+k}), i=1, 2 $ and
\begin{equation}\label{decomposi}
\Sigma_{s, k}={\mathbf G}_1 \bigcirc_{s, k} {\mathbf G}_2.
\end{equation}
Then, ${\mathbf G}_i, i=1, 2$ are both Rayleighian distributions fufilling the condition that there exist nonnegative numbers $\lambda_{i, r}, i=1, 2\mbox{ and } r=1, 2,\ldots, k$
such that for each i=1, 2 the number of non-zero coefficients ${\lambda_{i, r}}'s$ among $\lambda_{i, 1}, \lambda_{i, 2},\ldots, \lambda_{i, k}$ are equal to the dimension of ${\mathbf G}_i,$ respectively. Moreover,
\begin{equation}
\lambda_{1, r}^2+\gamma_{2, r}^2=1,\qquad r=1, 2, ..., k
\end{equation}
and
\begin{equation}\label{form i}
{ \mathbf G}_i =T_{\lambda_{i, 1}}\sigma_s\times T_{\lambda_{i, 2}}\sigma_s\times \ldots\times T_{\lambda_{i, k}} \sigma_s
\end{equation}
\end{theorem}
\begin{proof}
Suppose that the equation (\ref{decomposi}) holds. Using the map $F_{s, k}$ we have
\begin{equation*}
F_{s, k}(\Sigma_{s, k})=F_{s, k}({\mathbf G}_1)\big.\ast F_{s, k}({\mathbf G}_2)
\end{equation*}
which, by virtue of (\ref{Gaussian-Rayleigh}), implies that
\begin{equation*}
N({\mathbf 0}, 2(s+1){\mathbf I})=F_{s, k}({\mathbf G}_1)\big.\ast F_{s, k}({\mathbf G}_2).
\end{equation*}
By the well-known L\'evy-Cram\'er Theorem on $\mathbb R^k$ (cf. Linnik and Ostrovskii \cite{LiOst}), that they are both symmetric Gaussian distributions on $\mathbb R^k.$ Consequently, they must be of the form
(\ref{form i}) and the coefficients ${\lambda_{i, r}}'s$ satisfy
the above stated conditions.
\end{proof}
\end{document} |
\begin{document}
\title{The chromatic polynomial for cycle graphs}
\author[J. Lee]{Jonghyeon Lee}
\address[Jonghyeon Lee]{
Department of Mathematics
\\Inha University
\\Incheon 22212, Korea}
\email{[email protected]}
\author[H. Shin]{Heesung Shin$^\dagger$}
\address[Heesung Shin]{
Department of Mathematics
\\Inha University
\\Incheon 22212, Korea}
\email{[email protected]}
\date{\today}
\thanks{$\dagger$ Corresponding author. This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2017R1C1B2008269).}
\begin{abstract}
Let $P(G,\lambda)$ denote the number of proper vertex colorings of $G$ with $\lambda$ colors.
The chromatic polynomial $P(C_n,\lambda)$ for the cycle graph $C_n$ is well-known as
$$P(C_n,\lambda) = (\lambda-1)^n+(-1)^n(\lambda-1)$$
for all positive integers $n\ge 1$.
Also its inductive proof is widely well-known by the \emph{deletion-contraction recurrence}.
In this paper, we give this inductive proof again and three other proofs of this formula of the chromatic polynomial for the cycle graph $C_n$.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:intro}
The number of proper colorings of a graph with finite colors was introduced only for planar graphs by George David Birkhoff \cite{Bir12} in 1912, in an attempt to prove the four color theorem, where the formula for this number was later called by the chromatic polynomial.
In 1932, Hassler Whitney \cite{Whi32} generalized Birkhoff's formula from the planar graphs to general graphs.
In 1968, Ronald Cedric Read \cite{Rea68} introduced the concept of chromatically equivalent graphs and asked which polynomials are the chromatic polynomials of some graph, that remains open.
\subsection*{Chromatic polynomial}
For a graph $G$, a \emph{coloring} means almost always a \emph{(proper) vertex coloring}, which is a labeling of vertices of $G$ with colors such that no two adjacent vertices have the same colors. Let $P(G,\lambda)$ denote the number of (proper) vertex colorings of $G$ with $\lambda$ colors and $\chi(G)$ the least number $\lambda$ satisfying $P(G, \lambda)>0$, where $P(G, \lambda)$ and $\chi(G)$ are called a \emph{chromatic polynomial} and \emph{chromatic number} of $G$, respectively.
In fact, it is clear that the number of $\lambda$-colorings is a polynomial in $\lambda$ from a deletion-contraction recurrence.
\begin{prop}[Deletion-contraction recurrence]
For a given a graph $G$ and an edge $e$ in $G$, we have
\begin{align}
P(G,\lambda) = P(G-e,\lambda) - P(G/e,\lambda),
\label{eq:rec}
\end{align}
where $G-e$ is a graph obtained by deletion the edge $e$
and $G/e$ is a graph obtained by contraction the edge $e$.
\end{prop}
\begin{eg}
The chromatic polynomials of graphs in Figure~\ref{fig:rec} are
\begin{align*}
P(G,\lambda) &=\lambda(\lambda-1)^2(\lambda-2),\\
P(G-e,\lambda)&=\lambda^2(\lambda-1)(\lambda-2), and\\
P(G/e,\lambda)&=\lambda(\lambda-1)(\lambda-2).
\end{align*}
It is confirmed that \eqref{eq:rec} is true for the graph $G$ and the edge $e$ in Figure~\ref{fig:rec}.
\end{eg}
\begin{figure}
\caption{$G$ , $G-e$ and $G/e$}
\label{fig:rec}
\end{figure}
\subsection*{Cycle graph}
A \emph{cycle graph $C_n$} is a graph that consists of a single cycle of length $n$, which could be drown by a $n$-polygonal graph in a plane.
The chromatic polynomial for cycle graph $C_n$ is well-known as follows.
\begin{thm}
\label{thm:main}
For a positive integer $n\ge1$, the chromatic polynomial for cycle graph $C_n$ is
\begin{align}
P(C_n,\lambda) = (\lambda-1)^n+(-1)^n(\lambda-1)
\label{eq:main}
\end{align}
\end{thm}
\begin{eg}
For an integer $n \le 3$, it is easily checked that the chromatic polynomials of $C_n$ are from \eqref{eq:main} as follows.
\begin{align*}
P(C_1, \lambda) &= (\lambda-1)+(-1)(\lambda-1) = 0 ,\\
P(C_2, \lambda) &= (\lambda-1)^2+(-1)^2(\lambda-1) = \lambda(\lambda-1),\\
P(C_3, \lambda) &= (\lambda-1)^3+(-1)^3(\lambda-1) = \lambda(\lambda-1)(\lambda-2).
\end{align*}
As shown in Figure~\ref{fig:cycle}, the cycle graph $C_1$ is a graph with one vertex and one loop and $C_1$ cannot be colored, that means $P(C_1, \lambda) = 0$.
The cycle graph $C_2$ is a graph with two vertices, where two edges between two vertices,
and $C_2$ can have colorings by assigning two vertices with different colors, that means $P(C_2, \lambda) = \lambda(\lambda-1)$.
The cycle graph $C_3$ is drawn by a triangle and $C_3$ can have colorings by assigning all three vertices with different colors, that means $P(C_3, \lambda) = \lambda(\lambda-1)(\lambda-2)$.
\begin{figure}
\caption{$C_n$ $(1 \leq n \leq 5)$}
\label{fig:cycle}
\end{figure}
\end{eg}
\section{Four proofs of Theorem~\ref{thm:main}}
In this section, we show the formula~\eqref{eq:main} in four different ways.
\subsection{Inductive proof}
This inductive proof is widely well-known.
A \emph{path graph $P_n$} is a connected graph in which $n-1$ edges connect $n$ vertices of vertex degree at most $2$, which could be drawn on a single straight line.
The chromatic polynomial for path graph $P_n$ is easily obtained by coloring all vertices $v_1, \dots, v_n$ where $v_i$ and $v_{i+1}$ have different colors for $i=1,
\dots, n-1$.
\begin{lem}
\label{lem:path}
For a positive integer $n\ge1$, the chromatic polynomial for path graph $P_n$ is
\begin{align}
P(P_n,\lambda)=\lambda(\lambda-1)^{n-1}.
\label{eq:path}
\end{align}
\end{lem}
We use an induction on the number $n$ of vertices by the deletion-contraction recurrence and the above lemma for path graph:
It is already shown that \eqref{eq:main} is true for $n\le3$ by the example in Section~\ref{sec:intro}.
Assume that \eqref{eq:main} is true for a positive integer $n$.
Using \eqref{eq:rec} and \eqref{eq:path}, we have
\begin{align*}
P(C_{n+1},\lambda)
&=P(C_{n+1}-e,\lambda)-P(C_{n+1}/e,\lambda)
\tag*{by \eqref{eq:rec}} \\
&=P(P_{n+1},\lambda)-P(C_n,\lambda) \\
&=\lambda(\lambda-1)^n-\left( (\lambda-1)^n+(-1)^n(\lambda-1) \right)
\tag*{by \eqref{eq:path}}\\
&=(\lambda-1)^{n+1}+(-1)^{n+1}(\lambda-1).
\end{align*}
\begin{figure}
\caption{$C_{n+1}
\end{figure}
Thus, \eqref{eq:main} is true for all positive integers $n\ge1$.
\subsection{Proof by inclusion-exclusion principle}
The \emph{inclusion-exclusion principle} is a technique of counting the size of the union of finite sets.
\begin{prop}[Inclusion-exclusion principle]
Let $A_1, A_2, \dots, A_n$ be subsets of a finite set $U$.
Then number of elements excluding their union is as follows
\begin{align*}
\abs{\bigcap_{i=1}^n {\overline{A_i}}}
&=\sum_{I \subset [n]} (-1)^{\abs{I}} \abs{\bigcap_{i\in I} A_i}\\
&=\abs{U} - \sum_{i=1}^{n} \abs{A_i} + \sum_{i<j} \abs{A_i \cap A_j} - \dots +(-1)^n \abs{A_1 \cap \cdots \cap A_n}
\end{align*}
where $\overline{A}$ is the complement of $A$ in $U$.
\end{prop}
Considering every condition to assign different colors to two adjacent vertices, for each edge $e$, we define a finite sets of arbitrary (including improper) colorings to assign same color to two adjacent vertices by the edge $e$.
Let $A_i$ be a set of colorings such that two vertices $v_i$ and $v_{i+1}$ are of same color, where $v_{n+1}$ is regarded as $v_1$.
Applying the inclusion-exclusion principle, we can write the following
\begin{align*}
P(C_n,\lambda)
&=\vert U \vert
- \sum_{i=1}^{n} \abs{A_i}
+ \sum_{i<j} \abs{A_i \cap A_j}
+ \cdots
+ (-1)^n \abs{A_1 \cap \dots \cap A_n} \\
&= \lambda^n
- \binom{n}{1} \lambda^{n-1}
+ \binom{n}{2} \lambda^{n-2}
+ \cdots
+ (-1)^{n-1} \binom{n}{n-1} \lambda
+ (-1)^n \lambda \\
&= (\lambda-1)^n - (-1)^n +(-1)^n \lambda \\
&= (\lambda-1)^n+(-1)^n(\lambda-1).
\end{align*}
Thus, \eqref{eq:main} is true for all positive integers $n\ge1$.
\subsection{Algebric proof}
Let us consider a case of $n=5$ and $\lambda=4$, that is, to assign the vertices of $C_5$ in four colors: red, blue, yellow, and green. Also let us consider a complete graph $K_4$ with vertex names red, blue, yellow, and green, see Figure~\ref{fig:one-one}.\\
\begin{figure}
\caption{A cycle graph $C_5$ and a graph $K_4$ with names of colors}
\label{fig:one-one}
\end{figure}
When red-blue-red-yellow-green is assigned in order from the vertex $v_1$ to the vertex $v_5$ in $C_5$, it is corresponding to a closed walk of length $5$ in $K_4$ which begins and ends at red, that is, it is red-blue-red-yellow-green-red in $K_4$.
By generalizing it, we have a correspondence between $\lambda$-colorings of $C_n$ and closed walks of length $n$ in $K_\lambda$. By this correspondence, it is enough to count the number of closed walks of length $n$ in $K_\lambda$, instead of the number of $\lambda$-colorings of $C_n$.
For a graph $G$ with vertex set $\set{v_1, \dots, v_n}$,
the \emph{adjacency matrix} of $G$ is an $n \times n$ square matrix $A$
such that its element $A_{ij}$ is one when there is an edge between two vertices $v_i$ and $v_j$, and zero when there is no edge between $v_i$ and $v_j$.
\begin{figure}
\caption{A graph $G$ and its adjacency matrix $A$}
\end{figure}
The following related to an adjacency matrix is well-known.
\begin{prop}
\label{prop:adj}
Let $A$ be the adjacency matrix of the graph $G$ on $n$ vertices $v_1, \dots, v_n$.
Then the $(i,j)$th entry of the matrix $A^n$ is
the number of the walk of length $n$ beginning at $v_i$ and ending at $v_j$.
\end{prop}
By Proposition~\ref{prop:adj}, we can calculate the number of closed walk of length $n$ in the complete graph $K_\lambda$: Let $A$ be an adjacency matrix of $K_\lambda$. Then $A$ is a $\lambda \times \lambda$ matrix as follows
\begin{align*}
A = \left( a_{ij} \right) =
\begin{pmatrix}
0 & 1 & \cdots & 1 & 1 \\
1 & 0 & \cdots & 1 & 1 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
1 & 1 & \cdots & 0 & 1 \\
1 & 1 & \cdots & 1 & 0
\end{pmatrix},
\end{align*}
where $a_{ij}=0$ if $i=j$, and otherwise $a_{ij}=1$.
So the number of closed walks of length $n$ in $K_\lambda$ is enumerated by $tr(A^n)$,
which equals the sum of all eigenvalues of $A^n$.
Also let all eigenvalues of the matrix $A$ be denoted by $u_1, \dots, u_\lambda$, then all eigenvalues of the matrix $A^n$ are $u_1^n, \dots, u_\lambda^n$.
\begin{align*}
A =
\begin{pmatrix}
0 & 1 & \cdots & 1 & 1 \\
1 & 0 & \cdots & 1 & 1 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
1 & 1 & \cdots & 0 & 1 \\
1 & 1 & \cdots & 1 & 0
\end{pmatrix}
\sim
\begin{pmatrix}
\lambda-1 & 0 & \cdots & 0 & 0 \\
0 & -1 & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & -1 & 0 \\
0 & 0 & \cdots & 0 & -1
\end{pmatrix},
\end{align*}
Since the matrix $A$ have $\lambda$ eigenvalues
$u_1 = \lambda-1$ and $u_2 =\dots=u_\lambda = -1$,
we have
\begin{align*}
tr(A^n) =\sum_{i=1}^{\lambda}u_i^n
= (\lambda-1)^n + \underbrace{(-1)^n + \dots + (-1)^n}_{\text{$\lambda-1$ times}}.
\end{align*}
Thus, \eqref{eq:main} is true for all positive integers $n\ge1$.
\subsection{Bijective proof}
Let $X_n$ denote the set of $\lambda$-colorings of $C_n$ and
$[\lambda-1]^n$ be the set of $n$-tuples of positive integers less than $\lambda$,
where $[\lambda-1]$ means $\set{1, \dots, \lambda-1}$.
We consider a mapping $\varphi$ from $\lambda$-colorings of $C_n$ in $X_n$
to $n$-tuples in $[\lambda-1]^n$.
\subsubsection*{A mapping $\varphi$ from $X_n$ to $[\lambda-1]^n$}
The mapping $\varphi:X_n \to [\lambda-1]^n$ is defined as follows:
Let $\omega$ be a $\lambda$-coloring of $C_n$ in $X_n$,
we write $\omega=(\omega_1, \dots, \omega_n)$ where $\omega_i$ is the color of $v_i$ in $C_n$
and it is obvious that $\omega_i \neq \omega_{i+1}$ for $1 \le i \le \lambda$, where $\omega_{n+1}$ is regarded as $\omega_1$.
An entry $\omega_i$ is called a \emph{cyclic descent} of $C$
if $\omega_i > \omega_{i+1}$ for $1\leq i\leq \lambda$.
Then we define $\varphi(\omega) = \sigma = (\sigma_1, \dots, \sigma_n)$ with
\begin{align*}
\sigma_i =
\begin{cases}
\omega_i - 1, & \mbox{\text{if $\omega_i$ is a cyclic descent}}\\
\omega_i, & \mbox{\text{otherwise}}.
\end{cases}
\end{align*}
Given a $\lambda$-coloring $\omega$,
if $\omega_i = \lambda$ then $\omega_{i+1} < \lambda$,
so $\omega_i=\lambda$ should be a cyclic descent.
Thus we have $\sigma_i<\lambda$ for all $1\le i \le n$
and $\varphi(\omega)$ belongs to $[\lambda-1]^n$.
For example, in a case of $n=9$ and $\lambda=4$,
$\omega=(1,2,1,3,2,3,1,4,2) \in X_9$ is given as an example of $4$-colorings of $C_9$.
Here $\omega_2=2$, $\omega_4=3$, $\omega_6 = 3$, $\omega_8=4$, and $\omega_9=2$ are cyclic descents of $\omega$. So we have
$$\varphi(\omega)= \sigma = (1,1,1,2,2,2,1,3,1) \in [3]^9.$$
\subsubsection*{A mapping $\psi$ as the inverse of $\varphi$}
Let $Z_n$ be the set of $n$-tuples $\sigma = (\sigma_1, \sigma_2, \dots, \sigma_n)$ in $[\lambda-1]^n$ with $$\sigma_1 = \sigma_2 = \dots = \sigma_n$$
and it is obvious that the size of $Z_n$ is $\lambda-1$.
We would like to describe a mapping
$\psi: \left( [\lambda-1]^n\setminus Z_n \right) \to X_n$
in order to satisfy $\varphi\circ\psi$ is the identity on $[\lambda-1]^n\setminus Z_n$ as follows:
Given a $\sigma \in [\lambda-1]^n\setminus Z_n$,
we define $\overline{\sigma}=(\overline{\sigma}_1, \dots, \overline{\sigma}_n)$ with
\begin{align*}
\overline{\sigma}_i =
\begin{cases}
\sigma_i + 1, & \mbox{\text{if $\sigma_i$ is a cyclic descent}}\\
\sigma_i, & \mbox{\text{otherwise}}.
\end{cases}
\end{align*}
Since $\overline{\sigma}$ may have consecutive same entries,
we define $\psi(\sigma) = \omega = (\omega_1, \dots, \omega_n)$
from $\overline\sigma$
with $\omega_i = \overline\sigma_i + 1$
for any entry $\overline\sigma_i$ of $\overline\sigma$
with a finite positive even integer $\ell$ satisfying
\begin{align*}
\overline\sigma_{i} = \overline\sigma_{i+1} = \dots = \overline\sigma_{i+\ell-1} \neq \overline\sigma_{i+\ell},
\end{align*}
where $\overline\sigma_{n+k}$ is regarded as $\overline{\sigma}_k$ for $1\le k \le n$,
and $\omega_i = \overline\sigma_i$, otherwise.
Thus $\omega$ has no consecutive same entries
and $1 \le \omega_i \le \lambda$ for all $1\le i \le n$,
so $\psi(\sigma)=\omega$ belongs to $X_n$.
Moreover, it is obvious that $\sigma_i \le \omega_i \le \sigma_i+1$ for all $1\le i \le n$
and if $\omega_i = \sigma_i + 1$ for some $1 \le i \le n$ then $\omega_i$ is a cyclic descent in $\omega$.
Hence $\varphi(\omega) = \sigma$ and $\sigma \in [\lambda-1]^n\setminus Z_n$
if and only if $\psi(\sigma)=\omega$.
In a previous example,
$\sigma = (1,1,1,2,2,2,1,3,1)$ is denoted as an example of $9$-tuples in $[3]^9$.
Here $\sigma_6 = 2$, $\sigma_8=3$ are cyclic descents of $\sigma$
and we obtain $\overline\sigma=(1,1,1,2,2,3,1,4,1)$.
And then there exist only three entries $\overline\sigma_2$, $\overline\sigma_4$, and $\overline\sigma_9$ in $\overline\sigma$ satisfying the following
\begin{align*}
k=2:& \quad \overline\sigma_2 = \overline\sigma_3 \neq \overline\sigma_4 \quad (\ell=2),\\
k=4:& \quad \overline\sigma_4 = \overline\sigma_5 \neq \overline\sigma_6 \quad (\ell=2), \text{ and }\\
k=9:& \quad \overline\sigma_9 = \overline\sigma_1 = \overline\sigma_2 = \overline\sigma_3 \neq \overline\sigma_4 \quad (\ell=4),
\end{align*}
so we get $\omega_2 = \overline\sigma_2 + 1 = 2$, $\omega_4= \overline\sigma_4 + 1 =3$, $\omega_9= \overline\sigma_9 + 1 =2$, and
$$\psi(\sigma)= \omega = (1,2,1,3,2,3,1,4,2) \in X_9.$$
Let $Y_n$ be the set of $\lambda$-colorings $\omega$ in $X_n$ with $\varphi(\omega) \in Z_n$.
Since two mapping $\varphi$ and $\psi$ are bijections between $X_n \setminus Y_n$ and $[\lambda-1]^n \setminus Z_n$,
the size of the set $X_n \setminus Y_n$ is same with the size of the $[\lambda-1]^n \setminus Z_n$, which is equal to $(\lambda-1)^n - (\lambda-1).$
When $n$ is even, for any $1\le i \le \lambda-1$,
there exist only two $n$-tuples in $X_n$
\begin{align*}
\omega &= (i+1, i, i+1, i, \dots, i+1, i) \quad \text{ and } \quad
\omega = (i, i+1, i, i+1, \dots, i, i+1)
\end{align*}
satisfying $\varphi(\omega) = (i, i, \dots, i) \in Z_n$.
If $n$ is even, the size of $Y_n$ is equal to $2(\lambda-1)$ and we obtain
\begin{align}
P(C_n,\lambda) &= \abs{X_n}
= \abs{X_n \setminus Y_n} + \abs{Y_n} \tag*{} \\
&= \left[ (\lambda-1)^n - (\lambda-1) \right] + 2(\lambda-1).
\label{eq:even}
\end{align}
When $n$ is odd,
there is no $n$-tuples satisfying $\varphi(\omega) \in Z_n$ and the set $Y_n$ is empty.
If $n$ is odd, we obtain
\begin{align}
P(C_n,\lambda) &= \abs{X_n}
= \abs{X_n \setminus Y_n} + \abs{Y_n} \tag*{} \\
&= \left[ (\lambda-1)^n - (\lambda-1) \right] + 0.
\label{eq:odd}
\end{align}
Therefore, \eqref{eq:main} yields from \eqref{eq:even} and \eqref{eq:odd} for all positive integers $n\ge1$.
\end{document} |
\begin{document}
\title{Network Formation under Random Attack and Probabilistic Spread}
\begin{abstract}
We study a network formation game where agents receive
benefits by forming connections to other agents but also incur
both direct and indirect costs from the formed connections.
Specifically, once the agents have
purchased their connections, an attack starts at a randomly
chosen vertex in the
network and spreads according to the independent cascade
model with a fixed probability,
destroying any infected agents. The utility or welfare of an agent
in our game is defined to be
the expected size of the agent's connected component
post-attack minus her expenditure in
forming connections.
Our goal is to understand the properties of the equilibrium
networks formed in this game.
Our first result concerns the edge density of equilibrium networks.
A network connection increases
both the likelihood of remaining connected to other agents after
an attack as well the likelihood of
getting infected by a cascading spread of infection. We show that
the latter concern primarily prevails and
any equilibrium network in our game contains only $O(n\log n)$
edges where $n$ denotes the number of
agents. On the other hand, there are equilibrium networks that
contain $\Omega(n)$ edges showing that our
edge density bound is tight up to a logarithmic factor.
Our second result shows that the presence of attack and its
spread through a cascade does not significantly
lower social welfare as long as the network is not too dense.
We show that any non-trivial equilibrium
network with $O(n)$ edges has $\Theta(n^2)$ social welfare,
asymptotically similar to the social welfare guarantee
in the game without any attacks.
\end{abstract}
{\bf s}ection{Introduction}
\label{sec:intro}
We study a network formation game where strategic
agents (vertices on a graph)
receive both benefits and costs from forming connections
to other agents. While
various benefit functions exist in the
literature~\ensuremath{c_{\textrm{e}}\xspace}ite{BalaG00, FabrikantLMPS03},
we focus on the \emph{reachability network benefit}.
Here, the benefit of an agent is
the size of her connected component in the collectively
formed graph. This models
settings where reachability (rather than centrality) motivates
joining the network, e.g.
when transmitting packets over technological networks
such as the Internet.
Most previous works feature a direct edge cost $\ensuremath{c_{\textrm{e}}\xspace} > 0$
for forming a
link.~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} depart from this notion by
studying a game
where forming links introduces an additional
\emph{indirect} cost by exposing agents
to contagious network shocks. These indirect
costs can model scenarios such
as virus spread through technological or biological
networks.
Our work continues this investigation of direct and
indirect connection costs.
To model the indirect cost we assume that, after
network formation, an adversary attacks a
single vertex uniformly at random. The attack then
kills the vertex and spreads through the
network via the independent cascade model
according to parameter $p$~\ensuremath{c_{\textrm{e}}\xspace}ite{KempeKT03}.
This random attack and probabilistic spread captures
the epidemiological quality of virus spread
in both biological and technological networks.
At a high level, our work is most closely related
to two previous works.~\ensuremath{c_{\textrm{e}}\xspace}itet{BalaG00} study a
reachability network game without attacks and show a
sharp characterization of equilibrium networks:
every tree and the empty network can form in
equilibria.~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} study a reachability
network formation game where an adversary
inspects the formed network and then deliberately
attacks a single vertex in the network. The attack
then spreads deterministically to neighboring vertices
according to a known rule, while agents may immunize
against the attack for a fixed cost. Our
game is most similar to the latter setting under a
random adversary and high immunization cost.
However, in our setting attacks spread probabilistically
(through independent cascades) rather
than deterministically. This yields an arguably more
realistic model of infection spread but incurs
additional complexity: computing the expected
connectivity benefit of an agent in a given network
is now \#P-complete~\ensuremath{c_{\textrm{e}}\xspace}ite{WangCW12}.
\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} show that while more diverse
equilibrium networks, including ones with multiple cycles,
can emerge in addition to trees and the empty
graph, the equilibrium networks with $n$ agents will have
at most $2n-4$ edges;
less than twice the number of edges that can form in the
equilibria of the attack-free game.
Furthermore, they show that
the social welfare is at least $n^2-o(n^{5/3})$ in non-trivial
equilibrium networks.~Asymptotically, this is the maximum welfare possible which is
achieved in any nonempty equilibrium of the attack-free game.
In the regime where the cost of immunization is high,
the game of \ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} only admits disconnected
and fragmented equilibrium networks due to deterministic
spread of the attack, and the social welfare of the
resulting networks may be as low as
$\Theta(n)$.
$\newline$
\noindent\textbf{Our Results and Techniques}
In our game, computing utilities or even verifying
network equilibrium is computationally hard.
We circumvent this difficulty by proving structural
properties for equilibrium networks. First,
we provide an upper bound on the edge density in equilibria.
\begin{thm}[Statement of Theorem~\ref{thm:density1}]
\label{thm:1}
Any equilibrium network on $n$ vertices has
$O\left(n\log n/p\right)$ edges.
\end{thm}
For constant $p$ this upper bound is tight up to a
logarithmic factor. The possibility of over-building
therefore differentiates our game from those
of~\ensuremath{c_{\textrm{e}}\xspace}itet{BalaG00} and~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16}, but the extent
of over-building is limited.
To prove Theorem~\ref{thm:1}, we first show
that any equilibrium network
with more than $\Omega(n\log(n/p))$ edges
contains an induced subgraph
with large minimum cut size. We then show that
if a network has large minimum cut
size, in \emph{every} attack (with high probability),
either almost all vertices in the network
will die or almost all vertices in the network will survive.
As a result, any vertex in the induced
subgraph can beneficially deviate by dropping an edge.
Together, these observations allows us to prove the claimed
edge density bound.
Next, we show that any equilibrium network that is
nontrivial (i.e. contains at least one edge)
also contains a large connected component.
Moreover, as long as the
network is not too dense, it achieves a constant
approximation to the best welfare
possible of the attack-free game.
\begin{thm}[Informal Statement of
Theorems~\ref{lem:largest}~and~\ref{thm:welfare}]
\label{thm:welf}
Any non-trivial equilibrium network over $n$
vertices contains a connected component
of size at least $n/3$. Furthermore, if the
number of edges in the network is
$O(n/p)$, then the social welfare is $\Omega(n^2)$.
\end{thm}
To prove Theorem~\ref{thm:welf}, we first show
that any agent in a small connected
component can increase her connectivity benefits
by purchasing an edge to a larger
component without significantly increasing her
attack risk. This implies the existence
of a large connected component. We then
use the large component to argue
that when the equilibrium network is sparse,
the surviving network post-attack still contains
a large connected component. This guarantees
large social welfare.
While~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} show robustness
of the structural properties of
the original reachability game
of~\ensuremath{c_{\textrm{e}}\xspace}itet{BalaG00} to a variation with attack,
deterministic spread and the option of
immunization for players, we show
robustness in another variant that involves a
cascading attack but disallows
immunization. However, on the technical front,
the tools that we use
to prove these robustness results are very
different from the analysis
of both of these previous games.
\noindent\textbf{Organization}
We introduce our model
and discuss the related work in Section~\ref{sec:model}.
In Section~\ref{sec:examples} we present examples of
equilibrium networks of our game.
Sections~\ref{sec:density}~and~\ref{sec:welfare} are devoted
to the characterization of the edge density and social welfare.
We conclude with directions for future work in Section~\ref{sec:future}.
{\bf s}ection{Model}
\label{sec:model}
We start by formalizing our model and borrow most of our
notation and terminology from~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16}. We assume the $n$ vertices of a graph
(network) correspond to individual players. Each player has the choice
to purchase edges to other players at a \emph{fixed} cost of $\ensuremath{c_{\textrm{e}}\xspace}>0$ per edge.
Throughout we assume that $\ensuremath{c_{\textrm{e}}\xspace}$ is a constant independent of $n$.
Furthermore, we use the term \emph{high probability} to refer to probability
at least $1-o(1/n)$ henceforth.
A (pure) \emph{strategy} $s_i{\bf s}ubseteq [n]$ for player $i$
consists of a subset of players to whom player $i$ purchased an edge.
We assume that edge purchases are unilateral i.e. players do not need
approval to purchase an edge to another player
but that the connectivity benefits and risks are bilateral.\footnote{As
an example of a scenario where the consequences are bilateral
even though the link formation is unilateral,
consider the spread of a disease in a social network where the links
are formed as a result of physical proximity of individuals.
The social benefits and potential risks of a contagious disease are
bilateral in this case although the link formation as a result of proximity
is unilateral. We leave the study of the bilateral edge formation
for future work.}
Let ${\bf s} = (s_1,\ldots,s_n)$ denote the strategy profile for all the
players. Fixing ${\bf s}$, the set of edges purchased by all the
players induces an undirected graph. We denote a \emph{game}
\emph{graph} as a graph $G$, where $G=(V,E)$ is the undirected
graph induced by the edge purchases of all players.
Fixing a game graph $G$, the adversary selects a \emph{single}
vertex $v\in V$ uniformly at random to start the attack.
The attack kills $v$ and then spreads according to the
independent cascade model
with probability $p\in(0,1)$~\ensuremath{c_{\textrm{e}}\xspace}ite{KempeKT03}.\footnote{Throughout we assume
that $p$ is a constant independent of the number of players $n$. We discuss
the regime in which $p$ decreases as the number of players increases
in Section~\ref{sec:edge-small-p}.}
In the independent cascade model, in the first round, the attack spreads independently
killing each of the neighbors of the initially attacked vertex $v$ with probability $p$.
In the next round, the spread continues from all the neighbors of $v$ that were killed in the
previous round.
The spread stops when no new vertex was killed in the last round or when all the vertices are killed.
The adversary's attack can be alternatively described as follows. Fixing a game graph $G$,
let $G[p]$ denote the \emph{random} graph
obtained by retaining each edge of $G$ independently with probability $p$.
The adversary picks a vertex $v$ uniformly at random to start the attack.
The attack kills $v$ and all the vertices in the connected component of
$G[p]$ that contains $v$.
Let $\ensuremath{CC\xspace}_i(v)$ denote the \emph{expected} size of the connected component of player $i$
post-attack to a vertex $v$ and we define $\ensuremath{CC\xspace}_v(v)$ to be $0$. Then the expected utility (utility for short) of
player $i$ in strategy profile ${\bf s}$ denoted by $u_i({\bf s})$ is precisely
\[
u_i({\bf s}) =\frac{1}{|V|}{\bf s}um_{v\in V}\ensuremath{CC\xspace}_i\left(v\right)-|s_i| \ensuremath{c_{\textrm{e}}\xspace}.
\]
We refer to the sum of utilities of all the
players playing a strategy profile ${\bf s}$ as the \emph{social welfare} of ${\bf s}$.
\ensuremath{c_{\textrm{e}}\xspace}itet{WangCW12} show that computing the exact spread of the attack in the independent cascade model
is \#P-complete in general. This implies that, given a strategy profile ${\bf s}$, computing the expected
size of the connected component of all vertices (and hence the expected utility of all vertices)
is \#P-complete. However, an approximation of these quantities can be obtained
by Monte Carlo simulation.
We model each of the $n$ players as
strategic agents who deterministically choose which edges to purchase.
A strategy profile ${\bf s}$ is a
\emph{pure strategy Nash equilibrium} if,
for any player $i$, fixing the behavior of the other players to be
${\bf s}m{i}$, the expected utility for $i$, $u_i({\bf s})$, cannot strictly increase when playing any strategy $s'_i$
over $s_i$. We focus our attention to pure strategy Nash equilibrium (or equilibrium) in this work.
Since computing the expected utilities in our game is \#P-complete, even verifying
that a strategy profile is an equilibrium is \#P-complete. Hence as our main contribution,
we prove structural properties for the equilibrium networks regardless of this computational barrier.
{\bf s}ubsection{Related Work}
\label{sec:related}
There are two lines of work closely related to ours. First,~\ensuremath{c_{\textrm{e}}\xspace}itet{BalaG00}
study the attack-free version of our game. They show that equilibrium networks are either
trees or the empty network. Also since there is no attack, the social welfare in nonempty equilibrium
networks is asymptotically $n^2-o(n^2)$.
Second,~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} study a network formation game where players in addition to having the option of
purchasing edges can also purchase immunization from the attack. Since we do not study the effect of immunization
purchases in our game, our game corresponds to the regime of parameters in their game
where the cost of immunization is so high that no vertex would purchase immunization in equilibria.
Moreover, they study several different adversarial attack models and our attack model coincides
with their \emph{random attack adversary}.
The main difference between our work and theirs is that they assume the attack spreads
deterministically while we assume the attack spreads according to the independent cascade
model~\ensuremath{c_{\textrm{e}}\xspace}ite{KempeKT03}. In many real world scenarios e.g. the spread of
contagious disease over the network of people, the spread is \emph{not deterministic}. Hence our
work can be seen as a first attempt to make the model of \ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} closer to real world
applications. However, the change in the spread of attack comes with a significant increase in the
complexity of the game as even computing the utilities of the players in our game is \#P-complete.
While \ensuremath{c_{\textrm{e}}\xspace}itet{FriedrichIKLNS17} have shown that best responses for players can be computed in polynomial
time under various attack models, the question of whether best response dynamics
converges to an equilibrium network is open in the model of
\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16}.
Similar to \ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} we show that diverse equilibrium networks
can form in our game. While they show that
all equilibrium networks over $n\ge 4$ players have at most $2n-4$ edges, we show
that the number of edges in any equilibrium network is at most $O(n\log n)$
and this bound is tight up to a logarithmic factor.
Furthermore, \ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} show that the social welfare is asymptotically $n^2-o(n^2)$
in non-trivial equilibrium networks. Their definition of non-trivial networks requires
the network to have at least one immunized vertex and one edge.
In the regime where the cost of immunization is high,
the game of \ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} only admits disconnected and fragmented
equilibrium networks due to the deterministic spread of the attack.
Such networks (even excluding the empty graph) can have social welfare
as low as $\Theta(n)$.
We show that any low density equilibrium network of our game
enjoys a social welfare of $\Theta(n^2)$ as long as the network contains at least one edge.
\ensuremath{c_{\textrm{e}}\xspace}itet{Kliemann11} introduced a network formation game with
reachability benefits and an attack on the formed network that destroys exactly one link
with no further spread. Their equilibrium networks are sparse and also admit high social
welfare as removing an edge can create at most two connected components.
\ensuremath{c_{\textrm{e}}\xspace}itet{KliemannSS17} extend this to allow attacks on vertices while focusing
on swapstable equilibria.
\ensuremath{c_{\textrm{e}}\xspace}itet{BlumeEKKT11} introduce a network formation game with bilateral
edge formation. They assume both edge and link failures can happen simultaneously
but independent of the failures so far in the network. These differences make it hard to
directly compare the two models. They show a tension between optimal and
stable networks and exploring such properties in depth in our model is an interesting
direction.
Finally, network formation games, with a variety of different connectivity benefit models,
have been studied extensively in computer science see e.g.~\ensuremath{c_{\textrm{e}}\xspace}ite{BalaG00, BlumeEKKT11, Kliemann11}.
We refer the reader to the related work section of~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} for a comprehensive
summary of other related work especially on the topic of optimal security choices for networks.
{\bf s}ection{Examples of Equilibrium Networks}
\label{sec:examples}
In this section we show that a diverse set of topologies can emerge in the equilibrium
of our game. Similar to the models of~\ensuremath{c_{\textrm{e}}\xspace}itet{BalaG00}~and~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} the empty graph can
form in the equilibrium of our game when $\ensuremath{c_{\textrm{e}}\xspace}\geq1$.
Moreover, similar to both models, trees can form in equilibria (See the left panel of Figure~\ref{fig:eq-examples}).
Finally, while~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16} show that in the regime
of their game where the cost of immunization is high (so no vertex would immunize)
no connected network can form in equilibria due to the deterministic spread of the attack,
we show that connected networks indeed can form in the equilibria of our game
(See Figure~\ref{fig:eq-examples}).
\begin{figure}\label{fig:eq-examples}
\end{figure}
We remark that pure strategy equilibria exist in all parameter regimes of our game.
When $\ensuremath{c_{\textrm{e}}\xspace} \geq 1$, the empty network can form in equilibria for all $p$. When $\ensuremath{c_{\textrm{e}}\xspace}<1$
a cycle or two disconnected hub-spoke structure of size $n/2$ can form in equilibria
depending on whether $p$ is far or close to 1 ($(1-\omega(1/n))$ and $(1-o(1))$,
respectively).
Examples in Figure~\ref{fig:eq-examples} show that denser networks can form in equilibria
compared to the model of~\ensuremath{c_{\textrm{e}}\xspace}itet{BalaG00} and the high immunization cost regime of the
model of~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16}. So it is natural to ask how dense equilibrium networks
can be. We study this question in Section~\ref{sec:density} and show an upper bound of
$O(n \log n)$ on the density of the equilibrium networks. Since the examples in Figure~\ref{fig:eq-examples}
have $\Theta(n)$ edges, our upper bound is tight up to a logarithmic factor.
Moreover, while all the
equilibrium networks in Figure~\ref{fig:eq-examples} are connected, there might still exist
equilibrium networks in our game that are highly disconnected. In Section~\ref{sec:welfare}
we show that any equilibrium network with at least one edge contains a large connected component.
However, even with the guarantee of a large connected component, there might still be
concerns that the equilibrium networks can become highly fragmented after the attack. In Section~\ref{sec:welfare}
we show that as long as the equilibrium network is not too dense, the social welfare is lower bounded
by $\Theta(n^2)$ i.e. a constant fraction of the social welfare achieved in the attack-free game.
We obtain these structural results even tough we cannot
compute utilities nor even verify that an equilibrium has reached
due to computational barriers. We view these results as are our
most significant technical contributions.
{\bf s}ection{Edge Density}
\label{sec:density}
We now analyze the edge density of equilibrium networks.
\begin{thm}
\label{thm:density1}
Any equilibrium network on $n$ vertices has
$O\left(n\log n/p\right)$ edges.
\end{thm}
The proof of Theorem~\ref{thm:density1} is due
to the following observations which
we formally state and prove next. At a high level,
we first show that if $G$ has large
enough edge density, then $G$ contains an induced
subgraph $H$ whose minimum cut size is large. We
then show a large minimum cut size implies that $H[p]$
is connected with high probability. This
means that in almost all attacks that infect a vertex in $H$,
all vertices in $H$ will get infected.
So a vertex in $H$ would have a beneficial deviation in
the form of dropping an edge; which
contradicts the assumption that $G$ was an equilibrium
network. This proves that equilibrium networks cannot be
too dense.
More formally,
we first show in Lemma~\ref{lem:cut} that if $G$ is \emph{dense enough}
it contains a subgraph $H$ with a
minimum cut size, denoted by $\alpha(H)$, of at least
$\Omega\left(\log n/p\right)$.
\begin{lemma}
\label{lem:cut}
Let $G=(V,E)$ be a graph on $n$ vertices. There exists a
constant $k$ such that if $|E| \geq k n \log n/p$
then $G$ contains an induced subgraph $H$
with $\alpha(H)\geq k\log n/p$.
\end{lemma}
\begin{proof}
If $\alpha(G)\geq k\log n/p$ then $G$ is the desired
graph. Otherwise, there is a cut of size
less than $k \log n/p$ that partitions $G$ into two graphs
$G_1$ and $G_2$. Repeat this process at
$G_1$ and $G_2$, and build a decomposition tree $T$ in
this manner. Any leaf of this tree $T$ is
either a singleton vertex or a graph where the minimum
cut size is at least $k \log n/p$. If
at least one leaf in $T$ satisfies
the latter property, then we are done and this is our
desired graph $H$. We now argue that it can not be the case
that all leaf vertices in $T$ are singletons.
To see this, note that there can be at most $n-1$
internal vertices in $T$ and each internal
vertex in $T$ corresponds to removing up to
$k \log n/p$ edges from $G$. Thus the
decomposition process removes at most
$k(n-1)\log n/p$ edges. On the other hand,
$G$ has at least $k n \log n/p$ edges.
It follows that not all leaves of $T$ can
be singleton vertices.
\end{proof}
We then show that if $\alpha(G)$ is
$\Omega\left(\log n/p\right)$ then
with high probability $G[p]$ is connected.
\begin{lemma}[\ensuremath{c_{\textrm{e}}\xspace}itet{noga}]
\label{lem:noga}
Let $G=(V,E)$ be a graph on $n$ vertices.
Then for any constant $b>0$ there exists
a constant $k(b)$ such that
if $\alpha(G) \geq k(b) \log n/p$ then with
probability at least $1-n^{-b}$,
$G[p]$ is connected.\footnote{The statement in \ensuremath{c_{\textrm{e}}\xspace}itet{noga} requires
$\alpha(G[p]) \geq k(b) \log n$. Since $\alpha(G[p])=\alpha(G)p$
this translates to the condition stated in Lemma~\ref{lem:noga}.}
\end{lemma}
We now define a property which we call
\emph{almost certain infection} and show
that no equilibrium network can contain an
induced subgraph satisfying this property.
\begin{definition}
\label{def:certain}
Let $G=(V,E)$ be a graph on $n$ vertices and
let $H$ be a subgraph of $G$ on more than one vertex.
$H$ has the \emph{almost certain infection} property if whenever
any vertex in $H$ is attacked, then with probability at least
$1-o(1/n)$ the attack spreads to every vertex in $H$.
\end{definition}
\begin{lemma}
\label{lem:certain}
Let $G=(V,E)$ be an equilibrium network on $n$ vertices.
$G$ cannot contain an induced subgraph $H=(V', E')$ such that
$H$ satisfies the almost infection property.
\end{lemma}
\begin{proof}
Consider any equilibrium graph $G$ that
violates the assertion of the claim. Let $H$
be an induced subgraph of $G$ with the
almost certain infection property.
We first prove that $H$ contains a cycle.
Assume by the way of contradiction that $H$ does not
have a cycle, so $H$ is a collection of trees.
Let $u$ be any leaf in $H$. Then $u$ is
incident to at most one edge in $H$.
Therefore, with probability $1-p$, this edge is not
in $G[p]$ and $u$ is not connected to any other
vertices in $G[p]$. So $H$ cannot have the almost
certain infection property
(Recall that we assumed $p$ is a constant
independent of the number of players $n$).
This means that $H$ contains a cycle $C$. Let $(u, v)$
be an edge on the cycle $C$. Assume
without loss of generality that $u$ purchased the edge $(u,v)$.
Now let $\xi$ be the event that an attack
propagates to some vertex in $H$ after the attack.
Then conditioned on $\xi$, with probability at least
$1-o(1/n)$, all vertices in $H$
die. Hence vertex $u$ in $H$ has negative utility. On
the other hand, if $\xi$ does not occur, then the utility of $u$
remains unchanged even if we remove the edge $(u, v)$.
Thus vertex $u$ can strictly improve her utility in this case
by dropping the edge $(u, v)$.
A contradiction to the fact that $G$ is an equilibrium network.
\end{proof}
We are now ready to prove Theorem~\ref{thm:density1}.
\noindent\textit{Proof of Theorem~\ref{thm:density1}.}
Assume by way of contradiction that $G$ has more
than $kn\log n/p$ edges where $k$ is the constant
in Lemma~\ref{lem:cut}. Then
by Lemma~\ref{lem:cut}, $G$ contains a subgraph
$H=(V', E')$ such that $\alpha(H)\geq k\log n/p$.
Since $|E'|\geq |V'|$, by Lemma~\ref{lem:noga},
$H$ has the almost certain infection property.
However, $G$ cannot be an equilibrium network
by Lemma~\ref{lem:certain}.
\qed
The most interesting regime for the probability of spread $p$
is when $p$ is a constant independent of $n$. While the upper
bound in Theorem~\ref{thm:density1} holds for all $p$, it becomes
vacuous as $p$ gets small i.e. it becomes bigger than the
trivial bound of $n^2/2$ when $p\leq k\log n/n$
for constant $k$. In Section~\ref{sec:edge-small-p}
we analyze the edge density of equilibrium networks
in the regime where $p < 1/n$. We
show that the number of edges in any equilibrium
network is bounded by $O(n)$ in this regime.
To prove the density result we utilize properties of
the Galton-Watson branching process and
random graph model of Erd{\"{o}}s-R\'enyi, as well
as tools from extremal graph theory.
{\bf s}ubsection{Small $p$ Regime}
\label{sec:edge-small-p}
In this section we focus on the regime where $p < 1/n$
and prove the following upper bound on the edge density.
\begin{thm}
\label{thm:smallp}
Let $p=\kappa/n$ for some constant $\kappa <1$.
Let $G=(V,E)$ be an equilibrium network over $n$ vertices.
Then for sufficiently large $n$, $|E|\le \max\{1/\ensuremath{c_{\textrm{e}}\xspace}, 24000\} n$.
\end{thm}
We prove Theorem~\ref{thm:smallp} by contradiction and show that
if the equilibrium graph has more than $\max\{1/\ensuremath{c_{\textrm{e}}\xspace}, 24000\} n$ edges,
there exists a beneficial deviation in the form of dropping an edge
for one of the players.
In order to prove Theorem~\ref{thm:smallp}, we need structural results
stated in Lemmas~\ref{lem:cut2}~and~\ref{lem:logc}.
First, consider an edge $(u, v)$ purchased by vertex $u$. Purchasing this edge
would not have increased the connectivity benefit of $u$ unless, after
some attack, the edge $(u,v)$ is the only path connecting $u$ to
$v$ (and possibly other vertices that are only reachable through $v$).
In Lemma~\ref{lem:cut2} (which we will prove later) we show that if a
graph is dense enough, then there exists an edge $(u, v)$ such that
many vertices should be deleted in order to make $(u,v)$ the
only remaining path connecting $u$ and $v$.
\begin{lemma}
\label{lem:cut2}
Let $G=(V,E)$ be a graph on $n>3 \gamma$ vertices with
$|E| \geq 2.5\gamma(n-\gamma)-1$ for some $\gamma \in \field{N}$. Then
there exist two vertices $v_1$ and $v_2 \in V$ such that $(v_1, v_2)\in E$,
and at least $\gamma+1$ vertices need to be deleted so that the only
path from $v_1$ to $v_2$ in $G$ is through the direct edge $(v_1, v_2)$.
\end{lemma}
Second, as described in Section~\ref{sec:model}, the number of vertices that are
killed in any attack is the size of the connected component in $G[p]$ that
contains the initially attacked vertex.
Lemma~\ref{lem:logc} (which we will prove later) bounds the size of a randomly
chosen connected component in $G[p]$.
\begin{lemma} \label{lem:logc}
Let $G=(V,E)$ be an equilibrium network over $n$ vertices
with $|E|=kn$ and $\max\{1/\ensuremath{c_{\textrm{e}}\xspace}, 24000\}\le k = O(\log n)$.
When $p<1/n$ and $n$ is sufficiently large,
the size of the connected component of a randomly chosen
vertex $v$ in $G[p]$ is at most
$k/3$ with probability at least $1-2\ensuremath{c_{\textrm{e}}\xspace}/(3n)$.
\end{lemma}
We now give the formal proof of Theorem~\ref{thm:smallp}.
\begin{proof}[Proof of Theorem~\ref{thm:smallp}]
Assume by way of contradiction that $G$ is an equilibrium network with
$kn$ edges where $k> \max\{1/\ensuremath{c_{\textrm{e}}\xspace},24000\}$.
Let $\gamma = \lfloor k/2.5 \rfloor$, we have $kn>2.5\gamma(n-\gamma)-1$.
By Lemma~\ref{lem:cut2}, there exists an edge $(u,v)$ such that in order to
make $(u,v)$ the only remaining path connecting $u$ and $v$, we need to delete
at least $\gamma+1$ vertices. Without loss of generality assume that
$u$ has purchased the edge $(u,v)$. If in any attack at most $\gamma$
vertices are killed, $u$ will not lose any connectivity benefit
after dropping this edge but decrease her expenditure by $\ensuremath{c_{\textrm{e}}\xspace}$.
Consider the size of the largest connected component in $G[p]$. Since $\kappa<1$,
the size of the largest connected component in $G[p]$ is at most $\beta \log n$ with probability $1-o(1/n)$
for sufficiently large constant $\beta$ which only depends on $\kappa$. When
$G$ is the complete graph,
$G[p]$ corresponds to the random graph generated by the
Erd{\"{o}}s-R\'enyi model. In such case, the size of the largest
component of $G[p]$ is $O(\log n)$, with high probability, when
$p n = \kappa <1$ \ensuremath{c_{\textrm{e}}\xspace}ite{erdos1960evolution}.
If $\gamma>\beta \log n$, then with probability at most $o(1/n)$,
the attack kills more than $\gamma$ vertices, in which case the connectivity
benefit of $u$ can decrease by at most $n$ after
dropping the edge $(u,v)$.
So the expected connectivity benefit of $u$ decreases by at most $o(1)$
after the deviation but her expenditure also decreases by $\ensuremath{c_{\textrm{e}}\xspace}$.
Hence, after the deviation, the expected change in the utility of $u$ is at least $\ensuremath{c_{\textrm{e}}\xspace} - o(1)>0$
which contradicts the assumption that $G$ is an equilibrium network
(recall that we have assumed $\ensuremath{c_{\textrm{e}}\xspace}$ is a constant independent of $n$).
If $\gamma \le \beta \log n$, then by definition,
$k = O(\log n)$. Since $k$ is also at least $\max \{ 24000,1/\ensuremath{c_{\textrm{e}}\xspace} \}$,
by Lemma~\ref{lem:logc}, with probability at least $1-2\ensuremath{c_{\textrm{e}}\xspace}/(3n)$,
the attack kills at most $k/3<\gamma$ vertices, in which case the connectivity
benefit of $u$ remains unchanged after dropping the edge $(u,v)$.
With probability at most $2\ensuremath{c_{\textrm{e}}\xspace}/(3n)$, more than $k/3$ vertices are killed
in which case the connectivity benefit of $u$ can decrease by at most $n$.
So the expected connectivity benefit of $u$ decreases by at most $2\ensuremath{c_{\textrm{e}}\xspace}/3$ after
the deviation but her expenditure also decreases by $\ensuremath{c_{\textrm{e}}\xspace}$.
Hence, after the deviation, the expected change in the utility of $u$ is at least $\ensuremath{c_{\textrm{e}}\xspace}/3>0$,
which contradicts the assumption that $G$ is an equilibrium network.
\end{proof}
We now proceed to prove Lemmas~\ref{lem:cut2}~and~\ref{lem:logc}.
\begin{proof} [Proof of Lemma~\ref{lem:cut2}]
Assume by way of contradiction that $G$ is the graph with
smallest number of vertices such that
$G$ has $n>3\gamma$ vertices and at least $2.5\gamma(n-\gamma)-1$
edges.
Therefore, $G$ has two vertices $v_1$ and $v_2\in V$ such that $(v_1, v_2)\in E$
and there exists a vertex set $S{\bf s}ubset V$ with at most $\gamma$ vertices, such that
after deleting $S$ the only
path from $v_1$ to $v_2$ in $G$ is through the edge $(v_1, v_2)$.
If $S$ has less than $\gamma$ vertices, then we add arbitrary vertices from $V$
(but not $v_1$ or $v_2$) to $S$ so that $S$ has exactly $\gamma$ vertices
(we can always do so because $G$ has more than $3\gamma$ vertices).
Consider the graph where the edge $(v_1, v_2)$ and the vertices in $S$ are removed,
$v_1$ and $v_2$ are not connected in this graph. Let $C_1$ be the connected component
that contains $v_1$ and $C_2 = V {\bf s}etminus S {\bf s}etminus C_1$. By definition,
$(v_1,v_2)$ is the only edge between $C_1$ and $C_2$ in $G$.
Define two graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ as subgraphs of $G$
induced by $C_1 \ensuremath{c_{\textrm{e}}\xspace}up S$
and $C_2 \ensuremath{c_{\textrm{e}}\xspace}up S$. Suppose $G_1$ has $n_1$
vertices and $G_2$ has $n_2$ vertices where $\gamma+1\leq n_1\leq n-1$ and $\gamma+1\leq n_2\leq n-1$.
Also without loss of generality assume $n_1\geq n_2$.
We have that $n_1+n_2 = n+ \gamma$.
On the other hand, $G_1$ and $G_2$ have at least $2.5\gamma(n_1+n_2-2\gamma)-2$
edges in total (any edge which is not ($v_1$,$v_2$) is either in $G_1$ or $G_2$).
So either $G_1$ has at least $2.5\gamma(n_1-\gamma)-1$ edges or $G_2$ has at least $2.5\gamma(n_2-\gamma)-1$ edges.
Also by the property of $G$,
for any pair of vertices $v_1', v'_2\in V_1$ (or $v_1', v'_2\in V_2$) such that $(v'_1, v'_2)\in E_1$
(or $(v'_1, v'_2)\in E_2$) we only need to delete
at most $\gamma$ vertices so that the only
path from $v'_1$ to $v'_2$ is through the direct edge $(v'_1, v'_2)$.
We claim that there exists a graph $G'$ (which is either $G_1$ or $G_2$) with $n'>2\gamma$
vertices that has at least $2.5\gamma(n'-\gamma)-1$ edges.
Note that if $G_1$ has at least $2.5\gamma(n_1-\gamma)-1$ edges then we are done
since $n_1 \geq n_2$ and $n_1+n_2 > 4 \gamma$ imply that $n_1 > 2\gamma$.
So suppose
$G_1$ has less than $2.5\gamma(n_1-\gamma)-1$ edges. Therefore,
$G_2$ has at least $2.5\gamma(n_2-\gamma)-1$ edges. Again if $n_2 > 2\gamma$
we are done so suppose $n_2\leq 2\gamma$.
Consider the edges which are in $G_2$ but not in $G_1$.
These edge have at least one endpoint in $C_2$, so there are at most
\begin{align*}
\frac{(n_2-\gamma)(n_2-\gamma-1)}{2}+\gamma(n_2-\gamma) &< (n_2-\gamma)(\frac{n_2}{2}+\frac{\gamma}{2})\\
&\leq \frac{3\gamma}{2}(n_2-\gamma) \\
& \leq 2.5\gamma(n_2-\gamma) - 1
\end{align*}
such edges. But this would imply
$G_1$ has at least $2.5\gamma(n_1-\gamma)-1$.
If $G'$ has strictly more than $3\gamma$ vertices, it contradicts our assumption that $G$ is
the smallest graph with the property stated in the lemma. If $G'$ has at most $3\gamma$ vertices,
then $G'$ has
at most
\begin{align*}
\frac{n'(n'-1)}{2} &= \frac{(n'-2\gamma)(n'+2\gamma)}{2} +2\gamma^2 - \frac{n'}{2} \\
&\le2.5 \gamma(n'-2\gamma) + 2 \gamma^2 - \frac{n'}{2}\\
& < 2.5 \gamma (n'-\gamma)-1
\end{align*}
edges, which is a contradiction.
\end{proof}
Before proving Lemma~\ref{lem:logc},
let us introduce some notation. Let $H{\bf s}ubseteq V$
be the set of vertices in $G$ with degree at least $n^{3/4}$. Also let $L=V{\bf s}etminus H$ be the
set of vertices in $G$ with degree strictly less than $n^{3/4}$. Hence, $H$ and $L$ correspond
to vertices with \emph{high} and \emph{low} degrees in $G$, respectively.
Recall that $G[p]$ is a random graph where each edge of $G$ is sampled independently
to be retained in $G[p]$ with probability $p$.
Using $H$ and $L$, the creation of $G[p]$ can be describe as a three step sampling process.
In the first step, edges with both endpoints in $H$ are sampled to be retained.
In the second step, edges with both endpoints in $L$ are sampled to be retained.
Finally, in the third step, edges with one endpoint in $H$ and the other endpoint in $L$
are sampled to be retained.
In Lemma~\ref{lem:higd}, we first show
that with high probability the size of the largest connected component created
by the first step and second step of the sampling process is at most 5 and 12, respectively.
These connected components can then be connected together in the third step of the sampling
to create larger connected components in $G[p]$.
We show that with high probability, the third step would not connect more than
3 of the connected components of high degree vertices (which were created in the first step).
This implies that the number
of high degree vertices in any connected component of $G[p]$ is at most $15$ with
high probability.
We then show in Lemma~\ref{lem:degdist} that for any $\alpha>0.001$, the expected
number of vertices with at least $\alpha k$ edges
in $G[p]$ is at most $\ensuremath{c_{\textrm{e}}\xspace}/(2^{1200\alpha}k)$. We then use the structural results of
Lemma~\ref{lem:higd} and~\ref{lem:degdist} to show that
with probability at least $1-2\ensuremath{c_{\textrm{e}}\xspace}/(3n)$, the size of the connected component of a
randomly chosen vertex in $G[p]$ is at most $k/3$ .
\begin{lemma} \label{lem:higd}
Let $G=(V,E)$ be an equilibrium network over $n$ vertices
with $|E|=kn$ with $k \ge 24000$ and $k=O(\log n)$.
Suppose $p<1/n$ and $n$
is sufficiently large. Then, with probability at least $1-o(1/n)$, (1) the connected
components generated in the first step and the second step of the sampling process
of creating $G[p]$ have size
at most $5$ and $12$, respectively and (2)
no component in $G[p]$ has more than 15 vertices from the set $H$.
\end{lemma}
\begin{proof}
Recall that in the first two
steps of creating the $G[p]$, we sample
edges to retain independently with probability
$p$ from the graphs induced
by only $H$ and $L$, respectively. Since
$|E|=kn$ the number of high
degree vertices is at most $\tilde{O}(n^{1/4})$
(where the notation $\tilde{O}$
hides logarithmic dependencies on $n$).
So any vertex in the graph induced
by $H$ has degree at most
$\tilde{O}(n^{1/4})$. Moreover, by definition, the
vertices in the graph induced by $L$
have degree at most $n^{3/4}$. By
Corollary~\ref{cor:gw} (in Appendix~\ref{sec:useful-lem}), with high probability, the random
components formed by the first step and
second step of the sampling process have size at most 5 and 12, respectively.
We refer to the set of the components of $G[p]$ that are
formed by step one and two of the sampling
by $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively.
Consider a component $C_2\in \mathcal{C}_2$. The probability
that there is an edge in $G$ between any vertex in $C_2$
and a specific high degree vertex is bounded by $12/n$.
This means that the probability that the
vertices in $C_2$ are connected to more than one
high degree vertex is at most
$\tbinom{|H|}{2}(12/n)^2 = \tilde{O}(n^{-1.5})$.
Similarly the probability that the
vertices in $C_2$ are connected to more than two
high degree vertices is at most
$\tbinom{|H|}{3} (12/n)^3 = o(n^{-2})$.
Therefore, with high probability, there is no
component $C_2\in \mathcal{C}_2$ that is connected to three high degree vertices.
Moreover, the probability that there are three connected components
in $\mathcal{C}_2$ that is connected to $2$ high degree vertices
is at most $\tbinom{n}{3} (\tilde{O}(n^{-1.5}))^3 = o(n^{-1})$.
In the third step of creating the $G[p]$, components
from $\mathcal{C}_1$ and $\mathcal{C}_2$ would become connected
by sampling the edges in between $H$ and $L$.
As we showed there are
most two components in $\mathcal{C}_1$ that will be
connected in the $G[p]$ by the edges sampled
in the third step.
This means, with high probability, no component in
$G[p]$ will include more than
3 components from $\mathcal{C}_1$; so, with high probability,
no component in $G[p]$ has more than 15 high degree vertices.
\end{proof}
\begin{lemma} \label{lem:degdist}
Let $G=(V,E)$ be an equilibrium network over $n$ vertices
with $|E|=kn$ and $k\ge \max\{24000,1/\ensuremath{c_{\textrm{e}}\xspace}\}$.
For any $\alpha>0.001$, when $p<1/n$, the
expected number of vertices
that have at least $\alpha k$ adjacent edges in
$G[p]$ is at most $\ensuremath{c_{\textrm{e}}\xspace}/(2^{1200\alpha}k)$
for sufficiently large $n$.
\end{lemma}
\begin{proof}
For a vertex with degree $d>\alpha k$, the probability that
she has $\alpha k$ edges in $G[p]$ is
at most
\begin{align*}
\tbinom{d}{\alpha k} n^{-\alpha k}
&< (\frac{de}{n\alpha k})^{\alpha k}
< \frac{d}{n} (\frac{e}{\alpha k})^{\alpha k}\\
&< \frac{d}{n} \ensuremath{c_{\textrm{e}}\xspace}dot 8^{-\alpha k}
\le \frac{d}{n} \ensuremath{c_{\textrm{e}}\xspace}dot 2^{-2\alpha k} \ensuremath{c_{\textrm{e}}\xspace}dot 2^{-24000\alpha}\\
&< \frac{d}{n} \ensuremath{c_{\textrm{e}}\xspace}dot k^{-3} \ensuremath{c_{\textrm{e}}\xspace}dot 2^{-24000\alpha}
< \frac{d}{n} \ensuremath{c_{\textrm{e}}\xspace}dot \frac{\ensuremath{c_{\textrm{e}}\xspace}}{2k^2} 2^{-1200\alpha}.
\end{align*}
The first inequality is due to Stirling's formula.
Other inequalities are due to
$k\alpha\ge 24$, $k\ge \max\{24000,1/\ensuremath{c_{\textrm{e}}\xspace}\}$ and $e/(\alpha k) < 1/8$.
Adding up the probabilities for all the vertices
(using linearity of expectation)
and using the fact that the sum of
the degrees of all the vertices is $2kn$ will
conclude the proof.
\end{proof}
We now have all the background to prove
Lemma~\ref{lem:logc}.
\noindent\textit{Proof of Lemma \ref{lem:logc}.}
If we randomly choose a vertex $v$, the probability
that the connected
component of $v$ in $G[p]$ has size at least $k/3$
is upper bounded by $1/n$ times the sum of the sizes
of components
with size at least $k/3$ in $G[p]$.
This is in turn upper bounded by
$$\frac{1}{n}{\bf s}um_{i=1}^n (i+1) \frac{k}{3} (x_i-x_{i+1})
< \frac{1}{n}{\bf s}um_{i=1}^n k x_i,$$
where $x_i$ is the number of components with size at
least $ik/3$.
Recall that we partitioned the vertices of $G$ into high
and low degree vertex sets $H$
and $L$ based on the degree. We described a
three step sampling process for creating $G[p]$
and referred to the set of connected components of $G[p]$ that are
formed by step one and two of the sampling
by $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively.
Let $\xi$ be the event such that each $C_2\in \mathcal{C}_2$
has at most 12 vertices
and each component in $G[p]$ has at most
$15$ high degree vertices.
By Lemma~\ref{lem:higd}, $\mathrm{Pr}[\bar{\xi}] =
1-\mathrm{Pr}[\xi]=o(1/n)$. Let $y_i$ be the number of vertices
with at least $ik/600$ edges in $G[p]$. By
Lemma~\ref{lem:degdist}, $\mathbb{E}[y_i]\leq \ensuremath{c_{\textrm{e}}\xspace}/(2^{2i} k) \le \ensuremath{c_{\textrm{e}}\xspace}/(2^{i+1}k)$.
Fix a component of $G[p]$.
Conditioned on event $\xi$, if all the high degree vertices in the component have
degree at most $i k/600$, then each high degree vertex
is connected to at most $ik/600$ of the components in $\mathcal{C}_2$.
So the size of this component is at most $15 \ensuremath{c_{\textrm{e}}\xspace}dot 12 \ensuremath{c_{\textrm{e}}\xspace}dot (ik/600) + 15 = 3ik/10 + 15 < (ik/3)$.
Therefore, each component with size at least $ik/3$ contains at least one
vertex with at least $i k/600$ adjacent edges in $G[p]$.
This means that $\mathbb{E}[x_i|\xi] \le \mathbb{E}[y_i|\xi]$. So
\begin{align*}
\mathbb{E}[x_i|\xi] &\le \mathbb{E}[y_i|\xi]
= \frac{\mathbb{E}[y_i]- \mathbb{E}[y_i|\bar{\xi}] \mathrm{Pr}[\bar{\xi}]}{\mathrm{Pr}[\xi]}\\
&\le \frac{\mathbb{E}[y_i]}{\mathrm{Pr}[\xi]}
\le \frac{\ensuremath{c_{\textrm{e}}\xspace}}{(2^{i+1} k)(1-o(1/n))} \\
&= \frac{\ensuremath{c_{\textrm{e}}\xspace}}{2^{i+1}k} \ensuremath{c_{\textrm{e}}\xspace}dot (1+o(1/n)).
\end{align*}
So the probability that $v$ is in a component of size at least $k/3$ given $\xi$ is
\begin{align*}
\frac{1}{n}\mathbb{E}[{\bf s}um_{i=1}^n k x_i|\xi]
&\le \frac{1}{n}{\bf s}um_{i=1}^n k \mathbb{E}[x_i| \xi] \\
&= (1+o(1/n))\frac{1}{n}{\bf s}um_{i=1}^n \frac{\ensuremath{c_{\textrm{e}}\xspace}}{2^{i+1}}\\
&= (1+o(1/n))\frac{\ensuremath{c_{\textrm{e}}\xspace}}{2n}.
\end{align*}
So the overall probability that a vertex is in a component with
size at least $k/3$ is at most $(1+o(1/n))\ensuremath{c_{\textrm{e}}\xspace}/(2n)+\mathrm{Pr}[\bar{\xi}]<2\ensuremath{c_{\textrm{e}}\xspace}/(3n)$.
\qed
{\bf s}ection{Social Welfare}
\label{sec:welfare}
In this section we provide a lower bound on the social welfare of equilibrium networks.
Similar to other reachability games, the empty graph can form in
equilibrium~\ensuremath{c_{\textrm{e}}\xspace}ite{BalaG00, GoyalJKKM16}.
Hence without any further assumptions,
no meaningful guarantee on the social welfare can be made. Hence, we focus on non-trivial
equilibrium networks defined as follows.
\begin{definition}
\label{def:non-trivial}
An equilibrium network is non-trivial if it contains at least one edge.
\end{definition}
Definition~\ref{def:non-trivial} rules out the empty network but it is still possible that
a non-trivial equilibrium network contains many small connected components or
becomes highly fragmented after the attack. In this section we show that none
of these concerns materialize. In particular, in
Theorem~\ref{lem:largest}, we first show that
any non-trivial equilibrium network contains at least one large connected
component. We then show in Lemma~\ref{lem:small-components}
that when the network is not too dense,
the equilibrium network cannot become highly fragmented after the attack.
These two observations allows us to prove our social welfare
lower bound as stated in Theorem~\ref{thm:welfare}.
We start by showing that any non-trivial equilibrium network contains a
large connected component.
\begin{thm} \label{lem:largest}
Let $G=(V,E)$ be a non-trivial equilibrium network over $n$ vertices.
Then, for sufficiently large $n$, the largest connected component of $G$ has at least
$n/3$ vertices.
\end{thm}
\begin{proof}
Throughout we require $n > \max\{15\ensuremath{c_{\textrm{e}}\xspace}+15\ensuremath{c_{\textrm{e}}\xspace}^2, (9+3\ensuremath{c_{\textrm{e}}\xspace})/2, \ensuremath{c_{\textrm{e}}\xspace}^2+4\ensuremath{c_{\textrm{e}}\xspace}+5\}$.
Consider two cases: (1) when $G$ contains no isolated vertices\footnote{An
isolated vertex is a vertex with no incident edges.}
and (2) when $G$ contains at least one isolated vertex.
In the first case, assume by way of contradiction that $G$ has at least $4$ connected components.
Let $C_0$ be a smallest connected component in $G$, say of size $n_0$.
Also let $C_1$ be a second-smallest connected component in $G$, say of size $n_1$.
By construction, $n_0\le n_1$.
Since $C_1$ has at least one edge and this edge has cost $\ensuremath{c_{\textrm{e}}\xspace}$,
the size of $C_1$ is at least $\ensuremath{c_{\textrm{e}}\xspace} +1$.
Otherwise the vertex who bought this edge can improve her utility by dropping this edge.
Consider the deviation that a vertex $v$ in $C_0$ adds an edge to an arbitrary vertex in $C_1$.
We show that the increase in the connectivity benefit of $v$ is more than $\ensuremath{c_{\textrm{e}}\xspace}$.
If the attack does not start at $C_0$ or $C_1$, which occurs with probability $(n-n_0-n_1)/n$,
then the connectivity benefit of $v$ increases by $n_1$.
If the attack starts at $C_0$, then the only way for the attack to reach $C_1$ is through
the newly added edge by $v$. Hence, in this case the connectivity benefit of $v$ does not decrease.
Therefore, the only scenario in which the connectivity benefit of $v$ can decrease is when
the attack starts at $C_1$
(which occurs with probability $n_1/n$). In this case, the connectivity benefit of $v$ can decrease
by at most $n_0$. So the change in the connectivity benefit of $v$ is at least
$$\Delta \geq \frac{(n-n_0-n_1)}{n}n_1-\frac{n_1}{n}n_0 = \frac{n_1(n-2n_0-n_1)}{n},$$
after the deviation. We show that $\Delta > \ensuremath{c_{\textrm{e}}\xspace}$ which is a contradiction to $G$ being an equilibrium network.
We consider two sub-cases based on the value of $n_1$: (1a) $n_1>5\ensuremath{c_{\textrm{e}}\xspace}$ and (1b) $n_1\le 5\ensuremath{c_{\textrm{e}}\xspace}$.
First consider case (1a) where $n_1>5\ensuremath{c_{\textrm{e}}\xspace}$.
Since $G$ has at least $4$ connected components, then $n_0\le n/4$,
$n_0+n_1\le n/2$ and $2n_0+n_1\le 3n/4$.
So $$n_1(n-2n_0-n_1)>5\ensuremath{c_{\textrm{e}}\xspace} \frac{n}{4} > \ensuremath{c_{\textrm{e}}\xspace} n \implies \Delta > \ensuremath{c_{\textrm{e}}\xspace}.$$
Next consider case (1b) where $n_1\le 5\ensuremath{c_{\textrm{e}}\xspace}$. Since $n-2n_0-n_1 \ge n-15\ensuremath{c_{\textrm{e}}\xspace}$
and $n_1\geq \ensuremath{c_{\textrm{e}}\xspace}+1$ then
\begin{align*}
n_1(n-2n_0-n_1) &\ge (\ensuremath{c_{\textrm{e}}\xspace}+1)(n-15\ensuremath{c_{\textrm{e}}\xspace}) \\&= \ensuremath{c_{\textrm{e}}\xspace} n + n -15\ensuremath{c_{\textrm{e}}\xspace} - 15\ensuremath{c_{\textrm{e}}\xspace}^2 > \ensuremath{c_{\textrm{e}}\xspace} n\\
&\implies \Delta > \ensuremath{c_{\textrm{e}}\xspace},
\end{align*}
when $n>15\ensuremath{c_{\textrm{e}}\xspace}+15\ensuremath{c_{\textrm{e}}\xspace}^2$.
Therefore, the deviation of adding an edge by $v$ will increase the connectivity benefit
of $v$ by strictly more than $\ensuremath{c_{\textrm{e}}\xspace}$ which is a contradiction. So $G$ contains at most $3$
connected components in case (1) and hence
the largest connected component of $G$ contains at least $n/3$ vertices.
In case (2), we show that the largest connected component of $G$ contains at
least $n-\ensuremath{c_{\textrm{e}}\xspace}-3$ vertices. Therefore, the largest connected component
of $G$ contains at least $n/3$ vertices when $n > (9+3\ensuremath{c_{\textrm{e}}\xspace})/2$.
Let $v$ be an isolated vertex and let $C^{{\bf s}tar}$ be a largest connected component of $G$, say of size
$n^{{\bf s}tar}$. Consider the deviation where $v$ adds an edge to an arbitrary vertex in $C^{{\bf s}tar}$.
If the attack does not start neither in $C^{{\bf s}tar}$ nor at $v$ (which occurs with probability $(n-n^{{\bf s}tar}-1)/n)$,
then the connectivity benefit of $v$ increases by $n^{{\bf s}tar}$.
The only scenario in which the connectivity benefit of $v$ can decrease is when the attack starts at $C^{{\bf s}tar}$
(which occurs with probability $n^{{\bf s}tar}/n$).
In this case, the connectivity benefit of $v$ can decrease by at most $1$. So the change $\Delta$ in the
connectivity benefit of $v$ after the deviation is at least
$$\Delta \geq \frac{(n-n^{{\bf s}tar}-1)}{n}n^*-\frac{n^{{\bf s}tar}}{n} = \frac{n^*(n-n^*-2)}{n}.$$
Note that $\Delta \leq \ensuremath{c_{\textrm{e}}\xspace}$, since $G$ is an equilibrium network.
This implies that $n^*(n-n^*-2) \le \ensuremath{c_{\textrm{e}}\xspace} n$. To show that $n^* > n-\ensuremath{c_{\textrm{e}}\xspace}-3$,
consider the function $f(x) = x(n-x-2)$. $f$ is increasing when $x\leq (n-2)/2$
and decreasing when $x\geq (n-2)/2$. Moreover,
$$f(\ensuremath{c_{\textrm{e}}\xspace}+1) = f(n-\ensuremath{c_{\textrm{e}}\xspace}-3) = (\ensuremath{c_{\textrm{e}}\xspace}+1)(n-\ensuremath{c_{\textrm{e}}\xspace}+3) > \ensuremath{c_{\textrm{e}}\xspace} n$$
when $n > \ensuremath{c_{\textrm{e}}\xspace}^2+4\ensuremath{c_{\textrm{e}}\xspace}+5$. So $f(n^*)$ is always larger than $\ensuremath{c_{\textrm{e}}\xspace} n$
when $\ensuremath{c_{\textrm{e}}\xspace}+1\leq n^*\leq n-3-\ensuremath{c_{\textrm{e}}\xspace}$. Since we showed $f(n^*) = n^*(n-n^*-2) \le \ensuremath{c_{\textrm{e}}\xspace} n$
in equilibrium it most be the case that either $n^* < \ensuremath{c_{\textrm{e}}\xspace}+1$ or $n^* > n-3-\ensuremath{c_{\textrm{e}}\xspace}$.
The former cannot happen because we assumed $G$ is non-empty, so $C^*$
must have at least one edge and therefore $n^* \geq \ensuremath{c_{\textrm{e}}\xspace}+1$. Hence, $n^* > n-3-\ensuremath{c_{\textrm{e}}\xspace}$
as claimed.
\end{proof}
We next present Lemma~\ref{lem:small-components} that describes
the relationship between the expected size of the largest connected component of $G[p]$
and the connectivity benefits of the vertices in $G$.
\begin{lemma}
\label{lem:small-components}
Let $G=(V,E)$ be an equilibrium network over $n$ vertices. Let
$C$ be any connected component in $G$ of size $n_C$.
If the expected size of the largest component of $C[p]$ is at most $n_C(1-\epsilon)$,
then the expected sum of the connectivity benefits of the vertices in $C$
is at least $(n-n_C)n_C^2/n+\epsilon n_C^3/(3n)$.
\end{lemma}
\begin{proof}
With probability $(n-n_C)/n$, the attack starts at a vertex outside of $C$. In this
case the sum of the connectivity benefits of vertices in $C$ is $n_C^2$
(the first term in the lower bound).
Otherwise, with probability $n_C/n$, the attack starts at a vertex in $C$.
We claim that in this case, the sum of the connectivity benefits of the vertices
in $C$ is at least $\epsilon n_C^2/3$ (the second term in the lower bound).
To prove the claim it suffices to show that if the largest component in $C[p]$ has size $X$,
then the sum of connectivity benefits of the vertices in $C$ is at least $(n_C-X) n_C/3$.
The claim would then follow by taking the expectation and using the assumption of the theorem
that $X=n_C(1-\epsilon)$.
Assume by way of contradiction that the sum of connectivity benefits of vertices in
$C$ is less than $(n_C-X)n_C/3$.
Then there exists a connected component $C_0$ in $C[p]$ such that if we delete the vertices in $C_0$,
then the sum of connectivity benefits of the vertices when the attack
destroys $C_0$ is less than $(n_C-X) n_C/3$.
Suppose $C_0$ has size $n_0$ and we know $n_0\le X$.
\begin{figure}\label{fig:proof}
\end{figure}
Let $C_1, C_2,\dots, C_k$ be the connected components in the subgraph
of $G$ induced by $C {\bf s}etminus C_0$ and
let $n_1,n_2,\dots,n_k$ denote their sizes, respectively (see Figure~\ref{fig:proof}).
Then by the assumption on the sum of connectivity benefits of the vertices after deleting $C_0$,
when the attack destroys $C_0$ we have that
\begin{equation}
\label{eq:x2}
{\bf s}um_{i=1}^k n_i^2 < \frac{(n_C-X) n_C}{3}.
\end{equation}
If the attack starts at a vertex in a component $C_i$, then the vertices in $C {\bf s}etminus C_i$
will still remain connected.
This means the sum of connectivity benefits of the vertices in $C$ is at least
\begin{align*}
{\bf s}um_{i=1}^k \frac{n_i}{n_C} (n_C-n_i)^2 &= {\bf s}um_{i=1}^k \frac{n_i}{n_C}(n_C^2-2n_Cn_i+n_i^2) \\
&\ge {\bf s}um_{i=1}^k n_Cn_i - 2{\bf s}um_{i=1}^k n_i^2\\
&=n_C{\bf s}um_{i=1}^k n_i - 2{\bf s}um_{i=1}^k n_i^2 .
\end{align*}
Since ${\bf s}um_{i=1}^k n_i=n_C-n_0 \ge n_C-X$ and ${\bf s}um_{i=1}^k n_i^2 < (n_C-X) n_C/3$
by Equation~(\ref{eq:x2}), the sum of the connectivity
benefits of the vertices in $C$ is at least $(n_C-X) n_C/3$; which is a contradiction.
\end{proof}
Theorem~\ref{lem:largest}~and~Lemma~\ref{lem:small-components} allow us
to prove a lower bound on the social welfare of
non-trivial equilibrium networks.
\begin{thm}
\label{thm:welfare}
Let $G=(V,E)$ be a non-trivial equilibrium network over $n$ vertices.
For any $\epsilon\in(0,1/8)$ and sufficiently large $n$, if $|E| < (1-2\epsilon)n\log(1/\epsilon)/p$,
then the social welfare of $G$ is at least $\epsilon n^2/3 - O(n/p)$.
\end{thm}
\begin{proof}
We show the expected sum of the connectivity benefits of the vertices
in $G$ is at least $\epsilon n^2/3$.
Subtracting off the cumulative expenditure for edge purchases which is $|E|\ensuremath{c_{\textrm{e}}\xspace} = O(n/p)$
then imply the statement of the theorem.
Suppose the largest connected component of $G$ say $C^*$
has size $n^{{\bf s}tar}$. By Theorem~\ref{lem:largest},
$n^{{\bf s}tar}\ge n/3$. We consider two cases based
on the size of $n^*$: (1) $n^{{\bf s}tar} \le (1-2\epsilon)n$ and (2) $n^* > (1-2\epsilon)n$.
In case (1), where $n^{{\bf s}tar} \leq (1-2\epsilon)n$, the sum of the connectivity
benefits of the vertices in the largest connected component is at least $(n-n^{{\bf s}tar})n^{{\bf s}tar 2}/n$.
This is because with probability of $(n-n^{{\bf s}tar})/n$ the attack starts outside of the component
and all the vertices in the component survive; in such case the sum of connectivity benefits
of the vertices in $C^*$ is $n^{{\bf s}tar 2}$. Moreover, the derivative of $(n-n^{{\bf s}tar})n^{{\bf s}tar 2}/n$
with respect to $n^{{\bf s}tar}$ is $(-3{n^{{\bf s}tar}}^2+2n n^{{\bf s}tar})/n$, which is positive when $0<n^{{\bf s}tar}<2n/3$
and negative when $n^{{\bf s}tar}>2n/3$. Since $n^{{\bf s}tar} \in [n/3,(1-2\epsilon)n]$, the minimum value of
$(n-n^{{\bf s}tar})n^{{\bf s}tar 2}/n$ should be at one of the end points which correspond to values
$2n^2/27$ or $2\epsilon(1-2\epsilon)^2n^2$, respectively. Both of these values
are larger than $\epsilon n^3/3$ when $\epsilon<1/8$, which means the sum of connectivity benefits
is at least $\epsilon n^2/3$ in this case.
In case (2), where $n^* > (1-2\epsilon)n$, the number of edges in the connected component $C^*$ is
at most $n^{{\bf s}tar}\log(1/\epsilon)/p$ (which occurs when all the edges are in this component).
Let us denote the vertices in $C^*$ by numbers from $1$ to $n^{{\bf s}tar}$. Let $d_i$ denote the
degree of vertex $i$. We first bound the expected number of isolated
vertices of $C^*$ in $G[p]$.
A vertex becomes isolated in $G[p]$ if none of the edges
adjacent to it are sampled to be retained. This event occurs with probability $(1-p)^{d_i}$
for vertex $i$. So we can derive a lower bound on the expected number of isolated vertices of $C^*$ in $G[p]$
as follows.
\begin{align*}
{\bf s}um_{i=1}^{n^{{\bf s}tar}} (1-p)^{d_i}\geq n^* (1-p)^{\frac{2|E|}{n^{{\bf s}tar}}} > \epsilon n^{{\bf s}tar},
\end{align*}
where the first inequality is by inequality of arithmetic and geometric means and the second inequality is
by the assumption that $|E| < n^{{\bf s}tar}\log(1/\epsilon)/p$. Thus
the expected size of the largest connected component in $C^*[p]$ is at most $n^*(1-\epsilon)$.
We can now apply Lemma~\ref{lem:small-components}
to show that the expected sum of the connectivity benefits of the vertices in $C^*$ is at least
$$\frac{\epsilon n^{{\bf s}tar 3}}{3n}+ \frac{(n-n^{{\bf s}tar})n^{{\bf s}tar 2}}{n} = \frac{n^{{\bf s}tar 2}n-(1-\epsilon/3)n^{{\bf s}tar 3}}{n}.$$
This is strictly decreasing in $n^*$ as the derivative with respect to $n^*$ is negative. So
the expected sum of connectivity benefits of the vertices in $C^*$ (and hence in $G$) is
at least $\epsilon n^2/3$ (when $n^{{\bf s}tar}=n$).
\end{proof}
Finally, we remark that unlike the models of~\ensuremath{c_{\textrm{e}}\xspace}itet{BalaG00}~and~\ensuremath{c_{\textrm{e}}\xspace}itet{GoyalJKKM16},
achieving a social welfare of $n^2-o(n^2)$ is impossible in our game even when restrciting
to sparse and non-trivial
equilibrium networks. This is formalized in Proposition~\ref{lem:upper-welfare}.
\begin{pro}
\label{lem:upper-welfare}
There exists a non-trivial equilibrium network $G=(V,E)$ over $n$ vertices with $O(n)$ edges
such that the social welfare of $G$ is $kn^2$ for $k <1 $.
\end{pro}
\begin{proof}
The hub-spoke equilibrium (see Figure~\ref{fig:eq-examples})
with $p = 0.6$ satisfies the condition of Proposition~\ref{lem:upper-welfare}
with $k=0.4$.
\end{proof}
{\bf s}ection{Conclusions}
\label{sec:future}
We studied a natural network formation game where each network connection
has the potential to both bring additional utility to an agent as well add to her
risk of being infected by a cascading infection attack. We showed that the
equilibria resulting from these competing concerns are essentially sparse
and containing at most $O(n \log n)$ edges. We also showed that any
non-trivial equilibrium network in our game achieves the highest possible
social welfare of $\Theta(n^2)$ whenever the equilibrium network has only
$O(n)$ edges.
The Price of Anarchy in our model is $\Theta(n)$.
To illustrate, consider the $\ensuremath{c_{\textrm{e}}\xspace} \geq 1$ regime. A central planner can built
a cycle or two disconnected hub-spoke structures of size $n/2$ depending on
whether the probability of spread of the attack $p$ is low or high, respectively
(and both of these structures can also form in equilibrium). Such networks
have social welfare of $\Theta(n^2)$. However, the empty network is an
equilibrium network in this regime implying a Price of Anarchy of at least
$\Theta(n)$ -- the worst Price of Anarchy possible. The Price of Stability
in our model is $\Theta(1)$ since the social welfare is trivially bounded by $n^2$
and either of the two equilibrium networks above achieve a social welfare of $\Theta(n^2)$.
Our results suggest several natural questions for future work. Our upper
bound of $O(n \log n)$ is a logarithmic factor higher than the densest equilibrium
network that we can create. Narrowing this gap is the most interesting open
question. Improving our network density upper bound to $O(n)$ edges
would immediately imply that all non-trivial equilibrium networks achieve
$\Omega(n^2)$ social welfare. Another direction for future work is to analyze
how network density and social welfare evolves when agents additionally
have an option to invest in immunization that protects them from infections.
\appendix
{\bf s}ection{Galton-Watson Branching Process}
\label{sec:useful-lem}
The Galton-Watson branching process was introduced by Galton as a mathematical model
for the propagation of family names. In the process, a population of individuals
(e.g. people) evolve over discrete time $n=1, 2, \ldots$. Each $n$th generation individual
(i.e. individuals who are produced at time $n$) produce a random number of individuals
(called \emph{offsprings}) independently according to some
distribution $\xi$ (called the \emph{offspring distribution}) for the $n+1$th generation.
The goal is to study the number of individuals in the future generations.
The process can go to extinction when after $n$ generation, with high probability,
the number of individuals in generation $n+1$ is 0. This can happen for example when
we start from 1 individual and $\mathbb{E}[\xi]<1$. In case the process goes
to extinction, we are interested
in characterizing how fast this happens or how
many individuals are generated before extinction.
\begin{lemma}[Galton-Watson process~\ensuremath{c_{\textrm{e}}\xspace}ite{DraiefM09}]
\label{lem:gw}
Let $\xi$ denote the offspring distribution of each individual in the Galton-Watson process
when starting with one individual. Furthermore, let $h=\text{sup}_{\theta\ge 0}\{\theta-\log \mathbb{E}[e^{\theta\xi}]\}$.
Let $\mathcal{T}$ denote the set of of total individuals created by process
when the process goes to extinction. Then
$\mathrm{Pr}[\mathcal{|T|}>k] \le e^{-kh}$ for all $k\in \mathbb{N}$.
\end{lemma}
\begin{cor} \label{cor:gw}
Let $\epsilon > 0$ and $n\in\field{N}$.
In the Galton-Watson process, suppose the offspring distribution $\xi$
is the sum of $m=O(n^{1-\epsilon})$ Bernoulli random variables with probability at most $1/n$.
Then with probability at least $1-o(n^{-2})$, the number of individuals created by the process
is at most $3/\epsilon$.
\end{cor}
\begin{proof}
\begin{align*}
\mathbb{E}[e^{\theta\xi}] & = {\bf s}um_{i=1}^m \mathrm{Pr}[\xi=i] e^{\theta i}
\le {\bf s}um_{i=1}^m \frac{1}{n^i} \tbinom{m}{i} e^{\theta i}
\le {\bf s}um_{i=1}^m \frac{m^i e^{\theta i}}{n^ii!}\\
&\le {\bf s}um_{i=1}^m (\frac{me^{\theta}}{n})^i
\end{align*}
Let $\theta'=\epsilon \log n - 1$,
$$\mathbb{E}[e^{\theta' \xi}] \le {\bf s}um_{i=1}^m (\frac{1}{e})^i \le \frac{e}{e-1} $$
So \begin{align*}h&=\text{sup}_{\theta\ge 0}\{\theta-\log \mathbb{E}[e^{\theta\xi}]\}\geq \theta'-\mathbb{E}[e^{\theta' \xi}] \\&\geq \epsilon\log n -1 - \frac{1}{e-1} > \epsilon \log n -3.\end{align*}
Therefore by Lemma~\ref{lem:gw}, $$\mathrm{Pr}[|\mathcal{T}|>\frac{3}{\epsilon}]\le \frac{e^{9/\epsilon}}{n^3} = o(n^{-2}),$$
as claimed.
\end{proof}
\end{document} |
\begin{document}
\title{Robust Inference with Variational Bayes}
\section{Introduction}\label{sec:intro}
In Bayesian analysis, the posterior follows from the data and a choice of a
prior and a likelihood. One hopes that the posterior is robust to reasonable
variation in the choice of prior and likelihood, since this choice is made by
the modeler and is necessarily somewhat subjective. For example, the process of
prior elicitation may be prohibitively time-consuming, two practitioners may
have irreconcileable subjective prior beliefs, or the model may be so complex
and high-dimensional that humans cannot reasonably express their prior beliefs
as formal distributions. All of these circumstances might give rise to a range
of reasonable prior choices. If the posterior changes substantially with these
choices of prior, then the analysis lacks objectivity. Measuring the sensitivity
of the posterior to variation in the likelihood and prior is the central concern
of the field of \emph{robust Bayes}. A robust posterior is one that does not
depend strongly on reasonable variation in the choice of model or prior, and
robust Bayes provides methods for quantifying posterior robustness
\citep{insua:2012:robust}.
Despite the fundamental importance of the problem and a considerable body of
literature, the tools of robust Bayes are not commonly used in practice. This is
in large part due to the difficulty of calculating robustness measures from MCMC
draws\citep{berger:2012:robust, roos:2015:sensitivity}. Although methods for
computing robustness measures from MCMC draws exist, they lack generality and
often require additional coding or computation \footnote{See \app{mcmc} for a
literature review.}. Consequently, formal robust Bayes methods are least used in
complex, hierarchical models, exactly when they are needed most. Instead,
modelers are tempted to either compute ad-hoc robustness estimates (e.g. by
manually changing the priors and re-running their chain) or to ignore the
problem altogether.
In contrast to MCMC, variational Bayes (VB) techniques are readily amenable to
robustness analysis. The derivative of a posterior expectation with respect to
a prior or data perturbation is a measure of \emph{local robustness} to the
prior or likelihood \citep{gustafson:2012:localrobustnessbook}. Because VB casts
posterior inference as an optimization problem, its methodology is built on the
ability to calculate derivatives of posterior quantities with respect to model
parameters, even in very complex models. Variational methods for posterior
approximation are increasingly providing a scalable alternative to MCMC for
posterior approximation, and this offers the opportunity to bring fast,
easy-to-use robustness measures into common practice.
In the present work, we develop local prior robustness measures for
\emph{mean-field variational Bayes} (MFVB), a VB technique which imposes a
particular factorization assumption on the variational posterior approximation.
In past work \citep{giordano:2015:lrvb}, we demonstrated that a MFVB analysis can
be quickly and straightforwardly augmented to provide information about local
perturbations of the posterior variational approximation using linear response
methods from statistical physics. We show that this framework can be extended
to provide fast, easy-to-use prior robustness measures for posterior inference
and thereby bring robustness analysis into common Bayesian practice.
In the remainder of the present work, we start by outlining existing local prior
measures of robustness in \mysec{robustness_measures}. We extend the linear
response techniques of \citep{giordano:2015:lrvb} in \mysec{lrvb_formulas}. In
\mysec{lrvb_robustness} we use these results to derive closed-form measures of
the sensitivity of mean-field variational posterior approximation to prior
specification. In \mysec{experiments} we demonstrate our method on a
meta-analysis of randomized controlled interventions in access to microcredit in
developing countries.
\section{Robustness measures}\label{sec:robustness_measures}
Denote our $N$ data points by $x = (x_1, \ldots, x_N)$ with $x_n \in
\mathbb{R}^{D}$. Denote our parameter by the vector $\theta \in \mathbb{R}^{K}$.
We denote the prior parameters by $\alpha$, where either $\alpha \in
\mathbb{R}^{M}$ or $\alpha$ may be function-valued. Let $\pthetapost$ denote the
posterior distribution of $\theta$, as given by Bayes' Theorem:
\begin{eqnarray*}
\pthetapost\left(\theta\right) :=
p\left(\theta \vert x, \alpha \right) =
\frac{p\left(x \vert \theta \right) p\left(\theta \vert \alpha \right)}
{p\left(x\right)}.
\end{eqnarray*}
A typical end product of a Bayesian analysis might be a posterior expectation of
some function $g\left(\theta\right)$ (e.g., a mean or variance): $\epgtheta$, which is a
functional of $g$. We suppose that we have determined that the prior parameter
$\alpha$ belongs to some set $\mathcal{A}$, perhaps after expert prior
elicitation. Finding the extrema of $\epgtheta$ as $\alpha$ ranges over all of
$\mathcal{A}$ is intractable or difficult except in special cases
\citep{moreno:2012:globalrobustness}. An alternative is to examine how much
$\epgtheta$ changes locally in response to small perturbations in the value of
$\alpha$:
\begin{eqnarray}\label{eq:local_robustness}
\left. \frac{d\epgtheta}{d\alpha} \right|_{\alpha} \Delta \alpha
\end{eqnarray}
That is, we consider \emph{local robustness}
\citep{gustafson:2012:localrobustnessbook} properties in lieu of global ones.
When $\alpha$ is function-valued, we take \eq{local_robustness} to be a Gateaux
derivative. By calculating \eq{local_robustness} for all
$\Delta \alpha \in \mathcal{A} - \alpha$, we can estimate the robustness
of $\epgtheta$ in a small neighborhood of $\alpha$.
\section{Linear response variational Bayes and extensions}
\label{sec:lrvb_formulas}
We next review and extend linear response perturbations to a mean-field
variational Bayes posterior approximation \citep{giordano:2015:lrvb} in order to
quickly and easily evaluate \eq{local_robustness}. Let $\qthetapost$ denote the
variational approximation to posterior $\pthetapost$. Recall that $\qthetapost$
is an approximate distribution selected to minimize the Kullback-Liebler
divergence between $\pthetapost$ and $q$ across distributions $q$ in some class
$\mathcal{Q}$. We consider the case where the variational family, $\mathcal{Q}$,
is a class of products of exponential family distributions
\citep{bishop:2006:pattern}:
\begin{eqnarray}\label{eq:kl_minimization}
\qthetapost &:=&
\textrm{argmin}_{q \in \mathcal{Q}} \left\{S - L\right\}
\quad \textrm{for} \quad
\mathcal{Q} = \left\{q: q(\theta) = \prod_{k=1}^K q(\theta_k); \quad \forall k,
q(\theta_k) \propto \exp(\eta_k ^T \theta_k) \right\}
\nonumber\\
L &:=& \mathbb{E}q\left[ \log p\left(x \vert \theta \right)\right] +
\mathbb{E}q\left[ \log p\left(\theta \vert \alpha \right) \right]
,\; \quad
S := \mathbb{E}q\left[ \log q \left(\theta \right) \right]
\end{eqnarray}
We assume that $\qthetapost$, the solution to \eq{kl_minimization}, has interior
exponential family parameter $\eta_k$. In this case, $\qthetapost$ can be
completely characterized by its mean parameters, $m := \expectq{\theta}$
\citep{wainwright2008graphical}. One can perturb the objective in
\eq{kl_minimization} in the direction of a function $f$ of the mean parameter
$m$ by some amount $t$, where $t$ is a vector with length equal to the output of
$f$:
\begin{eqnarray}\label{eq:perturbed_elbo}
q_t & := &
\textrm{argmin}_{q \in \mathcal{Q}} \left\{S - L + f(m)^T t \right\}
\end{eqnarray}
\citep{giordano:2015:lrvb} showed that when $f(m) = m$, we can calculate the
local change in the mean of $q_t$ as $t$ varies:
\begin{eqnarray}\label{eq:basic_lrvb}
\left. \frac{d\mathbb{E}_{q_t}\left[\theta\right]}{dt^T} \right|_{t=0} =
\left(I - VH\right)^{-1} V =: \hat{\Sigma},
\quad \textrm{where } V := \textrm{Cov}_{\qthetapost}\left(\theta\right)
\textrm{ and } H := \frac{\partial^2 L}{\partial m \partial m^T}.
\end{eqnarray}
As shown in \app{functions}, if $f(m)$ and $h(m)$ are both smooth functions of
$m$, then
\begin{eqnarray}\label{eq:function_covariance}
\frac{d h(m_t)}{dt} = \nabla h ^T \hat\Sigma \nabla f
\end{eqnarray}
\eq{basic_lrvb} is the special case of \eq{function_covariance} where $h(m) =
f(m) = m$. In \citep{giordano:2015:lrvb}, the goal was to calculate a posterior
covariance estimate $\hat{\Sigma}$. Here, our goal is to calculate a measure of
robustness to changes in $\alpha$. Let $\alpha_t = \alpha + \Delta \alpha t$ be
the value of $\alpha$ perturbed in direction $\Delta \alpha$ by an infinitesimal
scalar amount $t$. $\Delta \alpha$ may be vector- or function-valued. Note that
$\mathbb{E}q\left[\log(p(\theta \vert \alpha))\right]$ from \eq{kl_minimization} is a
function of $m$, since it is an expectation with respect to $q$, which is
completely parameterized by $m$. Assuming that $p(\theta \vert \alpha)$ is a
smooth function of $\alpha$, a Taylor expansion in $\Delta \alpha t$ gives
\begin{eqnarray}\label{eq:f_definition}
\mathbb{E}q\left[\log(p(\theta \vert \alpha_t))\right] &=&
\mathbb{E}q\left[\log(p(\theta \vert \alpha))\right] +
\frac{d}{d\alpha^T} \mathbb{E}q\left[\log(p(\theta\vert\alpha))\right]
\Delta \alpha t + O(t^2) \Rightarrow \nonumber\\
f(m) &:=& \frac{d}{d\alpha^T} \mathbb{E}q\left[\log(p(\theta\vert\alpha))\right]
\Delta \alpha
\quad \textrm{ and } \quad h(m) := \expectq{g(\theta)}
\end{eqnarray}
With $f(m)$ and $h(m)$ defined as in \eq{f_definition}, \eq{function_covariance}
gives us the robustness measure \eq{local_robustness}. As in LRVB, these
derivatives are in fact the exact robustness of the variational posterior
expectations to prior perturbation. The extent to which it represents the true
prior sensitivity depends on the extent to which the MFVB means are good
estimates of the true posterior means.
\section{Robustness measures from LRVB}\label{sec:lrvb_robustness}
We now turn to calculating $f(m)$ from \eq{f_definition} for some common
cases. For simplicity, we will take $g(\theta) = \theta$.
First, consider a prior in the exponential family with sufficient statistics
$\pi(\theta)$.
\begin{eqnarray}\label{eq:finite_dim_perturbation}
\log p(\theta \vert \alpha) &=& \alpha^T \pi(\theta) \Rightarrow
f(m) = \expectq{\pi(\theta)} \Delta \alpha
\end{eqnarray}
Here, $\pi(\theta)$ is a vector of the same length as $\alpha$.
Note that $f(m)$ may be known exactly or estimated using Monte Carlo
simulation. The simplest case is when the priors are conditionally conjugate
for $p(x \vert \theta)$. In that case, $\pi(\theta) = \theta$, and
$\frac{d \expectq{\theta_i}}{d \alpha_j} = \hat\Sigma_{ij}$.
A more complex non-conjugate example is the LKJ prior on a covariance
matrix, which we explore in \mysec{experiments}.
Next, we consider changing the functional form of $p(\theta \vert \alpha)$,
taking $\Delta \alpha$ to be function-valued. We will focus on perturbations to
the prior marginals, since local robustness properties of functional
neighborhoods of the full posterior have bad asymptotic properties
\citep{gustafson:1996:localposterior}. Let $\theta_i$ be a subvector of $\theta$
whose marginal we will perturb. We assume that both the prior and variational
distribution factor across $\theta_i$:
\begin{eqnarray*}
\qthetapost(\theta) = q(\theta_i) q(\theta_{-i})
\quad\textrm{ and }\quad
p(\theta \vert \alpha) =
p(\theta_i \vert \alpha_i) p(\theta_{-i} \vert \alpha_{-i})
\end{eqnarray*}
where $-i$ denotes ${1,...,K} \setminus i$.
For simplicity of notation, assume without loss of generality
that the $i$ indices come first: $\theta^T = (\theta_i^T, \theta_{-i}^T)$
(Both $\qthetapost$ and the prior may factorize still further.)
In order to ensure that the perturbed prior is properly normalized,
we will shift an infinitesimal amount of prior mass from the original
$p(\theta_i \vert \alpha)$ to a density $p_c(\theta_i)$:
\begin{eqnarray}\label{eq:epsilon_contamination}
p(\theta_i \vert \alpha_i, \epsilon) =
(1 - \epsilon) p(\theta_i \vert \alpha_i) + \epsilon p_c(\theta_i)
\end{eqnarray}
This is known as $\epsilon$-contamination, and
its construction guarantees that the perturbed prior is properly normalized
\footnote{$\epsilon$-contamination is principally adopted for analytic
convenience, though it is an expressive class of perturbations
\citep{gustafson:1996:localposterior}. For more exotic perturbation classes,
which we do not consider here, see\citep{zhu:2011:bayesian}.}. By taking
$p_c(\theta_i) = \delta(\theta_i - \theta_{i0})$ to be a Dirac
delta function at $\theta_{i0}$, \eq{function_covariance} and
\eq{f_definition} give (see \app{robust_derivations}):
\begin{eqnarray}\label{eq:delta_function_sensitivity}
\frac{d \mathbb{E}q[\theta]}{ d \epsilon} &=&
\frac{\qthetapost(\theta_{i0})}{p(\theta_{i0} \vert \alpha)}
(I - VH)^{-1}
\left(\begin{array}{c}
\theta_{i0} - m_{i}\\
0
\end{array}\right)
\end{eqnarray}
This is known as an ``influence function''
\citep{gustafson:2012:localrobustnessbook}.
Note that $p(\theta_{i0} \vert \alpha)$ is known \textit{a priori}, and that
$\qthetapost(\theta_{i0})$ is a function of the moment parameters $m$, since $m$
entirely specifies $\qthetapost$. Viewed as a function of $\theta_0$,
\eq{delta_function_sensitivity} characterizes how much each moment parameter,
$m$, is affected by adding an infinitesimal amount of prior mass at
$\theta_{i0}$. By the linearity of the
derivative, one can use weighted combinations of delta functions and
\eq{delta_function_sensitivity} to estimate the sensitivity to any prior
function
\footnote{A closed form for $p_c(\theta_i)$ other than weighted
combinations of Dirac delta functions is given in \app{function_sensitivity}.
The influence function is closely related to the worst-case prior perturbation
within a metric ball in the space of prior functions
\citep{gustafson:1996:localposterior}. We show in \app{extreme} that LRVB also
gives a closed form for this worst-case perturbation.
\app{mcmc_intuition_comparison} provides some intuition by comparing the LRVB
results to the corresponding formulas for exact inference.}.
\section{Experiments}\label{sec:experiments}
We applied the methods above to a hierarchical model of microcredit
interventions in development economics \citep{meager:2015:microcredit}. One
output of the model is $\mu$ and $\tau$, top level parameters in a hierarchical
model that measure average site profitability and the effectiveness of
microcredit interventions, respectively. Here, we present the sensitivity of
these parameters to $\Lambda$, the information matrix of a normal prior on
$(\mu,\tau)$, and $\eta$, the concentration parameter in a non-conjugate LKJ
prior\citep{lewandowski:2009:lkj} on the covariance of $(\mu,\tau)$.
The left panel of \fig{MicrocreditMainText} shows the estimates from
\eq{finite_dim_perturbation} normalized by the posterior standard deviation. The
results are robust to $\eta$ but extremely non-robust to $\Lambda$. The second
panel compares the prediction of \eq{finite_dim_perturbation} to the actual
change in MCMC means to a small change in $\Lambda_{11}$. The results match
closely. The third panel shows \eq{delta_function_sensitivity}, the influence
function of the prior for $(\mu, \tau)$ on $\tau$. The ``X'' is the posterior
mean. Adding prior mass on only one side of the mean would be highly
influential, though it is hard to imagine such a prior representing an
\emph{a priori} belief.
We formed the LRVB estimates using JuMP\citep{JuMP:LubinDunningIJOC} and used
STAN\citep{stan-manual:2015} to generate MCMC samples. The VB and MCMC results are
nearly identical, indicating that the assumptions necessary for LRVB
hold. Generating one set of MCMC draws took 15 minutes, and the LRVB estimates,
including calculating all the reported sensitivity measures, took 45 seconds.
For more details, see \app{microcredit_experiment}.
\begin{knitrout}
\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{figure}\label{fig:MicrocreditMainText}
\end{figure}
\end{knitrout}
{\small
}
\appendix
\appendixpage
\section{Robust Bayes with MCMC}\label{app:mcmc}
There is an extensive literature on Robust Bayesian techniques, surveyed
in \citep{insua:2012:robust}. We focus on local robustness
techniques
\citep{gustafson:2012:localrobustnessbook, gustafson:1996:localposterior,
gustafson:1996:localmarginals}. In the original papers, many
authors focused on either theoretical results or models with special
structure that rendered robustness measures tractable.
Of MCMC, one of the founders of the field of Bayesian Robustness writes:
``The MCMC methodology was not directly compatible with many of the
robust Bayesian techniques that had been developed, so that it was
unclear how formal robust Bayesian analysis could be incorporated into the
future `Bayesian via MCMC' world.
Paradoxically, MCMC has dramatically increased the need for consideration
of Bayesian robustness, in that the modeling that is now routinely
utilized in Bayesian analysis is of such complexity that inputs (such as priors)
can be elicited only in a very casual fashion.''\citep{berger:2012:robust}
Another recent author adds:
``Surprisingly, despite considerable theoretical advances in formal
sensitivity analysis, it is barely used in every-day practice...
a formal robustness methodology which is feasible, fairly quick,
operating with low extra computing effort and provided by
default in a dedicated software, is strongly required.''
\citep{roos:2015:sensitivity}
A number of papers have proposed methods for performing robustness analyses
using MCMC techniques. \citep{gustafson:1996:localmarginals}, following many
previous theoretical works \citep{gustafson:2012:localrobustnessbook}, exchanges
the integral in a posterior expectation with the derivative with respect to
prior perturbations, giving a robustness estimate that can be evaluated from
MCMC samples. \citep{perez:2006:mcmc, perez:2006:sensitivity} extends this idea.
These approaches exploit importance sampling and / or closed forms for
derivatives or posterior densities, and care must be taken to control the
variance of the MCMC estimates. The papers
\citep{kass:1989:approximate,mcculloch:1989:local} make second-order
approximations to the log posterior and employ numerical techniques to calculate
robustness measures. \citep{roos:2015:sensitivity} uses a sophisticated
methodology to choose a grid of prior points at which they numerically estimate
the sensitivity using estimates of the posterior density.
\citep{bornn:2010:sequential} proposes a distinctive method based on particle
filtering in which particle weights are re-adjusted to produce draws from a
perturbed prior. The authors are unaware of any previous work applying robust
Bayes techniques in the context of variational methods.
The advantage of using robust Bayes with LRVB over these MCMC-based techniques
is simplicity and computational ease. Little extra code and no extra
approximations or assumptions beyond that required for LRVB are required to
compute the robustness measures below. LRVB robustness measures are the exact
sensitivity of the variational solution to changes in the prior, and they will
be accurate to the extent that the variational approximation to the posterior
mean of interest is accurate \citep{giordano:2015:lrvb}.
\section{LRVB covariance of functions}\label{app:functions}
Let us consider LRVB estimates of the covariances of functions of
natural parameters rather than the natural parameters themselves.
Suppose we have a function $\phi\left(\eta\right)$, and a variational
solution $q\left(m\right)$ where $m=\mathbb{E}q\left[\eta\right]$. Since
$q$ is fully parameterized by $m$, we can write
\begin{eqnarray*}
\mathbb{E}q\left[\phi\left(\eta\right)\right] & = & f\left(m\right)
\end{eqnarray*}
for some continuous $f\left(m\right)$. We can consider a perturbed
log likelihood that also includes $f\left(m\right)$:
\begin{eqnarray*}
\log p_{t} & = & \log p+t_{0}^{T}m+t_{f}f\left(m\right):=\log p+t^{T}m_{f}\\
t & := & \left(\begin{array}{c}
t_{0}\\
t_{f}
\end{array}\right)\\
m_{f} & := & \left(\begin{array}{c}
m\\
f\left(m\right)
\end{array}\right)
\end{eqnarray*}
As in \citep{giordano:2015:lrvb}, we use the fixed point equations:
\begin{eqnarray*}
E_{t} & := & E+t^{T}m_{f}\\
\frac{dE_{t}}{dm} & = & 0\Rightarrow\\
\frac{dE}{dm}+\left(\begin{array}{cc}
I & \nabla f\end{array}\right)\left(\begin{array}{c}
t_{0}\\
t_{f}
\end{array}\right) & = & 0\\
M\left(m\right) & := & \frac{\partial E}{\partial m}+m\\
M_{t}\left(m\right) & := & M\left(m\right)+\left(\begin{array}{cc}
I & \nabla f\end{array}\right)\left(\begin{array}{c}
t_{0}\\
t_{f}
\end{array}\right)\\
M_{t}\left(m^{*}\right) & := & m^{*}\textrm{ (definition of }m^{*}\textrm{)}\\
\frac{dm_{t}^{*}}{dt^{T}} & = &
\left.\frac{\partial M_{t}}{\partial m^{T}}\right|_{_{m=m_{t}^{*}}}
\frac{dm_{t}^{*}}{dt^{T}}+\frac{\partial M_{t}}{\partial t^{T}}\\
& = & \left(\left.\frac{\partial M}{\partial m^{T}}\right|_{_{m=m_{t}^{*}}}+
\frac{\partial}{\partial m^{T}}\left(\begin{array}{cc}
I & \nabla f\end{array}\right)\left(\begin{array}{c}
t_{0}\\
t_{f}
\end{array}\right)\right)\frac{dm^{*}}{dt^{T}}+\left(\begin{array}{cc}
I & \nabla f\end{array}\right)
\end{eqnarray*}
The term $\frac{\partial}{\partial m^{T}}\left(\begin{array}{cc}
I & \nabla f\end{array}\right)\left(\begin{array}{c}
t_{0}\\
t_{f}
\end{array}\right)$ is awkward, but it disappears when we evaluate at $t=0$,
giving
\begin{eqnarray*}
\frac{dm_{t}^{*}}{dt^{T}} & = &
\left(\left.\frac{\partial M}{\partial m^{T}}\right|_{_{m=m_{t}^{*}}}\right)
\frac{dm^{*}}{dt^{T}}+\left(\begin{array}{cc}
I & \nabla f\end{array}\right)\\
& = & \left(\frac{\partial^{2}E}{\partial m\partial m^{T}}+
I\right)\frac{dm^{*}}{dt^{T}}+\left(\begin{array}{cc}
I & \nabla f\end{array}\right)\\
\frac{dm^{*}}{dt^{T}} & = &
-\left(\frac{\partial^{2}E}{\partial m\partial m^{T}}\right)^{-1}
\left(\begin{array}{cc}I & \nabla f\end{array}\right)
\end{eqnarray*}
Recalling that
\begin{eqnarray*}
\frac{dm^{*}}{dt_{0}^{T}} & := & \hat{\Sigma}
\end{eqnarray*}
We can plug in to see that
\begin{eqnarray*}
\frac{dm^{*}}{dt_{f}^{T}} & = & \hat{\Sigma}\nabla f
\end{eqnarray*}
This means that the covariance of the natural sufficient statistics
with the function $\phi\left(\eta\right)$ are determined by a linear
combination of the LRVB covariance matrix.
A similar conclusion can be reached by considering the response of
the expectation of a quantity other than a natural parameter to a
generic perturbation. Consider perturbing the log likelihood by some
function $t_{g}g\left(m\right)$. Then by the reasoning above,
\begin{eqnarray*}
\frac{df\left(m\right)}{dt_{g}} & = & \frac{df}{dm^{T}}\frac{dm}{dt_{g}}
= \nabla f^{T}\hat{\Sigma}\nabla g
\end{eqnarray*}
This is \eq{function_covariance}, and represents the LRVB covariance between two
quantities with variational expectation $f\left(m\right)$ and $g\left(m\right)$
respectively. As in the present, that covariance can also be interpreted as the
sensitivity of $g(m)$ to a perturbation of the objective by $g(m)$.
\section{Robustness Derivations} \label{app:robust_derivations}
In this section, we derive results stated in \mysec{lrvb_robustness}.
For generality, when possible we will derive results for the full
vector $\theta$ rather than the sub-vector $\theta_i$ when the proof
would be identical for the subvector under the assumption that
$q(\theta) = q(\theta_i) q(\theta_{-i})$.
\subsection{Sensitivity to $\epsilon-$contamination}
For a given $p_c(\theta)$ in \eq{epsilon_contamination}, we can consider
$p(\theta \vert \alpha, \epsilon)$ to be a class of priors parameterized
by $(\alpha, \epsilon)$, and take $\epsilon = \Delta \alpha$ in
\eq{f_definition}. We then need to calculate
\begin{eqnarray*}
\left. \frac{d}{d\epsilon}
\mathbb{E}q\left[\log p(\theta \vert \alpha, \epsilon) \right] \right|_{\epsilon = 0} &=&
\mathbb{E}q\left[\left. \frac{d}{d\epsilon}
\log \left((1 - \epsilon) p(\theta \vert \alpha) + \epsilon p_c(\theta)\right)
\right|_{\epsilon = 0} \right] \\
&=& \mathbb{E}q\left[\frac{p_c(\theta)}{p(\theta \vert \alpha)} - 1\right] \\
\end{eqnarray*}
Since the variational solution is unaffected by adding constants to the ELBO,
we can take
\begin{eqnarray}\label{eq:epsilon_f_formula}
f(m) &:=& \mathbb{E}q\left[\frac{p_c(\theta)}{p(\theta \vert \alpha)}\right]
\end{eqnarray}
\subsection{Sensitivity to a function}\label{app:function_sensitivity}
We will calculate $\nabla f(m)$ using \eq{epsilon_f_formula} for a general
function $p_c(\theta)$ and then use this result to derive
\eq{delta_function_sensitivity} as a special case. In this section,
we rely on the fact that the variational distribution, $q(\theta)$, is
in the exponential family.
The directional derivative for a perturbation $p_c(\theta)$ is given by
the Taylor expansion of $\mathbb{E}q\left[\frac{p_c(\theta)}{p\left(\theta
\vert \alpha\right)}\right]$ in terms of the exponential family moment
parameters:
\begin{eqnarray}\label{eq:exepcted_general_function_perturb}
\frac{d}{dm}\mathbb{E}q\left[\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)}\right] &=&
V^{-1}\frac{d}{d\eta}\int\exp\left(\eta^{T}\theta-A\left(\theta\right)\right)
\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)}d\theta \nonumber\\
&=& V^{-1}\int q\left(\theta\right)\left(\theta-m\right)
\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)}d\theta \nonumber\\
&=& V^{-1}\mathbb{E}q\left[\left(\theta-m\right)
\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)}\right]
\end{eqnarray}
Taking $p_c(\theta) = \delta\left(\theta_{i} = \theta_{i0}\right)$ to
be a Dirac delta function gives \eq{delta_function_sensitivity}.
\subsection{Extremal derivative}\label{app:extreme}
The influence function is closely related to the worst-case prior perturbation
in a metric ball around the original prior, $p(\theta_i \vert \alpha_i)$. We
refer the reader to \citep{gustafson:1996:localposterior} for the background.
Given \eq{exepcted_general_function_perturb}, the proof for the variational case
is essentially identical.
First, to match \citep{gustafson:1996:localposterior}, let $p_c(\theta)$ be a
signed measure and consider perturbations of the form
\begin{eqnarray*}
p(\theta \vert \alpha, \epsilon) = p(\theta \vert \alpha) + \epsilon p_c(\theta)
\end{eqnarray*}
Because the variational solution is invariant to constants, the variational
sensitivity to this perturbation is identical to that of $\epsilon-$
contamination. Consequently, the sensitivity is given by
\eq{exepcted_general_function_perturb} and \eq{function_covariance}:
\begin{eqnarray*}
\frac{d\mathbb{E}q\left[g(\theta)\right]}{dt} & = &
\nabla h ^{T}\hat{\Sigma}V^{-1}\mathbb{E}q\left[\left(\theta-m\right)
\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)}\right]\\
& = & \mathbb{E}q\left[\nabla h ^{T} \left(I-VH\right)^{-1}\left(\theta-m\right)
\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)}\right]
\end{eqnarray*}
Define
\begin{eqnarray*}
a\left(\theta\right) & = &
\nabla h ^{T} \left(I-VH\right)^{-1}\left(\theta-m\right)
\frac{q\left(\theta\right)}{p\left(\theta \vert \alpha\right)}
\end{eqnarray*}
As in \citep{gustafson:1996:localposterior},
for $p \in [1, \infty]$ and $\frac{1}{p}+\frac{1}{q}=1$, define the size
of a perturbation as
\begin{eqnarray} \label{eq:perturb_size}
\left(\int \left| \frac{ p_c(\theta)}{p(\theta \vert \alpha)} \right| ^p
d\Pi\right)^{\frac{1}{p}}
\end{eqnarray}
...where $\Pi$ is the measure on $\theta$ induced by $p(\theta \vert \alpha)$
and $p \in [1, \infty]$. Let $(\cdot)^{+}$ denote the positive part and
$(\cdot)^{-}$ the negative part of the term in the parentheses.
\begin{eqnarray*}
\mathbb{E}q\left[\left|R^{T}\left(I-VH\right)^{-1}\left(\theta-m\right)
\frac{q\left(\theta\right)}{p\left(\theta \vert \alpha\right)}\right|^{+}\right] & = &
\int\left|a\left(\theta\right)^{+}
\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)}\right|d\Pi\\
& \le & \left(\int\left|a\left(\theta\right)^{+}\right|^{q}
d\Pi\right)^{\frac{1}{q}}
\left(\int\left|\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)}
\right|^{p}d\Pi\right)^{\frac{1}{p}}\\
& = & \left(\int\left|a\left(\theta\right)^{+}\right|^{q}d\Pi\right)^{\frac{1}{q}}
\end{eqnarray*}
Since we are taking $p_c(\theta)$ such that
$\|\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)};\Pi\|_{p}=1$.
This is maximized when
\begin{eqnarray*}
\left|a\left(\theta\right)^{+}\right|^{q} & \propto &
\left|\frac{p_c(\theta)}{p\left(\theta \vert \alpha\right)}\right|^{p}\\
p_c(\theta) & = & \pi\left|\left(R^{T}\left(I-VH\right)^{-1}
\left(\theta-m\right)\right)^{+}
\frac{q\left(\theta\right)}{p\left(\theta \vert \alpha\right)}\right|^{\frac{q}{p}}
\end{eqnarray*}
A similar analysis follows for $a(\theta)^{-}$, and it follows that the
worst-case prior perturbation in a $p-$neighborhood of $p(\theta \vert \alpha)$
is given by
\begin{eqnarray}\label{eq:extremal_perturbation}
p_c(\theta) &=&
p(\theta \vert \alpha)
\max\left\{\left|a(\theta)^{+}\right|^{\frac{1}{p-1}},
\left|a(\theta)^{-}\right|^{\frac{1}{p-1}} \right\}
\end{eqnarray}
\subsection{Comparison with Exact Results}\label{app:mcmc_intuition_comparison}
Comparing \eq{extremal_perturbation} with
\citep[Equation~6]{gustafson:1996:localposterior} lends some intuition.
In our notation, the exact extremal perturbation is given by
\eq{gustafson_extremal} by the same expression as \eq{extremal_perturbation}
but with a different $a(\theta)$:
\begin{eqnarray}\label{eq:gustafson_extremal}
a_p\left(\theta\right) &=&
g(\theta) \left(\theta - \expectp{\theta} \right)
\frac{p\left(\theta \vert x\right)}{p(\theta \vert \alpha)}
\end{eqnarray}
Here, $\qthetapost$ plays the role of the marginal posterior $p(\theta \vert x)$,
and $\expectq{g(\theta)}^T (I-VH)^{-1}$ plays the role of $g(\theta)$.
Note that a principal difficulty of using
\eq{gustafson_extremal} is that \eq{gustafson_extremal}
requires knowledge of ratio of the posterior density to the prior
density, which is not automatically available from MCMC draws. The
MFVB solution circumvents this difficulty by providing an explicit parametric
approximation to the posterior density.
\section{Microcredit Model}\label{app:microcredit_experiment}
\marginpar{NEW}command{15.01}{15.01}
\marginpar{NEW}command{0.02}{0.02}
\marginpar{NEW}command{20.01}{20.01}
\marginpar{NEW}command{20.01}{20.01}
\marginpar{NEW}command{2.01}{2.01}
\marginpar{NEW}command{2.01}{2.01}
We will reproduce a variant of the analysis performed in
\citep{meager:2015:microcredit}, though with somewhat different prior choices.
Randomized controlled trials were run in seven different sites to try to measure
the effect of access to microcredit on various measures of business success.
Each trial was found to lack power individually for various reasons, so there
could be some benefit to pooling the results in a simple hierarchical model. For
the purposes of demonstrating robust Bayes techniques with VB, we will focus on
the simpler of the two models in \citep{meager:2015:microcredit} and ignore
covariate information.
We will index sites with $k=1,..,K$ (here, $K=7$) and business within a site by
$i=1,...,N_k$. In site $k$ and business $i$ we observe whether the business was
randomly selected for increased access to microcredit, denoted $T_{ik}$, and the
profit after intervention, $y_{ik}$. We follow \citep{rubin:1981:estimation} and
assume that each site has an idiosyncratic average profit, $\mu_k$ and average
improvement in profit, $\tau_k$, due to the intervention. Given $\mu_k$,
$\tau_k$, and $T_{ik}$, the observed profit is assumed to be generated according
to
\begin{eqnarray*}
y_{ik} \vert \mu_k, \tau_k, T_{ik}, \sigma_{k} &\sim&
N\left(\mu_k + T_{ik} \tau_k, \sigma^2_{k} \right)
\end{eqnarray*}
The site effects, $(\mu_k, \tau_k)$, are assumed to come from an overall
pool of effects and may be correlated:
\begin{eqnarray*}
\left( \begin{array}{c} \mu_k \\ \tau_k \end{array}\right) &\sim&
N\left(
\left( \begin{array}{c} \mu \\ \tau \end{array}\right), C \right) \\
C &:=&
\left( \begin{array}{cc}
\sigma^2_\mu & \sigma_{\mu\tau} \\
\sigma_{\mu\tau} & \sigma^2_\tau
\end{array}\right)
\end{eqnarray*}
The effects $\mu$, $\tau$, and the covariance matrix $V$ are unknown parameters
that require priors. For $(\mu, \tau)$ we simply use a bivariate normal prior.
However, choosing an appropriate prior for a covariance matrix can be
conceptually difficult \citep{barnard:2000:modeling}. Following the recommended
practice of the software package STAN\citep{stan-manual:2015}, we derive a
variational model to accommodate the non-conjugate LKJ prior
\cite{lewandowski:2009:lkj}, allowing the user to model the covariance and
marginal variances separately. Specifically, we use
\begin{eqnarray*}
C &=:& SRS\\
S &=& \textrm{Diagonal matrix}\\
R &=& \textrm{Covariance matrix}\\
S_{kk} &=& \sqrt{\textrm{diag}(C)_k}\\
\end{eqnarray*}
We can then put independent priors on the scale of the variances, $S_{kk}$,
and on the covariance matrix, $R$. We model the inverse of $C$ with a Wishart
variational distribution, and use the following priors:
\begin{eqnarray*}
q\left(C^{-1}\right) &=& \textrm{Wishart}(V_\Lambda, n)\\
p\left(S\right) &=& \prod_{k=1}^{2} p(S_{kk})\\
S_{kk}^{2} &\sim& \textrm{InverseGamma}(\alpha_{scale}, \beta_{scale})\\
\log p(R) &=& (\eta - 1) \log |R| + C\\
\end{eqnarray*}
The necessary expectations have closed forms with the Wishart variational
approximation, as derived in \app{lkj}.
In addition, we put a normal prior on $(\mu, \tau)^T$ and an inverse
gamma prior on $\sigma_k^2$:
\begin{eqnarray*}
\left(\begin{array}{cc} \mu \\ \tau \end{array}\right)
&\sim& N \left(\left(\begin{array}{cc} 0 \\ 0 \end{array}\right), \Lambda^{-1} \right)\\
\sigma_{k}^{2} &\sim&
\textrm{InverseGamma}(\alpha_\tau, \beta_\tau) \\
\end{eqnarray*}
The prior parameters used were:
\begin{eqnarray*}
\Lambda &=& \left( \begin{array}{cc}
0.02 & 0 \\
0 & 0.02
\end{array}\right) \\
\eta &=& 15.01 \\
\sigma_{k}^{-2} &\sim&
\textrm{InverseGamma}(2.01, 2.01) \\
\alpha_{scale} &=& 20.01 \\
\beta_{scale} &=& 20.01 \\
\alpha_{\tau} &=& 2.01 \\
\beta_{\tau} &=& 2.01
\end{eqnarray*}
\subsection{Results}
First, note that the the MCMC results match the VB means very closely,
indicating that the assumptions underlying LRVB are satisfied. The least-
well estimated parameters are $C^{-1}$.
\begin{knitrout}
\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{figure}\label{fig:MicrocreditMCMCComparison}
\end{figure}
\end{knitrout}
We will focus on the robustness of $\mu$ and $\tau$, since as the higher-level
parameters in the hierarchical model, they are both more susceptible to prior
influence and more generally interpretable (as the average profit and
the causal effect of microcredit, respectively).
The sensitivity of $(\mu, \tau)$ to $\Lambda$ and $\eta$ is shown in
the left panel of \fig{MuGraph} as a proportion of the LRVB posterior
standard deviation. The parameters can be seen to be quite sensitive
to changes in $\Lambda$. For example, if the upper left component
of $\Lambda$, $\Lambda_{11}$, were to increase by $0.04$, $\mathbb{E}q[\mu]$ would
be expected to increase by two posterior standard deviations. If $0.06$
is a subjectively reasonable value for $\Lambda_{11}$, then the ordinary
posterior confidence interval for $\mu$ is quite inadequate in capturing
the subjective range of beliefs that might be assigned to $\mu$. In contrast,
the sensitivty to $\eta$, the LKJ parameter, is quite small.
The right panel of \fig{MuGraph} shows the influence function of $(\mu, \tau)$
on $\tau$. The $X$ marks the posterior mean. Recall that the prior mean
is $(0,0)$ and relatively diffuse. The numbers are quite large,
indicating that adding a small amount of prior mass precisely near the
posterior could influence the posterior considerably. However, such
a prior perturbation would have to have informed by the data -- adding
mass nearly anywhere else would have a much smaller effect. What kind
of prior perturbation is reasonable remains a subjective decision of the
modeler.
\begin{knitrout}
\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{figure}\label{fig:MuGraph}
\end{figure}
\end{knitrout}
Finally, \fig{MicrocreditPerturbation} shows the effects of changing
$\Lambda_{11}$ on a re-run MCMC chain compared with the effects predicted
by LRVB robustness measurements. The results are very good for all except
$C^{-1}$, which was not estimated well by the VB model. Even for
$C^{-1}$, the LRVB estimates are directionally correct.
\begin{knitrout}
\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{figure}\label{fig:MicrocreditPerturbation}
\end{figure}
\end{knitrout}
\section{LKJ Priors for Covariance Matrices in Mean Field Variational Inference}\label{app:lkj}
In this section we briefly derive closed form expressions for using an
LKJ prior with a Wishart variational approximation.
We want to estimate a multivariate normal covariance matrix with flexible
priors. For simplicity, let us study in isolation the model:
\begin{eqnarray*}
\log p\left(y\vert\Lambda\right) & = & -\frac{1}{2}y^{T}\Lambda y+\frac{1}{2}\log\Lambda\\
\Lambda & = & \Sigma^{-1}\\
\Sigma & =: & SRS\\
S_{k} & := & \sqrt{diag\left(\Sigma\right)_{k}}\\
\log p\left(S\right) & = & \sum_{k=1}^{K}\log p\left(S_{k}\right)\\
\log p\left(R\right) & = & \log\left(C\left|R\right|^{\eta-1}\right)\\
& = & \left(\eta-1\right)\log\left|R\right|+C\\
& = & \textrm{ (LKJ prior)}
\end{eqnarray*}
Let us use a Wishart variational distribution for $\Lambda$:
\begin{eqnarray*}
q\left(\Lambda\right) & = & \textrm{Wishart}\left(V,n\right)\\
E_{q}\left[\Lambda\right] & = & nV\\
E_{q}\left[\log\left|\Lambda\right|\right] & = & \psi_{p}\left(\frac{n}{2}\right)+\log\left|V\right|+K\log2\\
\psi_{p}\left(n\right) & = & \sum_{i=1}^{p}\psi\left(\frac{2n+1-i}{2}\right)
\end{eqnarray*}
Then $\Sigma$ has an inverse Wishart distribution:
\begin{eqnarray*}
E_{q}\left[\Sigma\right] & = & \frac{V^{-1}}{n-K-1}\\
\Sigma_{kk} & \sim & \textrm{InverseWishart}\left(\left(V^{-1}\right)_{kk},n-K+1\right)\\
E_{q}\left[\Sigma_{kk}\right] & = & \frac{\left(V^{-1}\right)_{kk}}{n-K+1-2}=\frac{\left(V^{-1}\right)_{kk}}{n-K-1}\\
\log p\left(\Sigma_{kk}\right) & = & -\left(\frac{\left(n-K+1\right)+1+1}{2}\right)\log\Sigma_{kk}-\frac{1}{2}\frac{\left(V^{-1}\right)_{kk}}{\Sigma_{kk}}+C\\
& = & \left(-\frac{n-K+1}{2}-1\right)\log\Sigma_{kk}-\frac{\frac{1}{2}\left(V^{-1}\right)_{kk}}{\Sigma_{kk}}+C\\
& = & \log\left(\textrm{InvGamma}\left(\frac{n-K+1}{2},\frac{1}{2}\left(V^{-1}\right)_{kk}\right)\right)\Rightarrow\\
E_{q}\left[\log\Sigma_{kk}\right] & = & \log\left(\frac{1}{2}\left(V^{-1}\right)_{kk}\right)-\psi\left(\frac{n-K+1}{2}\right)
\end{eqnarray*}
We'll also need the expectation of the square root of an inverse gamma
distributed variable.
\begin{eqnarray*}
p\left(x\right) & = & \frac{\beta^{\alpha}}{\Gamma\left(\alpha\right)}x^{-\alpha-1}\exp\left(\frac{-\beta}{x}\right)\\
E\left[x^{\frac{1}{2}}\right] & = & \int\frac{\beta^{\alpha}}{\Gamma\left(\alpha\right)}x^{-\alpha-1+\frac{1}{2}}\exp\left(\frac{-\beta}{x}\right)dx\\
& = & \int\frac{\beta^{\alpha}}{\Gamma\left(\alpha\right)}\frac{\beta^{\alpha-\frac{1}{2}}}{\Gamma\left(\alpha-\frac{1}{2}\right)}\frac{\Gamma\left(\alpha-\frac{1}{2}\right)}{\beta^{\alpha-\frac{1}{2}}}x^{-\left(\alpha-\frac{1}{2}\right)-1}\exp\left(\frac{-\beta}{x}\right)dx\\
& = & \frac{\beta^{\alpha}}{\Gamma\left(\alpha\right)}\frac{\Gamma\left(\alpha-\frac{1}{2}\right)}{\beta^{\alpha-\frac{1}{2}}}\\
& = & \beta^{\frac{1}{2}}\frac{\Gamma\left(\alpha-\frac{1}{2}\right)}{\Gamma\left(\alpha\right)}
\end{eqnarray*}
Thus
\begin{eqnarray*}
E_{q}\left[\sqrt{\Sigma_{kk}}\right]=E_{q}\left[S_{k}\right] & = & \sqrt{\frac{1}{2}\left(V^{-1}\right)_{kk}}\frac{\Gamma\left(\frac{n-K}{2}\right)}{\Gamma\left(\frac{n-K+1}{2}\right)}
\end{eqnarray*}
This means we have a closed form expectation of the LKJ prior. For
the scale parameters, we can use a gamma prior distribution:
\begin{eqnarray*}
\log p\left(S_{k}\right) & = & \log\Gamma\left(\alpha,\beta\right)\\
& = & -\beta S_{k}+\left(\alpha-1\right)\log S_{k}+C\\
& = & -\beta S_{k}+\frac{\left(\alpha-1\right)}{2}\log S_{k}^{2}+C
\end{eqnarray*}
Finally, these expectations are given in terms of the natural parameters, but
for LRVB we need derivatives with respect to the mean parameters. In the Wishart
distribution, the mapping from mean parameters to natural parameters does not
have a closed form. \eq{basic_lrvb} requires the derivatives of the likelihood
with respect to the moment parameters, and the Hessian must be transformed
before use. Note that the Hessian of the likelihood is not necessarily at a
maximum, so the transform requires a third-order tensor product.
\end{document} |
\begin{document}
\title[{Some summation theorems for truncated Clausen series}]{Some summation theorems for truncated Clausen series and applications}
\author[M.I. Qureshi, Saima Jabee$^{*}$ and Dilshad Ahamad]{M.I. Qureshi, Saima Jabee$^{*}$ and Dilshad Ahamad}
\address{M.I. Qureshi: Department of Applied Sciences and Humanities,
Faculty of Engineering and Technology,
Jamia Millia Islamia (A Central University),
New Delhi 110025, India.}
\email{miqureshi\[email protected]}
\address{Saima Jabee: Department of Applied Sciences and Humanities,
Faculty of Engineering and Technology,
Jamia Millia Islamia (A Central University),
New Delhi 110025, India. }
\email{[email protected]}
\address{Dilshad Ahamad: Department of Applied Sciences and Humanities,
Faculty of Engineering and Technology,
Jamia Millia Islamia (A Central University),
New Delhi 110025, India.}
\email{[email protected]}
\keywords{Watson summation theorem; Whipple summation theorem; Dixon summation theorem; Saalsch\"{u}tz summation theorem; Truncated series; Hypergeometric summation theorems; Mellin transforms.}
\subjclass[2010]{33C05, 33C20, 44A10.}
\thanks{*Corresponding author}
\begin{abstract}
The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen's hypergeometric series with unit argument, when one numerator parameter and one denominator parameter are negative integers. Further, using our truncated summation theorems, we obtain the Mellin transforms of the product of exponential function and Goursat's truncated hypergeometric function.
\end{abstract}
\begin{center}
\today
\end{center}
\maketitle
{
\section{Introduction}
In our investigations, we shall use the following standard notations:\\
$\mathbb{N}:=\{1,2,3,\dots\}$; $\mathbb{N}_0:=\mathbb{N}\bigcup\{0\}$; $\mathbb{Z}_0^-:=\mathbb{Z}^-\bigcup\{0\}=\{0,-1,-2,-3,\dots\}$.\\
The symbols $\mathbb{C}$, $\mathbb{R}$, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}^+$ and $\mathbb{R}^-$ denote the sets of complex numbers, real numbers, natural numbers, integers, positive and negative real numbers respectively.\\
The Pochhammer symbol $(\alpha)_{p}$ ~$(\alpha, p \in\mathbb{C})$ (\cite[p.22 eq(1), p.32 Q.N.(8) and Q.N.(9)]{Rainville}, see also \cite[p.23, eq(22) and eq(23)]{Srivastava3}), is defined by
\begin{equation}\label{f.eq(g1)}
(\alpha)_{p}:=\frac{\Gamma(\alpha+p)}{\Gamma(\alpha)}=
\begin{cases}
$1$ & ;(p=0; \alpha \in \mathbb {C}\setminus \{0\})\\
\alpha (\alpha+1)\ldots (\alpha+n-1) & ;(p=n \in \mathbb {N}; \alpha \in \mathbb {C}\setminus {\mathbb{Z}_0^-})\\
\frac{(-1)^{n}k!}{(k-n)!} & ;(\alpha=-k; p=n; n,k \in \mathbb{N}_0; {0}\leq{n}\leq{k})\\
$0$ & ;(\alpha=-k; p=n; n,k \in \mathbb{N}_0;{n}>{k})\\
\frac{(-1)^{n}}{(1-\alpha)_n} & ;(p=-n; n\in \mathbb{N}; \alpha \in \mathbb{C}\setminus \mathbb{Z}),
\end{cases}
\end{equation}
it being understood conventionally that $(0)_0=1$ and assumed tacitly that the Gamma quotient exists.\\
The generalized hypergeometric function ${_p}F_q$ (\cite[Art.44, pp.73-74]{Rainville}, see also \cite{Bailey}), is defined by
\begin{eqnarray}\label{f.eq(g2)}
{_p}F_{q}\left[\begin{array}{r} \alpha_1, \alpha_2, \dots, \alpha_p;\\
~\\ \beta_1, \beta_2, \dots, \beta_q;\end{array}\ z\right]={_p}F_{q}\left[\begin{array}{r} (\alpha_p);\\
~\\ (\beta_q);\end{array}\ z\right]=\sum_{n=0}^{\infty}\frac{\displaystyle\prod_{j=1}^{p}(\alpha_j)_n}{\displaystyle\prod_{j=1}^{q}(\beta_j)_n}\frac{z^n}{n!}.
\end{eqnarray}
By convention, a product over the empty set is unity.\\
$\big(p, q \in \mathbb{N}_0;~ p\leqq{q+1}~;~ p\leqq{q}~ \text{and}~ |z|<\infty ;\big.$
$~\big. p=q+1 ~\text{and}~ |z|<1;~ p=q+1, |z|=1~\text{and}~\Re(\omega)>0;~p=q+1, |z|=1, z\neq 1~\text{and}~ -1< \Re(\omega) \leq0\big)$,\\
where \[\omega:=\sum_{j=1}^{q}{\beta}_j-\sum_{j=1}^{p}{\alpha}_j, \] \[\big(\alpha_j\in \mathbb{C}~(j=1, 2,\dots,p ); \beta_j\in\mathbb{C}\setminus\mathbb{Z}_0^-(j=1, 2, \dots, q) \big),\]
where $\Re$ denotes the real part of complex number throughout the paper.\\
A finite series identity (reversal of the order of terms in finite summation) is given by
\begin{eqnarray}\label{eq(g14)}
\sum_{n=0}^{m}\Phi(n)=\sum_{n=0}^{m}\Phi(m-n);\quad{m\in\mathbb{N}_0}.
\end{eqnarray}
The truncated hypergeometric series is given by:
\begin{eqnarray}\label{f.eq(g16)}
&&\text{The sum of the first (m+1)-terms of infinite series } {_{p}}F_{q}\left[\begin{array}{r} (\alpha_p);\\
~\\ (\beta_q);\end{array}\ z\right]\nonumber\\
&&\qquad\qquad={_{p}}F_{q}\left[\begin{array}{r} (\alpha_p);\\
~\\ (\beta_q);\end{array}\ z\right]_{m}=\sum_{n=0}^{m}\frac{\displaystyle\prod_{j=1}^{p}(\alpha_j)_n}{\displaystyle\prod_{j=1}^{q}(\beta_j)_n}\frac{z^n}{n!}\nonumber\\
&&\qquad\qquad=\frac{[(\alpha_p)]_{m}z^{m}}{[(\beta_q)]_{m}m!}{_{q+2}}F_{p}\left[\begin{array}{r} -m, 1-(\beta_{q})-m,1;\\
~\\ 1-(\alpha_p)-m;\end{array}\ \frac{(-1)^{p+q+1}}{z}\right],
\end{eqnarray}
where $(\alpha_{p}), (\beta_{q}), 1-(\alpha_{p})-m, 1-(\beta_{q})-m\in\mathbb{C}\setminus\mathbb{Z}_0^-$; $m\in\mathbb{N}_0$, and
\begin{eqnarray}
[(\alpha_p)]_{m}=(\alpha_1)_{m}(\alpha_2)_{m}\dots(\alpha_p)_{m}=\prod_{i=1}^{p}(\alpha_i)_{m}=\prod_{i=1}^{p}\frac{\Gamma(\alpha_i+m)}{\Gamma(\alpha_i)},
\end{eqnarray}
with similar interpretation for others.\\
The terminating hypergeometric series (the hypergeometric polynomial) is given by
\begin{eqnarray}\label{eq(g15)}
{_{p+1}}F_{q}\left[\begin{array}{r} -m, (\alpha_p);\\
~\\ (\beta_q);\end{array}\ z\right]&=&\frac{[(\alpha_p)]_{m}(-z)^{m}}{[(\beta_q)]_{m}}{_{q+1}}F_{p}\left[\begin{array}{r} -m, 1-(\beta_{q})-m;\\
~\\ 1-(\alpha_p)-m;\end{array}\ \frac{(-1)^{p+q}}{z}\right],\nonumber\\
\end{eqnarray}
where $(\alpha_{p}), (\beta_{q}), 1-(\alpha_{p})-m, 1-(\beta_{q})-m\in\mathbb{C}\setminus\mathbb{Z}_0^-$ and $m\in\mathbb{N}_0$.\\
If $\ell> m$; $m,\ell\in\mathbb{N}; \alpha,\beta,\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^{-}$, then series ${_3}F_2\left[\begin{array}{r} -m, \alpha,\beta;\\
~\\ -\ell,\gamma;\end{array}\ z\right]$ is an infinite series and is given by the following series representation (see for example \cite[p.41, eq.(3.1.26); p.42, eq.(3.2.6)]{Luke} and \cite[p.438, eq.(7.2.3.5)]{Prudnikov2})
\begin{eqnarray}
{_3}F_2\left[\begin{array}{r} -m, \alpha,\beta;\\
~\\ -\ell,\gamma;\end{array}\ z\right]&=&\sum_{r=0}^{m}\frac{(-m)_r(\alpha)_r(\beta)_rz^r}{(-\ell)_r(\gamma)_rr!}+\sum_{r=\ell+1}^{\infty}\frac{(-m)_r(\alpha)_r(\beta)_rz^r}{(-\ell)_r(\gamma)_rr!}\nonumber\\
&&={_3}F_2\left[\begin{array}{r} -m, \alpha,\beta;\\
~\\ -\ell,\gamma;\end{array}\ z\right]_m+\sum_{r=\ell+1}^{\infty}\frac{(-m)_r(\alpha)_r(\beta)_rz^r}{(-\ell)_r(\gamma)_rr!}.
\end{eqnarray}
In original notation, the higher order Goursat hypergeometric function is represented by double integral \cite[p. 286]{Goursat}. So we have
\begin{eqnarray}\label{eq(1)}
G\left(\begin{array}{r} \alpha,\beta;\\ \gamma, \delta;\end{array}z\right)&=&\frac{\Gamma{(\gamma)}\Gamma{(\delta)}}{\Gamma{(\alpha)}\Gamma{(\beta)}\Gamma{(\gamma-\alpha)}\Gamma{(\delta-\beta)}}\times\nonumber\\
&&\times\int_{0}^{1}\int_{0}^{1}u^{\alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\delta-\beta-1}e^{zuv}dudv,
\end{eqnarray}
where $\Re{(\gamma)}>\Re{(\alpha)}>0$, $\Re{(\delta)}>\Re{(\beta)}>0$,\\
and
\begin{eqnarray*}
G\left(\begin{array}{r} \alpha,\beta;\\ \gamma, \delta;\end{array}z\right)&=&1+\sum_{n=1}^{\infty}\frac{(\alpha)_n(\beta)_n z^n}{(\gamma)_n(\delta)_n n!}\\
&=&{_2}F_{2}\left[\begin{array}{r} \alpha,\beta;\\ \gamma, \delta;\end{array}z\right],
\end{eqnarray*}
where $\gamma,\delta\in\mathbb{C}\setminus\mathbb{Z}_0^{-}$ and $|z|<\infty$.\\
It is also well known that, under certain conditions, the Goursat's function \cite[p. 286]{Goursat} ${_2}F_2(\alpha, \beta; \gamma, \delta; z)$ is defined by
\begin{eqnarray}\label{eq(2)}
{_2F_2} \left[\begin{array}{r} \alpha,\beta;\\ \gamma, \delta;\end{array}z\right]=\frac{\Gamma{(\delta)}}{\Gamma{(\alpha)}\Gamma{(\delta-\alpha)}}\int_{0}^{1}v^{\alpha-1}(1-v)^{\delta-\alpha-1}{_1F_1} \left[\begin{array}{r} \beta;\\ \gamma;\end{array}zv\right]dv,
\end{eqnarray}
where $\Re{(\delta)}>\Re{(\alpha)}>0$ and ${_1}F_{1}(\cdot)$ is Kummer's confluent hypergeometric function.\\
An integral transform that may be considered as the multiplicative form of the two-sided Laplace transform is known as Mellin transform, which is closely related to the Fourier transform, Laplace transform and other transforms. The Mellin transform is defined by
\begin{eqnarray}\label{eq(1.9)}
\mathcal{M}\{f(t);s\}=\int_{0}^{\infty}t^{s-1}f(t)dt=g(s),
\end{eqnarray}
where $s$ is a complex variable, above integral exists with suitable convergence conditions.\\
Until 1990, only few classical summation theorems for ${_2}F_1$ and ${_3}F_2$ were known. Subsequently, some progress has been made in generalizing these classical summation theorems (see \cite{Kim, Lavoie1, Lavoie2, Lavoie3, Miller, Rakha1, Rakha2}).\\
\section{Summation theorems for non-terminating, terminating and truncated clausen series }
In this section, we have verified the following terminating and truncated Clausen summation theorems by taking suitable values of parameters. So, without any loss of convergence, we can relax convergence conditions in some cases.\\
The classical Watson's summation theorem for non-terminating Clausen's hypergeometric series of unit argument \cite[p.16, section 3.3(1)]{Bailey} takes the form
\begin{eqnarray}\label{eq(2.1)}
{_3F_2} \left[\begin{array}{r} \alpha,\beta, \gamma;\\ \frac{1+\alpha+\beta}{2}, 2\gamma;\end{array}1\right] &=& \frac{\Gamma{\left(\frac{1}{2}\right)}\Gamma{\left(\gamma+\frac{1}{2}\right)}\Gamma{\left(\frac{1+\alpha+\beta}{2}\right)}\Gamma{\left(\gamma+\frac{1-\alpha-\beta}{2}\right)}}{\Gamma{\left(\frac{1+\alpha}{2}\right)}\Gamma{\left(\frac{1+\beta}{2}\right)}\Gamma{\left(\gamma+\frac{1-\alpha}{2}\right)}\Gamma{\left(\gamma+\frac{1-\beta}{2}\right)}},
\end{eqnarray}
provided $\Re(\gamma+\frac{1-\alpha-\beta}{2})>0; \frac{1+\alpha+\beta}{2}, \gamma, 2\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^{-}$ and parameters are adjusted in such a way that the series on the left-hand side is well defined.\\
When $\alpha=-2m$ in equation \eqref{eq(2.1)}, we get a Watson's summation theorem for terminating hypergeometric series (containing (2m+1)-terms)
\begin{eqnarray}\label{eq(2.2)}
{_3F_2} \left[\begin{array}{r} -2m,\beta, \gamma;\\ \frac{1-2m+\beta}{2}, 2\gamma;\end{array}1\right] &=& \frac{\left(\frac{1}{2}\right)_m\left(\gamma+\frac{1-\beta}{2}\right)_m}{\left(\gamma+\frac{1}{2}\right)_m\left(\frac{1-\beta}{2}\right)_m},
\end{eqnarray}
where $\beta,\gamma, 2\gamma, \frac{1+\beta}{2}-m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}$; $m\in\mathbb{N}$.\\
When $\alpha=-2m-1$ in equation \eqref{eq(2.1)}, we get another Watson's summation theorem for terminating hypergeometric series (containing-(2m+2) terms)
\begin{eqnarray}\label{eq(2.3)}
{_3F_2} \left[\begin{array}{r} -2m-1,\beta, \gamma;\\ \frac{\beta-2m}{2}, 2\gamma;\end{array}1\right] &=& 0,
\end{eqnarray}
where $\beta, \gamma, 2\gamma, \frac{\beta-2m}{2}\in\mathbb{C}\setminus\mathbb{Z}_0^{-}$; $m\in\mathbb{N}$.\\
We recall a Watson's summation theorem for truncated Clausen's series (containing (m+1)-terms) \cite[p.238, eq(2.2)]{Bailey1}
\begin{eqnarray}\label{eq(2.4)}
{_3F_2} \left[\begin{array}{r} -m,\alpha,\beta;\\ -2m,\frac{1+\alpha+\beta}{2};\end{array}1\right]_m &=& \frac{\left(\frac{1+\alpha}{2}\right)_m\left(\frac{1+\beta}{2}\right)_m}{\left(\frac{1}{2}\right)_m\left(\frac{1+\alpha+\beta}{2}\right)_m},
\end{eqnarray}
where $\alpha,\beta,\frac{1+\alpha+\beta}{2}\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m\in\mathbb{N}$.\\
On setting $\gamma=-m-k-\frac{1}{2}$ in equation \eqref{eq(2.2)}, we obtain Watson's summation theorem for truncated Clausen's series (containing-(2m+1) terms) is given by
\begin{eqnarray}\label{eq(2.5)}
{_3F_2} \left[\begin{array}{r} -2m,\beta,-m-k-\frac{1}{2};\\ -2m-2k-1, \frac{1+\beta}{2}-m;\end{array}1\right]_{2m}&=& \frac{\left(\frac{1}{2}\right)_m\left(\frac{2+\beta+2k}{2}\right)_m}{\left(\frac{1-\beta}{2}\right)_m\left(1+k\right)_m},
\end{eqnarray}
where $\beta,\frac{1+\beta}{2}-m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
On setting $\gamma=-m-k-\frac{1}{2}$ in equation \eqref{eq(2.3)}, we obtain another Watson's summation theorem for truncated Clausen's series (containing-(2m+2) terms) is given by
\begin{eqnarray}\label{eq(2.6)}
{_3F_2} \left[\begin{array}{r} -2m-1,\beta,-m-k-\frac{1}{2};\\ -2m-2k-1, \frac{\beta}{2}-m;\end{array}1\right]_{2m+1}&=&0,
\end{eqnarray}
where $\beta,\frac{\beta}{2}-m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
The following summation theorem for Clausen's non-terminating series due to Saalsch\"{u}tz's (\cite[p.21,section 3.8(2)]{Bailey}, \cite[p.534, Entry 12]{Prudnikov2}, see also \cite[p.73(2.4.4.4) and p.246(III.31)]{Slater}) is given by
\begin{eqnarray}\label{eq(2.7)}
&&{_3F_2} \left[\begin{array}{r} \alpha,\beta, \gamma+\delta-\alpha-
\beta-1;\\ \gamma, \delta;\end{array}1\right]\nonumber\\
&&=\frac{\Gamma{(\gamma)}\Gamma{(\delta)}\Gamma{(\gamma-\alpha-\beta)}\Gamma{(\delta-\alpha-\beta)}}{\Gamma{(\gamma-\alpha)}\Gamma{(\gamma-\beta)}\Gamma{(\delta-\alpha)}\Gamma{(\delta-\beta)}}+\frac{1}{(\alpha+\beta-\gamma)}\frac{\Gamma{(\gamma)}\Gamma{(\delta)}}{\Gamma{(\alpha)}\Gamma{(\beta)}\Gamma{(\gamma+\delta-\alpha-\beta)}}\times\nonumber\\
&&\times{_3F_2} \left[\begin{array}{r} \gamma-\alpha,\gamma-\beta, 1;\\ \gamma-\alpha-\beta+1, \gamma+\delta-\alpha-\beta;\end{array}1\right],
\end{eqnarray}
where $\Re{(\delta-\alpha-\beta)}>0$ and $\Re{(\gamma-\alpha-\beta)}>0$.\\
If we set $\delta=-m+1-\gamma+\alpha+\beta$, $m$ being positive integer, in the right-hand side of equation \eqref{eq(2.7)}, we obtain Saalsch\"{u}tz's summation theorem for Clausen's terminating series (\cite[p.9, section 2.2(1)]{Bailey}, see also \cite[p.87, Th 29]{Rainville})
\begin{eqnarray}\label{eq(2.8)}
{_3F_2} \left[\begin{array}{r} \alpha,\beta, -m;\\ \gamma, 1+\alpha+\beta-\gamma-m;\end{array}1\right] &=& \frac{\left(\gamma-\alpha\right)_m\left(\gamma-\beta\right)_m}{\left(\gamma\right)_m\left(\gamma-\alpha-\beta\right)_m},
\end{eqnarray}
where $ \alpha,\beta,\gamma, 1+\alpha+\beta-\gamma-m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m\in\mathbb{N}$.\\
On setting $\gamma=-m-k$ in equation \eqref{eq(2.8)}, we get Saalsch\"{u}tz's summation theorem for truncated series
\begin{eqnarray}\label{eq(2.8c)}
{_3F_2} \left[\begin{array}{r} -m,\alpha,\beta;\\ -m-k, 1+\alpha+\beta+k;\end{array}1\right]_m &=& \frac{\left(1+\alpha+k\right)_m\left(1+\beta+k\right)_m}{\left(1+k\right)_m\left(1+\alpha+\beta+k\right)_m},
\end{eqnarray}
where $\alpha,\beta,1+\alpha+\beta+k\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
Next we recall another Saalsch\"{u}tz's summation theorem for Clausen's terminating series (\cite[p.24]{Bailey1}, see also \cite[p.87, Theorem 30]{Rainville})
\begin{eqnarray}\label{eq(2.8a)}
{_3F_2} \left[\begin{array}{r} -m, \alpha+m, 1+\alpha-\beta-\gamma;\\ 1+\alpha-\beta, 1+\alpha-\gamma;\end{array}1\right] &=& \frac{\left(\beta\right)_m\left(\gamma\right)_m}{\left(1+\alpha-\beta\right)_m\left(1+\alpha-\gamma\right)_m},
\end{eqnarray}
where $\alpha+m, 1+\alpha-\beta-\gamma,1+\alpha-\beta,1+\alpha-\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m\in\mathbb{N}$.\\
If we set $1+\alpha-\beta=-m-k$ in equation \eqref{eq(2.8a)}, we obtain the following Saalsch\"{u}tz's summation theorem for truncated series
\begin{eqnarray}\label{eq(2.8b)}
{_3F_2} \left[\begin{array}{r} -m, \beta-k-1, -m-k-\gamma;\\ -m-k, \beta-\gamma-m-k;\end{array}1\right]_m &=& \frac{\left(\beta\right)_m\left(\gamma\right)_m}{\left(1+k\right)_m\left(1+k+\gamma-\beta\right)_m},
\end{eqnarray}
where $\beta-k-1,-m-k-\gamma,\beta-\gamma-m-k\in\mathbb{C}\setminus\mathbb{Z}_0; m,k\in\mathbb{N}$.\\
Next we recall Whipple's summation theorem for non-terminating Clausen's series \cite[p.16, section 3.4(1)]{Bailey}
\begin{eqnarray}\label{eq(2.9)}
&&{_3F_2} \left[\begin{array}{r} \alpha,1- \alpha, \beta;\\ \gamma, 2\beta-\gamma+1;\end{array}1\right]\nonumber\\
&&=\frac{\pi \Gamma{(\gamma)}\Gamma{(2\beta-\gamma+1)}}
{2^{2\beta-1}\Gamma{\left(\frac{\alpha+2\beta-\gamma+1}{2}\right)}\Gamma{\left(\frac{\alpha+\gamma}{2}\right)}\Gamma{\left(\frac{2-\alpha+2\beta-\gamma}{2}\right)}\Gamma{\left(\frac{1-\alpha+\gamma}{2}\right)}},
\end{eqnarray}
where $\Re{(\beta)}>0,\gamma,2\beta-\gamma+1\in\mathbb{C}\setminus\mathbb{Z}_0^{-}$.\\
On setting $\alpha=-2m$ in equation \eqref{eq(2.9)}, we get Whipple's summation theorem for terminating series
\begin{eqnarray}\label{eq(2.10)}
{_3F_2} \left[\begin{array}{r} -2m,1+2m, \beta;\\ \gamma, 1+2\beta-\gamma;\end{array}1\right]=\frac{\left(\frac{2-\gamma}{2}\right)_m\left(\frac{1-2\beta+\gamma}{2}\right)_m}{\left(\frac{1+\gamma}{2}\right)_m\left(\frac{2+2\beta-\gamma}{2}\right)_m},
\end{eqnarray}
where $\beta,\gamma,1+2\beta-\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^{-}$; $m\in\mathbb{N}$.\\
On setting $\alpha=-2m-1$ in equation \eqref{eq(2.9)}, we get another Whipple's summation theorem for terminating series
\begin{eqnarray}\label{eq(2.11)}
{_3F_2} \left[\begin{array}{r} -2m-1,2+2m, \beta;\\ \gamma, 1+2\beta-\gamma;\end{array}1\right]=\frac{(\gamma-1)(2\beta-\gamma)\left(\frac{3-\gamma}{2}\right)_m\left(\frac{2-2\beta+\gamma}{2}\right)_m}{(\gamma)(1+2\beta-\gamma)\left(\frac{2+\gamma}{2}\right)_m\left(\frac{3+2\beta-\gamma}{2}\right)_m},
\end{eqnarray}
where $\beta,\gamma,1+2\beta-\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^{-}$; $m\in\mathbb{N}$.\\
Another Whipple's summation theorem for Clausen's terminating series is given by (\cite[p. 157, eq(3.1)]{Qureshi4}, see also \cite[p. 190, eq(2)]{Dzhrbashyan} and \cite[p. 238, eq(3.1)]{Bailey1})
\begin{eqnarray}\label{eq(2.13)}
&&{_3F_2} \left[\begin{array}{r} -m,\alpha, 1-\alpha;\\ \gamma, 1-\gamma-2m;\end{array}1\right]=\frac{\left(\frac{\gamma+\alpha}{2}\right)_{m} \left(\frac{\gamma-\alpha+1}{2}\right)_{m}}{\left(\frac{\gamma}{2}\right)_{m} \left(\frac{\gamma+1}{2}\right)_{m}},
\end{eqnarray}
where $\alpha,1-\alpha, \gamma, 1-\gamma-2m, \frac{\alpha+\gamma+2m}{2},\frac{1-\alpha+\gamma+2m}{2}\in\,\mathbb{C}\setminus\mathbb{Z}_0^-$; $m\in\mathbb{N}$.\\
If we set $\gamma=-2m-k$ in equation \eqref{eq(2.13)}, we get Whipple summation theorem for truncated series containing (m+1)-terms
\begin{eqnarray}\label{eq(2.14a)}
{_3F_2} \left[\begin{array}{r} -m,\alpha, 1-\alpha;\\ -2m-k, 1+k;\end{array}1\right]_m=\frac{\left(\frac{2-\alpha+k}{2}\right)_{m} \left(\frac{1+\alpha+k}{2}\right)_{m}}{\left(\frac{2+k}{2}\right)_{m} \left(\frac{1+k}{2}\right)_{m}},
\end{eqnarray}
where $\alpha,1-\alpha\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
On setting $\gamma=-2m-2k$ in equation \eqref{eq(2.10)}, we get Whipple summation theorem for truncated series containing (2m+1)-terms
\begin{eqnarray}\label{eq(2.15)}
&&{_3F_2} \left[\begin{array}{r} -2m,1+2m,\beta;\\ -2m-2k, 2\beta+2m+2k+1;\end{array}1\right]_{2m}=\frac{(1+2\beta+2k)_{2m}\left(1+k\right)_{2m}}{(1+2k)_{2m}\left(1+\beta+k\right)_{2m}},
\end{eqnarray}
where $\beta,2\beta+2m+2k+1\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
On setting $\gamma=-2m-2k-1$ in equation \eqref{eq(2.10)}, we get Whipple summation theorem for truncated series containing (2m+1)-terms
\begin{eqnarray}\label{eq(2.15a)}
&&{_3F_2} \left[\begin{array}{r} -2m,1+2m,\beta;\\ -2m-2k-1, 2\beta+2+2m+2k;\end{array}1\right]_{2m}=\frac{(2+2\beta+2k)_{2m}\left(\frac{3+2k}{2}\right)_{2m}}{(2+2k)_{2m}\left(\frac{3+2\beta+2k}{2}\right)_{2m}},\nonumber\\
\end{eqnarray}
where $\beta,2\beta+2m+2k+2\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
If we set $\gamma=-2m-2k-1$ in equation \eqref{eq(2.11)}, we get Whipple summation theorem for truncated series containing (2m+2)-terms
\begin{eqnarray}\label{eq(2.16)}
&&{_3F_2} \left[\begin{array}{r} -2m-1,2+2m,\beta;\\ -2m-2k-1, 2\beta+2m+2k+2;\end{array}1\right]_{2m+1}\nonumber\\
&&=\frac{(k+1)(2\beta+2m+2k+1)(2\beta+2k+1)_{2m}\left(2+k\right)_{2m}}{(2m+2k+1)(\beta+k+1)(2k+1)_{2m}\left(2+\beta+k\right)_{2m}},
\end{eqnarray}
where $\beta,2\beta+2m+2k+2\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
If we set $\gamma=-2m-2k-2$ in equation \eqref{eq(2.11)}, we get Whipple summation theorem for truncated series containing (2m+2)-terms
\begin{eqnarray}\label{eq(2.16a)}
&&{_3F_2} \left[\begin{array}{r} -2m-1,2+2m,\beta;\\ -2m-2k-2, 2\beta+2m+2k+3;\end{array}1\right]_{2m+1}\nonumber\\
&&=\frac{(2k+3)(\beta+m+k+1)(2\beta+2k+2)_{2m}\left(\frac{5+2k}{2}\right)_{2m}}{(m+k+1)(2\beta+2k+3)(2k+2)_{2m}\left(\frac{5+2\beta+2k}{2}\right)_{2m}},
\end{eqnarray}
where $\beta,2\beta+2m+2k+3\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
The classical Dixon's summation theorem for Clausen's non-terminating series \cite[p.13, section 3.1(1)]{Bailey} is given by
\begin{eqnarray}\label{eq(2.18)}
&&{_3F_2} \left[\begin{array}{r} \alpha,\beta, \gamma;\\ 1+\alpha-\beta, 1+\alpha-\gamma;\end{array}1\right]\nonumber\\
&&=\frac{\Gamma{\left(1+\frac{\alpha}{2}\right)}\Gamma{(1+\alpha-\beta)}\Gamma{(1+\alpha-\gamma)}\Gamma{\left(1+\frac{\alpha}{2}-\beta-\gamma\right)}}{\Gamma{\left(1+\alpha\right)}\Gamma{\left(1+\frac{\alpha}{2}-\beta\right)}\Gamma{\left(1+\frac{\alpha}{2}-\gamma\right)}\Gamma{\left(1+\alpha-\beta-\gamma\right)}},
\end{eqnarray}
where $\Re{(\alpha-2\beta-2\gamma)}>-2; \alpha,\beta,\gamma\in\mathbb{C};1+\alpha-\beta,1+\alpha-\gamma,1+\frac{\alpha}{2},1+\frac{\alpha}{2}-\beta-\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^-$.\\
Equation \eqref{eq(2.18)} can be written as
\begin{eqnarray}\label{eq(2.19)}
&&{_3F_2} \left[\begin{array}{r} \alpha,\beta, \gamma;\\ 1+\alpha-\beta, 1+\alpha-\gamma;\end{array}1\right]\nonumber\\
&&=\frac{\cos\left(\frac{\pi \alpha}{2}\right)\Gamma{(1-\alpha)}\Gamma{(1+\alpha-\beta)}\Gamma{(1+\alpha-\gamma)}\Gamma{\left(1+\frac{\alpha}{2}-\beta-\gamma\right)}}{\Gamma{\left(1-\frac{\alpha}{2}\right)}\Gamma{\left(1+\frac{\alpha}{2}-\beta\right)}\Gamma{\left(1+\frac{\alpha}{2}-\gamma\right)}\Gamma{\left(1+\alpha-\beta-\gamma\right)}}\\
&&=\frac{\cos\left(\frac{\pi \alpha}{2}\right)\Gamma{\left(\beta-\frac{\alpha}{2}\right)}\Gamma{(\gamma-\frac{\alpha}{2})}\Gamma{(1-\alpha)}\Gamma{\left(\beta+\gamma-\alpha\right)}}{\Gamma{\left(\beta-\alpha\right)}\Gamma{(\gamma-\alpha)}\Gamma{\left(1-\frac{\alpha}{2}\right)}\Gamma{\left(\beta+\gamma-\frac{\alpha}{2}\right)}}\times\nonumber\\
&&\times\frac{\sin\{\pi \left(\beta-\frac{\alpha}{2}\right)\}\sin\{\pi \left(\gamma-\frac{\alpha}{2}\right)\}\sin\{\pi \left(\beta+\gamma-\alpha\right)\}}{\sin\{\pi \left(\beta-\alpha\right)\}\sin\{\pi \left(\gamma-\alpha\right)\}\sin\{\pi \left(\beta+\gamma-\frac{\alpha}{2}\right)\}}.
\end{eqnarray}
On setting $\alpha=-2m$ in equation \eqref{eq(2.18)}, we obtain Dixon's summation theorem for terminating series
\begin{eqnarray}\label{eq(2.20)}
{_3F_2} \left[\begin{array}{r} -2m,\beta, \gamma;\\ 1-2m-\beta, 1-2m-\gamma;\end{array}1\right]=\frac{(\beta)_{m}(\gamma)_{m}2^{2m}\left(\frac{1}{2}\right)_m(\beta+\gamma)_{2m}}{(\beta)_{2m}(\gamma)_{2m}(\beta+\gamma)_{m}},
\end{eqnarray}
where $\beta, \gamma\in \mathbb{C}\setminus\mathbb{Z}; m\in\mathbb{N}$.\\
On setting $\alpha=-2m-1$ in equation \eqref{eq(2.18)}, we obtain another Dixon's summation theorem for terminating series
\begin{eqnarray}\label{eq(2.21)}
&&{_3F_2} \left[\begin{array}{r} -2m-1,\beta, \gamma;\\ -2m-\beta, -2m-\gamma;\end{array}1\right]=0,
\end{eqnarray}
where $\beta, \gamma\in \mathbb{C}\setminus\mathbb{Z}; m\in\mathbb{N}$.\\
On setting $\beta=1+k$ in equation \eqref{eq(2.20)}, we obtain Dixon's summation theorem for truncated series
\begin{eqnarray}\label{eq(2.22)}
{_3F_2} \left[\begin{array}{r} -2m,1+k, \gamma;\\ -2m-k, 1-2m-\gamma;\end{array}1\right]_{2m}=\frac{(1+k)_{m}(\gamma)_{m}2^{2m}\left(\frac{1}{2}\right)_m(1+k+\gamma)_{2m}}{(1+k)_{2m}(\gamma)_{2m}(1+k+\gamma)_{m}},
\end{eqnarray}
where $\gamma,1-2m-\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
On setting $\gamma=1+k$ in equation \eqref{eq(2.22)}, we obtain another Dixon's summation theorem for truncated series
\begin{eqnarray}\label{eq(2.23)}
{_3F_2} \left[\begin{array}{r} -2m,1+k, 1+k;\\ -2m-k, -2m-k;\end{array}1\right]_{2m}=\frac{(1+k)_{m}(1+k)_{m}2^{2m}\left(\frac{1}{2}\right)_m(2+2k)_{2m}}{(1+k)_{2m}(1+k)_{2m}(2+2k)_{m}},
\end{eqnarray}
where $m,k\in\mathbb{N}$.\\
On setting $\beta=1+k$ in equation \eqref{eq(2.21)}, we obtain Dixon's summation theorem for truncated series
\begin{eqnarray}\label{eq(2.24)}
&&{_3F_2} \left[\begin{array}{r} -2m-1,1+k, \gamma;\\ -2m-1-k, -2m-\gamma;\end{array}1\right]_{2m+1}=0,
\end{eqnarray}
where $\gamma,-2m-\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
On setting $\gamma=1+k$ in equation \eqref{eq(2.24)}, we obtain another Dixon's summation theorem for truncated series
\begin{eqnarray}\label{eq(2.25)}
&&{_3F_2} \left[\begin{array}{r} -2m-1,1+k, 1+k;\\ -2m-1-k, -2m-1-k;\end{array}1\right]_{2m+1}=0,
\end{eqnarray}
where $m,k\in\mathbb{N}$.\\
On setting $\beta=-2m$ in equation \eqref{eq(2.18)}, we obtain Dixon's summation theorem for terminating series
\begin{eqnarray}\label{eq(2.20a)}
{_3F_2} \left[\begin{array}{r} -2m,\alpha, \gamma;\\ 1+\alpha+2m, 1+\alpha-\gamma;\end{array}1\right]=\frac{(1+\alpha)_{2m}\left(1+\frac{\alpha}{2}-\gamma\right)_{2m}}{\left(1+\frac{\alpha}{2}\right)_{2m}(1+\alpha-\gamma)_{2m}},
\end{eqnarray}
where $\alpha, \gamma\in \mathbb{C}\setminus\mathbb{Z}; m\in\mathbb{N}$.\\
On setting $\beta=-2m-1$ in equation \eqref{eq(2.18)}, we obtain another Dixon's summation theorem for terminating series
\begin{eqnarray}\label{eq(2.20b)}
{_3F_2} \left[\begin{array}{r} -2m-1,\alpha, \gamma;\\ 2+\alpha+2m, 1+\alpha-\gamma;\end{array}1\right]=\frac{(1+\alpha)(2+\alpha-2\gamma)(2+\alpha)_{2m}\left(2+\frac{\alpha}{2}-\gamma\right)_{2m}}{(2+\alpha)(1+\alpha-\gamma)\left(2+\frac{\alpha}{2}\right)_{2m}(2+\alpha-\gamma)_{2m}},
\end{eqnarray}
where $\alpha, \gamma\in \mathbb{C}\setminus\mathbb{Z}; m\in\mathbb{N}$.\\
On setting $\gamma=1+\alpha+2m+k$ in equation \eqref{eq(2.20a)}, we obtain Dixon's summation theorem for truncated series
\begin{eqnarray}\label{eq(2.20c)}
{_3F_2} \left[\begin{array}{r} -2m,\alpha, 1+\alpha+2m+k;\\ -2m-k, 1+\alpha+2m;\end{array}1\right]_{2m}=\frac{(1+\alpha)_{2m}\left(1+\frac{\alpha}{2}+k\right)_{2m}}{\left(1+\frac{\alpha}{2}\right)_{2m}(1+k)_{2m}},
\end{eqnarray}
where $\alpha,1+\alpha+2m+k,1+\alpha+2m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
On setting $\gamma=2+\alpha+2m+k$ in equation \eqref{eq(2.20b)}, we obtain another Dixon's summation theorem for truncated series
\begin{eqnarray}\label{eq(2.20d)}
&&{_3F_2} \left[\begin{array}{r} -2m-1,\alpha, 2+\alpha+2m+k;\\ -2m-k-1, 2+\alpha+2m;\end{array}1\right]_{2m+1}\nonumber\\
&&=\frac{(1+\alpha)(2+2k+\alpha+4m)(2+\alpha)_{2m}\left(1+\frac{\alpha}{2}+k\right)_{2m}}{(2+\alpha)(1+2m+k)\left(2+\frac{\alpha}{2}\right)_{2m}(1+k)_{2m}},
\end{eqnarray}
where $\alpha,2+\alpha+2m+k,2+\alpha+2m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$.\\
Also, on setting $\gamma=-m$ in equation \eqref{eq(2.19)}, we obtain Dixon's theorem for Clausen's terminating series
\begin{eqnarray}\label{eq(2.26)}
&&{_3F_2} \left[\begin{array}{r} \alpha, \beta, -m;\\ 1+\alpha-\beta, 1+\alpha+m;\end{array}1\right]\nonumber\\
&&=\frac{\cos\left(\frac{\pi \alpha}{2}\right)\Gamma{(1-\alpha)}\Gamma{(1+\alpha-\beta)}\Gamma{(1+\alpha+m)}\Gamma{\left(1+\frac{\alpha}{2}-\beta+m\right)}}{\Gamma{\left(1-\frac{\alpha}{2}\right)}\Gamma{\left(1+\frac{\alpha}{2}-\beta\right)}\Gamma{\left(1+\frac{\alpha}{2}+m\right)}\Gamma{\left(1+\alpha-\beta+m\right)}},
\end{eqnarray}
where $\alpha,\beta,1+\alpha-\beta,1+\alpha+m\in\mathbb{C}\setminus\mathbb{Z}_0^-; m\in \mathbb{N}$.\\
On setting $\beta=-m$ in equation \eqref{eq(2.18)}, we get Dixon's summation theorem for Clausen's terminating series
\begin{eqnarray}\label{eq(2.27)}
{_3F_2} \left[\begin{array}{r} -m, \alpha, \gamma;\\ 1+\alpha+m, 1+\alpha-\gamma;\end{array}1\right]=\frac{(1+\alpha)_m\left(1+\frac{\alpha}{2}-\gamma\right)_m}{\left(1+\frac{\alpha}{2}\right)_m(1+\alpha-\gamma)_m},
\end{eqnarray}
where $\alpha,\gamma,1+\alpha+m,1+\alpha-\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^-; m\in \mathbb{N}$.\\
In section 3, we discuss the applications of some summation theorems for truncated Clausen hypergeometric series in Mellin transforms of the product of exponential function and truncated Goursat hypergeometric function.
\section{Applications in Mellin transforms}
In this section, we obtain Mellin transforms of the product of exponential function and truncated Goursat's function ${_2F_2}(\cdot)$ (when one numerator and one denominator parameters are negative integers),
\begin{eqnarray}\label{eq(7.1)}
\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -m, a;\\ -m-\ell, b;\end{array}\lambda t\right]_m;s\right\}&=&\int_{0}^{\infty}t^{s-1}e^{-\mu t}{_2F_2} \left[\begin{array}{r} -m, a;\\ -m-\ell, b;\end{array}\lambda t\right]_mdt\nonumber\\
&&=\frac{\Gamma(s)}{\mu^s}{_3F_2} \left[\begin{array}{r} -m, a,s;\\ -m-\ell, b;\end{array}\frac{\lambda}{\mu}\right]_m,
\end{eqnarray}
where $ \Re{(s)}>0$; $\Re{(\mu)}>0$ and $m,\ell\in\mathbb{N}$.\\
We derive some new results for Mellin transform as applications of summation theorems discussed in previous section.\\
\textbf{Case I.} On setting $\ell=m, a=\alpha, b=\frac{1+\alpha+\beta}{2}, \lambda=\mu$ and $s=\beta$ in equation \eqref{eq(7.1)} and using Watson's truncated summation theorem \eqref{eq(2.4)}, we obtain
\begin{eqnarray}\label{eq(7.2)}
\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -m,\alpha;\\ -2m,\frac{1+\alpha+\beta}{2};\end{array}\mu t\right]_m;\beta\right\}=\frac{\Gamma(\beta)}{\mu^{\beta}}\frac{\left(\frac{1+\alpha}{2}\right)_m\left(\frac{1+\beta}{2}\right)_m}{\left(\frac{1}{2}\right)_m\left(\frac{1+\alpha+\beta}{2}\right)_m},
\end{eqnarray}
where $\alpha,\frac{1+\alpha+\beta}{2}\in\mathbb{C}\setminus\mathbb{Z}_0^{-}$; $m\in\mathbb{N}$; $\Re(\beta)>0,\Re(\mu)>0$.\\
\textbf{Case II.} Replacing $m$ by $2m$ and after that setting $\ell=2k+1,a=-m-k-\frac{1}{2}, b=\frac{1+\beta}{2}-m, \lambda=\mu$ and $s=\beta$ in equation \eqref{eq(7.1)} and using Watson's truncated summation theorem \eqref{eq(2.5)}, we obtain
\begin{eqnarray}\label{eq(7.14)}
\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m,-m-k-\frac{1}{2};\\ -2m-2k-1, \frac{1+\beta}{2}-m;\end{array}\mu t\right]_{2m};\beta\right\}=\frac{\Gamma(\beta)}{\mu^{\beta}}\frac{\left(\frac{1}{2}\right)_m\left(\frac{2+\beta+2k}{2}\right)_m}{\left(\frac{1-\beta}{2}\right)_m\left(1+k\right)_m},
\end{eqnarray}
where $\left(\frac{1+\beta}{2}\right)-m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N};$ $\Re(\beta)>0, \Re(\mu)>0$.\\
\textbf{Case III.} Replacing $m$ by $2m+1$ and after that setting $\ell=2k,a=-m-k-\frac{1}{2}, b=\frac{\beta}{2}-m, \lambda=\mu$ and $s=\beta$ in equation \eqref{eq(7.1)} and using Watson's truncated summation theorem \eqref{eq(2.6)}, we obtain
\begin{eqnarray}\label{eq(7.15)}
\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m-1,-m-k-\frac{1}{2};\\ -2m-2k-1, \frac{\beta}{2}-m;\end{array}\mu t\right]_{2m+1};\beta\right\}=0,
\end{eqnarray}
where $\frac{\beta}{2}-m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N};$ $\Re(\beta)>0, \Re(\mu)>0$.\\
\textbf{Case IV.} On setting $\ell=k,a=\alpha, b=1+\alpha+\beta+k, \lambda=\mu$ and $s=\beta$ in equation \eqref{eq(7.1)} and using Saalsch\"{u}tz's truncated summation theorem \eqref{eq(2.8c)}, we obtain
\begin{eqnarray}\label{eq(7.3)}
\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -m,\alpha;\\ -m-k, 1+\alpha+\beta+k;\end{array}\mu t\right]_m;\beta\right\}= \frac{\Gamma(\beta)}{\mu^{\beta}}\frac{\left(1+\alpha+k\right)_m\left(1+\beta+k\right)_m}{\left(1+k\right)_m\left(1+\alpha+\beta+k\right)_m},\nonumber\\
\end{eqnarray}
where $\alpha,1+\alpha+\beta+k\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N};$ $\Re(\beta)>0, \Re(\mu)>0$.\\
\textbf{Case V.} On setting $\ell=k,a=\beta-k-1, b=\beta-\gamma-m-k, \lambda=\mu$ and $s=-m-k-\gamma$ in equation \eqref{eq(7.1)} and using Saalsch\"{u}tz's truncated summation theorem \eqref{eq(2.8b)}, we obtain
\begin{eqnarray}\label{eq(g7.4)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -m,\beta-k-1;\\ -m-k, \beta-\gamma-m-k;\end{array}\mu t\right]_m;-m-k-\gamma\right\}\nonumber\\
&&\qquad=\frac{\Gamma(-m-k-\gamma)}{\mu^{-m-k-\gamma}}\frac{\left(\beta\right)_m\left(\gamma\right)_m}{\left(1+k\right)_m\left(1+k+\gamma-\beta\right)_m},
\end{eqnarray}
where $\beta-k-1,\beta-\gamma-m-k\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N};$ $\Re(-m-k-\gamma)>0, \Re(\mu)>0$.\\
\textbf{Case VI.} On setting $\ell=m+k,a=1-\alpha, b=1+k, \lambda=\mu$ and $s=\alpha$ in equation \eqref{eq(7.1)} and using Whipple's truncated summation theorem \eqref{eq(2.14a)}, we obtain
\begin{eqnarray}\label{eq(7.5)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -m,1-\alpha;\\ -2m-k, 1+k;\end{array}\mu t\right]_m;\alpha\right\}=\frac{\Gamma(\alpha)}{\mu^{\alpha}}\frac{\left(\frac{2-\alpha+k}{2}\right)_{m} \left(\frac{1+\alpha+k}{2}\right)_{m}}{\left(\frac{2+k}{2}\right)_{m} \left(\frac{1+k}{2}\right)_{m}},
\end{eqnarray}
where $1-\alpha\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$ and $\Re(\alpha)>0,\Re(\mu)>0$.\\
\textbf{Case VII.} Replacing $m$ by $2m$ and after that setting $\ell=2k,a=1+2m, b=1+2m+2k+2\beta, \lambda=\mu$ and $s=\beta$ in equation \eqref{eq(7.1)} and using Whipple's truncated summation theorem \eqref{eq(2.15)}, we obtain
\begin{eqnarray}\label{eq(7.6)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m,1+2m;\\ -2m-2k, 2\beta+1+2m+2k;\end{array}\mu t\right]_{2m};\beta\right\}\nonumber\\
&&\qquad=\frac{\Gamma(\beta)}{\mu^{\beta}}\frac{(1+2\beta+2k)_{2m}\left(1+k\right)_{2m}}{(1+2k)_{2m}\left(1+\beta+k\right)_{2m}},
\end{eqnarray}
where $1+2m+2k+2\beta\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$ and $\Re(\beta)>0,\Re(\mu)>0$.\\
\textbf{Case VIII.} Replacing $m$ by $2m$ and after that setting $\ell=2k+1,a=1+2m, b=2+2m+2k+2\beta, \lambda=\mu$ and $s=\beta$ in equation \eqref{eq(7.1)} and using Whipple's truncated summation theorem \eqref{eq(2.15a)}, we obtain
\begin{eqnarray}\label{eq(7.7)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m,1+2m;\\ -2m-2k-1, 2\beta+2+2m+2k;\end{array}\mu t\right]_{2m};\beta\right\}\nonumber\\
&&\qquad=\frac{\Gamma(\beta)}{\mu^{\beta}}\frac{(2+2\beta+2k)_{2m}\left(\frac{3+2k}{2}\right)_{2m}}{(2+2k)_{2m}\left(\frac{3+2\beta+2k}{2}\right)_{2m}},
\end{eqnarray}
where $2+2m+2k+2\beta\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$ and $\Re(\beta)>0,\Re(\mu)>0$.\\
\textbf{Case IX.} Replacing $m$ by $2m+1$ and after that setting $\ell=2k,a=2+2m, b=2+2m+2k+2\beta, \lambda=\mu$ and $s=\beta$ in equation \eqref{eq(7.1)} and using Whipple's truncated summation theorem \eqref{eq(2.16)}, we obtain
\begin{eqnarray}\label{eq(7.8)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m-1,2+2m;\\ -2m-2k-1, 2\beta+2m+2k+2;\end{array}\mu t\right]_{2m+1};\beta\right\}\nonumber\\
&&=\frac{\Gamma(\beta)}{\mu^{\beta}}\frac{(k+1)(2\beta+2m+2k+1)(2\beta+2k+1)_{2m}\left(2+k\right)_{2m}}{(2m+2k+1)(\beta+k+1)(2k+1)_{2m}\left(2+\beta+k\right)_{2m}},
\end{eqnarray}
where $2+2m+2k+2\beta\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$ and $\Re(\beta)>0,\Re(\mu)>0$.\\
\textbf{Case X.} Replacing $m$ by $2m+1$ and after that setting $\ell=2k+1,a=2+2m, b=3+2m+2k+2\beta, \lambda=\mu$ and $s=\beta$ in equation \eqref{eq(7.1)} and using Whipple's truncated summation theorem \eqref{eq(2.16a)}, we obtain
\begin{eqnarray}\label{eq(7.9)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m-1,2+2m;\\ -2m-2k-2, 2\beta+2m+2k+3;\end{array}\mu t\right]_{2m+1};\beta\right\}\nonumber\\
&&=\frac{\Gamma(\beta)}{\mu^{\beta}}\frac{(2k+3)(\beta+m+k+1)(2\beta+2k+2)_{2m}\left(\frac{5+2k}{2}\right)_{2m}}{(m+k+1)(2\beta+2k+3)(2k+2)_{2m}\left(\frac{5+2\beta+2k}{2}\right)_{2m}},
\end{eqnarray}
where $3+2m+2k+2\beta\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$ and $\Re(\beta)>0,\Re(\mu)>0$.\\
\textbf{Case XI.} Replacing $m$ by $2m$ and after that setting $\ell=k,a=1+k, b=1-2m-\gamma, \lambda=\mu$ and $s=\gamma$ in equation \eqref{eq(7.1)} and using Dixon's truncated summation theorem \eqref{eq(2.22)}, we obtain
\begin{eqnarray}\label{eq(7.10)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m,1+k;\\ -2m-k, 1-2m-\gamma;\end{array}\mu t\right]_{2m};\gamma\right\}\nonumber\\
&&=\frac{\Gamma(\gamma)}{\mu^{\gamma}}\frac{(1+k)_{m}(\gamma)_{m}2^{2m}\left(\frac{1}{2}\right)_m(1+k+\gamma)_{2m}}{(1+k)_{2m}(\gamma)_{2m}(1+k+\gamma)_{m}},
\end{eqnarray}
where $1-2m-\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$ and $\Re(\gamma)>0,\Re(\mu)>0$.\\
\textbf{Case XII.} Replacing $m$ by $2m$ and after that setting $\ell=k,a=1+k, b=-2m-k, \lambda=\mu$ and $s=1+k$ in equation \eqref{eq(7.1)} and using Dixon's truncated summation theorem \eqref{eq(2.23)}, we obtain
\begin{eqnarray}\label{eq(7.11)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m,1+k;\\ -2m-k, -2m-k;\end{array}\mu t\right]_{2m};1+k\right\}\nonumber\\
&&=\frac{\Gamma(1+k)}{\mu^{1+k}}\frac{(1+k)_{m}(1+k)_{m}2^{2m}\left(\frac{1}{2}\right)_m(2+2k)_{2m}}{(1+k)_{2m}(1+k)_{2m}(2+2k)_{m}},
\end{eqnarray}
where $m,k\in\mathbb{N}$ and $\Re(\mu)>0$.\\
\textbf{Case XIII.} Replacing $m$ by $2m+1$ and after that setting $\ell=k,a=\gamma, b=-2m-\gamma, \lambda=\mu$ and $s=1+k$ in equation \eqref{eq(7.1)} and using Dixon's truncated summation theorem \eqref{eq(2.24)}, we obtain
\begin{eqnarray}\label{eq(7.16)}
\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m-1, \gamma;\\ -2m-1-k, -2m-\gamma;\end{array}\mu t\right]_{2m+1};1+k\right\}=0,
\end{eqnarray}
where $\gamma,-2m-\gamma\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$ and $\Re(\mu)>0$.\\
\textbf{Case XIV.} Replacing $m$ by $2m+1$ and after that setting $\ell=k,a=1+k, b=-2m-k-1, \lambda=\mu$ and $s=1+k$ in equation \eqref{eq(7.1)} and using Dixon's truncated summation theorem \eqref{eq(2.25)}, we obtain
\begin{eqnarray}\label{eq(7.17)}
\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m-1,1+k;\\ -2m-1-k, -2m-1-k;\end{array}\mu t\right]_{2m+1};1+k\right\}=0,
\end{eqnarray}
where $m,k\in\mathbb{N}$ and $\Re(\mu)>0$.\\
\textbf{Case XV.} Replacing $m$ by $2m$ and after that setting $\ell=k,a=\alpha, b=1+\alpha+2m, \lambda=\mu$ and $s=1+\alpha+2m+k$ in equation \eqref{eq(7.1)} and using Dixon's truncated summation theorem \eqref{eq(2.20c)}, we obtain
\begin{eqnarray}\label{eq(7.12)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m,\alpha;\\ -2m-k, 1+\alpha+2m;\end{array}\mu t\right]_{2m};1+\alpha+2m+k\right\}\nonumber\\
&&=\frac{\Gamma(1+\alpha+k+2m)}{\mu^{1+\alpha+k+2m}}\frac{(1+\alpha)_{2m}\left(1+\frac{\alpha}{2}+k\right)_{2m}}{\left(1+\frac{\alpha}{2}\right)_{2m}(1+k)_{2m}},
\end{eqnarray}
where $\alpha,1+\alpha+2m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$ and $\Re(1+\alpha+2m+k)>0,\Re(\mu)>0$.\\
\textbf{Case XVI.} Replacing $m$ by $2m+1$ and after that setting $\ell=k,a=\alpha, b=2+\alpha+2m, \lambda=\mu$ and $s=2+\alpha+2m+k$ in equation \eqref{eq(7.1)} and using Dixon's truncated summation theorem \eqref{eq(2.20d)}, we obtain
\begin{eqnarray}\label{eq(7.13)}
&&\mathcal{M}\left\{e^{-\mu t}{_2F_2} \left[\begin{array}{r} -2m-1,\alpha;\\ -2m-k-1, 2+\alpha+2m;\end{array}\mu t\right]_{2m+1};2+\alpha+2m+k\right\}\nonumber\\
&&=\frac{\Gamma(2+\alpha+2m+k)}{\mu^{2+\alpha+2m+k}}\frac{(1+\alpha)(2+2k+\alpha+4m)(2+\alpha)_{2m}\left(1+\frac{\alpha}{2}+k\right)_{2m}}{(2+\alpha)(1+2m+k)\left(2+\frac{\alpha}{2}\right)_{2m}(1+k)_{2m}},\nonumber\\
\end{eqnarray}
where $\alpha,2+\alpha+2m\in\mathbb{C}\setminus\mathbb{Z}_0^{-}; m,k\in\mathbb{N}$ and $\Re(2+\alpha+2m+k)>0,\Re(\mu)>0$.\\
\textbf{Remark.} In the next communication \cite{Qureshi2}, we shall obtain the Mellin transform of the product of exponential function and infinite Goursat series ${_2F_2} \left[\begin{array}{r} -m,\alpha;\\ -m-\ell, \beta;\end{array}\lambda t\right]$.
\section*{Concluding remarks}
In previous sections, we have derived some summation theorems for Clausen's terminating and truncated hypergeometric series ${_3F_2}$ when one numerator and one denominator parameters are negative integers. In the sequel of this paper, we have derived some summation formulae for Gauss' hypergeometric series ${_2F_1}$, Clausen hypergeometric series ${_3F_2}$ and have discussed their applications (see for example \cite{Qureshi2,Qureshi3}). It is expected that these summation formulae will be of wide interest and will help to advance research in the field of special functions.\\
We conclude our present investigation by observing that several hypergeometric summation theorems can be derived from a known summation theorem in an analogous manner.}
\end{document} |
\begin{equation}gin{document}
\author{Fabio Cavalletti}\thanks{Universit\`a degli Studi di Pavia, Dipartimento di Matematica, email: [email protected]}
\title[An Overview of $L^{1}$ optimal transportation]{An Overview of $L^{1}$ optimal transportation \\ on metric measure spaces}
\keywords{optimal transport; Monge problem; Ricci curvature; curvature dimension condition}
\begin{equation}gin{abstract}
The scope of this note is to make a self-contained survey of the recent developments and achievements of the theory of $L^{1}$-Optimal Transportation on metric measure spaces.
Among the results proved in the recent papers \cite{CM1,CM2} where the author, together with A. Mondino,
proved a series of sharp (and in some cases rigid) geometric and functional inequalities in the setting of
metric measure spaces enjoying a weak form of Ricci curvature lower bound, we
review the proof of the L\'evy-Gromov isoperimetric inequality.
\end{abstract}
\maketitle
\text{\varrhom span}ection{Introduction}
The scope of this note is to make a self-contained survey of the recent developments and achievements of the theory of $L^{1}$-Optimal Transportation on metric measure spaces.
We will focus on the general scheme adopted in the recent papers \cite{CM1,CM2} where the author, together with A. Mondino,
proved a series of sharp (and in some cases even rigid and stable) geometric and functional inequalities in the setting of
metric measure spaces enjoying a weak form of Ricci curvature lower bound.
Roughly the general scheme consists in reducing the initial problem to a family of easier one-dimensional problems;
as it is probably the most relevant result obtained with this technique, we will review in detail how to proceed to obtain the L\'evy-Gromov isoperimetric inequality for metric measure spaces
verifying the Riemmanian Curvature Dimension condition (or, more generally, essentially non-branching
metric measure spaces verifying the Curvature Dimension condition).
In \cite{biacava:streconv, cava:MongeRCD} a fine analysis of the Monge problem in the metric setting was done treating, with a different perspective,
similar questions whose answers were later used also in \cite{CM1,CM2}.
We therefore believe the Monge problem and V.N. Sudakov's approach to it (see \cite{sudakov}) is a good starting point for our review and to see
how $L^{1}$-Optimal Transportation naturally yields a reduction of the problem to a family of one-dimensional problems.
It is worth stressing that the dimensional reduction proposed by V.N. Sudakov to solve the Monge problem is only one of the strategy to attack the problem.
Monge problem has a long story and many different authors contributed to obtain solutions in different frameworks with different approaches;
here we only mention that the first existence result for the Monge problem was independently obtained in \cite{caffa:Monge} and in \cite{trudi:Monge}.
We also mention the subsequent generalizations obtained in \cite{Ambrosio:Monge,AKP,feldMccann} and we refer to the monograph \cite{Vil:topics}
for a more complete list of results.
\text{\varrhom span}ubsection{Monge problem}
The original problem posed by Monge in 1781 can be restated in modern language as follows:
given two Borel probability measures $\mu_{0}$ and $\mu_{1}$ over $\mathbb{R}^{d}$, called marginal measures,
find the optimal manner of transporting $\mu_{0}$ to $\mu_{1}$; the transportation of $\mu_{0}$ to $\mu_{1}$ is understood as a map $T : \mathbb{R}^{d} \to \mathbb{R}^{d}$
assigning to each particle $x$ a final position $T(x)$ fulfilling the following compatibility condition
\begin{equation}gin{equation}\bigl\langlebel{E:transportmap}
\mathfrak quad T_{\text{\varrhom span}harp} \, \mu_{0} = \mu_{1}, \mathfrak quad \textrm{i.e. }\quad \mu_{0}(T^{-1}(A)) = \mu_{1}(A), \quad \varphiorall \, A \ \textrm{Borel set};
\end{equation}
any map $T$ verifying the previous condition will be called a transport map.
The optimality requirement is stated as follows:
\begin{equation}gin{equation}\bigl\langlebel{E:MongeEuclid}
\int_{\mathbb{R}^{d}} |T(x) - x| \, \mu_{0}(dx) \leq \int_{\mathbb{R}^{d}} |\hat T(x) -x | \, \mu_{0}(dx),
\end{equation}
for any other $\hat T$ transport map.
In proving the existence of a minimizer,
the first difficulty appears studying the domain of the minimization, that is the set of maps $T$ verifying \eqref{E:transportmap}. Suppose
$\mu_{0} = f_{0} \mathcal{L}^{d}$ and $\mu_{1} = f_{1} \mathcal{L}^{d}$ where $\mathcal{L}^{d}$ denotes the $d$-dimensional Lebesgue measure;
a smooth injective map $T$ is then a transport map if and only if
$$
f_{1} (T(x)) |\partialt (DT)(x) | = f_{0}(x), \mathfrak quad \mu_{0}\textrm{-a.e.} \ x \in \mathbb{R}^{d},
$$
showing a strong non-linearity of the constrain.
The first big leap in optimal transportation theory was achieved by Kantorovich considering a suitable relaxation of the problem:
associate to each transport map the probability measure $(Id,T)_{\text{\varrhom span}harp} \mu_{0}$ over $\mathbb{R}^{d}\times \mathbb{R}^{d}$ and introduce the set
of \emph{transport plans}
$$
\Pi(\mu_{0},\mu_{1}) : = \left\{ \pi \in \mathcal{P}(\mathbb{R}^{d}\times \mathbb{R}^{d}) \colon P_{1\,\text{\varrhom span}harp} \pi = \mu_{0},\ P_{2\,\text{\varrhom span}harp} \pi = \mu_{1} \varrhoight\};
$$
where $P_{i} : \mathbb{R}^{d} \times \mathbb{R}^{d} \to \mathbb{R}^{d}$ is the projection on the $i$-th component, with $i =1,2$.
By definition $(Id,T)_{\text{\varrhom span}harp} \mu_{0} \in \Pi(\mu_{0},\mu_{1})$ and
$$
\int_{\mathbb{R}^{d}} |T(x) - x| \, \mu_{0}(dx) = \int_{\mathbb{R}^{d}\times \mathbb{R}^{d}} |x-y| \, \left( (Id, T)_{\text{\varrhom span}harp} \mu_{0}\varrhoight)(dxdy);
$$
then it is natural to consider the minimization of the following functional (called Monge-Kantorovich minimization problem)
\begin{equation}gin{equation}\bigl\langlebel{E:MK}
\Pi(\mu_{0},\mu_{1}) \ni \pi \longmapsto \mathcal{I}(\pi) : = \int_{\mathbb{R}^{d}\times\mathbb{R}^{d}} |x-y| \, \pi(dxdy).
\end{equation}
The big advantage being now that $\Pi(\mu_{0},\mu_{1})$ is a convex subset of $\mathcal{P}(\mathbb{R}^{d}\times \mathbb{R}^{d})$ and it is compact
with respect to the weak topology.
Since the functional $\mathcal{I}$ is linear, the existence of a minimizer follows straightforwardly.
Then a strategy to obtain a solution of the original Monge problem is to start from an optimal transport plan $\pi$ and prove
that it is indeed concentrated on the graph of a Borel map $T$; the latter is equivalent to $\pi = (Id,T)_{\text{\varrhom span}harp} \mu_{0}$.
To run this program one needs to deduce from optimality some condition on the geometry of the support of the transport plan.
This was again obtained by Kantorovich introducing a dual formulation of \eqref{E:MK} and finding out that for any probability measures
$\mu_{0}$ and $\mu_{1}$ with finite first moment, there exists a $1$-Lipschitz function $\varphi : \mathbb{R}^{d} \to \mathbb{R}$ such that
$$
\Pi(\mu_{0},\mu_{1}) \ni \pi \ \textrm{is optimal} \quad \iff \quad \pi \big(\{ (x,y) \in \mathbb{R}^{2d} \colon \varphi(x) - \varphi(y) = |x-y| \} \big) = 1.
$$
At this point one needs to focus on the structure of the set
\begin{equation}gin{equation}\bigl\langlebel{E:transportproduct}
\mathcal{G}amma : = \big\{ (x,y) \in \mathbb{R}^{2d} \colon \varphi(x) - \varphi(y) = |x-y| \big\}.
\end{equation}
\begin{equation}gin{definition}
A set $\mathcal{L}ambda \text{\varrhom span}ubset \mathbb{R}^{2d}$ is $|\cdot|$-cyclically monotone if and only if for any finite subset of $\mathcal{L}ambda$,
$\{ (x_{1},y_{1}), \dots, (x_{N},y_{N})\} \text{\varrhom span}ubset \mathcal{L}ambda$ it holds
$$
\text{\varrhom span}um_{1\leq i\leq N} |x_{i} - y_{i}| \leq \text{\varrhom span}um_{1 \leq i\leq N} |x_{i} - y_{i+1}|,
$$
where $y_{N+1} : = y_{1}$.
\end{definition}
Almost by definition, the set $\mathcal{G}amma$ is $|\cdot|$-cyclically monotone
and whenever $(x,y) \in \mathcal{G}amma$ considering $z_{t} : = (1-t) x + t y$ with $t \in [0,1]$ it holds that $(z_{s},z_{t}) \in \mathcal{G}amma$,
for any $s \leq t$. In particular this suggests that $\mathcal{G}amma$ produces a family of disjoint lines of $\mathbb{R}^{d}$ along where the optimal transportation should move.
This can be made rigorous considering the following ``relation'' between points:
a point $x$ is in relation with $y$ if, using optimal geodesics selected by the above optimal transport problem, one can travel from $x$ to $y$ or viceversa.
That is, consider $R : = \mathcal{G}amma \cup \mathcal{G}amma^{-1}$ and define $x \text{\varrhom span}im y$ if and only if $(x,y) \in R$.
Then $\mathbb{R}^{d}$ will be decomposed (up to a set of Lebesgue-measure zero) as $\mathcal{T} \cup Z$
where $\mathcal{T}$ will be called the \emph{transport set} and $Z$ the set of points not moved by the optimal transportation problem.
The important property of $\mathcal{T}$ being that
$$
\mathcal{T} = \bigcup_{q\in Q} X_{q}, \mathfrak quad X_{q} \textrm{ straight line}, \mathfrak quad X_{q} \cap X_{q'} = \emptyset, \quad \textrm{if } q \neq q'.
$$
Here $Q$ is a set of indices; a convenient way to index a straight line $X_{q}$ is to select an element of $X_{q}$ and call it, with an abuse of notation, $q$.
With this choice the set $Q$ can be understood as a subset of $\mathbb{R}^{d}$.
Once a partition of the space is given, one obtains via Disintegration Theorem a corresponding decomposition of marginal measures:
$$
\mu_{0} = \int_{Q} \mu_{0\,q} \, \mathfrak q(dq), \mathfrak quad \mu_{1} = \int_{Q} \mu_{1\, q} \, \mathfrak q(dq);
$$
where $\mathfrak q$ is a Borel probability measure over the set of indices $Q \text{\varrhom span}ubset \mathbb{R}^{d}$. If $Q$ enjoys a measurability condition (see Theorem \textrm{Re}\,f{T:disintr} for details),
the conditional measures $\mu_{0\,q}$ and $\mu_{1\,q}$ are concentrated on the straight line with index $q$, i.e. $\mu_{0\,q} (X_{q}) = \mu_{1\,q} (X_{q})= 1$, for $\mathfrak q$-a.e. $q \in Q$.
Then a classic way to construct an optimal transport maps is to
\begin{equation}gin{itemize}
\item[-] consider $T_{q}$ the monotone rearrangement along $X_{q}$ of $\mu_{0\, q}$ to $\mu_{1\,q}$;
\item[-] define the transport map $T$ as $T_{q}$ on each $X_{q}$.
\end{itemize}
The map $T$ will be then an optimal transport map moving $\mu_{0}$ to $\mu_{1}$; it is indeed easy to check that
$(Id,T)_{\text{\varrhom span}harp}\mu_{0} \in \Pi(\mu_{0},\mu_{1})$ and $(x,T(x)) \in \mathcal{G}amma$ for $\mu_{0}$-a.e. $x$.
So the original Monge problem has been reduced to the following family of one-dimensional problems:
for each $q \in Q$ find a minimizer of the following functional
$$
\Pi(\mu_{0\,q},\mu_{1\,q}) \ni \pi \longmapsto \mathcal{I}(\pi) : = \int_{X_{q}\times X_{q}} |x-y| \,\pi(dxdy),
$$
that is concentrated on the graph of a Borel function.
As $X_{q}$ is isometric to the real line, whenever $\mu_{0\,q}$ does not contain any atom (i.e $\mu_{0\,q} (x) = 0$, for all $x \in X_{q}$),
the monotone rearrangement $T_{q}$ exists and the existence of an optimal transport map $T$ constructed as before follows.
The existence of a solution has been reduced therefore to a regularity property of the disintegration of $\mu_{0}$.
As already stressed before, this approach to the Monge problem, mainly due to V.N. Sudakov, was proposed in \cite{sudakov}
and was later completed in the subsequent papers \cite{caffa:Monge} and in \cite{trudi:Monge}.
See also \cite{caravenna} for a complete Sudakov approach to Monge problem when the Euclidean distance is replaced by any strictly convex norm and
\cite{biadaneri} where any norm is considered.
In all these papers, assuming $\mu_{0}$ to be absolutely continuous with respect to $\mathcal{L}^{d}$ give the sufficient regularity to solve the problem.
The Monge problem can be actually stated, and solved, in a much more general framework.
Given indeed two Borel probability measures $\mu_{0}$ and $\mu_{1}$ over a complete and
separable metric space $(X,\text{\varrhom span}fd)$, the notion of transportation map perfectly makes
sense and the optimality condition \eqref{E:MongeEuclid} can be naturally formulated using the distance $\text{\varrhom span}fd$ as a cost function instead of the Euclidean norm:
\begin{equation}gin{equation}\bigl\langlebel{E:Mongemetric}
\int_{\mathbb{R}^{d}} \text{\varrhom span}fd(T(x), x) \, \mu_{0}(dx) \leq \int_{\mathbb{R}^{d}} \text{\varrhom span}fd(\hat T(x),x) \, \mu_{0}(dx).
\end{equation}
The problem can be relaxed to obtain a transport plan $\pi$ solution of the corresponding Monge-Kantorovich minimization problem.
Also the Kantorovich duality applies yielding the existence of a $1$-Lipschitz function $\varphi : X \to \mathbb{R}$ such that
$$
\Pi(\mu_{0},\mu_{1}) \ni \pi \ \textrm{is optimal} \quad \iff \quad \pi \big( \mathcal{G}amma \big) = 1,
$$
where $\mathcal{G}amma := \{ (x,y) \in X\times X \colon \varphi(x) - \varphi(y) = \text{\varrhom span}fd(x,y) \}$ is $\text{\varrhom span}fd$-cyclically monotone. \\
All the strategy proposed for the Euclidean problem can be adopted: produce a decomposition of $X$ as
$\mathcal{T} \cup Z$ where $Z$ is the set of points not moved by the optimal transportation problem and
$\mathcal{T}$ is the transport set and it is partitioned, up to a set of measure zero, by a family of geodesics $\{ X_{q} \}_{q \in Q}$;
via Disintegration Theorem one obtains as before a reduction of the Monge problem to a family of one-dimensional problems
$$
\Pi(\mu_{0\,q},\mu_{1\,q}) \ni \pi \longmapsto \mathcal{I}(\pi) : = \int_{X_{q}\times X_{q}} \text{\varrhom span}fd(x,y) \,\pi(dxdy).
$$
Therefore, since $X_{q}$ with distance $\text{\varrhom span}fd$ is isometric to an interval of the real line with Euclidean distance,
the problem is reduced to proving that for $\mathfrak q$-a.e. $q \in Q$ the conditional measure $\mu_{0\,q}$ does not have any atoms.
Clearly in showing such a result, besides the regularity of $\mu_{0}$ itself, the regularity of the ambient space $X$ does play a crucial role.
In particular, together with the localization of the Monge problem to $X_{q}$, it should come a localization of the regularity of the space.
This is the case when the metric space $(X,\text{\varrhom span}fd)$ is endowed with a reference probability measure $\mathfrak m$ and the resulting
metric measure space $(X,\text{\varrhom span}fd,\mathfrak m)$ verifies a weak Ricci curvature lower bound.
In \cite{biacava:streconv} we in fact observed that if $(X,\text{\varrhom span}fd,\mathfrak m)$ verifies the so-called measure contraction property $\mathsf{MCP}$, then
for $\mathfrak q$-a.e. $q \in Q$ the one-dimensional metric measure space $(X_{q},\text{\varrhom span}fd,\mathfrak m_{q})$ verifies $\mathsf{MCP}$ as well,
where $\mathfrak m_{q}$ is the conditional measure of $\mathfrak m$ with respect to the family of geodesics $\{ X_{q}\}_{q \in Q}$.
Now the assumption $\mu_{0}\ll \mathfrak m$ is sufficient to solve the Monge problem.
It is worth mentioning that \cite{biacava:streconv} was the first contribution where regularity of conditional measures were obtained in a purely non-smooth framework.
The techniques introduced in \cite{biacava:streconv} permitted also to threat such regularity issues in the infinite dimensional setting of Wiener space; see \cite{cava:Wiener}.
This short introduction should suggest that $L^{1}$-Optimal Transportation permits to obtain an efficient dimensional reduction
together with a localization of the ``smoothness'' of the space for very general metric measure spaces.
We now make a short introduction also to the L\'evy-Gromov isoperimetric inequality.
\text{\varrhom span}ubsection{L\'evy-Gromov isoperimetric inequality}
The L\'evy-Gromov isoperimetric inequality \cite[Appendix C]{Gro} can be stated as follows: if $E$ is a (sufficiently regular)
subset of a Riemannian manifold $(M^N,g)$ with dimension $N$ and Ricci bounded below by $K>0$, then
\begin{equation}gin{equation}\bigl\langlebel{eq:LevyGromov}
\varphirac{|\partial E|}{|M|}\geq \varphirac{|\partial B|}{|S|},
\end{equation}
where $B$ is a spherical cap in the model sphere $S$, i.e. the $N$-dimensional sphere with constant Ricci curvature equal to $K$, and $|M|,|S|,|\partial E|, |\partial B|$ denote the appropriate $N$ or $N-1$ dimensional volume, and where $B$ is chosen so that
$|E|/|M|=|B|/|S|$. As $K >0$ both $M$ and $S$ are compact and their volume is finite; hence the previous equality and \eqref{eq:LevyGromov} makes sense.
In other words, the L\'evy-Gromov isoperimetric inequality states that isoperimetry in $(M,g)$ is at least as strong as in the model space $S$.
A general introduction on the isoperimetric problem goes beyond the scopes of this note; here it is worth mentioning
that a complete description of isoperimetric inequality in spaces admitting singularities is quite an hard task and the bibliography reduces to \cite{MilRot, MR, MorPol}.
See also \cite[Appendix H]{EiMe} for more details.
We also include the following reference to the isoperimetric problem corresponding to different approaches: for a geometric measure theory approach see \cite{Mor};
for the point of view of optimal transport see \cite{FiMP, Vil}; for the connections with convex and integral geometry see \cite{BurZal};
for the recent quantitative forms see \cite{CL, FuMP} and finally for an overview of the more geometric aspects see \cite{Oss, Rit, Ros}.
Coming back to L\'evy-Gromov isoperimetric inequality, it makes sense naturally also in the broader class of metric measure spaces, i.e. triples $(X,\text{\varrhom span}fd,\mathfrak m)$ where
$(X,\text{\varrhom span}fd)$ is complete and separable and $\mathfrak m$ is a Radon measure over $X$.
Indeed the volume of a Borel set is replaced by its $\mathfrak m$-measure, $\mathfrak m(E)$;
the boundary area of the smooth framework instead can be replaced by the Minkowski content:
\begin{equation}gin{equation}\bigl\langlebel{def:MinkCont}
\mathfrak m^+(E):=\liminf_{\textbf{X}repsilon\downarrow 0} \varphirac{\mathfrak m(E^\textbf{X}repsilon)- \mathfrak m(E)}{\textbf{X}repsilon},
\end{equation}
where $E^{\textbf{X}repsilon}:=\{x \in X \,:\, \exists y \in E \, \text{ such that } \, \text{\varrhom span}fd(x,y)< \textbf{X}repsilon \}$ is the $\textbf{X}repsilon$-neighborhood of $E$ with respect to the metric $\text{\varrhom span}fd$;
the natural analogue of ``dimension $N$ and Ricci bounded below by $K>0$'' is encoded in the so-called Riemannian Curvature Dimension condition,
$\mathbb{R}CD^{*}(K,N)$ for short. As normalization factors appears in \eqref{eq:LevyGromov}, it is also more convenient
to directly consider the case $\mathfrak m(X) = 1$.
So the L\'evy-Gromov isoperimetric problem for a m.m.s. $(X,\text{\varrhom span}fd,\mathfrak m)$ with $\mathfrak m(X) = 1$ can be formulated as follows: \\
\noindent
\emph{
Find the largest function $\mathcal{I}_{K,N}:[0,1]\to \mathbb{R}^+$ such that for every Borel subset $E\text{\varrhom span}ubset X$ it holds
$$
\mathfrak m^{+}(E) \geq \mathcal{I}_{K,N}(\mathfrak m(E)),
$$
with $\mathcal{I}_{K,N}$ depending on $N, K \in \mathbb{R}$ with $K>0$ and $N>1$.
}
Then in \cite{CM1} (Theorem 1.2) the author with A. Mondino proved the non-smooth L\'evy-Gromov isoperimetric inequality \eqref{eq:LevyGromov}
\begin{equation}gin{theorem}[L\'evy-Gromov in $\mathbb{R}CD^*(K,N)$-spaces, Theorem 1.2 of \cite{CM1}] \bigl\langlebel{thm:LG}
Let $(X,\text{\varrhom span}fd,\mathfrak m)$ be an $\mathbb{R}CD^*(K,N)$ space for some $N\in \mathbb{N}$ and $K>0$ and $\mathfrak m(X)=1$. Then for every Borel subset $E\text{\varrhom span}ubset X$ it holds
$$
\mathfrak m^+(E)\geq \varphirac{|\partial B|}{|S|},
$$
where $B$ is a spherical cap in the model sphere $S$ (the $N$-dimensional sphere with constant Ricci curvature equal to $K$) chosen so that $|B|/|S|=\mathfrak m(E)$.
\end{theorem}
We refer to Theorem 1.2 of \cite{CM1} (or Theorem 6.6) for the more general statement.
The link between Theorem \textrm{Re}\,f{thm:LG} and the first part of the Introduction, where the Monge problem was discussed, stands in the techniques used to prove Theorem \textrm{Re}\,f{thm:LG}.
The main obstacle to L\'evy-Gromov type inequalities in the non-smooth metric measure spaces setting is that the
previously known proofs rely on regularity properties of isoperimetric regions and on powerful results of geometric measure theory (see for instance \cite{Gro,Mor})
that are out of disposal in the framework of metric measure spaces.
The recent paper of B. Klartag \cite{klartag} permitted to obtain a proof of the L\'evy-Gromov isoperimetric inequality, still in the framework of smooth Riemannian manifolds,
avoiding regularity of optimal shapes and using instead an optimal transportation argument involving $L^1$-Optimal Transportation and ideas of convex geometry.
This approach goes back to Payne-Weinberger \cite{PW} and was later developed by Gromov-Milman \cite{GrMi},
Lov\'asz-Simonovits \cite{LoSi} and Kannan-Lov\'asz-Simonovits \cite{KaLoSi};
it consists in reducing a multi-dimensional problem, to easier one-dimensional problems. B. Klartag's contribution was to observe that
a suitable $L^{1}$-Optimal Transportation problem produces what he calls a \emph{needle decomposition} (in our terminology will be called disintegration)
that localize (or reduce) the proof of the isoperimetric inequality to the proof of a family of one-dimensional isoperimetric inequalities;
also the regularity of the space is localized.
The approach of \cite{klartag} does not rely on the regularity of the isoperimetric region, nevertheless it still heavily
makes use of the smoothness of the ambient space to obtain the localization; in particular
it makes use of sharp properties of the geodesics in terms of Jacobi fields and estimates on the second fundamental forms of suitable level sets,
all objects that are still not enough understood in general metric measure space in order to repeat the same arguments.
Hence to apply the localization technique to the L\'evy-Gromov isoperimetric inequality in singular spaces,
structural properties of geodesics and of $L^1$-optimal transportation
have to be understood also in the general framework of metric measure spaces.
Such a program already started in the previous work of the author with S. Bianchini \cite{biacava:streconv} and of the
author \cite{cava:MongeRCD, cava:decomposition}.
Finally with A. Mondino in \cite{CM1} we obtained the general result permitting to obtained the L\'evy-Gromov isoperimetric inequality.
\text{\varrhom span}ubsection{Outline}
The outline of the paper goes as follows:
Section \textrm{Re}\,f{S:preliminaries} contains all the basic material on Optimal Transportation and the theory of Lott-Sturm-Villani spaces, that is
metric measure spaces verifying the Curvature Dimension condition, $\mathbb{C}D(K,N)$ for short. It also covers some basics on isoperimetric inequality,
Disintegration Theorem and selection theorems we will use during the paper.
In Section \textrm{Re}\,f{S:transportset} we prove all the structure results on the building block
of $L^{1}$-Optimal Transportation, the $\text{\varrhom span}fd$-cyclically monotone sets. Here no curvature assumption enters.
In Section \textrm{Re}\,f{S:cyclically} we show that the aforementioned sets induce a partition of almost all transport, provided the space
enjoies a stronger form of the essentially non-branching condition; we also show that each element of the partition is a geodesic (and therefore a one-dimensional set).
Section \textrm{Re}\,f{S:ConditionalMeasures} contains all the regularity results of conditional
measures of the disintegration induced by the $L^{1}$-Optimal Transportation problem. In particular we will present three assumptions, each one implying the previous one,
yielding three increasing level of regularity of the conditional measures. Finally in Section \textrm{Re}\,f{S:application} we collect the
consequences of the regularity results of Section \textrm{Re}\,f{S:ConditionalMeasures}; in particular we first show the existence of a
solution of the Monge problem under very general regularity assumption (Theorem \textrm{Re}\,f{T:mongeff}) and finally we go back to the L\'evy-Gromov
isoperimetric inequality (Theorem \textrm{Re}\,f{T:iso}).
\text{\varrhom span}ection{Preliminaries}\bigl\langlebel{S:preliminaries}
In what follows we say that a triple $(X,\text{\varrhom span}fd, \mathfrak m)$ is a metric measure space, m.m.s. for short,
if $(X, \text{\varrhom span}fd)$ is a complete and separable metric space and $\mathfrak m$ is positive Radon measure over $X$.
For this paper we will only be concerned with m.m.s. with $\mathfrak m$ probability measure, that is $\mathfrak m(X) =1$.
The space of all Borel probability measures over $X$ will be denoted by $\mathcal{P}(X)$.
A metric space is a geodesic space if and only if for each $x,y \in X$
there exists $\gamma \in \mathcal{G}eo(X)$ so that $\gamma_{0} =x, \gamma_{1} = y$, with
$$
\mathcal{G}eo(X) : = \{ \gamma \in C([0,1], X): \text{\varrhom span}fd(\gamma_{s},\gamma_{t}) = |s-t| \text{\varrhom span}fd(\gamma_{0},\gamma_{1}), \text{ for every } s,t \in [0,1] \}.
$$
It follows from the metric version of the Hopf-Rinow Theorem (see Theorem 2.5.28 of \cite{BBI}) that for complete geodesic spaces,
local completeness is equivalent to properness (a metric space is proper if every closed ball is compact).
So we assume the ambient space $(X,\text{\varrhom span}fd)$ to be proper and geodesic, hence also complete and separable.
Moreover we assume $\mathfrak m$ to be a proability measure, i.e. $\mathfrak m(X)=1$.
We denote by $\mathcal{P}_{2}(X)$ the space of probability measures with finite second moment endowed with the $L^{2}$-Wasserstein distance $W_{2}$ defined as follows: for $\mu_0,\mu_1 \in \mathcal{P}_{2}(X)$ we set
\begin{equation}gin{equation}\bigl\langlebel{eq:Wdef}
W_2^2(\mu_0,\mu_1) = \inf_{ \pi} \int_{X\times X} \text{\varrhom span}fd^2(x,y) \, \pi(dxdy),
\end{equation}
where the infimum is taken over all $\pi \in \mathcal{P}(X \times X)$ with $\mu_0$ and $\mu_1$ as the first and the second marginal, called the set of transference plans.
The set of transference plans realizing the minimum in \eqref{eq:Wdef} will be called the set of optimal transference plans.
Assuming the space $(X,\text{\varrhom span}fd)$ to be geodesic, also the space $(\mathcal{P}_2(X), W_2)$ is geodesic.
Any geodesic $(\mu_t)_{t \in [0,1]}$ in $(\mathcal{P}_2(X), W_2)$ can be lifted to a measure $\nu \in {\mathcal {P}}(\mathcal{G}eo(X))$,
so that $({\varrhom e}_t)_\text{\varrhom span}harp \, \nu = \mu_t$ for all $t \in [0,1]$.
Here for any $t\in [0,1]$, ${\varrhom e}_{t}$ denotes the evaluation map:
$$
{\varrhom e}_{t} : \mathcal{G}eo(X) \to X, \mathfrak quad {\varrhom e}_{t}(\gamma) : = \gamma_{t}.
$$
Given $\mu_{0},\mu_{1} \in \mathcal{P}_{2}(X)$, we denote by
$\mathrm{OptGeo}(\mu_{0},\mu_{1})$ the space of all $\nu \in \mathcal{P}(\mathcal{G}eo(X))$ for which $({\varrhom e}_0,{\varrhom e}_1)_\text{\varrhom span}harp\, \nu$
realizes the minimum in \eqref{eq:Wdef}. If $(X,\text{\varrhom span}fd)$ is geodesic, then the set $\mathrm{OptGeo}(\mu_{0},\mu_{1})$ is non-empty for any $\mu_0,\mu_1\in \mathcal{P}_2(X)$.
It is worth also introducing the subspace of $\mathcal{P}_{2}(X)$
formed by all those measures absolutely continuous with respect with $\mathfrak m$: it is denoted by $\mathcal{P}_{2}(X,\text{\varrhom span}fd,\mathfrak m)$.
\text{\varrhom span}ubsection{Geometry of metric measure spaces}\bigl\langlebel{Ss:geom}
Here we briefly recall the synthetic notions of lower Ricci curvature bounds, for more detail we refer to \cite{BS10,lottvillani:metric,sturm:I, sturm:II, Vil}.
In order to formulate the curvature properties for $(X,\text{\varrhom span}fd,\mathfrak m)$ we introduce the following distortion coefficients: given two numbers $K,N\in \mathbb{R}$ with $N\geq0$, we set for $(t,\theta) \in[0,1] \times \mathbb{R}_{+}$,
\begin{equation}gin{equation}\bigl\langlebel{E:sigma}
\text{\varrhom span}igma_{K,N}^{(t)}(\theta):=
\begin{equation}gin{cases}
\infty, & \textrm{if}\ K\theta^{2} \geq N\pi^{2}, \crcr
\displaystyle \varphirac{\text{\varrhom span}in(t\theta\text{\varrhom span}qrt{K/N})}{\text{\varrhom span}in(\theta\text{\varrhom span}qrt{K/N})} & \textrm{if}\ 0< K\theta^{2} < N\pi^{2}, \crcr
t & \textrm{if}\ K \theta^{2}<0 \ \textrm{and}\ N=0, \ \textrm{or if}\ K \theta^{2}=0, \crcr
\displaystyle \varphirac{\text{\varrhom span}inh(t\theta\text{\varrhom span}qrt{-K/N})}{\text{\varrhom span}inh(\theta\text{\varrhom span}qrt{-K/N})} & \textrm{if}\ K\theta^{2} \leq 0 \ \textrm{and}\ N>0.
\end{cases}
\end{equation}
We also set, for $N\geq 1, K \in \mathbb{R}$ and $(t,\theta) \in[0,1] \times \mathbb{R}_{+}$
\begin{equation}gin{equation} \bigl\langlebel{E:tau}
\tau_{K,N}^{(t)}(\theta): = t^{1/N} \text{\varrhom span}igma_{K,N-1}^{(t)}(\theta)^{(N-1)/N}.
\end{equation}
As we will consider only the case of essentially non-branching spaces, we recall the following definition.
\begin{equation}gin{definition}\bigl\langlebel{D:essnonbranch}
A metric measure space $(X,\text{\varrhom span}fd, \mathfrak m)$ is \emph{essentially non-branching} if and only if for any $\mu_{0},\mu_{1} \in \mathcal{P}_{2}(X)$,
with $\mu_{0}$ absolutely continuous with respect to $\mathfrak m$, any element of $\mathrm{OptGeo}(\mu_{0},\mu_{1})$ is concentrated on a set of non-branching geodesics.
\end{definition}
A set $F \text{\varrhom span}ubset \mathcal{G}eo(X)$ is a set of non-branching geodesics if and only if for any $\gamma^{1},\gamma^{2} \in F$, it holds:
$$
\exists \; \bar t\in (0,1) \text{ such that } \ \varphiorall t \in [0, \bar t\,] \quad \gamma_{ t}^{1} = \gamma_{t}^{2}
\quad
\mathcal{L}ongrightarrow
\quad
\gamma^{1}_{s} = \gamma^{2}_{s}, \quad \varphiorall s \in [0,1].
$$
\begin{equation}gin{definition}[$\mathbb{C}D$ condition]\bigl\langlebel{D:CD}
An essentially non-branching m.m.s. $(X,\text{\varrhom span}fd,\mathfrak m)$ verifies $\mathsf{CD}(K,N)$ if and only if for each pair
$\mu_{0}, \mu_{1} \in \mathcal{P}_{2}(X,\text{\varrhom span}fd,\mathfrak m)$ there exists $\nu \in \mathrm{OptGeo}(\mu_{0},\mu_{1})$ such that
\begin{equation}gin{equation}\bigl\langlebel{E:CD}
\varrho_{t}^{-1/N} (\gamma_{t}) \geq \tau_{K,N}^{(1-t)}(\text{\varrhom span}fd( \gamma_{0}, \gamma_{1}))\varrho_{0}^{-1/N}(\gamma_{0})
+ \tau_{K,N}^{(t)}(\text{\varrhom span}fd(\gamma_{0},\gamma_{1}))\varrho_{1}^{-1/N}(\gamma_{1}), \mathfrak quad \nu\text{-a.e.} \, \gamma \in \mathcal{G}eo(X),
\end{equation}
for all $t \in [0,1]$, where $({\varrhom e}_{t})_\text{\varrhom span}harp \, \nu = \varrho_{t} \mathfrak m$.
\end{definition}
For the general definition of $\mathbb{C}D(K,N)$ see \cite{lottvillani:metric, sturm:I, sturm:II}.
\begin{equation}gin{remark}\bigl\langlebel{R:CDN-1}
It is worth recalling that if $(M,g)$ is a Riemannian manifold of dimension $n$ and
$h \in C^{2}(M)$ with $h > 0$, then the m.m.s. $(M,g,h \, vol)$ verifies $\mathbb{C}D(K,N)$ with $N\geq n$ if and only if (see Theorem 1.7 of \cite{sturm:II})
$$
Ric_{g,h,N} \geq K g, \mathfrak quad Ric_{g,h,N} : = Ric_{g} - (N-n) \varphirac{\nabla_{g}^{2} h^{\varphirac{1}{N-n}}}{h^{\varphirac{1}{N-n}}}.
$$
In particular if $N = n$ the generalized Ricci tensor $Ric_{g,h,N}= Ric_{g}$ makes sense only if $h$ is constant.
Another important case is when $I \text{\varrhom span}ubset \mathbb{R}$ is any interval, $h \in C^{2}(I)$
and $\mathcal{L}^{1}$ is the one-dimensional Lebesgue measure; then the m.m.s. $(I ,|\cdot|, h \mathcal{L}^{1})$ verifies $\mathbb{C}D(K,N)$ if and only if
\begin{equation}gin{equation}\bigl\langlebel{E:CD-N-1}
\left(h^{\varphirac{1}{N-1}}\varrhoight)'' + \varphirac{K}{N-1}h^{\varphirac{1}{N-1}} \leq 0,
\end{equation}
and verifies $\mathbb{C}D(K,1)$ if and only if $h$ is constant. Inequality \eqref{E:CD-N-1} has also a non-smooth counterpart; if we drop the smoothness assumption on $h$
it can be proven that the m.m.s. $(I ,|\cdot|, h \mathcal{L}^{1})$ verifies $\mathbb{C}D(K,N)$ if and only if
\begin{equation}gin{equation}\bigl\langlebel{E:curvdensmmR}
h( (1-s) t_{0} + s t_{1} )^{1/(N-1)}
\geq \text{\varrhom span}igma^{(1-s)}_{K,N-1}(t_{1} - t_{0}) h (t_{0})^{1/(N-1)} + \text{\varrhom span}igma^{(s)}_{K,N-1}(t_{1} - t_{0}) h (t_{1})^{1/(N-1)},
\end{equation}
that is the formulation in the sense of distributions of the differential inequality
$$
\left(h^{\varphirac{1}{N-1}}\varrhoight)'' + \varphirac{K}{N-1}h^{\varphirac{1}{N-1}} \leq 0.
$$
Recall indeed that $s \mapsto \text{\varrhom span}igma^{(s)}_{K,N-1}(\theta)$ solves in the classical sense $f'' + (t_{1}-t_{0})^{2} \varphirac{K}{N-1}f = 0$.
\end{remark}
We also mention the more recent Riemannian curvature dimension condition $\mathbb{R}CD^{*}(K,N)$. In the infinite dimensional case, i.e. $N = \infty$, it was introduced \cite{AGS11b}.
The class $\mathbb{R}CD^{*}(K,N)$ with $N<\infty$ has been proposed in \cite{G15} and deeply investigated
in \cite{AGS, EKS} and \cite{AMS}. We refer to these papers and references therein for a general account
on the synthetic formulation of Ricci curvature lower bounds for metric measure spaces.
Here we only mention that $\mathbb{R}CD^{*}(K,N)$ condition
is an enforcement of the so called reduced curvature dimension condition, denoted by $\mathbb{C}D^{*}(K,N)$, that has been introduced in \cite{BS10}:
in particular the additional condition is that the Sobolev space $W^{1,2}(X,\mathfrak m)$ is an Hilbert space, see \cite{G15, AGS11a, AGS11b}.
The reduced $\mathbb{C}D^{*}(K,N)$ condition asks for the same inequality \eqref{E:CD} of $\mathbb{C}D(K,N)$ but the
coefficients $\tau_{K,N}^{(t)}(\text{\varrhom span}fd(\gamma_{0},\gamma_{1}))$ and $\tau_{K,N}^{(1-t)}(\text{\varrhom span}fd(\gamma_{0},\gamma_{1}))$
are replaced by $\text{\varrhom span}igma_{K,N}^{(t)}(\text{\varrhom span}fd(\gamma_{0},\gamma_{1}))$ and $\text{\varrhom span}igma_{K,N}^{(1-t)}(\text{\varrhom span}fd(\gamma_{0},\gamma_{1}))$, respectively.
Hence while the distortion coefficients of the $\mathbb{C}D(K,N)$ condition
are formally obtained imposing one direction with linear distortion and $N-1$ directions affected by curvature,
the $\mathbb{C}D^{*}(K,N)$ condition imposes the same volume distortion in all the $N$ directions.
For both definitions there is a local version that is of some relevance for our analysis. Here we state only the local formulation $\mathsf{CD}(K,N)$,
being clear what would be the one for $\mathsf{CD}^{*}(K,N)$.
\begin{equation}gin{definition}[$\mathbb{C}D_{loc}$ condition]\bigl\langlebel{D:loc}
An essentially non-branching m.m.s. $(X,\text{\varrhom span}fd,\mathfrak m)$ satisfies $\mathbb{C}D_{loc}(K,N)$ if for any point $x \in X$ there exists a neighborhood $X(x)$ of $x$ such that for each pair
$\mu_{0}, \mu_{1} \in \mathcal{P}_{2}(X,\text{\varrhom span}fd,\mathfrak m)$ supported in $X(x)$
there exists $\nu \in \mathrm{OptGeo}(\mu_{0},\mu_{1})$ such that \eqref{E:CD} holds true for all $t \in [0,1]$.
The support of $({\varrhom e}_{t})_\text{\varrhom span}harp \, \nu$ is not necessarily contained in the neighborhood $X(x)$.
\end{definition}
One of the main properties of the reduced curvature dimension condition is the globalization one:
under the essentially non-branching property, $\mathsf{CD}^{*}_{loc}(K,N)$ and $\mathsf{CD}^{*}(K,N)$ are equivalent (see \cite[Corollary 5.4]{BS10}), i.e. the
$\mathsf{CD}^{*}$-condition verifies the local-to-global property.
We also recall a few relations between $\mathbb{C}D$ and $\mathbb{C}D^{*}$.
It is known by \cite[Theorem 2.7]{GigliMap} that, if $(X,\text{\varrhom span}fd,\mathfrak m)$ is a non-branching metric measure space
verifying $\mathbb{C}D(K,N)$ and $\mu_{0}, \mu_{1} \in \mathcal{P}(X)$ with $\mu_{0}$ absolutely continuous with respect to $\mathfrak m$,
then there exists a unique optimal map $T : X \to X$ such $(id, T)_\text{\varrhom span}harp\, \mu_{0}$ realizes the minimum in \eqref{eq:Wdef} and the set
$\mathrm{OptGeo}(\mu_{0},\mu_{1})$ contains only one element. The same proof holds if one replaces the non-branching assumption with the more general
one of essentially non-branching, see for instance \cite{GRS2013}.
\text{\varrhom span}ubsection{Isoperimetric profile function}
Given a m.m.s. $(X,\text{\varrhom span}fd,\mathfrak m)$ as above and a Borel subset $A\text{\varrhom span}ubset X$, let $A^{\textbf{X}repsilon}$ denote the $\textbf{X}repsilon$-tubular neighborhood
$$
A^{\textbf{X}repsilon}:=\{x \in X \,:\, \exists y \in A \text{ such that } \text{\varrhom span}fd(x,y) < \textbf{X}repsilon \}.
$$
The Minkowski (exterior) boundary measure $\mathfrak m^+(A)$ is defined by
\begin{equation}gin{equation}\bigl\langlebel{eq:MinkCont}
\mathfrak m^+(A):=\liminf_{\textbf{X}repsilon\downarrow 0} \varphirac{\mathfrak m(A^\textbf{X}repsilon)-\mathfrak m(A)}{\textbf{X}repsilon}.
\end{equation}
The \emph{isoperimetric profile}, denoted by ${\mathcal{I}}_{(X,\text{\varrhom span}fd,\mathfrak m)}$, is defined as the point-wise maximal function so that $\mathfrak m^+(A)\geq \mathcal{I}_{(X,\text{\varrhom span}fd,\mathfrak m)}(\mathfrak m(A))$
for every Borel set $A \text{\varrhom span}ubset X$, that is
\begin{equation}gin{equation}\bigl\langlebel{E:profile}
\mathcal{I}_{(X,\text{\varrhom span}fd,\mathfrak m)}(v) : = \inf \big\{ \mathfrak m^{+}(A) \colon A \text{\varrhom span}ubset X \, \textrm{ Borel}, \, \mathfrak m(A) = v \big\}.
\end{equation}
If $K>0$ and $N\in \mathbb{N}$, by the L\'evy-Gromov isoperimetric inequality \eqref{eq:LevyGromov} we know that, for $N$-dimensional smooth manifolds having Ricci $\geq K$,
the isoperimetric profile function is bounded below by the one of the $N$-dimensional round sphere of the suitable radius.
In other words the \emph{model} isoperimetric profile function is the one of ${\mathbb S}^N$.
For $N\geq 1, K\in \mathbb{R}$ arbitrary real numbers the situation is more complicated, and just recently E. Milman \cite{Mil} discovered what is the model isoperimetric profile.
We refer to \cite{Mil} for all the details. Here we just recall the relevance of isoperimetric profile functions for m.m.s. over $(\mathbb{R}, |\cdot|)$:
given $K\in \mathbb{R}, N\in[1,+\infty)$ and $D\in (0,+\infty]$, consider the function
\begin{equation}gin{equation}\bigl\langlebel{defcI}
\mathcal{I}_{K,N,D}(v) : = \inf \left\{ \mu^{+}(A) \colon A\text{\varrhom span}ubset \mathbb{R}, \,\mu(A) = v, \, \mu \in \mathcal{F}_{K,N,D} \varrhoight\},
\end{equation}
where $\mathcal{F}_{K,N,D}$ denotes the set of $\mu \in \mathcal{P}(\mathbb{R})$ such that $\text{\varrhom span}upp(\mu) \text{\varrhom span}ubset [0,D]$
and $\mu = h \cdot \mathcal{L}^{1}$ with $h \in C^{2}((0,D))$ satisfying
\begin{equation}gin{equation}\bigl\langlebel{eq:DiffIne}
\left( h^{\varphirac{1}{N-1}} \varrhoight)'' + \varphirac{K}{N-1} h^{\varphirac{1}{N-1}} \leq 0 \quad \text{if }N \in (1,\infty), \quad h\equiv \textrm{const} \quad \text{if }N=1.
\end{equation}
Then from \cite[Theorem 1.2, Corollary 3.2]{Mil} it follows that for $N$-dimensional smooth manifolds having Ricci $\geq K$, with $K\in \mathbb{R}$
arbitrary real number, and diameter $D$, the isoperimetric profile function is bounded below by $\mathcal{I}_{K,N,D}$ and the bound is sharp.
This also justifies the notation.
Going back to non-smooth metric measure spaces (what follows is taken from \cite{CM1}), it is necessary to consider the following broader family of measures:
\begin{equation}gin{eqnarray}
\mathcal{F}^{s}_{K,N,D} : = \{ \mu \in \mathcal{P}(\mathbb{R}) : &\text{\varrhom span}upp(\mu) \text{\varrhom span}ubset [0,D], \, \mu = h_{\mu} \mathcal{L}^{1},\,
h_{\mu}\, \textrm{verifies} \, \eqref{E:curvdensmmR} \ \textrm{and is continuous if } N\in (1,\infty), \nonumber \\
& \quad h_{\mu}\equiv \textrm{const} \text{ if }N=1 \},
\end{eqnarray}
and the corresponding comparison \emph{synthetic} isoperimetric profile:
$$
\mathcal{I}^{s}_{K,N,D}(v) : = \inf \left\{ \mu^{+}(A) \colon A\text{\varrhom span}ubset \mathbb{R}, \,\mu(A) = v, \, \mu \in \mathcal{F}^{s}_{K,N,D} \varrhoight\},
$$
where $\mu^{+}(A)$ denotes the Minkowski content defined in \eqref{eq:MinkCont}.
The term synthetic refers to $\mu \in \mathcal{F}^{s}_{K,N,D}$ meaning that the Ricci curvature bound is satisfied in its synthetic formulation:
if $\mu = h \cdot \mathcal{L}^{1}$, then $h$ verifies \eqref{E:curvdensmmR}.
We have already seen that $\mathcal{F}_{K,N,D} \text{\varrhom span}ubset \mathcal{F}^{s}_{K,N,D}$; actually one can prove that
$\mathcal{I}^{s}_{K,N,D}$ coincides with its smooth counterpart $\mathcal{I}_{K,N,D}$ for every volume $v \in [0,1]$ via a smoothing argument.
We therefore need the following approximation result. In order to state it let us recall that a standard mollifier in $\mathbb{R}$ is a non negative $C^\infty(\mathbb{R})$
function $\psi$ with compact support in $[0,1]$ such that $\int_{\mathbb{R}} \psi = 1$.
\begin{equation}gin{lemma}[Lemma 6.2, \cite{CM1}]\bigl\langlebel{lem:approxh}
Let $D \in (0,\infty)$ and let $h:[0,D] \to [0,\infty)$ be a continuous function. Fix $N\in (1,\infty)$ and for $\textbf{X}repsilon>0$ define
\begin{equation}gin{equation}
h_{\textbf{X}repsilon}(t):=[h^{\varphirac{1}{N-1}}\ast \psi_{\textbf{X}repsilon} (t)]^{N-1} := \left[ \int_{\mathbb{R}} h(t-s)^{\varphirac{1}{N-1}} \psi_{\textbf{X}repsilon} (s) \, d s\varrhoight]^{N-1}
= \left[ \int_{\mathbb{R}} h(s)^{\varphirac{1}{N-1}} \psi_{\textbf{X}repsilon} (t-s) \, d s\varrhoight]^{N-1},
\end{equation}
where $\psi_\textbf{X}repsilon(x)=\varphirac{1}{\textbf{X}repsilon} \psi(x/\textbf{X}repsilon)$ and $\psi$ is a standard mollifier function. The following properties hold:
\begin{equation}gin{enumerate}
\item $h_{\textbf{X}repsilon}$ is a non-negative $C^\infty$ function with support in $[-\textbf{X}repsilon, D+\textbf{X}repsilon]$;
\item $h_{\textbf{X}repsilon}\to h$ uniformly as $\textbf{X}repsilon \downarrow 0$, in particular $h_{\textbf{X}repsilon} \to h$ in $L^{1}$.
\item If $h$ satisfies the convexity condition \eqref{E:curvdensmm} corresponding to the above fixed $N>1$
and some $K \in \mathbb{R}$ then also $h_{\textbf{X}repsilon}$ does. In particular $h_{\textbf{X}repsilon}$ satisfies the differential inequality \eqref{eq:DiffIne}.
\end{enumerate}
\end{lemma}
Using this approximation one can prove the following
\begin{equation}gin{theorem}[Theorem 6.3, \cite{CM1}]\bigl\langlebel{thm:I=Is}
For every $v\in [0,1]$, $K \in \mathbb{R}$, $N\in [1,\infty)$, $D\in (0,\infty]$ it holds $\mathcal{I}^{s}_{K,N,D}(v)=\mathcal{I}_{K,N,D}(v)$.
\end{theorem}
\text{\varrhom span}ubsection{Disintegration of measures}
We include here a version of Disintegration Theorem that we will use.
We will follow Appendix A of \cite{biacara:extreme} where a self-contained approach (and a proof) of Disintegration Theorem in countably generated measure spaces can be found.
An even more general version of Disintegration Theorem can be found in Section 452 of \cite{Fre:measuretheory4}.
Recall that a $\text{\varrhom span}igma$-algebra is \emph{countably generated} if there exists a countable family of sets so that
the $\text{\varrhom span}igma$-algebra coincide with the smallest $\text{\varrhom span}igma$-algebra containing them.
Given a measurable space $(X, \mathscr{X})$, i.e. $\mathscr{X}$ is a $\text{\varrhom span}igma$-algebra of subsets of $X$,
and a function $\mathbb{Q}Q : X \to Q$, with $Q$ general set, we can endow $Q$ with the \emph{push forward $\text{\varrhom span}igma$-algebra} $\mathscr{Q}$ of $\mathscr{X}$:
$$
C \in \mathscr{Q} \quad \mathcal{L}ongleftrightarrow \quad \mathbb{Q}Q^{-1}(C) \in \mathscr{X},
$$
which could be also defined as the biggest $\text{\varrhom span}igma$-algebra on $Q$ such that $\mathbb{Q}Q$ is measurable.
Moreover given a probability measure $\mathfrak m$ on $(X,\mathscr{X})$, define a probability
measure $\mathfrak q$ on $(Q,\mathscr{Q})$ by push forward via $\mathbb{Q}Q$, i.e. $\mathfrak q := \mathbb{Q}Q_\text{\varrhom span}harp \, \mathfrak m$.
This general scheme fits with the following situation: given a measure space $(X,\mathscr{X},\mathfrak m)$, suppose
a partition of $X$ is given in the form $\{ X_{q}\}_{q\in Q}$, $Q$ is the set of indices and $\mathbb{Q}Q : X \to Q$ is the quotient map, i.e.
$$
q = \mathbb{Q}Q(x) \iff x \in X_{q}.
$$
Following the previous scheme, we can consider also the quotient $\text{\varrhom span}igma$-algebra $\mathscr{Q}$ and the quotient measure $\mathfrak q$ obtaining
the quotient measure space $(Q, \mathscr{Q}, \mathfrak q)$.
\begin{equation}gin{definition}
\bigl\langlebel{defi:dis}
A \emph{disintegration} of $\mathfrak m$ \emph{consistent with} $\mathbb{Q}Q$ is a map
$$
Q \ni q \longmapsto \mathfrak m_{q} \in \mathcal{P}(X,\mathscr{X})
$$
such that the following hold:
\begin{equation}gin{enumerate}
\item for all $B \in \mathscr{X}$, the map $\mathfrak m_{\cdot}(B)$ is $\mathfrak q$-measurable;
\item for all $B \in \mathscr{X}, C \in \mathscr{Q}$ satisfies the consistency condition
$$
\mathfrak m \left(B \cap \mathbb{Q}Q^{-1}(C) \varrhoight) = \int_{C} \mathfrak m_{q}(B)\, \mathfrak q(dq).
$$
\end{enumerate}
A disintegration is \emph{strongly consistent with respect to $\mathbb{Q}Q$} if for $\mathfrak q$-a.e. $q \in Q$ we have $\mathfrak m_{q}(\mathbb{Q}Q^{-1}(q))=1$.
The measures $\mathfrak m_q$ are called \emph{conditional probabilities}.
\end{definition}
When the map $\mathbb{Q}Q$ is induced by a partition of $X$ as before, we will directly say that the disintegration is consistent with the partition, meaning that
the disintegration is consistent with the quotient map $\mathbb{Q}Q$ associated to the partition $X = \cup_{q\in Q} X_{q}$.
We now report Disintegration Theorem.
\begin{equation}gin{theorem}[Theorem A.7, Proposition A.9 of \cite{biacara:extreme}]\bigl\langlebel{T:disintegrationgeneral}
\bigl\langlebel{T:disintr}
Assume that $(X,\mathscr{X},\varrhoho)$ is a countably generated probability space and $X = \cup_{q \in Q}X_{q}$ is a partition of $X$.
Then the quotient probability space $(Q, \mathscr{Q},\mathfrak q)$ is essentially countably generated and
there exists a unique disintegration $q \mapsto \mathfrak m_{q}$ consistent with the partition $X = \cup_{q\in Q} X_{q}$.
The disintegration is strongly consistent if and only if there exists a $\mathfrak m$-section $S \in \mathscr{X}$ such that the $\text{\varrhom span}igma$-algebra $\mathscr{S}$ contains $\mathcal{B}(S)$.
\end{theorem}
We expand the statement of Theorem \textrm{Re}\,f{T:disintegrationgeneral}.\\
In the measure space $(Q, \mathscr{Q},\mathfrak q)$, the $\text{\varrhom span}igma$-algebra $\mathscr{Q}$ is \emph{essentially countably generated} if, by definition,
there exists a countable family of sets $Q_{n} \text{\varrhom span}ubset Q$ such that for any $C \in \mathscr{Q}$ there exists $\hat C \in \hat{\mathscr{Q}}$,
where $\hat{\mathscr{Q}}$ is the $\text{\varrhom span}igma$-algebra generated by $\{ Q_{n} \}_{n \in \mathbb{N}}$, such that $\mathfrak q(C\, \mathcal{D}elta \, \hat C) = 0$.
Uniqueness is understood in the following sense: if $q\mapsto \mathfrak m^{1}_{q}$ and $q\mapsto \mathfrak m^{2}_{q}$ are two consistent disintegrations then
$\mathfrak m^{1}_{q}=\mathfrak m^{2}_{q}$ for $\mathfrak q$-a.e. $q \in Q$.
Finally, a set $S$ is a section for the partition $X = \cup_{q}X_{q}$ if for any $q \in Q$ there exists a unique $x_{q} \in S \cap X_{q}$.
A set $S_{\mathfrak m}$ is an $\mathfrak m$-section if there exists $Y \text{\varrhom span}ubset X$ with $\mathfrak m(X \text{\varrhom span}etminus Y) = 0$ such that the partition $Y = \cup_{q} (X_{q} \cap Y)$
has section $S_{\mathfrak m}$. Once a section (or an $\mathfrak m$-section) is given, one can obtain the measurable space $(S,\mathscr{S})$ by pushing forward the $\text{\varrhom span}igma$-algebra
$\mathscr{X}$ on $S$ via the map that associates to any $X_{q} \ni x \mapsto x_{q} = S \cap X_{q}$.
\text{\varrhom span}ection{Transport set}\bigl\langlebel{S:transportset}
The following setting is fixed once for all:
\begin{equation}gin{center}
$(X,\text{\varrhom span}fd,\mathfrak m)$ is a fixed metric measure space with $\mathfrak m(X)=1$ such that \\
the ambient metric space $(X, \text{\varrhom span}fd)$ is geodesic and proper (hence complete and separable).
\end{center}
Let $\varphi : X \to \mathbb{R}$ be any $1$-Lipschitz function. Here we present some useful results (all of them already presented in \cite{biacava:streconv})
concerning the $\text{\varrhom span}fd$-cyclically monotone set associated with $\varphi$:
\begin{equation}gin{equation}\bigl\langlebel{E:Gamma}
\mathcal{G}amma : = \{ (x,y) \in X\times X : \varphi(x) - \varphi(y) = \text{\varrhom span}fd(x,y) \},
\end{equation}
that can be seen as the set of couples moved by $\varphi$ with maximal slope.
Recall that a set $\mathcal{L}ambda \text{\varrhom span}ubset X \times X$ is said to be $\text{\varrhom span}fd$-cyclically monotone if for any finite set of points $(x_{1},y_{1}),\dots,(x_{N},y_{N})$ it holds
$$
\text{\varrhom span}um_{i = 1}^{N} \text{\varrhom span}fd(x_{i},y_{i}) \leq \text{\varrhom span}um_{i = 1}^{N} \text{\varrhom span}fd(x_{i},y_{i+1}),
$$
with the convention that $y_{N+1} = y_{1}$.
The following lemma is a consequence of the $\text{\varrhom span}fd$-cyclically monotone structure of $\mathcal{G}amma$.
\begin{equation}gin{lemma}\bigl\langlebel{L:cicli}
Let $(x,y) \in X\times X$ be an element of $\mathcal{G}amma$. Let $\gamma \in \mathcal{G}eo(X)$ be such that $\gamma_{0} = x$
and $\gamma_{1}=y$. Then
$$
(\gamma_{s},\gamma_{t}) \in \mathcal{G}amma,
$$
for all $0\leq s \leq t \leq 1$.
\end{lemma}
\begin{equation}gin{proof}
Take $0\leq s \leq t \leq 1$ and note that
\begin{equation}gin{align*}
\varphi(\gamma_{s})& - \varphi(\gamma_{t})\crcr
= &~ \varphi(\gamma_{s}) - \varphi(\gamma_{t}) + \varphi(\gamma_{0}) - \varphi(\gamma_{0}) + \varphi(\gamma_{1}) - \varphi(\gamma_{1})\crcr
\geq &~ \text{\varrhom span}fd(\gamma_{0},\gamma_{1}) - \text{\varrhom span}fd(\gamma_{0},\gamma_{s}) - \text{\varrhom span}fd(\gamma_{t},\gamma_{1}) \crcr
= &~ \text{\varrhom span}fd(\gamma_{s},\gamma_{t}).
\end{align*}
The claim follows.
\end{proof}
It is natural then to consider the set of geodesics $G \text{\varrhom span}ubset \mathcal{G}eo(X)$ such that
$$
\gamma \in G \iff \{ (\gamma_{s},\gamma_{t}) : 0\leq s \leq t \leq 1 \} \text{\varrhom span}ubset \mathcal{G}amma,
$$
that is $G : = \{ \gamma \in \mathcal{G}eo(X) : (\gamma_{0},\gamma_{1}) \in \mathcal{G}amma \}$.
We now recall some basic definitions of the $L^{1}$-optimal transportation theory that will be needed to describe the structure of $\mathcal{G}amma$.
\begin{equation}gin{definition}\bigl\langlebel{D:transport}
We define the set of \emph{transport rays} by
$$
R := \mathcal{G}amma \cup \mathcal{G}amma^{-1},
$$
where $\mathcal{G}amma^{-1}:= \{ (x,y) \in X \times X : (y,x) \in \mathcal{G}amma \}$.
The set of \emph{initial points} and \emph{final points} are defined respectively by
\begin{equation}gin{align*}
{\mathfrak a} :=& \{ z \in X: \nexists \, x \in X, (x,z) \in \mathcal{G}amma, \text{\varrhom span}fd(x,z) > 0 \}, \crcr
{\mathfrak b} :=& \{ z \in X: \nexists \, x \in X, (z,x) \in \mathcal{G}amma, \text{\varrhom span}fd(x,z) > 0 \}.
\end{align*}
The set of \emph{end points} is ${\mathfrak a} \cup {\mathfrak b}$.
We define the subset of $X$, \emph{transport set with end points}:
$$
\mathcal{T}_{e} = P_{1}(\mathcal{G}amma \text{\varrhom span}etminus \{ x = y \}) \cup
P_{1}(\mathcal{G}amma^{-1}\text{\varrhom span}etminus \{ x=y \}).
$$
where $\{ x = y\}$ stands for $\{ (x,y) \in X^{2} : \text{\varrhom span}fd(x,y) = 0 \}$.
\end{definition}
Few comments are in order. Notice that $R$ coincide with $\{(x,y) \in X \times X \colon |\varphi(x) -\varphi(y)| = \text{\varrhom span}fd(x,y) \}$;
the name transport set with end points for $\mathcal{T}_{e}$ is motivated by the fact that later on we will consider a more regular subset of $\mathcal{T}_{e}$ that will be called transport set;
moreover if $x \in X$ for instance is moved forward but not backward by $\varphi$, this is translated in $x \in \mathcal{G}amma$ and $x \notin \mathcal{G}amma^{-1}$; anyway it belongs to $\mathcal{T}_{e}$.
We also introduce the following notation that will be used throughout the paper;
we set $\mathcal{G}amma(x):=P_2(\mathcal{G}amma\cap(\{x\}\times X))$ and $\mathcal{G}amma^{-1}(x):=P_2(\mathcal{G}amma^{-1}\cap(\{x\} \times X))$.
More in general if $F \text{\varrhom span}ubset X \times X$, we set $F(x) = P_2(F \cap (\{x\}\times X))$.
\begin{equation}gin{remark}\bigl\langlebel{R:regularity}
Here we discuss the measurability of the sets introduced in Definition \textrm{Re}\,f{D:transport}.
Since $\varphi$ is $1$-Lipschitz, $\mathcal{G}amma$ is closed and therefore $\mathcal{G}amma^{-1}$ and $R$ are closed as well.
Moreover by assumption the space is proper, hence the sets $\mathcal{G}amma, \mathcal{G}amma^{-1}, R$ are $\text{\varrhom span}igma$-compact (countable union of compact sets).
Then we look at the set of initial and final points:
$$
{\mathfrak a} = P_{2} \left( \mathcal{G}amma \cap \{ (x,z) \in X\times X : \text{\varrhom span}fd(x,z) > 0 \} \varrhoight)^{c}, \mathfrak quad
{\mathfrak b} = P_{1} \left( \mathcal{G}amma \cap \{ (x,z) \in X\times X : \text{\varrhom span}fd(x,z) > 0 \} \varrhoight)^{c}.
$$
Since $\{ (x,z) \in X\times X : \text{\varrhom span}fd(x,z) > 0 \} = \cup_{n} \{ (x,z) \in X\times X : \text{\varrhom span}fd(x,z) \geq 1/n \}$, it follows that
it follows that both ${\mathfrak a}$ and ${\mathfrak b}$ are the complement of $\text{\varrhom span}igma$-compact sets.
Hence ${\mathfrak a}$ and ${\mathfrak b}$ are Borel sets. Reasoning as before, it follows that $\mathcal{T}_{e}$ is a $\text{\varrhom span}igma$-compact set.
\end{remark}
\begin{equation}gin{lemma} \bigl\langlebel{L:mapoutside}
Let $\pi \in \Pi(\mu_{0},\mu_{1})$ with $\pi(\mathcal{G}amma) = 1$, then
$$
\pi(\mathcal{T}_e \times \mathcal{T}_e \cup \{x = y\}) = 1.
$$
\end{lemma}
\begin{equation}gin{proof}
It is enough to observe that if $(z,w) \in \mathcal{G}amma$ with $z \neq w$, then $w \in \mathcal{G}amma(z)$ and $z \in \mathcal{G}amma^{-1}(w)$
and therefore
$$
(z,w) \in \mathcal{T}_{e}\times \mathcal{T}_{e}.
$$
Hence $\mathcal{G}amma \text{\varrhom span}etminus \{x = y\} \text{\varrhom span}ubset \mathcal{T}_e \times \mathcal{T}_e$. Since $\pi(\mathcal{G}amma) =1$,
the claim follows.
\end{proof}
As a consequence, $\mu_{0}(\mathcal{T}_e) = \mu_{1}(\mathcal{T}_e)$ and any optimal map $T$ such that
$T_\text{\varrhom span}harp \mu_{0} \llcorner_{\mathcal{T}_e}= \mu_{1} \llcorner_{\mathcal{T}_e}$
can be extended to an optimal map $T'$ with $ T^{'}_\text{\varrhom span}harp \mu_{0} = \mu_{1}$ with the same cost by setting
\begin{equation}gin{equation}
\bigl\langlebel{E:extere}
T'(x) =
\begin{equation}gin{cases}
T(x), & \textrm{if } x \in \mathcal{T}_e \crcr
x, & \textrm{if } x \notin \mathcal{T}_e.
\end{cases}
\end{equation}
It can be proved that the set of transport rays $R$ induces an equivalence relation on a subset of $\mathcal{T}_{e}$.
It is sufficient to remove from $\mathcal{T}_{e}$
the branching points of geodesics. Then using curvature properties of the space, one can prove that such branching points all have $\mathfrak m$-measure zero.
\text{\varrhom span}ubsection{Branching structures in the Transport set}
What follows was first presented in \cite{cava:MongeRCD}.
Consider the sets of respectively forward and backward branching points
\begin{equation}gin{align}\bigl\langlebel{E:branchingpoints}
A_{+}: = &~\{ x \in \mathcal{T}_{e} : \exists z,w \in \mathcal{G}amma(x), (z,w) \notin R \}, \nonumber \\
A_{-} : = &~\{ x \in \mathcal{T}_{e} : \exists z,w \in \mathcal{G}amma(x)^{-1}, (z,w) \notin R \}.
\end{align}
The sets $A_{\pm}$ are $\text{\varrhom span}igma$-compact sets. Indeed since $(X,\text{\varrhom span}fd)$ is proper, any open set is $\text{\varrhom span}igma$-compact.
The main motivation for the definition of $A_{+}$ and $A_{-}$ is contained in the next
\begin{equation}gin{theorem}\bigl\langlebel{T:equivalence}
The set of transport rays $R\text{\varrhom span}ubset X \times X$ is an equivalence relation on the set
$$
\mathcal{T}_{e} \text{\varrhom span}etminus \left( A_{+} \cup A_{-} \varrhoight).
$$
\end{theorem}
\begin{equation}gin{proof}
First, for all $x \in P_{1}(\mathcal{G}amma)$, $(x,x) \in R$. If $x,y \in \mathcal{T}_{e}$ with $(x,y) \in R$, then by definition of $R$, it follows straightforwardly that $(y,x) \in R$.
So the only property needing a proof is transitivity. Let $x,z,w \in \mathcal{T}_{e} \text{\varrhom span}etminus \left( A_{+} \cup A_{-} \varrhoight)$
be such that $(x,z), (z,w) \in R$ with $x,z$ and $w$ distinct points. The claim is $(x,w) \in R$.
So we have 4 different possibilities: the first one is
\[
z\in \mathcal{G}amma(x), \quad w \in \mathcal{G}amma(z).
\]
This immediately implies $w \in \mathcal{G}amma(x)$ and therefore $(x,w) \in R$.
The second possibility is
\[
z\in \mathcal{G}amma(x), \quad z \in \mathcal{G}amma(w),
\]
that can be rewritten as $(z,x), (z,w) \in \mathcal{G}amma^{-1}$. Since $z \notin A_{-}$, necessarily $(x,w) \in R$.
Third possibility:
\[
x\in \mathcal{G}amma(z), \quad w \in \mathcal{G}amma(z),
\]
and since $z \notin A_{+}$ it follows that $(x,w) \in R$.
The last case is
\[
x\in \mathcal{G}amma(z), \quad z \in \mathcal{G}amma(w),
\]
and therefore $x \in \mathcal{G}amma(w)$, hence $(x,w) \in R$ and the claim follows.
\end{proof}
Next, we show that each equivalence class of $R$ is formed by a single geodesic.
\begin{equation}gin{lemma}\bigl\langlebel{L:singlegeo}
For any $x \in \mathcal{T}$ and $z,w \in R(x)$ there exists $\gamma \in G \text{\varrhom span}ubset \mathcal{G}eo(X)$ such that
$$
\{ x, z,w \} \text{\varrhom span}ubset \{ \gamma_{s} : s\in [0,1] \}.
$$
If $\hat \gamma \in G$ enjoys the same property, then
\[
\big( \{ \hat \gamma_{s} : s \in [0,1] \} \cup \{ \gamma_{s} : s \in [0,1] \} \big) \text{\varrhom span}ubset \{ \tilde \gamma_{s} : s \in [0,1] \}
\]
for some $\tilde \gamma \in G$.
\end{lemma}
Since $G = \{ \gamma \in \mathcal{G}eo(X) : (\gamma_{0},\gamma_{1}) \in \mathcal{G}amma \}$, Lemma \textrm{Re}\,f{L:singlegeo} states that
as soon as we fix an element $x$ in $\mathcal{T}_{e} \text{\varrhom span}etminus (A_{+} \cup A_{-})$ and we pick two elements $z,w$ in the same equivalence class of $x$, then these three points are aligned on
a geodesic $\gamma$ whose image is again all contained in the same equivalence class $R(x)$.
\begin{equation}gin{proof}
Assume that $x, z$ and $w$ are all distinct points otherwise the claim follows trivially.
We consider different cases.
\noindent
\emph{First case:} $z \in \mathcal{G}amma(x)$ and $w \in \mathcal{G}amma^{-1}(x)$. \\
By $\text{\varrhom span}fd$-cyclical monotonicity
$$
\text{\varrhom span}fd(z,w) \leq \text{\varrhom span}fd(z,x) + \text{\varrhom span}fd(x,w) = \varphi(w) - \varphi(z) \leq \text{\varrhom span}fd(z,w).
$$
Hence $z,x$ and $w$ lie on a geodesic.
\noindent
\emph{Second case:} $z,w \in \mathcal{G}amma(x)$. \\
Without loss of generality $\varphi(x) \geq \varphi(w) \geq \varphi(z)$. Since in the proof of
Lemma \textrm{Re}\,f{L:geoingamma} we have already excluded the case $\varphi(w) = \varphi(z)$, we assume $\varphi(x) > \varphi(w) > \varphi(z)$.
Then if there would not exist any geodesics $\gamma \in G$ with $\gamma_{0} = x$ and $\gamma_{1} = z$ and $\gamma_{s} = w$,
there will be $\gamma \in G$ with $(\gamma_{0},\gamma_{1}) = (x,z)$ and $s \in (0,1)$ such that
$$
\varphi(\gamma_{s}) = \varphi(w), \mathfrak quad \gamma_{s} \in \mathcal{G}amma(x), \mathfrak quad \gamma_{s} \neq w.
$$
As observed in the proof of Lemma \textrm{Re}\,f{L:geoingamma}, this would imply that $(\gamma_{s},w) \notin R$ and
since $x \notin A_{+}$ this would be a contradiction. Hence the second case follows.
The remaining two cases follow with the same reasoning, exchanging the role of $\mathcal{G}amma(x)$ with the one of $\mathcal{G}amma^{-1}(x)$.
The second part of the statement follows now easily.
\end{proof}
\text{\varrhom span}ection{Cyclically monotone sets}\bigl\langlebel{S:cyclically}
Following Theorem \textrm{Re}\,f{T:equivalence} and Lemma \textrm{Re}\,f{L:singlegeo}, the next step is to prove that both $A_{+}$ and $A_{-}$ have $\mathfrak m$-measure zero,
that is branching happens on rays with zero $\mathfrak m$-measure.
Already from the statement of this property, it is clear that some regularity assumption on $(X,\text{\varrhom span}fd,\mathfrak m)$ should play a role. We will indeed assume the space to
enojoy a stronger form of essentially non-branching.
Recall that the latter is formulated in terms of geodesics of $(\mathcal{P}_{2}(X),W_{2})$ hence of $\text{\varrhom span}fd^{2}$-cyclically monotone set, while we need
regularity for the $\text{\varrhom span}fd$-cyclically monotone set $\mathcal{G}amma$.
Hence it is necessary to include $\text{\varrhom span}fd^{2}$-cyclically monotone sets as subset of $\text{\varrhom span}fd$-cyclically monotone sets.
We present here a strategy introduced by the author in \cite{cava:decomposition, cava:MongeRCD} from where all the material presented in this section is taken.
Section \textrm{Re}\,f{Ss:structure} contains results from \cite{biacava:streconv} while Section \textrm{Re}\,f{Ss:balanced} is taken from \cite{CM1}.
\begin{equation}gin{lemma}[Lemma 4.6 of \cite{cava:decomposition}]\bigl\langlebel{L:12monotone}
Let $\mathcal{D}elta \text{\varrhom span}ubset \mathcal{G}amma$ be any set so that:
$$
(x_{0},y_{0}), (x_{1},y_{1}) \in \mathcal{D}elta \quad \mathbb{R}ightarrow \quad (\varphi(y_{1}) - \varphi(y_{0}) )\cdot (\varphi(x_{1}) - \varphi(x_{0}) ) \geq 0.
$$
Then $\mathcal{D}elta$ is $\text{\varrhom span}fd^{2}$-cyclically monotone.
\end{lemma}
\begin{equation}gin{proof}
It follows directly from the hypothesis of the lemma that the set
$$
\mathcal{L}ambda: = \{ (\varphi(x), \varphi(y) ) : (x,y) \in \mathcal{D}elta \} \text{\varrhom span}ubset \mathbb{R}^{2},
$$
is monotone in the Euclidean sense. Since $\mathcal{L}ambda \text{\varrhom span}ubset \mathbb{R}^{2}$, it is then a standard fact that $\mathcal{L}ambda$ is
also $|\cdot|^{2}$-cyclically monotone, where $|\cdot|$ denotes the modulus.
We anyway include a short proof:
there exists a maximal monotone multivalued function $F$ such that $\mathcal{L}ambda \text{\varrhom span}ubset \textrm{graph} (F)$ and its domain is an interval, say $(a,b)$ with $a$ and $b$ possibly infinite;
moreover, apart from countably many $x \in \mathbb{R}$, the set $F(x)$ is a singleton.
Then the following function is well defined:
$$
\Psi(x) : = \int_{c}^{x} F(s) ds,
$$
where $c$ is any fixed element of $(a,b)$. Then observe that
$$
\Psi(z) - \Psi(x) \geq y(z-x), \mathfrak quad \varphiorall \ z,x \in (a,b),
$$
where $y$ is any element of $F(x)$. In particular this implies that $\Psi$ is convex and $F(x)$ is a subset of its sub-differential.
In particular $\mathcal{L}ambda$ is $|\cdot |^{2}$-cyclically monotone. \\
Then for $\{(x_{i},y_{i})\}_{ i \leq N} \text{\varrhom span}ubset \mathcal{D}elta$, since $\mathcal{D}elta \text{\varrhom span}ubset \mathcal{G}amma$,
it holds
\begin{equation}gin{align*}
\text{\varrhom span}um_{i=1}^{N} \text{\varrhom span}fd^{2}(x_{i},y_{i}) = &~ \text{\varrhom span}um_{i =1}^{N}|\varphi(x_{i}) - \varphi(y_{i})|^{2} \crcr
\leq&~ \text{\varrhom span}um_{i =1}^{N}|\varphi(x_{i}) - \varphi(y_{i+1})|^{2} \crcr
\leq &~ \text{\varrhom span}um_{i=1}^{N} \text{\varrhom span}fd^{2}(x_{i},y_{i+1}),
\end{align*}
where the last inequality is given by the 1-Lipschitz regularity of $\varphi$. The claim follows.
\end{proof}
To study the set of branching points is necessary to relate point of branching to geodesics.
In the next Lemma, using Lemma \textrm{Re}\,f{L:cicli}, we observe that once a branching happens
there exist two distinct geodesics, both contained in $\mathcal{G}amma(x)$,
that are not in relation in the sense of $R$.
\begin{equation}gin{lemma}\bigl\langlebel{L:geoingamma}
Let $x \in A_{+}$.
Then there exist two distinct geodesics $\gamma^{1},\gamma^{2} \in G$
such that
\begin{equation}gin{itemize}
\item[-] $(x,\gamma_{s}^{1}), (x,\gamma_{s}^{2}) \in \mathcal{G}amma$ for all $s \in [0,1]$;
\item[-] $(\gamma_{s}^{1},\gamma^{2}_{s}) \notin R$ for all $s \in [0,1]$;
\item[-] $\varphi(\gamma^{1}_{s}) = \varphi(\gamma^{2}_{s})$ for all $s \in [0,1]$.
\end{itemize}
Moreover both geodesics are non-constant.
\end{lemma}
\begin{equation}gin{proof}
From the definition of $A_{+}$ there exists $z,w \in \mathcal{T}_{e}$ such that $z,w \in \mathcal{G}amma(x)$ and $(z,w) \notin R$.
Since $z,w \in \mathcal{G}amma(x)$, from Lemma \textrm{Re}\,f{L:cicli} there exist two geodesics $\gamma^{1},\gamma^{2} \in G$ such that
$$
\gamma^{1}_{0} = \gamma^{2}_{0} = x, \quad \gamma^{1}_{1} = z, \quad \gamma^{2}_{1} = w.
$$
Since $(z,w) \notin R$, necessarily both $z$ and $w$ are different from $x$ and $x$ is not a final point, that is $x \notin {\mathfrak b}$.
So the previous geodesics are not constant.
Since $z$ and $w$ can be exchanged, we can also assume that $\varphi(z) \geq \varphi(w)$.
Since $z \in \mathcal{G}amma(x)$, $\varphi(x) \geq \varphi(z)$ and by continuity there exists
$s_{2} \in (0,1]$ such that
$$
\varphi(z) = \varphi(\gamma^{2}_{s_{2}}).
$$
Note that $z \neq \gamma^{2}_{s_{2}}$, otherwise $w \in \mathcal{G}amma(z)$ and therefore $(z,w) \in R$.
Moreover still $(z,\gamma^{2}_{s_{2}}) \notin R$. Indeed if the contrary was true, then
$$
0= |\varphi(z) - \varphi(\gamma^{2}_{s_{2}}) | = \text{\varrhom span}fd(z,\gamma^{2}_{s_{2}}),
$$
that is a contradiction with $z \neq \gamma^{2}_{s_{2}}$.
So by continuity there exists $\partiallta > 0$ such that
$$
\varphi (\gamma^{1}_{1-s} ) = \varphi (\gamma^{2}_{s_{2}(1-s)} ), \mathfrak quad \text{\varrhom span}fd(\gamma^{1}_{1-s}, \gamma^{2}_{s_{2}-s}) > 0,
$$
for all $0 \leq s \leq \partiallta$.
Hence reapplying the previous argument $(\gamma^{1}_{1-s}, \gamma^{2}_{s_{2}(1-s)}) \notin R$.
The curve $\gamma^{1}$ and $\gamma^{2}$ of the claim are then obtained
properly restricting and rescaling the geodesic $\gamma^{1}$ and $\gamma^{2}$ considered so far.
\end{proof}
The previous correspondence between branching points and couples of branching geodesics can be proved to be measurable.
We will make use of the following selection result, Theorem 5.5.2 of \cite{Sri:courseborel}. We again refer to \cite{Sri:courseborel} for some preliminaries on analytic sets.
\begin{equation}gin{theorem}
\bigl\langlebel{T:vanneuma}
Let $X$ and $Y$ be Polish spaces, $F \text{\varrhom span}ubset X \times Y$ analytic, and $\mathcal{A}$ be the $\text{\varrhom span}igma$-algebra generated by the analytic subsets of X.
Then there is an $\mathcal{A}$-measurable section $u : P_1(F) \to Y$ of $F$.
\end{theorem}
Recall that given $F \text{\varrhom span}ubset X \times Y$, a \emph{section $u$ of $F$} is
a function from $P_1(F)$ to $Y$ such that $\textrm{graph}(u) \text{\varrhom span}ubset F$.
\begin{equation}gin{lemma}\bigl\langlebel{L:selectiongeo}
There exists an $\A$-measurable map $u : A_{+} \mapsto G \times G$ such
that if $u(x) = (\gamma^{1},\gamma^{2})$ then
\begin{equation}gin{itemize}
\item[-] $(x,\gamma_{s}^{1}), (x,\gamma_{s}^{2}) \in \mathcal{G}amma$ for all $s \in [0,1]$;
\item[-] $(\gamma_{s}^{1},\gamma^{2}_{s}) \notin R$ for all $s \in [0,1]$;
\item[-] $\varphi(\gamma^{1}_{s}) = \varphi(\gamma^{2}_{s})$ for all $s \in [0,1]$.
\end{itemize}
Moreover both geodesics are non-constant.
\end{lemma}
\begin{equation}gin{proof}
Since $G = \{ \gamma \in \mathcal{G}eo(X) : (\gamma_{0},\gamma_{1}) \in \mathcal{G}amma \}$
and $\mathcal{G}amma \text{\varrhom span}ubset X \times X$ is closed, the set $G$ is a complete and separable metric space.
Consider now the set
\begin{equation}gin{align*}
F:= &~ \{ (x,\gamma^{1},\gamma^{2}) \in \mathcal{T}_{e}\times G \times G : (x,\gamma^{1}_{0}), (x,\gamma^{2}_{0}) \in \mathcal{G}amma \} \crcr
&~ \cap\left( X\times \{ (\gamma^{1},\gamma^{2}) \in G\times G : \text{\varrhom span}fd(\gamma^{1}_{1},\gamma^{2}_{1})>0 \} \varrhoight) \crcr
&~ \cap\left( X\times \{ (\gamma^{1},\gamma^{2}) \in G\times G : \text{\varrhom span}fd(\gamma^{1}_{0},\gamma^{2}_{0})>0 \} \varrhoight) \crcr
&~ \cap\left( X\times \{ (\gamma^{1},\gamma^{2}) \in G\times G : \text{\varrhom span}fd(\gamma^{1}_{0},\gamma^{1}_{1})>0 \} \varrhoight) \crcr
&~ \cap\left( X\times \{ (\gamma^{1},\gamma^{2}) \in G\times G : \varphi(\gamma^{1}_{i}) = \varphi(\gamma^{2}_{i}), \, i =0,1 \} \varrhoight).
\end{align*}
It follows from Remark \textrm{Re}\,f{R:regularity} that $F$ is $\text{\varrhom span}igma$-compact. To avoid possible intersections
in interior points of $\gamma^{1}$ with $\gamma^{2}$ we consider the following map:
\begin{equation}gin{align*}
h : G \times G &~ \to ~ [0,\infty) \crcr
(\gamma^{1},\gamma^{2}) & ~ \mapsto ~ h(\gamma^{1},\gamma^{2}) : = \min_{s\in [0,1]} \, \text{\varrhom span}fd ( \gamma^{1}_{s},\gamma^{2}_{s}).
\end{align*}
From compactness of $[0,1]$, we deduce the continuity of $h$. Therefore
$$
\hat F : = F \cap \{ (x,\gamma^{1},\gamma^{2} )\in X \times G\times G : h(\gamma^{1},\gamma^{2}) > 0\}
$$
is a Borel set and from Lemma \textrm{Re}\,f{L:geoingamma},
$$
\hat F \cap \left( \{x\} \times G\times G \varrhoight) \neq \emptyset
$$
for all $x \in A_{+}$. By Theorem \textrm{Re}\,f{T:vanneuma} we infer the existence of an $\mathcal{A}$-measurable selection $u$ of $\hat F$.
Since $A_{+} = P_{1}(\hat F)$ and if $u(x) = (\gamma^{1},\gamma^{2})$, then
$$
\text{\varrhom span}fd( \gamma^{1}_{s},\gamma^{2}_{s}) > 0, \mathfrak quad \varphi(\gamma^{1}_{s}) = \varphi(\gamma^{2}_{s}),
$$
for all $s \in [0,1]$, and therefore $(\gamma^{1}_{s},\gamma^{2}_{s}) \notin R$ for all $s \in [0,1]$.
The claim follows.
\end{proof}
We are ready to prove the following
\begin{equation}gin{proposition}\bigl\langlebel{P:nobranch}
Let $(X,\text{\varrhom span}fd,\mathfrak m)$ be a m.m.s. such that for any $\mu_{0},\mu_{1} \in \mathcal{P}(X)$ with $\mu_{0} \ll \mathfrak m$ any optimal transference plan for $W_{2}$ is concentrated on the
graph of a function. Then
$$
\mathfrak m(A_{+}) = \mathfrak m(A_{-}) = 0.
$$
\end{proposition}
\begin{equation}gin{proof}
{\bf Step 1.} \\
Suppose by contradiction that $\mathfrak m(A_{+})>0$.
By definition of $A_{+}$, thanks to Lemma \textrm{Re}\,f{L:geoingamma} and Lemma \textrm{Re}\,f{L:selectiongeo},
for every $x \in A_{+}$ there exist two non-constant geodesics $\gamma^{1},\gamma^{2} \in G$
such that
\begin{equation}gin{itemize}
\item[-] $(x,\gamma_{s}^{1}), (x,\gamma_{s}^{2}) \in \mathcal{G}amma$ for all $s \in [0,1]$;
\item[-] $(\gamma_{s}^{1},\gamma^{2}_{s}) \notin R$ for all $s \in [0,1]$;
\item[-] $\varphi(\gamma^{1}_{s}) = \varphi(\gamma^{2}_{s})$ for all $s \in [0,1]$.
\end{itemize}
Moreover the map $A_{+} \ni x \mapsto u(x) : = (\gamma^{1},\gamma^{2}) \in G^{2}$ is $\A$-measurable.
By inner regularity of compact sets (or by Lusin's Theorem), possibly selecting a subset of $A_{+}$ still with strictly positive $\mathfrak m$-measure,
we can assume that the previous map is continuous and in particular the functions
$$
A_{+} \ni x \mapsto \varphi(\gamma^{i}_{j}) \in \mathbb{R}, \mathfrak quad i =1,2, \ j = 0,1
$$
are all continuous. Put
$$
\alpha_{x} : = \varphi(\gamma^{1}_{0}) = \varphi(\gamma^{2}_{0}), \mathfrak quad \begin{equation}ta_{x} : = \varphi(\gamma^{1}_{1}) = \varphi(\gamma^{2}_{1})
$$
and note that $\alpha_{x} > \begin{equation}ta_{x}$.
Now we want to show the existence of a subset $B \text{\varrhom span}ubset A_{+}$, still with $\mathfrak m(B) > 0$, such that
$$
\text{\varrhom span}up_{x \in B} \begin{equation}ta_{x} < \inf_{x\in B} \alpha_{x}.
$$
By continuity of $\alpha$ and $\begin{equation}ta$, a set $B$ verifying the previous inequality
can be obtained considering the set $A_{+} \cap B_{r}(x)$, for $x \in A_{+}$ with $r$ sufficiently small.
Since $\mathfrak m(A_{+})>0$, for $\mathfrak m$-a.e. $x \in A_{+}$ the set $A_{+}\cap B_{r}(x)$ has positive $\mathfrak m$-measure.
So the existence of $B \text{\varrhom span}ubset A_{+}$ enjoying the aforementioned properties follows.
{\bf Step 2.} \\
Let $I = [c,d]$ be a non trivial interval such that
$$
\text{\varrhom span}up_{x \in B} \begin{equation}ta_{x} < c < d <\inf_{x\in B} \alpha_{x}.
$$
Then by construction for all $x \in B$ the image of the composition of the geodesics $\gamma^{1}$ and $\gamma^{2}$ with $\varphi$
contains the interval $I$:
$$
I \text{\varrhom span}ubset \{ \varphi(\gamma^{i}_{s}) : s \in [0,1] \}, \mathfrak quad i = 1,2.
$$
Then fix any point inside $I$, say $c$ and consider for any $x \in B$ the value $s(x)$ such that $\varphi(\gamma^{1}_{s(x)}) = \varphi(\gamma^{2}_{s(x)}) = c$.
We can now define on $B$ two transport maps $T^{1}$ and $T^{2}$ by
$$
B \ni x \mapsto T^{i}(x) : = \gamma^{i}_{s(x)}, \mathfrak quad i =1,2.
$$
Accordingly we define the transport plan
$$
\eta : = \varphirac{1}{2} \left( (Id, T^{1})_{\text{\varrhom span}harp} \mathfrak m_{B} + (Id, T^{2})_{\text{\varrhom span}harp} \mathfrak m_{B} \varrhoight),
$$
where $\mathfrak m_{B} : = \mathfrak m(B)^{-1} \mathfrak m \llcorner_{B}$.
{\bf Step 3.} \\
The support of $\eta$ is $\text{\varrhom span}fd^{2}$-cyclically monotone. To prove it we will use Lemma \textrm{Re}\,f{L:12monotone}.
The measure $\eta$ is concentrated on the set
\[
\mathcal{D}elta : = \{ (x,\gamma^{1}_{s(x)}) : x \in B \} \cup \{ (x,\gamma^{2}_{s(x)}) : x \in B \} \text{\varrhom span}ubset \mathcal{G}amma.
\]
Take any two couples $(x_{0},y_{0}), (x_{1},y_{1}) \in \mathcal{D}elta$ and notice that by definition:
\[
\varphi(y_{1}) - \varphi(y_{0}) = 0,
\]
and therefore trivially $\left( \varphi(y_{1}) - \varphi(y_{0}) \varrhoight) \left( \varphi(x_{1}) - \varphi(x_{0}) \varrhoight) = 0$,
and Lemma \textrm{Re}\,f{L:12monotone} can be applied to $\mathcal{D}elta$.
Hence $\eta$ is optimal with $(P_{1})_{\text{\varrhom span}harp}\eta \ll \mathfrak m$ and is not induced by a map; this is a contradiction with
the assumption. It follows that $\mathfrak m(A_{+})=0$. The claim for $A_{-}$ follows in the same manner.
\end{proof}
\begin{equation}gin{remark}\bigl\langlebel{R:nobranch}
If the space is itself non-branching, then Proposition \textrm{Re}\,f{P:nobranch} can be proved more directly under the assumption \textrm{Re}\,f{A:1}, that will be introduced
at the beginning of Section \textrm{Re}\,f{S:ConditionalMeasures}. Recall that $(X,\text{\varrhom span}fd,\mathfrak m)$ is non-branching if for any $\gamma^{1},\gamma^{2} \in \mathcal{G}eo$ such that
$$
\gamma^{1}_{0} = \gamma^{2}_{0}, \mathfrak quad \gamma^{1}_{t} = \gamma^{2}_{t},
$$
for some $t \in (0,1)$, implies that $\gamma^{1}_{1} = \gamma^{2}_{1}$. In particular the following statement holds
\noindent
\emph{ Let $(X,\text{\varrhom span}fd,\mathfrak m)$ be non-branching and assume moreover \textrm{Re}\,f{A:1} to hold. Then
$$
\mathfrak m(A_{+}) = \mathfrak m(A_{-}) = 0.
$$
}
For the proof of this statement (that goes beyond the scope of this note) we refer to \cite{biacava:streconv}, Lemma 5.3.
The same comment will also apply to the next Theorem \textrm{Re}\,f{T:RCD}.
\end{remark}
To summarize what proved so far introduce also the following notation: the set
\begin{equation}gin{equation}\bigl\langlebel{E:transportset}
\mathcal{T} : = \mathcal{T}_{e} \text{\varrhom span}etminus (A_{+} \cup A_{-})
\end{equation}
will be called the \emph{transport set}. Since $\mathcal{T}_{e}, A_{+}$ and $A_{-}$ are $\text{\varrhom span}igma$-compact sets,
notice that $\mathcal{T}$ is countable intersection of $\text{\varrhom span}igma$-compact sets and in particular Borel.
\begin{equation}gin{theorem}[Theorem 5.5, \cite{cava:MongeRCD}]\bigl\langlebel{T:RCD}
Let $(X,\text{\varrhom span}fd,\mathfrak m)$ be such that for any $\mu_{0},\mu_{1} \in \mathcal{P}(X)$ with $\mu_{0} \ll \mathfrak m$ any optimal transference plan for $W_{2}$ is concentrated on the
graph of a function. Then the set of transport rays $R\text{\varrhom span}ubset X \times X$ is an equivalence relation on the transport set $\mathcal{T}$
and
$$
\mathfrak m(\mathcal{T}_{e} \text{\varrhom span}etminus \mathcal{T} ) = 0.
$$
\end{theorem}
To recap, we have shown that given a $\text{\varrhom span}fd$-monotone set
$\mathcal{G}amma$, the set of all those points moved by $\mathcal{G}amma$, denoted with $\mathcal{T}_{e}$,
can be written, neglecting a set of $\mathfrak m$-measure zero, as the union of a family of disjoint geodesics.
The next step is to decompose the reference measure $\mathfrak m$ restricted to $\mathcal{T}$
with respect to the partition given by $R$, where each equivalence class is given by
$$
[x] = \{ y \in \mathcal{T}: (x,y) \in R \}.
$$
Denoting the set of equivalence classes with $Q$, we can apply Disintegration Theorem (see Theorem \textrm{Re}\,f{T:disintegrationgeneral})
to the measure space $(\mathcal{T}, \mathcal{B}(\mathcal{T}), \mathfrak m)$ and obtain
the disintegration of $\mathfrak m$ consistent with the partition of $\mathcal{T}$ in rays:
$$
\mathfrak m\llcorner_{\mathcal{T}} = \int_{Q} \mathfrak m_{q} \, \mathfrak q(dq),
$$
where $\mathfrak q$ is the quotient measure.
\text{\varrhom span}ubsection{Structure of the quotient set}\bigl\langlebel{Ss:structure}
In order to use the strength of Disintegration Theorem to localize the measure, one needs to obtain a \emph{strongly consistent} disintegration.
Following the last part of Theorem \textrm{Re}\,f{T:disintegrationgeneral}, it is necessary to build a section $S$ of $\mathcal{T}$ together with a measurable quotient map with image $S$.
\begin{equation}gin{proposition}[$Q$ is locally contained in level sets of $\varphi$]\bigl\langlebel{P:Qlevelset}
It is possible to construct a Borel quotient map $\mathbb{Q}Q: \mathcal{T} \to Q$ such that the quotient set $Q \text{\varrhom span}ubset X$
can be written locally as a level set of $\varphi$ in the following sense:
$$
Q = \bigcup_{i\in \mathbb{N}} Q_{i}, \mathfrak quad Q_{i} \text{\varrhom span}ubset \varphi^{-1}(\alpha_{i}),
$$
where $\alpha_i \in \mathbb{Q}$, $Q_{i}$ is analytic and $Q_{i} \cap Q_{j} = \emptyset$, for $i\neq j$.
\end{proposition}
\begin{equation}gin{proof}
{\bf Step 1.}\\
For each $n \in \mathbb{N}$, consider the set $\mathcal{T}_{n}$ of those points $x$ having ray $R(x)$ longer than $1/n$, i.e.
$$
\mathcal{T}_{n} : = P_{1} \{ (x,z,w) \in \mathcal{T}_{e} \times \mathcal{T}_{e} \times \mathcal{T}_{e} \colon z,w \in R(x), \, \text{\varrhom span}fd(z,w) \geq 1/n \} \cap \mathcal{T}.
$$
It is easily seen that $\mathcal{T}=\bigcup_{n \in \mathbb{N}} \mathcal{T}_n$ and that $\mathcal{T}_{n}$ is Borel: the set $\mathcal{T}_{e}$ is $\text{\varrhom span}igma$-compact and therefore its projection is again $\text{\varrhom span}igma$-compact.
Moreover if $x \in \mathcal{T}_{n}, y \in \mathcal{T}$ and $(x,y) \in R$ then also $y \in \mathcal{T}_{n}$: for $x \in \mathcal{T}_{n}$ there exists $z,w \in \mathcal{T}_{e}$ with $z,w\in R(x)$ and $\text{\varrhom span}fd(z,w)\geq 1/n$.
Since $x\in \mathcal{T}$ necessarily $z,w \in \mathcal{T}$. Since $R$ is an equivalence relation on $\mathcal{T}$ and $y \in \mathcal{T}$, it follows that $z,w \in R(y)$. Hence $y \in \mathcal{T}_{n}$.
In particular, $\mathcal{T}_{n}$ is the union of all those maximal rays of $\mathcal{T}$ with length at least $1/n$.
Using the same notation, we have $\mathcal{T} = \cup_{n\in \mathbb{N}} \mathcal{T}_{n}$ with $\mathcal{T}_{n}$ Borel, saturated with respect to $R$, each ray of $\mathcal{T}_{n}$ is longer than $1/n$
and $\mathcal{T}_{n} \cap \mathcal{T}_{n'} = \emptyset$ as soon as $n \neq n'$.
Now we consider the following saturated subsets of $\mathcal{T}_{n}$: for $\alpha \in \mathbb{Q}$
\begin{equation}gin{equation}\bigl\langlebel{eq:defTnalpha}
\mathcal{T}_{n,\alpha}:= P_{1} \Big( R \cap \Big \{ (x,y) \in \mathcal{T}_{n} \times \mathcal{T}_{n} \colon \varphi(y) = \alpha - \varphirac{1}{3n}\Big \} \Big)
\cap P_{1} \Big( R \cap \Big \{ (x,y) \in \mathcal{T}_{n} \times \mathcal{T}_{n} \colon \varphi(y) = \alpha+ \varphirac{1}{3n} \Big\} \Big),
\end{equation}
and we claim that
\begin{equation}gin{equation} \bigl\langlebel{eq:Tnalpha}
\mathcal{T}_{n} = \bigcup_{\alpha \in \mathbb{Q}} \mathcal{T}_{n,\alpha} .
\end{equation}
We show the above identity by double inclusion. First note that $(\text{\varrhom span}upset)$ holds trivially. For the converse inclusion $(\text{\varrhom span}ubset)$ observe that
for each $\alpha \in \mathbb{Q}$, the set $ \mathcal{T}_{n,\alpha}$ coincides with the family of those rays $R(x) \cap \mathcal{T}_{n}$ such that there exists $y^{+},y^{-} \in R(x)$ such that
\begin{equation}gin{equation}\bigl\langlebel{eq:ypm}
\varphi(y^{+}) = \alpha - \varphirac{1}{3n}, \mathfrak quad \varphi(y^{-}) = \alpha + \varphirac{1}{3n}.
\end{equation}
Then we need to show that any $x \in \mathcal{T}_{n}$, also verifies $x \in \mathcal{T}_{n,\alpha}$ for a suitable $\alpha \in \mathbb{Q}$. So fix $x \in \mathcal{T}_{n}$ and since $R(x)$ is longer than $1/n$, there exist $z,y^{+},y^{-} \in R(x) \cap \mathcal{T}_{n}$ such that
$$
\varphi(y^{-}) -\varphi(z) = \varphirac{1}{2n}, \mathfrak quad \varphi(z) -\varphi(y^{+})= \varphirac{1}{2n}.
$$
Consider now the geodesic $\gamma \in G$ such that $\gamma_{0} = y^{-}$ and $\gamma_{1} = y^{+}$. By continuity of
$[0,1] \ni t \mapsto \varphi(\gamma_{t})$ it follows the existence of $0 < s_{1}< s_{2} < s_{3} <1$ such that
$$
\varphi(\gamma_{s_{3}}) = \varphi(\gamma_{s_{2}})- \varphirac{1}{3n}, \mathfrak quad \varphi(\gamma_{s_{1}}) = \varphi(\gamma_{s_{2}}) + \varphirac{1}{3n}, \mathfrak quad \varphi \in \mathbb{Q}.
$$
This concludes the proof of the identity \eqref{eq:Tnalpha}.
{\bf Step 2.}\\
By the above construction, one can check that for each $\alpha \in \mathbb{Q}$, the level set $\varphi^{-1}(\alpha)$ is a quotient set for $\mathcal{T}_{n,\alpha}$, i.e.
$\mathcal{T}_{n,\alpha}$ is formed by disjoint geodesics each one intersecting $\varphi^{-1}(\alpha)$ in exactly one point. Equivalently, $\varphi^{-1}(\alpha)$ is a section for the partition of $\mathcal{T}_{n}$
induced by $R$.
Moreover $\mathcal{T}_{n,\alpha}$ is obtained as the projection of a Borel set and it is therefore analytic. \\
Since $\mathcal{T}_{n,\alpha}$ is saturated with respect to $R$ either $\mathcal{T}_{n,\alpha} \cap \mathcal{T}_{n,\alpha'} = \emptyset$ or $\mathcal{T}_{n,\alpha} = \mathcal{T}_{n,\alpha'}$.
Hence, removing the unnecessary $\alpha$, we can assume that
$\mathcal{T} = \bigcup_{n \in \mathbb{N}, \alpha\in \mathbb{Q}} \mathcal{T}_{n,\alpha}$, is a partition.
Then we characterize $\mathbb{Q}Q : \mathcal{T} \to \mathcal{T}$ defining its graph as follows:
$$
\textrm{graph}(\mathbb{Q}Q) := \bigcup_{n \in \mathbb{N}, \alpha\in \mathbb{Q}} \mathcal{T}_{n,\alpha} \times \left( \varphi^{-1}(\alpha) \cap\mathcal{T}_{n,\alpha} \varrhoight).
$$
Notice that $\textrm{graph}(\mathbb{Q}Q)$ is analytic and therefore $\mathbb{Q}Q: \mathcal{T} \to Q$ is Borel (see Theorem 4.5.2 of \cite{Sri:courseborel}). The claim follows.
\end{proof}
\begin{equation}gin{corollary}
The following strongly consistent disintegration formula holds true:
\begin{equation}gin{equation}\bigl\langlebel{E:disint}
\mathfrak m \llcorner_{\mathcal{T}} =
\int_{Q} \mathfrak m_{q} \, \mathfrak q(dq), \mathfrak quad \mathfrak m_{q}(\mathbb{Q}Q^{-1}(q)) = 1, \ \mathfrak q\text{-a.e.}\ q \in Q.
\end{equation}
\end{corollary}
\begin{equation}gin{proof}
From Proposition \textrm{Re}\,f{P:Qlevelset} there exists an analytic quotient set $Q$ with Borel quotient map $\mathbb{Q}Q : \mathcal{T} \to Q$. In particular $Q$ is a section and the push-forward $\text{\varrhom span}igma$-algebra
of $\mathcal{B}(\mathcal{T})$ on $Q$ contains $\mathcal{B}(Q)$.
From Theorem \textrm{Re}\,f{T:disintr} \eqref{E:disint} follows.
\end{proof}
\begin{equation}gin{remark}\bigl\langlebel{R:regulardisint}
One can improve the regularity of the disintegration formula \eqref{E:disint} as follows.
From inner regularity of Borel measures there exists $S \text{\varrhom span}ubset Q$ $\text{\varrhom span}igma$-compact, such that
$\mathfrak q(Q \text{\varrhom span}etminus S) = 0$. The subset $R^{-1}(S) \text{\varrhom span}ubset \mathcal{T}$ is again $\text{\varrhom span}igma$-compact, indeed
\begin{equation}gin{align*}
R^{-1}(S)
= &~ \{ x\in \mathcal{T} \colon (x,q) \in R, \, q \in S \} = P_{1} (\{ (x,q) \in \mathcal{T} \times S \colon (x,q) \in R \} ) \crcr
= &~ P_{1} (\mathcal{T} \times S \cap R) = P_{1} (\mathcal{T}_{e} \times S \cap R).
\end{align*}
and the regularity follows. Notice that $R^{-1}(S)$ is formed by non-branching rays and $\mathfrak m(\mathcal{T} \text{\varrhom span}etminus R^{-1})(S)) = \mathfrak q(Q \text{\varrhom span}etminus S) = 0$.
Hence we have proved that the transport set with end points $\mathcal{T}_{e}$ admits a saturated, partitioned by disjoint rays, $\text{\varrhom span}igma$-compact subset of full measure
with $\text{\varrhom span}igma$-compact quotient set.
Since in what follows we will not use the definition \eqref{E:transportset}, we will denote this set with $\mathcal{T}$ and its quotient set with $Q$.
\end{remark}
For ease of notation $X_{q} : = \mathbb{Q}Q^{-1}(q)$. The next goal will be to deduce regularity properties for the conditional measures $\mathfrak m_{q}$.
The next function will be of some help during the note.
\begin{equation}gin{definition}[Definition 4.5, \cite{biacava:streconv}][Ray map]
\bigl\langlebel{D:mongemap}
Define the \emph{ray map}
$$
g: \textrm{Dom}(g) \text{\varrhom span}ubset Q \times \mathbb{R} \to \mathcal{T}
$$
via the formula:
\begin{equation}gin{align*}
\textrm{graph} (g) : = &~ \Big\{ (q,t,x) \in Q \times [0,+\infty) \times \mathcal{T}: (q,x) \in \mathcal{G}amma, \, \text{\varrhom span}fd(q,x) = t \Big\} \crcr
&~ \cup \Big\{ (q,t,x) \in Q \times (-\infty,0] \times \mathcal{T} : (x,q) \in \mathcal{G}amma, \, \text{\varrhom span}fd(x,q) = t \Big\} \crcr
= &~ \textrm{graph}(g^+) \cup \textrm{graph}(g^-).
\end{align*}
\end{definition}
Hence the ray map associates to each $q \in Q$ and $t\in \textrm{Dom\,}(g(q, \cdot))\text{\varrhom span}ubset \mathbb{R}$ the
unique element $x \in \mathcal{T}$ such that $(q,x) \in \mathcal{G}amma$ at distance $t$ from $q$
if $t$ is positive or the unique element $x \in \mathcal{T}$ such that $(x,q) \in \mathcal{G}amma$ at distance $-t$ from $q$
if $t$ is negative. By definition $\textrm{Dom}(g) : = g^{-1}(\mathcal{T})$.
Notice that from Remark \textrm{Re}\,f{R:regulardisint} it is not restrictive to assume $\textrm{graph} (g)$ to be $\text{\varrhom span}igma$-compact.
In particular the map $g$ is Borel.
Next we list few (trivial) regularity properties enjoyed by $g$.
\begin{equation}gin{proposition} \bigl\langlebel{P:gammaclass}
The following holds.
\begin{equation}gin{itemize}
\item[-] $g$ is a Borel map.
\item[-] $t \mapsto g(q,t)$ is an isometry and if $s,t \in \textrm{Dom\,}(g(q,\cdot))$ with $s \leq t$ then $( g(q,s), g(q,t) ) \in \mathcal{G}amma$;
\item[-] $\textrm{Dom}(g) \ni (q,t) \mapsto g(q,t)$ is bijective on $\mathbb{Q}Q^{-1}(Q) = \mathcal{T}$, and its inverse is
$$
x \mapsto g^{-1}(x) = \big( \mathbb{Q}Q(x),\pm \text{\varrhom span}fd(x,\mathbb{Q}Q(x)) \big)
$$
where $\mathbb{Q}Q$ is the quotient map previously introduced and the positive or negative sign depends on
$(x,\mathbb{Q}Q(x)) \in \mathcal{G}amma$ or $(\mathbb{Q}Q(x),x) \in \mathcal{G}amma$.
\end{itemize}
\end{proposition}
Observe that from Lemma \textrm{Re}\,f{L:cicli}, $\textrm{Dom\,} (g(q,\cdot))$ is a convex subset of $\mathbb{R}$ (i.e. an interval),
for any $q \in Q$. Using the ray map $g$, we will review in Section \textrm{Re}\,f{S:ConditionalMeasures} how
to prove that $\mathfrak q$-a.e. conditional measure $\mathfrak m_{q}$ is absolutely continuous with respect to the $1$-dimensional Hausdorff measure on $X_{q}$,
provided $(X,\text{\varrhom span}fd,\mathfrak m)$ enjoys weak curvature properties. The other main use of the ray map $g$ was presented in Section 7 of \cite{biacava:streconv} where it was used
to build a 1-dimensional metric currents in the sense of Ambrosio-Kirchheim (see \cite{AK}) associated to $\mathcal{T}$.
It is worth also noticing that so far, besides the assumption of Proposition \textrm{Re}\,f{P:nobranch}, no extra assumption on the geometry of the space was used.
In particular, given two probability measures $\mu_{0}$ and $\mu_{1}$ with finite first moment, the associated transport set permits
to decompose the reference measure $\mathfrak m$ in one-dimensional conditional measures $\mathfrak m_{q}$, i.e. formula \eqref{E:disint} holds.
\text{\varrhom span}ubsection{Balanced transportation}\bigl\langlebel{Ss:balanced}
Here we want underline that the disintegration (or one-dimensional localization) of $\mathfrak m$ induced
by the $L^{1}$-Optimal Transportation problem between $\mu_{0}$ and $\mu_{1}$ is actually a localization of the Monge problem.
We will present this fact considering a function $f : X \to \mathbb{R}$ such that
$$
\int_{X} f(x)\,\mathfrak m(dx) = 0, \mathfrak quad \int_{X}|f(x)|\text{\varrhom span}fd(x,x_{0}) \, \mathfrak m(dx) < \infty,
$$
and considering $\mu_{0} : = f_{+}\,\mathfrak m$ and $\mu_{1} : = f_{-}\,\mathfrak m$, where $f_{\pm}$ denotes the positive and the negative part of $f$.
We can also assume $\mu_{0}, \mu_{1} \in \mathcal{P}(X)$ and study the Monge minimization problem between $\mu_{0}$ and $\mu_{1}$.
This setting is equivalent to study the general Monge problem assuming both $\mu_{0},\mu_{1} \ll \mathfrak m$;
note indeed that $\mu_{0}$ and $\mu_{1}$ can always be assumed to be concentrated on disjoint sets (see \cite{biacava:streconv} for details).
If $\varphi$ is an associated Kantorovich potential producing as before the transport set $\mathcal{T}$, we have a disintegration of $\mathfrak m$ as follows:
$$
\mathfrak m\llcorner_{\mathcal{T}} = \int_{Q} \mathfrak m_{q} \, \mathfrak q(dq), \mathfrak quad \mathfrak m_{q}(X_{q}) =1,\ \mathfrak q\textrm{-a.e.} \, q \in Q.
$$
Then the natural localization of the Monge problem would be to consider for every $q \in Q$ the Monge minimization problem between
$$
\mu_{0\, q} : = f_{+} \,\mathfrak m_{q},\quad \mu_{1\, q} : = f_{-} \,\mathfrak m_{q},
$$
in the metric space $(X_{q},\text{\varrhom span}fd)$ (that is isometric via the ray map $g$ to an interval of $\mathbb{R}$ with the Euclidean distance).
To check that this family of problems makes sense we need to prove the following
\begin{equation}gin{lemma}\bigl\langlebel{L:balancing}
It holds that for $\mathfrak q$-a.e. $q \in Q$ one has $\int_{X} f \, \mathfrak m_{q} = 0$.
\end{lemma}
\begin{equation}gin{proof}
Since for both $\mu_{0}$ and $\mu_{1}$ the set $\mathcal{T}_{e} \text{\varrhom span}etminus \mathcal{T}$ is negligible ($\mu_{0},\mu_{1} \ll \mathfrak m$),
for any Borel set $C \text{\varrhom span}ubset Q$
\begin{equation}gin{eqnarray}\bigl\langlebel{E:mu0=mu1}
\mu_{0}(\mathbb{Q}Q^{-1}(C)) &= & \pi \Big( (\mathbb{Q}Q^{-1}(C) \times X) \cap \mathcal{G}amma \text{\varrhom span}etminus \{ x = y\} \Big) \nonumber \\
&= & \pi \Big( ( X \times \mathbb{Q}Q^{-1}(C)) \cap \mathcal{G}amma \text{\varrhom span}etminus \{ x = y\} \Big) \nonumber \\
&= & \mu_{1}(\mathbb{Q}Q^{-1}(C)), \bigl\langlebel{eq:mu0=mu1}
\end{eqnarray}
where the second equality follows from the fact that $\mathcal{T}$ does not branch: indeed since $\mu_{0}(\mathcal{T}) = \mu_{1}(\mathcal{T}) = 1$,
then $\pi \big( (\mathcal{G}amma \text{\varrhom span}etminus \{ x= y\}) \cap \mathcal{T} \times \mathcal{T} \big) =1$ and therefore if $x,y \in \mathcal{T}$ and $(x,y)\in \mathcal{G}amma$,
then necessarily $\mathbb{Q}Q(x) = \mathbb{Q}Q(y)$, that is they belong to the same ray. It follows that
$$
(\mathbb{Q}Q^{-1}(C) \times X) \cap (\mathcal{G}amma \text{\varrhom span}etminus \{ x = y\}) \cap (\mathcal{T} \times \mathcal{T}) =
( X\times \mathbb{Q}Q^{-1}(C) ) \cap (\mathcal{G}amma \text{\varrhom span}etminus \{ x = y\}) \cap (\mathcal{T} \times \mathcal{T}),
$$
and \eqref{E:mu0=mu1} follows.
Since $f$ has null mean value it holds $\int_X f_{+}(x) \mathfrak m(dx)= - \int_X f_{-}(x) \mathfrak m(dx)$,
which combined with \eqref{E:mu0=mu1} implies that for each Borel $C \text{\varrhom span}ubset Q$
\begin{equation}gin{align*}
\int_{C} \int_{X_{q}} f(x) \mathfrak m_{q}(dx) \mathfrak q(dq) = &~ \int_{C} \int_{X_{q}} f_{+}(x) \mathfrak m_{q}(dx) \mathfrak q(dq) - \int_{C} \int_{X_{q}} f_{-}(x) \mathfrak m_{q}(dx) \mathfrak q(dq) \crcr
= &~ \left( \int_{X} f_{+}(x) \mathfrak m(dx) \varrhoight)^{-1} \left( \mu_{0}(\mathbb{Q}Q^{-1}(C)) - \mu_{1}(\mathbb{Q}Q^{-1}(C)) \varrhoight) \crcr
= &~ 0.
\end{align*}
Therefore for $\mathfrak q$-a.e. $q \in Q$ the integral $\int f \, \mathfrak m_{q}$ vanishes and the claim follows.
\end{proof}
It can be proven in greater generality and without assuming $\mu_{1}\ll \mathfrak m$ that the Monge problem is localized once
a strongly consistent disintegration of $\mathfrak m$ restricted to the transport ray is obtained. See \cite{biacava:streconv} for details.
\text{\varrhom span}ection{Regularity of conditional measures} \bigl\langlebel{S:ConditionalMeasures}
We now review regularity and curvature properties of $\mathfrak m_{q}$. What contained in this section is a collection of results spread across
\cite{biacava:streconv, cava:decomposition, cava:MongeRCD} and \cite{CM1}. We try here to give a unified presentation.
We will inspect three increasing level of regularity: for $\mathfrak q$-a.e. $q \in Q$
\begin{equation}gin{enumerate}[label=(\textbf{R.\arabic*})]
\item \bigl\langlebel{R:1} $\mathfrak m_{q}$ has no atomic part, i.e. $\mathfrak m_{q}(\{x\}) = 0$, for any $x \in X_{q}$;
\item \bigl\langlebel{R:2} $\mathfrak m_{q}$ is absolutely continuous with respect to $\mathcal{H}^{1}\llcorner_{X_{q}} = g(q,\cdot)_{\text{\varrhom span}harp} \mathcal{L}^{1}$;
\item \bigl\langlebel{R:3} $\mathfrak m_{q} = g(q,\cdot)_{\text{\varrhom span}harp} (h_{q}\,\mathcal{L}^{1})$ verifies $\mathbb{C}D(K,N)$, i.e. the m.m.s. $(\mathbb{R}, |\cdot|, h_{q} \, \mathcal{L}^{1})$ verifies $\mathbb{C}D(K,N)$.
\end{enumerate}
We will review how to obtain \textrm{Re}\,f{R:1}, \textrm{Re}\,f{R:2}, \textrm{Re}\,f{R:3} starting from the following three \emph{increasing} regularity assumptions on the space:
\begin{equation}gin{enumerate}[label=(\textbf{A.\arabic*})]
\item \bigl\langlebel{A:1} if $C \text{\varrhom span}ubset \mathcal{T}$ is compact with $\mathfrak m(C)> 0$, then $\mathfrak m(C_{t}) > 0$ for uncountably many $t \in \mathbb{R}$;
\item \bigl\langlebel{A:2} if $C \text{\varrhom span}ubset \mathcal{T}$ is compact with $\mathfrak m(C)> 0$, then $\mathfrak m(C_{t}) > 0$ for a set of $t \in \mathbb{R}$ with $\mathcal{L}^{1}$-positive measure;
\item \bigl\langlebel{A:3} the m.m.s. $(X,\text{\varrhom span}fd,\mathfrak m)$ verifies $\mathbb{C}D(K,N)$.
\end{enumerate}
Given a compact set $C \text{\varrhom span}ubset X$, we indicate with $C_{t}$ its translation along the transport set at distance with sign $t$, see the following
Definition \textrm{Re}\,f{D:evolution}.
We will see that: \textrm{Re}\,f{A:1} implies \textrm{Re}\,f{R:1}, \textrm{Re}\,f{A:2} implies \textrm{Re}\,f{R:2} and \textrm{Re}\,f{A:3} implies \textrm{Re}\,f{R:3}.
Actually we will also show a variant of $\textrm{Re}\,f{A:3}$ (assuming $\mathsf{MCP}$ instead of $\mathbb{C}D$) implies a variant of \textrm{Re}\,f{R:3} ($\mathsf{MCP}$ instead of $\mathbb{C}D$).
Even if we do not to state it each single time, assumptions \textrm{Re}\,f{A:1} and \textrm{Re}\,f{A:2} are not hypothesis on the smoothness of the space but
on the regularity of the set $\mathcal{G}amma$ and therefore on the Monge problem itself; they should both be read as:
\emph{for $\mu_{0}$ and $\mu_{1}$ probability measures over $X$, assume the existence of a 1-Lipschitz Kantorovich potential $\varphi$ such that
the associated transport set $\mathcal{T}$ verifies} \textrm{Re}\,f{A:1} \emph{(or }\textrm{Re}\,f{A:2}\emph{)}.
\text{\varrhom span}ubsection{Atomless conditional probabilities}
The results presented here are taken from \cite{biacava:streconv}.
\begin{equation}gin{definition}\bigl\langlebel{D:evolution}
Let $C \text{\varrhom span}ubset \mathcal{T}$ be a compact set. For $t \in \mathbb{R}$ define the \emph{$t$-translation $C_{t}$ of $C$} by
$$
C_t := g \big( \{ (q,s +t) \colon (q,s) \in g^{-1}(C) \} \big).
$$
\end{definition}
\noindent
Since $C \text{\varrhom span}ubset \mathcal{T}$ is compact, $g^{-1}(C)\text{\varrhom span}ubset Q \times \mathbb{R}$ is $\text{\varrhom span}igma$-compact ($\textrm{graph} (g)$ is $\text{\varrhom span}igma$-compact) and the same holds true for
$$
\{ (q,s +t) \colon (q,s) \in g^{-1}(C) \}.
$$
Since
$$
C_{t} = P_{3}( \textrm{graph}(g) \cap \{ (q,s +t) \colon (q,s) \in g^{-1}(C) \} \times \mathcal{T} ),
$$
it follows that $C_{t}$ $\text{\varrhom span}igma$-compact (projection of $\text{\varrhom span}igma$-compact sets is again $\text{\varrhom span}igma$-compact). \\
Moreover the set $B : = \{ (t,x) \in \mathbb{R} \times \mathcal{T} \colon x \in C_{t} \}$ is Borel and therefore by Fubini's Theorem
the map $t \mapsto \mathfrak m(C_t)$ is Borel. It follows that \textrm{Re}\,f{A:1} makes sense.
\begin{equation}gin{proposition}[Proposition 5.4, \cite{biacava:streconv}]\bigl\langlebel{P:nonatoms}
Assume \emph{\textrm{Re}\,f{A:1}} to hold and the space to be non-branching.
Then \emph{\textrm{Re}\,f{R:1}} holds true, that is for $\mathfrak q$-a.e. $q \in Q$ the conditional measure $\mathfrak m_{q}$ has no atoms.
\end{proposition}
\begin{equation}gin{proof}
The partition in trasport rays and the associated disintegration are well defined, see Remark \textrm{Re}\,f{R:nobranch}.
From the regularity of the disintegration and the fact that $\mathfrak q(Q) = 1$,
we can assume that the map $q \mapsto \mathfrak m_q$ is weakly continuous on a compact set $K \text{\varrhom span}ubset Q$ with $\mathfrak q(Q \text{\varrhom span}etminus K) < \textbf{X}repsilon$ such
that the length of the ray $X_{q}$, denoted by $L(X_{q})$, is strictly larger than $\textbf{X}repsilon$ for all $q \in K$. It is enough to prove the proposition on $K$.
{\bf Step 1.}\\
From the continuity of $K \ni q \mapsto \mathfrak m_q \in \mathcal{P}(X)$ w.r.t. the weak topology, it follows that the map
$$
q \mapsto C(q) := \big\{ x \in X_{q}: \mathfrak m_q(\{x\}) > 0 \big\} = \cup_n \big\{ x \in X_{q}: \mathfrak m_q(\{x\}) \geq 2^{-n} \big\}
$$
is $\text{\varrhom span}igma$-closed, i.e. its graph is countable union of closed sets:
in fact, if $(q_m,x_m) \to (y,x)$ and $\mathfrak m_{q_m}(\{x_m\}) \geq 2^{-n}$, then $\mathfrak m_q(\{x\}) \geq 2^{-n}$
by upper semi-continuity on compact sets.
Hence it is Borel, and by Lusin Theorem (Theorem 5.8.11 of \cite{Sri:courseborel}) it is the countable union of Borel graphs:
setting in case $c_i(q) = 0$, we can consider them as Borel functions on $K$ and order them w.r.t. $\mathcal{G}amma$ in the following sense:
$$
\mathfrak m_{q,\textrm{atomic}} = \text{\varrhom span}um_{i \in \mathbb{Z}} c_i(q) \partiallta_{x_i(q)}, \quad (x_i(q), x_{i+1}(q)) \in \mathcal{G}amma, \ i \in \mathbb{Z},
$$
with $K \ni q \mapsto x_{i}(q)$ Borel.
{\bf Step 2.} \\
Define the sets
$$
S_{ij}(t) := \Big\{ q \in K: x_i(q) = g \big( g^{-1}(x_j(q)) + t \big) \Big\},
$$
Since $K \text{\varrhom span}ubset Q$, to define $S_{ij}(t)$ we are using the $\textrm{graph} (g) \cap Q \times \mathbb{R} \times \mathcal{T}$, which is $\text{\varrhom span}igma$-compact:
hence $\textrm{graph}(S_{ij})$ is analytic.
For $A_j := \{x_j(q), q \in K\}$ and $t \in \mathbb{R}^+$ we have that
\begin{equation}gin{align*}
\mathfrak m((A_j)_t) =&~ \int_K \mathfrak m_q((A_j)_t)\, \mathfrak q(dq) = \int_K \mathfrak m_{q,\textrm{atomic}}((A_j)_t) \, \mathfrak q(dq) \crcr
=&~ \text{\varrhom span}um_{i \in \mathbb{Z}} \int_K c_i(q) \partiallta_{x_i(q)} \big( g(g^{-1}(x_j(q)) + t) \big)\, \mathfrak q(dq) = \text{\varrhom span}um_{i \in \mathbb{Z}} \int_{S_{ij}(t)} c_i(q) \, \mathfrak q(dq),
\end{align*}
and we have used that $A_j \cap X_{q}$ is a singleton. Then for fixed $i,j \in \mathbb{N}$, again from the fact that $A_j \cap X_{q}$ is a singleton
$$
S_{ij}(t) \cap S_{ij}(t') =
\begin{equation}gin{cases}
S_{ij}(t) & t = t', \crcr
\emptyset & t \not= t',
\end{cases}
$$
and therefore the cardinality of the set $\big\{ t : \mathfrak q(S_{ij}(t)) > 0 \big\}$ has to be countable.
On the other hand,
$$
\mathfrak m((A_j)_t) > 0 \quad \mathcal{L}ongrightarrow \quad t \in \bigcup_i \big\{ t : \mathfrak q(S_{ij}(t)) > 0 \big\},
$$
contradicting \textrm{Re}\,f{A:1}.
\end{proof}
\text{\varrhom span}ubsection{Absolute continuity}
\bigl\langlebel{Ss:regolarita'}
The results presented here are taken from \cite{biacava:streconv}.
The condition \textrm{Re}\,f{A:2} can be stated also in the following way: for every compact set $C \text{\varrhom span}ubset \mathcal{T}$
$$
\mathfrak m(C) > 0 \quad \mathcal{L}ongrightarrow \quad \int_{\mathbb{R}} \mathfrak m(C_t) dt > 0.
$$
\begin{equation}gin{lemma}
\bigl\langlebel{Lem:dec}
Let $\mathfrak m$ be a Radon measure and
$$
\mathfrak m_q = r_{q}\, g(q,\cdot)_\text{\varrhom span}harp \mathcal{L}^1 + \omega_q, \quad \omega_q \perp g(q,\cdot)_\text{\varrhom span}harp \mathcal{L}^1
$$
be the Radon-Nikodym decomposition of $\mathfrak m_q$ w.r.t. $g(q,\cdot)_\text{\varrhom span}harp \mathcal{L}^1$. Then there exists a Borel set $C\text{\varrhom span}ubset X$ such that
$$
\mathcal{L}^{1} \Big(P_{2}\big( g^{-1} (C) \cap (\{q\} \times \mathbb{R}) ) \big) \Big)= 0,
$$
and $\omega_q = \mathfrak m_q \llcorner_C$ for $\mathfrak q$-a.e. $q \in Q$.
\end{lemma}
\begin{equation}gin{proof}
Consider the measure $\bigl\langlembda = g_\text{\varrhom span}harp (\mathfrak q \otimes \mathcal L^1)$,
and compute the Radon-Nikodym decomposition
\[
\mathfrak m = \varphirac{D \mathfrak m}{D \bigl\langlembda} \bigl\langlembda + \omega.
\]
Then there exists a Borel set $C$ such that $\omega = \mathfrak m \llcorner_C$ and $\bigl\langlembda(C)=0$. The set $C$ proves the Lemma. Indeed
$C = \cup_{q \in Q} C_{q}$ where $C_{q} = C \cap R(q)$ is such that
$\mathfrak m_q \llcorner_{C_{q}} = \omega_{q} $ and $g(q,\cdot)_{\text{\varrhom span}harp}\mathcal{L}^{1}(C_{q})=0$ for $\mathfrak q$-a.e. $q \in Q$.
\end{proof}
\begin{equation}gin{theorem}[Theorem 5.7, \cite{biacava:streconv}]\bigl\langlebel{teo:a.c.}
Assume \emph{\textrm{Re}\,f{A:2}} to hold and the space to be non-branching. Then \emph{\textrm{Re}\,f{R:2}} holds true, that is for $\mathfrak q$-a.e. $q \in Q$ the conditional measure $\mathfrak m_{q}$
is absolute continuous with respect to $g(q,\cdot)_{\text{\varrhom span}harp}\mathcal{L}^{1}$.
\end{theorem}
The proof is based on the following simple observation.
\noindent Let $\eta$ be a Radon measure on $\mathbb{R}$. Suppose that for all $A \text{\varrhom span}ubset \mathbb{R}$ Borel with $\eta(A)>0$ it holds
\[
\int_{\mathbb{R}^+} \eta(A+t) dt = \eta \otimes\mathcal{L}^1 \big( \{ (x,t): t \geq 0, x - t \in A \} \big) > 0.
\]
Then $\eta \ll \mathcal{L}^1$.
\begin{equation}gin{proof}
The proof will use Lemma \textrm{Re}\,f{Lem:dec}: take $C$ the set constructed in Lemma \textrm{Re}\,f{Lem:dec} and suppose by contradiction that
$$
\mathfrak m(C) > 0 \quad \text{and} \quad \mathfrak q \otimes \mathcal{L}^1 (g^{-1}(C)) = 0.
$$
In particular, for all $t \in \mathbb{R}$ it follows that
$$
\mathfrak q \otimes \mathcal{L}^1 (g^{-1}(C_t)) = 0.
$$
By Fubini-Tonelli Theorem
\begin{equation}gin{align*}
0< &~ \int_{\mathbb{R}^+} \mathfrak m(C_t) \,dt = \int_{\mathbb{R}^+} \bigg( \int_{g^{-1}(C_t)} (g^{-1})_\text{\varrhom span}harp \mathfrak m(dq\,d\tau) \bigg) dt \crcr
= &~ \big( (g^{-1})_\text{\varrhom span}harp \mathfrak m \otimes \mathcal{L}^1 \big) \Big( \Big\{ (q,\tau,t): (q,\tau) \in g^{-1}(\mathcal{T}), (q,\tau-t) \in g^{-1}(C) \Big\} \Big) \crcr
\leq &~ \int_{Q \times \mathbb{R}} \mathcal{L}^1 \big( \big\{\tau - g^{-1}(C \cap \mathbb{Q}Q^{-1}(q)) \big\} \big) \, (g^{-1})_{\text{\varrhom span}harp} \mathfrak m (dq\, d\tau) \crcr
= &~ \int_{Q \times \mathbb{R}} \mathcal{L}^1 \big( g^{-1}(C \cap \mathbb{Q}Q^{-1}(q)) \big) \,(g^{-1})_{\text{\varrhom span}harp} \mathfrak m (dq\,d\tau) \crcr
= &~ \int_{Q} \mathcal{L}^1 \big( g^{-1}(C \cap \mathbb{Q}Q^{-1}(y)) \big)\, \mathfrak q(dy) = 0.
\end{align*}
That gives a contradiction.
\end{proof}
The proof of Theorem \textrm{Re}\,f{teo:a.c.} inspired the definition of \emph{inversion points} and of \emph{inversion plan} as presented in \cite{CM0}, in particular see
{\bf Step 2.} of the proof of Theorem 5.3 of \cite{CM0}.
\text{\varrhom span}ubsection{Weak Ricci curvature bounds: $\mathsf{MCP}(K,N)$}
The presentation of the following results is taken from \cite{cava:MongeRCD}. The same results were already proved in \cite{biacava:streconv}
using more involved arguments and different notation.
In this section we additionally assume the metric measure space to satisfy the measure contraction property $\mathsf{MCP}(K,N)$. Recall that the space
is also assumed to be non-branching.
\begin{equation}gin{lemma}\bigl\langlebel{L:evo1}
For each Borel $C \text{\varrhom span}ubset \mathcal{T}$ and $\partiallta \in \mathbb{R}$ the set
$$
\left( C \times \{ \varphi= \partiallta \} \varrhoight) \cap \mathcal{G}amma,
$$
is $\text{\varrhom span}fd^{2}$-cyclically monotone.
\end{lemma}
\begin{equation}gin{proof} The proof follows straightforwardly from Lemma \textrm{Re}\,f{L:12monotone}. The set
$\left( C \times \{ \varphi = c \} \varrhoight) \cap \mathcal{G}amma$ is trivially a subset of $\mathcal{G}amma$ and whenever
$$
(x_{0},y_{0}), (x_{1},y_{1}) \in \left( C \times \{ \varphi = \partiallta \} \varrhoight) \cap \mathcal{G}amma,
$$
then $(\varphi (y_{1}) - \varphi(y_{0}) ) \cdot (\varphi(x_{1}) - \varphi(x_{0}) ) = 0$.
\end{proof}
We can deduce the following
\begin{equation}gin{corollary}\bigl\langlebel{C:evo1}
For each Borel $C \text{\varrhom span}ubset \mathcal{T}$ and $\partiallta \in \mathbb{R}$ define
$$
C_{\partiallta}: =P_{1}(\left( C \times \{ \varphi = \partiallta \} \varrhoight) \cap \mathcal{G}amma).
$$
If $\mathfrak m(C_{\partiallta}) > 0$, there exists a unique $\nu \in \mathrm{OptGeo}$ such that
\begin{equation}gin{equation}\bigl\langlebel{E:12mappa}
\left( e_{0} \varrhoight)_{\text{\varrhom span}harp} \nu = \mathfrak m( C_{\partiallta} )^{-1} \mathfrak m\llcorner_{C_{\partiallta}},
\mathfrak quad (e_{0},e_{1})_{\text{\varrhom span}harp}( \nu ) \Big( \left( C \times \{ \varphi = \partiallta \} \varrhoight) \cap \mathcal{G}amma \Big) = 1.
\end{equation}
\end{corollary}
\noindent
From Corollary \textrm{Re}\,f{C:evo1}, we infer the existence of a map $T_{C,\partiallta}$ depending on $C$ and $\partiallta$ such that
$$
\left( Id,T_{C,\partiallta} \varrhoight)_{\text{\varrhom span}harp} \left( \mathfrak m( C_{\partiallta} )^{-1} \mathfrak m\llcorner_{C_{\partiallta}} \varrhoight) = (e_{0},e_{1})_{\text{\varrhom span}harp} \nu.
$$
Taking advantage of the ray map $g$, we define a convex combination between the identity map and
$T_{C,\partiallta}$ as follows:
$$
C_{\partiallta} \ni x \mapsto \left(T_{C,\partiallta}\varrhoight)_{t}(x) \in \{ z \in \mathcal{G}amma(x) : \text{\varrhom span}fd(x,z) = t \cdot \text{\varrhom span}fd(x,T_{C,\partiallta}(x)) \}.
$$
Since $C \text{\varrhom span}ubset \mathcal{T}$, the map $\left(T_{C,\partiallta}\varrhoight)_{t}$ is well defined for all $t\in [0,1]$.
We then define the evolution of any subset $A$ of $C_{\partiallta}$ in the following way:
$$
[0,1] \ni t \mapsto \left(T_{C,\partiallta}\varrhoight)_{t}(A).
$$
In particular from now on we will adopt the following notation:
$$
A_{t} : = \left(T_{C,\partiallta} \varrhoight)_{t}(A), \mathfrak quad \varphiorall A \text{\varrhom span}ubset C_{\partiallta}, \ A \ \textrm{ compact}.
$$
So for any Borel $C \text{\varrhom span}ubset \mathcal{T}$ compact and $\partiallta \in \mathbb{R}$ we have defined an evolution
for compact subsets of $C_{\partiallta}$. The definition of the evolution depends both on $C$ and $\partiallta$.
\begin{equation}gin{remark}\bigl\langlebel{R:regularity2}
Here we spend a few lines on the measurability of the maps involved in the definition of evolution of sets assuming for simplicity $C$ to be compact.
First note that since $\mathcal{G}amma$ is closed and $C$ is compact, we can prove that also $C_{\partiallta}$ is compact.
Indeed from compactness of $C$ we obtain that $\varphi$ is bounded on $C$ and then, since $C$ is bounded, it follows that also
$C \times \{\varphi = c \} \cap \mathcal{G}amma$ is bounded. Since $X$ is proper, compactness follows.
Moreover
$$
\textrm{graph} (T_{C,\partiallta}) = \left( C \times \{ \varphi= \partiallta \} \varrhoight) \cap \mathcal{G}amma,
$$
hence $T_{C,\partiallta}$ is continuous. Moreover
$$
\left(T_{C,\partiallta}\varrhoight)_{t}(A) = P_{2} \left( \{(x,z) \in \mathcal{G}amma \cap (A \times X) : \text{\varrhom span}fd(x,z) = t\cdot \text{\varrhom span}fd(x,T_{C,\partiallta}(x)) \}\varrhoight),
$$
hence if $A$ is compact, the same holds for $\left(T_{C,\partiallta}\varrhoight)_{t}(A)$ and
$$
[0,1] \ni t \mapsto \mathfrak m(\left(T_{C,\partiallta}\varrhoight)_{t}(A))
$$
is $\mathfrak m$-measurable.
\end{remark}
The next result gives quantitative information on the behavior of the map $t \mapsto \mathfrak m(A_{t})$.
The statement will be given assuming the lower bound on the generalized Ricci curvature $K$ to be positive.
Analogous estimates holds for any $K \in \mathbb{R}$.
\begin{equation}gin{proposition}\bigl\langlebel{P:mcp}
For each compact $C \text{\varrhom span}ubset \mathcal{T}$ and $\partiallta \in \mathbb{R}$ such that $\mathfrak m(C_{\partiallta}) >0$, it holds
\begin{equation}gin{equation}\bigl\langlebel{E:mcp}
\mathfrak m( A_{t} ) \geq (1-t) \cdot \inf_{x\in A} \left(\varphirac{\text{\varrhom span}in \left( (1-t) \text{\varrhom span}fd(x,T_{C,\partiallta}(x) )\text{\varrhom span}qrt{K/(N-1)} \varrhoight) }
{\text{\varrhom span}in \left( \text{\varrhom span}fd(x,T_{C,\partiallta}(x))\text{\varrhom span}qrt{K/(N-1)} \varrhoight)} \varrhoight)^{N-1} \mathfrak m(A),
\end{equation}
for all $t \in [0,1]$ and $A \text{\varrhom span}ubset C_{\partiallta}$ compact set.
\end{proposition}
\begin{equation}gin{proof}
The proof of \eqref{E:mcp} is obtained by the standard method of approximation with Dirac deltas of the second marginal.
Even though similar arguments already appeared many times in literature, in order to be self-contained, we include all the details.
For ease of notation $T = T_{C,\partiallta}$ and $C = C_{\partiallta}$.
{\bf Step 1.}\\
Consider a sequence $\{ y_{i} \}_{i \in \mathbb{N}} \text{\varrhom span}ubset \{\varphi = \partiallta\}$ dense in $T(C)$.
For each $I \in \mathbb{N}$, define the family of sets
$$
E_{i,I} : = \{ x \in C : \text{\varrhom span}fd(x,y_{i}) \leq \text{\varrhom span}fd(x,y_{j}) , j =1,\dots, I \},
$$
for $i =1, \dots, I$. Then for all $I \in \mathbb{N}$, by the same argument of Lemma \textrm{Re}\,f{L:evo1}, the set
$$
\mathcal{L}ambda_{I}: = \bigcup_{i =1}^{I} E_{i,I}\times \{ y_{i} \} \text{\varrhom span}ubset X \times X,
$$
is $\text{\varrhom span}fd^{2}$-cyclically monotone. Consider then $A_{i,I} : = A \cap E_{i,I}$ and the approximate evolution
$$
A_{i,I,t} : = \{ z \in X \colon \text{\varrhom span}fd(z,y_{i}) = (1-t)\text{\varrhom span}fd(x,y_{i}), \ x \in A_{i,I}\};
$$
and notice that $A_{i,I,0} = A_{i,I}$. Then by $\mathsf{MCP}(K,N)$ it holds
$$
\mathfrak m( A_{i,I,t} ) \geq (1-t) \cdot \inf_{x\in A_{i,I}} \left(\varphirac{\text{\varrhom span}in \left( (1-t) \text{\varrhom span}fd(x,x_{i} )\text{\varrhom span}qrt{K/(N-1)} \varrhoight) }
{\text{\varrhom span}in \left( \text{\varrhom span}fd(x,x_{i})\text{\varrhom span}qrt{K/(N-1)} \varrhoight)} \varrhoight)^{N-1} \mathfrak m(A_{i,I}).
$$
Taking the sum over $i \leq I$ in the previous inequality implies
$$
\text{\varrhom span}um_{i \leq I} \mathfrak m( A_{i,I,t} ) \geq (1-t) \cdot \inf_{x\in A} \left(\varphirac{\text{\varrhom span}in \left( (1-t) \text{\varrhom span}fd(x,T_{I}(x) )\text{\varrhom span}qrt{K/(N-1)} \varrhoight) }
{\text{\varrhom span}in \left( \text{\varrhom span}fd(x,T_{I}(x))\text{\varrhom span}qrt{K/(N-1)} \varrhoight)} \varrhoight)^{N-1} \mathfrak m(A),
$$
where $T_{I}(x) : = y_{i}$ for $x \in E_{i,I}$.
From $\text{\varrhom span}fd^{2}$-cyclically monotonicity and the non-branching of the space, up to a set of measure zero, the map $T_{I}$ is well defined,
i.e. $\mathfrak m(E_{i,I} \cap E_{j,I}) = 0$ for $i \neq j$.
It follows that for each $I \in \mathbb{N}$ we can remove a set of measure zero from $A$ and obtain
$$
A_{i,I,t} \cap A_{j,I,t} = \emptyset, \quad i\neq j.
$$
As before consider also the interpolated map $T_{I,t}$ and observe that $A_{I,t} = T_{I,t} (A)$. Since also $A$ is compact we obtain
$$
\mathfrak m( A_{I,t} ) \geq (1-t) \cdot \min_{x\in A} \left(\varphirac{\text{\varrhom span}in \left( (1-t) \text{\varrhom span}fd(x,T_{I}(x) )\text{\varrhom span}qrt{K/(N-1)} \varrhoight) }
{\text{\varrhom span}in \left( \text{\varrhom span}fd(x,T_{I}(x))\text{\varrhom span}qrt{K/(N-1)} \varrhoight)} \varrhoight)^{N-1} \mathfrak m(A).
$$
{\bf Step 2.}\\
Since $C$ is a compact set, for every $I \in \mathbb{N}$
the set $\mathcal{L}ambda_{I}$ is compact as well and it is a subset of $C \times \{ \varphi = \partiallta \}$ that
can be assumed to be compact as well. By compactness, there exists a subsequence $I_{n}$ and
a compact set $\mathcal{T}heta \text{\varrhom span}ubset C \times \{ \varphi = \partiallta \}$ compact such that
$$
\lim_{n \to \infty} \text{\varrhom span}fd_{\mathcal{H}}(\mathcal{L}ambda_{I_{n}}, \mathcal{T}heta) = 0,
$$
where $\text{\varrhom span}fd_{\mathcal{H}}$ is the Hausdorff distance.
Since the sequence $\{y_{i}\}_{i\in \mathbb{N}}$ is dense in $\{\varphi = \partiallta \}$ and $C \text{\varrhom span}ubset \mathcal{T}$ is compact, by definition of $E_{i,I}$,
necessarily for every $(x,y) \in \mathcal{T}heta$ it holds
$$
\varphi(x)+ \varphi(y) = \text{\varrhom span}fd(x,y), \quad \varphi(y) = \partiallta.
$$
Hence $\mathcal{T}heta \text{\varrhom span}ubset \mathcal{G}amma \cap C\times \{\varphi = \partiallta\}$ and this in particular implies, by upper semicontinuity of $\mathfrak m$
along converging sequences of closed sets, that
$$
\mathfrak m(A_{t}) \geq \limsup_{n \to \infty} \mathfrak m(A_{I_{n},t})\, .
$$
The claim follows.
\end{proof}
As the goal is to localize curvature conditions, we first need to prove that almost every conditional probability is absolutely continuous
with respect to the one dimensional Hausdorff measure restricted to the correct geodesic.
One way is to prove that Proposition \textrm{Re}\,f{P:mcp} implies \textrm{Re}\,f{A:2} and then apply Theorem \textrm{Re}\,f{teo:a.c.} to obtain \textrm{Re}\,f{R:2} (approach used in \cite{biacava:streconv}).
Another option is to repeat verbatim the proof of Theorem \textrm{Re}\,f{teo:a.c.} substituting the translation with the evolution considered in
Proposition \textrm{Re}\,f{P:mcp} and to observe that the claim follows (approach used in \cite{cava:MongeRCD}).
So we take for granted the following
\begin{equation}gin{proposition}\bigl\langlebel{P:MCP-1}
Assume the non-branching m.m.s. $(X,\text{\varrhom span}fd,\mathfrak m)$ to satisfy $\mathsf{MCP}(K,N)$.
Then \emph{\textrm{Re}\,f{R:2}} holds true, that is for $\mathfrak q$-a.e. $q \in Q$ the conditional measure $\mathfrak m_{q}$
is absolute continuous with respect to $g(q,\cdot)_{\text{\varrhom span}harp}\mathcal{L}^{1}$.
\end{proposition}
To fix the notation, we now have proved the existence of a Borel function $h : \textrm{Dom\,}(g) \to \mathbb{R}_{+}$ such that
\begin{equation}gin{equation}\bigl\langlebel{E:definitionh}
\mathfrak m\llcorner\mathcal{T} = g_{\text{\varrhom span}harp} \left( h \, \mathfrak q \otimes \mathcal{L}^{1} \varrhoight)
\end{equation}
Using standard arguments, estimate \eqref{E:mcp} can be localized at the level of the density $h$:
for each compact set $A \text{\varrhom span}ubset \mathcal{T}$
\begin{equation}gin{align*}
\int_{P_{2}(g^{-1}(A_{t}) )} & h(q,s) \mathcal{L}^{1}(ds) \crcr
\geq (1-t) & \left( \inf_{\tau \in P_{2}(g^{-1}(A))} \varphirac{\text{\varrhom span}in( (1-t) |\tau - \text{\varrhom span}igma| \text{\varrhom span}qrt{K/(N-1)} ) }{\text{\varrhom span}in( |\tau - \text{\varrhom span}igma| \text{\varrhom span}qrt{K/(N-1)} )} \varrhoight)^{N-1}
\int_{P_{2}(g^{-1}(A))} h(q,s) \mathcal{L}^{1}(ds),
\end{align*}
for $\mathfrak q$-a.e. $q \in Q$ such that $g(q,\text{\varrhom span}igma) \in \mathcal{T}$. Then using change of variable, one obtains that for $\mathfrak q$-a.e. $q \in Q$:
$$
h(q,s+|s-\text{\varrhom span}igma| t ) \geq \left(
\varphirac{\text{\varrhom span}in( (1-t) |s - \text{\varrhom span}igma| \text{\varrhom span}qrt{K/(N-1)} ) }{\text{\varrhom span}in( |s - \text{\varrhom span}igma| \text{\varrhom span}qrt{K/(N-1)} )} \varrhoight)^{N-1}
h(y,s),
$$
for $\mathcal{L}^{1}$-a.e. $s \in P_{2}(g^{-1}(R(q)))$ and $\text{\varrhom span}igma \in \mathbb{R}$ such that $s + |\text{\varrhom span}igma -s| \in P_{2}(g^{-1}(R(q)))$.
We can rewrite the estimate in the following way:
$$
h(q, \tau ) \geq \left(
\varphirac{\text{\varrhom span}in( ( \text{\varrhom span}igma - \tau ) \text{\varrhom span}qrt{K/(N-1)} ) }{\text{\varrhom span}in( ( \text{\varrhom span}igma - s ) \text{\varrhom span}qrt{K/(N-1)} )} \varrhoight)^{N-1} h(q,s),
$$
for $\mathcal{L}^{1}$-a.e. $s \leq \tau \leq \text{\varrhom span}igma$ such that $g(q,s), g(q,\tau), g(q,\text{\varrhom span}igma) \in \mathcal{T}$.
Since evolution can be also considered backwardly, we have proved the next
\begin{equation}gin{theorem}[Localization of $\mathsf{MCP}$, Theorem 9.5 of \cite{biacava:streconv}]\bigl\langlebel{T:densityestimates}
Assume the non-branching m.m.s. $(X,\text{\varrhom span}fd,\mathfrak m)$ to satisfy $\mathsf{MCP}(K,N)$.
For $\mathfrak q$-a.e. $q \in Q$ it holds:
$$
\left( \varphirac{\text{\varrhom span}in( ( \text{\varrhom span}igma_{+} - \tau ) \text{\varrhom span}qrt{K/(N-1)} ) }{\text{\varrhom span}in( ( \text{\varrhom span}igma_{+} - s ) \text{\varrhom span}qrt{K/(N-1)} )} \varrhoight)^{N-1}
\leq \varphirac{h(q, \tau )} {h(q,s)}
\leq \left( \varphirac{\text{\varrhom span}in( ( \tau - \text{\varrhom span}igma_{-} ) \text{\varrhom span}qrt{K/(N-1)} ) }{\text{\varrhom span}in( (s - \text{\varrhom span}igma_{-} ) \text{\varrhom span}qrt{K/(N-1)} )} \varrhoight)^{N-1},
$$
for $\text{\varrhom span}igma_{-} < s \leq \tau < \text{\varrhom span}igma _{+}$ such that their image via $g(q,\cdot)$ is contained in $R(q)$.
\end{theorem}
\noindent
In particular from Theorem \textrm{Re}\,f{T:densityestimates} we deduce that
\begin{equation}gin{equation}\bigl\langlebel{E:regularityh}
\{ t \in \textrm{Dom\,}(g(q,\cdot)) \colon h(q,t) > 0 \} = \textrm{Dom\,}(g(q,\cdot)),
\end{equation}
in particular such set is convex and $t \mapsto h(q,t)$ is locally Lipschitz continuous.
\text{\varrhom span}ubsection{Weak Ricci curvature bounds: $\mathbb{C}D(K,N)$}
The results presented here are taken from \cite{CM1}.
We now turn to proving that the conditional probabilities inherit the synthetic Ricci curvature lower bounds, that is, \textrm{Re}\,f{A:3} implies \textrm{Re}\,f{R:3}.
Actually it is enough to assume the space to verify such a lower bound only locally to obtain
globally the synthetic Ricci curvature lower bound on almost every 1-dimensional metric measure spaces.
Since under the essentially non-branching condition $\mathbb{C}D_{loc}(K,N)$ implies $\mathsf{MCP}(K,N)$ and existence and uniqueness of optimal transport maps,
see \cite{cavasturm:MCP}, we can already assume \eqref{E:definitionh} and
\eqref{E:regularityh} to hold. In particular $t \mapsto h_{q}(t)$ is locally Lipschitz continuous, where, for easy of notation $h_{q} = h(q,\cdot)$.
\begin{equation}gin{theorem}[Theorem 4.2 of \cite{CM1}]\bigl\langlebel{T:CDKN-1}
Let $(X,\text{\varrhom span}fd,\mathfrak m)$ be an essentially non-branching m.m.s. verifying the $\mathbb{C}D_{loc}(K,N)$ condition for some $K\in \mathbb{R}$ and $N\in [1,\infty)$.
Then for any 1-Lipschitz function $\varphi:X\to \mathbb{R}$, the associated transport set $\mathcal{G}amma$ induces a disintegration
of $\mathfrak m$ restricted to the transport set verifying the following inequality: if $N> 1$
for $\mathfrak q$-a.e. $q \in Q$ the following curvature inequality holds:
\begin{equation}gin{equation}\bigl\langlebel{E:curvdensmm}
h_{q}( (1-s) t_{0} + s t_{1} )^{1/(N-1)}
\geq \text{\varrhom span}igma^{(1-s)}_{K,N-1}(t_{1} - t_{0}) h_{q} (t_{0})^{1/(N-1)} + \text{\varrhom span}igma^{(s)}_{K,N-1}(t_{1} - t_{0}) h_{q} (t_{1})^{1/(N-1)},
\end{equation}
for all $s\in [0,1]$ and for all $t_{0}, t_{1} \in \textrm{Dom\,}(g(q,\cdot))$ with $t_{0} < t_{1}$. If $N =1$, for $\mathfrak q$-a.e. $q \in Q$ the density $h_{q}$ is constant.
\end{theorem}
\begin{equation}gin{proof}
We first consider the case $N>1$.
{\bf Step 1.} \\
Thanks to Proposition \textrm{Re}\,f{P:Qlevelset}, without any loss of generality we can assume that the quotient set $Q$ (identified with the set $\{g(q,0) : q \in Q\}$)
is locally a subset of a level set of the map $\varphi$ inducing the transport set, i.e.
there exists a countable partition $\{ Q_{i}\}_{i\in \mathbb{N}}$ with $Q_{i} \text{\varrhom span}ubset Q$ Borel set such that
$$
\{ g(q,0) : q \in Q_{i} \} \text{\varrhom span}ubset \{ x \in X : \varphi(x) = \alpha_{i} \}.
$$
It is clearly sufficient to prove \eqref{E:curvdensmm} on each $Q_{i}$; so fix $\bar i \in \mathbb{N}$ and for ease of notation assume $\alpha_{\bar i} = 0$ and $Q = Q_{\bar i}$.
As $\textrm{Dom\,}(g(q,\cdot))$ is a convex subset of $\mathbb{R}$, we can also restrict to a uniform subinterval
$$
(a_0,a_1) \text{\varrhom span}ubset \textrm{Dom\,}(g(q,\cdot)), \mathfrak quad \varphiorall \ q \ \in Q_{i},
$$
for some $a_0,a_1 \in \mathbb{R}$. Again without any loss of generality we also assume $a_0 < 0 < a_1$.
Consider any $a_{0} <A_{0} < A_{1} < a_{1}$ and $L_{0}, L_{1} >0$ such that $A_{0} + L_{0} < A_{1}$ and $A_{1} + L_{1} < a_{1}$.
Then define the following two probability measures
$$
\mu_{0} : = \int_{Q} g(q,\cdot)_\text{\varrhom span}harp \left( \varphirac{1}{L_{0}} \mathcal{L}^{1}\llcorner_{ [A_{0},A_{0}+L_{0}] } \varrhoight) \, \mathfrak q(dq), \mathfrak quad
\mu_{1} : = \int_{Q} g(q,\cdot)_\text{\varrhom span}harp \left( \varphirac{1}{L_{1}} \mathcal{L}^{1}\llcorner_{ [A_{1},A_{1}+L_{1}] } \varrhoight) \, \mathfrak q(dq).
$$
Since $g(q,\cdot)$ is an isometry one can also represent $\mu_{0}$ and $\mu_{1}$ in the following way:
$$
\mu_{i} : = \int_{Q} \varphirac{1}{L_{i}} \mathcal{H}^{1}\llcorner_{ \left\{g(q,t) \colon t \in [A_{i},A_{i}+L_{i}] \varrhoight\} } \, \mathfrak q(dq)
$$
for $i =0,1$. Both $\mu_{i}$ are absolutely continuous with respect to $\mathfrak m$ and $\mu_{i} = \varrho_{i} \mathfrak m$
with
$$
\varrho_{i} (g(q,t)) = \varphirac{1}{L_{i}} h_{q}(t)^{-1}, \mathfrak quad \varphiorall \, t \in [A_{i},A_{i}+L_{i}].
$$
Moreover from Lemma \textrm{Re}\,f{L:12monotone} it follows that the curve $[0,1] \ni s \mapsto \mu_{s} \in \mathcal{P}(X)$ defined by
$$
\mu_{s} : = \int_{Q} \varphirac{1}{L_{s}} \mathcal{H}^{1}\llcorner_{ \left\{g(q,t) \colon t \in [A_{s},A_{s}+L_{s}] \varrhoight\} } \, \mathfrak q(dq)
$$
where
$$
L_{s} : = (1 - s)L_{0} + sL_{1}, \mathfrak quad A_{s} : = (1-s ) A_{0} + s A_{1}
$$
is the unique $L^{2}$-Wasserstein geodesic connecting $\mu_{0}$ to $\mu_{1}$. Again one has $\mu_{s} = \varrho_{s} \mathfrak m$ and can also write its density in the following way:
$$
\varrho_{s} (g(q,t)) = \varphirac{1}{L_{s}} h_{q}(t)^{-1}, \mathfrak quad \varphiorall \, t \in [A_{s},A_{s}+L_{s}].
$$
{\bf Step 2.}\\
By $\mathbb{C}D_{loc}(K,N)$ and the essentially non-branching property one has: for $\mathfrak q$-a.e. $q \in Q_{i}$
$$
(L_{s})^{\varphirac{1}{N}} h_{q}( (1-s) t_{0} + s t_{1} )^{\varphirac{1}{N}}
\geq \tau_{K,N}^{(1-s)}(t_{1}-t_{0}) (L_{0})^{\varphirac{1}{N}} h_{q}( t_{0} )^{\varphirac{1}{N}}+ \tau_{K,N}^{(s)}(t_{1}-t_{0}) (L_{1})^{\varphirac{1}{N}} h_{q}( t_{1} )^{\varphirac{1}{N}},
$$
for $\mathcal{L}^{1}$-a.e. $t_{0} \in [A_{0},A_{0} + L_{0}]$ and $t_{1}$ obtained as the image of $t_{0}$ through the monotone rearrangement of $[A_{0},A_{0}+L_{0}]$ to
$[A_{1},A_{1}+L_{1}]$ and every $s \in [0,1]$. If $t_{0} = A_{0} + \tau L_{0}$, then $t_{1} = A_{1} + \tau L_{1}$. Also $A_{0}$ and $A_{1} +L_{1}$ should be taken close enough to
verify the local curvature condition.
Then we can consider the previous inequality only for $s = 1/2$ and include the explicit formula for $t_{1}$ and obtain:
\begin{equation}gin{align*}
(L_{0} + L_{1})^{\varphirac{1}{N}} &h_{q}(A_{1/2} + \tau L_{1/2})^{\varphirac{1}{N}} \\
& \geq
\text{\varrhom span}igma^{(1/2)}_{K,N-1}( A_{1} - A_{0} + \tau |L_{1} - L_{0}| )^{\varphirac{N-1}{N}} \left\{ (L_{0})^{\varphirac{1}{N}} h_{q}(A_{0} + \tau L_{0})^{\varphirac{1}{N}}
+ (L_{1})^{\varphirac{1}{N}} h_{q}(A_{1} + \tau L_{1})^{\varphirac{1}{N}} \varrhoight\},
\end{align*}
for $\mathcal{L}^{1}$-a.e. $\tau \in [0,1]$, where we used the notation $A_{1/2}:=\varphirac{A_0+A_1}{2}, L_{1/2}:=\varphirac{L_0+L_1}{2}$.
Now observing that the map $s \mapsto h_{q}(s)$ is continuous, the previous inequality also holds for $\tau =0$:
\begin{equation}gin{equation}\bigl\langlebel{E:beforeoptimize}
(L_{0} + L_{1})^{\varphirac{1}{N}} h_{q}(A_{1/2} )^{\varphirac{1}{N}}
\geq
\text{\varrhom span}igma^{(1/2)}_{K,N-1}( A_{1} - A_{0})^{\varphirac{N-1}{N}}
\left\{ (L_{0})^{\varphirac{1}{N}} h_{q}(A_{0})^{\varphirac{1}{N}} + (L_{1})^{\varphirac{1}{N}} h_{q}(A_{1})^{\varphirac{1}{N}} \varrhoight\},
\end{equation}
for all $A_{0} < A_{1}$ with $A_{0},A_{1}\in (a_0, a_1)$, all sufficiently small $L_{0}, L_{1}$ and $\mathfrak q$-a.e. $q\in Q$,
with exceptional set depending on $A_{0},A_{1},L_{0}$ and $L_{1}$.
Noticing that \eqref{E:beforeoptimize} depends in a continuous way on $A_{0},A_{1},L_{0}$ and $L_{1}$, it follows that there
exists a common exceptional set $N \text{\varrhom span}ubset Q$ such that $\mathfrak q(N) = 0$ and for each $q \in Q\text{\varrhom span}etminus N$ for all
$A_{0},A_{1},L_{0}$ and $L_{1}$ the inequality \eqref{E:beforeoptimize} holds true.
Then one can make the following (optimal) choice
$$
L_{0} : = L \varphirac{h_{q}(A_{0})^{\varphirac{1}{N-1}} }{h_{q}(A_{0})^{\varphirac{1}{N-1}} + h_{q}(A_{1})^{\varphirac{1}{N-1}} }, \mathfrak quad
L_{1} : = L \varphirac{h_{q}(A_{1})^{\varphirac{1}{N-1}} }{h_{q}(A_{0})^{\varphirac{1}{N-1}} + h_{q}(A_{1})^{\varphirac{1}{N-1}} },
$$
for any $L > 0$ sufficiently small, and obtain that
\begin{equation}gin{equation}\bigl\langlebel{E:CDKN-1}
h_{q}(A_{1/2} )^{\varphirac{1}{N-1}}
\geq
\text{\varrhom span}igma^{(1/2)}_{K,N-1}( A_{1} - A_{0})
\left\{ h_{q}(A_{0})^{\varphirac{1}{N-1}} + h_{q}(A_{1})^{\varphirac{1}{N-1}} \varrhoight\}.
\end{equation}
Now one can observe that \eqref{E:CDKN-1} is precisely the inequality requested for $\mathbb{C}D^{*}_{loc}(K,N-1)$ to hold.
As stated in Section \textrm{Re}\,f{Ss:geom}, the reduced curvature-dimension condition verifies the local-to-global property.
In particular, see \cite[Lemma 5.1, Theorem 5.2]{cavasturm:MCP}, if a function verifies \eqref{E:CDKN-1} locally,
then it also satisfies it globally.
Hence $h_{q}$ also verifies the inequality requested for $\mathbb{C}D^{*}(K,N-1)$ to hold, i.e. for $\mathfrak q$-a.e. $q \in Q$, the density $h_{q}$ verifies \eqref{E:curvdensmm}.
\\
{\bf Step 3.}\\
For the case $N =1$, repeat the same construction of {\bf Step 1.} and obtain for $\mathfrak q$-a.e. $q \in Q$
$$
(L_{s}) h_{q}( (1-s) t_{0} + s t_{1} ) \geq (1-s) L_{0} h_{q}( t_{0} )+ s L_{1} h_{q}( t_{1} ),
$$
for any $s \in [0,1]$ and $L_{0}$ and $L_{1}$ sufficiently small. As before, we deduce for $s = 1/2$ that
$$
\varphirac{L_{0} + L_{1}}{2} h_{q}( A_{1/2} ) \geq \varphirac{1}{2} \left(L_{0} h_{q}( A_{0} )+ L_{1}h_{q}( A_{1} ) \varrhoight).
$$
Now taking $L_{0} = 0$ or $L_{1} = 0$, it follows that necessarily $h_{q}$ has to be constant.
\end{proof}
Accordingly to Remark \textrm{Re}\,f{R:CDN-1}, Theorem \textrm{Re}\,f{T:CDKN-1} can be alternatively stated as follows. \\
\noindent
\emph{
If $(X,\text{\varrhom span}fd,\mathfrak m)$ is an essentially non-branching m.m.s. verifying $\mathbb{C}D_{loc}(K,N)$
and $\varphi : X \to \mathbb{R}$ is a 1-Lipschitz function, then the corresponding decomposition of the space
in maximal rays $\{ X_{q}\}_{q\in Q}$ produces a disintegration $\{\mathfrak m_{q} \}_{q\in Q}$ of $\mathfrak m$ so that for $\mathfrak q$-a.e. $q\in Q$,
$$
\textrm{the m.m.s. }( \textrm{Dom\,}(g(q,\cdot)), |\cdot|, h_{q} \mathcal{L}^{1}) \quad \textrm{verifies} \quad \mathbb{C}D(K,N).
$$
}
Accordingly, one says that the disintegration $q \mapsto \mathfrak m_{q}$ is a $\mathbb{C}D(K,N)$ disintegration.
The disintegration obtained with $L^{1}$-Optimal Transportation is also balanced in the sense of Section \textrm{Re}\,f{Ss:balanced}.
This additional information together with what proved so far is collected in the next
\begin{equation}gin{theorem}[Theorem 5.1 of \cite{CM1}]\bigl\langlebel{T:localize}
Let $(X,\text{\varrhom span}fd, \mathfrak m)$ be an essentially non-branching metric measure space verifying the $\mathbb{C}D_{loc}(K,N)$ condition for some $K\in \mathbb{R}$ and $N\in [1,\infty)$.
Let $f : X \to \mathbb{R}$ be $\mathfrak m$-integrable such that $\int_{X} f\, \mathfrak m = 0$ and assume the existence
of $x_{0} \in X$ such that $\int_{X} | f(x) |\, \text{\varrhom span}fd(x,x_{0})\, \mathfrak m(dx)< \infty$.
Then the space $X$ can be written as the disjoint union of two sets $Z$ and $\mathcal{T}$ with $\mathcal{T}$ admitting a partition
$\{ X_{q} \}_{q \in Q}$ and a corresponding disintegration of $\mathfrak m\llcorner_{\mathcal{T}}$, $\{\mathfrak m_{q} \}_{q \in Q}$ such that:
\begin{equation}gin{itemize}
\item For any $\mathfrak m$-measurable set $B \text{\varrhom span}ubset \mathcal{T}$ it holds
$$
\mathfrak m(B) = \int_{Q} \mathfrak m_{q}(B) \, \mathfrak q(dq),
$$
where $\mathfrak q$ is a probability measure over $Q$ defined on the quotient $\text{\varrhom span}igma$-algebra $\mathcal{Q}$.
\item For $\mathfrak q$-almost every $q \in Q$, the set $X_{q}$ is a geodesic and $\mathfrak m_{q}$ is supported on it.
Moreover $q \mapsto \mathfrak m_{q}$ is a $\mathbb{C}D(K,N)$ disintegration.
\item For $\mathfrak q$-almost every $q \in Q$, it holds $\int_{X_{q}} f \, \mathfrak m_{q} = 0$ and $f = 0$ $\mathfrak m$-a.e. in $Z$.
\end{itemize}
\end{theorem}
The proof is just a collection of already proven statements. We include it for readers convenience.
\begin{equation}gin{proof}
Consider
$$
\mu_{0} : = f_{+} \mathfrak m \varphirac{1}{\int f_{+}\mathfrak m}, \mathfrak quad \mu_{1} : = f_{-}\mathfrak m \varphirac{1}{\int f_{-}\mathfrak m},
$$
where $f_{\pm}$ stands for the positive and negative part of $f$, respectively.
From the summability assumption on $f$ it follows the existence of $\varphi : X \to \mathbb{R}$, $1$-Lipschitz Kantorovich potential for the couple of marginal probability $\mu_{0}, \mu_{1}$.
Since the m.m.s. $(X,\text{\varrhom span}fd,\mathfrak m)$ is essentially non-branching, the transport set $\mathcal{T}$ is partitioned by the rays:
$$
\mathfrak m_{\mathcal{T}} = \int_{Q} \mathfrak m_{q}\, \mathfrak q(dq),\mathfrak quad \mathfrak m_{q}(X_{q}) = 1, \quad \mathfrak q-\textrm{a.e. } q \in Q;
$$
moreover $(X,\text{\varrhom span}fd,\mathfrak m)$ verifies $\mathbb{C}D_{loc}$ and therefore Theorem \textrm{Re}\,f{T:CDKN-1} implies that $q \mapsto \mathfrak m_{q}$ is a $\mathbb{C}D(K,N)$ disintegration.
Lemma \textrm{Re}\,f{L:balancing} implies that
$$
\int_{X_{q}} f(x) \, \mathfrak m_{q}(dx) = 0.
$$
To conclude moreover note that in $X \text{\varrhom span}etminus \mathcal{T}$ necessarily $f$ has to be zero.
Take indeed any $B \text{\varrhom span}ubset X\text{\varrhom span}etminus \mathcal{T}$ compact with $\mathfrak m(B) > 0$ and assume $f \neq 0$ over $B$. Then possibly taking a subset, we can assume $f > 0$ over $B$
and therefore $\mu_{0}(B) > 0$. Since
$$
\mu_{0} = \int_{Q} \mu_{0\,q} \mathfrak q(dq), \mathfrak quad \mu_{0\,q} (X_{q}) = 1,
$$
necessarily $B$ cannot be a subset of $X\text{\varrhom span}etminus \mathcal{T}$ yielding a contradiction. All the claims are proved.
\end{proof}
\text{\varrhom span}ection{Applications}\bigl\langlebel{S:application}
Here we will collect some applications of the results proved so far, in particular of Proposition \textrm{Re}\,f{P:nonatoms} and Theorem \textrm{Re}\,f{T:CDKN-1}
\text{\varrhom span}ubsection{Solution of the Monge problem}
Here we review how regularity of conditional probabilities of the one-dimensional disintegration studied so far permits to construct a solution to the Monge
problem. In particular we will see how Proposition \textrm{Re}\,f{P:nonatoms} allows to construct an optimal map $T$.
As the plan is to use the one-dimensional reduction, first we recall the one dimensional result for the Monge problem \cite{Vil}.
\begin{equation}gin{theorem}
\bigl\langlebel{T:oneDmonge}
Let $\mu_{0}, \mu_{1}$ be probability measures on $\mathbb{R}$, $\mu_{0}$ with no atoms, and let
$$
H(s) := \mu_{0}((-\infty,s)), \quad F(t) := \mu_{1}((-\infty,t)),
$$
be the left-continuous distribution functions of $\mu_{0}$ and $\mu_{1}$ respectively. Then the following holds.
\begin{equation}gin{enumerate}
\item The non decreasing function $T : \mathbb{R} \to \mathbb{R} \cup [-\infty,+\infty)$ defined by
$$
T(s) := \text{\varrhom span}up \big\{ t \in \mathbb{R} : F(t) \leq H(s) \big\}
$$
maps $\mu_{0}$ to $\mu_{1}$. Moreover any other non decreasing map $T'$ such that $T'_\text{\varrhom span}harp \mu_{0} = \mu_{1}$ coincides
with $T$ on the support of $\mu_{0}$ up to a countable set.
\item If $\phi : [0,+\infty] \to \mathbb{R}$ is non decreasing and convex, then $T$ is an optimal transport relative to the cost $c(s,t) = \phi(|s-t|)$.
Moreover $T$ is the unique optimal transference map if $\phi$ is strictly convex.
\end{enumerate}
\end{theorem}
\begin{equation}gin{theorem}[Theorem 6.2 of \cite{biacava:streconv}]\bigl\langlebel{T:mongeff}
Let $(X,\text{\varrhom span}fd,\mathfrak m)$ be a non-branching metric measure space and consider $\mu_{0}, \mu_{1} \in \mathcal{P}(X)$ with finite first moment.
Assume the existence of a Kantorovich potential $\varphi$ such that the associated transport set $\mathcal{T}$ verifies \emph{\textrm{Re}\,f{A:1}}.
Assume $\mu_{0} \ll \mathfrak m$.
Then there exists a Borel map $T: X \to X$ such that
$$
\int_{X} \text{\varrhom span}fd(x,T(X)) \, \mu_{0} (dx) = \min_{\pi \in \Pi(\mu_{0},\mu_{1})} \int_{X\times X} \text{\varrhom span}fd(x,y) \, \pi(dxdy).
$$
\end{theorem}
Theorem \textrm{Re}\,f{T:mongeff} was presented in \cite{biacava:streconv} assuming the space to be non-branching, while here we assume essentially non-branching.
\begin{equation}gin{proof}
{\bf Step 1.} One dimensional reduction of $\mu_{0}$. \\
Let $\varphi : X \to \mathbb{R}$ be the Kantorovich potential from the assumptions and $\mathcal{T}$ the corresponding transport set. Accordingly
$$
\mathfrak m\llcorner_{\mathcal{T}} = \int_{Q} \mathfrak m_{q}\,\mathfrak q(dq),
$$
with $\mathfrak m_{q}(X_{q}) = 1$ for $\mathfrak q$-a.e. $q \in Q$. Moreover from \textrm{Re}\,f{A:1} for $\mathfrak q$-a.e. $q \in Q$
the conditional $\mathfrak m_{q}$ has no atoms, i.e. $\mathfrak m_{q}(\{z \}) = 0$ for all $z \in X$.
From Lemma \textrm{Re}\,f{L:mapoutside}, we can assume that $\mu_{0}(\mathcal{T}_{e}) = \mu_{1}(\mathcal{T}_{e}) = 1$. Since $\mu_{0} = \varrho_{0} \mathfrak m$, with $\varrho_{0} : X \to [0,\infty)$,
from Theorem \textrm{Re}\,f{T:equivalence} we have $\mu_{0}(\mathcal{T}) = 1$. Hence
$$
\mu_{0} = \int_{Q} \varrho_{0} \mathfrak m_{q} \, \mathfrak q(dq) = \int_{Q} \mu_{0 \, q} \, \mathfrak q_{0}(dq), \mathfrak quad
\mu_{0\, q} : = \varrho_{0} \mathfrak m_{q} \left(\int_{X} \varrho_{0}(x) \mathfrak m_{q}(dx) \varrhoight)^{-1},
$$
and $\mathfrak q_{0} = \mathbb{Q}Q_{\text{\varrhom span}harp} \mu_{0}$. In particular $\mu_{0,\, q}$ has no atoms and $\mu_{0\,q}(X_{q}) = 1$.
{\bf Step 2.} One dimensional reduction of $\mu_{1}$. \\
As we are not making any assumption on $\mu_{1}$ we cannot exclude that $\mu_{1}(\mathcal{T}_{e} \text{\varrhom span}etminus \mathcal{T}) >0$
and therefore to localize $\mu_{1}$ one cannot proceed as for $\mu_{0}$.
Consider therefore an optimal transport plan $\pi$ with $\pi(\mathcal{G}amma) = 1$. Since $\pi (\mathcal{T} \times \mathcal{T}_{e}) = 1$ and a partition of $\mathcal{T}$ is given,
we can consider the following family of sets $\{ X_{q} \times \mathcal{T}_{e}\}_{q\in Q}$ as a partition of $\mathcal{T} \times \mathcal{T}_{e}$;
note indeed that $X_{q} \times \mathcal{T}_{e} \cap X_{q'} \cap \mathcal{T}_{e} = \emptyset$ as soon as $q \neq q'$.
The domain of the quotient map $\mathbb{Q}Q : \mathcal{T} \to Q$ can be trivially extended to $\mathcal{T} \times \mathcal{T}_{e}$ by saying that $\mathbb{Q}Q(x,z) = \mathbb{Q}Q(x)$
and observe that
$$
\mathbb{Q}Q_{\text{\varrhom span}harp} \,\pi(I) = \pi \left( \mathbb{Q}Q^{-1} (I) \varrhoight) =\pi \left( \mathbb{Q}Q^{-1}(I) \times \mathcal{T}_{e} \varrhoight) =\mu_{0}(\mathbb{Q}Q^{-1}(I)) =\mathfrak q_{0}(I).
$$
In particular this implies that
$$
\pi = \int_{Q} \pi_{q} \, \mathfrak q_{0}(dq), \mathfrak quad \pi_{q} (X_{q} \times \mathcal{T}_{e}) = 1, \quad \textrm{for } \mathfrak q_{0}\textrm{-a.e. } q \in Q.
$$
Then applying the projection
$$
\mu_{0} = P_{1\,\text{\varrhom span}harp} \pi = \int_{Q} P_{1\,\text{\varrhom span}harp}(\pi_{q}) \, \mathfrak q_{0}(dq),
$$
and by uniqueness of disintegration $P_{1\,\text{\varrhom span}harp}(\pi_{q}) = \mu_{0\,\mathfrak q}$ for $\mathfrak q_{0}$-a.e. $q\in Q$. Then we can find a localization of $\mu_{1}$ as follows:
$$
\mu_{1} = P_{2\,\text{\varrhom span}harp} \pi = \int_{Q} P_{2\,\text{\varrhom span}harp}(\pi_{q}) \, \mathfrak q_{0}(dq) = \int_{Q} \mu_{1\,q} \, \mathfrak q_{0}(dq),
$$
where by definition we posed $\mu_{1\, q} : = P_{2\,\text{\varrhom span}harp}(\pi_{q})$ and by construction $\mu_{1\, q} (X_{q}) = \mu_{0\,q}(X_{q}) = 1$.
{\bf Step 3.} Solution to the Monge problem.\\
For each $q \in Q$ consider the distribution functions
$$
H(q,t) := \mu_{0\,q}((-\infty,t)), \quad F(q, t) := \mu_{1\,q }((-\infty,t)),
$$
where for ease of notation $\mu_{i\,q} = g(q,\cdot)^{-1}_{\text{\varrhom span}harp} \mu_{i\,q}$ for $i = 0,1$.
Then define $\hat T$, as Theorem \textrm{Re}\,f{T:oneDmonge} suggests, by
$$
\hat T(q,s) := \Big(q, \text{\varrhom span}up \big\{ t : F(q,t) \leq H(q,s) \big\} \Big).
$$
Note that since $H$ is continuous ($\mu_{0\,q}$ has no atoms), the map $s \mapsto \hat T(q,s)$ is well-defined.
Then define the transport map $T: \mathcal{T} \to X$ as $g\circ \hat T \circ g^{-1}$.
It is fairly easy to observe that
$$
T_{\text{\varrhom span}harp} \,\mu_{0} = \int_{Q} \left(g\circ \hat T \circ g^{-1}\varrhoight)_{\text{\varrhom span}harp} \mu_{0\, q}\, \mathfrak q_{0}(dq) = \int_{Q} \mu_{1\, q} \,\mathfrak q_{0}(dq) = \mu_{1};
$$
moreover $(x,T(x)) \in \mathcal{G}amma$ and therefore the graph of $T$ is $\text{\varrhom span}fd$-cyclically monotone and therefore the map $T$ is optimal.
Extend $T$ to $X$ as the identity.
It remains to show that it is Borel. First observe that, possibly taking a compact subset of $Q$
the map $q \mapsto (\mu_{0\,q},\mu_{1\,q})$ can be assumed to be weakly continuity; it follows that the maps
$$
\textrm{Dom\,}(g) \ni (q,t) \mapsto H(q,t) := \mu_{0\,q}((-\infty,t)), \ \ (q,t) \mapsto F(q,t) := \mu_{1\,q}((-\infty,t))
$$
are lower semicontinuous. Then for $A$ Borel,
$$
\hat T^{-1}(A \times [t,+\infty)) = \big\{ (q,s) : q \in A, H(q,s) \geq F(q,t) \big\} \in \mathcal{B}(Q \times \mathbb{R}),
$$
and therefore the same applies for $T$.
\end{proof}
If $(X,\text{\varrhom span}fd,\mathfrak m)$ verifies $\mathsf{MCP}$ then it also verifies \textrm{Re}\,f{A:1}, see Proposition \textrm{Re}\,f{P:MCP-1}. So we have the following
\begin{equation}gin{corollary}[Corollary 9.6 of \cite{biacava:streconv}]\bigl\langlebel{C:MCP-Monge}
Let $(X,\text{\varrhom span}fd,\mathfrak m)$ be a non-branching metric measure space verifying $\mathsf{MCP}(K,N)$. Let $\mu_{0}$ and $\mu_{1}$ be probability measures
with finite first moment and $\mu_{0}\ll \mathfrak m$. Then there exists a Borel optimal transport map $T: X \to X$ solution to the Monge problem.
\end{corollary}
Corollary \textrm{Re}\,f{C:MCP-Monge} in particular implies the existence of solutions to the Monge problem in the Heisenberg group when $\mu_{0}$ is assumed
to be absolutely continuous with respect to the left-invariant Haar measure.
\begin{equation}gin{theorem}[Monge problem in the Heisenberg group]
Consider $(\mathbb{H}^n, \text{\varrhom span}fd_c,\mathcal{L}^{2n+1} )$, the $n$-dimensional Heisenberg group endowed with the Carnot-Carath\'eodory distance $\text{\varrhom span}fd_{c}$ and
the $(2n+1)$-Lebesgue measure that coincide with the Haar measure on $(\mathbb{H}^n, \text{\varrhom span}fd_c)$ under the identification $\mathbb{H}^n\text{\varrhom span}imeq \mathbb{R}^{2n+1}$.
Let $\mu_{0}$ and $\mu_{1}$ be two probability measures with finite first moment and $\mu_{0}\ll \mathcal{L}^{2n+1}$.
Then there exists a Borel optimal transport map $T: X \to X$ solution to the Monge problem.
\end{theorem}
\begin{equation}gin{remark}
The techniques used so far were successfully used also to threat the more general case of infinite dimensional spaces with curvature bound, see \cite{cava:Wiener} where
the existence of solutions for the Monge minimization problem in the Wiener space is proved.
Note that the material presented in the previous sections can be obtained also without assuming the existence of a $1$-Lipschitz Kantorovich potential (e.g. the Wiener space);
the decomposition of the space in geodesics and the associated disintegration of the reference measures can be obtained starting from a generic $\text{\varrhom span}fd$-cyclically monotone set.
For all the details see \cite{biacava:streconv}.
\end{remark}
\text{\varrhom span}ubsection{Isoperimetric inequality}
We now turn to the second main application of techniques reviewed so far,
the L\'evy-Gromov isoperimetric inequality in singular spaces. The results of this section are taken from \cite{CM1,CM2}.
\begin{equation}gin{theorem}[Theorem 1.2 of \cite{CM1}]\bigl\langlebel{T:iso}
Let $(X,\text{\varrhom span}fd,\mathfrak m)$ be a metric measure space with $\mathfrak m(X)=1$, verifying the essentially non-branching property and $\mathbb{C}D_{loc}(K,N)$ for some $K\in \mathbb{R},N \in [1,\infty)$.
Let $D$ be the diameter of $X$, possibly assuming the value $\infty$.
Then for every $v\in [0,1]$,
$$
\mathcal{I}_{(X,\text{\varrhom span}fd,\mathfrak m)}(v) \ \geq \ \mathcal{I}_{K,N,D}(v),
$$
where $\mathcal{I}_{K,N,D}$ is the model isoperimetric profile defined in \eqref{defcI}.
\end{theorem}
\begin{equation}gin{proof}
First of all we can assume $D<\infty$ and therefore $\mathfrak m \in \mathcal{P}_{2}(X)$: indeed from the Bonnet-Myers Theorem if $K>0$ then $D<\infty$, and if $K\leq 0$ and $D=\infty$ then the model isoperimetric profile \eqref{defcI} trivializes, i.e. $\mathcal{I}_{K,N,\infty}\equiv 0$ for $K\leq 0$.
For $v=0,1$ one can take as competitor the empty set and the whole space respectively, so it trivially holds
$$
\mathcal{I}_{(X,\text{\varrhom span}fd,\mathfrak m)}(0)=\mathcal{I}_{(X,\text{\varrhom span}fd,\mathfrak m)}(1)= \mathcal{I}_{K,N,D}(0)=\mathcal{I}_{K,N,D}(1)=0.
$$
Fix then $v\in(0,1)$ and let $A\text{\varrhom span}ubset X$ be an arbitrary Borel subset of $X$ such that $\mathfrak m(A)=v$.
Consider the $\mathfrak m$-measurable function $f(x) : = \chi_{A}(x) - v$ and notice that $\int_{X} f \, \mathfrak m = 0$.
Thus $f$ verifies the hypothesis of Theorem \textrm{Re}\,f{T:localize} and noticing that $f$ is never null,
we can decompose $X = Y \cup \mathcal{T}$ with
$$
\mathfrak m(Y)=0, \mathfrak quad \mathfrak m\llcorner_{\mathcal{T}} = \int_{Q} \mathfrak m_{q}\, \mathfrak q(dq),
$$
with $\mathfrak m_{q} = g(q,\cdot)_\text{\varrhom span}harp \left( h_{q} \cdot \mathcal{L}^{1}\varrhoight)$;
moreover, for $\mathfrak q$-a.e. $q \in Q$, the density $h_{q}$ verifies \eqref{E:curvdensmm} and
$$
\int_{X} f(z) \, \mathfrak m_{q}(dz) = \int_{\textrm{Dom\,}(g(q,\cdot))} f(g(q,t)) \cdot h_{q}(t) \, \mathcal{L}^{1}(dt) = 0.
$$
Therefore
\begin{equation}gin{equation}\bigl\langlebel{eq:volhq}
v=\mathfrak m_{q} ( A \cap \{ g(q,t) : t\in \mathbb{R} \} ) = (h_{q}\mathcal{L}^1) (g(q,\cdot)^{-1}(A)), \quad \text{ for $\mathfrak q$-a.e. $q \in Q$}.
\end{equation}
For every $\textbf{X}repsilon>0$ we then have
\begin{equation}gin{align*}
\varphirac{\mathfrak m(A^\textbf{X}repsilon)-\mathfrak m(A)}{\textbf{X}repsilon} &~ = \varphirac{1}{\textbf{X}repsilon} \int_{\mathcal{T}} \chi_{A^\textbf{X}repsilon\text{\varrhom span}etminus A} \,\mathfrak m(dx)
= \varphirac{1}{\textbf{X}repsilon} \int_{Q} \left( \int_{X} \chi_{A^\textbf{X}repsilon\text{\varrhom span}etminus A} \, \mathfrak m_{q} (dx) \varrhoight)\, \mathfrak q(dq) \crcr
&~ = \int_{Q} \varphirac{1}{\textbf{X}repsilon} \left( \int_{\textrm{Dom\,}(g(q,\cdot))} \chi_{A^\textbf{X}repsilon\text{\varrhom span}etminus A} \,h_{q}(t) \, \mathcal{L}^{1}(dt) \varrhoight)\, \mathfrak q(dq) \crcr
&~ = \int_{Q} \left( \varphirac{(h_{q}\mathcal{L}^1)(g(q,\cdot)^{-1}(A^\textbf{X}repsilon))- (h_{q}\mathcal{L}^1)(g(q,\cdot)^{-1}(A))}{\textbf{X}repsilon} \varrhoight)\, \mathfrak q(dq) \crcr
&~ \geq \int_{Q} \left( \varphirac{(h_{q}\mathcal{L}^1)((g(q,\cdot)^{-1}(A))^\textbf{X}repsilon)- (h_{q}\mathcal{L}^1)(g(q,\cdot)^{-1}(A))}{\textbf{X}repsilon} \varrhoight)\, \mathfrak q(dq), \crcr
\end{align*}
where the last inequality is given by the inclusion $ (g(q,\cdot)^{-1}(A))^\textbf{X}repsilon \cap \text{\varrhom span}upp(h_q) \text{\varrhom span}ubset g(q,\cdot)^{-1}(A^\textbf{X}repsilon)$. \\
Recalling \eqref{eq:volhq} together with $h_{q}\mathcal{L}^1\in \mathcal{F}^{s}_{K,N,D}$, by Fatou's Lemma we get
\begin{equation}gin{align*}
\mathfrak m^+(A) &~ = \liminf_{\textbf{X}repsilon\downarrow 0} \varphirac{\mathfrak m(A^\textbf{X}repsilon)-\mathfrak m(A)}{\textbf{X}repsilon} \crcr
&~\geq \int_{Q} \left( \liminf_{\textbf{X}repsilon\downarrow 0} \varphirac{(h_{q}\mathcal{L}^1)((g(q,\cdot)^{-1}(A))^\textbf{X}repsilon) -
(h_{q}\mathcal{L}^1)(g(q,\cdot)^{-1}(A))}{\textbf{X}repsilon} \varrhoight)\, \mathfrak q(dq) \crcr
&~ = \int_{Q} \left( (h_{q}\mathcal{L}^1)^+(g(q,\cdot)^{-1}(A)) \varrhoight)\, \mathfrak q(dq) \crcr
&~ \geq \int_{Q} \mathcal{I}^s_{K,N,D} (v) \, \mathfrak q(dq) \crcr
&~ = \mathcal{I}_{K,N,D} (v),
\end{align*}
where in the last equality we used Theorem \textrm{Re}\,f{thm:I=Is}.
\end{proof}
From the definition of $\mathcal{I}_{K,N,D}$, see \eqref{defcI}, and the smooth results of E. Milman in \cite{Mil}, the estimates proved in Theorem \textrm{Re}\,f{T:iso} are sharp.
Furthermore, 1-dimensional localization technique permits to obtain rigidity in the following sense:
if for some $v \in (0,1)$ it holds $\mathcal{I}_{(X,\text{\varrhom span}fd,\mathfrak m)}(v)= \mathcal{I}_{K,N,\pi}(v)$, then $(X,\text{\varrhom span}fd,\mathfrak m)$ is a spherical suspension.
It is worth underlining that to obtain such a result $(X,\text{\varrhom span}fd,\mathfrak m)$ is assumed to be in the more regular class of $\mathbb{R}CD$-spaces.
Even more, one can prove an almost rigidity statement: if $(X,\text{\varrhom span}fd,\mathfrak m)$ is an $\mathbb{R}CD^*(K,N)$ space such that $\mathcal{I}_{(X,\text{\varrhom span}fd,\mathfrak m)}(v)$ is close to $\mathcal{I}_{K,N,\pi}(v)$
for some $v \in (0,1)$, this force $X$ to be close, in the measure-Gromov-Hausdorff distance, to a spherical suspension. What follows is Corollary 1.6 of \cite{CM1}.
\begin{equation}gin{theorem}[Almost equality in L\'evy-Gromov implies mGH-closeness to a spherical suspension] \bigl\langlebel{cor:AlmRig}
For every $N\in [2, \infty) $, $v \in (0,1)$, $\textbf{X}repsilon>0$ there exists $\bar{\partiallta}=\bar{\partiallta}(N,v,\textbf{X}repsilon)>0$ such that the following hold. For every $\partiallta \in [0, \bar{\partiallta}]$, if $(X,\text{\varrhom span}fd,\mathfrak m)$ is an $\mathbb{R}CD^*(N-1-\partiallta,N+\partiallta)$ space satisfying
$$
\mathcal{I}_{(X,\text{\varrhom span}fd,\mathfrak m)}(v)\leq \mathcal{I}_{N-1,N,\pi}(v)+\partiallta,
$$
then there exists an $\mathbb{R}CD^*(N-2,N-1)$ space $(Y, \text{\varrhom span}fd_Y, \mathfrak m_Y)$ with $\mathfrak m_Y(Y)=1$ such that
$$
\text{\varrhom span}fd_{mGH}(X, [0,\pi] \times_{\text{\varrhom span}in}^{N-1} Y) \leq \textbf{X}repsilon.
$$
\end{theorem}
We refer to \cite{CM1} for the precise rigidity statement (Theorem 1.4, \cite{CM1}) and for the proof of Theorem 1.4 and Corollary 1.6 of \cite{CM1}.
See also \cite{CM1} for the precise definition of spherical suspension.
We conclude by recalling that 1-dimensional localization was used also in \cite{CM2} to obtain sharp version of several functional inequalities
(e.g. Brunn-Minkowski, spectral gap, Log-Sobolev etc.) in the class of $\mathbb{C}D(K,N)$-spaces. See \cite{CM2} for details.
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\end{document} |
\begin{document}
\title{Equivalent definitions for (degree one) Cameron-Liebler classes of generators in finite classical polar spaces}
\begin{abstract}
In this article, we study \emph{degree one Cameron-Liebler} sets of generators in all finite classical polar spaces, which is a particular type of a Cameron-Liebler set of generators in this polar space, \cite{CLpolar}. These degree one Cameron-Liebler sets are defined similar to the Boolean degree one functions, \cite{Ferdinand.}. We summarize the equivalent definitions for these sets and give a classification result for the degree one Cameron-Liebler sets in the polar spaces $W(5,q)$ and $Q(6,q)$.
\end{abstract}
\textbf{Keywords}: Cameron-Liebler set, finite classical polar space, Boolean degree one function.
\par
\textbf{MSC 2010 codes}: 51A50, 05B25, 05E30, 51E14, 51E30.
\section{Introduction}
The investigation of Cameron-Liebler sets of generators in polar spaces is inspired by the research on Cameron-Liebler sets in finite projective spaces. This research started with Cameron-Liebler sets of lines in $\PG(3,q)$, defined by P. Cameron and R. Liebler in \cite{begin}. A set $\mathcal{L} $ of lines in $\PG(3,q)$ is a Cameron-Liebler set of lines if and only if the number of lines in $\mathcal{L}$ disjoint to a given line $l$ only depends on whether $l \in \mathcal{L}$ or not.
We also find Cameron-Liebler sets in the theory of tight sets for graphs: Cameron-Liebler sets are the tight sets of the type $I$ \cite{bart}.
After many results about those Cameron-Liebler line sets in $\PG(3,q)$, the Cameron-Liebler set concept has been generalized to many other contexts: Cameron-Liebler line sets in $\PG(n,q)$ \cite{phdDrudge}, Cameron-Liebler sets of $k$-spaces in $\PG(2k+1,q)$ \cite{CLkclas}, Cameron-Liebler sets of $k$-spaces in $\PG(n,q)$ \cite{CLksetn}, Cameron-Liebler classes in finite sets \cite{CLset,eenextra,tweeextra} and Cameron-Liebler sets of generators in finite classical polar spaces \cite{CLpolar} were defined. The central problem for Cameron-Liebler sets is to find for which parameter $x$ a Cameron-Liebler set exists, and finding examples with this parameter \cite{phdDrudge,feng,CL20,CL21,Klaus,CL26}.
In this article, we will investigate Cameron-Liebler sets in finite classical polar spaces.
The finite classical polar spaces are the hyperbolic quadrics $Q^+(2d-1,q)$, the parabolic quadrics $Q(2d,q)$, the elliptic quadrics $Q^-(2d+1,q)$, the hermitian polar spaces $H(2d-1,q^2)$ and $H(2d,q^2)$, and the symplectic polar spaces $W(2d-1,q)$, with $q$ prime power.
Here we investigate the sets of generators defined by the following definition, where $A$ is the incidence matrix of points and generators, and we call these sets \emph{degree one Cameron-Liebler sets}.
\begin{definition}\label{defspecialCL}
A degree one Cameron-Liebler set of generators in a finite classical polar space $\mathcal{P}$ is a set of generators in $\mathcal{P}$, with characteristic vector $\chi$ such that $\chi \in \im(A^T)$.
\end{definition}
This definition corresponds with the definition of Boolean degree one functions for generators in polar spaces, in \cite{Ferdinand.} by Y. Filmus and F. Ihringer. In their article, they define Boolean degree one functions, or Cameron-Liebler sets in projective and polar spaces by the fact that the corresponding characteristic vector lies in $V_0\perp V_1$, which are eigenspaces of the related association scheme (see Section \ref{section2}).
In \cite{CLpolar}, M. De Boeck, M. Rodgers, L. Storme and A. \v{S}vob introduced Cameron-Liebler sets of generators in the finite classical polar spaces. In this article, Cameron-Liebler set of generators in the polar spaces are defined by the \emph{disjointness-definition} and the authors give several equivalent definitions for these Cameron-Liebler sets.
\begin{definition}[{\cite{CLpolar}}]\label{defCL}
Let $\mathcal{P}$ be a finite classical polar space with parameter $e$ and rank $d$. A set $\mathcal{L}$ of generators in $\mathcal{P}$ is a Cameron-Liebler set of generators in $\mathcal{P}$ if and only if for every generator $\pi$ in $\mathcal{P}$, the number of elements of $\mathcal{L}$, disjoint from $\pi$ equals $(x-\chi(\pi))q^{\binom{d-1}{2}+e(d-1)}$.
\end{definition}
\begin{table}[h]\begin{center}
\begin{tabular}{ | c | c| c| }
\hline
Type $I$ & Type $II$ & Type $III$ \\ \hline
$Q^-(2d+1,q)$ & $Q^+(2d-1,q)$, $d$ even& $Q(4n+2,q)$ \\
$Q(2d,q)$, $d$ even & & $W(4n+1,q)$ \\
$Q^+(2d-1,q)$, $d$ odd & & \\
$W(2d-1,q)$, $d$ even & & \\
$H(2d-1,q)$, $q$ square & & \\
$H(2d,q)$, $q$ square & & \\ \hline
\end{tabular}
\caption{Three types of polar spaces}\label{tabeltype}
\end{center}\end{table}
In this article, we consider three different types of polar spaces, see Table \ref{tabeltype}. Type $I$ and $II$ corresponds with type $I$ and $II$ respectively, defined in \cite{CLpolar}, while type $III$ in this paper corresponds with the union of type $III$ and $IV$ in $\cite{CLpolar}$, as we handle the symplectic polar spaces $W(4n+1,q)$, for both $q$ odd and $q$ even, in the same way. Definition \ref{defCL} and Definition \ref{defspecialCL} are equivalent for the polar spaces of type $I$ by \cite[Theorem 3.7, Theorem 3.15]{CLpolar}. For the polar spaces of type $II$ we can consider the (degree one) Cameron-Liebler sets of one class of generators; we see that Cameron-Liebler sets and degree one Cameron-Liebler sets coincide when we only consider one class (see \cite[Theorem 3.16]{CLpolar}). For the polar spaces of type $III$, this equivalence no longer applies and for these polar spaces, any degree one Cameron-Liebler set is also a regular Cameron-Liebler set, but not vice versa.
Cameron-Liebler sets were introduced by a group-theoretical argument: a set $\mathcal{L}$ of lines is a Cameron-Liebler set of lines in $\PG(3,q)$ if and only if PGL$(3,q)$ has the same number of orbits on the lines of $\mathcal{L}$ and on the points of $\PG(3,q)$.
If the incidence matrix $A$ of points and generators of a polar space $\mathcal{P}$ has trivial kernel, then we also find a group-theoretical definition for degree one Cameron-Liebler sets of generators in $\mathcal{P}$. This theorem follows from {\cite[Lemma 3.3.11]{phdfred}}.
\begin{theorem}
Let $X$ be the set of points in a classical polar space $\mathcal{P}$, let $M$ be the set of generators in $\mathcal{P}$ and let $A$ be the point-generator matrix of $\mathcal{P}$. Consider an automorphismgroup $G$ acting on the sets $X$ and $M$ with orbits $O_1, \dots, O_n$ and $O'_1, \dots, O'_m$, respectively. If $A$ has trivial kernel, then each $O'_i$ is a degree one Cameron-Liebler set in $\mathcal{P}$ if and only if $n=m$.
\end{theorem}
In Section $2$ we give some preliminaries about the classical polar spaces and we discuss several properties of the eigenvalues of the association scheme for generators of finite classical polar spaces. In Section $3$, we give an overview of the equivalent definitions and several properties of degree one Cameron-Liebler sets in polar spaces. In Section $4$ we give an equivalent definition for Cameron-Liebler sets in the hyperbolic quadrics $Q^+(2d-1,q)$, $d$ even and in Section $5$ we end with some classification results for degree one Cameron-Liebler sets, especially in the polar spaces $W(5,q)$ and $Q(6,q)$.
\section{Preliminaries}\label{section2}
For an extensive and detailed introduction about distance-regular graphs, polar spaces and association schemes for generators of finite classical polar spaces, we refer to \cite{CLpolar}. For more general information about association schemes of distance regular graphs, we refer to \cite{bose, brouwer}. We only repeat the necessary definitions and information. Note that in this article, we will work in a projective context and all mentioned dimensions are projective dimensions.
\subsection{Finite classical polar spaces}\label{setcionfcps}
We start with the definition of finite classical polar spaces.
\begin{definition}
Finite classical polar spaces are incidence geometries consisting of subspaces that are totally isotropic with respect to a non-degenerate quadratic or non-degenerate reflexive sesquilinear form on a vector space $\mathbb{F}_q^{n+1}$.
\end{definition}
In this article all polar spaces we will handle are the finite classical polar spaces, so we will call them the polar spaces.
We also give the definition of the rank and the parameter $e$ of a polar space.
\begin{definition}
A generator of a polar space is a subspace of maximal dimension and the rank $d$ of a polar space is the projective dimension of a generator plus $1$. The parameter of a polar space $\mathcal{P}$ over $\mathbb{F}_q$ is defined as the number $e$ such that the number of generators through a $(d-2)$-space of $\mathcal{P}$ equals $q^e+1$.
\end{definition}
In Table \ref{tabele} we give the parameter $e$ of the polar spaces.
\begin{table}[h]\begin{center}
\begin{tabular}{ | c | c| }
\hline
Polar space & $e$ \\ \hline \hline
$Q^+(2d-1,q)$ & $0$ \\ \hline
$H(2d-1,q)$ & $1/2$ \\ \hline
$W(2d-1,q)$ & $1$ \\ \hline
$Q(2d,q)$ & $1$ \\ \hline
$H(2d,q)$ & $3/2$ \\ \hline
$Q^-(2d+1,q)$ & $2$ \\ \hline
\end{tabular}
\caption{The parameter of the polar spaces}\label{tabele}
\end{center}\end{table}
To ease the notations, we will work with the \emph{Gaussian binomial coefficient} $\begin{bmatrix}a\\b\end{bmatrix}_q$ for positive integers $a,b$ and prime power $q\geq 2$:
\begin{align*}
\begin{bmatrix}a\\b\end{bmatrix}_q=\prod_{i=1}^b \frac{q^{a-b+i}-1}{q^i-1} = \frac{(q^a-1)\dots (q^{a-b+1}-1)}{(q^b-1)\dots (q-1)}.
\end{align*}
We will write $\qbin ab$ if the field size $q$ is clear from the context. The number $\qbin ab_q$ equals the number of $(b-1)$-spaces in $\PG(a-1,q)$, and the equality $\qbin ab = \qbin{a}{a-b}$ follows immediately from duality.\\
We end this section by defining several substructures in polar spaces. A \emph{spread} in a polar space $\mathcal{P}$ is a set $S$ of generators in $\mathcal{P}$ such that every point of $\mathcal{P}$ is contained in precisely one element of $S$. A \emph{point-pencil} in a polar space $\mathcal{P}$ with vertex $P\in \mathcal{P}$ is the set of all generators in $\mathcal{P}$ through the point $P$. \\
\subsection{Eigenvalues of the association scheme for generators in polar spaces}
First of all, remark that in this article vectors are regarded as column vectors and we define $\textbf{\textit{j}}_n$ to be the all one vector of length $n$. We write $\textbf{\textit{j}}$ for $\textbf{\textit{j}}_n$ if the length is clear from the context.\\
Let $\mathcal{P}$ be a finite classical polar space of rank $d$ and let $\Omega$ be its set of generators. The relations $R_i$ on $\Omega$ are defined as follows: $(\pi,\pi') \in R_i$ if and only if $\dim(\pi \cap \pi') =d-i-1$, for generators $\pi,\pi'\in \Omega$ with $i=0, ...,d$. We define $A_i$ as the incidence matrix of the relation $R_i$. By the theory of association schemes we know that there is an orthogonal decomposition $V_0 \perp V_1 \perp \cdots \perp V_d$ of $\mathbb{R}^\Omega$ in common eigenspaces of $A_0,A_1,...,A_d$.
\begin{lemma}[{\cite[Theorem 4.3.6]{phdfred}}]\label{eigenvallem}
In the association scheme of a polar space over $\mathbb{F}_q$ of rank $d$ and parameter $e$, the eigenvalue $P_{ji}$ of the relation $R_i$ corresponding to the subspace $V_j$ is given by:
\begin{align*}
P_{ji} = \sum\limits_{s=\max{(0,j-i)}}^{\min{(j,d-i)}} (-1)^{j+s}\begin{bmatrix} j\\s\end{bmatrix} \begin{bmatrix}d-j \\ d-i-s\end{bmatrix} q^{e(i+s-j)+\binom{j-s}{2}+\binom{i+s-j}{2}}.
\end{align*}.
\end{lemma}
Before we start with investigating the Cameron-Liebler sets of generators in finite classical polar spaces, we give an important lemma about the eigenvalues $P_{ji}$.\\
\begin{lemma} \label{lemma2}
In the association scheme of polar spaces, the eigenvalue $P_{1i}$ of $A_i$ corresponds only with the eigenspace $V_1$ for $i\neq 0$, except in the following cases.
\begin{enumerate}
\item The hyperbolic quadrics $Q^+(2d-1,q)$. Here $P_{1i}=P_{d-1,i}$ for $i$ even, so $P_{1i}$ also corresponds with $V_{d-1}$, for every relation $R_i$, $i$ even.
\item The parabolic quadrics $Q(4n+2,q)$ and the symplectic quadrics $W(4n+1,q)$. Here $P_{1d} = P_{dd}$, so $P_{1d}$ also corresponds with $V_d$ for the disjointness relation $R_d$.
\end{enumerate}
\end{lemma}
\begin{proof}
We need to prove, given a fixed $i$ and $j\neq 1$, that $P_{1i} \neq P_{ji}$ for $q$ a prime power. For $j=0$ and for all $i\neq 0$, it is easy to calculate that $P_{1i}\neq P_{0i}$, so we can suppose that $j>1$.
For $i=1$ we can directly compare the eigenvalues $P_{11}$ and $P_{j1}$.
\begin{align*}
P_{11} = P_{j1} &\Leftrightarrow \begin{bmatrix}d-1 \\ 1\end{bmatrix} q^e -1= \begin{bmatrix}d-j \\ 1\end{bmatrix} q^e -\begin{bmatrix}j \\ 1 \end{bmatrix}\\
& \Leftrightarrow \frac{-q+1+(q^{d-1}-1)q^e}{q-1}=\frac{-q^j+1+(q^{d-j}-1)q^e}{q-1}\\
& \Leftrightarrow (q^{d-j+e-1}+1)(q^{j-1}-1)=0
\end{align*}
Since $j>1$ the last equation gives a contradiction for any $q$.
For $i\geq 2$ we introduce $\phi_i(j) = \max\{k\mid \mid q^k|P_{ji} \}$, the exponent of
$q$ in $P_{ji}$. If $P_{ij}=0$, we put $\phi_i(j)=\infty$. We will show that $\phi_i(j)$ is different from $\phi_i(1)$ for most values of $i$ and $j$. For $j=1$, we find that
\begin{align*}
P_{1i}=-\begin{bmatrix}
d-1 \\ d-i\end{bmatrix} q^{\binom{i-1}{2}+e(i-1)}+\begin{bmatrix}
d-1 \\ i\end{bmatrix} q^{\binom{i}{2}+ei}=q^{\binom{i-1}{2}+e(i-1)} \left(\qbin{d-1}{i}q^{i-1+e} -\qbin{d-1}{i-1} \right).
\end{align*}
We can see that $\phi_i(1)=\binom{i-1}{2}+e(i-1)$, since $i-1+e\geq 1$ and $\qbin ab=1 \pmod q$ for all $0\leq b\leq a$.
In Lemma \ref{eigenvallem} we see that ${\phi_i(j)}$ depends on the last factor of every term in the sum.
To find $\phi_i(j)$ we first need to find $z$ such that $q^{e(i+z-j)+\binom{j-z}{2}+\binom{i+z-j}{2}}$ is a factor of every term in the sum, or equivalently, such that $f_{ji}(s)=e(i+s-j)+\binom{j-s}{2}+\binom{i+s-j}{2}$ reaches its minimum for $s=z$. So for most cases, we have that $\phi_i(j)=f_{ij}(z)$, but in some cases it occurs that two values of $z$ correspond with opposite terms with factor $q^{\phi_i(j)}$. These cases, we have to investigate separately. \\
We can check that $z$ is the integer or integers in $[\max\{0,j-i\}, \dots, \min\{j,d-i\}]$, closest to $j-\frac{i}{2}-\frac{e}{2}$.
Since $i\geq 2$ we have three possibilities for the value of $z$, as we always have $j-i\leq j-\frac{i}{2}-\frac{e}{2}<j$:
\begin{itemize}
\item $z=0$ if $j-\frac{i}{2}-\frac{e}{2}<0$,
\item $z \in \{j-\frac{i}{2}-\frac{e}{2}, j-\frac{i}{2}-\frac{e}{2}\pm \frac{1}{2}\}$ if $0\leq j-\frac{i}{2}-\frac{e}{2}\leq d-i$,
\item $z=d-i$ if $j-\frac{i}{2}-\frac{e}{2}>d-i$.
\end{itemize}
Now we handle these three cases.
\begin{itemize}
\item If $j-\frac{i}{2}-\frac{e}{2} < 0$, we see that $f_{ji}$ is minimal for the integer $z=0$.
We remark that in this case there is only $1$ value of $s$, namely $0$, for which the corresponding term is divisible by $q^{\phi_i(j)}$ but not by $q^{\phi_i(j)+1}$. This is important to exclude the case where $2$ terms with factor $q^{\phi_i(j)}$ would be each others opposite.
We find that $\phi_i(j)=f_{ji}(0)=\binom{i}{2}+(j-i)(j-e)$, and since $\phi_i(1)=\binom{i-1}{2}+e(i-1)$, the values $\phi_i(j)$ and $\phi_i(1)$ are equal if and only if $j= 1 \vee j= i+e-1$. We only have to check the latter case, and recall that $ j-\frac{i}{2}-\frac{e}{2} < 0$. It follows that $i+e< 2$, a contradiction since we supposed $i\geq 2$.
\item If $0 \leq j-\frac{i}{2}-\frac{e}{2} \leq d-i$ we see that $f_{ji}$ is minimal for the integer $z$ closest to $j-\frac{i}{2}-\frac{e}{2}$.
\def\vrule height 5ex depth 3.5ex width 0pt{\vrule height 5ex depth 3.5ex width 0pt}
\pagestyle{empty}
\begin{table}[hp]
\centering
\renewcommand{2.1}{2.1}
\setlength\extrarowheight{-2pt}
\begin{tabular}{|>{\centering\arraybackslash}p{6mm}|p{20mm}|p{26mm}|p{45mm}|>{\centering\arraybackslash}p{20mm}|>{\centering\arraybackslash}p{6mm}|}\hline
\boldmath{$e$} & \boldmath{$i$} & \boldmath{$z$} & \boldmath{$\phi_i(j)=f_{ji}(z)$} &\boldmath{$\phi_i(1)$}& \boldmath{$S$}\\ \hline
\hline \multicolumn{6}{|c|}{$Q^+ (2d-1,q)$}\\ \hline
\multirow{2}{*}{$0$}
& even & $j-\frac{i}{2}$ & $\frac{i(i-2)}{4}$&$\frac{(i-1)(i-2)}{2}$ & $\{2\}$\\ \cline{2-6}
& odd & $j-\frac{i}{2}\pm \frac{1}{2}$ & $\begin{cases} \frac{(i-1)^2}{4} &\mbox{if } j\neq \frac{d}{2} \\
\infty & \mbox{if } j=\frac{d}{2} \end{cases}$&$\frac{(i-1)(i-2)}{2}$ & $\{3\}$\\ \hline
\hline \multicolumn{6}{|c|}{$\mathcal{H}(2d-1,q)$, with $q$ square}\\ \hline
\multirow{2}{*}{$\frac{1}{2}$}
& even & $j-\frac{i}{2}$ & $\frac{i(i-1)}{4}$&$\frac{(i-1)^2}{2}$ & $\{2\}$\\ \cline{2-6}
& odd & $j-\frac{i}{2}-\frac{1}{2}$ & $\frac{i(i-1)}{4}$&$\frac{(i-1)^2}{2}$ & $\emptyset$\\ \hline
\hline \multicolumn{6}{|c|}{$Q(2d,q)$, $W(2d-1,q)$, with $d\not\equiv 0 \pod{4}$}\\ \hline
\multirow{2}{*}{$1$}
& even & $j-\frac{i}{2}-\frac{1}{2}\pm \frac{1}{2}$ & $ \frac{i^2}{4}$&$\frac{i(i-1)}{2}$ & $\{2\}$\\ \cline{2-6}
& odd & $j-\frac{i}{2}-\frac{1}{2}$ & $\frac{i^2-1}{4}$&$\frac{i(i-1)}{2}$ & $\emptyset$\\ \hline
\hline \multicolumn{6}{|c|}{$Q(2d,q)$, $W(2d-1,q)$, with $d\equiv 0 \pod{4}$}\\ \hline
\multirow{3}{*}{$1$}
& even, $i\neq\frac{d}{2}$ & $j-\frac{i}{2}-\frac{1}{2}\pm \frac{1}{2}$ & $\frac{i^2}{4}$&$\frac{i(i-1)}{2}$ & $\{2\}$\\ \cline{2-6}
& $i=\frac{d}{2}$ & $j-\frac{i}{2}-\frac{1}{2}\pm \frac{1}{2}$ & \vrule height 5ex depth 3.5ex width 0pt $\begin{cases} \infty &\text{if $j= \frac{d}{2}+1$}\\ \frac{i^2}{4} & \text{else} \end{cases}$&$\frac{i(i-1)}{2}$ & $\{2\}$\\ \cline{2-6}
& odd & $j-\frac{i}{2}-\frac{1}{2}$ & $\frac{i^2-1}{4}$&$\frac{i(i-1)}{2}$ & $\emptyset$\\ \hline
\hline \multicolumn{6}{|c|}{$\mathcal{H}(2d,q)$, with $q$ square}\\ \hline
\multirow{2}{*}{$\frac{3}{2}$}
& even & $j-\frac{i}{2}-1$ & $\frac{(i-1)(i+2)}{4}$&$\frac{i^2-1}{2}$ & $\emptyset$\\ \cline{2-6}
& odd & $j-\frac{i}{2}-\frac{1}{2}$ & $\frac{(i-1)(i+2)}{4}$ &$\frac{i^2-1}{2}$ & $\emptyset$\\ \hline
\hline \multicolumn{6}{|c|}{$Q^- (2d+1,q)$, with $d\not\equiv 2 \pod{4}$}\\ \hline
\multirow{2}{*}{$2$}
& even & $j-\frac{i}{2}-1$ & $\frac{i^2}{4}+\frac{i}{2}-1$&$\frac{(i-1)(i+2)}{2}$ & $\emptyset$\\ \cline{2-6}
& odd & $j-\frac{i}{2}-1\pm\frac{1}{2}$ & $\frac{(i-1)(i+3)}{4}$&$\frac{(i-1)(i+2)}{2}$ & $\emptyset$\\ \hline
\hline \multicolumn{6}{|c|}{$Q^- (2d+1,q)$, with $d\equiv 2 \pod{4}$}\\ \hline
\multirow{3}{*}{$2$}
& even & $j-\frac{i}{2}-1$ & $\frac{i^2}{4}+\frac{i}{2}-1$&$\frac{(i-1)(i+2)}{2}$ & $\emptyset$\\ \cline{2-6}
& odd, $i\neq\frac{d}{2}$ & $j-\frac{i}{2}-1\pm \frac{1}{2}$ & $\frac{(i-1)(i+3)}{4}$&$\frac{(i-1)(i+2)}{2}$ & $\emptyset$\\ \cline{2-6}
& $i=\frac{d}{2}$ & $j-\frac{i}{2}-1\pm\frac{1}{2}$ & \vrule height 5ex depth 3.5ex width 0pt $\begin{cases} \infty &\text{if $j= \frac{d}{2}+2$} \\ \frac{(i-1)(i+3)}{4} & \text{else} \end{cases}$&$\frac{(i-1)(i+2)}{2}$ & $\emptyset$\\ \hline
\end{tabular}
\caption{For $0\leq j-\frac{i}{2}-\frac{e}{2} \leq d-i$, with $S=\{ i \geq 2 \mid\phi_i(j)=\phi_i(1)\}$. }
\label{tabel}
\end{table}
In Table \ref{tabel} we list the different cases depending on $e$ and the parity of $i$. Note that we have to check, for $e=0, i$ odd, for $e=1, i$ even, and for $e=2, i$ odd, that the two values of $z$ do not correspond with two opposite terms with factor $q^{\phi_i(j)}$. By calculating and taking in account the conditions $0\leq j-\frac{i}{2}-\frac{e}{2}\leq d-i$, we find out that those cases do not correspond with two opposite terms, except in the following cases:
\begin{itemize}
\item $e=0, j=\frac{d}{2}$ and $i$ odd,
\item $e=1, j=\frac{d}{2}+1, i=\frac{d}{2}$ and $i$ even,
\item $e=2,j=\frac{d}{2}+2, i=\frac{d}{2}$ and $i$ odd.
\end{itemize}
In these cases, $P_{ij}=0$, so $\phi_i(j)=\infty\neq \phi_i(1)$.
Remark that for every $e$, $i$ and $j>1$, $\phi_i(j)=f_{ij}(z)$ is independent of $j$, see the fifth column in Table \ref{tabel}. In the last column we give the values of $i$ for which $\phi_i(j)=\phi_i(1)$. As we supposed $i\geq 2$, we see that we have to check the eigenvalues for $i=2$ if $e\in \{0,\frac{1}{2}, 1\}$ and for $i=3$ if $e=0$ in detail.
\begin{itemize}
\item Case $i=2$ and $e\in\{0,\frac{1}{2},1\}$:\small {\begin{align*}
& P_{12} = P_{j2} \\
& \Leftrightarrow -\begin{bmatrix} d-1 \\ 1\end{bmatrix}q^e+ \begin{bmatrix}d-1 \\ 2\end{bmatrix} q^{1+2e} =\begin{bmatrix} j\\2 \end{bmatrix}q -\begin{bmatrix} d-j\\ 1 \end{bmatrix}\begin{bmatrix}j \\ 1 \end{bmatrix}q^e+\begin{bmatrix}d-j \\ 2\end{bmatrix} q^{1+2e} \\
& \Leftrightarrow \left(\qbin{d-1}{2}-\qbin{d-j}{2} \right) q^{2e} + \qbin{d-j-1}{1}\qbin{j-1}{1} q^{e} =\begin{bmatrix} j\\2 \end{bmatrix}.
\end{align*}
For $e=\frac{1}{2}$ and $e=1$, we see that the right and left hand side of the last equation are different modulo $q$. So we can assume $e=0$.
\begin{align*}
& P_{12} = P_{j2}\\
&\Leftrightarrow \frac{(q^{d-1}-1)(q^{d-2}-1)}{(q^2-1)(q-1)}-\frac{(q^{d-j}-1)(q^{d-j-1}-1)}{(q^2-1)(q-1)}+\frac{(q^{d-j-1}-1)(q^{j-1}-1)}{(q-1)(q-1)} = \frac{(q^j-1)(q^{j-1}-1)}{(q^2-1)(q-1)} \\
& \Leftrightarrow q^{2d-3}-q^{2d-2j-1}+q-q^{2j-1}=0 \\
& \Leftrightarrow q(q^{2j-2}-1)(q^{2(d-j-1)}-1)=0
\end{align*}}
Since $j>1$, we see that $P_{12}=P_{j2}$ if and only if $j=d-1$. This corresponds with the first exception in the lemma with $i=2$.
\item Case $i=3$ and $e=0$.
\begin{align*}
&P_{13} = P_{j3} \\
&\Leftrightarrow -\begin{bmatrix} d-1 \\ 2\end{bmatrix}q+ \begin{bmatrix}d-1 \\ 3\end{bmatrix} q^{3} =-\begin{bmatrix}j \\ 3 \end{bmatrix}q^3+\begin{bmatrix} j\\2 \end{bmatrix}\begin{bmatrix} d-j\\ 1 \end{bmatrix}q -\begin{bmatrix}j \\ 1 \end{bmatrix}\begin{bmatrix} d-j\\ 2 \end{bmatrix}q+\begin{bmatrix}d-j \\ 3\end{bmatrix} q^{3} \\
&\Leftrightarrow -\begin{bmatrix} d-1 \\ 2\end{bmatrix}+ \begin{bmatrix}d-1 \\ 3\end{bmatrix} q^{2} =-\begin{bmatrix}j \\ 3 \end{bmatrix}q^2+\begin{bmatrix} j\\2 \end{bmatrix}\begin{bmatrix} d-j\\ 1 \end{bmatrix} -\begin{bmatrix}j \\ 1 \end{bmatrix}\begin{bmatrix} d-j\\ 2 \end{bmatrix}+\begin{bmatrix}d-j \\ 3\end{bmatrix} q^{2}
\end{align*}
Since the right and left hand side of the last equation are different modulo $q$, we see that $P_{13} \neq P_{j3}$ for $j>1$. Recall that $\qbin ab =1 \pmod q$.
\end{itemize}
\item If $ j-\frac{i}{2}-\frac{e}{2} > d-i$, we see that $f_{ji}$ is minimal for the integer $z=d-i$. Remark again that there is only one value of $s$ for which the corresponding term is divisible by $q^{\phi_i(j)}$ but not by $q^{\phi_i(j)+1}$. This excludes the case where $2$ terms with factor $q^{\phi_i(j)}$ would be each others opposite.
We find that $\phi_i(j)=f_{ji}(d-i)=(j-e-d+1)(j-d+i-1)+\binom{i-1}{2}+e(i-1)$, and we know that $\phi_i(1)=\binom{i-1}{2}+e(i-1)$. These two values $\phi_i(j)$ and $\phi_i(1)$ are equal if and only if $j = e+d-1$ or $j= d-i+1$.
\begin{itemize}
\item Suppose $j=d+e-1$. As $j,d \in \mathbb{Z}$, we know that $e \in \mathbb{Z}$. If $e=2$, then $j=d+1 > d$, a contradiction. For $e=1$, we find that $P_{1i}=P_{di}$ if and only if $i=d$ and $d$ odd. This corresponds to the polar spaces $Q(4n+2,q)$ and $ W(4n+1,q)$. For $e=0$ and $j=d-1$, we find that $P_{1i}=P_{d-1,i}$ for $i$ even. This corresponds to the exception for the polar spaces $Q^+(2d-1,q)$ and $i$ even.
\item Suppose $j=d-i+1$. Since $j-\frac{i}{2}-\frac{e}{2} > d-i$, we know that $i+e<2$, which gives a contradiction as we supposed $i\geq 2$.\qedhere
\end{itemize}
\end{itemize}
\end{proof}
We continue with well-known theorems that will be useful in the following sections. The first theorem follows from \cite[Theorem 2.14]{CLpolar} which was originally proven in \cite{delsarte}. For the second theorem we add a proof for completeness. The ideas are already present in \cite[Lemma 2]{bamberg} and \cite[Lemma 2.1.3]{phdfred}.
\begin{theorem}\label{lemmaV0V1}
Let $\mathcal{P}$ be a finite classical polar space of rank $d$ and parameter $e$, and let $\Omega$ be the set of all generators of $\mathcal{P}$. Consider the eigenspace decomposition $\mathbb{R}^\Omega =V_0\perp V_1 \perp \dots \perp V_d$ related to the association scheme, and using the classical order. Let $A$ be the point-generator incidence matrix of $\mathcal{P}$, then $\im(A^T) = V_0 \perp V_1$ and $V_0 = \langle j \rangle$.
\end{theorem}
\begin{theorem} \label{stellingalgemeen}
Let $R_i$ be a relation of an association scheme on the set $\Omega$ with adjacency matrix $A_i$ and let $\mathcal{L} \subset \Omega$ be a set, with characteristic vector $\chi$, such that for any $\pi \in \Omega$, we have that
\begin{align*}
|\{x\in \mathcal{L} | (x,\pi) \in R_i \}| = \begin{cases} \alpha_i \ \text{If } \pi \in \mathcal{L} \\ \beta_i \ \text{If } \pi \notin \mathcal{L}
\end{cases}
\end{align*}
with $\alpha_i -\beta_i=P$ an eigenvalue of $A_i$ for the eigenspace $V$, then $v_i= \chi + \frac{\beta_i}{P-P_{0i}}\textbf{\textit{j}} \in V$.
\end{theorem}
Remark that the eigenspace $V$ in the previous theorem can be the direct sum of several eigenspaces of the association scheme. Note that an association scheme is not necessary in this theorem, a regular relation suffices.
\begin{proof}
We show that $v_i = \chi + \frac{\beta_i}{P - P_{0i}}\textbf{\textit{j}}$, with $P=\alpha_i-\beta_i$ is an eigenvector for the matrix $A_i$ with eigenvalue $P$:
\begin{align*}
A_i\left(\chi + \frac{\beta_i}{P - P_{0i}}\textbf{\textit{j}}\right) =& \alpha_i \chi + \beta_i (\textbf{\textit{j}}-\chi) + \frac{\beta_i}{P - P_{0i}}P_{0i} \textbf{\textit{j}} \\
=& P\left(\chi + \frac{\beta_i}{P - P_{0i}}\textbf{\textit{j}}\right).
\end{align*}\\
So we find that $\chi + \frac{\beta_i}{P - P_{0i}}\textbf{\textit{j}} \in V$.
\end{proof}\\
\section{Degree one Cameron-Liebler sets}
In this section we investigate the degree one Cameron-Liebler sets and give an equivalent definition.
Recall that for polar spaces of type $I$ Cameron-Liebler sets and degree one Cameron-Liebler coincide.
Using Lemma \ref{lemma2} and Theorem \ref{stellingalgemeen}, we can give a new equivalent definition for these degree one Cameron-Liebler sets of generators in polar spaces. Remark that the following theorem is an extension of Lemma $4.9$ in \cite{CLpolar}.
\begin{theorem} \label{stelling}
Let $\mathcal{P}$ be a finite classical polar space, of rank $d$ with parameter $e$, let $\mathcal{L}$ be a set of generators of $\mathcal{P}$ and $i$ be an integer with $1\leq i\leq d$. Denote $\frac{|\mathcal{L}|}{\prod_{i=0}^{d-2}(q^{e+i}+1)}$ by $x$.
If $\mathcal{L}$ is a degree one Cameron-Liebler set of generators in $\mathcal{P}$ then the number of elements of $\mathcal{L}$ meeting a generator $\pi$ in a $(d-i-1)$-space equals
\begin{align}\label{formulelang}
\left\{ \begin{matrix}
\left( (x-1) \begin{bmatrix}d-1 \\i-1\end{bmatrix} +q^{i+e-1}\begin{bmatrix}d-1 \\ i\end{bmatrix} \right) q^{\binom{i-1}{2}+ (i-1)e} & \mbox{If } \pi \in \mathcal{L}\\ x \begin{bmatrix} d-1 \\ i-1\end{bmatrix} q^{\binom{i-1}{2}+(i-1)e} & \mbox{If }\pi \notin \mathcal{L}.
\end{matrix}\right.
\end{align}
If this propery holds for a polar space $\mathcal{P}$ and an integer $i$ such that
\begin{itemize}
\item $i$ is odd for $\mathcal{P}=Q^+(2d-1,q)$,
\item $i\neq d$ for $\mathcal{P}=Q(2d,q)$ or $\mathcal{P}=W(2d-1,q)$ both with $d$ odd or
\item $i$ is arbitrary otherwise,
\end{itemize}
then $\mathcal{L}$ is a degree one Cameron-Liebler set with parameter $x$.
\end{theorem}
\begin{proof}
Consider first a degree one Cameron-Liebler set $\mathcal{L}$ of generators in the polar space $\mathcal{P}$ with characteristic vector $\chi$. As $\chi \in V_0\perp V_1$, we have $\chi = v+a\textbf{\textit{j}}$ for some $v\in V_1$ and some $a\in \mathbb{R}$. Since $|\mathcal{L}| = \langle j,\chi \rangle = x\prod_{i=0}^{d-2}(q^{i+e}+1)$, we find that $a=\frac{x}{q^{d+e-1}+1}$, hence $\chi = \frac{x}{q^{d+e-1}+1}\textbf{\textit{j}}+v$.
Recall that the matrix $A_i$ is the incidence matrix of the relation $R_i$, which describes whether the dimension of the intersection of two generators equals $d-i-1$ or not. This implies that the vector $A_i \chi$, on the position corresponding to a generator $\pi$, gives the number of generators in $\mathcal{L}$, meeting $\pi$ in a $(d-i-1)$-space. We have
\begin{align*}
A_i \chi =& A_iv+\frac{x}{q^{d+e-1}+1}A_i \textbf{\textit{j}}= P_{1i}v+\frac{x}{q^{d+e-1}+1}P_{0i}\textbf{\textit{j}} \\
=& \left( \begin{bmatrix}d-1 \\i \end{bmatrix}q^{\binom{i}{2}+ei}-\begin{bmatrix}d-1 \\ i-1\end{bmatrix}q^{\binom{i-1}{2}+e(i-1)} \right) v+ \frac{x}{q^{d+e-1}+1}\begin{bmatrix}d \\ i\end{bmatrix}q^{\binom{i}{2}+ei} \textbf{\textit{j}}\\
=& \left( \begin{bmatrix}d-1 \\i \end{bmatrix}q^{\binom{i}{2}+ei}-\begin{bmatrix}d-1 \\ i-1\end{bmatrix}q^{\binom{i-1}{2}+e(i-1)} \right) \left(\chi - \frac{x}{q^{d+e-1}+1}\textbf{\textit{j}} \right)+ \frac{x}{q^{d+e-1}+1}\begin{bmatrix}d \\ i\end{bmatrix}q^{\binom{i}{2}+ei} \textbf{\textit{j}}\\
=&\frac{xq^{\binom{i-1}{2}+e(i-1)}}{q^{d+e-1}+1}\left(\begin{bmatrix}d-1 \\i-1 \end{bmatrix}-\begin{bmatrix}d-1 \\i \end{bmatrix}q^{i+e-1}+\begin{bmatrix}d \\i \end{bmatrix}q^{i+e-1} \right)\textbf{\textit{j}} \\ & + q^{\binom{i-1}{2}+e(i-1)}\left(\begin{bmatrix}d-1 \\i \end{bmatrix}q^{i+e-1}-\begin{bmatrix}d-1 \\i-1 \end{bmatrix} \right) \chi \\
=& q^{\binom{i-1}{2}+e(i-1)} \left(x\begin{bmatrix}d-1 \\i-1 \end{bmatrix}\textbf{\textit{j}} + \left(\begin{bmatrix}d-1 \\i \end{bmatrix}q^{i+e-1} -\begin{bmatrix}d-1 \\i-1 \end{bmatrix}\right) \chi \right),
\end{align*}
which proves the first implication.\\
For the proof of the other implication, suppose that $\mathcal{L}$ is a set of generators in $\mathcal{P}$ with the property described in the statement of the theorem. We apply Theorem \ref{stellingalgemeen} with $\Omega$ the set of all generators in $\mathcal{P}$, $R_i$ the relation $\{(\pi, \pi')|\dim(\pi \cap\pi') = d-i-1\}$, and
\begin{align*}
\alpha_i &= \left( (x-1) \begin{bmatrix}d-1 \\i-1\end{bmatrix} +q^{i+e-1}\begin{bmatrix}d-1 \\ i\end{bmatrix} \right) q^{\binom{i-1}{2}+ (i-1)e},\\
\beta_i &= x \begin{bmatrix} d-1 \\ i-1\end{bmatrix} q^{\binom{i-1}{2}+(i-1)e}.\end{align*}
As $\alpha_i - \beta_i = P_{1i}$, we find that $v_i = \chi + \frac{\beta_i}{P_{1i} - P_{0i}}\textbf{\textit{j}} \in V_1$, for the admissible values of $i$, by Lemma \ref{lemma2}. Hence by Definition \ref{defspecialCL}, $\mathcal{L}$ is a degree one Cameron-Liebler set in $\mathcal{P}$.
\end{proof}\\
Remark that this definition is also a new equivalent definition for Cameron-Liebler sets of generators in polar spaces of type $I$, as for these polar spaces, degree one Cameron-Liebler sets and Cameron-Liebler sets coincide.
In the following lemma, we give some properties of degree one Cameron-Liebler sets in a polar space.
\begin{lemma}\label{lemmapropdegree oneCL}
Let $\mathcal{L}$ be a degree one Cameron-Liebler set of generators in a polar space $\mathcal{P}$ and let $\chi$ be the characteristic vector of $\mathcal{L}$. Denote $\frac{|\mathcal{L}|}{\prod_{i=0}^{d-2}(q^{e+i}+1)}$ again by $x$. Then $\mathcal{L}$ has the following properties:
\begin{enumerate}
\item $\chi = \frac{x}{q^{d+e-1}+1}\textbf{\textit{j}} +v $ with $v \in V_1$,
\item $\chi - \frac{x}{q^{d+e-1}+1}\textbf{\textit{j}}$ is an eigenvector with eigenvalue $P_{1i}$ for all adjacency matrices $A_i$ in the association scheme,
\item if $\mathcal{P}$ admits a spread, then $|\mathcal{L}\cap S|=x$ for every spread $\mathcal{S}$ of $\mathcal{P}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first property follows from the first part of the proof of Theorem \ref{stelling}. The second property follows from the first property since $\chi - \frac{1}{q^{d+e-1}+1}\textbf{\textit{j}}\in V_1$.
Consider now a spread $S$ in $\mathcal{P}$ with characteristic vector $\chi_S$ and let $A$ be the point-generator incidence matrix of $\mathcal{P}$. Since $\chi\in \im(A^T) = \ker(A)^\perp$ and by \cite[Lemma 3.6(i), $m=1$]{CLpolar}, which gives that $u=\chi_S-\frac{1}{\prod_{i=0}^{d-2}(q^{e+i}+1)}\textbf{\textit{j}}\in \ker(A)$, we find, by taking the inner product of $u$ and $\chi$, that
\begin{align*}
|\mathcal{L}\cap S| = \langle \chi_S, \chi \rangle = \frac{1}{\prod_{i=0}^{d-2}(q^{e+i}+1)}\langle \textbf{\textit{j}},\chi \rangle = \frac{1}{\prod_{i=0}^{d-2}(q^{e+i}+1)} |\mathcal{L}|= x.
\end{align*}\qedhere \end{proof}
We also give some properties of degree one Cameron-Liebler sets of generators in polar spaces that can easily be proved.
\begin{lemma} \label{basislemma4}
Let $\mathcal{L}$ and $\mathcal{L}'$ be two degree one Cameron-Liebler sets of generators in a polar space $\mathcal{P}$ with parameters $x$ and $x'$ respectively, then the following statements are valid.
\begin{enumerate}
\item $0 \leq x,x' \leq q^{d-1+e}+1$.
\item $|\mathcal{L}|=x\prod_{i=0}^{d-2}(q^{i+e}+1)$.
\item The set of all generators in the polar space $\mathcal{P}$ not in $\mathcal{L}$ is a degree one Cameron-Liebler set of generators in $\mathcal{P}$ with parameter $q^{d-1+e}+1-x$.
\item If $\mathcal{L} \cap \mathcal{L}' = \emptyset$ then $\mathcal{L} \cup \mathcal{L}'$ is a degree one Cameron-Liebler set of generators in $\mathcal{P}$ with parameter $x+x'$.
\item If $\mathcal{L} \subseteq \mathcal{L}'$ then $\mathcal{L} \setminus \mathcal{L}'$ is a degree one Cameron-Liebler set of generators in $\mathcal{P}$ with parameter $x-x'$.
\end{enumerate}
\end{lemma}
\begin{lemma}[{\cite[Lemma 2.3]{Ferdinand.}}]\label{lemmaferdi}
Let $\mathcal{P}$ be a polar space of rank $d$ and let $\mathcal{P'}$ be a polar space, embedded in $\mathcal{P}$ with the same rank $d$. If $\mathcal{L}$ is a degree one Cameron-Liebler set in $\mathcal{P}$, then the restriction of $\mathcal{L}$ to $\mathcal{P'}$ is again a degree one Cameron-Liebler set.
\end{lemma}
Note that Theorem \ref{stelling} does not hold for some values of $i$, dependent on the polar space $\mathcal{P}$, since for these cases, we cannot apply Lemma \ref{lemma2}. We will now show that there are examples of generator sets that admit the property of Theorem \ref{stelling} for the non-admitted values of $i$, but that are not degree one Cameron-Liebler sets. These are however Cameron-Liebler sets in the sense of \cite{CLpolar}.
\begin{remark}\label{commentvb4.6}
By investigating \cite[Example $4.6$]{CLpolar}, we find an example of a Cameron-Liebler set in a polar space of type $III$ with $d=3$, that is not a degree one Cameron-Liebler set: a base-plane. A \emph{base-plane} in a polar space $\mathcal{P}$ of rank $3$ with base the plane $\pi$ is the set of all planes in $\mathcal{P}$, intersecting $\pi$ in at least a line.
Let $\mathcal{P}$ be a polar space of type $III$ of rank $3$, so $\mathcal{P}=W(5,q)$ or $\mathcal{P}=Q(6,q)$. Let $\pi$ be a plane and let $\mathcal{L}$ be the base-plane with base $\pi$. This set $\mathcal{L}$ is a Cameron-Liebler set in $\mathcal{P}$, but not a degree one Cameron-Liebler set. This follows from Theorem \ref{stelling} with $i=1$: The number of generators of $\mathcal{L}$, meeting a plane $\alpha$ of $\mathcal{L}$ in a line depends on whether $\alpha$ equals $\pi$ or not. As those two numbers, for $\alpha = \pi$ and $\alpha \neq \pi$ are different, the property in Theorem \ref{stelling} does not hold. This implies that the set $\mathcal{L}$ is no degree one Cameron-Liebler set. By similar arguments, we can also use Theorem \ref{stelling} with $i=2$, to show that a base-plane is not a degree one Cameron-Liebler set. However, the equalities for $i=3$ in Theorem \ref{stelling} hold.
\end{remark}
\begin{remark}\label{vb1}
A hyperbolic class is the set of all generators of one class of a hyperbolic quadric $Q^+(4n+1,q)$ embedded in a polar space $\mathcal{P}$ with $\mathcal{P}=Q(4n+2,q)$ or $\mathcal{P}=W(4n+1,q)$, $q$ even. We know that this set is a Cameron-Liebler set, see \cite[Remark 3.25]{CLpolar}, but we can prove that this set is not a degree one Cameron-Liebler set, by considering $\im(B^T)$, where $B$ is the incidence matrix of hyperbolic classes and generators. Every hyperbolic class corresponds to a row in the matrix $B$. If the characteristic vectors of all hyperbolic classes would lie in $V_0 \perp V_1$, then $\im(B^T)\subseteq V_0\perp V_1$. This gives a contradiction since $\im(B^T)= V_0\perp V_1\perp V_d$ by \cite[Lemma 3.26]{CLpolar}.\\
Remark that for the polar spaces $W(4n+1,q)$, $q$ odd, we do not have this example as there is no hyperbolic quadric $Q^+(4n+1,q)$ embedded in these symplectic polar spaces.
\end{remark}
In the previous remark we found that one class of a hyperbolic quadric $Q^+(4n+1,q)$ embedded in a $Q(4n+2,q)$ or $W(4n+1,q)$, $q$ even is no degree one Cameron-Liebler set. In the next example we show that an embedded hyperbolic quadric (so both classes) is a degree one Cameron-Liebler set in the polar spaces $Q(4n+2,q)$ and $W(4n+1,q)$, $q$ even.
\begin{example}[{\cite[Example $4.4$]{CLpolar}}] \label{vb}
Consider a polar space $\mathcal{P}$ as in Remark \ref{vb1}.
By Lemma \ref{lemmaferdi} we know
that this set of generators is a degree one Cameron-Liebler set, hence also a Cameron-Liebler set.
\end{example}
\begin{table}[h]\begin{center}
\begin{tabular}{ | l |c|c| }
\hline
Example & CL & degree one CL \\ \hline \hline
All generators of $\mathcal{P}$ &$\times$& $\times$ \\ \hline
Point-pencil (defined in Section \ref{setcionfcps} ). & $\times$& $\times$ \\ \hline
Base-plane (defined in Example \ref{commentvb4.6}). & $\times$ & \\ \hline
Hyperbolic class (defined in Comment \ref{vb1}). & $\times$& \\ \hline
Embedded hyperbolic quadric (defined in Example \ref{vb}). &$\times$& $\times$ \\ \hline
\end{tabular}
\caption{Examples of Cameron-Liebler and degree one Cameron-Liebler sets.}\label{tabelexample}
\end{center}\end{table}
Remark by Lemma \ref{basislemma4}(3), that also the complements of the sets in Table \ref{tabelexample} are Cameron-Liebler sets or degree one Cameron-Liebler sets respectively.
\section{Polar spaces $Q^+(2d-1,q)$, $d$ even}
In the previous section we introduced degree one Cameron-Liebler sets while in this section we handle Cameron-Liebler sets in polar spaces. We focus on Cameron-Liebler in one class of generators in the polar spaces $Q^+(2d-1,q)$, $d$ even. These Cameron-Liebler sets were introduced in \cite[Section 3]{CLpolar} and are defined in only one class of generators, in contrast to the (degree one) Cameron-Liebler sets in other polar spaces.
It is known that the generators of a hyperbolic quadric $Q^+(2d-1,q)$ can be divided in two classes such that for any two generators $\pi$ and $\pi'$ we have $\dim(\pi\cap\pi')\equiv d\pmod{2}$ iff $\pi$ and $\pi'$ belong to the same class. By restricting the classical association scheme of the hyperbolic quadric $Q^+(2d-1,q)$ to the even relations, we define an association scheme for one class of generators. For more information, see \cite[Remark $2.18$, Lemma $3.12$]{CLpolar}.
Let $R'_i$ and $A'_i$ be $R_{2i}$ and $A_{2i}$ respectively, restricted to the rows and columns corresponding to the generators of this class. Let $V'_j$ be $ V_j \perp V_{d-j}$, also restricted to the subspace corresponding to these generators.
For the polar spaces $Q^+(2d-1,q)$, $d$ even, we thus have the relations $R_i'$, $i = 0, \dots ,\frac{d}{2}$, and the eigenspaces $V_j'$, $j = 0, \dots, \frac{d}{2}$. For this association scheme on one class of generators we give the analogue of Lemma \ref{lemma2}.
\begin{lemma} \label{lemma4}
The eigenvalue $P_{1,2i}$ of $A_i' = A_{2i}$ corresponds only with the eigenspace $V_1' = V_1 \perp V_{d-1}$ for the classical polar spaces $Q^+(2d-1,q)$, $d$ even.
\end{lemma}
\begin{proof}
This lemma follows from Lemma \ref{lemma2} as for the hyperbolic quadrics $Q^+(2d-1,q)$ we found that $P_{1k}=P_{d-1,k}$ for $j$ even. This implies that the eigenvalue $P_{1,2i}$ corresponds with $V_1 \perp V_{d-1}$.
\end{proof}
Here again, we find a new equivalent definition.
\begin{theorem} \label{stelling3}
Let $\mathcal{G}$ be a class of generators of the hyperbolic quadric $Q^+(2d-1,q)$ of even rank $d$ and let $\mathcal{L}$ be a set of generators of $\mathcal{G}$. The set $\mathcal{L}$ is a Cameron-Liebler set of generators in $\mathcal{G}$ if and only if for every generator $\pi$ in $\mathcal{G}$, the number of elements of $\mathcal{L}$ meeting $\pi$ in a $(d-2i-1)$-space equals \\
\begin{align*}
\left\{ \begin{matrix}
\left( (x-1) \begin{bmatrix}d-1 \\2i-1\end{bmatrix} +q^{2i-1}\begin{bmatrix}d-1 \\ 2i\end{bmatrix} \right) q^{(2i-1)(i-1)} & \mbox{If } \pi \in \mathcal{L}\\ x \begin{bmatrix} d-1 \\ 2i-1\end{bmatrix} q^{(2i-1)(i-1)} & \mbox{If }\pi \notin \mathcal{L}.
\end{matrix}\right.
\end{align*}
\end{theorem}
\begin{proof}
Let $\mathcal{L}$ be a set of generators in $\mathcal{G}$ with the property described in the theorem, then the first implication is a direct application of Theorem \ref{stellingalgemeen} with $\Omega$ the set of all generators in $\mathcal{G}$, $R_i$ the relation $R'_i=\{(\pi, \pi')|\dim(\pi \cap\pi') = d-2i-1\}$, and \begin{align*}
\alpha_i &= \left( (x-1) \begin{bmatrix}d-1 \\2i-1\end{bmatrix} +q^{2i-1}\begin{bmatrix}d-1 \\ 2i\end{bmatrix} \right) q^{(2i-1)(i-1)},\\
\beta_i &= x \begin{bmatrix} d-1 \\ 2i-1\end{bmatrix} q^{(2i-1)(i-1)}.\end{align*}
As $\alpha_i - \beta_i = P_{1,2i}$, we find that $v_i = \chi + \frac{\beta_i}{P_{1,2i} - P_{0,2i}}\textbf{\textit{j}} \in V'_1$, hence $\chi \in V'_0 \perp V'_1 $ and by Lemma $3.15$ in \cite{CLpolar} we know that $\chi \in$ $\im(A^T)$.
Now it follows from \cite[Definition 3.16(iv)]{CLpolar} that $\mathcal{L}$ is a (degree one) Cameron-Liebler set of $\mathcal{G}$. The other implication is \cite[Lemma $4.10$]{CLpolar}.\end{proof}\\
\section{Classification results}
We try to use the ideas from the classification results for Cameron-Liebler sets of polar spaces of type $I$ and the polar spaces $Q^+(2d-1,q)$, $d$ even, in \cite[Section 6]{CLpolar}, to find classification results for degree one Cameron-Liebler sets in polar spaces.\\
We start with some definitions and a lemma that proves that the parameter $x$ is always an integer.
\begin{definition}
A \emph{partial ovoid} is a set of points in a polar space such that each generator contains at most one
point of this set. It is called an \emph{ovoid} if each generator contains precisely one point of the set.
\end{definition}
\begin{definition}
An Erd\H{o}s-Ko-Rado (EKR) set of $k$-spaces is a set of k-spaces which
are pairwise not disjoint.
\end{definition}
\begin{lemma}
If $\mathcal{L}$ is a degree one Cameron-Liebler set in a polar space $\mathcal{P}$ with parameter $x$, then $x\in \mathbb{N}$
\end{lemma}
\begin{proof}
For all polar spaces, except the hyperbolic quadrics $Q^+(2d-1,q)$, $d$ even, we refer to \cite[Lemma 4.8]{CLpolar}.
Suppose now that $\mathcal{L}$ is a degree one Cameron-Liebler set in $\mathcal{P}=Q^+(2d-1,q)$, $d$ even, with parameter $x$. Then $\mathcal{L}$ is also a Cameron-Liebler set in $\mathcal{P}$ with parameter $x$. If $\Omega_1$ and $\Omega_2$ are the two classes of generators in $\mathcal{P}$, then $\mathcal{L}\cap \Omega_1$ and $\mathcal{L}\cap \Omega_2$ are Cameron-Liebler sets of $\Omega_1$ and $\Omega_2$ with parameter $x$, by \cite[Theorem 3.20]{CLpolar}. Hence $x$ is the parameter of a Cameron-Liebler set in one class of generators of $Q^+(2d-1,q)$, $d$ even. This implies, by \cite[Lemma 4.8]{CLpolar}, that $x\in \mathbb{N}$. \end{proof}\\
Now we continue with a classification result for degree one Cameron-Liebler sets with parameter $1$ in all polar spaces.
\begin{theorem}
A degree one Cameron-Liebler set in a polar space $\mathcal{P}$ of rank $d$ with parameter $1$ is a point-pencil.
\end{theorem}
\begin{proof}
For the polar spaces of type $I$ and $III$, the theorem follows from \cite[Theorem 6.4]{CLpolar} as any degree one Cameron-Liebler set is a Cameron-Liebler set and since a base-plane, a base-solid and a hyperbolic class, are no degree one Cameron-Liebler sets (see Remark \ref{commentvb4.6} and Remark \ref{vb1}).
The theorem for the polar spaces of type $I$ follows from \cite[Theorem 6.4]{CLpolar} as degree one Cameron-Liebler sets and Cameron-Liebler sets coincide for these polar spaces.
Let $\mathcal{L}$ be a degree one Cameron-Liebler set with parameter $1$ in a polar space $\mathcal{P}$ and remark that $\mathcal{L}$ is an EKR set with size $\prod_{i=0}^{d-2}(q^{i+e}+1)$. For the polar spaces $Q(4n+2,q)$ and $W(4n+1,q)$, we find by \cite[Theorem 23, Theorem 40]{pepe} that $\mathcal{L}$ is a point-pencil, a hyperbolic class or a base-plane if $n=1$. Using Remark \ref{vb1} and Remark \ref{commentvb4.6}, we find that the last two possibilities are no degree one Cameron-Liebler sets. Hence $\mathcal{L}$ is a point-pencil.
Suppose now that $\mathcal{P}$ is the hyperbolic quadric $Q^+(4n-1,q)$ with $\Omega_1$ and $\Omega_2$ the two classes of generators. By \cite[Theorem 3.20]{CLpolar}, we know that $\mathcal{L}\cap \Omega_1$ and $\mathcal{L}\cap \Omega_2$ are Cameron-Liebler sets in $\Omega_1, \Omega_2$ respectively, with parameter $1$. Using \cite[Theorem 6.4]{CLpolar} we see that $\mathcal{L}\cap \Omega_i$ is a point-pencil or a base-solid if $n=2$ for $i=1,2$. A base-solid is the set of all $3$-spaces intersecting a fixed 3-space (the base) in precisely a plane. Note that all elements of the base-solid belong to a different class of the hyperbolic quadric than the base itself.
If $n=2$, so $d=4$, and $\mathcal{L}\cap \Omega_1$ or $\mathcal{L}\cap \Omega_2$ is a base-solid with base $\pi$, then there are at least $(q+1)(q^2+1)$ elements of $\mathcal{L}$ meeting $\pi$ in a plane. This contradicts Theorem \ref{stelling}, whether $\pi\in \mathcal{L}$ or not. So we find that $\mathcal{L}\cap \Omega_1$ and $\mathcal{L}\cap \Omega_2$ are both point-pencils with vertex $v_1$ and $v_2$ respectively. Now we show that $v_1=v_2$. Suppose $v_1\neq v_2$. Consider a generator $\alpha \in \Omega_2\setminus \mathcal{L}$ through $v_1$. Then $\alpha$ intersects $q^2+q+1$ generators of $\mathcal{L}\cap \Omega_1$ in a plane through $v_1$. This gives a contradiction with Theorem \ref{stelling}, which proves that $v_1=v_2$. Hence $\mathcal{L}$ is a point-pencil through $v_1=v_2$. \end{proof} \\
The classification result in \cite[Theorem 6.7]{CLpolar} for polar spaces of type $I$ is also valid for degree one Cameron-Liebler sets in all polar spaces.
\begin{theorem}
Let $\mathcal{P}$ be a finite classical polar space of rank $d$ and parameter $e$, and let $\mathcal{L}$ be a degree one Cameron-Liebler set of $\mathcal{P}$ with parameter $x$. If $x\leq q^{e-1}+1$ then $\mathcal{L}$ is the union of $x$ point-pencils whose vertices are pairwise non-collinear or $x=q^{e-1}+1$ and $\mathcal{L}$ is the set of generators in an embedded polar space of rank $d$ and with parameter $e-1$.
\end{theorem}
\begin{proof}
In Lemma $6.5$, Theorem $6.6$ and Theorem $6.7$ of \cite{CLpolar}, the authors use \cite[Lemma $4.9$]{CLpolar} to prove the classification result. We can use the same proof as we can use Theorem \ref{stelling} instead of \cite[Lemma $4.9$]{CLpolar}. \end{proof}\\
Remark that the last possibility corresponds to an embedded hyperbolic quadric $Q^+(4n+1,q)$ if $\mathcal{P}=Q(4n+2,q)$ or $\mathcal{P}=W(4n+1,q)$ with $q$ even. If $\mathcal{P}=W(4n+1,q)$ with $q$ odd, then $\mathcal{P}$ admits no embedded polar space with rank $n$ and parameter $e-1=0$.\\
For the symplectic polar space $W(5,q)$ and the parabolic quadric $Q(6,q)$ we give a stronger classification result.
Remark the polar spaces $Q(6,q)$ and $W(5,q)$ are isomorphic for $q$ even.
We find $W(5,q)$, for $q$ even, by a projection of $Q(6,q)$ from the nucleus $N$ of $Q(6,q)$ to a hyperplane not through $N$ in the ambient projective space $\PG(6,q)$. In this way, there is a one-one connection between the planes of $W(5,q)$ and the planes of $Q(6,q)$.
We start with some lemmas.
\begin{lemma} \label{lemmas1s2d1d2}
Let $\mathcal{L}$ be a degree one Cameron-Liebler set of generators (planes) in $W(5,q)$ or $Q(6,q)$ with parameter $x$.
\begin{enumerate}
\item For every $\pi \in \mathcal{L}$, there are $s_1$ elements of $\mathcal{L}$ meeting $\pi$.
\item For skew $\pi, \pi'\in \mathcal{L}$, there exist exactly $d_2$ subspaces in $\mathcal{L}$ that are skew to both $\pi$ and $\pi'$ and there exist $s_2$ subspaces in $\mathcal{L}$ that meet both $\pi$ and $\pi'$.
Here, $d_2$, $s_1$ and $s_2$ are given by:
\begin{align*}
d_2(q,x) &= (x-2)q^2(q-1)\\
s_1(q,x) &= x(q^2+1)(q+1)-(x-1)q^3 = q^3+x(q^2+q+1)\\
s_2(q,x) &= x(q^2+1)(q+1)-2(x-1)q^3 +d_2(q,x).
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}Let $\mathcal{P}$ be the polar space $W(5,q)$ or $Q(6,q)$, hence $d=3$ and $e=1$.
\begin{enumerate}
\item This follows directly from Theorem \ref{stelling}, for $i=d$ and $|\mathcal{L}|=x(q^2+1)(q+1)$.
\item Let
$\chi_\pi$ and $\chi_{\pi'}$ be the characteristic vectors of $\{\pi\}$ and $\{\pi'\}$, respectively. Let $\mathcal{Z}$ be the set of all planes in $\mathcal{P}$ disjoint to $\pi$ and $\pi'$, and let $\chi_\mathcal{Z}$ be its characteristic vector. Furthermore, let $v_\pi$ and $v_{\pi'}$ be the incidence vectors of $\pi$ and $\pi'$, respectively, with their positions corresponding to the points of $\mathcal{P}$. Note that $A\chi_\pi = v_\pi$ and $A\chi_{\pi'} = v_{\pi'}$. \\
The number of planes through a point $P\notin \pi \cup \pi'$ and disjoint to $\pi$ and $\pi'$ is the number of lines in $P^\perp$, disjoint to the lines corresponding to $\pi$ and $\pi'$. By \cite[Corollary $19$]{kms} this number equals $q^2(q-1)$, and we find:
\begin{align*}
A\chi_\mathcal{Z} &=q^2(q-1)(\textbf{\textit{j}}-v_\pi-v_{\pi'}) \\
&=q^2(q-1)\left(A\frac{\textbf{\textit{j}}}{(q^2+1)(q+1)}-A\chi_\pi-A\chi_{\pi'}\right)\\
\Leftrightarrow\qquad&\chi_\mathcal{Z}-q^2(q-1)\left(\frac{\textbf{\textit{j}}}{(q^2+1)(q+1)}-\chi_\pi-\chi_{\pi'}\right) \in \ker(A).
\end{align*}
We know that the characteristic vector $\chi$ of $\mathcal{L}$ is included in $\ker(A)^\perp$. This implies:
\begin{align*}
&&\chi_\mathcal{Z} \cdot \chi &=q^2(q-1)\left(\frac{\textbf{\textit{j}}\cdot \chi}{(q^2+1)(q+1)}-\chi(\pi)-\chi(\pi')\right) \\
&\Leftrightarrow & |\mathcal{Z}\cap \mathcal{L}| &=(x-2)q^2(q-1)
\end{align*}
which gives the formula for $d_2(q,x)$. The formula for $s_2(q,x)$ follows from the inclusion-exclusion principle. \qedhere
\end{enumerate} \end{proof}
In the following lemma, corollary and theorem, we will use $s_1,s_2,d_1,d_2$ for the values $s_1(q,x),s_2(q,x),$ $d_1(q,x), d_2(q,x)$ if the field size $q$ and the parameter $x$ are clear from the context. For the definition of these values, we refer to the previous lemma.
The following lemma is a generalization of Lemma $2.4$ in \cite{Klaus}.
\begin{lemma}\label{lemmaklaus}
If $c$ is a nonnegative integer such that
\begin{align*}
(c+1)s_1-\binom{c+1}{2}s_2 > x(q^2+1)(q+1)\;,
\end{align*}
then no degree one Cameron-Liebler set of generators in $W(5,q)$ or $Q(6,q)$ with parameter $x$ contains $c+1$ mutually skew generators.
\end{lemma}
\begin{proof}
Let $\mathcal{P}$ be the polar space $W(5,q)$ or $Q(6,q)$ and assume that $\mathcal{P}$ has a degree one Cameron-Liebler set $\mathcal{L}$ of generators with parameter $x$ that contains $c+1$ mutually disjoint subspaces $\pi_0,\pi_1,\dots,\pi_c$. Lemma \ref{lemmas1s2d1d2} shows that $\pi_i$, meets at least $s_1(q,x)-i\cdot s_2(q,x)$ elements of $\mathcal{L}$ that are skew to $\pi_0, \pi_1, \dots,\pi_{i-1}$. Hence $x(q^2+1)(q+1) = |\mathcal{L}| \geq (c+1) s_1-\sum_{i=0}^c i s_2$ which contradicts the assumption.
\end{proof}
\begin{gevolg}\label{gevolgklaus}
A degree one Cameron-Liebler set of generators in $W(5,q)$ or $Q(6,q)$ with parameter $2\leq x\leq \sqrt[3]{2q^2}-\frac{\sqrt[3]{4q}}{3}+\frac{1}{6}$ contains at most $x$ pairwise disjoint generators.
\end{gevolg}
\begin{proof}
Let $\mathcal{L}$ be a degree one Cameron-Liebler set of generators in $W(5,q)$ or $Q(6,q)$ with parameter $x$. Using Lemma \ref{lemmaklaus} for $e=1,d=3, c=x$ we find that if $q^3-q^2x+\frac{q+1}{2}x^2-\frac{q+1}{2}x^3>0$, then $\mathcal{L}$ contains at most $x$ pairwise disjoint generators. Since $f_q(x)=q^3-q^2x-\frac{q+1}{2}x^2(x-1)$ is decreasing on $[1,+\infty[$, we find that it is sufficient that $f_q\left(\sqrt[3]{2q^2}-\frac{\sqrt[3]{4q}}{3}+\frac{1}{6}\right)>0$, as we only consider the values of $x$ in $[2,\dots,\sqrt[3]{2q^2}-\frac{\sqrt[3]{4q}}{3}+\frac{1}{6}]$. Using a computer algebra packet, it can be checked that $f_q\left(\sqrt[3]{2q^2}-\frac{\sqrt[3]{4q}}{3}+\frac{1}{6}\right)>0$ for all $q\geq 2$.
\end{proof}
\begin{theorem}
A degree one Cameron-Liebler set $\mathcal{L}$ of generators in $W(5,q)$ or $Q(6,q)$ with parameter $2\leq x\leq \sqrt[3]{2q^2}-\frac{\sqrt[3]{4q}}{3}+\frac{1}{6}$ is the union of $\alpha$ embedded disjoint hyperbolic quadrics $Q^+(5,q)$ and $x-2\alpha$ point-pencils whose vertices are pairwise disjoint and not contained in the $\alpha$ hyperbolic quadrics $Q^+(5,q)$. For the polar space $Q(6,q)$ or $W(5,q)$ with $q$ even, $\alpha\in \{0,...,\lfloor \frac{x}{2} \rfloor\}$, for the polar space $W(5,q)$ with $q$ odd, $\alpha=0$.
\end{theorem}
\begin{proof}
Let $\mathcal{P}$ be the polar space $W(5,q)$ or $Q(6,q)$ and $\mathcal{L}$ be a degree one Cameron-Liebler set in $\mathcal{P}$. Note that the generators in these polar spaces are planes. By Corollary \ref{gevolgklaus}, there are $c$ pairwise disjoint planes $\pi_1,\pi_2,\dots, \pi_c$ with $c\leq x$ in $\mathcal{L}$. Let $S_i$ be the set of planes in $\mathcal{L}$ intersecting $\pi_i$ and not intersecting $\pi_j$ for all $j\neq i$. By Lemma \ref{lemmas1s2d1d2} there are, for a fixed $i$, at least $s_1-(c-1)s_2\geq s_1-(x-1)s_2 =q^3-(x-2)q^2-(x^2-2x)(q+1)$ planes in $S_i$. As $S_i$ is an EKR set by Corollary \ref{gevolgklaus}, $S_i$ has to be a part of a point-pencil (PP), a base plane (BP) or one class of an embedded hyperbolic quadric $Q^+(5,q)$ (CEHQ). Remark that if $\mathcal{P}$ is $W(5,q)$ with $q$ odd, then $\mathcal{P}$ cannot contain a CEHQ, so for this polar space, the only possibilities are a PP of BP,
by \cite[Lemmas $3.3.7$, $3.3.8$ and $3.3.16$]{phdmaarten}. Using Theorem \ref{stelling}, we can prove that if the set $S_i$ is a part of a PP, BP or CEHQ, then $\mathcal{L}$ has to contain all planes of this PP, BP or CEHQ. We show this for the case where the set of planes forms a part of a PP. So assume $S_i$ is a subset of the point-pencil with vertex $P$, and there is a plane $\gamma \notin \mathcal{L}$ through $P$. This would imply that $\gamma$ meets at least $q^3-(x-2)q^2-(x^2-2x)(q+1)$ planes in $\mathcal{L}$ non-trivially. This gives a contradiction by Theorem \ref{stelling} for $i=1$ and $i=2$, as $\gamma \notin \mathcal{L}$ intersects with precisely $x(q^2+q+1)<q^3-(x-2)q^2-(x^2-2x)(q+1)$ planes of $\mathcal{L}$.\\
This argument also works for the BP and CEHQ, so we can conclude that if $\mathcal{L}$ contains an $S_i$ which is a part of a PP, BP or CEHQ, then $\mathcal{L}$ has to contain the whole PP, BP or CEHQ respectively, which we will call $\mathcal{L}_i$. \\
Remark first that $\mathcal{L}$ cannot contain a BP with base $\pi$ as then $\pi \in \mathcal{L}$ intersects $q^3+q^2+q>q^2+q+x-1$ planes of $\mathcal{L}$ in a line, which gives a contradiction with Theorem \ref{stelling}. This implies that all sets $\mathcal{L}_i$ are PP's or CEHQ's. Now we show that every two sets $\mathcal{L}_i$ and $\mathcal{L}_j$ are disjoint. Suppose first that $\mathcal{L}_i$ and $\mathcal{L}_j$ are two PP's with vertices $P_i$ and $P_j$ respectively, that are not disjoint. Then there are at most $q+1$ planes in $\mathcal{L}_i \cap \mathcal{L}_j$ and let $\alpha$ be one of them. Now we see that $\alpha$ meets at least $2(q^3+q^2+q+1)-(q+1)$ elements of $\mathcal{L}$, contradicting Theorem \ref{stelling}.
If $\mathcal{L}_i$ and $\mathcal{L}_j$ are two non-disjoint CEHQ's or a CEHQ and a PP that are non-disjoint, then we can use the same arguments as above: In both cases, there are at most $q+1$ planes in $\mathcal{L}_i \cap \mathcal{L}_j$, which implies that a plane $\alpha \in \mathcal{L}_i \cap \mathcal{L}_j$ meets at least $2(q^3+q^2+q+1)-(q+1)$ elements of $\mathcal{L}$ non-trivially, contradicting Theorem \ref{stelling}.
Now we know that $\mathcal{L}$ contains the disjoint union of $c\leq x$ sets $\mathcal{L}_i$ of planes, where every set is a PP or CEHQ. As the number of planes in a PP or CEHQ equals $(q^2+1)(q+1)$, and the total number of planes in $\mathcal{L}$ equals $x(q^2+1)(q+1)$ (see Lemma \ref{basislemma4}(2)), we see that $\mathcal{L}$ equals the disjoint union of $x$ sets $\mathcal{L}_i$. \\
To end this proof, we want to show that the only possible composition of $\mathcal{L}$ exists of PP's and embedded hyperbolic quadrics. If $\mathcal{L}$ contains one class of an embedded hyperbolic quadric, then $\mathcal{L}$ also contains the other class of this hyperbolic quadric. This also follows from Theorem \ref{stelling}: suppose $\mathcal{L}$ contains only one class of an embedded hyperbolic quadric and let $\pi$ be a plane of the other class of this embedded hyperbolic quadric. Then we can show that $\pi$ is also a plane of $\mathcal{L}$: we know that $\pi$ meets $q^2+q+1$ planes of the hyperbolic quadric in a line, so at least so many planes of $\mathcal{L}$, in a line. But if $\pi \notin \mathcal{L}$, then by Theorem \ref{stelling}, $\pi$ can only meet $x< \sqrt[3]{2q^2}$ planes of $\mathcal{L}$ in a line, a contradiction.
This implies that $\mathcal{L}$ has to be the disjoint union of point-pencils and embedded hyperbolic quadrics.
Remark that two point-pencils are disjoint if the corresponding vertices are non-collinear. As there exists a partial ovoid of size $q+1$ in $\mathcal{P}$, we can find $x$ pairwise disjoint point-pencils.\\
Note that for $q$ odd and $\mathcal{P} = W(5,q)$, there are no embedded hyperbolic quadrics, so in this case $\mathcal{L}$ is the disjoint union of $x$ point-pencils.
We end the proof by showing that, for $\mathcal{P} = Q(6,q)$ or $\mathcal{P} = W(5,q)$ and $q$ even, there exist disjoint embedded hyperbolic quadrics in $\mathcal{P}$.
It suffices to show this only for $\mathcal{P} = Q(6,q)$, by the connection between $Q(6,q)$ and $W(5,q)$ for $q$ even.
Consider two embedded hyperbolic quadrics $Q^+(5,q)$ in $Q(6,q)$, that intersect in a parabolic quadric $Q(4,q)$. These two hyperbolic quadrics have no planes in common as the generators of $Q(4,q)$ are lines, so these two embedded hyperbolic quadrics are disjoint.
Remark that the disjoint union of point-pencils and the disjoint union of embedded hyperbolic quadrics is a degree one Cameron-Liebler set by Lemma \ref{basislemma4}$(4)$, as a point-pencil is a degree one Cameron-Liebler set and for $\mathcal{P}\neq W(5,q)$ or $q$ even, an embedded hyperbolic quadric of the same rank is also a degree one Cameron-Liebler set.
\end{proof}
This theorem agrees with Conjecture $5.1.3$ in \cite{Ferdinand.}, as this conjecture says that every degree one Cameron-Liebler set in $W(5,q)$ is the disjoint union of non-degenerate hyperplane sections and point-pencils.
\begin{remark}
Recall that the disjoint union of point-pencils and disjoint embedded hyperbolic quadrics is also an example of a degree one Cameron-Liebler set of generators in the other polar spaces of type $III$ (see Lemma \ref{basislemma4} and Example \ref{vb}).
We also remark that we could not generalize this classification result to other classical polar spaces, as for these polar spaces, there is not enough information known about large EKR sets in these polar spaces. For the polar spaces $Q^+(4n+1,q)$ there are some EKR results in \cite{maarten2}. Since in this case, the large examples of EKR sets have much more elements than the largest known Cameron-Liebler sets, we cannot use these results.
\end{remark}
\section*{Summary of properties for type \texorpdfstring{$III$}{III}}
In Table \ref{tabeldef} we give an overview of properties where we distinguish sufficient properties, necessary properties and characteristic properties, or definitions for Cameron-Liebler sets and for degree one Cameron-Liebler sets.
Suppose in this table that $\mathcal{L}$ is a set of generators in the polar space $\mathcal{P}$ of type $III$, with characteristic vector $\chi$. Suppose also that $\pi$ is a generator in $\mathcal{P}$, not necessarily in $\mathcal{L}$.\\
\begin{table}[h]\begin{center}
\begin{tabular}{ | l l |c|c| }
\hline
Property && CL & degree one CL \\ \hline \hline
$\chi \in V_0 \perp V_1$. && $S$& $C$ \\ \hline
$\forall \pi \in \mathcal{P}$: $|\{\tau \in \mathcal{L}| \tau \cap \pi = \emptyset\}| = (x-\chi(\pi))q^{\binom d 2}$. && $C$& $N$ \\ \hline
$\chi-\frac{x}{q^d+1}\textbf{\textit{j}}$ is an eigenvector of $A_d$ with eigenvalue $-q^{\binom d 2}$. && $C$ & $N$\\ \hline
$\forall \pi \in \mathcal{P}, |\{\tau \in \mathcal{L}| \dim(\tau \cap \pi) = d-i-1 \}|=$ (\ref{formulelang}), for $0\leq i <d$ &&$S$& $C$ \\ \hline
If $\mathcal{P}$ admits a spread, then $|\mathcal{L}\cap S| = x$ $\forall$ spread $S\in \mathcal{P}$. && $C$ &$N$ \\ \hline
\end{tabular}
\caption{Overview of the sufficient ($S$), necessary ($N$) and characterising ($C$) properties.}\label{tabeldef}
\end{center}\end{table}
\end{document} |
\begin{document}
\title{Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values}
\begin{abstract}
We introduce adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values. They are two variants of cyclotomic multiple zeta values, closely related to each other. They arise as key tools for the study of $p$-adic cyclotomic multiple zeta values. Moreover, cyclotomic multiple harmonic values provide an adelic lift to a cyclotomic generalization of finite multiple zeta values. We establish certain standard properties of these two objects. We consider two types of properties : some related to double shuffle relations, and some related to associator and Kashiwara-Vergne relations.
This is Part II-1 of \emph{$p$-adic cyclotomic multiple zeta values and $p$-adic pro-unipotent harmonic actions}.
\end{abstract}
\tableofcontents
\section{Introduction}
\subsection{The algebraic theory of cyclotomic multiple zeta values}
Let $d$ be a positive integer, $(n_{i})_{d}=(n_{1},\ldots,n_{d})$ be a sequence of positive integers and $(\xi_{i})_{d}=(\xi_{1},\ldots,\xi_{d})$ be a sequence of roots of unity in $\mathbb{C}$, with $(\xi_{d},n_{d}) \not= (1,1)$. The following complex number is called a cyclotomic multiple zeta value (abbreviated as MZV$\mu_{N}$, where $N$ is a positive integer such that $\xi_{i}^{N}=1$ for all $i$) :
\begin{equation} \label{eq:multizetas} \zeta \big( (n_{i})_{d};(\xi_{i})_{d} \big) = \sum_{0<m_{1}<\ldots<m_{d}} \frac{\big( \frac{\xi_{2}}{\xi_{1}}\big)^{m_{1}} \ldots \big( \frac{1}{\xi_{d}}\big)^{m_{d}}}{m_{1}^{n_{1}} \ldots m_{d}^{n_{d}}} .
\end{equation}
The adjective cyclotomic and the notation $\mu_{N}$ are omitted if $N=1$. One says that $n=n_{1}+\ldots+n_{d}$ is the weight of $\big( (n_{i})_{d};(\xi_{i})_{d} \big)$ and that $d$ is its depth. Denoting by $(\epsilon_{n},\ldots,\epsilon_{1})=(\underbrace{0,\ldots,0}_{n_{d}-1},\xi_{d},\ldots,\underbrace{0,\ldots,0}_{n_{1}-1},
\xi_{1})$, we have
\begin{equation} \label{eq:multizetas integral} \zeta \big( (n_{i})_{d};(\xi_{i})_{d} \big) = (-1)^{d}\int_{0}^{1} \frac{dt_{n}}{t_{n}-\epsilon_{n}} \int_{0}^{t_{n-1}} \ldots \int_{0}^{t_{2}} \frac{dt_{1}}{t_{1}-\epsilon_{1}}.
\end{equation}
\indent This shows that MZV$\mu_{N}$'s are Betti - de Rham periods of the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$ (\cite{Deligne Goncharov}, \S5.16).
\newline\indent Let $p$ be a prime number which does not divide $N$. $p$-adic cyclotomic multiple zeta values ($p$MZV$\mu_{N}$'s) are defined as $p$-adic analogues of the integrals in (\ref{eq:multizetas integral})
\cite{Deligne Goncharov} \cite{Furusho 1} \cite{Furusho 2} \cite{Yamashita} \cite{Unver MZV} \cite{U2}, \cite{I-1}, \cite{I-3}. They are elements $\zeta_{p,\alpha}\big((n_{i})_{d};(\xi_{i})_{d}\big)$ of the field $K_{p}$ generated by $\mathbb{Q}_{p}$ and a primitive $N$-th root of unity, where $\alpha \in \mathbb{Z} \cup \{\pm \infty\} - \{0\}$ is the number of iterations of the Frobenius of the crystalline pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$. They are reductions of $p$-adic periods by \cite{Yamashita}.
\newline\indent In the framework of $\pi_{1}^{\un}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$, MZV$\mu_{N}$'s resp. $p$MZV$\mu_{N}$'s often appear via their non-commutative generating series $\Phi_{\KZ}$ resp. $\Phi_{p,\alpha}$, which is an element of the $\mathbb{C}$-algebra, resp. $K_{p}$-algebra of non-commutative power series over the formal variables $e_{x}$ where $x$ is $0$ or a $N$-th root of unity. The coefficient of $e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}$ in $\Phi_{\KZ}$, resp. in $\Phi_{p,\alpha}$ is $(-1)^{d}\zeta\big((n_{i})_{d};(\xi_{i})_{d}\big)$, resp. $(-1)^{d}\zeta_{p,\alpha}\big((n_{i})_{d};(\xi_{i})_{d}\big)$.
\newline\indent According to the philosophy of periods, one wants to study the algebra generated by MZV$\mu_{N}$'s resp. $p$MZV$\mu_{N}$'s over the $N$-th cyclotomic field $k_{N}$, using the motivic nature of $\pi_{1}^{\un}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$ established in \cite{Deligne Goncharov}. There are many known examples of families of polynomial equations in that algebra. Three among them are particularly meaningful and well-known : these are the regularized double shuffle equations, the associator equations, and the Kashiwara-Vergne equations. We will refer to them as the standard equations. At least in the $N=1$ case, it is conjectured that each of them generates all polynomial equations satisfied by MZV$\mu_{N}$'s, resp. $p$MZV$\mu_{N}$'s. Moreover, one can show that the set of solutions to these equations have structures of torsors for the Ihara action which is a byproduct of the motivic Galois action on $\pi_{1}^{\un}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$ (\cite{Deligne Goncharov} \S5.12).
\subsection{A computation of $p$-adic cyclotomic multiple zeta values which keeps track of the motivic Galois action}
In \cite{I-1} \cite{I-2} \cite{I-3}, we have found formulas for $p$-adic cyclotomic multiple zeta values, which are analogues of the expression of sums of series (\ref{eq:multizetas}).
\newline\indent In these formulas, $\Phi_{p,\alpha}$ is involved via $\Ad_{\Phi_{p,\alpha}}(e_{1}) = \Phi_{p,\alpha}^{-1}e_{1}\Phi_{p,\alpha}$ and its images $\Ad_{\Phi_{p,\alpha}^{(\xi)}}(e_{\xi})$, $\xi \in \mu_{N}(K_{p})$, by the automorphisms $(z \mapsto \xi z)_{\ast}$ of $\pi_{1}^{\un,\dR}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$. The correspondence between the coefficients of $\{\Phi_{p,\alpha}^{(\xi)}\}^{-1}e_{\xi}\Phi^{(\xi)}_{p,\alpha}$ and the coefficients of $\Phi_{p,\alpha}$ will be discussed in a subsequent paper.
\newline\indent Concretely, computing $p$MZV$\mu_{N}$'s means computing the Frobenius structure of the KZ differential equation (equation (\ref{eq: nabla KZ})) associated with $\pi_{1}^{\un,\dR}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$, whose solutions are called multiple polylogarithms (equation (\ref{eq:MPL})) and admit the following power series expansion at $0$ :
\begin{equation} \label{eq:Li 0} \Li\big((n_{i})_{d};(\xi_{i})_{d} \big)(z) = \sum_{0<m_{1}<\ldots<m_{d}} \frac{\big( \frac{\xi_{2}}{\xi_{1}} \big)^{m_{1}} \ldots \big( \frac{z}{\xi_{d}} \big)^{m_{d}}}{m_{1}^{n_{1}} \ldots m_{d}^{n_{d}}}.
\end{equation}
\noindent The computation of $p$MZV$\mu_{N}$'s arises as a characterization of the coefficients of each $\Ad_{\Phi_{p,\alpha}^{(\xi)}}(e_{\xi})$ in terms of weighted cyclotomic multiple harmonic sums, which are essentially the coefficients above (below, $m$ is any positive integer and the other indices are as above) :
\begin{equation} \label{eq:mult har sums} \text{har}_{m} \big((n_{i})_{d} ;(\xi_{i})_{d+1} \big) = m^{n_{d}+\ldots+n_{1}} \sum_{0<m_{1}<\ldots<m_{d}<m}
\frac{\big( \frac{\xi_{2}}{\xi_{1}} \big)^{m_{1}} \ldots \big(\frac{\xi_{d+1}}{\xi_{d}}\big)^{m_{d}}
\big(\frac{1}{\xi_{d+1}}\big)^{m}}{m_{1}^{n_{1}}\ldots m_{d}^{n_{d}}}.
\end{equation}
\indent The computation gives a formula for the coefficients of $\Ad_{\Phi_{p,\alpha}^{(\xi)}}(e_{\xi})$, as sums of series whose terms are linear combinations over $k_{N}$ of the numbers $\text{har}_{p^{\alpha}} \big((n_{i})_{d} ;(\xi_{i})_{d+1} \big)$ which we call the prime weighted cyclotomic multiple harmonic sums (\cite{I-1}, Definition B.0.1). It also gives a converse of that formula, which is the following ; below, the notation $f[w]$ means the coefficient of a word $w$ in a non-commutative formal power series $f$, and the sum of series converges in $K_{p}$ :
\begin{multline} \label{eq:formula for n=1}
\har_{p^{\alpha}} \big( (n_{i})_{d};(\xi_{i})_{d+1} \big) =
(-1)^{d} \sum_{\xi \in \mu_{N}(K)} \sum_{l=0}^{\infty} \xi^{-p^{\alpha}} \Ad_{\Phi_{p,\alpha}^{(\xi)}}(e_{\xi}) \Big[ e_{0}^{l} e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}} \Big].
\end{multline}
\indent We have a total of three ways of viewing prime weighted multiple harmonic sums : (\ref{eq:Li 0}), (\ref{eq:mult har sums}) and (\ref{eq:formula for n=1}) ; this will make three frameworks of computation, which we represent respectively by the symbols $\int$, $\Sigma$ and $\int_{1,0}$. One of the central ideas in \cite{I-2} and \cite{I-3} was to express certain byproducts of the Frobenius both in a framework of integrals and the framework of series and to use that the two expressions were equal ; this gave formulas for $p$MZV$\mu_{N}$'s in terms of series.
\newline\indent Our computation of $p$MZV$\mu_{N}$'s keeps track of the motivic nature of $\pi_{1}^{\un}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$ : the main formulas are expressed by means of new group actions which we call pro-unipotent harmonic actions, which are certain $p$-adic byproducts of the motivic Galois action and of the Ihara action evoked above.
\subsection{Relating the explicit formulas for $p$-adic cyclotomic multiple zeta values and their algebraic theory}
The purpose of this paper and of the two subsequent ones \cite{II-2} and \cite{II-3} is to relate the explicit formulas for $p$MZV$\mu_{N}$'s obtained in \cite{I-1}, \cite{I-2} and \cite{I-3} to the algebraic theory of $p$MZV$\mu_{N}$'s.
\newline\indent The main reason for doing it is the fact that our formulas for $p$MZV$\mu_{N}$'s keep track of the motivic Galois action (via the pro-unipotent harmonic actions), and that they are explicit. Another reason for doing it is Kaneko-Zagier's notion of finite multiple zeta values \cite{Kaneko Zagier}, which are in $(\underset{p\text{ prime}}{\prod} \mathbb{Z}/p\mathbb{Z}) \big/ (\underset{p\text{ prime}}{\bigoplus} \mathbb{Z}/p\mathbb{Z})$ \cite{Kaneko Zagier}, and Rosen's lift of finite multiple zeta values, called truncated multiple zeta values, which are elements of the complete topological ring $\varprojlim_{n} (\underset{p\text{ prime}}{\prod} \mathbb{Z}/p^{n}\mathbb{Z}) \big/ (\underset{p\text{ prime}}{\bigoplus} \mathbb{Z}/p^{n}\mathbb{Z})$ \cite{Rosen}. In this paper, certain of our results will recover certain known results on these two notions, and interpret them in terms of $\pi_{1}^{\un}(\mathbb{P}^{1} - \{0,1,\infty\})$.
\newline\indent We are going to consider intrinsically the numbers which appear in the right-hand side of (\ref{eq:formula for n=1}) (as well as their complex analogues) and the numbers $\har_{p^{\alpha}}(w)$ involved in the formulas of \cite{I-1}, \cite{I-2}, \cite{I-3}, and turn them into notions called \emph{adjoint $p$-adic cyclotomic multiple zeta values} (Ad$p$MZV$\mu_{N}$'s) (Definition \ref{def adjoint}), and \emph{cyclotomic multiple harmonic values} (MHV$\mu_{N}$'s) (Definition \ref{def harmonic}). We are going to view them as two types of ``periods'' (in a generalized sense for the second case, as will be explained in \cite{II-3}).
Since each $\har_{p^{\alpha}}(w)$ is only an algebraic number, in order to turn it into an object which can be regarded as an interesting ``period'' comparable to MZV's, we are going to consider all $p$'s at the same time (as in the work of Kaneko-Zagier and Rosen), or all $\alpha$'s at the same time, or both ; and our cyclotomic multiple harmonic values will lift both Kaneko-Zagier's finite multiple zeta values and Rosen's truncated multiple zeta values. We will establish in \S2.2 the setting for computations on these objects, in the three frameworks $\int_{1,0}$, $\int$ and $\Sigma$.
\newline\indent We adopt the following principle, for this paper and all the subsequent ones : for each question on $p$MZV$\mu_{N}$'s which we want to study by means of explicit formulas, we find its analogue for the adjoint variant of $p$MZV$\mu_{N}$'s, we solve that adjoint variant, and it remains to pass from Ad$p$MZV$\mu_{N}$'s to $p$MZV$\mu_{N}$'s ; this last step is delayed to other works. In the present sequence of papers, we only consider the Ad$p$MZV$\mu_{N}$'s which are the natural version of $p$MZV$\mu_{N}$'s adapted to dealing with explicit formulas.
\newline\indent With the above ideas, the first step of our explicit version of the algebraic theory of $p$MZV$\mu_{N}$'s is thus to develop intrinsically the properties of adjoint $p$-adic cyclotomic multiple zeta values and cyclotomic multiple harmonic values, and this is the goal of this paper. In the end, our explicit version of the algebraic theory of $p$MZV$\mu_{N}$'s will be formulated as a ``comparison'' between the properties of Ad$p$MZV$\mu_{N}$'s and those of MHV$\mu_{N}$'s, where MHV$\mu_{N}$'s are defined by explicit formulas.
\subsection{Summary of the paper}
We are going to find ``adjoint'' and ``harmonic'' analogues of the standard properties of cyclotomic multiple zeta values.
The analogues of algebraic relations for cyclotomic multiple harmonic values which we find involve infinite sums of prime weighted cyclotomic multiple harmonic sums, which will be convergent both for the weight-adic topology and for the $p$-adic topology, uniformly with respect to $(p,\alpha)$, and with certain bounds on the $p$-adic norms of the rational coefficients. At first sight, the presence of infinite sums makes us leave the framework of algebraic geometry and periods. However, we are going to prove a close relation between the ``adjoint'' and the ``harmonic'' variants of the algebraic properties of MZV$\mu_{N}$'s, namely, a torsor structure using the pro-unipotent harmonic action defined in \cite{I-3}.
The paper is organized as follows.
In \S2 we review prerequisites and we define formally the main objects that we are going to study : adjoint cyclotomic multiple zeta values, cyclotomic multiple harmonic values, and related objects. We discuss the meaning of the definitions and we establish the setting for making computations with these objects.
In \S3 we focus on double shuffle relations and we establish some natural adjoint and harmonic variants of them. Below, $\DS_{\mu}$ is the double shuffle scheme defined by Racinet in \cite{Racinet}, $\circ^{\smallint_{1,0}}$ is the Ihara action, $\circ^{\smallint_{1,0}}_{\Ad}$ is the adjoint Ihara action introduced in \cite{I-2}, Definition 1.1.3, $\circ_{\har}^{\smallint_{1,0}}$ is the pro-unipotent harmonic action introduced in \cite{I-3}, Definition 2.1.2 ; $\mathcal{O}^{\mathcyr{sh}}$ is the shuffle Hopf algebra over the alphabet $e_{0\cup \mu_{N}}$, graded by the number of letters of words called their weight.
The map $\comp^{\har,\Ad}$ is defined in \S2.2.1.
\newline\newline
\textbf{Theorem 1.} \emph{
\newline\indent (i) (adjoint double shuffle) For any $\mu$, there exists an explicit affine subscheme
$\DS_{\mu,\Ad}$ of $\Spec(\mathcal{O}^{\mathcyr{sh}})$. One has a canonical morphism $\Ad(e_{1}) : \DS_{\mu} \rightarrow \DS_{\mu,\Ad}$, which is an isomorphism on its image, and sends Racinet's torsor to a torsor for the adjoint Ihara product.
\newline\indent In particular, the non-commutative generating series of Ad$p$MZV$\mu_{N}$'s resp. AdMZV$\mu_{N}$'s is in $\DS_{0,\Ad}(K_{p})$, resp. $\DS_{2i\pi,\Ad}(\mathbb{C})$.
\newline\indent (ii) (harmonic double shuffle) There exists an explicit affine ind-scheme $\DS_{\har}$, sub-ind-scheme of $\Spf(\mathcal{O}^{\mathcyr{sh}})$, which can be obtained by each of the three frameworks : ``$\int_{1,0}$'', ``$\int$'' and ``$\Sigma$''.
\newline\indent The non-commutative generating series MHV$\mu_{N}$'s is in $\DS_{\har}((\prod_{p}K_{p})^{\mathbb{N}})$.
\newline\indent (iii) (relation between adjoint and harmonic) The map $\comp^{\har,\Ad}$ sends $\DS_{0,\Ad} \mapsto \DS_{\har}$ and its image is a torsor under the pro-unipotent harmonic action $\circ_{\har}^{\smallint_{1,0}}$ of the group $(\Ad_{\DS_{0}}(e_{1}),\circ^{\smallint_{1,0}}_{\Ad})$.
\newline\indent (iv) More generally, adjoint double shuffle equations are satisfied by adjoint multiple polylogarithms and harmonic double shuffle equations are satisfied by harmonic multiple polylogarithms.}
\newline\newline
We note that in (ii) above, the frameworks of computations give harmonic double shuffle equations which look different at first sight but we can prove that they are equivalent.
In \S4 we focus on associator and Kashiwara-Vergne equations. We consider an analogy between the passage from associator equations to Kashiwara-Vergne equations constructed in \cite{AT} and \cite{AET} and the passage from double shuffle relations to adjoint double shuffle relations constructed in \S3. The main result is the following.
\newline
\newline
\textbf{Theorem 2.} (rough version)
\newline\indent \emph{(i) Kashiwara-Vergne equations arise naturally as a property of ($p$-adic, complex) adjoint MZV's rather than as a property of ($p$-adic, complex) MZV's and naturally amount to certain polynomial equations on adjoint ($p$-adic, complex) MZV's.
\newline \indent (ii) There are equations satisfied by harmonic multiple polylogarithms which are of the same source with associator equations and Kashiwara-Vergne equations (functoriality of the KZ equation).}
\newline
\newline
In \S5, as a corollary, we explain that this paper recasts the study finite and symmetric (or symmetrized) multiple zeta values (see \cite{NoteCRAS}) as a particular case and a byproduct of the study of adjoint MZV's and MHV's, which is the theme arising naturally from the study of $p$MZV$\mu_{N}$'s via explicit formulas. Indeed, finite multiple zeta values are reductions of our multiple harmonic values modulo large primes, and symmetric multiple zeta values are a particular case of our adjoint multiple zeta values ; thus they satisfy a particular case of the equations of Theorem 1 and Theorem 2. We introduce finite cyclotomic multiple zeta values and finite multiple polylogarithms, and in which we will more generally relate our study of $p$MZV$\mu_{N}$'s via explicit formulas to these objects.
In \cite{II-2} we will explain how to recover certain properties of Ad$p$MZV$\mu_{N}$'s by the explicit formulas and the properties of MHV$\mu_{N}$'s, which will mostly answer to a question of Deligne and Goncharov. In \cite{II-3} we will formalize the fact that MHV$\mu_{N}$'s can be regarded as periods in a generalized sense, using motivic multiple zeta values. In \cite{III-1} we will study adjoint and harmonic distribution relations.
An open question is whether $\Ad(e_{1}) : \DS_{\mu} \rightarrow \DS_{\mu,\Ad}$ is an isomorhism. We see this question as analogous to the conjecture made in \cite{AT} of an isomorphism relating associators and solutions to the Kashiwara-Vergne problem.
\emph{Acknowledgments.} I thank Masanobu Kaneko, Ivan Marin and Pierre Cartier for useful discussions, and an anonymous referee whose remarks and suggestions enabled me to improve this paper. This work has been done at Universit\'{e} Paris Diderot with support of ERC grant 257638, Universit\'{e} de Strasbourg with support of Labex IRMIA, and Universit\'{e} de Gen\`{e}ve with support of NCCR SwissMAP.
\section{Definitions and setting for computations\label{review}}
We review some material on pro-unipotent fundamental groupoids (\S2.1), we define adjoint cyclotomic multiple zeta values, in the $p$-adic case (\S2.2) and in the complex case (\S2.3), we define cyclotomic multiple harmonic values (\S2.4) and their ``overconvergent'' variants (\S2.5). At the same time we establish the setting for making computations with all these objects and we show that replacing cyclotomic multiple zeta values by their adjoint variants does not change the algebra that they generate. Finally we define more generally adjoint and harmonic analogues of multiple polylogarithms (\S2.6), which admit respectively adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values as special values.
In all this text, we denote by $\mathbb{N}$ resp. $\mathbb{N}^{\ast}$ the set of nonnegative, resp. positive integers ; $d$ and $n_{i}$ ($1 \leqslant i \leqslant d$) denote positive integers, $\xi_{i}$ ($1 \leqslant i \leqslant d$ or $1 \leqslant i \leqslant d+1$ depending on the context) are $N$-th roots of unity.
\numberwithin{equation}{subsection}
\subsection{Review on pro-unipotent fundamental groupoids}
\subsubsection{Generalities on the Betti and de Rham realizations of pro-unipotent fundamental groupoids}
Let $X$ be a smooth algebraic variety over a field $K$ of characteristic zero, with $X= \overline{X} - D$ where $\overline{X}$ is proper and smooth and $D$ is a normal crossings divisor.
\newline\indent The de Rham pro-unipotent fundamental groupoid $\pi_{1}^{\un,\dR}\big( X\big)$ (\cite{Deligne}, \S10.27, \S10.30 (ii)) is a groupoid over $X$ in the category of affine schemes over $K$, whose base-points are the points of $X$, and the points of the punctured tangent spaces at points of $D$ (\cite{Deligne}, \S15). Assuming that $H^{1}(\overline{X},\mathcal{O}_{\overline{X}})=0$, which holds in the examples of this paper, one also has a canonical base-point $\omega_{\dR}$ (\cite{Deligne}, \S12.4) with, for any couple of base-points $(x,y)$, an isomorphism of schemes $\pi_{1}^{\un,\dR}(X,y,x)\simeq \pi_{1}^{\un,\dR}(X,\omega_{\dR})$, and these isomorphisms are compatible with the groupoid structure. The bundle $\pi_{1}^{\un,\dR}(X,\omega_{\dR}) \times X$ carries the universal unipotent integrable connection on $X$.
\newline\indent Let us now assume that we have an embedding $K \hookrightarrow \mathbb{C}$. Then we also have the Betti pro-unipotent fundamental groupoid $\pi_{1}^{\un,\B}(X \times_{\Spec(K)} \Spec(\mathbb{C}))$, which is another groupoid in the category of affine schemes over $X \times_{\Spec(K)} \Spec(\mathbb{C})$, defined as the Malcev completion of the topological fundamental groupoid of $X(\mathbb{C})$ \cite{Deligne}.
\newline\indent Chen's theorem \cite{Chen} or, equivalently, the Riemann-Hilbert correspondence \cite{Deligne equa diff} restricted to unipotent objects, gives a natural isomorphism
\begin{equation} \label{eq:isomorphism} \pi_{1}^{\un,\B}(X) \times_{\Spec(K)} \Spec(\mathbb{C}) \buildrel \sim \over \longrightarrow \pi_{1}^{\un,\dR}(X) \times_{\Spec(K)} \Spec(\mathbb{C}).
\end{equation}
Its coefficients are iterated path integrals on $X$ in Chen's sense \cite{Chen}. If $X$ is defined over a number field, they are periods, called the Betti-de Rham periods of $\pi_{1}^{\un}(X)$.
\subsubsection{The de Rham pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$}
Let $X=(\mathbb{P}^{1} - \{0,\mu_{N},\infty\}) / K$ where $K$ contains a primitive $N$-th root of unity.
\newline\indent By \cite{Deligne}, \S12, the affine scheme $\pi_{1}^{\un,\dR}(X,\omega_{\dR})$ is canonically isomorphic to $\Spec(\mathcal{O}^{\mathcyr{sh}})\times_{\Spec(\mathbb{Q})} \Spec(K)$, where $\mathcal{O}^{\mathcyr{sh}}$ is the shuffle Hopf algebra on the alphabet $e_{0\cup \mu_{N}} = \{e_{x}\text{ }|\text{ }x \in \{0\} \cup \mu_{N}(K)\}$. By definition, $\mathcal{O}^{\mathcyr{sh}}$ is the $\mathbb{Q}$-vector space $\mathbb{Q}\langle e_{0\cup \mu_{N}} \rangle$ which admits as a basis the set of words on $e_{0 \cup \mu_{N}}$ including the empty word, graded by the number of letters of words called their weight, endowed with the following operations : the shuffle product $\mathcyr{sh}$ defined by $(e_{i_{l+l'}}\ldots e_{i_{l+1}})\text{ }\mathcyr{sh}\text{ }(e_{i_{l}} \ldots e_{i_{1}}) =
\sum\limits_{\sigma}
e_{i_{\sigma^{-1}(l+l')}} \ldots e_{i_{\sigma^{-1}(1)}}$ where the sum is over permutations $\sigma$ of $\{1,\ldots,l+l'\}$ such that $\sigma(1)<\ldots<\sigma(l)$ and $\sigma(l+1)<\ldots<\sigma(l+l')$ ; the deconcatenation coproduct $\Delta_{\dec}$ defined by $\Delta_{\dec}(e_{i_{l}}\ldots e_{i_{1}}) = \sum\limits_{l'=0}^{l} e_{i_{l}}\ldots e_{i_{l'+1}} \otimes e_{i_{l'}} \ldots e_{i_{1}}$ ; the counit $\epsilon$ equal to the augmentation map ; the antipode $S$ defined by $S(e_{i_{l}}\ldots e_{i_{1}}) = (-1)^{l} e_{i_{1}}\ldots e_{i_{l}}$.
\begin{Notation} Let $K\langle\langle e_{0 \cup \mu_{N}} \rangle\rangle$ be the $K$-algebra of non-commutative formal power series over the variables $e_{x}$, $x \in \{0\} \cup \mu_{N}(K)$ and, for $f\in K\langle\langle e_{0 \cup \mu_{N}} \rangle\rangle$ and $w$ a word over the alphabet
$\{e_{x}\text{ }|\text{ }x \in \{0\} \cup \mu_{N}(K)\}$, let $f[w] \in K$ be the coefficient of $w$ in $f$.
\end{Notation}
The group scheme $\Spec(\mathcal{O}^{\mathcyr{sh}})$ is pro-unipotent, and the completed dual of the Hopf algebra $\mathcal{O}^{\mathcyr{sh}}$ is $K \langle \langle e_{0\cup \mu_{N}}\rangle \rangle$ viewed as the Hopf algebra obtained as the completion of the universal enveloping algebra of the free Lie algebra in the variables $e_{x}$, $x \in \{0\} \cup \mu_{N}(K)$. We will denote by $\Delta_{\mathcyr{sh}}$ its coproduct. We have
\begin{equation} \label{eq:shuffle equation} \begin{array}{ll} \Spec(\mathcal{O}^{\mathcyr{sh}})(K) & = \{ f \in K\langle\langle e_{0 \cup \mu_{N}} \rangle\rangle \text{ }|\text{ }\forall w,w' \text{ words on }e_{0 \cup \mu_{N}}, f[w\text{ }\mathcyr{sh}\text{ }w']=f[w]f[w'],\text{ and }f[\emptyset] = 1 \}
\\ & = \{ f \in K\langle\langle e_{0 \cup \mu_{N}} \rangle\rangle \text{ }|\text{ }\Delta_{\mathcyr{sh}}(f) = f \otimes f \},
\end{array}
\end{equation}
\begin{equation} \label{eq:shuffle equation modulo products}
\begin{array}{ll}
\Lie(\Spec(\mathcal{O}^{\mathcyr{sh}})(K)) & = \{ f \in K \langle\langle e_{0 \cup \mu_{N}} \rangle\rangle \text{ }|\text{ }\forall w,w' \text{ words on }e_{0 \cup \mu_{N}},\text{ }f[w\text{ }\mathcyr{sh}\text{ }w']=0\}
\\ & = \{ f \in K \langle\langle e_{0 \cup \mu_{N}} \rangle\rangle \text{ }|\text{ }\Delta_{\mathcyr{sh}}(f) = f \otimes 1 + 1 \otimes f \}.
\end{array}
\end{equation}
\newline\indent The canonical connection on $\pi_{1}^{\un,\dR}(X,\omega_{\dR})\times X$ in the sense of \cite{Deligne}, \S12 is the Knizhnik-Zamolodchikov (KZ) connection :
\begin{equation} \label{eq: nabla KZ} \nabla_{\KZ} : f \mapsto df - \bigg( e_{0} f \frac{dz}{z} + \sum_{\xi \in \mu_{N}(K)} e_{\xi} f \frac{dz}{z-\xi} \bigg).
\end{equation}
\indent
\begin{Notation} \label{la premiere notation}
(\cite{Deligne Goncharov}, \S5), let $\Pi = \pi_{1}^{\un,\dR}(X,\omega_{\dR})$ and $\Pi_{1,0} = \pi_{1}^{\un,\dR}(X,-\vec{1}_{1},\vec{1}_{0})$ where $\vec{v}_{x}$ means the tangent vector $\vec{v}$ at $x$.
\end{Notation}
\subsubsection{Comparison between the Betti and de Rham pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$}
Following \cite{Deligne Goncharov}, \S5.16, let $dch \in \pi_{1}^{\un,\B}(X,\vec{1}_{0},\vec{1}_{1})(\mathbb{C})$ be the image, by the Malcev completion map
$ \pi_{1}(X,\vec{1}_{0},\vec{1}_{1}) \rightarrow \pi_{1}^{\un,\B}(X,\vec{1}_{0},\vec{1}_{1})(\mathbb{C})$,
of the homotopy class of $\gamma : t \in [0,1] \mapsto t \in [0,1]$ ; let
\begin{equation} \label{eq:Phi KZ}
\Phi_{\KZ} = \comp_{\B,\dR}(dch) \in \Pi_{1,0}(\mathbb{C}),
\end{equation}
which appeared first in \cite{Drinfeld}, \S2 in the $N=1$ case ; $\Phi_{\KZ}$ is the non-commutative generating series of MZV$\mu_{N}$'s : indeed, given the above definition, the formula for cyclotomic multiple zeta values as iterated integrals (\ref{eq:multizetas}) amounts to :
\begin{equation} \label{eq:coefficient}
\zeta \big( (n_{i})_{d};(\xi_{i})_{d} \big) = (-1)^{d}\Phi_{\KZ}[e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}].
\end{equation}
\indent Multiple polylogarithms, abbreviated as MPL's \cite{Goncharov} are multivalued holomorphic functions on $X(\mathbb{C})$, defined as iterated integrals in the sense of Chen \cite{Chen}, such that the non-commutative generating series
$\Li = 1 + \sum\limits_{\substack{n \in \mathbb{N}^{\ast} \\ \epsilon_{1},\ldots,\epsilon_{n} \in \{0,1\}^{n}}} \Li\big( e_{\epsilon_{n}} \cdots e_{\epsilon_{1}} \big)e_{\epsilon_{n}} \cdots e_{\epsilon_{1}}$
defines a solution to the $\KZ$ equation (\ref{eq: nabla KZ}). For $\gamma$ a differentiable topological path on $\mathbb{P}^{1}(\mathbb{C})$ such that $\gamma\big((0,1)\big) \subset (\mathbb{P}^{1} - \{0,\mu_{N},\infty\})(\mathbb{C})$, and $\gamma'(0)\not= 0$ and $\gamma'(1)\not= 0$,
\begin{equation} \label{eq:MPL} \displaystyle\Li\big( e_{\epsilon_{n}} \cdots e_{\epsilon_{1}} \big)(\gamma) =
\int_{t_{n}=0}^{1} \gamma^{\ast}(\frac{dz}{z-\epsilon_{n}})(t_{n}) \int_{t_{n-1}=0}^{t_{n}} \ldots \gamma^{\ast}(\frac{dz}{z-\epsilon_{2}})(t_{2}) \int_{t_{1}=0}^{t_{2}} \gamma^{\ast}(\frac{dz}{z-\epsilon_{1}})(t_{1}).
\end{equation}
When that integral diverges, (\ref{eq:MPL}) means the regularized iterated integral defined by considering the similar integral on $\gamma([\epsilon,1-\epsilon'])$, its asymptotic expansion when $\epsilon,\epsilon' \rightarrow 0$ which is in $\mathbb{C}[[\epsilon,\epsilon']][\log(\epsilon),\log(\epsilon')]$, and finally the coefficient of this asymptotic expansion. It depends only of the homotopy class of $\gamma$, in the extended sense which includes tangential base-points, i.e. it depends on $\gamma'(0)$ and $\gamma'(1)$.
\newline\indent If we take $\gamma(0)=0$, the formal power series expansion of the iterated integrals (\ref{eq:MPL}) at $z=0$ is convergent for $|z|<1$ :
\begin{equation} \label{eq:multiple polylogarithms power series expansion}
\Li[e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}](z) = (-1)^{d} \sum_{0<m_{1}<\ldots <m_{d}}
\frac{\big( \frac{\xi_{2}}{\xi_{1}} \big)^{m_{1}} \ldots \big(\frac{1}{\xi_{d}}\big)^{m_{d}}}{m_{1}^{n_{1}}\ldots m_{d}^{n_{d}}}.
\end{equation}
\subsubsection{The crystalline Frobenius of the de Rham pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$ over $K_{p}$}
The differential equation $\nabla_{\KZ}$ (\ref{eq: nabla KZ}) has a crystalline Frobenius structure over $K_{p}$ (\cite{Deligne}, \S11).
The theory of Coleman integration, which relies on this Frobenius structure, enables to define $p$-adic analogues of MPL's and MZV$\mu_{N}$'s \cite{Furusho 1} \cite{Furusho 2} \cite{Yamashita}. In particular, one has $\Li_{p,X_{K}}^{\KZ}$ resp. $\Li_{p,X_{K}^{(p^{\alpha})}}^{\KZ}$, the generating series of $p$-adic multiple polylogarithms, Coleman functions characterized as the solution to $\nabla_{\KZ}$ resp. its pull-back $\nabla_{\KZ}^{(p^{\alpha})}$ by the Frobenius $\sigma$ of $K$ iterated $\alpha$ times, equivalent to $e^{\log_{p}(z)e_{0}}$ at $\vec{1}_{0}$ (defined in \cite{Furusho 1}, \cite{Furusho 2} for $N=1$ and \cite{Yamashita} for any $N$).
We consider the Frobenius iterated $\alpha$ times $(\alpha \in \mathbb{N}^{\ast})$ in the sense of \cite{I-1}, \S1.
Then, another type of $p$-adic analogue of MPL's and MZV$\mu_{N}$'s can be defined in a more ad hoc way, using canonical de Rham paths \cite{Deligne Goncharov} \cite{Unver MZV} \cite{U2} \cite{I-1} \cite{I-3}. In the end, with \cite{I-1} and \cite{I-3}, we have for each $\alpha \in \mathbb{Z}\cup \{\pm \infty\} - \{0\}$, an element $\Phi_{p,\alpha} \in \Pi_{1,0}(K)$, which characterizes the Frobenius at base-points $(\vec{1}_{1},\vec{1}_{0})$ iterated $\alpha$ times, where $\alpha=-\infty$ corresponds to Coleman integration. $p$-adic cyclotomic multiple zeta values are defined as its coefficients, namely :
$$ \zeta_{p,\alpha}\big((n_{i})_{d};(\xi_{i})_{d}\big) = (-1)^{d} \Phi_{p,\alpha}[e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}]. $$
One also has $\Li_{p,\alpha}^{\dagger}$, the non-commutative generating series of overconvergent $p$-adic multiple polylgarithms (\cite{I-1}, \S1), which are overconvergent analytic functions on the affinoid analytic space $U_{0\infty}^{\an}=\mathbb{P}^{1,\an} - \underset{\xi \in \mu_{N}(K)}{\cup} \B(\xi,1)$, where $\B(x,r)$ means the open ball of center $x$ and radius $r$. It is characterized by $\Li_{p,\alpha}^{\dagger}(0)=1$ and the following differential equation (\cite{I-1}, Proposition 2.1)
\begin{equation}
\label{eq:horizontality equation}
d\Li_{p,\alpha}^{\dagger} = \bigg( p^{\alpha}e_{0}\omega_{0}(z) + \sum_{\xi \in \mu_{N}(K)} p^{\alpha} \omega_{\xi}(z) e_{\xi} \bigg) \Li_{p,\alpha}^{\dagger} - \Li_{p,\alpha}^{\dagger} \bigg( \omega_{0}(z^{p^{\alpha}})e_{0} + \sum_{\xi \in \mu_{N}(K)} \omega_{z_{0}^{p^{\alpha}}}(z^{p^{\alpha}}) \Ad_{\Phi^{(\xi)}_{p,\alpha}} (e_{\xi}) \bigg)
\end{equation}
\noindent where $\Phi_{p,\alpha}^{(\xi)} = (x \mapsto \xi x)_{\ast}(\Phi_{p,\alpha})$. Equivalently, it is characterized by
\begin{multline} \label{eq:horizontality1}
\Li_{p,\alpha}^{\dagger}(z)(e_{0},(e_{\xi})_{\xi \in \mu_{N}(K)})
\times\Li_{p,X_{K}^{(p^{\alpha})}}^{\KZ}(z^{p^{\alpha}})\big(e_{0},(\Ad_{\Phi^{(\xi)}_{p,\alpha}}(e_{\xi}))_{\xi \in \mu_{N}(K)} \big)
\\ = \Li_{p,X_{K}}^{\KZ}(z) \big(p^{\alpha}e_{0},(p^{\alpha}e_{\xi})_{\xi \in \mu_{N}(K)} \big)
\end{multline}
We note that $\Li_{p,X_{K}}^{\KZ}$ and $\Li_{p,X_{K}^{(p^{\alpha})}}^{\KZ}$ depend on the choice of a branch of the $p$-adic logarithm, but not $\Li_{p,\alpha}^{\dagger}$, nor $p$MZV$\mu_{N}$'s.
\subsubsection{The motivic pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$}
The motivic pro-unipotent fundamental groupoid $\pi_{1}^{\un,\text{mot}}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$ is defined and studied in \cite{Deligne Goncharov}. Let $G_{\omega}$ be the fundamental group associated with the Tannakian category of mixed Tate motives over $k_{N}$ which are unramified at primes $p$ prime to $N$, and the canonical fiber functor $\omega$. It is a semi-direct product $G_{\omega}= \mathbb{G}_{m} \ltimes U_{\omega}$ where $U_{\omega}$ is pro-unipotent. It acts on $\Pi_{1,0}$, and this action encodes the algebraic theory of MZV$\mu_{N}$'s and $p$MZV$\mu_{N}$'s according to the conjecture of periods (\cite{Deligne Goncharov}, \S5). The action of $\mathbb{G}_{m}$ encodes the weight grading and is
\begin{equation} \label{eq:tau} \tau : \begin{array}{cc} \mathbb{G}_{m} \times \Pi_{1,0} \rightarrow \Pi_{1,0}
\\ \big( \lambda,f(e_{0},(e_{\xi})_{\xi \in \mu_{N}(K)})\big) \mapsto f(\lambda e_{0},(\lambda e_{\xi})_{\xi \in \mu_{N}(K)})
\end{array}.
\end{equation}
The action of $U_{\omega}$ has been computed by Goncharov \cite{Goncharov}. The image of this action by a certain morphism is isomorphic to the Ihara product (our notation below is not standard)
\begin{equation} \label{eq:Ihara} \circ^{\smallint_{1,0}} :
\begin{array}{cc} \Pi_{1,0} \times \Pi_{1,0} \rightarrow \Pi_{1,0}
\\
(g,f) \mapsto g \circ^{\smallint_{1,0}} f = g(e_{0},(e_{\xi})_{\xi \in \mu_{N}(K)}) \times f\big(e_{0},(\Ad_{g^{(\xi)}}(e_{\xi}))_{\xi \in \mu_{N}(K)}\big)
\end{array} .
\end{equation}
\subsubsection{The Betti and de Rham pro-unipotent fundamental groupoid of a more general $\mathbb{P}^{1} - D$}
By \cite{Deligne}, the description of the Betti and de Rham realizations of $\pi_{1}^{\un}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$ of \S2.1.2 remains true in the case of an arbitrary punctured projective line $\mathbb{P}^{1} - D$ over a subfield of $\mathbb{C}$, provided that we replace the alphabet $e_{0 \cup \mu_{N}}$ by the alphabet $\{e_{x}\text{ }|\text{ }x \in D - \{\infty\}\}$. It can also be deduced from the description of $\pi_{1}^{\un}(\mathcal{M}_{0,n})$ of \S2.1.3.
We write the words on that alphabet on the form $e_{0}^{n_{d}-1}e_{z_{d}}\ldots e_{0}^{n_{1}-1}e_{z_{1}}e_{0}^{n_{0}-1}$ with $d$ and $n_{i}$ positive integers and $z_{i} \in D - \{0,\infty\}$. For most computations, we can restrict to words such that $n_{d}\geqslant 2$ and $n_{0}=1$. The word above is then denoted also by $((n_{i})_{d},(z_{i})_{d})$.
\newline\indent The multiple polylogarithms on $\mathcal{M}_{0,n}$ (\S2.1.3) induce multiple polylogarithms on $\mathbb{P}^{1} - \{0,x_{1},\ldots,x_{r},\infty\}$, whose power series expansion are given by (\ref{eq:Li series bis}) as functions of $z$, which are solution to the KZ equation, generalization of (\ref{eq: nabla KZ}) :
\begin{equation} \label{eq: nabla KZ prime} \nabla_{\KZ} : f \mapsto df - \bigg( e_{0} f \frac{dz}{z} + \sum_{\xi \in \mu_{N}(K)} e_{\xi} f \frac{dz}{z-\xi} \bigg) .
\end{equation}
The weighted multiple harmonic sums, obtained by the coefficients of the power series expansion (\ref{eq:Li series bis}), are now the following numbers (where the $z_{i}$'s are in $D - \{0,\infty\}$) :
$$ \har_{m}((n_{i})_{d},(z_{i})_{d+1}) =
\sum_{0<m_{1} <\ldots < m_{d}<m}
\frac{\big( \frac{z_{2}}{z_{1}} \big)^{m_{1}} \ldots \big(\frac{z_{d+1}}{z_{d}}\big)^{m_{d}} \big(\frac{1}{z_{d+1}}\big)^{m}}{m_{1}^{n_{1}} \ldots m_{d}^{n_{d}}} . $$
\subsubsection{The Betti and de Rham pro-unipotent fundamental groupoid of the moduli spaces $\mathcal{M}_{0,n}$}
Let, for $n \in \mathbb{N}^{\ast}$, the scheme
$\mathcal{M}_{0,n+3} = \{ (x_{1},\ldots,x_{n+3}) \in (\mathbb{P}^{1})^{n+3} \text{ } |\text{ } x_{i}\not= x_{j} \text{ for all i} \not= \text{j} \} / \PGL_{2}$ over $\mathbb{Q}$, and let $\overline{\mathcal{M}}_{0,n+3}$ be its Deligne-Mumford compactification, which is a smooth projective variety such that $\overline{\mathcal{M}}_{0,n+3} - \mathcal{M}_{0,n+3}$ is a normal crossings divisor. The $x_{i}$'s are called the canonical coordinates. The homography of $\mathbb{P}^{1}$ which sends $(x_{n+1},x_{n+2},x_{n+3})$ to $(0,1,\infty)$ induces an isomorphism between $\mathcal{M}_{0,n+3}$ and the affine variety
$\{(y_{1},y_{2},\ldots,y_{n}) \in (\mathbb{P}^{1} - \{0,1,\infty\})^{n} \text{ }|\text{ for all i,j}, y_{i} \not= y_{j} \}$, defined by $\displaystyle y_{i} = \frac{x_{i} - x_{n+1}}{x_{i}-x_{n+3}} \frac{x_{n+2} - x_{n+3}}{x_{n+2}-x_{n+1}}$, $(1 \leqslant i \leqslant n)$ ; the $y_{i}$'s are called simplicial coordinates. In particular, this gives $\mathcal{M}_{0,4} \simeq \mathbb{P}^{1} - \{0,1,\infty\}$, $\overline{\mathcal{M}_{0,4}} =\mathbb{P}^{1}$ ; $\overline{\mathcal{M}_{0,5}}$ is obtained blowing up $(\mathbb{P}^{1})^{2} \supset \mathcal{M}_{0,5}$ at the three points where $(\mathbb{P}^{1})^{2} - \mathcal{M}_{0,5}$ is not normal crossings, namely, in simplicial coordinates, $(0,0)$, $(1,1)$ and $(\infty,\infty)$.
\newline\indent By \cite{Deligne} \S12, $\Lie \big(\pi_{1}^{\un,\dR}(\mathcal{M}_{0,n+3},\omega_{\dR}) \big)$ is the pro-nilpotent Lie algebra with generators $e_{ij}$, $1 \leq i \not= j \leqslant n+3$,
\noindent and relations $e_{ij} = e_{ji}$, for all $i,j$,
$\sum\limits_{j=1}^{n} e_{ij} = 0$ for all $i$, and $[e_{ij},e_{kl}] = 0$ for all $i,j,k,l$ pairwise distinct. This determines the pro-unipotent affine group scheme $\pi_{1}^{\un,\dR}(\mathcal{M}_{0,n},\omega_{\dR})$.
\newline\indent The canonical connection on $\pi_{1}^{\un,\dR}(\mathcal{M}_{0,n},\omega_{\dR}) \times \mathcal{M}_{0,n}$ in the sense of \cite{Deligne}, \S12, is the KZ connection in several variables :
\begin{equation} \label{eq:nablaKZ M0,n} \nabla_{\KZ} : f \mapsto df - \sum_{1 \leqslant i<j \leqslant n+3} e_{ij} d\log(x_{i}-x_{j}) f ,
\end{equation}
i.e., in the cubic coordinates $c_{i}$ defined by
$y_{i} = c_{i} \ldots c_{n}$, $(1 \leqslant i \leqslant n)$,
\begin{equation} \nabla_{\KZ} : f \mapsto df
- \bigg( \sum_{u=1}^{r} \frac{dc_{u}}{c_{u}} \sum_{u\leqslant i<j \leqslant n} e_{ij}
- \sum_{\substack{1\leqslant v \leqslant v' \leqslant n \\ 2 \leqslant i \leqslant j }} \frac{d(c_{v} \ldots c_{v'})}{c_{v} \ldots c_{v'} -1} e_{v-1,v'}
- \sum_{\substack{1\leqslant v \leqslant v' \leqslant n \\ 1 \leqslant i \leqslant j}} \frac{d(c_{v} \ldots c_{v'})}{c_{v} \ldots c_{v'} -1} e_{v',n-1} \bigg) f.
\end{equation}
\indent
The groupoid $\pi_{1}^{\un,\B}(\mathcal{M}_{0,n})$ can be computed as follows by induction on $n$ \cite{FR}. Each forgetful map $\mathcal{M}_{0,n+1} \rightarrow \mathcal{M}_{0,n}$ is a fibration and induces a long exact sequence in homotopy ; given that the $\mathcal{M}_{0,n}$'s are $K(\pi,1)$ spaces, that long exact sequence amounts to a short exact sequence :
$1 \rightarrow \pi_{1}^{\text{top}}\big(F(\mathbb{C}) \big)
\rightarrow \pi_{1}^{\text{top}}(\mathcal{M}_{0,n+1}(\mathbb{C}))
\rightarrow \pi_{1}^{\text{top}}(\mathcal{M}_{0,n}(\mathbb{C})) \rightarrow 1$ where $F$ is the fiber of the forgetful map ; moreover, that short exact sequence is split. By the exactness of the functor of Malcev completion, it gives rise to a short split exact sequence $1 \rightarrow \pi_{1}^{\un,\B}\big(F(\mathbb{C}) \big)
\rightarrow \pi_{1}^{\un,\B}(\mathcal{M}_{0,n+1}(\mathbb{C}))
\rightarrow \pi_{1}^{\un,\B}(\mathcal{M}_{0,n}(\mathbb{C})) \rightarrow 1$.
\newline\indent Multiple polylogarithms in several variables are defined as follows \cite{Goncharov}. One consider first the following family of iterated integrals : for $a_{0},\ldots,a_{n+1} \in \mathbb{C}$, and $\gamma$ a path in $\mathbb{C} - \{a_{1},\ldots,a_{n}\}$ such that $\gamma(0)=a_{0}$ and $\gamma(1)=a_{n+1}$ :
\begin{equation} \label{eq:Li} I\big( a_{n+1} ; a_{n},\ldots,a_{1} ; a_{0} \big)(\gamma) = \int_{t_{n}=0}^{1}\gamma^{\ast}\bigg(\frac{dz}{z-a_{n}}\bigg)(t_{n}) \int_{t_{n-1}=0}^{t_{n}} \ldots \gamma^{\ast}\bigg(\frac{dz}{z-a_{2}}\bigg)(t_{2}) \int_{t_{1}=0}^{t_{2}} \gamma^{\ast}\big( \frac{dz}{z-a_{1}}\big)(t_{1}) .
\end{equation}
One usually writes $(a_{n},\ldots,a_{1})$ as
$(\overbrace{0,\ldots,0}^{n_{d}-1},z_{d},\ldots,\overbrace{0,\ldots,0}^{n_{1}-1},z_{1},\overbrace{0,\ldots,0}^{n_{0}-1})$ and $a_{n+1}=z$, and an affine change of variable allows to assume that $a_{0}=0$. Finally one writes $(x_{1},\ldots,x_{d})= (\frac{z_{2}}{z_{1}},\ldots,\frac{z}{z_{d}})$. The functions obtained after these transformations are multiple polylogarithms in several variables. They define a solution to the KZ equation (\ref{eq:nablaKZ M0,n}). If $n_{0}=1$, one writes
$(\underbrace{0,\ldots,0}_{n_{d}-1},z_{d},\ldots,\underbrace{0,\ldots,0}_{n_{1}-1},z_{1})=\big( (n_{i})_{d}; (z_{i})_{d} \big)$. MPL's have the following power series expansion :
\begin{equation} \label{eq:Li series bis} \Li\big( (n_{i})_{d};(z_{i})_{d} \big)(z) = \sum_{0<m_{1} <\ldots < m_{d}}
\frac{\big( \frac{z_{2}}{z_{1}} \big)^{m_{1}} \ldots \big(\frac{z}{z_{d}}\big)^{m_{d}}}{m_{1}^{n_{1}}\ldots m_{d}^{n_{d}}} .
\end{equation}
\subsection{Adjoint $p$-adic cyclotomic multiple zeta values}
For each prime number $p$ which does not divide $N$, $K_{p}$ is the extension of $\mathbb{Q}_{p}$ generated by the $N$-th roots of unity in $\overline{\mathbb{Q}_{p}}$.
\newline\indent Let $\mathcal{O}^{\ast}$ be the $\mathbb{Q}$-vector space generated by the empty word and the words of the form
$\big((n_{i})_{d};(\xi_{i})_{d} \big)$, identified to elements of $\mathcal{O}^{\mathcyr{sh}}$ (defined in \S2.1.2) by
$\big((n_{i})_{d};(\xi_{i})_{d} \big) = e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}$.
\subsubsection{Definition}
In addition with the notations above, $b$ is a non-negative integer, and $\Lambda$ is a formal variable.
\begin{Definition} \label{def adjoint}(i) Let adjoint $p$-adic cyclotomic multiple zeta values (Ad$p$MZV$\mu_{N}$'s) be the numbers
\begin{multline} \label{eq:zeta adjoint}
\zeta^{\Ad}_{p,\alpha} \big( (n_{i})_{d};b;(\xi_{i})_{d+1}\big) = (-1)^{d} \sum_{\xi \in \mu_{N}(K)} \xi^{-p^{\alpha}} \Ad_{\Phi^{(\xi)}_{p,\alpha}}(e_{\xi}) \big[e_{0}^{b}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}} \big]
\\ = (-1)^{d} \sum_{d'=0}^{d} \bigg( \prod_{i=d'+1}^{d} {-n_{i} \choose l_{i}} \bigg)\text{ }\xi_{d'}^{-p^{\alpha}}\text{ }
\zeta_{p,\alpha}^{(\xi_{d'})}\big( (n_{d-i}+l_{d-i});(\xi_{d+1-i}) \big)_{0 \leq i \leq d-d'}\text{ }
\zeta_{p,\alpha}^{(\xi_{d'})}\big((n_{i});(\xi_{i})\big)_{1 \leq i \leq d'-1} .
\end{multline}
\noindent (ii) Let the $\Lambda$-adjoint $p$-adic cyclotomic multiple zeta values ($\Lambda$Ad$p$MZV$\mu_{N}$'s) be the following power series :
\begin{multline} \zeta^{\Lambda \Ad}_{p,\alpha} \big( (n_{i})_{d};(\xi_{i})_{d+1} \big) = \Lambda^{n_{1}+\ldots+n_{d}} \sum_{b=0}^{\infty} \Lambda^{b} \zeta_{p,\alpha}^{\Ad} \big((n_{i})_{d};b;(\xi_{i})_{d+1} \big)
\\
= (-1)^{d} \sum_{\xi \in \mu_{N}(K)} \xi^{-p^{\alpha}} \Ad_{\Phi^{(\xi)}_{p,\alpha}}(e_{\xi})
\bigg[ \frac{\Lambda^{n_{1}+\ldots+n_{d}}}{1-\Lambda e_{0}} e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}} \bigg] .
\end{multline}
\end{Definition}
In the case of $\mathbb{P}^{1} - \{0,1,\infty\}$, adjoint $p$-adic multiple zeta values are the numbers
$\zeta^{\Ad}_{p,\alpha}((n_{i})_{d};b) = (\Phi_{p,\alpha}^{-1}e_{1}\Phi_{p,\alpha})[e_{0}^{b}e_{1}e_{0}^{n_{d}-1}e_{1}\ldots e_{0}^{n_{1}-1}e_{1}] \in \mathbb{Q}_{p}$.
\newline\indent Let $\mathcal{O}^{\ast}_{\Ad}$ be the $\mathbb{Q}$-vector space generated by the words of the form $ \big( \big((n_{i})_{d};b;(\xi_{i})_{d+1} \big)\big)$ : $b$ is written separately because it will play a particular role. Moreover, we will see that the $b=0$ and $b>0$ cases will sometimes have different properties (the former corresponds to finite cyclotomic multiple zeta values, see \S6), as already suggested by the formulas for $p$MZV$\mu_{N}$'s in \cite{I-2} in which this distinction exists.
\begin{Definition} Let $K\langle\langle Y_{N}^{\Ad} \rangle\rangle$ be the set of non-commutative formal power series of the following type : $f=\sum\limits_{w=\big( (n_{i})_{d};(\xi_{i})_{d+1}\big)}\sum\limits_{b\in \mathbb{N}} f[w;b](w;b)$ with $f[w;b] \in K$.
\end{Definition}
\subsubsection{Relation with $p$-adic cyclotomic multiple zeta values}
We now show that $p$MZV$\mu_{N}$'s and Ad$p$MZV$\mu_{N}$'s generate the same $k_{N}$-algebra, compatibly with the weight and depth.
Let $K$ be a field of characteristic $0$. Let $\Theta(K)$ be the group of characters of $\mu_{N}$ with values in $K^{\ast}$, i.e. the set of multiplicative morphisms $\mu_{N}(K) \rightarrow (K^{\ast},\ast)$. Let $\mathcal{O}^{\mathcyr{sh}}_{n,\leq d}$
be the $k_{N}$-vector space generated by the elements $\mathcyr{sh}_{i=1}^{r}w_{i}$ where $r\geq 1$ and $w_{i}$ are words such that $\sum\limits_{i=1}^{r}\weight(w_{i})=n$ and $\sum\limits_{i=1}^{r}\depth(w_{i})\leq d$. Below we implicitly view $\Phi$ and $\Phi_{\Ad}$ as functions $\mathcal{O}^{\mathcyr{sh}} \rightarrow K$.
\begin{Definition} (i) For any $\chi \in \Theta$, let $\Moy_{\chi} : K \langle\langle e_{0\cup\mu_{N}} \rangle\rangle \rightarrow K \langle\langle e_{0\cup\mu_{N}} \rangle\rangle$ the map which sends $f \mapsto \sum\limits_{\xi \in \mu_{N}(K)} \chi(\xi) f^{(\xi)}$.
(ii) For any $\Phi \in \tilde{\Pi}_{1,0}(K)$,
let $\Phi_{\Ad,\chi} = \Moy_{\chi} \Ad_{\Phi}(e_{1}) = \sum\limits_{\xi \in \mu_{N}(K)} \chi(\xi) {\Phi^{(\xi)}}^{-1}e_{\xi}\Phi^{(\xi)}$.
\end{Definition}
\begin{Proposition} The maps $\tilde{\Pi}_{1,0}(K) \rightarrow K \langle \langle e_{0 \cup \mu_{N}} \rangle\rangle$, $\Phi \mapsto \Ad_{e_{1}}(\Phi)$ and $\tilde{\Pi}_{1,0}(K) \times \Theta(K) \rightarrow \Theta \langle \langle e_{0 \cup \mu_{N}} \rangle\rangle$, $(\Phi,\chi) \mapsto \Phi_{\Ad,\chi}$, are injective. Moreover, for all $n \geq d\geq 0$, we have $\Phi(\mathcal{O}^{\mathcyr{sh}}_{n,\leq d}) = \Phi_{\Ad,\chi}(\mathcal{O}^{\mathcyr{sh}}_{n+1,\leq d+1})$.
\end{Proposition}
\begin{proof} The injectivity of $\Phi \mapsto \Phi^{-1}e_{1}\Phi$ follows from $K \langle \langle e_{1}\rangle\rangle \cap \tilde{\Pi}_{1,0}(K)= \{1\}$ and the implication $f e_{1} = e_{1}f \Rightarrow f \in K \langle \langle e_{1}\rangle\rangle$. That implication is proved as follows : if $w$ is a word which contains at least a letter different from $e_{1}$, one shows that $f[w]=0$, by writing $w=e_{1}^{n}e_{x}z$ with $x \not=1$ and by an induction on $n$.
Let us prove the rest of the statement. We start with a few preliminary properties :
\noindent\newline\indent (a) for all $x_{1},\ldots,x_{n} \in \{0\} \cup \mu_{N}$, $\Phi^{(\xi)}[e_{x_{n}} \ldots e_{x_{1}}] = \Phi [e_{\xi^{-1}x_{n}} \ldots e_{\xi^{-1}x_{1}}] $. Thus, for all $n\geq d \geq 1$, the $k_{N}$-vector space generated by the $\Phi^{(\xi)}[w]$'s with $w$ a word of weight $n$ and depth $d$ is independent of $\xi$.
\newline\indent (b) For $f \in \tilde{\Pi}_{1,0}(K)$, writing $f^{-1}f=1$ we get by induction that, for all $n\geq d \geq 1$,
we have $f(\mathcal{O}^{\mathcyr{sh}}_{n,\leq d}) = f^{-1}(\mathcal{O}^{\mathcyr{sh}}_{n,\leq d})$ and we also have $f^{-1}[w] \equiv - f[w] \mod f(\mathcal{O}^{\mathcyr{sh}}_{n,\leq d-1})$ for all words $w$.
\newline\indent (c) For $f$ a solution to the shuffle equation or to the shuffle equation modulo products and such that $f[e_{0}] = 0$, any $f[w]$ with $w$ a word of weight $n$ and depth $d$ is a $\mathbb{Z}$-linear combination of the $f[w']$ with $w'$ a word of weight $n$ and depth $d$ of the form $w'=e_{0}w''e_{\xi}$ with $\xi \in \mu_{N}(K)$.
\newline\indent (d) By (a) and the shuffle equation, for $\Phi \in \tilde{\Pi}_{1,0}(K)$, we have $\Phi^{(\xi)}[e_{0}^{l}] = \Phi^{(\xi)}[e_{\xi}] = 0$ for all $\xi \in \mu_{N}(K)$ and $l\geq 1$.
\newline\indent The inclusion $\Phi(\mathcal{O}^{\mathcyr{sh}}_{n,\leq d}) \supset \Phi_{\Ad}(\mathcal{O}^{\mathcyr{sh}}_{n+1,\leq d+1})$ is clear by the shuffle equation for $\Phi$, and we want to prove the converse inclusion and the injectivity, by an induction on $d$.
\newline\indent The only non-zero coefficient in $\Phi$ in depth $0$ is the coefficient of the empty word (weight 0), equal to 1. The only non-zero coefficients in $\Phi_{\Ad,\chi}$ in depth $\leq 1$ are the coefficients $\Phi_{\Ad,\chi}[e_{\xi}]$ (weight 1), equal to $\chi(\xi)$. They are in $k_{N} \subset K$. This determines $\chi$ in terms of $\Phi_{\Ad,\chi}$. We have $\Phi(\mathcal{O}^{\mathcyr{sh}}_{n,\leq 0}) = \Phi_{\Ad,\chi}(\mathcal{O}^{\mathcyr{sh}}_{n+1,\leq 1})$ for all $n$. This proves the result for $d=0$.
\newline\indent Now for any $d>0$, we consider a word of the form
$w=e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1} \ldots e_{0}^{n_{2}-1}e_{\xi_{2}}e_{\xi_{1}}$ with $l>0$, and the $\xi_{i}$'s in $\mu_{N}(K)$ (i.e. the special case $n_{1}=1$ and $l>0$ in usual words). We have by (d)
$$ \Phi_{\Ad,\chi}[e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1} \ldots e_{0}^{n_{2}-1}e_{\xi_{2}}e_{\xi_{1}}] \equiv \chi(\xi_{0}) {\Phi^{(\xi_{0})}}^{-1}[e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1} \ldots e_{0}^{n_{2}-1}e_{\xi_{2}}] \mod \Phi(\mathcal{O}^{\mathcyr{sh}}_{n,\leq d-1}) $$
By induction on $d$ and (a), (b), (c) above, this determines $\Phi^{(\xi)}$, thus $\Phi$, in depth $\leq d$, in terms of $\Phi_{\Ad}$, thus it proves the injectivity in depth $\leq d$, and this also proves the inclusion $\Phi(\mathcal{O}^{\mathcyr{sh}}_{n,\leq d}) \subset \Phi_{\Ad,\chi}(\mathcal{O}^{\mathcyr{sh}}_{n,\leq d+1})$.
\end{proof}
\begin{Corollary} The $p$MZV$\mu_{N}$'s and Ad$p$MZV$\mu_{N}$'s generate the same $k_{N}$-algebra.
More precisely, for any $n \geq d\geq 1$, the two following vector spaces are equal : the $k_{N}$-vector space of generated by products
$\prod_{i=1}^{r}\zeta^{\Ad}_{p,\alpha}(w)$ with $r \geq 1$,
$\sum\limits_{i=1}^{r} \weight(w_{i}) = n$ and $\sum\limits_{i=1}^{r} \depth(w_{i}) \leq d$, and The $k_{N}$-vector space generated by products $\prod_{i=1}^{r}\zeta^{\Ad}_{p,\alpha}(w)$ with $r \geq 1$,
$\sum\limits_{i=1}^{r} (\weight(w_{i})-1) = n$ and $\sum\limits_{i=1}^{r} (\depth(w_{i})-1) \leq d$.
\end{Corollary}
\begin{proof} We apply Proposition 2.2.4 to $\Phi = \Phi_{p,\alpha}$ and $\chi : \xi \mapsto \xi^{-p^{\alpha}}$.
\end{proof}
\subsection{Adjoint complex cyclotomic multiple zeta values}
\subsubsection{Definition}
The most direct complex analogue of Ad$p$MZV$\mu_{N}$'s (Definition \ref{def adjoint}) is the following.
Let a character $\chi : \mu_{N}(\mathbb{C}) \rightarrow \mathbb{C}^{\ast}$ and let $\Phi_{\KZ,\Ad,\chi} = \sum\limits_{\xi \in \mu_{N}(K)} \chi(\xi) {\Phi_{\KZ}^{-1}}^{(\xi)}e_{\xi}{\Phi_{\KZ}}^{(\xi)}$.
\begin{Definition} \label{def adjoint complex}(i) Let the adjoint cyclotomic multiple zeta values be the numbers :
\begin{equation} \label{eq:complex01} \zeta^{\Ad}\big( (n_{i})_{d};(\xi_{i})_{d+1};l;\chi) = \Phi_{\KZ,\Ad,\chi} [e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}] .
\end{equation}
(ii) Let the $\Lambda$-adjoint cyclotomic multiple zeta values be the numbers :
\begin{equation} \label{eq:complex02} \zeta^{\Lambda \Ad}((n_{i})_{d};(\xi_{i})_{d+1};\chi) = \Phi_{\KZ,\Ad,\chi} \big[ \frac{1}{1-\Lambda e_{0}}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}} \big] .
\end{equation}
\end{Definition}
Whereas in the crystalline setting, one has the Frobenius automorphism, whose restriction to $\Lie(\Pi_{0,0})$ sends $(e_{0},e_{1}) \mapsto (\frac{e_{0}}{p},\Phi_{p,1}^{-1}\frac{e_{1}}{p}\Phi_{p,1})$, in the Betti-de Rham setting, one has the automorphism of $\Pi_{0,0}$ which expresses the monodromy of the KZ connection (\ref{eq: nabla KZ}), which sends $(e^{e_{0}},e^{e_{1}}) \mapsto (e^{2i\pi e_{0}}, \Phi_{\KZ}^{-1}e^{2i\pi e_{1}}\Phi_{\KZ})$. Let $\comp^{B,DR}$ be the Betti-de Rham comparison isomorphism of $\Pi_{0,0}$ ; let $\gamma$ be the straight path $[0,1] \rightarrow \mathbb{C}$, $t\mapsto t$ ; let $c_{0}$, resp. $c_{1}$ be a simple loop around $0$, resp. $1$ positively oriented ; we have
$$ (e^{2i\pi e_{0}},\Phi_{\KZ}^{-1}e^{2i\pi e_{1}}\Phi_{\KZ}) = \big(\comp^{\B,\dR}(c_{0}), \comp^{\B,\dR} (\gamma^{-1}c_{1}\gamma) \big) . $$
It is thus also natural to consider the following :
\begin{Definition} Let
\begin{equation} \label{eq:complex01 bis} \zeta_{\exp}^{\Ad}\big( (n_{i})_{d};(\xi_{i})_{d+1};l;\chi) = \sum\limits_{\xi \in \mu_{N}(\mathbb{C})} \chi(\xi) ({\Phi_{\KZ}^{(\xi)}}^{-1}e^{2i\pi e_{\xi}}\Phi^{(\xi)}_{\KZ}) [e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}] .
\end{equation}
\end{Definition}
We define similarly $\zeta_{\exp}^{\Lambda,\Ad}$. Finally, since unlike in the $p$-adic case there is no privileged choice of $\chi$, it is also natural to consider the following :
\begin{Definition} Let
\begin{equation} \label{eq:complex01 bis} \zeta_{\exp,1}^{\Ad}\big( (n_{i})_{d};(\xi_{i})_{d+1};l) = ({\Phi_{\KZ}^{(\xi)}}^{-1}e^{2i\pi e_{\xi}}\Phi^{(\xi)}_{\KZ}) [e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}] .
\end{equation}
\end{Definition}
We define similarly $\zeta_{\exp,1}^{\Lambda,\Ad}$
We have ${\Phi^{(\xi)}_{\KZ}}^{-1}e^{2i\pi e_{\xi}}\Phi^{(\xi)}_{\KZ} = 1 + 2i\pi {\Phi^{(\xi)}_{\KZ}}^{-1}e_{\xi}\Phi^{(\xi)}_{\KZ} + \zeta(2)
{\Phi_{\KZ}^{(\xi)}}^{-1}\sum\limits_{n\geq 2} \frac{(2i\pi)^{n-2}e_{1}^{n}}{n!} {\Phi_{\KZ}^{(\xi)}}^{-1}$.
Thus, the numbers (\ref{eq:complex01 bis}) divided by $2 \pi i$ are congruent to the numbers (\ref{eq:complex01}) modulo $\zeta(2)$ and they satisfy the results of \S3 and \S4.
One has a variant of cyclotomic multiple zeta values defined as follows.
Let $\phi_{\infty}$ be the Frobenius at infinity of $\pi_{1}^{\un,\dR}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$ ;
it induces the automorphism of $\Lie(\Pi_{0,0})$ which sends $(e_{0},e_{1}) \mapsto (-e_{0},-\Phi_{-}^{-1}e_{1}\Phi_{-})$, where $\Phi_{-} = \phi_{\infty}({}_{\vec{1}_{1}} 1 _{\vec{1}_{0}}) \in \Pi_{1,0}(\mathbb{R})$. $\Phi_{-}$ is $h_{\KZ}$ in \cite{Enriquez}, \S11.1 whereas the $N=1$ case of $\Phi_{-}$ is $g_{\KZ}$ in \cite{Enriquez}, \S11.1. The coefficients of $\Phi_{-}$ are denoted by
$\zeta_{-}\big((n_{i})_{d};(\xi_{i})_{d}\big) = \Phi_{-}[e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}]$. They generate a $\mathbb{Q}$-vector subspace of the one of MZV$\mu_{N}$'s. (For details in the $N=1$ case see \cite{Furusho 2}, \S2.2).
\begin{Definition} Let the adjoint variants of the numbers $\zeta_{-}$ be :
\begin{equation} \zeta_{-}^{\Ad} \big( (n_{i})_{d};(\xi_{i})_{d+1};b \big) = (-1)^{d}
\sum\limits_{\xi \in \mu_{N}(\mathbb{C})} \xi
({\Phi^{(\xi)}_{-}}^{-1}e_{\xi}\Phi^{(\xi)}_{-})
\big[ e_{0}^{b} e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}} \big] .
\end{equation}
\end{Definition}
In the $N=1$ case, $\Phi_{-}$ is in $\GRT_{1}(\mathbb{R})$ and in $\DS_{0}(\mathbb{R})$, \cite{Furusho 2} (in particular $\Phi_{-}[e_{0}e_{1}]=0$). In the general case, we have a similar fact by \cite{Enriquez}, \S11.1 and \cite{Racinet}.
\subsubsection{Relation with cyclotomic multiple zeta values}
\begin{Proposition} (i) The respective $k_{N}[2\pi i]$-algebras generated by the numbers $\zeta^{\Ad}(.)$, $\zeta_{\exp}^{\Ad}(.)$, $\zeta_{\exp,1}^{\Ad}(.)$ and $\zeta(.)$ are equal.
\newline (ii) The respective $k_{N}$-algebras generated by the numbers $\zeta_{-}^{\Ad}(.)$ and $\zeta^{-}(.)$ are equal.
\end{Proposition}
\begin{proof} Similar to the proof of Proposition 2.2.4 plus the inversion of a Vandermonde linear system.
By inverting a Vandermonde linear system, the data of (\ref{eq:complex01}) resp. (\ref{eq:complex01 bis}) is equivalent respectively to the data of the sequences
$\displaystyle \bigg( ({\Phi_{\KZ}^{(\xi)}}^{-1} e_{\xi}\Phi^{(\xi)}_{\KZ}) [e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}] \bigg)_{\xi \in \mu_{N}(K)}$, and
\newline
$\bigg( ({\Phi_{\KZ}^{(\xi)}}^{-1}e^{2i\pi e_{\xi}}\Phi^{(\xi)}_{\KZ}) [e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}]\bigg)_{\xi \in \mu_{N}(K)}$. Moreover, we have $\exp \big( {\Phi_{\KZ}^{(\xi)}}^{-1} e_{\xi}\Phi^{(\xi)}_{\KZ}) \big) = {\Phi_{\KZ}^{(\xi)}}^{-1} \exp(2\pi i e_{\xi}) \Phi^{(\xi)}_{\KZ}$. We also note that in the two above sequences, all the terms can be obtained one from another by applying an automorphism $(z \mapsto \xi z)_{\ast}$ of
$\pi_{1}^{\un,\dR}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$.
\end{proof}
Note that, as in the $p$-adic case, the proof gives more precise information on the fact that the above equalities are compatible with depth filtration. They are also of course compatible with the weight.
In the end, in the complex case there are several natural objects which can be called adjoint variants of cyclotomic multiple zeta values, and they can be considered as equivalent. In function of the property that we are looking at, it can be more convenient to consider a version or another of this object.
\subsection{Cyclotomic multiple harmonic values}
\subsubsection{Definition\label{mhv definition}}
\begin{Definition} \label{def harmonic}For any index $w=((n_{i})_{d};(\xi_{i})_{d+1})$, let
\newline\indent (i) for each $p \in \mathcal{P}_{N}$, $\har_{p^{\mathbb{N}}}(w)
= \big( \har_{p^{\alpha}} (w)\big)_{\alpha \in \mathbb{N}} \in K_{p}^{\mathbb{N}}$, which we call a $p$-adic cyclotomic multiple harmonic value.
\newline\indent (ii) for each $\alpha \in \mathbb{N}^{\ast}$, $\har_{\mathcal{P}^{\alpha}}(w) = \big( \har_{p^{\alpha}} (w)\big)_{p \in \mathcal{P}_{N}} \in \prod_{p\in\mathcal{P}_{N}} K_{p}$, which we call an adelic cyclotomic multiple harmonic value.
\newline\indent (iii) $\har_{\mathcal{P}_{N}^{\mathbb{N}}}(w) =
\big( \har_{p^{\alpha}} (w)\big)_{(p,\alpha) \in \mathcal{P}_{N} \times \mathbb{N}} \in \big( \prod_{p\in\mathcal{P}_{N}} K_{p} \big)^{\mathbb{N}}$, which we call a ($p$-adic $\times$ adelic) cyclotomic multiple harmonic value.
\end{Definition}
We will view (i) as the natural explicit $p$-adic substitute to $p$MZV$\mu_{N}$'s ; (ii) as the natural lift of the cyclotomic finite MZV's, defined in \S6 ; (iii) as the natural way to formulate the algebraic properties of (i) and (ii) in a unified way. We will refer most of the time to (iii) and omit the adjective ``$p$-adic $\times$ adelic''. We will abbreviate cyclotomic multiple harmonic values as MHV$\mu_{N}$'s.
\newline\indent Let $\mathcal{O}^{\ast}_{\har}$ be the $\mathbb{Q}$-vector space generated by the empty word and the words of the form $\big((n_{i})_{d};(\xi_{i})_{d+1} \big)$.
\subsubsection{Setting for computations, integrals at (1,0)\label{setting infinite sums}}
In the framework of integrals at $(1,0)$, represented by the symbol $\int_{1,0}$, we will transfer algebraic properties from MZV$\mu_{N}$'s to $\Lambda$-adic AdMZV$\mu_{N}$'s, and the $\Lambda=1$ case will give properties of MHV$\mu_{N}$'s, via equation (\ref{eq:formula for n=1}).
By that equation, the index $\big((n_{i})_{d};(\xi_{i})_{d+1} \big)$ of MHV$\mu_{N}$'s is identified to $\frac{1}{1-e_{0}}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}$.
The map $\tau(\Lambda) : \mathcal{O}^{\mathcyr{sh}} \rightarrow \mathcal{O}^{\mathcyr{sh}}[\Lambda] ; w \mapsto \Lambda^{\weight(w)}w$ sends an element to its orbit under the motivic Galois action $\tau$ of $\mathbb{G}_{m} (\ref{eq:tau})$.
\begin{Notation} \label{definition sigma inv DR}
(i) Let $\comp^{\text{har},\Ad} : \mathcal{O}^{\mathcyr{sh}} \rightarrow \widehat{\mathcal{O}^{\mathcyr{sh}}}$, $w \mapsto \frac{1}{1-e_{0}}w$.
\newline (ii) Let $\comp^{\Lambda \Ad,\Ad} : \mathcal{O}^{\mathcyr{sh}} \rightarrow \widehat{\tau(\Lambda)\mathcal{O}^{\mathcyr{sh}}}$, $w \mapsto \frac{1}{1-\Lambda e_{0}} w$.
\newline (iii) Let $\comp_{\Lambda}^{\Lambda \Ad,\har} : \widehat{\mathcal{O}^{\mathcyr{sh}}} \mapsto \widehat{\tau(\Lambda)\mathcal{O}^{\mathcyr{sh}}}$, $\frac{1}{1-e_{0}}w \mapsto \frac{1}{1-\Lambda e_{0}}w$.
\end{Notation}
We also denote in the same way the duals of these maps. The map $\comp^{\text{har},\Ad}$ was called $\comp^{\Sigma \smallint}$, ``comparison from integrals to series'', in \cite{I-2}. In this paper, it seems more relevant to call it the ``comparison from adjoint (analogues of MZV$\mu_{N}$'s) to harmonic (analogues of MZV$\mu_{N}$'s)''. We note that $\comp^{\Lambda \Ad, \Ad} = \widehat{\tau(\Lambda)}^{-1} \circ \big(\frac{1}{\Lambda}\comp^{\har,\Ad}\big) \circ \widehat{\tau(\Lambda)}$.
\newline\indent The strategy of transferring algebraic properties stated above is motivated by the notions of adjoint Ihara action $\circ_{\Ad}^{\smallint_{1,0}}$ (\cite{I-2} Definition 1.1.3) and pro-unipotent harmonic action $\circ_{\har}^{\smallint_{1,0}}$ (\cite{I-3} Definition 2.1.2), which are pushforwards of the usual Ihara action (\ref{eq:Ihara}) by $\Ad(e_{1})$ and $\comp^{\text{har},\Ad}$ which appears implicitly in equation (\ref{eq:formula for n=1}) ; and with the fact that the Ihara action is compatible with the algebraic relations of MZV$\mu_{N}$'s.
\newline\indent $\Ad(e_{1})$ restricted to $ \tilde{\Pi}_{1,0}(K) = \{f \in \Pi_{1,0}(K) \text{ }|\text{ }f[e_{0}] = f[e_{1}] = 0\}$ is injective. The map $\comp^{\Sigma \smallint}$, viewed here as $\comp^{\text{har},\Ad}$, is not injective, but taking into account all the relations of iteration of the Frobenius enables to replace it by an injective map (\cite{I-3}, \S4.2).
Thus we consider that applying the push-forward by $\Ad(e_{1})$ and $\comp^{\text{har},\Ad}$ should not lose information.
\subsubsection{Setting for computations, power series expansions of integrals at $0$}
In that framework, we will view prime weighted cyclotomic multiple harmonic sums via equation (\ref{eq:Li 0}). Let us write more precisely how they are connected to multiple polylogarithms.
\begin{Notation} \label{notation coefficient of power series}For $S = \sum\limits_{(n_{1},\ldots,n_{r}) \in \mathbb{N}^{r}} c_{n_{1},\ldots,n_{r}} x_{1}^{n_{1}} \ldots x_{r}^{n_{r}} \in R[[x_{1},\ldots,x_{r}]]$ a formal power series, where $R$ is a ring, for all $(n_{1},\ldots,n_{r}) \in \mathbb{N}^{r}$, we write $c_{n_{1},\ldots,n_{r}}= S[x_{1}^{n_{1}} \ldots x_{r}^{n_{r}}]$.
\end{Notation}
By the power series expansion of multiple polylogarithms (\ref{eq:Li series bis}), we have two slightly different formulas :
\begin{equation} \har_{m}\big( (n_{i})_{d};(\xi_{i})_{d+1} \big) = m^{n_{1}+\ldots+n_{d}+l}\Li[e_{0}^{l-1}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}][z^{m}];
\end{equation}
\begin{equation} \label{eq:har et Li 2} \har_{m}((n_{i})_{d};(\xi_{i})_{d+1}) = m^{n_{1}+\ldots+n_{d}} \sum_{0<m'<m} \xi_{d+1}^{m-m'} \Li[ e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}][z^{m'}].
\end{equation}
\indent We consider multiple polylogarithms restricted to a function on a $(\mathbb{P}^{1} - Z)^{2} - \text{diagonal}$ where $Z$ is a finite subset of $\mathbb{P}^{1}$, and we choose $O = (\vec{1}_{0},\vec{1}_{0})$ in cubic coordinates as the origin of paths of integration.
\newline\indent We use the notation $\omega_{z}=\frac{dy}{y-z}$ for $z \in Z- \{0,\infty\}$. By (\ref{eq:Li series bis}), for $(y_{1},y_{2})$ on a neighbourhood of the chosen reference base-point $O$, for $\gamma$ the straight path from $O$ to $(y_{1},y_{2})$, for any two words
$w = \big( (n_{i})_{d}:(z_{i})_{d} \big)$,
$\tilde{w} = \big( (\tilde{n}_{i})_{d}:(\tilde{z}_{i})_{d} \big)$, we have :
\begin{multline}
\int_{\gamma} \big( \omega_{0}^{t_{d'}-1}\omega_{z_{d'}} \ldots \omega_{0}^{t_{1}-1}\omega_{z_{1}}\big)(y_{2}) \big(\omega_{0}^{n_{d}-1}\omega_{z_{d}} \ldots \omega_{0}^{n_{1}-1}\omega_{z_{1}}\big)(y_{1}y_{2})
\\ =
\sum_{0<m_{1}<\ldots<m_{d}<m'_{1}<\ldots<m'_{d'}=m}
\frac{\big( \frac{z_{2}}{z_{1}} \big)^{m_{1}} \ldots \big(\frac{y_{1}}{z_{d}}\big)^{m}
\big( \frac{\tilde{z}_{2}}{\tilde{z}_{1}} \big)^{m_{1}} \ldots \big(\frac{y_{2}}{\tilde{z}_{d'}}\big)^{m}} {m_{1}^{n_{1}}\ldots m_{d}^{n_{d}} m_{1}^{\tilde{n}_{1}}\ldots m_{d'-1}^{\tilde{n}_{d'-1}} m_{d}^{\tilde{n}_{d'}}}.
\end{multline}
\subsubsection{Setting for computations, series}
In this framework, we will view prime weighted multiple harmonic sums via equation (\ref{eq:mult har sums}). We consider them as functions of the upper bound $m$ of their domain of summation, $\mathbb{N} \rightarrow \mathbb{C}$, resp. $\mathbb{N} \rightarrow K_{p}$. We will study them by using only the structure of topological field of $\mathbb{C}$ and $\mathbb{C}_{p}$, both at the source and at the target.
\newline\indent If $c : \mathbb{N} \rightarrow \mathbb{C}$ is a function such that $c(0)=0$, the Newton series of $c$ is the function : $\displaystyle z \mapsto \sum\limits_{m \in\mathbb{N}} (\nabla c)_{m} {z \choose m}$, where
$\nabla c = \Big(\sum\limits_{m'=0}^{m} (-1)^{m'}{m \choose m'} c(m')\Big)_{m \in \mathbb{N}}$ and $\displaystyle {z \choose m} = \frac{z(z-1)\ldots(z-m+1)}{m!}$. It is usually defined on a half-plane of the type $\{ \text{Re}(z) > \rho \}$ of $\mathbb{C}$. This notion appears in certain proofs of results on multiple harmonic sums by several other authors, which we will generalize and interpret in terms of cyclotomic multiple harmonic values.
\subsection{``Overconvergent'' variants of cyclotomic multiple harmonic values}
\subsubsection{Definition}
For any word $w$ on $e_{0 \cup \mu_{N}}$, the power series expansion of the functions $\Li_{p,X_{K}}^{\KZ}[w]$ at $0$ in $k_{N}[[z]][\log(z)]$ in is identical to the one of the complex multiple polylogarithm $\Li[w]$ of (\ref{eq:multiple polylogarithms power series expansion}). Thus, the coefficients of the power series expansions of $\Li_{p,\alpha}^{\dagger}$ can be written in terms of multiple harmonic sums and cyclotomic $p$-adic multiple zeta values. In the next statement, we use Notation \ref{notation coefficient of power series}.
\begin{Definition}
\noindent (i) Let the overconvergent variants of cyclotomic multiple harmonic sums be the numbers $\frak{h}_{m}^{\dagger_{p,\alpha}}(w) = \Li_{p,\alpha}^{\dagger}[w][z^{m}] \in K$, where $w$ is a word on $e_{0 \cup \mu_{N}}$ and $m \in \mathbb{N}$.
\newline (ii) Let the overconvergent variants of weighted cyclotomic multiple harmonic sums be the numbers $\har_{m}^{\dagger_{p,\alpha}}(w) = m^{\weight(w)}\Li_{p,\alpha}^{\dagger}[w][z^{m}]$, where $w$ is a word on $e_{0 \cup \mu_{N}}$ and $m \in \mathbb{N}$.
\newline (iii) Let overconvergent prime weighted cyclotomic multiple harmonic sums be the numbers
$\har_{p^{\alpha}}^{\dagger_{p,\alpha}}(w)$, where $w$ is a word on $e_{0 \cup \mu_{N}}$.
\end{Definition}
The following is a variant of Definition \ref{def harmonic}, from which we take the same notations. We now consider all $p$'s resp. all $\alpha$'s at the same time.
\begin{Definition} \label{variant harmonic}For any index $w= \big( (n_{i})_{d};(\xi_{i})_{d+1} \big)$, we call respectively
\newline (i) for each $p \in \mathcal{P}_{N}$, $\har^{\dagger}_{p^{\mathbb{N}}}(w)
= \big( \har^{\dagger_{p,\alpha}}_{p^{\alpha}} (w)\big)_{\alpha \in \mathbb{N}} \in K_{p}^{\mathbb{N}}$, overconvergent variants of adelic cyclotomic multiple harmonic values.
\newline (ii) for each $\alpha \in \mathbb{N}^{\ast}$, $\har^{\dagger}_{\mathcal{P}^{\alpha}}(w) = \big( \har^{\dagger_{p,\alpha}}_{p^{\alpha}} (w)\big)_{p \in \mathcal{P}_{N}} \in \underset{p\in\mathcal{P}_{N}}{\prod} K_{p}$, overconvergent variants of $p$-adic cyclotomic multiple harmonic values.
\newline (iii) $\har^{\dagger}_{\mathcal{P}_{N}^{\mathbb{N}}}(w)=\big( \har^{\dagger_{p,\alpha}}_{p^{\alpha}} (w)\big)_{(p,\alpha) \in \mathcal{P}_{N} \times \mathbb{N}} \in \big( \underset{p\in\mathcal{P}_{N}}{\prod} K_{p} \big)^{\mathbb{N}}$, overconvergent variants of ($p$-adic $\times$ adelic) cyclotomic multiple harmonic values.
\end{Definition}
As in the previous sections, we will usually refer to (iii) and omit the adjective ``$p$-adic $\times$ adelic''. We will use the abbreviation MHV$\mu_{N}^{\dagger}$.
\subsubsection{Setting for computations}
In the previous parts, we have used three settings for computations (\S2.2.2, \S2.2.3, \S2.2.4) for studying prime weighted multiple harmonic values, corresponding to the frameworks $\int_{1,0}$, $\int$ and $\Sigma$ respectively. We now add to them a complement :
\begin{Proposition} \label{prop formula dagger} (i) ($\int_{1,0}$) We have :
\begin{equation} \label{eq:overconv expression}
\har_{p^{\alpha}}^{\dagger_{p,\alpha}}[e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}]
= \sum_{b=l}^{\infty} \sum_{\xi \in \mu_{N}(K)} \xi^{-p^{\alpha}}
\Ad_{{\Phi_{p,\alpha}^{(\xi)}}}(e_{\xi}) [e_{0}^{b}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}] .
\end{equation}
\noindent (ii) ($\int$) \label{lemma power series expansion Li dagger} We have $ \Li_{p,\alpha}^{\dagger}[w][z^{p^{\alpha}}] = \har^{\dagger}_{p,\alpha}$ and, for $m \in \{1,\ldots,p^{\alpha}-1\}$, $\Li_{p,\alpha}^{\dagger}[w][z^{m}] = (p^{\alpha})^{\weight(w)}\Li(z)[z^{m}] = \frac{1}{m^{l}}\frak{h}_{m} \big((n_{i})_{d};(\xi_{i})_{d+1} \big)$.
\newline\noindent (iii) ($\Sigma$) We can write a formula as sums of series for each $\har_{p^{\alpha}}^{\dagger_{p,\alpha}}(w)$ by composing (\ref{eq:overconv expression}) with the formula for $\Ad_{\Phi_{p,\alpha}^{(\xi)}}(e_{\xi})$ as sums of series obtained in the main theorem of \cite{I-2}.
\end{Proposition}
\begin{proof} (i) For any word $w=e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}$ and let us consider the coefficient $[w][z^{p^{\alpha}}]$ of equation (\ref{eq:horizontality1}) ; we obtain :
\begin{multline} \label{eq:remainder} \har_{p^{\alpha}}^{\dagger_{p,\alpha}}
(e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}})
\\ = \har_{p^{\alpha}} \big( (n_{i})_{d};(\xi_{i})_{d+1} \big) -
\sum_{b=0}^{l} \sum_{\xi \in \mu_{N}(K)} \xi^{-1}
\Ad_{{\Phi_{p,\alpha}^{(\xi)}}}(e_{\xi}) [e_{0}^{b}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}] .
\end{multline}
If we combine this and equation (\ref{eq:formula for n=1}) proven in \cite{I-2}, we obtain the result. (In \cite{I-2}, we actually obtained equation (\ref{eq:formula for n=1}) by proving that $\har_{p^{\alpha}}^{\dagger_{p,\alpha}}(e_{0}^{l}e_{z_{i_{d+1}}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}) \displaystyle \underset{l \rightarrow \infty}{\rightarrow} 0$, which was an immediate consequence of the main result of \cite{I-1}.)
\newline (ii) Follows from equation (\ref{eq:horizontality1}) and from that we have $\Li(z^{p^{\alpha}})[z^{m}]=0$ for all $m \in \{1,\ldots,p^{\alpha}-1\}$.
\newline (iii) Immediate.
\end{proof}
In particular, (\ref{eq:overconv expression}) means that the numbers $\har_{p^{\alpha}}^{\dagger_{p,\alpha}}(w)$ are the remainders of the sums of series of (\ref{eq:formula for n=1}) which express $\har_{p^{\alpha}}$ in terms of cyclotomic $p$-adic multiple zeta values. This leads us to a variant of Definition \ref{def adjoint} (ii) :
\begin{Definition} \label{def over adjoint}
Let, for all words, the overconvergent variants of $\Lambda$-adjoint $p$-adic cyclotomic multiple zeta values ($\Lambda$Ad$p$MZV$\mu_{N}^{\dagger}$'s) be
\begin{multline} \zeta^{\Lambda,\Ad,\dagger}_{p^{\alpha}} (e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{i_{1}}})
\\ = \sum_{b \geqslant l} \sum_{\xi \in \mu_{N}(K)} \xi^{-p^{\alpha}} \Lambda^{b+n_{d}+\ldots+n_{1}}
\Ad_{\Phi_{p,\alpha}^{(\xi)}}(e_{\xi})
[e_{0}^{b}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}}\ldots e_{0}^{n_{1}-1}e_{\xi_{1}}] .
\end{multline}
\end{Definition}
In particular, the $l=0$ case of $\Lambda$Ad$p$MZV$\mu_{N}^{\dagger}$'s are just $\Lambda$Ad$p$MZV$\mu_{N}$'s. For $l\geqslant 0$, they are remainders of $\Lambda$Ad$p$MZV$\mu_{N}$'s viewed as power series in $\Lambda$.
\subsection{Adjoint multiple polylogarithms and harmonic multiple polylogarithms}
Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values are particular values of functions which retain a lot of their properties, and which we introduce now.
\subsubsection{Adjoint multiple polylogarithms}
In this context, the generalization of cyclotomic multiple zeta values are multiple polylogarithms, with the only restriction that we assume them to be iterated integrals on the straight path from $0$ to $1$.
\newline\indent This suggests the following generalization of Definition \ref{def adjoint} :
\begin{Definition} Let adjoint multiple polylogarithms be
\begin{multline*} \Li^{\Ad}\big( (n_{i})_{d},l;(z_{i})_{d+1} \big) = (-1)^{d} \sum_{z \in Z} z^{-1} ({\Li^{(z)}}^{-1}e_{z}\Li^{(z)})\big[e_{0}^{l}e_{z_{d+1}}e_{0}^{n_{d}-1}e_{z_{d}}\ldots e_{0}^{n_{1}-1}e_{z_{1}} \big] \\ = \sum_{d'=1}^{d+1} \prod_{i=d'}^{d} {-n_{i} \choose l_{i}}
z_{d'}^{-1} (-1)^{n_{d'}+\ldots+n_{d}}
\Li^{(z_{d'})} \big( (n_{d-i}+l_{d-i})_{d-d'},(z_{d-i})_{d-d'} \big)
\Li^{(z_{d'})} \big( (n_{i})_{d'-1},(z_{i})_{d'-1} \big)
\end{multline*}
where $\Li^{(z)}$ denotes an iterated integral on a straight path from $0$ to $z$.
\end{Definition}
Of course, as in \S7.1 we also have a variant where $e_{z}$ above is replaced by $e^{2i\pi e_{z}}$. We note that we have the following adjoint variant of the KZ equation (\ref{eq: nabla KZ}) : for all $u \in \Lie(\Pi_{0,0})$,
$$ d\Ad_{\Li}(u) =
\Ad_{\Li_{p,X_{K}}^{\KZ}} \bigg( \ad_{u} \Big( e_{0} \omega_{0} + \sum_{z_{0} \in D- \{0,\infty\}} e_{z_{0}}\omega_{z_{0}} \Big) \bigg) . $$
Moreover, in the case of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$, we have an adjoint variant of the differential equation satisfied by the overconvergent $p$-adic multiple polylogarithms (\ref{eq:horizontality equation}) : for all $u \in \Lie(\Pi_{0,0})$,
\begin{multline*}
d\Ad_{\Li_{p,\alpha}^{\dagger}}(u) =
\\
\Ad_{\Li_{p,\alpha}^{\dagger}} \bigg( \ad_{u} \Big( \omega_{0}(z^{p^{\alpha}})e_{0} + \sum_{\xi \in \mu_{N}(K)} \omega_{\xi^{p^{\alpha}}}(z^{p^{\alpha}})e_{\xi^{p^{\alpha}}} \Big) \bigg) -
\ad_{\Ad_{\Li_{p,\alpha}^{\dagger}}(u)} \Big(
\omega_{0}(z) e_{0} + \sum_{\xi \in \mu_{N}(K)} \omega_{\xi}(z)\Ad_{\Phi^{(\xi^{p^{\alpha}})}_{p,\alpha}}(e_{\xi^{p^{\alpha}}}) \Big) .
\end{multline*}
\subsubsection{A generalization of cyclotomic multiple harmonic values and finite cyclotomic multiple zeta values}
We now assume that $\mathbb{P}^{1} - D$ is defined over a number field, which is embedded in $\mathbb{C}_{p}$ for all primes $p$.
\begin{Definition} Let multiple harmonic polylogarithms be
$$ \har_{\mathcal{P}^{\mathbb{N}}} =
\bigg( \sum_{0<m_{1} <\ldots < m_{d}<p^{\alpha}}
\frac{\big( \frac{z_{2}}{z_{1}} \big)^{m_{1}} \ldots \big(\frac{z_{d+1}}{z_{d}}\big)^{m_{d}}\big(\frac{1}{z_{d+1}}\big)^{p^{\alpha}}}{m_{1}^{n_{1}}\ldots m_{d}^{n_{d}}} \bigg) \in \prod_{p \in \mathcal{P}} \mathbb{C}_{p}^{\mathbb{N}^{\ast}} . $$
\end{Definition}
Unlike in \S2.4 and \S2.5, we do not claim a crystalline meaning for this notion.
\section{Around double shuffle equations\label{double shuffle}}
We review the regularized double shuffle equations (\S3.1), we construct adjoint double shuffle equations (\S3.2) and harmonic double shuffle relations (\S3.3), in the three frameworks $\int_{1,0}$, $\int$, $\Sigma$, which proves the theorem stated in \S1.3. We discuss a particular consequence of the double shuffle equation, which we call the ``reversal'' equation, and we find its adjoint and harmonic counterparts (\S3.4).
\subsection{Review on the double shuffle equations}
Let $K$ be a field of characteristic zero which contains a primitive $N$-th root of unity. Let the alphabet $Y_{N} =\{ y_{n}^{(\xi)}\text{ }|\text{ }n\geqslant 1, \xi \in \mu_{N}(K)\}$. A word on $e_{0\cup \mu_{N}}$ of the form $e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}$ can be viewed as the word $y_{n_{1}}^{(\xi_{1})} \ldots y_{n_{d}}^{(\xi_{d})}$ on $Y_{N}$.
\newline\indent Racinet has defined \cite{Racinet}, for each $\lambda\in K$ the set $\DS_{\lambda}(K)$ : this is the set of couples $(\psi_{\mathcyr{sh}},\psi_{\ast}) \in K \langle\langle e_{0\cup \mu_{N}} \rangle\rangle \times K \langle\langle Y_{N} \rangle\rangle$ such that $\psi_{\mathcyr{sh}}$ satisfies the shuffle equation (equation (\ref{eq:shuffle eq}) below), $\psi_{\ast}$ satisfies the quasi-shuffle equation (equation (\ref{eq:stuffle eq}) below) ; a certain relation between $\psi_{\mathcyr{sh}}$ and $\psi_{\ast}$ (equation (\ref{eq:rel between reg}) below) ; as well as $\psi_{\mathcyr{sh}}[e_{0}] = \psi_{\mathcyr{sh}}[e_{1}]=0$, and $\psi_{\mathcyr{sh}}[e_{0}e_{1}]=-\lambda^{2}/24$. This defines $\DS_{\lambda}$ as a subscheme of $\Pi_{1,0}$. Racinet proves that the Ihara product (\ref{eq:Ihara}) restricts to a group law on $\DS_{0}$ and to an action of $\DS_{0}$ on $\DS_{\lambda}$ which makes it a torsor.
\newline\indent We have $\Phi_{\KZ} \in \DS_{2\pi i}(\mathbb{C})$, and $\Phi_{p,\alpha} \in \DS_{0}(K_{p})$ : for $\alpha=1$ and $\alpha=-\infty$, this follows from \cite{Besser Furusho} ($N=1$), \cite{Yamashita} (any $N$); by the relations of iteration of the Frobenius (\cite{I-3}, equations (1.11), (1.12), (1.13) and Proposition 1.5.2), and by Racinet's theorem, it follows that this is true for any $\alpha \in \mathbb{Z} \cup \{\pm \infty\} - \{0\}$.
\subsubsection{The shuffle relation of iterated integrals}
The shuffle equation appears in equation (\ref{eq:shuffle equation}) : an element $f \in k \langle \langle e_{0 \cup \mu_{N}} \rangle\rangle$ satisfies the shuffle equation if :
\begin{equation} \label{eq:shuffle eq} \forall w,w' \text{ words on }e_{0\cup \mu_{N}}, \text{ we have } f[w]f[w'] = f[w\text{ }\mathcyr{sh}\text{ }w']
\end{equation}
where $\mathcyr{sh}$ is the shuffle product reviewed in \S2.1.1, characterized by induction on the weight by $w\text{ }\mathcyr{sh}\text{ }1 = 1\text{ }\mathcyr{sh}\text{ }w = w$ ($1$ is the empty word) and $e_{x}w \text{ } \mathcyr{sh} \text{ } e_{x'}w'
= e_{x}(w \text{ }\mathcyr{sh}\text{ } e_{x'}w') + e_{x'}(e_{x}w \text{ } \mathcyr{sh}\text{ } w')$ or equivalently
$w e_{x} \text{ } \mathcyr{sh}\text{ } w'e_{x}
= (w\text{ } \mathcyr{sh}\text{ } w'e_{x'})e_{x} + (w\text{ } \mathcyr{sh}\text{ } w'e_{x})e_{x'}$ for all $x,x' \in \{0\} \cup \mu_{N}(K)$. The shuffle equation amounts to
\begin{equation} \label{eq:shuffle eq bis} \hat{\Delta}_{\mathcyr{sh}}(f) = f \hat{\otimes} f
\end{equation}
where $\hat{\Delta}_{\mathcyr{sh}}$ is the coproduct in $K \langle \langle e_{0 \cup \mu_{N}} \rangle\rangle$ viewed as the completed dual of the Hopf algebra $\mathcal{O}^{\mathcyr{sh}}$.
\newline\indent The fact that the generating series $\Phi_{\KZ}$, resp. $\Phi_{p,\alpha}$ satisfies the shuffle equation follows directly from their definition as points of $\Pi_{1,0} \simeq \Spec(\mathcal{O}^{\mathcyr{sh},e_{0\cup \mu_{N}}})$ and from equation (\ref{eq:shuffle equation}), and amounts to a family of relations on MZV$\mu_{N}$'s, resp. $p$MZV$\mu_{N}$'s. The coefficient of $\Phi_{\KZ}$ at a word $e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}e_{0}^{n_{0}-1}=e_{x_{n}}\ldots e_{x_{1}}$ which does not necessarily satisfy the hypothesis $n_{d}\geqslant 2$ and $n_{0}=1$, is the regularized iterated integral
$\displaystyle \underset{\epsilon,\epsilon' \rightarrow 0}{\Reg\lim} \int_{\epsilon}^{1-\epsilon'} \frac{dt_{n}}{t_{n}-x_{n}} \ldots \frac{dt_{1}}{t_{1}-x_{1}}$
defined as the constant term in the asymptotic expansion of $\int_{\epsilon}^{1-\epsilon'} \frac{dt_{n}}{t_{n}-x_{n}} \ldots \frac{dt_{1}}{t_{1}-x_{1}}$, which is in $\mathbb{C}[[\epsilon,\epsilon']][\log(\epsilon),\log(\epsilon')]$. That coefficient is denoted by $\zeta_{\mathcyr{sh}}(e_{x_{n}}\ldots e_{x_{1}})$ and is called a (shuffle-)regularized MZV$\mu_{N}$. Similarly, the coefficient of $\Phi_{p,\alpha}$ at such a word is denoted by $(\zeta_{p,\alpha})_{\mathcyr{sh}}(e_{x_{n}}\ldots e_{x_{1}})$ and called a (shuffle-)regularized $p$MZV$\mu_{N}$. The shuffle equation for $\Phi_{\KZ}$ also follows from the identity
\begin{equation} \label{eq:eq shuffle source} \displaystyle \int_{\epsilon<t_{1}< \ldots < t_{n}<1-\epsilon'} \times \int_{\epsilon<t_{n+1}<\ldots t_{n+n'}<1-\epsilon'} = \sum_{\substack{\sigma \text{ permutation of }\{1,\ldots,l+l'\} \\\text{s.t. } \sigma(1)<\ldots<\sigma(n)\\ \text{and } \sigma(n+1)<\ldots<\sigma(n+n')}} \int_{\epsilon<t_{\sigma^{-1}(1)} < \ldots<t_{\sigma^{-1}(n+n')}<1-\epsilon'},
\end{equation}
which follows from an equality between domains of integration.
The definition of multiple polylogarithms as coefficients of a point of $\pi_{1}^{\un,\dR}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$ and equation (\ref{eq:shuffle equation}) imply that they satisfy the shuffle equation ; this also follows from their definition as iterated integrals (\ref{eq:MPL}) and equation (\ref{eq:eq shuffle source}).
\subsubsection{The quasi-shuffle relation of iterated series}
An element $f \in K \langle\langle Y_{N} \rangle\rangle$ satisfies the quasi-shuffle equation if :
\begin{equation} \label{eq:stuffle eq}
\forall w,w' \text{ words on }Y_{N}, \text{ we have }
f[w]f[w'] = f[w \ast w']
\end{equation}
where $\ast$, the quasi-shuffle product on $\mathbb{Q} \langle Y_{N} \rangle$, is defined as follows, by induction on the depth : $1 \ast w = w \ast 1 = w$ ($1$ is the empty word) and
$w y^{(\xi)}_{n} \ast w'y^{(\xi')}_{n'} = (w \ast w'y^{(\xi')}_{n'})y^{(\xi)}_{n} + (wy^{(\xi)}_{n} \ast w') y^{(\xi')}_{n'} + (w \ast w')y^{(\xi\xi')}_{n+n'}$.
The quasi-shuffle product makes $K \langle\langle Y_{N} \rangle\rangle$ into a commutative algebra. The quasi-shuffle equation amounts to
\begin{equation} \label{eq:stuffle eq bis}
\hat{\Delta}_{\ast}(f) = f \hat{\otimes} f
\end{equation}
where $\hat{\Delta}_{\ast}$ is the coproduct in $K \langle \langle Y_{N} \rangle\rangle$ viewed as the completed dual coalgebra of $\mathbb{Q} \langle Y_{N} \rangle$. Actually, one has actually a natural structure of Hopf algebra $\mathcal{O}^{\ast,e_{0\cup \mu_{N}}}$ on $\mathbb{Q} \langle Y_{N} \rangle$ in which the product is $\ast$ (\cite{Hoffman} for $N=1$).
\newline\indent For a word $w=\big((n_{i})_{d};(\xi_{i})_{d}\big)$ such that we do not necessarily have the hypothesis $(n_{d},\xi_{d})\not=(1,1)$ of (\ref{eq:multizetas}), let $\Reg \displaystyle \sum\limits_{0<m_{1}<\ldots<m_{d}} \frac{\big( \frac{\xi_{2}}{\xi_{1}} \big)^{m_{1}} \ldots \big(\frac{1}{\xi_{d}}\big)^{m_{d}}}{m_{1}^{n_{1}}\ldots m_{d}^{n_{d}}}$ be the constant term in the asymptotic expansion of $\displaystyle \sum\limits_{0<m_{1}<\ldots<m_{d}<m} \frac{\big( \frac{\xi_{2}}{\xi_{1}} \big)^{m_{1}} \ldots \big(\frac{1}{\xi_{d}}\big)^{m_{d}}}{m_{1}^{n_{1}}\ldots m_{d}^{n_{d}}}$ when $m \rightarrow \infty$, which is in $\mathbb{C}[\log(m)][[\frac{1}{m}]]$, in which the terms involving the Euler-Mascheroni constant are ``removed'' in a canonical way (see \cite{Racinet}). We denote it by $\zeta_{\ast}(w)$ and we call it a regularized MZV$\mu_{N}$. Let $(\Phi_{\KZ})_{\ast}= 1+ \sum\limits_{w\text{ word on }Y_{N}} \zeta_{\ast}(w)w$ ; $(\Phi_{\KZ})_{\ast}$ satisfies the quasi-shuffle equation ; this follows from the identities
\begin{equation} \label{eq:quasi shuffle eq source} \sum_{0<m_{1}<\ldots<m_{d}<m} \times \sum_{0<m'_{1}<\ldots<m'_{d'}<m} = \sum_{\text{quasi-shuffle elements}}\text{ }\sum_{0<m''_{1}<\ldots<m''_{r}<m}
\end{equation}
where a quasi-shuffle element is a way to order $m_{1},\ldots,m_{d}$ and $m'_{1},\ldots,m'_{d'}$, which determines an integer $r \in \{\max(d,d'),\ldots,d+d'\}$ and variables $m''_{1},\ldots,m''_{r}$ : for example, for $d=d'=1$, there are three quasi-shuffle elements : $\{m_{1}<m'_{1}\}$, $\{m_{1}=m'_{1}\}$ and $\{m_{1}>m'_{1}\}$.
\newline\indent In the $p$-adic case, the fact that $p$MZV$\mu_{N}$'s satisfy the quasi-shuffle equations \cite{Besser Furusho} is proved by using formal properties of Coleman functions. Moreover, an analogue of the quasi-shuffle regularization which still satisfies the quasi-shuffle equation has been defined in \cite{Furusho Jafari} in the case of $N=1$. We can deduce easily a similar definition for any $N$.
\newline\indent One has a variant $\ast_{\har}$ on $\mathcal{O}_{\har}^{\ast}$ (defined in \S2.2.1) adapted to cyclotomic multiple harmonic sums. For any words $w = \big((n_{i})_{d};(\xi_{i})_{d+1})$ and
$w' = \big((\tilde{n}_{i})_{d};(\tilde{\xi}_{i})_{d+1})$, $w \ast_{\har} w'$ is the sum, indexed by the set of quasi-shuffle elements $(u_{i})_{d''}$ of
$\big((n_{i})_{d},(\tilde{n}_{i})_{d}\big)$, of the sequences
$\big((u_{i})_{d''}, (\xi_{a_{i}} \tilde{\xi}_{b_{i}})_{d''+1} \big)$ defined as follows : $a_{1} = b_{1} = 1$ and,
for $2 \leqslant i \leqslant d''$,
$ (a_{i},b_{i}) = \left\{ \begin{array}{ll} (a_{i-1}+1 ,b_{i-1})&
\text{ if } \exists l\text{ }|\text{ } u_{i-1}=n_{l},
\\ (a_{i-1}, b_{i-1}+1) &\text{ if }\exists l'\text{ }|\text{ } u_{i-1}=\tilde{n}_{l'},\text{ }
\\ (a_{i-1}+1,b_{i-1}+1)&\text{ if }\exists l,l' \text{ }|\text{ } u_{i-1}=n_{l}+\tilde{n}_{l'},\text{ } \end{array}\right.$. This makes $\mathcal{O}_{\har}^{\ast}$ into a a commutative algebra. We have natural morphisms of algebras $i : \mathcal{O}^{\ast} \hookrightarrow \mathcal{O}_{\har}^{\ast}$, $\big( (n_{i})_{d};(\xi_{i})_{d} \big) \mapsto \big( (n_{i})_{d};((\xi_{i})_{d},1) \big)$ and $r : \mathcal{O}_{\har}^{\ast} \twoheadrightarrow \mathcal{O}^{\ast}$,
$\big( (n_{i})_{d};(\xi_{i})_{d+1} \big) \mapsto \big( (n_{i})_{d}; (\frac{\xi_{i}}{\xi_{d+1}})\big) $, with $r \circ i=\id$.
\newline\indent By equation (\ref{eq:quasi shuffle eq source}), the power series expansion of multiple polylogarithms (\ref{eq:multiple polylogarithms power series expansion}) satisfy a version of the quasi-shuffle relation. It can be encoded by means of $\ast_{\har}$ ; we will use it in \S3.3.2.
\subsubsection{The relation between the two regularizations of MZV$\mu_{N}$'s}
The two regularizations of MZV$\mu_{N}$'s are related as follows. Let $\pr : K\langle \langle e_{0\cup \mu_{N}} \rangle\rangle \rightarrow K\langle \langle Y_{N} \rangle\rangle$ be the unique continuous (for the weight-adic topology) and linear map which sends a word $w$ to itself if its rightmost letter is not $e_{0}$ and to $0$ if it is $e_{0}$, where we view words on $Y_{N}$ as words on $e_{0\cup \mu_{N}}$ as usual.
Let the maps $\textbf{p},\textbf{q} : \mathcal{O}^{\mathcyr{sh}} \rightarrow \mathcal{O}^{\mathcyr{sh}}$ defined as follows (\cite{Racinet}, \S2.2.3) :
$$ \textbf{p}(e_{0}^{n_{d}-1}e_{\sigma_{d}} \cdots e_{0}^{n_{1}-1}e_{\sigma_{1}}e_{0}^{n_{0}-1} ) = e_{0}^{n_{d}-1}e_{\sigma_{d}^{-1}} \cdots
e_{0}^{n_{2}-1}e_{(\sigma_{d}\cdots \sigma_{2})^{-1}} e_{0}^{n_{1}-1}e_{(\sigma_{d}\cdots \sigma_{1})^{-1}}e_{0}^{n_{0}-1} $$
$$ \textbf{q} (e_{0}^{n_{d}-1}e_{\xi_{d}} \cdots e_{0}^{n_{1}-1}e_{\xi_{1}}e_{0}^{n_{0}-1} ) = e_{0}^{n_{d}-1}e_{\xi_{d}^{-1}} \cdots e_{0}^{n_{1}-1}e_{\xi_{3}^{-1}\xi_{2}} e_{0}^{n_{1}-1}e_{\xi_{2}^{-1}\xi_{1}}e_{0}^{n_{0}-1} $$
By duality they define maps $K \langle\langle e_{0\cup \mu_{N}} \rangle\rangle \rightarrow K \langle\langle e_{0\cup \mu_{N}} \rangle\rangle$, which we will also denote by $\textbf{p}$ and $\textbf{q}$ and, by restriction, they define maps $K \langle \langle Y_{N} \rangle\rangle K \langle \langle Y_{N} \rangle\rangle$. We have (\cite{Racinet}, Corollaire 2.24, D\'{e}finition 3.1) :
\begin{equation} \label{eq:rel between reg} \textbf{q}\pr(\Phi_{\KZ}) = \exp \bigg( \sum\limits_{n=2}^{\infty} \frac{(-1)^{n}}{n}\zeta(n)y_{1}^{n}
\bigg)(\Phi_{\KZ})_{\ast} .
\end{equation}
This formula can be compared with equations (\ref{eq:multizetas}) and (\ref{eq:multizetas integral}).
\newline\indent The $p$-adic analogue of this formula has been proved in \cite{Furusho Jafari}, Theorem 0.1, (iii), in the $N=1$ case, and this can be easily adapted to any $N$.
\subsection{Adjoint $p$-adic double shuffle equations}
For $\Phi \in \DS_{0}(K)$, we are going to find some ``adjoint double shuffle equations'' for $\Phi_{\Ad,\chi}$, as defined in Definition 2.2.3. In particular, we will obtain adjoint double shuffle equations for adjoint $p$MZV$\mu_{N}$'s.
\subsubsection{Expression of $\Phi_{\Ad,\chi}$ in terms of $K \langle \langle Y_{N}\rangle\rangle$}
The first step is to rewrite $\pr(f_{\Ad})$ in terms of operations on $K\langle\langle Y_{N}\rangle\rangle$.
\newline\indent Below the hat denotes the completion with respect to the weight grading. The notation below refers to the word ``shifting'', this terminology will be explained in a subsequent paper.
\begin{Definition} \label{def shft} (i) Let
$\mathcyr{sh}ft_{\ast} : \mathcal{O}^{\ast} \rightarrow \widehat{\mathcal{O}^{\ast}}$,
$w(e_{0},(e_{\xi})_{\xi \in \mu_{N}(K)}) \mapsto w (\frac{1}{1+e_{0}}e_{0},(\frac{1}{1+e_{0}}e_{\xi})_{\xi \in \mu_{N}(K)})$.
\newline (ii) For any $l \in \mathbb{N}$, let
$\mathcyr{sh}ft_{l} : \mathcal{O}^{\ast} \mapsto \mathcal{O}^{\ast}$,
$\displaystyle e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}} \mapsto \sum\limits_{\substack{l_{1}+\ldots+l_{d}=l \\ l_{1},\ldots,l_{d} \geqslant 0}} \prod_{i=1}^{d} {-n_{i} \choose l_{i}} e_{0}^{n_{d}+l_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}+l_{1}-1}e_{\xi_{1}}$.
\newline Let $\mathcyr{sh}ft^{\vee}_{l} : K \langle \langle Y_{N}\rangle\rangle \rightarrow K \langle \langle Y_{N}\rangle\rangle$, defined by for all $w$,
$(\mathcyr{sh}ft_{l}^{\vee}G)[w] = G[\mathcyr{sh}ft_{l}(w)]$.
\newline (iii) \label{def SY} For any $G \in K \langle \langle Y_{N}\rangle\rangle$ and $\xi \in \mu_{N}(K)$, let $G^{\inv,\xi} \in K \langle\langle Y_{N}\rangle\rangle$ be defined by, for all $d\geq 0$,
$G^{\inv,\xi}[y_{l+1}^{(\xi_{d+1})}y_{n_{d}}^{(\xi_{d})} \ldots y_{n_{1}}^{(\xi_{1})}] = \left\{ \begin{array}{ll} (-1)^{n_{1}+\ldots+n_{d}}G[y_{n_{1}}^{(\xi_{2})} \ldots y_{n_{d}}^{(\xi_{d+1})}] & \xi_{1}=\xi
\\ 0 & \xi_{1}\not=\xi \end{array} \right.$.
\end{Definition}
We note that $\mathcyr{sh}ft_{l}$ is the coefficient of $\Lambda^{l}$ in $\widehat{\tau(\Lambda)}^{-1} \mathcyr{sh}ft_{\ast} \tau(\Lambda)$.
\begin{Proposition} \label{prop adjoint star} For any $f\in \tilde{\Pi}_{1,0}(K)$, then we have
$$ \pr(f_{\Ad,\chi}) = \sum_{l\geq 0} \sum_{w_{l}} \sum_{\xi \in \mu_{N}(K)} \bigg( \chi(\xi)(-1)^{l} \big( \mathcyr{sh}ft^{\vee}_{\ast,l} \pr(f^{(\xi)})\big)^{\inv,\xi} \pr(f^{(\xi)}) \bigg) [w_{l}] w_{l} $$
\end{Proposition}
\begin{proof} Consider a word whose rightmost letter is not $e_{0}$, and write it as $w=e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}$ ($\xi_{i}$'s are roots of unity, $l\geq 0$, $n_{i}\geq 1$). Then, for $f$ a solution to the shuffle equation such that $f[e_{0}]=0$, we have
$$ \sum_{\xi \in \mu_{N}(K)} \chi(\xi)({f^{(\xi)}}^{-1}e_{\xi}f^{(\xi)})[w] =
\sum_{d'=1}^{d+1} \chi(\xi_{d'})
{f^{(\xi_{d'})}}^{-1}[e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1} \ldots e_{0}^{n_{d'}-1}]
f^{(\xi_{d'})}[e_{0}^{n_{d'-1}-1}e_{\xi_{d'-1}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}]
$$
By using the formula for the antipode of the shuffle Hopf algebra we have
$$ {f^{(\xi_{d'})}}^{-1}[e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1} \ldots e_{0}^{n_{d'}-1}] = (-1)^{n_{d'}+\ldots+n_{d}+l}
f^{(\xi_{d'})}[e_{0}^{n_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{d}-1}e_{\xi_{d+1}}e_{0}^{l}]
$$
By using a well-known consequence of the shuffle equation and of $f[e_{0}]=0$, we have
$$
f^{(\xi_{d'})}[e_{0}^{n_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{d}-1}e_{\xi_{d+1}}e_{0}^{l}] =
\sum_{l_{d'}+\ldots+l_{d}=l}
\prod_{i=d'}^{d} {-n_{i} \choose l_{i}}
f^{(\xi_{d'})}[e_{0}^{n_{d'}+l_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{d}+l_{d}-1}e_{\xi_{d+1}}] $$
By the definition of $\mathcyr{sh}ft_{l}$ (Definition 3.2.1 (iii)), we have
$$
\sum_{l_{d'}+\ldots+l_{d}=l}
\prod_{i=d'}^{d} {-n_{i} \choose l_{i}}
f^{(\xi_{d'})}[e_{0}^{n_{d'}+l_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{d}+l_{d}-1}e_{\xi_{d+1}}] =
(\mathcyr{sh}ft_{l}^{\vee}f^{(\xi_{d'})})[(e_{0}^{n_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{d}-1}e_{\xi_{d+1}})] $$
Moreover,
$$ (-1)^{l+n_{d'}+\ldots+n_{d}}
(pr f^{(\xi_{d'})})[\mathcyr{sh}ft_{l}(e_{0}^{n_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{d}-1}e_{\xi_{d+1}})] = (-1)^{l} \big( \mathcyr{sh}ft^{\vee}_{\ast,l} \pr(f^{(\xi)})\big) ^{\inv,\xi_{d'}}[e_{0}^{l}e_{\xi_{d+1}}\ldots e_{0}^{n_{d'}-1}e_{\xi_{d'}}] $$
and, finally, since all the words involved have their rightmost letter not equal to $e_{0}$ we can replace everywhere $f^{(\xi)}$ by $\pr(f^{(\xi)})$. In the end, we have
\begin{multline} \sum_{\xi \in \mu_{N}(K)} \chi(\xi)({f^{(\xi)}}^{-1}e_{\xi}f^{(\xi)})[w]
\\
\begin{array}{l} = \displaystyle \sum_{d'=1}^{d+1} \chi(\xi_{d'}) (-1)^{l} \big( \mathcyr{sh}ft^{\vee}_{l} \pr(f^{(\xi)}) \big)^{\inv,\xi_{d'}}[e_{0}^{l}e_{\xi_{d+1}}\ldots e_{0}^{n_{d'}-1}e_{\xi_{d'}}] \pr(f^{(\xi_{d'})})[e_{0}^{n_{d'-1}-1}e_{\xi_{d'-1}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}]
\\ \displaystyle = \Big( \sum_{d'=1}^{d+1} \chi(\xi_{d'}) (-1)^{l} (\mathcyr{sh}ft^{\vee}_{l} \pr(f^{(\xi_{d'})}))^{\inv,\xi_{d'}} \pr(f^{(\xi_{d'})}) \Big) [w]
\\ \displaystyle = \Big( \sum_{\xi \in \mu_{N}(K)} \chi(\xi) (-1)^{l} (\mathcyr{sh}ft^{\vee}_{\ast,l} \pr(f^{(\xi)}))^{\inv,\xi} \pr(f^{(\xi)}) \Big) [w]
\end{array}
\end{multline}
\end{proof}
\subsubsection{Relation between the two regularizations}
The second step is to observe that the relation between the two regularizations becomes trivial in the adjoint setting. This is an aspect of the proximity between adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic sums which we are using in this work.
\newline\indent In view of the Lemma \ref{prop adjoint star}, we now define an analogue of $\pr(f_{\Ad})$ in which $f$ is replaced by $f_{\ast}$ defined by the equalities :
\begin{equation} \label{eq:passage a star 1} E_{f} = \exp(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n} f[e_{0}^{n-1}e_{1}] e_{1}^{n})
\end{equation}
\begin{equation} \label{eq:passage a star 2} \textbf{q} \pr(f) = E_{f} f_{\ast}
\end{equation}
\begin{Definition} For any $f \in \tilde{\Pi}_{1,0}(K)$, let
$\displaystyle f_{\Ad,\chi,\ast} = \sum_{l\geq 0} \sum_{w_{l}} \sum_{\xi \in \mu_{N}(K)} \chi(\xi) \big((-1)^{l} \big( \mathcyr{sh}ft^{\vee}_{\ast,l}(f_{\ast}^{(\xi)})\big)^{\inv,\xi}f_{\ast}^{(\xi)} \big)[w_{l}]w_{l}$
where the sum is over $w_{l}$ of the form $e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}=y_{l+1}^{(\xi_{d+1})}y_{n_{d}}^{(\xi_{d})} \ldots y_{n_{1}}^{(\xi_{1})}$, $d\geq 0$.
\end{Definition}
In \cite{I-2}, Definition 1.1.3 and Proposition 1.1.4, we have defined the adjoint Ihara product on $\Ad_{\Pi_{1,0}}(e_{1})$ by the formula $(g,f) \mapsto g \circ_{\Ad}^{\smallint_{1,0}} f = f (e_{0},(g^{(\xi)})_{\xi \in \mu_{N}(K)})$, and we have proved that it is a group law on $\Ad_{\Pi_{1,0}}(e_{1})$.
\newline\indent In the next statement, (i) says that the two regularizations (integrals and series) give the same result for adjoint $p$-adic cyclotomic multiple zeta values : this is coherent with the fact that their adjoint quasi-shuffle relation ((iii) below) can be understooed via cyclotomic multiple harmonic sums, as we are going to see in the subsequent paper.
\begin{Proposition} \label{comparaison reg adjoint}We have
\begin{equation} \label{eq:comparison of regularisations} \textbf{q}\pr(\Phi_{\Ad,\chi}) = \Phi_{\Ad,\chi,\ast}
\end{equation}
\end{Proposition}
\begin{proof} (i) We are going to simplify the expression of $pr(\Phi)$ given by Lemma \ref{prop adjoint star}. By equation (\ref{eq:passage a star 2}) we have, for each $\xi \in \mu_{N}(K)$ and $l\geq 0$,
$$ (-1)^{l}\mathcyr{sh}ft^{\vee}_{l} (pr(\Phi^{(\xi)})^{\inv,\xi} ) \pr(\Phi^{(\xi)}) = (-1)^{l}\mathcyr{sh}ft^{\vee}_{l}\big( E^{(\xi)}(\Phi_{\ast}^{(\xi)})^{\inv,\xi}) \big) E^{(\xi)} \Phi_{\ast}^{(\xi)} $$
Let now a word $w=e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}$. By the definition of the map $G \mapsto G^{\inv,\xi}$ (Definition \ref{def shft} (iii)), for all $d'$ ($1 \leq d' \leq d+1$), assuming $\xi=\xi_{d'}$, we have
\begin{multline*} (-1)^{l}\big( \mathcyr{sh}ft_{l}^{\vee}(E_{\Phi}^{(\xi)}\Phi_{\ast}^{(\xi)}) \big)^{\inv,\xi}[e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{d'}-1}e_{\xi_{d'}}]
\\ =
(-1)^{l+n_{d}+\ldots+n_{d'}}
\big( \mathcyr{sh}ft_{l}^{\vee}(E^{(\xi)}\Phi_{\ast}^{(\xi)})\big) [e_{0}^{n_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{d}-1}e_{\xi_{d+1}}]
\end{multline*}
by the definition of $\mathcyr{sh}ft_{l}$ (Definition 3.2.1, (iii)), and by the fact that $E_{\Phi}^{(\xi)}$ has non-zero coefficients only at certain words which are of the form $e_{\xi}^{n}$, we have
\begin{multline} \label{eq:3} \big(\mathcyr{sh}ft^{\vee}_{\ast,l}\big( E_{\Phi}^{(\xi)}(\Phi_{\ast}^{(\xi)}) \big)[e_{0}^{n_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{d}-1}e_{\xi_{d+1}}]
\\
\begin{array}{l} \displaystyle
= \big(E_{\Phi}^{(\xi)}
\mathcyr{sh}ft^{\vee}_{\ast,l}(\Phi_{\ast}^{(\xi)}) \big) [e_{0}^{n_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{d}-1}e_{\xi_{d+1}}]
\\ \displaystyle
= \sum_{r=d'}^{d+1} E_{\Phi}^{(\xi)}[e_{0}^{n_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{r}-1}e_{\xi_{r+1}}]
\mathcyr{sh}ft^{\vee}_{\ast,l}(\Phi_{\ast}^{(\xi)}) [e_{0}^{n_{r+1}-1}e_{\xi_{r+2}} \ldots e_{0}^{n_{d}-1}e_{\xi_{d+1}}]
\end{array}
\end{multline}
By the definition of the map $G \mapsto G^{\inv,\xi}$ (Definition \ref{def shft} (iii)), we have
$$ (-1)^{n_{r+1}+\ldots+n_{d}} \mathcyr{sh}ft^{\vee}_{l}(\Phi_{\ast}^{(\xi)}) [e_{0}^{n_{r+1}-1}e_{\xi_{r+2}} \ldots e_{0}^{n_{d}-1}e_{\xi_{d+1}}] = \big(
\mathcyr{sh}ft^{\vee}_{l}(\Phi_{\ast}^{(\xi)}) \big)^{(\inv,\xi_{r+1})} [e_{0}^{l}e_{\xi_{d+1}} \ldots e_{0}^{n_{r+2}-1}e_{\xi_{r+1}}] $$
Since $E_{\Phi}^{(\xi)}$ has non-zero coefficients only at words of the form $e_{\xi}^{n}$, in the sum over $r$ in (\ref{eq:3}), a non-zero term can appear only if $\xi_{r+1}=\xi_{d'}$, and we can write
$$ E_{\Phi}^{(\xi)}[e_{0}^{n_{d'}-1}e_{\xi_{d'+1}} \ldots e_{0}^{n_{r}-1}e_{\xi_{r+1}}] = E_{\Phi}^{(\xi)}[e_{\xi_{r+1}}e_{0}^{n_{r}-1} \ldots e_{\xi_{d'+1}}e_{0}^{n_{d'}-1}]
= E_{\Phi}^{(\xi)}[e_{0}^{n_{r}-1} \ldots e_{0}^{n_{d'}-1}e_{\xi_{d'}}] $$
Below we use the notation $\tau$ introduced in (\ref{eq:tau}). By the two previous equalities, (\ref{eq:3}) multiplied by $(-1)^{l+n_{d}+\ldots+n_{d'}}$ is equal to
\begin{multline*} \sum_{r=d'}^{d+1} (-1)^{l}
\big(\mathcyr{sh}ft^{\vee}_{\ast,l}(\Phi_{\ast}^{(\xi_{d})}) \big)^{(\inv,\xi_{d'})} [e_{0}^{l}e_{\xi_{d+1}} \ldots e_{0}^{n_{r+1}-1}e_{\xi_{r+1}}] \big( \tau(-1)E_{\Phi}^{(\xi)}[e_{0}^{n_{r}-1} \ldots e_{0}^{n_{d'}-1}e_{\xi_{d'}}]
\\ =
(-1)^{l}\bigg( \big(\mathcyr{sh}ft^{\vee}_{\ast,l}(\Phi_{\ast}^{(\xi_{d})}) \big)^{(\inv,\xi_{d'})} (\tau(-1)E^{(\xi)}) \bigg)[e_{0}^{l}e_{\xi_{d+1}} \ldots e_{\xi_{d'+1}} e_{0}^{n_{d'}-1}e_{\xi_{d'}}]
\end{multline*}
This formula combined to Lemma \ref{prop adjoint star} shows that
\begin{multline} \label{eq:avant derniere} \pr(\Phi)[w]
= \sum_{d'=1}^{d+1} \bigg( (-1)^{l} \xi_{d'}^{-1}
\big(\mathcyr{sh}ft^{\vee}_{\ast,l}(\Phi_{\ast}^{(\xi_{d'})}) \big)^{(\inv,\xi_{d'})} (\tau(-1)E^{(\xi_{d'})}) E^{(\xi_{d'})} \Phi_{\ast}^{(\xi_{d'})} \bigg)[w]
\\ = \sum_{\xi \in \mu_{N}(K)} \bigg( (-1)^{l} \xi^{-1}
\big(\mathcyr{sh}ft^{\vee}_{\ast,l}(\Phi_{\ast}^{(\xi)}) \big)^{(\inv,\xi)} (\tau(-1)E^{(\xi)}) E^{(\xi)} \Phi_{\ast}^{(\xi)} \bigg)[w]
\end{multline}
For any $S \in K\langle \langle e_{0\cup \mu_{N}}\rangle\rangle$, and $n \geq 0$, we have $(\tau(-1)S)^{n} = \tau(-1)(S^{n})$ because for any words $w_{i}$, $\displaystyle \sum_{i=1}^{n}\weight(w_{i}) = \weight(w_{1}\ldots w_{n})$. Thus, if the coefficient of the empty word in $S$ is $0$, we can write
$\displaystyle \tau(-1)\exp(S) = \sum_{n=0}^{\infty} \tau(-1)\frac{S^{n}}{n!} = \sum_{n=0}^{\infty} \frac{(\tau(-1)S)^{n}}{n!} = \exp (\tau(-1)S)$. In particular, by (\ref{eq:passage a star 1}) $\displaystyle \tau(-1)E_{\Phi} =
\exp \big(\sum_{n=2}^{\infty} \frac{1}{n}\Phi[e_{0}^{n-1}e_{1}]e_{1}^{n} \big)$. Thus
$$(\tau(-1)E_{\Phi}) E_{\Phi} = \exp \big(\sum_{n=2}^{\infty} \frac{1}{n}\Phi[e_{0}^{n-1}e_{1}]e_{1}^{n} \big) \exp \big(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n}\Phi[e_{0}^{n-1}e_{1}]e_{1}^{n} \big) =
\exp \big(\sum_{n=2}^{\infty} \frac{(1+(-1)^{n})}{n}\Phi[e_{0}^{n-1}e_{1}]e_{1}^{n} \big) $$
For $n$ odd, we have $1+(-1)^{n}=0$ ; for $n$ even, since $\Phi \in \DS_{0}(K)$ we have $\Phi[e_{0}^{n-1}e_{1}]=0$. This proves
\begin{equation} \label{eq:derniere} \tau(-1)(E_{\Phi})E_{\Phi} = 1
\end{equation}
The result follows from (\ref{eq:avant derniere}) and (\ref{eq:derniere}).
\end{proof}
\subsubsection{Definition of the adjoint double shuffle relations}
\begin{Proposition} \label{221}Let $\Phi \in \DS_{0}(K)$ and $\chi \in \Theta$.
\newline (i) We have
\begin{equation} \label{eq:ds adjoint 1}
\text{for all non-empty words }w,w',\text{ } \Phi_{\Ad,\chi}[w \text{ }\mathcyr{sh} \text{ }w'] = 0 .
\end{equation}
\noindent (ii) We have
\begin{multline} \label{eq:ds adjoint 2}
\text{for all words }w,w'\text{ and }L \in \mathbb{N},
\text{ }\sum_{\substack{l,l'\geqslant 0 \\ l+l'=L}} \Phi_{\Ad,\chi,\ast}[w;l] \Phi_{\Ad,\chi,\ast}[w';l'] = \Phi_{\Ad,\chi,\ast}[ w \ast_{\har} w';L] .
\end{multline}
\end{Proposition}
\begin{proof} (i) Let $\Delta_{\mathcyr{sh}}$ be the shuffle coproduct. For all $\xi \in \mu_{N}(K)$, we have
$\Delta_{\mathcyr{sh}}(e_{\xi})= e_{\xi} \otimes 1 + 1 \otimes e_{\xi}$, and $\Delta_{\mathcyr{sh}}(\Phi^{(\xi)})=\Phi^{(\xi)} \otimes \Phi^{(\xi)}$, whence $\Delta( {\Phi^{(\xi)}}^{-1} e_{\xi} \Phi^{(\xi)}) = {\Phi^{(\xi)}}^{-1} e_{\xi} \Phi^{(\xi)} \otimes 1 + 1 \otimes {\Phi^{(\xi)}}^{-1} e_{\xi} \Phi^{(\xi)}$.
This implies $\Delta_{\mathcyr{sh}}(\Phi_{\Ad}) = \Phi_{\Ad} \otimes 1 + 1 \otimes \Phi_{\Ad}$.
\newline\indent (ii) Let us prove that $\mathcyr{sh}ft_{\ast}$ is a morphism of quasi-shuffle algebras. For simplicity, we do the proof for $\mathbb{P}^{1} - \{0,1,\infty\}$. The dual of $\mathcyr{sh}ft_{\ast}$ is the concatenation algebra morphism $\imath_{\ast}^{\vee}$ defined by
$$ y_{n} \mapsto \Lambda^{n} \sum_{l=0}^{n-1} \Lambda^{l} {n-1 \choose l} (-1)^{n-l} y_{n-l}
= \Lambda^{n} \sum_{l=1}^{n} \Lambda^{n-l} (-1)^{l} y_{l} {n-1 \choose n-l} . $$
\noindent We have
$(\imath^{\vee} \otimes \imath^{\vee})\Delta_{\ast}(y_{n})
=
1 \otimes \imath^{\vee}(y_{n})
+
\imath^{\vee}(y_{n}) \otimes 1
+
\sum_{k=1}^{n-1} \imath^{\vee}(y_{k}) \otimes \imath^{\vee}(y_{n-k})$ ; the third term of this sum is
$$ \Lambda^{n} \sum_{k=1}^{n-1} \big(
\sum_{l=1}^{k} \Lambda^{k-l} {k-1 \choose k-l} (-1)^{l} y_{l}
\big)
\otimes
\big(
\sum_{l=1}^{n-k} \Lambda^{n-k-l} {n-k-1 \choose n-k-l'} (-1)^{l'} y_{l'}
\big) $$
$$ = \Lambda^{n}
\sum_{L=2}^{n} \Lambda^{n-L}(-1)^{n-L}
\big(
\sum_{\substack{l+l'=L\\ l,l'\geqslant 1}} y_{l} \otimes y_{l'}
\big)
\sum_{\substack{l\leqslant k \leqslant n-L+l}} {k-1 \choose k-l} {n-k-1 \choose n-k-l'}
$$
\noindent and for all $l,l'$ such that $l+l'=L$, we have
$$
\sum_{\substack{l\leqslant k \leqslant n-L+l}} {k-1 \choose k-l} {n-k-1 \choose n-k-l'}
=
\sum_{k'=0}^{n-L} {k' + l-1 \choose k'} {n-L-k'+l'-1 \choose n-L-k'}
= {n-L+L-1 \choose n-L} .
$$
\noindent On the other hand, $S_{Y}$ is an anti-morphism of quasi-shuffle algebras. This gives the result.
\end{proof}
\begin{Definition} We call (\ref{eq:ds adjoint 1}) the \emph{adjoint shuffle equation} and (\ref{eq:ds adjoint 2}) the \emph{adjoint quasi-shuffle equation} ; and we call the collection of (\ref{eq:ds adjoint 1}) and (\ref{eq:ds adjoint 2}) the \emph{adjoint double shuffle equations}.
We denote by $\DS_{0,\Ad}(K)$ the set of $\psi \in K \langle \langle e_{0\cup\mu_{N}} \rangle\rangle$ which satisfy the adjoint shuffle equation and such that for all $\chi \in \Theta$, $\textbf{q} \pr \Moy_{\chi}(\psi)$ satisfies the adjoint quasi-shuffle equation.
These equations define an affine scheme $\DS_{0,\Ad}$ over $k_{N}$, which we call the \emph{adjoint double shuffle scheme}.
\end{Definition}
By the previous propositions, we have proved that if $\Phi \in \DS_{0}(K)$ then $\Ad_{\Phi}(e_{1})$ is in $\DS_{0,\Ad}(K)$.
We note that the adjoint shuffle equation is also known as the linearlized shuffle equation, or the shuffle equation modulo products, and amounts to say that $\Ad_{\Phi}(e_{1})$ is primitive for the shuffle coproduct $\Delta_{\mathcyr{sh}}$. We also note that the adjoint quasi-shuffle equation can be reformulated as saying that $\Ad_{\Phi}(e_{1})$ is a "grouplike" element for a certain adjoint variant $\Delta_{\ast}^{\Ad}$ of $\Delta_{\ast}$.
\subsubsection{Stability by the Ihara product}
We now deduce from Racinet's theorem that $\DS_{0}$ is a group for the Ihara product \cite{Racinet} an adjoint variant of this statement.
\begin{Proposition} The image of the map $\Ad(e_{1}) : \DS_{0} \mapsto \DS_{0,\Ad}$ is an algebraic group with the adjoint Ihara product $\circ_{\Ad}^{\smallint_{1,0}}$.
\end{Proposition}
\begin{proof} By Racinet's theorem \cite{Racinet}, $(\DS_{0},\circ^{\smallint_{1,0}})$ is an algebraic group. By \cite{I-2}, Proposition 1.1.4, $\Ad(e_{1})$ is a morphism of algebraic groups $(\tilde{\Pi}_{1,0},\circ^{\smallint_{1,0}}) \buildrel \sim \over \longrightarrow (\Ad_{\tilde{\Pi}_{1,0}}(e_{1}),\circ_{\Ad}^{\smallint_{1,0}})$.
\end{proof}
The following question seems interesting, as we will explain in the next section :
\begin{Question} \label{question} Is $\Ad(e_{1}) : \DS_{0} \mapsto \DS_{0,\Ad}$ an isomorphism ?
\end{Question}
\subsection{Adjoint complex double shuffle equations}
\subsubsection{Adjoint double shuffle equation\label{paragraph lifts}}
Let $K$ be a field of characteristic $0$. For $\mu \in K - \{0\}$, let $\DS_{\mu}$ be the scheme of regularized double shuffle relations with parameter $\mu$ defined by Racinet \cite{Racinet}.
In order to write the relation between the two regularizations of adjoint cyclotomic multiple zeta values, we have to consider a slightly more general object.
Let $\rho_{\Ad}$ be the linear map $K[T_{1},T_{2}] \rightarrow K[T_{1},T_{2}]$ defined by the formula
$$ \rho (e^{T_{2}e_{1}} e^{T_{1}e_{1}}) \mapsto (-1)^{\weight}(e^{T_{1}e_{1}})(-1)^{weight}(E)E e^{T_{2}e_{1}} $$
We now define the regularized versions of adjoint complex cyclotomic multiple zeta values. Let the two regularizations $\Phi_{\KZ}(T)$ and $\Phi_{\KZ,\ast}(T)$ of $\Phi_{\KZ}$ resp. $\Phi_{\KZ,\ast}$ which can be found in \cite{Racinet} ($T$ is a formal variable). We define their adjoint analogues, the two regularizations of $\Phi_{\KZ,\Ad,\chi}$ :
(a) The regularization in the sense of integrals :
$\displaystyle \Phi_{\KZ,\Ad,\chi}(T) = \sum\limits_{\xi \in \mu_{N}(K)} \chi(\xi) {\Phi(T)^{-1}}^{(\xi)}e_{\xi}\Phi(T)^{(\xi)} $
(b) The regularization in the sense of series (in view of Proposition \ref{prop adjoint star}) :
\newline
$\displaystyle \Phi_{\KZ,\Ad,\chi,\ast}(T) = \sum_{l\geq 0} \sum_{w_{l}} \sum_{\xi \in \mu_{N}(K)} \chi(\xi) \big((-1)^{l} \big( \mathcyr{sh}ft^{\vee}_{\ast,l}(\Phi_{\ast}(T)^{(\xi)})\big)^{\inv,\xi} \Phi_{\ast}^{(\xi)}(T) \big)[w_{l}]w_{l} \in \mathbb{C} \langle\langle Y_{N} \rangle\rangle$
By considering their coefficients, we deduce the regularized version of adjoint MZV$\mu_{N}$'s :
\begin{Definition} (a) The regularized (in the sense of integrals) AdMZV$\mu_{N}$'s are
$$ \zeta^{\Ad}\big( (n_{i})_{d};(\xi_{i})_{d+1};l;\chi)(T) =
\Phi_{\KZ,\Ad,\chi}(T) [e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}]. $$
(b) The regularized (in the sense of series) AdMZV$\mu_{N}$'s are
$$ \zeta^{\Ad}\big( (n_{i})_{d};(\xi_{i})_{d+1};l;\chi)(T) =
\Phi_{\KZ,\Ad,\chi,\ast}(T) [e_{0}^{l}e_{\xi_{d+1}}e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1}e_{\xi_{1}}]. $$
And similarly for the $\Lambda$-adic AdMZV$\mu_{N}$'s.
\end{Definition}
We generalize the previous definitions as follows :
$$ \Phi_{\KZ,\chi,\Ad}(T_{1},T_{2}) = \sum_{\xi \in \mu_{N}(K)} \chi(\xi) {\Phi(T_{2})^{-1}}^{(\xi)}e_{\xi}\Phi(T_{1})^{(\xi)} $$
$$ \Phi_{\KZ,\ast,\chi,\Ad}(T_{1},T_{2}) = \sum_{l\geq 0} \sum_{w_{l}} \sum_{\xi \in \mu_{N}(K)} \chi(\xi) \big((-1)^{l} \big( \mathcyr{sh}ft^{\vee}_{\ast,l}(\Phi_{\ast}(T_{2})^{(\xi)})\big)^{\inv,\xi} \Phi_{\ast}^{(\xi)}(T_{1}) \big)[w_{l}]w_{l} $$
\begin{Proposition} If $\Phi \in \DS_{\mu}(K)$, we have
\newline (i) $\Phi_{\Ad,\chi}(T)$ satisfies the adjoint shuffle equation
\newline (ii) $\Phi_{\Ad,\chi,\ast}(T)$ satisfies the adjoint quasi-shuffle equation
\newline (iii) We have $\textbf{q} \pr \rho (\Phi_{\Ad,\chi}(T_{1},T_{2})) = \Phi_{\Ad,\chi,\ast}(T_{1},T_{2})$,
$\Phi_{\Ad,\chi}(T) = \Phi_{\Ad,\chi}(T,T)$, $ \Phi_{\Ad,\chi,\ast}(T) = \Phi_{\Ad,\chi,\ast}(T,T)$
\newline (iv) We have $\Phi_{\Ad,\chi}[e_{1}e_{0}e_{1}] = (\Phi^{-1}e_{1}\Phi)[e_{1}e_{0}e_{1}] = 2\Phi[e_{0}e_{1}]$.
\end{Proposition}
\begin{proof} Similar to the proofs in the $p$-adic case (\S3.2). (iv) follows from $\Phi^{-1}[e_{1}e_{0}] = (-1)^{2}\Phi[e_{0}e_{1}]$ and $({\Phi^{-1}}^{(\xi)}e_{\xi}\Phi^{(\xi)})[e_{1}e_{0}e_{1}]=0$ for $\xi\not=1$.
\end{proof}
\begin{Definition} Let $\DS_{\Ad,\eta}(\mathbb{C})$ be the set of
$(\Phi_{\Ad,\chi}(T_{1},T_{2}),\Phi_{\Ad,\chi,\ast}(T_{1},T_{2})) \in \mathbb{C}(T)\langle \langle e_{0\cup \mu_{N}} \rangle\rangle^{2}$ which satisfy the above equations and
$\Phi_{\Ad,\chi}[e_{1}e_{0}e_{1}]=2 (-\mu^{2}/24)$.
\newline The above equations define an affine scheme which we call the \emph{scheme of adjoint regularized double shuffle relations}.
\end{Definition}
We deduce :
\begin{Corollary} The map $\Ad(e_{1})$ defines a morphism $\DS_{\mu} \rightarrow \DS_{\mu,\Ad}$.
By the adjoint Ihara action, the image of the map $\DS_{\mu} \rightarrow \DS_{\mu,\Ad}$ is a torsor under the group defined as the image of the map $\DS_{0} \rightarrow \DS_{0,\Ad}$.
\end{Corollary}
\begin{proof} This follows from Racinet's theorem that $\DS_{\mu}$ is a torsor under the group $\DS_{0}$ for the Ihara action, and that $\Ad(e_{1})$ is a morphism of algebraic groups $(\tilde{\Pi}_{1,0},\circ^{\smallint_{1,0}}) \buildrel \sim \over \longrightarrow (\Ad_{\tilde{\Pi}_{1,0}}(e_{1}),\circ_{\Ad}^{\smallint_{1,0}}))$ (\cite{I-2}, Proposition 1.1.4).
\end{proof}
Particular cases of our adjoint regularized double shuffle relations can be found in \cite{Hi} and \cite{HMS} which appeared recently on the arXiv.
\begin{Remark} We note that $$ (-1)^{\weight}(E)E= \exp(\sum_{n\geq 2,\text{even}} \frac{2}{n} \Phi[e_{0}^{n-1}e_{1}]e_{1}^{n} ) = \exp(\sum_{n\geq 1} \frac{1}{n} \Phi[e_{0}^{2n-1}e_{1}]e_{1}^{2n} ) $$
In the context of \S3, this was equal to 1. Here, in the case where $\Phi=\Phi_{\KZ}$, we have
$\Phi[e_{0}^{2n-1}e_{1}] = \frac{(-1)^{n-1} B_{2n}}{2.(2n)!} (24\Phi[e_{0}e_{1}])^{n}$. It follows from \cite{Ihara Kaneko Zagier}, Theorem 7 that this relation holds for all solutions to the double shuffle equations. Thus
$$ (-1)^{\weight}(E)E = \exp(\sum_{n\geq 1} \frac{1}{n} \frac{(-1)^{n-1} B_{2n}}{2.(2n)!} (24\Phi[e_{0}e_{1}])^{n}e_{1}^{2n} )= \exp(\sum_{n\geq 1} \frac{- B_{2n}}{2n.(2n)!} (-24\Phi[e_{0}e_{1}]e_{1}^{2})^{n}) $$
This term appears implicitly in the relation between the two regularizations of adjoint cyclotomic multiple zeta values.
\end{Remark}
\subsubsection{Other aspects of the adjoint double shuffle relations\label{paragraph lifts}}
We give two $\Lambda$-adjoint formulations of the above adjoint shuffle equation.
\begin{Proposition} \label{4.8} For all $w,w' \in \mathcal{O}^{\ast}$, and for $\Lambda, \Lambda'$ formal variables, we have, for all $w,w' \in \mathcal{O}^{\ast}$, and for any $\xi, \xi' \in \mu_{N}(\mathbb{C})$
\begin{multline}
(\Phi_{\KZ}^{-1}e^{2\pi i e_{1}} \Phi_{\KZ}) (\frac{1}{1-\Lambda e_{0}}e_{\xi}w)\text{ }
(\Phi_{\KZ}^{-1}e^{2\pi i e_{1}} \Phi_{\KZ})(\frac{1}{1-\Lambda' e_{0}}e_{\xi'}w')
\\ =
(\Phi_{\KZ}^{-1}e^{2\pi i \xi} \Phi_{\KZ}) \bigg( \frac{1}{1-(\Lambda+\Lambda') e_{0}} e_{\xi} \big( w \text{ }\mathcyr{sh} \text{ } (\frac{1}{1-\Lambda' e_{0}}e_{\xi'}w') \big)
+ e_{\xi'} \big( (\frac{1}{1-\Lambda e_{0}}e_{\xi}w) \text{ }\mathcyr{sh} \text{ }w' \big) \bigg),
\end{multline}
\begin{equation}
(\Phi_{\KZ}^{-1}e_{1} \Phi_{\KZ}) \bigg( \frac{1}{1-(\Lambda+\Lambda') e_{0}} e_{\xi}( w \text{ }\mathcyr{sh} \text{ } (\frac{1}{1-\Lambda' e_{0}}e_{\xi'}w'))
+ e_{\xi'}((\frac{1}{1-\Lambda e_{0}}e_{\xi}w) \text{ }\mathcyr{sh} \text{ }w')\bigg) = 0.
\end{equation}
\end{Proposition}
\begin{proof} Below we use using Notation \ref{definition sigma inv DR} for $\comp^{\Lambda \Ad,\Ad}$. (a) Let us show that
\begin{multline} \label{eq:thing to prove} \comp^{\Lambda \Ad,\Ad}e_{\xi}w
\text{ }\mathcyr{sh}\text{ }\comp^{\Lambda'\Ad,\Ad}e_{\xi'}w' =
\\
\comp^{(\Lambda+\Lambda')\Ad, \Ad}e_{\xi}( w \text{ }\mathcyr{sh} \text{ } \comp^{\Lambda'\Ad,\Ad}e_{\xi'}w')
+
\comp^{(\Lambda+\Lambda')\Ad,\Ad}
e_{\xi'}(\comp^{\Lambda\Ad,\Ad}e_{\xi}w \text{ }\mathcyr{sh} \text{ }w') .
\end{multline}
\noindent Let $L$ be the left-hand side in (\ref{eq:thing to prove}). We compute the image of $\partial_{e_{x}}(L)$ for all $x \in \{0\} \cup \mu_{N}(K)$ (for the definition of $\partial_{e_{x}}$, see the proof of Proposition \ref{prop 2.3.1}). Given that $\partial_{e_{\xi}}$ is a derivation for $\mathcyr{sh}$, we obtain (where $\delta$ is Kronecker's symbol) :
\begin{equation*}
\begin{array}{l}
\partial_{e_{0}}L =
(\Lambda+\Lambda')L,
\\
\partial_{e_{\xi}}L = (w\text{ }\mathcyr{sh}\text{ } \comp^{\Lambda'\Ad,\Ad}e_{\xi'}w') + \delta_{\xi,\xi'}( \comp^{\Lambda\Ad,\Ad}e_{\xi}w \text{ }\mathcyr{sh}\text{ }w'),
\\
\partial_{e_{\xi'}}L = \delta_{\xi,\xi'} (w\text{ } \mathcyr{sh}\text{ } \comp^{\Lambda'\Ad,\Ad}e_{\xi'}w') + ( \comp^{\Lambda\Ad,\Ad}e_{\xi}w\text{ }\mathcyr{sh}\text{ } w'),
\\
\partial_{e_{x}}L = 0 \text{ }\text{if}\text{ }x \not\in \{0,\xi,\xi'\}.
\end{array}
\end{equation*}
We deduce (\ref{eq:thing to prove}) by $L=\sum\limits_{x\in \{0\} \cup \mu_{N}(K)} e_{x}\partial_{e_{x}}(L)$.
\newline\indent (b) On the other hand, $f^{-1}e^{2i \pi e_{1}} f$ resp. $f^{-1}e_{1} f$ satisfies the shuffle equation, resp. the shuffle equation modulo products. We deduce the result.
\end{proof}
\subsubsection{Adjoint and harmonic double shuffle equations for multiple polylogarithms}
The shuffle relation of multiple polylogarithms is true by their definition in terms of the iterated integrals (\ref{eq:Li}).
\newline\indent A quasi-shuffle relation for multiple polylogarithms is still true by their power series expansion (\ref{eq:Li series bis}). However, it involves several curves at the same time : if $D,D'$ are two finite subsets of $\mathbb{P}^{1}(K)$, both containing $0$ and $\infty$, the quasi-shuffle equation expressing product of multiple polylogarithms associated respectively with $\mathbb{P}^{1} - D$ and $\mathbb{P}^{1} - D'$ will involve $\mathbb{P}^{1} - DD'$ where $DD' = \{0,\infty\} \cup \{xx'\text{ | }\text{ } x \in D - \{0,\infty\} , x' \in D' - \{0,\infty\} \}$. We leave the exact definitions to the reader.
\newline\indent In the end, by imitating the proofs of \S3, we obtain :
\begin{Proposition} (rough version)
The adjoint multiple polylogarithms and multiple harmonic polylogarithms resp. finite multiple polylogarithms satisfy a generalization of the double shuffle equations of \S3 resp. \S6.
\end{Proposition}
\subsection{Harmonic double shuffle equations}
We define harmonic double shuffle equations for cyclotomic multiple harmonic values of Definition \ref{def harmonic}, in the frameworks $\smallint_{1,0}$ (\S3.3.1), $\int$ (\S3.3.2), $\Sigma$ (\S3.3.3), and we prove that they are equivalent (\S3.3.4).
\subsubsection{In the framework $\smallint_{1,0}$}
We use the adjoint double shuffle equations of \S3.2 to define the harmonic double shuffle equations in the framework of $\smallint_{1,0}$. In \cite{I-3}, Definition 2.1.2, we have introduced $\circ_{\har}^{\smallint_{1,0}}$, the pro-unipotent harmonic action of integrals at (1,0).
\begin{Proposition} \label{prop 2.3.1}Let $\psi \in K \langle \langle e_{0 \cup \mu_{N}} \rangle\rangle$ such that $\psi[e_{0}]=0$. Let $h=\comp^{\Lambda \Ad,\Ad}\psi$, i.e. $h= \sum\limits_{w} \psi[\frac{1}{1-\Lambda e_{0}}w]w$ where the sum is over words $w$ of the form $e_{\xi_{d+1}} e_{0}^{n_{d}-1}e_{\xi_{d}} \ldots e_{0}^{n_{1}-1} e_{\xi_{1}}$.
\newline (i) $\psi$ satisfies the adjoint quasi-shuffle equation if and only if $h$ satisfies
\begin{equation} \label{eq: DS har int1,0 1}
\text{for all words }w,w',\text{ }h(w \ast w') = h(w) \text{ }h(w') .
\end{equation}
\noindent (ii) $\psi$ satisfies the adjoint shuffle equation if and only if $h$ satisfies
\begin{equation} \label{eq: DS har int1,0 2}
\text{for all words }w,w'\text{ and for all }n \in \mathbb{N}^{\ast},\text{ }
h (e_{\xi'} (e_{0}^{n-1}e_{\xi} w \text{ } \mathcyr{sh} \text{ } w' )) =
h ( e_{\xi} ( w \text{ }\mathcyr{sh} \text{ }\frac{1}{1 - \Lambda e_{0}} e_{0}^{n-1}e_{\xi'}w') ) .
\end{equation}
\noindent (iii) The set
$\{ \comp^{\Lambda \Ad,\Ad}\psi \text{ }|\text{ }\psi \in \Ad_{\DS_{0}(K)}(e_{1})\}$ is a torsor under the group $(\Ad_{\DS_{0}(K)}(e_{1}),\circ_{\Ad}^{\smallint_{1,0}})$ for the pro-unipotent harmonic action $\circ_{\har}^{\smallint_{1,0}}$.
\end{Proposition}
\begin{proof} (i) The adjoint quasi shuffle equation amounts to
$$ \big(\sum_{L \geqslant 0} \psi_{\ast}[w;l]\Lambda^{l} \big) \big(\sum_{L\geqslant 0} \psi_{\ast}[w';l']\Lambda^{l'} \big) = \sum_{L \geqslant 0} \Lambda^{L} \sum_{l+l'=L} \psi[w;l] \psi[w';l'] = \sum_{L \geqslant 0}\Lambda^{L} \psi[ w \ast w';L], $$
i.e. $h(w)h(w') = h(w \ast w')$.
\newline (ii) Let us prove that, for all $w,w'$ words, and $n \in \mathbb{N}^{\ast}$, we have :
\begin{multline} \label{eq:equation 4.10}
-\comp^{\Lambda \Ad,\Ad} \bigg( (e_{0}^{n-1}e_{\xi}w) \text{ }\mathcyr{sh}\text{ } w' - w \text{ }\mathcyr{sh}\text{ } \mathcyr{sh}ft_{\ast}(e_{0}^{n-1}e_{\xi'})w') \bigg)
=
\\
\sum_{t=0}^{n-1} \big( e_{0}^{t}e_{\xi} w \big) \text{ }\mathcyr{sh}\text{ } \big( (-1)^{n-t}(\mathcyr{sh}ft_{\ast}(e_{0}^{n-1-t}e_{\xi'})w' \big).
\end{multline}
For any $x \in \{0\} \cup \mu_{N}(K)$, let $\partial_{e_{x}}$, resp. $\tilde{\partial}_{e_{x}}$
be the unique linear maps $\mathcal{O}^{\mathcyr{sh}} \rightarrow \mathcal{O}^{\mathcyr{sh}}$ that send the empty word to $0$ and defined on the other words by
$\partial_{e_{x}}(e_{x'}w) =\tilde{\partial}_{e_{x}}(we_{x'}) = \left\{ \begin{array}{l} w \text{ if }x = x'
\\ 0 \text{ if }x \not= x'
\end{array} \right.$. For all $w \in \mathcal{O}^{\mathcyr{sh}}$, we have $w = \sum\limits_{x \in \{0\} \cup \mu_{N}(K)} e_{x}\partial_{e_{x}}(w) = \sum\limits_{x \in \{0\} \cup \mu_{N}(K)} \tilde{\partial}_{e_{x}}(w)e_{x}$, and $\partial_{e_{x}}$ and $\tilde{\partial}_{e_{x}}$ are derivations for the shuffle product.
\newline\indent Let $R$ be the right-hand side in (\ref{eq:equation 4.10}). We have $R= e_{0} \partial_{e_{0}}R + \sum\limits_{\xi \in \mu_{N}(K)} e_{\xi}\partial_{e_{\xi}}(R)$. In order to prove (\ref{eq:equation 4.10}) it suffices to show the equalities :
if $\xi \not= \xi'$,
$\left\{\begin{array}{l}\partial_{e_{\xi}}(R) = w \text{ }\mathcyr{sh}\text{ } (-1)^{n}\mathcyr{sh}ft_{\ast}(e_{0}^{n-1}e_{\xi'}w')
\\ \partial_{e_{\xi'}}(R) = - (e_{0}^{n-1}e_{\xi}w)\text{ }\mathcyr{sh}\text{ } w'\end{array} \right.$ ; if $\xi=\xi'$, $\partial_{e_{\xi}}(R) = w \text{ }\mathcyr{sh}\text{ } (-1)^{n}\mathcyr{sh}ft_{\ast}(e_{0}^{n-1}e_{\xi}w') - (e_{0}^{n-1}e_{\xi}w) \text{ }\mathcyr{sh}\text{ } w'$, and $\partial_{e_{0}}(R) = \Lambda R$.
\newline The two first ones are clear ; let us show the last one. One has :
\begin{multline} \label{eq:e0x}
\partial_{e_{0}}(R) =
\sum_{t=1}^{n-1} (e_{0}^{t-1}e_{\xi}w)
\text{ }\mathcyr{sh}\text{ }
\frac{1}{(\Lambda e_{0}-1)^{n-t}} e_{0}^{n-1-t} e_{\xi'}w'
\\
+ \sum_{t=0}^{n-2} (e_{0}^{t}e_{\xi}w)
\text{ }\mathcyr{sh}\text{ }
\frac{1}{(\Lambda e_{0}-1)^{n-t}} e_{0}^{n-2-t} e_{\xi'}w'
+ (e_{0}^{n-1}e_{\xi}w)
\mathcyr{sh}
\frac{\Lambda}{\Lambda e_{0}-1} e_{\xi'}w' .
\end{multline}
The sum of the two first terms of the right hand side of (\ref{eq:e0x}) equals
$$ \sum_{t=0}^{n-2} (e_{0}^{t}e_{\xi}w)
\text{ }\mathcyr{sh}\text{ }
\bigg[1 + \frac{1}{\Lambda e_{0}-1}\bigg]
\frac{1}{(\Lambda e_{0}-1)^{n-1-t}}
e_{0}^{n-2-t}e_{\xi'}w' = \Lambda \sum_{t=0}^{n-2} (e_{0}^{t}e_{\xi}w)
\text{ }\mathcyr{sh}\text{ }
\frac{1}{(\Lambda e_{0}-1)^{n-t}}
e_{0}^{n-1-t}e_{\xi'}w' $$
This and the third term of (\ref{eq:e0x}) are, respectively, the $0 \leqslant t\leqslant n-2$ terms and the $t=n-1$ term of $\Lambda R$. This proves (\ref{eq:equation 4.10}) which implies the result.
\newline (iii) By \cite{I-3}, equation (2.1), $\circ_{\har}^{\smallint_{1,0}}$ is characterized by an equation which we can rewrite with the notation of this paper as
$\comp^{\har,\Ad}(g \circ_{\Ad}^{\smallint_{1,0}} f) = g \circ_{\har}^{\smallint_{1,0}} \comp^{\har,\Ad}f$. This gives the result.
\end{proof}
\begin{Definition}
We call (\ref{eq: DS har int1,0 1}) the \emph{harmonic quasi-shuffle equation} and (\ref{eq: DS har int1,0 2}) the \emph{harmonic shuffle equation of the framework $\int_{1,0}$} ; and we call the collection of (\ref{eq: DS har int1,0 1}) and (\ref{eq: DS har int1,0 2}) the \emph{harmonic double shuffle equations} of the framework $\int_{1,0}$.
\newline Let $\DS_{\har}^{\smallint_{1,0}}$ be the affine ind-scheme defined by the equations (\ref{eq: DS har int1,0 1}) and (\ref{eq: DS har int1,0 2}), which we call the \emph{harmonic double shuffle equations} of the framework $\int_{1,0}$.
\end{Definition}
\subsubsection{In the framework $\smallint$}
We construct the harmonic double shuffle equations in the framework of $\smallint$, i.e. by considering power series expansions of multiple polylogarithms.
\begin{Proposition} \label{lemma shuffle rien} (i) For any words $w,\tilde{w}$, we have
\begin{equation} \label{eq: DS har int 1} \har_{\mathcal{P}^{\mathbb{N}}}(w)\har_{\mathcal{P}^{\mathbb{N}}}(\tilde{w}) = \har_{\mathcal{P}^{\mathbb{N}}}(w \ast_{\har} \tilde{w}) .
\end{equation}
\noindent (ii) For any words $w= \big( (n_{i})_{d};(\xi_{i})_{d} \big)$, $\tilde{w} =\big( (\tilde{n}_{i})_{d};(\tilde{\xi}_{i})_{d}\big)$,
we have
\begin{multline} \label{eq: DS har int 2}
\har_{\mathcal{P}^{\mathbb{N}}}(w\text{ }\mathcyr{sh}\text{ }\tilde{w}) = \har_{\mathcal{P}^{\mathbb{N}}}\big( (\mathcyr{sh}ft_{\ast}S_{Y})(\tilde{w}) w\big)
\\ = \sum_{l_{1},\ldots,l_{d} \in \mathbb{N}}
\prod_{i=1}^{d'} {-\tilde{n}_{i} \choose l_{i}} (-1)^{\tilde{n}_{i}}
\har_{\mathcal{P}^{\mathbb{N}}}\big(
n_{1},\ldots,n_{d-1},n_{d}+\tilde{n}_{d}+l_{d'},\tilde{n}_{d'-1}+l_{d'-1},\ldots,\tilde{n}_{1}+l_{1};
\xi_{1},\ldots,\xi_{d},\tilde{\xi}_{d'},\ldots,\tilde{\xi}_{1} \big) .
\end{multline}
\end{Proposition}
\begin{proof} (i) This amounts to the quasi-shuffle relation for prime weighted cyclotomic multiple harmonic sums, $\har_{p^{\alpha}}(w) \har_{p^{\alpha}}(w') \har_{p^{\alpha}}(w \ast w')$, which is a consequence of the known double shuffle relations for multiple polylogarithms in two variables.
\newline (ii) The shuffle equation for multiple polylogarithms in one variable gives $\Li[w\text{ }\mathcyr{sh}\text{ }\tilde{w}] = \Li[w] \Li[\tilde{w}]$. Let $m \in \mathbb{N}^{\ast}$ ; by equation (\ref{eq:multiple polylogarithms power series expansion}) for all $m' \in \{1,\ldots,m-1\}$,
$$ \Li[\tilde{w}][z^{m-m'}] = \sum_{0<m_{1}<\ldots <m_{d} = m-m'} \frac{(\frac{\xi_{2}}{\xi_{1}})^{m_{1}} \ldots (\frac{1}{\xi_{d}})^{m_{d}}}{m_{1}^{n_{1}} \ldots m_{d}^{n_{d}}} = \sum_{m'=m'_{d}<\ldots<m'_{1}<m}
\frac{(\frac{\xi_{2}}{\xi_{1}})^{m'-m'_{1}} \ldots (\frac{1}{\xi_{d}})^{m'-m'_{d}}}{(m'-m'_{d})^{n_{d}} \ldots (m'-m'_{1})^{n_{1}}} $$
and
\begin{equation*} \label{eq:intermediate}
(\Li[w]\Li[\tilde{w}])[z^{m}] = \sum_{0<m_{1}<\ldots<m_{d} < m' < m'_{d'}<\ldots<m'_{1}<m}
\frac{\big( \frac{\xi_{2}}{\xi_{1}} \big)^{m_{1}} \ldots \big( \frac{\xi_{d+1}}{\xi_{d}} \big)^{m_{d}}
\big( \frac{\xi_{d'+1}}{\xi_{d+1}} \big)^{m'}
\big( \frac{\xi_{d'}}{\xi_{d+1}} \big)^{m'_{d'}} \ldots \big(\frac{\xi_{2}}{\xi_{1}} \big)^{m_{1}} }
{ m_{1}^{n_{1}}\ldots {m'}_{d}^{n_{d}} (m-{m'}_{d'})^{\tilde{n}_{d'}} \ldots (m-{m'}_{1})^{\tilde{n}_{1}}} .
\end{equation*}
Now assume that $m = p^{\alpha}$. For all $m'_{i} \in \{1,\ldots,p^{\alpha}-1\}$. We have $v_{p}(m'_{i})< v_{p}(p^{\alpha})$, thus $\displaystyle\frac{1}{(m'_{i}-p^{\alpha})^{\tilde{n}_{i}}}= m^{-\tilde{n}_{i}} \sum\limits_{l\geqslant 0} {-\tilde{n}_{i} \choose l} \big( \frac{p^{\alpha}}{m'_{i}} \big)^{l} \in \mathbb{Z}_{p}$, whence :
\begin{multline*}
(\Li[w]\Li[\tilde{w}])[z^{p^{\alpha}}] =
\\
\sum_{l_{1},\ldots,l_{d'} \in \mathbb{N}} \prod_{i=1}^{d} {-{n'}_{i} \choose l_{i}} (-1)^{t'_{i}} \frac{1}{z_{j_{1}}^{p^{\alpha}}} \sum_{\substack{0<m_{1} < \ldots < m_{d-1} < m_{d} = m \\ = m_{d} < m_{d-1} < \ldots m_{1} < p^{\alpha}}}
\frac{ (\frac{z_{i_{2}}}{z_{i_{1}}})^{n_{1}} \ldots (\frac{1}{z_{i_{d}}})^{n} y_{d'}^{n} (\frac{z_{j_{d'-1}}}{z_{j_{d'}}})^{m_{d-1}} \ldots (\frac{z_{j_{2}}}{z_{i_{1}}})^{n_{1}}}
{m_{1}^{n_{1}} \ldots l_{d}^{n_{d}+t_{d'}+l_{d'}} l_{d-1}^{n_{d-1}+l_{d-1}} \ldots l_{1}^{n_{1}+l_{1}}} .
\end{multline*}
On the other hand, $w\text{ }\mathcyr{sh}\text{ } \tilde{w}$ is a linear combination of words whose rightmost letter is not $e_{0}$ and, by equation (\ref{eq:har et Li 2}), we have
$$ \sum_{0<m'<p^{\alpha}} \Li[w\text{ }\mathcyr{sh}\text{ }\tilde{w}][m'] = \har_{p^{\alpha}}[w\text{ }\mathcyr{sh}\text{ }\tilde{w}] . $$
\noindent We deduce the result, given the definition of $\mathcyr{sh}ft_{\ast}$ and $S_{Y}$ (Definition \ref{def SY}).
\end{proof}
\begin{Definition}
We call (\ref{eq: DS har int 1}) the
\emph{harmonic quasi-shuffle equation}
and (\ref{eq: DS har int 2}) the \emph{harmonic shuffle equation of the framework $\int$} ; and we call the collection of (\ref{eq: DS har int 1}) and (\ref{eq: DS har int 2}) the \emph{harmonic double shuffle equations} of the framework $\int$.
\newline Let $\DS_{\har}^{\smallint}$ be the affine ind-scheme defined by the equations (\ref{eq: DS har int 1}) and (\ref{eq: DS har int 2}), which we call the \emph{harmonic double shuffle equations} of the framework $\int$.
\end{Definition}
\subsubsection{In the framework $\Sigma$}
Given that the power series expansion of multiple polylogarithms is written in terms of sums of series, we get directly the same harmonic double shuffle equations in the framework $\Sigma$. Indeed, it is known that the integral shuffle relation of cyclotomic multiple zeta values can be understood purely in terms of their formula as iterated series : it is a generalization of a proof which goes back to Euler, who defined multiple zeta values in depth $1$ and $2$ by the series formula in (\ref{eq:multizetas}), and found the double shuffle relations in this particular case. The same proof gives the shuffle relation for multiple harmonic sums : for $w=\big((n_{i})_{d};(\xi_{i})_{d+1}\big)$, $w'=\big(({n'}_{i})_{d};({\xi'}_{i})_{d+1}\big)$,
$$ \har_{m}(w\text{ }\mathcyr{sh}\text{ }w') = \sum_{0<m_{1}<\ldots<m_{d} < m' < m'_{d'}<\ldots<m'_{1}<m}
\frac{\big( \frac{\xi_{2}}{\xi_{1}} \big)^{m_{1}} \ldots \big( \frac{\xi_{d+1}}{\xi_{d}} \big)^{m_{d}}
\big( \frac{\xi_{d'+1}}{\xi_{d+1}} \big)^{m'}
\big( \frac{\xi_{d'}}{\xi_{d+1}} \big)^{m'_{d'}} \ldots \big(\frac{\xi_{2}}{\xi_{1}} \big)^{m_{1}} }
{ m_{1}^{n_{1}}\ldots {m'}_{d}^{n_{d}} (m-{m'}_{d'})^{\tilde{n}_{d'}} \ldots (m-{m'}_{1})^{\tilde{n}_{1}}} , $$
from which one can deduce the harmonic shuffle relation defined in \S3.3.2.
As concerns the quasi-shuffle relation for cyclotomic multiple harmonic values, it is of course immediate in terms of series. Thus we can directly define :
\begin{Definition} Let $\DS_{\har}^{\Sigma} = \DS_{\har}^{\smallint}$.
\end{Definition}
\subsubsection{Comparison between the results of the three frameworks}
We now show that the definitions of \S3.3.1 and \S3.3.2 are equivalent.
\begin{Lemma} \label{lemma for comparison of ds} Let $F$ a function $\mathcal{O}^{\ast} \rightarrow K$ and $\tilde{\imath}$, a function $\mathcal{O}^{\ast} \rightarrow \mathcal{O}^{\ast}[[\Lambda]]$ satisfying, for all $a,b$ words in $\mathcal{O}^{\ast}$, $\tilde{\imath}(ab)= \tilde{\imath}(b)\tilde{\imath}(a)$. We have an equivalence between :
\newline (i) $\forall n \in \mathbb{N}^{\ast}$, $\forall \xi \in \mu_{N}(K)$, $\forall w,w'$ words in $\mathcal{O}^{\ast}$, $F\big((e_{0}^{n-1}e_{\xi}w) \mathcyr{sh} w'\big) = F\big(w \mathcyr{sh} (\tilde{\imath}(e_{0}^{n-1}e_{\xi})w')\big)$
\newline (ii) $\forall u,w,w'$ words in $\mathcal{O}^{\ast}$, $F\big((uw) \mathcyr{sh} w'\big) = F\big(w \mathcyr{sh} (\tilde{\imath}(u)w')\big)$
\newline (iii) $\forall w,w'$ words in $\mathcal{O}^{\ast}$, $F(w\text{ }\mathcyr{sh}\text{ }w') = F(\tilde{\imath}(w) w')$.
\end{Lemma}
\begin{proof} (i) $\Rightarrow$ (ii) : we write $u$ as a concatenation of words of the form $e_{0}^{n_{i}-1}e_{1}$ and we iterate (iii).
\newline (ii) $\Rightarrow$ (iii) : we take $w = \emptyset$.
\newline (iii) $\Rightarrow$ (i) : we apply (iii) to each member of (i).
\end{proof}
\begin{Proposition} We have $\DS_{\har}^{\smallint_{1,0}} = \DS_{\har}^{\smallint}$.
\end{Proposition}
\begin{proof} The harmonic quasi-shuffle relations in $ \DS_{\har_{\mathcal{P}^{\mathbb{N}^{\ast}}}}^{\smallint}$ and $ \DS_{\har_{\mathcal{P}^{\mathbb{N}^{\ast}}}}^{\smallint_{1,0}}$, equations (\ref{eq: DS har int1,0 1}) and (\ref{eq: DS har int 1}), are identical. The equivalence between the harmonic shuffle equations (\ref{eq: DS har int1,0 2}) and (\ref{eq: DS har int 2}) follows from Lemma \ref{lemma for comparison of ds}.
\end{proof}
\begin{Definition} Let us call denote by
$\DS_{\har}$ the common value of $\DS_{\har}^{\smallint_{1,0}}$, $\DS_{\har}^{\smallint}$ and $\DS_{\har}^{\Sigma}$ and call its equations the prime harmonic double shuffle equations.
\end{Definition}
\begin{Remark} We see that comparing the results in the frameworks $\int_{1,0}$ and $\int$ requires a proof, whereas comparing the results in the frameworks $\int$ and $\Sigma$ is trivial, and this is because the power series expansion of multiple polylogarithms is expressed in terms of series. By contrast, in \cite{I-2} and \cite{I-3}, comparing the pro-unipotent harmonic actions $\circ_{\har}^{\smallint}$ and $\circ_{\har}^{\Sigma}$ required a proof, whereas comparing the pro-unipotent harmonic actions $\circ_{\har}^{\smallint_{1,0}}$ and $\circ_{\har}^{\smallint}$ was simple.
\end{Remark}
We now write the "overconvergent" variants of the previous results.
\begin{Proposition} \label{prop other view on }
(i) ($\int_{1,0}$) $\Lambda$Ad$p$MZV$\mu_{N}^{\dagger}$'s satisfy equations obtained as remainders (in the sense of power series in $\Lambda$) of the equations of $\Lambda$Ad$p$MZV$\mu_{N}$'s. By taking $\Lambda=1$, we deduce equations satisfied by MHV$\mu_{N}^{\dagger}$'s.
\newline (ii) ($\int$) (a) For all words $w,w'$ on $e_{0\cup \mu_{N}}$ we have :
$$ \har_{p^{\alpha}}^{\dagger_{p,\alpha}}(w \text{ }\mathcyr{sh}\text{ }w') = \har_{p^{\alpha}}^{\dagger_{p,\alpha}}(w) +
\har_{p^{\alpha}}^{\dagger_{p,\alpha}}(w') + \har_{p^{\alpha}}(\mathcyr{sh}ft_{\ast}(S_{Y}(w))w') . $$
(b) For any automorphism $\sigma$ of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$ which fixes $0$, we have an equation satisfied by $\har_{p^{\alpha}}^{\dagger_{p,\alpha}}$ obtained by
$$ \Li_{p,\alpha}^{\dagger}(\sigma(z)) = \sigma_{\ast} \har_{p^{\alpha}}^{\dagger_{p,\alpha}}(z) . $$
\end{Proposition}
\begin{proof} (i) This is immediate by Definition \ref{def over adjoint}.
\newline (ii) (a) This follows from the shuffle relation $\Li_{p,\alpha}^{\dagger}[w\text{ }\mathcyr{sh}\text{ }w'] = \Li_{p,\alpha}^{\dagger}[w]\Li_{p,\alpha}^{\dagger}[w']$ specialized to the coefficient $[z^{p^{\alpha}}]$, combined with the differential equation (\ref{eq:horizontality equation}) characterizing $\Li_{p,\alpha}^{\dagger}$, Lemma \ref{lemma power series expansion Li dagger}, and the proof of Lemma \ref{lemma shuffle rien}.
\newline (b) This follows from the definition of $\Li_{p,\alpha}^{\dagger}$ in terms of the value of the Frobenius on the canonical de Rham path (\cite{I-1}, \S1) and the functoriality of the Frobenius.
\end{proof}
We see that, as in \S3.3, the formulas for the harmonic shuffle relation in the framework $\int_{1,0}$ and in the framework $\int$ are different. We leave to the reader to check that they are equivalent, following \S3.3.3. Using the framework $\Sigma$ here is beyond the scope of this paper : this requires to uses the formulas for Ad$p$MZV$\mu_{N}$'s found in \cite{I-2}, and this will be the subject of \cite{II-2}.
\begin{Definition} Let $\DS_{\har}^{\dagger}$, resp. $\M_{\har}^{\dagger}$ be the affine ind-scheme defined by the equations obtained in Proposition \ref{prop other view on } as variants of those of $\DS_{\har}$, resp. $\M_{\har}$ defined in \S3 and \S4 respectively.
\end{Definition}
We have canonical isomorphisms
$\DS_{\har}^{\dagger} \simeq \DS_{\har}$, resp. $\M_{\har}^{\dagger} \simeq \M_{\har}$.
\begin{Remark} An extension of Proposition \ref{prop other view on } (ii), which would include a quasi-shuffle equation for $\har_{p^{\alpha}}^{\dagger_{p,\alpha}}$ and the equations obtained by the functoriality with respect to automorphisms of
$\mathcal{M}_{0,5}^{(N)}$, could be obtained by using the variant of $\Li_{p,\alpha}^{\dagger}$ on $\mathcal{M}_{0,5}^{(N)}$, defined by the Frobenius of $\pi_{1}^{\un,\crys}(\mathcal{M}_{0,5}^{(N)})$. Here, in simplicial coordinates, $\mathcal{M}_{0,5}^{(N)}$ is
$\{ (y_{1},y_{2},\ldots,y_{n}) \in (\mathbb{P}^{1} - \{0,1,\infty\})^{n} \text{ }|\text{ }\forall i,j, \forall \xi \text{ s.t. }\xi^{N}=1, y_{i} \not= \xi y_{j} \}$. We leave it to the reader.
\end{Remark}
\subsection{A corollary of the shuffle equation : the (usual, adjoint and harmonic) reversal equations\label{paragraph reflexion}}
Let $f \in \Pi_{1,0}(K)$ ; since $f$ satisfies the shuffle equation, denoting by $S$ the antipode of the shuffle Hopf algebra, we have $\hat{S}^{\vee}(f) = f^{-1}$. By writing $f^{-1} = (1-(1-f))^{-1} = \sum\limits_{l\geqslant 0} (1-f)^{l}$, we deduce polynomial equations on the coefficients of $f$ :
\begin{equation} \label{eq: reflexion}
\text{for all words }w,\text{ }
f[S(w)] = \sum_{\substack{l\geqslant 0 \\ w_{1}\ldots w_{l}=w}} (-1)^{l}\prod_{i=1}^{l}f[w_{i}] .
\end{equation}
\begin{Definition} \label{def reflexion} We call (\ref{eq: reflexion}) the reversal equation.
\end{Definition}
Applying it to $f=\Phi_{\KZ}$ resp. $f=\Phi_{p,\alpha}$ gives a family of polynomial equation on MZV$\mu_{N}$'s, resp. $p$MZV$\mu_{N}$'s. Our terminology ``reversal equation'' in Definition \ref{def reflexion} is motivated by Rosen's ``asymptotic reversal theorem'', \cite{Rosen} which we are going to recover below, as a particular case of Proposition \ref{harmonic reflexion}, as particular cases of our results, and interpret in terms of $\pi_{1}^{\un}(\mathbb{P}^{1} - \{0,1,\infty\})$. We are going to see that the reversal equation has adjoint and harmonic analogues which are given by relatively simple formulas and which are quite natural.
\begin{Proposition} \label{reflexion adjoint} Let $\psi \in K\langle \langle e_{0\cup \mu_{N}}\rangle\rangle$ satisfying the shuffle equation modulo products (equation (\ref{eq:ds adjoint 1})). We have :
\begin{multline} \label{eq:reflexion adjoint}
\text{for any positive integers }
n_{i}, n'_{i'} \text{ and N-th roots of unity } \xi_{i},\xi'_{i},
\\
\sum_{\substack{l \geqslant 0 \\
l_{i} \geqslant 0 (1 \leqslant i \leqslant d)\\
l+l_{1}+\ldots+l_{d}=L}}
\prod_{i=1}^{d} {-n_{i} \choose l_{i}} \psi\big( \big( (n'_{i})_{d'},(n_{i}+l_{i})_{d};({\xi'}_{i})_{d'},(\xi_{i})_{d},\xi \big),l\big)
\\ = \sum_{\substack{l' \geqslant 0 \\
{l'}_{i} \geqslant 0 (1 \leqslant i \leqslant d') \\
l'+{l'}_{1}+\ldots+{l'}_{d'}=L}}
\prod_{i=1}^{d'} {-{n'}_{i} \choose {l'}_{i}}
\psi\big( \big( (n_{i})_{d'},({n'}_{i}+{l'}_{i})_{d};\xi,({\xi}_{i})_{d'},({\xi'}_{i})_{d}\big),l'\big) .
\end{multline}
\end{Proposition}
\begin{proof} The result amounts to the following equality :
\begin{multline} \label{eq: 4 2 5} (-1)^{\sum\limits_{i=1}^{d}n_{i}} \psi[ \frac{1}{1-\Lambda e_{0}}e_{\xi} \frac{e_{0}^{n_{1}-1}}{(1-\Lambda e_{0})^{n_{1}}}e_{\xi_{1}} \ldots
\frac{e_{0}^{n_{d}-1}}{(1-\Lambda e_{0})^{n_{d}}}e_{\xi_{d}}e_{0}^{{n'}_{d'}-1}e_{{\xi'}_{d'}}\ldots e_{0}^{{n'}_{1}-1}e_{{\xi'}_{1}}] \\ =(-1)^{\sum\limits_{i'=1}^{d'}{n'}_{i'}}
\psi \big[\frac{1}{1-\Lambda e_{0}}e_{\xi_{1}} \ldots \frac{e_{0}^{{n'}_{1}}-1}{(1-\Lambda e_{0})^{n_{1}}}e_{\xi_{d'}}
\frac{e_{0}^{{n'}_{d'}}-1}{(1-\Lambda e_{0})^{{n'}_{d'}}}e_{\xi_{d}}e_{0}^{n_{d}-1}\ldots e_{\xi_{1}}e_{0}^{n_{1}-1}e_{\xi} \big] .
\end{multline}
(a) By the hypothesis, $\psi$ is primitive for the shuffle coproduct. Thus, let $S$ be the antipode of $\mathcal{O}^{\mathcyr{sh}}$ ; we have $\hat{S}^{\vee}(\psi)=-\psi$ ; and the first line of (\ref{eq: 4 2 5}) is equal to
\begin{equation} \label{eq:this} (-1)^{\sum\limits_{j=1}^{d'}{n'}_{i}} \psi\big[e_{{\xi'}_{1}}e_{0}^{{n'}_{1}-1}\ldots e_{{\xi'}_{d'}}e_{0}^{{n'}_{d'}-1}e_{{\xi}_{d}}
\frac{e_{0}^{n_{d}-1}}{(1+\Lambda e_{0})^{n_{d}}} \ldots e_{\xi_{1}} \frac{e_{0}^{n_{1}-1}}{(1+\Lambda e_{0})^{n_{1}}}e_{\xi}\frac{1}{1+\Lambda e_{0}}\big] .
\end{equation}
(b) Let $f \in K \langle \langle e_{0\cup \mu_{N}} \rangle\rangle$ such that, for all words $w$, we have $f[w\text{ }\mathcyr{sh}\text{ }e_{0}] = f[w]f[e_{0}]$ ; let $T,U_{1},\ldots,U_{d}$ formal variables ; we have :
\begin{multline}
\label{eq: shuffle coefficients e0 TU}
f[\frac{e_{0}^{{n''}_{d}-1}}{(1-U_{d}e_{0})^{{n''}_{d}}}e_{{\xi''}_{d}} \ldots
\frac{e_{0}^{{n''}_{1}-1}}{(1-U_{1}e_{0})^{{n''}_{1}}}e_{{\xi''}_{1}} \frac{1}{1 - Te_{0}}]
\\ =
f[\frac{e_{0}^{{n''}_{d''}-1}}{(1 - (U_{d}-T)e_{0})^{{n''}_{d''}}}e_{{\xi''}_{d''}}\ldots
\frac{e_{0}^{{n''}_{1}-1}}{(1 - (U_{1}-T)e_{0})^{{n''}_{1}}}
e_{{\xi''}_{1}}] e^{f[e_{0}]T} .
\end{multline}
We apply this to the particular case where $f[e_{0}]=0$ and where the first line of (\ref{eq: shuffle coefficients e0 TU}) is (\ref{eq:this}). In that case, the second line of (\ref{eq: shuffle coefficients e0 TU}) becomes equal to the second line of (\ref{eq: 4 2 5}).
\end{proof}
\begin{Definition} We call (\ref{eq:reflexion adjoint}) the adjoint reversal equation.
\end{Definition}
\begin{Proposition} \label{harmonic reflexion}We can prove by the three frameworks $\int_{1,0}$, $\int$ and $\Sigma$ that :
\begin{equation} \label{eq:first eq of corollary}
\text{for all words }w,w',\text{ }
\text{har}_{\mathcal{P}^{\mathbb{N}}}( (\mathcyr{sh}ft_{\ast}S_{Y})(w')w) = \text{har}_{\mathcal{P}^{\mathbb{N}}}( (\mathcyr{sh}ft_{\ast}S_{Y})(w)w') .
\end{equation}
\end{Proposition}
\begin{proof} (i) In the framework $\int_{1,0}$ : we take $f=\sum\limits_{\xi \in \mu_{N}(K)} \xi^{-p^{\alpha}} {\Phi_{p,\alpha}^{(\xi)}}^{-1}e_{\xi}\Phi_{p,\alpha}^{(\xi)}$ and $\Lambda=1$ in equation (\ref{eq: 4 2 5}) and we use equation (\ref{eq:formula for n=1}).
\newline (ii) In the framework $\smallint$ : this is a direct consequence of the harmonic shuffle equation (equation \ref{eq: DS har int 2}).
\newline (iii) In the framework $\Sigma$ : this is an easy generalization of Rosen's proof of the ``asymptotic reversal theorem'' \cite{Rosen} which corresponds to the $N=1$, $\alpha=1$ and $w'=\emptyset$ case.
\end{proof}
\begin{Definition} We call (\ref{eq:first eq of corollary}) the harmonic reversal equation.
\end{Definition}
We also note that, by choosing $w'$ to be the empty word in (\ref{eq:the last equation}) is deduced by choosing $w'$ equal to the empty word in (\ref{eq:first eq of corollary}), we deduce the simpler equation
\begin{equation} \label{eq:the last equation}
\text{ for all words }w,\text{ }
\text{har}_{p^{\alpha}}((\mathcyr{sh}ft_{\ast}S_{Y})(w)) = \text{har}_{p^{\alpha}}(w) .
\end{equation}
\section{Around associator equations and Kashiwara-Vergne equations\label{double shuffle}}
We review briefly associator equations and Kashiwara-Vergne equations and the known relation between them (\S4.1) then we explain that Kashiwara-Vergne equations can be formulated as a property of adjoint MZV's more naturally than MZV's (\S4.2) and we explain equations satisfied by multiple harmonic values, and more generally harmonic multiple polylogarithms, which are related to Kashiwara-Vergne equations (\S4.3). In this section, we take $N=1$ most of the time, because to our knowledge there is no cyclotomic Kashiwara-Vergne theory.
\subsection{Review on associators and Kashiwara-Vergne equations}
\subsubsection{Associators}
The notion of associators has been introduced in \cite{Drinfeld}. Let $k$ be a field of characteristic $0$ and let $\mu \in k$. By \cite{Furusho pentagon}, the set of associators $M_{\mu}(k)$ is the set of elements $\Phi \in \Pi(k)$ (we are using Notation \ref{la premiere notation}, and assuming $N=1$), such that $\mu=\pm \sqrt{24\Phi[e_{0}e_{1}]}$ and
$$ \phi(e_{12},e_{23}+e_{24})\phi(e_{13}+e_{23},e_{34}) = \phi(e_{23},e_{34}) \phi(e_{12}+e_{13},e_{24}+e_{34})
\phi(e_{12},e_{23}) $$
where $e_{ij}$ are the generators of $\Lie(\pi_{1}^{\un,\dR}(\mathcal{M}_{0,5},\omega_{\dR}))$ (\S2.1.3). The definition in \cite{Drinfeld} uses several equations : (2.12), (2.13) and a rescaled version of equation (5.3) of \cite{Drinfeld}.
\newline\indent On the other hand one has the scheme $\GRT_{1}$ defined in \cite{Drinfeld}, (equations (5.12), (5.13), (5.14), (5.15) of \cite{Drinfeld}) ; by \cite{Drinfeld}, Proposition 5.9, $\GRT_{1}$ is isomorphic to $M_{0}$. We note that equation (5.15) of \cite{Drinfeld} is
\begin{equation} \label{eq:sum of residues} e_{0} + \phi(e_{0},e_{1})^{-1}e_{1}\phi(e_{0},e_{1}) + \phi(e_{0},e_{\infty})^{-1}e_{\infty}\phi(e_{0},e_{\infty}) = 0
\end{equation}
where $e_{0}+e_{1}+e_{\infty}=0$ : the $e_{x}$'s generate $\Lie(\Pi)$.
\newline\indent The Ihara product (\ref{eq:Ihara}) restricts to a group law on $\GRT_{1}$ (\cite{Drinfeld}, equation (5.16)) and to an action of $\GRT_{1}$ on $M_{\mu}$ which makes $M_{\mu}$ into a $\GRT_{1}$-torsor (\cite{Drinfeld}, Proposition 5.5).
\newline\indent The generating series of multiple zeta values, $\Phi_{\KZ} \in \Pi_{1,0}(\mathbb{R})$ (equation (\ref{eq:Phi KZ})) has been first introduced in \cite{Drinfeld} \S2, and we have $\Phi_{\KZ} \in M_{2i\pi}(\mathbb{R})$ by \cite{Drinfeld}. By \cite{Unver Drinfeld}, combined with the fact that $\Phi_{p,1}$ is in the commutator subgroup of $\Pi_{1,0}(\mathbb{Q}_{p})$, proved in \cite{Furusho 2}, \S3, we have $\GRT_{1}(\mathbb{Q}_{p})$ ; this implies that $\Phi_{p}^{\KZ} \in \GRT_{1}(\mathbb{Q}_{p})$ (\cite{Furusho 2}, proof of Proposition 3.1). By the relations of iteration of the Frobenius (\cite{I-3}, equations (1.11), (1.12), (1.13) and Proposition 1.5.2), which involve the Ihara product, this implies that $\Phi_{p,\alpha} \in \GRT_{1}(\mathbb{Q}_{p})$ for all $\alpha \in \mathbb{Z} \cup \{\pm \infty\} - \{0\}$.
\subsubsection{Review on Kashiwara-Vergne equations, according to Alekseev, Enriquez and Torossian \cite{AT}, \cite{AET}}
Let $k$ be a field of characteristic $0$.
Let $\lie_{n}$ be the free Lie algebra over $k$ on $n$ variables $x_{1},\ldots,x_{n}$. Let $\widehat{\lie}_{n}$ be its degree completion (where the $x_{i}$'s have degree 1).
\newline\indent For any $u_{1},\ldots,u_{n}$ in $\widehat{\lie}_{n}$, let $[[u_{1},\ldots,u_{n}]]$ be the derivation of $\lie_{n}$ defined by $x_{i} \mapsto [x_{i},u_{i}]$ for all $i$. Such a derivation is called tangential and the set of tangential derivations is denoted by $\tder_{n}$.
\newline\indent For any $U_{1},\ldots,U_{n}$ in $\exp(\widehat{\lie}_{n})$, let $[[U_{1},\ldots,U_{n}]]$ be the automorphism of $\widehat{\lie}_{n}$ defined by $x_{i} \mapsto U_{i}x_{i}U_{i}^{-1}$ for all $i$. Such an automorphism is called tangential and the set of tangential automorphism is denoted by $\TAut_{n}$.
\newline\indent Let $A_{n}$ be the universal enveloping algebra of $f_{n}$ and $T_{n}=A_{n}/[A_{n},A_{n}]$. The image of an element $S$ by the map $A_{n} \rightarrow T_{n}$ is denoted by $\langle S \rangle$. Let $\partial_{k} : A_{n} \rightarrow A_{n}$ be defined by $x= x_{0}+\sum_{k=1}^{n} \partial_{k}(x)x_{k}$ et $x_{0} \in k$. Let $j : \text{tder}_{n} \rightarrow \hat{T}_{n}$, $[[u_{1},\ldots,u_{n}]] \mapsto \langle \sum_{k=1}^{n} x_{k} \partial_{k}(u_{k})\rangle$. It integrates into $J: \TAut_{n} \mapsto \hat{T}_{n}$ (\cite{AT}, Proposition 5.1).
\newline\indent Below, $F_{2}(k)$ be the pro-unipotent completion of the free-group on two generators $X,Y$ and $\sim$ means ``is conjugated to''. Following \cite{AET} \S2.1, let
\begin{multline*} \Sol \KV =
\\ \{ \mu : F_{2}(k) \buildrel \sim \over \longrightarrow \Aut(\exp(\widehat{\lie}_{2})) \text{ }|\text{ }\mu(X) \sim e^{x},\mu(Y) \sim e^{y}, \mu(XY)=e^{x+y}, \exists r \in u^{2}k[[u]], j(a) = \langle r(x+y)-r(x)-r(y)\rangle \}
\end{multline*}
where $r$ is uniquely determined by $\mu$. Following \cite{AET}, \S2.2, let
$$ \KRV = \{ a \in \Aut(\widehat{\lie}_{2}) \text{ }|\text{ }a(x) \sim x,a(y) \sim y, a(x+y)=x+y,
\\ \exists s \in u^{2}k[[u]], J(a) = \langle s(x+y)-s(x)-s(y)\rangle \} . $$
\indent By \cite{AET}, Theorem 2.1, the Ihara action (\ref{eq:Ihara}) makes $\KRV$ into a group and $\Sol \KV$ a torsor under that group, and there is a morphism of torsors $M_{1}(K) \rightarrow \Sol\KV(K)$, which sends $\Phi$ to
\begin{equation} \label{eq:XY}
\mu_{\Phi} : \begin{array}{l}
X \mapsto \Phi(x,-x-y)^{-1}e^{x}\Phi(x,-x-y) \\
Y \mapsto e^{-(x+y)/2}\Phi(y,-x-y)^{-1}e^{y} \Phi(y,-x-y)e^{(x+y)/2}
\end{array} .
\end{equation}
The element $r$ associated with $\mu_{\Phi}$ is equal to $-\log(\Gamma_{\Phi})$ where $\Gamma_{\Phi}= \exp \big(\sum\limits_{n\geq 2} \frac{(-1)^{n}}{n} \zeta_{\Phi}(n)u^{n} \big)$ (\cite{AET}, Proposition 2.2). We note that
$\zeta_{\Phi}(2n) = -\frac{1}{2}\frac{B_{2n}}{(2n)!}$ for all $n$, which is independent of $\Phi$.
\subsection{The Kashiwara-Vergne equations as a property of adjoint multiple zeta values}
We restrict the study to $N=1$ because to our knowledge there is no cyclotomic Kashiwara-Vergne theory. There is a cyclotomic associator theory \cite{Enriquez}.
We are going to see that Kashiwara-Vergne equations can be naturally viewed as a property of adjoint MZV's rather than as a property of MZV's : this gives much simpler equations. This gives an analogy between the passage from associator equations to Kashiwara-Vergne equations constructed in \cite{AET} and \cite{AT} and the passage from double shuffle equations to adjoint double shuffle equations that we have constructed in \S3.
In particular, our question \ref{question} is an analogue of a conjecture of Alekseev-Torossian \cite{AT} which compares Drinfeld associators and solutions to the Kashiwara-Vergne problem.
\subsubsection{In the $p$-adic case}
We consider that the Kashiwara-Vergne equations are the adjoint version of associator equations. Let us rewrite them - or rather the equations of KRV - in terms of Ad$p$MZV's. By the above discussion, we consider the following equation :
\begin{equation} \label{eq:KV} \langle j(\mu_{\Phi_{p,\alpha}})\rangle = \langle \log(\Gamma_{\Phi_{p,\alpha}}(x)) + \log(\Gamma_{\Phi_{p,\alpha}}(y)) - \log(\Gamma_{\Phi_{p,\alpha}})(x+y) \rangle .
\end{equation}
Moreover, by the isomorphism $\lie_{2} \simeq \Lie \Pi_{0,0}$ given by $(x,y) \leftrightarrow (e_{1},e_{\infty})$ where $e_{0}+e_{1}+e_{\infty}=0$,
$\mu_{\Phi_{p,\alpha}}$ sends $\left\{ \begin{array}{l}
e_{0} \mapsto e_{0}
\\ e_{1} \mapsto \Phi_{p,\alpha}(e_{0},e_{1})^{-1} e_{1} \Phi_{p,\alpha}(e_{0},e_{1})
\\ e_{\infty} \mapsto \Phi_{p,\alpha}(e_{0},e_{\infty})^{-1} e_{\infty} \Phi_{p,\alpha}(e_{0},e_{\infty}) \end{array} \right.$. This recovers equation (\ref{eq:sum of residues}).
\newline This is obtained by applying
$\underset{\mu \rightarrow 0}{\lim} \frac{1}{\mu} \frac{d}{d\mu}$ to the automorphism
$\left\{ \begin{array}{l} e^{e_{1}} \mapsto \Phi(e_{0},e_{1})^{-1} e^{\mu e_{1}} \Phi(e_{0},e_{1})
\\ e^{e_{\infty}} \mapsto e^{\frac{\mu}{2}e_{0}} \Phi(e_{0},e_{\infty})^{-1} e^{\mu e_{\infty}} \Phi(e_{0},e_{\infty}) e^{-\frac{\mu}{2} e_{0}}
\end{array} \right.$ obtained by rescaling (\ref{eq:XY}) : this is the variant of $\Sol KV$ and of the map $M_{1} \rightarrow \Sol \KV$ obtained by choosing $M_{\mu}$ instead of $M_{1}$.
\begin{Proposition} \label{KV explicit}The equation (\ref{eq:KV}) amounts to explicit linear equations on adjoint $p$MZV's.
\end{Proposition}
\begin{proof} Equation (\ref{eq:KV}) amounts to say that
$$ j(\mu_{\Phi}) - \log(\Gamma_{\Phi}(e_{1})) - \log(\Gamma_{\Phi}(e_{\infty})) + \log(\Gamma_{\Phi})(e_{1}+e_{\infty}) \rangle $$
has image zero by the quotient map $k\langle\langle e_{1},e_{\infty}\rangle\rangle \rightarrow k
\langle\langle e_{1},e_{\infty} \rangle\rangle / [k
\langle\langle e_{1},e_{\infty} \rangle\rangle,k
\langle\langle e_{1},e_{\infty} \rangle\rangle]$.
\newline\indent Let $\tilde{\partial}_{e_{1}}^{(e_{1},e_{\infty})}, \tilde{\partial}_{e_{\infty}}^{(e_{1},e_{\infty})} :
k\langle \langle e_{1},e_{\infty}\rangle\rangle \rightarrow k\langle\langle e_{1},e_{\infty} \rangle\rangle$ be defined, by, for any word $w$ on the alphabet $e_{1},e_{\infty}$,
$w = \partial_{e_{1}}^{(e_{1},e_{\infty})}(w)e_{1} +
\partial_{e_{\infty}}^{(e_{1},e_{\infty})}(w)e_{\infty}$.
\newline One can compute $j(\mu_{\Phi})$ as follows. We have
\begin{equation} \label{eq:jmuPhi} j(\mu_{\Phi}) = e_{1} \tilde{\partial}_{e_{1}}^{(e_{1},e_{\infty})} \big(\Phi^{-1}e_{1}\Phi\big)(e_{0},e_{1}) + e_{\infty}\tilde{\partial}_{e_{\infty}}^{(e_{1},e_{\infty})}\big(\Phi^{-1}e_{1}\Phi\big)(e_{0},e_{\infty})
\end{equation}
and the right-hand side of (\ref{eq:jmuPhi}) can be expressed using what follows. Let $w(e_{0},e_{1}) = e_{0}^{n_{d}-1}e_{1}\ldots e_{1}e_{0}^{n_{0}-1}$.
\newline\indent
If $n_{0}\geq 2$ we have
$\left\{
\begin{array}{l} \tilde{\partial}_{e_{1}}^{(e_{1},e_{\infty})}(w(e_{1},e_{\infty})) = e_{0}^{n_{d}-1}e_{1}\ldots e_{1}e_{0}^{n_{0}-1}e_{1}(-e_{1})
\\ \tilde{\partial}_{e_{\infty}}^{(e_{1},e_{\infty})}(w(e_{1},e_{\infty})) = e_{0}^{n_{d}-1}e_{1}\ldots e_{1}e_{0}^{n_{0}-1}e_{1}(-e_{\infty})
\end{array} \right.$.
\newline\indent
If $n_{0}=1$, we have
$\left\{
\begin{array}{l} \tilde{\partial}_{e_{1}}^{(e_{1},e_{\infty})}(w(e_{1},e_{\infty})) = e_{0}^{n_{d}-1}e_{1}\ldots e_{1}e_{0}^{n_{0}-1}
\\ \tilde{\partial}_{e_{\infty}}^{(e_{1},e_{\infty})}(w(e_{1},e_{\infty})) = 0 \end{array} \right. $
\newline\indent On the other hand, $\log(\Gamma_{\Phi}(e_{1})) + \log(\Gamma_{\Phi}(e_{\infty})) - \log(\Gamma_{\Phi})(e_{1}+e_{\infty}) \rangle$ is equal to :
$$ \sum_{n\geqslant 2} \frac{(-1)^{n}}{n}\zeta_{\Phi}(n) ( e_{1}^{n} + e_{\infty}^{n}-(e_{1}+e_{\infty})^{n}) = \sum_{n\geqslant 2} \frac{(-1)^{n}\zeta_{\Phi}(n)}{n} e_{1}^{n} +
\sum_{n\geqslant 2} \frac{\zeta_{\Phi}(n)}{n} \big( (e_{0}+e_{1})^{n} - e_{0}^{n} \big) . $$
\indent We have a basis of $k\langle \langle e_{1},e_{\infty}\rangle\rangle$ formed by the words on the alphabet $\{e_{1},e_{\infty}\}$. It is sent to a generating family of $k\langle \langle e_{1},e_{\infty}\rangle\rangle/[k\langle \langle e_{1},e_{\infty}\rangle\rangle,k\langle \langle e_{1},e_{\infty}\rangle\rangle]$ from which we can extract a basis. A simple computation gives a partition of the set of words on $\{e_{1},e_{\infty}\}$ according to their image in $k\langle \langle e_{1},e_{\infty}\rangle\rangle/[k\langle \langle e_{1},e_{\infty}\rangle\rangle,k\langle \langle e_{1},e_{\infty}\rangle\rangle]$.
\newline\indent Finally we use the isomorphism $k\langle \langle e_{0},e_{1}\rangle\rangle \simeq k\langle \langle e_{1},e_{\infty}\rangle\rangle$, $f(e_{0},e_{1}) \mapsto f(e_{1},e_{\infty})$ defined by $e_{\infty}=-e_{0}-e_{1}$.
\end{proof}
\begin{Remark} Equation (\ref{eq:sum of residues}) can be regarded as a part of the Kashiwara-Vergne equations. It also amounts to an equation on $p$MZV's : for all words,
\begin{equation} \label{eq:KV dim 1} \forall w,\text{ } \zeta_{p,\alpha}^{\Ad}(w) + \zeta_{p,\alpha}^{\Ad}(w(e_{0}-e_{1},-e_{1})) = 0 .
\end{equation}
Indeed, we have $(\phi^{-1}e_{1}\phi)(e_{0},e_{\infty}) = \sum\limits_{w\text{ word on }\{e_{0},e_{1}\}}
(\phi^{-1}e_{1}\phi)[w(e_{0}-e_{1},-e_{1})]w$.
\end{Remark}
\subsubsection{In the complex case}
The automorphism of $\Pi_{0,0}(\mathbb{C})$ which sends $(e^{e_{0}},e^{e_{1}}) \mapsto (e^{2i\pi e_{0}}, \Phi_{\KZ}^{-1}e^{2i\pi e_{1}}\Phi_{\KZ})$ satisfies the Kashiwara-Vergne equations rescaled by $\tau(2\pi i)$ where $\tau$ is as in equation (\ref{eq:tau}). This can be easily generalized to any $N$, using \cite{AKKN}. The computations of \S4 can be repeated with the equations instead of the equations of the group $\KRV$. As concerns the equations of dimension 1, the equation
$e_{0} + (\Phi_{\KZ}^{-1}e_{1}\Phi_{\KZ})(e_{0},e_{1}) + (\Phi_{\KZ}^{-1}e_{1}\Phi_{\KZ})(e_{0},e_{\infty}) \equiv 0 \mod (\zeta(2))$ is
$$ e^{2i\pi e_{0}} . \big( \Phi(e_{0},e_{1})^{-1} e^{2i\pi e_{1}} \Phi(e_{0},e_{1}) \big).
\big( e^{\frac{\mu}{2}e_{0}} \Phi(e_{0},e_{\infty})^{-1} e^{2i\pi e_{\infty}} \Phi(e_{0},e_{\infty}) e^{-\frac{\mu}{2} e_{0}} \big) = 1 . $$
We leave the details to the reader.
\subsection{Related properties of harmonic multiple polylogarithms}
We find properties of multiple harmonic values and harmonic multiple polylogarithms, in the three frameworks $\int_{1,0}$, $\int$ and $\Sigma$, which are related to the previosu considerations. We prove that the equations arising from $\pi_{1}^{\un}(\mathbb{P}^{1} - \{0,1,\infty\})$ in these three frameworks are equivalent.
\subsubsection{In the framework $\int_{1,0}$}
\begin{Proposition} \label{harmonic duality DR}
(i) $\Phi^{-1}e_{1}\Phi$ satisfies equation (\ref{eq:KV dim 1}) if and only if
$h(w) = \comp^{\Lambda \Ad,\Ad} (-1)^{\depth}\Phi^{-1}e_{1}\Phi$ satisfies
$$ \forall w,\text{ } h( w(e_{0}+e_{1},-e_{1})) = - \sum_{d'\geqslant 1, \text{ }z = e_{0}^{t_{d'}-1}e_{1}\ldots e_{0}^{t_{1}-1}e_{1}} (-1)^{d'} h(z.w) . $$
(ii) $\Phi^{-1}e_{1}\Phi$ satisfies equation (\ref{eq:KV}) if and only if $\comp^{\Lambda\Ad,\Ad}(\Phi^{-1}e_{1}\Phi)$ satisfies certain equations on $\Lambda$-adjoint $p$MZV's.
\end{Proposition}
Of course, in this statement, we can replace $\comp^{\Lambda \Ad,\Ad}$ by $\comp^{\har,\Ad}$, and $\Lambda$-adjoint $p$MZV's by MHV's.
\begin{proof} (i) We have, for all words $w$ :
$(1-\Lambda(e_{0}+e_{1}))^{-1}e_{1}w = (1-\Lambda e_{0})^{-1}e_{1}w + (1-\Lambda e_{0})^{-1}\Lambda e_{1}(1-\Lambda (e_{0}+e_{1}))^{-1}e_{1}w$.
Indeed, we have $1 = (1 - \Lambda e_{0} - \Lambda e_{1} )(1 - \Lambda e_{0} - \Lambda e_{1} )^{-1} =
(1 - \Lambda e_{0} )(1 - \Lambda e_{0} - \Lambda e_{1} )^{-1} - \Lambda e_{1} (1 - \Lambda e_{0} - \Lambda e_{1} )^{-1}$. Left multiplication by $(1 - \Lambda e_{0})^{-1}$ gives $(1-\Lambda(e_{0}+e_{1}))^{-1} = (1-\Lambda e_{0})^{-1} + (1-\Lambda e_{0})^{-1}\Lambda e_{1}(1-\Lambda (e_{0}+e_{1}))^{-1}$. Whence the equality. This implies the result.
\newline (ii) Follows from the definitions and from the translation of equation (\ref{eq:KV}) in the proof of Proposition \ref{KV explicit}.
\end{proof}
\begin{Definition} Let $\M_{\har}$ the ind-scheme defined by the equations among MHV's resp. $\Lambda$-adjoint $p$MZV's obtained in Proposition \ref{harmonic duality DR}.
\end{Definition}
\subsubsection{In the framework $\int$ \label{pre-associator paragraph}}
For any $n \geqslant 4$, any automorphism of $\mathcal{M}_{0,n}$ induces by functoriality an automorphism of $\pi_{1}^{\un,\dR}(\mathcal{M}_{0,n})$ equipped with $\nabla_{\KZ}$. The functoriality of $\nabla_{\KZ}$ gives a relation of the type
\begin{equation} \label{eq: LtildeCL} \tilde{L} = C L
\end{equation}
where $L$ and $\tilde{L}$ are two different branchs of multiple polylogarithms on $\mathcal{M}_{0,n}$, defined as the unique solutions to $\nabla_{\KZ}$ with prescribed asymptotics at a chosen base-point, and $C \in \pi_{1}^{\un,\dR}(\mathcal{M}_{0,5},\omega_{\dR})$
\begin{Definition} Let us call pre-associator equations the equations of the form (\ref{eq: LtildeCL}).
\end{Definition}
It is sufficient to restrict to $n \in \{4,5\}$.
The associator equations are deduced from the pre-associator equations by writing $C$ in terms of $\Phi_{\KZ}$ and by using that the automorphisms which are involved are of finite order.
\newline\indent Let us denote by $O$ the tangential base-point $(\vec{1}_{0},\vec{1}_{0})$ in cubic coordinates on $\overline{\mathcal{M}}_{0,5}$ as well as its image by $\overline{\mathcal{M}}_{0,5} \rightarrow \overline{\mathcal{M}}_{0,4}$, which we choose as the origin of the paths of integration. Let $\Stab_{O}^{\mathcal{M}_{0,4}}$ and $\Stab_{O}^{\mathcal{M}_{0,5}}$ be the stabilizers of $O$ in $\Aut(\mathcal{M}_{0,4}) = S_{3}$ and $\Aut(\mathcal{M}_{0,5})= S_{5}$.
\newline\indent The associator equations (duality, hexagon and pentagon) can be obtained by the pre-associator equations associated with the automoprhisms $(z \mapsto 1-z)_{\ast}$, $(z \mapsto \frac{1}{z})_{\ast}$ resp. $\big(\sigma: (x_{1},x_{2},x_{3},x_{4},x_{5}) \mapsto ( x_{5},x_{4},x_{1},x_{3},x_{2})\big)_{\ast} = \big((c_{1},c_{2}) \mapsto (c_{2}, \frac{1 -c_{2}}{1 - c_{1}c_{2}})\big)_{\ast}$ in cubic coordinates, which are elements of $\Aut(\mathcal{M}_{0,4}) - \Stab_{O}^{\mathcal{M}_{0,4}}$, resp. $\Aut(\mathcal{M}_{0,4}) - \Stab_{O}^{\mathcal{M}_{0,5}}$.
\newline\indent We now write consequences of the pre-associator equations associated with elements of $\Stab_{O}^{\mathcal{M}_{0,4}}$ and $\Stab_{O}^{\mathcal{M}_{0,5}}$. We consider the involution $\sigma : z \mapsto \frac{z}{z-1}$ which is in $\Aut(\mathcal{M}_{0,4})$, and the element of $\Aut(\mathcal{M}_{0,5})$ written in cubic coordinates as $\rho : (c_{1},c_{2}) \mapsto \big( -c_{1} \frac{1-c_{2}}{1-c_{1}}, - c_{2} \frac{1-c_{1}}{1-c_{2}}\big)$.
\begin{Proposition} \label{harmonic duality DR RT} (i) The pre-associator equation associated with $\sigma : z\mapsto \frac{z}{z-1}$ implies :
$$ \har_{\mathcal{P}^{\mathbb{N}}}( w(e_{0}+e_{1},-e_{1})) = - \sum_{\substack{d'\geqslant 1,\text{ }z = e_{0}^{t_{d'}-1}e_{1}\ldots e_{0}^{t_{1}-1}e_{1}}} (-1)^{\depth(z)} \har_{\mathcal{P}^{\mathbb{N}}}(z.w) . $$
\noindent (ii) \label{5 32}The pre-associator equation associated with $\rho : (c_{1},c_{2}) \mapsto \big( -c_{1} \frac{1-c_{2}}{1-c_{1}}, - c_{2} \frac{1-c_{1}}{1-c_{2}}\big)$ implies a family of equations on multiple harmonic values.
\end{Proposition}
\begin{proof} (i) See more generally the proof of Proposition \ref{proposition 5 30}, which we can apply to $(a,c,d)=(1,1,-1)$, knowing that $\displaystyle \sigma_{1,1,-1}^{\ast}\big( \frac{dz}{z} \big) = \frac{dz}{z} - \frac{dz}{z-1}$ and $\displaystyle\sigma_{1,1,-1}^{\ast} \big( \frac{dz}{z-1} \big) = \frac{dz}{z-1}$.
\newline (ii) We have, for any sequence of coefficients $(a_{m,n})$,
$$ \sum_{n,m\geq 0} a_{n,m}
\bigg(-\frac{c_{1}(1-c_{2})}{1-c_{1}}\bigg)^{n}
\bigg(-\frac{c_{2}(1-c_{1})}{1-c_{2}}\bigg)^{m}
= \bigg( \sum_{n<m} + \sum_{n=m} + \sum_{n>m} \bigg) a_{n,m}
\bigg(-\frac{c_{1}(1-c_{2})}{1-c_{1}}\bigg)^{n}
\bigg(-\frac{c_{1}(1-c_{2})}{1-c_{1}}\bigg)^{m} . $$
The subsum $\displaystyle \sum\limits_{n=m}$ is
$\sum_{n \geq 0} a_{n,n} \big(c_{1}c_{2}\big)^{n} $.
The subsum $\displaystyle \sum\limits_{n<m}$ is, after writing $(n,m)=(n_{1},n_{1}+n_{2})$,
$$ \sum_{\substack{0\leq n_{1} \\ 1\leq n_{2}}} a_{n_{1},n_{1}+n_{2}} (-c_{1})^{n_{1}}(-c_{2})^{n_{1}+n_{2}} \big( \frac{1-c_{1}}{1-c_{2}} \big)^{n_{2}} =
\sum_{\substack{0\leq n_{1}\\ 1\leq n_{2}}}\sum_{\substack{0\leq l_{1} \leq n_{2} \\ 0 \leq l_{2}}} a_{n_{1},n_{1}+n_{2}}
{n_{2} \choose l_{1}}{-n_{2} \choose l_{2}}
(-c_{1})^{n_{1}+l_{1}}(-c_{2})^{n_{1}+n_{2}+l_{2}}
$$
and after writing $(N,M)=(n_{1}+l_{1},n_{1}+n_{2}+l_{2})$,
$$ = \sum_{N,M\geq 0} c_{2}^{N}c_{1}^{M} \sum_{\substack{0 \leq l_{1} \leq n_{2} \leq M \\0 \leq l_{2} \leq N}} a_{n_{1},n_{1}+n_{2}} (-1)^{M-n_{1}+n_{2}}
{n_{2} \choose M-n_{1}} {-n_{2} \choose N-n_{1}-n_{2}} $$
\begin{equation} \label{eq:interm} = \sum_{N,M\geq 0} c_{2}^{N}c_{1}^{M} (-1)^{N+M} \sum_{\substack{0 \leq \leq n_{1} \leq N \\0 \leq n_{1}+n_{2} \leq M}} a_{n_{1},n_{1}+n_{2}}
{n_{2} \choose M-n_{1}}
{N-n_{1}-1 \choose n_{2}-1} .
\end{equation}
On the other hand, for all integers $0 \leqslant a \leqslant b$, we have
$$ {b \choose a} = \frac{b}{a} \bigg( \frac{b}{1}-1 \bigg) \ldots \bigg( \frac{b}{a-1}-1 \bigg) = \frac{b}{a}
\sum_{r \geq 0} b^{r} \frak{h}_{a}(\underbrace{1,\ldots,1}_{r})(-1)^{b-r}$$
where, for any positive integer $m$ and any word $w$, $\frak{h}_{m}(w)$ is defined by $\har_{m}(w)=m^{\weight(w)}\frak{h}_{m}(w)$.
Whence (\ref{eq:interm}) equals
\begin{multline*} \sum_{N,M\geq 0} c_{2}^{N}c_{1}^{M} (-1)^{N+M} \sum_{\substack{0 \leq l_{1} \leq n_{2} \leq M \\0 \leq l_{2} \leq N \\ n_{1}=N-l_{1}}} a_{n_{1},n_{1}+n_{2}}
\\
\sum_{R_{1}\geq 0} n_{2}^{R_{1}+1}\frac{(-1)^{n_{2}-R_{1}}}{M-n_{1}}
\frak{h}_{M-n_{1}}(\underbrace{1,\ldots,1}_{R_{1}})
\sum_{R_{2} \geq 0} \frac{(-1)^{N-n_{1}-1-R_{2}} }{n_{2}-1} (N-n_{1}-1)^{r+1} \frak{h}_{n_{2}-1}(\underbrace{1,\ldots,1}_{R_{2}}) .
\end{multline*}
We now assume that $N=M=p^{\alpha}$ (a more general version of the computation in which $M$ and $N$ are any powers of $p$). In the domain of summation
$\{(m_{1},\ldots,m_{R_{1}}) \in \mathbb{N}^{r}\text{ }|\text{ }0<m_{1}<\ldots<m_{R_{1}} \}$ of $\frak{h}_{M-n_{1}}(\underbrace{1,\ldots,1}_{R_{1}})$, we make the change of variable $m'_{i}=M-R_{1}$. We conclude by using the property of delocalization of cyclotomic multiple harmonic sums established in \cite{I-2}, Proposition-Definition 4.2.2.
\end{proof}
\begin{Definition} Let $M_{\har}^{\smallint}$ be the affine ind-scheme defined by the equations of Proposition \ref{harmonic duality DR RT}.
\end{Definition}
Let us consider all homographies of $\mathbb{P}^{1}$ which preserve $0$. For $a,c,d \in K$ with $K=\mathbb{C}$ or $K=\mathbb{C}_{p}$, let $\sigma_{a,c,d} : z \in \mathbb{P}^{1}(K)\mapsto \frac{az}{cz+d} \in \mathbb{P}^{1}(K)$.
\begin{Proposition} \label{proposition 5 30}
The horizontality of the morphisms of the form $(\sigma_{a,c,d})_{\ast} : \pi_{1}^{\un,\dR}(\mathbb{P}^{1} - D) \rightarrow \pi_{1}^{\un,\dR}(\mathbb{P}^{1} - D')$ with respect to $\nabla_{\KZ}$ gives an explicit family of relations between prime weighted multiple harmonic sums.
\end{Proposition}
\begin{proof} Let $\big((n_{i})_{d}; (z_{i})_{d+1} \big)$ be any index. For $z$ close to $0$, we consider the following equality of iterated integrals :
\begin{equation} \label{eq:formula 0} \int_{0}^{\sigma_{a,c,d}(z)} \omega_{0}^{n_{d}-1}\omega_{z_{d}} \ldots \omega_{0}^{n_{1}-1}\omega_{z_{1}} =
\int_{0}^{z} (\sigma_{a,c,d})^{\ast} \big( \omega_{0}^{n_{d}-1}\omega_{z_{d}} \ldots \omega_{0}^{n_{1}-1}\omega_{z_{1}}\big) .
\end{equation}
\noindent The left-hand side of (\ref{eq:formula 0}) is $\Li \big((n_{i})_{d};(z_{i})_{d+1}\big) \big(\frac{az}{cz+d} \big)$ ; let us write its power series expansion at $0$. For $z \in \mathbb{C}$ on a neighborhood of $0$, we have, for all $n \in \mathbb{N}$ :
$\big(\frac{az}{cz+d} \big)^{n}
= \big(\frac{az}{d} \big)^{n} \sum\limits_{l\geqslant 0} {-n_{d} \choose l} \big(\frac{c}{d}z\big)^{l}$. Thus, by the power series expansion of multiple polylogarithms (\ref{eq:Li series bis}),
\begin{equation} \label{eq:first step} \Li \big( (n_{i})_{d};(z_{i})_{d} \big) \bigg(\frac{az}{cz+d} \bigg) = \sum_{0<m_{1}<\ldots<m_{d}<m}
\frac{ (\frac{z_{i_{2}}}{z_{i_{1}}})^{m_{1}} \ldots (\frac{z_{i_{d}}}{z_{i_{d-1}}})^{m_{d-1}} (\frac{a}{c z_{i_{d}}})^{m_{d}}}{m_{1}^{n_{1}} \ldots m_{d}^{n_{d}}} \big(\frac{c}{d}\big)^{m} {m-1 \choose m - m_{d}} .
\end{equation}
\noindent For all $m \in \mathbb{N}^{\ast}$, and $\tilde{m} \in \{1,\ldots,m-1\}$, we have
$\displaystyle {m - 1 \choose m - \tilde{m}} =
\frac{(m-1)(m-2) \ldots (m - (m-\tilde{m}))}{1 \times 2 \times \ldots \times (m-\tilde{m})}
= \big(\frac{m}{1}-1\big)\big(\frac{m}{2}-1\big) \ldots \big(\frac{m}{m-\tilde{m}}-1\big)$ ; expanding this product gives : $\displaystyle {m-1 \choose m - \tilde{m}} = \sum\limits_{r \geqslant 0} m^{r} (-1)^{m-\tilde{m}-r} \frak{h}_{m-\tilde{m}} (\underset{r}{\underbrace{1,\ldots,1}})$ where the sum over $r$ is finite. By a change of variable, we have, for all $r \in \mathbb{N}$ : $\displaystyle \frak{h}_{m-\tilde{m}} (\underset{r}{\underbrace{1,\ldots,1}}) = \sum\limits_{\tilde{m}<j_{r}< \ldots <j_{1}<m} \frac{(-1)^{r}}{(m-j_{r}) \ldots (m-j_{1})}$. Now, we assume that $m = p^{\alpha}$ with $p$ a prime number and $\alpha \in \mathbb{N}^{\ast}$. By factorizing by $ \frac{1}{(-j_{1})\ldots (-j_{r})}$ and expanding into $p$-adic series factors of the form $\frac{1}{1-x}$ with $|x|_{p}<1$, we obtain
\begin{equation} \label{eq:second step} {p^{\alpha} - 1 \choose p^{\alpha} - \tilde{m}} = (-1)^{\tilde{m}-1}
\sum_{\substack{ r \geqslant 0 \\ l_{1},\ldots,l_{r} \geqslant 0}} (p^{\alpha})^{r + \sum\limits_{i=1}^{r} l_{i} }\sum_{\tilde{m}<j_{r}<\ldots <j_{1}<p^{\alpha}} \frac{1}{j_{r}^{1+l_{r}} \ldots j_{1}^{1+l_{1}}}
\end{equation}
\noindent By (\ref{eq:first step}) and (\ref{eq:second step}) we have :
\begin{multline} \label{eq: 5 4 4} (p^{\alpha})^{n_{1}+\ldots+n_{d}}\Li \Big( (n_{i})_{d};(z_{i})_{d} \Big) \big(\frac{az}{cz+d} \big)[z^{p^{\alpha}}] \\
= - \sum_{\substack{ r \geqslant 0 \\ l_{1},\ldots,l_{r} \geqslant 0}}
(p^{\alpha})^{\sum\limits_{j=1}^{d}n_{j} + r + \sum\limits_{i=1}^{r} l_{i}}\sum_{0<n_{1}<\ldots<n_{d}<j_{r}<\ldots<j_{1}<p^{\alpha}}
\frac{ (\frac{z_{i_{2}}}{z_{i_{1}}})^{m_{1}} \ldots (\frac{z_{i_{d}}}{z_{i_{d-1}}})^{m_{d-1}} (-\frac{a}{c z_{i_{d}}})^{m_{d}}}{m_{1}^{n_{1}} \ldots m_{d}^{n_{d}} j_{r}^{l_{r}+1} \ldots j_{1}^{l_{1}+1}} \big(\frac{c}{d} \big)^{p^{\alpha}}
\\ = -\sum_{\substack{ r \geqslant 0 \\ l_{1},\ldots,l_{r} \geqslant 0}} \har_{p^{\alpha}}
\bigg( (n_{i})_{d};(1+l_{i})_{d}; (z_{i_{j}})_{d},\big(\frac{-a}{c}\big)_{r},
\frac{-ad}{c^{2}} \bigg)
\end{multline}
\noindent The right-hand side of (\ref{eq:formula 0}) can be expressed in terms of multiple harmonic sums via (\ref{eq:Li series bis}) and the fact that, for all $y\in \mathbb{C}$, we have :
$$ \sigma_{a,c,d}^{\ast}\bigg(\frac{dz}{z - y}\bigg) = \bigg( \frac{(a- yc)}{(a-y c)z + (-yd)} - \frac{c}{cz+d} \bigg) dz $$
\end{proof}
\subsubsection{In the framework $\Sigma$}
\begin{Proposition} \label{harmonic duality RT} We can prove in the framework $\Sigma$ that, for all words $w$ :
\begin{equation}
\label{eq: harmonic duality RT}
\har_{\mathcal{P}^{\mathbb{N}}}( w(e_{0}+e_{1},-e_{1})) = - \sum_{\substack{d'\geqslant 1 \\ z = e_{0}^{t_{d'}-1}e_{1}\ldots e_{0}^{t_{1}-1}e_{1}}} (-1)^{\depth(z)} \har_{\mathcal{P}^{\mathbb{N}}}(z.w) .
\end{equation}
\end{Proposition}
This is a generalization of a result of Rosen called the ``asymptotic duality theorem'' \cite{Rosen}, which is equivalent to the $\alpha=1$ case of this result, although it is formulated differently.
\begin{proof} In Rosen's proof of the asymptotic duality theorem \cite{Rosen}, which we generalize from $\alpha=1$ to any $\alpha \in \mathbb{N}^{\ast}$, let us modify the last step by writing, for all $n_{d} \in \{1,\ldots,p^{\alpha}-1\}$, ${p^{\alpha} - 1 \choose m_{d} - 1} = {p^{\alpha} - 1 \choose p^{\alpha} - m_{d}}$ and applying the canonical expansion of binomial coefficients in terms of multiple harmonic sums to ${p^{\alpha} - 1 \choose p^{\alpha} - m_{d}}$ instead of ${p^{\alpha} - 1 \choose m_{d}-1}$. Instead of obtaining quantities of the form $\sum_{0<n<p^{\alpha}} \frak{h}_{m}(w') \frak{h}_{m}(w'')$ and linearizing them by the quasi-shuffle formula as in \cite{Rosen}, we obtain quantities of the form $\sum_{0<m<p^{\alpha}} \frak{h}_{m}(w')\frak{h}_{p^{\alpha}-m}(w'')$, and we make a change of variable $m_{i}\mapsto p^{\alpha}-m_{i}$ in the domain of summation of $\frak{h}_{p^{\alpha}-m}(w'')$, which gives an infinite sums of prime weighted multiple harmonic sums $\har_{p^{\alpha}}(w''')$.
\end{proof}
The asymptotic duality theorem in \cite{Rosen} is that for all indices $w$, we have, for all primes $p$,
\begin{equation} \label{eq: remarque 1}
\har_{p}\big( w(e_{0}+e_{1},-e_{1})\big) = \har_{p}\big( w + (w \ast (\frac{1}{1+y_{1}}) \big)
\end{equation}
It is obtained as a $p$-adic lift of a theorem of Hoffman \cite{Hoffman 2}, about multiple harmonic sums $\frak{h}_{p}(w) \mod p$, called the "duality theorem", which relies on the Newton series of multiple harmonic sums, in the sense of \S2.2.4. The proof in \cite{Hoffman 2} and \cite{Rosen} remains true if one replaces $\har_{p}$ by $\har_{p^{\alpha}}$ for all $\alpha \in \mathbb{N}^{\ast}$. Our alternative version (\ref{eq: harmonic duality RT}) does not use the quasi-shuffle equation. However, given the quasi-shuffle relation for $\har_{p^{\alpha}}$, equations (\ref{eq: remarque 1}) and the generalization of (\ref{eq: harmonic duality RT}) to any $\alpha$ are equivalent. Indeed, we have, for all $w = e_{0}^{n_{d}-1}e_{1}\ldots e_{0}^{n_{1}-1}e_{1}$,
\begin{equation} \label{eq: remarque 2}
\har_{p^{\alpha}}(w(e_{0}+e_{1},-e_{1}))
= \har_{p^{\alpha}}(e_{0}^{n_{d}}e_{1} - e_{0}^{n_{d}-1}e_{1})(e_{0}^{n_{d-1}-1}e_{1}\ldots e_{0}^{n_{1}-1}e_{1} \ast \frac{1}{1+e_{1}}) .
\end{equation}
\noindent Finally, for all words $w$ and all $n \in \mathbb{N}$, we have :
$$ \har_{p^{\alpha}}\big( e_{0}^{n}e_{1} ( w \ast \frac{1}{1+e_{1}}) \big)
= - \sum_{\substack{d\geqslant 1 \\ x_{d},\ldots,x_{1} \geqslant 1 \\ z=e_{0}^{x_{d}-1}e_{1}\ldots e_{0}^{n+x_{1}-1}e_{1}}} (-1)^{\depth(z)}\har_{p^{\alpha}}(z.w) . $$
\subsubsection{Comparison between the results of the three frameworks}
We observe that Proposition \ref{harmonic duality DR} (i), Proposition 4.3.4 (i) and Proposition 4.3.6, obtained respectively in the frameworks $\int_{1,0}$, $\int$ and $\Sigma$ are the same results. This is the part of the harmonic associator equations obtained by using $\pi_{1}^{\un}(\mathcal{M}_{0,4})$.
\section{Finite cyclotomic multiple zeta values and finite multiple polylogarithms}
We define ``finite'' analogues of adjoint cyclotomic multiple zeta values and study their algebraic properties. This gives a generalization of the notion of finite multiple zeta values of Kaneko and Zagier and an interpretation of that notion in terms of the crystalline pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$.
\subsection{Review on finite multiple zeta values}
Let $\mathcal{P}$ be the set of prime numbers. The following ring is a $\mathbb{Q}$-algebra, it is denoted by $\mathcal{A}$ by Kaneko and Zagier and is called the ring of integers modulo infinitely large primes \cite{Kontsevich} :
\begin{equation}
\label{eq:integers mod infinitely large}
\mathbb{F}_{p\rightarrow\infty} =
\mathbb{Z}/_{p\rightarrow\infty} = \big( \prod_{p \in \mathcal{P}} \mathbb{Z}/p\mathbb{Z} \big) / \big( \bigoplus_{p \in \mathcal{P}} \mathbb{Z}/p\mathbb{Z} \big) .
\end{equation}
Kaneko and Zagier have defined finite multiple zeta values as the following numbers, where $d$ and $n_{i}$ $(1\leqslant i \leqslant d)$ are any positive integers (see also \cite{Hoffman 2} and \cite{Zhao} for earlier almost identical notions) :
\begin{equation} \zeta_{\mathcal{A}} \big((n_{i})_{d} \big) = \left(\sum_{0<m_{1}<\ldots<m_{d}<p} \frac{1}{m_{1}^{n_{1}} \ldots m_{d}^{n_{d}}} \mod p \right)_{p \in \mathcal{P}} \in \mathcal{A} .
\end{equation}
They conjecture that the following formula defines an isomorphism between the $\mathbb{Q}$-algebra generated by fMZV's and the $\mathbb{Q}$-algebra generated by MZV's moded out by the ideal $(\zeta(2))$ : the explicit formula is striking
\begin{equation} \label{eq:Kaneko Zagier iso}
\zeta_{\mathcal{A}}\big( (n_{i})_{d} \big) \mapsto \sum_{d'=0}^{d} (-1)^{n_{d'+1}+\ldots+n_{d}}
\zeta(n_{1},\ldots,n_{d'})\zeta(n_{d},\ldots,n_{d'+1}) \mod \zeta(2) .
\end{equation}
According to the period conjectures on MZV's and $p$MZV's, it is equivalent to make the same conjecture with $p$MZV's instead of MZV's modulo $\zeta(2)$. With Definition \ref{def adjoint}, the $p$-adic analogue of the right-hand side of (\ref{eq:Kaneko Zagier iso}) is $\zeta_{p,\alpha}^{\Ad}\big((n_{i})_{d};0\big)$. Their complex analogues are sometimes called symmetric or symmetrized multiple zeta values \cite{NoteCRAS}, or finite real multiple zeta values.
\subsection{Finite cyclotomic multiple zeta values}
\subsubsection{Variants of the ring of integers modulo infinitely large primes}
We introduce some variants of the ring $\mathbb{F}_{p\rightarrow \infty}$ (\ref{eq:integers mod infinitely large}) :
\begin{Definition} \label{def mod p infinitely large}(i) For each $\alpha \in \mathbb{N}^{\ast}$, let $\mathbb{F}_{p^{\alpha} \rightarrow \infty}=\big( \prod_{p \in \mathcal{P}} \mathbb{F}_{p^{\alpha}} \big) / \big( \bigoplus_{p \in \mathcal{P}} \mathbb{F}_{p^{\alpha}}\big)$.
\newline (ii) Let $\overline{\mathbb{F}}_{p\rightarrow \infty} = \big( \prod_{p \in \mathcal{P}} \overline{\mathbb{F}_{p}} \big) / \big( \bigoplus_{p \in \mathcal{P}} \overline{\mathbb{F}_{p}} \big)$.
\newline (iii) Let the Frobenius of $\mathbb{F}_{p^{\alpha}\rightarrow \infty}$ resp. $\overline{\mathbb{F}}_{p\rightarrow \infty}$ be the automorphism defined as $\sigma : (x_{p})_{p} \mapsto (x_{p}^{p})_{p}$.
\end{Definition}
These rings are $\mathbb{Q}$-algebras. We have an inclusion
$\mathbb{F}_{p^{\alpha}\rightarrow \infty} \hookrightarrow \overline{\mathbb{F}}_{p\rightarrow \infty}$, and, if $\alpha'$ divides $\alpha''$, we have an inclusion $\mathbb{F}_{p^{\alpha'}\rightarrow \infty} \hookrightarrow \mathbb{F}_{p^{\alpha''}\rightarrow \infty}$, and these inclusions are compatible.
\subsubsection{Finite cyclotomic multiple zeta values}
We now generalize the definition of finite multiple zeta values. Below, for any root of unity $\xi \in \overline{\mathbb{Q}}$ and for any prime $p$, we denote by $\overline{\xi}$ the image of $\xi$ in $\overline{\mathbb{F}_{p}}$.
\begin{Definition} \label{def cycl mzv} Let finite cyclotomic multiple zeta values (fMZV$\mu_{N}$'s) be the following numbers : for any positive integers $d$ and $n_{i}$ ($1\leqslant i \leqslant d$) and roots of unity $\xi_{i}$ ($1\leqslant i \leqslant d+1$),
\begin{equation} \zeta_{f}
\big( (n_{i})_{d};(\xi_{i})_{d+1} \big) = \left(\sum_{0<m_{1}<\ldots<m_{d}<p} \frac{\big( \frac{\bar{\xi}_{2}}{\bar{\xi}_{1}} \big)^{m_{1}} \ldots \big(\frac{\bar{\xi}_{d+1}}{\bar{\xi}_{d}}\big)^{m_{d}} \big(\frac{1}{\bar{\xi}_{d+1}} \big)^{p}}{m_{1}^{n_{1}} \ldots m_{d}^{n_{d}}} \right)_{p \in \mathcal{P}} \in \overline{\mathbb{F}}_{p \rightarrow \infty} .
\end{equation}
\end{Definition}
\noindent We note that, letting $N$ be the lcm of the orders of the $\xi_{i}$'s as roots of unity, then, for $p$ large enough, $p$ does not divide $N$, and the crystalline realization of $\pi_{1}^{\un}(\mathbb{P}^{1} \setminus \{0,\mu_{N},\infty\})$ is defined.
\newline\indent By extrapolating on Kaneko-Zagier's conjecture (\S6.1), we can speculate an equivalence between the properties of the numbers $\zeta_{f}(n_{i})_{d};(\xi_{i})_{d+1}$ and the numbers
\begin{equation} \label{eq: analogue of finite} \zeta_{p,\alpha}^{\Ad} \big( (n_{i})_{d};0;(\xi_{i})_{d}\big) .
\end{equation}
We take again the notations of Definition \ref{def mod p infinitely large}. For any $z \in D \setminus \{0,\infty\}$, we denote by $\bar{z}$ the reduction of $z$ modulo $p$ large enough.
\begin{Definition} Let finite multiple polylogarithms be the numbers :
$$ \Li_{f}
\big( (n_{i})_{d};(z_{i})_{d+1} \big) = \left(\sum_{0<m_{1}<\ldots<m_{d}<p}
\frac{\big( \frac{\bar{z}_{2}}{\bar{z}_{1}} \big)^{m_{1}} \ldots \big(\frac{\bar{z}_{d+1}}{\bar{z}_{d}} \big)^{m_{d}} \big(\frac{1}{\bar{z}_{d+1}}\big)^{p} }{m_{1}^{n_{1}} \ldots m_{d}^{n_{d}}} \mod p \right)_{p \in \mathcal{P}} \in \overline{\mathbb{F}}_{p\rightarrow \infty} . $$
\end{Definition}
\subsubsection{Equations satisfied by finite cyclotomic multiple zeta values}
We have
$$ \overline{\mathbb{F}}_{p\rightarrow \infty} = \bigg\{ (x_{p})_{p} \in \prod_{p\in \mathcal{P}}\mathbb{C}_{p} \text{ }|\text{ } v_{p}(x_{p}) \geqslant 0 \text{ for p large} \bigg\} \text{ }\bigg/\text{ }\bigg\{ (x_{p})_{p} \in \prod_{p\in \mathcal{P}} \mathbb{C}_{p} \text{ }|\text{ } v_{p}(x_{p}) \geqslant 1 \text{ for p large} \bigg\} . $$
\indent Finite cyclotomic multiple zeta values are expressed as reductions of cyclotomic multiple harmonic values, up to the Frobenius of Definition \ref{def mod p infinitely large} (iii) : for any harmonic word $w$, we have
\begin{equation} \label{eq:red mod cycl har} \sigma^{\alpha-1}\zeta_{f}\big(w\big) =
(p^{-\weight(w)}\har_{p^{\alpha}}(w) \mod p)_{p,\alpha} .
\end{equation}
Indeed, with the notations of (\ref{eq:mult har sums}), the subsum of $\har_{p^{\alpha}}\big( (n_{i})_{d};(\xi_{i})_{d+1}\big)$ on the subdomain $(m_{1},\ldots,m_{d}) \in p^{\alpha-1}\mathbb{N}^{\ast} \times \ldots \times p^{\alpha-1}\mathbb{N}^{\ast}$ is equal, by the change of variable $m_{i}=p^{\alpha-1}m'_{i}$, to $\har_{p} \big( (n_{i})_{d};(\xi^{p^{\alpha}-1}_{i})_{d+1}) \big)$, which is of valuation $\geqslant 0$, and the subsum on the complementary domain has $p$-adic valuation $\geqslant 1$.
\newline\indent On the other hand, the numbers (\ref{eq: analogue of finite}) are reductions of the $\Lambda$-adjoint $p$-adic cyclotomic multiple zeta values :
\begin{equation} \label{eq:modulo Lambda}
\zeta_{p,\alpha}^{\Lambda \Ad}\big( (n_{i})_{d};0;(\xi_{i})_{d+1}\big)= \zeta_{p,\alpha}^{\Lambda \Ad} \big((n_{i})_{d};(\xi_{i})_{d+1} \big) \mod \Lambda^{\weight(w)+1} .
\end{equation}
In the $N=1$ case, Akagi-Hirose-Yasuda have proved an integrality property of $p$MZV's and deduced a finite variant of equation (\ref{eq:formula for n=1}) \cite{AHY} :
\begin{equation} \label{eq: AHY} \zeta_{\mathbb{Z}_{/p\rightarrow \infty}}\big((n_{i})_{d}\big) = \big( p^{-\sum_{i=1}^{d}n_{i}}\zeta_{p,1}^{\Ad}\big((n_{i})_{d};0\big) \mod p \big)_{p\text{ prime}} \in \mathcal{A} .
\end{equation}
By the relations of iteration of the Frobenius (\cite{I-3}, equations (1.11), (1.12), (1.13) and Proposition 1.5.2), equation (\ref{eq: AHY}) remains true if $\zeta_{p,1}^{\Ad}$ is replaced by $\zeta_{p,\alpha}^{\Ad}$. Moreover, this can be generalized to any $N$ by using the integrality property of $p$-adic iterated integrals proved in \cite{Chatzis}.
\newline By equations (\ref{eq:red mod cycl har}), (\ref{eq:modulo Lambda})
and the cyclotomic generalization of (\ref{eq: AHY}), we deduce immediately from \S3 and \S4 some equations satisfied by fMZV$\mu_{N}$'s and their analogues (\ref{eq: analogue of finite}). This leads to the following definition :
\begin{Definition} (i) Let $\DS_{f}$ the scheme of finite double shuffle equations, be the pro-affine scheme defined the term of lowest weight in the equations defining $\DS_{\har}$ of \S3.3.
\newline (ii) Let $\GRT_{f}$, the scheme of finite associator equations, be the pro-affine scheme defined by the term of lowest weight in the equations defining $\M_{\har}$ of \S4.3.
\end{Definition}
Some of these equations already appeared in the literature. In the $N=1$ case, the finite double shuffle equations appear in \cite{Kaneko}, \cite{Kaneko Zagier} and in \cite{NoteCRAS}. To our knowledge, the method of proof in \cite{Kaneko}, \cite{Kaneko Zagier} uses the framework $\int$.
The one dimensional part of the finite associator equations appears in \cite{Hoffman 2}, where it is called the "duality theorem".
The finite version of the reversal equation (\ref{paragraph reflexion}) in the $N=1$ case is the formula
$\zeta_{\mathcal{A}}(n_{1},\ldots,n_{d}) = (-1)^{n_{1}+\ldots+n_{d}}\zeta_{\mathcal{A}}(n_{1},\ldots,n_{d}) \mod p$, which appears in \cite{Zhao}, lemma 3.3 and \cite{Hoffman 2}, theorem 4.5.
\subsection{A generalization : finite analogues of adjoint cyclotomic multiple zeta values}
We now associate a ``finite'' analogue to all adjoint cyclotomic multiple zeta values $\zeta_{p,\alpha}^{\Ad}\big((n_{i})_{d};b;(\xi_{i})_{d+1}\big)$, not only the case $b=0$. We are going to use the overconvergent cyclotomic multiple harmonic values (Definition \ref{variant harmonic}) :
\begin{Definition} Let the adjoint finite cyclotomic multiple zeta values (Ad$f$MZV$\mu_{N}$'s) be the numbers
$$ \zeta^{\Ad}_{f,\alpha} \big( (n_{i})_{d};l;(\xi_{i})_{d+1} \big) =
(p^{-(n_{1}+\ldots+n_{d}+l)}\har_{p,\alpha}^{\dagger}( (n_{i})_{d};l;(\xi_{i})_{d+1}) \mod p)_{p} \in \overline{\mathbb{F}}_{p\rightarrow \infty} . $$
\end{Definition}
By the integrality of $p$-adic iterated integrals on punctured projective lines proved in \cite{Chatzis}, and by the expression of
$\har_{p,\alpha}^{\dagger}\big((n_{i})_{d};l;(\xi_{i})_{d+1}\big)$ as an infinite sums of Ad$p$MZV$\mu_{N}$'s (Proposition \ref{prop formula dagger} (i)), the numbers $\zeta^{\Ad}_{f,\alpha} \big( (n_{i})_{d};l;(\xi_{i})_{d+1} \big)$ are well defined and satisfy a generalization of equation (\ref{eq: AHY}) :
$$ \zeta_{f,\alpha} \big( (n_{i})_{d};l;(\xi_{i})_{d+1} \big) = \big( \zeta^{\Ad}_{p,\alpha} \big((n_{i})_{d};l;(\xi_{i})_{d+1}\big) \mod p \big) \in \overline{\mathbb{F}}_{p\rightarrow\infty} .
$$
We also have a generalization of equation (\ref{eq:modulo Lambda}), which refers to the overconvergent variant of $\Lambda$Ad$p$MZV$\mu_{N}$'s (Definition \ref{def over adjoint}) :
$$ \zeta^{\Ad}_{p,\alpha} \big((n_{i})_{d};l;(\xi_{i})_{d+1}\big) = \zeta_{p,\alpha}^{\Lambda \Ad \dagger} \big((n_{i})_{d};(\xi_{i})_{d+1}\big) \mod \Lambda^{\weight(w)+1} $$
In the $N=1$ case, the numbers $\zeta_{f,\alpha} \big( (n_{i})_{d};l\big)$ are in the $\mathbb{Q}$-vector space generated by finite multiple zeta values. This follows from equation (\ref{eq: AHY}) and the fact that, by \cite{Yasuda}, that the numbers $\zeta^{\Ad}_{p,\alpha}\big((n_{i})_{d};0\big)$ generate the space of $p$MZV's. We guess that this remains true for any $N$, up to the Frobenius $\sigma$ of $\overline{\mathbb{F}}_{p\rightarrow \infty}$ (Definition \ref{def mod p infinitely large}). We generalize again the conjecture of Kaneko and Zagier :
\begin{Conjecture} The numbers
$\zeta^{\Ad}_{f,\alpha} \big( (n_{i})_{d};l;(\xi_{i})_{d+1}\big)$ and $\zeta_{p,\alpha}^{\Ad} \big( (n_{i})_{d};l;(\xi_{i})_{d+1}\big)$ satisfy the same algebraic properties.
\end{Conjecture}
In a summary, starting with MHV$\mu_{N}$'s, one can on the one hand consider the associated graded for the weight filtration which gives Ad$p$MZV$\mu_{N}$'s, and on the other hand consider the associated graded for the filtration defined by the uniform topology on $\underset{p\in\mathcal{P}}{\prod} \mathbb{C}_{p}$ which arises from the $p$-adic topologies (for $p$ infinitely large), which gives Ad$f$MZV$\mu_{N}$'s ; the conjecture says that these two grading operations give equivalent results. Thus, this conjecture states a certain adelic case of equality in the inequality between the slopes of the Frobenius and the Hodge filtration on the log-crystalline cohomology which represents $\pi_{1}^{\un,\crys}(\mathbb{P}^{1} \setminus\{0,\mu_{N},\infty\},\vec{1}_{1},\vec{1}_{0})$. As a conclusion, we have interpreted Kaneko-Zagier's notion of finite multiple zeta values in crystalline terms. We will study it further in a subsequent paper.
\end{document} |
\begin{document}
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\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\centerline{\Large \bf {Some Connections Between
(Sub)Critical}}
\vskip 2mm
\centerline{\Large \bf Branching Mechanisms
and Bernstein Functions}
\vskip 1cm
\centerline{\Large \bf Jean Bertoin$^{(1)}$, Bernard Roynette$^{(2)}$, and
Marc Yor$^{(1)}$}
\vskip 1cm
\noindent
\noindent
(1) {\sl Laboratoire de Probabilit\'es et Mod\` eles Al\'eatoires
and Institut universitaire de France,
Universit\'e Pierre et Marie Curie, 175, rue du Chevaleret,
F-75013 Paris, France.}
\vskip 2mm
\noindent
(2) {\sl {Institut Elie Cartan, Campus Scientifique,
BP 239, Vandoeuvre-l\` es-Nancy Cedex
F-54056, France
}}
\begin{abstract}
We describe some connections, via composition, between two functional
spaces: the space of (sub)critical branching mechanisms and the space of
Bernstein functions. The functions ${\bf e}_\alpha: x\to x^{\alpha}$ where
$x\geq0$ and
$0<\alpha\leq 1/2$, and in particular the critical parameter
$\alpha=1/2$, play a distinguished role.
\end{abstract}
\section{Introduction} This note is a prolongation of \cite{RY} where the
following remarkable property of the function ${\bf e}_\alpha: x\to
x^{\alpha}$ was pointed at for $\alpha=1/2$: if ${\math P}si$ is a (sub)critical
branching mechanism, then ${\math P}si\circ {\bf e}_{1/2} $ is a Bernstein
function (see the next section for the definition of these notions). In
the present work, we first show that this property extends to every
$\alpha\in]0,1/2]$. Then we characterize the class of so-called internal
functions, i.e. that of Bernstein functions ${\math P}hi$ such that the compound
function
${\math P}si\circ {\math P}hi$ is again a Bernstein function for every
(sub)critical branching mechanism ${\math P}si$. In the final section,
we gather classical results on transformations of completely monotone
functions, Bernstein functions and (sub)critical branching mechanisms
which are used in our analysis.
\section{Some functional spaces}
\subsection{Completely monotone functions}
For every Radon measure $\mu$
on $[0,\infty[$, we associate the function
$\L_{\mu}: ]0,\infty[\to[0,\infty]$ defined by
\begin{equation}\label{eq0}
\L_{\mu}(q):=\int_{]0,\infty[}\e^{-qx}\mu(dx)\,,
\end{equation}
i.e. $\L_{\mu}$ is the Laplace transform of $\mu$.
We denote by
\begin{equation}\label{eq1}
{\bf CM}:=\left\{\L_\mu: \L_\mu(q)<\infty\hbox{ for all }q>0\right\}\,,
\end{equation}
which is an algebraic convex cone (i.e. a convex cone which is
further stable under inner product). The celebrated theorem of Bernstein
(see for instance Theorem 3.8.13 in \cite{Jacob}) identifies
${\bf CM}$ with the space of completely monotone functions, i.e.
functions $f: ]0,\infty[
\to[0,\infty[$ of class ${\cal C}^{\infty}$ such
that for every integer $n\geq1$, the $n$-th derivative $f^{(n)}$ of $f$
has the same sign as $(-1)^n$.
Recall from monotone convergence that $\L_\mu$ has a (possibly infinite) limit at $0+$
which coincides with the total mass of $\mu$.
We shall focus on two natural sub-cones of ${\bf CM}$:
\begin{equation}\label{eq2}
\B_1:=\left\{\L_{\mu}: \int_{]0,\infty[}(1\wedge
x^{-1})\mu(dx)<\infty\right\}
\end{equation}
We further denote by $\B_1^{\downarrow}$ the sub-space of functions in
$\B_1$ which are the Laplace transforms of absolutely
continuous measures with a decreasing density :
\begin{equation}\label{eq4}
\B_1^{\downarrow}:=\left\{\L_\mu: \mu(dx)=g(x)dx, g \hbox{ decreasing
and }
\int_{0}^{\infty}(1\wedge x^{-1})g(x)dx<\infty\right\}.
\end{equation}
Note that the density $g$ then has limit $0$ at infinity.
\subsection{Bernstein functions}
For every triple $(a,b,\Lambda)$ with $a,b\geq0$ and $\Lambda$ a
positive measure on
$]0,\infty[$ such that
\begin{equation}\label{eq5}
\int_{]0,\infty[}(x\wedge 1) \Lambda(dx)<\infty\,,
\end{equation}
we associate the function ${\math P}hi_{a,b,\Lambda}: ]0,\infty[\to [0,\infty[$
defined by
\begin{equation}\label{eq6}
{\math P}hi_{a,b,\Lambda}(q):=a +bq+\int_{]0,\infty[}(
1-\e^{-q x})\Lambda(dx)\,,
\end{equation}
and call ${\math P}hi_{a,b,\Lambda}$ the Bernstein function with characteristics
$(a,b,\Lambda)$.
We denote the convex cone of Bernstein functions by
\begin{equation}\label{eq7}
\B_2:=\left\{{\math P}hi_{a,b,\Lambda}: a,b\geq0\hbox{ and $\Lambda$
positive measure fulfilling (\ref{eq5})}\right\}.
\end{equation}
It is well-known that $\B_2$ can be identified
with the space of real-valued ${\cal C}^{\infty}$ functions $f:
]0,\infty[\to[0,\infty[$ such that for every integer $n\geq1$, the $n$-th
derivative $f^{(n)}$ of $f$ has the same sign as $(-1)^{n-1}$. See
Definition 3.9.1 and Theorem 3.9.4 in \cite{Jacob}.
Bernstein functions appear as Laplace exponents of subordinators,
see e.g. Chapter 1 in \cite{Besf},
Chapter 6 in \cite{Sato}, or Section 3.9 in \cite{Jacob}. This means that
${\math P}hi\in\B_2$ if and only if there exists an increasing process
$\sigma=(\sigma_t, t\geq0)$ with values in $[0,\infty]$ ($\infty$ serves
as absorbing state) with independent and stationary increments as long as
$\sigma_t<\infty$, such that for every $t\geq0$
$${\math E}(\exp(-q\sigma_t))\,=\,\exp(-t{\math P}hi(q))\,,\qquad q>0.$$
In this setting, $a$ is known as the killing rate, $b$ as the drift coefficient, and
$\Lambda$ as the L\'evy measure.
We shall further denote by $\B_2^{\downarrow}$ the subspace of
Bernstein functions for which
the L\'evy measure is absolutely continuous with a monotone
decreasing density, viz.
$$\B_2^{\downarrow}:=\left\{{\math P}hi_{a,b,\Lambda}: a,b\geq0
\hbox{ and }\Lambda(dx)=g(x)dx, g\geq0
\hbox{ decreasing and }
\int_{0}^{\infty}(x\wedge 1)g(x)dx<\infty\right\}.$$
\subsection{ (Sub)critical branching mechanisms}
For every triple $(a,b,{\math P}i)$ with $a,b\geq0$ and ${\math P}i$ positive measure on
$]0,\infty[$ such that
\begin{equation}\label{eq8}
\int_{]0,\infty[}(x\wedge x^2) {\math P}i(dx)<\infty
\end{equation}
we associate the function
${\math P}si_{a,b,{\math P}i}: ]0,\infty[\to[0,\infty[$ defined by
\begin{equation}\label{eq9}
{\math P}si_{a,b,{\math P}i}(q):=a q +bq^2+\int_{]0,\infty[}(
\e^{-q x}-1 +q x){\math P}i(dx)\,,
\end{equation}
and denote the convex cone of such functions by
\begin{equation}\label{eq10}
\B_3:=\left\{{\math P}si_{a,b,{\math P}i}: a,b\geq0\hbox{ and ${\math P}i$ a positive measure
such that (\ref{eq8}) holds}\right\}
\end{equation}
Functions in $\B_3$ are convex increasing functions of class ${\cal
C}^{\infty}$ that vanish at $0$; they coincide with the class
of branching mechanisms for (sub)critical continuous state branching
processes, where (sub)critical means critical or sub-critical. See Le
Gall \cite{LG} on page 132.
Alternatively, functions in the space $\B_3$ can also be viewed as
Laplace exponents
of L\'evy processes with no positive jumps that do not drift to
$-\infty$ (or, equivalently, with nonnegative mean). In this setting, $a$
is the drift coefficient,
$2b$ the Gaussian coefficient, and ${\math P}i$ the image of the L\'evy measure
by the map $x\to-x$.
See e.g. Chapter VII in \cite{Belp}.
\section{Composition with ${\bf e}_{\alpha}$}
Stable subordinators correspond to a remarkable one-parameter family
of Bernstein functions denoted here by
$({\bf e}_{\alpha}, 0<\alpha<1)$, where
$$
{\bf e}_{\alpha}(q):=q^{\alpha}\,=\,
{\alpha\over
\Gamma(1-\alpha)}
\int_{0}^{\infty}
(1-\e^{-q x}) x^{-1-\alpha}dx\,, \qquad q>0\,.
$$
\begin{theorem}\label{T1} The following assertions are equivalent:
\noindent {\rm (i)} $\alpha\in]0,1/2]$.
\noindent {\rm (ii)} For every ${\math P}si\in \B_3$, ${\math P}si\circ {\bf
e}_{\alpha}\in\B_2$.
\end{theorem}
The implication (ii) ${\math R}ightarrow$ (i) is immediate.
Indeed, ${\math P}si_{0,1,0}: q\to q^2$ belongs to $\B_3$, but
${\bf e}_{2\alpha}={\math P}si_{0,1,0}\circ {\bf e}_{\alpha}$ is in
$\B_2$ if and only if $2\alpha\leq 1$.
However, the converse (i) ${\math R}ightarrow$ (ii) is not straightforward and
relies on the following technical lemma, which appears as Lemma VI.1.2 in
\cite{RY}. Here, for the sake of completeness, we provide a proof.
\begin{lemma}\label{L1} For $\alpha\in ]0,1/2]$,
let $\sigma^{(\alpha)}=(\sigma^{(\alpha)}_x, x\geq0)$ be a stable subordinator
with index $\alpha$ with Laplace transform
$${\math E}\left(\exp\left(-{q} \sigma^{(\alpha)}_x\right)\right)
=\exp(-x q^{\alpha})\,,\qquad x,q>0\,.$$
Denote by $p^{(\alpha)}(x,t)$ the density of the law of
$\sigma^{(\alpha)}_x$. Then for every $,x,t>0$, we have
$$p^{(\alpha)}(x,t)\leq {\alpha\over \Gamma(1-\alpha)} x
t^{-(1+\alpha)}\,.$$
\end{lemma}
\noindent{\bf Remark :} The bound in Lemma \ref{L1} is sharp, as it is
well-known that for any $0<\alpha<1$ and each fixed $t>0$
$$p^{(\alpha)}(x,t) \,\sim\,{\alpha\over \Gamma(1-\alpha)} x
t^{-(1+\alpha)}\,,\qquad x\to\infty.$$
More precisely, there is a series representation of $p^{(\alpha)}(x,t)$, see Formula
(2.4.7) on page 90
in Zolotarev \cite{Zol}:
$$p^{(\alpha)}(x,1)={1\over \pi}\sum_{n=1}^{\infty}(-1)^{n-1}{\Gamma(n\alpha+1)\over \Gamma(n+1))}\sin(\pi n \alpha) x^{-n\alpha -1}\,.$$
Using the identity
$$\Gamma(\alpha)\Gamma(1-\alpha)={\pi \over\sin (\alpha \pi)}\,,$$
this agrees of course with the above estimate. It is interesting to note
that the second leading term in the expansion,
$$-{\Gamma(2\alpha+1)\over 2\pi}\sin(2\pi \alpha) x^{-2\alpha -1},$$
is negative for $\alpha<1/2$, but positive for $\alpha >1/2$.
So the bound in Lemma \ref{L1} would fail for $\alpha>1/2$.
\noindent{\bf Proof:}\hskip10pt In the case $\alpha=1/2$, there is an explicit expression for the
density
$$p^{(1/2)}(x,t)\,=\,{x\over 2\sqrt{\pi t^3}}\exp\left(-{x^2\over
4t}\right)\,,$$
from which the claim is obvious (recall that $\Gamma(1/2)=\sqrt \pi$).
In the case $\alpha<1/2$, we start from the identity
$$\exp(-xq^{\alpha})
\,=\,\int_{0}^{\infty}\e^{-qt} p^{(\alpha)}(x,t)dt
\,, $$
and take the derivative in the variable $q$ to get
$$\alpha q^{\alpha-1}\exp(-xq^{\alpha})
\,=\,\int_{0}^{\infty}\e^{-qt} {t\over x}
p^{(\alpha)}(x,t)dt\,,$$
and then
$$\alpha q^{\alpha-1}\left(1-\exp(-xq^{\alpha})\right)
\,=\,\int_{0}^{\infty}\e^{-qt}\left({\alpha\over \Gamma(1-\alpha)}
t^{-\alpha}- {t\over x} p^{(\alpha)}(x,t)\right)dt\,.$$
Denote the left hand-side by $g(x,q)$, and take the derivative in the
variable $x$. We obtain
$${\partial g(x,q)\over \partial x}
\,=\,\alpha q^{2\alpha-1}\e^{-xq^{\alpha}}
\,=\,\alpha q^{2\alpha-1}\int_{0}^{\infty}\e^{-qt}
p^{(\alpha)}(x,t)dt\,.$$
On the other hand, since $1-2\alpha>0$,
$$q^{2\alpha-1}\,=\,{1\over
\Gamma(1-2\alpha)}\int_{0}^{\infty}\e^{-qs}s^{-2\alpha}ds\,,$$
and hence
$${\partial g(x,q)\over \partial x}
\,=\,{\alpha\over
\Gamma(1-2\alpha)}\int_{0}^{\infty}{ds\over
s^{2\alpha}}\int_{0}^{\infty}dt
\e^{-q(s+t)}
p^{(\alpha)}(x,t)\,.$$
The change of variables $u=t+s$ yields
$${\partial g(x,q)\over \partial x}
\,=\,{\alpha\over
\Gamma(1-2\alpha)}\int_{0}^{\infty}du\int_{0}^{u}{ds\over
s^{2\alpha}}
\e^{-qu}
p^{(\alpha)}(x,u-s)\,;$$
and since $g(0,t)=0$, we finally obtain the identity
\begin{eqnarray*}
& &\int_{0}^{\infty}\e^{-qt}\left({\alpha\over \Gamma(1-\alpha)}
t^{-\alpha}- {t\over x} p^{(\alpha)}(x,t)\right)dt\\
&=&
{\alpha\over
\Gamma(1-2\alpha)}\int_{0}^{x}dy\int_{0}^{\infty}du\int_{0}^{u}{ds\over
s^{2\alpha}}
\e^{-qu}
p^{(\alpha)}(x,u-s)\,.
\end{eqnarray*}
Inverting the Laplace transform, we conclude that
$${\alpha\over \Gamma(1-\alpha)}
t^{-\alpha}- {t\over x} p^{(\alpha)}(x,t)
\,=\,
{\alpha\over
\Gamma(1-2\alpha)}\int_{0}^{x}dy\int_{0}^{t}{ds\over
s^{2\alpha}}
p^{(\alpha)}(x,t-s)\,,$$
which entails our claim.
\vrule height 1.5ex width 1.4ex depth -.1ex \vskip20pt
We are now able to prove Theorem \ref{T1}.
\noindent{\bf Proof:}\hskip10pt Let ${\math P}si_{a,b,{\math P}i}\in\B_3$.
Since both $a{\bf e}_{\alpha}$ and $b{\bf e}_{2\alpha}$ are Bernstein
functions, there is no loss of generality in assuming that
$a=b=0$. Set for $t>0$
$$\nu_{\alpha}(t):=
{\alpha\over
\Gamma(1-\alpha) t^{1+\alpha}}\int_{0}^{\infty}{\math P}i(dx) x
\left(1-{\Gamma(1-\alpha) t^{1+\alpha}\over \alpha x}
p^{(\alpha)}(x,t)\right)\,.$$
It follows from Lemma \ref{L1} that $\nu_{\alpha}(t)\geq0$.
We have for every $q>0$
\begin{eqnarray*}
& &\int_{0}^{\infty}(1-\e^{-qt})\nu_{\alpha}(t)dt\\
&=&\int_{0}^{\infty}{\math P}i(dx) x \int_{0}^{\infty}dt
\left({\alpha (1-\e^{-qt})\over
\Gamma(1-\alpha) t^{1+\alpha}}
-{p^{(\alpha)}(x,t)\over x}+
\e^{-qt}
{p^{(\alpha)}(x,t)\over x}\right)\\
&=&\,\int_{0}^{\infty}{\math P}i(dx) x\left(q^{\alpha}-{1\over
x}+{\e^{-q^{\alpha}x}\over x}\right)\\
&=& {\math P}si_{0,0,{\math P}i}({\bf e}_{\alpha}(q))\,.
\end{eqnarray*}
As this quantity is finite for every $q>0$, this shows that
${\math P}si_{0,0,{\math P}i}\circ{\bf e}_{\alpha}\in\B_2$.
\vrule height 1.5ex width 1.4ex depth -.1ex \vskip20pt
\noindent {\bf Remark :} The proof gives a stronger
result than that stated in Theorem \ref{T1}. Indeed, we specified the
L\'evy measure
$\nu_{\alpha}$ of ${\math P}si_{0,0,{\math P}i}\circ{\bf e}_{\alpha}$.
Furthermore, in the case $\alpha=1/2$, this expression shows that
${\math P}si_{0,0,{\math P}i}\circ{\bf
e}_{1/2}\in\B_2^{\downarrow}$. It is interesting to combine this
observation with the forthcoming Proposition \ref{P2} : for every
${\math P}si\in\B_3$, ${\math P}si\circ{\bf
e}_{1/2}\in\B_2^{\downarrow}$, thus ${\rm Id}\times ({\math P}si\circ{\bf
e}_{1/2}): q\to q{\math P}hi(\sqrt q)$ is again in $\B_3$, and in turn
${\bf e}_{1/2}\times ({\math P}si\circ{\bf
e}_{1/4})\in\B_2^{\downarrow}$. More generally, we have by iteration
that for every integer $n$
$${\bf e}_{2-2^{1-n}}\times ({\math P}si\circ{\bf
e}_{2^{-n}})\in\B_3\,,$$
and
$${\bf e}_{1-2^{-n}}\times ({\math P}si\circ{\bf
e}_{2^{-n-1}})\in\B_2^{\downarrow}\,.$$
\section{Internal functions}
It is well-known that the cone ${\bf CM}$ of completely monotone
functions and the cone $\B_2$ of Bernstein functions are both stable by
right composition with a Bernstein function; see Proposition \ref{P3}
below. Theorem
\ref{T1} incites us to consider also compositions
of (sub)critical branching mechanisms and Bernstein functions; we make
the following definition :
\begin{definition} A Bernstein function ${\math P}hi\in\B_2$ is said
{\rm internal} if ${\math P}si\circ {\math P}hi\in\B_2$ for every ${\math P}si\in\B_3$.
\end{definition}
Theorem \ref{T1} shows that the functions ${\bf e}_{\alpha}$ are internal
if and only if $\alpha\in]0,1/2]$. The critical parameter $\alpha=1/2$
plays a distinguished role. Indeed, we could also prove Theorem \ref{T1}
using the following alternative route. First, we check that ${\bf
e}_{1/2}$ is internal (see \cite{RY}), and then we deduce by subordination
that for every
$\alpha<1/2$ that ${\math P}si\circ {\bf e}_{\alpha}=
{\math P}si\circ {\bf e}_{1/2}\circ {\bf e}_{2\alpha}$ is again a Bernstein
function for every ${\math P}si\in\B_3$. Developing this argument, we easily
arrive at the following characterization of internal functions :
\begin{theorem}\label{T2} Let ${\math P}hi={\math P}hi_{a,b,\Lambda}\in\B_2$
be a Bernstein function. The following assertions are then equivalent:
\noindent{ \rm (i)} ${\math P}hi$ is internal,
\noindent{ \rm (ii)} ${\math P}hi^2\in\B_2$,
\noindent{ \rm (iii)} $b=0$ and there exists a subordinator $\sigma=
(\sigma_t, t\geq0)$ such that
$$\Lambda(dx)\,=\,c\int_{0}^{\infty} t^{-3/2} {\math P}(\sigma_t\in dx)dt\,.$$
\end{theorem}
\noindent{\bf Proof:}\hskip10pt (i) ${\math R}ightarrow$ (ii) is obvious as ${\math P}si_{0,1,0}\circ {\math P}hi
={\math P}hi^2$.
(ii) ${\math R}ightarrow$ (i). We know from Theorem \ref{T1} or \cite{RY} that
for every ${\math P}si\in\B_3$, ${\math P}si\circ {\bf e}_{1/2}\in\B_2$.
It follows by subordination that for every Bernstein function
$\kappa\in\B_2$, ${\math P}si\circ {\bf e}_{1/2}\circ \kappa\in\B_2$.
Take $\kappa={\math P}hi^2$, so $ {\bf e}_{1/2}\circ \kappa={\math P}hi$,
and hence ${\math P}hi$ is internal.
(iii) ${\math R}ightarrow$ (ii) Let $\kappa$ denote the Bernstein function of
$\sigma$. We have
\begin{eqnarray*}
{\math P}hi(q)\,&=&\,a +\int_{]0,\infty[}(1-\e^{-qx})\Lambda(dx)\\
&=&\,a + c \int_{]0,\infty[}\int_{0}^{\infty}dt (1-\e^{-qx}) t^{-3/2}
{\math P}(\sigma_t\in dx)\\
&=&\,a + c \int_{0}^{\infty}dt (1-\e^{-t\kappa(q)}) t^{-3/2}\,.
\end{eqnarray*}
The change of variables $t\kappa(q)=u$ yields
$${\math P}hi(q)\,=\,a+c'\sqrt{\kappa(q)}$$
and hence
$${\math P}hi^2(q)\,=\,a^2 + 2ac'\sqrt{\kappa(q)} +c'^2\kappa(q)\,.$$
Since $\kappa^{1/2}={\bf e}_{1/2}\circ \kappa$ is again a Bernstein
function, we thus see that ${\math P}hi^2\in\B_2$.
(ii) ${\math R}ightarrow$ (iii) Recall that the drift coefficient $b$ of
${\math P}hi_{a,b,\Lambda}$ is given by
$$\lim_{q\to\infty} {\math P}hi_{a,b,\Lambda}(q)/q\,=\,b\,;$$
see e.g. page 7 in \cite{Besf}. It follows immediately that
$b=0$ whenever $\kappa:={\math P}hi_{a,b,\Lambda}^2\in\B_2$.
Recall from Sato \cite{Sato} on page 197-8 that if $\tau^{(1)}$ and
$\tau^{(2)}$ are two independent subordinators with respective Bernstein
functions ${\math P}hi^{(1)}$ and ${\math P}hi^{(2)}$, then the compound process
$\tau^{(1)}\circ \tau^{(2)}:=\tau^{(3)}$ is again a subordinator with
Bernstein function ${\math P}hi^{(3)}:={\math P}hi^{(2)}\circ {\math P}hi^{(1)}$; moreover
its L\'evy measure $\Lambda^{(3)}$ is given by
$$\Lambda^{(3)}(dx)=\int_{0}^{\infty}{\math P}(\tau^{(1)}_t\in
dx)\Lambda^{(2)}(dt)\,,$$
where $\Lambda^{(2)}$ denotes the L\'evy measure of $\tau^{(2)}$.
As ${\math P}hi_{a,b,\Lambda}={\bf e}_{1/2}\circ \kappa$, and the L\'evy measure
of ${\bf e}_{1/2}$ is $ct^{-3/2}dt$ with $c=1/(2\sqrt \pi)$,
we deduce that
$$\Lambda(dx)\,=\,c\int_{0}^{\infty} {\math P}(\sigma_t\in dx) t^{-3/2}dt\,.$$
The proof of Theorem \ref{T2} is now complete.
\vrule height 1.5ex width 1.4ex depth -.1ex \vskip20pt
It is noteworthy that if ${\math P}hi_{a,b,\Lambda}$ is internal
and $\Lambda\not\equiv 0$,
then
$$\int_{]0,\infty}x \Lambda(dx)\,=\,\infty\,.$$
Indeed,
$$\int_{]0,\infty}x \Lambda(dx)
\,=\,c\int_{0}^{\infty}\int_{]0,\infty[} x {\math P}(\sigma_t\in dx)
t^{-3/2}dt
\,=\,c\int_{0}^{\infty}{\math E}(\sigma_1)
t^{-1/2}dt\,=\,\infty\,.$$
For instance, the Bernstein function $q\to \log(1+q)$ of the gamma
subordinator is not internal.
\begin{corollary}\label{C1} For every ${\math P}si\in\B_3$,
wewrite ${\math P}hi$ for the inverse function of ${\math P}si$
and then ${\math P}hi'$ for its derivative. Then
$1/{\math P}hi'$ is internal.
\end{corollary}
\noindent{\bf Proof:}\hskip10pt
It is known (see Corollary \ref{C2} below) that
$1/{\math P}hi'$ is a Bernstein function; let us check that its
square is also a Bernstein function.
We know that ${\math P}si''\in \B_1$ (Proposition \ref{P1} below)
and
${\math P}hi\in \B_2$ (Proposition \ref{P4} below); we deduce
from Proposition \ref{P3} that ${\math P}si''\circ {\math P}hi\in\B_1$.
If we write $I(f): x\to\int_{0}^{x}f(y)dy$ for
every locally integrable function $f$, then again by Proposition \ref{P1},
we get that $I({\math P}si''\circ {\math P}hi)$
is a Bernstein function.
Now
$${\math P}si''\,=\,-{{\math P}hi''\circ {\math P}si\over
({\math P}hi'\circ {\math P}si)^3}\,,$$
so
$${\math P}si''\circ {\math P}hi\,=\,-{{\math P}hi''\over
({\math P}hi')^3}\,,$$
and we conclude that
$${1\over 2({\math P}hi')^2}\,=\,I({\math P}si''\circ {\math P}hi)\in
\B_2\,.$$
\vrule height 1.5ex width 1.4ex depth -.1ex \vskip20pt
\section{Some classical results and their consequences}
For convenience, this section gathers some classical transformations
involving $\B_j$, $j\in\{1,2,3\}$ and related subspaces, which
have been used in the preceding section. We start by considering
derivatives and indefinite integrals. The following statement is
immediate.
\begin{proposition}\label{P1} Let $j=2,3$ and $f:]0,\infty[\to[0,\infty[$
be a ${\cal C}^{\infty}$-function with derivative
$f'$. For $j=3$, we further suppose that
$\lim_{q\to0}f(q)=0$.
There is the equivalence
$$f\in\B_j \ \Longleftrightarrow \ f'\in \B_{j-1}\,.$$
\end{proposition}
The next statement is easily checked using integration by parts.
\begin{proposition}\label{P2} Let $j=2,3$ and consider two functions
$f,g:]0,\infty[\to[0,\infty[$ which are related by the identity
$f(q)=qg(q)$.
Then there is the equivalence
$$f\in\B_j \hbox{ and } \lim_{q\to0} f(q)=0 \ \Longleftrightarrow \ g\in
\B_{j-1}^{\downarrow}
\,.$$
\end{proposition}
Proposition \ref{P2} has well-known probabilistic interpretations.
First, let $\sigma$ be a subordinator with Bernstein function $f\in \B_2$
with unit mean, viz. ${\math E}(\sigma_1)=1$, which is equivalent to $f'(0+)=1$.
Then the completely monotone function $g(q):=f(q)/q$
is the Laplace transform of a probability measure on ${\math R}_+$. The
latter appears in the renewal theorem for subordinators (see e.g.
\cite{BvHS}); in particular it describes the weak limit of the so-called
age process
$A(t)=t-g_t$ as
$t\to\infty$, where
$g_t:=\sup\left\{\sigma_s: \sigma_s<t\right\}.$
Second, let $X$ be a L\'evy process with no positive jumps and Laplace
exponent $f\in \B_3$. The L\'evy process reflected at its infimum,
$X_t-\inf_{0\leq s \leq t} X_s$, is Markovian; and if $\tau$ denotes its
inverse local time at $0$, then $\sigma=-X\circ \tau$ is a subordinator
called the descending ladder-height
process. The Bernstein function
of the latter is then given by
$g(q)=f(q)/q$; ; see e.g. Theorem VII.4(ii) in \cite{Belp}.
We next turn our attention to composition of functions;
here are some classical properties
\begin{proposition}\label{P3}
Consider two functions $f,g:]0,\infty[\to[0,\infty[$. Then we have the
implications
$$f,g\in\B_2 \ \Longrightarrow \ f\circ g\in \B_2\,,$$
$$f\in{\bf CM} \hbox{ and }g\in \B_2 \ \Longrightarrow \ f\circ g\in
{\bf CM}\,,$$
$$f\in\B_1 \hbox{ and }g\in \B_2 \ \Longrightarrow \ f\circ g\in
\B_1\,.$$
\end{proposition}
The first statement in Proposition \ref{P3} is related to the
celebrated subordination of Bochner (see, e.g. Section 3.9 in
\cite{Jacob} or Chapter 6 in
\cite{Sato}); more precisely if $\sigma$ and $\tau$ are two independent
subordinators with respective Bernstein functions $f_{\sigma}$ and
$f_{\tau}$, then $\sigma\circ \tau$ is again a subordinator whose
Bernstein function is $f_{\tau}\circ f_{\sigma}$. The second
statement is a classical result which can be found as Criterion 2 on page
441 in Feller \cite{Feller}; it is also related to Bochner's
subordination.
Finally we turn our attention to inverses.
\begin{proposition}\label{P4} Consider a
function $f:]0,\infty[\to]0,\infty[$. Then
$$f\in\B_2\cup \B_3 \ \Longrightarrow \ 1/f\in
{\bf CM}\,.$$
Further, if $f^{-1}$ denotes the inverse of $f$ when the latter is a
bijection, then
$$f\in\B_3\,,f\not\equiv 0 \ \Longrightarrow \ f^{-1}\in \B_2\,.$$
\end{proposition}
We mention that
if $f\in\B_3$, the completely monotone function $1/f$ is the Laplace
transform of the so-called scale function of the L\'evy process $X$ with
no positive jumps which has Laplace exponent $f$. See Theorem VII.8 in
\cite{Belp}. On the other hand,
$f^{-1}$ is the Bernstein function of the subordinator
of first-passage times $T_t:=\inf\left\{s\geq0: X_s>t\right\}$;
see e.g. Theorem VII.1 in \cite{Belp}. Finally, in the case when
$f\in\B_2$ is a Bernstein function, the completely monotone function $1/f$
is the Laplace transform of the renewal measure $U(dx)=\int_{0}^{\infty}
{\math P}(\sigma_t\in dx)dt$, where $\sigma$ is a subordinator with Bernstein
function $f$.
\begin{corollary}\label{C2} Let ${\math P}si\not\equiv 0$ be a function in
$\B_3$, and denote by ${\math P}hi={\math P}si^{-1}\in\B_2$ its inverse bijection.
Then $q\to1/{\math P}hi'(q)$ and ${\rm Id}/{\math P}hi: q\to q/{\math P}hi(q)$ are Bernstein
functions. Furthermore ${1\over {\math P}hi {\math P}hi'}: q\to1/({\math P}hi(q) {\math P}hi'(q))$ is
completely monotone.
\end{corollary}
\noindent{\bf Proof:}\hskip10pt We know from Propositions
\ref{P1} and \ref{P4} that both ${\math P}hi$ and ${\math P}si'$ are Bernstein
functions. We conclude from Proposition \ref{P3}
that $1/{\math P}hi'={\math P}si'\circ {\math P}hi$ is again in $\B_2$.
Similarly, we know from Proposition \ref{P2} that
$q\to {\math P}si(q)/q$ is a Bernstein function, and composition on the right
by the Bernstein function ${\math P}hi$ yields ${\rm Id}/{\math P}hi$ that is again
in $\B_2$.
Finally, we can write $1/({\math P}hi {\math P}hi')=f\circ {\math P}hi$ where
$f(q)={\math P}si'(q)/q$. We know from Proposition \ref{P1} that ${\math P}si'\in\B_2$,
so $f\in{\rm CM}$ by Proposition \ref{P2}. Since ${\math P}hi\in\B_2$, we
conclude from Proposition \ref{P3} that $f\circ {\math P}hi\in{\bf CM}$.
\vrule height 1.5ex width 1.4ex depth -.1ex \vskip20pt
If ${\math P}hi={\math P}si^{-1}$ is the Bernstein
function given by the inverse of a function ${\math P}si\in\B_3$, the Bernstein
function $1/{\math P}hi'$ is the exponent of the subordinator $L^{-1}$ defined
as the inverse of the local time at $0$ of the L\'evy process with no
positive jumps and Laplace exponent ${\math P}si$. See e.g. Exercise VII.2 in
\cite{Belp}. On the other hand, ${\rm Id}/{\math P}hi$ is then the Bernstein
function of the decreasing ladder times, see Theorem VII.4(ii) in
\cite{Belp}. The interested reader is also referred to \cite{Be1} for further
factorizations for Bernstein functions which arise naturally for
L\'evy processes with no positive jumps, and their probabilistic
interpretations.
Next, recall that a function $f:]0,\infty[\to{\math R}_+$ is called a Stieltjes
transform if it can be expressed in the form
$$f(q)\,=\,b+\int_{[0,\infty[}{\nu(dt)\over t+q}\,,\qquad q>0\,,$$
where $b\geq0$ and $\nu$ is a Radon measure on ${\math R}_+$ such that
$\int_{[0,\infty[}(1\wedge t^{-1})\nu(dt)<\infty$.
Equivalently, a Stieltjes
transform is the Laplace transform of a Radon
measure $\mu$ on ${\math R}_+$ of the type $\mu(dx)=b\delta_0(dx)+h(x)dx$,
where $b\geq0$ and $h$ is a completely monotone function which belongs to
$L^1(\e^{-qx}dx)$ for every $q>0$; see e.g. Section 3.8 in \cite{Jacob}.
\begin{corollary}\label{C3} Let $f\in \B_2$ be a Bernstein
function such that its derivative $f'$ is a Stieltjes transform.
Then for every Bernstein function $g\in B_2$, the function
$f\circ {1\over g}$ is completely monotone.
\end{corollary}
\noindent{\bf Proof:}\hskip10pt We can write
$$f(q)\,=\,a+ bq + \int_{0}^{q}dr\int_{0}^{\infty}dx\e^{-rx}h(x)\,,\qquad
q>0\,,$$
where $a,b\geq0$ and $h\in\B_1$. Thus
$$f(q)\,=\,a+ bq + \int_{0}^{\infty}dx(1-\e^{-qx}){h(x)\over x}\,,\qquad
q>0\,,$$
and then
$$f\circ {1\over g}(q)\,=\,a+{b\over
g(q)}+\int_{0}^{\infty}dx(1-\e^{-x/g(q)}){h(x)\over x}\,.$$
We already know from Proposition \ref{P4} that $a+b/g \in {\bf CM}$.
The change of variable $y=x/g(q)$ yields
$$\int_{0}^{\infty}dx(1-\e^{-x/g(q)}){h(x)\over x}
\,=\,\int_{0}^{\infty}(1-\e^{-y})h(yg(q)){dy\over y}\,.$$
For each fixed $y>0$, $yg$ is a Bernstein function, so by Proposition
\ref{P3}, the function $q\to h(yg(q))$ is completely monotone.
We conclude that for every integer $n\geq
0$,
$$(-1)^n{\partial^n\over \partial q^n}(f\circ {1\over g})(q)
\,=\,\int_{0}^{\infty}(-1)^n{\partial^n\over \partial q^n}(h(yg(\cdot))(q)
(1-\e^{-y}){dy\over y}\,\geq\,0\,,$$
which establishes our claim.
\vrule height 1.5ex width 1.4ex depth -.1ex \vskip20pt
\end{document} |
\begin{eqnarray}gin{document}
\title{Quantum Mechanics of Consecutive Measurements}
\author{Jennifer R. Glick}
\author{Christoph Adami}
\email[Corresponding author.\\]{[email protected]}
\affiliation{Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA}
\begin{eqnarray}gin{abstract}Consecutive quantum measurements performed on the same system can reveal fundamental insights into quantum theory's causal structure, and probe different aspects of the quantum measurement problem. According to the Copenhagen interpretation, measurements affect the quantum system in such a way that the quantum superposition
collapses after the measurement, erasing any knowledge of the prior state. We show that a sequence of measurements in a collapse picture is equivalent to a quantum Markov chain, and that considering the unitary evolution of quantum wavefunctions interacting consecutively with more than two detectors reveals an experimentally measurable difference between a collapse and unitary picture. The non-Markovian nature of sequential measurements that we report is consistent with earlier discoveries in optimal quantum state discrimination.
\end{abstract}
\maketitle
{\em Introduction.}---The physics of consecutive (sequential) measurements on the same quantum system has enjoyed increased attention as of late, as it probes the causal structure of quantum mechanics~\cite{Brukner2014}. It is of interest to researchers concerned about the apparent lack of time-reversal invariance of Born's rule~\cite{Rovelli2015,OreshkovCerf2015}, as well as to those developing a consistent formulation of covariant quantum mechanics~\cite{ReisenbergerRovelli2002,OlsonDowling2007}, which does not allow for a time variable to define the order of (possibly non-commuting) projections~\cite{OreshkovCerf2014}.
Consecutive measurements can be seen to challenge our understanding of quantum theory in an altogether different manner, however. According to standard theory, a measurement causes the state of a quantum system to ``collapse'', re-preparing it as an eigenstate of the measured operator so that after multiple consecutive measurements on the quantum system any information about the initial preparation is erased. However, recent investigations of sequential measurements on a single quantum system with the purpose of optimal state discrimination have already hinted that quantum information survives the collapse~\cite{Nagalietal2012,Bergouetal2013}. In this Letter, we ask if it is possible to consistently assume that consecutive measurements form a Markov chain~\cite{HaydenMarkov2004,Datta2015}, that is, whether measurements ``wipe the slate clean".
While the suggestion that the relative state description of quantum measurement~\cite{Everett1957} (see also ~\cite{Zeh1973,Deutsch1984,CerfAdami1996,CerfAdami1998}) and the Copenhagen interpretation are at odds and may lead to measurable differences has been made before~\cite{Zeh1973,Deutsch1984}, here we frame the problem of consecutive measurements in the language of quantum information theory. The formulation we present suggests experimentally measurable differences between a collapse and unitary picture, in the density matrices and von Neumann entropies of the joint state of detectors. We will argue that a collapse picture of quantum measurement is therefore untenable for multiple consecutive measurements.
We start by carefully constructing the initial state as an arbitrary mixed quantum state $\rho_Q$. We can ``purify" $\rho_Q$ by defining a pure state where $Q$ is entangled with a reference system $R$~\cite{NielsenChuang_Book}
\begin{eqnarray}
|QR\rangle=\sum_{n=1}^d\alpha_n|r_n\rangle |n\rangle_R\;. \langlebel{state}
\end{eqnarray}
Here, $d$ is the dimension of the quantum state's Hilbert space, the $\alpha_n$ are arbitrary complex coefficients, and the states of $R$ are denoted by $|n\rangle_R$ while those of $Q$ are labeled $|r_n\rangle$. In what follows, we first assume that $Q$ is an ``unprepared'' or ``unknown'' state with maximum entropy so that in (\ref{state}) it is maximally entangled with $R$, i.e., $\alpha_n=1/\sqrt d$. With such an assumption, we do not bias any subsequent measurements~\cite{Wootters2006}. We later discuss what happens in consecutive measurements on a known (that is, prepared) quantum state.
{\em Measurement of unprepared quantum states.}---To measure $Q$ with a detector (or ancilla) $A$~\footnote{We focus here on orthogonal measurements, a special case of the more general POVMs (positive operator-valued measures) that use non-orthogonal states. What follows can be extended to POVMs, while at the same time Neumark's theorem guarantees that any POVM can be realized by an orthogonal measurement in an extended Hilbert space.}
we rewrite the quantum state in terms of the detector's eigenstates $|i\ranglengle_A$, which automatically serve as the ``interpretation basis"~\cite{Deutsch1984}, using the unitary matrix $V_{ni}=\langlengle a_i|r_n\ranglengle$ (letters $a,b,\ldots$ with subscripts $i,j,\ldots$ indicate the basis $Q$ is written in, while $i,j,\ldots$ label the ancilla's basis states). We then entangle $Q$ with $A$, in initial state $|0\rangle_A$, using a unitary entangling operation $\hat{U}_E$~\cite{CerfAdami1998} .
\begin{eqnarray}
|QRA \ranglengle = \hat{U}_E |QR\ranglengle |0\ranglengle_A = \frac{1}{\sqrt d} \sum_{ni} V_{ni} |a_i\ranglengle |n\ranglengle_R |i\ranglengle_A \;. ~~~
\end{eqnarray}
We can rewrite the reference's states
in terms of the $A$ basis by defining $|i\ranglengle_R = \sum_n V^\intercal_{in}|n\ranglengle_R$ with the transpose of $V$, so that the joint system $QRA$ appears as
(we omit from now on the labels from the ancilla states)
\begin{eqnarray}
|QRA\rangle=\frac1{\sqrt d}\sum_i|a_i\rangle |i\rangle_R |i \rangle\;. \langlebel{schmidt}
\end{eqnarray}
Tracing out the reference from the full density matrix $\rho_{QRA} = |QRA\rangle\langle QRA|$, we note that the detector is correlated with the quantum system~\cite{CerfAdami1998} and each has maximum entropy $S_Q \!=\! S_A \!=\! \log d$. The von Neumann entropy is defined as $S_X = S(\rho_X) = - \mathrm{Tr} (\rho_X \log \rho_X)$ for a density matrix $\rho_X$. We also note in passing that $R$ can be thought of as representing all previous measurements of the quantum system that have occurred before $A$.
We now measure $Q$ again, but in a rotated basis $|a_i\rangle=\sum_jU_{ij}|b_j\rangle$, by entangling it with an ancilla $B$. Unitarity implies that $\sum_j|U_{ij}|^2 \! = \! \sum_i|U_{ij}|^2 \! = \! 1$. Then, with $|j\rangle$ the basis states of ancilla $B$ and $|ij\rangle = |i\rangle|j\rangle$,
\begin{eqnarray}
|QRAB\rangle=\frac1{\sqrt d}\sum_{ij}U_{ij}|b_j\rangle |i\rangle_R |ij\rangle \;.
\end{eqnarray}
It is easy to show that $\rho_A \!=\! \rho_B \!=\! 1/d\sum_i | i\ranglengle\langlengle i |$, so that both detectors have maximum entropy $\log d$, while
\begin{eqnarray}
\rho_{AB}=\frac1d\sum_{i}|i\rangle\langle i| \otimes \sum_j |U_{ij}|^2 |j\ranglengle\langlengle j| \;. \langlebel{rhoab}
\end{eqnarray}
Equation~\eqref{rhoab} immediately implies that if the quantum system is measured repeatedly in the same basis ($U_{ij} = \delta_{ij}$) by independent detectors, all of those detectors will be perfectly correlated (they reflect the same outcome), creating the illusion of a wavefunction collapse~\cite{CerfAdami1996,CerfAdami1998}.
The preceding results are entirely consistent with the standard formalism for orthogonal measurements~\cite{Peres1995,Holevo2011}, where the conditional probability $p_{j|i}$ to observe outcome $j$, given that the previous measurement yielded outcome $i$, is
\begin{eqnarray}
p_{j|i}=|U_{ij}|^2\;. \langlebel{cond}
\end{eqnarray}
Indeed, our findings thus far are fully consistent with a picture in which a measurement collapses the quantum state (or alternatively, where a measurement recalibrates an observer's ``catalogue of expectations"~\cite{Schroedinger1935,Englert2013,Fuchsetal2014}). To see this,
we write the joint density matrix of detector $A$ that records outcome $i$ with probability $1/d$ and detector $B$ that measures the same quantum state at an angle determined by the rotation $U$
\begin{eqnarray}
\rho^{\rm coll}_{AB}=\frac1d\sum_i |i\ranglengle \langlengle i| \otimes \rho^i_B ~ ,
\end{eqnarray}
with $\rho^i_B$ defined using projection operators $P_i = |i\rangle\langle i|$
\begin{eqnarray}
\rho^i_B=\frac{{\rm Tr}_A\left(P_i\rho_{AB}P_i^\dagger\right)}{ {\rm Tr}_{AB}\left(P_i\rho_{AB}P_i^\dagger\right) }=\sum_j |U_{ij}|^2|j\ranglengle\langlengle j|\;. \langlebel{proj}
\end{eqnarray}
Let us perform another measurement of the quantum system using an ancilla $C$ such that $|b_j\rangle=\sum_kU'_{jk}|c_k\rangle$.
We then find (here and before, indices $i$ refer to $A$, $j$ to $B$, and now $k$ to $C$)
\begin{eqnarray}
|QRABC\rangle=\frac1{\sqrt d}\sum_{ijk}U_{ij}U'_{jk}|c_k\ranglengle |i\ranglengle_R |ijk\rangle\;. \langlebel{qabc}
\end{eqnarray}
Tracing out $Q$ and $R$ from the density matrix $\rho_{QRABC}$, as we do not observe either the quantum system nor reference, leads to the joint state of three detectors
\begin{eqnarray}gin{equation}
\rho_{ABC}= \! \frac1d \! \sum_{i}|i\rangle\langle i| \otimes \sum_{jj'} U_{ij}U^*_{ij'}|j\rangle\langle j'|\otimes\sum_kU'_{jk}U^{\prime *}_{j'k}|k\rangle\langle k| \: .\langlebel{full}
\end{equation}
Tracing this expression over $C$ recovers $\rho_{AB}$ in Eq.~(\ref{rhoab}) as it should because the measurement $C$ does not affect the joint state of the past measurements $A$ and $B$. Tracing over $B$ gives
\begin{eqnarray}
\rho_{AC}=\frac1d \sum_{i} |i\rangle\langle i|\otimes\sum_{jk}|U_{ij}|^2 \, |U'_{jk}|^2 \, |k\rangle\langle k|\;, \langlebel{ac}
\end{eqnarray}
while tracing over $A$ yields
\begin{eqnarray}
\rho_{BC}=\frac1d \sum_{j} |j\rangle\langle j|\otimes\sum_k|U'_{jk}|^2 \, |k\rangle\langle k|\;. \langlebel{bc}
\end{eqnarray}
All three pairwise density matrices are diagonal in the detector product basis (see Theorem 1 in the Supplementary Material~\cite{supp}). We can take ``diagonal in the detector product basis" to be synonymous with ``classical". At the same time, each detector has classical information about the quantum system
(Theorem 2 in Supplementary Material~\cite{supp}).
Furthermore, all pairwise density matrices discussed here, and their corresponding entropies, are identical to those in a collapse picture.
However, the joint state of all three detectors when assuming a collapse picture is incoherent
\begin{eqnarray}gin{equation}\langlebel{coll}
\rho_{ABC}^{\rm coll} \!=\! \frac1d \! \sum_{i} \! |i\rangle\langle i| \otimes \! \sum_j \! |U_{ij}|^2|j\rangle\langle j| \otimes \! \sum_k \! |U'_{jk}|^2|k\rangle\langle k|,
\end{equation}
unlike expression~(\ref{full}) obtained in the unitary picture, which is coherent due to the non-diagonality of the $B$ subsystem. The presence of these additional terms in~(\ref{full}) has fundamental consequences for our understanding of the measurement process. After all, the three measurements were implemented as projective measurements, which according to the traditional view
``reduce" the wavefunction of the system. Indeed, such an apparent collapse has taken place after the second consecutive measurement (\ref{rhoab}) as the corresponding density matrix has no off-diagonal terms. However, the third measurement seemingly {\em undoes} this projection, as can be seen from the appearance of off-diagonal terms in~\eqref{full}. This ``reversal" is different from protocols that can ``un-collapse" weak measurements~\cite{korotkov2006,Jordan2010}, because it is clear that the wavefunction~\eqref{qabc} underlying the density matrix is and remains unprojected.
\setlength\begin{eqnarray}lowcaptionskip{-1ex}
\begin{eqnarray}gin{figure}[t]
\includegraphics[width=1.0\linewidth]{fig1.pdf}
\caption{Entropy Venn diagram for the joint state of three qudit detectors $A$, $B$, $C$ that consecutively measure an unprepared quantum system $Q$ (entropies with logarithm to base $d$). (a) A unitary description of quantum measurement where the middle detector $B$ is fully known given the past $A$ and future $C$ (grey shaded area). (b) A collapse description, where $A$ and $C$ are independent when given $B$ (grey shaded area).}
\langlebel{fig:venn1}
\end{figure}
To determine if the reported survival
of the quantum superposition has measurable consequences, we calculate the entropy of each pair of detectors and of the joint state of all three detectors. Taking logarithms to base $d$, the pairwise entropies follow directly from~\eqref{rhoab},~\eqref{bc}, and~\eqref{ac}:
\begin{eqnarray}\langlebel{pair_entropies}
S_{AB} \! & = & \! 1 \! - \frac{1}{d} \sum_{ij} |U_{ij}|^2 \log |U_{ij}|^2, \langlebel{SAB}\\
S_{BC} \! & = & \! 1 \! - \frac{1}{d} \sum_{jk} |U'_{jk}|^2 \log |U'_{jk}|^2, \langlebel{SBC}\\
S_{AC} \! & = & \! 1 \! - \frac{1}{d} \sum_{ik} \! \Big( \! \sum_j |U_{ij}|^2 |U'_{jk}|^2 \Big) \! \log \! \Big( \! \sum_{j'} |U_{ij'}|^2 |U'_{j'k}|^2 \Big). \noindentonumber \\ \langlebel{SAC}
\end{eqnarray}
Furthermore, $S_{AC}$ is equal to $S_{ABC}$, the entropy of $\rho_{ABC}$ (this holds for any three consecutive detectors, see Corollary 1.1 in Supplementary Material~\cite{supp}). From the definition of conditional entropy~\cite{CerfAdami_PRL1997} it follows that, given the measurement $A$ in the past and the measurement $C$ in the future, detector $B$'s state is fully determined
(see grey area in Fig.~\ref{fig:venn1}(a))
\begin{eqnarray}
S(B|AC) = S_{ABC} - S_{AC} = 0 \: .
\end{eqnarray}
It can be shown quite generally that this quantity does not vanish in a collapse picture (Corollary 1.1 in Supplementary Material~\cite{supp}).
We now briefly show that the measurement chain in a collapse picture is Markovian, as defined in~\cite{HaydenMarkov2004} (see also~\cite{Datta2015} and references therein). From~\eqref{coll}, the joint entropy of all three detectors (using $H$ to distinguish collapse entropies from $S$, the entropies in the unitary picture) is
\begin{eqnarray}gin{equation}\langlebel{entropy_ABC_collapse}
H_{ABC} = 1 \! - \frac{1}{d} \! \sum_{ij} \! |U_{ij}|^2 \! \log |U_{ij}|^2 - \frac{1}{d} \! \sum_{jk} \! |U'_{jk}|^2 \! \log |U'_{jk}|^2 ,
\end{equation}
or, $H_{ABC} = S_{A} + H(B|A) + H(C|B)$. Using the chain rule for entropies~\cite{CerfAdami1998}, $H_{ABC} = S_A + H(B|A) + H(C|BA)$, we see immediately that $H(C|BA) = H(C|B)$, the Markov property for entropies~\cite{HaydenMarkov2004,Datta2015}.
This further implies that subsystems $A$ and $C$
are independent from the perspective of $B$, since the conditional mutual entropy~\cite{CerfAdami1998} vanishes (see grey area in Fig.~\ref{fig:venn1}(b))
\begin{eqnarray}
H(A:C|B) & = & H(C|B) - H(C|BA) = 0 ~ .
\end{eqnarray}
\noindent The equivalent quantity does not vanish in the unitary formalism, reflecting the fundamentally non-Markovian nature of the quantum chain of measurements (see Theorem 3 in Supplementary Material~\cite{supp} for a derivation in an arbitrarily long chain). Figure~\ref{fig:venn1} uses quantum entropy Venn diagrams (see, e.g.,~\cite{CerfAdami1998}) to highlight these key differences between the two pictures.
\begin{eqnarray}gin{figure}[t]
\includegraphics[width=0.9\linewidth]{fig2.pdf}
\caption{Entropy Venn diagram for the joint state of three qubit detectors $A$, $B$, $C$. Detector $B$ measures $Q$ at an angle $\theta=\pi/4$ relative to the basis of $A$, and $C$ measures at $\theta'=\pi/4$ relative to the basis of $B$. Venn diagram based on (a) unitary evolution of the wavefunction, and (b) according to the collapse picture. (c) In both cases any two detectors $D_i$ and $D_j$ are uncorrelated when the third is traced out.}
\langlebel{fig:venn2}
\end{figure}
We can readily apply this formalism to the specific case of qubits ($d \!=\! 2$). Measurements with detector $B$ at an angle $\theta$ relative to the previous measurement $A$, and $C$ at an angle $\theta'$ to $B$, can each, without loss of generality, be implemented with a rotation matrix of the form
\begin{eqnarray}
U&=&\left( \! \begin{eqnarray}gin{array}{cc}\cos(\theta)& -\sin(\theta)\noindentonumber\\
\sin(\theta)& \;\;\; \cos(\theta)\noindentonumber
\end{array} \! \right) \: .
\end{eqnarray}
For measurements at $\theta = \theta' =\pi/4$ for example, we have $|U_{ij}|^2=|U^\prime_{ij}|^2=1/2$, and we expect the outcome of each measurement to be random, that is, $S_A \! = \! S_B \! = \! S_C \! = \! 1$ bit. The joint entropy of each pair of detectors is two bits, as can be read off of Eqs.~(\ref{SAB}-\ref{SAC}). Because of the non-diagonal nature of (\ref{full}), the joint density matrix of the three detectors (using $\sigma_z$, the third Pauli matrix, and $\mathbb{I}$, the identity of dimension 2)
\begin{eqnarray}
\rho_{ABC} = \frac{1}{8} \langlebel{three}
\begin{eqnarray}gin{pmatrix}
\mathbb{I} & -\sigma_z & 0 & 0 \\
-\sigma_z & \mathbb{I} & 0 & 0 \\
0 & 0 & \mathbb{I} & \sigma_z \\
0 & 0 & \sigma_z & \mathbb{I}
\end{pmatrix} ,
\end{eqnarray}
has entropy $S_{ABC}=2$ bits, as can be checked by finding the eigenvalues of (\ref{three}). The collapse density matrix (\ref{coll}) on the other hand gives $H_{ABC}=3$ bits, as can be verified from~\eqref{entropy_ABC_collapse}. Figure~\ref{fig:venn2} summarizes the entropic relationships for qubits in the two pictures.
It is instructive to note that the Venn diagram in Fig.~\ref{fig:venn2}(a) is the same as the one obtained for a one-time binary cryptographic pad where two classical binary variables (the source and the key) are combined to a third (the message) via a controlled-NOT operation~\cite{Schneidman2003} (the density matrices underlying the Venn diagrams are very different, however). Still, it implies that the state of any one of the three detectors can be predicted from knowing the joint state of the two others, in clear violation of the collapse postulate that a measurement wipes clean the history of the quantum state.
However, the prediction of $C$ cannot be achieved using expectation values from $B$'s and $A$'s states separately, as the diagonal of (\ref{full}) corresponds to a uniform probability distribution. For the qubit case, the difference between the density matrix $\rho_{ABC}$ and the collapse version can be ascertained by revealing the off-diagonal terms via quantum state tomography (see, e.g.,~\cite{Whiteetal1999}), or by measuring just a single moment~\cite{Tanaka2014} of the density matrix, such as ${\rm Tr} (\rho_{ABC}^2)$.
{\em Measurement of prepared quantum states.}---Suppose a quantum system is prepared in the known state
\begin{eqnarray}
\rho_Q=\sum_{j=1}^d p_j |a_j\rangle\langle a_j|\;, \langlebel{prep}
\end{eqnarray}
which we already wrote in the basis of ancilla $A$, as this will be the first measurement. We can always prepare a state like~\eqref{prep} by measuring an unknown quantum state in a given, but arbitrary, basis. Then, a second measurement at a relative angle $\theta$ gives rise to a projected state for $Q$ that is equivalent to the density matrix (\ref{proj}). If we choose for the state preparation only the outcome $i=0$, for example, then $p_j=|U_{0j}|^2$ provides the probability distribution.
The purification of (\ref{prep}) in terms of $A$'s basis is
\begin{eqnarray}\langlebel{rhoa}
|QA\rangle=\sum_{i}\sqrt p_i|a_i\rangle |i\rangle\;,
\end{eqnarray}
creating a correlated state with $S_Q \!=\! S_A \!=\! -\sum_i p_i\log p_i$. We now proceed as before. Introduce detector $B$ with its eigenbasis $\langle b_j|a_i\rangle=U_{ij}$ (this $U$ is not to be confused with $U$ in the definition of $p_j$ above). After entanglement,
\begin{eqnarray}
|QAB\rangle=\sum_{ij}\sqrt{p_i} ~ U_{ij}|b_j\rangle |ij\rangle \: ,
\end{eqnarray}
giving rise to
\begin{eqnarray}
\rho_{AB}&=&\sum_{ii'}\sqrt{ p_i p_{i'}} ~ |i\rangle\langle i'|\otimes \sum_j U_{ij} U^*_{i'j} |j\rangle\langle j|\;, \\
\rho_{B}&=&\sum_{ij}p_i|U_{ij}|^2 |j\rangle\langle j|\;. \langlebel{rhob}
\end{eqnarray}
The entropy of $B$ is naturally $S_B=-\sum_j q_j\log q_j$, where $q_j=\sum_i p_i |U_{ij}|^2$ is the marginal probability obtained from the joint probability $p_{ij}=p_i \: p_{j|i}$. The conditional probability $p_{j|i}$ of obtaining outcome $j$ with $B$, given that we had obtained outcome $i$ with $A$, was defined in (\ref{cond}). Introducing detector $C$, the joint density matrix for all three detectors is ($\rho^{\rm coll}_{ABC}$ has no such off-diagonal terms)
\begin{eqnarray}gin{equation}\begin{eqnarray}gin{split}
\rho_{ABC}=\sum_{ii'}\sqrt{p_i p_{i'}} |i\rangle\langle i'| & \otimes \sum_{jj'} U_{ij}U^*_{i'j'}|j\rangle\langle j'| \\
& \otimes\sum_k U'_{jk} U^{\prime *}_{j'k}|k\rangle\langle k| \:. \langlebel{full=k}
\end{split}\end{equation}
From~\eqref{full=k}, detector $C$'s entropy is $S_C = - \sum_k q'_k \log q'_k$, where $q'_k = \sum_{ij} p_i \: |U_{ij}|^2 \: |U'_{jk}|^2$ is obtained from the joint probability $p_{ijk}=p_i \: p_{j|i} \: p_{k|j}$, and $p_{k|j}=|U'_{jk}|^2$.
\begin{eqnarray}gin{figure}[t]
\begin{eqnarray}gin{center}
\includegraphics[width=0.7\linewidth]{fig3.pdf} ~~~
\end{center}
\vspace*{-5mm}
\caption{Conditional entropies $S(D|C)$ and $S(C|B)$ in the unitary picture and $H(D|C)$ in the collapse picture, for three consecutive measurements on the prepared state~\eqref{prep} as a function of the state preparation $p$. Each detector is at an angle $\pi/8$ relative to the previous detector. For these angles, $H(D|C)=H(C|B)$.
}
\langlebel{fig:qubits}
\end{figure}
We apply these results once more to qubits,
with three measurements $B$, $C$, and $D$ of the quantum system after the preparation with $A$, each at an angle $\pi/8$. In Fig.~\ref{fig:qubits}, we show that
in a unitary description all measurements prior to $C$ leave a trace: the conditional entropies $S(C|B)$ and $S(D|C)$ retain a dependence on the preparation $p$. The collapse entropy $H(D|C)$, on the contrary, is independent of $p$. This formalism can also be used to succinctly describe the quantum Zeno~\cite{HomeWhitaker1997,Peres1995} and anti-Zeno~\cite{KaulakysGontis1997,LewensteinRzazewski2000,Luis2003} effects (see Supplementary Material~\cite{supp}).
{\em Conclusions.}---Conventional wisdom in quantum mechanics dictates that the measurement process ``collapses'' the state of a quantum system so that the probability a particular detector fires depends only on the state preparation and the measurement chosen.
Using a quantum-information-theoretic approach,
we have argued that a collapse picture makes predictions that differ from those of the unitary (relative state) approach if multiple consecutive measurements are considered. Should future experiments corroborate the manifestly unitary formulation we have outlined, such results would further support the notion of the reality of the quantum state~\cite{Pusey2012} and that the wavefunction is not merely a bookkeeping device that summarizes an observer's knowledge about the system~\cite{Englert2013,Fuchsetal2014}. We hope that moving discussions about the nature of quantum reality from philosophy into the empirical realm will ultimately lead to a more complete (and satisfying) understanding of quantum physics.
\begin{eqnarray}gin{acknowledgments}
CA would like to thank N. J. Cerf and S. J. Olson for discussions, and acknowledges support by the Army Research Offices grant \# DAAD19-03-1-0207. Financial support by a Michigan State University fellowship to JRG is gratefully acknowledged.
\end{acknowledgments}
\begin{eqnarray}gin{thebibliography}{35}
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the wavefunction by undoing quantum measurements}},}\ }\href@noop {}
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\end{document} |
\begin{document}
\title{Singular loci of sparse resultants}
\begin{abstract}
We study the singularity locus of the sparse resultant of two univariate polynomials, and apply our results to estimate singularities of a coordinate projection of a generic spatial complete intersection curve.
\end{abstract}
\tableofcontents
\section{Introduction}
Given a pair of finite sets $B_1$ and $B_2\subset\mathbb Z$ with at least two elements in each, one can consider the space of pairs of sparse Laurent polynomials supported at $B_1$ and $B_2$:
$$\mathbb C^{B_1}\times\mathbb C^{B_2}=\left\{\left(f_1(x)=\sum_{k\in B_1}f^1_kx^k,\,f_2(x)=\sum_{k\in B_2}f^2_kx^k\right)\right\}.$$
The sparse resultant $R_B\subset\mathbb C^{B_1} \times \mathbb C^{B_2}$ is the closure of the set
$$H(1)=\{(f_1,f_2)\in\mathbb C^{B_1} \times \mathbb C^{B_2}\,|\,f_1(x)=f_2(x)=0\mbox{ has at least one non-zero solution in } \mathbb C^*\}.$$
\begin{rem}
We need the closure here, because the roots can tend to $0$ and infinity.
\end{rem}
This algebraic hypersurface is actively studied starting from the paper \cite{S94} and the book \cite{GKZ94}. We aim at describing its singular locus $\mathop{\rm sing}\nolimits R_B$.
In the classical case $B_i=\{0,1,2,\ldots,d_i\}$ the sparse resultant $R_B$ is the usual resultant, that is the hypersurface defined by Sylvester matrix. If $d_i > 2$, the singular locus has one irreducible component, and its codimension is 2. The irreducible component is
the closure of a (locally closed) irreducible set
$$H(1,1)=\{(f_1,f_2) \in\mathbb C^{B_1} \times \mathbb C^{B_2}\,|\,f_1(x)=f_2(x)=0\mbox{ has at least two non-zero solutions}\}.$$
Moreover, in the classical case we know how singular the resultant is at a generic point of its singular locus: its transversal singularity type at a generic point $(f_1,f_2)\in \mathop{\rm sing}\nolimits R_B$ (i.e. the type of the singularity of the intersection of $R_B$ with a germ of a 2-dimensional plane transversal to $\mathop{\rm sing}\nolimits R_B$ at $(f_1,f_2)$) is $\mathcal{A}_1$ (i.e. that of the union of two transversal lines in the plane).
Our main result describes the conditions on $B_1$ and $B_2$ under which similar description holds true.
\begin{theor}\label{theormain}
i) There exists a codimension 3 subset $\Sigma\subset \mathbb C^{B_1}\times\mathbb C^{B_2}$ such that at every point of $\mathop{\rm sing}\nolimits R_B\setminus\Sigma$ the transversal singularity type of $R_B$ is $\mathcal{A}_1$, unless at least one of the following conditions holds:
\begin{enumerate}
\item One can shift $B_1$ and $B_2$ to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$.
\item One can split one of $B_i$'s (say, $B_1$) into $B'\sqcup B''$ so that $B'$, $B''$ and $B_2$ can be shifted to the same sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 3$.
\item For every $i=1,2$, the two leftmost elements of $B_i$ differ by more than 1.
\item For every $i=1,2$, the two rightmost elements of $B_i$ differ by more than 1.
\item The number of elements in one of $B_i$'s (say, $B_1$) is 2, and $\max B_2-\min B_2 > 2$.
\end{enumerate}
ii) The singular locus of the resultant $R_B$ consists of several irreducible components of codimension 2, unless the following condition holds:
\begin{enumerate}
\item[(6)] $B_1=\{i,i+k\}$ and $B_2=\{j,j+k\}$ for some $i,j \in Z$ and $k \in \mathbb N$.
\end{enumerate}
\end{theor}
In fact, the condition (1) is redundant, because it implies the conditions (3) and (4), but it is convenient to the narrative. The following examples illustrate what happens to the singular locus once one of the conditions (1-6) takes place.
\begin{exa}\label{exaintro}
First let us consider part (i).
1) If condition (1) takes place, then for every pair $(f_1, f_2)$ of Laurent polynomials with a non-zero common root $x$ there is also $(k-1)$ more non-zero common roots $x \cdot \epsilon_k$, where $\epsilon_k$ are $k$-th roots of unity.
In this case, the study of the resultant $R_B$ can be reduced to the study of a resultant $R_{B'}$ for a smaller pair of support sets $B_1'$ and $B_2'$ such that $B_1=k \cdot B_1'+m_1=\{kb+m: b \in B_1'\}$ and $B_2=k \cdot B_2'+m_2$. Indeed, $\mathbb C^{B_1} \times \mathbb C^{B_2} \cong \mathbb C^{B_1'} \times \mathbb C^{B_2'}$, and, moreover, $N_B(1)=N_{B'}(1)$, thus $R_B \cong R_{B'}$.
2) If condition (2) takes place, then, for some component of the singular locus of the resultant, the transversal singularity type at a generic point is expected to have $k \geqslant 3$ components, thus to differ from $\mathcal{A}_1$.
For instance, consider $B_1=\{0,1,3\}$ and $B_2=\{0,3\}$. Let us denote the polynomials by $ax^3+bx+c$ and $dx^3+e$. Then the resultant is given by the equation
\begin{equation}\label{eqae_cd}
(ae-cd)^3+b^3d^2e=0
\end{equation}
This equation is homogeneous in $(a,b,c)$ and in $(d,e)$, so let us restrict it to $a=d=1$. Then the singular locus of the restriction is given by the equation $b=c-e=0$, which corresponds to a component of the singular locus of the subset defined by Formula (\ref{eqae_cd}). Choosing $b=\epsilon_1,\; c=1+\epsilon_2,\; e=1$ as a transversal plane germ to this component, we see that the transversal singularity of this component is given by $\epsilon_1^3-\epsilon_2^3=0$, thus it is the union of three transversal lines intersecting in a common point.
3) If condition (3) or (4) takes place, then, for some component of the singular locus of the resultant, the transversal singularity type at a generic point is expected to have a singular component, thus to differ from $\mathcal{A}_1$.
For instance, consider the same example $B_1=\{0,1,3\}$ and $B_2=\{0,3\}$. There are two more components of the singularity locus which are given by $a=d=0$ and $d=e=0$. For the first one, the transversal singularity is the cusp $x^2=y^3$.
The case when condition (4) takes place can be reduced to (3) by the change of coordinate $\tilde x=x^{-1}$, because the sets $B_1$ and $B_2$ can be replaced by the sets $B_1'=-B_1=\{-b: b \in B_1\}$ and $B_2'=-B_2$.
4) If condition (5) takes place, then again, for some component of the singular locus of the resultant, the transversal singularity type at a generic point is expected to have $k \geqslant 3$ components, thus to differ from $\mathcal{A}_1$.
For instance, suppose that $|B_1|=2$ and $B_2=\{0,1,\ldots,d_2\}$ with $d_2>2$. There is one more component of the singularity locus which is given by the equation $f \equiv 0$. This component has codimension 2, and in its general point one of the polynomials is identically 0 and the second has $d_2$ different roots, thus the transversal singularity type has $d_2>2$ components.
The same happens with the component $d=e=0$ from the previous case.
Now let us consider part (ii).
5) If condition (6) takes place, then the singular locus consists of one point $(0,0) \in \mathbb C^{B_1} \times\mathbb C^{B_2}$, thus has codimension 4. Indeed, without loss of generality we can replace $B_1$ and $B_2$ by $B'_1=B'_2=\{0,k\}$. The singular locus for $B'_1$ and $B'_2$ is the same as for $B''_1=B''_2=\{0,1\}$, because we consider them as subsets, not counting the multiplicities. Now let us denote the polynomials by $ax+b$ and $cx+d$. The resultant is given by the equation $ad=bc$, thus it has just one singular point $(0,0)$.
\end{exa}
\begin{rem}
Actually, once the condition (2), (3), (4) or (5) takes place in the setting of the theorem, the singular locus of the resultant is always expected to have a component, at whose generic point the transversal singularity type differs from $\mathcal{A}_1$. However, the study of its singularity type (and the proof that it differs from $\mathcal{A}_1$) is non-trivial and will be done in a separate paper.
\end{rem}
\begin{rem}
While conditions similar to (1), (3) and (4) are familiar to the experts in tropical geometry and Newton polytopes, the condition (2) is new (to the best of our knowledge). The source of this condition is the subsequent theorem \ref{theorstratumranks21for111} describing degenerations of certain matrices of Vandermonde type.
\end{rem}
\begin{rem}
It would be important to obtain a version of this theorem for $A$-discriminants: their singular locus is studied e.g. in \cite{E13}, \cite{DHT16} and \cite{V21} under the assumption that it has the expected dimension and transversal singularity type, but so far there are no known criteria for these assumption to hold true even for univariate polynomials (except for some sufficient conditions in Section 3.4 of \cite{E13}).
\end{rem}
\begin{rem}
In this article, we prove Theorem \ref{theormain} over the field of complex numbers. This setting is important in our proof of Lemma \ref{lem3minor}, where one of the key steps essentially makes appeal to elementary geometry in the complex line (see Figure \ref{figinscrangles}). We shall address the case of arbitrary field in a subsequent paper, but we do not know to what extent it remains valid in the finite characteristics.
\end{rem}
Most of the paper is devoted to the proof of Theorem \ref{theormain}. The proof consists of two parts: in Sections \ref{secfiltr}--\ref{secstratrootspace}, we systematically study the natural stratifications of the resultant and certain related objects. The aim of this study is to reduce Theorem \ref{theormain} to several nontrivial facts about ranks of certain Vandermonde-type matrices. These facts (Lemma \ref{lemstratumranks11} and especially Theorem \ref{theorstratumranks21for111}) are then proved in Sections \ref{secstrats11forn111}--\ref{secstrats12forn111}.
In the last section, as an application of Theorem \ref{theormain}, we prove the following fact, partially answering the question given in the Remark 1.2 of \cite{V19}.
Given two finite sets $A_1$ and $A_2$ in $\mathbb Z^3$ and a pair of generic complex (or real) Laurent polynomials $f_1$ and $f_2$ supported at these sets, the equations $f_1=f_2=0$ define a smooth spatial algebraic curve in $\mathbb CC^3$ (or $(\mathbb R^*)^3$). The closure $C$ of its projection to the first coordinate plane is not in general smooth: at least it may have singularities of type $\mathcal{A}_1$ at the points having two preimages. Such $\mathcal{A}_1$ singularities are stable under local perturbations of the smooth spatial curve, similarly to the transversal self-intersections of a knot diagram.
\begin{theor}\label{theorproj} The curve $C$ has no other singularity types, unless the projections $B_1$ and $B_2$ of $A_1$ and $A_2$ to the last coordinate line satisfy one of the five conditions of the part i) of the main theorem \ref{theormain}.
\end{theor}
\begin{rem}
Even if the sets $A_1$ and $A_2$ in this theorem are the sets of all lattice points in prescribed Newton polytopes, their projections $B_1$ and $B_2$ rarely consist of several consequtive integers. They usually have gaps. This explains why we need Theorem 1.2 for arbitrary support sets, and not just for the classical case $B_i=\{0,1,2,\ldots,d_i\}$.
\end{rem}
\section{A filtration of the sparse resultant}\label{secfiltr}
This section is devoted to the first reductions of part (i) of the main theorem, as a result we reduce it to the much more concrete Theorem \ref{theorcodimNpre}, which we prove in the subsequent sections.
\subsection{A branched covering of the sparse resultant}
Let us recall that the sparse resultant $R_B\subset\mathbb C^{B_1} \times \mathbb C^{B_2}$ is the closure of the set
$$H(1)=\{(f_1,f_2)\in\mathbb C^{B_1} \times \mathbb C^{B_2}\,|\,f_1(x)=f_2(x)=0\mbox{ has at least one non-zero solution}\}.$$
Let us define the subset $\widetilde R_B \subset \mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb CP^1$ to be the closure of
$$\widetilde H(1) = \{(f,g,x) \in \mathbb C^{B_1} \times \mathbb C^{B_2} \times \mathbb C^\times \,|\, f(x)=g(x)=0\}.$$
We would like to study $R_B$ using $\widetilde R_B$ and the tautological projection $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb CP^1 \to \mathbb C^{B_1}\times\mathbb C^{B_2}$.
The subset $\widetilde H(1)$ is given by the equations $f(x)=\sum_{i \in B_1} f_i x^i=0$ and $g(x)=\sum_{j \in B_2} g_j x^j=0$. They are equations in variables $f_i$ for $i \in B_1$, variables $g_j$ for $j \in B_2$ and a variable $x$. Its closure $\widetilde R_B$ is the projectivization with respect to $x$, thus it is given by the equations
\begin{equation}\label{eqsheaf}
\widehat f(x,y)=\sum_{i \in B_1} f_i x^{i-\min B_1}y^{\max B_1-i}=0 \quad \text{ and } \quad \widehat g(x,y)=\sum_{i \in B_2} g_j x^{j-\min B_2}y^{\max B_2-j}=0,
\end{equation}
where $(x:y)$ are homogeneous coordinates on $\mathbb CP^1$.
\begin{lemma}\label{lemwidehatRBsmooth}
The set $\widetilde R_B$ is smooth.
\end{lemma}
\begin{proof}
{\ifshort
It follows from the fact that the space $\widetilde R_B$ is a vector bundle over $\mathbb CP^1$.
\fi}
{\iflong
Let us consider the subset $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y$, where $\mathbb C_y \subset \mathbb CP^1$ is defined by $y \neq 0$. As $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y \cong \mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C$, we have a canonical coordinate chart on it. For the subset $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C^\times$, where $\mathbb C^\times \subset \mathbb CP^1$ is defined by $x,y \neq 0$, it is true that the subset $\widetilde R_B \cap (\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C^\times)=\widetilde H(1)$ is defined exactly by the equations $f(x)=g(x)=0$.
Let us suppose that $\min B_1=\min B_2=0$. Then for the canonical coordinate chart on $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y$ it is true that the subset $\widetilde R_B \cap (\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y)$ is defined exactly by the equations $f(x)=g(x)=0$.
The differentials in this canonical coordinate chart are
$$df(x)=\sum_{i \in B_1} x^i df_i + \sum_{j \in B_2} 0\,dg_j + f'(x) dx,$$
$$dg(x)=\sum_{i \in B_1} 0\,df_i + \sum_{j \in B_2} x^j dg_j + g'(x) dx.$$
We supposed that $\min B_1=0$ and $\min B_2=0$, thus the differentials contain the summands $df_0$ and $dg_0$ and thus they are linearly independent at each point $(f,g,x) \in \mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y$, so that $\widetilde R_B \cap (\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y)$ is smooth.
In general case, we can shift $B_1$ and $B_2$ inside $\mathbb Z$ and obtain the case when $\min B_1=\min B_2=0$.
To prove the smoothness of $\widetilde R_B \cap (\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_x)$, where $\mathbb C_x \subset \mathbb CP^1$ is defined by $x \neq 0$, let us notice that it is equal to $\widetilde R_{B'} \cap (\mathbb C^{B_1'}\times\mathbb C^{B_2'}\times\mathbb C_y)$ for $B_1'=-B_1=\{-b_1 \,|\, b_1\in B_1\}$ and $B_2'=-B_2$. Consequently, the whole $\widetilde R_B \subset \mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb CP^1$ is smooth.
\fi}
\end{proof}
\begin{lemma}
The subset $R_B$ is the image of $\widetilde R_B$ under the tautological projection $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb CP^1 \to \mathbb C^{B_1}\times\mathbb C^{B_2}$.
\end{lemma}
\begin{proof}
The tautological projection maps $\widetilde H(1)$ to $H(1)$, thus it maps $\widetilde R_B$, which is the closure of $\widetilde H(1)$, to $R_B$, which is the closure of $H(1)$. By the definition of $\widetilde H(1)$ the restriction $\pi: \widetilde R_B \to R_B$ maps onto $H(1)$; the tautological projection is proper, thus the image of $\widetilde R_B$ is closed, thus the image is the whole $R_B$; that is, $\pi$ is surjective and $R_B$ is the image of $\widetilde R_B$.
\end{proof}
We will denote the restriction of the tautological projection as $\pi: \widetilde R_B \to R_B$.
As the set $\widetilde R_B$ is given by Formula (\ref{eqsheaf}), for each $(f,g) \neq (0,0)$ there is a finite number of preimages $(f,g,x)$. Consequently, $R_B$ can be non-smooth at the point $(f,g) \neq (0,0)$ in one of the following two cases: either there is more than one preimage $(f,g,x)$, or the map $\pi$ is not a local embedding at $(f,g,x)$. To detail it, we will need some definitions.
\subsection{Multiplicities of the roots}
Let $B \subset \mathbb Z$ be a finite set. Let $f(x)=\sum_{b\in B}c_bx^b \in\mathbb C^B$ be a (Laurent) polynomial with the support $B$ (below we will usually write simply "polynomial" instead of "Laurent polynomial" for conciseness). From now on, we shall identify $f(x)$ with the section
$$s_f=\sum_{b\in B}c_bx^{b-\min B}y^{\max B-b}$$
of the invertible sheaf $\mathcal{O}(\max B-\min B)$ on $\mathbb CP^1$ with homogeneous coordinates $x$ and $y$.
Notice that the formula is the same as Formula (\ref{eqsheaf}).
The purpose is to be able to speak of the multiplicities of $f$ at $0$ and $\infty\in\mathbb CP^1$.
\begin{defin}\label{defmultipl1}
Let $f(x) \in \mathbb C^B$ be a polynomial, and $x \in \mathbb CP^1$ be a point. The multiplicity of $f$ at $x$ is the multiplicity of the root of the section $s_f$ at $x$. It is denoted $\mathop{\rm ord}\nolimits_x(f)$.
\end{defin}
{\iflong
Explicitly, if $B=\{b_1, b_2, \ldots, b_m\}$ with $b_1 < b_2 < \ldots < b_m$, the multiplicity of $f$ at $x \in \mathbb C^\times$ is equal to the multiplicity of the root $x$ of the Laurent polynomial $f(x)$, and to calculate the multiplicity of $f$ at $0$ or $\infty$ we write down $c_{b_1}, 0, \ldots, 0, c_{b_2}, 0, \ldots, 0, c_{b_m}$, where $c_{b_i}$ is written on $(b_i-\min B_i+1)$-th position, and count the number of zeroes on the left or on the right respectively. If $f$ is identically zero, then the multiplicity of $f$ at any $x \in \mathbb CP^1$ is infinite.
\fi}
\begin{rem}
If $f$ is a polynomial, not just a Laurent polynomial, then under this identification the multiplicity of the section $s_f$ at 0 is different from the multiplicity of the root of $f$ at 0 when $f$ is considered as a polynomial. {\iflong In particular, if we shift $B_i$ to the right by $n$ and multiply $f$ by $x^n$, then the multiplicity of $s_f$ at $0$ would not change, but the multiplicity of $f$ at 0 when $f$ is considered as a polynomial would increase by $n$.\fi}
\end{rem}
{\iflong
\begin{rem}
To study the multiplicity at 0, we could just shift $B$ such that $\min B=0$. But we also need the multiplicity at infinity, and the definition above allows us to do study both and in a systematic way.
\end{rem}
\fi}
\begin{defin}
Let $f_1(x) \in \mathbb C^{B_1}$ and $f_2(c) \in \mathbb C^{B_2}$ be two polynomials, and $x \in \mathbb CP^1$ be a point. The multiplicity of a pair $(f_1, f_2)$ at $x$ is the minimum of the multiplicities of the sections $s_{f_1}$ and $s_{f_2}$ at $x$. It is denoted $\mathop{\rm ord}\nolimits_x(f_1, f_2)$.
\end{defin}
In particular, we say that $(f_1,f_2)$ has a common root at $x \in \mathbb CP^1$ if $\mathop{\rm ord}\nolimits_x(f_1,f_2) \geqslant 1$.
\begin{rem}
{\iflong For $x \in \mathbb C^\times$ it coincides with the usual notion of the common root of a pair of Laurent polymonials $f_1$ and $f_2$.\fi} For $x=0$ ($x=\infty$) it means that both $f_1$ and $f_2$ have zero leftmost (rightmost) coefficients respectively.
\end{rem}
\subsection{Singular locus of the sparse resultant}\label{subsetsingloc}
The definition of common root of $(f,g)$ in $\mathbb CP^1$ allows us to see the set $\widetilde R_B$ as the set of triples $(f,g,x)$ where $x$ is a common root of $(f,g)$. As the sparse resultant $R_B$ is the image of $\widetilde R_B$, it consists of pairs $(f,g)$ that has a common root. This definition, more simple then the definition as the closure of $H(1)$, it possible due to introduction of the notion of common roots at 0 and infinity.
Now we also to see the map $\pi: \widetilde R_B \to R_B$ differently: the preimage $\pi^{-1}(f,g)$ consists of all $(f,g,x)$ such that $x$ is a common root of $(f,g)$. Thus if $(f,g)$ has $k$ common roots, then $R_B$ locally at $(f,g)$ consists of $k$ (or less) local branches corresponding to these roots.
\begin{lemma}\label{lemrbnonlocdiff}
The map $\widetilde R_B \to R_B$ is not local embedding at $(f,g,x)$ if and only if $x$ is a common root of $(f,g)$ of multiplicity more than one.
\end{lemma}
\begin{proof}
{\ifshort
It is a special case of Thom's transversality lemma.
\fi}
{\iflong
The map $\pi: \widetilde R_B \to R_B$ is local embedding at the point $(f,g,x)$ if and only if the tangent map $\pi_*: T_{(f,g,x)} \widetilde R_B \to T_{(f,g)}(\mathbb C^{B_1} \times \mathbb C^{B_2})$ is injective.
Suppose that $x \neq \infty$. Let us as in Lemma \ref{lemwidehatRBsmooth} consider the subset $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y$, where $\mathbb C_y \subset \mathbb CP^1$ is defined by $y \neq 0$, and the canonical coordinate chart on it. The tangent map of the projection $\mathbb C^{B_1} \times \mathbb C^{B_2} \times \mathbb C \to \mathbb C^{B_1} \times \mathbb C^{B_2}$ is the quotient by $\partial/\partial x$, and the tangent space $T_{(f,g,x)} \widetilde R_B$ is $V(df(x), dg(x))$, thus the tangent map $\pi_*$ is non-injective at $(f,g,x)$ if and only if $\partial/\partial x$ is the root of both $df(x)$ and $dg(x)$ at $(f,g,x)$.
From the explicit formulas for $df(x)$ and $dg(x)$ we see that it means $f'(x)=g'(x)=0$. The point $(f,g,x)$ is in $\widetilde R_B$, thus $f(x)=g(x)=0$, thus the map $\pi: \widetilde R_B \to R_B$ is not a local embedding at $(f,g,x)$ if and only if $x$ is a root of $(f, g)$ of multiplicity at least two.
If $x = \infty$, then we can replace $B_i$ with $B_i'=-B_i$ and prove the same.
\fi}
\end{proof}
\begin{sledst}\label{corsing}
Suppose that $B_1$ and $B_2$ can not be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$. The subvariety $R_B$ is singular at the point $(f,g)\in \mathbb C^{B_1}\times\mathbb C^{B_2}$
if and only if
1) $(f,g)$ has a root in $\mathbb CP^1$ of multiplicity at least 2, or
2) $(f,g)$ has at least two roots in $\mathbb CP^1$.
\end{sledst}
\begin{proof}
Indeed, by Lemma \ref{lemwidehatRBsmooth} $R_B$ is the image of a smooth set $\widetilde R_B$. The set $R_B$ can be non-smooth at the point $(f,g)$ in one of the following two cases: either the map $\pi$ is not a local embedding at $(f,g,x)$, or there is more than one preimage $(f,g,x)$. It corresponds to a root of multiplicity at least 2 and two roots respectively.
Vice versa, suppose that $(f,g) \neq (0,0)$ has a root of multiplicity at least 2. Then the map $\pi$ is not a local embedding at $(f,g,x)$. This map has finite fibers (expect for $(0,0)$), thus the image $R_B$ is singular at $(f,g)$.
Now suppose that $(f,g) \neq (0,0)$ has at least two roots. If some of them have multiplicity at least 2, then $R_B$ is singular at $(f,g)$. If all have multiplicity 1, then $R_B$ at the point $(f,g)$ has several local branches corresponding to the roots, each of which is smooth. They can not coincide. Indeed, if two branches coindice in a small neighborhood $U$ of $(f,g)$ in $R_B$, then every pair $(f_1,g_1)\in U$ has two common roots, which is possible only if $B_1$ and $B_2$ can be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$.
At last, if $(f,g)=(0,0)$, then then the conical space $R_B$ can be smooth if and only if it is linear, that can not happen. {\iflong
Indeed, for a linear set the equations $df(x)$ and $dg(x)$ that define the tangent space to $R_B$ should be constant, which means $|B_1|=|B_2|=1$.
\fi}
\end{proof}
\begin{lemma}\label{lemtransvers}
Suppose that $B_1$ and $B_2$ can not be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$. Let $x_1$ and $x_2$ be different roots of $(f,g)$ of multiplicity 1. Then the corresponding local branches of $R_B$ are transversal hypersurfaces, unless $x_1$ and $x_2$ are both from $\mathbb C^\times$ and either both have multiplicity at least 2 for $f$, or both have multiplicity at least 2 for $g$.
\end{lemma}
\begin{proof}
Suppose that neither $x_1$, nor $x_2$ is equal to the infinity. {\ifshort
Let us shift $B_1$ and $B_2$ so that $\min B_1=\min B_2=0$, and consider the subset $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y$, where $\mathbb C_y \subset \mathbb CP^1$ is defined by $y \neq 0$. As $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y \cong \mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C$, we have a canonical coordinate chart on it. In this coordinate chart, the subset $\widetilde R_B \cap (\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y)$ is defined exactly by the equations $f(x)=g(x)=0$.
\fi}
{\iflong
Let us as in Lemma \ref{lemwidehatRBsmooth} shift $B_1$ and $B_2$ so that $\min B_1=\min B_2=0$, and consider the subset $\mathbb C^{B_1}\times\mathbb C^{B_2}\times\mathbb C_y$, where $\mathbb C_y \subset \mathbb CP^1$ is defined by $y \neq 0$, and the canonical coordinate chart on it.
\fi}
{\ifshort
The tangent space to $R_B$ is given by the equations
$$df(x)=\sum_{i \in B_1} x^i df_i + \sum_{j \in B_2} 0\,dg_j + f'(x) dx,$$
$$dg(x)=\sum_{i \in B_1} 0\,df_i + \sum_{j \in B_2} x^j dg_j + g'(x) dx.$$
So now consider the map $\pi: \widetilde R_B \to R_B$ and its tangent map $\pi_*: V(df(x), dg(x)) \to T_{(f,g)}(\mathbb C^{B_1} \times \mathbb C^{B_2})$ at the points $(f,g,x_k)$, $k=1,2$.
\fi}
{\iflong
Similarly to Lemma \ref{lemrbnonlocdiff}, consider the map $\pi: \widetilde R_B \to R_B$ and its tangent map $\pi_*: V(df(x), dg(x)) \to T_{(f,g)}(\mathbb C^{B_1} \times \mathbb C^{B_2})$ at the points $(f,g,x_k)$, $k=1,2$.
\fi}
We have that either $f'(x_k) \neq 0$ or $g'(x_k) \neq 0$, thus the image is given by the linear equation
$$g'(x_k) \cdot df(x)-f'(x_k) \cdot dg(x)=g'(x_k) \cdot \sum_{i \in B_1} x_k^i df_i - f'(x_k) \cdot \sum_{j \in B_2} x_k^j dg_j.$$
Suppose that the hypersurfaces are not transversal, thus such covectors for $x_1$ and $x_2$ are proportional and non-zero. The covectors have summands $f'(x_1)x_1^mdg_m$ and $f'(x_2)x_2^mdg_m$ with $m > 0$, thus if $x_1=0$ and $x_2 \neq 0$, then $f'(x_2)=0$, and similarly $g'(x_2)=0$. The point $(f,g,x_2)$ is in $\widetilde R_B$, thus $f(x_2)=g(x_2)=0$ and so $x_2$ is a root of $(f,g)$ of multiplicity at least 2, but that contradicts the conditions of the lemma. Thus $x_1=0$ and $x_2 \neq 0$ can not happen, and as $x_1 \neq x_2$ and the conditions are simmetric in $x_1$ and $x_2$, we have that $x_1 \neq 0$ and $x_2 \neq 0$.
The covectors have summands $f'(x_1)dg_0$ and $f'(x_2)dg_0$, thus either both $f'(x_1)$ and $f'(x_2)$ vanish, or both are non-zero. The points $(f,g,x_k)$ are in $\widetilde R_B$, thus $f(x_1)=f(x_2)=0$. By the conditions of the lemma, neither $x_1$ and $x_2$ have multiplicity at least 2 for $f$, thus $f'(x_1) \neq 0$ and $f'(x_2) \neq 0$. Consequently, the covectors $\sum_{j \in B_2} x_1^j dg_j$ and $\sum_{j \in B_2} x_2^j dg_j$ are proportional, thus $x_1/x_2$ is a $k$-th root of unity with $k\geqslant 2$ and one can shift $B_2$ to a proper sublattice $k\mathbb Z\subset\mathbb Z$. Similarly with $g'$ and $B_1$, thus $B_1$ and $B_2$ can be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$, which contradicts the conditions of the lemma.
If neither $x_1$, nor $x_\infty$ is equal to 0, then we can replace $B_i$ with $B_i'=-B_i$ and prove the same.
If $x_1=0$ and $x_2=\infty$, then the equations of the hypersurfaces are
$$g'(x_1)df_{\min B_1}-f'(x_1)dg_{\min B_2} \text{ and } g'(x_2)df_{\max B_1}-f'(x_2)dg_{\max B_2},$$
thus they are also transversal.
\end{proof}
\begin{sledst}\label{cortranstype}
Suppose that $B_1$ and $B_2$ can not be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$. The transversal singularity type of $R_B$ at the singular point $(f,g) \in \mathbb C^{B_1}\times\mathbb C^{B_2}$ is $\mathcal A_1$ unless
1) $(f,g)$ has three roots in $\mathbb CP^1$,
2) $(f,g)$ has two roots in $\mathbb C^\times$, both of which have multiplicity at least two for $f$ (or, similarly, for $g$),
3) $(f,g)$ has a root of multiplicity at least 2.
\end{sledst}
In particular, it takes into consideration the case $(f,g)=(0,0)$.
\subsection{The filtration}
The definition of the multiplicities allows us to filter the space $\mathbb C^{B_1} \times \mathbb C^{B_2}$, containing the resultant $R_B$, into the subsets $N(p)$ defined by the orders of common roots. It will be more convenient to narrow them down to
$$\mathbb C^{B_1\times} \times \mathbb C^{B_2\times}=\left\{ (f_1,f_2) \in \mathbb C^{B_1} \times \mathbb C^{B_2} \,|\, f_1 \not\equiv 0 \text{ and } f_2 \not\equiv 0 \right\},$$
because if $f_i$ is identically zero, then it has more roots than is expected. In particular, the subsets $N(1)$ and $N(1,1)$ defined below are the intersections of the subsets $H(1)$ and $H(1,1)$ from the introduction and the beginning of this section with the subset $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$.
\begin{defin}\label{defN}
A symmetric filtration subset
$$N_{j_0}^{j_\infty}(j_1,\ldots,j_k),\quad j_0, j_\infty \geqslant 0, \quad j_1,\ldots,j_k \geqslant 1, \quad k \geqslant 0$$
consists of $(f_1,f_2) \in \mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$ such that
\begin{enumerate}
\item $\mathop{\rm ord}\nolimits_0(f_1, f_2) \geqslant j_0$ and $\mathop{\rm ord}\nolimits_\infty(f_1, f_2) \geqslant j_\infty$;
\item $f_1=f_2=0$ has at least $k$ distinct solutions $x_1,\ldots,x_k$ in $\mathbb C^\times$;
\item at the $m$-th solution $x_m$ holds $\mathop{\rm ord}\nolimits_{x_m}(f_1, f_2) \geqslant j_m$.
\end{enumerate}
\end{defin}
The numbering of the symmetric filtration subsets is defined up to a permutation of $(j_1, \ldots, j_k)$: for example, $N_1^2(3,4,5)$ is equal to $N_1^2(4,5,3)$, but in general not to $N_2^1(3,4,5)$.
If $k=0$, the symmetric filtration subset is denoted simply $N_{j_0}^{j_\infty}$. We will also denote $N_0^0(j_1,\ldots,j_k)$ by $N(j_1,\ldots,j_k)$.
The subset $N(1)$ is $H(1) \cap (\mathbb C^{B_1\times}\times\mathbb C^{B_2\times})$, thus the closure of $N(1)$ is $R_B \cap (\mathbb C^{B_1\times}\times\mathbb C^{B_2\times})$.
Now we can reformulate the $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$-part of Corollary \ref{corsing} as follows:
\begin{sledst}
Suppose that $B_1$ and $B_2$ can not be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$. Then the intersection of the singular locus $\mathop{\rm sing}\nolimits R_B$ with $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$ is
$$N(2) \cup N^2_0 \cup N^0_2 \cup N(1,1) \cup N^1_0(1) \cup N^0_1(1).$$
\end{sledst}
Moreover, we can generalize the last definition for the case of two separate lines of indices.
\begin{defin}\label{defNN}
A general filtration subset
$$N\;_{j_0}^{j_\infty}\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right),\quad j_0,j_\infty \geqslant 0, \quad j^1_1,\ldots,j^1_k,j^2_1,\ldots,j^2_k \geqslant 1$$
consists of $(f_1,f_2) \in \mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$ such that
\begin{enumerate}
\item $\mathop{\rm ord}\nolimits_0(f_1, f_2) \geqslant j_0$ and $\mathop{\rm ord}\nolimits_\infty(f_1, f_2) \geqslant j_\infty$;
\item $f_1=f_2=0$ has at least $k$ distinct solutions $x_1,\ldots,x_k$ in $\mathbb C^\times$;
\item at the $m$-th solution $x_m$ holds $\mathop{\rm ord}\nolimits_{x_m}(f_1) \geqslant j^1_m$ and $\mathop{\rm ord}\nolimits_{x_m}(f_2) \geqslant j^2_m$.
\end{enumerate}
The numbering of the general filtration subsets is defined up to a simultaneous permutation of $(j^1_1, \ldots, j^1_k)$ and $(j^2_1, \ldots, j^2_k)$.
We will denote $N\;_0^0\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$ by $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$.
\end{defin}
Here we have that
$$N_{j_0}^{j_\infty}(j_1,\ldots,j_k)=N\;_{j_0}^{j_\infty}\left(\,^{j_1,\ldots,j_k}_{j_1,\ldots,j_k}\right).$$
Now we can similarly reformulate the $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$-part of Corollary \ref{cortranstype}:
\begin{sledst}\label{cortranstype2}
The transversal singularity type of $R_B$ at the point $(f,g) \in \mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$ is $\mathcal A_1$ unless $(f,g)$ lies in one of the following filtration subsets:
1) $N(1,1,1)$, $N^0_1(1,1)$, $N^1_0(1,1)$ or $N_1^1(1)$;
2) $N(2)$, $N_2^0$ or $N_0^2$;
3) $N\left(\,^{2,2}_{1,1}\right)$ or $N\left(\,^{1,1}_{2,2}\right)$
\end{sledst}
\begin{rem}\label{remideaofmainthm}
One can easily see that almost all of that subsets $N(p)$ listed above usually have codimension at least 3. Moreover, we have that $N\left(\,^{2,2}_{1,1}\right) \subset N\left(\,^{2,1}_{1,1}\right)$ and $N\left(\,^{1,1}_{2,2}\right) \subset N\left(\,^{1,1}_{2,1}\right)$. Consequently, if the subsets $N(1,1,1)$, $N\left(\,^{2,1}_{1,1}\right)$ and $N\left(\,^{1,1}_{2,1}\right)$ have codimension at least 3, then there exists a codimension 3 subset $\Sigma\subset \mathbb C^{B_1}\times\mathbb C^{B_2}$ such that at every point of $(\mathop{\rm sing}\nolimits R_B\setminus\Sigma) \cap (\mathbb C^{B_1\times}\times\mathbb C^{B_2\times})$ the transversal singularity type of $R_B$ is $\mathcal{A}_1$. It is the part (i) of the main theorem \ref{theormain} expect for the subset $(\{0\} \times \mathbb C^{B_2}) \cup (\mathbb C^{B_1} \times \{0\})$.
\end{rem}
\subsection{Expected codimensions of the filtration subsets}
The relation of the subsets $N(p)$ to the singular locus of the resultant $R_B$ motivates our interest in their codimension.
\begin{defin}
The expected codimension of a general fitration subset $N\;_{j_0}^{j_\infty}\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$ is the number
$$2j_0+2j_\infty+\sum_{m=1}^k (j^1_m+j^2_m-1).$$
In particular, the expected codimensions of a symmetric filtration subset $N_{j_0}^{j_\infty}(j_1,\ldots,j_k)$ is
$$2j_0+2j_\infty+\sum_{m=1}^{k} (2j_m-1).$$
\end{defin}
{\iflong
\begin{rem}
Indeed, to fix orders of the root at 0, one should impose $2j_0$ conditions on $(f_1, f_2)$: $j_0$ conditions that $f_1$ has a root of order at least $1, 2, \ldots, j_0$ at 0, and the same with $f_2$; similarly with $\infty$; and to fix a common root that has order $j^1_m$ for $f_1$ and $j^2_m$ for $f_2$ and is lying in $\mathbb C^\times$, one should first impose the condition of the existence of a common root $x_m$, then impose $(j^1_m-1)$ conditions that $f_1$ has a root of order at least $2, \ldots, j_m$ at $x_m$, and the same with $f_2$. We expect that each condition increases the codimension by 1.
\end{rem}
\fi}
\begin{exa}
The symmetric filtration subsets of expected codimensions 1, 2 and 3 are
\begin{enumerate}
\item $N(1)$;
\item $N_1^0$, $N_0^1$ and $N(1,1)$;
\item $N_1^0(1)$, $N_0^1(1)$, $N(2)$ and $N(1,1,1)$.
\end{enumerate}
\end{exa}
\begin{rem}
The filtration subsets
$N\left(\,^{2,1}_{1,1}\right)$ and $N\left(\,^{1,1}_{2,1}\right)$ from Remark \ref{remideaofmainthm} have expected codimension 3.
\end{rem}
\begin{rem}\label{remcodimreduc}
An algebraic subset can consist of several irreducible components of different dimensions, thus we should be accurate when talking about its codimension. When we say that an algebraic subset has codimension equal to the given number, we mean that the codimension of every its irreducible component is equal to the given number. When we say that the codimension of an algebraic subset is at most (at least) the given number, we mean that the codimension of every its irreducible component is at most (at least) the given number. In particular, in all of the mentioned cases it may be empty, although some sources consider the empty set to have codimension $\infty$.
\end{rem}
\subsection{Proof of the part (i) of the main theorem}
We have a following theorem:
\begin{theor}\label{theorcodimNpre}
Unless $(B_1,B_2)$ satisfy one of the five conditions of the part (i) of the main theorem \ref{theormain}, every symmetric filtration subset $N(p)$ of the expected codimension 1, 2 or 3 has actual codimension at least 1, 2 or 3 respectively.
Moreover, under these conditions, the filtration subsets $N\left(\,^{2,1}_{1,1}\right)$ and $N\left(\,^{1,1}_{2,1}\right)$ also have codimension at least 3.
\end{theor}
\begin{rem}
Each of the conditions of the part (i) of the main theorem \ref{theormain} is only used to estimate the codimensions of some of the strata. The exact details see in the theorem
\ref{theorcodimN}.
\end{rem}
We will prove this theorem in Sections \ref{secstratrootspace}-\ref{secstrats12forn111}. Now we will use this theorem and Corollary \ref{cortranstype2} to prove the part (i) of the main theorem \ref{theormain}.
\begin{proof}[Proof of the main theorem \ref{theormain}, part (i)]
Let us first consider the $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$-part. By Theorem \ref{theorcodimNpre}, in the conditions of the part (i) of the main theorem the filtration subsets
$$N(1,1,1), \quad N(2), \quad N\left(\,^{2,1}_{1,1}\right), \quad N\left(\,^{1,1}_{2,1}\right), \quad N_0^1(1) \; \text{ and } \; N_1^0(1)$$
have codimensions at least three, thus the same is true for their subsets
$$N(1,1,1), \quad N(2), \quad N\left(\,^{2,2}_{1,1}\right), \quad N\left(\,^{1,1}_{2,2}\right), \quad N_0^1(1,1) \; \text{ and } \; N_1^0(1,1).$$
The same is obviously true for the rest of the filtration subsets of Corollary \ref{cortranstype2}, namely the subsets
$$N_1^1(1), \quad N_2^0 \; \text{ and } \; N_0^2,$$
thus by Corollary \ref{cortranstype2} there exists a codimension 3 subset $\Sigma'\subset \mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$ such that at every point of $(\mathop{\rm sing}\nolimits R_B \cap (\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}))\setminus\Sigma'$ the transversal singularity type of $R_B$ is $\mathcal{A}_1$.
Let us now consider the $(\{0\} \times \mathbb C^{B_2}) \cup (\mathbb C^{B_1} \times \{0\})$-part. We need to find a codimension 3 subset $\Sigma''\subset (\{0\} \times \mathbb C^{B_2}) \cup (\mathbb C^{B_1} \times \{0\})$ such that at every point of $(\mathop{\rm sing}\nolimits R_B \cap (\mathbb C^{B_1}\times\mathbb C^{B_2})) \setminus\Sigma''$ the transversal singularity type of $R_B$ is $\mathcal{A}_1$.
By Corollary \ref{corsing}, the subset $\{0\}\times\mathbb C^{B_2\times}$ appears in $\mathop{\rm sing}\nolimits R_B$ if and only if $B_2 \neq \{j, j+1\}$. Its codimension if $|B_1|$, thus if $|B_1|>2$, then we can include $(\{0\}\times\mathbb C^{B_2\times})$ into $\Sigma''$. Otherwise $|B_1|=2$ and it is a subset of codimension 2. By Lemma \ref{lemtransvers}, at its general point $(0,g)\in \{0\}\times\mathbb C^{B_2\times}$ (such that $g$ has no multiple roots) the subset $R_B$ has $\max B_2-\min B_2$ smooth transversal branches, thus if $\max B_2-\min B_2=2$, we can still add to $\Sigma''$ a part of $\{0\}\times\mathbb C^{B_2\times}$ containing polynomials with multiple roots. The case of $|B_1|=2$ and $\max B_2-\min B_2>2$ is excluded by the condition (5).
The same can be proven about $\mathbb C^{B_1\times} \times \{0\}$.
As for the subset $\{(0,0)\}$, it has codimension at least 4.
Thus we get a subset $\Sigma''$ as necessary.
We can now take $\Sigma$ to be $\Sigma' \cup \Sigma''$.
\end{proof}
\section{A stratification of the sparse resultant}
\subsection{The stratification}
In this section, we will study the strata $M(p)$ of a stratification related to the filtration $N$ introduced in Section \ref{secfiltr}. It will be more convenient again to narrow them down to
$$\mathbb C^{B_1\times} \times \mathbb C^{B_2\times}=\left\{ (f_1,f_2) \in \mathbb C^{B_1} \times \mathbb C^{B_2} \,|\, f_1 \not\equiv 0 \text{ and } f_2 \not\equiv 0 \right\},$$
because if $f_i$ is identically zero, then it has more roots than is expected.
\begin{defin}
A stratum
$$M_{j_0}^{j_\infty}(j_1,\ldots,j_k),\quad j_0, j_\infty \geqslant 0, \quad j_1,\ldots,j_k \geqslant 1, \quad k \geqslant 0$$
consists of $(f_1,f_2) \in \mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$ such that
\begin{enumerate}
\item $\mathop{\rm ord}\nolimits_0(f_1, f_2) = j_0$ and $\mathop{\rm ord}\nolimits_\infty(f_1, f_2) = j_\infty$;
\item $f_1=f_2=0$ has exactly $k$ distinct solutions $x_1,\ldots,x_k$ in $\mathbb C^\times$;
\item at the $m$-th solution $x_m$ holds $\mathop{\rm ord}\nolimits_{x_m}(f_1, f_2) = j_m$.
\end{enumerate}
\end{defin}
The strata numbering is defined up to a permutation of $(j_1, \ldots, j_k)$: for example, $M_1^2(3,4,5)$ is equal to $M_1^2(4,5,3)$, but not to $M_2^1(3,4,5)$. The strata that do no differ by the permutation of $(j_1, \ldots, j_k)$ do not intersect. The space $\mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$ is the disjoint union of such strata.
If $k=0$, a stratum is denoted simply as $M_{j_0}^{j_\infty}$.
To study these sets, let us first notice that these sets are indeed strata in the sense that they are locally (Zariski) closed. To prove this, we introduce the following partial order $\succcurlyeq$ on the tuples $I=(j_0; j_\infty; j_1, \ldots, j_k)$ indexing these strata.
\begin{defin}\label{defstratorder}
Let us consider two strata
$$M(p)=M_{j_0}^{j_\infty}(j_1,\ldots,j_k) \quad \text{and} \quad M(q)=M_{g_0}^{g_\infty}(g_1,\ldots,g_l).$$
We say that $q \succcurlyeq p$ if, informally, the common roots of a pair $(h_1,h_2)\in M(q)$ in $\mathbb CP^1$ can be obtained from the common roots of a pair $(f_1,f_2)\in M(p)$ in $\mathbb CP^1$ by
\begin{enumerate}
\item increasing their multiplicities,
\item gluing some of the roots in $\mathbb C^\times$ together,
\item moving some of the roots from $\mathbb C^\times$ to $0$ ot $\infty$ (but not moving the roots at $0$ or $\infty$ anywhere),
\item adding some new common roots.
\end{enumerate}
Formally, $q \succcurlyeq p$ if for each $i \in \{0; \infty; 1, \ldots, k\}$ there is $r(i)\in \{0; \infty; 1, \ldots, l\}$ such that $r(0)=0$, $r(\infty)=\infty$ and for any $s \in \{0, \infty, 1, \ldots, l\}$ holds
$$\left(\sum_{i: \; r(i)=s} j_i\right) \leqslant g_s.$$
\end{defin}
\begin{defin}\label{defmwidehat}
Let us also define closed strata to be
$\widehat M(p)=\bigsqcup_{q \succcurlyeq p} M(q)$.
\end{defin}
\begin{exa}\label{exawidehat0011}
The closed stratum $\widehat M(1)$ consists of all pairs $(f_1, f_2)$ (of non-zero polynomials) that have at least one common root in $\mathbb CP^1$ of multiplicity at least 1. By the remark at the beginning of Subsection \ref{subsetsingloc}, $\widehat M(1)=R_B \cap (\mathbb C^{B_1\times}\times\mathbb C^{B_2\times})$. Moreover, in terms of Section \ref{secfiltr}, $\widehat M(1)=N(1)\cup N^1\cup N_1$.
Similarly, the closed stratum $\widehat M(1,1)$ consists of all pairs $(f_1, f_2)$ that have either at least two common roots in $\mathbb CP^1$ or at least one common root in $\mathbb CP^1$ of multiplicity at least 2. By Corollary \ref{corsing}, if $B_1$ and $B_2$ can not be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$, then $\widehat M(1,1)=\mathop{\rm sing}\nolimits R_B \cap (\mathbb C^{B_1\times}\times\mathbb C^{B_2\times})$. In terms of Section \ref{secfiltr}, $\widehat M(1,1)=N(2) \cup N^2 \cup N_2 \cup N(1,1) \cup N^1(1) \cup N_1(1)$.
\end{exa}
In Subsection \ref{subsecclosstrat} we will prove the following lemma, see Corollary \ref{corcodimwidehatM}:
\begin{lemma} The closed strata
$\widehat M(p)=\bigsqcup_{q \succcurlyeq p} M(q)$ are indeed (Zariski) closed in $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$.
\end{lemma}
\begin{sledst}
The sets $M(p)=\widehat M(p)\setminus\bigcup_{q \succ p} \widehat M(q)$ are locally closed.
\end{sledst}
\begin{rem}\label{remb1ab}
The closed stratum $\widehat M(p)$ is not always equal to the closure of $M(p)$.
For example, let us consider $B_1=\{a, b\}$ and any $B_2$ (with at least two elements, as usual). A polynomial $f_1 \in \mathbb C^{B_1\times}$ has form $c_1x^a+c_2x^b$ and thus can not have two equal roots from $\mathbb C^\times$, thus the stratum $M_0^0(2)=\varnothing$. At the same time $\widehat M_0^0(2)$ contains $\widehat M_2^0$, which is non-empty if $b-a \geqslant 2$.
As we can see in this example, a stratum may be empty even if its expected codimension is less than the dimension of ambient space. Indeed, $M_0^0(2)$ has expected codimension 3 and lies in the space $\mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$ of the dimension at least 4.
\end{rem}
\begin{rem}
The symmetric filtration subsets $N(p)$ are different from $M(p)$ in the fact that the conditions are inequalities, not equalities. But they are also different from $\widehat M(p)$, because, saying informally, inside $\widehat M(p)$ one can move points to 0 or $\infty$ and glue points. For example, $M_1^0(1)$ and $M(2)$ lie in $\widehat M_0^0(1,1)$, but usually not in $N_0^0(1,1)$.
\end{rem}
{\iflong
\begin{rem}
The subset $M(p)$ may be empty, while $N(p)$ is not empty. For example, if $B_1=kB_1'$ and $B_2=kB_2'$ with $k \geqslant 2$, then similarly to Example \ref{exaintro}, part 1), $M_0^0(1)=\varnothing$ and $N_0^0(1)=N_0^0(k)$ has codimension 1.
The subset $N(p)$ may be empty, while $\widehat M(p)$ is not empty. For example, in the conditions of Remark \ref{remb1ab} $N_0^0(2)=\varnothing$ and $\widehat M_0^0(2) \supset \widehat M_2^0$, which is non-empty if $b-a \geqslant 2$. In particular, $\widehat M(p)$ is not always equal to the closure of $N(p)$.
\end{rem}
\fi}
\subsection{Expected codimensions of the strata}
\begin{defin}
Just like for $N_{j_0}^{j_\infty}(j_1,\ldots,j_k)$, the expected codimension of the subset $M_{j_0}^{j_\infty}(j_1,\ldots,j_k)$ is the number
$$2j_0+2j_\infty+\sum_{m=1}^{k} (2j_m-1).$$
\end{defin}
\begin{rem}
If $q \succcurlyeq p$, then the expected codimension of $M(q)$ is not greater than the expected codimension of $M(p)$. If $q \succcurlyeq p$ and the expected codimensions or $M(q)$ and $M(p)$ are equal, then $q=p$.
\end{rem}
\begin{lemma}\label{lemcodimclassic}
In the classical case $B_i=\{0,1,2,\ldots,d_i\}$, the codimension of the stratum $M(p)$ (at every its irreducible component) is equal to the expected one.
\end{lemma}
For the sake of completeness, we provide the proof of this classical fact by parameterizing naturally a finite cover of the set $M(p)$.
\begin{proof}
First, let us consider the case of $M(p)=M^0_0(j_1,\ldots,j_k)$. The idea is that, as neither $f_1$, nor $f_2$ is identically zero, the condition $(f_1, f_2) \in M^0_0(j_1,\ldots,j_k)$ is equivalent to the condition that they can be written as
$$f_1(t)=\prod_{i=1}^k (t-z_i)^{j_i} \prod_{i=1}^{e_1} (t-x_i) \text{ and } f_2(t)=\prod_{i=1}^k (t-z_i)^{j_i} \prod_{i=1}^{e_2} (t-y_i)$$
with some open conditions on $x_i$, $y_i$ and $z_i$, and that there is only a finite number of ways to write $(f_1, f_2)$ in this form. It works only for $B_i$ without gaps, because otherwise not all $(f_1, f_2)$ of the form above will lie in $\mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$.
{\iflong
Formally, consider the subset
$$P \subset \mathbb C^{e_1} \times \mathbb C^{e_2} \times \mathbb C^k, \quad e_1=d_1-\sum_{m=1}^k j_m, \quad e_2=d_2-\sum_{m=1}^k j_m$$
of tuples $(x_1,\ldots,x_{e_1}; \; y_1,\ldots,y_{e_2}; \; z_1,\ldots,z_k)$ such that $z_i$ are pairwise different and non-zero, and that for each $m=1,\ldots,k$ at least one of the polynomials $\prod_{i=1}^{e_1}(x_i-z_m)$ and $\prod_{i=1}^{e_1}(x_i-z_m)$ is non-zero. It is an open subset of $\mathbb C^{d_1} \times \mathbb C^{e_2} \times \mathbb C^k$ and thus it has the dimension $d_1+d_2-\sum_{m=1}^{k} (2j_m-1)$.
There is a map $P \to M(p)$ of the form
$$\quad (x_1,\ldots,x_{e_1}; \; y_1,\ldots,y_{e_2}; \; z_1,\ldots,z_k) \mapsto \left( \prod_{i=1}^{e_1} (t-x_i) \prod_{i=1}^k (t-z_i)^{j_i}, \; \prod_{i=1}^{e_2} (t-y_i) \prod_{i=1}^k (t-z_i)^{j_i} \right),$$
which is surjective and has finite fibers, thus $\mathop{\rm \bold d}\nolimitsim P=\mathop{\rm \bold d}\nolimitsim M(p)$ and
$$\mathop{\rm codim}\nolimits M(p)=d_1+d_2-\mathop{\rm \bold d}\nolimitsim M(p)=\sum_{m=1}^{k} (2j_m-1),$$
which is the expected codimension.
To prove the general case, let us notice that the stratum
$$M_{j_0}^{j_\infty}(j_1,\ldots,j_k) \subset \mathbb C^{B_1\times} \times \mathbb C^{B_2\times} \text{ with } B_i=\{0,\ldots,d_i\}$$
is identified with the stratum
$$M_0^0(j_1,\ldots,j_k) \subset \mathbb C^{B_1'\times} \times \mathbb C^{B_2'\times} \text{ with } B_i'=\{j_0,\ldots,d_i-j_\infty\}$$
under the natural embedding $\mathbb C^{B_1'\times} \times \mathbb C^{B_2'\times} \hookrightarrow \mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$, and that the last stratum can identified with the stratum
$$M_0^0(j_1,\ldots,j_k) \subset \mathbb C^{B_1''\times} \times \mathbb C^{B_2''\times} \text{ with } B_i''=\{0,\ldots,d_i-j_0-j_\infty\}$$
by the inclusion $\mathbb C^{B_1''\times} \times \mathbb C^{B_2''\times} \hookrightarrow \mathbb C^{B_1'\times} \times \mathbb C^{B_2'\times}$ induced by the pair of maps $B_i'' \to B_i'$, $b \mapsto b+j_0$.
Thus
\begin{equation*}
\begin{split}
&\mathop{\rm codim}\nolimits M_{j_0}^{j_\infty}(j_1,\ldots,j_k)=d_1+d_2-\mathop{\rm \bold d}\nolimitsim M_{j_0}^{j_\infty}(j_1,\ldots,j_k)=\\
&=d_1+d_2-((d_1-j_0-j_\infty)+(d_2-j_0-j_\infty)-\mathop{\rm codim}\nolimits M_0^0(j_1,\ldots,j_k)=\\
&=2j_0+2j_\infty+\sum_{m=1}^{k} (2j_m-1),
\end{split}
\end{equation*}
which is the expected codimension.
\fi}
\end{proof}
\begin{rem}
If we would define the stratum $M(p)$ not inside $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$, but inside $\mathbb C^{B_1}\times\mathbb C^{B_2}$, it would be wrong even in the classical case $B_i=\{0,1,2,\ldots,d_i\}$. Indeed, consider the stratum $M_0^0(\underbrace{1,\ldots,1}_{d_2})$. A general polynomial $f_2 \in \mathbb C^{B_2\times}$ has $d_2$ different roots, all of which are the roots of $f_1\equiv 0$, thus the subset $M_0^0(1,\ldots,1) \cap (\{0\} \times \mathbb C^{B_2\times})$ is open in $\{0\} \times \mathbb C^{B_2\times}$ and has codimension $(d_1+1)$. If $d_1 + 1 < d_2$, then the codimension of $M_0^0(1,\ldots,1) \cap (\{0\} \times \mathbb C^{B_2\times})$ is less than $d_2$, which is the expected codimension of $M_0^0(1,\ldots,1)$.
\end{rem}
The lemma survives as an estimate in the general case.
\begin{sledst}\label{corcodimMatmost}
In general, the codimension of the stratum $M(p)$ at every its irreducible component is at most the expected one.
\end{sledst}
Thus $M(p)$ can have irreducible components of larger dimension than expected, but can not have irreducible components of smaller dimension.
\begin{proof}
Equivalently, we can prove that the codimension of the stratum $M(p)$ at every its point is at most the expected one.
Consider the natural embedding $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}\to \mathbb C^{\mathop{\rm conv}\nolimits B_1\times}\times\mathbb C^{\mathop{\rm conv}\nolimits B_2\times}$, where $\mathop{\rm conv}\nolimits B_i=\{ \min B_i, \ldots, \max B_i \}$ is the convex hull of $B_i$. The stratum $M(p)$ in $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$ is the preimage of the corresponding stratum in $\mathbb C^{\mathop{\rm conv}\nolimits B_1\times}\times\mathbb C^{\mathop{\rm conv}\nolimits B_2\times}$. Thus at every point of the stratum $M(p)$ in $\mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$ its codimension is not greater than the codimension of the corresponding stratum in $\mathbb C^{\mathop{\rm conv}\nolimits B_1\times}\times\mathbb C^{\mathop{\rm conv}\nolimits B_2\times}$ at the image point, which is equal to the expected one.
\end{proof}
The inverse estimate is in general not true, it requires some conditions on $B_1$ and $B_2$.
\begin{exa}
For example, if one can shift $B_1$ and $B_2$ to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$, then $\widehat M(1)=\widehat M(1,1)=\ldots=\widehat M(1,\ldots,1)$, where the last $\widehat M(1,\ldots,1)$ has $k$ ones. But the expected codimensions of $\widehat M(1)$ and $\widehat M(1,\ldots,1)$ are 1 and $k$ respectively, thus some of the strata $M(p)$ do not have the expected codimension (usually $M(1,\ldots,1)$ has codimension 1 instead of $k$).
\end{exa}
\subsection{Proof of the part (ii) of the main theorem}\label{subsecclosstrat}
Using the methods of Lemma \ref{lemcodimclassic}, one can prove the similar facts about the closed strata.
\begin{lemma}
In the classical case $B_i=\{0,1,2,\ldots,d_i\}$, the set $\widehat M(p)$ is closed and its codimension of $\widehat M(p)$ is equal to the expected codimension (of $M(p)$).
\end{lemma}
{\iflong
\begin{proof}
Consider a subset $\mathbb P \widehat M(p) \subset \mathbb P(\mathbb C^{B_1}) \times \mathbb P(\mathbb C^{B_2})$ which is a projectivization of $\widehat M(p) \subset \mathbb C^{B_1\times}\times\mathbb C^{B_2\times}$. One can similarly to the proof of Lemma \ref{lemcodimclassic} construct a closed subset $\widehat P \subset (\mathbb CP^1)^{e_1} \times (\mathbb CP^1)^{e_2} \times (\mathbb CP^1)^k$ and a surjective map $\widehat P \to \mathbb P \widehat M(p)$ with finite fibers, thus $\mathbb P \widehat M(p)$ is closed and its codimension is equal to the expected one (at every point, and thus at every irreducible component). Moreover, the same holds for $\widehat M(p)$.
\end{proof}
\fi}
\begin{sledst}\label{corcodimwidehatM}
In general, the closed stratum $\widehat M(p)$ is indeed closed and its codimension (at every its irreducible component) is at most the expected one.
\end{sledst}
{\iflong
It can be deduced from the lemma above similarly to Corollary \ref{corcodimMatmost}.
\fi}
\begin{lemma}\label{lemclosm}
If $|B_1|>2$ and $B_2\neq\{j,j+1\}$, then $\{0\} \times \mathbb C^{B_2}$ lies in the closure of $\widehat M(1,1)$.
\end{lemma}
Notice that both conditions in the lemma are necessary. If $|B_1|=2$, then both $\{0\} \times \mathbb C^{B_2}$ and $\widehat M(1,1)$ have codimension 2 (and $\widehat M(1,1)$ may be empty). If $B_2=\{i,i+1\}$, then $\widehat M(1,1)$ is empty.
\begin{proof}
To show that $(0,g) \in \{0\} \times \mathbb C^{B_2}$ lies in this closure, it is enough to find $f \in \mathbb C^{B_1\times}$ such that $f$ and $g$ has two common roots, thus we would have that $(f, g) \in \widehat M(1,1)$ and $(0,g)=\lim_{t\to 0}(tf, g)$.
Suppose that $g$ has two distinct non-zero roots $x_1$ and $x_2$. The system
$$f(x_1)=\sum_{i \in B_1} f_i x_1^i=0 \ \text{ and } \ f(x_2)=\sum_{i \in B_1} f_i x_2^i=0$$
is a system of two homogeneous equations on $|B_2|>2$ variables, thus it has a non-zero solution $f$. This solution is a polynomial $f \in \mathbb C^{B_1\times}$ with two common roots with $g$ that is necessary to the proof.
If $B_2=\{j,j+1\}$ for some $i \in \mathbb Z$, then any $g \in \mathbb C^{B_2\times}$ has only one non-zero root. Otherwise a general polynomial $g \in \mathbb C^{B_2\times}$ has at least two different roots, thus $(0,g)$ lies in the closure of $\widehat M(1,1)$. As it is true for a general $g$, the subset $\{0\} \times \mathbb C^{B_2}$ also lies there.
\end{proof}
\begin{proof}[Proof of the main theorem \ref{theormain}, part (ii)]
We would like to have not only the condition (6) of the main theorem \ref{theormain}, but also the condition (1). Thus let us take the largest $k$ such that $B_1=k \cdot B_1'+m_1$ and $B_2=k \cdot B_2'+m_2$ for some $B_1'$ and $B_2' \subset \mathbb Z$: now we can not shift $B_1'$ and $B_2'$ to the same proper sublattice $l\mathbb Z\subset\mathbb Z$ with $l\geqslant 2$. As we know from Example \ref{exaintro}, part 1), $R_B \cong R_{B'}$.
By Lemma \ref{corcodimwidehatM} the subset $\widehat M(1,1)$ is closed in $\mathbb C^{B_1'\times}\times\mathbb C^{B_2'\times}$ and has codimension at most two (at every its irreducible component). But by Example \ref{exawidehat0011} $\widehat M(1,1)=\mathop{\rm sing}\nolimits R_{B'} \cap (\mathbb C^{B_1'\times}\times\mathbb C^{B_2'\times})$. As $B_1'$ and $B_2'$ can not be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$, the subset $R_{B'}$ has codimension at least 1, thus the subset $\widehat M(1,1)=\mathop{\rm sing}\nolimits R_{B'} \cap (\mathbb C^{B_1'\times}\times\mathbb C^{B_2'\times})$ has codimension at least 2. Thus $\widehat M(1,1)$ has codimension exactly 2.
Let us now notice that by Corollary \ref{corsing}
$$\mathop{\rm sing}\nolimits R_{B'}=\begin{cases}
\widehat M(1,1) \cup \{(0,0)\} \quad \text{ if } \ B_1'=\{i,i+1\} \ \text { and } \ B_2'=\{j,j+1\} \\
\widehat M(1,1) \cup (\{0\} \times \mathbb C^{B_2'}) \quad \text{ if } \ B_1'=\{i,i+1\} \ \text { and } \ B_2'\neq\{j,j+1\} \\
\widehat M(1,1) \cup (\mathbb C^{B_1'} \times \{0\}) \quad \text{ if } \ B_1'\neq\{i,i+1\} \ \text { and } \ B_2'=\{j,j+1\} \\
\widehat M(1,1) \cup (\{0\} \times \mathbb C^{B_2'}) \cup (\mathbb C^{B_1'} \times \{0\}) \quad \text{ otherwise }
\end{cases}$$
If $|B_1'|=2$, then the subset $\{0\} \times \mathbb C^{B_2}$ has codimension 2, which is ok. Otherwise, we have $|B_1'|>2$ and $B_2' \neq \{j,j+1\}$, thus Lemma \ref{lemclosm} says that $\{0\} \times \mathbb C^{B_2'}$ lies in the closure of $\widehat M(1,1)$, so it does not give a component of $\mathop{\rm sing}\nolimits R_{B'}$ of larger codimension.
The same can be proven about $\mathbb C^{B_1'} \times \{0\}$.
As for the subset $\{(0,0)\}$, it appears by itself only in the case $B_1'=\{i,i+1\}$ and $B_2'=\{j,j+1\}$. Then $B_1=k \cdot B_1'+m_1=\{i',i'+k\}$ and $B_2=k \cdot B_2'+m_2=\{j',j'+k'\}$, which is excluded by the condition (6) of the main theorem \ref{theormain}.
\end{proof}
\section{A stratification of the solution space}\label{secstratrootspace}
In this section, we will prove Theorem \ref{theorcodimNpre} modulo Lemma \ref{lemstratumranks11}, Theorem \ref{theorstratumranks21for111} and Lemma \ref{lem3minor2} from the subsequent sections (the proofs of Lemmas \ref{lemcork2simple} and \ref{lemcork2} refer to these facts), thus finishing the proof of the main theorem \ref{theormain}.
\subsection{The relation to the stratication of the sparse resultant}
Suppose that $j_0=j_\infty=0$ and study the filtration subsets $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right) \subset \mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$ (in particular, $N(j_1,\ldots,j_k)=N\left(\,^{j_1,\ldots,j_k}_{j_1,\ldots,j_k}\right)$; see Definition \ref{defNN}). Fix $B_1$ and $B_2 \subset \mathbb Z$ and a subset $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$.
Consider the solution space
$$Z_k=\left\{ (x_1, \ldots, x_k) \subset (\mathbb C^\times)^k \,|\, x_i \neq x_j \; \text{for} \; i \neq j \right\}$$
of tuples of different non-zero numbers.
We would like to define a stratification on it that is related with the structure of $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$.
\begin{defin}\label{defmatrsimple}
First let us for each tuple $x=(x_1,\ldots,x_k)$ define a pair of polynomials
$$h^1_x(t)=\prod_{m=1}^k (t-x_m)^{j_m^1} \quad \text{ and } \quad h^2_x(t)=\prod_{m=1}^k (t-x_m)^{j_m^2}.$$
Then consider a pair of maps
$$\psi^1_x: \mathbb C^{B_1} \to \mathbb C[t]/(h^1_x(t)) \quad \text{ and } \quad \psi^2_x: \mathbb C^{B_2} \to \mathbb C[t]/(h^2_x(t))$$
given by
$$f(t) \mapsto f(t) \mod h^1_x(t) \quad \text{ and } \quad g(t) \mapsto g(t) \mod h^2_x(t).$$
(c.f. \cite{E13}*{Def. 3.21}).
We can now define the strata
$$S^1_{n_1} = \left\{ (x_1, \ldots, x_k) \subset Z_k \,|\, \mathop{\rm codim}\nolimits \mathop{\rm Im}\nolimits \psi^1_x = n_1 \right\},$$
$$S^2_{n_2} = \left\{ (x_1, \ldots, x_k) \subset Z_k \,|\, \mathop{\rm codim}\nolimits \mathop{\rm Im}\nolimits \psi^2_x = n_2 \right\},$$
$$S_{n_1, n_2} = S^1_{n_1} \cap S^2_{n_2} = \left\{ (x_1, \ldots, x_k) \subset Z_k \,|\, \mathop{\rm codim}\nolimits \mathop{\rm Im}\nolimits \psi^1_x = n_1 \text{ and } \mathop{\rm codim}\nolimits \mathop{\rm Im}\nolimits \psi^2_x = n_2 \right\}.$$
To sum it up, we fix $B_1$ and $B_2 \subset \mathbb Z$, fix a subset $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$ and define a stratification
$$Z_k = \bigsqcup_{n_1, n_2} S_{n_1, n_2} \subset (\mathbb C^\times)^k.$$
\end{defin}
\begin{rem}\label{remM}
The polynomial $h^i_x(t)$ vanishes on $x_m$: $h^i_x(x_m)=0$; moreover, a number of its derivatives also vanish: $(h^i_x)^{(d^i_m)}(x_m)=0$ for $d^i_m<j^i_m$; as a consequence, the space $\mathbb C[t]/(h^i_x(t))$ has a system of coordinate functions sending the class $f(t) \mod h^i_x(t)$ to
$$f(x_1), f'(x_1), \ldots, f^{(j^i_1-1)}(x_1), \quad f(x_2), f'(x_2), \ldots, f^{(j^i_2-1)}(x_2), \quad \ldots, \quad f(x_k), f'(x_k), \ldots, f^{(j^i_k-1)}(x_k).$$
These functions form a basis of the dual space. Using this basis, we can write down the matrix $M_i$ of the map $\psi^i_x$.
For example, suppose that $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)=N(1,1,1)$. Then the matrix of the map $\psi^i_x$ has the form
\begin{equation}\label{eqM}
M_i=M_i(x,y,z) = \begin{pmatrix}
x^{b^i_1} & x^{b^i_2} & x^{b^i_3} & \ldots & x^{b^i_{s_i}} \\
y^{b^i_1} & y^{b^i_2} & y^{b^i_3} & \ldots & y^{b^i_{s_i}} \\
z^{b^i_1} & z^{b^i_2} & z^{b^i_3} & \ldots & z^{b^i_{s_i}}
\end{pmatrix}
\end{equation}
where the columns are numbered by the elements of $B_i$ and $x_1,x_2,x_3$ are replaced by $x,y,z$.
Moreover, we have
$$\mathop{\rm codim}\nolimits \mathop{\rm Im}\nolimits \psi^i_x=\mathop{\rm cork}\nolimits M_i,$$
where $M_i$ is some matrix depending on $x_1,\ldots,x_k$.
\end{rem}
\begin{rem}
Now consider the subsets
$$\widehat S^i_{n_i} = \bigcup_{m_i \geqslant n_i} S^i_{m_i} = \left\{ (x_1, \ldots, x_k) \subset Z_k \,|\, \mathop{\rm codim}\nolimits \mathop{\rm Im}\nolimits \psi^i_x \geqslant n_i \right\}.$$
By the previous remark,
$$\widehat S^i_{n_i}= \left\{ (x_1, \ldots, x_k) \subset Z_k \,|\, \mathop{\rm cork}\nolimits M_i \geqslant n_i \right\},$$
thus they are closed. Consequently, the strata $S^i_{n_i}$ and $S_{n_1, n_2}=S^1_{n_1} \cap S^2_{n_2}$ are locally closed.
\end{rem}
Thus theorem \ref{theorcodimNpre} reduces to estimating the codimension of $S_{n_1,n_2}$.
Our interest in these strata is due to the following lemma:
\begin{lemma}\label{lemcodimS}
If for each $S_{n_1, n_2}$ the codimension (of every its irreducible component) is at least $n_1 + n_2$, then the codimension of (every irreducible component of) $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$ is at least the expected codimension $\sum_{m=1}^k(j^1_m+j^2_m-1)$.
\end{lemma}
Typically, we would have a simple set of conditions implying that $\mathop{\rm codim}\nolimits S^1_{n_1} \geqslant n_1$ and $\mathop{\rm codim}\nolimits S^2_{n_2} \geqslant n_2$, and the difficult part would be to estimate the codimension of their intersection.
\begin{proof}
Consider the subset $\widetilde N=\widetilde N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$ consisting of $(f_1;\ f_2; x_1, \ldots, x_k) \in \mathbb C^{B_1\times} \times \mathbb C^{B_2\times} \times Z_k$ such that for $m=1,\ldots,k$ holds $\mathop{\rm ord}\nolimits_{x_m}(f_1) \geqslant j^1_m$ and $\mathop{\rm ord}\nolimits_{x_m}(f_2) \geqslant j^2_m$.
The image of $\widetilde N$ under the projection $p: \mathbb C^{B_1\times} \times \mathbb C^{B_2\times} \times Z_k \to \mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$ is exactly $N=N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$, and the preimage of any point of $N$ consists of a finite number of points. We have that $\mathop{\rm \bold d}\nolimitsim\ (\mathbb C^{B_1\times} \times \mathbb C^{B_2\times} \times Z_k) = \mathop{\rm \bold d}\nolimitsim\ (\mathbb C^{B_1\times} \times \mathbb C^{B_2\times}) + k$. Thus if we prove that the codimension of (every irreducible component of) $\widetilde N$ is at least $\sum_{m=1}^k(j^1_m+j^2_m)$, we will prove that the codimension of (every irreducible component of) $N$ is at least the expected codimension $\sum_{m=1}^k(j^1_m+j^2_m-1)$.
Now consider the projection $q: \mathbb C^{B_1\times} \times \mathbb C^{B_2\times} \times Z_k \to Z_k$. The fiber over the point $(x_1, \ldots, x_k) \in Z_k$ is a subset in $\mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$ of all pairs of non-zero polynomials $(f_1, f_2)$ such that at the point $x_m$ the polynomial $f_1$ has a root of multiplicity at least $j^1_m$ and the polynomial $f_2$ has a root of multiplicity at least $j^2_m$. It is the direct product of the spaces
$$\ker \psi^1_x \setminus \{0\} = \left\{ f_1 \in \mathbb C^{B_1\times} \,|\, \mathop{\rm ord}\nolimits_{x_m} f_1 \geqslant j^1_m \right\} \text{ and } \ker \psi^2_x \setminus \{0\} = \left\{ f_2 \in \mathbb C^{B_2\times} \,|\, \mathop{\rm ord}\nolimits_{x_m} f_2 \geqslant j^2_m \right\}.$$
Let us notice that
$$\mathop{\rm codim}\nolimits (\ker \psi^i_x \setminus \{0\}) \geqslant \mathop{\rm codim}\nolimits \ker \psi^i_x=\mathop{\rm \bold d}\nolimitsim \mathop{\rm Im}\nolimits \psi^i_x=\sum_{m=1}^{k} j^i_m-\mathop{\rm codim}\nolimits \mathop{\rm Im}\nolimits \psi^i_x=\sum_{m=1}^{k} j^i_m-n_i,$$
thus for the point $(x_1, \ldots, x_k) \in S_{n_1, n_2} \subset Z_k$ the codimension of the fiber
$$q^{-1}(x_1, \ldots, x_k)=(\ker \psi^1_x \setminus \{0\}) \times (\ker \psi^2_x \setminus \{0\}) \subset \mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$$
is at least $\sum_{m=1}^{k} (j^1_m+j^2_m) - n_1 - n_2$.
If the codimension of (every irreducible component of) $S_{n_1,n_2}$ is at least $n_1+n_2$, then the codimension of (every irreducible component of) $\widetilde N \cap q^{-1}(S_{n_1,n_2})$ is at least $\sum_{m=1}^{k} (j^1_m+j^2_m)$. Consequently, the codimension of (every irreducible component of) $\widetilde N$ is at least $\sum_{m=1}^{k} (j^1_m+j^2_m)$, and the codimension of (every irreducible component of) $N$ is at least $\sum_{m=1}^{k} (j^1_m+j^2_m-1)$, as necessary.
\end{proof}
\subsection{The stratification for one polynomial}
Let us fix $i=1$ or $2$ and first consider only one of the stratifications $S^i_{n_i}$. We will take a subset $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right) \subset \mathbb C^{B_1\times} \times \mathbb C^{B_2\times}$ and study the corresponding stratifications $Z_k = \bigsqcup_{n_i} S^i_{n_i}$ for small $k$.
We wonder how the structure of $Z_k = \bigsqcup_{n_i} S^i_{n_i}$ depends on $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$ and $B_i$. It depends only on the coefficients $j^i_1,\ldots,j^i_k$ for the fixed $i$, not on the whole $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$, thus we will simply write "for $(j^i_1,\ldots,j^i_k)$" instead of "for $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$".
To simplify the notations, we need a following definition:
\begin{defin}
For $B_i=\{ b^i_1, \ldots, b^i_k \}$ let us define
$$\phi(B_i) := (b^i_2-b^i_1,\ b^i_3-b^i_2,\ \ldots,\ b^i_k-b^i_{k-1})$$
to be the greatest common divisor of the differences.
\end{defin}
Thus the condition (1) of the main theorem \ref{theormain} can be written as $(\phi(B_1), \phi(B_2))\geqslant 2$, where $(\phi(B_1), \phi(B_2))$ is the greatest common divisor of $\phi(B_1)$ and $\phi(B_2)$.
First we will study the case of $(1,1,1)$. Here we use the matrices $M_i$ from Formula (\ref{eqM}) of Remark \ref{remM}.
\begin{lemma}\label{lemcork1simple}
The subset $S^i_2$ for $(1,1,1)$ is non-empty if and only if $\phi(B_i)\geqslant 3$.
Namely, all such triples $(x,y,z)$ that $\mathop{\rm cork}\nolimits M_i(x,y,z)=2$ have the form $(c,ct,cu)$, where $t$ and $u$ are two $\phi(B_i)$-th roots of unity such that $t \neq u$, $t \neq 1$ and $u \neq 1$.
In particular, if $S^i_2$ is non-empty, it has codimension 2.
\end{lemma}
\begin{proof}
If $c$ is a non-zero number, then the matrices $M_i(x,y,z)$ and $M_i(cx,cy,cz)$ differ by the multiplication of their $j$-th columns on non-zero numbers $c^{b_j^i}$, thus their ranks are the same and $(x,y,z) \in S^i_{n_i}$ if and and only if $(cx,cy,cz) \in S^i_{n_i}$.
Thus we can replace $M_i(x,y,z)$ with $M_i(1,t,u)$, where $t=y/x$ and $u=z/x$. The matrix
$$M_i(1,t,u)=\begin{pmatrix}
1 & 1 & 1 & ... & 1 \\
t^{b^i_1} & t^{b^i_2} & t^{b^i_3} & ... & t^{b^i_{s_i}} \\
u^{b^i_1} & u^{b^i_2} & u^{b^i_3} & ... & u^{b^i_{s_i}}
\end{pmatrix}$$
has corank 2 if and only if all its $2 \times 2$-minors are degenerate, thus
$$t^{b_1^i}=t^{b_2^i}=t^{b_3^i}=\ldots=t^{b_{s_i}^i} \quad \text{ and } \quad u^{b_1^i}=u^{b_2^i}=u^{b_3^i}=\ldots=u^{b_{s_i}^i}.$$
But $t \neq 0$, thus $t$ is a $k_1$-th root of unity for some $k_1$ and all $(b^i_q-b^i_p)$ should be divisible by $k_1$. Similarly, $u$ is a $k_2$-th root of unity for some $k_2$ and all $(b^i_q-b^i_p)$ should be divisible by $k_2$. Now $t$ and $u$ are both $k$-th roots of unity for the greatest common divisor $k=(k_1, k_2)$, and all $(b^i_q-b^i_p)$ should be divisible by $k$. Moreover, $t \neq u$, $t \neq 1$ and $u \neq 1$, thus there are at least three different $k$-th roots of unity, thus $k \geqslant 3$. To sum it up, $\phi(B_i) \geqslant 3$.
Vice versa, if $\phi(B) = k \geqslant 3$, then we can choose three different $k$-th roots of unity, namely 1, $t$ and $u$, and the matrix $M_i(1,t,u)$ will be of the corank 2.
\end{proof}
\begin{sledst}\label{corcodim111}
Consider a decomposition $Z_3=S^i_0 \cup S^i_1 \cup S^i_2$ for $(1,1,1)$:
\begin{itemize}
\item If $|B_i| \geqslant 3$, then holds $Z_3=\begin{cases}
S^i_0 \cup S^i_1 \cup S^i_2 \text{ with } \mathop{\rm codim}\nolimits S^i_1 \geqslant 1 \\
\quad \quad \quad \quad \quad \text { and } S^i_2 \text{ non-empty of codim 2 if } \phi(B_i) \geqslant 3 \\
S^i_0 \cup S^i_1 \text{ with } \mathop{\rm codim}\nolimits S^i_1 \geqslant 1 \text{ otherwise }
\end{cases}$
\item If $|B_i|=2$, then holds $Z_3=\begin{cases}
S^i_1 \cup S^i_2 \text{ with } S^i_2 \text{ non-empty of codim 2 if } \phi(B_i) \geqslant 3 \\
S^i_1 \text { otherwise }
\end{cases}$
\end{itemize}
\end{sledst}
\begin{proof}
The subset $S^i_0$ consists of $(x,y,z)$ such that the matrix $M_i(x,y,z)$ from Formula (\ref{eqM}) of Remark \ref{remM} has a non-degenerate $3 \times 3$-minor.
Suppose that $|B_i|\geqslant 3$: then the subset $S^i_0$ is open. Its complement $S^i_1 \cup S^i_2$ is a subset of codimension at least 1, but by Lemma \ref{lemcork1simple} the subset $S^i_2$ is of codimension 2 (or empty), thus $S^i_1$ is also of codimension at least 1. Moreover, by Lemma \ref{lemcork1simple}, the subset $S^i_2$ is non-empty if and only if $\phi(B_i)\geqslant 3$.
Now suppose that $|B_i|=2$: then there is no $3 \times 3$-minors in $M_i$, thus the subset $S^i_0$ is empty. Thus $Z_3=S^i_1 \cup S^i_2$, but $S^i_2$ is a subset of codimension 2 (or empty), thus now $S^i_1$ is open.
\end{proof}
Now let us consider the other filtration subsets $N(j_1,\ldots,j_k)$ of expected codimension 1, 2 or 3. The other cases are proven relatively straightforward compared to $N(1,1,1)$, which is why we devoted a separate statement to $N(1,1,1)$. We have a following lemma (cf. \cite{N19}*{Th. 1.1}):
\begin{lemma}\label{lemcork1}
i) For $(k)$ there are following equations:
\begin{itemize}
\item If $|B_i| \geqslant k$, then holds $Z_1=S^i_0$.
\item If $|B_i| \leqslant k-1$, then holds $Z_1=S^i_{k-|B_i|}$
\end{itemize}
ii) For $(k,1)$ there are following decompositions:
\begin{itemize}
\item If $|B_i|\geqslant k+1$, then holds $Z_2=S^i_0 \cup S^i_1$ with $\mathop{\rm codim}\nolimits S^i_1 \geqslant 1$
\item If $|B_i|\leqslant k$, then holds $Z_2=S^i_1$
\end{itemize}
Moreover, for $(1,1)$ we have that $S^i_1$ is non-empty if and only if $\phi(B_i) \geqslant 2$.
\end{lemma}
In particular, this lemma covers the cases of $N(1)$, $N(2)$ and $N(1,1)$, but it also covers the case of $N\left(\,^{2,1}_{1,1}\right)$.
\subsection{The stratification for two polynomials}
Now let us fix a subset $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right) \subset \mathbb C^{B_1} \times \mathbb C^{B_2}$ and consider the stratification $Z_k = \bigsqcup_{n_1,n_2} S_{n_1,n_2}$ with $S_{n_1, n_2}=S^1_{n_1} \cap S^2_{n_2}$. We wonder how its structure depends on $N\left(\,^{j^1_1,\ldots,j^1_k}_{j^2_1,\ldots,j^2_k}\right)$ and $(B_1, B_2)$.
\begin{lemma}\label{lemcork2simple}
Suppose that $|B_1|\geqslant 3$ and $|B_2|\geqslant 3$. For $N(1,1,1)$ there is a decomposition
$$Z_3=\bigsqcup_{\genfrac{}{}{0pt}{2}{0\leqslant i\leqslant 2}{0\leqslant j\leqslant 2}} S_{ij}$$
such that
\begin{itemize}
\item $S_{00}$ is open,
\item $\mathop{\rm codim}\nolimits S_{01}$ and $\mathop{\rm codim}\nolimits S_{10} \geqslant 1$,
\item $\mathop{\rm codim}\nolimits S_{02}$ and $\mathop{\rm codim}\nolimits S_{20} \geqslant 2$,
\item if $B_1 \setminus \{\max(B_1)\}$ and $B_2 \setminus \{\max(B_2)\}$ can \textbf{not} be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$, and $B_1 \setminus \{\min(B_1)\}$ and $B_2 \setminus \{\min(B_2)\}$ can \textbf{not} be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$, then $\mathop{\rm codim}\nolimits S_{11} \geqslant 2$,
\item if $B_1$ can be split into $B'\sqcup B''$ so that $B'$, $B''$ and $B_2$ can be shifted to the same sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 3$, but $B_1$ and $B_2$ can \textbf{not} be shifted to the same sublattice $k\mathbb Z$, then $\mathop{\rm codim}\nolimits S_{12} = 2$ and it is non-empty, otherwise it is empty (and symmetrically with $S_{21}$),
\item if $B_1$ and $B_2$ can be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 3$, then $\mathop{\rm codim}\nolimits S_{22} = 2$ and it is non-empty, otherwise it is empty,
\end{itemize}
and all the strata can be empty if not mentioned otherwise.
\end{lemma}
To simplify the proof, we can use homogeneity considerations, that is replace
$$Z_3=\left\{ (x, y, z) \subset (\mathbb C^\times)^3 \,|\, x \neq y \neq z \neq x \right\}$$
with
$$\mathbb P Z_3=\left\{ (1:t:u) \subset \mathbb CP^2 \,|\, t,u \neq 0 \text{ and } t \neq u, t \neq 1, u \neq 1 \right\},$$
and replace the strata $S^i_{n_i}$ (and $S_{n_1,n_2}$) with their projectivizations $\mathbb P S^i_{n_i}$ (and $\mathbb P S_{n_1,n_2}$ respectively), having the same codimensions of components.
\begin{proof}
The cases of $S_{00}$, $S_{01}$, $S_{10}$, $S_{02}$ and $S_{20}$ follow from Corollary \ref{corcodim111}.
The case of $S_{11}=S_1^1 \cap S_1^2$
follows from Lemma \ref{lemstratumranks11} of Section \ref{secstrats11forn111}. Indeed, by Lemma \ref{lemcork1simple} all the irreducible components of $\mathbb P\widehat S_1^1=\mathbb P S_1^1 \ \cup \ \mathbb P S_2^1$ and of $\mathbb P\widehat S_1^2$ have codimension at least 1. If $\mathbb P S_1^1$ has an irreducible component of codimension 1, then $\mathbb P\widehat S_{11} = \mathbb P\widehat S_1^1 \cap \mathbb P\widehat S_1^2 = \mathbb P S_{11} \cup \mathbb P S_{12} \cup \mathbb P S_{21} \cup \mathbb P S_{22}$ has an irreducible component $W \subset \mathbb P Z_3$ of codimension 1. The subset $\mathbb P\widehat S_{11} \subset \mathbb P Z_3$ is defined by $3 \times 3$-minors of $M_1$ and $M_2$, that is by the polynomials
$$\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}(t,u)=\begin{pmatrix}
1 & 1 & 1 \\
t^a & t^b & t^c\\
u^a & u^b & u^c
\end{pmatrix},$$
where we take the triples of elements $\{a, b, c\} \subset B_1$ or $\{a, b, c\} \subset B_2$ such that $a < b < c$. Let $S$ be a set of such polynomials.
We would like apply Lemma \ref{lemstratumranks11} to $S$, where $f_{a,b,c}(t,u)\equiv 0$ for each $\mathop{\rm \bold d}\nolimitset_{a,b,c}+f_{a,b,c} \in S$. The polynomial $GCD(S)$ vanishes on $\mathbb P\widehat S_{11} \subset \mathbb P Z_3$, where $\mathbb P Z_3$ is a complement of $V(tu(t-1)(u-1)(t-u)) \subset \mathbb C^2$, and $\mathbb P\widehat S_{11}$ is non-empty, thus $GCD(S)$ does not divide a power of $tu(t-1)(u-1)(t-u)$.
Suppose that the first alternative of Lemma \ref{lemstratumranks11} holds: $k=GCD(\{b-a \,|\, \mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c} \in S\}) > 1$. Then all the differences $(b^1_i-b^1_j)$ where $b^1_i \neq \max B_1$ and $b^1_j \neq \max B_1$ are divisible by $k$, and all the differences $(b^2_i-b^2_j)$ where $b^2_i \neq \max B_2$ and $b^2_j \neq \max B_2$ are divisible by $k$, thus $B_1 \setminus \{\max(B_1)\}$ and $B_2 \setminus \{\max(B_2)\}$ can be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$.
Suppose that the second alternative of Lemma \ref{lemstratumranks11} holds: similarly, $B_1 \setminus \{\min(B_1)\}$ and $B_2 \setminus \{\min(B_2)\}$ can be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$.
The case of $S_{12}=S_1^1 \cap S_2^2$ (and similarly $S_{21}=S_2^1 \cap S_1^2$)
is the most difficult part of the lemma. It follows from Theorem \ref{theorstratumranks21for111} of Section \ref{secstrats12forn111}. Indeed, by Lemma \ref{lemcork1simple} the codimension of $S^2_2$ is 2 (maybe empty), thus the codimension of $S_{12}$ is at least 2, while it is at most 2, because it is equal to the codimension of $\mathbb P S^2_2$. Thus we should check the conditions when $S_{12}$ is non-empty.
If $S_{12}$ is non-empty, then we can take $(1:t:u) \in \mathbb P S_{12}$ and apply Lemma \ref{lemcork1simple}. As $(1:t:u) \in \mathbb P S_2^2$, then $t$ and $u$ are $n$-th roots of unity, where $n=\phi(B_2)$. Moreover, $(1:t:u) \in \mathbb P \widehat S_1^1$, thus we can apply Theorem \ref{theorstratumranks21for111} to $x=t$, $y=u$ and $B=B_1$ and get that $B_1$ can be split into $B'\sqcup B''$ so that $\phi(B')$, $\phi(B'')$ and $n=\phi(B_2)$ have a common divisor $k\geqslant 3$. It means exactly that $B'$, $B''$ and $B_2$ can be shifted to the same sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 3$, as necessary. Moreover, suppose that $B_1$ and $B_2$ can be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$. Then $(1:t:u) \in \mathbb P S_{22}$, thus $(1:t:u) \not\in \mathbb P S_{12}$, a contradiction.
Vice versa, if $B_1$ can be split into $B'\sqcup B''$ so that $B'$, $B''$ and $B_2$ can be shifted to the same sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 3$, then we can choose three different $k$-th roots of unity, namely 1, $t$ and $u$. As $k$ divides $\phi(B_1)$, we have that $(1:t:u) \in \mathbb P S^2_2$ by Lemma \ref{lemcork1simple}. The subset $\mathbb P \widehat S_1^1$ is defined by $3 \times 3$-minors of $M_1$, that is by the polynomials
$$\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}(t,u)=\begin{pmatrix}
1 & 1 & 1 \\
t^a & t^b & t^c\\
u^a & u^b & u^c
\end{pmatrix},$$
where we take the triples of elements $\{a, b, c\} \subset B_1$. As $B_1 = B'\sqcup B''$, there are two of $a$, $b$ and $c$ from the same subset. Without loss of generality we can suppose that $a$ and $b \in B'$. As $k$ divides $\phi(B')$, we have that $t^a=t^b$ and $u^a=u^b$, thus the $3 \times 3$-minor $\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}$ has two equal columns and is zero. To sum it up, $(1:t:u) \in \mathbb P \widehat S^1_1$ and $(1:t:u)\in \mathbb P \widehat S_{12}$, thus $\widehat S_{12}=S_{12} \cup S_{22}$ is non-empty.
Moreover, we have the condition that $B_1$ and $B_2$ can not be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$. Let $l$ be the largest number such that $B_1$ and $B_2$ can be shifted to the same proper sublattice $l\mathbb Z\subset\mathbb Z$. As $B_1=B'\sqcup B''$, we have that $k$ is divisible by $l$. Moreover, $k > l$, thus we can choose $t$ and $u$ in such a way that they are not both $l$-th roots of unity. It means that $(1:t:u)$ lies not in $S_{22}$, but in $S_{12}$, and thus $S_{12}$ is non-empty.
The case of $S_{22}=S_2^1 \cap S_2^2$
is proven as follows. By Lemma \ref{lemcork1simple}, $\mathbb P S_2^i$ consists of $(1:t:u)$, where $t$ and $u$ are two $\phi(B_i)$-th roots of unity such that $t \neq u$, $t \neq 1$ and $u \neq 1$. Thus if $(1:t:u) \in \mathbb P S_{22}=\mathbb P S_2^1 \cap \mathbb P S_2^2$, then $t$ and $u$ are $\phi(B_1)$-th roots of unity and $\phi(B_2)$-th roots of unity, thus they are $k=(\phi(B_1),\phi(B_2))$-th roots of unity. Let us recall that $t \neq 1$, $u \neq 1$ and $t \neq u$, thus $k \geqslant 3$. In other words, $B_1$ and $B_2$ can be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 3$. Both $\mathbb P S_2^1$ and $\mathbb P S_2^2$ has codimension 2 in the space $\mathbb P Z_3$ of dimension 2, thus $\mathbb P S_{22}$ is also of codimension 2.
Vice versa, if $B_1$ and $B_2$ can be shifted to the proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 3$, we can take a number $t$ which is not equal to 1 and is a $k$-th root of unity, then $(1:t) \in \mathbb P S_{11}$ and thus $S_{11}$ is non-empty.
\end{proof}
Now we can prove the most difficult part of Theorem \ref{theorcodimNpre}.
\begin{sledst}\label{codimNsimple}
Under the conditions of the part (i) of the main theorem \ref{theormain}, the codimension of (every irreducible component of) $N(1,1,1)$ is at least the expected codimension 3.
\end{sledst}
\begin{proof}
By the condition (5) of the main theorem, we have that either $|B_1|=2$ and $\max B_2-\min B_2\leqslant 2$ (in which case $N(1,1,1)$ is empty), or $|B_1|>2$, and the same with $B_2$.
Thus we can use Lemma \ref{lemcork2simple} and prove that $\mathop{\rm codim}\nolimits S_{n_1,n_2}\geqslant n_1+n_2$. Indeed, the condition for $S_{11}$ follows from the conditions (3) and (4) of the main theorem, while the condition for $S_{12}$ and $S_{21}$ follows from the condition (2) of the main theorem.
Now by Lemma \ref{lemcodimS} we have $\mathop{\rm codim}\nolimits N(1,1,1) \geqslant 3$.
\end{proof}
Now let us consider the other filtration subsets $N(j_1,\ldots,j_k)$ of expected codimension 1, 2 or 3.
\begin{lemma}\label{lemcork2}
There are following decompositions:
\begin{itemize}
\item For $N(k)$ holds $Z_1=S_{00}$.
\item For $N(1,1)$ holds $Z_2=S_{00} \cup S_{01} \cup S_{10} \cup S_{11}$, where
\begin{enumerate}
\item $\mathop{\rm codim}\nolimits S_{01} = \mathop{\rm codim}\nolimits S_{10} = 1$ (and they may be empty),
\item if $B_1$ and $B_2$ can be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$, then $\mathop{\rm codim}\nolimits S_{11} = 1$ and it is non-empty, otherwise it is empty.
\end{enumerate}
\item If $|B_1|>2$, then for $N\left(\,^{2,1}_{1,1}\right)$ holds $Z_3=S_{00} \cup S_{01} \cup S_{10} \cup S_{11}$, where
\begin{enumerate}
\item $\mathop{\rm codim}\nolimits S_{01}$ and $\mathop{\rm codim}\nolimits S_{10} \geqslant 1$ (and they may be empty),
\item if $B_1$ and $B_2$ can be shifted to the same proper sublattice $k\mathbb Z\subset\mathbb Z$ with $k\geqslant 2$, then $\mathop{\rm codim}\nolimits S_{11} = 1$ and it is non-empty, otherwise it is empty.
\end{enumerate}
\item If $|B_1|=2$, then for $N\left(\,^{2,1}_{1,1}\right)$ holds $Z_3= S_{10} \cup S_{11}$, where $\mathop{\rm codim}\nolimits S_{11}=1$.
\end{itemize}
\end{lemma}
Here we list those symmetric filtration subsets $N(j_1,\ldots,j_k)$ whose expected codimension is 1, 2 or 3, and also $N\left(\,^{2,1}_{1,1}\right)$. We write $S_{ij}$ instead of $S_{i,j}$ for shortness.
\begin{proof}
Most of the cases trivially follow from Lemmas \ref{lemcork1simple} and \ref{lemcork1}.
The case of $S_{11}=S_2^1 \cap S_2^1$ for $N(1,1)$ can be proven similarly to the case of $S_{22}$ for $N(1,1,1)$.
The case of $S_{11}=S_1^1 \cap S_1^2$ for $N\left(\,^{2,1}_{1,1}\right)$ follows from Lemma \ref{lem3minor2} similarly to how the case of $S_{11}$ for $N(1,1,1)$ follows from Lemma \ref{lem3minor}.
\end{proof}
Now we can formulate Theorem \ref{theorcodimNpre} in full detail:
\begin{theor}\label{theorcodimN}
The following strata have codimensions (of every their irreducible components) more then or equal to their expected codimensions unless the following conditions of the main theorem \ref{theormain} hold:
\begin{itemize}
\item $\mathop{\rm codim}\nolimits N(1) \geqslant 1$ always.
\item $\mathop{\rm codim}\nolimits N(1,1) \geqslant 2$ unless (1).
\item $\mathop{\rm codim}\nolimits N^0_1 \geqslant 2$ always.
\item $\mathop{\rm codim}\nolimits N^1_0 \geqslant 2$ always.
\item $\mathop{\rm codim}\nolimits N(2) \geqslant 3$ always.
\item $\mathop{\rm codim}\nolimits N(1,1,1) \geqslant 3$ unless (1)-(5).
\item $\mathop{\rm codim}\nolimits N_0^1(1) \geqslant 3$ always.
\item $\mathop{\rm codim}\nolimits N_1^0(1) \geqslant 3$ always.
\item $\mathop{\rm codim}\nolimits N\left(\,^{2,1}_{1,1}\right) \geqslant 3$ unless (1).
\item $\mathop{\rm codim}\nolimits N\left(\,^{1,1}_{2,1}\right) \geqslant 3$ unless (1).
\end{itemize}
\end{theor}
It is deduced from Lemmas \ref{lemcork2simple} and \ref{lemcork2} directly case-by-case.
\section{The stratum S(1,1) of the solution space for the filtration subset N(1,1,1)}\label{secstrats11forn111}
In this section we will study the stratum $S_{1,1}$ for $N(1,1,1)$, although without refering to it as such. This section is independent from the other parts of the text.
We will work inside the space
$$\mathbb P Z_3=\left\{ (t, u) \subset (\mathbb C^\times)^2 \,|\, t \neq 1, \ u \neq 1, \ t \neq u \right\}.$$
\begin{defin}
Consider a triple of integers $a<b<c$. It defines a (Laurent) polynomial
$$\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}(t,u)=\mathop{\rm \bold d}\nolimitset \begin{pmatrix}
1 & 1 & 1 \\
t^a & t^b & t^c\\
u^a & u^b & u^c
\end{pmatrix}$$
for $a < b < c$. Its support is a hexagon $N_{det_{a,b,c}}$ with vertices $(a,b)$, $(a,c)$, $(b,c)$, $(c,b)$, $(c,a)$ and $(b,a)$, see Figure \ref{fignewtdet}.
Now take a polynomial $f_{a,b,c}(t,u)$ such that its support lies inside $N_{det_{a,b,c}}$ and such that it is divisible by $det_{0,1,2}$. We will call such a polynomial \textit{good} polynomial.
\end{defin}
\begin{lemma}\label{lemstratumranks11}
Let $S$ be a (not necessary finite) set of good polynomials $\mathop{\rm \bold d}\nolimitset_{a,b,c}+f_{a,b,c}$. Suppose that their greatest common divisor $GCD(S)$ does not divide a power of $tu(t-1)(u-1)(t-u)$. Then at least one of the numbers
$$GCD(\{b-a \,|\, \mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c} \in S\}) \quad \text{ and } \quad GCD(\{c-b \,|\, \mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c} \in S\})$$
is greater than 1.
\end{lemma}
\begin{proof}
Let us decompose $GCD(S)$ as $g(t,u) \cdot (t-1)^\alpha \cdot (u-1)^\beta \cdot (t-u)^\gamma$, where $g(t,u)$ is not divisible by $(t-1)$, $(u-1)$ or $(t-u)$. Here we will consider divisibility in the ring of (Laurent) polynomials.
For a polynomial $p(t,u)=\sum p_{ij}t^iu^j$ its Newton polygon $N_p \subset \mathbb Z^2$ is defined as the convex hull of all $(i,j) \in \mathbb Z^2$ such that $p_{ij} \neq 0$. By the conditions of the lemma, $g(t,u)$ is not a monomial, thus $N_g$ has a non-trivial edge $v=(v_1, v_2)$. We will use this edge to find the number $k>1$ which divides all the necessary $(b-a)$ or $(c-b)$.
Suppose that $\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c} \in S$. It is divisible by $GCD(S)$ and, moreover, by $g$. The polynomial $\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c}$ is divisible by $\mathop{\rm \bold d}\nolimitset\nolimits_{0,1,2}$, while $g$ is coprime with it, thus $(\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c}) / \mathop{\rm \bold d}\nolimitset\nolimits_{0,1,2}$ is divisible by $g$, that is
$$(\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c})/\mathop{\rm \bold d}\nolimitset\nolimits_{0,1,2} = g \cdot h,$$
where $h(t,u)$ is a (Laurent) polynomial. By the properties of the Newton polygon holds
$$N_{\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c}} = N_g + N_h + N_{\mathop{\rm \bold d}\nolimitset\nolimits_{0,1,2}},$$
where $+$ is the Minkowski sum $A+B=\{a+b: a\in A, b\in B\}$.
The Minkowski sum of two convex polygons in the plane can be computed as follows. Let us orientate the edges of each polygon counterclockwise. Then the edges of the Minkowski sum are sums of the parallel edges of the two convex polygons going in the same direction. Thus the edge $v$ is a part of some edge of $N_{\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c}}$.
The polynomial
$$\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}(t,u)+f_{a,b,c}(t,u)=t^au^b-t^bu^a+t^bu^c-t^cu^b+t^cu^a-t^au^c+f_{a,b,c}(t,u)$$
has Newton polytope as shown on Figure \ref{fignewtdet}, and its edges oriented counterclockwise are proportional with positive coefficients to the vectors $(1,0)$, $(0,1)$, $(-1,1)$, $(-1,0)$, $(0,-1)$ and $(1,-1)$.
\begin{figure}
\caption{The Newton polytope of $\mathop{\rm \bold d}
\label{fignewtdet}
\end{figure}
Let us now consider a weighted degree function $\mathop{\rm \bold d}\nolimitseg_v(t^iu^j)=v_2 \cdot i - v_1 \cdot j$, for which the largest monomials of $N_g$ are the monomials that lie in the edge $v$. For a polynomial $p(t,u)=\sum p_{ij}t^iu^j$ let its degree with respect to $v$ be $\mathop{\rm \bold d}\nolimitseg_v(p)=\max_{p_{ij} \neq 0} \mathop{\rm \bold d}\nolimitseg_v(t^iu^j)$, and let its highest part with respect to $v$ be
$$p^v(t,u)=\sum_{\mathop{\rm \bold d}\nolimitseg_v(t^iu^j)=\mathop{\rm \bold d}\nolimitseg_v(p)} p_{ij}t^iu^j.$$
In particular, $g^v(t,u)=\sum_{(i,j) \in v} g_{ij}t^iu^j$, that is the highest part of $g(t,u)$ consists exactly of the monomials of $v$.
The highest part is multiplicative, thus
$$(\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}(t,u)+f_{a,b,c}(t,u))^v = g^v(t,u) \cdot h_1^v(t,u) \cdot \mathop{\rm \bold d}\nolimitset\nolimits_{0,1,2}^v(t,u).$$
If $v$ is proportional with positive coefficient to $(1,0)$, then $$(\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}(t,u)+f_{a,b,c}(t,u))^v=-t^bu^a+t^cu^a=t^bu^a(t^{c-b}-1) \text{ and } \mathop{\rm \bold d}\nolimitset\nolimits^v_{0,1,2}(t,u)=tu(t-1),$$
thus we have that $(t^{c-b}-1)/(t-1)$ is divisible by $g^v(t,u)$ (in the ring of Laurent polynomials).
We can now take the smallest natural number $k$ such that $(t^k-1)/(t-1)$ is divisible by $g^v(t,u)$ and denote it by $k$. The edge $v$ is non-empty, thus $g^v(t,u)$ is not a monomial and thus is non-invertible in the ring of Laurent polynomials, therefore $k \geqslant 2$.
The greatest common divisor of $(t^\alpha-1)$ and $(t^\beta-1)$ is $(t^{(\alpha, \beta)}-1)$. If for some $a$, $b$ and $c$ we have $\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c} \in S$, then the polynomial $(t^{c-b}-1)/(t-1)$ is divisible by $g^v(t,u)$, thus $(t^{(c-b,k)}-1)/(t-1)$ is also divisible by $g^v(t,u)$, thus $(c-b,k) \geqslant k$, therefore $(c-b)$ is divisible by $k$. Consequently, $GCD(\{c-b \,|\, \mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c} \in S\}) > 1$.
Now if $v$ is proportional with positive coefficient to $(-1,0)$, then $$(\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}(t,u)+f_{a,b,c}(t,u))^v=t^bu^c-t^au^c=t^au^c(t^{b-a}-1).$$
We can similarly define $k$ as the smallest natural number such that $(t^k-1)/(t-1)$ is divisible by $g^v(t,u)$, and similarly prove that if for some $a$, $b$ and $c$ we have $\mathop{\rm \bold d}\nolimitset\nolimits_{a,b,c}+f_{a,b,c} \in S$, then $(b-a)$ is divisible by $k$.
The other four cases can be proven similarly.
\end{proof}
\section{The strata S(1,2) and S(2,1) of the solution space for the filtration subset N(1,1,1)}\label{secstrats12forn111}
In this section we will study the stratum $S_{1,2}$ for $N(1,1,1)$, although without refering to it as such. This section is independent from the other parts of the text.
\begin{defin}
For a finite collection of integer numbers $n_1, n_2, \ldots, n_k$ let us define the function
$$\phi(n_1, n_2, \ldots, n_k) := (n_2-n_1, n_3-n_2, \ldots, n_k-n_{k-1})$$
to be the greatest common divisor of their differences.
\end{defin}
\begin{theor}\label{theorstratumranks21for111}
Fix $n \in \mathbb Z$. Let $x$ and $y$ be two $n$-th roots of unity, such that $x \neq 1$, $y \neq 1$, $x \neq y$. Let $B=\{ a, b, c, d, \ldots\} \subset \mathbb Z$ be a finite set of integer numbers. Consider a matrix
$$M(B;x,y) = \begin{pmatrix}
1 & 1 & 1 & 1 & ...\\
x^a & x^b & x^c & x^d & ...\\
y^a & y^b & y^c & y^d & ...
\end{pmatrix},$$
where the columns are parametrized by $B$. If all the $3 \times 3$-minors of $M(B,x,y)$ are degenerate, then $B$ can be split into $B=B' \sqcup B''$ such that $\phi(B')$, $\phi(B'')$ and $n$ have a common divisor $k \geqslant 3$.
\end{theor}
This section is devoted to the proof of this theorem.
\subsection{Lemmas on 3x3-minors}
First, let us consider one $3 \times 3$-minor.
\begin{lemma}\label{lem3minor}
Let $x$ and $y$ be complex numbers of absolute value $1$, let $a$, $b$ and $c$ be integer numbers and consider a matrix
$$M=M(a,b,c;x,y) = \begin{pmatrix}
1 & 1 & 1\\
x^a & x^b & x^c \\
y^a & y^b & y^c
\end{pmatrix}$$
If this matrix is degenerate, then it has either two proportional rows or two proportional columns.
\end{lemma}
\begin{rem}
The determinant of this matrix is obviously related to a simplest Schur polynomial. In a forthcoming joint work with A. Voorhaar and A. Esterov, we shall use this relation in both directions: to apply known results on zero loci of Schur polynomials to a further study of singularities of resultants and discriminants, and to derive new results on zero loci of Schur polynomials from what we do here.
\end{rem}
\begin{proof}
Let us write the equation $\mathop{\rm \bold d}\nolimitset M=0$ as
$$(x^a-x^b)(y^a-y^c)=(x^a-x^c)(y^a-y^b).$$
Suppose that $x^a=x^b$. Then either $x^a=x^c$ and $M$ has two proportional rows or $y^a=y^b$ and $M$ has two proportional (even equal) columns.
As the conditions of the lemma are symmetric in $a$, $b$ and $c$, and are symmetric in $x$ and $y$, we can now suppose that the numbers $x^a$, $x^b$, $x^c$ are pairwise different and that the numbers $y^a$, $y^b$, $y^c$ are also pairwise different. Thus we can write the condition above as
$$\frac{x^a-x^b}{x^a-x^c}=\frac{y^a-y^b}{y^a-y^c} \neq 0.$$
In particular, if we consider $x^a$, $x^b$ and $x^c$ as points in the plane, the following angles are equal: $\angle(x^b,x^a,x^c)=\angle(y^b,y^a,y^c)$.
\begin{figure}
\caption{The points $x^a$, $x^b$ and $x^c$ on the unit circle}
\label{figinscrangles}
\end{figure}
But the numbers $x$ and $y$ has absolute value $1$, thus the points $x^a$, $x^b$, $x^c$, $y^a$, $y^b$, $y^c$ lie on the unit circle and these angles are the inscribed angles on the arcs $(x^b, x^c)$ and $(y^b, y^c)$. An inscribed angle is half of the central angle on the same arc (see Figure \ref{figinscrangles}), thus
$$\angle(x^b,0,x^c)=2\angle(x^b,x^a,x^c)=2\angle(y^b,y^a,y^c)=\angle(y^b,0,y^c).$$
But $|x^b|=|x^c|=|y^b|=|y^c|=1$, thus $\angle(x^b,0,x^c)=\angle(y^b,0,y^c)$ implies $x^b/x^c=y^b/y^c$. As the conditions of the lemma are symmetric in $a$, $b$ and $c$, the matrix $M$ has two proportional rows.
\end{proof}
\iffalse
\begin{rem}\label{remlem3minor}
We can prove the second part of the lemma algerbraically. Indeed, let $\overline x$ be a complex conjugate to $x$. Then
$$\frac{x^a-x^b}{x^a-x^c} \Big/ \frac{\overline x^a-\overline x^b}{\overline x^a-\overline x^c}=\frac{x^a-x^b}{\overline x^a-\overline x^b} \cdot \frac{\overline x^a-\overline x^c}{x^a-x^c}=(-x^ax^b) \cdot (-x^{-a}x^{-c})=x^b/x^c,$$
thus $x^b/x^c=y^b/y^c$.
\end{rem}
\fi
Let us also prove a similar lemma for $J(2,1)$.
\begin{lemma}\label{lem3minor2}
Let $x$ be a complex number of absolute value $1$, let $a$, $b$ and $c$ be different integer numbers and consider a matrix
$$M=M(a,b,c;x,y) = \begin{pmatrix}
1 & 1 & 1\\
a & b & c \\
x^a & x^b & x^c
\end{pmatrix}$$
If this matrix is degenerate, then it has two equal rows, namely the rows 1 and 3.
\end{lemma}
\begin{proof}
Let us write the equation $\mathop{\rm \bold d}\nolimitset M=0$ as
$$(a-b)(x^a-x^c)=(a-c)(x^a-x^b).$$
Suppose that $x^a=x^b$. Then $x^a=x^c$ and $M$ has two equal rows.
As the conditions of the lemma are symmetric in $a$, $b$ and $c$, we can now suppose that the numbers $x^a$, $x^b$, $x^c$ are pairwise different. Thus we can write the condition above as
$$\frac{x^a-x^b}{x^a-x^c}=\frac{a-b}{a-c} \neq 0.$$
The number on the right hand side is real, thus the plane vectors $x^a-x^b$ and $x^a-x^c$ are proportional. As the points $x^a$, $x^b$ and $x^c$ lie on the unit circle, it is possible only in the case $x^b=x^c$, which is a contradiction.
\end{proof}
\subsection{Proportionality of columns and rows}
\begin{lemma}\label{lemprop}
Consider a $3 \times n$-matrix $M$ with non-zero entries. Suppose that every its $3 \times 3$-minor has either two proportional rows or two proportional columns. Then either $M$ has two proportional rows or its columns can be divided into two groups of mutually proportional columns.
\end{lemma}
\begin{rem}
The lemma can be formulated in a more symmetric way. Let $M$ be a matrix with non-zero entries. Suppose that for every its $3 \times 3$-minor either its rows or its columns form at most two classes of equivalence with respect to proportionality. Then either the rows or the columns of the whole $M$ form at most two classes of equivalence with respect to proportionality. Moreover, one can prove it using the same logic even without the restriction that the dimensions of $M$ are $3\times n$.
\end{rem}
\begin{proof}
We will limit ourselves with the case when $M$ is $3 \times n$-matrix. Suppose that $M$ has three mutually non-proportional columns. Without loss of generality let them be the columns 1, 2 and 3. Consider the $3 \times 3$-minor given by these columns. By the conditions of the lemma it has two proportional rows. Without loss of generality let them be the rows 1 and 2. We would like to prove that the rows 1 and 2 of the whole $M$ are also proportional.
Consider the columns 1 and 2 which are triples of non-zero numbers. They are not proportional, while their 1-st and 2-nd entries are. Thus their 3-rd entries can not be proportional neither to their 1-st entries, nor to their 2-nd entries. Similarly, for the columns 1 and 3 (or 2 and 3) their 3-rd entries are not proportional neither to their 1-st entries, nor to their 2-nd entries.
Now consider the rows 1 and 2 of the whole $M$ and let us prove that they are proportional. Take their $k$-th elements. The column $k$ is proportional at most to one of the columns 1, 2 and 3. Without loss of generality suppose that it is not proportional neither to the column 1, nor to the column 2. Consider $3 \times 3$-minor given by the columns 1, 2 and $k$. Its columns are not proportional. The 1-st and 2-nd entries of its rows 1 and 3 are also not proportional, and the 1-st and 2-nd entries of its rows 2 and 3 are also not proportional. Thus by the conditions of the lemma the rows 1 and 2 of this minor should be proportional.
Consequently, the rows 1 and 2 of the whole $M$ are proportional.
\end{proof}
Now let us prove Theorem \ref{theorstratumranks21for111}.
\begin{proof}
By Lemma \ref{lem3minor}, the conditions of Lemma \ref{lemprop} are satisfied, thus either
$$M(B;x,y) = \begin{pmatrix}
1 & 1 & 1 & 1 & ...\\
x^a & x^b & x^c & x^d & ...\\
y^a & y^b & y^c & y^d & ...
\end{pmatrix}$$
has two proportional rows or its columns can be divided into two groups of mutually proportional columns.
It the rows 1 and 2 are proportional, then $x^a=x^b=x^c=x^d=\ldots$ and $y^a=y^b=y^c=y^d=\ldots$, thus (as in the proof of Lemma \ref{lemcork1simple} above) $x$ and $y$ are $\phi(B)$-th roots of unity. But they are also $n$-th roots of unity, thus they are $k=(\phi(B),n)$-th roots of unity. Moreover, as $x \neq y$ and they are different from 1, we have that $k \geqslant 3$. Thus we can define $B'=B$, $B''=\varnothing$.
If the columns, which are enumerated by $B$, can be divided into two groups of mutually proportional columns, then let the groups be $B'$ and $B''$. Then $x^{b_i}=x^{b_j}$ for $i, j \in B'$, thus $x$ is $\phi(B')$-th root of unity. Similarly, $x$ is $\phi(B'')$-th root of unity, and $y$ is $\phi(B')$-th and $\phi(B'')$-th root of unity. To sum it up, $x$ and $y$ are $k$-th roots of unity such that $\phi(B')$, $\phi(B'')$ and $n$ are divisible by $k$. Moreover, as $x \neq y$ and they are different from 1, we have that $k \geqslant 3$.
\end{proof}
\section{The projection of a spatial complete intersection curve}
In this section, we prove Theorem \ref{theorproj}.
Let $f_i$ be a generic linear combination of monomials from a finite subset $A_i$ of the monomial lattice $\mathbb Z^3$.
Then we have a map
$$F:\mathbb CC^2 \to \mathbb C^{B_1} \times \mathbb C^{B_2}, \quad (x,y) \mapsto (f_1(x,y,\cdot), f_2(x,y,\cdot)),$$
and the sought closure $C$ of the image of $V(f_1,f_2)$ under the coordinate projection $\mathbb CC^3\to\mathbb CC^2$ equals the preimage of the resultant $R_B \subset \mathbb C^{B_1}\times\mathbb C^{B_2}$ under this map.
This observation allows to learn about singularities of $C$ from the singularities of $R_B$, using the following fact (see e.g. \cite{E08}, \cite{K16} or Theorem 10.7 in \cite{EL}).
\begin{utver}\label{ptransv}
For an algebraic set $V\subset\mathbb CC^n$ and generic linear combinations $h_i$ of monomials from non-empty finite sets $H_i\subset\mathbb Z^k,\,i=1,\ldots,k$, the intersection of $V$ and the zero locus $h_1=\cdots=h_k=0$ is a codimension $k$ subset in $V$, and this intersection is transversal at every smooth point of $V$.
In particular, the intersection is empty if $\mathop{\rm \bold d}\nolimitsim V<k$.
\end{utver}
Let us take any $x\in C=F^{-1}(R_B)$ and study $C$ in its neighborhood, depending on (1) what part of $R_B$ the pont $F(x)$ belongs to, and (2) what coordinate plane in $\mathbb C^{B_1} \times \mathbb C^{B_2}$ it belongs to.
More specifically, for every $I\subset B_1\sqcup B_2$, consider the coordinate plane $$\mathbb C^I:=\left\{\left(\sum_{b\in B_1\cap I}c_{1,b}t^b, \sum_{b\in B_2\cap I}c_{2,b}t^b\right)\right\}\subset\mathbb C^{B_1} \times \mathbb C^{B_2}.$$
Choose the unique $I$ such that $F(x)\in\mathbb CC^I$.
Recall that, by the part (i) of the main theorem \ref{theormain}, there exists a codimension 3 subset $\Sigma\subset \mathop{\rm sing}\nolimits R_B$, such that at every point of $\mathop{\rm sing}\nolimits R_B\setminus\Sigma$ the resultant is locally the union of two transversal smooth hypersurfaces, and, in particular, $\mathop{\rm sing}\nolimits R_B$ is smooth at this point.
a) Assume $F(x)\in\mathbb CC^I\cap\Sigma$. This cannot happen by the dimension count, applying Proposition \ref{ptransv} to $\mathbb CC^n:=\mathbb CC^2\times\mathbb CC^I$ with coordinates $x_1,x_2,y_i,i\in I$, the set $V:=\mathbb CC^2\times(\mathbb CC^I\cap\Sigma)$, and the polynomials $h_i(x,y)=f_i(x)-y_i,\,i\in I$ and $h_i=f_i,\,i\notin I$.
b) Otherwise, assume $F(x)\in\mathbb CC^I\cap\mathop{\rm sing}\nolimits R_B$. Since $\mathop{\rm sing}\nolimits(\mathbb CC^I\cap\mathop{\rm sing}\nolimits R_B)$ has codimension at least 3, by the same dimension count, $F(x)$ is a regular point of $\mathbb CC^I\cap\mathop{\rm sing}\nolimits R_B$. Moreover, at this point $F$ transversally intersects $\mathbb CC^I\cap\mathop{\rm sing}\nolimits R_B$, applying Proposition \ref{ptransv} to the same $\mathbb CC^n$ and $h_i$'s and the set $V:=\mathbb CC^2\times(\mathbb CC^I\cap\mathop{\rm sing}\nolimits R_B)$. Thus $F$ transversally intersects $\mathop{\rm sing}\nolimits R_B$ at $F(x)\notin\Sigma$, thus $C=F^{-1}(R_B)$ is locally the union of two transversal smooth curves.
c) Otherwise, assume $F(x)\in\mathbb CC^I\cap R_B$. Then, by the same dimension count, $F(x)$ is a regular point of $\mathbb CC^I\cap R_B$. Moreover, at this point $F$ transversally intersects $\mathbb CC^I\cap R_B$, applying Proposition \ref{ptransv} to the same $\mathbb CC^n$ and $h_i$'s and the set $V:=\mathbb CC^2\times(\mathbb CC^I\cap R_B)$. Thus $F$ transversally intersects $ R_B$ at $F(x)\notin\mathop{\rm sing}\nolimits R_B$, thus $C=F^{-1}(R_B)$ is locally a smooth curve.
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\begin{document}
\title{Experimental test of error-tradeoff uncertainty relation using a
continuous-variable entangled state}
\author{Yang Liu$^{1,2}$}
\author{Zhihao Ma$^{3}$}
\author{Haijun Kang$^{1,2}$}
\author{Dongmei Han$^{1,2}$}
\author{Meihong Wang$^{1,2}$}
\author{Zhongzhong Qin$^{1,2}$}
\author{Xiaolong~Su$^{1,2}$}
\email{[email protected]}
\author{Kunchi Peng$^{1,2}$}
\affiliation{$^1$State Key Laboratory of Quantum Optics and Quantum Optics Devices,
Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China\\
$^2$Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan, Shanxi 030006, China\\
$^3$Department of Mathematics, Shanghai Jiaotong University, Shanghai,
200240, China\\
}
\begin{abstract}
Heisenberg's original uncertainty relation is related to measurement
effect, which is different from the preparation uncertainty
relation. However, it has been shown that Heisenberg's
error-disturbance uncertainty relation can be violated in some cases.
We experimentally test the error-tradeoff uncertainty relation by
using a continuous-variable Einstein-Podolsky-Rosen (EPR) entangled
state. Based on the quantum correlation between the two entangled
optical beams, the errors on amplitude and phase quadratures of one
EPR optical beam coming from joint measurement are estimated
respectively, which are used to verify the error-tradeoff relation.
Especially, the error-tradeoff relation for error-free measurement
of one observable is verified in our experiment. We also verify the
error-tradeoff relations for nonzero errors and mixed state by
introducing loss on one EPR beam. Our experimental results
demonstrate that Heisenberg's error-tradeoff uncertainty relation is
violated in some cases for a continuous-variable system, while the
Ozawa's and Brainciard's relations are valid.
\end{abstract}
\maketitle
\section{ Introduction}
As one of the cornerstones of quantum mechanics, uncertainty relation
describes the measurement limitation on two incompatible observables \cite
{Heisenberg}. It should be emphasized that the uncertainty relation actually
states an intrinsic property of a quantum system, rather than a statement
about the observational success of current technology. Uncertainty relation
has deep connection with many special characters in quantum mechanics, such
as Bell non-locality and entanglement \cite{Bell2,EPR2}, which cannot occur
in classical world. With rapid progress in quantum technology, such as
quantum communication and quantum computation \cite{COMMUN,FURU}, in recent
years, it is important for us to know the fundamental limitations in the
achievable accuracy of quantum measurement.
Note that there are two different types of uncertainty relations, one is the
preparation uncertainty relation, which studies the minimal dispersion of
two quantum observables before measurement \cite{Kennard,Rob29}. The
Robertson uncertainty relation \cite{Rob29}, reads as $\sigma (x)\sigma
(p)\geq \hbar /2$, is a typical example in this sense, where $\sigma (x)$
and $\sigma (p)$ are the standard deviations of position and momentum of a
particle. For such uncertainty relation, the measurements of $x$ and $p$ are
performed on an ensemble of identically prepared quantum systems. While in
the original spirit of Heisenberg's idea \cite{Heisenberg}, the Heisenberg's
uncertainty principle should be based on the observer's effect, which means
that measurement of a certain system cannot be made without affecting the
system. So this leads to the second type of uncertainty relation:
measurement uncertainty relation, which studies to what extent the accuracy
of position measurement of a particle is related to the disturbance of the
particle's momentum, so called the error-disturbance uncertainty relation
\cite{Ozawa03}. It is also called the error-tradeoff relation in the
approximate joint measurements of two incompatible observables \cite
{Ozawa04,Branciard}.
Heisenberg's error-tradeoff uncertainty relation for joint measurement is generally
expressed as
\begin{equation}
\varepsilon (A)\varepsilon (B)\geq C_{AB} \label{H}
\end{equation}
where $C_{AB}=\left\vert \left\langle [A,B]\right\rangle \right\vert /2$, $
[A,B]=AB-BA$. However, it has been shown that this relation is not valid in
some cases \cite{Balllentine}. For this reason, Ozawa and Hall proposed new
measurement uncertainty relations which have been theoretically proven to be
universally valid for any incompatible observables, respectively \cite
{Ozawa03,Ozawa04,Hall04}. After that, Branciard proposed a new uncertainty
relation, which is universally valid and tighter than the Ozawa's relation
\cite{Branciard}. There are also other types of measurement uncertainty
relations generalizing Heisenberg's original idea, which can be found in
Refs. \cite
{Werner2,PhysRevA022106,PhysRevA032,PhysRevLett050401,lu,Barchielli2017}.
Experimental tests of the measurement uncertainty relations have been
demonstrated in photonic \cite
{EXPphotons1,EXPphotons2,EXPphotons3,EXPphotons4,EXPphotons5,EXPphotons6},
spin \cite{EXPpolarizedneutrons1,EXPphotons7,EXPphotons8,EXPW1}, and ion
trap systems \cite{EXPW2}. All of these experiments are limited in
discrete-variable systems. Up to now, experimental test of the measurement
uncertainty relation based on continuous-variable system has not been
reported.
\begin{figure}
\caption{ (a) Schematic of the test principle for
error-tradeoff relation by joint measurement on a continuous-variable
entangled state. A quantum state $\protect\rho $ is measured in a joint
measurement apparatus $M$, where two compatible observables $C$ and $D$ are
measured simultaneously to approximate two incompatible observables $A$ and $
B$, respectively. The right inset describes the joint measurement apparatus
for the error-free measurement of observable $A.$ (b) Schematic of
experimental setup. An EPR entangled state is produced by a NOPA operating
in the state of deamplification. The two modes of EPR state are used as the
signal state $\protect\rho $ and the meter state $\protect\rho _{M}
\end{figure}
In this paper, we present the first experimental test of the error-tradeoff relation
for two incompatible variables, amplitude and phase quadratures of an
optical mode, using a continuous-variable EPR entangled state. Based on
quantum correlations of the EPR entangled beams, the error-tradeoff relation
with zero error (error-free) of one observable is verified directly by
performing joint measurement on two EPR beams. In this case, Heisenberg's error-tradeoff uncertainty
relation is violated, while Ozawa's and Branciard's relations are valid. We
also test the error-tradeoff relations for nonzero errors and mixed state by
introducing loss on signal mode. Our experimental test of the
continuous-variable error-tradeoff relations makes the test of the
measurement uncertainty relation more complete.
\section{Theoretical framework}
One mode of EPR entangled state is used as signal state $\rho $ and two
incompatible observables are taken as $A=\hat{x}_{1}$ and $B=\hat{p}_{1}$,
respectively [Fig. 1(a)], where $\hat{x}_{1}=(\hat{a}+\hat{a}^{\dag })/2$ and $
\hat{p}_{1}=(\hat{a}-\hat{a}^{\dag })/2i$ denote the amplitude and phase
quadratures of $\rho $, respectively. Another mode of EPR entangled state is
used as the meter state $\rho _{M}$. Two compatible observables $C$ and $D$
are measured simultaneously to approximate $A$ and $B.$ The quality of the
approximations are characterized by defining the root-mean-square errors $
\varepsilon (A)=\langle (C-A)^{2}\rangle ^{1/2}$ and $\varepsilon
(B)=\langle (D-B)^{2}\rangle ^{1/2}$. Ozawa's error-tradeoff relation is
expressed by \cite{Ozawa03,Ozawa04}
\begin{equation}
\varepsilon (A)\varepsilon (B)+\varepsilon (A)\sigma (B)+\sigma
(A)\varepsilon (B)\geqslant C_{AB} \label{O}
\end{equation}
where $\sigma (A)$\ is the standard deviation of observable $A$. The
Branciard's error-tradeoff relation is given by \cite{Branciard}
\begin{eqnarray}
&&\big[\varepsilon ^{2}(A)\sigma ^{2}(B)+\sigma ^{2}(A)\varepsilon ^{2}(B)
\notag \label{B} \\
&&+2\varepsilon (A)\varepsilon (B)\sqrt{\sigma ^{2}(A)\sigma
^{2}(B)-C_{AB}^{2}}\big]^{1/2}\geqslant C_{AB}
\end{eqnarray}
where the parameter $C_{AB}=1/4$ denote that $A$ and $B$ cannot be jointly
measured on $\rho $ simultaneously. The variances of the amplitude and phase
quadratures of two EPR beams are expressed as $\sigma ^{2}(\hat{x}
_{1})=\sigma ^{2}(\hat{p}_{1})=\sigma ^{2}(\hat{x}_{2})=\sigma ^{2}(\hat{p}
_{2})=(e^{2r}+e^{-2r})/8$, where $r$ is the squeezing parameter \cite{FURU}.
In the experiment, we test Heisenberg's, Ozawa's and Branciard's error-tradeoff uncertainty relations
in three cases, i.e., error-free measurement of one observable, nonzero
error and mixed state cases.
\section{Experimental implementation and results}
In the experiment, an EPR entangled state with $-$2.9 dB squeezing
and 3.9 dB antisqueezing is prepared by a nondegenerate optical
parametric amplifier (NOPA), as shown in Fig. 1(b), which consists
of an $a$-cut type-II KTP crystal and a concave mirror \cite{EPRSU}.
The front face of the KTP crystal is used as the input coupler, and
the concave mirror with 50 mm curvature serves as the output
coupler. The front face of the KTP crystal is coated with the
transmission of 42\% at 540 nm and high reflectivity at 1080 nm. The
end face of the KTP crystal is antireflection coated for both 540 nm
and 1080 nm. In the measurement, a sample size of 5$\times
10^{5\text{ }}$data points is used for all quadrature measurements
with sampling rate of 500 K/s. The interference efficiency between
signal and local oscillatior is 99\% and the quantum efficiency of
photodiodes are 99.6\%.
At first, we consider a situation that the observable $A$ is measured
accurately (error-free measurement of observable $A$), i.e., the optimal
estimation $C=A$. The measured phase quadrature $D=\hat{p}_{2}$ is used to
approximate the observable $B$. Because the amplitude quadrature $\hat{x}_{1}
$ of $\rho $ and the phase quadrature $\hat{p}_{2}$ of $\rho _{M}$ are
compatible, they can be measured simultaneously. The errors for
approximating $A$ and $B$ are expressed as $\varepsilon (A)=\sqrt{\langle
(C-A)^{2}\rangle }=0$, and $\varepsilon (B)=\sqrt{\langle (D-B)^{2}\rangle }=
\sqrt{\sigma ^{2}(\hat{p}_{2}-\hat{p}_{1})}=e^{-r}/\sqrt{2}$, respectively.
Since $\varepsilon (A)=0$ and $\varepsilon (B)<\infty $, we have
\begin{equation}
\varepsilon (A)\varepsilon (B)=0.
\end{equation}
It is obvious that Heisenberg's error-tradeoff uncertainty relation is violated. The Ozawa's and
Branciard's relations are the same for $\varepsilon (A)=0$, which are
\begin{equation}
\sigma (A)\varepsilon (B)=\sqrt{1+e^{-4r}}/4\geqslant 1/4.
\label{error-free}
\end{equation}
The amplitude quadrature $\hat{x}_{1}$ of the signal state is measured by a
homodyne detector HD1 in the time domain, as shown in Fig. 1(b). To evaluate
the error $\varepsilon (B)$, we experimentally measure the observables $B$
and $D$, i.e. the phase quadratures $\hat{p}_{1}$ and $\hat{p}_{2}$, by two
homodyne detectors (HD1 and HD2) simultaneously.
\begin{figure}
\caption{Results of the uncertainty relation in case of
error-free measurement of observable $A$. (a) The error $\protect
\varepsilon (B)$ as a function of the relative phase. (b) The LHS of
the Ozawa's and Branciard's relation as a function of the relative phase.
The right hand side of the relations $C_{AB}
\end{figure}
\begin{figure}
\caption{Results of the uncertainty relations in case of
nonzero errors. (a) The errors $\protect\varepsilon (A)$ (red curve)
and $\protect\varepsilon (B)$ (blue curve) as functions of the transmission
efficiency. (b) The LHS of the relations as functions of the
transmission efficiency. Blue curve: the Heisenberg's relation in Eq. (1).
Yellow curve: the Ozawa's relation in Eq. (2). Black curve: the Branciard's
relation in Eq. (3). The right hand side of the relations $C_{AB}
\end{figure}
In our experiment, the achievable lower bound is limited by the quantum
correlation of the EPR entangled state [Eq.~(\ref{error-free})]. In order to
demonstrate this property, we change the quantum correlation of signal state
and meter state by changing the relative phase $\theta $ between the two
mode of EPR entangled state. Thus, the error $\varepsilon (B)=\sqrt{\sigma
^{2}(e^{i\theta }\hat{p}_{2}-\hat{p}_{1})}$ is measured in experiment. When
the relative phase $\theta =0^{\circ }$ and $\theta =360^{\circ }$, the
minimum error is obtained [Fig. 2(a)] and the left-hand-side (LHS) of the
relation reaches its minimum value [Fig. 2(b)], which is determined by the
present squeezing level. When$\ \theta =180^{\circ }$, the maximum error is
obtained, which corresponds to the measurement of anti-correlated noise $
\sqrt{\sigma ^{2}(\hat{p}_{2}+\hat{p}_{1})}$. The results confirm that the
Ozawa's and Branciard's relations are the same and valid for the error-free
measurement of observable $A$.
\begin{figure}
\caption{ Uncertainty relation for mixed state. (a)
The error $\protect\varepsilon (B)$ as a function of the transmission
efficiency. (b) The LHS of the Ozawa's and Branciard's relation as a
function of the transmission efficiency. The right hand side of the
relations $C_{AB}
\end{figure}
Then, we test the error-tradeoff relation with nonzero errors. When both
errors are not equal to zero, Ozawa's and Branciard's relations are
different. In the experiment, we apply a linear operation on the signal
mode, which is done by transmitting the signal mode through a lossy channel,
as shown in the inset of Fig. 1(b). In this case, the amplitude and phase
quadratures of the signal mode are changed to $\hat{x}_{1}^{^{\prime }}=
\sqrt{T}\hat{x}_{1}+\sqrt{1-T}\hat{x}_{v}$ and $\hat{p}_{1}^{^{\prime }}=
\sqrt{T}\hat{p}_{1}+\sqrt{1-T}\hat{p}_{v}$, respectively, after transmitted
over the lossy channel, where $\hat{x}_{v}$ and $\hat{p}_{v}$\ represent the
amplitude and phase quadratures of vacuum. By choosing $C=\hat{x}
_{1}^{^{\prime }}$ and $D=\hat{p}_{2}$, which are compatible, the errors for
the two incompatible observables $A=\hat{x}_{1}$ and $B=\hat{p}_{1}$ are $
\varepsilon (A)=\sqrt{\sigma ^{2}(\hat{x}_{1}^{^{\prime }}-\hat{x}_{1})}$
and $\varepsilon (B)=\sqrt{\sigma ^{2}(\hat{p}_{2}-\hat{p}_{1})}$,
respectively.
In this case, the error $\varepsilon (A)$ increases with the decreasing of
channel efficiency, while the error $\varepsilon (B)$ is not affected by the
channel efficiency [Fig. 3(a)]. Heisenberg's error-tradeoff unceratinty relation is violated when the
transmission efficiency is higher than 0.3. While the Ozawa's and
Branciard's relations are always valid [Fig. 3(b)]. By comparing the LHS of
Ozawa's and Branciard's relation, we confirm that Branciard's relation is
tighter than Ozawa's relation.
\begin{figure}
\caption{Lower bounds of the error-tradeoff relations. Blue
dashed curve: the Heisenberg's bound. Yellow dotted curve: the Ozawa's
bound. Gray solid curve: the Branciard's bound. Red circles: experimental
data for error free measurement of observable $A$ as shown in Fig. 2. Black
diamonds: experimental data for nonzero errors condition as shown in Fig. 3.
}
\end{figure}
Finally, we demonstrate the error-tradeoff relation for mixed state, i.e.,
the state $\rho $ transmitted over a lossy channel. Here, observables $C=A=
\hat{x}_{1}^{^{\prime }}$, $B=\hat{p}_{1}^{^{\prime }}$, and $D=\hat{p}_{2}$
are chosen, and thus errors for the mixed state are $\varepsilon (A)=0$ and $
\varepsilon (B)=\sqrt{\sigma ^{2}(\hat{p}_{2}-\hat{p}_{1}^{^{\prime }})}$,
respectively. In this case, Ozawa's and Branciard's relations are the same.
The error $\varepsilon (B)$ and the LHS of the relation increase along with
the decreasing of transmission efficiency as shown in Fig. 4(a) and 4(b),
respectively. The error and LHS of the relation get the minimum value when
the transmission efficiency is unit.
The predicted lower bounds for Heisenberg's [Eq. (\ref{H})], Ozawa's [ Eq. (
\ref{O})] and Brinciard's [Eq. (\ref{B})] error-tradeoff relations are
compared in the plane ($\varepsilon (A),\varepsilon (B)$), as shown in Fig.
5. For the Heisenberg's error-tradeoff uncertainty relation (bounded by the blue dashed curve), one of
the error must be infinite when the other goes to zero. While in our
experiment, for the case of error $\varepsilon (A)=0$, the finite error $
\varepsilon (B)$ is observed (red circles), which violates the
Heisenberg's error-tradeoff uncertainty relation, yet satisfies the
Ozawa's and Branciard's relation. For the case of nonzero errors,
only one of the observed values satisfies the Heisenberg's
error-tradeoff uncertainty relation (the data with 0.2 transmission
efficiency). Our experimental data do not reach the lower bound of
the relations for the limitation of the experiment condition, for
example the limited squeezing parameter.
\section{Conclusion}
We experimentally test the Heisenberg's, Ozawa's and Branciard's
error-tradeoff relations for continuous-variable observables, i.e.,
amplitude and phase quadratures of an optical mode. Especially, we
investigate the error-tradeoff relation in case of zero error by using
Gaussian EPR entangled state. Three different measurement apparatus are
applied in our experiment, which are used to test the error-tradeoff
relation for three different cases. The results demonstrate that the
Heisenberg's error-tradeoff uncertainty relation is violated in some cases while the Ozawa's and the
Brinciard's relations are valid. Our work is useful not only in understanding
fundamentals of physical measurement but also in developing of continuous
variable quantum information technology.
\section*{ACKNOWLEDGMENTS}
This research was supported by the NSFC (Grant Nos. 11834010,and 61601270), the program of Youth Sanjin Scholar, the Applied
Basic Research Program of Shanxi Province (Grant No. 201601D202006),
National Basic Research Program of China (Grant No. 2016YFA0301402), and the
Fund for Shanxi ``1331 Project" Key Subjects Construction.
Y.L. and Z.M. contributed equally to this work.
\end{document} |
\begin{document}
\title{Correlation and Entanglement of Multipartite States}
\author{Y. B. Band and I. Osherov}
\affiliation{Departments of Chemistry and Electro-Optics and the Ilse
Katz Center for Nano-Science, \\
Ben-Gurion University, Beer-Sheva 84105, Israel}
\date{\today}
\begin{abstract}
We derive a classification and a measure of classical- and
quantum-correlation of multipartite qubit, qutrit, and in general,
$n$-level systems, in terms of SU$(n)$ representations of density
matrices. We compare the measure for the case of bipartite
correlation with concurrence and the entropy of entanglement. The
characterization of correlation is in terms of the number of nonzero
singular values of the correlation matrix, but that of mixed state
entanglement requires additional invariant parameters in the density
matrix. For the bipartite qubit case, the condition for mixed state
entanglement is written explicitly in terms of the invariant paramters
in the density matrix. For identical particle systems we analyze the
effects of exchange symmetry on classical and quantum correlation.
\end{abstract}
\pacs{03.67.-a, 03.67.Mn, 03.65.Ud}
\maketitle
Quantum entanglement is an information resource; it plays an important
role in many protocols for quantum-information processing, including
quantum computation \cite{PShor_97_Grover_96}, quantum cryptography
\cite{Ekert_91}, teleportation \cite{Bennett_93}, superdense coding
\cite{Bennett_92}, and quantum error correction protocols \cite{qec}.
Techniques for characterizing the bipartite entanglement and
correlation of pure and mixed quantum states have enabled many
advances in quantum information and the study of decoherence
\cite{White_01}. Many quantum information protocols use bipartite
entanglement, but multipartite entanglement \cite{GHZ_89}, also has
quantum-information applications, e.g., controlled secure direct
communication \cite{cont-com}, quantum error correction
\cite{Calderbank_96}, controlled teleportation \cite{Karlsson_98} and
secret sharing \cite{Hillery_99}. It has been shown that any
inseparable two-qubit states can be distilled to a singlet-state form
with enough copies of the qubit-pairs \cite{Horodecki_97} and
algorithms for multi-copy entanglement distillation for pairs of
qubits have been developed \cite{Dehaene}. Moreover, multipartite
entanglement offers a means of enhancing interferometric precision
beyond the standard quantum limit and is therefore relevant to
increasing the precision of atomic clocks by decreasing projection
noise in spectroscopy \cite{Wineland_94}. Here we use a
representation of the density matrix for qubit, qutrit, and more
generally, $n$-level systems containing 2, 3, \ldots, and $N$-parts,
in terms of the correlations between the subsystems to quantify the
classical and quantum correlation of multipartite systems. Our
classification of correlation is in terms of the correlation matrix
and its singular values, and our classification of entanglement of
mixed states \cite{Werner_89} is associated with the Peres-Horodecki
criterion \cite{Peres_06}, which we express in terms of additional
invariant parameters in the density matrix. Separate measures of
bipartite, tripartite, etc., correlation are required, since general
mixed states can have bipartite correlation as well as higher
subsystem-number-correlation.
Werner \cite{Werner_89} defined a mixed state of an $N$-partite system
as {\em separable}, i.e., {\em classically-correlated}, if it can be
written as a convex sum,
\begin{equation} \label{Eq:1}
\rho = \sum_k p_k \, \rho^A_k \rho^B_k \ldots \rho^N_k ~, \quad p_k
> 0 ~, \quad \sum_k p_k = 1 ~,
\end{equation}
where $\rho^A_k$ is a valid density matrix of subsystem $A$, etc.
Otherwise, Werner defined it to be {\em entangled}, i.e., {\em
quantum-correlated}. Unfortunately, this definition of entanglement
for mixed states is not constructive, since, in general, it cannot be
used to decide whether a given density matrix is separable or
entangled. Moreover, a quantitative measure of entanglement of
multi-partite systems has proven to be difficult to devise. Note that
studies of the best separable approximation to an arbitrary density
have been carried out and have led to a proposal of a measure for
entanglement \cite{sep_app}. Furthermore, aspects of the geometry of
separability and entanglement based on Schmidt decomposition have been
studied and led to an analysis of the question of separability for the
two-qubit case \cite{Leinaas_06}.
In what follows, we categorize classically-correlated and
quantum-correlated states and characterize their correlation in terms
the number of nonzero singular values \cite{lin_alg}, $\{d_i\}$, of
the correlation matrix ${\bf C}$, and characterize entanglement of
bitpartite qubit systems using the Peres-Horodecki criterion
\cite{Peres_06} which is reformulated totally in terms of the
parameters used in forming the density matrix.
First, let us consider a bipartite qubit system. For two uncorrelated
qubits, call them $A$ and $B$, we can write the density matrix as a
product, $\rho_{AB} = \rho_{A} \rho_{B}$, where the individual qubit
density matrices can be written as $\rho_J = \frac{1}{2} \, (1 + {\bf
n}_J \cdot {\boldsymbol \sigma}_J)$, where $J = A,B$, the
${\boldsymbol \sigma}_J$ are Pauli matrices for particle $J$ and the
Bloch vectors are ${\bf n}_J = \langle {\boldsymbol \sigma}_J \rangle
= \mathrm{Tr} \, {\boldsymbol \sigma}_J \rho_J$ \cite{Fano_83}. For
two correlated qubits,
\begin{equation} \label{Eq:2}
\rho_{AB} = \frac{1}{4} \, \left[(1 + {\bf n}_A \cdot {\boldsymbol
\sigma}_A) \, (1 + {\bf n}_B \cdot {\boldsymbol \sigma}_B) +
{\boldsymbol \sigma}_A \cdot {\bf C}^{AB} \cdot {\boldsymbol
\sigma}_B \right] ,
\end{equation}
where the tensor ${\bf C}^{AB}$ specifies the qubit correlations,
\begin{equation} \label{Eq:3}
C^{AB}_{ij} \equiv \langle \sigma_{i,A} \sigma_{j,B} \rangle - \langle
\sigma_{i,A} \rangle \langle \sigma_{j,B} \rangle = \langle
\sigma_{i,A} \sigma_{j,B} \rangle - n_{i,A} \, n_{j,B} .
\end{equation}
The density matrix $\rho_{AB}$ is a 4$\times$4 Hermitian matrix with
trace unity, so 15 parameters are required to parameterize it. The 3
components of ${\bf n}_A$, the 3 components of ${\bf n}_B$, and the 9
components $C_{ij}$ of the 3$\times$3 matrix {\bf C}, where we have no
longer explicitly shown the subsystem superscripts, are sufficient for
this purpose.
Similarly for the bipartite qutrit case. The 3$\times$3 density
matrix of a single qutrit can be written as $\rho = \frac{1}{3} \,
\left(1 + \frac{3}{2} \langle \lambda_{i} \rangle \lambda_{i}\right)$
where the $\lambda_i$ are the eight traceless Hermitian Gellman
matrices familiar from SU(3) \cite{Georgi_99}, and $\langle
\lambda_{i} \rangle = \mathrm{Tr} \, \lambda_{i} \rho$. A bipartite
qutrit density matrix can be parameterized in the form
\begin{equation} \label{Eq:4}
\rho_{AB} = \frac{1}{9} \, [(1 + \frac{3}{2} \langle
\lambda_{i,A} \rangle \lambda_{i,A} ) \, (1 +
\frac{3}{2} \langle \lambda_{j,B} \rangle \lambda_{j,B} ) +
\frac{9}{4} \lambda_{i,A} C_{ij} \lambda_{j,B} ] ,
\end{equation}
\begin{equation} \label{Eq:5}
C_{ij} \equiv \langle \lambda_{i,A} \lambda_{j,B} \rangle - \langle
\lambda_{i,A} \rangle \langle \lambda_{j,B} \rangle ~,
\end{equation}
where $C_{ij}$ specifies the correlation between $\lambda_{i,A}$ and
$\lambda_{j,B}$. Here, $\rho_{AB}$ is a 9$\times$9 Hermitian matrix
with trace unity, so 80 parameters are required to parameterize it.
The eight components of $\langle \lambda_{i,A} \rangle$, eight
components of $\langle \lambda_{i,B} \rangle$, and 64 components
$C_{ij}$ of the 8$\times$8 matrix ${\bf C}$ are sufficient for this
purpose. The same procedure can be used for bipartite 4-level systems
using the 15 traceless 4$\times$4 Hermitian generator matrices for
SU(4), and bipartite $n$-level systems with the $n^2-1$ traceless
$n$$\times$$n$ Hermitian matrices. Likewise, a general qubit-qutrit
6$\times$6 density matrix takes the form $\rho_{AB} = \frac{1}{6} \,
[(1 + n_{i,A} \sigma_{i,A} ) \, (1 + \frac{3}{2} \langle \lambda_{j,B}
\rangle \lambda_{j,B} ) + \frac{6}{4} \sigma_{i,A} C_{ij}
\lambda_{j,B}]$ with $C_{ij} \equiv \langle \sigma_{i,A} \lambda_{j,B}
\rangle - \langle \sigma_{i,A} \rangle \langle \lambda_{j,B} \rangle$,
$i = 1, 2, 3$ and $j= 1, \ldots, 8$.
Our bipartite correlation measure for an $n$-level and $m$-level
system is based on the $(n^2-1)$$\times$$(m^2-1)$ correlation matrix
${\bf C}$:
\begin{equation} \label{Eq:E_C}
{\cal E}_{C} \equiv \frac{n_<^2}{4(n_<^2-1)} \mathrm{Tr} \, {\bf C}
{\bf C}^T = \frac{n_<^2}{4(n_<^2-1)} \sum_{i,j} C_{ij} C_{ji}^T ~,
\end{equation}
where $n_< = {\mathrm{min}}(n,m)$. ${\cal E}_{C} = \frac{n_<^2}
{4(n_<^2-1)} \mathrm{Tr} \, (\rho_{AB} - \rho_{A}\rho_{B})^2$ is a
nonnegative real number. If ${\bf C}$ is a normal matrix
\cite{lin_alg}, $\mathrm{Tr} \, {\bf C} {\bf C}^T$ equals to the sum
of the squares of its eigenvalues, but ${\bf C}$ need not be normal.
${\cal E}_{C}$ is basis-independent; any rotation in Hilbert space
leaves it unchanged. The normalization factor $n_<^2/[4(n_<^2-1)]$ in
(\ref{Eq:E_C}) is such that the maximum possible value of ${\cal
E}_{C}$ is unity. ${\cal E}_{C}$ measures both classical- and
quantum-correlation. This measure of bipartite correlation was
suggested in Ref.~\cite{Schlienz_95} for pure states and $n=m$.
The correlation matrix ${\bf C}$ quantifies the correlation and the
entanglement of bipartite states. For pure two-qubit states, the
number of nonzero singular values (NSVs) of ${\bf C}$ is zero for
non-entangled states (${\bf C}$ vanishes), and three for entangled
states. For classically-correlated states with two terms in the sum
[see Eq.~(\ref{Eq:qubit_CC})], only one NSV occurs, two NSVs occur for
three terms, three NSVs occur for four or more terms, and for
entangled (i.e., quantum-correlated) mixed states there are three
NSVs. These cases are summarized in Fig.~\ref{Fig.two-qubit-class}.
Entangled mixed states can be differentiated from
classically-correlated states with 3 NSVs by applying the
Peres-Horodecki (PH) partial transposition condition \cite{Peres_06}
[which corresponds to changing the sign of $n_{y,B}$ and the matrix
elements $C^{AB}_{iy}$ that multiply $\sigma_{y,B}$ in (\ref{Eq:2}),
and determining whether the resulting $\rho$ is still a genuine
density matrix --- if it is, the state is classically correlated,
i.e., unentangled but correlated] to the density matrices with 3 NSVs.
The {\underline{{\em only}} categories that cannot be distinguished
without use of the PH condition are the mixed-entangled and the
classically correlated states with $\ge 4$ NSVs.
\begin{figure}
\caption{Classification of two-qubit states. Categories can be
experimentally distinguished by measuring ${\bf n}
\label{Fig.two-qubit-class}
\end{figure}
Similarly, for a two qutrit pure state, the number of NSVs of ${\bf
C}$ is zero for non-entangled states (the ${\bf C}$ matrix vanishes),
three, if only two basis states are present in the entangled state,
five, if one of the qutrits contains only two basis states but the
other contains three, and eight if all three basis states are present.
For classically correlated qutrit states, there are 1, 2, \ldots, 8
NSVs for 2, 3, \ldots, and 9 or more terms in the sum, etc. A similar
classification in terms of the number of NSVs exists for qubit-qutrit
and $n$-level systems.
A general three-qubit density matrix can be written as
\[
\rho_{ABC} = \frac{1}{8} \, [ (1 + {\bf n}_A \cdot {\boldsymbol
\sigma}_A) \, (1 + {\bf n}_B \cdot {\boldsymbol \sigma}_B) \, (1 +
{\bf n}_C \cdot {\boldsymbol \sigma}_C)
\]
\[
\; \; + {\boldsymbol \sigma}_A \cdot {\bf C}^{AB} \cdot
{\boldsymbol \sigma}_B + {\boldsymbol \sigma}_A \cdot {\bf C}^{AC}
\cdot {\boldsymbol \sigma}_C + {\boldsymbol \sigma}_B \cdot {\bf
C}^{BC} \cdot {\boldsymbol \sigma}_C
\]
\begin{equation} \label{Eq:6}
\; \; + \sum_{ijk} \sigma_{i,A}
\sigma_{j,B} \sigma_{k,C} D_{ijk} ]~,
\end{equation}
where ${\bf C}^{AB}$, ${\bf C}^{AC}$, and ${\bf C}^{BC}$ are the
bipartite correlation matrices and the tensor that specifies the
tripartite correlations is
\begin{equation} \label{Eq:7}
D_{ijk} \equiv \langle \sigma_{i,A} \sigma_{j,B} \sigma_{k,C} \rangle -
\langle \sigma_{i,A} \rangle \langle \sigma_{j,B} \rangle \langle
\sigma_{k,C} \rangle ~.
\end{equation}
A tripartite qutrit state can be similarly parameterized:
\[
\rho_{ABC} = \frac{1}{27} \, [ \prod_{I} (1 + \frac{3}{2} \sum_i
\langle \lambda_{i,I} \rangle \lambda_{i,I}) + \frac{9}{4}
\sum_{I,J} \sum_{i,j} \lambda_{i,I} C_{ij,IJ} \lambda_{j,J}
\]
\begin{equation} \label{Eq:8}
\; \; + \frac{27}{8} \sum_{I,J,K} \sum_{i,j,k} \lambda_{i,A}
\lambda_{j,B} \lambda_{k,C} D_{ijk,IJK} ]~,
\end{equation}
\begin{equation} \label{Eq:9}
D_{ijk,IJK} \equiv \langle \lambda_{i,I} \lambda_{j,J} \lambda_{k,K}
\rangle - \langle \lambda_{i,I} \rangle \langle \lambda_{j,J}
\rangle \langle \lambda_{k,K} \rangle ~.
\end{equation}
Our tripartite correlation measure ${\cal E}_D$ is based on the
correlation matrix ${\bf D}$, ${\cal E}_{D} \equiv K \sum_{i,j,k}
D_{ijk}^2$, which can also be written as
\begin{equation} \label{Eq:E_D}
{\cal E}_{D} = K \, \mathrm{Tr} \, (\rho_{ABC} - \rho_{A}\rho_{B}
\rho_{C} - \sum_{I,J(I \ne J)} \sum_{i,j} C_{ij}^{I,J}
\sigma_{i,I} \sigma_{j,J} )^2~,
\end{equation}
where $K = 1/4$ for qubits, and $K = 27/160$ for qutrits with
$\sigma$s replaced by $\lambda$s. ${\cal E}_D$ is also a
basis-independent nonnegative real number; any rotation in Hilbert
space leaves it unchanged. A tripartite system may have bipartite- as
well as tripartite-correlation. The bipartite correlation of a
tripartite system is the sum of the correlation for the three
bipartite pairs,
\begin{equation} \label{Eq:E_C_3}
{\cal E}_{C} \equiv \frac{n^2}{4(n^2-1)} \sum_{I,J (I \ne J)}
\mathrm{Tr} \, {\bf C}^{I,J} ({\bf C}^{I,J})^T ~,
\end{equation}
where $I,J = A, B, C$. The density matrices of four-particle and
higher qubit, qutrits, and $n$-level system states can be constructed
similarly, but with increased complexity. For example, it is clear
from Eq.~(\ref{Eq:E_D}) how to generalize and obtain the four-particle
correlation of four-particle systems: ${\cal E}_{E} \equiv K'
\sum_{i,j,k,l} E_{ijkl}^2$, where the four-particle-correlation term
of the four-qubit density matrix $\rho_{ABCD}$ is $\sum_{ijkl}
\sigma_{i,A} \sigma_{j,B} \sigma_{k,C} \sigma_{l,D} E_{ijkl}$ and $K'
= 1/8$.
We now present some examples of qubit and qutrit bipartite and
tripartite correlated states. The maximally entangled bipartite qubit
states are the Bell states,
\begin{equation} \label{Eq:Bell_States}
|\Psi^{\pm} \rangle \! = \! \frac{1}{\sqrt{2}} [ |\! \!
\uparrow \downarrow \rangle \pm |\! \! \downarrow\uparrow
\rangle ] ~, \quad |\Phi^{\pm}\rangle \! = \! \frac{1}{\sqrt{2}}
[ |\! \! \uparrow \uparrow \rangle \pm |\! \! \downarrow
\downarrow \rangle ] ~.
\end{equation}
For all these states, $\langle
{\boldsymbol \sigma}_{A} \rangle = \langle {\boldsymbol \sigma}_{B}
\rangle = {\bf 0}$, i.e., ${\bf n}_A = {\bf n}_B = {\bf 0}$. For the
singlet, $\langle \sigma_{i,A} \sigma_{j,B} \rangle = -\delta_{ij}$
(the spins are oppositely polarized). The density matrices of the
Bell states are:
\begin{eqnarray} \label{Eq:Bell_States_rho}
\rho_{\Psi^{-}} &=& \frac{1}{4} \, (1_A 1_B - {\boldsymbol
\sigma}_A \cdot {\boldsymbol \sigma}_B) ~, \nonumber \\
\rho_{\Psi^{+}} &=& \frac{1}{4} \left( 1_A 1_B + {\boldsymbol
\sigma}_A \cdot {\boldsymbol \sigma}_B - 2\sigma_{z,A} \sigma_{z,B}
\right) ~, \nonumber \\
\rho_{\Phi^{+}} &=& \frac{1}{4} \left( 1_A 1_B + {\boldsymbol
\sigma}_A \cdot {\boldsymbol \sigma}_B - 2\sigma_{y,A} \sigma_{y,B}
\right) ~, \nonumber \\
\rho_{\Phi^{-}} &=& \frac{1}{4} \left( 1_A 1_B + {\boldsymbol
\sigma}_A \cdot {\boldsymbol \sigma}_B - 2\sigma_{x,A} \sigma_{x,B}
\right) ~.
\end{eqnarray}
The correlation matrices of the Bell's states are diagonal and the
correlation measure is ${\cal E}_C=1$, i.e., they are maximally
entangled.
Let us now consider the Rashid pure states \cite{Rashid_78},
\begin{equation} \label{Eq:Rashid_States}
|\phi^+\rangle = (2 \mathrm{cosh}(2\theta))^{-1/2} \, (e^{-\theta}
|\!\!\uparrow \uparrow \rangle + e^{\theta} |\!\!\downarrow
\downarrow \rangle) ~,
\end{equation}
whose density matrix is $\rho_{\phi^+} = \frac{1}{4} \, \{ [1_A -
\tanh(2\theta) \sigma_{z,A}] [1_B - \tanh(2\theta) \sigma_{z,B}] +
\mathrm{sech}(2\theta) ( \sigma_{x,A} \sigma_{x,B} - \sigma_{y,A}
\sigma_{y,B} ) + \mathrm{sech}^2(2\theta) \sigma_{z,A} \sigma_{z,B}
\}$. When $\theta = 0$, $|\phi^+\rangle = |\Phi^+\rangle$, and as
$\theta \to \pm \infty$, an unentangled state results. The
nonvanishing correlation matrix elements are: $C_{xx} =
\mathrm{sech}(2\theta), C_{yy} = -\mathrm{sech}(2\theta), C_{zz} =
\mathrm{sech}^2(2\theta)$. Using (\ref{Eq:E_C}) we obtain the
correlation measure ${\cal E}_C(|\phi^+ \rangle) = \frac{1}{3}
\mathrm{Tr}\, {\bf C} {\bf C}^T = \frac{1}{3}(2 \,
\mathrm{sech}^2(2\theta) + \mathrm{sech}^4(2\theta))$. The
concurrence ${\cal C}$ \cite{Wootters_98,Peres_06} is $$ {\cal
C}(|\phi^+ \rangle) = \sqrt{2\left(1-\mathrm{Tr} \,
\left[\rho _A{}^2\right]\right)} = \mathrm{sech}(2\theta) ~,$$
since
$$\rho_A = \frac{1}{2} \, \left(\! \! \begin{array}{cc}
1-\mathrm{tanh}(2\theta) & 0 \\ 0 &
1+\mathrm{tanh}(2\theta) \end{array} \! \! \right) ~,$$
and the entanglement entropy is $S \equiv - \mathrm{Tr} \, [\rho_A
\log_2 \rho_A]$. These results are graphically presented in
Fig.~\ref{Fig.Corr_measures}. All the measures equal unity for
$\theta = 0$ and decrease rapidly vs.~$\theta$.
\begin{figure}
\caption{(color online) Comparison of the ${\cal E}
\label{Fig.Corr_measures}
\end{figure}
Two-qubit classically-correlated states take the form
\begin{equation} \label{Eq:qubit_CC}
{\rho}^{\mathrm{CC}} = \frac{1}{4} \sum_{k \ge 2} p_k \, (1 +
{\bf n}_{A,k} \cdot {\boldsymbol \sigma}_A) \, (1 + {\bf n}_{B,k}
\cdot {\boldsymbol \sigma}_B) ~,
\end{equation}
with $\sum_k p_k = 1$ and $p_k > 0$. The density matrix for the
classically-correlated state can be written in the form of
Eq.~(\ref{Eq:2}) with Bloch vectors
\begin{equation} \label{Eq:qubit_CC_1}
{\bf n}_{A} = \sum_k p_k \, {\bf n}_{A,k} ~, \quad {\bf n}_{B} =
\sum_k p_k \, {\bf n}_{B,k} ~,
\end{equation}
and correlation matrix
\begin{equation} \label{Eq:qubit_CC_2}
C_{ij} = \sum_k p_k \, n_{i,A,k} \left[ n_{j,B,k} - \sum_l p_l
\, n_{j,B,l} \right] ~.
\end{equation}
For example, for classically-correlated mixed states of the form
$\rho^{\mathrm{CC}} = (2 \, \mathrm{sech}^2(2\theta))^{-1/2}
(e^{-\theta} |\! \! \downarrow \uparrow \rangle \, \langle
\downarrow \uparrow \! \! |+ e^{\theta}|\!\!\uparrow
\downarrow\rangle \, \langle\uparrow \downarrow \! \! |)$, we find
that all the correlation coefficients vanish, except for $C_{zz} = -
\mathrm{sech}^2(2\theta)$, the density matrix in representation
(\ref{Eq:2}) is $\rho^{\mathrm{CC}} = \frac{1}{4} \left( 1_A 1_B -
\mathrm{sech}^2(2\theta) \, \sigma_{z,A} \sigma_{z,B}\right)$, and the
classical-correlation measure is ${\cal E}_C^{\mathrm{CC}} =
\frac{1}{3} \,\mathrm{sech}^4(2\theta)$.
It is elucidating to consider the Werner two-qubit density matrix
composed of a sum of a singlet state and the maximally mixed state,
$\rho^{W} = p |\Psi^{-} \rangle \langle \Psi^{-} | + \frac{1-p}{4}
{\bf 1}$, or, the more general Werner two-qubit density matrix,
\begin{equation} \label{Eq:GW}
\rho^{GW} = p \, |\psi^- \rangle \langle \psi^- | + \frac{1-p}{4}
{\bf 1} ~,
\end{equation}
where $|\psi^- \rangle = (2 \mathrm{cosh}(2\theta))^{-1/2} \,
(e^{-\theta} |\!\!\uparrow \downarrow \rangle - e^{\theta}
|\!\!\downarrow \uparrow \rangle)$. $\rho^{GW}$ reduces to $\rho^{W}$
for $\theta = 0$. For $\rho^{GW}$,
\begin{equation} \label{Eq:GW_n}
{\bf n}_A = - {\bf n}_B = p \, \mathrm{tanh}(2\theta) \, {\hat{\bf
z}} ~,
\end{equation}
and
\begin{equation} \label{Eq:14}
{\bf C}^{GW} = -p \left( \! \!
\begin{array}{ccc}
{\mathrm{sech}(2\theta)} & 0 & 0 \\
0 & {\mathrm{sech}(2\theta)} & 0 \\
0 & 0 & {1-p + p \, \mathrm{sech}^2(2\theta)}
\end{array} \! \! \right).
\end{equation}
The PH entanglement criterion \cite{Peres_06} shows that this state is
entangled if $p [(1 + 2 \mathrm{sech}(2 \theta)] \ge 1$. Figure
\ref{Ec_vs_p_theta} plots the PH criterion limit and the correlation
measure, ${\cal E}_C(p,\theta) = \sum_i d_i^2 = 1 - p + (2p^2+p)
\mathrm{sech}^2(2\theta)$, for the generalized Werner state. Note that
the PH criterion is not obtainable from ${\bf C}$ alone, but can be
obtained using the invariant parameters $\xi \equiv \sum_i d_i -
\frac{{\bf n}_A \cdot {\bf C} \cdot {\bf n}_B}{{\bf n}_A \cdot {\bf
n}_B}$ and ${\bf n}_A \cdot {\bf n}_B$. More explicitly, $p [1 + 2
\mathrm{sech}(2 \theta)] = -\xi + \sqrt{\xi^2/4 - {\bf n}_A \cdot {\bf
n}_B}$, so the PH condition reads
\begin{equation} \label{Eq:PH}
-\frac{\xi}{2} + \frac{-\xi + \sqrt{\xi^2 - 4 \, {\bf n}_A \cdot
{\bf n}_B}}{2} \ge 1 ~,
\end{equation}
which can be written as the condition: the largest root of the
quadratic equation, $(x + \xi/2)^2 + \xi (x + \xi/2) + {\bf n}_A \cdot
{\bf n}_B = 0$, is greater than unity. Thus, mixed state entanglement
is determined not only by ${\bf C}$ but by additional invariant
characteristics of the density matrix, i.e., invariant characteristics
composed of the parameters ${\bf C}$, ${\bf n}_A$ and ${\bf n}_B$ used
to form the density matrix (whereas the correlation is determined only
in terms of ${\bf C}$). The physical significance of the scalar
product ${\bf n}_A \cdot {\bf n}_B$ as the projection of the
expectation value of the spin of one qubit on the other, is clear, as
is the physical significance of $\xi$ as a specific projection of the
singular values of the correlation matrix that depends on the average
spins ${\bf n}_A$ and ${\bf n}_B$ \cite{xi}. However, the physical
significance of the PH entanglement criterion is not yet clear; i.e.,
the physical interpretation of Eq.~(\ref{Eq:PH}) [or the quadratic
equation] remains to be uncovered. But at least the PH condition is
now expressed only in terms of the physical parameters appearing in
the density matrix, rather than by the partial transposition
condition, which is more removed from physical interpretation.
\begin{figure}
\caption{(color online) ${\cal E}
\label{Ec_vs_p_theta}
\end{figure}
As an example of a tripartite pure qutrit state, consider
\begin{equation} \label{Eq:3_qutrit_1}
|\psi^{E3}\rangle = \frac{e^{\theta_1} e^{\theta_2} | v_1 v_1 v_1
\rangle + e^{-\theta_1} | v_2 v_2 v_2 \rangle + e^{-\theta_2} |
v_3 v_3 v_3 \rangle}{\sqrt{e^{2\theta_1} e^{2\theta_2} +
e^{-2\theta_1} + e^{-2\theta_2}}} ,
\end{equation}
where $| v_1 \rangle = (1,0,0)^T$, $| v_2 \rangle = (0,1,0)^T$, $| v_3
\rangle = (0,0,1)^T$. The qutrit bipartite correlation ${\cal E}_C$
and tripartite correlation ${\cal E}_D$ are plotted in
Fig.~\ref{Fig3.3_qutrit_ED_EC}. The maximum of ${\cal E}_D$ is at
$\theta_1 = \theta_2 = 0$, where ${\cal E}_D = 1$, but the bipartite
correlation dips there. Three ridges of high bipartite and tripartite
correlation occur, one at $\theta_1 = \theta_2 =$ negative, and two
others at 120 degrees rotation from the first.
\begin{figure}
\caption{(color online) The bipartite and tripartite correlation
measures, ${\cal E}
\label{Fig3.3_qutrit_ED_EC}
\end{figure}
In most quantum information systems, qubits are distinguishable, so
there is no need to account for exchange symmetry, but for {\em
identical} bosonic or fermionic systems, the density matrix must be
properly symmetrized, e.g., $\rho_{AB}^{\mathrm{sym}} = {\cal S}
\rho_{AB} {\cal S}$. For two qubits, the symmetrization operator is
${\cal S} = \frac{1}{2} (1 + P_{AB}) = \frac{3}{4} +
\frac{1}{4}{\boldsymbol \sigma}_A \cdot {\boldsymbol \sigma}_B$ and
the antisymmetrization operator is ${\cal A} = \frac{1}{2} (1 -
P_{AB}) = \frac{1}{4} - \frac{1}{4}{\boldsymbol \sigma}_A \cdot
{\boldsymbol \sigma}_B$, hence,
$\rho_{AB}^{\mathrm{sym}} = \left(\! \frac{3}{4} +
\frac{1}{4}{\boldsymbol \sigma}_A \cdot {\boldsymbol \sigma}_B
\! \right) \rho_{AB} \left(\! \frac{3}{4} + \frac{1}{4}{\boldsymbol
\sigma}_A \cdot {\boldsymbol \sigma}_B \! \right)$,
$\rho_{AB}^{\mathrm{anti}} = \left(\! \frac{1}{4} -
\frac{1}{4}{\boldsymbol \sigma}_A \cdot {\boldsymbol \sigma}_B
\! \right) \rho_{AB} \left(\! \frac{1}{4} - \frac{1}{4}{\boldsymbol
\sigma}_A \cdot {\boldsymbol \sigma}_B \! \right)$.
If the qubits are antisymmetric, $\rho_{AB} =
\rho_{AB}^{\mathrm{anti}} = {\cal A} \rho_{AB} {\cal A}$, the state
must be pure singlet, $\rho_{AB}^{\mathrm{anti}} = \frac{1}{4} (1 -
{\boldsymbol \sigma}_{A} \cdot {\boldsymbol \sigma}_{B})$; it cannot
be a mixed state, as opposed to $\rho_{AB}^{\mathrm{sym}}$ which can
be mixed. If spatial degrees of freedom need to be included in the
description, in addition to the internal degrees of freedom, a
bipartite density matrix can always be written as a product of an
internal (i.e., spin) part and an external (i.e., space) part. Hence,
a symmetric density matrix $\rho_{AB}^{\mathrm{sym}}$ for the internal
degrees of freedom must be multiplied by a symmetric [antisymmetric]
spatial density matrix for the spatial degrees of freedom $\{{\bf
r}_A$, ${\bf r}_B\}$ for bosons [fermions], and
$\rho_{AB}^{\mathrm{anti}}$ must be multiplied by an antisymmetric
[symmetric] spatial density matrix for bosons [fermions], so that the
full density matrix has the right exchange symmetry. We show
elsewhere that this has relevance to collisional shifts in atomic
clocks \cite{YB_IO_10}.
In summary, we have developed a classification of correlation for
multipartite $N$-level quantum systems by writing their density
matrices in terms of SU$(N)$ generators, and we defined a measure of
correlation for such systems, based upon their correlation matrices.
The entanglement involves not just the correlation matrix but also
other invariant parameters in the density matrix for the system. This
formulation can now be used for a variety of applications, e.g., in
the optimization of quantum gates and in the calculation of
collisional clock shifts.
This work was supported in part by grants from the U.S.-Israel
Binational Science Foundation (No.~2006212), the Israel Science
Foundation (No.~29/07), and the James Franck German-Israel Binational
Program.
\end{document} |
\begin{document}
\global\long\,def\,d{\,d}
\global\long\,def\mathrm{tr}{\mathrm{tr}}
\global\long\,def\operatorname{spt}{\operatorname{spt}}
\global\long\,def\,div{\operatorname{div}}
\global\long\,def\operatorname{osc}{\operatorname{osc}}
\global\long\,def\esssup{\esssup}
\global\long\,def\dashint{\,dashint}
\global\long\,def\esssinf{\esssinf}
\global\long\,def\esssliminf{\esssliminf}
\global\long\,def|{|}
\global\long\,def\operatorname{sgn}{\operatorname{sgn}}
\global\long\,def\,dist{\operatorname{dist}}
\global\long\,def\,diam{\operatorname{diam}}
\global\long\,defp_{\text{min}}{p_{\text{min}}}
\global\long\,defp_{\text{max}}{p_{\text{max}}}
\excludeversion{details}\keywords{non-divergence form equation, normalized equation, $p$-Laplace, Hölder gradient regularity, viscosity solution, inhomogeneous equation}\subjclass[2020]{35J92, 35J70, 35J75, 35D40}\,date{October 2021}
\title[hölder gradient regularity]{Hölder gradient regularity for the inhomogeneous normalized $p(x)$-Laplace
equation}
\begin{abstract}
We prove the local gradient Hölder regularity of viscosity solutions
to the inhomogeneous normalized $p(x)$-Laplace equation
\[
-\Delta_{p(x)}^{N}u=f(x),
\]
where $p$ is Lipschitz continuous, $\inf p>1$, and $f$ is continuous
and bounded.
\end{abstract}
\author{Jarkko Siltakoski}
\maketitle
\section{Introduction}
We study the \textit{inhomogeneous normalized $p(x)$-Laplace equation}
\begin{equation}
-\Delta_{p(x)}^{N}u=f(x)\quad\text{in }B_{1},\label{eq:normalized p(x)}
\end{equation}
where
\[
-\Delta_{p(x)}^{N}u:=-\Delta u-(p(x)-2)\frac{\left\langle D^{2}uDu,Du\right\rangle }{\left|Du\right|^{2}}
\]
is the \textit{normalized $p(x)$-Laplacian}, $p:B_{1}\rightarrow\mathbb{R}$
is Lipschitz continuous, $1<p_{\min}:=\inf_{B_{1}}p\leq\sup_{B_{1}}p=:p_{\max}$
and $f\in C(B_{1})$ is bounded. Our main result is that viscosity
solutions to (\ref{eq:normalized p(x)}) are locally $C^{1,\alpha}$-regular.
Normalized equations have attracted a significant amount of interest
during the last 15 years. Their study is partially motivated by their
connection to game theory. Roughly speaking, the value function of
certain stochastic tug-of-war games converges uniformly up to a subsequence
to a viscosity solution of a normalized equation as the step-size
of the game approaches zero \cite{peresShefield08,manfrediParviainenRossi10,manfrediParviainenRossi12,banerjeeGarofalo15,blancRossi19}.
In particular, a game with space-dependent probabilities leads to
the normalized $p(x)$-Laplace equation \cite{arroyoHeinoParviainen17}
and games with running pay-offs lead to inhomogeneous equations \cite{ruosteenoja16}.
In addition to game theory, normalized equations have been studied
for example in the context of image processing \cite{does11,elmoatazToutainTenbrinck15}.
The variable $p(x)$ in (\ref{eq:normalized p(x)}) has an effect
that may not be immediately obvious: If we formally multiply the equation
by $\left|Du\right|^{p(x)-2}$ and rewrite it in a divergence form,
then a logarithm term appears and we arrive at the expression
\begin{equation}
-\,div(\left|Du\right|^{p(x)-2}Du)+\left|Du\right|^{p(x)-2}\log(\left|Du\right|)Du\cdot Dp=\left|Du\right|^{p(x)-2}f(x).\label{eq:strong p(x)}
\end{equation}
For $f\equiv0$, this is the so called \textit{strong $p(x)$-Laplace
equation} introduced by Adamowicz and Hästö \cite{adamowiczHasto10,adamowiczHasto11}
in connection with mappings of finite distortion. In the homogeneous
case viscosity solutions to (\ref{eq:normalized p(x)}) actually coincide
with weak solutions of (\ref{eq:strong p(x)}) \cite{siltakoski18},
yielding the $C^{1,\alpha}$-regularity of viscosity solutions as
a consequence of a result by Zhang and Zhou \cite{strongpx_regularity}.
In the present paper our objective is to prove $C^{1,\alpha}$-regularity
of solutions to (\ref{eq:normalized p(x)}) directly using viscosity
methods. The Hölder regularity of solutions already follows from existing
general results, see \cite{krylovSafonov79,krylovSafonov80,caffarelli89,caffarelliCabre}.
More recently, Imbert and Silvestre \cite{imbertSilvestre12} proved
the gradient Hölder regularity of solutions to the elliptic equation
\[
\left|Du\right|^{\gamma}F(D^{2}u)=f,
\]
where $\gamma>0$ and Imbert, Jin and Silvestre \cite{jinsilvestre17,imbertJinSilvestre16}
obtained a similar result for the parabolic equation
\[
\partial_{t}u=\left|Du\right|^{\gamma}\Delta_{p}^{N}u,
\]
where $p>1$, $\gamma>-1$. Furthermore, Attouchi and Parviainen \cite{attouchiParv}
proved the $C^{1,\alpha}$-regularity of solutions to the inhomogeneous
equation $\partial_{t}u-\Delta_{p}^{N}u=f(x,t)$. Our proof of Hölder
gradient regularity for solutions of (\ref{eq:normalized p(x)}) is
in particular inspired by the papers \cite{jinsilvestre17} and \cite{attouchiParv}.
We point out that recently Fang and Zhang \cite{fangZhang21b} proved
the $C^{1,\alpha}$-regularity of solutions to the parabolic normalized
$p(x,t)$-Laplace equation
\begin{equation}
\partial_{t}u=\Delta_{p(x,t)}^{N}u,\label{eq:parabolic normalized p(x)}
\end{equation}
where $p\in C_{\text{loc}}^{1}$. The equation (\ref{eq:parabolic normalized p(x)})
naturally includes (\ref{eq:normalized p(x)}) if $f\equiv0$. However,
in this article we consider the inhomogeneous case and only suppose
that $p$ is Lipschitz continuous. More precisely, we have the following
theorem.
\begin{thm}
\label{thm:main-1} Suppose that $p$ is Lipschitz continuous in $B_{1}$,
$p_{\min}>1$ and $f\in C(B_{1})$ is bounded. Let $u$ be a viscosity
solution to
\[
-\Delta_{p(x)}^{N}u=f(x)\quad\text{in }B_{1}.
\]
Then there is $\alpha(N,p_{\min},p_{\max},p_{L})\in(0,1)$ such that
\[
\left\Vert u\right\Vert _{C^{1,\alpha}(B_{1/2})}\leq C(N,p_{\min},p_{\max},p_{L},\left\Vert f\right\Vert _{L^{\infty}(B_{1})},\left\Vert u\right\Vert _{L^{\infty}(B_{1})}),
\]
where $p_{L}$ is the Lipschitz constant of $p$.
\end{thm}
The proof of Theorem \ref{thm:main-1} is based on suitable uniform
$C^{1,\alpha}$-regularity estimates for solutions of the regularized
equation
\begin{equation}
-\Delta v-(p_{\varepsilon}(x)-2)\frac{\left\langle D^{2}vDv,Dv\right\rangle }{\left|Dv\right|^{2}+\varepsilon^{2}}=g(x),\label{eq:intro regularized}
\end{equation}
where it is assumed that $g$ is continuous and $p_{\varepsilon}$
is smooth. In particular, we show estimates that are independent of
$\varepsilon$ and only depend on $N$, $\sup p$, $\inf p$, $\left\Vert Dp_{\varepsilon}\right\Vert _{L^{\infty}}$
and $\left\Vert g\right\Vert _{L^{\infty}}$. To prove such estimates,
we first derive estimates for the perturbed homogeneous equation
\begin{equation}
-\Delta v-(p_{\varepsilon}(x)-2)\frac{\left\langle D^{2}v(Dv+q),Dv+q\right\rangle }{\left|Dv\right|^{2}+\varepsilon^{2}}=0,\label{eq:intro homogeneous}
\end{equation}
where $q\in\mathbb{R}^{N}$. Roughly speaking, $C^{1,\alpha}$-estimates
for solutions of (\ref{eq:intro homogeneous}) are based on ``improvement
of oscillation'' which is obtained by differentiating the equation
and observing that a function depending on the gradient of the solution
is a supersolution to a linear equation. The uniform $C^{1,\alpha}$-estimates
for solutions of (\ref{eq:intro homogeneous}) then yield uniform
estimates for the inhomogeneous equation (\ref{eq:intro regularized})
by an adaption of the arguments in \cite{imbertSilvestre12,attouchiParv}.
With the \textit{a priori} regularity estimates at hand, the plan
is to let $\varepsilon\rightarrow0$ and show that the estimates pass
on to solutions of (\ref{eq:normalized p(x)}). A problem is caused
by the fact that, to the best of our knowledge, uniqueness of solutions
to (\ref{eq:normalized p(x)}) is an open problem for variable $p(x)$
and even for constant $p$ if $f$ is allowed to change signs. To
deal with this, we fix a solution $u_{0}\in C(\overline{B}_{1})$
to (\ref{eq:normalized p(x)}) and consider the Dirichlet problem
\begin{equation}
-\Delta_{p(x)}^{N}u=f(x)-u_{0}(x)-u\quad\text{in }B_{1}\label{eq:intro dirichlet}
\end{equation}
with boundary data $u=u_{0}$ on $\partial B_{1}$. For this equation
the comparison principle holds and thus $u_{0}$ is the unique solution.
We then consider the approximate problem
\begin{equation}
-\Delta u_{\varepsilon}-(p_{\varepsilon}(x)-2)\frac{\left\langle D^{2}u_{\varepsilon}Du_{\varepsilon},Du_{\varepsilon}\right\rangle }{\left|Du_{\varepsilon}\right|^{2}+\varepsilon^{2}}=f_{\varepsilon}(x)-u_{0,\varepsilon}(x)-u_{\varepsilon}\label{eq:intro regularized 2}
\end{equation}
with boundary data $u_{\varepsilon}=u_{0}$ on $\partial B_{1}$ and
where $p_{\varepsilon},f_{\varepsilon},u_{0,\varepsilon}\in C^{\infty}(B_{1})$
are such that $p\rightarrow p_{\varepsilon}$, $f_{\varepsilon}\rightarrow f$
and $u_{0,\varepsilon}\rightarrow u_{0}$ uniformly in $B_{1}$ and
$\left\Vert Dp_{\varepsilon}\right\Vert _{L^{\infty}(B_{1})}\leq\left\Vert Dp\right\Vert _{L^{\infty}(B_{1})}$.
As the equation (\ref{eq:intro regularized 2}) is uniformly elliptic
quasilinear equation with smooth coefficients, the solution $u_{\varepsilon}$
exists in the classical sense by standard theory. Since $u_{\varepsilon}$
also solves (\ref{eq:intro regularized}) with $g(x)=f_{\varepsilon}(x)-u_{0,\varepsilon}(x)-u_{\varepsilon}(x)$,
it satisfies the uniform $C^{1,\alpha}$-regularity estimate. We then
let $\varepsilon\rightarrow0$ and use stability and comparison principles
to show that $u_{0}$ inherits the regularity estimate.
For other related results, see for example the works of Attouchi,
Parviainen and Ruosteenoja \cite{OptimalC1} on the normalized $p$-Poisson
problem $-\Delta_{p}^{N}u=f$, Attouchi and Ruosteenoja \cite{attouchiRuosteenoja18,attouchiRuosteenoja20,attouchi20}
on the equation $-\left|Du\right|^{\gamma}\Delta_{p}^{N}u=f$ and
its parabolic version, De Filippis \cite{deflippis21} on the double
phase problem $(\left|Du\right|^{q}+a(x)\left|Du\right|^{s})F(D^{2}u)=f(x)$
and Fang and Zhang \cite{fangZhang21} on the parabolic double phase
problem $\partial_{t}u=(\left|Du\right|^{q}+a(x,t)\left|Du\right|^{s})\Delta_{p}^{N}u$.
We also mention the paper by Bronzi, Pimentel, Rampasso and Teixeira
\cite{bronziPimentelRampassoTeixeira} where they consider fully nonlinear
variable exponent equations of the type $\left|Du\right|^{\theta(x)}F(D^{2}u)=0$.
The paper is organized as follows: Section 2 is dedicated to preliminaries,
Sections 3 and 4 contain $C^{1,\alpha}$-regularity estimates for
equations (\ref{eq:intro homogeneous}) and (\ref{eq:intro regularized 2}),
and Section 5 contains the proof of Theorem (\ref{thm:main-1}). Finally,
the Appendix contains an uniform Lipschitz estimate for the equations
studied in this paper and a comparison principle for equation (\ref{eq:intro dirichlet}).
\section{Preliminaries}
\subsection{Notation}
We denote by $B_{R}\subset\mathbb{R}^{N}$ an open ball of radius
$R>0$ that is centered at the origin in the $N$-dimensional Euclidean
space, $N\geq1$. The set of symmetric $N\times N$ matrices is denoted
by $S^{N}$. For $X,Y\in S^{N}$, we write $X\leq Y$ if $X-Y$ is
negative semidefinite. We also denote the smallest eigenvalue of $X$
by $\lambda_{\min}(X)$ and the largest by $\lambda_{\max}(X)$ and
set
\[
\left\Vert X\right\Vert :=\sup_{\xi\in B_{1}}\left|X\xi\right|=\sup\left\{ \left|\lambda\right|:\lambda\text{ is an eigenvalue of }X\right\} .
\]
We use the notation $C(a_{1},\ldots,a_{k})$ to denote a constant
$C$ that may change from line to line but depends only on $a_{1},\ldots,a_{k}$.
For convenience we often use $C(\hat{p})$ to mean that the constant
may depend on $p_{\min}$, $p_{\max}$ and the Lipschitz constant
$p_{L}$ of $p$.
For $\alpha\in(0,1)$, we denote by $C^{\alpha}(B_{R})$ the set of
all functions $u:B_{R}\rightarrow\mathbb{R}$ with finite Hölder norm
\[
\left\Vert u\right\Vert _{C^{\alpha}(B_{R})}:=\left\Vert u\right\Vert _{L^{\infty}(B_{R})}+\left[u\right]_{C^{\alpha}(B_{R})},\text{\ensuremath{\quad}where }\left[u\right]_{C^{\alpha}(B_{R})}:=\sup_{x,y\in B_{R}}\frac{\left|u(x)-u(y)\right|}{\left|x-y\right|^{\alpha}}.
\]
Similarly, we denote by $C^{1,\alpha}(B_{R})$ the set of all functions
for which the norm
\[
\left\Vert u\right\Vert _{C^{1,\alpha}(B_{R})}:=\left\Vert u\right\Vert _{C^{\alpha}(B_{R})}+\left\Vert Du\right\Vert _{C^{\alpha}(B_{R})}
\]
is finite.
\subsection{Viscosity solutions}
Viscosity solutions are defined using smooth test functions that touch
the solution from above or below. If $u,\varphi:\mathbb{R}^{N}\rightarrow\mathbb{R}$
and $x\in\mathbb{R}^{N}$ are such that $\varphi(x)=u(x)$ and $\varphi(y)<u(y)$
for $y\not=x_{0}$, then we say that \textit{$\varphi$ touches $u$
from below at $x_{0}$}.
\begin{defn}
\label{def:viscosity solutions} Let $\Omega\subset\mathbb{R}^{N}$
be a bounded domain. Suppose that $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$
is continuous. A lower semicontinuous function $u:\Omega\rightarrow\mathbb{R}$
is a \textit{viscosity supersolution} to
\[
-\Delta_{p(x)}^{N}u\geq f(x,u)\quad\text{in }\Omega
\]
if the following holds: Whenever $\varphi\in C^{2}(\Omega)$ touches
$u$ from below at $x\in\Omega$ and $D\varphi(x)\not=0$, we have
\[
-\Delta\varphi(x)-(p(x)-2)\frac{\left\langle D^{2}\varphi(x)D\varphi(x),D\varphi(x)\right\rangle }{\left|D\varphi(x)\right|^{2}}\geq f(x,u(x))
\]
and if $D\varphi(x)=0$, then
\[
-\Delta\varphi(x)-(p(x)-2)\left\langle D^{2}\varphi(x)\eta,\eta\right\rangle \ge f(x,u(x))\quad\text{for some }\eta\in\overline{B}_{1}.
\]
Analogously, a lower semicontinuous function $u:\Omega\rightarrow\mathbb{R}$
is a viscosity subsolution if the above inequalities hold reversed
whenever $\varphi$ touches $u$ from above. Finally, we say that
$u$ is a \textit{viscosity solution} if it is both viscosity sub-
and supersolution.
\end{defn}
\begin{rem*}
The special treatment of the vanishing gradient in Definition \ref{def:viscosity solutions}
is needed because of the singularity of the equation. Definition \ref{def:viscosity solutions}
is essentially a relaxed version of the standard definition in \cite{userguide}
which is based on the so called semicontinuous envelopes. In the standard
definition one would require that if $\varphi$ touches a viscosity
supersolution $u$ from below at $x$, then
\[
\begin{cases}
-\Delta_{p(x)}^{N}\varphi(x)\geq f(x,u(x)) & \text{if }D\varphi(x)\not=0,\\
-\Delta\varphi(x)-(p(x)-2)\lambda_{\min}(D^{2}\varphi(x))\geq f(x,u(x)) & \text{if }D\varphi(x)=0\text{ and }p(x)\geq2,\\
-\Delta\varphi(x)-(p(x)-2)\lambda_{\max}(D^{2}\varphi(x))\geq f(x,u(x)) & \text{if }D\varphi(x)=0\text{ and }p(x)<2.
\end{cases}
\]
Clearly, if $u$ is a viscosity supersolution in this sense, then
it is also a viscosity supersolution in the sense of Definition \ref{def:viscosity solutions}.
\end{rem*}
\section{Hölder gradient estimates for the regularized homogeneous equation\label{sec:regularized homogeneous}}
In this section we prove $C^{1,\alpha}$-regularity estimates for
solutions to the equation
\begin{equation}
-\Delta u-(p(x)-2)\frac{\left\langle D^{2}u(Du+q),Du+q\right\rangle }{\left|Du+q\right|^{2}+\varepsilon^{2}}=0\quad\text{in }B_{1},\label{eq:regularized homogeneous}
\end{equation}
where $p:B_{1}\rightarrow B_{1}$ is Lipschitz, $p_{\min}>1$, $\varepsilon>0$
and $q\in\mathbb{R}^{N}$. Our objective is to obtain estimates that
are independent of $q$ and $\varepsilon$. Observe that (\ref{eq:regularized homogeneous})
is a uniformly elliptic quasilinear equation with smooth coefficients.
Viscosity solutions to (\ref{eq:regularized homogeneous}) can be
defined in the standard way and they are smooth if $p$ is smooth.
\begin{prop}
\label{prop:c infty} Suppose that $p$ is smooth. Let $u$ be a viscosity
solution to (\ref{eq:regularized homogeneous}) in $B_{1}$. Then
$u\in C^{\infty}(B_{1})$.
\end{prop}
It follows from classical theory that the corresponding Dirichlet
problem admits a smooth solution (see \cite[Theorems 15.18 and 13.6]{gilbargTrudinger01}
and the Schauder estimates \cite[Theorem 6.17]{gilbargTrudinger01}).
The viscosity solution $u$ coincides with the smooth solution by
a comparison principle \cite[Theorem 3]{kawohlNikolai98}.
\subsection{Improvement of oscillation}
Our regularity estimates for solutions of (\ref{eq:regularized homogeneous})
are based on improvement of oscillation. We first prove such a result
for the linear equation
\begin{equation}
-\mathrm{tr}(G(x)D^{2}u)=f\quad\text{in }B_{1},\label{eq:linear equation}
\end{equation}
where $f\in C^{1}(B_{1})$ is bounded, $G(x)\in S^{N}$ and there
are constants $0<\lambda<\varLambda<\infty$ such that the eigenvalues
of $G(x)$ are in $[\lambda,\varLambda]$ for all $x\in B_{1}$. The
result is based on the following rescaled version of the weak Harnack
inequality found in \cite[Theorem 4.8]{caffarelliCabre}. Such Harnack
estimates for non-divergence form equations go back to at least Krylov
and Safonov \cite{krylovSafonov79,krylovSafonov80}.
\begin{lem}[Weak Harnack inequality]
\label{lem:weak harnack} Let $u\geq0$ be a continuous viscosity
supersolution to (\ref{eq:linear equation}) in $B_{1}$. Then there
are positive constants $C(\lambda,\varLambda,N)$ and $q(\lambda,\varLambda,N)$
such that for any $\tau<\frac{1}{4\sqrt{N}}$ we have
\begin{equation}
\tau^{-\frac{N}{q}}\left(\int_{B_{\tau}}\left|u\right|^{q}\,d x\right)^{1/q}\leq C\left(\inf_{B_{2\tau}}u+\tau\left(\int_{B_{4\sqrt{N}\tau}}\left|f\right|^{N}\,d x\right)^{1/N}\right).\label{eq:weak harnack}
\end{equation}
\end{lem}
\begin{proof}
Suppose that $\tau<\frac{1}{4\sqrt{N}}$ and set $S:=8\tau$. Define
the function $v:B_{\sqrt{N}/2}\rightarrow\mathbb{R}$ by
\begin{align*}
v(x) & :=u(Sx)
\end{align*}
and set
\[
\tilde{G}(x):=G(Sx)\quad\text{and}\quad\tilde{f}(x):=S^{2}f(Sx).
\]
Then, if $\varphi\in C^{2}$ touches $v$ from below at $x\in B_{\sqrt{N}/2}$,
the function $\phi(x):=\varphi(x/S)$ touches $u$ from below at $Sx$.
Therefore
\[
-\mathrm{tr}(G(Sx)D^{2}\phi(Sx))\geq f(Sx).
\]
Since $D^{2}\phi(Sx)=S^{-2}D^{2}\varphi(x)$, this implies that
\[
-\mathrm{tr}(G(Sx)D^{2}\varphi(x))\geq S^{2}f(Sx).
\]
Thus $v$ is a viscosity supersolution to
\[
-\mathrm{tr}(\tilde{G}(x)D^{2}v)\geq\tilde{f}(x)\quad\text{in }B_{\sqrt{N}/2}.
\]
We denote by $Q_{R}$ a cube with side-length $R/2$. Since $Q_{1}\subset B_{\sqrt{N}/2}$,
it follows from \cite[Theorem 4.8]{caffarelliCabre} that there are
$q(\lambda,\varLambda,N)$ and $C(\lambda,\varLambda,N)$ such that
\begin{align*}
\left(\int_{B_{1/8}}\left|v\right|^{q}\,d x\right)^{1/q}\leq\left(\int_{Q_{1/4}}\left|v\right|^{q}\,d x\right)^{1/q} & \leq C\left(\inf_{Q_{1/2}}v+\left(\int_{Q_{1}}|\tilde{f}|^{N}\,d x\right)^{1/N}\right)\\
& \leq C\left(\inf_{B_{1/4}}v+\left(\int_{B_{\sqrt{N}/2}}|\tilde{f}|^{N}\,d x\right)^{1/N}\right).
\end{align*}
By the change of variables formula we have
\begin{align*}
\int_{B_{1/8}}\left|v\right|^{q}\,d x= & \int_{B_{1/8}}\left|u(Sx)\right|^{q}\,d x=S^{-N}\int_{B_{S/8}}\left|u(x)\right|^{q}\,d x
\end{align*}
and
\[
\int_{B_{\sqrt{N}/2}}|\tilde{f}|^{N}\,d x=S^{2N}\int_{B_{\sqrt{N}/2}}\left|f(Sx)\right|^{N}\,d x=S^{N}\int_{B_{S\sqrt{N}/2}}\left|f(x)\right|^{N}\,d x.
\]
Recalling that $S=8\tau$, we get
\[
8^{-\frac{N}{q}}\tau^{-\frac{N}{q}}\left(\int_{B_{\tau}}\left|u(x)\right|^{q}\,d x\right)^{1/q}\leq C\left(\inf_{B_{2\tau}}u+8\tau\left(\int_{B_{S\sqrt{N}/2}}\left|f(x)\right|^{N}\,d x\right)^{1/N}\right).
\]
Absorbing $8^{\frac{N}{q}}$ into the constant, we obtain the claim.
\end{proof}
\begin{lem}[Improvement of oscillation for the linear equation]
\label{lem:imposc linear} Let $u\geq0$ be a continuous viscosity
supersolution to $\eqref{eq:linear equation}$ in $B_{1}$ and $\mu,l>0$.
Then there are positive constants $\tau(\lambda,\varLambda,N,\mu,l,\left\Vert f\right\Vert _{L^{\infty}(B_{1})})$
and $\theta(\lambda,\varLambda,N,\mu,l)$ such that if
\begin{equation}
\left|\left\{ x\in B_{\tau}:u\ge l\right\} \right|>\mu\left|B_{\tau}\right|,\label{eq:improvement of oscillation 1}
\end{equation}
then we have
\[
u\geq\theta\quad\text{in }B_{\tau}.
\]
\end{lem}
\begin{proof}
By the weak Harnack inequality (Lemma \ref{lem:weak harnack}) there
exist constants $C_{1}(\lambda,\varLambda,N)$ and $q(\lambda,\varLambda,N)$
such that for any $\tau<1/(4\sqrt{N}),$ we have
\begin{equation}
\inf_{B_{2\tau}}u\geq C_{1}\tau^{\frac{-N}{q}}\left(\int_{B_{\tau}}\left|u\right|^{q}\,d x\right)^{1/q}-\tau\left(\int_{B_{4\sqrt{N}\tau}}\left|f\right|^{N}\,d x\right)^{1/N}.\label{eq:improvement of oscillation 2}
\end{equation}
In particular, this holds for
\[
\tau:=\min\left(\frac{1}{4\sqrt{N}},\sqrt{\frac{C_{1}\left|B_{1}\right|^{\frac{1}{q}-\frac{1}{N}}\mu^{\frac{1}{q}}l}{2\cdot4\sqrt{N}(\left\Vert f\right\Vert _{L^{\infty}(B_{1})}+1)}}\right).
\]
We continue the estimate (\ref{eq:improvement of oscillation 2})
using the assumption (\ref{eq:improvement of oscillation 1}) and
obtain
\begin{align*}
\inf_{B_{\tau}}u\geq\inf_{B_{2\tau}}u & \geq C_{1}\tau^{-\frac{N}{q}}\left(\left|\left\{ x\in B_{\tau}:u\geq l\right\} \right|l^{q}\right)^{1/q}-\tau\left(\int_{B_{4\sqrt{N}\tau}}\left|f\right|^{N}\,d x\right)^{1/N}\\
& \geq C_{1}\tau^{-\frac{N}{q}}\mu^{\frac{1}{q}}\left|B_{\tau}\right|^{\frac{1}{q}}l-\tau\left|B_{4\sqrt{N}\tau}\right|^{\frac{1}{N}}\left\Vert f\right\Vert _{L^{\infty}(B_{1})}\\
& =C_{1}\left|B_{1}\right|^{\frac{1}{q}}\mu^{\frac{1}{q}}l\tau^{-\frac{N}{q}}\tau^{\frac{N}{q}}-4\sqrt{N}\left|B_{1}\right|^{\frac{1}{N}}\left\Vert f\right\Vert _{L^{\infty}(B_{1})}\tau^{2}\\
& =C_{1}\left|B_{1}\right|^{\frac{1}{q}}\mu^{\frac{1}{q}}l-4\sqrt{N}\left|B_{1}\right|^{\frac{1}{N}}\left\Vert f\right\Vert _{L^{\infty}(B_{1})}\tau^{2}.\\
& \geq\frac{1}{2}C_{1}\left|B_{1}\right|^{\frac{1}{q}}\mu^{\frac{1}{q}}l,\\
& =:\theta,
\end{align*}
where the last inequality follows from the choice of $\tau$.
\end{proof}
We are now ready to prove an improvement of oscillation for the gradient
of a solution to (\ref{eq:regularized homogeneous}). We first consider
the following lemma, where the improvement is considered towards a
fixed direction. We initially also restrict the range of $\left|q\right|$.
The idea is to differentiate the equation and observe that a suitable
function of $Du$ is a supersolution to the linear equation (\ref{eq:linear equation}).
Lemma \ref{lem:imposc linear} is then applied to obtain information
about $Du$.
\begin{lem}[Improvement of oscillation to direction]
\label{lem:imposc dir} Suppose that $p$ is smooth. Let $u$ be
a smooth solution to (\ref{eq:regularized homogeneous}) in $B_{1}$
with $\left|Du\right|\leq1$ and either $q=0$ or $\left|q\right|>2$.
Then for every $0<l<1$ and $\mu>0$ there exist positive constants
$\tau(N,\hat{p},l,\mu)<1$ and $\gamma(N,\hat{p},l,\mu)<1$ such that
\[
\left|\left\{ x\in B_{\tau}:Du\cdot d\leq l\right\} \right|>\mu\left|B_{\tau}\right|\quad\text{implies\ensuremath{\quad}}Du\cdot d\leq\gamma\text{ in }B_{\tau}
\]
whenever $d\in\partial B_{1}$.
\end{lem}
\begin{proof}
To simplify notation, we set
\begin{align*}
A_{ij}(x,\eta) & :=\,delta_{ij}+(p(x)-2)\frac{(\eta_{i}+q_{i})(\eta_{j}+q_{j})}{\left|\eta+q\right|^{2}+\varepsilon^{2}}.
\end{align*}
We also denote the functions $\mathcal{A}_{ij}:x\mapsto A_{ij}(x,Du(x))$,
$\mathcal{A}_{ij,x_{k}}:x\mapsto(\partial_{x_{k}}A_{ij})(x,Du(x))$
and $\mathcal{A}_{ij,\eta_{k}}:x\mapsto(\partial_{\eta_{k}}A_{ij})(x,Du(x))$.
Then, since $u$ is a smooth solution to (\ref{eq:regularized homogeneous})
in $B_{1}$, we have in Einstein's summation convention
\[
-\mathcal{A}_{ij}u_{ij}=0\quad\text{pointwise in }B_{1}.
\]
Differentiating this yields
\begin{align}
0=(\mathcal{A}_{ij}u_{ij})_{k} & =\mathcal{A}_{ij}u_{ijk}+(\mathcal{A}_{ij})_{k}u_{ij}\nonumber \\
& =\mathcal{A}_{ij}u_{ijk}+\mathcal{A}_{ij,\eta_{m}}u_{ij}u_{km}+\mathcal{A}_{ij,x_{k}}u_{ij}\quad\text{for all }k=1,\ldots N.\label{eq:imposc dir 2}
\end{align}
Multiplying these identities by $d_{k}$ and summing over $k$, we
obtain
\begin{align}
0 & =\mathcal{A}_{ij}u_{ijk}d_{k}+\mathcal{A}_{ij,\eta_{m}}u_{ij}u_{km}d_{k}+\mathcal{A}_{ij,x_{k}}u_{ij}d_{k}\nonumber \\
& =\mathcal{A}_{ij}(Du\cdot d-l)_{ij}+\mathcal{A}_{ij,\eta_{m}}u_{ij}(Du\cdot d-l)_{m}+\mathcal{A}_{ij,x_{k}}u_{ij}d_{k}.\label{eq:imposc dir 3}
\end{align}
Moreover, multiplying (\ref{eq:imposc dir 2}) by $2u_{k}$ and summing
over $k$, we obtain
\begin{align}
0 & =2\mathcal{A}_{ij}u_{ijk}u_{k}+2\mathcal{A}_{ij,\eta_{m}}u_{ij}u_{km}u_{k}+2\mathcal{A}_{ij,x_{k}}u_{ij}u_{k}\nonumber \\
& =\mathcal{A}_{ij}(2u_{ijk}u_{k}+2u_{kj}u_{ki})-2\mathcal{A}_{ij}u_{kj}u_{ki}+2\mathcal{A}_{ij,\eta_{m}}u_{ij}u_{km}u_{k}+2\mathcal{A}_{ij,x_{k}}u_{ij}u_{k}\nonumber \\
& =\mathcal{A}_{ij}(u_{k}^{2})_{ij}-2\mathcal{A}_{ij}u_{kj}u_{ki}+\mathcal{A}_{ij,\eta_{m}}u_{ij}(u_{k}^{2})_{m}+2\mathcal{A}_{ij,x_{k}}u_{ij}u_{k}\nonumber \\
& =\mathcal{A}_{ij}(\left|Du\right|^{2})_{ij}+\mathcal{A}_{ij,\eta_{m}}u_{ij}(\left|Du\right|^{2})_{m}+2\mathcal{A}_{ij,x_{k}}u_{ij}u_{k}-2\mathcal{A}_{ij}u_{kj}u_{ki}.\label{eq:imposc dir 4}
\end{align}
We will now split the proof into the cases $q=0$ or $\left|q\right|>2$,
and proceed in two steps: First we check that a suitable function
of $Du$ is a supersolution to the linear equation (\ref{lem:imposc linear})
and then apply Lemma \ref{lem:imposc linear} to obtain the claim.
\textbf{Case $q=0$, Step 1: }We denote $\Omega_{+}:=\left\{ x\in B_{1}:h(x)>0\right\} $,
where
\[
h:=(Du\cdot d-l+\frac{l}{2}\left|Du\right|^{2})^{+}.
\]
If $\left|Du\right|\leq l/2$, we have
\[
Du\cdot d-l+\frac{l}{2}\left|Du\right|^{2}\leq-\frac{l}{2}+\frac{l^{3}}{8}<0.
\]
This implies that $\left|Du\right|>l/2$ in $\Omega_{+}$. Therefore,
since $q=0$, we have in $\Omega_{+}$
\begin{align}
\left|\mathcal{A}_{ij,\eta_{m}}\right| & =\left|p(x)-2\right|\left|\frac{\,delta_{im}(u_{j}+q_{j})+\,delta_{jm}(u_{i}+q_{i})}{\left|Du+q\right|^{2}+\varepsilon^{2}}-\frac{2(u_{m}+q_{m})(u_{i}+q_{i})(u_{j}+q_{j})}{(\left|Du+q\right|^{2}+\varepsilon^{2})^{2}}\right|\nonumber \\
& \leq8l^{-1}\left\Vert p-2\right\Vert _{L^{\infty}(B_{1})},\label{eq:imposc dir 5}\\
\left|\mathcal{A}_{ij,x_{k}}\right| & =\left|Dp(x)\right|\left|\frac{(\eta_{i}+q_{i})(\eta_{j}+q_{j})}{\left|\eta+q\right|^{2}+\varepsilon^{2}}\right|\leq p_{L}.\label{eq:imposc dir 6}
\end{align}
Summing up the equations (\ref{eq:imposc dir 3}) and (\ref{eq:imposc dir 4})
multiplied by $2^{-1}l$, we obtain in $\Omega_{+}$
\begin{align*}
0=\ & \mathcal{A}_{ij}(Du\cdot d-l)_{ij}+\mathcal{A}_{ij,\eta_{m}}u_{ij}(Du\cdot d-l)_{m}+\mathcal{A}_{ij,x_{k}}u_{ij}d_{k}\\
& +2^{-1}l\big(\mathcal{A}_{ij}(\left|Du\right|^{2})_{ij}+\mathcal{A}_{ij,\eta_{m}}u_{ij}(\left|Du\right|^{2})_{m}+2\mathcal{A}_{ij,x_{k}}u_{ij}u_{k}-2\mathcal{A}_{ij}u_{kj}u_{ki}\big)\\
=\ & \mathcal{A}_{ij}h_{ij}+\mathcal{A}_{ij,\eta_{m}}u_{ij}h_{m}+\mathcal{A}_{ij,x_{k}}u_{ij}d_{k}+l\mathcal{A}_{ij,x_{k}}u_{ij}u_{k}-l\mathcal{A}_{ij}u_{kj}u_{ki}\\
\leq\ & \mathcal{A}_{ij}h_{ij}+\left|\mathcal{A}_{ij,\eta_{m}}u_{ij}\right|\left|h_{m}\right|+\left|\mathcal{A}_{ij,x_{k}}u_{ij}\right|\left|d_{k}+lu_{k}\right|-l\mathcal{A}_{ij}u_{kj}u_{ki}.
\end{align*}
Since $\left|Du\right|\leq1$, we have $\left|d_{k}+lu_{k}\right|^{2}\leq4$
and by uniform ellipticity $\mathcal{A}_{ij}u_{kj}u_{ki}\geq\min(p_{\min}-1,1)\left|u_{ij}\right|^{2}$.
Therefore, by applying Young's inequality with $\epsilon>0$, we obtain
from the above display
\begin{align*}
0 & \leq\mathcal{A}_{ij}h_{ij}+N^{2}\epsilon^{-1}(\left|h_{m}\right|^{2}+\left|d_{k}+lu_{k}\right|^{2})+\epsilon(\left|\mathcal{A}_{ij,\eta_{m}}\right|^{2}+\left|\mathcal{A}_{ij,x_{k}}\right|^{2})\left|u_{ij}\right|^{2}-l\mathcal{A}_{ij}u_{kj}u_{ki}\\
& \leq\mathcal{A}_{ij}h_{ij}+N^{2}\epsilon^{-1}(\left|Dh\right|^{2}+4)+\epsilon C(N,\hat{p})(l^{-2}+1)\left|u_{ij}\right|^{2}-l\min(p_{\text{min}}-1,1)\left|u_{ij}\right|^{2},
\end{align*}
where in the second estimate we used (\ref{eq:imposc dir 5}) and
(\ref{eq:imposc dir 6}). By taking $\epsilon$ small enough, we obtain
\begin{equation}
0\leq\mathcal{A}_{ij}h_{ij}+C_{0}(N,\hat{p})\frac{\left|Dh\right|^{2}+1}{l^{3}}\quad\text{in }\Omega_{+},\label{eq:imposc dir 7}
\end{equation}
Next we define
\begin{equation}
\overline{h}:=\frac{1}{\nu}(1-e^{\nu(h-H)}),\quad\text{where}\quad H:=1-\frac{l}{2}\quad\text{and}\quad\nu:=\frac{C_{0}}{l^{3}\min(p_{\text{min}}-1,1)}.\label{eq:imposc dir 10}
\end{equation}
Then by (\ref{eq:imposc dir 7}) and uniform ellipticity we have in
$\Omega_{+}$
\begin{align*}
-\mathcal{A}_{ij}\overline{h}{}_{ij} & =\mathcal{A}_{ij}(h_{ij}e^{\nu(h-H)}+\nu h_{i}h_{j}e^{\nu(h-H)})\\
& \geq e^{\nu(h-H)}(-C_{0}\frac{\left|Dh\right|^{2}}{l^{3}}-\frac{C_{0}}{l^{3}}+\nu\min(p_{\text{min}}-1,1)\left|Dh\right|^{2})\\
& \geq-\frac{C_{0}}{l^{3}}.
\end{align*}
Since the minimum of two viscosity supersolutions is still a viscosity
supersolution, it follows from the above estimate that $\overline{h}$
is a non-negative viscosity supersolution to
\begin{equation}
-\mathcal{A}_{ij}\overline{h}_{ij}\ge\frac{-C_{0}}{l^{3}}\quad\text{in }B_{1}.\label{eq:imposc dir 8}
\end{equation}
\textbf{Case $q=0$, Step 2: }We set $l_{0}:=\frac{1}{\nu}(1-e^{\nu(l-1)})$.
Then, since $\overline{h}$ solves (\ref{eq:imposc dir 8}), by Lemma
\ref{lem:imposc linear} there are positive constants $\tau(N,p,l,\mu)$
and $\theta(N,p,l,\mu)$ such that
\[
\left|\left\{ x\in B_{\tau}:\overline{h}\geq l_{0}\right\} \right|>\mu\left|B_{\tau}\right|\quad\text{implies}\quad\overline{h}\geq\theta\quad\text{in }B_{\tau}.
\]
If $Du\cdot d\leq l$, we have $\overline{h}\geq l_{0}$ and therefore
\[
\left|\left\{ x\in B_{\tau}:\overline{h}\geq l_{0}\right\} \right|\geq\left|\left\{ x\in B_{\tau}:Du\cdot d\leq l\right\} \right|>\mu\left|B_{\tau}\right|,
\]
where the last inequality follows from the assumptions. Consequently,
we obtain
\[
\overline{h}\geq\theta\quad\text{in }B_{\tau}.
\]
Since $h-H\leq0$, by convexity we have $H-h\geq\overline{h}$. This
together with the above estimate yields
\[
1-2^{-1}l-(Du\cdot d-l+2^{-1}l\left|Du\right|^{2})\geq\theta\quad\text{in }B_{\tau}
\]
and so
\[
Du\cdot d+2^{-1}l(Du\cdot d)^{2}\leq Du\cdot d+2^{-1}l\left|Du\right|^{2}\leq1+2^{-1}l-\theta\quad\text{in }B_{\tau}.
\]
Using the quadratic formula, we thus obtain the desired estimate
\[
Du\cdot d\leq\frac{-1+\sqrt{1+2l(1+2^{-1}l-\theta)}}{l}=\frac{-1+\sqrt{(1+l)^{2}-2l\theta}}{l}=:\gamma<1\quad\text{in }B_{\tau}.
\]
\textbf{Case $\left|q\right|>2$: }Computing like in (\ref{eq:imposc dir 5})
and (\ref{eq:imposc dir 6}), we obtain this time in $B_{1}$
\[
\left|\mathcal{A}_{ij,\eta_{m}}\right|\leq4\left\Vert p-2\right\Vert _{L^{\infty}(B_{1})}\quad\text{and}\quad\left|\mathcal{A}_{ij,x_{k}}\right|\leq p_{L}
\]
Moreover, this time we set simply
\[
h:=Du\cdot d-l+2^{-1}l\left|Du\right|^{2}.
\]
Summing up the identities (\ref{eq:imposc dir 3}) and (\ref{eq:imposc dir 4})
and using Young's inequality similarly as in the case $\left|q\right|=0$,
we obtain in $B_{1}$
\begin{align*}
0 & \leq\mathcal{A}_{ij}h_{ij}+N^{2}\epsilon^{-1}(\left|h_{m}\right|^{2}+\left|d_{k}+lu_{k}\right|^{2})+\epsilon(\left|\mathcal{A}_{ij,\eta_{m}}\right|^{2}+\left|\mathcal{A}_{ij,x_{k}}\right|^{2})\left|u_{ij}\right|^{2}-l\mathcal{A}_{ij}u_{kj}u_{ki}\\
& \leq\mathcal{A}_{ij}h_{ij}+N^{2}\epsilon^{-1}(\left|Dh\right|^{2}+4)+\epsilon C(\hat{p})\left|u_{ij}\right|^{2}-lC(\hat{p})\left|u_{ij}\right|^{2}.
\end{align*}
By taking small enough $\epsilon$, we obtain
\[
0\leq\mathcal{A}_{ij}h_{ij}+C_{0}(N,\hat{p})\frac{\left|Dh\right|^{2}+1}{l}\quad\text{in }B_{1}.
\]
Next we define $\overline{h}$ and $H$ like in (\ref{eq:imposc dir 10}),
but set instead $\nu:=C_{0}/(l\min(p_{\min}-1,1))$. The rest of the
proof then proceeds in the same way as in the case $q=0$.\begin{details}
\[
\overline{h}:=\frac{1}{\nu}(1-e^{\nu(h-H)}),\quad\text{where}\quad H:=1-\frac{l}{2}\quad\text{and}\quad\nu:=\frac{C_{0}}{l\min(p_{\text{min}}-1,1)}.
\]
Then we have in $B_{1}$
\begin{align}
-A_{ij}\overline{h}_{ij} & =A_{ij}(h_{ij}e^{\nu(h-H)}+\nu h_{i}h_{j}e^{\nu(h-H)})\nonumber \\
& \geq e^{\nu(h-H)}(-C_{0}\frac{\left|Dh\right|^{2}}{l}-C_{0}+\nu\min(p_{\text{min}}-1,1)\left|Dh\right|^{2})\nonumber \\
& \geq-C_{0}.\label{eq:imposc dir 9}
\end{align}
\textbf{(Case $\left|q\right|>2$, Step 2)} We set $l_{0}:=\frac{1}{\nu}(1-e^{\nu(l-1)})$.
Then, since $\overline{h}$ solves (\ref{eq:imposc dir 9}), by Lemma
(\ref{eq:imposc dir 9}) there are positive constants $\tau(p,N,l,\mu)$
and $\theta(p,N,l,\mu)$ such that
\[
\left|\left\{ x\in B_{\tau}:\overline{h}\geq l_{0}\right\} \right|>\mu\left|B_{\tau}\right|\quad\text{implies\ensuremath{\quad}}\overline{h}\ge\theta\quad\text{in }B_{\tau}.
\]
If $Du\cdot d\leq l$, we have $\overline{h}\geq l_{0}$ and therefore
\[
\left|\left\{ x\in B_{\tau}:h\geq l_{0}\right\} \right|\geq\left|\left\{ x\in B_{\tau}:Du\cdot d\leq l\right\} \right|>\mu\left|B_{\tau}\right|,
\]
where the last estimate follows from the assumptions. Consequently
we obtain
\[
\overline{h}\geq\theta\quad\text{in }B_{\tau}.
\]
By convexity we have $H-h\geq\overline{h}.$ This together with the
above estimate yields
\[
1-2^{-1}l-(Du\cdot d-l+2^{-1}l\left|Du\right|^{2})\geq\theta\quad\text{in }B_{\tau}
\]
and so
\[
Du\cdot d+2^{-1}l(Du\cdot d)^{2}\leq Du\cdot d+2^{-1}l\left|Du\right|^{2}\leq1+2^{-1}l-\theta\quad\text{in }B_{\tau}.
\]
\end{details}
\end{proof}
Next we inductively apply the previous lemma to prove the improvement
of oscillation.
\begin{thm}[Improvement of oscillation]
\label{thm:imposc} Suppose that $p$ is smooth. Let $u$ be a smooth
solution to (\ref{eq:regularized homogeneous}) in $B_{1}$ with $\left|Du\right|\leq1$
and either $q=0$ or $\left|q\right|>2$. Then for every $0<l<1$
and $\mu>0$ there exist positive constants $\tau(N,\hat{p},l,\mu)<1$
and $\gamma(N,\hat{p},l,\mu)<1$ such that if
\begin{equation}
\left|\left\{ x\in B_{\tau^{i+1}}:Du\cdot d\leq l\gamma^{i}\right\} \right|>\mu\left|B_{\tau^{i+1}}\right|\quad\text{for all }d\in\partial B_{1},\ i=0,\ldots,k,\label{eq:imposc cnd}
\end{equation}
then
\begin{equation}
\left|Du\right|\leq\gamma^{i+1}\quad\text{in }B_{\tau^{i+1}}\quad\text{for all }i=0,\ldots,k.\label{eq:imposc cnd2}
\end{equation}
\end{thm}
\begin{proof}
Let $k\geq0$ be an integer and suppose that (\ref{eq:imposc cnd})
holds. We proceed by induction.\textbf{ }
\textbf{Initial step:} Since (\ref{eq:imposc cnd}) holds for $i=0$,
by Lemma \ref{lem:imposc dir} we have $Du\cdot d\leq\gamma$ in $B_{\tau}$
for all $d\in\partial B_{1}$. This implies (\ref{eq:imposc cnd2})
for $i=0$.
\textbf{Induction step:} Suppose that $0<i\leq k$ and that (\ref{eq:imposc cnd2})
holds for $i-1$. We define
\[
v(x):=\tau^{-i}\gamma^{-i}u(\tau^{i}x).
\]
Then $v$ solves
\[
-\Delta v-(p(\tau^{i}x)-2)\frac{\left\langle D^{2}v(Dv+\gamma^{-i}q),Dv+\gamma^{i}q\right\rangle }{\left|Dv+\gamma^{-i}q\right|^{2}+(\gamma^{-i}\varepsilon)^{2}}=0\quad\text{in }B_{1}.
\]
Moreover, by induction hypothesis $\left|Dv(x)\right|=\gamma^{-i}\left|Du(\tau^{i}x)\right|\leq\gamma^{-i}\gamma^{i}=1$
in $B_{1}$. Therefore by Lemma \ref{lem:imposc dir} we have that
\begin{equation}
\left|\left\{ x\in B_{\tau}:Dv\cdot d\leq l\right\} \right|>\mu\left|B_{\tau}\right|\quad\text{implies}\quad Dv\cdot d\leq\gamma\text{ in }B_{\tau}\label{eq:imposc 1}
\end{equation}
whenever $d\in\partial B_{1}$. Since
\[
\left|\left\{ x\in B_{\tau}:Dv\cdot d\leq l\right\} \right|>\mu\left|B_{\tau}\right|\iff\left|\left\{ x\in B_{\tau^{i+1}}:Du\cdot d\leq l\gamma^{i}\right\} \right|>\mu\left|B_{\tau^{i+1}}\right|,
\]
we have by (\ref{eq:imposc cnd}) and (\ref{eq:imposc 1}) that $Dv\cdot d\leq\gamma$
in $B_{\tau}$. This implies that $Du\cdot d\leq\gamma^{i+1}$ in
$B_{\tau^{i+1}}$. Since $d\in\partial B_{1}$ was arbitrary, we obtain
(\ref{eq:imposc cnd2}) for $i$.
\end{proof}
\subsection{Hölder gradient estimates}
In this section we apply the improvement of oscillation to prove $C^{1,\alpha}$-estimates
for solutions to (\ref{eq:regularized homogeneous}). We need the
following regularity result by Savin \cite{savin07}.
\begin{lem}
\label{lem:small perturbation}Suppose that $p$ is smooth. Let $u$
be a smooth solution to (\ref{eq:regularized homogeneous}) in $B_{1}$
with $\left|Du\right|\leq1$ and either $q=0$ or $\left|q\right|>2$.
Then for any $\beta>0$ there exist positive constants $\eta(N,\hat{p},\beta)$
and $C(N,\hat{p},\beta)$ such that if
\[
\left|u-L\right|\leq\eta\quad\text{in }B_{1}
\]
for some affine function $L$ satisfying $1/2\leq\left|DL\right|\leq1$,
then we have
\[
\left|Du(x)-Du(0)\right|\leq C\left|x\right|^{\beta}\quad\text{for all }x\in B_{1/2}.
\]
\end{lem}
\begin{proof}
Set $v:=u-L$. Then $v$ solves
\begin{equation}
-\Delta u-\frac{(p(x)-2)\left\langle D^{2}u(Du+q+DL),Du+q+DL\right\rangle }{\left|Du+q+DL\right|^{2}+\varepsilon^{2}}=0\quad\text{in }B_{1}.\label{eq:small perturbation}
\end{equation}
Observe that by the assumption $1/2\leq\left|DL\right|\leq1$ we have
$\left|Du+q+DL\right|\geq1/4$ if $\left|Du\right|\leq1/4$. It therefore
follows from \cite[Theorem 1.3]{savin07} (see also \cite{wang13})
that $\left\Vert v\right\Vert _{C^{2,\beta}(B_{1/2})}\leq C$ which
implies the claim.
\end{proof}
We also use the following simple consequence of Morrey's inequality.
\begin{lem}
\label{lem:morrey lemma}Let $u:B_{1}\rightarrow\mathbb{R}$ be a
smooth function with $\left|Du\right|\leq1$. For any $\theta>0$
there are constants $\varepsilon_{1}(N,\theta),\varepsilon_{0}(N,\theta)<1$
such that if the condition
\[
\left|\left\{ x\in B_{1}:\left|Du-d\right|>\varepsilon_{0}\right\} \right|\leq\varepsilon_{1}
\]
is satisfied for some $d\in S^{N-1}$, then there is $a\in\mathbb{R}$
such that
\[
\left|u(x)-a-d\cdot x\right|\leq\theta\text{ for all }x\in B_{1/2}.
\]
\end{lem}
\begin{proof}
By Morrey's inequality (see for example\ \cite[Theorem 4.10]{measuretheoryevans})
\begin{align*}
\underset{x\in B_{1/2}}{\operatorname{osc}}(u(x)-d\cdot x) & =\sup_{x,y\in B_{1/2}}\left|u(x)-d\cdot x-u(y)+d\cdot y\right|\\
& \leq C(N)\Big(\int_{B_{1}}\left|Du-d\right|^{2N}\,d x\Big)^{\frac{1}{2N}}\\
& \leq C(N)(\varepsilon_{1}^{\frac{1}{2N}}+\varepsilon_{0}).
\end{align*}
Therefore, denoting $a:=\inf_{x\in B_{1/2}}(u(x)-d\cdot x)$, we have
for any $x\in B_{1/2}$
\[
\left|u(x)-a-d\cdot x\right|\leq\operatorname{osc}_{B_{1/2}}(u(x)-d\cdot x)\leq C(N)(\varepsilon_{1}^{\frac{1}{2N}}+\varepsilon_{0})\leq\theta,
\]
where the last inequality follows by taking small enough $\varepsilon_{0}$
and $\varepsilon_{1}$.
\end{proof}
We are now ready to prove a Hölder estimate for the gradient of solutions
to (\ref{eq:regularized homogeneous}). We first restrict the range
of $\left|q\right|$.
\begin{lem}
\label{thm:regularized apriori} Suppose that $p$ is smooth. Let
$u$ be a smooth solution to (\ref{eq:regularized homogeneous}) in
$B_{1}$ with $\left|Du\right|\leq1$ and either $q=0$ or $\left|q\right|>2$.
Then there exists a constant $\alpha(N,\hat{p})\in(0,1)$ such that
\[
\left\Vert Du\right\Vert _{C^{\alpha}(B_{1/2})}\leq C(N,\hat{p}).
\]
\end{lem}
\begin{proof}
For $\beta=1/2,$ let $\eta>0$ be as in Lemma \ref{lem:small perturbation}.
For $\theta=\eta/2$, let $\varepsilon_{0},\varepsilon_{1}$ be as
in Lemma \ref{lem:morrey lemma}. Set
\[
l:=1-\frac{\varepsilon_{0}^{2}}{2}\quad\text{and}\quad\mu:=\frac{\varepsilon_{1}}{\left|B_{1}\right|}.
\]
For these $l$ and $\mu$, let $\tau,\gamma\in(0,1)$ be as in Theorem
\ref{thm:imposc}. Let $k\geq0$ be the minimum integer such that
the condition (\ref{eq:imposc cnd}) does not hold.
\textbf{Case $k=\infty$:} Theorem \ref{thm:imposc} implies that
\[
\left|Du\right|\leq\gamma^{i+1}\quad\text{in }B_{\tau^{i+1}}\text{ for all }i\geq0.
\]
Let $x\in B_{\tau}\setminus\left\{ 0\right\} $. Then $\tau^{i+1}\leq\left|x\right|\leq\tau^{i}$
for some $i\geq0$ and therefore
\[
i\leq\frac{\log\left|x\right|}{\log\tau}\leq i+1.
\]
We obtain
\begin{equation}
\left|Du(x)\right|\leq\gamma^{i}=\frac{1}{\gamma}\gamma^{i+1}\leq\frac{1}{\gamma}\gamma^{\frac{\log\left|x\right|}{\log\tau}}=\frac{1}{\gamma}\gamma^{\frac{\log\left|x\right|}{\log\gamma}\cdot\frac{\log\gamma}{\log\tau}}=:C\left|x\right|^{\alpha},\label{eq:regularized apriori 0}
\end{equation}
where $C=1/\gamma$ and $\alpha=\log\gamma/\log\tau$.
\textbf{Case $k<\infty$:} There is $d\in\partial B_{1}$ such that
\begin{equation}
\left|\left\{ x\in B_{\tau^{k+1}}:Du\cdot d\leq l\gamma^{k}\right\} \right|\leq\mu\left|B_{\tau^{k+1}}\right|.\label{eq:regularized apriori 1}
\end{equation}
We set
\[
v(x):=\tau^{-k-1}\gamma^{-k}u(\tau^{k+1}x).
\]
Then $v$ solves
\[
-\Delta v-(p(\tau^{k+1}x)-2)\frac{\left\langle D^{2}v(Dv+\gamma^{-k}q),Dv+\gamma^{-k}q\right\rangle }{\left|Dv+\gamma^{-k}q\right|^{2}+\gamma^{-2k}\varepsilon^{2}}=0\quad\text{in }B_{1}
\]
and by (\ref{eq:regularized apriori 1}) we have
\begin{align}
\left|\left\{ x\in B_{1}:Dv\cdot d\leq l\right\} \right| & =\left|\left\{ x\in B_{1}:Du(\tau^{k+1}x)\cdot d\leq l\gamma^{k}\right\} \right|\nonumber \\
& =\tau^{-N(k+1)}\left|\left\{ x\in B_{\tau^{k+1}}:Du(x)\cdot d\leq l\gamma^{k}\right\} \right|\nonumber \\
& \leq\tau^{-N(k+1)}\mu\left|B_{\tau^{k+1}}\right|=\mu\left|B_{1}\right|=\varepsilon_{1}.\label{eq:regularized apriori 2}
\end{align}
Since either $k=0$ or (\ref{eq:imposc cnd}) holds for $k-1$, it
follows from Theorem \ref{thm:imposc} that $\left|Du\right|\leq\gamma^{k}$
in $B_{\tau^{k}}$. Thus
\begin{equation}
\left|Dv(x)\right|=\gamma^{-k}\left|Du(\tau^{k+1}x)\right|\leq1\quad\text{in }B_{1}.\label{eq:regularized apriori 3}
\end{equation}
For vectors $\xi,d\in B_{1}$, it is easy to verify the following
fact
\[
\left|\xi-d\right|>\varepsilon_{0}\implies\xi\cdot d\leq1-\varepsilon_{0}^{2}/2=l.
\]
Therefore, in view of (\ref{eq:regularized apriori 2}) and (\ref{eq:regularized apriori 3}),
we obtain
\[
\left|\left\{ x\in B_{1}:\left|Dv-d\right|>\varepsilon_{0}\right\} \right|\leq\varepsilon_{1}.
\]
Thus by Lemma \ref{lem:morrey lemma} there is $a\in\mathbb{R}$ such
that
\[
\left|v(x)-a-d\cdot x\right|\leq\theta=\eta/2\quad\text{for all }x\in B_{1/2}.
\]
Consequently, by applying Lemma \ref{lem:small perturbation} on the
function $2v(2^{-1}x)$, we find a positive constant $C(N,\hat{p})$
and $e\in\partial B_{1}$ such that
\[
\left|Dv(x)-e\right|\leq C\left|x\right|\quad\text{in }B_{1/4}.
\]
Since $\left|Dv\right|\leq1$, we have also
\[
\left|Dv(x)-e\right|\leq C\left|x\right|\quad\text{in }B_{1}.
\]
Recalling the definition of $v$ and taking $\alpha^{\prime}\in(0,1)$
so small that $\gamma/\tau^{\alpha^{\prime}}<1$ we obtain
\begin{equation}
\left|Du(x)-\gamma^{k}e\right|\leq C\gamma^{k}\tau^{-k-1}\left|x\right|\leq\frac{C}{\tau^{\alpha^{\prime}}}\left(\frac{\gamma}{\tau^{\alpha^{\prime}}}\right)^{k}\left|x\right|^{\alpha^{\prime}}\leq C\left|x\right|^{\alpha^{\prime}}\quad\text{in }B_{\tau^{k+1}},\label{eq:regularized apriori 4}
\end{equation}
where we absorbed $\tau^{\alpha^{\prime}}$ into the constant. On
the other hand, we have
\[
\left|Du\right|\leq\gamma^{i+1}\quad\text{in }B_{\tau^{i+1}}\text{ for all }i=0,\ldots,k-1
\]
so that, if $\tau^{i+2}\leq\left|x\right|\leq\tau^{i+1}$ for some
$i\in\left\{ 0,\ldots,k-1\right\} $, it holds that
\[
\left|Du(x)-\gamma^{k}e\right|\leq2\gamma^{i+1}\frac{\left|x\right|^{\alpha^{\prime}}}{\left|x\right|^{\alpha^{\prime}}}\leq\frac{2}{\tau^{\alpha^{\prime}}}\left(\frac{\gamma}{\tau^{\alpha^{\prime}}}\right)^{i+1}\left|x\right|^{\alpha^{\prime}}\le C\left|x\right|^{\alpha^{\prime}}.
\]
Combining this with (\ref{eq:regularized apriori 4}) we obtain
\begin{equation}
\left|Du(x)-\gamma^{k}e\right|\leq C\left|x\right|^{\alpha^{\prime}}\quad\text{in }B_{\tau}.\label{eq:regularized apriori 5}
\end{equation}
The claim now follows from (\ref{eq:regularized apriori 0}) and (\ref{eq:regularized apriori 5})
by standard translation arguments. \begin{details}
Let $x_{0}\in B_{1}$. We set
\[
v(x):=2u(\frac{1}{2}(x-x_{0})).
\]
Since $v$ solves
\[
\Delta v+(p(\frac{1}{2}(x-x_{0}))-2)\frac{\left\langle D^{2}v(Dv+q),Dv+q\right\rangle }{\left|Dv+q\right|^{2}+\varepsilon^{2}}=0\quad\text{in }B_{1},
\]
by Lemma \ref{thm:regularized apriori} there exists $C(p,N)$ and
$\alpha(p,N)$ such that
\[
\left|Dv(x)-Dv(0)\right|\leq C\left|x\right|^{\alpha}\quad\text{for all }x\in B_{1/2}.
\]
In other words, we have
\[
\left|Du(\frac{1}{2}(x-x_{0}))-Du(\frac{1}{2}x_{0})\right|\leq C\left|x\right|^{\alpha}\quad\text{for all }x\in B_{1/2},x_{0}\in B_{1},
\]
from which it follows that
\[
\left|Du(x-x_{0})-Du(x_{0})\right|\le C\left|x\right|^{\alpha}\quad\text{for all }x\in B_{1/4},x_{0}\in B_{1/2}.
\]
This implies the Hölder estimate
\[
\left\Vert Du\right\Vert _{C^{\alpha}(B_{1/2})}\leq C.
\]
\end{details}
\end{proof}
\begin{thm}
\label{cor:h=0000F6lder estimate for regularized} Let $u$ be a bounded
viscosity solution to (\ref{eq:regularized homogeneous}) in $B_{1}$
with $q\in\mathbb{R}^{N}$. Then
\begin{equation}
\left\Vert u\right\Vert _{C^{1,\alpha}(B_{1/2})}\leq C(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})})\label{eq:h=0000F6lder estimate for regularized 1}
\end{equation}
for some $\alpha(N,\hat{p})\in(0,1)$.
\end{thm}
\begin{proof}
Suppose first that $p$ is smooth. Let $\nu_{0}(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})})$
and $C_{0}(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})})$
be as in the Lipschitz estimate (Theorem \ref{thm:Lipschitz estimate}
in the Appendix) and set
\[
M:=2\max(\nu_{0},C_{0}).
\]
If $\left|q\right|>M$, then by Theorem \ref{lem:lipschitz lemma}
we have
\[
\left|Du\right|\leq C_{0}\quad\text{in }B_{1/2}.
\]
We set $\tilde{u}(x):=2u(x/2)/C_{0}$. Then $\left|D\tilde{u}\right|\leq1$
in $B_{1}$ and $\tilde{u}$ solves
\[
-\Delta\tilde{u}-(p(x/2)-2)\frac{\left\langle D^{2}\tilde{u}(D\tilde{u}+q/C_{0}),D\tilde{u}+q/C_{0}\right\rangle }{\left|D\tilde{u}+q/C_{0}\right|^{2}+(\varepsilon/C_{0})^{2}}=0\quad\text{in }B_{1},
\]
where $q/C_{0}>2$. Thus by Theorem \ref{thm:regularized apriori}
we have
\[
\left\Vert D\tilde{u}\right\Vert _{C^{\alpha}(B_{1/2})}\leq C(N,\hat{p}),
\]
which implies (\ref{eq:h=0000F6lder estimate for regularized 1})
by standard translation arguments.
If $\left|q\right|\leq M$, we define
\[
w:=u-q\cdot x.
\]
Then by Theorem \ref{thm:Lipschitz estimate} we have
\[
\left|Dw\right|\leq C(N,\hat{p},\left\Vert w\right\Vert _{L^{\infty}(B_{1})})=:C^{\prime}(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})})\quad\text{in }B_{1/2}.
\]
We set $\tilde{w}(x):=2w(x/2)/C^{\prime}.$ Then $\left|D\tilde{w}\right|\leq1$
and so by Theorem \ref{lem:small perturbation} we have
\[
\left\Vert D\tilde{w}\right\Vert _{C^{\alpha}(B_{1/2})}\leq C(N,\hat{p}),
\]
which again implies (\ref{eq:h=0000F6lder estimate for regularized 1}).
Suppose then that $p$ is merely Lipschitz continuous. Take a sequence
$p_{j}\in C^{\infty}(B_{1})$ such that $p_{j}\rightarrow p$ uniformly
in $B_{1}$ and $\left\Vert Dp_{j}\right\Vert _{L^{\infty}(B_{1})}\leq\left\Vert Dp\right\Vert _{L^{\infty}(B_{1})}$.
For $r<1$, let $u_{j}$ be a solution to the Dirichlet problem
\[
\begin{cases}
-\Delta u_{j}-(p_{j}(x)-2)\frac{\left\langle D^{2}u(Du_{j}+q),Du_{j}+q\right\rangle }{\left|Du_{j}+q\right|^{2}+\varepsilon^{2}}=0 & \text{in }B_{r},\\
u_{j}=u & \text{on }B_{r}.
\end{cases}
\]
As observed in Proposition \ref{prop:c infty}, the solution exists
and we have $u_{j}\in C^{\infty}(B_{r})$. By comparison principle
$\left\Vert u_{j}\right\Vert _{L^{\infty}(B_{r})}\leq\left\Vert u\right\Vert _{L^{\infty}(B_{1})}$.
Then by the first part of the proof we have the estimate
\[
\left\Vert u_{j}\right\Vert _{C^{1,\beta}(B_{r/2})}\leq C(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})}).
\]
By \cite[Theorem 4.14]{caffarelliCabre} the functions $u_{j}$ are
equicontinuous in $B_{1}$ and so by the Ascoli-Arzela theorem we
have $u_{j}\rightarrow v$ uniformly in $B_{1}$ up to a subsequence.
Moreover, by the stability principle $v$ is a solution to (\ref{eq:regularized homogeneous})
in $B_{r}$ and thus by comparison principle \cite[Theorem 2.6]{kawohlKutev07}
we have $v\equiv u$. By extracting a further subsequence, we may
ensure that also $Du_{j}\rightarrow Du$ uniformly in $B_{r/2}$ and
so the estimate $\left\Vert Du\right\Vert _{C^{1,\beta}(B_{r/2})}\leq C(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})})$
follows.
\end{proof}
\section{Hölder gradient estimates for the regularized inhomogeneous equation\label{sec:sec 2}}
In this section we consider the inhomogeneous equation
\begin{equation}
-\Delta u-(p(x)-2)\frac{\left\langle D^{2}u(Du+q),Du+q\right\rangle }{\left|Du\right|^{2}+\varepsilon^{2}}=f(x)\quad\text{in }B_{1},\label{eq:non-homogeneous reguralized}
\end{equation}
where $p:B_{1}\rightarrow\mathbb{R}$ is Lipschitz continuous, $p_{\min}>1$,
$\varepsilon>0$, $q\in\mathbb{R}^{N}$ and $f\in C(B_{1})$ is bounded.
We apply the $C^{1,\alpha}$-estimates obtained in Theorem \ref{cor:h=0000F6lder estimate for regularized}
to prove regularity estimates for solutions of (\ref{eq:non-homogeneous reguralized})
with $q=0$. Our arguments are similar to those in \cite[Section 3]{attouchiParv},
see also \cite{imbertSilvestre12}. The idea is to use the well known
characterization of $C^{1,\alpha}$-regularity via affine approximates.
The following lemma plays a key role: It states that if $f$ is small,
then a solution to (\ref{eq:non-homogeneous reguralized}) can be
approximated by an affine function. This combined with scaling properties
of the equation essentially yields the desired affine functions.
\begin{lem}
\label{lem:non-homogeneous regularized first lemma}There exist constants
$\epsilon(N,\hat{p})$,$\tau(N,\hat{p})\in(0,1)$ such that the following
holds: If $\left\Vert f\right\Vert _{L^{\infty}(B_{1})}\leq\epsilon$
and $w$ is a viscosity solution to (\ref{eq:non-homogeneous reguralized})
in $B_{1}$ with $q\in\mathbb{R}^{N}$, $w(0)=0$ and $\operatorname{osc}_{B_{1}}w\leq1$,
then there exists $q^{\prime}\in\mathbb{R}^{N}$ such that
\[
\operatorname{osc}_{B_{\tau}}(w(x)-q^{\prime}\cdot x)\leq\frac{1}{2}\tau.
\]
Moreover, we have $\left|q^{\prime}\right|\le C(N,\hat{p})$.
\end{lem}
\begin{proof}
Suppose on the contrary that the claim does not hold. Then, for a
fixed $\tau(N,\hat{p})$ that we will specify later, there exists
a sequence of Lipschitz continuous functions $p_{j}:B_{1}\rightarrow\mathbb{R}$
such that
\[
p_{\min}\leq\inf_{B_{1}}p_{j}\leq\sup_{B_{1}}p_{j}\leq p_{\max}\quad\text{and}\quad(p_{j})_{L}\leq p_{L},
\]
functions $f_{j}\in C(B_{1})$ such that $f_{j}\rightarrow0$ uniformly
in $B_{1}$, vectors $q_{j}\in\mathbb{R}^{N}$ and viscosity solutions
$w_{j}$ to
\[
-\Delta w_{j}-(p_{j}(x)-2)\frac{\left\langle D^{2}w_{j}(Dw_{j}+q_{j}),Dw_{j}+q_{j}\right\rangle }{\left|Dw_{j}+q_{j}\right|^{2}+\varepsilon^{2}}=f_{j}(x)\quad\text{in }B_{1}
\]
such that $w_{j}(0)=0$, $\operatorname{osc}_{B_{1}}w_{j}\leq1$ and
\begin{equation}
\operatorname{osc}_{B_{\tau}}(w_{j}(x)-q^{\prime}\cdot x)>\frac{\tau}{2}\quad\text{for all }q^{\prime}\in\mathbb{R}^{N}.\label{eq:first 0}
\end{equation}
By \cite[Proposition 4.10]{caffarelliCabre}, the functions $w_{j}$
are uniformly Hölder continuous in $B_{r}$ for any $r\in(0,1)$.
Therefore by the Ascoli-Arzela theorem, we may extract a subsequence
such that $w_{j}\rightarrow w_{\infty}$ and $p_{j}\rightarrow p_{\infty}$
uniformly in $B_{r}$ for any $r\in(0,1)$. Moreover, $p_{\infty}$
is $p_{L}$-Lipschitz continuous and $p_{\min}\leq p_{\infty}\leq p_{\max}$.
It then follows from (\ref{eq:first 0}) that
\begin{equation}
\operatorname{osc}_{B_{\tau}}(w_{\infty}(x)-q^{\prime}\cdot x)>\frac{\tau}{2}\quad\text{for all }q^{\prime}\in\mathbb{R}^{N}.\label{eq:first 1}
\end{equation}
We have two cases: either $q_{j}$ is bounded or unbounded.
\textbf{Case $q_{j}$ is bounded: }In this case $q_{j}\rightarrow q_{\infty}\in\mathbb{R}^{N}$
up to a subsequence. It follows from the stability principle that
$w_{\infty}$ is a viscosity solution to
\begin{equation}
-\Delta w_{\infty}-(p_{\infty}(x)-2)\frac{\left\langle D^{2}w_{\infty}(Dw_{\infty}+q_{\infty}),Dw_{\infty}+q_{\infty}\right\rangle }{\left|Dw_{\infty}+q_{\infty}\right|^{2}+\varepsilon^{2}}=0\quad\text{in }B_{1}.\label{eq:first -1}
\end{equation}
Hence by Theorem \ref{cor:h=0000F6lder estimate for regularized}
we have $\left\Vert Dw_{\infty}\right\Vert _{C^{\beta_{1}}(B_{1/2})}\leq C(N,\hat{p})$
for some $\beta_{1}(N,\hat{p})$. The mean value theorem then implies
the existence of $q^{\prime}\in\mathbb{R}^{N}$ such that
\[
\operatorname{osc}_{B_{r}}(u-q^{\prime}\cdot x)\leq C_{1}(N,\hat{p})r^{1+\beta_{1}}\quad\text{for all }r\leq\frac{1}{2}.
\]
\textbf{Case $q_{j}$ is unbounded:} In this case we take a subsequence
such that $\left|q_{j}\right|\rightarrow\infty$ and the sequence
$d_{j}:=d_{j}/\left|d_{j}\right|$ converges to $d_{\infty}\in\partial B_{1}$.
Then $w_{j}$ is a viscosity solution to
\[
-\Delta w_{j}-(p_{j}(x)-2)\frac{\left\langle D^{2}w_{j}(| q_{j}|^{-1}Dw_{j}+d_{j}),| q_{j}|^{-1}Dw_{j}+d_{j}\right\rangle }{\left|\left|q_{j}\right|^{-1}Dw_{j}+d_{j}\right|^{2}+\left|q_{j}\right|^{-2}\varepsilon^{2}}=f_{j}(x)\quad\text{in }B_{1}.
\]
It follows from the stability principle that $w_{\infty}$ is a viscosity
solution to
\[
-\Delta w_{j}-(p_{\infty}(x)-2)\left\langle D^{2}w_{\infty}d_{\infty},d_{\infty}\right\rangle =0\quad\text{in }B_{1}.
\]
By \cite[Theorem 8.3]{caffarelliCabre} there exist positive constants
$\beta_{2}(N,\hat{p})$, $C_{2}(N,\hat{p})$, $r_{2}(N,\hat{p})$
and a vector $q^{\prime}\in\mathbb{R}^{N}$ such that
\[
\operatorname{osc}_{B_{r}}(w_{\infty}-q^{\prime}\cdot x)\leq C_{2}r^{1+\beta_{2}}\quad\text{for all }r\leq r_{2}.
\]
We set $C_{0}:=\max(C_{1},C_{2})$ and $\beta_{0}:=\min(\beta_{1},\beta_{2})$.
Then by the two different cases there always exists a vector $q^{\prime}\in\mathbb{R}^{N}$
such that
\[
\operatorname{osc}_{B_{r}}(w_{\infty}-q^{\prime}\cdot x)\leq C_{0}r^{1+\beta_{0}}\quad\text{for all }r\leq\min(\frac{1}{2},r_{2}).
\]
We take $\tau$ so small that $C_{0}\tau^{\beta_{0}}\leq\frac{1}{4}$
and $\tau\leq\min(\frac{1}{2},r_{2})$. Then, by substituting $r=\tau$
in the above display, we obtain
\begin{equation}
\operatorname{osc}_{B_{\tau}}(w_{\infty}-q^{\prime}\cdot x)\leq C_{0}\tau^{\beta_{0}}\tau\leq\frac{1}{4}\tau,\label{eq:first 2}
\end{equation}
which contradicts (\ref{eq:first 1}).
The bound $\left|q^{\prime}\right|\le C(N,\hat{p})$ follows by observing
that (\ref{eq:first 2}) together with the assumption $\operatorname{osc}_{B_{1}}w\leq1$
yields $\left|q^{\prime}\right|\leq C$. Thus the contradiction is
still there even if (\ref{eq:first 1}) is weakened by requiring additionally
that $\left|q^{\prime}\right|\leq C$.
\end{proof}
\begin{lem}
\label{lem:second lemma} Let $\tau(N,\hat{p})$ and $\epsilon(N,\hat{p})$
be as in Lemma \ref{lem:non-homogeneous regularized first lemma}.
If $\left\Vert f\right\Vert _{L^{\infty}(B_{1})}\leq\epsilon$ and
$u$ is a viscosity solution to (\ref{eq:non-homogeneous reguralized})
in $B_{1}$ with $q=0$, $u(0)=0$ and $\operatorname{osc}_{B_{1}}u\leq1$, then
there exists $\alpha\in(0,1)$ and $q_{\infty}\in\mathbb{R}^{N}$
such that
\[
\sup_{B_{\tau^{k}}}\left|u(x)-q_{\infty}\cdot x\right|\leq C(N,\hat{p})\tau^{k(1+\alpha)}\quad\text{for all }k\in\mathbb{N}.
\]
\end{lem}
\begin{proof}
\textbf{Step 1:} We show that there exists a sequence $(q_{k})_{k=0}^{\infty}\subset\mathbb{R}^{N}$
such that
\begin{equation}
\operatorname{osc}_{B_{\tau^{k}}}(u(x)-q_{k}\cdot x)\leq\tau^{k(1+\alpha)}.\label{eq:second lemma 1}
\end{equation}
When $k=0$, this estimate holds by setting $q_{0}=0$ since $u(0)=0$
and $\operatorname{osc}_{B_{1}}\leq1$. Next we take $\alpha\in(0,1)$ such that
$\tau^{\alpha}>\frac{1}{2}$. We assume that $k\ge0$ and that we
have already constructed $q_{k}$ for which (\ref{eq:second lemma 1})
holds. We set
\[
w_{k}(x):=\tau^{-k(1+\alpha)}(u(\tau^{k}x)-q_{k}\cdot(\tau^{k}x))
\]
and
\[
f_{k}(x):=\tau^{k(1-\alpha)}f(\tau^{k}x).
\]
Then by induction assumption $\operatorname{osc}_{B_{1}}(w_{k})\leq1$ and $w_{k}$
is a viscosity solution to
\[
-\Delta w_{k}-\frac{(p(\tau^{k}x)-2)\left\langle D^{2}w_{k}(Dw_{k}+\tau^{-k\alpha}q_{k}),Dw_{k}+\tau^{-k\alpha}q_{k}\right\rangle }{\left|Dw_{k}+\tau^{-k\alpha}q_{k}\right|^{2}+(\tau^{-k\alpha}\varepsilon)^{2}}=f_{k}(x)\quad\text{in }B_{1}.
\]
By Lemma \ref{lem:non-homogeneous regularized first lemma} there
exists $q_{k}^{\prime}\in\mathbb{R}^{N}$ with $\left|q_{k}^{\prime}\right|\leq C(N,\hat{p})$
such that
\[
\operatorname{osc}_{B_{\tau}}(w_{k}(x)-q_{k}^{\prime}\cdot x)\leq\frac{1}{2}\tau.
\]
Using the definition of $w_{k}$, scaling to $B_{\tau^{k+1}}$ and
dividing by $\tau^{-k(\alpha+1)}$, we obtain from the above
\[
\operatorname{osc}_{B_{\tau^{k+1}}}(u(x)-(q_{k}+\tau^{k\alpha}q_{k}^{\prime})\cdot x)\leq\frac{1}{2}\tau^{1+k(1+\alpha)}\leq\tau^{(k+1)(1+\alpha)}.
\]
Denoting $q_{k+1}:=q_{k}+\tau^{k\alpha}q_{k}^{\prime}$, the above
estimate is condition (\ref{eq:second lemma 1}) for $k+1$ and the
induction step is complete.
\textbf{Step 2:} Observe that whenever $m>k$, we have
\[
\left|q_{m}-q_{k}\right|\leq\sum_{i=k}^{m-1}\tau^{i\alpha}\left|q_{i}^{\prime}\right|\leq C(N,\hat{p})\sum_{i=k}^{m-1}\tau^{i\alpha}.
\]
Therefore $q_{k}$ is a Cauchy sequence and converges to some $q_{\infty}\in\mathbb{R}^{N}$.
Thus
\[
\sup_{x\in B_{\tau^{k}}}(q_{k}\cdot x-q_{\infty}\cdot x)\leq\tau^{k}\left|q_{k}-q_{\infty}\right|\leq\tau^{k}\sum_{i=k}^{\infty}\tau^{i\alpha}q_{i}^{\prime}\leq C(N,\hat{p})\tau^{k(1+\alpha)}.
\]
This with (\ref{eq:second lemma 1}) implies that
\[
\sup_{x\in B_{\tau^{k}}}\left|u(x)-q_{\infty}\cdot x\right|\leq C(N,\hat{p})\tau^{k(1+\alpha)}.\qedhere
\]
\end{proof}
\begin{thm}
\label{cor:main corollary}Suppose that $u$ is a viscosity solution
to (\ref{eq:non-homogeneous reguralized}) in $B_{1}$ with $q=0$
and $\operatorname{osc}_{B_{1}}\leq1$. Then there are constants $\alpha(N,\hat{p})$
and $C(N,\hat{p},\left\Vert f\right\Vert _{L^{\infty}(B_{1})})$ such
that
\[
\left\Vert u\right\Vert _{C^{1,\alpha}(B_{1/2})}\leq C.
\]
\end{thm}
\begin{proof}
Let $\epsilon(N,\hat{p})$ and $\tau(N,\hat{p})$ be as in Lemma \ref{lem:second lemma}.
Set
\[
v(x):=\kappa u(x/4)
\]
where $\kappa:=\epsilon(1+\left\Vert f\right\Vert _{L^{\infty}(B_{1})})^{-1}$.
For $x_{0}\in B_{1}$, set
\[
w(x):=v(x+x_{0})-v(x_{0}).
\]
Then $\operatorname{osc}_{B_{1}}w\leq1$, $w(0)=0$ and $w$ is a viscosity solution
to
\[
-\Delta w-\frac{(p(x/4+x_{0}/4)-2)\left\langle D^{2}wDw,Dw\right\rangle }{\left|Dw\right|^{2}+\varepsilon^{2}\kappa^{2}/4^{2}}=g(x)\quad\text{in }B_{1},
\]
where $g(x):=\kappa f(x/4+x_{0}/4)/4^{2}$. Now $\left\Vert g\right\Vert _{L^{\infty}(B_{1})}\leq\epsilon$
so by Lemma \ref{lem:second lemma} there exists $q_{\infty}(x_{0})\in\mathbb{R}^{N}$
such that
\[
\sup_{x\in B_{\tau^{k}}}\left|w(x)-q_{\infty}(x_{0})\cdot x\right|\leq C(N,\hat{p})\tau^{k(1+\alpha)}\quad\text{for all }k\in\mathbb{N}.
\]
Thus we have shown that for any $x_{0}\in B_{1}$ there exists a vector
$q_{\infty}(x_{0})$ such that
\[
\sup_{x\in B_{r}(x_{0})}\left|v(x)-v(x_{0})-q_{\infty}(x_{0})\cdot(x-x_{0})\right|\leq C(N,\hat{p})r{}^{1+\alpha}\quad\text{for all }r\in(0,1].
\]
This together with a standard argument (see for example \cite[Lemma A.1]{attouchiParv})
implies that $[Dv]_{C^{\alpha}(B_{1})}\leq C(N,\hat{p})$ and so by
defintion of $v$, also $[Du]_{C^{\alpha}(B_{1/4})}\leq C(N,\hat{p},\left\Vert f\right\Vert _{L^{\infty}(B_{1})})$.
The conclusion of the theorem then follows by a standard translation
argument.
\end{proof}
\section{Proof of the main theorem}
In this section we finish the proof our main theorem.
\begin{proof}[Proof of Theorem \ref{thm:main-1}]
We may assume that $u\in C(\overline{B}_{1})$. By Comparison Principle
(Lemma \ref{lem:comparison principle} in the Appendix) $u$ is the
unique viscosity solution to
\begin{equation}
\begin{cases}
-\Delta v-\frac{(p(x)-2)\left\langle D^{2}vDv,Dv\right\rangle }{\left|Dv\right|^{2}}=f(x)+u-v & \text{in }B_{1},\\
v=u & \text{on }\partial B_{1}.
\end{cases}\label{eq:main 1}
\end{equation}
By \cite[Theorem 15.18]{gilbargTrudinger01} there exists a classical
solution $u_{\varepsilon}$ to the approximate problem
\[
\begin{cases}
-\Delta u_{\varepsilon}-\frac{(p_{\varepsilon}(x)-2)\left\langle D^{2}u_{\varepsilon}Du_{\varepsilon},Du_{\varepsilon}\right\rangle }{\left|Du_{\varepsilon}\right|^{2}+\varepsilon^{2}}=f_{\varepsilon}(x)+u-u_{\varepsilon} & \text{in }B_{1},\\
u_{\varepsilon}=u & \text{on }\partial B_{1},
\end{cases}
\]
where $p_{\varepsilon},f_{\varepsilon},u_{\varepsilon}\in C^{\infty}(B_{1})$
are such that $p_{\varepsilon}\rightarrow p$, $f_{\varepsilon}\rightarrow f$
and $u_{\varepsilon}\rightarrow u_{0}$ uniformly in $B_{1}$ as $\varepsilon\rightarrow0$
and $\left\Vert Dp_{\varepsilon}\right\Vert _{L^{\infty}(B_{1})}\leq\left\Vert Dp\right\Vert _{L^{\infty}(B_{1})}$.
The maximum principle implies that $\left\Vert u_{\varepsilon}\right\Vert _{L^{\infty}(B_{1})}\leq2\left\Vert f\right\Vert _{L^{\infty}(B_{1})}+2\left\Vert u\right\Vert _{L^{\infty}(B_{1})}$.
By \cite[Proposition 4.14]{caffarelliCabre} the solutions $u_{\varepsilon}$
are equicontinuous in $\overline{B}_{1}$ (their modulus of continuity
depends only on $N$, $p$, $\left\Vert f\right\Vert _{L^{\infty}(B_{1})}$,
$\left\Vert u\right\Vert _{L^{\infty}(B_{1})}$ and modulus of continuity
of $u$). Therefore by the Ascoli-Arzela theorem we have $u_{\varepsilon}\rightarrow v\in C(\overline{B}_{1})$
uniformly in $\overline{B}_{1}$ up to a subsequence. By the stability
principle, $v$ is a viscosity solution to (\ref{eq:main 1}) and
thus by uniqueness $v\equiv u$.
By Corollary \ref{cor:main corollary} we have $\alpha(N,\hat{p})$
such that
\begin{equation}
\left\Vert Du_{\varepsilon}\right\Vert _{C^{\alpha}(B_{1/2})}\leq C(N,\hat{p},\left\Vert f\right\Vert _{L^{\infty}(B_{1})},\left\Vert u\right\Vert _{L^{\infty}(B_{1})})\label{eq:main 2}
\end{equation}
and by the Lipschitz estimate \ref{thm:Lipschitz estimate} also
\[
\left\Vert Du_{\varepsilon}\right\Vert _{L^{\infty}(B_{1/2})}\leq C(N,\hat{p},\left\Vert f\right\Vert _{L^{\infty}(B_{1})},\left\Vert u\right\Vert _{L^{\infty}(B_{1})}).
\]
Therefore by the Ascoli-Arzela theorem there exists a subsequence
such that $Du_{\varepsilon}\rightarrow\eta$ uniformly in $B_{1/2}$,
where the function $\eta:B_{1/2}\rightarrow\mathbb{R}^{N}$ satisfies
\[
\left\Vert \eta\right\Vert _{C^{\alpha}(B_{1/2})}\leq C(N,\hat{p},\left\Vert f\right\Vert _{L^{\infty}(B_{1})},\left\Vert u\right\Vert _{L^{\infty}(B_{1})}).
\]
Using the mean value theorem and the estimate (\ref{eq:main 2}),
we deduce for all $x,y\in B_{1/2}$
\begin{align*}
& \left|u(y)-u(x)-(y-x)\cdot\eta(x)\right|\\
& \ \leq\left|u_{\varepsilon}(x)-u_{\varepsilon}(y)-(y-x)\cdot Du_{\varepsilon}(x)\right|\\
& \ \ \ \ +\left|u(y)-u_{\varepsilon}(y)-u(x)+u_{\varepsilon}(x)\right|+\left|x-y\right|\left|\eta(x)-Du_{\varepsilon}(x)\right|\\
& \leq C(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})})\left|x-y\right|^{1+\alpha}+o(\varepsilon)/\varepsilon.
\end{align*}
Letting $\varepsilon\rightarrow0$, this implies that $Du(x)=\eta(x)$
for all $x\in B_{1/2}$.
\end{proof}
\appendix
\section{Lipschitz estimate}
In this section we apply the method of Ishii and Lions \cite{ishiiLions90}
to prove a Lipschitz estimate for solutions to the inhomogeneous normalized
$p(x)$-Laplace equation and its regularized or perturbed versions.
We need the following vector inequality.
\begin{lem}
\label{lem:lipschitz lemma}Let $a,b\in\mathbb{R}^{N}\setminus\left\{ 0\right\} $
with $a\not=b$ and $\varepsilon\geq0$. Then
\[
\left|\frac{a}{\sqrt{\left|a\right|^{2}+\varepsilon^{2}}}-\frac{b}{\sqrt{\left|b\right|^{2}+\varepsilon^{2}}}\right|\leq\frac{2}{\max\left(\left|a\right|,\left|b\right|\right)}\left|a-b\right|.
\]
\end{lem}
\begin{proof}
We may suppose that $\left|a\right|=\max(\left|a\right|,\left|b\right|)$.
Let $s_{1}:=\sqrt{\left|a\right|^{2}+\varepsilon^{2}}$ and $s_{2}:=\sqrt{\left|b\right|^{2}+\varepsilon^{2}}$.
Then
\begin{align*}
\left|\frac{a}{s_{1}}-\frac{b}{s_{2}}\right|=\frac{1}{s_{1}}\left|a-b+\frac{b}{s_{2}}(s_{2}-s_{1})\right| & \leq\frac{1}{s_{1}}(\left|a-b\right|+\frac{\left|b\right|}{s_{2}}\left|s_{2}-s_{1}\right|)\\
& \leq\frac{1}{\left|a\right|}(\left|a-b\right|+\left|s_{2}-s_{1}\right|).
\end{align*}
Moreover
\begin{align*}
\left|s_{2}-s_{1}\right| & =\left|\sqrt{\left|a\right|^{2}+\varepsilon^{2}}-\sqrt{\left|b\right|^{2}+\varepsilon^{2}}\right|=\frac{\left|\left|a\right|^{2}-\left|b\right|^{2}\right|}{\sqrt{\left|a\right|^{2}+\varepsilon^{2}}+\sqrt{\left|b\right|^{2}+\varepsilon^{2}}}\\
& \leq\frac{(\left|a\right|+\left|b\right|)\left|\left|a\right|-\left|b\right|\right|}{\left|a\right|+\left|b\right|}\leq\left|a-b\right|\qedhere.
\end{align*}
\end{proof}
$ $
\begin{thm}[Lipschitz estimate]
\label{thm:Lipschitz estimate} Suppose that $p:B_{1}\rightarrow\mathbb{R}$
is Lipschitz continuous, $p_{\min}>1$ and that $f\in C(B_{1})$ is
bounded. Let $u$ be a viscosity solution to
\[
-\Delta u-(p(x)-2)\frac{\left\langle D^{2}u(Du+q),Du+q\right\rangle }{\left|Du+q\right|^{2}+\varepsilon^{2}}=f(x)\quad\text{in }B_{1},
\]
where $\varepsilon\geq0$ and $q\in\mathbb{R}^{N}$. Then there are
constants $C_{0}(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})},\left\Vert f\right\Vert _{L^{\infty}(B_{1})})$
and $\nu_{0}(N,\hat{p})$ such that if $\left|q\right|>\nu_{0}$ or
$\left|q\right|=0$, then we have
\[
\left|u(x)-u(y)\right|\leq C_{0}\left|x-y\right|\quad\text{for all }x,y\in B_{1/2}.
\]
\end{thm}
\begin{proof}
We let $r(N,\hat{p})\in(0,1/2)$ denote a small constant that will
be specified later. Let $x_{0},y_{0}\in B_{r/2}$ and define the function
\[
\Psi(x,y):=u(x)-u(y)-L\varphi(\left|x-y\right|)-\frac{M}{2}\left|x-x_{0}\right|^{2}-\frac{M}{2}\left|y-y_{0}\right|^{2},
\]
where $\varphi:[0,2]\rightarrow\mathbb{R}$ is given by
\[
\varphi(s):=s-s^{\gamma}\kappa_{0},\quad\kappa_{0}:=\frac{1}{\gamma2^{\gamma+1}},
\]
and the constants $L(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})}),M(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})})>0$
and $\gamma(N,\hat{p})\in(1,2)$ are also specified later. Our objective
is to show that for a suitable choice of these constants, the function
$\Psi$ is non-positive in $\overline{B_{r}}\times\overline{B_{r}}$.
By the definition of $\varphi$, this yields $u(x_{0})-u(y_{0})\leq L\left|x_{0}-y_{0}\right|$
which implies that $u$ is $L$-Lipschitz in $B_{r}$. The claim of
the theorem then follows by standard translation arguments.
Suppose on contrary that $\Psi$ has a positive maximum at some point
$(\hat{x},\hat{y})\in\overline{B_{r}}\times\overline{B_{r}}$. Then
$\hat{x}\not=\hat{y}$ since otherwise the maximum would be non-positive.
We have
\begin{align}
0 & <u(\hat{x})-u(\hat{y})-L\varphi(\left|\hat{x}-\hat{y}\right|)-\frac{M}{2}\left|\hat{x}-x_{0}\right|^{2}-\frac{M}{2}\left|\hat{y}-y_{0}\right|^{2}\nonumber \\
& \leq\left|u(\hat{x})-u(\hat{y})\right|-\frac{M}{2}\left|\hat{x}-x_{0}\right|^{2}.\label{eq:lipschitz est 1}
\end{align}
Therefore, by taking
\begin{equation}
M:=\frac{8\operatorname{osc}_{B_{1}}u}{r^{2}},\label{eq:lipschitz est M}
\end{equation}
we get
\[
\left|\hat{x}-x_{0}\right|\leq\sqrt{\frac{2}{M}\left|u(\hat{x})-u(\hat{y})\right|}\leq r/2
\]
and similarly $\left|\hat{y}-y_{0}\right|\leq r/2$. Since $x_{0},y_{0}\in B_{r/2}$,
this implies that $\hat{x},\hat{y}\in B_{r}$.
By \cite[Proposition 4.10]{caffarelliCabre} there exist constants
$C^{\prime}(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})},\left\Vert f\right\Vert _{L^{\infty}(B_{1})})$
and $\beta(N,\hat{p})\in(0,1)$ such that
\begin{equation}
\left|u(x)-u(y)\right|\leq C^{\prime}\left|x-y\right|^{\beta}\quad\text{for all }x,y\in B_{r}.\label{eq:lipschitz est 2}
\end{equation}
It follows from (\ref{eq:lipschitz est 1}) and (\ref{eq:lipschitz est 2})
that for $C_{0}:=\sqrt{2C^{\prime}}\sqrt{M}$ we have
\begin{align}
M\left|\hat{x}-x_{0}\right|\leq C_{0}\left|\hat{x}-\hat{y}\right|^{\beta/2},\nonumber \\
M\left|\hat{y}-y_{0}\right|\leq C_{0}\left|\hat{x}-\hat{y}\right|^{\beta/2}.\label{eq:lipschitz est 3}
\end{align}
Since $\hat{x}\not=\hat{y}$, the function $(x,y)\mapsto\varphi(\left|x-y\right|)$
is $C^{2}$ in a neighborhood of $(\hat{x},\hat{y})$ and we may invoke
the Theorem of sums \cite[Theorem 3.2]{userguide}. For any $\mu>0$
there exist matrices $X,Y\in S^{N}$ such that
\begin{align*}
(D_{x}(L\varphi(\left|x-y\right|))(\hat{x},\hat{y}),X) & \in\overline{J}^{2,+}(u-\frac{M}{2}\left|x-x_{0}\right|^{2})(\hat{x}),\\
(-D_{y}(L\varphi(\left|x-y\right|))(\hat{x},\hat{y}),Y) & \in\overline{J}^{2,-}(u+\frac{M}{2}\left|y-y_{0}\right|^{2})(\hat{y}),
\end{align*}
which by denoting $z:=\hat{x}-\hat{y}$ and
\begin{align*}
a & :=L\varphi^{\prime}(\left|z\right|)\frac{z}{\left|z\right|}+M(\hat{x}-x_{0}),\\
b & :=L\varphi^{\prime}(\left|z\right|)\frac{z}{\left|z\right|}-M(\hat{y}-y_{0}),
\end{align*}
can be written as
\begin{equation}
(a,X+MI)\in\overline{J}^{2,+}u(\hat{x}),\quad(b,Y-MI)\in\overline{J}^{2,-}u(\hat{y}).\label{eq:lipschitz est 6}
\end{equation}
By assuming that $L$ is large enough depending on $C_{0}$, we have
by (\ref{eq:lipschitz est 3}) and the fact $\varphi^{\prime}\in\left[\frac{3}{4},1\right]$
\begin{align}
\left|a\right|,\left|b\right| & \leq L\left|\varphi^{\prime}(\left|\hat{x}-\hat{y}\right|)\right|+C_{0}\left|\hat{x}-\hat{y}\right|^{\beta/2}\leq2L,\label{eq:lipschitz est 4}\\
\left|a\right|,\left|b\right| & \geq L\left|\varphi^{\prime}(\left|\hat{x}-\hat{y}\right|)\right|-C_{0}\left|\hat{x}-\hat{y}\right|^{\beta/2}\geq\frac{1}{2}L.\label{eq:lipschitz est 5}
\end{align}
Moreover, we have
\begin{align}
-(\mu+2\left\Vert B\right\Vert )\begin{pmatrix}I & 0\\
0 & I
\end{pmatrix} & \leq\begin{pmatrix}X & 0\\
0 & -Y
\end{pmatrix}\nonumber \\
& \leq\begin{pmatrix}B & -B\\
-B & B
\end{pmatrix}+\frac{2}{\mu}\begin{pmatrix}B^{2} & -B^{2}\\
-B^{2} & B^{2}
\end{pmatrix},\label{eq:lipschitz est 7}
\end{align}
where
\begin{align*}
B & =L\varphi^{\prime\prime}(\left|z\right|)\frac{z}{\left|z\right|}\otimes\frac{z}{\left|z\right|}+\frac{L\varphi^{\prime}(\left|z\right|)}{\left|z\right|}\left(I-\frac{z}{\left|z\right|}\otimes\frac{z}{\left|z\right|}\right),\\
B^{2} & =BB=L^{2}(\varphi^{\prime\prime}(\left|z\right|))^{2}\frac{z}{\left|z\right|}\otimes\frac{z}{\left|z\right|}+\frac{L^{2}(\varphi^{\prime}(\left|z\right|))^{2}}{\left|z\right|^{2}}\left(I-\frac{z}{\left|z\right|}\otimes\frac{z}{\left|z\right|}\right).
\end{align*}
Using that $\varphi^{\prime\prime}(\left|z\right|)<0<\varphi^{\prime}(\left|z\right|)$
and $\left|\varphi^{\prime\prime}(\left|z\right|)\right|\leq\varphi^{\prime}(\left|z\right|)/\left|z\right|$,
we deduce that
\begin{equation}
\left\Vert B\right\Vert \leq\frac{L\varphi^{\prime}(\left|z\right|)}{\left|z\right|}\quad\text{and}\quad\text{\ensuremath{\left\Vert B^{2}\right\Vert \leq\frac{L^{2}(\varphi^{\prime}(\left|z\right|))^{2}}{\left|z\right|^{2}}}}.\label{eq:lipschitz b est}
\end{equation}
Moreover, choosing
\[
\mu:=4L\left(\left|\varphi^{\prime\prime}(\left|z\right|)\right|+\frac{\left|\varphi^{\prime}(\left|z\right|)\right|}{\left|z\right|}\right),
\]
and using that $\varphi^{\prime\prime}(\left|z\right|)<0$, we have
\begin{align}
\left\langle B\frac{z}{\left|z\right|},\frac{z}{\left|z\right|}\right\rangle +\frac{2}{\mu}\left\langle B^{2}\frac{z}{\left|z\right|},\frac{z}{\left|z\right|}\right\rangle =L\varphi^{\prime\prime}(\left|z\right|)+\frac{2}{\mu}L^{2}\left|\varphi^{\prime\prime}(\left|z\right|)\right| & \leq\frac{L}{2}\varphi^{\prime\prime}(\left|z\right|).\label{eq:lipschitz est 8}
\end{align}
We set $\eta_{1}:=a+q$ and $\eta_{2}:=b+q$. By (\ref{eq:lipschitz est 4})
and (\ref{eq:lipschitz est 5}) there is a constant $\nu_{0}(L)$
such that if $\left|q\right|=0$ or $\left|q\right|>\nu_{0}$, then
\begin{equation}
\left|\eta_{1}\right|,\left|\eta_{2}\right|\geq\frac{L}{2}.\label{eq:lipschitz est 9}
\end{equation}
We denote $A(x,\eta):=I+(p(x)-2)\eta\otimes\eta$ and $\overline{\eta}:=\frac{\eta}{\sqrt{\left|\eta\right|^{2}+\varepsilon^{2}}}$.
Since $u$ is a viscosity solution, we obtain from (\ref{eq:lipschitz est 6})
\begin{align}
0 & \leq\mathrm{tr}(A(\hat{x},\overline{\eta}_{1})(X+MI))-\mathrm{tr}(A(\hat{y},\overline{\eta}_{2})(Y-MI))+f(\hat{x})-f(\hat{y})\nonumber \\
& =\mathrm{tr}(A(\hat{y},\overline{\eta}_{2})(X-Y))+\mathrm{tr}((A(\hat{x},\overline{\eta}_{2})-A(\hat{y},\overline{\eta}_{2}))X)\nonumber \\
& \ \ \ +\mathrm{tr}((A(\hat{x},\overline{\eta}_{1})-A(\hat{x},\overline{\eta}_{2}))X)+M\mathrm{tr}(A(\hat{x},\overline{\eta}_{1})+A(\hat{y},\overline{\eta}_{2}))\nonumber \\
& \ \ \ +f(\hat{x})-f(\hat{y})\nonumber \\
& =:T_{1}+T_{2}+T_{3}+T_{4}+T_{5}.\label{eq:lipschitz est 11}
\end{align}
We will now proceed to estimate these terms. The plan is to obtain
a contradiction by absorbing the other terms into $T_{1}$ which is
negative by concavity of $\varphi$.
\textbf{Estimate of $T_{1}$: }Multiplying (\ref{eq:lipschitz est 7})
by the vector $(\frac{z}{\left|z\right|},-\frac{z}{\left|z\right|})$
and using (\ref{eq:lipschitz est 8}), we obtain an estimate for the
smallest eigenvalue of $X-Y$
\begin{align*}
\lambda_{\min}(X-Y) & \leq\left\langle (X-Y)\frac{z}{\left|z\right|},\frac{z}{\left|z\right|}\right\rangle \\
& \leq4\left\langle B\frac{z}{\left|z\right|},\frac{z}{\left|z\right|}\right\rangle +\frac{8}{\mu}\left\langle B^{2}\frac{z}{\left|z\right|},\frac{z}{\left|z\right|}\right\rangle \leq2L\varphi^{\prime\prime}(\left|z\right|).
\end{align*}
The eigenvalues of $A(\hat{y},\overline{\eta}_{2})$ are between $\min(1,p_{\text{min}}-1)$
and $\max(1,p_{\max}-1)$. Therefore by \cite{theobald75}
\begin{align*}
T_{1}=\mathrm{tr}(A(\hat{y},\overline{\eta}_{2})(X-Y)) & \leq\sum_{i}\lambda_{i}(A(\hat{y},\overline{\eta}_{2}))\lambda_{i}(X-Y)\\
& \leq\min(1,p_{\text{min}}-1)\lambda_{\min}(X-Y)\\
& \leq C(\hat{p})L\varphi^{\prime\prime}(\left|z\right|).
\end{align*}
\textbf{Estimate of $T_{2}$: }We have
\[
T_{2}=\mathrm{tr}((A(\hat{x},\overline{\eta}_{2})-A(\hat{y},\overline{\eta}_{2}))X)\leq\left|p(\hat{x})-p(\hat{y})\right|\left|\left\langle X\overline{\eta}_{2},\overline{\eta}_{2}\right\rangle \right|\leq C(\hat{p})\left|z\right|\left\Vert X\right\Vert ,
\]
where by (\ref{eq:lipschitz est 7}) and (\ref{eq:lipschitz b est})
\begin{align}
\left\Vert X\right\Vert \leq\left\Vert B\right\Vert +\frac{2}{\mu}\left\Vert B\right\Vert ^{2} & \leq\frac{L\left|\varphi^{\prime}(\left|z\right|)\right|}{\left|z\right|}+\frac{2L^{2}(\varphi^{\prime}(\left|z\right|))^{2}}{4L(\left|\varphi^{\prime\prime}(\left|z\right|)\right|+\frac{\left|\varphi^{\prime}(\left|z\right|)\right|}{\left|z\right|})\left|z\right|^{2}}\nonumber \\
& \leq\frac{2L\varphi^{\prime}(\left|z\right|)}{\left|z\right|}.\label{eq:lipschitz est 12}
\end{align}
\textbf{Estimate of $T_{3}$: }From Lemma \ref{lem:lipschitz lemma}
and the estimate (\ref{eq:lipschitz est 9}) it follows that
\begin{align}
\left|\overline{\eta}_{1}-\overline{\eta}_{2}\right| & \leq\frac{2\left|\eta_{1}-\eta_{2}\right|}{\max(\left|\eta_{1}\right|,\left|\eta_{2}\right|)}\leq\frac{4}{L}\left|\eta_{1}-\eta_{2}\right|=\frac{4}{L}\left|a-b\right|\nonumber \\
& \leq\frac{4}{L}(M\left|\hat{x}-x_{0}\right|+M\left|\hat{y}-y_{0}\right|)\leq\frac{8C_{0}}{L}\left|z\right|^{\beta/2},\label{eq:lipschitz est 10}
\end{align}
where in the last inequality we used (\ref{eq:lipschitz est 3}).
Observe that
\[
\left\Vert \overline{\eta}_{1}\otimes\overline{\eta}_{1}-\overline{\eta}_{2}\otimes\overline{\eta}_{2}\right\Vert =\left\Vert (\overline{\eta}_{1}-\overline{\eta}_{2})\otimes\overline{\eta}_{1}-\overline{\eta}_{2}\otimes(\overline{\eta}_{2}-\overline{\eta}_{1})\right\Vert \leq(\left|\overline{\eta}_{1}\right|+\left|\overline{\eta}_{2}\right|)\left|\overline{\eta}_{1}-\overline{\eta}_{2}\right|.
\]
Using the last two displays, we obtain by \cite{theobald75} and (\ref{eq:lipschitz est 12})
\begin{align*}
T_{3}=\mathrm{tr}((A(\hat{x},\overline{\eta}_{1})-A(\hat{x},\overline{\eta}_{2}))X) & \leq N\left\Vert A(x_{1},\overline{\eta}_{1})-A(x_{1},\overline{\eta}_{2})\right\Vert \left\Vert X\right\Vert \\
& \leq N\left|p(x_{1})-2\right|(\left|\overline{\eta}_{1}\right|+\left|\overline{\eta}_{2}\right|)\left|\overline{\eta}_{1}-\overline{\eta}_{2}\right|\left\Vert X\right\Vert \\
& \leq\frac{C(N,\hat{p})C_{0}}{L}\left|z\right|^{\beta/2}\left\Vert X\right\Vert \\
& \leq C(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}},\left\Vert f\right\Vert _{L^{\infty}})\sqrt{M}\varphi^{\prime}(\left|z\right|)\left|z\right|^{\beta/2-1}.
\end{align*}
\textbf{Estimate of $T_{4}$ and $T_{5}$: }By Lipschitz continuity
of $p$ we have
\begin{align*}
T_{4} & =M\mathrm{tr}(A(\hat{x},\overline{\eta}_{1})+A(\hat{y},\overline{\eta}_{2}))\leq2MC(N,\hat{p}).
\end{align*}
We have also
\[
T_{5}=f(\hat{x})-f(\hat{y})\leq2\left\Vert f\right\Vert _{L^{\infty}(B_{1})}.
\]
Combining the estimates, we deduce the existence of positive constants
$C_{1}(N,\hat{p})$ and $C_{2}(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})},\left\Vert f\right\Vert _{L^{\infty}(B_{1})})$
such that
\begin{align}
0 & \leq C_{1}L\varphi^{\prime\prime}(\left|z\right|)+C_{2}\big(L\varphi^{\prime}(\left|z\right|)+\sqrt{M}\varphi^{\prime}(\left|z\right|)\left|z\right|^{\frac{\beta}{2}-1}+M+1\big)\nonumber \\
& \leq C_{1}L\varphi^{\prime\prime}(\left|z\right|)+C_{2}(L+\sqrt{M}\left|z\right|^{\frac{\beta}{2}-1}+M+1)\label{eq:lipschitz est 15}
\end{align}
where we used that $\varphi^{\prime}(\left|z\right|)\in[\frac{3}{4},1]$.
We take $\gamma:=\frac{\beta}{2}+1$ so that we have
\[
\varphi^{\prime\prime}(\left|z\right|)=\frac{1-\gamma}{2^{\gamma+1}}\left|z\right|^{\gamma-2}=\frac{-\beta}{2^{\frac{\beta}{2}+3}}\left|z\right|^{\frac{\beta}{2}-1}=:-C_{3}\left|z\right|^{\frac{\beta}{2}-1}.
\]
We apply this to (\ref{eq:lipschitz est 15}) and obtain
\begin{align}
0 & \leq(C_{2}\sqrt{M}-C_{1}C_{3}L)\left|z\right|^{\frac{\beta}{2}-1}+C_{2}(L+M+1)\label{eq:lipschitz est 155}
\end{align}
We fix $r:=\frac{1}{2}\left(\frac{6C_{2}}{C_{1}C_{3}}\right)^{\frac{1}{\frac{\beta}{2}-1}}$.
By (\ref{eq:lipschitz est M}) this will also fix $M=(N,\hat{p},\left\Vert u\right\Vert _{L^{\infty}(B_{1})})$.
We take $L$ so large that
\[
L>\max(\frac{2C_{2}\sqrt{M}}{C_{1}C_{3}},M+1).
\]
Then by (\ref{eq:lipschitz est 155}) we have
\begin{align*}
0<-\frac{1}{2}C_{1}C_{3}L\left|z\right|^{\frac{\beta}{2}-1}+2C_{2}L & \leq L(-\frac{1}{2}C_{1}C_{3}(2r)^{\frac{\beta}{2}-1}+2C_{2})\\
& =-LC_{2}\leq0,
\end{align*}
which is a contradiction.
\end{proof}
\section{Stability and comparison principles}
\begin{lem}
Suppose that $p\in C(B_{1})$, $p_{\min}>1$ and that $f:B_{1}\times\mathbb{R}\rightarrow\mathbb{R}$
is continuous. Let $u_{\varepsilon}$ be a viscosity solution to
\[
-\Delta u_{\varepsilon}-(p_{\varepsilon}(x)-2)\frac{\left\langle D^{2}u_{\varepsilon}Du_{\varepsilon},Du_{\varepsilon}\right\rangle }{\left|Du_{\varepsilon}\right|^{2}+\varepsilon^{2}}=f_{\varepsilon}(x,u(x))\quad\text{in }B_{1}
\]
and assume that $u_{\varepsilon}\rightarrow u\in C(B_{1})$, $p_{\varepsilon}\rightarrow p$
and $f_{\varepsilon}\rightarrow f$ locally uniformly as $\varepsilon\rightarrow0$.
Then $u$ is a viscosity solution to
\[
-\Delta u-(p(x)-2)\frac{\left\langle D^{2}uDu,Du\right\rangle }{\left|Du\right|^{2}}=f(x,u(x))\quad\text{in }B_{1}.
\]
\end{lem}
\begin{proof}
It is enough to consider supersolutions. Suppose that $\varphi\in C^{2}$
touches $u$ from below at $x$. Since $u_{\varepsilon}\rightarrow u$
locally uniformly, there exists a sequence $x_{\varepsilon}\rightarrow x$
such that $u_{\varepsilon}-\varphi$ has a local minimum at $x_{\varepsilon}$.
We denote $\eta_{\varepsilon}:=D\varphi(x_{\varepsilon})/\sqrt{\left|D\varphi(x_{\varepsilon})\right|^{2}+\varepsilon^{2}}.$
Then $\eta_{\varepsilon}\rightarrow\eta\in\overline{B}_{1}$ up to
a subsequence. Therefore we have
\begin{align}
0 & \leq-\Delta\varphi(x_{\varepsilon})-(p_{\varepsilon}(x_{\varepsilon})-2)\left\langle D^{2}\varphi(x_{\varepsilon})\eta_{\varepsilon},\eta_{\varepsilon}\right\rangle -f_{\varepsilon}(x_{\varepsilon},u_{\varepsilon}(x_{\varepsilon}))\nonumber \\
& \rightarrow-\Delta\varphi(x)-(p(x)-2)\left\langle D^{2}\varphi(x_{\varepsilon})\eta,\eta\right\rangle -f(x,u(x)),\label{eq:stability 1}
\end{align}
which is what is required in Definition \ref{def:viscosity solutions}
in the case $D\varphi(x)=0$. If $D\varphi(x)\not=0$, then $D\varphi(x_{\varepsilon})\not=0$
when $\varepsilon$ is small and thus $\eta=D\varphi(x)/\left|D\varphi(x)\right|$.
Therefore \ref{eq:stability 1} again implies the desired inequality.
\end{proof}
\begin{lem}
\label{lem:comparison principle}Suppose that $p:B_{1}\rightarrow\mathbb{R}$
is Lipschitz continuous, $p_{\min}>1$ and that $f\in C(B_{1})$ is
bounded. Assume that $u\in C(\overline{B}_{1})$ is a viscosity subsolution
to $-\Delta_{p(x)}^{N}u\leq f-u$ in $B_{1}$ and that $v\in C(\overline{B}_{1})$
is a viscosity supersolution to $-\Delta_{p(x)}^{N}v\geq f-v$ in
$B_{1}$. Then
\[
u\leq v\quad\text{on }\partial B_{1}
\]
implies
\[
u\leq v\quad\text{in }B_{1}.
\]
\end{lem}
\begin{proof}
\textbf{Step 1:} Assume on the contrary that the maximum of $u-v$
in $\overline{B}_{1}$ is positive. For $x,y\in\overline{B}_{1}$,
set
\[
\Psi_{j}(x,y):=u(x)-v(y)-\varphi_{j}(x,y),
\]
where $\varphi_{j}(x,y):=\frac{j}{4}\left|x-y\right|^{4}$. Let $(x_{j},y_{j})$
be a global maximum point of $\Psi_{j}$ in $\overline{B}_{1}\times\overline{B}_{1}$.
Then
\[
u(x_{j})-v(y_{j})-\frac{j}{4}\left|x_{j}-y_{j}\right|^{4}\geq u(0)-v(0)
\]
so that
\[
\frac{j}{4}\left|x_{j}-y_{j}\right|^{4}\leq2\left\Vert u\right\Vert _{L^{\infty}(B_{1})}+2\left\Vert v\right\Vert _{L^{\infty}(B_{1})}<\infty.
\]
By compactness and the assumption $u\leq v$ on $\partial B_{1}$
there exists a subsequence such that $x_{j},y_{j}\rightarrow\hat{x}\in B_{1}$
and $u(\hat{x})-v(\hat{x})>0$. Finally, since $(x_{j},y_{j})$ is
a maximum point of $\Psi_{j}$, we have
\[
u(x_{j})-v(x_{j})\leq u(x_{j})-v(y_{j})-\frac{j}{4}\left|x_{j}-y_{j}\right|^{4},
\]
and hence by continuity
\begin{equation}
\frac{j}{4}\left|x_{j}-y_{j}\right|^{4}\leq v(x_{j})-v(y_{j})\rightarrow0\label{eq:comparison convergence}
\end{equation}
as $j\rightarrow\infty$.
\textbf{Step 2:} If $x_{j}=y_{j}$, then $D_{x}^{2}\varphi_{j}(x_{j},y_{j})=D_{y}^{2}\varphi_{j}(x_{j},y_{j})=0$.
Therefore, since the function $x\mapsto u(x)-\varphi_{j}(x,y_{j})$
reaches its maximum at $x_{j}$ and $y\mapsto v(y)-(-\varphi_{j}(x_{j},y))$
reaches its minimum at $y_{j}$, we obtain from the definition of
viscosity sub- and supersolutions that
\[
0\leq f(x_{j})-u(x_{j})\quad\text{and}\quad0\geq f(y_{j})-v(y_{j}).
\]
That is $0\leq f(x_{j})-f(y_{j})+v(y_{j})-u(x_{j}),$ which leads
to a contradiction since $x_{j},y_{j}\rightarrow\hat{x}$ and $v(\hat{x})-u(\hat{x})<0$.
We conclude that $x_{j}\not=y_{j}$ for all large $j$. Next we apply
the Theorem of sums \cite[Theorem 3.2]{userguide} to obtain matrices
$X,Y\in S^{N}$ such that
\[
(D_{x}\varphi(x_{j},y_{j}),X)\in\overline{J}^{2,+}u(x_{j}),\quad(-D_{y}\varphi(x_{j},y_{j}),Y)\in\overline{J}^{2,-}v(y_{j})
\]
and
\begin{equation}
\begin{pmatrix}X & 0\\
0 & -Y
\end{pmatrix}\leq D^{2}\varphi(x_{j},y_{j})+\frac{1}{j}(D^{2}(x_{j},y_{j}))^{2},\label{eq:matrix ineq}
\end{equation}
where
\[
D^{2}(x_{j},y_{j})=\begin{pmatrix}M & -M\\
-M & M
\end{pmatrix}
\]
with $M=j(2(x_{j}-y_{j})\otimes(x_{j}-y_{j})+\left|x_{j}-y_{j}\right|^{2}I)$.
Multiplying the matrix inequality (\ref{eq:matrix ineq}) by the $\mathbb{R}^{2N}$
vector $(\xi_{1},\xi_{2})$ yields
\begin{align*}
\left\langle X\xi_{1},\xi_{1}\right\rangle -\left\langle Y\xi_{2},\xi_{2}\right\rangle & \leq\left\langle (M+2j^{-1}M^{2})(\xi_{1}-\xi_{2}),\xi_{1}-\xi_{2}\right\rangle \\
& \leq(\left\Vert M\right\Vert +2j^{-1}\left\Vert M\right\Vert ^{2})\left|\xi_{1}-\xi_{2}\right|^{2}.
\end{align*}
Observe also that $\eta:=D_{x}\varphi(x_{j},y_{j})=-D_{y}(x_{j},y_{j})=j\left|x_{j}-y_{j}\right|^{2}(x_{j}-y_{j})\not=0$
for all large $j$. Since $u$ is a subsolution and $v$ is a supersolution,
we thus obtain
\begin{align*}
& f(y_{j})-f(x_{j})+u(x_{j})-v(y_{j})\\
& \ \leq\mathrm{tr}(X-Y)+(p(x_{j})-2)\left\langle X\frac{\eta}{\left|\eta\right|},\frac{\eta}{\left|\eta\right|}\right\rangle -(p(y_{j})-2)\left\langle Y\frac{\eta}{\left|\eta\right|},\frac{\eta}{\left|\eta\right|}\right\rangle \\
& \ \leq(p(x_{j})-1)\left\langle X\frac{\eta}{\left|\eta\right|},\frac{\eta}{\left|\eta\right|}\right\rangle -(p(y_{j})-1)\left\langle Y\frac{\eta}{\left|\eta\right|},\frac{\eta}{\left|\eta\right|}\right\rangle \\
& \ \leq(\left\Vert M\right\Vert +2j^{-1}\left\Vert M\right\Vert ^{2})\Big|\sqrt{p(x_{j})-1}-\sqrt{p(y_{j})-1}\Big|^{2}\\
& \ \leq Cj\left|x_{j}-y_{j}\right|^{2}\frac{\left|p(x_{j})-p(y_{j})\right|^{2}}{\left(\sqrt{p(x_{j})-1}+\sqrt{p(y_{j})-1}\right)^{2}}\\
& \leq C(\hat{p})j\left|x_{j}-y_{j}\right|^{4}.
\end{align*}
This leads to a contradiction since the left-hand side tends to $u(\hat{x})-v(\hat{y})>0$
and the right-hand side tends to zero by (\ref{eq:comparison convergence}).
\add
\end{proof}
\end{document} |
\begin{document}
\baselineskip 16pt
\title{Some new characterizations of $PST$-groups}
\author{Xiaolan Yi \\
{\small Department of Mathematics, Zhejiang Sci-Tech University,}\\
{\small Hangzhou 310018, P.R.China}\\
{\small E-mail:
[email protected]}\\ \\
{ Alexander N. Skiba }\\
{\small Department of Mathematics, Francisk Skorina Gomel State University,}\\
{\small Gomel 246019, Belarus}\\
{\small E-mail: [email protected]}}
\date{}
\maketitle
\begin{abstract} Let $H$ and $B$ be subgroups of a finite group
$G$ such that $G=N_{G}(H)B$.
Then we say that $H$ is \emph{quasipermutable} (respectively
\emph{$S$-quasipermutable}) in $G$
provided $H$ permutes with $B$ and
with every subgroup (respectively with every Sylow subgroup)
$A$ of $B$ such that $(|H|, |A|)=1$.
In this paper we analyze the influence of
$S$-quasipermutable and quasipermutable
subgroups on the structure of $G$. As an application, we give new
characterizations of soluble $PST$-groups.
\end{abstract}
\let\emptyorig\empty
\renewcommand{\empty}{\empty}
\footnotetext{Keywords: finite group, quasipermutable subgroup,
$PST$-group,
Hall subgroup, supersoluble group,
Gasch\"utz subgroup, Carter subgroup, saturated formation.}
\footnotetext{Mathematics Subject Classification (2010): 20D10,
20D15, 20D20}
\let\empty\emptyorig
\section{Introduction}
Throughout this paper, all groups are finite and $G$ always denotes
a finite group. Moreover $p$ is always supposed to be a prime and
$\pi$ is a subset of the set $\Bbb{P}$ of all primes; $\pi (G)$ denotes the
set of all primes dividing $|G|$.
A subgroup $H$ of $G$ is said to be \emph{quasinormal} or
\emph{permutable} in $G$ if $H$ permutes with every subgroup $A$ of $G$, that is, $HA=AH$;
$H$ is said to be \emph{$S$-permutable}
in $G$ if $H$ permutes with every Sylow subgroup of $G$.
A group $G$ is called a \emph{$PT$-group} if
permutability is a
transitive relation on $G$, that is, every permutable subgroup of
a permutable subgroup
of $G$ is permutable in $G$.
A group $G$ is called a \emph{$PST$-group} if
$S$-permutability is a transitive relation on $G$.
As well as $T$-groups, $PT$-groups and $PST$-groups possess many interesting
properties (see Chapter 2 in \cite{prod}).
The general description of
$PT$-groups and $PST$-groups were firstly obtained by Zacher
\cite{G.Zacher} and Agrawal \cite{Agr},
for the soluble case, and
by Robinson in \cite{217}, for the general case. Nevertheless,
in the further publications, the authors (see for example recent papers\cite{78}--
\cite{khaledII}) have found out and described
many other interesting characterizations of soluble $PT$ and $PST$-groups.
In this paper we give new "Hall"-characterizations of soluble
$PST$-groups on the basis of the following
{\bf Definition 1.1.} We say that
a subgroup $H$ is \emph{quasipermutable} (respectively
\emph{$S$-quasipermutable}) in $G$
provided $H$ permutes with $B$ and
with every subgroup (respectively with every Sylow subgroup)
$A$ of $B$ such that $(|H|, |A|)=1$.
Examples and some applications of quasipermutable
subgroups were discussed in our
papers \cite{Bull} and \cite{proble} (see also remarks in Section 5 below).
In this paper,
we give the following result, which we consider as one more
motivation for introducing the concept of quasipermutability.
{\bf Theorem A.} {\sl Let $D=G^{\cal N} $ and $\pi =\pi (D)$.
Then the following
statements are equivalent:}
(i) {\sl $D$ is a Hall subgroup of $G$ and
every Hall subgroup of $G$ is quasipermutable in $G$.}
(ii) {\sl $G$ is a soluble $PST$-group.}
(iii) {\sl Every subgroup of $G$ is quasipermutable in $G$.}
(iv) {\sl Every $\pi$-subgroup of $G$
and some minimal supplement of $D$ in
$G$ are quasipermutable in $G$.}
In the proof Theorem A
we use the next three our results.
A subgroup $S$ of $G$ is called a \emph{Gasch\"utz} subgroup
of $G$ (L.A. Shemetkov \cite[IV, 15.3]{26}) if $S$ is supersoluble
and for any subgroups $K \leq H$ of $G$, where $S\leq K$, the number $|H:K|$
is not prime.
{\bf Theorem B.} {\sl The following statements are equivalent:}
(I) {\sl $G$ is soluble, and if $S$ is a Gasch\"utz
subgroup of $G$, then
every Hall subgroup $H$ of $G$ satisfying $\pi (H)\subseteq \pi (S)$
is quasipermutable in $G$.}
(II) {\sl $G$ is supersoluble and the following hold: }
(a) {\sl $G=DC$, where $D=G^{\cal N}$
is an abelian complemented subgroup of $G$ and $C$ is a Carter subgroup of
$G$;}
(b) {\sl $D\cap C$ is normal in $G$ and
$(p, |D/D\cap C|)=1$ for all prime divisors $p$ of $|G|$ satisfying $(p-1, |G|)=1$.}
(c) {\sl For any non-empty set $\pi $ of primes, every $\pi $-element of any
Carter subgroup of $G$ induces a power automorphism on the
Hall $\pi'$-subgroup of $D$.}
(III) {\sl Every Hall subgroup of $G$ is quasipermutable in $G$.}
Let $\cal F$ be a class of groups. If $1\in {\cal F}$, then we write $G^{\cal F}$ to denote the
intersection of all normal subgroups $N$ of $G$ with $G/N\in {\cal
F}$. The class $\cal F$ is said to be a \emph{formation} if either
${\cal F}= \varnothing $ or $1\in {\cal F}$ and every
homomorphic image of $G/G^{\cal F}$ belongs to $ {\cal F}$ for any group $G$.
The formation ${\cal F}$
is said to be \emph{saturated} if $G\in {\cal F}$ whenever $G/\Phi (G) \in {\cal
F}$. A subgroup $H$ of $G$ is said to be an \emph{$\cal F$-projector} of $G$
provided $H\in {\cal F}$ and $E=E^{\cal F}H$ for any subgroup $E$ of $G$
containing $H$. By the Gasch\"utz's theorem \cite[VI, 9.5.4 and 9.5.6]{hupp}, for any saturated formation
$\cal F$, every soluble group $G$ has an $\cal F$-projector and any two
$\cal F$-projectors of $G$ are conjugate.
{\bf Theorem C.} {\sl Let $\cal F$ be a saturated formation containing
all nilpotent groups. Suppose that $G$ is soluble and let $\pi =\pi (C)
\cap \pi( G^{\cal F})$, where $C$ is an $\cal F$-projector of $G$. If
every maximal subgroup of every Sylow $p$-subgroup of $G$ is
$S$-quasipermutable in $G$ for all $p\in \pi $, then $G^{\cal F}$ is a
Hall subgroup of $G$.}
{\bf Theorem D.} {\sl Let $\cal F$ be a saturated formation containing
all supersoluble groups and $\pi =\pi (F^{*}(G^{\cal F}))$. If
$G^{\cal F}\ne 1$, then for some $p\in \pi $ some maximal subgroup of a Sylow
$p$-subgroup of $G$ is not $S$-quasipermutable in $G$.}
In this theorem $F^{*}(G^{\cal F})$ denotes the generalized Fitting
subgroup of $G^{\cal F}$, that is, the product of all normal quasinilpotent subgroups of
$G^{\cal F}$.
The main tool in the proofs of Theorems C and D is the following our result.
{\bf Proposition.} {\sl Let $E$ be a normal subgroup of $G$ and
$P$ a Sylow $p$-subgroup of $E$ such that $|P| > p$. }
(i) {\sl If every
number $V$ of some fixed ${\cal M}_{\phi}(P)$ is $S$-quasipermutable in
$G$, then $E$ is $p$-supersoluble. }
(ii) {\sl If
every maximal subgroup of $P$ is $S$-quasipermutable in
$G$, then every chief factor of $G$ between $E$ and $O_{p'}(E)$
is cyclic. }
(iii) {\sl If every maximal subgroup of every Sylow subgroup of $E$ is $S$-quasipermutable
in $G$, then every chief factor of $G$ below $E$ is cyclic. }
In this proposition we write ${\cal M}_{\phi}(G)$, by analogy with \cite{shirong},
to denote a set of maximal subgroups of $G$
such that ${\Phi}(G)$ coincides with the intersection of all
subgroups in ${\cal M}_{\phi}(G)$.
Note that Proposition may be independently interesting because this result
unifies and generalize
many known results, and in particular, Theorems 1.1--1.5 in
\cite{shirong} (see Section 5). In Section 5 we discus also some
further applications of the results.
All unexplained notation and terminology are standard. The reader is
referred to \cite{26}, \cite{DH}, or \cite{Bal-Ez} if necessary.
\section{Basic Propositions}
Let $H$ be a subgroup of $G$.
Then we say, following \cite{Bull}, that $H$ is \emph{propermutable}
(respectively \emph{$S$-propermutable})
in $G$ provided there is a subgroup $B$ of
$G$ such that $G=N_{G}(H)B$ and $H$ permutes with all subgroups (respectively with all
Sylow subgroups) of $B$.
{\bf Proposition 2.1.} {\sl Let $H\leq G$ and $N$ a normal subgroup of $G$. Suppose that
$H$ is quasipermutable ($S$-quasipermutable) in $G$.}
(1) {\sl If either $H$ is a Hall subgroup of $G$ or
for every prime $p$ dividing $|H|$ and for every Sylow $p$-subgroup $H_{p}$
of $H$ we have $H_{p}\nleq N$, then $HN/N$ is quasipermutable
($S$-quasipermutable, respectively) in $G/N$. }
(2) {\sl If $\pi =\pi (H)$ and $G$ is $\pi$-soluble, then $H$
permutes with some Hall $\pi'$-subgroup of $G$. }
(3) {\sl $H$ permutes with some Sylow $p$-subgroup
of $G$ for every prime $p$ dividing $|G|$ such that $(p, |H| )=1$.}
(4) {\sl $|G:N_{G}(H\cap N)|$ is a
$\pi$-number, where $\pi = \pi (N)\cup \pi (H)$.}
(5) {\sl If $H$ is propermutable ($S$-propermutable) in $G$, then
$HN/N$ is propermutable ($S$-propermutable, respectively) in $G/N$. }
(6) {\sl If $H$ is $S$-propermutable in $G$, then $H$
permutes with some Sylow $p$-subgroup
of $G$ for any prime $p$ dividing $|G|$. }
(7) {\sl Suppose that $G$ is $\pi$-soluble. If $H$ is a Hall $\pi$-subgroup
of $G$, then $H$ is
propermutable ($S$-propermutable, respectively) in $G$. }
{\bf Proof.} By hypothesis, there is a subgroup $B$ of $G$ such that $G=N_{G}(H)B$
and $H$ permutes with $B$ and with all
subgroups (with all Sylow subgroups, respectively) $A$ of $B$
such that $(|H|, |A|)=1$.
(1) It is clear that
$$G/N=(N_{G}(H)N/N)(BN/N)=N_{G/N}(HN/N)(BN/N).$$
Let $K/N$ be any subgroup (any Sylow subgroup, respectively) of
$BN/N$ such that $(|HN/N|, |K/N|)=1$.
Then $K=(K\cap B)N$. Let $B_{0}$ be a minimal supplement of $K\cap B\cap N$ to $K\cap B$.
Then $K/N=(K\cap
B)N/N =B_{0}(K\cap B\cap N)N/N=B_{0}N/N$ and $K\cap B\cap N\cap B_{0}=N\cap B_{0}\leq \Phi (B_{0})$.
Therefore $\pi (K/N)=\pi (K\cap B/K\cap B\cap N)=\pi (B_{0})$, so
$(|HN/N|, |B_{0}|)=1$. Suppose that some prime $p\in \pi (B_{0})$
divides $|H|$, and let $H_{p}$ be
a Sylow $p$-subgroup of $H$. We shall show that
$H_{p}\nleq N$. In fact, we may suppose that
$H$ is a Hall subgroup of $G$. But in this case,
$H_{p}$ is
a Sylow $p$-subgroup of $G$. Therefore, since $p\in \pi (B_{0})\subseteq \pi (G/N)$,
$H_{p}\nleq N$. Hence $p$
divides $|HN/N|$, a contradiction. Thus $(|H|, |B_{0}|)=1$, so in the case, when $H$ is
quasipermutable in $G$, we have $HB_{0}=B_{0}H$ and hence $HN/N$
permutes with $K/N=B_{0}N/N$. Thus $HN/N$ is quasipermutable in
$G/N$.
Finally, suppose that $H$ is $S$-quasipermutable in $N$. In this case,
$B_{0}$ is a $p$-subgroup of $B$, so
for some Sylow $p$-subgroup $B_{p}$ of $B$ we have $B_{0}\leq B_{p}$ and $(|H|, p)=1$.
Hence
$K/N=B_{0}N/N\leq B_{p}N/N$, which implies that $K/N= B_{p}N/N$.
But $H$ permutes with $B_{p}$ by hypothesis, so $HN/N$
permutes with $K/N$.
Therefore $HN/N$ is $S$-quasipermutable in $G/N$.
(2) By \cite[VI, 4.6]{hupp}, there are
Hall $\pi'$-subgroups $E_{1}$, $E_{2}$ and $E$ of $N_{G}(H)$,
$B$ and $G$, respectively, such that $E=E_{1}E_{2}$.
Then $H$ permutes with all Sylow subgroups of $E_{2}$ by hypothesis, so
$$HE=H(E_{1}E_{2})=(HE_{1})E_{2}=(E_{1}H)E_{2}=$$ $$
E_{1}(HE_{2})=E_{1}(E_{2}H)=(E_{1}E_{2})H=EH$$
by \cite[A, 1.6]{DH}.
(3) See the proof of (2).
(4) Let $p$ be a prime such that $p\not \in \pi $. Then by (3),
there is a Sylow $p$-subgroup $P$ of $G$ such that $HP=PH$ is a
subgroup of $G$. Hence $HP\cap N=H\cap N$ is a normal subgroup of
$HP$. Thus $p$ does not divide $|G:N_{G}(H\cap N)|$.
(5) See the proof of (1).
(6) See the proof of (2).
(7) Since $G$ is
$\pi$-soluble, $B$ is $\pi$-soluble. Hence by \cite[VI, 1.7]{hupp},
$B=B_{\pi}B_{\pi'}$
where $B_{\pi}$ is a Hall $\pi$-subgroup of $B$ and $B_{\pi'}$ is a
Hall $\pi'$-subgroup of $B$. By \cite[VI, 4.6]{hupp}, there are
Hall $\pi$-subgroups $N_{\pi}$, $B_{\pi}$ and $G_{\pi}$ of $N_{G}(H)$,
$B$ and $G$, respectively, such that $G_{\pi}=N_{\pi}B_{\pi}$. But since
$H\leq N_{\pi}$,
$N_{\pi}$ is a Hall $\pi$-subgroup of $G$. Therefore
$G_{\pi}=N_{\pi}B_{\pi}=N_{\pi}$, so $B_{\pi}\leq N_{\pi}$.
Hence
$G=N_{G}(H)B=N_{G}(H)B_{\pi}B_{\pi'}=N_{G}(H)B_{\pi'}$, so $H$ is
propermutable ($S$-propermutble, respectively) in $G$.
A group $G$ is said to be a \emph{$C_{\pi }$-group} provided $G$ has a Hall
${\pi }$-subgroup and any two Hall ${\pi }$-subgroups of $G$ are
conjugate.
On the basis of Proposition 2.1 the following two results are proved.
{\bf Proposition 2.2.} {\sl Let $H$ be
a Hall $S$-quasipermutable subgroup of $G$. If $\pi = \pi (|G:H|)$, then $G$
is a $C_{\pi }$-group.}
{\bf Proposition 2.3.} {\sl Let $E$ be a normal subgroup of $G$ and $H$
a Hall $\pi$-subgroup of $E$.
If $H$ is nilpotent and $S$-quasipermutable in $G$, then $E$ is $\pi$-soluble.}
\section{Groups with a Hall quasipermutable subgroup }
A group $G$ is said to be \emph{$\pi$-separable} if every chief factor of
$G$ is either a $\pi$-group or a $\pi'$-group. Every $\pi$-separable
group $G$ has a series $$1=P_{0}(G)\leq M_{0}(G) < P_{1}(G) < M_{1}(G)
< \ldots < P_{t}(G)\leq M_{t}(G)=G $$ such that
$$M_{i}(G)/P_{i}(G) =O_{\pi'}(G/P_{i}(G))$$ ($i=0, 1, \ldots , t$)
and $$P_{i+1}(G)/M_{i}(G)= O_{\pi}(G/M_{i}(G))$$
($i=1, \ldots , t$)
The number $t$ is called the \emph{$\pi$-length} of $G$ and denoted by
$l_{\pi}(G)$ (see \cite[p. 249]{rob}).
One more result, which we use use in the proof of our main results, is the
following
{\bf Theorem 3.1.} {\sl Let $H$ be a
Hall subgroup of $G$ and $\pi =\pi (H)$. Suppose that $H$ is
quasipermutable in $G$. }
(I) {\sl If $ p > q $ for all primes $p$ and $q$ such that $p\in \pi $ and
$q$ divides $|G:N_{G}(H)|$, then $H$ is normal in $G$.}
(II) {\sl If $H$ is supersoluble, then $G$ is $\pi$-soluble.}
(III) {\sl If $H$ is $\pi$-separable, then the following fold:}
(i) {\sl $H'\leq O_{\pi}(G)$. If, in addition,
$N_{G}(H)$ is nilpotent, then $G'\cap H \leq O_{\pi}(G)$.}
(ii) {\sl $l_{\pi}(G) \leq 2$ and $l_{\pi'}(G) \leq 2$. }
(iii) {\sl If for some prime $p\in \pi'$ a Hall $\pi'$-subgroup $E$
of $G$
is $p$-supersoluble, then $G$ is $p$-supersoluble. }
Let $\cal M$ and $\cal H$ be non-empty formations. Then the
\emph{product} ${\cal M} {\cal H}$ of these formations is the class of all
groups $G$ such that $G^{\cal H}\in {\cal M}$. It is well-known that such an operation
on the set of all non-empty formations is associative (Gasch\"utz). The symbol
${\cal M}^{t}$ denotes the product of $t$ copies of ${\cal M}$.
We shall need following well-known
Gasch\"utz-Shemetkov's theorem \cite[Corollary 7.13]{100}.
{\bf Lemma 3.2}. {\sl The product of any two non-empty
saturated formations is also a saturated formation.}
In in the proof of Theorem 3.1 we use the following
{\bf Lemma 3.3.} {\sl The class $\cal F$ of all $\pi$-separable groups $G$
with $l_{\pi}(G) \leq t$ is a saturated formation.}
{\bf Proof.} It is not difficult to show that for any non-empty
set $\omega \subseteq \Bbb{P}$ the
class ${\cal G}_{\omega}$ of all $\omega$-groups is a saturated
formation and that ${\cal F}=({\cal G}_{\pi'}{\cal G}_{\pi})^{t}{\cal G}_{\pi'}$.
Hence ${\cal F}$ is a saturated formation by Lemma 3.2.
{\bf Lemma 3.4.} {\sl Suppose that $G$ is separable. If Hall
$\pi$-subgroups
of $G$ are abelian, then $l_{\pi}(G) \leq 1$.}
{\bf Proof.} Suppose that this lemma is false and let
$G$ be a counterexample of minimal order. Let $N$ be a minimal normal subgroup of $G$.
Since $G$ is
$\pi$-separable, $N$ is a $\pi$-group or a $\pi'$-group.
It is clear that the hypothesis holds for $G/N$, so $l_{\pi}(G/N) \leq 1$ by the choice of $G$.
By Lemma 3.3, the class of all $\pi$-soluble groups with $l_{\pi}(G) \leq 1$ is
a saturated formation. Therefore $N$ is a unique minimal normal subgroup
of $G$, $N\nleq \Phi (G)$ and $N$ is not a $\pi'$-group. Hence $N$ is a $\pi$-group and
$N=C_{G}(N)$ by \cite[A, 15.2]{DH}. Therefore $N\leq H$, where $H$ is a Hall
$\pi$-subgroup of $G$. But since $H$ is abelian, $N=H$ is a Hall
$\pi$-subgroup of $G$. Hence
$l_{\pi}(G) \leq 1$.
A group $G$ is called \emph{$\pi$-closed} provided $G$ has a normal
Hall $\pi$-subgroup.
{\bf Lemma 3.5.} {\sl Let $H$ be a Hall $\pi$-subgroup of $G$. If $G$ is
$\pi$-separable and $H\leq Z(N_{G}(H))$, then $G$ is $\pi'$-closed. }
{\bf Proof.} Suppose that this lemma is false and let
$G$ be a counterexample of minimal order. Then $G\ne H$.
The class $\cal F$ of all $\pi'$-closed groups coincides with the
product ${\cal G}_{\pi'}{\cal G}_{\pi}$. Hence $\cal F$ is a saturated formation by
Lemma 3.2.
Let $N$ be a minimal normal subgroup of $G$. Since $G$ is
$\pi$-separable, $N$ is a $\pi$-group or a $\pi'$-group. Moreover, $G$
is a $C_{\pi}$-group by \cite[9.1.6]{rob}), so the hypothesis holds for $G/N$.
Hence $G/N$ is
$\pi'$-closed by the choice of $G$.
Therefore $N$ is the only minimal normal subgroup of $G$, $N\nleq \Phi
(G) $ and $N$ is a $\pi$-group. Therefore $N\leq H$ and $N=C_{G}(N)$ by
\cite[A, 15.2]{DH}. Since $H\leq Z(N_{G}(H))$ and $H$ is
a Hall $\pi$-subgroup of $G$, $N=H$. Therefore $N\leq Z(G)$, which
implies that $N=H=G$. This contradiction completes the proof of the lemma.
\section{Proof of Theorem A}
\
Recall that $G$ is a $PST$-group if and only if $G=D\rtimes M$, where $D=G^{\cal N
}$
is abelian Hall subgroup of $G$ and
every element $x\in M$ induces a power automorphism on $D$ \cite{Agr}. Therefore
the implication
(i) $\Rightarrow$ (ii) is a direct corollary of Theorem B.
Now suppose that $G=D\rtimes M$, where $D=G^{\cal N
}$, is a soluble
$PST$-group. Let $H$ be any subgroup of $G$ and $S$ a Hall $\pi '$-subgroup of $H$.
Since $G$ is soluble, we may
assume without loss of generality that $S\leq M$. Hence
$H=(D\cap H)(M\cap H)=(D\cap H)S$ and $D\cap H$ is normal in $G$.
Let $\pi _{1}= \pi (S)$. Let $A$ be a Hall $\pi _{1}$-subgroup
of $M$ and $E$ a complement to $A$ in $M$. Then $E\leq
C_{G}(S)$. Therefore $G= DM=DAE=N_{G}(H)(DA)$ and every subgroup $L$ of $DA$ satisfying
$(|H|, |L|)=1 $ is contained in $D$. Thus $H$
is quasipermutablein $G$. Thus (ii) $\Rightarrow$ (iii).
(iv) $\Rightarrow$ (ii) By Theorems C and D, $G$ is supersoluble and $D$
is a Hall subgroup of $G$.
Therefore $G=D\rtimes W$, where $W$ is a Hall $\pi'$-subgroup of $G$. By hypothesis, $W$ is
quasipermutable in $G$. Now arguing similarly as in the proof of Theorem
B one can show that $D$ is abelian and every subgroup of $D$ is normal in $G$. Therefore
$G$ is a $PST$-group.
\section{Final remarks }
\
1. The subgroup $S_{3}$ is quasipermutable, $S$-propermutable and not
propermutable in $S_{4}$. If $H$ is the subgroup of order 3 in $S_{3}$,
then $H$ is $S$-quasipermutable and not quasipermutable in $S_{4}$.
2. Arguing similarly to the proof of Theorem A one can prove the following fact.
{\bf Theorem 5.1.} {\sl Suppose that $G$ is soluble and let
$\pi =\pi (G^{\cal N})$. Then $G$ is a $PST$-group if and only if
every subnormal $\pi$-subgroup and a Hall $\pi'$-subgroup of $G$
are propermutable in $G$. }
3. If $G$ is metanilpotent, that is $G/F(G)$ is nilpotent,
then for every Hall subgroup $E$ of $G$ we have $G=N_{G}(E)F(G)$.
Therefore, in this case, every characteristic
subgroup of every Hall subgroup of $G$ is $S$-propermutable in $G$. In
particular, every Hall subgroup of every supersoluble group is
$S$-propermutable. This observation makes natural the following
question: {\sl What is the structure of $G$ under the hypothesis
that every Hall subgroup of $G$ is propermutable in $G$ ?} Theorem B
gives an answer to this question.
4. Every maximal subgroup of a supersoluble group is quasipermutable. Therefore, in fact,
Theorem A shows that the class of all soluble groups in which
quaipermutability is a transitive relation coincides with the
class of all soluble $PST$-groups.
5. We say that $G$ is a \emph{$SQT$-group} if $S$-quasipermutability is
a transitive relation in $G$. Arguing similarly to the proof of
Theorem A one can prove the following fact.
{\bf Theorem 5.2.} {\sl A soluble group $G$ is an $SQT$-group if and only if
$G=D\rtimes M $ is supersoluble,
where $D$ and $M$
are Hall nilpotent subgroups of $G$ and the index
$|G:DN_{G}(H\cap D)|$ is a
$\pi (H)$-number for every subgroup $H$ of $G$. }
6. A subgroup $H$ of $G$ is called \emph{$SS$-quasinormal} \cite{shirong}
(\emph{semi-normal} \cite{Su}) in $G$ provided $G$ has a subgroup $B$
such that $HB=G$ and $H$ permutes with all Sylow
subgroups ($H$ permutes with all subgroups, respectively) of $B$.
It is clear that every $SS$-quasinormal subgroup is
$S$-propermutable and every semi-normal subgroup is propermutable.
Moreover, there are simple examples (consider, for example, the group
$C_{7}\rtimes \text{Aut} (C_{7})$, where $C_{7}$ is a group of order 7) which show that,
in general, the class of all
$S$-propermutable subgroups of $G$ is wider than the class of all its
$SS$-quasinormal subgroups and the class of all
propermutable subgroups of $G$ is wider than the class of all its
semi-normal subgroups.
Therefore Proposition covers main results (Theorems 1.1--1.5) in \cite{shirong}.
7. Theorem 3.1 is used in the proof of Theorem B.
From this result we also get
{\bf Corollary 5.3} (See \cite[Theorem 5.4]{8}). {\sl Let $H$ be
a Hall semi-normal subgroup
of $G$. If $p
> q $ for all primes $p$ and $q$ such that $p$ divides $|H|$ and $q$
divides $|G:H|$, then $H$ is normal in $G$. }
{\bf Corollary 5.4} (See \cite[Theorem]{GuoS}). {\sl Let $P$ be a
Sylow $p$-subgroup of $G$. If $P$ is semi-normal in $G$, then the
following statements hold: }
(i) {\sl $G$ is $p$-soluble and $P'\leq O_{p}(G)$.}
(ii) {\sl $l_{p}(G) \leq 2$. }
(iiii {\sl If for some prime $q\in \pi'$ a Hall $p'$-subgroup
of $G$
is $q$-supersoluble, then $G$ is $q$-supersoluble. }
{\bf Corollary 5.5} (See \cite[Theorem 3]{podg}). {\sl If a
Sylow $p$-subgroup $P$ of
$G$, where $p$ is the largest prime dividing $|G|$,
is semi-normal in $G$, then $P$ is normal in $G$.}
\end{document} |
\begin{document}
\baselineskip=17pt
\subjclass[2020]{11B68, 11D41}
\keywords{Diophantine equations, exponential equations, Bernoulli polynomials}
\title[On equal values of products and power sums...]{On equal values of products and power sums of consecutive elements in an arithmetic progression}
\author[A. Bazs\'o, D. Kreso, F. Luca and \'A. Pint\'er]{A. Bazs\'o, D. Kreso, F. Luca, \'A. Pint\'er, and Cs. Rakaczki}
\address{A. Bazs\'o \newline
\indent Institute of Mathematics \newline
\indent University of Debrecen \newline
\indent P.O. Box 400, H-4002 Debrecen, Hungary \newline
\indent and \newline
\indent ELKH-DE Equations, Functions, Curves and their Applications Research Group}
\email{[email protected]}
\address{D. Kreso \newline
\indent Institute f\"ur Mathematik \newline
\indent Technische Universit\"at Graz \newline
\indent Steyrergasse 30, 8010 Graz Austria}
\email{[email protected]}
\address{F. Luca \newline
\indent School of Mathematics \newline
\indent Wits University \newline
\indent 1 Jan Smuts, Brammfontein, 2000 Johannesburg, South Africa, \newline
\indent Research Group in Algebraic Structures and Applications \newline
\indent King Abdulaziz University \newline
\indent Abdulah Sulayman, Jeddah 22254, Saudi Arabia, \newline
\indent and \newline
\indent Mathematical Institute \newline
\indent UNAM Ap. Postal 61-3 (Xangari) \newline
\indent CP 58 089. Morelia, Michoac\'an, Mexico}
\email{[email protected]}
\address{\'A. Pint\'er \newline
\indent Institute of Mathematics \newline
\indent U
niversity of Debrecen \newline
\indent P.O. Box 400, H-4002 Debrecen, Hungary}
\email{[email protected]}
\address{Cs. Rakaczki\newline
\indent Institute of Mathematics\newline
\indent University of Miskolc \newline
\indent H-3515 Miskolc Campus, Hungary}
\email{ [email protected]}
\thanks{}
\date{}
\begin{abstract}
In this paper we study the Diophantine equation
\begin{align*}
b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\
&y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right),
\end{align*}
where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$ and obtain a general ineffective result based on Bilu-Tichy method.
\end{abstract}
\maketitle
\section{Introduction}
The polynomials
\begin{equation}
S_{a,b}^k \left(x\right) = b^k + \left(a+b\right)^k + \left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k \label{pol:skabx}
\end{equation}
and
\begin{equation}
R_c^{\ell} \left(x\right) = x\left(x+c\right) \left(x+2c\right) \ldots \left(x+ \left(\ell-1\right)c\right),
\end{equation}
are natural generalizations of the widely studied polynomials $S_k (x) = S_{1,0}^k (x)$ and $R_{\ell} (x) =R_1 ^{\ell} (x)$, respectively.
Various Diophantine equations concerning $R_{\ell} (x)$ and $S_k (x)$ have been extensively investigated. See e.g. \cite{BBKPT} and the references given there.
It is easy to see that involving Bernoulli polynomials, the polynomial defined above by \eqref{pol:skabx} can be rewritten as
\begin{equation} \label{eq:mainI}
S_{a,b}^k \left(x\right) = \frac{a^k}{k+1} \left(B_{k+1} \left(x+ \frac{b}{a}\right) - B_{k+1} \left(\frac{b}{a}\right)\right).
\end{equation}
In \cite{BBKPT}, Bilu, Brindza, Kirschenhofer, Pint\'er and Tichy proved that for $k \geq 1, \ell \geq 2$, and $(k,\ell) \neq (1,2)$, the equation $S_k (x) = R_{\ell} (y)$ has at most finitely many integer solutions. They also proved a similar result for the equation $S_k (x) = S_{\ell} (y)$. Both of these results were ineffective, since their proofs were mainly based on the general finiteness criterion of Bilu and Tichy \cite{BiluTichy} for Diophantine equations of the form $f(x) = g(y)$. In certain special cases they also proved effective finiteness results for the corresponding equations.
In our earlier paper \cite{BKLP}, using a slightly modified approach, we generalized the above result of Bilu, Brindza, Kirschenhofer, Pint\'er and Tichy \cite{BBKPT} concerning the equation $S_k (x) = S_{\ell} (y)$ by proving that the more general equation $S_{a,b}^k (x) = S_{c,d}^{\ell} (y)$ has at most finitely many solutions in rational integers $x,y$. This theorem is also ineffective, but it is made effective in some special cases.
The purpose of this paper is to study the equation
\begin{equation} \label{eq:RS}
S_{a,b}^k (x) = R_{c}^{\ell} (y)
\end{equation}
in integers $x,y$.
As first result we prove a generalization of original Sch\"affer problem on the power values of power sums.
\begin{theorem} \label{thm:eff1}
Let $a,b$ be rational integers with $a>0$ and suppose that $c=0$. We consider equation
\begin{equation}\label{eq:c=0}
S_{a,b}^k (x) =y^{\ell}
\end{equation}
in integers $x,y>1$ and $\ell\geq 2$. If $(k,a,b)\neq (1,2,1 )$ then $\ell<C_1$ where $C_1$ is an effectively computable constant depending only on $k,a$ and $b$. Further, apart from the cases
$$(k,\ell,a,b)\in\{ (1,2,a,0),(3,2,a,0),(3,2,2,1),(3,4,a,0), (5,2,a,0) \}$$
equation (\ref{eq:c=0}) implies $\max(|x|,|y|)<C_2$ where $C_2$ is an effectively computable constant depending on $k,\ell,a$ and $b$.
\end{theorem}
In degenerate case $(k,a,b)\neq (1,2,1 )$ we have $x^2=y^{\ell}$, thus we can not give an upper bound for $\ell$. From the 5 exceptional cases 4 are the well-known examples by Sch\"affer ($b=0)$, and if $(k,\ell,a,b)=(3,2,2,1)$ we get the equation $x^2(2x^2-1)=y^2$, and the theory of Pell equations yields infinitely many integer solutions in $x,y$. We remark that Bazs\'o \cite{B} considered a more general case for shifted power values of power sums.
One can treat the cases when the parameters $k$ or $\ell$ are small. Set
$I_1=\{(1,2),(1,4),(3,2),(3,4) \}$ and $I_2=I_1\cup \{(5,2),(5,4)\}$. We obtain
\begin{theorem} \label{thm:eff2}
Let $\ell\geq 2$ be a rational integer and $k\in\{1,3\}$ with $(k,\ell)\notin I_1$, or $k\geq 1$ is a rational integer and $\ell\in \{2,4\}$ with $(k,\ell)\notin I_2$. Equation (\ref{eq:RS}) has only finitely many solutions in integers $x$ and $y$, and $\max(|x|,|y|)$ is bounded by an effectively computable constant depending on $a,b,c$ and $\ell$ or $k$, respectively.
\end{theorem}
When $(k,\ell)=(1,2)$ from (\ref{eq:RS}) we have the Pellian equation
$$
(2ax+2b-a)^2-2a(2y+c)^2=(2b-a)^2-2ac^2.
$$
Now, there are infinitely many solutions in positive integers $a,b,c$ of the equation $(2b-a)^2-2c^2=1$, and similarly for fixed triplet $(a,b,c)$ there exist infinitely many solutions $x,y$ for the previous Pellian equation.
Focusing on the exceptional cases we have
\begin{theorem} \label{thm:eff3}
Apart from the cases
\begin{itemize}
\item $(k,\ell,a,b,c)=(1,4,2,b,c)$ with $b=\pm 2c^2+1$,
\item $(k,\ell,a,b,c)=(3,2,a,b,c)$ with $\frac{c^2}{a^3}-B_4\left( \frac{b}{a} \right)=\frac{1}{30}$ or $-\frac{7}{240}$,
\item $(k,\ell,a,b,c)=(3,4,1,b,c)$ with $b(b-1)=2c^2$,
\item $(k,\ell,a,b,c)=(5,2,a,b,c)$ with $\frac{3c^2}{2a^5}-B_6\left( \frac{b}{a} \right)=-\frac{1}{42}$ or $-\frac{1}{189}$,
and
\item $ (k,\ell,a,b,c)=(5,4,a,b,c)$ with $\frac{6c^4}{a^5}-B_6\left( \frac{b}{a} \right)=-\frac{1}{42}$ or $-\frac{1}{189},$
\end{itemize}
the equations
$$S_{a,b}^1 (x) = R_{c}^{4} (y), S_{a,b}^3 (x) = R_{c}^{2} (y),S_{a,b}^3 (x) = R_{c}^{4} (y),S_{a,b}^5 (x) = R_{c}^{2} (y),$$
and $S_{a,b}^5 (x) = R_{c}^{4} (y)$, respectively,
in integers $x,y$ imply $\max(|x|,|y|)<C_3,$ where $C_3$ is an effectively computable constant depending on $a,b$ and $c$.
\end{theorem}
Our main result is the following general analogue of Theorem 1.1 in \cite{BBKPT}.
\begin{theorem} \label{thm:main}
Let $k,\ell$ be rational integers with $k\geq 2, k\notin\{3,5\}$ and $\ell=3$ or $\ell\geq 5$. Then for all nonzero integers $a,b,c$ with $\gcd(a,b)=1$ equation (\ref{eq:RS}) has only finitely many solutions $(x,y)$.
\end{theorem}
\section{Auxiliary results}
We denote by $\mathbb{C}[x]$ the ring of polynomials in the variable $x$ with complex coefficients. A decomposition of a polynomial $F(x) \in \mathbb{C}[x]$ is an equality of the following form
$$
F(x) = G_1 (G_2 (x)) \ \ \ (G_1 (x), G_2 (x) \in \mathbb{C}[x]),
$$
which is nontrivial if
$$
\deg G_1 (x) > 1 \ \ \ \text{and} \ \ \ \deg G_2 (x) > 1.
$$
Two decompositions $F(x) = G_1 (G_2 (x))$ and $F(x) = H_1 (H_2 (x))$ are said to be equivalent if there exists a linear polynomial $\ell (x) \in \mathbb{C}[x]$ such that $G_1 (x) = \ell (H_1 (x))$ and $H_2 (x) = \ell (G_2 (x))$. The polynomial $F(x)$ is called decomposable if it has at least one nontrivial decomposition; otherwise it is said to be indecomposable.
Bazs\'o, Pint\'er and Srivastava \cite{BPS} recently proved the following theorem about the decomposition of the polynomial $S_{a,b}^k \left(x\right)$.
\begin{lemma} \label{thm:BPS}
The polynomial $S_{a,b}^k \left(x\right)$ is indecomposable for even $k$. If $k=2v-1$ is odd, then any nontrivial decomposition of $S_{a,b}^k \left(x\right)$ is equivalent to the following decomposition:
\begin{equation}
S_{a,b}^k \left(x\right) = \widehat{S}_v \left(\left(x+\frac{b}{a} - \frac{1}{2}\right)^2\right).
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{thm:BPS}]
This is Theorem 2 of \cite{BPS}.
\end{proof}
For classifying the decompositions of the polynomial $R_c^{\ell} (x)$ we need the following lemma.
\begin{lemma} \label{lemma:Rlx}
The polynomial $R_{\ell} (x) =R_1 ^{\ell} (x)$ is indecomposable if $k$ is odd. If $\ell=2m$ is
even then any nontrivial decomposition of $R_k(x)$ is equivalent to
\begin{equation}
R_{\ell}(x)=\widehat{R}_m((x+(\ell-1)/2)^2),
\end{equation}
where
$$\widehat{R}_m(x)=\left(x-\frac{1}{4}\right)\left(x-\frac{9}{4}\right)\cdots \left(x-\frac{(2m-1)^2}{4}\right).$$
In particular, the polynomial $\widehat{R}_m(x)$ is indecomposable for any $m$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:Rlx}]
See Theorem 4. 3 in \cite{BBKPT}.
\end{proof}
The proof of the general case is based on the previous lemma and on the easy observation
\begin{equation} \label{eq:obser}
R_c^{\ell} (x)=c^{\ell} R_{\ell} \left(\frac{x}{c}\right).
\end{equation}
\begin{lemma} \label{lemma:Rclx}
The polynomial $R_c^{\ell} (x)$ is indecomposable if $\ell$ is odd. If $\ell=2m$ is even, then any nontrivial decomposition of $R_c^{\ell} (x)$ is equivalent to
$$
R_c^{\ell} (x) = \widehat{R}_c^m \left(\left(x + \frac{(\ell-1)c}{2}\right)^2\right),
$$
where
$$
\widehat{R}_c^m (x) = \left(x - \frac{c^2}{4}\right) \left(x - \frac{9c^2}{4}\right) \ldots \left(x - \frac{((2m-1)c)^2}{4}\right).
$$
and the polynomial $\widehat{R}_c^m (x)$ is indecomposable for any $m$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:Rclx}]
Let $\ell$ be an odd integer with $\ell\geq 1$. On supposing the contrary we obtain
$$R_c^{\ell} (x)=f_1(f_2(x)),$$
where $\deg f_1>1$ and $\deg f_2>1$. Using (\ref{eq:obser}) we have
$$c^{\ell} R_{\ell} \left(\frac{x}{c}\right)=f_1(f_2(x))$$
and
$$R_{\ell}(x)=\frac{1}{c^{\ell}}f_1(f_2(cx))$$
which is a contradiction. In the even case, from Lemma \ref{lemma:Rlx} and (\ref{eq:obser}) we get
$$f_2(x)=\left(\frac{x}{c}+\frac{\ell-1}{2}\right)^2$$
and our lemma is proved.
\end{proof}
Our next lemma provides information on the structure of the zeros of Bernoulli polynomials.
\begin{lemma} \label{eff:1}
(i) For every $d\in \mathbb Q$ and rational integer $k\geq 3$ the polynomial $B_k(x)+d$ has at least three simple zeros apart from the cases $(k,d)\in\{(4,\frac{1}{30}),(4,-\frac{7}{240}),(6,-\frac{1}{42}),(6,-\frac{1}{189})\}$.
(ii) For every $d\in \mathbb Q$ and rational integer $k\geq 7$, the polynomial $B_k(x)+d$ has at least one complex nonreal zero.
(iii) The zeros of $B_k(x)$ are all simple.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{eff:1}]
For $d=0$ and odd values of $k\geq 3$ Part (i) is a consequence of a theorem by Brillhart \cite[Corollary of Theorem 6]{Bril}. For non-zero rational $d$ and odd $k$ with $k\geq 3$ and for even values of $k\geq 4$ our lemma follows from \cite[Theorem]{pr}, and \cite[Theorem 2.3]{raka} and the subsequent remarks, respectively.
For (ii) assume that all the zeros of $B_k(x)+d$ are real. Then also all the zeros of its derivative
$$(B_k(x)+d)'=kB_{k-1}(x)$$
are real. By induction, all the roots of $B_{k-1}(x), B_{k-2}(x),\ldots $ are real. Since $k\geq 7$ and $B_6(x)$ has a complex nonreal root, we obtain a contradiction. Part (iii) was proved in \cite{dilcher}.
\end{proof}
Let $q$ be a rational number and put
$$f_{\ell,q}(x)=x(x+1)\cdots (x+\ell-1)+q.$$
\begin{lemma} \label{Rl+q}
Suppose that $\ell\geq 3$. Then $f_{\ell,q}(x)$ has at least three simple zeros apart from the cases $f_{4,1}(x)=x(x+1)(x+2)(x+3)+1$ and $f_{4,-\frac{9}{16}}(x)=x(x+1)(x+2)(x+3)-\frac{9}{16}$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Rl+q}] This is a reformulation of Theorem 2 in \cite{yuan}.
\end{proof}
Our next auxiliary result is an easy consequence of an effective theorem concerning the $S$-integer solutions of so-called hyperelliptic equations.
\begin{lemma}\label{lem:hyper}
Let $f(x)$ be a polynomial with rational coefficients and with at least two distinct zeros and $u,v$ be fixed positive rational numbers. Then the equation
$$f\left(\frac{x}{u}\right)=vy^z$$
in integers $x, y>1$ and $z>1$ implies $z<C_3$, where $C_3$. Further, if the polynomial $f$ has at least two simple zeros, then all the solutions $x$ and $y$ of the equation
$$f\left(\frac{x}{u}\right)=vy^m, m\geq 3$$
satisfy $\max(|x|,y)<C_4$, and if $f$ possesses at least three simple zeros then all the solutions $x.y$ of the equation
$$f\left(\frac{x}{u}\right)=vy^2$$
are bounded by $C_5$. Here $C_3,C_4$ and $C_5$ are effectively computable constants depending on the parameters of $f, u$ and $v$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:hyper}]
This lemma is an easy consequence of a classical theorem of Schinzel and Tijdeman \cite{ST} and the main result of \cite{brindza}.
\end{proof}
The ineffective statement of this paper is mainly based on the following lemma, which is analogous to Theorem 4.4 in \cite{BBKPT}.
\begin{lemma} \label{lemma:mainineff}
Let $k\geq 2$ be a rational integer with $k\notin\{3,5\}$. There exist no polynomial $p(x)$ and $\alpha, \beta, \gamma, \delta\in \mathbb C$ such that
$$S_{a,b}^{k}(x)=R_{c}^{\ell}(p(x)\sqrt{\alpha x^2+\beta x+\gamma}+\delta).$$
\end{lemma}
To prove this lemma we need the next result.
\begin{lemma} \label{lemma:aux}
Assume that $f(x), g(x)\in \mathbb Q[x]$ and that $f(x)=g(\lambda x+\nu)$. Further, suppose that all the zeros of $g(x)$ are rational and that $f(x)$ vanishes at $\beta\in \mathbb Q$ but it is not of the form $h((x-\beta)^d)$, where $h(x)\in \mathbb Q[x]$ and $d>1$. Then $\lambda, \nu\in \mathbb Q$.
\end{lemma}
\begin{proof} This is Lemma 4.5 in \cite{BBKPT}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lemma:mainineff}]
We have
$$R_{c}^{\ell}(p(x)\sqrt{\alpha x^2+\beta x+\gamma}+\delta)=c^{\ell}R_{\ell}\left((p(x)/c\sqrt{\alpha x^2+\beta x+\gamma}+\delta/c\right),$$
so up to replacing $p(x)$ and $\delta$ by $p(x)/c$ and $\delta/c$, we may work with the polynomial $c^{\ell}R_{\ell}(x)$ instead of the polynomial $R_{c}^{\ell}(x)$. We follow the proof of Theorem 4.4 in \cite{BBKPT}. We start with the particular case for $k, \ell \geq 2$ there exist no polynomial $p(x)$ such that
$$S_{a,b}^{k}(x)=c^{\ell} R_{\ell}(p(x)).$$
Assume on the contrary, Lemma \ref{thm:BPS} implies that $\deg p(x)\leq 2$. Suppose first that $\deg p(x)=1$. Then $p(x)=\lambda x + \nu$ and $\ell=k+1$. Suppose first that $\frac{b}{a}=\frac{1}{2}$ and that $k$ is odd. Then $b=1, a=2$ and
$$S_{2,1}^{k}(x)=\frac{2^k}{k+1}\left(B_{k+1}\left(x+\frac{1}{2}\right)-B_{k+1}\left(\frac{1}{2}\right)\right)=$$
$$\frac{2^k}{k+1}\left(x^{k+1}-\frac{(k+1)k}{24}x^{k-1}+\frac{(k+1)k(k-1)(k-2)}{384}x^{k-3}+\ldots \right)$$
for all $k\geq 5$. The zeros of $R_{\ell}(\lambda x+\nu)$ are
$$-\frac{j+\nu}{\lambda}$$
for $j=0, \ldots, \ell$, and their sum must be $0$ because $x^k$ appears with coefficient equal to zero in $S_{2,1}^{k}(x)$, therefore
$$0=-\frac{1}{\lambda}\sum_{j=0}^{k}(j+\nu),$$
so
$$\nu=-\frac{k}{2}=-\frac{\ell-1}{2}.$$
Thus we get that
$$S_{2,1}^{k}(x)=c^{k+1}R_{k+1}\left(\lambda x-\frac{k}{2}\right)=c^{k+1}\widehat{R}_{(k+1)/2}((\lambda x)^2)=$$
$$=c^{k+1}\left((\lambda x)^2-\frac{1}{4}\right)\left((\lambda x)^2-\frac{9}{4}\right)\cdots \left((\lambda x)^2-\frac{k^2}{4}\right)=$$
$$=c^{k+1}\left( (\lambda x)^{k+1}-\frac{k(k+1)(k+2)}{24}(\lambda x)^{k-1}+\right.$$
$$\left. +\frac{k(k^2-1)(k^2-4)(5k+12)}{5760}(\lambda x)^{k-3}+\ldots \right).$$
Identifying the first three nonzero coefficients above, we get
$$\frac{2^k}{k+1}=(c\lambda)^{k+1},$$
$$\frac{2^k k}{24}=\frac{c^{k+1}\lambda^{k-1}k(k+1)(k+2)}{24},$$
$$\frac{2^k k(k-1)(k-2)}{384}=\frac{c^{k+1}\lambda^{k-3}k(k^2-1)(k^2-4)(5k+12)}{5760}.$$
Dividing the first equation by the second one we have
$$\lambda^2=k+2$$
and dividing the second equation by the third one we obtain
$$\lambda^2=\frac{5k+12}{15},$$
giving $k=-1.8$, contradiction.
From now on, we assume that either $b/a\neq 1/2$ or $b/a=1/2$ but $k$ is even. Then the argument in \cite{BBKPT} applies. Namely, $S_{a,b}^{k}(x)$ has a zero at $x=0$ and it is not of the form $h(x^d)$ for some $d>1$ and polynomial $h(x)$ by Lemma \ref{thm:BPS}, so $\lambda, \nu \in \mathbb Q$ by Lemma \ref{lemma:aux}. In particular, all the zeros of the polynomial $R_{k+1}(\lambda x+\nu)$ are real. By Lemma \ref{eff:1}, we deduce that $k\leq 5$. So, we have to check the impossibility of the identity
$$S_{a,b}^k(x)=c^{k+1}R_{k+1}(\lambda x+\nu)$$
for some $a,b,c\in \mathbb N$ with $\gcd(a,b)=1$, $\lambda, \nu\in \mathbb Q$ and $k\in \{2,3,4,5\}$. We give the details only for $k=2$ the calculations in other cases are very similar and we leave them to the reader.
For $k=2$ we have
$$S_{a,b}^{2}(x)=\frac{a^2}{3}x^3+\frac{a(2b-a)}{2}x^2+\left(\frac{a^2}{6}-ab+b^2\right)x$$
and
$$c^3R_3(\lambda x+\nu)=c^3\lambda^3 x^3+3c^3\lambda^2(\nu+1)x^2+c^3(2\lambda+6\lambda\nu+3\lambda \nu^2)x+c^3(\nu^3+3\nu^2+2\nu).$$
On comparing the corresponding coefficients we get
\begin{equation} \label{comp:1}
\frac{a^2}{3}=c^3\lambda^3
\end{equation}
\begin{equation} \label{comp:2}
\frac{a(2b-a)}{2}=3c^3\lambda^2 (\nu+1)
\end{equation}
\begin{equation} \label{comp:3}
\frac{a^2}{6}-ab+b^2=c^3\lambda (3\nu^2+6\nu+2)
\end{equation}
and
\begin{equation} \label{comp:4}
0=c^3 \nu (\nu+1)(\nu+2).
\end{equation}
From (\ref{comp:4}) we obtain $\nu \in \{0,-1,-2\}$. Suppose first that $\nu=0$. Then dividing (\ref{comp:1}) by (\ref{comp:2}) and dividing (\ref{comp:2}) by (\ref{comp:3}) we get
$$\lambda=\frac{2a}{2b-a}$$
and
$$\lambda=\frac{a(2b-a)}{\frac{a^2}{2}-3ab+3b^2}.$$
These relations yield
$$a^2-6ab+6b^2=(2b-a)^2$$
and
$$2b(b-a)=0,$$
thus $a=b$ or $b=0$, a contradiction. Now assume that $\nu=-1$. Then from (\ref{comp:2}), $a(2b-a)=0$, we get a contradiction again. Finally, if $\nu=-2$, using the previous argument, and obtaining $3b^2+2ab=0$ we arrive at a contradiction.
We now assume that that $\deg p(x)=2$, in which case $k+1=2\ell$. By Lemma \ref{thm:BPS}, the decomposition $S_{a,b}^k (x)=c^{\ell}R_{\ell}(p(x))$ is equivalent to
$$S_{a,b}^{k} (x)= \widehat{S}_{(k+1)/2} \left(\left(x+\frac{b}{a} - \frac{1}{2}\right)^2\right)$$
which means that
$$p(x)=\lambda\left(x+\frac{b}{a} - \frac{1}{2}\right)^2 + \nu \quad \text{and} \quad \widehat{S}_{(k+1)/2}(x)=c^{\ell} R_{\ell}(\lambda x+\nu).$$
If $\ell=2$, we get $k=3$, however this is an easily excludable case. Thus we may assume that $\ell\geq 3$. The polynomial $\widehat{S}_{k}(x)$, vanishes at $x_0=(1/2-b/a)^2$, because
$$\widehat{S}_{m}\left(\left(\frac{1}{2}-\frac{b}{a}\right)^2\right)=S_{a,b}^{2m-1}\left(1-\frac{2b}{a}\right)=$$
$$=\frac{a^{2\ell-1}}{2\ell}\left(B_{2\ell}\left(1-\frac{2b}{a}+\frac{b}{a}\right)-B_{2\ell}\left(\frac{b}{a}\right)\right)=0,$$
where we used the fact that $B_{2\ell}(1-y)=B_{2\ell}(y)$ with $y=b/a$. This polynomial is not of the form $h((x-x_0)^d)$ for some $d>1$ by the argument from the footnote of page 181 on \cite{BBKPT}. Indeed, if it were, by the indecomposability of $\widehat{S}_{\ell}(x)$ (see Lemma \ref{thm:BPS}), we would get that
$$h(x)=\frac{a^{2\ell-1}}{(x-x_0)^m},$$
so
$$\frac{a^{2\ell-1}}{2\ell}\left(B_{2\ell}\left(x+\frac{b}{a}-B_{2\ell}\left(\frac{b}{a}\right)\right)\right)=\frac{a^{2\ell-1}}{2\ell}\left(\left(x+\frac{b}{a}-\frac{1}{2}\right)^2-x_0\right)^{\ell},$$
so
$$B_{2\ell}(x)=(x^2-x_0)^{\ell}+C,$$
where
$$C=\frac{2\ell}{a^{2\ell-1}}B_{2\ell}\left(\frac{b}{a}\right).$$
Taking the derivative in the above formula and using the fact that $k\geq 3$, we conclude that $\pm \sqrt{x_0}$ are double roots of $B_{2\ell}'(x)=2mB_{2\ell-1}(x)$, which is impossible by Lemma \ref{eff:1}, part (iii). Hence, $\lambda, \nu \in \mathbb Q$. It remains to identify coefficients. It is easy to see that the polynomial $\widehat{S}_{\ell}(x)$ and the polynomial $\widetilde{B_{\ell}}(x)$ of \cite{BBKPT} are related via the formula
$$\widehat{S}_{\ell}(x)=\frac{a^{2\ell-1}}{2\ell}\widetilde{B_{\ell}}(x)+D,$$
with
$$D=\frac{a^{2\ell-1}}{2\ell}\left(B_{2\ell}-B_{2\ell}\left(\frac{b}{a}\right)\right).$$
Thus, we get, from a previous calculation (with the change of variable $k+1=2\ell$),
$$\widehat{S}_{\ell}(x)=\frac{a^{2\ell-1}}{2\ell}\left(x^\ell-\frac{2\ell(2\ell-1)}{24}x^{\ell-1}+\right.$$
$$\left.+\frac{2\ell(2\ell-1)(2\ell-2)(2\ell-3)}{384}x^{\ell-2}+\ldots \right).$$
Writing
$$c^{\ell}R_{\ell}(\lambda x+\nu)=c^{\ell}\left((\lambda x+\nu)^{\ell}+\frac{\ell(\ell-1)}{2}(\lambda x+\nu)^{\ell-1}+\right.$$
$$\left. +\frac{\ell(\ell-1)(2\ell-1)}{6}(\lambda x+\nu)^{\ell-2}+\ldots \right),$$
and identifying the corresponding coefficients, we get
$$\frac{a^{2\ell-1}}{2\ell}=c^\ell \lambda^m;$$
$$-\frac{a^{2\ell-1}(2\ell-1)}{24}=c^{\ell} \lambda^{\ell-1}\ell\left(\nu+\frac{\ell-1}{2}\right);$$
and
$$\frac{a^{2\ell-1}(2\ell-1)(2\ell-2)(2\ell-3)}{384}=$$
$$\frac{c^{\ell}\lambda^{\ell-2}\ell(\ell-1)}{2}\left(\nu^2+(\ell-1)\nu+\frac{2\ell-1}{3} \right).$$
Taking ratios of the first two equations and then the next two equations, we get
$$\frac{\lambda}{\nu+(\ell-1)}=-\frac{12}{2\ell-1};$$
$$\frac{\lambda(\nu+(\ell-1)/2}{\nu^2+(\ell-1)\nu+(2\ell-1)/3}=-\frac{4}{2\ell-3}.$$
Dividing the second equation above equation by the first, we get
$$\frac{(\nu+(\ell-1)/2)^2}{(\nu+(\ell-1)/2)^2-(3\ell^2-14\ell+7)/12}=\frac{2\ell-1}{3(2\ell-3)}.$$
This gives
$$\frac{x^2}{x^2-(3\ell^2-14\ell+7)/12}=\frac{2\ell-1}{3(2\ell-3)}$$
with $x=\nu+(\ell-1)/2$, so
$$(\ell-2)x^2=-\frac{(2\ell-1)(3\ell^2-14\ell+7)}{12}.$$
This can be checked to be false for $\ell=2,3,4$ and for $\ell\geq 5$, the left-hand side is positive and the right-hand side is negative.
This shows that indeed it is not possible that $S_{a,b}^k(x)=c^{\ell}R_{\ell}(p(x))$ for some polynomial $p(x)$.
Now we can prove the theorem in its full generality. Assume that
$$r(x)=\alpha x^2+\beta x+\gamma$$
is not a complete square, otherwise $p(x)\sqrt{r(x)}+\delta$ is a polynomial, case which has already been treated. The argument from \cite{BBKPT} applies to say that
$$c^{\ell}R_{\ell}(p(x)\sqrt{r(x)}+\delta)=c^{\ell} r(x)^{\ell/2}p(x)^{\ell}+$$
$$+c^{\ell} r(x)^{(\ell-1)/2}p(x)^{\ell-1}\left(\ell\delta+\frac{\ell(\ell-1)}{2}\right)+\ldots $$
is a polynomial so $\ell$ must be even. Furthermore,
$$\ell\delta+\frac{\ell(\ell-1)}{2}=0,$$
that is $\delta=-\frac{\ell-1}{2}$.
But then
$$R_{\ell}(p(x)\sqrt{r(x)}+\delta)=R_{\ell}\left(p(x)\sqrt{r(x)}-\frac{\ell-1}{2}\right)=\widehat{R}_{\ell/2}(r(x)p(x)^2).$$
Thus, $S_{a,b}^k(x)=\widehat{R}_{\ell/2}(\widetilde{p}(x))$, where $\widetilde{p}(x)=r(x)p(x)^2$. The case $\ell=2$ leads to
$$S_{a,b}^k(x)=cr(x)p(x)^2-\frac{c}{4},$$
so
$$\frac{a^k}{k+1}\left( B_{k+1}\left(x+\frac{b}{a}\right)-B_{k+1}\left(\frac{b}{a}\right) \right)=cr(x)p(x)^2-\frac{c}{4},$$
so
$$B_{k+1}(x)=\frac{c(k+1)}{a^k}r\left(x-\frac{b}{a}\right)p\left(x-\frac{b}{a}\right)^2+\left(B_{k+1}\left(\frac{b}{a}\right) -\frac{c(k+1)}{4a^k}\right).$$
By Lemma \ref{eff:1}, we get $k\in \{3,5\}$, however, these cases are excluded by the condition of our lemma.
So, it must be the case that $\ell\geq 4$. We have $S_{a,b}^k(x)=\widehat R_{\ell/2}(\widetilde{p}(x))$. By Lemma \ref{thm:BPS} and the fact that $r(x)$ is not a complete square, it follows that in fact $r(x)$ is a linear polynomial. Say $r(x)=\lambda x+\nu$. Assume first that $b/a=1/2$ and $n$ is even. We then get
$$\widehat{R}_{\ell/2}(r(x))=S_{a,b}^k(x)=\widehat{S}_{(k+1)/2}\left(\left(x+\frac{b}{a}-\frac{1}{2}\right)^2\right),$$
with a linear polynomial $r(x)$, contradicting the indecomposability of $\widehat{R}_{\ell/2}(x)$, see Lemma \ref{lemma:Rlx}. So either $b/a\neq 1/2$ or $b/a=1/2$ but $k$ is not even. Then $S_{a,b}^{k}(x)$ has $x=0$ as a zero but it is not of the form $h(x^d)$ for any $d>1$ by Lemma \ref{thm:BPS} and $\widehat{R}_{\ell}(x)$ has rational zeros. So, from Lemma \ref{lemma:aux}, $\lambda$ and $\nu$ are rational. In particular, all zeros of $\widehat{R}_{\ell/2}(r(x))$ are real. Thus, $S_{a,b}^{k}$ has only real roots showing that $k\in \{2,3,4,5\}$. Considering these small cases, on comparing the corresponding coefficients we obtain a contradiction.
\end{proof}
We will introduce some notation to recall the finiteness criterion by Bilu and Tichy. In what follows $\alpha$ and $\beta$ are nonzero rational numbers, $\mu,\nu$ and $q$ are positive integers, $p$ is a nonnegative integer and $\nu(X)\in \mathbb Q[X]$ is a nonzero polynomial (which may be constant).
A standard pair of the first kind is $(X^q,\alpha X^{p}\nu(X)^q)$ or switched, $(\alpha X^{p}\nu(X)^q, X^q)$, where $0\leq p<q, (p,q)=1$ and $p+\deg \nu(X)>0$.
A standard pair of the second kind is $(x^2,(\alpha x^2+\beta)\nu(x)^2)$ (or switched).
Denote by $D_{\mu}(x,\delta)$ the $\mu$th Dickson polynomial, defined by the functional equation $D_{\mu}(z+\delta/z,\delta)=z^{\mu}+(\delta/z)^{\mu}$or by the explicit formula
$$D_{\mu}(x,\delta)=\sum_{i=0}^{[\mu/2]}d_{\mu,i}x^{\mu-2i},$$
with
$$d_{\mu,i}=\frac{\mu}{\mu-i}{\binom{\mu-i}{i}}(-\delta)^i.$$
A standard pair of the third kind is $D_{\mu}(x,\alpha^{\nu}),D_{\nu}(x,\alpha^{\mu}$, where $\gcd(\mu,\nu)=1$.
A standard pair of the fourth kind is $\left(\alpha^{-\mu/2} D_{\mu}(x,\alpha), -\beta^{-\nu/2}D_{\nu}D_{\nu}(x,\beta)\right),$ where $\gcd(\mu,\nu)=2$.
A standard pair of the fifth kind is $((\alpha x^2-1)^3, 3x^4-4x^3)$ (or switched).
\begin{lemma} \label{lemma:BT}
Let $Rx),S(x)$ be nonconstant polynomials such that the equation $R(x)=S(y)$ has infinitely many solutions in rational integers $x,y$. Then $R(x)=\phi(f(\kappa(x)))$ and $S(x)=\phi(g(\lambda(x)))$ where $\kappa(x),\lambda(x)\in \mathbb Q[x]$ are linear polynomials, $\phi(x)\in \mathbb Q[x]$, and $(f(x),g(x))$ is a standard pair.
\end{lemma}
We need the analogs of Lemmata 5.2 and 5.3 in \cite{BBKPT}. Both of them follow immediately from the analogous results in \cite{BBKPT}, so their proofs are omitted.
\begin{lemma} \label{lemma:npower}
None of the polynomials $S_{a,b}^n(a_1x+a_0)$ or $c^mR_m(b_1x+b_0)$ or is of the form $e_1x^q+e_0$ with $q\geq 3$.
\end{lemma}
\begin{lemma} \label{lemma:ndickson}
The polynomial $S_{a,b}^{n}(a_1 x+a_0)$ is not of the form $e_1D_{t}(x,\delta)+e_0$, where $D_t(x,\delta)$ is the Dickson polynomial with $t>4$ and $\delta\in \mathbb Q\setminus \{0\}$.
\end{lemma}
\section{Proofs of the Theorems}
\begin{proof}[Proof of Theorem \ref{thm:eff1}]
If $b=0$ we essentialy obtain the original Sch\"affer equation so in the sequel we assume $b\neq 0$.
For $k\in \{2,4\}$ or $k\geq 6$, our theorem is an easy consequence of (\ref{eq:mainI}), Lemmata \ref{eff:1} and \ref{lem:hyper}. Suppose that $k=1$ and consider the equation
$$S_{a,b}^1 (x) =\frac{1}{2}x(ax+2b-a)=y^{\ell}.$$
Since $(a,b)=1$, one can see that the quadratic polynomial on the left hand side has two simple zeros apart from the case $a=2,b=1$. For $k=3$ and $k=5$ the discriminant of $S_{a,b}^k (x)$
is
$$\frac{1}{256}a^6b^4(a-b)^4(a-2b)^2(a^2+4ab-4b^2)$$
and
$$\frac{1}{967458816}a^{20}b^4(a-b)^4(a-2b)^2(a^2+3ab-3b^2)^4(a^2-6ab+6b^2)^2\times$$
$$\times (a^2+4ab-4b^2)(a^2+2ab-2b^2)^2(3a^4+12a^3b+4a^2b^2-32ab^3+16b^4), $$
respectively.
We have two critical cases (i. e. $\frac{a}{b}$ is a rational zero of discriminants,) $a=b=1$ and $a=2,b=1$. In the first case
$$S_{1,1}^{k}(x)=S_{k}(x+1),$$
where $S_k(x)$ denotes the usual Sh\"affer's sum of $k$th powers. If $a=2, b=1$ then we get
$$S_{2,1}^{3}(x)=2x^4-x^2\,\,\mbox{and}\,\, S_{2,1}^{5}(x)=\frac{1}{3}x^2(16x^4-20x^2+7),$$
and these polynomials have two and four simple zeros, respectively.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:eff2}]
First we consider our equation
$$
S_{a,b}^{k}(x)=R_{c}^{\ell}(y)
$$
in integers $x$ and $y$, where $\ell\geq 2$ and $k\in\{1,3\}$. Formulas (\ref{eq:mainI}) and (\ref{eq:obser})
give
$$ R_{c}^{\ell}(y)=c^{\ell}R_{\ell}\left(\frac{y}{c}\right)=\frac{1}{2}x(ax+2b-a),$$
$$8ac^{\ell}R_{\ell}\left(\frac{y}{c}\right)+(2b-a)^2=(2ax+2b-a)^2,$$
and
$$ R_{c}^{\ell}(y)=c^{\ell}R_{\ell}\left(\frac{y}{c}\right)=\frac{1}{4}x(ax+2b-a)\times$$
$$\times (a^2x^2+(2ba-a^2)x+2b^2-2ab),$$
$$4ac^{\ell}R_{\ell}\left(\frac{y}{c}\right)=X(X+2b^2-2ab)=(X+(b^2-ab))^2-(b^2-ab)^2,$$
where $X=a^2x^2+(2ab-b^2)x$, respectively, and Lemmas \ref{Rl+q} and \ref{lem:hyper} complete the proof for $(k,\ell)\notin \{(1,2),(3,2),(1,4),(3,4)\}$.
Now, if $\ell\in\{2,4\}$ we have
$S_{a,b}^{k}(x)=y(y+c), 4S_{a,b}^{k}(x)+c^2=(2y+c)^2$
and
$$S_{a,b}^{k}(x)=y(y+c)(y+2c)(y+3c)=(y^2+3cy+c^2)^2-c^4,$$
respectively, and our result is proved by (\ref{eq:mainI}) and Lemmas \ref{eff:1} and \ref{lem:hyper} for
$$(k,\ell)\notin \{(1,2),(3,2),(1,4),(3,4),(5,2),(5,4)\}.$$
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:eff3}]
For $(k,\ell)=(1,4)$ we get
$$\frac{1}{2}x(ax+2b-a)=y(y+c)(y+2c)(y+3c),$$
and
$$8ac^4\frac{y}{c}\left(\frac{y}{c}+1 \right)\left(\frac{y}{c} +2 \right)\left( \frac{y}{c}+3 \right)=(2ax+2b-a)^2-(2b-a)^2.$$
Since $\frac{(2b-a)^2}{8ac^4}$ is non-negative, we cannot guarantee three simple zeros when this fraction is 1 (cf. Lemma \ref{Rl+q}). If
$(2b-a)^2=8ac^4$, we have $a=2$. Indeed, by the parities $a\neq 1$. Denote by $p$ an arbitrary prime divisor of $a$, so $p|a$ and thus $p|2b$, and $p=2$. Now, if $a=2^{\alpha}$, where $\alpha\geq 2$, then $\mbox{ord}_2(2b-a)=1$ which is a contradiction, so $a=2$ and $(b-1)^2=4c^4$. For $(k,\ell)=(3,4)$ we can apply a very similar argument, and here
$$\frac{(b^2-ab)^2}{4ac^4}=1,$$
and this yields $a=1,\left(b(b-1)\right)^2=4c^4$.
For $(k,\ell)=(3,2),(5,2)$ and $(5,4)$ we follow the same idea, and give the details only for case $(k,\ell=(3,2)$.
Consider the equation
$$S_{a,b}^{3}(x)=\frac{a^3}{4}\left(B_4\left(x+\frac{b}{a}\right)-B_4\left(\frac{b}{a}\right)\right)=R_{c}^{2}(y)=y(y+c),$$
and
$$a^3\left(B_4\left(x+\frac{b}{a}\right)-B_4\left(\frac{b}{a}\right)+\frac{c^2}{a^3}\right)=(2y+c)^2.$$
Finally, Lemma \ref{eff:1} completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main}] We follow Section 5.3 of \cite{BBKPT}. In view of the small cases treated and
the fact that we have proved the analog of Theorem 4.4 in \cite{BBKPT}, the argument
from Page 184 shows that we may assume that $(f(x), g(x))$ do not form
a pair of second or fifth kind. If it is of the first kind, we get the same
contradiction based on Lemma \ref{lemma:npower}, and if it is of the fourth kind, we get
again the contradiction based on Lemma \ref{lemma:ndickson}. So, we only need to revisit the
argument in \cite{BBKPT} for the pairs of the third kind. For this, we just notice that,
with the notations from there, all coefficients $s_i$ get multiplied by $a^n = a_2$
(except for the last one which also gets shifted but hopefully we shall not
get to it), and all the coefficients $r_j$ get multiplied by $c_m$. So, the analogs of
(26)-(29) in \cite{BBKPT} become
$$s_3=\frac{b_{1}^{3}a^2}{3}=e_1,$$
$$s_1=-\frac{b_1a^2}{24}=-3e_1\alpha^m,$$
$$r_m=a_{1}^{m}c^m=e_1,$$
$$r_{m-2}=-a_{1}^{m-2}c^m\frac{m(m-1)(m+1)}{24}=-e_1m\alpha^3.$$
So we get
$$\alpha^m=\frac{b_{1}^{-2}}{24}, \alpha^3=a_{1}^{-2}\frac{m^2-1}{24}, c^ma_{1}^{m}=\frac{a^2b_{1}^{3}}{3}.$$
Hence,
$$\frac{b_{1}^{-6}}{24^3}=a_{1}^{-2m}\left(\frac{m^2-1}{24}\right)^m=(a^2c^{-m})^{-2}9b_{1}^{-6}\left(\frac{m^2-1}{24}\right)^m,$$
giving
$$\frac{1}{2^9 3^5}=\left(\frac{c^m}{a^2}\right)^2\left(\frac{m^2-1}{24}\right)^m.$$
If $m$ is even, the number on the right above is a square of a rational number, whereas the number on the left is not. So, $m$ is odd. Now, we get
$$\frac{1}{2^9 3^5}=\left(\frac{c^m}{a^2}\right)^2\left(\frac{m^2-1}{24}\right)^{m-1}\left(\frac{m^2-1}{24}\right),$$
or
$$\frac{1}{m^2-1}=2^6 3^4\left(\frac{c^m}{a^2}\right)^2\left(\frac{m^2-1}{24}\right)^{m-1},$$
and the right-hand side above is a square of a rational number, therefore so
is the left-hand side, so $m^2-1$ is a square, contradiction.
The theorem is proved.
\end{proof}
\end{document} |
\begin{document}
\thispagestyle{plain}
\title[Compositions into Powers of $b$]{Compositions into Powers of $b$: \\
Asymptotic Enumeration and Parameters}
\author{Daniel Krenn}
\address{Daniel Krenn \\
Institute of Analysis and Computational Number Theory (Math A) \\
Graz University of Technology \\
Steyrergasse 30 \\
8010 Graz \\
Austria
}
\email{\href{mailto:[email protected]}{[email protected]} \textit{or}
\href{mailto:[email protected]}{[email protected]}}
\author{Stephan Wagner}
\address{Stephan Wagner \\
Department of Mathematical Sciences \\
Stellenbosch University \\
Private Bag X1 \\
Matieland 7602 \\
South Africa
}
\email{\href{mailto:[email protected]}{[email protected]}}
\thanks{This material is based upon work supported by the National Research
Foundation of South Africa under grant number 70560.}
\thanks{Daniel Krenn is supported by the Austrian Science Fund (FWF): P24644
and by the Austrian Science Fund (FWF): W1230, Doctoral Program ``Discrete
Mathematics''.}
\thanks{The authors would like to thank Christian Elsholtz for pointing us at the problems discussed in this paper.}
\date{\today}
\keywords{compositions, powers of $2$,
infinite transfer matrices, asymptotic enumeration}
\begin{abstract}
For a fixed integer base $b\geq2$, we consider the number of compositions
of~$1$ into a given number of powers of~$b$ and, related, the maximum number
of representations a positive integer can have as an ordered sum of powers
of~$b$.
We study the asymptotic growth of those numbers and give precise asymptotic
formulae for them, thereby improving on earlier results of Molteni. Our
approach uses generating functions, which we obtain from infinite transfer
matrices.
With the same techniques the distribution of the largest denominator and the
number of distinct parts are investigated.
\end{abstract}
\maketitle
\section{Introduction}
Representations of integers as sums of powers of $2$ occur in various contexts, most notably of course in the usual binary representation.
\emph{Partitions} of integers into powers of $2$, i.e., representations of the
form
\begin{equation}\ellbel{eq:partition}
\ell = 2^{a_1} + 2^{a_2} + \cdots + 2^{a_n}
\end{equation}
with nonnegative integers $a_1 \geq a_2 \geq \cdots \geq a_n$ (not necessarily distinct!) are also known as
\emph{Mahler partitions} (see \cite{Knuth:1966:almost, Bruijn:1948:mahler,
Pennington:1953:Mahler-part-prob, Mahler:1940:spec-functional-eq}).
The number of such partitions exhibits interesting periodic fluctuations. The
situation changes, however, when \emph{compositions} into powers of $2$ are considered,
i.e., when the summands are arranged in an order. In other words, we consider
representations of the form~\eqref{eq:partition} without further restrictions
on the exponents $a_1$, $a_2$, \ldots, $a_n$ other than being nonnegative.
Motivated by the study of the exponential sum
\begin{equation*}
s(\xi) = \sum_{r=1}^{\tau} \xi^{2^r},
\end{equation*}
where $\xi$ is a primitive $q$th root of unity and $\tau$ is the order of $2$
modulo $q$ (see \cite{Molteni:2010:canc-exp-sum}),
Molteni~\cite{Molteni:2012:repr-2-powers-asy} recently studied the maximum
number of representations a positive integer can have as an ordered sum of $n$
powers of $2$. More generally, fix an integer $b\geq2$, let
\begin{equation}\ellbel{eq:def-Ub}
\f{\mathcal{U}_b}{\ell,n} = \card*{\set*{(a_1,a_2,\ldots,a_n) \in \N_0^n}{
b^{a_1} + b^{a_2} + \dots + b^{a_n} = \ell}}
\end{equation}
be the number of representations of $\ell$ as an ordered sum of $n$ powers of
$b$, and let $\f{\mathcal{W}_b}{s,n}$ be the maximum of $\f{\mathcal{U}_b}{\ell,n}$ over all
positive integers $\ell$ with $b$\nbd-ary sum of digits equal to $s$. It was
shown in \cite{Molteni:2010:canc-exp-sum} that
\begin{equation}\ellbel{eq:ws_equation}
\frac{\f{\mathcal{W}_2}{s,n}}{n!}
= \sum_{\substack{k_1,k_2,\ldots,k_s \geq 1 \\ k_1+k_2+\cdots+k_s = n}}
\prod_{j=1}^s \frac{\mathcal{W}_2(1,k_j)}{k_j!},
\end{equation}
which generalizes in a straightforward fashion to arbitrary bases~$b$. So
knowledge of $\f{\mathcal{W}_b}{1,n}$ is the key to understanding $\f{\mathcal{W}_b}{s,n}$ for
arbitrary $s$.
For the moment, let us consider the case $b=2$. There is an equivalent
characterisation of $\f{\mathcal{W}_2}{1,n}$ in terms of compositions of $1$. To this
end, note that the number of representations of $2^h\ell$ as a sum of powers
of $2$ is the same as the number of representations of $\ell$ for all integers $h$
if negative exponents are allowed as well (simply multiply/divide everything by
$2^h$). Therefore, $\f{\mathcal{W}_2}{1,n}$ is also the number of solutions to the
Diophantine equation
\begin{equation}\ellbel{eq:maineq}
2^{-k_1} + 2^{-k_2} + \cdots + 2^{-k_n} = 1
\end{equation}
with nonnegative integers $k_1,k_2,\ldots,k_n$, i.e., the number of
\emph{compositions} of $1$ into powers of $2$. This sequence starts with
\begin{equation*}
1, 1, 3, 13, 75, 525, 4347, 41245, 441675, 5259885, 68958747, \dots
\end{equation*}
and is \OEIS{A007178} in the On-Line Encyclopedia of Integer
Sequences~\cite{OEIS:2014}.
The main goal of this paper is to determine precise asymptotics for the number
of such binary compositions as $n \to \infty$. Lehr, Shallit and
Tromp~\cite{Lehr-Shallit-Tromp:1996:vec-spc-autom-reals} encountered these
compositions in their work on automatic sequences and gave a first bound,
namely
\begin{equation*}
\mathcal{W}_2(1,n)/n! \leq K \cdot 1.8^n
\end{equation*}
for some constant $K$. It was mainly based on an asymptotic formula for the
number of \emph{partitions} of $1$ into powers of $2$, which was derived
independently in different contexts, cf.\@ \cite{Boyd:1975,
Flajolet-Prodinger:1987:level, Komlos-Moser-Nemetz:1984} (or see
the recent paper of Elsholtz, Heuberger and
Prodinger~\cite{Elsholtz-Heuberger-Prodinger:2013:huffm} for a detailed
survey). This bound was further improved by Molteni, who gave the inequalities
\begin{equation*}
0.3316 \cdot (1.1305)^n \leq \mathcal{W}_2(1, n)/n! \leq (1.71186)^{n-1} \cdot n^{-1.6}
\end{equation*}
in \cite{Molteni:2010:canc-exp-sum}. Giorgilli and
Molteni~\cite{Giorgilli-Molteni:2013:repr-2-powers-rec} provided an efficient
recursive formula for $\mathcal{W}_2(1,n)$ and used it to prove an intriguing congruence
property. In his recent paper~\cite{Molteni:2012:repr-2-powers-asy}, Molteni
succeeded in proving the following result, thus also disproving a conjecture of
Knuth on the asymptotic behaviour of $\mathcal{W}_2(1,n)$.
\begin{thm}[Molteni~\cite{Molteni:2012:repr-2-powers-asy}]\ellbel{thm:molteni}
The limit
\begin{equation*}
\gamma = \lim_{n \to \infty} (\mathcal{W}_2(1,n)/n!)^{1/n}
= 1.192674341213466032221288982528755\ldots
\end{equation*}
exists.
\end{thm}
Molteni's argument is quite sophisticated and involves the study of the
spectral radii of certain matrices. The aim of this paper will be to present a
different approach to the asymptotics of $\mathcal{W}_2(1,n)$ (and more generally,
$\f{\mathcal{W}_2}{s,n}$) by means of generating functions that allows us to obtain more
precise information. Our main theorem reads as follows.
\begin{thm}\ellbel{thm:asymptotics}
There exist constants $\alpha = 0.2963720490\dots$, $\gamma =
1.1926743412\dots$ (as in Theorem~\ref{thm:molteni}) and $\kappa =
2/(3\gamma) < 1$ such that
\begin{equation*}
\frac{\mathcal{W}_2(1,n)}{n!} = \alpha \gamma^{n-1} (1 + \Oh{\kappa^n}).
\end{equation*}
More generally, for every fixed $s$, there exists a polynomial $\f{P_s}{n}$
with leading term
\begin{equation*}
(\alpha/\gamma)^s n^{s-1}/(s-1)!
\end{equation*}
such that
\begin{equation*}
\frac{\f{\mathcal{W}_2}{s,n}}{n!} = \f{P_s}{n} \gamma^{n} (1 + \Oh{\kappa^n}).
\end{equation*}
\end{thm}
We also prove a more general result for arbitrary bases instead of $2$.
Consider the Diophantine equation
\begin{equation}\ellbel{eq:maineq_general}
b^{-k_1} + b^{-k_2} + \cdots + b^{-k_n} = 1.
\end{equation}
Multiplying by the common denominator and taking the equation modulo $b-1$, we
see that there can only be solutions if $n \equiv 1 \bmod b-1$, i.e., $n =
(b-1)m+1$ for some nonnegative integer $m$. We write $q_b(m)$ for the number of
solutions ($n$-tuples of nonnegative integers
satisfying~\eqref{eq:maineq_general}) in this case. Note that $q_b(m)$ is also
the maximum number of representations of an arbitrary power of $b$ as an
ordered sum of $n = (b-1)m+1$ powers of $b$. We have the following general
asymptotic formula.
\begin{thm}\ellbel{thm:asymptotics:general_base}
For every positive integer $b \geq 2$, there exist constants $\alpha =
\alpha_b$, $\gamma = \gamma_b$ and $\kappa = \kappa_b < 1$ such that the
number $q_b(m)$ of compositions of $1$ into $n = (b-1)m+1$ powers of $b$,
which is also the maximum number $\f{\mathcal{W}_b}{1,n}$ of representations of a
power of $b$ as an ordered sum of $n$ powers of $b$, satisfies
\begin{equation*}
\frac{\f{\mathcal{W}_b}{1,n}}{n!} = \frac{q_b(m)}{n!}
= \alpha \gamma^m (1 + \Oh{\kappa^m}).
\end{equation*}
More generally, the maximum number $\f{\mathcal{W}_b}{s,n}$ of representations of a
positive integer with $b$-ary sum of digits $s$ as an ordered sum of $n =
(b-1)m+s$ powers of $b$ is asymptotically given by
\begin{equation*}
\frac{\f{\mathcal{W}_b}{s,n}}{n!} = \f{P_{b,s}}{m} \gamma^m (1 + \Oh{\kappa^m}),
\end{equation*}
where $\f{P_{b,s}}{m}$ is a polynomial with leading term $\alpha^s
m^{s-1}/(s-1)!$.
\end{thm}
The key idea is to equip every \emph{partition} of $1$ into powers of $2$ (or
generally $b$) with a weight that essentially gives the number of ways it can
be permuted to a composition, and to apply the recursive approach that was used
to count partitions of $1$: if $p_2(n)$ denotes the number of such partitions
into $n$ summands, then the remarkable generating function identity
\begin{equation}\ellbel{eq:partitionsgf}
\sum_{n=1}^{\infty} p_2(n)x^n =
\frac{\sum_{j=0}^{\infty} (-1)^j x^{2^j-1}
\prod_{i=1}^j \frac{x^{2^i-1}}{1-x^{2^i-1}}}{\sum_{j=0}^{\infty} (-1)^j
\prod_{i=1}^j \frac{x^{2^i-1}}{1-x^{2^i-1}}}
\end{equation}
holds, and this can be generalised to arbitrary bases $b$, see the recent paper of
Elsholtz, Heuberger and
Prodinger~\cite{Elsholtz-Heuberger-Prodinger:2013:huffm}. In our case, we do
not succeed to obtain a similarly explicit formula for the generating function,
but we can write it as the quotient of two determinants of infinite matrices
and infer analytic information from it. The paper is organised as follows: we
first describe the combinatorial argument that yields the generating function,
a priori only within the ring of formal power series. We then study the
expression obtained for the generating function in more detail to show that it
can actually be written as the quotient of two entire functions. The rest of
the proof is a straightforward application of residue calculus (using the
classical Flajolet--Odlyzko singularity
analysis~\cite{Flajolet-Odlyzko:1990:singul}).
Furthermore, we consider the maximum of $\f{\mathcal{U}_b}{\ell,n}$ over all $\ell$, for
which we write
\begin{equation*}
\f{M_b}{n} = \max_{\ell \geq 1}\, \f{\mathcal{U}_b}{\ell,n}
= \max_{s \geq 1}\, \f{\mathcal{W}_b}{s,n}.
\end{equation*}
This means that $\f{M_b}{n}$ is the maximum possible number of representations
of a positive integer as a sum of exactly $n$ powers of $b$. Equivalently,
it is the largest coefficient in the power series expansion of
\begin{equation*}
\big( x + x^b + x^{b^2} + \cdots \big)^n.
\end{equation*}
When $b=2$, Molteni~\cite{Molteni:2012:repr-2-powers-asy} obtained the
following bounds for this quantity:
\begin{equation*}
(1.75218)^n \ll \f{M_2}{n}/n! \leq (1.75772)^n.
\end{equation*}
The gap between the two estimates is already very small; we improve this a
little further by providing the constant of exponential growth as well as a
precise asymptotic formula.
\begin{thm}\ellbel{thm:maximum}
For a certain constant $\nu = 1.7521819\ldots$ (defined precisely in
Section~\ref{sec:thm-max}), we have
\begin{equation*}
\f{M_2}{n}/n! \leq \nu^n
\end{equation*}
for all $n \geq 1$, and the constant is optimal: we have the more precise
asymptotic formula
\begin{equation*}
\f{M_2}{n}/n! \sim \ellmbda n^{-1/2} \nu^n
\end{equation*}
with $\ellmbda = 0.2769343\ldots$.
\end{thm}
Again, Theorem~\ref{thm:maximum} holds for arbitrary integer bases $b\geq2$ for
some constants $\nu=\nu_b$ and $\ellmbda=\ellmbda_b$ (it will be explained
precisely how they are obtained). This is formulated as
Theorem~\ref{thm:maximum-full} in Section~\ref{sec:thm-max}.
The final section contains the analysis of some parameters. We study the
exponent of the largest denominator and the number of distinct parts in a
composition of~$1$. In both cases a central limit theorem is shown; mean and
variance are linear in the number of summands, cf.\@ Theorems~\ref{thm:largest}
and~\ref{thm:distinct}.
\section{The Recursive Approach}
\ellbel{sec:rec}
For our purposes, it will be most convenient to work in the setting of
compositions of $1$, i.e., we are interested in the number $q_b(m)$ of
(ordered) solutions to the Diophantine equation~\eqref{eq:maineq_general},
where $n = (b-1)m+1$, as explained in the introduction. Our first goal is to
derive a recursion for $q_b(m)$ and some related quantities, which leads to a
system of functional equations for the associated generating functions.
Let $\mathbf{k} = (k_1,k_2,\ldots,k_n)$ be a solution to the Diophantine
equation~\eqref{eq:maineq_general} with $k_1 \geq k_2 \geq \cdots \geq k_n$. We
will refer to such an $n$-tuple as a ``partition'' (although technically the
$k_i$ are only the exponents in a partition). We denote by $\mathsf{c}(\mathbf{k})$ the
number of ways to turn it into a composition. If $w_0$ is the number of zeros,
$w_1$ the number of ones, etc.\@ in $\mathbf{k}$, then we clearly have
\begin{equation*}
\mathsf{c}(\mathbf{k}) = \frac{n!}{\prod_{j \geq 0} w_j!}.
\end{equation*}
The \emph{weight} of a partition $\mathbf{k}$, denoted by $\mathsf{w}(\mathbf{k})$, is now
simply defined as
\begin{equation*}
\mathsf{w}(\mathbf{k}) = \frac{1}{\prod_{j \geq 0} w_j!} = \frac{\mathsf{c}(\mathbf{k})}{n!}.
\end{equation*}
Now let
\begin{multline*}
\mathcal{P}_m = \Big\{ \mathbf{k} = (k_1,k_2,\dots,k_n) \,\Big\vert\,
\text{$n = (b-1)m+1$, } \\
\text{$b^{-k_1}+b^{-k_2} + \dots + b^{-k_n} = 1$,
$k_1 \geq k_2 \geq \dots \geq k_n$} \Big\}
\end{multline*}
be the set of all partitions of $1$ with $n = (b-1)m+1$ terms and, likewise,
\begin{equation*}
\mathcal{C}_m = \Big\{ \mathbf{k} = (k_1,k_2,\dots,k_n) \,\Big\vert\,
\text{$n = (b-1)m+1$, $b^{-k_1}+b^{-k_2} + \dots + b^{-k_n} = 1$} \Big\}
\end{equation*}
the set of compositions. We obtain the formula
\begin{equation*}
q_b(m) = \card*{\mathcal{C}_{m}}
= \sum_{\mathbf{k} \in \mathcal{P}_m} \mathsf{c}(\mathbf{k}) = n!\, \sum_{\mathbf{k} \in \mathcal{P}_m} \mathsf{w}(\mathbf{k})
\end{equation*}
for their number.
Our next step involves an important observation that is also used to obtain the
generating function~\eqref{eq:partitionsgf}. Consider an element $\mathbf{k}$ of
$\mathcal{P}_m$, and let $r$ be the number of times the greatest element $k_1$ occurs
(i.e., $k_1 = k_2 = \cdots = k_r > k_{r+1}$). This number must be divisible by
$b$ (as can be seen by multiplying~\eqref{eq:maineq_general} by $b^{k_1}$)
unless $\mathbf{k}$ is the trivial partition, so we can replace them by $r/b$
fractions with denominator $b^{k_1-1}$.
This process can be reversed. Given a partition $\mathbf{k}$ in which the largest
element occurs $r$ times, we can replace $s$, $1 \leq s \leq r$, of these
fractions by $bs$ fractions with denominator $b^{k_1+1}$. This recursive
construction can be illustrated nicely by a tree structure as in
Figure~\ref{fig:tree} for the case $b=2$. Each partition corresponds to a
so-called canonical tree (see \cite{Elsholtz-Heuberger-Prodinger:2013:huffm}),
and vice versa. Note that if $\mathbf{k} \in \mathcal{P}_m$, then the resulting partition
$\mathbf{k'}$ lies in $\mathcal{P}_{m+s}$, and we clearly have
\begin{equation}\ellbel{eq:weightrel}
\mathsf{w}(\mathbf{k'}) = \mathsf{w}(\mathbf{k}) \cdot \frac{r!}{(r-s)!\,(bs)!}.
\end{equation}
\begin{figure}
\caption{The canonical tree associated with the partition $1 =
3\cdot2^{-2}
\end{figure}
Now we can turn to generating functions. Let $\mathcal{P}_{m,r}$ be the subset of
$\mathcal{P}_m$ that only contains partitions for which $k_1 = k_2 = \cdots = k_r >
k_{r+1}$ (i.e., in~\eqref{eq:maineq_general}, the largest exponent occurs
exactly $r$ times), and let $\mathcal{C}_{m,r}$ be the set of compositions obtained by
permuting the terms of an element of $\mathcal{P}_{m,r}$. We define a generating
function by
\begin{equation*}
\f{Q_r}{x} = \sum_{m \geq 0} \frac{\card*{\mathcal{C}_{m,r}}}{((b-1)m+1)!} x^m
= \sum_{m \geq 0} \sum_{\mathbf{k} \in \mathcal{P}_{m,r}} \frac{\mathsf{c}(\mathbf{k})}{((b-1)m+1)!} x^m
= \sum_{m \geq 0} \sum_{\mathbf{k} \in \mathcal{P}_{m,r}} \mathsf{w}(\mathbf{k}) x^m.
\end{equation*}
We have $\f{Q_1}{x} = 1$ and $\f{Q_r}{x} = 0$ for all other $r$ not divisible by $b$. Moreover, for all $s \geq 1$ the recursive relation described above and in
particular~\eqref{eq:weightrel} yield
\begin{equation}\ellbel{eq:genfun_id}
\begin{split}
\f{Q_{bs}}{x} &= \sum_{m \geq 0} \sum_{\mathbf{k'} \in \mathcal{P}_{m,bs}} \mathsf{w}(\mathbf{k'})
x^m = \sum_{r \geq s} \sum_{m \geq s} \sum_{\mathbf{k} \in \mathcal{P}_{m-s,r}} \mathsf{w}(\mathbf{k})
\frac{r!}{(r-s)!\,(bs)!}x^m \\
&= x^s \sum_{r \geq s} \frac{r!}{(r-s)!\,(bs)!} \sum_{m \geq s}
\sum_{\mathbf{k} \in \mathcal{P}_{m-s,r}} \mathsf{w}(\mathbf{k})x^{m-s} = x^s \sum_{r \geq s}
\frac{r!}{(r-s)!\,(bs)!} \f{Q_r}{x}.
\end{split}
\end{equation}
This can be seen as an infinite system of linear equations. Define the infinite
(column-)vector $\mathbf{V}(x) =
(\f{Q_b}{x},\f{Q_{2b}}{x},\f{Q_{3b}}{x},\ldots)^T$, and the infinite matrix
$\mathbf{M}(x)$ by its entries
\begin{equation*}
m_{ij} = \begin{cases}
\frac{(bj)!\,x^i}{(bj-i)!\,(bi)!} & \text{if } i \leq bj, \\
0 & \text{otherwise.}
\end{cases}
\end{equation*}
Then the identity~\eqref{eq:genfun_id} above turns into the matrix identity
\begin{equation}\ellbel{eq:matrixid}
\mathbf{V}(x) = \mathbf{M}(x) \mathbf{V}(x) + \frac{x}{b!} \mathbf{e}_1,
\end{equation}
where $\mathbf{e}_1 = (1,0,0,\ldots)^T$ denotes the first unit vector. Within the
ring of formal power series, this readily yields
\begin{equation}\ellbel{eq:vexplicit}
\mathbf{V}(x) = \frac{x}{b!} (\mathbf{I}-\mathbf{M}(x))^{-1} \mathbf{e}_1,
\end{equation}
and the generating function
\begin{equation*}
\f{Q}{x} = \sum_{r \geq 1} \f{Q_r}{x}
= \sum_{m \geq 0} \frac{q_b(m)}{((b-1)m+1)!} x^m
\end{equation*}
(recall that $q_b(m)$ is the number of compositions of $1$ into $n = (b-1)m+1$
powers of $b$) is given by
\begin{equation*}
\f{Q}{x} = 1 + \mathbf{1}^T \mathbf{V}(x)
= 1 + \frac{x}{b!} \mathbf{1}^T (\mathbf{I}-\mathbf{M}(x))^{-1} \mathbf{e}_1.
\end{equation*}
For our asymptotic result, we will need the dominant singularity of $\f{Q}{x}$,
i.e., the zero of $\det(\mathbf{I}-\mathbf{M}(x))$ that is closest to $0$. A priori, it is not
even completely obvious that this determinant is well-defined, but the
reasoning is similar to a number of comparable problems.
As mentioned earlier, the determinant $\f{T}{x} = \det(\mathbf{I}-\mathbf{M}(x))$
exists a priori within the ring of formal power series, as the limit of the
principal minor determinants. We can write it as
\begin{equation}\ellbel{eq:detexpression}
\det(\mathbf{I}-\mathbf{M}(x)) = \sum_{h\geq0} (-1)^h
\sum_{\substack{1\leq i_1<i_2<\dots<i_h \\ i_1,\dots,i_h\in\N}} x^{i_1+i_2+\cdots+i_h}
\sum_{\sigma} (\operatorname{sgn} \sigma)
\prod_{k=1}^h \frac{(b\f{\sigma}{i_k})!}{(b\f{\sigma}{i_k}-i_k)!\,(bi_k)!},
\end{equation}
where the inner sum is over all permutations~$\sigma$ of
$\{i_1,i_2,\ldots,i_h\}$. Using Eaves' sufficient condition,
cf.~\cite{Eaves:1970}, we get at least convergence for $\abs{x}<1$.
We can even
show that the formal power series~$T$ given by~\eqref{eq:detexpression} defines an
entire function. This is proven in Section~\ref{sec:bounds}.
The same is true (by the same argument) for
\begin{equation*}
\f{S}{x} = \mathbf{1}^T \operatorname{adj}(\mathbf{I}-\mathbf{M}(x)) \mathbf{e}_1
= \det(\mathbf{M}^*(x)),
\end{equation*}
where $\mathbf{M}^*$ is obtained from $\mathbf{I}-\mathbf{M}(x)$ by replacing the first row
by $\mathbf{1}$. Hence we can write the generating function $\f{Q}{x}$ as
\begin{equation}\ellbel{eq:gen-A-num-den}
\f{Q}{x} = 1 + \frac{x}{b!} \frac{\f{S}{x}}{\f{T}{x}},
\end{equation}
where $\f{S}{x}$ and $\f{T}{x}$ are both entire functions. The singularities of
$\f{Q}{x}$ are thus all poles, and it remains to determine the dominant
singularity, i.e., the zero of $T(x) = \det(\mathbf{I}-\mathbf{M}(x)) $ with smallest
modulus.
\section{Bounds and Entireness}
\ellbel{sec:bounds}
In this section the two formal power series
\begin{equation*}
\f{T}{x} = \sum_{n\geq 0} t_n x^n
= \det(I-\mathbf{M}(x))
\end{equation*}
and
\begin{equation*}
\f{S}{x} = \sum_{n\geq 0} s_n x^n
= \mathbf{1}^T \operatorname{adj}(I-\mathbf{M}(x)) \mathbf{e}_1
\end{equation*}
of Section~\ref{sec:rec} (in particular cf.\ Equations~\eqref{eq:detexpression}
and~\eqref{eq:gen-A-num-den}) are analyzed. Other (similar) functions arising
on the way can be dealt with in a similar fashion.
Note that $\f{S}{x}$ is the
determinant of a matrix, which is obtained by replacing the first row of
$I-\mathbf{M}(x)$ by $\mathbf{1}$.
We find bounds for the coefficients $t_n$ and $s_n$, which will be needed for
numerical calculations with guaranteed error estimates as well. Further, those
bounds will tell us that the two functions $\f{T}{x}$ and $\f{S}{x}$ are
entire.
\begin{lemma}\ellbel{lem:bound-det-coeff}
The coefficients $t_n$ satisfy the bound
\begin{equation*}
\abs{t_n} \leq \exp\left(-\frac{b-1}{2}n\log n - cn + n\f{g}{n} \right)
\end{equation*}
with $c = (b-1)\left(\log\frac{b-1}{\sqrt{2}}-1\right)$ and with a
decreasing function $\f{g}{n}$, which tends to zero as
$n\to\infty$. In particular, the formal power series $T$ defines an entire
function. The same is true for the formal power series~$S$. More precisely,
we have
\begin{equation*}
\abs{s_n} \leq \left((b-1)!+1\right)
\exp\left(-\frac{b-1}{2}n\log n - cn + (n+1) \f{g}{n} \right).
\end{equation*}
\end{lemma}
\begin{proof}
Recall expression~\eqref{eq:detexpression} for the determinant, namely
\begin{equation*}
\det(\mathbf{I}-\mathbf{M}(x)) = \sum_{h\geq0} (-1)^h
\sum_{\substack{1\leq i_1<i_2<\dots<i_h \\ i_1,\dots,i_h\in\N}} x^{i_1+i_2+\cdots+i_h}
\sum_{\sigma} (\operatorname{sgn} \sigma)
\prod_{k=1}^h \frac{(b\f{\sigma}{i_k})!}{(b\f{\sigma}{i_k}-i_k)!\,(bi_k)!}.
\end{equation*}
Write
$n = i_1 + i_2 + \cdots + i_h$ for the exponent of $x$, and note that
\begin{equation*}
\prod_{k=1}^h \frac{(b\f{\sigma}{i_k})!}{(bi_k)!} = 1,
\end{equation*}
which is independent of the permutation~$\sigma$. We also have
\begin{equation*}
\sum_{k = 1}^h (b\f{\sigma}{i_k}-i_k) = (b-1)\sum_{k = 1}^h i_k = (b-1)n.
\end{equation*}
Since $a! \geq \exp(a(\log a - 1))$ for all positive integers $a$ and $f(x) =
x(\log x - 1)$ is a convex function, we have
\begin{align*}
\prod_{k = 1}^h (b\f{\sigma}{i_k}-i_k)!
&\geq \exp \left( \sum_{k=1}^h (b\f{\sigma}{i_k}-i_k)
\left( \log (b\f{\sigma}{i_k}-i_k) - 1 \right) \right) \\
&\geq \exp \left( h \frac{(b-1)n}{h}
\left( \log \frac{(b-1)n}{h} - 1 \right) \right) \\
&= \exp \left( (b-1)n \left( \log \frac{(b-1)n}{h} - 1 \right) \right).
\end{align*}
Since $i_1,i_2,\ldots,i_h$ have to be distinct, we also have
\begin{equation*}
n = i_1 + i_2 + \cdots + i_h \geq 1 + 2 + \cdots + h = \frac{h(h+1)}{2} \geq \frac{h^2}{2}.
\end{equation*}
Thus $h \leq \sqrt{2n}$, which means that
\begin{equation*}
\prod_{k = 1}^h (b\f{\sigma}{i_k}-i_k)!
\geq \exp \left( \frac{b-1}{2}n\log n +
(b-1)n \left(\log\frac{b-1}{\sqrt{2}} - 1\right) \right).
\end{equation*}
Now that we have an estimate for each term in~\eqref{eq:detexpression}, let
us also determine a bound for the number of terms corresponding to each
exponent $n$.
It is well known that the number of partitions $q(n)$ of $n$ into distinct
parts is asymptotically equal to $\exp\!\big( \pi \sqrt{n/3} + \Oh{\log n}
\big)$. In Robbins's
paper~\cite{Robbins:2007:upper-bound-partitions-distinct} we can find the
explicit upper bound\footnote{Note that in the published version
of~\cite{Robbins:2007:upper-bound-partitions-distinct} a constant in the
main theorem is printed incorrectly.}
\begin{equation*}
q(n) \leq \frac{\pi}{\sqrt{12n}}
\exp\left(\frac{\pi}{\sqrt{3}}\sqrt{n} + \frac{\pi^2}{12}\right).
\end{equation*}
For each choice of $\{i_1,i_2,\ldots,i_h\}$, there are at most $h!$
permutations~$\sigma$ that contribute, which can be bounded by means of
Stirling's formula (using also $h \leq \sqrt{2n}$ again). This gives
\begin{equation*}
h! \leq \exp\left( h \log h - h + \tfrac12 \log h + 1 \right)
\leq \exp \left((\sqrt{2n} + \tfrac12)\log(\sqrt{2n}) - \sqrt{2n}+1\right).
\end{equation*}
It follows that the coefficient $t_n$ of $T$ is bounded (in absolute values)
by
\begin{multline*}
\frac{\exp\left(
\frac{\pi}{\sqrt{3}}\sqrt{n} + \frac{\pi^2}{12}
+ \log\pi - \frac12 \log (12n)
+ (\sqrt{2n} + \tfrac12)\log(\sqrt{2n}) - \sqrt{2n}+1
\right)}{\exp\left(
\frac{b-1}{2}n\log n +
(b-1)n \left(\log\frac{b-1}{\sqrt{2}} - 1\right)
\right)} \\
=\exp\left(-\frac{b-1}{2}n\log n - cn + \Oh{\sqrt{n} \log n} \right),
\end{multline*}
which proves the theorem for a suitable choice of $g(n)$. A possible explicit
bound (relevant for our numerical calculations, see
Section~\ref{sec:numerical}) is
\begin{equation*}
\abs{t_n} \leq \exp\left(-\frac{b-1}{2}n\log n - cn
+ \sqrt{\frac{n}2}\log n + \sqrt{n} + 3\right).
\end{equation*}
Since this bound decays superexponentially, the determinant $T
=\det(\mathbf{I}-\mathbf{M}(x))$ is an entire function.
The same argument works for $S$. There, we split up into the summands
where we have $i_1=1$ and all other summands. For the second part (the summands
with $i_1>1$), the terms are the same as in the determinant that defines $T$, so it is bounded by the same expression.
Each of the summands with $i_1=1$ equals a summand of $\det(I-\mathbf{M}(x))$ multiplied by the factor
\begin{equation*}
-\frac{(b\f{\sigma}{i_1}-i_1)!\,(bi_1)!}{(b\f{\sigma}{i_1})!\, x}
= -\frac{b!}{x} \frac{(b\f{\sigma}{1}-1)!}{(b\f{\sigma}{1})!}
= -\frac{(b-1)!}{x \f{\sigma}{1}}
\end{equation*}
or is zero (when $\f{\sigma}{i_1}=1$). Therefore, the sum of these terms can be bounded by $(b-1)!$ times the bound we
obtained for the coefficient of $x^{n+1}$ in $\det(I-\mathbf{M}(x))$. This gives us
\begin{align*}
|s_n| &\leq \exp\left(-\frac{b-1}{2}n\log n - cn + n \f{g}{n} \right) \\
&\qquad + (b-1)!\exp\left(-\frac{b-1}{2}(n+1)\log (n+1) - c(n+1) + (n+1) \f{g}{n+1} \right) \\
&\leq (1+(b-1)!)\exp\left(-\frac{b-1}{2}n\log n - cn + (n+1) \f{g}{n} \right),
\end{align*}
which completes the proof.
\end{proof}
Lemma~\ref{lem:bound-det-coeff} immediately yields a simple estimate for the tails of the power series $S$ and $T$.
\begin{lemma}\ellbel{lem:bound-det-tail}
Let $N\in\N$ and $x\in\C$, and let $c$ and $\f{g}{n}$ be as in
Lemma~\ref{lem:bound-det-coeff}. Set
\begin{equation*}
q = \frac{e^{\f{g}{N}} \abs{x}}{e^c \sqrt{N^{b-1}}}
\end{equation*}
and suppose that $q<1$. Then we have the inequality
\begin{equation*}
\abs[Big]{\sum_{n \geq N} t_n x^n} \leq \frac{q^N}{1-q}
\end{equation*}
for the tails of the infinite sum in the determinant~$T$. For the tails of
the determinant $S$, we have the analogous inequality
\begin{equation*}
\abs[Big]{\sum_{n \geq N} s_n x^n} \leq ((b-1)!+1)e^{g(N)}\frac{q^N}{1-q}.
\end{equation*}
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:bound-det-coeff} we have
\begin{equation*}
\abs{t_n} \leq \exp\Big(-\frac{b-1}{2}n\log n - cn + n \f{g}{n} \Big).
\end{equation*}
Now we use monotonicity to obtain
\begin{equation*}
\abs[Big]{\sum_{n \geq N} t_n x^n}
\leq \sum_{n \geq N}
\left(\frac{e^{\f{g}{n}} \abs{x}}{e^c \sqrt{n^{b-1}}}\right)^n
\leq \sum_{n \geq N}
\left(\frac{e^{\f{g}{N}} \abs{x}}{e^c \sqrt{N^{b-1}}}\right)^n
= q^N \frac{1}{1-q}.
\end{equation*}
The second inequality follows in the same way.
\end{proof}
\section{Analyzing the Generating Function}
\ellbel{sec:poles-gf}
Infinite systems of functional equations appear quite frequently in the
analysis of combinatorial problems, see for example the recent work of Drmota,
Gittenberger and
Morgenbesser~\cite{Drmota-Gittenberger-Morgenbesser:2012:inf-systems}. Alas,
their very general theorems are not applicable to our situation as the infinite
matrix~$\mathbf{M}$ does not represent an $\ell_p$-operator (one of their main
requirements), due to the fact that its entries increase (and tend to $\infty$)
along rows. However, we can adapt some of their ideas to our setting.
The main result of this section is the following lemma.
\begin{lemma}\ellbel{lem:poles}
For every $b \geq 2$, the generating function $\f{Q}{x}$ has a simple pole at
a positive real point $\rho_b$ and no other poles with modulus $< \rho_b +
\epsilon_b$ for some $\epsilon_b > 0$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:poles}]
First of all, we rule out the possibility that $\f{Q}{x}$ is
entire by providing a lower bound for the coefficients $q_b(m)$. To this end,
consider compositions of $1$ consisting of $b-1$ copies of
$b^{-1},b^{-2},\ldots,b^{1-m}$ and $b$ copies of $b^{-m}$. Since there are
$\frac{((b-1)m+1)!}{((b-1)!)^{m-1}b!}$ possible ways to arrange them in an
order, we know that
\begin{equation*}
q_b(m) \geq \frac{((b-1)m+1)!}{((b-1)!)^{m-1}b!},
\end{equation*}
from which it follows that the radius of convergence of $\f{Q}{x}$ is at most
$(b-1)!$. Since all coefficients are positive, Pringsheim's theorem
guarantees that the radius of convergence, which we denote by $\rho_b$, is
also a singularity.
We already know that $\f{Q}{x}$ is meromorphic (being the
quotient of two entire functions), hence $\rho_b$ is a pole singularity. Let
$p$ be the pole order, and with $Q_{br}$ as in Section~\ref{sec:rec} set
\begin{equation*}
w_r = \lim_{x \to \rho_b^-} (\rho_b-x)^p
\f{Q_{br}}{x},
\end{equation*}
which must be nonnegative and real. Moreover, we have
\begin{equation*}
\f{Q_b}{x} = \frac{x}{b!} + x \sum_{r \geq 1}
\frac{r}{(b-1)!} \f{Q_{br}}{x}
\geq \frac{x}{b!} \Big(1 + \sum_{r \geq 1} \f{Q_{br}}{x} \Big)
= \frac{x}{b!} \f{Q}{x},
\end{equation*}
which shows that $w_1$ is even strictly positive. Multiplying the matrix
equation~\eqref{eq:matrixid} by $(\rho_b-x)^p$ and taking the limit, we see
that $\mathbf{w} = (w_1,w_2,\ldots)^T$ is a right eigenvector of
$\mathbf{M}(\rho_b)$. Since all entries in $\mathbf{M}(\rho_b)$ are nonnegative and those on
and above the main diagonal are strictly positive, it follows that $w_r > 0$
for all $r$, i.e., all functions $\f{Q_r}{x}$ have the same pole order (as
$\f{Q}{x}$).
Now we split the identity~\eqref{eq:matrixid}. Let $m_{11} = x/(b-1)!$ be the
first entry of $\mathbf{M}(x)$, $\mathbf{c}$ the rest of the first column, $\mathbf{r}$
the rest of the first row and $\overline{\mathbf{M}}$ the matrix obtained from
$\mathbf{M}$ by removing the first row and the first column. Moreover,
$\overline{\mathbf{V}}$ is obtained from $\mathbf{V}$ by removing the first entry
$\f{Q_b}{x}$. Now we have
\begin{equation}\ellbel{eq:a1eq}
\f{Q_b}{x} = m_{11} \f{Q_b}{x}
+ \mathbf{r}\, \overline{\mathbf{V}} + \frac{x}{b!}
\end{equation}
and
\begin{equation*}
\overline{\mathbf{V}} = \mathbf{c} \f{Q_b}{x}
+ \overline{\mathbf{M}}\, \overline{\mathbf{V}},
\end{equation*}
from which we obtain
\begin{equation}\ellbel{eq:vexpl}
\overline{\mathbf{V}} = (\mathbf{I} - \overline{\mathbf{M}})^{-1}
\mathbf{c} \f{Q_b}{x}.
\end{equation}
Once again, the inverse $(\mathbf{I} - \overline{\mathbf{M}})^{-1}$ exists a priori
in the ring of formal power series, but one can show that $\det(\mathbf{I} -
\overline{\mathbf{M}})$ is in fact an entire function, so the entries of the
inverse are all meromorphic (see again the calculations in
Section~\ref{sec:bounds}).
Moreover, $(\mathbf{I} - \overline{\mathbf{M}})^{-1} \mathbf{c}$ cannot have a
singularity at $\rho_b$ or at any smaller positive real number, because if
this was the case, the right hand side of~\eqref{eq:vexpl} would have a higher
pole order at that point than the left hand side. Since it has positive
coefficients only (the inverse can be expanded into a geometric series), its
entries must be analytic in a circle of radius $> \rho_b$ around $0$. Now we
substitute~\eqref{eq:vexpl} in~\eqref{eq:a1eq} to obtain
\begin{equation*}
\f{Q_b}{x} = m_{11} \f{Q_b}{x} + \mathbf{r} (\mathbf{I} -
\overline{\mathbf{M}})^{-1} \mathbf{c} \f{Q_b}{x} + \frac{x}{b!}
\end{equation*}
and thus
\begin{equation*}
\f{Q_b}{x} = \frac{x}{b!} \left( 1 - m_{11} - \mathbf{r}
(\mathbf{I} - \overline{\mathbf{M}})^{-1} \mathbf{c} \right)^{-1}.
\end{equation*}
Note that
\begin{equation*}
\f{R}{x} = m_{11} + \mathbf{r} (\mathbf{I}
- \overline{\mathbf{M}})^{-1} \mathbf{c}
\end{equation*}
has only positive coefficients, so $\f{R}{x} = 1$ has a unique positive real
solution, which must be $\rho_b$ (recalling that $\f{R}{x}$ is analytic in a
circle of radius $> \rho_b$ around $0$). Of course, $\f{R'}{\rho_b} > 0$, so
its multiplicity is $1$, which means that $\rho_b$ is a simple
pole. Moreover, by the triangle inequality there are no complex solutions of
$\f{R}{x} = 1$ with the same modulus, which means that there are no further
singularities of $\f{Q_b}{x}$ (and thus $\f{Q}{x}$) in a circle of radius
$\rho_b+\epsilon_b$ around $0$ for suitable $\epsilon_b > 0$.
\end{proof}
\section{Getting the Asymptotics}
\ellbel{sec:asymptotics}
In this section, we prove Theorems~\ref{thm:asymptotics}
and~\ref{thm:asymptotics:general_base}, which give us constants $\alpha_b$,
$\gamma_b$ and $\kappa_b < 1$ such that for $n=(b-1)m+1$
\begin{equation*}
\frac{\f{\mathcal{W}_b}{s,n}}{n!} = \f{P_{b,s}}{m} \gamma_b^m (1 + \Oh{\kappa_b^m})
\end{equation*}
holds, where $\f{P_{b,s}}{m}$ is a polynomial with leading term $\alpha_b^s
m^{s-1}/(s-1)!$. Numerical values of the $\alpha_b$ and $\gamma_b$ can be found
in Table~\ref{tab:values-thm-asy}. It is explained in the next section how
these numerical values are determined in a reliable way.
\begin{table}[htbp]
\centering
\begin{equation*}
\begin{array}{ccc}
b & \alpha & \gamma \\
\hline
2 & 0.296372 & 1.19268 \\
3 & 0.279852 & 0.534502 \\
4 & 0.236824 & 0.170268 \\
5 & 0.196844 & 0.0419317 \\
6 & 0.165917 & 0.00834837 \\
7 & 0.142679 & 0.00138959 \\
8 & 0.1249575 & 0.000198440 \\
\hline
\end{array}
\end{equation*}
\caption{Truncated decimal values for the constants of
Theorem~\ref{thm:asymptotics:general_base}. See Section~\ref{sec:numerical} for the method of computation.}
\ellbel{tab:values-thm-asy}
\end{table}
For easier reading, we skip the index~$b$ again, i.e., we set
$\alpha=\alpha_b$, $\gamma=\gamma_b$, and so on. The proof is the same for
all~$b$, except for the fact that different constants occur.
\begin{proof}[Proof of Theorem~\ref{thm:asymptotics:general_base}]
By now,
we know that the function $\f{Q}{x}$ can be written as the quotient of two
entire functions, cf.\ Section~\ref{sec:rec} and
Lemma~\ref{lem:bound-det-coeff}. More specifically, we use
\begin{equation*}
Q(x) = 1 + \frac{x}{b!} \frac{\f{S}{x}}{\f{T}{x}}.
\end{equation*}
As Lemma~\ref{lem:poles} shows, $Q(x)$ has exactly one pole~$\rho$ (which is
a simple pole) inside some disc with radius $\rho+\epsilon$, $\epsilon>0$,
around $0$. Thus we can directly apply singularity
analysis~\cite{Flajolet-Odlyzko:1990:singul} in the meromorphic setting
(cf.\@ Theorem IV.10 of~\cite{Flajolet-Sedgewick:ta:analy}) to obtain
\begin{align*}
\frac{q_b(m)}{((b-1)m+1)!}
&= - \frac{\f{S}{\rho}}{b!\f{T'}{\rho}} \rho^{-m}
+ \Oh{\left(\frac{1}{\rho+\epsilon}\right)^m}.
\end{align*}
This finishes the proof for $s=1$. Note that $\gamma=1/\rho$.
In the general case (arbitrary~$s$), we use the relation
\begin{equation*}
\sum_{n=1}^{\infty} \frac{\f{\mathcal{W}_b}{s,n}}{n!} x^n
= \bigg( \sum_{n=1}^{\infty} \frac{\f{\mathcal{W}_b}{1,n}}{n!} x^n \bigg)^s,
\end{equation*}
which follows from Equation~\eqref{eq:ws_equation} and gives us
\begin{equation*}
\sum_{m=0}^{\infty} \frac{\f{\mathcal{W}_b}{s,(b-1)m+s}}{((b-1)m+s)!} x^m
= Q(x)^s.
\end{equation*}
Once again, we make use of the fact here that the (exponential) generating
function is meromorphic, cf.\@ Section~\ref{sec:rec}. The singular expansion
of $Q(x)^s$ at $x = \rho = 1/\gamma$ is given by
\begin{equation*}
Q(x)^s = \Big( \frac{\alpha}{1-\gamma x}
+ \Oh{1} \Big)^s,
\end{equation*}
which has $\alpha^s / (1-\gamma x)^s$ as its main term. Once again,
singularity analysis~\cite{Flajolet-Odlyzko:1990:singul} yields the desired
asymptotic formula with main term as indicated in the statement of the
theorem.\end{proof}
\section{Reliable Numerical Calculations}
\ellbel{sec:numerical}
We want to calculate the constants obtained in the previous sections in a
reliable way. The current section is devoted to this task. Our main tool will
be interval arithmetic, which is performed by the computer algebra system
Sage~\cite{Stein-others:2014:sage-mathem-6.3}.
For the calculations, we need bounds for the tails of our infinite sums. We
start with the following two remarks, which improves the bound found in
Section~\ref{sec:bounds}.
\begin{remark}\ellbel{rem:bound-det-coeff}
The bounds of Lemma~\ref{lem:bound-det-coeff} for the
determinant~\eqref{eq:detexpression} can be tightened:
for an explicit $n$, we can calculate $\f{g}{n}$ more precisely by using the number of
partitions of $n$ into distinct parts (and not a bound for that number) and
similarly by using the factorial directly instead of Stirling's formula.
An even better, but less explicit bound for the $n$th coefficient of
$\det(I-\mathbf{M}(x))$ is given by
\begin{equation}\ellbel{eq:bound-det-coeff-specific-n}
\abs{t_n} \leq \sum_{h \geq 0} h! \sum_{\substack{1\leq i_1<i_2<\dots<i_h \\ i_1,\dots,i_h\in\N \\ i_1+i_2+\cdots+i_h = n}}
\exp \Big( - (b-1)n \Big( \log
\frac{(b-1)n}{h} - 1 \Big) \Big).
\end{equation}
Note that we do not know whether this bound is decreasing in $n$ or
not. However, for a specific $n$, one can calculate this bound, and it is
much smaller than the general bounds above. For example, for $b=2$, we have
$\abs{t_{60}} \leq 5.96\cdot 10^{-14}$ with this method, whereas
Lemma~\ref{lem:bound-det-coeff} would give the bound $0.00014$.
\end{remark}
\begin{remark}\ellbel{rem:bound-det-tail}
We can also get tighter bounds in Lemma~\ref{lem:bound-det-tail} using the
ideas presented in Remark~\ref{rem:bound-det-coeff}. We can even use
combinations of those bounds: For $M>N$, we separate
\begin{equation*}
\abs[Big]{\sum_{n \geq N} t_n x^n}
\leq \sum_{M > n \geq N} \abs{t_n} \abs{x}^n
+ \abs[Big]{\sum_{n \geq M} t_n x^n}
\end{equation*}
and use the bound~\eqref{eq:bound-det-coeff-specific-n} for $M > n \geq N$
and Lemma~\ref{lem:bound-det-tail} (tightened by some ideas from
Remark~\ref{rem:bound-det-coeff}) for the sum over $n \geq M$. For example,
again for $b=2$, we obtain the tail-bound
\begin{equation*}
\abs[Big]{\sum_{n \geq 60} t_n x^n} \leq 8.051\cdot 10^{-14}
+ 4.068\cdot 10^{-15}
\end{equation*}
for $\abs{x} \leq 1$, where $M=86$ was chosen. (We will denote the constant
on the right hand side of the inequality above by $B_{T_{60}}$, see the proof
of Lemma~\ref{lem:zeros-det-b2}.) Using Lemma~\ref{lem:bound-det-tail}
directly would just give $0.00103$.
\end{remark}
To get numerical values for the constants, we have to work with
\begin{equation*}
Q(x) = 1 + \frac{x \f{S}{x}}{2\f{T}{x}},
\end{equation*}
where the first few terms of these power series are given by
\begin{equation*}
\f{S}{x} = \mathbf{1}^T \operatorname{adj}(I-\mathbf{M}(x)) \mathbf{e}_1 = \det(M^*(x))
= 1 - \tfrac{5}{12} x^2 - \tfrac{1}{6} x^3
- \tfrac{1}{24} x^4 + \tfrac{1}{45} x^5 + \cdots
\end{equation*}
and
\begin{equation*}
\f{T}{x} = \det(I-\mathbf{M}(x))
= 1 - x - \tfrac{1}{2} x^2 + \tfrac{1}{6} x^3
+ \tfrac{1}{8} x^4 + \tfrac{3}{40} x^5 + \cdots,
\end{equation*}
cf.\@ Sections~\ref{sec:rec} and~\ref{sec:asymptotics}. We
obtain the following result for the denominator $\f{T}{x}$.
\begin{lemma}\ellbel{lem:zeros-det-b2}
For $b=2$, the function $\f{T}{x}$ has exactly one zero with $\abs{x} <
\frac{3}{2}$. This simple zero lies at $x_0 = 0.83845184342\dots$.
\end{lemma}
\begin{remark}
Note that $1/x_0 = \gamma = 1.192674341213\dots$, which is the constant found
in Molteni~\cite{Molteni:2012:repr-2-powers-asy}.
\end{remark}
\begin{proof}[Proof of Lemma~\ref{lem:zeros-det-b2}]
Denote the polynomials consisting of the first $N$ terms of $\f{T}{x}$ by
$\f{T_N}{x}$. We have $\abs{\f{T}{x} - \f{T_{60}}{x}} \leq B_{T_{60}}$ with
$B_{T_{60}} = 1.17\cdot 10^{-13}$, see Lemma~\ref{lem:bound-det-tail} and
Remark~\ref{rem:bound-det-tail}. On the other hand, we have
$\abs{\f{T_{60}}{x}} > 0.062$ for $\abs{x} = \frac32$ (the minimum is
attained on the positive real axis) by using a bisection method together with
interval arithmetic (in
Sage~\cite{Stein-others:2014:sage-mathem-6.3}). Therefore, the functions
$\f{T}{x}$ and $\f{T_{60}}{x}$ have the same number of zeros inside a disk
$\abs{x}<\frac32$ by Rouch\'e's theorem ($0.062>B_{T_{60}}$). This number
equals one, since there is only one zero, a simple zero, of $\f{T_{60}}{x}$
with absolute value smaller than $\frac32$.
To find the exact position of that zero consider $\f{T_{60}}{x} + B_{T_{60}}I$
with the interval $I = [-1,1]$. Again, using a bisection method (starting with
$\frac32 I$) plus interval arithmetic, we find an interval that contains
$x_0$. From this, we can extract correct digits of $x_0$.
\end{proof}
From the previous result, we can calculate all the constants. The values of
those for the first few $b$ can be found in Table~\ref{tab:values-thm-asy}. The
following remark gives some details.
\begin{remark}
As mentioned, to obtain reliable numerical values of all the constants
involved in the statement of our theorems, we use the bounds obtained in
Section~\ref{sec:bounds} together with interval arithmetic.
Let $b=2$ and denote, as above, the polynomials consisting of the first $N$
terms of $\f{S}{x}$ and $\f{T}{x}$, by $\f{S_N}{x}$ and $\f{T_N}{x}$
respectively. By the methods of Lemmas~\ref{lem:bound-det-coeff}
and~\ref{lem:bound-det-tail} and Remarks~\ref{rem:bound-det-coeff}
and~\ref{rem:bound-det-tail} we get, for instance, that $\abs{\f{T'}{x} -
\f{T'_{60}}{x}} \leq B_{T'_{60}}$ with $B_{T'_{60}} = 8.397\cdot
10^{-12}$. We also have $\abs{\f{S}{x} - \f{S_{60}}{x}} \leq B_{S_{60}}$ with
$B_{S_{60}} = 1.848\cdot 10^{-13}$ for the function in the numerator of
$\f{Q}{x}$. We plug $x_0$ into the approximations $S_{60}$ and $T'_{60}$ and
use these bounds to obtain precise values (with guaranteed error estimates)
for all the constants that occur in our formula.
\end{remark}
We finish this section with the following remark.
\begin{remark}
If one does not insist on such explicit error bounds for the numerical
approximations as above, one can get ``more precise'' numerical results (without formal proofs that all the digits are actually correct).
Here, specifically, the first three terms in the asymptotic expansion are as
follows: {\small
\begin{align*}
\f{\mathcal{W}_2}{1,n} / n! &= 0.296372049053529075588648642133 \cdot
1.192674341213466032221288982529^{n-1} \\
&\mathrel{\phantom{=}} +\,0.119736335383631653495068554245 \cdot
0.643427418149500070120570318509^{n-1} \\
&\mathrel{\phantom{=}} +\,0.0174783635210388007051384381833 \cdot
(-0.5183977738993377728627273570710)^{n-1} \\
&\mathrel{\phantom{=}} +\,\cdots
\end{align*}}
However, the numerical approximations lack the ``certifiability'' of
e.g.\@ those in Table~\ref{tab:values-thm-asy}.
\end{remark}
\section{Maximum Number of Representations}
\ellbel{sec:thm-max}
Let $\f{\mathcal{U}_b}{\ell,n}$ and $\f{\mathcal{W}_b}{s,n}$ be as defined in \eqref{eq:def-Ub}
in the introduction. In this section we analyze the function~$\f{M}{n} =
\f{M_b}{n}$, which equals the maximum of $\f{\mathcal{U}_b}{\ell,n}$ over all $\ell$,
i.e., we have
\begin{equation*}
\f{M}{n} = \max_{\ell \geq 1}\, \f{\mathcal{U}_b}{\ell,n}
= \max_{s \geq 1}\, \f{\mathcal{W}_b}{s,n}.
\end{equation*}
This gives the maximum number of representations any positive integer can have
as the sum of exactly $n$ powers of $b$.
Throughout this section, we use the generating function
\begin{equation*}
\f{W}{x} = \sum_{n=1}^{\infty} \frac{\f{\mathcal{W}_b}{1,n}}{n!} x^n.
\end{equation*}
Further, denote by $\theta$ the unique positive real solution (the power series
$W$ has real, nonnegative coefficients) with $\f{W}{\theta}=1$, and we set
$\nu=1/\theta$. We prove the following theorem, which is a generalized version
of Theorem~\ref{thm:maximum}.
\begin{thm}\ellbel{thm:maximum-full}
With the notions of $\f{W}{x}$, $\theta$ and $\nu$ as above, we have
\begin{equation}\ellbel{eq:M-upper-bound}
\f{M}{n}/n! \leq \nu^n
\end{equation}
for all $n \geq 1$, and the constant is optimal: We have the more precise
asymptotic formula
\begin{equation*}
\f{M}{n}/n! = \ellmbda n^{-1/2} \nu^n \big( 1 + \Oh[big]{n^{-1/2}} \big)
\end{equation*}
with $\ellmbda = (b-1) \left( \theta \f{W'}{\theta} \sigma \sqrt{2\pi}
\right)^{-1}$,
where $\sigma>0$ is defined by
\begin{equation*}
\sigma^2
= \frac{\f{W''}{\theta}}{\theta \f{W'}{\theta}^3}
+ \frac{1}{\theta^2 \f{W'}{\theta}^2} - \frac{1}{\theta \f{W'}{\theta}}.
\end{equation*}
Moreover, the maximum $\f{M}{n} = \max_{s\geq1}\, \f{\mathcal{W}_b}{s,n}$ is attained
at $s = \mu n + \Oh{1}$
with the constant $\mu = \left( \theta \f{W'}{\theta} \right)^{-1}$.
\end{thm}
In Table~\ref{tab:values-thm-max}, we are listing numerical values for the
constants of Theorem~\ref{thm:maximum-full}. These values are simply calculated by
using a finite approximation to $\f{W}{x}$, namely $\f{W_N}{x} = \sum_{n=1}^{N}
\frac{\f{\mathcal{W}_b}{1,n}}{n!} x^n$ for some precision~$N$.
\begin{table}
\centering
\begin{equation*}
\begin{array}{cccccc}
b & \ellmbda & \theta & \nu=1/\theta & \mu & \sigma^2 \\
\hline
2 & 0.27693430 & 0.57071698 & 1.75218196 & 0.44867215 & 0.41775807 \\
3 & 0.70656285 & 0.84340237 & 1.18567368 & 0.66924459 & 0.57114748 \\
4 & 1.70314663 & 0.95872521 & 1.04305174 & 0.87318716 & 0.37650717 \\
5 & 4.20099030 & 0.99167231 & 1.00839763 & 0.96645454 & 0.13477198 \\
6 & 10.61691472 & 0.99861115 & 1.00139078 & 0.99304650 & 0.03480989 \\
7 & 28.28286119 & 0.99980159 & 1.00019845 & 0.99880929 & 0.00714564 \\
8 & 80.09108610 & 0.99997520 & 1.00002480 & 0.99982638 & 0.00121534 \\
\hline
\end{array}
\end{equation*}
\caption{Values (numerical approximations) for the constants of
Theorem~\ref{thm:maximum-full}. In the calculations the approximation
$\f{W_{60}}{x}$ was used.}
\ellbel{tab:values-thm-max}
\end{table}
We start with the upper bound~\eqref{eq:M-upper-bound} of
Theorem~\ref{thm:maximum-full}, which is done in the following lemma.
\begin{lemma}\ellbel{lem:M-upper-bound}
We have
\begin{equation*}
\f{M}{n}/n! \leq \nu^n
\end{equation*}
for all $n \geq 1$.
\end{lemma}
\begin{proof}
Recall that Equation~\eqref{eq:ws_equation} gives us
\begin{equation*}
\sum_{n=1}^{\infty} \frac{\f{\mathcal{W}_b}{s,n}}{n!} x^n
= \bigg( \sum_{n=1}^{\infty} \frac{\f{\mathcal{W}_b}{1,n}}{n!} x^n \bigg)^s
= \f{W}{x}^s.
\end{equation*}
Since $\theta > 0$ was chosen such that $\f{W}{\theta} = 1$, it clearly
follows that
\begin{equation*}
\sum_{n=1}^{\infty} \frac{\f{\mathcal{W}_b}{s,n}}{n!} \theta^n = 1,
\end{equation*}
hence $\f{\mathcal{W}_b}{s,n}/n! \leq \theta^{-n}$ for all $s$ and $n$, and taking the
maximum over all $s \geq 1$ yields
\begin{equation*}
\f{M}{n}/n! = \max_{s \geq 1}\, \f{\mathcal{W}_b}{s,n}/n! \leq \theta^{-n} = \nu^n,
\end{equation*}
which is what we wanted to show.
\end{proof}
It remains to prove the asymptotic formula for $\f{M}{n}$. We first
gather some properties of the solution $x=\f{\theta}{u}$ of the functional
equation $\f{W}{x}=1/u$.
\begin{lemma}\ellbel{lem:alpha-minimal}
For $u\in\C$ with $\abs{u}\leq1$ and $\abs{\operatorname{Arg} u} \leq \frac{\pi}{b-1}$, each
root $x$ of $\f{W}{x} = 1/u$ satisfies the inequality $\abs{x} \geq \theta$,
where equality holds only if $x = \theta$ and $u =1$.
\end{lemma}
\begin{proof}
Let $u$ be as stated in the lemma. By the nonnegativity of the coefficients
of $W$ and the triangle inequality, we have
\begin{equation}\ellbel{eq:alpha-minimal:ineq}
\f{W}{\theta} = 1 \leq \abs{1/u}
= \abs{\f{W}{x}} \leq \f{W}{\abs{x}}.
\end{equation}
The first part of the lemma follows, since $W$ is increasing on the positive
real line. It remains to determine when equality holds, so we assume in the
following that $\abs{x} = \theta$.
Since the coefficients $\f{\mathcal{W}_b}{1,n}$ are nonzero only for $n\equiv1 \mod
b-1$, we can write $\f{W}{x} = x
\f{V}{x^{b-1}}$. From~\eqref{eq:alpha-minimal:ineq}, we obtain
\begin{equation*}
W(\theta) = \theta \f[big]{V}{\theta^{b-1}}
= \abs{x} \abs[big]{\f[big]{V}{x^{b-1}}} = \abs{W(x)}.
\end{equation*}
Since the coefficients of $V$ are indeed positive, the power series $V$ is
aperiodic\footnote{A power series is \emph{aperiodic} if the exponents whose
associated coefficients are not zero are not contained in $a+b\Z$ for any
$a$, $b$ with $b\geq2$.}. Therefore, the inequality
$\abs{\f[big]{V}{x^{b-1}}} \leq \f[big]{V}{\abs[big]{x^{b-1}}}$ is strict,
i.e., we have $\f[big]{V}{x^{b-1}} < \f[big]{V}{\abs[big]{x^{b-1}}}$ (which
would yield a contradiction to the assumption that $\abs{x}=\theta$) unless
$x^{b-1}$ is real and positive, which means that $x^{b-1} =
\theta^{b-1}$. When this is the case, we have
\begin{equation*}
\frac{\theta}{u} = \theta \f{W}{x} = \theta x \f[big]{V}{x^{b-1}}
= x \theta \f[big]{V}{\theta^{b-1}} = x \f{W}{\theta} = x,
\end{equation*}
so $\abs{\operatorname{Arg} x} = \abs{-\operatorname{Arg} u} \leq \frac{\pi}{b-1}$. This means that
$x^{b-1}$ can only be real and positive if $x$ is itself real and positive,
which implies that $x = \theta$ and $u = 1$.
\end{proof}
The following lemma tells us that the single dominant root of $\f{W}{x}=1$ is
the simple zero~$\theta$.
\begin{lemma}\ellbel{lem:alpha-dominant-root}
There exists exactly one root of $\f{W}{x}=1$ with $\abs{x}\leq\theta$,
namely $\theta$. Further, $\theta$ is a simple root, and there exists an $\epsilon>0$ such that $\theta$ is the
only root of $\f{W}{x}=1$ with absolute value less than
$\theta+\epsilon$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:alpha-minimal} with $u=1$, the positive real $\theta$ is
the unique root of $\f{W}{x}=1$ with minimal absolute value. This proves the
first part of the lemma.
Using Theorem~\ref{thm:asymptotics:general_base}, we get
\begin{equation*}
\abs{\f{W}{x}} = \f{O}{\sum_{m=0}^\infty \gamma^m \abs{x}^{(b-1)m}},
\end{equation*}
which is bounded for $\abs{x}<1/\gamma^{1/(b-1)}$. Therefore, the radius of
convergence $r$ of $W$ is at least $1/\gamma^{1/(b-1)}>\theta$, and so $W$ is
holomorphic inside a circle that contains $\theta$. Since zeros of holomorphic
functions do not accumulate, the existence of a suitable $\epsilon>0$ as desired
follows.
The root $\theta$ is simple, since $W(x)$ is strictly increasing on $(0,r)$.
\end{proof}
We are now ready to prove the asymptotic formula for $\f{M}{n}$. To this end,
we consider the bivariate generating function
\begin{equation*}
\f{G}{x,u} =
1 + \sum_{n=1}^{\infty} \sum_{s=1}^{\infty} \frac{\f{\mathcal{W}_b}{s,n}}{n!} x^nu^s
= \sum_{s=0}^{\infty} \f{W}{x}^su^s
= \frac{1}{1-u\f{W}{x}}.
\end{equation*}
In order to get $\max_{s\geq1}\, \f{\mathcal{W}_b}{s,n}$, we show that the
coefficients varying with $s$ fulfil a local limit law (as $n$ tends to
$\infty$). The maximum is then attained close to the mean.
\begin{proof}[Proof of Theorem~\ref{thm:maximum-full}]
Set
\begin{equation*}
\f{g_n}{u} = [x^n] \f{G}{x,u} = \sum_{s=1}^{\infty} \frac{\f{\mathcal{W}_b}{s,n}}{n!} u^s.
\end{equation*}
We extract $g_n$ from the bivariate generating function $\f{G}{x,u}$. In
order to do so, we proceed as in Theorem~IX.9 (singularity perturbation for
meromorphic functions) of Flajolet and
Sedgewick~\cite{Flajolet-Sedgewick:ta:analy}. An important detail here is the fact that $\frac{\f{\mathcal{W}_b}{s,n}}{n!} [u^sx^n] \f{G}{x,u}$ can only be nonzero if $s \equiv n \bmod (b-1)$, hence $g_n$ can also be expressed as
$$g_n(u) = u^r h_n(u^{b-1}),$$
where $r \in \{0,1,\ldots,b-2\}$ is chosen in such a way that $r \equiv n \bmod (b-1)$. This is also the reason why it was enough in Lemma~\ref{lem:alpha-minimal} to consider the case $|\operatorname{Arg} u| \leq \frac{\pi}{b-1}$.
Now we check that all requirements for applying the quasi-power theorem are fulfilled.
By Lemma~\ref{lem:alpha-dominant-root}, the function $\f{G}{x,1}$ has a dominant simple pole at $x=\theta$ and no other singularities with absolute values smaller than $\theta+\epsilon$. The denominator $1-u \f{W}{x}$ is analytic and not degenerated at $(x,u)=(\theta,1)$; the latter since its derivative with respect to $x$ is
$\f{W'}{\theta}\neq0$ ($\theta$ is a simple root of $F$) and its derivative
with respect to $u$ is $-\f{W}{\theta}=-1\neq0$.
Thus the function $\f{\theta}{u}$ which gives the solution to the equation $W(\theta(u)) = \theta$ with smallest modulus has the following properties: it is
analytic at $u=1$, it fulfils $\theta(1)=\theta$, and for some $\epsilon > 0$ and $u$ in a suitable neighbourhood of $1$, there is no $x \neq \theta(u)$ with $W(x) =
1/u$ and $|x| \leq \theta + \epsilon$.
Therefore, by Cauchy's integral formula and the residue theorem, we obtain
\begin{align*}
\f{g_n}{u} &= -\f{\operatorname{Res}}{\frac{1}{1-u \f{W}{x}} x^{-n-1},
x=\f{\theta}{u}}
+\frac{1}{2\pi i} \oint_{\abs{x}=\theta+\epsilon} \f{G}{z,u}\frac{\dd z}{z^{n+1}} \\
&= \frac{1}{u \f{\theta}{u} \f{W'}{\f{\theta}{u}}}
\left(\frac{1}{\f{\theta}{u}}\right)^n
+ \f{O}{(\theta+\epsilon)^{-n}}
\end{align*}
for $u$ in a suitable neighbourhood of $1$.
To get the results claimed in Theorem~\ref{thm:maximum-full}, we use a
local version of the quasi-power theorem, see Theorem IX.14
of~\cite{Flajolet-Sedgewick:ta:analy} or Hwang's original
paper~\cite{Hwang:1998:LLT}. Set
\begin{equation*}
\f{A}{u} = \left(u \f{\theta}{u} \f{W'}{\f{\theta}{u}} \right)^{-1}
\end{equation*}
and
\begin{equation*}
\f{B}{u} = \left( \f{\theta}{u} \right)^{-1},
\end{equation*}
so that
\begin{equation*}
g_n(u) = A(u) B(u)^n + \Oh{(\theta+\epsilon)^{-n}}.
\end{equation*}
In terms of $h_n$, this becomes
\begin{equation*}
h_n(v) = v^{-r/(b-1)} A(v^{1/(b-1)}) B(v^{1/(b-1)})^n + \Oh{(\theta+\epsilon)^{-n}}.
\end{equation*}
Here, $v^{1/(b-1)}$ is taken to be the principal $(b-1)$th root of $v$, which
satisfies $\abs{\operatorname{Arg} v^{1/(b-1)}} \leq \frac{\pi}{b-1}$.
Since $\f{\theta}{u} \neq 0$ for $u$ in a suitable neighbourhood of $0$, the
function $B$ is analytic at zero, and so is the function $A$ (since $W$ is
analytic in a neighbourhood of $\theta(1) = \theta$ as well and has a nonzero
derivative there). Moreover, we can use the fact that
$\abs{\f{\theta}{e^{i\varphi}}}$ has a unique minimum at $\varphi=0$ if we
assume that $\abs{\varphi} \leq \frac{\pi}{b-1}$ (which follows from
Lemma~\ref{lem:alpha-minimal}).
As a result, Theorem IX.14 of~\cite{Flajolet-Sedgewick:ta:analy} (slightly
adapted to account for the periodicity of $g_n$) gives us
\begin{equation}\ellbel{eq:max-llt}
\begin{split}
\frac{\f{\mathcal{W}_b}{s,n}}{n!} &=
\frac{(b-1)\f{A}{1} \f{B}{1}^n}{\sigma \sqrt{2\pi n}}
\f{\operatorname{exp}}{-\frac{z^2}{2\sigma^2}}
\left(1+ \f{O}{\frac{1}{\sqrt{n}}}\right) \\
&= \frac{(b-1)\nu^n}{\theta \f{W'}{\theta} \sigma \sqrt{2\pi n}}
\f{\operatorname{exp}}{-\frac{z^2}{2\sigma^2}} \left(1+
\f{O}{\frac{1}{\sqrt{n}}}\right),
\end{split}
\end{equation}
where $z=(s-\mu n)/\sqrt{n}$. Mean and variance can be calculated as follows.
We have
\begin{equation*}
\mu = \frac{\f{B'}{1}}{\f{B}{1}}
= - \frac{\f{\theta'}{1}}{\f{\theta}{1}}
= \frac{1}{\theta \f{W'}{\theta}},
\end{equation*}
and $\sigma>0$ is determined by
\begin{align*}
\sigma^2 &= \frac{\f{B''}{1}}{\f{B}{1}}
+ \frac{\f{B'}{1}}{\f{B}{1}}
- \left(\frac{\f{B'}{1}}{\f{B}{1}}\right)^2
= - \frac{\f{\theta''}{1}}{\f{\theta}{1}}
- \frac{\f{\theta'}{1}}{\f{\theta}{1}}
+ \left(\frac{\f{\theta'}{1}}{\f{\theta}{1}}\right)^2 \\
&= \frac{\f{W''}{\theta}}{\theta \f{W'}{\theta}^3}
- \frac{1}{\theta \f{W'}{\theta}} + \frac{1}{\theta^2 \f{W'}{\theta}^2},
\end{align*}
where we used implicit differentiation of $\f{W}{\f{\theta}{u}} = 1/u$ to get
expressions for $\f{\theta'}{u}$ and $\f{\theta''}{u}$.
The value $\f{\mathcal{W}_b}{s,n}/n!$ is maximal with respect to $s$ when $s = \mu n +
\Oh{1}$. Its asymptotic value can then be calculated by~\eqref{eq:max-llt}.
\end{proof}
\section{The Largest Denominator and
the Number of Distinct Parts}
\ellbel{sec:parameters}
In this last section we analyze some parameters of our compositions of~$1$. In particular, we will see that the exponent of the
largest denominator occurring in a random composition into a given number of powers of $b$ and
the number of distinct summands are both asymptotically normally distributed
and that their means and variances are of linear order.
Let us start with the largest denominator, for which we obtain the following theorem. Note
that we suppress the dependence on $b$ in all constants again.
\begin{thm}\ellbel{thm:largest}
The exponent of the largest denominator in a random composition of $1$ into
$m = (b-1)n+1$ powers of $b$ is asymptotically normally distributed with mean
$\mu_\ell n + \Oh{1}$ and variance $\sigma_\ell^2 n + \Oh{1}$.
\end{thm}
Numerical approximations to the values of $\mu_\ell$ and $\sigma_\ell^2$ can be
found in Table~\ref{tab:parameters}. The proof runs along the same lines as the proofs of Theorems~\ref{thm:asymptotics:general_base} and~\ref{thm:maximum-full}, so we only give a sketch here.
\begin{table}
\centering
\begin{equation*}
\begin{array}{ccccc}
b & \mu_\ell & \sigma_\ell^2 & \mu_d & \sigma_d^2 \\
\hline
2 & 0.81885148 & 2.38703164 & 0.71440975 & 2.13397882 \\
3 & 0.93352696 & 0.53468588 & 0.93318787 & 0.53600822 \\
4 & 0.97869416 & 0.15390515 & 0.97869416 & 0.15390519 \\
5 & 0.99366804 & 0.04335760 & 0.99366804 & 0.04335760 \\
6 & 0.99819803 & 0.01180985 & 0.99819803 & 0.01180985 \\
7 & 0.99950066 & 0.00315597 & 0.99950066 & 0.00315597 \\
8 & 0.99986404 & 0.00083471 & 0.99986404 & 0.00083471 \\
\hline
\end{array}
\end{equation*}
\caption{Values (numerical approximations) for the constants of
Theorems~\ref{thm:largest} and~\ref{thm:distinct}. In the numerical calculations
the power series were approximated by a polynomial consisting
of $40$ terms.}
\ellbel{tab:parameters}
\end{table}
\begin{proof}[Sketch of proof of Theorem~\ref{thm:largest}]
We start by considering a bivariate generating function for the investigated
parameter. In the recursive step described in Section~\ref{sec:rec} that led
us to the identity~\eqref{eq:genfun_id}, the exponent of the largest denominator
increases by $1$. Thus it is very easy to incorporate this parameter into the
generating function. Indeed, if $\ell(\mathbf{k})$ denotes the exponent of the
largest denominator that occurs in a composition (or partition) $\mathbf{k}$,
then the bivariate generating function
\begin{equation*}
\f{L_r}{x,y} = \sum_{n \geq 0} \sum_{\mathbf{k} \in C_{n,r}}
\frac{1}{((b-1)n+1)!}\, x^ny^{\ell(\mathbf{k})}
= \sum_{n \geq 0} \sum_{\mathbf{k} \in \mathcal{P}_{n,r}} \mathsf{w}(\mathbf{k})\, x^n y^{\ell(\mathbf{k})}
\end{equation*}
satisfies $\f{L_1}{x,y} = 1$ and
\begin{equation*}
\f{L_{bs}}{x,y} = x^s y \sum_{r \geq s} \frac{r!}{(r-s)!\,(bs)!} \f{L_r}{x,y}.
\end{equation*}
So if we set $\mathbf{V}(x,y) = (\f{L_b}{x,y}, \f{L_{2b}}{x,y}, \f{L_{3b}}{x,y},
\ldots)^T$, then we now have
\begin{equation*}
\mathbf{V}(x,y) = \frac{xy}{b!} (I-y\,\mathbf{M}(x))^{-1} \mathbf{e}_1
\end{equation*}
in analogy to~\eqref{eq:vexplicit} with the same infinite matrix as in
Section~\ref{sec:rec}. Moreover, we obtain
\begin{equation*}
\f{L}{x,y} = \sum_{r \geq 1} \f{L_r}{x}
= y + \mathbf{1}^T \mathbf{V}(x,y)
= 1 + \frac{xy}{b!} \mathbf{1}^T (I-y\,\mathbf{M}(x))^{-1} \mathbf{e}_1.
\end{equation*}
It follows by the same estimates as in Section~\ref{sec:bounds} that this is a meromorphic function in $x$ for $y$ in a suitable neighbourhood of $1$. Thus our bivariate generating function belongs to the meromorphic scheme as described in Section IX.6 of~\cite{Flajolet-Sedgewick:ta:analy}, and the asymptotics of mean and variance are obtained by standard tools of singularity analysis. Asymptotic normality follows by Hwang's quasi-power theorem~\cite{Hwang:1998}.
\end{proof}
For the number of distinct parts we prove the following result.
\begin{thm}\ellbel{thm:distinct}
The number of distinct parts in a random composition of $1$ into $m =
(b-1)n+1$ parts is asymptotically normally distributed with mean $\mu_d n +
\Oh{1}$ and variance $\sigma_d^2 n + \Oh{1}$.
\end{thm}
Approximations of the constants can be found in
Table~\ref{tab:parameters}. Again we only sketch the proof, since it uses the same ideas.
\begin{proof}[Sketch of proof of Theorem~\ref{thm:distinct}]
Again, we consider a bivariate generating function. In the recursive step, the number
of distinct parts increases by $1$, unless all fractions with highest
denominator are split. In this case, the number of distinct parts stays the
same. One can easily translate this to the world of generating functions: let
$d(\mathbf{k})$ be the number of distinct parts in $\mathbf{k}$, and let
$\f{D_r}{x,y}$ be the bivariate generating function, where $y$ now marks the
number of distinct parts, i.e., we use
\begin{equation*}
\f{D_r}{x,y} = \sum_{n \geq 0} \sum_{\mathbf{k} \in C_{n,r}}
\frac{1}{((b-1)n+1)!}\, x^ny^{d(\mathbf{k})}
= \sum_{n \geq 0} \sum_{\mathbf{k} \in \mathcal{P}_{n,r}} \mathsf{w}(\mathbf{k})\, x^n y^{d(\mathbf{k})}.
\end{equation*}
Then we have $\f{D_1}{x,y} = y$ and
\begin{equation*}
\f{D_{bs}}{x,y} = \frac{s!\, x^s}{(bs)!} \f{D_s}{x,y}
+ x^s y \sum_{r > s} \frac{r!}{(r-s)!\,(bs)!} \f{D_r}{x,y}.
\end{equation*}
Once again, we take the infinite vector $\mathbf{V}(x,y) = (\f{D_b}{x,y},
\f{D_{2b}}{x,y}, \f{D_{3b}}{x,y}, \ldots)^T$, and we define a modified
version $\mathbf{M}^*$ of the infinite matrix by its entries
\begin{equation*}
m_{ij}^* =
\begin{cases}
\frac{(bj)!\,x^iy}{(bj-i)!\,(bi)!} & \text{if } i < bj, \\
\frac{i!\,x^i}{(bi)!} & \text{if } i = bj, \\
0 & \text{otherwise.}
\end{cases}
\end{equation*}
Now
\begin{equation*}
\mathbf{V}(x,y) = \frac{xy}{b!} (I-\mathbf{M}^*(x))^{-1} \mathbf{e}_1
\end{equation*}
in analogy to~\eqref{eq:vexplicit}, and moreover
\begin{equation*}
\f{D}{x} = \sum_{r \geq 1} \f{D_r}{x}
= y + \mathbf{1}^T \mathbf{V}(x,y)
= y + \frac{xy}{b!} \mathbf{1}^T (I-\mathbf{M}^*(x))^{-1} \mathbf{e}_1.
\end{equation*}
Once again, we find that the bivariate function belongs to the meromorphic scheme, so that we can apply singularity analysis and the quasi-power theorem to obtain the desired result.
\end{proof}
\renewcommand{\MR}[1]{}
\end{document} |
\begin{document}
\begin{abstract} Let $X$ be a topological space. Let $X_0 \subseteq X$ be a second countable subspace. Also, assume that $X$ is first countable at any point of $X_0$. Then we provide
some conditions under which we ensure that $X_0$ is not Baire.
\end{abstract}
\title{ A criterion to specify the absence of Baire property } Subject Classification: 37B55, 54E52. \\
\title{ A criterion to specify the absence of Baire property } Keywords: Nonautonomous systems, topological transitivity, Baire space, Birkhoff theorem.
\title{ A criterion to specify the absence of Baire property }
\title{ A criterion to specify the absence of Baire property }
\section{Introduction }
A space $X$ is called Baire if the intersection of any sequence of dense open subsets of $X$ is dense in $X$.
Alternatively, this notation can be formulated in terms of second category sets.
The Baire category theory has numerous applications in Analysis and Topology.
Among these applications are, for instance, the open mapping, closed graph theorem and
the Banach-Steinhaus theorem in Functional Analysis \cite {{Aarts}, {Haworth}}.
The aim of this paper is to introduce a trick that concludes the absence of Baire property for some topological spaces using dynamical techniques and tools. Before
stating the main result, we establish some notations.
Let $(X,~\tau)$ be a topological space, $X_n$'s its subspaces and
$$ x_{n+1}=f_n(x_n), ~ n \in \mathbb{N} \cup \{0\}, $$ where
$ f_n: {X_n} \to { X_{n+1}}$ are continuous maps.
The family $\{f_n\}_{n=0}^{\infty}$ is called a nonautonomous discrete system \cite{{Shi}, {Shi-Chen}}.
For given $x_{0}\in X_{0}$, the orbit of $x_0$ is defined as
$$ orb(x_{0}):=\big\{ x_{0}, ~ f_{0}(x_{0}), ~ f_{1}\circ f_{0}(x_{0}), ~\cdots , ~f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}(x_{0}),~\cdots \big\},$$
and we say that this orbit starts from the point $x_0$. The topological structure of the orbit that starts from the point $x_0$ may be complex.
Here, we study the points of $ X_{0}$ whose orbits always intersect around $ X_{0}$. They are formulated as follows:
$$O:= \big\{x \in X_0 | ~~ {\overline{orb(x)}}^{{X}}\cap X_0=X_0 \big\}.$$
The system $\{f_n\}_{n=0}^{\infty}$ is called topologically transitive on $ X_{0}$ if for any two non-empty open sets $U_{0}$ and $ V_{0}$ in $ X_{0}, $ there
exists $ n \in \mathbb{N}$ such that $U_{n}\cap V_{0}\neq \phi $, where $ U_{i+1}=f_{i}(U_i)$ for $ 0 \leq i \leq n-1$, in other word
$(f_{n-1} \circ f_{n-2} \circ \cdots \circ f_{1} \circ f_{0})(U_{0}) \cap V_{0}\neq \phi$ \cite {Shi-Chen}.
Our main theorem is as follows:
\begin{theorem} \label{Theorem }
Let $X$ be a topological space. Let $X_0$ be a second countable subspace of $ X$ and let $X$ be first countable at any point of $X_0$. Also, suppose that
the system $\{f_n\}_{n=0}^{\infty}$ is topologically transitive on $X_{0}$ and $ \overline{O}\neq X_0$. Then $X_0$ can not be a Baire subspace.
\end{theorem}
Note that, if $X$ is a metric space, $X_n=X$, and $f_n=f $ for each $n$, then Theorem \ref{Theorem } will be obtained as a direct result of Birkhoff transitivity theorem. This fact was our motivation in writing the paper.
\section{Proof }
So as $X_{0}$ is a second countable subspace and $X$ first countable at any point of $X_0$, it is easy to show that
there exists a collection $\{U_m\}_{m \in N}$ of open sets in $X$ such that
\begin{itemize}
\item [$i$)] $ U_m \cap D_0 \neq\phi$,
\item [$ii$)] the family $\{U_m \cap D_0\}_{m \in \mathbb{N}}$ is a basis for $D_0$,
\item [$iii$)]for each $x_0\in D_0$, the family $\{U_m\}_{m \in {\mathbb{N}}}$ is a local basis for $x_0$ in $X$.
\end{itemize}
We claim that
$$O=\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty}{f_{n-1}}
\circ {f_{n-2}} \circ \cdots \circ {f_{1}} \circ {f_{0}}^{-1}(U_{m}). \eqno{(2.1)}$$
To prove the claim, put $O^*:=\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty}{f_{n-1}\circ f_{n-2} \circ \cdots \circ f_{1} \circ f_{0}}^{-1}(U_{m}).$
Firstly, we show that $O \subseteq O^*$. Suppose otherwise, there is $ x \in O $ such that $x \notin O^*$. So as $x \notin O^*$,
there exists $m \in \mathbb{N}$ such that for each $n \in \mathbb{N}$ we have
$$ {(f_{n-1}\circ f_{n-2}\circ \cdots \circ f_{1} \circ f_{0})}(x) \notin {U_{m}}.$$
Hence, $orb(x) \cap U_m=\phi$. Since $U_m\cap D_0 \neq \phi$, there exists an element $z\in U_m\cap X_0$, such that $z\notin {\overline{orb(x)}}^X.$
But $ z \in X_0$ and so $ {\overline{orb(x)}}^X\cap X_0 \neq D_0$. It is concluded that $x \notin O$ which contradicts the choice of $x$.
Now, it is shown that $O^* \subseteq O$. Let $x\in O^*$ but $x \notin O$. So as $x\in O^*$, concluded
for each $m \in \mathbb{N}$, there exists $n \in \mathbb{N}$ such that
$ {(f_{n-1}\circ f_{n-2} \circ \cdots \circ f_{1} \circ f_{0})}(x) \in U_m.$
Thus, $orb(x) \cap U_m \neq \phi$. Moreover, the relation $x \notin O$ indicates that there exists $ z\in X_0$ such that $z \notin \overline{orb(x)}^X.$ Consequently,
there exists $U_k$ containing $z$, such that $U_k \cap orb(x)= \phi$ that this contradicts with $ orb(x) \cap U_m \neq \phi$, for each $m\in\mathbb{N}$.
\\ By continuity of $ f_n: {X_n} \rightarrow { X_{n+1}},$
each set $\bigcup_{n=1}^{\infty}{(f_{n-1} \circ f_{n-2} \circ \cdots \circ f_{1} \circ f_{0})}^{-1}(U_{m})$ is open and because of transitivity, these open sets are dense in $X_{0}$. If $X_{0}$ be a Baire space, then (2.1) implies that $O$ is a dense $ G_{\delta}$-set. This is a contradict with $ \overline{O}\neq X_0$. Thus $X_0$ is not a Baire subspace, and the proof of the Theorem \ref{Theorem } is complete.
\section{Example }
\begin{example}
Consider $X=\mathbb H(\mathbb C)
=\big\{f:\mathbb C \rightarrow \mathbb C|~ f ~ is ~holomorphic \big\}$ endowed with the metric $d(f,g)=\displaystyle\sum_ {n=1}^{\infty}\frac {1}{2^n} min \big(~1,~p_n(f-g)\big),$
with $p_n(h) ={sup}_{|z| \leq n } |h(z)|$.
Then $X$ is a separable Banach space and besides that the differentiation operator $D:\mathbb H(\mathbb C) \rightarrow \mathbb H(\mathbb C)$ with $ D(f)=f^\prime$ is continuous \cite {Grosse-Erdmann}. Moreover, the space $\mathbb H(\mathbb C)$ is Baire and if we consider the dynamical system $ D:\mathbb H(\mathbb C ) \rightarrow \mathbb H(\mathbb C),$ then Birkhoff theorem guarantees the existence of functions that their orbit is dense in $\mathbb{ H} (\mathbb{ C})$.
Now, assume that
$$ X_0=\big\{\sum_{i=0}^{N} a_iz^i+\alpha g(z)\big |~ a_i , \alpha \in \mathbb{C} \big\}.$$
Then the subspace $ X_0$ is not Baire. To see this, take $\{\alpha_n\}_{n=0}^{\infty}$ be a subsequence
with $\alpha_0=0$ in this way that $D^{\alpha_n}(g)$ is convergent. We consider nonautonomous discrete system $\{f_{n}\}_{n=0}^{\infty}$ with $f_n=D^{\alpha_{n+1}-\alpha_n}$ where
$ X_{n}=\big\{\sum_{i=0}^{N} a_iz^i+\alpha g^{(\alpha_n)}(z)\big | a_i , \alpha \in \mathbb{C} \big\}$. By planning the arguments similar to what employed in the proof of Example 2.21 in \cite{Grosse-Erdmann}, we observe that the
system $\{f_n\}_{n=0}^{\infty}$ is topologically transitive. Now the assertion obtains by using Theorem \ref{Theorem } since the set $ O$ is empty.
\end{example}
\end{document} |
\begin{document}
\title{Rational singular loci of nilpotent varieties}
\author{William M. McGovern}
\subjclass{22E47,57S25}
\keywords{rational smoothness, intersection cohomology, nilpotent variety}
\begin{abstract}
We present two methods for computing the rational singular locus of the closure of a nilpotent orbit in a complex semisimple Lie algebra and give a number of interesting examples.
\end{abstract}
\maketitle
\section{Introduction}
Let $G$ be a complex reductive group with Lie algebra $\frak g$. Let $\mathcal N$ be the nilcone of $\frak g$ and let $X=\overline{G\cdot e}$ be a nilpotent variety in $\mathcal N$ (the closure of a nilpotent orbit $\mathcal O=G\cdot e$). Recall that a point $x$ of $X$ is said to be rationally smooth if for all $y$ in a neighborhood of $x$ in the complex topology we have $H_y^m(X) = 0$ for all $m\ne 2\dim X$ and $H_y^{2\dim X}(X) = \mathbb Q$, where $H_y$ denotes cohomology with support in $\{y\}$; here we can use either singular or intersection cohomology \cite{GM83}. The rational singular locus Rat Sing~$X$ consists of all points of $X$ at which $X$ is rationally singular (not smooth); this is a closed subset of $X$. Rational smoothness has played a major role in representation theory ever since the seminal work of Kazhdan and Lusztig in the seventies \cite{KL79}; it also provided one of the first applications of intersection cohomology. We will describe two methods of computing Rat Sing~$X$ and give some examples.
\section{First method--Brion}
Our first method applies to very general varieties $X$; they need not admit dense orbits under the action of any reductive group, but they must carry an action of a torus $T$ of dimension at least two. This method uses techniques of Brion first developed in \cite{Br99}.
\newtheorem*{theorem}{Theorem}
\begin{theorem}
Suppose that the variety $X$ admits an action of a torus $T$ such that $x\in X$ is an attractive fixed point of the $T$ action (so that all weights of $T$ on the tangent space $T_x X$ lie in an open half-space. Then $X$ is rationally smooth at $x$ if and only if
\begin{itemize}
\item Some punctured neighborhood of $x$ in $X$ is rationally smooth.
\item $X^{T'}$ is rationally smooth at $x$ for every subtorus $T'$ of codimension one in $T$.
\item $\dim_x X = \sum_{T'} \dim_x X^{T'}$ (sum over all subtori $T'$ of codimension one in $T$)
\end{itemize}
In addition, for any $T$-fixed point $x$ (attractive or not), the second condition is necessary for rational smoothness at $x$ and the third one becomes a necessary condition if $\dim_x X^T$ is subtracted from the left side and from every term in the right side.
\end{theorem}
For the proof see \cite[1.1,1.4]{Br99}. Here the first of these conditions is typically hard to verify, but essential: the second and third (in the absence of the first) might hold for one torus $T$ but fail for another, and in addition the first condition is needed to check that the cohomology vanishes at every point in a neighborhood of the one in question. We can apply these conditions in particular to a nilpotent variety $\overline{G\cdot e}$ and a point $x$ on it, provided that a torus $T$ of dimension at least two fixes $x$: this will occur whenever $x$ lies in the derived subalgebra of a Levi subalgebra of $\frak g$ of corank at least two. The case $x=0$ is especially easy; here we may take the torus to be the direct product of a maximal torus $T$ of $G$ and a 1-torus $\mathbb C^*$, the latter acting on $\frak g$ by scalar multiplication. Then $0$ is always an attractive fixed point for the action of this larger torus.
\newtheorem*{corollary}{Corollary}
\begin{corollary}
The full nilcone $\mathcal N$ is rationally smooth. Every nilpotent variety $\bar{\mathcal O}$ different from $\mathcal N$ and 0 is rationally singular at 0, with two exceptions, namely the closures of the minimal orbit in types $C_n$ and $G_2$.
\end{corollary}
The rational singularity result is a straightforward consequence of the last of Brion's criteria for rational smoothness: unless $\mathcal O$ is the minimal orbit, the subtori $T'$ contributing to the right side are exactly the centralizers of the positive root spaces in $\frak g$ and every nonzero term on the right side equals 2, whence the right side equals the dimension of $\mathcal N$ rather than $\bar{\mathcal O}$. Even if $\mathcal O$ is minimal, the right side winds up being too large, except in types $C_n$ and $G_2$. In these cases all three of Brion's criteria hold, the last two by direct calculation and the first one since there is only one singular point. Thus these varieties are rationally smooth. That $\mathcal N$ is rationally smooth is an old result of Borho and Macpherson \cite{BM83}; we will sketch their argument in the next seciton. This result can also be proved using Brion's techniques, by constructing a slice of $\mathcal N$ in the sense of \cite[2.1]{Br99}. This can be done inductively, starting with the full nilcone $\mathcal N'$ for $\frak{sl}_2$, which is well known to be rationally smooth.
It seems difficult in general to prove rational smoothness of nilpotent varieties using these techniques (because of the difficulty of verifying the first of Brion's conditions), but one can for example show that the rational singular locus of any spherical nilpotent variety in type $C$ lies one step below its boundary (in the chain of spherical varieties, ordered by inclusion; recall that a nilpotent orbit or its variety is called spherical if it admits a dense suborbit under the action of a Borel subgroup). The same result holds for the largest spherical variety in type $A$ in odd rank; for other spherical varieties in this type, the rational singular locus coincides with the boundary.
\section{Second method--Borho-MacPherson}
We turn now to the second method, which is due to Borho and MacPherson and applies specifically to nilpotent varieties \cite{BM83}. It uses the heavier machinery of intersection homology, but is able to compute the dimensions of the cohomology groups (rather than just determining whether they vanish outside the top degree). We begin by invoking the Springer correspondence: given $e$ the Springer fiber $\mathcal B_e$ of Borel subalgebras containing $e$ is such that its (singular) cohomology groups carry commuting actions of the component group $A(G\cdot e)$ of the centralizer of $e$ in $G$ and the Weyl group $W$. The representation $\sigma_e$ of $W$ on the $A(G\cdot e)$-fixed vectors in the top cohomology group is then irreducible \cite{S78}. Given now another nilpotent element $x$, the Borho-MacPherson Decomposition Theorem \cite{BM83} then implies that
$$
\dim IH^i_x(\bar{\mathcal O},\mathbb Q) = \dim\hom (\sigma_e,H^i(\mathcal B_x.\mathbb Q))
$$
\noindent where as usual $\bar{\mathcal O}$ denotes the closure of the orbit through $e$. Here the left hand side denotes the intersection homology groups of $\bar{\mathcal O}$, which are naturally isomorphic to its cohomology groups in the complementary degree. Hence $\bar{\mathcal O}$ is rationallly smooth at $x$ if and only if $IH^i_x(\bar{\mathcal O},\mathbb Q)$ vanishes in positive degree. If $\bar{\mathcal O} =\mathcal N$, then $\sigma_e$ is trivial and an old computation of Lusztig shows that $\sigma_e$ occurs once in $H^i(\mathcal B_x,\mathbb Q)$ if $i=0$ and not at all if $i>0$ (for any $x$), so $\mathcal N$ is rationally smooth. In general it is not easy to compute the module structure of $H^*(\mathcal B_x,\mathbb Q)$, but explicit tables are available in the exceptional cases \cite{BS84} while in the classical cases one has algorithms due to Shoji and Lusztig \cite{L81,FMM13,Sh83}. An easy special case occurs when $x$ is regular nilpotent in some proper Levi subalgebra $\frak l$ of $\frak g$: if $W_L\subset W$ is the Weyl group of $\frak l$, then $H^*(\mathbb B_x,\mathbb Q)$ is just the permutation representation of $W$ on $W/W_L$ \cite{AL82}. This permutation representation can be computed by the Littlewood-Richardson rule in the classical cases and tables of Alvis in the exceptional ones.
To determine rational smoothness at $x$ it remains to compute $IH^*_y$ for all $y$ in a neighborhood of $x$; by $G$-equivariance it suffices to compute this group for each of the finitely many orbits $G\cdot y$ whose closures lie between $\overline{G\cdot x}$ and $\bar{\mathcal O}$. One could ask whether cohomology vanishing at $x$ implies cohomology vanishing in a neighborhood (as it does for Schubert varieties). The answer is no, already in type $C_3$: there one computes that the cohomology of $\bar{\mathcal O}_{3,3}$ vanishes at points of $\mathcal O_{2,1^4}$, but not at points of $\mathcal O_{2^2,1^2}$, where $\mathcal O_{\lambda}$ denotes the orbit with partition $\lambda$ and exponents in partitions as usual denote repeated parts. It also fails in type $D_4$: the cohomology of $\bar{\mathcal O}_{5,3}$ vanishes at points of $\mathcal O_{3,2^2,1}$ but not at points of $\mathcal O_{3^2,1^2}$. But it holds in type $A$, as follows from the combinatorics of Kostka numbers and the fact that every nilpotent element in that type is regular in some Levi subalgebra.
We conclude with an example of a rational singular locus of codimension 2 (this cannot happen for Schubert varieties, or for closures of $K$-orbits in the flag variety $G/B$, where $K$ is a symmetric subgroup of $G$). Take $\mathcal O$ to be the orbit with Bala-Carter label $A_4$ in type $E_6$; this has dimension 60. Applying the second method and using \cite{BS84} we compute that the rational singular locus of $\bar{\mathcal O}$ coincides with its boundary and has dimension 58; this is the closure of the orbit $\mathcal O'$ with Bala-Carter label $D_4(a_1)$. In this case $\sigma_e$ occurs with multiplicity 3 in $H^*(\mathcal B_x)$ (where $e\in\mathcal O,x\in\mathcal O'$). In fact $\sigma_e $ is paired with the two-dimensional reflection representation of the component group $A(\mathcal O')$ in the Springer correspondence. The corresponding phenomenon also occurs for one the 42-dimensional orbits in type $F_4$ and the (unique) 40-dimensional orbit contained in its closure; there the component group of the smaller orbit is the symmetric group $S_4$ and so once again the multiplicity of the relevant Springer representation is larger than one.
\end{document} |
\begin{document}
\title{Realization of an Optimally Distinguishable Multi-photon Quantum
Superposition.}
\author{Francesco De Martini and Fabio Sciarrino}
\address{Dipartimento di Fisica and Istituto Nazionale di Fisica della\\
Materia, Universit\'{a} ''La Sapienza'', Roma 00185, Italy}
\maketitle
\begin{abstract}
We report the successful generation of an {\it entangled} multiparticle
quantum superposition of pure photon states. They result from a multiple
{\it universal} cloning of a single photon qubit by a high gain,
quantum-injected parametric amplifier. The {\it information preserving}
property of the process suggests for these states the name of ''{\it
multi-particle qubits''}.{\it \ }They are ideal objects for investigating
the emergence of the classical world in quantum systems with increasing
complexity, the decoherence processes and may allow the practical
implementation of the universal 2-qubit logic gates.
\end{abstract}
\pacs{PACS numbers: \tt\string}
\narrowtext
Since the golden years of quantum mechanics the interference of classically
distinguishable quantum states, first epitomized by the famous ''{\it
Schroedinger-Cat}'' apologue \cite{Schr35} has been the object of extensive
theoretical studies and recognized as a major conceptual paradigm of Physics
\cite{Cald83,Legg85}. In modern times the sciences of quantum information
(QI)\ and quantum computation deal precisely with collective processes
involving a multiplicity of interfering states, generally mutually entangled
and rapidly de-phased by decoherence \cite{Zure81}. In many respects the
implementation of this intriguing classical-quantum condition represents
today an unsolved problem in spite of recent successful studies carried out
mostly with atoms and ions \cite{Brun96,Monr96}. The present work reports on
a nearly decoherence-free all optical scheme based on the quantum-injected
optical parametric amplification (QI-OPA) of a single photon in a quantum
superposition state of polarization $(\pi )$, i.e. a $\pi -encoded$ {\it
qubit }\cite{DeMa98,DeMa01}.
Conceptually, the method consists of transferring the well accessible
condition of quantum superposition characterizing a single-photon qubit, $
N=1 $, to a {\it mesoscopic}, i.e. multi-photons amplified state $M>>1$,
here referred to as a ''{\it multi-particle qubit}'' ({\it M-qubit}). In
quantum optics this can be done by injecting in the QI-OPA the single-photon
{\it qubit, }$\alpha \left| H\right\rangle +\beta \left| V\right\rangle $,
here expressed in terms of two mutually orthogonal $\pi -$states, e.g.
horizontal and vertical linear $\pi ^{\prime }s$: $\left| H\right\rangle $, $
\left| V\right\rangle $. In virtue of the general \ {\it information
preserving} \ property of the OPA,\ the generated multi-particle state is
found to keep the {\it same} superposition character and the interfering
capabilities of the injected qubit, thus realizing the most relevant and
striking property of \ the {\it M-qubit} condition \cite{DeMa01}. Since the
present scheme basically realizes the {\it deterministic} $1\rightarrow M$
{\it universal} {\it optimal quantum} {\it cloning} {\it machine} (UOQCM),
i.e. able to copy {\it optimally} any unknown input qubit into $M>>1$ copies
with the same fidelity, the output state will be necessarily affected by
{\it squeezed-vacuum (SV)} noise. This one partially spoils the {\it exact},
i.e. ''classical'' distinguishability of the two interfering terms of the
superposition. This is but a further manifestation of \ the quantum\
''no-cloning theorem'' within the present \ {\it optimal} process. In a more
technical perspective, since any UOQCM can be designed to redistribute {\it
optimally }the initial information into many output channels, the present
scheme is expected to find useful QI\ applications, e.g. in quantum
cryptography \cite{Bech99}, \cite{Galv00} and in error correction schemes
\cite{Brub01}. In addition, a similar method was successfully adopted
recently to realize the first $1\rightarrow 2$ UOQCM and the first universal
quantum NOT-gate \cite{DeMa02,Pell03,Lama02}.
Let's refer to the apparatus: Fig. 1. The OPA active element was a nonlinear
(NL)\ crystal slab (BBO:\ $\beta -$barium borate), 1.5 mm thick cut for Type
II\ phase-matching, able to generate by spontaneous parametric down
conversion (SPDC) $\overrightarrow{\pi }-entangled$ $\ $pairs of photons.
Precisely, the OPA {\it intrinsic phase} was set as to generate by SPDC {\it
singlet} entangled states on the output modes, a condition assuring the {\it
universality} of the cloning transformation by the QI-OPA scheme \cite
{Lama02,DeMa98,DeMa02}. The photons of each pair were emitted with equal
wavelengths (wls) $\lambda =795nm$ over two spatial modes $-{\bf k}_{1}$ and
$-{\bf k}_{2}$ owing to a SPDC process excited by a coherent pump UV\ field
provided by a Ti:Sa Coherent MIRA mode-locked pulsed laser coupled to a
Second Harmonic Generator (SHG) and associated with a mode with wavevector
(wv) $-{\bf k}_{p}$ and wl $\lambda _{p}=397.5nm$. The average UV\ power was
$0.25W$, the {\it pulse repetition rate }${\it (}rep$-$r{\it )}$, $
r_{UV}=7.6\ast 10^{7}$ $s^{-1}$, and the {\it coherence time} of each UV
pulse as well of the generated single photon pulses were $\tau _{coh}=140$
fs. The SPDC process implied a 3-wave NL parametric interaction taking place
towards the right hand side (r.h.s.)\ of Fig. 1. The UV\ pump was
back-reflected over the mode ${\bf k}_{p}\ $onto the NL\ crystal by a
spherical mirror ${\bf M}_{p}$, with $\mu $-metrically adjustable position
{\bf Z}, thus exciting the main OPA ''cloning''\ process, towards the left
hand side (l.h.s.) of Fig.1. By the combined effect of two adjustable
optical UV waveplates (wp) $(\lambda /2$ + $\lambda /4)$ acting on the
projections of the linear polarization ${\bf \pi }_{p}$ of the UV field on
the fixed optical axis of the BBO crystal for the $-{\bf k}_{p}$ and ${\bf k}
_{p}$ counter propagating \ excitation processes, the SPDC\ excitation was
always kept at a very low level while the main OPA amplification could reach
{\it any large intensity, }as we shall see shortly. Precisely, by smartly
unbalancing the orientation angles $\vartheta _{
{\frac12}
}$ and $\vartheta _{
{\frac14}
\text{ }}$of the UV $wp^{^{\prime }}s$, the probability ratio of SPDC\
emission towards the r.h.s. of Fig.1, of two simultaneous correlated photon
pairs and of a single pair was always kept below $3\times 10^{-2}$ in any
{\it High Gain} condition. One of the photons of the SPDC\ emitted pair,
back-reflected by a fixed mirror ${\bf M}$, was {\it re-injected} onto the
amplifying NL\ crystal by the input mode ${\bf k}_{1}$, while the other
photon emitted over mode $(-{\bf k}_{2})\ $excited the detector $D_{T}$, the
{\it trigger }of the overall {\it conditional }experiment. The detectors $
(D)\;$were single-photon SPCM-AQR14. A proper setting\ of {\bf Z} secured
the space-time overlapping into the NL crystal of the interacting\
re-injected pulses with wl's $\lambda _{p}$ and $\lambda $, and then
determined the optimal QI-OPA condition. The time optical walk-off effects
due to the crystal birefringence were compensated by inserting in the modes $
{\bf k}_{1}$, ${\bf k}_{2}$ and $-{\bf k}_{2}$ three fixed X-cut quartz
plates $Q$ and one $\lambda /4\ $wp. Before re-injection into the NL\
crystal, the {\it pure} input qubit on mode ${\bf k}_{1}$, $\left| \Psi
\right\rangle _{in}$= $(\widetilde{\alpha }\left| \Psi \right\rangle
{\alpha \atop in}
+\widetilde{\beta }\left| \Psi \right\rangle
{\beta \atop in}
)$, $\left| \widetilde{\alpha }\right| ^{2}+\left| \widetilde{\beta }\right|
^{2}=1$, represented by the Bloch sphere shown in Fig. 1, underwent unitary $
SU(2)$ {\it rotation} $\Phi \ $transformations: $\widehat{U_{i}}\equiv \exp
(-i\sigma _{i}\Phi /2)\ $around the three Cartesian axes $i=x,y,z\ $by the
combined action of the $\lambda /2$ wp $WP_{T}$, of the adjustable {\it
Babinet Compensator} $B$ and of the polarizing beam-splitter $PBS_{T}$
acting on mode $(-{\bf k}_{2})$ in virtue of the nonlocality correlating the
modes $-{\bf k}_{1}$ and $-{\bf k}_{2}$. These $SU(2)$ transformations are
represented in Fig.1 (inset) by circles drawn on the surface of a
Bloch-sphere. The main OPA process, i.e. acting towards the l.h.s. of Fig.1
on the injected qubit $\left| \Psi \right\rangle _{in},\;$was characterized
by two different excitation regimes, establishing two corresponding {\it
sizes} of the output {\it M-qubit}:
A)\ \ {\it Low Gain} $(LG)$\ regime, characterized by a low excitation UV\
energy $(3.5nJ)$ per pulse, leading to a small value of the NL\ parametric ''
{\it gain}'' :\ $g=0.07.$
B)\ \ {\it High Gain}\ $(HG)$\ regime, characterized by a larger value (by a
factor $\approx 16$) of the {\it gain}: $g=1.13$.\ This condition was
attained by a further amplification of the UV ''pump'' beam by a Ti-Sa
regenerative amplifier Coherent-REGA\ operating at a pulse $rep$-$r$: $
r_{UV}=2.5\ast 10^{5}$ $s^{-1}$, with pulse duration: $180fs$:\ Fig.1.
Let us re-write the state $\left| \Psi \right\rangle _{in}$ by expressing
the interfering states as Fock product states: $\left| \Psi \right\rangle
{\alpha \atop in}
$= $\left| 1\right\rangle _{1h}\left| 0\right\rangle _{1v}\left|
0\right\rangle _{2h}\left| 0\right\rangle _{2v}\equiv \left|
1,0,0,0\right\rangle $; $\left| \Psi \right\rangle
{\beta \atop in}
$= $\left| 0,1,0,0\right\rangle $, accounting respectively for 1 photon with
horizontal $(h)$ polarization on the input ${\bf k}_{1}$, vacuum state on
the input ${\bf k}_{2}$, and 1 photon with vertical $(v)$ $\pi $ on ${\bf k}
_{1}$, vacuum state on the mode ${\bf k}_{2}$. The solution of the QI-OPA
dynamical equations is found to be expressed by the {\it M-qubit}: $\left|
\Psi \right\rangle \equiv (\widetilde{\alpha }\left| \Psi \right\rangle
^{\alpha }+\widetilde{\beta }\left| \Psi \right\rangle ^{\beta })$, with
\begin{equation}
\left| \Psi \right\rangle ^{\alpha }\equiv \gamma \sum\limits_{i,\
j=0}^{\infty }(-\Gamma )^{i}\Gamma ^{j}\sqrt{i+1}\ \left|
i+1,j,j,i\right\rangle ;\ \ \left| \Psi \right\rangle ^{\beta }\equiv \gamma
\sum\limits_{i,\ j=0}^{\infty }(-\Gamma )^{i}\Gamma ^{j}\sqrt{j+1\ }\left|
i,j+1,j,i\right\rangle
\end{equation}
where: $\gamma \equiv C^{-3}$, $C\equiv \cosh g$ \ and $\ \Gamma \equiv
\tanh g$ \cite{DeMa02}. These interfering {\it entangled}, multi-particle
states are {\it ortho-normal}, i.e. $\left| ^{i}\left\langle \Psi \right|
\left| \Psi \right\rangle ^{j}\right| ^{2}$= $\delta _{ij}$ $\left\{
i,j=\alpha ,\beta \right\} $\ and {\it pure, }i.e. fully represented by the
operators: $\rho ^{\alpha }=(\left| \Psi \right\rangle \left\langle \Psi
\right| )^{\alpha }$, $\rho ^{\beta }=(\left| \Psi \right\rangle
\left\langle \Psi \right| )^{\beta }$.\ Hence the {\it pure} state $\left|
\Psi \right\rangle $ is an entangled\ quantum superposition of two
multi-photon {\it pure} {\it states }and bears the {\it same} superposition
properties of the injected single-photon qubit. The genuine quantum
signature of this superposition, namely the {\it non-definite positive}
character of the Wigner function $W(\alpha ,\beta )$\ of the output $\left|
\Psi \right\rangle $ in the 8-dimensional complex phase-space $\alpha _{j}$,
$\beta _{j}$, $j=1,2$, was confirmed by a previous theoretical analysis \cite
{DeMa98}. For the sake of completeness, consider the overall output density
operator $\rho \equiv \left| \Psi \right\rangle \left\langle \Psi \right| \ $
and his {\it mixed-state} reductions over the $\overrightarrow{\pi }-vector$
spaces relative to the modes ${\bf k}_{1}$and ${\bf k}_{2}$: $\ \rho
_{1}=Tr_{2}\rho \ $;$\ \rho _{2}=Tr_{1}\rho $. These ones may be expanded as
a weighted superpositions of $p-square$ matrices of order $p=(n+2)$, the
relative weight $\Gamma ^{2}$ of each two successive matrices being
determined by the parametric {\it gain}. Note that $\Gamma ^{2}$ approaches
asymptotically the unit value for large $g$.\ In turn, the $p-square$
matrices may be expressed as sum of $2\times 2$ matrices as shown by the
following expressions:
\begin{equation}
\rho _{1}=\gamma ^{2}\sum\limits_{n=0}^{\infty }\Gamma ^{2n}\times
\sum\limits_{i=\ 0}^{n}\left[
\begin{array}{cc}
\left| \widetilde{\beta }\right| ^{2}(n-i+1) & \widetilde{\alpha }^{\ast }
\widetilde{\beta }\sqrt{(i+1)(n-i+1)} \\
\widetilde{\alpha }\widetilde{\beta }^{\ast }\sqrt{(i+1)(n-i+1)} & \left|
\widetilde{\alpha }\right| ^{2}(i+1)
\end{array}
\right]
\end{equation}
written in terms of the Fock basis: $\left\{ \left| i\right\rangle
_{1h}\left| n-i+1\right\rangle _{1v}\text{;\ }\left| i+1\right\rangle
_{1h}\left| n-i\right\rangle _{1v}\right\} $. Correspondingly:
\begin{equation}
\rho _{2}=\gamma ^{2}\sum\limits_{n=0}^{\infty }\Gamma ^{2n}\times
\sum\limits_{i=0}^{n+1}\left[
\begin{array}{cc}
\left| \widetilde{\beta }\right| ^{2}(n-i+1) & -\widetilde{\alpha }^{\ast }
\widetilde{\beta }\sqrt{(n-i+1)i} \\
-\widetilde{\alpha }\widetilde{\beta }^{\ast }\sqrt{(n-i+1)i} & \left|
\widetilde{\alpha }\right| ^{2}i
\end{array}
\right]
\end{equation}
in terms of the Fock basis:$\left\{ \left| n-i\right\rangle _{2h}\left|
i\right\rangle _{2v}\text{;\ }\left| n-i+1\right\rangle _{2h}\left|
i-1\right\rangle _{2v}\right\} $. Interestingly, the value $n$ appearing in
Eqs. 2, 3 coincides with the number of photon pairs generated by the QI-OPA\
amplification. Note the {\it non-diagonal }character{\it \ }of these
matrices\ implying the quantum superposition property of the overall state.
Furthermore the interfering states possess the {\it nonlocal} property of
all entangled quantum systems. It is worth noting that the Von Neumann
entropies $S(\rho _{j})=Tr(\rho _{j}\log _{2}\rho _{j})$, $j=1,2$ are equal,
thus implying the same {\it degree of mixedeness }on the 2 output channels
{\it :} $S(\rho _{1})=$ $S(\rho _{2})$ \cite{DeMa02,Benn96}.
The actual $\left| \Psi \right\rangle $ experimental detection of the output
states could be undertaken by measurements taken either (a) on the
injection, i.e. cloning mode ${\bf k}_{1}$, or (b) on the {\it anticloning}
mode ${\bf k}_{2}$, or (c) on both output modes. The option (b) was selected
since there the field on mode ${\bf k}_{2}$ is not affected by the input
qubit in absence of the QI-OPA\ action and then any registered {\it
interference} effect is by itself an unambiguous demonstration of the {\it
M-qubit} condition. This condition was carefully verified experimentally. In
addition, it was checked that only {\it non interfering} SV-noise was
detected on the output mode ${\bf k}_{2}$, in absence of the injection qubit
$\left| \Psi \right\rangle _{in}$.
Since the $1^{st}-$order interference property of any quantum object is
generally expressed by the $1^{st}$-$order$ {\it correlation-function} $
G^{(1)}$ \cite{Wall94}, this quantity was measured by two detectors $D_{2}\ $
and $D_{2^{\ast }}$ coupled to the mutually orthogonal fields $\widehat{c}
_{j}(t)$,$\ j=1,2$\ emerging from the polarizing beam-splitter $PBS_{2}$
inserted in the output ${\bf k}_{2}$: $G
{(1) \atop 2j}
\equiv \left\langle \Psi \right| \widehat{N}_{j}(t)\left| \Psi \right\rangle
$. In the present context the $G^{(1)}$ are expressed by ensemble averages
of the number operators $\widehat{N}_{j}(t)\equiv \widehat{c}_{j}^{\dagger
}(t)~\widehat{c}_{j}(t)$, written in terms of the detected fields: $\widehat{
c}_{1}(t)\equiv 2^{-
{\frac12}
}[\widehat{b}_{h}(t)+\widehat{b}_{v}(t)]$ , $\widehat{c}_{2}(t)\equiv 2^{-
{\frac12}
}[\widehat{b}_{h}(t)-\widehat{b}_{v}(t)]$. There the $\widehat{b}_{i}(t)$, $
i=h,v$, defined on the QI-OPA\ output ${\bf k}_{2}$, underwent a $45^{0}\pi
- $rotation by the $\lambda /2-$wp $WP_{2}$ before injection into $PBS_{2}$:
Fig. 1. The same operations transformed the basis $\{h,v\}$ into $\{H,V\}$.
By expressing the injected qubit as: $\left| \Psi \right\rangle _{in}\equiv
\left( \alpha \left| \Psi \right\rangle _{in}^{\alpha }+\beta e^{i\varphi
}\left| \Psi \right\rangle _{in}^{\beta }\right) $ with $\alpha $ and $\beta
$ real numbers, the $G
{(1) \atop 2j}
$ could be given as: $G
{(1) \atop 2H}
=\overline{n}+
{\frac12}
\overline{n}\left[ 1+2\alpha \beta \cos \varphi \right] $; $G
{(1) \atop 2V}
=\overline{n}+
{\frac12}
\overline{n}\left[ 1-2\alpha \beta \cos \varphi \right] $, showing the
superposition character of the output average field per mode with respect to
$\alpha ,\beta $ and $\varphi $, where $\overline{n}\equiv \sinh ^{2}g$. By
comparison of this result with the corresponding {\it non interfering}
averages $G
{(1) \atop 21,vac}
=G
{(1) \atop 22,vac}
=\overline{n}$ taken in absence of the injected qubit, i.e. over the {\it
input} {\it vacuum state,} the signal-to-noise ratio for the {\it M-qubit}
detection: was found:$\ S/N$ $=$ $2,$ for $\varphi =0$ and $\alpha =\beta
=2^{-
{\frac12}
}$.
The above result immediately suggests, as a demonstration of the $1^{st}$-$
order$\ ${\bf \pi -}$interference of the {\it M-qubit}, to draw the{\it \
fringe} patterns expressing the quantity: $\Delta G
{(1) \atop 2}
\equiv (G
{(1) \atop 2H}
-G
{(1) \atop 2V}
)$ and bearing the fringe ''visibility'': ${\cal V}=(G
{(1) \atop 2\max }
-G
{(1) \atop 2\min }
)/(G
{(1) \atop 2\max }
+G
{(1) \atop 2\min }
)=2\alpha \beta /3$. The experimental values of $G^{(1)}$ were obtained by
coincidence measurements involving the couples of detectors: $[D_{2}D_{T}]$
and $[D
{\ast \atop 2}
D_{T}]$. Note that the fringe patterns, reported in Fig. 2 and the
corresponding values of ${\cal V}$ were strongly affected by the
superposition character of the injection qubit and by the corresponding $
SU(2)$ transformations expressed by the closed paths drawn on the Bloch
sphere: Fig.1 (inset). In order to show more clearly the quantum efficiency
of the {\it M-qubit} for the two ''gain'' regimes, the {\it relative
coincidence rates,} $\xi =r_{SC}/r_{UV}$ were reported in Fig. 2, where $
r_{SC}$ expressed the values of $\Delta G
{(1) \atop 2}
\equiv (G
{(1) \atop 2H}
-G
{(1) \atop 2V}
)$. The experimental best-fit curves drawn in Figure 2 were found to
reproduce very closely the quantum theoretical results.
In the $LG$ excitation condition the {\it average} number of
quantum-interfering photon pairs was $\overline{N}\approx 0.009$ while the $
HG$ condition it was $\overline{N}\approx 4$, corresponding to the
realization of a much ''fatter'' {\it M-qubit}. The value of {\it average}
number of generated pairs was $\overline{N}=3\overline{n}$ in virtue of the
stimulated emission process. They were estimated by accounting for the
overall {\it quantum efficiency }of the $D$ detectors : $QE\approx 18\%$.
Calibrated attenuation filters placed in front of the $D^{\prime }s$ assured
the condition of single-photon detection in the $HG$ regime. By the Eqs. 2,
3 a straightforward evaluation of the output photon-pair distribution $p(n)$
could be carried out. By this one, the probability of generating a number of
pairs $N\geq 2\overline{N}=8$, i.e. a quantum superposition in excess of $16$
photons, was found to be $P(\overline{N})\equiv \sum\limits_{k=2\overline{N}
}^{\infty }p(k)\approx 14\%$. By moving the injected state with $\alpha
=\beta $ over the Bloch sphere, a visibility$\ V_{th}=33\%$ of the
fringe-pattern was expected from theory. Due to experimental imperfections,
mostly due to the mode selection before detection, we have measured $
V\approx 4\%$ $(4\%)$ with a 2-coincidence detection measurement scheme and $
V\approx 25\%$ $(7\%)$ with a 4-coincidence scheme respectively for the $LG$
and $HG$ regimes. The results above are expected to be {\it linearly scaled}
by adoption of a more efficient NL crystal and of a more powerful UV source.
Unlike other systems involving atoms or ions \cite{Monr96,Brun96} a
limitation of our method consists of the impossibility of controlling the
''distance'', $d$ on the {\it phase-space} of the interfering multi-particle
states by means of the QI-OPA parameters. This condition, implied by the
results of the Wigner function analysis \cite{DeMa98} can be expressed here
by the Hilbert-Schmidt $(d)$ of the interfering states: $d(\rho ^{\alpha
};\rho ^{\beta })=Tr\left[ (\rho ^{\alpha }-\rho ^{\beta })^{2}\right] $
\cite{Nie00}. This quantity, not affected by the amplification process, is
expressed in our case by: $d(\rho ^{\alpha };\rho ^{\beta })_{in}$= $d(\rho
^{\alpha };\rho ^{\beta })_{out}$= $2$. However, as a lucky counterpart of
this condition, the de-coherence of our system can be deemed generally
irrelevant as it is mostly determined by the stray reflection losses on the
surfaces of the optical components of the apparatus. Accordingly, a number
of photons in the range $(10^{2}\div 10^{3})$ could be made simultaneously
excited in quantum superposition without sizable decoherence effects: this
is but one of the commendable qualities of the device. Of course, the
injection into the OPA of a many particle state will results in a better $
S/N $ ratio. The M-qubit condition attained by injection into OPA\ \ of a
symmetrized 2 photon state is presently in progress in our Laboratory.
In summary, we have realized the interference of classically distinguishable
{\it multi-particle}, {\it orthonormal}, {\it pure} quantum states. These
characteristics indeed express the peculiar properties of the system
originally proposed by Schroedinger \cite{Schr35} with a relevant additional
feature: the {\it entanglement} of the interfering states. The adoption of
the OPA to {\it clone} {\it universally} a single qubit in a large gain
regime witnesses the scalability of the information preserving property of
this device\ in the quantum domain. Furthermore and very important, unlike
other systems involving atoms, e.g. in hardly accessible optical traps or
superconducting cavities \cite{Monr96,Brun96}, the {\it M-qubit} generated
by our system is directly accessible to measurement or to further
exploitation, e.g. by injection into a QI {\it gate}. \ In a more conceptual
perspective, the present realization could open a new trend of studies on
the persistence of the validity of several crucial laws of quantum mechanics
for entangled mixed-state systems of increasing complexity \cite{Brun96} and
on the violation of Bell-type inequalities in the multi-particles regime
\cite{Reid02}. This work has been supported by the FET European Network on
QI and Communication (Contract IST-2000-29681: ATESIT), by INFM (PRA\
''CLON'')\ and by MIUR (COFIN 2002). We thank V. Buzek, S. Popescu, for
stimulating discussions and D. Pelliccia for early experimental
collaboration.
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(1935).
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al.} Science {\bf 304}, 1478 (2004).
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most relevant results of the present experiment are reported by F. De
Martini, M. Ricci, F. Sciarrino in: ''Quantum State Estimation'', ed. by
M.G.A. Paris and J.Rehacek (Springer Verlag, Berlin 2004).
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Lett. {\bf 8}7, 150401 (2001). There the double-injection did not allow a
direct detection of the interference.
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802 (1982).
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022301 (2000).
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Phys. Rev. A {\bf 63}, 042308 (2001).
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(London{\it )} {\bf 419}, 815 (2002).
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Lett. {\bf 92}, 067901 (2004).
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Bouwmeester, Science\ {\bf 296}, 712 (2002).
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Wootters, Phys. Rev. A {\bf 54}, 3824 (1996).
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\centerline{\bf Figure Captions}
\vskip 8mm
\parindent=0pt
\parskip=3mm
Figure 1. Layout of the {\it quantum-injected OPA }apparatus. The device
{\bf A} represents the Regenerative Ti:Sa laser Amplifier to attain the High
Gain (HG)\ dynamical condition. INSET: Bloch sphere representation of the $
SU(2)$ unitary transformations applied to the input qubit.
Figure 2. Multi-photon interference fringe patterns expressed by the {\it
relative coincidence rate,} $\xi =\eta (r_{SC}/r_{UV})$ proportional to the
ratio of the 2-detector coincidence rates expressing the quantity: $\Delta G
{(1) \atop 2}
\equiv (G
{(1) \atop 2H}
-G
{(1) \atop 2V}
)$ and of the repetition rate of the UV pump excitation in $LG$ and $HG$
regimes $(\eta =$ $7.6\times 10^{7})$. The patterns correspond to the $SU(2)$
transformations shown in Fig.1 (inset) and affecting the injected qubit. $
G^{(1)}$ are the $1^{st}-order$ {\it correlation functions}.
\end{document} |
\begin{document}
\begin{abstract}
We will introduce formal frames of manifolds, which are a generalization of ordinary frames.
Their fundamental properties are discussed.
In particular, canonical forms are introduced, and torsions are defined in terms of them as a generalization of the structural equations.
It will be shown that the vanishing of torsions are equivalent to the realizability of given formal frames as ordinary frames.
We will also discuss deformations of linear connections on tangent bundles.
An application to deformations of foliations are then given.
\end{abstract}
\title{Formal frames and deformations of affine connections}
\setlength{\baselineskip}{16pt}
\section*{Introduction}
The theory of frames are well-developed.
It is based on the local diffeomorphisms of $\mathbb{R}^n$ which fix the origin, and their jets.
As a consequence, tensors, etc. are commutative in the lower indices.
For example, if $T$ is a related tensor and if $T^i{}_{jk}$ denote the components of $T$ with respect to a chart, then we have $T^i{}_{kj}=T^i{}_{jk}$.
The bundle of $2$-frames are quite related with connections.
The commutativity of the indices imply that these connections are torsion-free.
On the other hand, when we consider geometric structures, usually connections with torsions appear.
Garc\'\i a gave a framework which enables us to work on connections with torsions in~\cite{Garcia}.
These two frameworks are similar but differ at several points.
For example, the structural group in the Kobayashi construction~\cite{K_str} is much larger than Garc\'\i a's one and its action has a clear meaning.
On the other hand, Garc\'\i a's construction has an an advantage that any principal bundles can be treated, while the theory of frames work basically on manifolds and their frame bundles.
In this paper, we introduce a notion of formal frames, which is a kind of frames with non-commutative indices, and by which we can understand the both frameworks if we restrict ourselves to frame bundles.
We will introduce canonical forms like in the classical case (Definitions~\ref{defG1}, \ref{def4.4}).
Such canonical forms are studied by Garc\'\i a~\cite{Garcia} in basic cases.
Canonical forms in this article are generalizations of these classical ones.
Then, we will define torsions in terms of canonical forms (Definition~\ref{def5.2}).
They are a generalization of the structural equations in the classical setting.
It will be shown that a formal frame is actually a classical frame if and only if every torsion vanishes (Theorem~\ref{thm5.7}).
Integrability of formal frames under geometric structure can be also formulated on the bundle of formal frames in some cases (Remark~\ref{rem5.9}, Example~\ref{ex5.11}).
We then discuss deformations of linear connections on tangent bundles.
We will show that infinitesimal deformations of connections can be regarded as connections on certain principal bundles (Theorem~\ref{thm6.16}).
Finally, we will discuss deformations of foliations as an application.\par
Throughout this article, we will make use of the Einstein convention.
That is, a pair of upper and lower indices of the same letter is understood to run from $1$ to $n$ ($q$ in Section~\ref{sec6}) and be taken the sum.
For example, $f^i\pdif{}{x^i}$ means $\sum_{i=1}^nf^i\pdif{}{x^i}$.
When we compare representations of objects with respect to two charts, we represent one in plain letters and another one by adding `$\widehat{\;\ \;}$'.
For example, let $T$ be a tensor of type $(1,2)$.
If $(U,\varphi)$ and $(\widehat{U},\widehat{\varphi})$ are charts, then the components of $T$ with respect to $(U,\varphi)$ are represented by $T^i{}_{jk}$, and the ones with respect to $(\widehat{U},\widehat{\varphi})$ are represented by $\widehat{T}^i{}_{jk}$.
When we deal with a Lie group, its Lie algebra is represented by corresponding German letter, e.g.~if $G$ is a Lie group, then its Lie algebra is represented by $\mathfrak{g}$.
Finally, we always make use of the standard coordinates for $\mathbb{R}^n$.
\section{The bundle of formal $2$-frames}
\begin{notation}
Let $M$ be a manifold and $p\in M$.
If $f$ is a mapping defined on a neighborhood of $p$, then we say that $f$ is a mapping from $(M,p)$.
Precisely speaking, we consider the germ of $f$ at $p$.
If the target of $f$, say $N$, is specified, then we say that $f$ is a mapping from $(M,p)$ to $N$.
If moreover the image $f(p)$ is specified, then we say that $f$ is a mapping from $(M,p)$ to $(N,f(p))$.
If $f$ is a diffeomorphism to the image, we say that $f$ is a \textit{local diffeomorphism}.
\end{notation}
\begin{notation}
Let $V\to M$ and $W\to N$ be vector bundles.
If $F$ is a bundle morphism from $(V,M)$ to $(W,N)$, then the underlying map from $M$ to $N$ is represented by $f$.
The mapping induced on fibers are represented as $F_p\colon V_p\to W_{f(p)}$, where $V_p$ denotes the fiber of $V$ over $p\in M$.
When we consider bundle automorphisms, we do \textit{not} assume the underlying maps to be the identity.
\end{notation}
Hence a bundle morphism $F$ consists of a pair $(f,F_{\bullet})$.
\begin{definition}
\begin{enumerate}
\item
If $h=(h^i)$ is an $\mathbb{R}^m$-valued function on an open subset of $\mathbb{R}^n$, then we define an $M_{m,n}(\mathbb{R})$-valued function $Dh$ by setting $Dh^i{}_j=\pdif{h^i}{x^j}$, where $(x^1,\ldots,x^n)$ are the standard coordinates for $\mathbb{R}^n$.
\item
If $h$ is an $M_{m,n}(\mathbb{R})$-valued function, then we represent $h$ as $h^i{}_j$ and set $Dh^i{}_{jk}=\pdif{h^i{}_j}{x^k}$.
\end{enumerate}
\end{definition}
We recall the notion of frames of higher order~\cite{K_str}.
\begin{definition}
Let $r\geq1$.
For $p\in M$, an \textit{$r$-frame} at $p$ is the $r$-jet at $o\in\mathbb{R}^n$ of a local diffeomorphism, say $f$, from $(\mathbb{R}^n,o)$ to $(M,p)$ and is represented by $j^r_o(f)$.
The set of $r$-frames are represented by $P^r(M)$.
We set $\pi^r(j^r_o(f))=f(o)$.
If the target of $o$ is assumed to be a fixed point $p$, then we represent $P^r(M)$ as $P^r(M,p)$.
\end{definition}
The following is classical~\cite{K}.
\begin{theorem}
If we set $G^r_n=P^r(\mathbb{R}^n,o)$, then $G^r_n$ is a Lie group of dimension $n\sum_{i=1}^r\dbinom{n}{i}$ and $P^r(M)$ is naturally a principal $G^r_n$-bundle.
Indeed, the action is given by compositions.
\end{theorem}
\begin{remark}
The arguments and calculations in Sections~1, 2 are similar to those of~\cite{Kobayashi-Nagano}.
\end{remark}
Let now $\pi^1\colon P^1(M)\to M$ be the $1$-frame bundle, which is naturally isomorphic to the structure bundle of $TM$.
We represent $P^1(M)$ as $P^1$ if $M$ is clear.
Let $F=(f,F_\bullet)\colon T(\mathbb{R}^n,o)\to TM$ be a bundle isomorphism to the image such that $F_o=Df(o)$.
If $(U,\varphi)$ is a chart about $f(o)$, then $D\varphi\circ F$ is represented as $(D\varphi\circ F)_x=A^i{}_j(x)\pdif{}{y^i}_{f(x)}$, where $x\in\mathbb{R}^n$ is in a neighbourhood of $o$ and $(y^1,\dots,y^n)$ are coordinates for $\varphi(U)$.
Note that if we represent $\varphi\circ f$ as $\varphi\circ f(x)=\varphi(f(o))+a^i{}_jx^j+\overline{f}(x)$, where $\overline{f}$ is of order greater than one with respect to $x$, then we have $a^i{}_j=A^i{}_j(o)$.
Hence $j^1_o(F)$ can be represented by a triple $(a^i,a^i{}_j,a^i{}_{jk})\in\mathbb{R}^n\times\mathrm{GL}_n(\mathbb{R})\times\mathbb{R}^{n^3}$, where $a^i=(\varphi(f(o))^i$ and $a^i{}_{jk}=\pdif{}{x^k}A^i{}_j(o)$.
\begin{definition}
We set
\[
\widetilde{G}_n^2=\{j^1_o(F)\mid\text{$F$ is a bundle automorphism of $T(\mathbb{R}^n,o)$ such that $F_o=Df(o)$}\}.
\]
\end{definition}
Let $F,G\colon T(\mathbb{R}^n,o)\to T(\mathbb{R}^n,o)$.
The underlying map of $F\circ G$ is equal to $f\circ g$, and we have $(F\circ G)_x=F_{g(x)}\circ G_x$.
In particular, we have $f\circ g(o)=f(o)=o$ and that $(F\circ G)_o=F_{g(o)}\circ G_o=F_o\circ G_o$.
Hence $\widetilde{G}_n^2$ admits a group structure of which the product is the composition.
Indeed, $\widetilde{G}_n^2$ is a Lie group diffeomorphic to $\mathrm{GL}_n(\mathbb{R})\times\mathbb{R}^{n^3}$.
We will describe the product in Lemma~\ref{lem1.10}.
\begin{definition}
\label{def1.6}
Let
\[
\widetilde{P}^2(M)=\left\{j^1_o(F)\;\middle|\;\parbox[c]{18.4em}{$F\colon T(\mathbb{R}^n,o)\to TM$ is a bundle isomorphism\\ to the image such that $F_o=Df(o)$}\right\}.
\]
We call $\widetilde{P}^2(M)$ the \textit{bundle of formal $2$-frames} and also the formal $2$-frame bundle for short.
An element of $\widetilde{P}^2(M)$ is called a \textit{formal frame} of order $2$.
If $u=j^1_o(F)\in\widetilde{P}^2(M)$, then we set $\pi^2(u)=f(o)$.
We regard $F(o)$ as an element of $P^1(M)$ and set $\pi^2_1(u)=F(o)$.
If in addition $a=j^1_o(G)\in\widetilde{G}_n^2$, then we set $u.a=j^1_o(F\circ G)$.
\end{definition}
Note that we have $\pi^2=\pi^1\circ\pi^2_1$ and that $\widetilde{G}^2_n=\widetilde{P}^2(\mathbb{R}^n,o)$.
Let $p\in M$ and $(U,\varphi)$ be a chart about $p$.
If $u\in(\pi^2)^{-1}(U)$, then we represent $u=j^1_o(F)$, where $F$ is a bundle isomorphism.
We set $f_\varphi=\varphi\circ f$, $F_\varphi=D\varphi\circ F$ and associate with $u$ a triple $\left(f_\varphi(o)^i,F_\varphi(o)^i{}_j,\pdif{F_\varphi{}^i{}_j}{x^k}(o)\right)$, where $(x^1,\ldots,x^n)$ are the standard coordinates for $\mathbb{R}^n$.
We do not distinguish $F_\varphi(o)^i$ and $F_\varphi{}^i(o)$ and so on in what follows.
\begin{notation}
\label{not1.6}
We refer to the coordinates for $\widetilde{P}^2$ as above as the \textit{natural coordinates} for $\widetilde{P}^2$ associated with $\varphi$ after \cite{K}*{p.~140}.
\end{notation}
If $M=\mathbb{R}^n$, $\varphi={\mathrm{id}}$ and if $u\in\widetilde{G}_n^2$, then $F_\varphi(o)^i=0$ so that we can associate with $u$ a pair $\left(F_\varphi(o)^i{}_j,\pdif{F_\varphi{}^i{}_j}{x^k}(o)\right)$.
Let $a=j^1_o(G)\in\widetilde{G}_n^2$, where $g(o)=o$.
Then $u.a=j^1_o(F\circ G)$ is represented with respect to $\varphi$ as
\stepcounter{theorem}
\begin{align*}
\tag{\thetheorem}
\label{eqF4}
&\hphantom{{}={}}
\left(F_\varphi(o)^i,F_\varphi(g(o))^i{}_\alpha G(o)^\alpha{}_j,\pdif{F_\varphi{}^i{}_\alpha}{x^\beta}(g(o))G(o)^\alpha{}_jDg(o)^\beta{}_k+F_\varphi(o)^i{}_\alpha\pdif{G^\alpha{}_j}{x^k}(o)\right)\\*
&=\left(F_\varphi(o)^i,F_\varphi(o)^i{}_\alpha(o)G(o)^\alpha{}_j,\pdif{F_\varphi{}^i{}_\alpha}{x^\beta}(o)G(o)^\alpha{}_jG(o)^\beta{}_k+F_\varphi(o)^i{}_\alpha\pdif{G^\alpha{}_j}{x^k}(o)\right).
\end{align*}
where $Dg(o)^i{}_j=G(o)^i{}_j$ because $j^1_o(G)\in\widetilde{G}_n^2$.
Let $(U,\varphi)$ and $(U,\widehat{\varphi})$ be charts and $\phi=\widehat{\varphi}\circ\varphi^{-1}$ be the transition function.
Let $u=j^1_o(F)\in(\pi^2)^{-1}(U)\subset\widetilde{P}^2$ and represent $u$ as $(h^i,h^i{}_j,h^i{}_{jk})$ in the natural coordinates associated with $(U,\varphi)$.
If we represent $u$ with respect to $(U,\widehat{\varphi})$, then $F_{\widehat{\varphi}}=(D\phi\circ f)\circ F_{\varphi}$ so that we have
\begin{align*}
(\widehat{h}^i,\widehat{h}^i{}_j,\widehat{h}^i{}_{jk})&=(\phi(h^a)^i,D\phi(h^a)^i{}_\alpha h^\alpha{}_j,H\phi(h^a)^i{}_{\alpha\beta}h^\alpha{}_jh^\beta{}_k+D\phi(h^a)^i{}_\alpha h^\alpha{}_{jk})\\*
&=(\phi(h^a)^i,(D\phi(h^a)^i{}_j,H\phi(h^a)^i{}_{jk})(h^i{}_j,h^i{}_{jk})),
\end{align*}
where the product in the most right hand side is taken in $\widetilde{G}_n^2$.
Summing up, we have the following
\begin{lemma}
\label{lem1.10}
\begin{enumerate}
\item
Let $a,b\in\widetilde{G}_n^2$ and represent $a,b$ as $a=(a^i{}_j,a^i{}_{jk})$, $b=(b^i{}_j,b^i{}_{jk})$.
Then, we have
\[
ab=(a^i{}_\alpha b^\alpha{}_j,a^i{}_{\alpha\beta}b^\alpha{}_jb^\beta{}_k+a^i{}_\alpha b^\alpha{}_{jk}).
\]
\item
The bundle of formal $2$-frames $\widetilde{P}^2$ is a principal $\widetilde{G}_n^2$-bundle with the projection $\pi^2$.
\end{enumerate}
\end{lemma}
The bundles $\widetilde{P}^2$ and $P^2$, the groups $\widetilde{G}_n^2$ and $G_n^2$ are related as follows.
First we introduce the following
\begin{definition}
Let $u\in P^2$ and represent $u$ as $u=j^2_o(f)$, where $f\colon(\mathbb{R}^n,o)\to M$ is a local diffeomorphism.
We set $\epsilon(u)=j^1_o(Du)$.
\end{definition}
The map $\epsilon$ gives an inclusion of $G_n^2$ to $\widetilde{G}_n^2$.
There is also a projection from $\widetilde{P}^2$ to $P^2$.
First, if $j^1_o(F)\in\widetilde{P}^2$, then we may modify $f$ as follows.
Let $(U,\varphi)$ be a chart and let $(y^1,\ldots,y^n)$ be the standard coordinates for $\varphi(U)$.
We have
\begin{align*}
(\varphi\circ f(x))^i&=\varphi(f(o))^i+a^i{}_\alpha x^\alpha+\overline{f}(x)^i,\\*
F_\varphi(x)&=\pdif{}{y^i}(a^i{}_j+a^i{}_{j\alpha}x^\alpha+\overline{a^i{}_j}(x)),
\end{align*}
where $\overline{f}$ and $\overline{a^i{}_j}$ are of order greater than $1$ with respect to $x^i$.
As we are concerned with $1$-jets, we set $b^i{}_{jk}=\frac12(a^i{}_{jk}+a^i{}_{kj})$ and modify $\overline{f}$ as $\frac12b^i{}_{jk}x^jx^k+\overline{f}'(x)^i$, where $\overline{f}'(x)$ is of order greater than $2$.
After this modification, we have $H(\varphi\circ f)(o)^i{}_{jk}=b^i{}_{jk}$.
This property is stable in the following sense.
First, let $(U,\widehat{\varphi})$ be also a chart and set $\phi=\widehat{\varphi}\circ\varphi^{-1}$.
As we have just seen, we have
\[
(\widehat{a}^i{}_j,\widehat{a}^i{}_{jk})=(D\phi(p)^i{}_j,H\phi(p)^i{}_{jk})(a^i{}_j,a^i{}_{jk}),
\]
where $p=\varphi(f(o))$.
Since $H\phi(p)^i{}_{ml}=H\phi(p)^i{}_{lm}$, we have
\begin{align*}
\widehat{b}^i{}_{jk}&=\frac12(\widehat{a}^i{}_{jk}+\widehat{a}^i{}_{kj})\\*
&=\frac12(H\phi(p)^i{}_{\alpha\beta}a^\alpha{}_ja^\beta{}_k+D\phi(p)^i{}_\alpha a^\alpha{}_{jk}+H\phi(p)^i{}_{\alpha\beta}a^\alpha{}_ka^\beta{}_j+D\phi(p)^i{}_\alpha a^\alpha{}_{kj})\\*
&=H\phi(p)^i{}_{\alpha\beta}a^\alpha{}_ja^\beta{}_k+D\phi(p)^i{}_\alpha\left(\frac12(a^\alpha{}_{jk}+a^\alpha{}_{kj})\right)\\*
&=H\phi(p)^i{}_{\alpha\beta}a^\alpha{}_ja^\beta{}_k+D\phi(p)^i{}_\alpha b^\alpha{}_{jk}.
\end{align*}
On the other hand, we have
\stepcounter{theorem}
\begin{align*}
\tag{\thetheorem}
\label{eqF8}
&\hphantom{{}={}}
\phi\circ(\varphi\circ f)(x)\\*
&=\widehat{\varphi}(f(o))+D\phi(p)^i{}_\alpha a^\alpha{}_\beta x^\beta\\*
&\hphantom{{}={}}+\frac12\left(H\phi(p)^i{}_{\alpha\beta}a^\alpha{}_ja^\beta{}_k+D\phi(p)^i{}_\alpha b^\alpha{}_{jk}\right)x^jx^k+(\text{terms of order greater than $2$}).
\end{align*}
This means that the $2$-jets of modified mappings are independent of the choice of charts.
Similarly, we see that the modification is compatible with products with elements of $\widetilde{G}^2_n$.
Hence the following definitions make sense.
\begin{definition}
Let $j^1_o(F)\in\widetilde{P}^2$.
If we have $F_o=Df(o)$ and that $Hf(o)^i{}_{jk}=\frac12(DF(o)^i{}_{jk}+DF(o)^i{}_{kj})$, then we say $F$ is a \textit{normal} representative.
\end{definition}
Note that if we begin with a local diffeomorphism $f\colon(\mathbb{R}^n,o)\to M$ and if we consider $j^1_o(Df)$, then $Df$ is normal as a representative.
\begin{definition}
Let $u\in\widetilde{P}^2$.
We choose a normal representative $F$ for $u$ and set $\kappa(u)=j^2_o(f)$, where $F=(f,F_\bullet)$.
\end{definition}
From these arguments, we see the following
\begin{theorem}
\label{thm1.13}
\begin{enumerate}
\item
The mapping $\epsilon\colon P^2\to\widetilde{P}^2$ is well-defined and is an embedding of\/ $G_n^2$-bundles.
Moreover, we have $\widetilde{P}^2=\epsilon(P^2)\times_{G_n^2}\widetilde{G}_n^2$.
\item
The mapping $\kappa\colon\widetilde{P}^2\to P^2$ is well-defined bundle morphism as $G_n^2$-bundles.
If $(U,\varphi)$ is a chart about $\pi(u(o))$ and if we represent $u$ as $(h^i,h^i{}_j,h^i{}_{jk})$, then $\kappa(u)$ is represented as $\left(h^i,h^i{}_j,\frac{h^i{}_{jk}+h^i{}_{kj}}2\right)$.
\item
When regarded as a mapping from $\widetilde{G}_n^2$ to $G_n^2$, $\kappa$ is a homomorphism.
\end{enumerate}
\end{theorem}
\begin{remark}
\label{rem1.14}
If we replace $\mathrm{GL}_n(\mathbb{R})$ with a proper Lie subgroup, then the above averaging procedure does not work in general.
For example, if $G=\mathrm{SL}_n(\mathbb{R})$, then we have $h^i{}_{ik}=0$ but mostly we do \textit{not} have $h^i{}_{ji}=0$.
Even if the latter condition is satisfied, it is still not obvious that such a formal frame can be realized by a volume preserving local diffeomorphism.
On the other hand, we can replace $\mathbb{R}$ by $\mathbb{C}$, moreover, we can work in the holomorphic category so long as we stay in formal frames of finite order.
See also Remarks~\ref{rem4.9} and~\ref{rem5.9}.
\end{remark}
Let $u\in\widetilde{P}^2$ and we represent $u=j^1_o(F)$.
Then $F$ induces an isomorphism from $T_{{\mathrm{id}}}P^1(\mathbb{R}^n)$ to $T_u P^1$, where ${\mathrm{id}}$ denotes the identity map.
Indeed, let $X\in T_{{\mathrm{id}}}P^1(\mathbb{R})$ and represent $X$ by a family of local diffeomorphisms $g_t\colon(\mathbb{R}^n,o)\to\mathbb{R}^n$ such that $g_0={\mathrm{id}}$.
We can represent $X$ as $(X^i,X^i{}_j)$ by considering the natural coordinates.
Then, we have $\left.\pdif{g_t}{t}(o)\right|_{t=0}=X^i$ and that $\left.\pdif{Dg_t}{t}(o)\right|_{t=0}=X^i{}_j$.
We have
\stepcounter{theorem}
\begin{align*}
\tag{\thetheorem-1}
\label{eqF10-1}
&\hphantom{{}={}}
\left.\pdif{}{t}\right|_{t=0}f\circ g_t(o)=Df(o)\left.\pdif{g_t}{t}\right|_{t=0}(o)=Df(o)^i{}_lX^l=F(o)^i{}_\alpha X^\alpha,\\*
\tag{\thetheorem-2}
\label{eqF10-2}
&\hphantom{{}={}}
\left.\pdif{}{t}\right|_{t=0}(F\circ g_t)(o)Dg_t(o)\\*
&=DF(o)^i{}_{\alpha\beta}Dg_0(o)^\alpha{}_j\pdif{g_t}{t}(o)^\beta+F(o)^i{}_\alpha\left.\pdif{Dg_t}{t}(o)^\alpha{}_j\right|_{t=0}\\*
&=DF(o)^i{}_{j\alpha}X^\alpha+F(o)^i{}_\alpha X^\alpha{}_j.
\end{align*}
\begin{definition}
\label{defF11}
We represent the isomorphism from $T_{{\mathrm{id}}}P^1(\mathbb{R}^n)$ to $T_{\pi^1(u)}P^1$ obtained as above again by $u$ by abuse of notations.
\end{definition}
Similarly, we have an adjoint action of $\widetilde{G}_n^2$ on $T_{{\mathrm{id}}}P^1(\mathbb{R}^n)$.
Let $a\in\widetilde{G}_n^2$ and $X\in T_{{\mathrm{id}}}P^1(\mathbb{R}^n)$.
We represent $a$ by $F$ and $X$ by $g_t$, respectively.
Then, $\Ad_{a^{-1}}X$ is by definition the vector represented by $(f^{-1}\circ g_t\circ f,(F^{-1}\circ g_t\circ f)(Dg_t\circ f)F)$.
Concretely, we have
\begin{align*}
&\hphantom{{}={}}
\left.\pdif{}{t}\right|_{t=0}(f^{-1}\circ g_t\circ f)(o)\\*
&=(Df(o)^{-1})^i{}_lX^l,\\*
&\hphantom{{}={}}
\left.\pdif{}{t}\right|_{t=0}(F^{-1}\circ g_t\circ f)(o)(Dg_t\circ f)(o)F(o)\\*
&=-(F(o)^{-1})^i{}_\alpha DF(o)^\alpha{}_{\beta\gamma}(F(o)^{-1})^\beta{}_\delta X^\gamma F(o)^\delta{}_j+F^{-1}(o)^i{}_\alpha X^\alpha{}_\beta F(o)^\beta{}_j\\*
&=-(F(o)^{-1})^i{}_\alpha DF(o)^\alpha{}_{j\beta}X^\beta+F^{-1}(o)^i{}_\alpha X^\alpha{}_\beta F(o)^\beta{}_j.
\end{align*}
\section{The canonical form on the bundle of formal $2$-frames}
We are now in position to introduce the canonical form on $\widetilde{P}^2$ and its fundamental properties.
\begin{definition}
\label{defG1}
Let $u\in\widetilde{P}^2$ and $X\in T_u\widetilde{P}^2$.
We set $\theta(X)=u^{-1}(\pi^2_1{}_*X)\in T_{{\mathrm{id}}}P^1(\mathbb{R}^n)$ and call $\theta$ the \textit{canonical form}.
\end{definition}
\begin{definition}
Let $G$ be a Lie group.
Let $P$ be a principal $G$-bundle and $u\in P$.
We consider $u$ as a mapping from $G$ to $P$.
If $X\in\mathfrak{g}$, then we set $X^*_u=u_*X$.
We call $X^*$ the fundamental vector field associated with $X$.
\end{definition}
The following theorems directly follow from definitions.
\begin{theorem}
\label{thmG2}
We have the following.
\begin{enumerate}
\item
If $a\in\widetilde{G}_n^2$, then $R_a^*\theta=\Ad_{a^{-1}}\theta$, where $R_a$ and $\Ad_a$ denote the right action and the adjoint action of\/ $a$, respectively.
\item
If $X\in\widetilde{\mathfrak{g}}_n^2$, then $\theta^2(X^*)=\pi^2_1{}_*X$.
\end{enumerate}
\end{theorem}
\begin{theorem}
\label{thmG3}
Let $M,N$ be manifolds and $f\colon M\to N$ be a local diffeomorphism.
\begin{enumerate}
\item
We have $f^*\widetilde{P}^2(N)=\widetilde{P}^2(M)$.
\item
If $\theta_M,\theta_N$ denote the canonical forms on $M,N$, then we have $f^*\theta_N=\theta_M$.
\end{enumerate}
\end{theorem}
Note that $\theta$ is naturally represented as $(\theta^i,\theta^i{}_j)$.
We set $\theta^0=(\theta^i)$ and $\theta^1=(\theta^i{}_j)$.
\begin{definition}
We call $\theta^i$ the \textit{canonical form} of order $i+1$.
We set
\begin{align*}
\Theta&=d\theta^0+\theta^1\wedge\theta^0,\\*
\Omega&=d\theta^1+\theta^1\wedge\theta^1,
\end{align*}
and call them the \textit{torsion} and the \textit{curvature} of $\theta$.
\end{definition}
The canonical form is locally represented as follows by~\eqref{eqF10-2}.
\begin{lemma}
\label{lem2.6}
Let $(U,\varphi)$ be a chart and $u=(u^i,u^i{}_j,u^i{}_{jk})$ the associated natural coordinates.
If we set $(v^i{}_j)=(u^i{}_j)^{-1}$, then we have
\begin{align*}
\theta^0{}_u(X)&=v^i{}_\alpha du^\alpha,\\*
\theta^1{}_u(X)&=v^i{}_\alpha du^\alpha{}_j-v^i{}_\alpha u^\alpha{}_{j\beta}v^\beta{}_\gamma du^\gamma.
\end{align*}
\end{lemma}
The proof is straightforward and omitted.
\begin{theorem}
\label{thm2.6}
We have $P^2=\{u\in\widetilde{P}^2\mid \Theta_u=0\}$.
\end{theorem}
\begin{proof}
First note that we have $dv^i{}_j=-v^i{}_\alpha du^\alpha{}_\beta v^\beta{}_j$.
Hence we have
\begin{align*}
\Theta^i&=dv^i{}_\alpha\wedge du^\alpha+v^i{}_\alpha du^\alpha{}_\beta\wedge v^\beta{}_\gamma du^\gamma-v^i{}_\alpha u^\alpha{}_{\gamma\beta}v^\beta{}_\delta du^\delta\wedge v^\gamma{}_\epsilon du^\epsilon\\*
&=-v^i{}_\alpha du^\alpha{}_\beta v^\beta{}_\gamma\wedge du^\gamma+v^i{}_\alpha du^\alpha{}_\beta\wedge v^\beta{}_\gamma du^\gamma+v^i{}_\alpha u^\alpha{}_{\beta_1\beta_2}v^{\beta_1}{}_{\gamma_1}du^{\gamma_1}\wedge v^{\beta_2}{}_{\gamma_2}du^{\gamma_2}\\*
&=v^i{}_\alpha u^\alpha{}_{\beta_1\beta_2}v^{\beta_1}{}_{\gamma_1}du^{\gamma_1}\wedge v^{\beta_2}{}_{\gamma_2}du^{\gamma_2}.
\end{align*}
Therefore, $\Theta_u=0$ if and only if $u^\alpha{}_{\beta_1\beta_2}=u^\alpha{}_{\beta_2\beta_1}$, that is, $u\in P^2$.
\end{proof}
\begin{remark}
We have a kind of split exact sequence
\[
\xymatrix@1@C=30pt@M=6pt{
0 \ar[r] & P^2 \ar@<0.5ex>[r]^{\epsilon} & \widetilde{P}^2 \ar@<0,5ex>[l]^{\kappa} \ar[r]^-{\Theta} & M\times\mathbb{R}^n \ar[r] & 0.
}
\]
A generalization of Theorem~\ref{thm2.6} is given as Theorem~\ref{thm5.7}.
\end{remark}
The curvature $\Omega$ of $\theta$ is calculated as follows.
\begin{lemma}
We locally have
\begin{align*}
\Omega^i{}_j&=-v^i{}_\alpha du^\alpha{}_{j\beta}\wedge v^\beta{}_\gamma du^\gamma\\*
&\hphantom{{}={}}
+v^i{}_\alpha u^\alpha{}_{j\beta}v^\beta{}_\gamma du^\gamma{}_\delta\wedge v^\delta{}_\epsilon du^\epsilon
+v^i{}_\alpha u^\alpha{}_{\beta_1\beta_2}v^{\beta_1}{}_{\gamma_1}du^{\gamma_1}{}_j\wedge v^{\beta_2}{}_{\gamma_2}du^{\gamma_2}\\*
&\hphantom{{}={}}
-v^i{}_\alpha u^\alpha{}_{\beta_1\beta_2}v^{\beta_1}{}_{\gamma_1}u^{\gamma_1}{}_{j\delta}v^\delta{}_\epsilon du^\epsilon\wedge v^{\beta_2}{}_{\gamma_2}du^{\gamma_2}.
\end{align*}
\end{lemma}
\begin{proof}
We have
\begin{align*}
\Omega^i{}_j&=dv^i{}_\alpha\wedge du^\alpha{}_j\\*
&\hphantom{{}={}}
-dv^i{}_\alpha u^\alpha{}_{j\beta}v^\beta{}_\gamma\wedge du^\gamma-v^i{}_\alpha du^\alpha{}_{j\beta}v^\beta{}_\gamma\wedge du^\gamma-v^i{}_\alpha u^\alpha{}_{j\beta}dv^\beta{}_\gamma\wedge du^\gamma\\*
&\hphantom{{}={}}
+v^i{}_\alpha du^\alpha{}_\beta\wedge v^\beta{}_\gamma du^\gamma{}_j\\*
&\hphantom{{}={}}
-v^i{}_\alpha du^\alpha{}_\beta\wedge v^\beta{}_\gamma u^\gamma{}_{j\delta}v^\delta{}_\epsilon du^\epsilon-v^i{}_\alpha u^\alpha{}_{\beta_2\beta_1}v^{\beta_1}{}_{\gamma_1}du^{\gamma_1}\wedge v^{\beta_2}{}_{\gamma_2}du^{\gamma_2}{}_j\\*
&\hphantom{{}={}}
+v^i{}_\alpha u^\alpha{}_{\beta_2\beta_1}v^{\beta_1}{}_{\gamma_1}du^{\gamma_1}\wedge v^{\beta_2}{}_{\gamma_2}u^{\gamma_2}{}_{j\delta_2}v^{\delta_2}{}_{\epsilon_2}du^{\epsilon_2}.
\end{align*}
The first and the fifth, the second and the sixth terms cancel each other.
By rearranging the indices, we obtain the result.
\end{proof}
\begin{remark}
Both the torsion $\Theta$ and the curvature $\Omega$ involve $\theta^0$ rather than $du^i$.
We do not have characterizations of the curvature, however, it is related with the torsion of order $2$.
See Example~\ref{ex5.3}.
On the other hand, torsions have a clear meaning.
See Theorem~\ref{thm5.7}.
\end{remark}
\begin{remark}
We will introduce Lie groups $\widetilde{G}_n^r$ in Section~\ref{sec_Higheroder}, and there will be natural projections from $\widetilde{G}_n^r$ to $\widetilde{G}_n^{r-1}$.
On the other hand, we have an inclusion $\mathrm{GL}_n(\mathbb{R})$ into $\widetilde{G}_n^2$ defined by $a=(a^i{}_j)\mapsto(a^i{}_j,0)$.
If $u=(u^i,u^i{}_j,u^i{}_{jk})$ are the natural coordinates associated with a chart and if $a\in\mathrm{GL}_n(\mathbb{R})$, then we have $u.a=(u^i,u^i{}_la^l{}_j,u^i{}_{lm}a^l{}_ja^m{}_k)$ by~\eqref{eqF4}.
\end{remark}
We recall the notion of connections.
\begin{definition}
A $\mathfrak{gl}_n(\mathbb{R})$-valued $1$-form $\theta$ on $P^1$ is said to be a \textit{connection} if we have the following:
\begin{enumerate}
\item
If $a\in\mathrm{GL}_n(\mathbb{R})$, then $R_a^*\theta=\Ad_{a^{-1}}\omega$.
\item
If $X\in\mathfrak{gl}_n(\mathbb{R})$ and if $X^*$ denotes the fundamental vector field associated with $X$, then $\theta(X^*)=X$.
\end{enumerate}
\end{definition}
As in the classical cases, we have the following
\begin{theorem}[cf. Proposition~7.1 (p.~147) of~\cite{K}, Theorems~2 and~3 of~\cite{Garcia}]
\label{thmG11}
\ \linebreak
There is a one to one correspondence between the following objects\/\textup{:}
\begin{enumerate}
\item
Connections on $P^1$.
\item
Sections of $\pi^2_1\colon\widetilde{P}^2\to P^1$ which are equivariant under the $\mathrm{GL}_n(\mathbb{R})$-actions.
\item
Sections of $\widetilde{P}^2/\mathrm{GL}_n(\mathbb{R})\to M$.
\end{enumerate}
\end{theorem}
Before proving Theorem~\ref{thmG11}, we show the following
\begin{lemma}
\label{lemG12}
Sections of $\widetilde{P}^2/\mathrm{GL}_n(\mathbb{R})\to M$ is in one to one correspondence between $\mathrm{GL}_n(\mathbb{R})$-equivariant mappings from $\widetilde{P}^2$ to $\widetilde{G}_n^2/\mathrm{GL}_n(\mathbb{R})$, where $\widetilde{G}_n^2$ acts on $\widetilde{G}_n^2/\mathrm{GL}_n(\mathbb{R})$ on the right by $[g].a=[a^{-1}g]$.
\end{lemma}
\begin{proof}
We repeat a proof in Husem\"oller~\cite{Husemoller} for convenience.
First note that $\widetilde{P}^2/\mathrm{GL}_n(\mathbb{R})=\widetilde{P}^2\times_{\widetilde{G}_n^2}\widetilde{G}_n^2/\mathrm{GL}_n(\mathbb{R})$.
We represent elements of $\widetilde{P}^2\times_{\widetilde{G}_n^2}\widetilde{G}_n^2/\mathrm{GL}_n(\mathbb{R})$ as $[u,\alpha]$.
Let $\sigma\colon M\to\widetilde{P}^2/\mathrm{GL}_n(\mathbb{R})$ be a section.
If $u\in\widetilde{P}^2$, there uniquely exists an element, say $\alpha(u)\in\widetilde{G}_n^2/\mathrm{GL}_n(\mathbb{R})$, such that $\sigma(\pi^2(u))=[u,\alpha(u)]$.
If $a\in\widetilde{G}_n^2$, then we have $[u,\alpha(u)]=[u.a,\alpha(u).a]$.
On the other hand, we have $\sigma(\pi^2(u))=\sigma(\pi^2(u.a))=[u.a,\alpha(u.a)]$ so that $\alpha(u.a)=\alpha(u).a$.
Suppose conversely that $\alpha$ is given.
If $p\in M$, then we choose $u\in(\pi^2)^{-1}(p)$ and set $\sigma(p)=[u,\alpha(u)]\in\widetilde{P}^2/\mathrm{GL}_n(\mathbb{R})$.
If $a\in\widetilde{G}_n^2$, then we have $[u.a,\alpha(u.a)]=[u.a,\alpha(u).a]=[u,\alpha(u)]$ so that $\sigma$ is a well-defined section.
It is easy to see that this correspondence is one~to~one.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thmG11}]
The proof is almost identical to that of Proposition~7.1 of \cite{K}.
First let $\omega$ be a connection.
Let $(U,\varphi)$ be a chart and consider the associated natural coordinates $(u^i,u^i{}_j,u^i{}_{jk})$.
Then, $(u^i,u^i{}_j)$ are local coordinates for $P^1$.
Let $s$ be the local trivialization of $P^1$ which corresponds to these coordinates and set $\mu=s^*\omega$.
We represent $\mu=\pdif{}{x^i}\Gamma^i{}_{jk}dx^k$ using the Christoffel symbols.
Note that our symbol differs from the usual one, that is, the order of lower indices are reversed.
We set $\sigma(u^i,u^i{}_j)=(u^i,u^i{}_j,-\Gamma^i{}_{\alpha\beta}u^\alpha{}_ju^\beta{}_k)$.
If we set $\psi=(D\phi)^{-1}$, then we have
\stepcounter{theorem}
\[
\widehat{\Gamma}^i{}_{jk}=-H\phi^i{}_{\alpha\beta}\psi^\alpha{}_j\psi^\beta{}_k+D\phi^i{}_\alpha\Gamma^\alpha{}_{\beta\gamma}\psi^\beta{}_j\psi^\gamma{}_k.
\tag{\thetheorem}
\label{eq2.14}
\]
Hence we have
\begin{align*}
\widehat{\Gamma}^i{}_{\alpha\beta}\widehat{u}^\alpha{}_j\widehat{u}^\beta{}_k
&=-H\phi^i{}_{\alpha\beta}\psi^\alpha{}_\gamma\widehat{u}^\gamma{}_j\psi^\beta{}_\delta\widehat{u}^\delta{}_k+D\phi^i{}_\alpha\Gamma^\alpha{}_{\beta\gamma}\psi^\beta{}_\delta\widehat{u}^\delta{}_j\psi^\gamma{}_\epsilon\widehat{u}^\epsilon{}_k\\*
&=-H\phi^i{}_{\alpha\beta}u^\alpha{}_ju^\beta{}_k+D\phi^i{}_\alpha\Gamma^\alpha{}_{\beta\gamma}u^\beta{}_ju^\gamma{}_k.
\end{align*}
It follows that
\[
(\widehat{u}^i{}_j,-\widehat{\Gamma}^i{}_{\alpha\beta}\widehat{u}^\alpha{}_j\widehat{u}^\beta{}_k)=(D\phi^i{}_j,H\phi^i{}_{jk})(u^i{}_j,-\Gamma^i{}_{\alpha\beta}u^\alpha{}_ju^\beta{}_k).
\]
Therefore, locally defined $\sigma$ gives rise to a well-defined section of $\pi^2_1$ (cf.~\eqref{eqF8}).
If $a\in\mathrm{GL}_n(\mathbb{R})$, then we have
\begin{align*}
\sigma((u^i,u^i{}_j).a)&=\sigma(u^i,u^i{}_\alpha a^\alpha{}_j)\\*
&=(u^i,u^i{}_\alpha a^\alpha{}_j,-\Gamma^i{}_{\alpha\beta}u^\alpha{}_\gamma u^\beta{}_\delta a^\gamma{}_ja^\delta{}_k)\\*
&=\sigma(u^i,u^i{}_j).a
\end{align*}
so that the section is $\mathrm{GL}_n(\mathbb{R})$-equivariant.
Conversely, if $s$ is a $\mathrm{GL}_n(\mathbb{R})$-equivariant section of $\pi^2_1$, then $s^*\theta^2$ is a connection.
Next, let $\sigma\colon M\to\widetilde{P}^2/\mathrm{GL}_n(\mathbb{R})$ be a section.
By Lemma~\ref{lemG12}, $\sigma$ corresponds to a $\widetilde{G}_n^2$-equivariant map, say $\alpha$, from $\widetilde{P}^2$ to $\widetilde{G}_n^2/\mathrm{GL}_n(\mathbb{R})$.
We set $P=\alpha^{-1}([e])$, where $e\in\widetilde{G}_n^2$ denotes the unit.
Then $P$ is naturally a principal $\mathrm{GL}_n(\mathbb{R})$-bundle and the inclusion gives the desired section.
As $\widetilde{G}_n^2/\mathrm{GL}_n(\mathbb{R})\cong\mathbb{R}^{n^3}$ is contractible, thus obtained $P$ is isomorphic to each other.
Since there is a connection on $P^1$, we have a section from $P^1\to\widetilde{P}^2$ so that $P$ is isomorphic to $P^1$.
Conversely, let $\sigma\colon P^1\to\widetilde{P}^2$ be a $\mathrm{GL}_n(\mathbb{R})$-equivariant section.
By taking the quotients by $\mathrm{GL}_n(\mathbb{R})$-actions, we obtain a section $M\to\widetilde{P}^2/\mathrm{GL}_n(\mathbb{R})$.
We omit to show that these correspondences are one~to~one.
\end{proof}
\begin{remark}
Connections on $P^1$ correspond to affine connections on $TM$, and $\mathrm{GL}_n(\mathbb{R})$-equivariant sections of $\pi^2_1$ correspond to reductions of $\widetilde{P}^2$ to $\mathrm{GL}_n(\mathbb{R})$.
\end{remark}
\begin{remark}
The section associated with a connection corresponds to the geodesic equation although the lower indices of the Christoffel symbols need not commute.
\end{remark}
\begin{remark}
If $\nabla$ is a linear connection, then we can find a torsion free connection by modifying $\nabla$~\cite{KN}*{Proposition 7.9 (Chapter~3)}.
We can interpret this procedure as considering $\kappa_*\nabla$ instead of $\nabla$, where $\kappa_*\colon\widetilde{\mathfrak{g}}_n^2\to\mathfrak{g}_n^2$.
\end{remark}
\section{Comparison with Garc\'\i a's construction}
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$.
Let $M$ be a manifold and $\pi\colon P\to M$ a principal $G$-bundle over $M$.
Gac\'\i a gives in~\cite{Garcia} canonical forms on $1$-jet bundles associated with principal $G$-bundles.
\begin{definition}
Let $\mathcal{J}(P)$ the bundle of $1$-jets of germs of sections from $M$ to $P$.
If $s_p\in\mathcal{J}(P)$ is a jet at $p\in M$, then we set $\pi^1(s_p)=s_p(p)$ and $\overline{\pi}^1(s_p)=p$.
The bundle $\mathcal{J}(P)$ over $P$ is called the \textit{$\mathit{1}$-jet bundle} of $P$.
\end{definition}
Let $(U,\varphi)$ be a chart on $M$.
We assume that $P$ is trivial on $U$ and let $\psi\colon\pi^{-1}(U)\to U\times G$ be a trivialization.
Let $s$ be a section of $\pi\colon P\to M$ on $U\subset M$.
We represent $s$ by $\psi\circ s\circ\varphi^{-1}$ and $j^1_p(s)$ by $j^1_x(\psi\circ s\circ\varphi^{-1})$, where $x=\varphi(p)\in\varphi(U)$.
More concretely, let $\psi\circ s\circ\varphi^{-1}=({\mathrm{id}},h)$, where $h$ is a $G$-valued function on $\varphi(U)$.
Then, $j^1_p(s)$ is represented by $(x,h(x),Dh(x))$, which are the coordinates used by Garc\'\i a.
The following is obvious.
\begin{lemma}
The bundle $\mathcal{J}(P)$ admits a natural $G$-action on the right.
\end{lemma}
Let $u\in P$, $Y\in T_uP$ and suppose that $\pi_*Y=0$.
Then, there uniquely exists an element $Z\in\mathfrak{g}$ such that $Z^*_u=Y$, where $Z^*$ denotes the fundamental vector field associated with $Z$.
We represent $Z$ by $u^{-1}Y$.
\begin{definition}[Canonical form]
Let $p\in M$, $j^1_p(s)\in\mathcal{J}(P)$ and $X\in T_{j^1_p(s)}\mathcal{J}(P)$.
We set
\begin{align*}
d^\nu{}_{j^1_p(s)}X&=\pi^1{}_*X-s_*\overline{\pi}^1_*X,\\*
\theta_{j^1_p(s)}(X)&=s(p)^{-1}(d^\nu{}_{j^1_p(s)}X).
\end{align*}
We call $\theta$ the \textit{canonical form} on $\mathcal{J}(P)$.
\end{definition}
\begin{proposition}[cf. Theorem~\ref{thmG2}]
We have the following\textup{:}
\begin{enumerate}[\textup{\theenumi)}]
\item
If $g\in G$, then $R_g^*\theta=\Ad_{g^{-1}}\theta$.
\item
If $X\in\mathfrak{g}$ and if $X^*$ denotes the fundamental vector field associated with $X$, then $\theta(X^*)=\pi(X)$.
\end{enumerate}
\end{proposition}
In what follows, we assume that $\dim M=n$ and that $P$ is the frame bundle of $M$.
We have $G=\mathrm{GL}_n(\mathbb{R})$.
It might happen that $P$ admits a reduction, that is, $G$ might be a proper Lie subgroup of $G$.
Some of arguments work in such cases.
Let $j^2_o(f)\in P^2$ and $p=f(o)$.
It is easy to see that $j^1_p(Df\circ f^{-1})$ is well-defined.
In general, if $j^1_o(F)\in\widetilde{P}^2$, then $j^1_p(F\circ f^{-1})$ is also well-defined, where $p=f(o)$.
We set $\Phi(j^1_o(F))=j^1_p(F\circ f^{-1})$.
Conversely, let $p\in M$ and $s_p\in\mathcal{J}(P)$.
We regard $s(p)$ as a linear map from $T_o\mathbb{R}^n$ to $T_pM$ and choose a local diffeomorphism $f\colon(\mathbb{R}^n,o)\to(M,p)$ such that $Df(o)=s(p)$.
We consider the pair $(f,s\circ f)$ as a local isomorphism from $T(\mathbb{R}^n,o)$ to $T(M,p)$ and set $\Psi(s_p)=j^1_o(f,s\circ f)$.
\begin{lemma}
\label{lem3.5}
The mapping $\Psi$ is well-defined and we have $\Psi=\Phi^{-1}$.
\end{lemma}
\begin{proof}
First, $j^1_o(f,s\circ f)\in\widetilde{P}^2$ because $s\circ f(o)=s(p)=Df(o)$.
We have $j^1_o(f,s\circ f)=(f(o),s\circ f(o),Ds(f(o))Df(o))=(p,s(p),Ds(p))$ so that $\Psi$ is well-defined.
Finally, we have $\Psi=\Phi^{-1}$ by the definitions.
\end{proof}
If $(U,\varphi)$ is a chart about $p$, then mappings $\Phi$ and $\Psi$ are represented as follows.
First, if $j^1_o(F)\in\widetilde{P}^2$, then we set $F_\varphi=D\varphi\circ F$ and $f_\varphi=\varphi\circ f$.
If we set $q=\varphi(p)$, then $j^1_o(F)$ is represented by $(q,F_\varphi(o),DF_\varphi(o))$.
On the other hand, $j^1_p(F\circ f^{-1})$ is represented~by
\begin{align*}
j^1_q(F_\varphi\circ f_\varphi{}^{-1})&=(q,F_\varphi(o),DF_\varphi(o)(Df_\varphi(o))^{-1})\\*
&=(q,F_\varphi(o),DF_\varphi(o)^i{}_{j\alpha}((F_\varphi(o))^{-1})^\alpha{}_k)
\end{align*}
in the Garc\'\i a coordinates.
Therefore, if we make use of the natural coordinates on $\widetilde{P}^2$ and the Garc\'\i a coordinates on $\mathcal{J}(P)$, then we have
\begin{align*}
\Phi(u^i,u^i{}_j,u^i{}_{jk})&=(u^i,u^i{}_j,u^i{}_{jl}v^l{}_k),\\*
\Psi(x^i,y^i{}_j,z^i{}_{jk})&=(x^i,y^i{}_j,z^i{}_{jl}y^l{}_k),
\end{align*}
where $(v^i{}_j)=(u^i{}_j)^{-1}$.
On the other hand, $\widetilde{G}_n^2$ acts on $\mathcal{J}(P)$ on the right by Lemma~\ref{lem3.5}.
Indeed, if $s_p\in\mathcal{J}(P)$ and if $a\in\widetilde{G}_n^2$, then we can set $(s_p).a=\Phi(\Psi(s_p).a)$.
In the Garc\'\i a coordinates, we have
\[
(x^i,y^i{}_j,z^i{}_{jk})(a^i{}_j,a^i{}_{jk})=
(x^i,y^i{}_\alpha a^\alpha{}_j,z^i{}_{\alpha k}a^\alpha{}_j+y^i{}_\alpha a^\alpha{}_{j\beta}b^\beta{}_\gamma w^\gamma{}_k)
\]
where $(w^i{}_j)=(y^i{}_j)^{-1}$ and $(b^i{}_j)=(a^i{}_j)^{-1}$.
If $a\in\mathrm{GL}_n(\mathbb{R})$, namely, if $a^i{}_{jk}=0$, then we have
\[
(x^i,y^i{}_j,z^i{}_{jk})(a^i{}_j,a^i{}_{jk})=(x^i,y^i{}_\alpha a^\alpha{}_j,z^i{}_{\alpha k}a^\alpha{}_j).
\]
Hence the $\widetilde{G}_n^2$-action on $\mathcal{J}(P)$ is an extension of the $\mathrm{GL}_n(\mathbb{R})$-action.
The following is known.
We modify notations fitting to ours.
\begin{lemma}[\cite{Garcia}*{Section~3 and Lemma~1}]
Let $\theta'$ be the canonical form on $\mathcal{J}(P)$.
We have
\[
\theta'=w^i{}_\alpha(dy^\alpha{}_j-y^\alpha{}_{j\beta}dx^\beta).
\]
\end{lemma}
Therefore, we have the following.
\begin{theorem}
\begin{enumerate}
\item
The mapping $\Phi$ is an isomorphism of $\mathrm{GL}_n(\mathbb{R})$-bundles.
\item
If we define a $\widetilde{G}_n^2$-action on $\mathcal{J}(P)$ as above, then $\mathcal{J}(P)$ is a principal $\widetilde{G}_n^2$-bundle which is compatible the original $\mathrm{GL}_n(\mathbb{R})$-action on~$\mathcal{J}(P)$.
Moreover, $\Phi$ is an isomorphism of $\widetilde{G}_n^2$-bundles under these actions.
\item
If $\theta^1$ and $\theta'$ denote the canonical form of order~$2$ on $\widetilde{P}^2$ and the canonical form on $\mathcal{J}(P)$, respectively, then we have $\Phi^*\theta'=\theta^1$.
\end{enumerate}
\end{theorem}
\section{Bundles of formal frames of higher order}
\label{sec_Higheroder}
We can consider analogues of $P^r$, $r\geq3$.
We set $\widetilde{P}^0=M$, $\widetilde{P}^1=P^1$ and let $\pi^1_0$ denote the projection from $P^1$ to $M$.
Suppose that groups $\widetilde{G}^k_n$, $\widetilde{G}^k_n$-bundles $\widetilde{P}^k$ such that $\widetilde{G}^k_n=\widetilde{P}^k(\mathbb{R}^n,o)$, and projections $\pi^k_{k-1}\colon\widetilde{P}^k\to\widetilde{P}^{k-1}$ equivariant under the $\widetilde{G}^k_n$ and $\widetilde{G}^{k-1}_n$ actions are defined up to $k=r-1$.
This holds true for $r=2$.
Let $F=F^{r-1}\colon\widetilde{P}^{r-1}(\mathbb{R}^n,o)\to\widetilde{P}^{r-1}(M)$ be a locally defined isomorphism of $\widetilde{G}^{r-1}$-bundles and $F^k\colon\widetilde{P}^k(\mathbb{R}^n,o)\to\widetilde{P}^k(M)$, where $0\leq k\leq r-2$, be the underlying isomorphism in the sense that $F^0,F^1,\ldots,F^{r-1}$ are bundle isomorphisms and that
\[
\begin{CD}
\widetilde{P}^k(\mathbb{R}^n,o) @>{F^k}>> \widetilde{P}^k(M)\\
@V{\pi^k_l}VV @VV{\pi^k_l}V\\
\widetilde{P}^l(\mathbb{R}^n,o) @>>{F^l}> \widetilde{P}^l(M)\rlap{,}
\end{CD}
\]
where $\pi^k_l=\pi^{l+1}_l\circ\cdots\circ\pi^k_{k-1}$, is commutative for $1\leq l<k\leq r-1$.
\begin{definition}[cf. Definition~\ref{def1.6}]
We set
\begin{align*}
\widetilde{P}^r(M)&=\left\{j^1_o(F)\;\middle|\;\parbox[c]{160pt}{$F\colon\widetilde{P}^{r-1}(\mathbb{R}^n,o)\to\widetilde{P}^{r-1}(M)$,\\ $F^k(o)=j^1_o(F^{k-1})$ for $1\leq k\leq r$}\right\},\\*
\widetilde{G}_n^r&=\widetilde{P}^r(\mathbb{R}^n,o).
\end{align*}
We call $\widetilde{P}^r(M)$ as the \textit{bundle of formal frames} of order $r$, and elements of $\widetilde{P}^r(M)$ \textit{formal frames} of order $r$.
If $F\in\widetilde{P}^r(M)$, then we set $\pi^r_{r-1}(j^1_o(F))=F(o)$.
\end{definition}
We call formal frames of order greater than two as \textit{formal frames of higher order}.
We have the following.
The proof is easy and omitted.
\begin{lemma}
\begin{enumerate}
\item
Thus defined $\widetilde{G}_n^r$ is a Lie group of dimension $\sum_{l=1}^rn^l$.
\item
If we set $\pi^r=\pi^{r-1}\circ\pi^r_{r-1}$, then $\pi^r\colon\widetilde{P}^r(M)\to M$ is a principal $\widetilde{G}_n^r$-bundle.
\end{enumerate}
\end{lemma}
\begin{definition}
If $u=j^r_o(f)\in P^r(M)$, then we set $\epsilon(u)=j^1_o(D^{r-1}(f))$.
\end{definition}
Note that $\epsilon$ gives rise to an embedding of $G_n^r$ into $\widetilde{G}_n^r$.
\begin{definition}[cf. Definition~\ref{defG1}]
\label{def4.4}
Let $u\in\widetilde{P}^r(M)$ and $X\in T_u\widetilde{P}^r(M)$.
We set $\theta(X)=u^{-1}(\pi^r_{r-1*}X)\in T_{{\mathrm{id}}}\widetilde{P}^{r-1}(\mathbb{R}^n)$, where $u\colon T_{{\mathrm{id}}}\widetilde{P}^{r-1}(\mathbb{R})\to T_{\pi^r_{r-1}(u)}\widetilde{P}^{r-1}(M)$ is the isomorphism induced from the right action (cf. Definition~\ref{defF11}).
We call $\theta$ the \textit{canonical form}.
The canonical form is naturally represented as $(\theta^i,\theta^i{}_{j_1},\ldots,\theta^i{}_{j_1,\ldots,j_{r-1}})$.
We refer to $(\theta^i{}_{j_1,\ldots,j_{r-1}})$ as the canonical form of order $r$.
\end{definition}
The adjoint action of $\widetilde{G}_n^r$ on $T_{{\mathrm{id}}}\widetilde{P}^{r-1}(\mathbb{R}^n)$ is defined in a similar way as in the case of $r=2$.
\begin{theorem}[cf. Theorem~\ref{thmG2}]
We have the following.
\begin{enumerate}
\item
If $a\in\widetilde{G}_n^r$, then $R_a^*\theta=\Ad_{a^{-1}}\theta$.
\item
If $X\in\widetilde{\mathfrak{g}}_n^r$, then $\theta^r(X^*)=\pi^r_{r-1}{}_*X$, where $X^*$ denotes the fundamental vector field associated with $X$.
\end{enumerate}
\end{theorem}
The product in $\widetilde{G}_n^r$ remains similar to that of the group of $r$-frames $G_n^r$.
For example, if $r=3$, the product is given as follows.
Let $a,b\in\widetilde{G}_n^3$ and $a=j^1_o(F)$, $b=j^1_o(G)$.
Then, $ab=j^1_o(F\circ G)$.
If we represent $a$ as $a=(a^i{}_j,a^i{}_{jk},a^i{}_{jkl})$ and so on, we have
\begin{align*}
(ab)^i{}_j&=a^i{}_\alpha b^\alpha{}_j,\\*
(ab)^i{}_{jk}&=a^i{}_{\alpha\beta}b^\alpha{}_jb^\beta{}_k+a^i{}_\alpha b^\alpha{}_{jk},\\*
(ab)^i{}_{jkl}&=a^i{}_{\alpha\beta\gamma}b^{\alpha}{}_jb^{\beta}{}_kb^{\gamma}{}_l+a^i{}_{\alpha\beta}b^{\alpha}{}_{jl}b^{\beta}{}_k+a^i{}_{\alpha\beta}b^\alpha{}_jb^\beta{}_{kl}+a^i{}_{\alpha\beta}b^\alpha{}_{jk}b^\beta{}_l+a^i{}_{\alpha}b^{\alpha}{}_{jkl}.
\end{align*}
These formulae are obtained as follows.
Let $a,b\in\widetilde{G}_n^3$ and $a=j^1_o(F), b=j^1_o(G)$.
First, we represent $F(x)=(f^i(x),f^i{}_j(x),f^i{}_{jk}(x),f^i{}_{jkl}(x))$.
We have $F^0(x)=(f^i(x))$, $F^1(x)=(f^i(x),f^i{}_j(x))$, $F^2(x)=(f^i(x),f^i{}_j(x),f^i{}_{jk}(x))$ and $F^3(x)=F(x)$.
By the conditions required to $F$, we have
\begin{align*}
a^i{}_j&=\pdif{f^i}{x^j}(o)=f^i{}_j(o),\\*
a^i{}_{jk}&=\pdif{f^i{}_j}{x^k}(o)=f^i{}_{jk}(o),\\*
a^i{}_{jkl}&=\pdif{f^i{}_{jk}}{x^l}(o).
\end{align*}
We have $(F\circ G)^0(x)=f^i(g^m(x))$ so that $(ab)^i{}_j=a^i{}_\alpha b^\alpha{}_j$ holds in $\widetilde{G}_n^1$.
Hence we may assume that $(F\circ G)^1(x)=(f^i(g^m(x)),f^i{}_\alpha(g^m(x))g^\alpha{}_j(x))$ because we are concerned with jets at $o$.
It follows that
\[
(ab)^i{}_{jk}=(a^i{}_{\alpha\beta}b^\alpha{}_jb^\beta{}_k+a^i{}_\alpha b^\alpha{}_{jk})
\]
holds in $\widetilde{G}_n^2$.
Similarly, we may assume that
\begin{align*}
(F\circ G)^2(x)
&=(f^i(g^m(x)),f^i{}_\alpha(g^m(x))g^\alpha{}_j(x),\\*
&\hphantom{{}={}}\quad f^i{}_{\alpha\beta}(g^m(x))g^\alpha{}_j(x)g^\beta{}_k(x)+f^i{}_\alpha(g^m(x))g^\alpha{}_{jk}(x)).
\end{align*}
Hence we have
\begin{align*}
(ab)^i{}_{jkl}&=a^i{}_{\alpha\beta\gamma}b^\alpha{}_jb^\beta{}_kb^\gamma{}_l+a^i{}_{\alpha\beta}b^\alpha{}_{jl}b^\gamma{}_k+a^i{}_{\alpha\beta}b^\alpha{}_jb^\gamma{}_{kl}+a^i{}_{\alpha\beta}b^\beta{}_lb^\gamma{}_{jk}+a^i{}_\alpha b^\alpha{}_{jkl}
\end{align*}
in $\widetilde{G}_n^3$.
Note that if $F$ is actually derived from a local diffeomorphism, then we have $F(x)=a^i{}_jx^j+\frac12a^i{}_{jk}x^jx^k+\frac16a^i{}_{jkl}x^jx^kx^l+(\text{terms of order greater than $3$})$.
We come back to the bundle $\widetilde{P}^r$ and the canonical form on it.
We have the following
\begin{theorem}[cf.~Theorem~\ref{thmG3}]
\label{thm4.6}
Let $M,N$ be manifolds and $f\colon M\to N$ be a local diffeomorphism.
\begin{enumerate}
\item
We have $f^*\widetilde{P}^r(N)=\widetilde{P}^r(M)$.
\item
If $\theta_M,\theta_N$ denote the canonical forms on $M,N$, then we have $f^*\theta_N=\theta_M$.
\end{enumerate}
\end{theorem}
We can normalize elements of $\widetilde{P}^r(M)$ and $\widetilde{G}_n^r$ as follows (cf.~Theorem~\ref{thm1.13}).
Let $u=j^1_o(F)\in\widetilde{P}^r(M)$.
Let $(U,\varphi)$ be a chart about $\pi^r(u)$ and represent $u$ as $(h^i,h^i{}_{j_1},\ldots,h^i{}_{j_1,\ldots,j_r})$.
We set
\[
\overline{h}^i{}_{j_1,\ldots,j_k}=\frac1{k!}\sum_{\sigma\in\mathfrak{S}_k}h^i{}_{j_{\sigma(1)},\ldots,j_{\sigma(k)}},
\]
where $\overline{h}^i{}_{j_1,\ldots,j_0}$ is understood to be $\overline{h}^i=h^i$, and
\[
\overline{F}^0(x)=\sum_{k=0}^r\overline{h}^i{}_{j_1,\ldots,j_k}x^{j_1}\cdots x^{j_k}.
\]
Actually, we only consider the $r$-jet~of~$\overline{F}^0$ in what follows.
\begin{definition}
We set $\kappa(u)=j^r_o(\overline{F}^0)$.
\end{definition}
\begin{theorem}
\begin{enumerate}
\item
The mapping $\epsilon\colon P^r\to\widetilde{P}^r$ is well-defined and is an embedding of\/ $G_n^r$-bundles.
Moreover, we have $\widetilde{P}^r=\epsilon(P^r)\times_{G_n^r}\widetilde{G}_n^r$.
\item
The mapping $\kappa\colon\widetilde{P}^r\to P^r$ is well-defined bundle morphism as $G_n^r$-bundles.
\item
When regarded as a mapping from $\widetilde{G}_n^r$ to $G_n^r$, $\kappa$ is a homomorphism.
\end{enumerate}
\end{theorem}
\begin{proof}
We only show that the $r$-jet of $\overline{F}^0$ is well-defined.
We set $G(x)=h^i+h^i{}_{j_1}x^{j_1}+\frac12h^i{}_{j_1,j_2}+\cdots+\frac1{r!}h^i{}_{j_1,\ldots,j_r}x^{j_1}\cdots x^{j_r}$.
It is clear that $G$ is well-defined if we ignore terms of order higher than $r$.
On the other hand, we have $G(x)=\overline{F}^0(x)$.
\end{proof}
\begin{remark}
\label{rem4.9}
If we consider a Lie subgroup of $\mathrm{GL}_n(\mathbb{R})$, then we have the same kind of difficulties in the averaging process as in the case of $r=2$.
See also Remarks~\ref{rem1.14} and \ref{rem5.9}.
\end{remark}
We refer to \cite{Saunders} for more about jets.
\section{Torsions of higher order}
We begin with the following
\begin{theorem}[Structural equations~\cite{K_str}, \cite{Bott:Notes}]
\label{thm5.4}
Let $S=\{1,\ldots,k\}$.
Let $S_a=\{s_1,\ldots,s_a\}\subset S$ and $\{t_1,\ldots,t_b\}=S\setminus S_a$, where $S_a=\varnothing$ if $a=0$.
Then, on $P^r$, we have
\[
d\theta^i{}_{j_1,\ldots,j_k}+\sum_{S_a\subset S}\sum_{l=1}^n\theta^i{}_{j_{s_1},\ldots,j_{s_a},l}\wedge\theta^l{}_{j_{t_1},\ldots,j_{t_b}}=0.
\]
These equations are referred as the \textup{structural equations}.
\end{theorem}
Contractions appear in the structural equations.
On $P^r$, the lower indices commute so that we do not need to care about the order of indices.
It is not the case for $\widetilde{P}^r$ so that we should be aware of how we take contractions.
\begin{definition}
\label{def5.2}
Let $k\leq r-1$.
Let $p_0=1,p_1,\ldots,p_k\in\mathbb{N}$ be such that $1\leq p_a\leq a+1$ for $0\leq a\leq k$.
Let $S=(j_1,\ldots,j_k)$ be a ordered tuple of indices.
We set
\[
(\Theta^{k+1(p_1,\ldots,p_k)})^i{}_{j_1,\ldots,j_k}=d\theta^i{}_{j_1,\ldots,j_k}+\sum_{S_a\subset S}\sum_{l=1}^n\theta^i{}_{j_{s_1},\ldots,j_{s_{p_a-1}},\underset{\stackrel{\frown}{p_a}}{l},j_{s_{p_a}},\ldots,j_{s_a}}\wedge\theta^l{}_{j_{t_1},\ldots,j_{t_b}},
\]
where $S_a=(j_{s_1},\ldots,j_{s_a})$ is a ordered subset of $S$ and $(j_{t_1},\ldots,j_{t_b})=S\setminus S_a$ as ordered sets.
If $a=0$, then we set $S_0=\varnothing$.
We call $\Theta^{k(p_1,\ldots,p_{k-1})}$ the \textit{torsions} of order $k$ and of type $(p_1,\ldots,p_{k-1})$.
\end{definition}
\begin{example}
\label{ex5.3}
\begin{enumerate}
\item
The only torsion of order $1$ is $\Theta^1$.
\item
The torsions of order $2$ are $\Theta^{2(1)}$ and $\Theta^{2(2)}$.
We have
\begin{align*}
\Theta^{2(1)}&=d\theta^i{}_j+\theta^i{}_{\alpha j}\wedge\theta^\alpha+\theta^i{}_\alpha\wedge\theta^\alpha_j\\*
&=\Omega^i{}_j+\theta^i{}_{\alpha j}\wedge\theta^\alpha,\\*
\Theta^{2(2)}&=d\theta^i{}_j+\theta^i{}_{j\alpha}\wedge\theta^\alpha+\theta^i{}_\alpha\wedge\theta^\alpha{}_j\\*
&=\Omega^i{}_j+\theta^i{}_{j\alpha}\wedge\theta^\alpha.
\end{align*}
\item
The torsions of order $3$ are $\Theta^{3(1,1)}$, $\Theta^{3(2,1)}$, $\Theta^{3(1,2)}$, $\Theta^{3(2,2)}$, $\Theta^{3(1,3)}$ and $\Theta^{3(2,3)}$.
For example, we have
\begin{align*}
(\Theta^{3(1,3)})^i{}_{j_1j_2}&=d\theta^i{}_{j_1j_2}+\theta^i{}_\alpha\wedge\theta^\alpha{}_{j_1j_2}+\theta^i{}_{\alpha j_1}\wedge\theta^\alpha{}_{j_2}+\theta^i{}_{\alpha j_2}\wedge\theta^\alpha{}_{j_1}+\theta^i{}_{j_1j_2\alpha}\wedge\theta^\alpha,\\*
(\Theta^{3(2,2)})^i{}_{j_1j_2}&=d\theta^i{}_{j_1j_2}+\theta^i{}_\alpha\wedge\theta^\alpha{}_{j_1j_2}+\theta^i{}_{j_1\alpha}\wedge\theta^\alpha{}_{j_2}+\theta^i{}_{j_2\alpha}\wedge\theta^\alpha{}_{j_1}+\theta^i{}_{j_1\alpha j_2}\wedge\theta^\alpha.
\end{align*}
\item
In general, the number of the torsions of order $k$ is equal to $k!$.
\end{enumerate}
\end{example}
We have the following
\begin{theorem}
\label{thm5.7}
We have $P^r=\{u\in\widetilde{P}^r\mid\text{all torsions vanish at $u$}\}$.
\end{theorem}
We need some lemmata for proving Theorem~\ref{thm5.7}.
\begin{lemma}
\label{lem5.8}
Let $a=(a^i{}_{j_1},\ldots,a^i{}_{j_1,\ldots,j_r})$, $b=(b^i{}_{j_1},\ldots,b^i{}_{j_1,\ldots,j_r})\in\widetilde{G}_n^r$.
\begin{enumerate}
\item
If\/ $1\leq k\leq r$, then $(ab)^i{}_{j_1,\ldots,j_k}$ is represented by $a^i{}_{j_1,\ldots,j_l},b^i_{j_1,\ldots,j_l}$ with $l\leq k$.
\item
We have
\begin{align*}
(ab)^i{}_{j_1}&=a^i{}_\alpha b^\alpha_j,\\*
(ab)^i{}_{j_1,\ldots,j_r}&=a^i{}_{\alpha_1,\ldots,\alpha_r}b^{\alpha_1}{}_{j_1}\cdots b^{\alpha_r}{}_{j_r}\\*
&\hphantom{{}={}}+(\text{terms which do not involve $a^i{}_{j_1,\ldots,j_r}$ or $b^i{}_{j_1,\ldots,j_r}$})\\*
&\hphantom{{}={}}+a^i{}_\alpha b^\alpha{}_{j_1,\ldots,j_r}.
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}
By the construction, $(ab)^i{}_{j_1,\ldots,j_k}$ is obtained from $(ab)^i{}_{j_1,\ldots,j_{k-1}}$ as follows.
First regard $a,b$ as functions in $x$.
Then, we consider the derivative of $(ab)^i{}_{j_1,\ldots,j_{k-1}}$ with respect to $x$.
After replacing $\pdif{a^i{}_{j_1,\ldots,j_p}}{x^l}$ by $a^i{}_{j_1,\ldots,j_p,\alpha}b^\alpha{}_l$ and $\pdif{b^i{}_{j_1,\ldots,j_p}}{x^l}$ by $b^i{}_{j_1,\ldots,j_p,l}$, we obtain $(ab)^i{}_{j_1,\ldots,j_k}$.
In particular, the number of the lower indices of each terms of $(ab)^i{}_{j_1,\ldots,j_k}$ is equal to $k$.
Hence~1) and the first part of~2) hold.
On the other hand, terms in $(ab)^i{}_{j_1,\ldots,j_r}$ which involve $a^i{}_{j_1,\ldots,j_r}$ appear only if we take the `derivative' of terms in $(ab)^i_{j_1,\ldots,j_{r-1}}$ which involve $a^i{}_{j_1,\ldots,j_{r-1}}$.
In this case, we obtain $a^i{}_{\alpha_1,\ldots,\alpha_r}b^{\alpha_1}{}_{j_1}\cdots b^{\alpha_{r-1}}{}_{j_{r-1}}b^{\alpha_r}{}_{j_r}$.
Similarly, terms in $(ab)^i{}_{j_1,\ldots,j_r}$ which involve $b^i{}_{j_1,\ldots,j_r}$ appear only from the derivative of $a^i{}_\alpha b^\alpha{}_{j_1,\ldots,j_{r-1}}$ and we obtain $a^i{}_\alpha b^\alpha{}_{j_1,\ldots,j_{r-1},j_r}$.
\end{proof}
\begin{lemma}
\label{lem5.9}
Let $\theta=(\theta^0,\theta^1,\ldots,\theta^{r-1})$ be the canonical form on $\widetilde{P}^r$, where $r\geq2$.
In the natural coordinates \textup{(}Notation~\ref{not1.6}\textup{)}, we have
\begin{align*}
(\theta^{r-1})^i{}_{j_1,\ldots,j_{r-1}}&=-v^i{}_\alpha u^\alpha{}_{j_1,j_2,\ldots,j_{r-1}\beta}v^\beta{}_\gamma du^\gamma\\*
&\hphantom{{}={}}
\quad+(\text{terms which involve only $v^i{}_j$ and $u^i{}_{j_1,\ldots,j_l}$ with $l\leq r-2$}).
\end{align*}
\end{lemma}
\begin{proof}
Let $u\in\widetilde{P}^r$, $X\in T_u\widetilde{P}^r$ and $Y=\theta_u(X)\in T_{{\mathrm{id}}}\widetilde{P}^{r-1}(\mathbb{R}^n)$.
If we represent $Y$ as $Y=(Y^0,\ldots,Y^{r-1})$ using the natural coordinates, then, we have by Lemma~\ref{lem5.8}~that
\begin{align*}
&\hphantom{{}={}}
((uY)^{r-1})^i{}_{\alpha_1,\ldots,\alpha_{r-1}}\\*
&=u^i{}_{\alpha_1,\ldots,\alpha_{r-1},l}Y^l\\*
&\hphantom{{}={}}
+(\textit{terms which do not involve $u^i{}_{j_1,\ldots,j_r}$ or $Y^i{}_{j_1,\ldots,j_{r-1}}$})\\*
&\hphantom{{}={}}
+u^i{}_lY^l{}_{\alpha_1,\ldots,\alpha_{r-1}},
\end{align*}
where $Y^0=(Y^i)$ and $Y^{r-1}=(Y^i{}_{j_1,\ldots,j_{r-1}})$.
Since $Y^i=\theta^0(X)=v^i{}_\alpha X^\alpha$, we are done (see also~\eqref{eqF10-2} and Lemma~\ref{lem2.6}).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm5.7}]
Suppose that torsions of order less than $r$ vanish.
We may assume inductively that the lower indices of $u^i{}_{j_1,\ldots,j_k}$ commute for $k\leq r-1$.
By Lemma~\ref{lem5.9}, the only terms of $\Theta^{r(p_1,\ldots,p_{r-1})}$ which involve $u^i{}_{j_1,\ldots,j_r}$ is derived from $\theta^r\wedge\theta^0$, and we have
\begin{align*}
(\Theta^{r(p_1,\ldots,p_{r-1})})^i{}_{j_1,\ldots,j_{r-1}}&=-v^i{}_\alpha u^\alpha{}_{j_1,\ldots,\underset{\stackrel{\frown}{p_{r-1}}}{l},\ldots,j_{r-1},\beta}v^\beta{}_\gamma du^\gamma\wedge v^l{}_\delta du^\delta\\*
&\hphantom{{}={}}+(\textit{terms which do not involve $u^i{}_{j_1,\ldots,j_r}$}).
\end{align*}
On the other hand, by the structural equations (Theorem~\ref{thm5.4}), $\Theta^{r(p_1,\ldots,p_{r-1})}=0$ if every lower index commutes each other.
This implies that we have
\[
(\Theta^{r(p_1,\ldots,p_{r-1})})^i{}_{j_1,\ldots,j_{r-1}}=-v^i{}_\alpha u^\alpha{}_{j_1,\ldots,\underset{\stackrel{\frown}{p_{r-1}}}{l},\ldots,j_{r-1},\beta}v^\beta{}_\gamma du^\gamma\wedge v^l{}_\delta du^\delta.
\]
Therefore, we have $u^i{}_{j_1,\ldots,\underset{\stackrel{\frown}{p_{r-1}}}{l},\ldots,j_{r-1},\beta}=u^\alpha{}_{j_1,\ldots,\underset{\stackrel{\frown}{p_{r-1}}}{\beta},\ldots,j_{r-1},l}$ if $\Theta^{r(p_1,\ldots,p_{r-1})}=0$.
\end{proof}
\begin{remark}
If torsions of order less than $r$ vanish, then $\Theta^{r(p_1,\ldots,p_{r-1})}$ only depends on $p_{r-1}$.
Indeed, the proof of Theorem~\ref{thm5.7} can be read as a proof of the structural equation in the classical setting where lower indices are commutative.
\end{remark}
\begin{definition}
We set
\begin{align*}
P^\infty&=\varprojlim P^r=\left\{u=(u^r)_{r\in\mathbb{N}}\in\prod_{r\in\mathbb{N}}P^r\;\middle|\;\pi^r_su^r=u^s\ \text{if $s<r$}\right\},\\*
\widetilde{P}^\infty&=\varprojlim\widetilde{P}^r,\\*
G^\infty&=P^\infty(\mathbb{R}^n,o),\\*
\widetilde{G}^\infty&=\widetilde{P}^\infty(\mathbb{R}^n,o),
\end{align*}
and equip them with the limit topology.
To say about $P^\infty$ for example, this is the weakest topology with respect to which the natural mappings $P^\infty\to P^r$ are continuous.
We call $P^\infty$ as the \textit{bundle of frames of infinite order}, and $\widetilde{P}^\infty$ as the \textit{bundle of formal frames of infinite order}, respectively.
If $f\colon(\mathbb{R}^n,o)\to M$ is a local diffeomorphism, then the element of $P^\infty$ determined by $(j^r_o(f))_{r\in\mathbb{N}}$ is represented by $j^\infty_o(f)$.
Similarly, if $F=F^\infty=(F^r)_{r\in\mathbb{N}}$ is an infinite sequence of morphisms from $\widetilde{P}^r(\mathbb{R}^n)$ to $\widetilde{P}^r$ such that $\pi^r_{r-1}\circ F^r=F^{r-1}\circ\pi^r_{r-1}$ and that $F^r(o)=j^1_o(F^{r-1})$, then the element of $\widetilde{P}^\infty$ determined by $(j^1_o(F^r))_{r\in\mathbb{N}}$ is represented by $j^1_o(F)=j^1_o(F^\infty)$.
\end{definition}
Note that $G^\infty$ and $\widetilde{G}^\infty$ are topological groups.
\begin{theorem}
\label{thm5.8}
We have $P^\infty=\{u\in\widetilde{P}^\infty\mid\text{all the torsions vanish at $u$}\}$.
\end{theorem}
\begin{proof}
Suppose that $u\in\widetilde{P}^\infty$ and that all the torsions vanish at $u$.
By taking a chart on $M$, we find an infinite sequence $(u^i,u^i{}_j,\ldots,)$ of tensors of which the lower indices commute.
By \cite{Sternberg}*{Lemma~5}, we can find a local diffeomorphism $f\colon(\mathbb{R}^n,o)\to M$ of class $C^\infty$ which realizes this sequence, namely, we have $j^r_o(f)=(a^i,a^i{}_{j_1},\ldots,a^i{}_{j_1,\ldots,j_r})$ for any $r\in\mathbb{N}$.
If we set $\iota(u)=j^\infty_o(f)$, then $\iota$ gives an isomorphism.
\end{proof}
\begin{remark}
\label{rem5.9}
If we work in the analytic (real or complex) category, then Theorem~\ref{thm5.8} is no longer valid.
Indeed, we need a realization of a given Taylor series as an analytic function, which is in general impossible.
Similarly, if we work with geometric structures, then we can find a smooth function $f$ which realizes a jet at a point, however it is not certain if $f$ preserves the structure.
In order to that, we will need a kind of integrability conditions.
See also Remarks~\ref{rem1.14} and \ref{rem4.9}.
\end{remark}
\begin{example}
\label{ex5.11}
\begin{enumerate}
\item
Let $M$ be a $1$-dimensional M\"obius manifold in the sense that there exists an atlas $\{(U_\lambda,\varphi_\lambda)\}$ of $M$ such that the transition functions are linear fractional transformations which we call M\"obius ones.
In this case formal frames and frames concide and we have $\widetilde{P}^r(M)=P^r(M)$.
We can consider the bundle of formal frames which is defined by using only by M\"obius mappings and their derivatives, which we represent by $\mathcal{M}^r(M)$.
In other words, we consider reductions of $\widetilde{P}^r(M)$ such that the structural groups are derived from $\mathrm{PGL}_1(\mathbb{R})$.
We set now for a function of one variable $f$, $S(f)=\frac{f'''}{f'}-\frac32\left(\frac{f''}{f'}\right)^2$ which is called the \textit{Schwarzian derivative} of $f$.
It is well-known that $f$ is M\"obius if and only if we have $S(f)=0$.
On the other hand, it is also well-known that we have $S(\phi\circ f)=f^*S(\phi)+S(f)$ in general.
Therefore, if we locally set $S=(v^1{}_1u^1{}_{111}-\frac32(v^1{}_1u^1{}_{11})^2)du^1\otimes du^1$, where $v^1{}_1=1/u^1{}_1$, always considering atlases as above, then $S$ is well-defined on $\widetilde{P}^r(M)$.
We see that $j^r(f)\in\widetilde{P}^r(M)=P^r(M)$ is represented by a M\"obuis map from $(\mathbb{R}^1,o)$ to $M$ if and only if $S(f)=0$.
\item
If $n\geq2$, there also is a tensor $\Sigma$ of type $(1,2)$ called the \textit{Schwarzian derivative}, which is still a cocycle such that $\Sigma(f)=0$ if and only if $f$ is projectively linear~\cite{MolMor},~\cite{Oda}.
Hence if we work on projectively flat manifolds, then we can repeat the argument in~1) to obtain a reduction of $\widetilde{P}^r(M)$ and the Schwarzian derivative on it.
We can locally write down the Schwarzian derivative in terms of natural coordinates but we omit it because it is so involved (cf.~\cite{MolMor}*{\S2}).
In this case, we can normalize projective connections in terms of $P^r(M)$.
\item
We can ask if a given anti-symmetric $2$-tensor field is realized as the torsion of a connection.
If we consider projective structures, it is known that there is a connection which preserves the projective structure and of which the torsion is the given tensor field.
On the other hand, if we work on a symplectic manifold of dimension $2n$, where $n\geq2$, then, there exists tensor fields which cannot be realized as the torsion of connections which preserve the symplectic structure~\cite{Weyl},~\cite{KobNag}.
In any case, we need to consider $\widetilde{P}^r(M)$ in order to deal with non-trivial $2$-tensors.
\end{enumerate}
\end{example}
\begin{remark}
The bundle $\widetilde{P}^r(M)$ is quite related with geometric structures as $P^r(M)$ is so.
We will discuss relationship of $\widetilde{P}^2(M)$ and projective structures in~\cite{asuke:2022-2}.
\end{remark}
\section{Infinitesimal deformations of linear connections}
Related to the above constructions, we will discuss infinitesimal deformations of linear connections.
Some of fundamental references are \cite{Rag1}, \cite{Rag2}.
We fix a manifold $M$ and a connection $\nabla$ on $TM$.
Given a chart $(U,\varphi)$, let $\Gamma^i{}_{jk}$ denote the Christoffel symbols of $\nabla$ with respect to $(U,\varphi)$.
That is, if $(x^1,\ldots,x^n)$ denote the standard coordinates for $\varphi(U)$, then
\[
\nabla X=\Gamma^i{}_{jk}\pdif{}{x^i}X^j,
\]
where $\nabla X$ denotes the covariant derivative of $X$ and $X=X^i\pdif{}{x^i}$.
We remark again that the order of the lower indices of the Christoffel symbols are reversed.
Let $\{\nabla_t\}$ be a $1$-parameter smooth family of connections such that $\nabla_0=\nabla$.
Then the Christoffel symbols of $\nabla_t$ are represented as
\[
\Gamma_t{}^i{}_{jk}=\Gamma^i{}_{jk}+\mu^i{}_{jk}(t),
\]
where $\mu^i{}_{jk}(t)\pdif{}{x^i}\otimes dx^k$ is an element of $\mathrm{Hom}(TM,TM)$ for each $t$.
\begin{definition}
We call $\mu^i{}_{jk}(t)$ a \textit{deformation} of $\nabla$.
To be more precise, we call $\mu^i{}_{jk}(t)$ an \textit{actual deformation}.
\end{definition}
If we take the derivative at $t=0$, then we obtain an element of $\mathrm{Hom}(TM,TM)$.
This leads to the following
\begin{definition}
An \textit{infinitesimal deformation} of $\nabla$ is an element of $\mathrm{Hom}(TM,TM)$.
\end{definition}
It is clear that if $\mu^i{}_{jk}(t)$ is an actual deformation, then $\pdif{\mu^i{}_{jk}}{t}(0)$ is an infinitesimal deformation.
If $\mu$ is an infinitesimal deformation of $\nabla$, then $\mu$ is represented by a family $\{\mu^i{}_{jk}\}$ with respect to an atlas.
If $(U,\varphi)$, $(U,\widehat{\varphi})$ are charts and if $\phi$ denotes the transition function, then we have
\[
\widehat{\mu}^i{}_{\alpha\beta}(D\phi)^\alpha{}_j(D\phi)^\beta{}_k=(D\phi)^i{}_{\alpha}\mu^\alpha{}_{jk}.
\]
In what follows, we will show that a pair of a connection and its infinitesimal deformation can be understood as a connection on a certain bundle.
For this purpose, we will recall some basic notions.
Let $G$ be a Lie group and $TG$ the tangent bundle of $G$, which is also a Lie group called the tangent Lie group of $G$.
We consider the adjoint action of $G$ on $\mathfrak{g}$, where $\mathfrak{g}$ is regarded as the vector space of left invariant vector fields on $G$.
Then, it is well-known that $TG$ is isomorphic to $G\ltimes\mathfrak{g}$.
The product in $G\ltimes\mathfrak{g}$ is given by
\[
(A,X).(B,Y)=(AB,\Ad_{B^{-1}}X+Y).
\]
If $G$ is linear and if $G\subset\mathrm{GL}_n(\mathbb{R})$, then, we have a matrix representation of $TG$ to $\mathrm{GL}_{2n}(\mathbb{R})$ given by
\[
(A,X)\mapsto\begin{pmatrix}
A \\
AX & A
\end{pmatrix}.
\]
The Lie algebra of $G\ltimes\mathfrak{g}$ is described as follows.
We omit the proof.
\begin{lemma}
\label{lem6.3}
Let $\mathfrak{h}$ be the Lie algebra of $G\ltimes\mathfrak{g}$.
\begin{enumerate}
\item
The Lie algebra $\mathfrak{h}$ is a vector space $\mathfrak{g}\times\mathfrak{g}$ equipped with the bracket
\[
[(\dot{A},\dot{X}),(\dot{B},\dot{Y})]=([\dot{A},\dot{B}],\ad_{\dot{X}}\dot{B}+\ad_{\dot{A}}\dot{Y}).
\]
\item
The adjoint action of $G\ltimes\mathfrak{g}$ on $\mathfrak{h}$ is given by
\[
\Ad_{(A,X)}(\dot{B},\dot{Y})=(\Ad_A\dot{B},\Ad_A(\ad_X\dot{B}+\dot{Y})).
\]
\item
Suppose that $G$ is linear.
Then, an element $(\dot{A},\dot{B})\in\mathfrak{h}$ corresponds to $\begin{pmatrix}
\dot{A}\\
\dot{B} & \dot{A}
\end{pmatrix}\in\mathfrak{tg}$ under the matrix representation.
\end{enumerate}
\end{lemma}
Let $P$ be a principal $G$-bundle over $M$, and set $\underline{\mathfrak{g}}=M\times\mathfrak{g}$.
The projection from $\underline{\mathfrak{g}}$ to $M$ is represented by $\nu$.
\begin{definition}
Let $P\ltimes\underline{\mathfrak{g}}$ with the projection $\varpi$ to $M$ be a principal $TG$-bundle defined as follows.
We set $P\ltimes\underline{\mathfrak{g}}=\{(u,X)\in P\times\underline{\mathfrak{g}}\mid\pi(u)=\nu(X)\}$ and $\varpi(u,X)=\pi(u)$, namely, we consider the fiber product.
If $(B,Y)\in TG$, then we~set
\[
(u,X).(B,Y)=(u.B,\Ad_{B^{-1}}X+Y).
\]
\end{definition}
\begin{theorem}
\label{thm6.16}
There is a one-to-one correspondence between the following\textup{:}
\begin{enumerate}
\item
Pairs of connections on $TM$ and their infinitesimal deformations.
\item
Connections on $P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})$.
\item
Sections from $P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})$ to $\widetilde{P}^2(M)\ltimes\underline{\widetilde{\mathfrak{g}}^2_n}$ equivariant under the $\mathrm{GL}_n(\mathbb{R})\ltimes\mathfrak{gl}_n(\mathbb{R})$-action.
\end{enumerate}
\end{theorem}
\begin{proof}
We set $P=P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})$, $G=\mathrm{GL}_n(\mathbb{R})\ltimes\mathfrak{gl}_n(\mathbb{R})$ and let $\mathfrak{g}$ denote the Lie algebra of $G$.
First, the transition functions for $P$ is given as follows.
Let $(U,\varphi)$ and $(\widehat{U},\widehat{\varphi})$ be charts on $M$ and $\phi$ the transition function.
Then, $P$ is trivial on $U$.
Let $((x,g);X)=((x^i,g^i{}_j);X^i{}_j)\in\varphi(U)\times\mathrm{GL}_n(\mathbb{R})\times\mathfrak{gl}_n(\mathbb{R})$ be coordinates for $P|_U$ and $((\widehat{x},\widehat{g});\widehat{X})$ for $P|_{\widehat{U}}$.
We have $((\widehat{x},\widehat{g});\widehat{X})=((\phi(x),D\phi(x)g);X)$.
As $(D\phi(x)g,X)=(D\phi(x),0)(g,X)$ in $G=\mathrm{GL}_n(\mathbb{R})\ltimes\mathfrak{gl}_n(\mathbb{R})$, the transition function for $P$ is given by $(D\phi(x),0)$.
Let now $\{\omega\}$, where $\omega=(\theta,\mu)$, be a family of $\mathfrak{g}$-valued $1$-form on $M$.
This family represents a connection on $P$ if and only if we~have
\[
(\omega,\theta)=(D\phi,0)^{-1}d(D\phi,0)+\Ad_{(D\phi,0)^{-1}}(\widehat{\omega},\widehat{\theta}).
\]
This condition is equivalent to
\begin{align*}
\theta&=D\phi(x)^{-1}dD\phi+\Ad_{D\phi^{-1}}\widehat{\theta},\\*
\mu&=\Ad_{D\phi^{-1}}\widehat{\mu}.
\end{align*}
Hence $(\theta,\mu)$ represents a connection on $P$ if and only if $\mu$ is an infinitesimal deformation of $\theta$.
It is clear that the correspondence is one-to-one.
Next, we show that the conditions 2) and 3) are equivalent.
By theorems of Garc\'\i a (Theorems~2 and 3 of~\cite{Garcia}, cf. Theorem~\ref{thmG11}), connections on $P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})$ are in a one-to-one correspondence between $G$-equivariant sections of $\mathcal{J}(P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R}))\to P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})$.
Hence it suffices to show that $\mathcal{J}(P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R}))$ and $\widetilde{P}^2(M)\ltimes\underline{\widetilde{\mathfrak{g}}^2_n}$ are isomorphic as $G$-bundles.
Let $j^1_p(s)\in\mathcal{J}(P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R}))$, where $s$ is a section of $P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})\to M$ about $p$.
Let $(U,\varphi)$ be a chart such that $\varphi(p)=o$ and that $D\varphi^{-1}(o)=s(p)$.
We represent $s\circ\varphi=(s_1,s_2)$ and associate $j^1_p(s)$ with $(j^1_o(\varphi^{-1},s_1),j^1_o(s_2))$ (cf. Lemma~\ref{lem3.5}).
This gives a desired isomorphism.
Indeed, the correspondence is locally given as follows.
Let $(U,\varphi)$ be a chart and $(x^i,(a^i{}_j,b^i{}_j),(a^i{}_{jk},b^i{}_{jk}))$ be the representation of $s$ with respect to the Garc\'\i a coordinates.
We adopt as the coordinates for $\widetilde{P}^2\ltimes\underline{\widetilde{\mathfrak{g}}^2_n}$ the product of the natural coordinates for $\widetilde{P}^2$ and the trivial one for $\underline{\widetilde{\mathfrak{g}}^2_n}$.
Then, $j^1_p(s)$ is mapped to $(x^i,(a^i{}_j,a^i{}_{j\alpha}a^\alpha{}_k),(b^i{}_j,b^i{}_{jk}))$.
If $(g,X)\in\mathrm{GL}_n(\mathbb{R})\ltimes\mathfrak{gl}_n(\mathbb{R})$, then we have
\begin{align*}
&\hphantom{{}={}}
(x^i,(a^i{}_j,b^i{}_j),(a^i{}_{jk},b^i{}_{jk})).(g,X)\\*
&=(x^i,(a^i{}_\alpha g^\alpha{}_j,h^i{}_\alpha b^\alpha{}_\beta g^\beta{}_j+X^i{}_j),(a^i{}_{\alpha j}g^\alpha{}_k,h^i{}_\alpha b^\alpha{}_{\beta k}g^\beta{}_j)),
\end{align*}
where $(h^i{}_j)=(g^i{}_j)^{-1}$.
On the other hand, we have
\begin{align*}
&\hphantom{{}={}}
(x^i,(a^i{}_j,a^i{}_{j\alpha}a^\alpha{}_k),(b^i{}_j,b^i{}_{jk})).(g,X)\\*
&=(x^i,(a^i{}_\alpha g^\alpha{}_j,a^i{}_{\beta\alpha}g^\beta{}_ja^\alpha{}_\beta g^\beta{}_k),(h^i{}_\alpha b^\alpha{}_\beta g^\beta{}_j+X^i{}_j,h^i{}_\alpha b^\alpha{}_{\beta k}g^\beta{}_j)).
\end{align*}
Hence we obtained a morphism from $\mathcal{J}(P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R}))$ to $\widetilde{P}^2(M)\ltimes\underline{\widetilde{\mathfrak{g}}^2_n}$ equivariant under the $\mathrm{GL}_n(\mathbb{R})\ltimes\mathfrak{gl}_n(\mathbb{R})$-actions.
It is easy to see that this morphism is indeed an isomorphism.
\end{proof}
\begin{remark}
The canonical form on $\mathcal{J}(P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R}))$, which corresponds to the canonical form of order $2$ on $\widetilde{P}^2(M)\ltimes\underline{\widetilde{\mathfrak{g}}^2_n}$, is locally given~by
\[
(c^i{}_\alpha(da^\alpha{}_j-a^\alpha{}_{j\beta}dx^\beta),db^i{}_j-b^i{}_{j\alpha}dx^\alpha-b^i{}_\alpha c^\alpha{}_\beta(da^\beta{}_j-a^\beta{}_{j\gamma}dx^\gamma)+c^i{}_\alpha(da^\alpha{}_\beta-a^\alpha{}_{\beta\gamma}dx^\gamma)b^\gamma{}_j),
\]
where $(c^i{}_j)=(a^i{}_j)^{-1}$, with respect to the Garc\'\i a coordinates.
\end{remark}
An explanation of Theorem~\ref{thm6.16} can be given by using an auxiliary structure.
For this purpose, we recall $2$-tangent bundles (see~\cite{IY} for details).
\begin{definition}
We set $T^2M=T(TM)$ and call $T^2M$ as the \textit{$2$-tangent bundle}.
The projection from $T^2M$ to $TM$ is represented by $p^2$.
\end{definition}
Charts and transition functions on $T^2M$ are given as follows.
Let $(U,\varphi)$, $(\widehat{U},\widehat{\varphi})$ be charts of $M$ and $\phi$ the transition function from $U$ to $\widehat{U}$.
Then, $TM$ is trivial on $U$ and $\widehat{U}$.
If $(x,v)$ denote local coordinates for $TM$ on $p^{-1}(U)$, where $p\colon TM\to M$ is the projection, then the transition function is given as $(x,v)\mapsto(\phi(x),D\phi(x)v)$.
Further, $T^2M$ is trivial on $p^{-1}(U)$ and $p^{-1}(\widehat{U})$.
If $(x,v;\dot{x},\dot{v})$ denote local coordinates for $T^2M$ on $(p^2)^{-1}(p^{-1}(U))$, then the transition function is given as follows.
Let $\gamma\colon(-\epsilon,\epsilon)\to TM$ be a curve.
We represent this curve as $(x(t),v(t))$ using a chart.
We have $(\widehat{x}(t),\widehat{v}(t))=(\phi(x(t)),D\phi(x(t))v(t))$ so that
\stepcounter{theorem}
\[
(\widehat{x},\widehat{v};\dot{\widehat{x}},\dot{\widehat{v}})
=(\phi(x),D\phi(x)v;D\phi(x)\dot{x},H\phi(x)v\dot{x}+D\phi(x)\dot{v}),
\tag{\thetheorem}
\label{eq6.5}
\]
where $H\phi=D(D\phi)$ and $H\phi(x)v\dot{x}=(H\phi)^i{}_{\alpha\beta}v^\alpha\dot{x}^\beta$.
We refer to these coordinates as coordinates \textit{induced} by $(U,\varphi)$.
\begin{definition}
Let $(U,\varphi)$ be a chart on $M$ and consider induced coordinates.
If $u\in TM$, then we set
\[
{\left(\pdif{}{x^i}\right)^V}_u=\pdif{}{v^i}_u
\]
and call it the \textit{vertical lift} of $\pdif{}{x^i}_x$, where $x=p(u)$.
In general, we extend the vertical lift by linearity.
We set
\[
V=\{\text{vectors on $TM$ which are the vertical lifts of vectors on $M$}\}.
\]
\end{definition}
The following is easy.
\begin{lemma}
We have $T^2M/V\cong\pi^*TM$ as vector bundles over $TM$.
\end{lemma}
\begin{definition}
Let $\nabla$ be a connection on $TM$.
Let $(U,\varphi)$ be a chart on $M$ and consider induced coordinates.
Let $\Gamma^i{}_{jk}$ be the Christoffel symbols of $\nabla$ with respect to $(U,\varphi)$.
If $u\in TM$, then we set
\stepcounter{theorem}
\[
{\left(\pdif{}{x^i}\right)^H}_u=\pdif{}{x^i}_u-\Gamma(x)^\alpha{}_{i\beta}v^\beta\pdif{}{v^\alpha}_u
\tag{\thetheorem}
\label{eq6.6}
\]
and call it the \textit{horizontal lift} of $\pdif{}{x^i}_x$, where $x=p(u)$.
In general, we extend the horizontal lift by linearity.
Finally, we set
\[
H=\{\text{vectors on $TM$ which are the horizontal lifts of vectors on $M$}\}.
\]
\end{definition}
Note that $\pdif{}{x^i}$ on the left hand side of \eqref{eq6.6} refers to a vector on $M$, on the other hand, the same symbol on the right hand side refers to a vector on $TM$, we always considering charts and induced coordinates.
\begin{proposition}
The pairs $(V,p^2|_V)$ and $(H,p^2|_H)$ are isomorphic to $\pi^*TM$.
\end{proposition}
\begin{proof}
By~\eqref{eq6.5}, we have
\[
\left(\pdif{}{\widehat{x}^i},\pdif{}{\widehat{v}^i}\right)\begin{pmatrix}
D\phi(x)\\
H\phi(x)v & D\phi(x)
\end{pmatrix}=\left(\pdif{}{x^i},\pdif{}{v^i}\right).
\]
Therefore, we have
\begin{align*}
&\hphantom{{}={}}
\left(\left(\pdif{}{\widehat{x}^i}\right)^H,\left(\pdif{}{\widehat{x}^i}\right)^V\right)\begin{pmatrix}
I_n\\
\widehat{\Gamma}\widehat{v} & I_n
\end{pmatrix}\begin{pmatrix}
D\phi(x)\\
H\phi(x)v & D\phi(x)
\end{pmatrix}\begin{pmatrix}
I_n\\
-\Gamma v & I_n
\end{pmatrix}\\*
&=\left(\left(\pdif{}{x^i}\right)^H,\left(\pdif{}{x^i}\right)^V\right).
\end{align*}
By~\eqref{eq2.14}, we have
\[
\begin{pmatrix}
I_n\\
\widehat{\Gamma}\widehat{v} & I_n
\end{pmatrix}\begin{pmatrix}
D\phi(x)\\
H\phi(x)v & D\phi(x)
\end{pmatrix}\begin{pmatrix}
I_n\\
-\Gamma v & I_n
\end{pmatrix}=\begin{pmatrix}
D\phi(x)\\
& D\phi(x)
\end{pmatrix}.
\]
Hence $(V,p^2|_V)$ and $(H,p^2|_H)$ are isomorphic to $\pi^*TM$.
\end{proof}
\begin{remark}
It is known that the horizontal lifts recovers connections~\cite{IY} if connections are torsion free.
It remains valid in our setting as follows.
Let $X$ be a vector field on $M$ and $v\in T_pM$.
We regard $X$ as a mapping from $M$ to $TM$ and let $DX\colon TM\to T^2M$ be the derivative.
We represent $X=f^i\pdif{}{x^i}$ and $v=v^i\pdif{}{x^i}_p$ on a chart.
Then, we have
\[
\nabla_vX(p)=\left(\pdif{f^\alpha}{x^\beta}(p)v^\beta+\Gamma^\alpha{}_{\beta\gamma}(p)f^\beta(p)v^\gamma\right)\pdif{}{x^\alpha}_p.
\]
On the other hand, we have
\begin{align*}
DX(v)_p&=v^\alpha\pdif{}{x^\alpha}_{X(p)}+\pdif{f^\alpha}{x^\beta}(p)v^\beta\pdif{}{v^\alpha}_{X(p)},\\*
v^H_{X(p)}&=v^\alpha\pdif{}{x^\alpha}_{X(p)}-\Gamma^\alpha{}_{\beta\gamma}(p)v^\beta f^\gamma(p)\pdif{}{v^\alpha}_{X(p)}.
\end{align*}
It follows that we have
\[
(\nabla_vX(p)+T(v,X(p)))^V_{X(p)}=DX(v)_p-v^H_{X(p)},
\]
where $T$ denotes the torsion of $\nabla$.
\end{remark}
It is clear that $T^2M=H\oplus V$.
This identification can be seen as an isomorphism between $T^2M$ and $p^{2*}(TM)$.
Note that the horizontal lifts give sections of the projection from $T^2M$ to $TM$.
If $\nabla$ is a connection on $TM$ and $\mu$ its infinitesimal deformation, then we can define a connection, say $\widetilde\nabla$, on $H\oplus V$.
Indeed, let $\theta^i{}_j$, $\mu^i{}_j$ be matrix representations of $\nabla$ and $\mu$ with respect to $\pdif{}{x^i}$ on $M$.
On the other hand, we consider $\left(\left(\pdif{}{x^i}\right)^H,\left(\pdif{}{x^i}\right)^V\right)$ as a local frame for $T^2M=H\oplus V$.
Then locally defined $\mathfrak{tgl}_n(\mathbb{R})$-valued $1$-forms $\begin{pmatrix}
\theta^i{}_j \\
\mu^i{}_j & \theta^i{}_j
\end{pmatrix}$
give rise to a connection $\widetilde{\nabla}$.
The diagonal part correspond to the pull-back connection of $\nabla$ on $V$ and $H$ identified with $p^*(TM)$.
On $H$, we have an additional term valued in $V$ which is given by an element of $\mathrm{Hom}(H,V)$.
This corresponds to the pull-back of $\mu\in\mathrm{Hom}(TM,TM)$.
The connection $\widetilde\nabla$ is related with the one obtained by Theorem~\ref{thm6.16} as follows.
Let $\mathcal{P}^1(TM)$ be the principal bundle associated with $T^2M$, which is a $T\mathrm{GL}_n(\mathbb{R})$-bundle over $TM$ described as follows.
Let $(U,\varphi)$ be a chart of $M$ and $(x,v)$ be the associated coordinates for $TM|_U$.
Let $\{f,g\}\colon(\mathbb{R}^n\times\mathbb{R}^n,(o,o))\to TM$ be a mapping of the~form
\[
(\xi,\eta)\mapsto\{f,g\}(\xi,\eta)=(f(\xi),g(\xi)+Df(\xi)\eta)
\]
with respect to a chart.
If we ignore terms of order greater than $1$ with respect to $\eta$, this property is independent of charts.
Note also that such a mapping is a local diffeomorphism if and only if $Df(o)\in\mathrm{GL}_n(\mathbb{R})$.
\begin{definition}
We set
\begin{align*}
\mathcal{T}(M)&=\{\text{$\{f,g\}$ as above}\},\\*
\mathcal{T}_0(M)&=\{\{f,g\}\in\mathcal{T}\mid g(o)=o\},\\*
\mathcal{P}'(TM)&=\{j^1_{(o,o)}(\{f,g\})\mid\{f,g\}\in\mathcal{T}(M)\},\ \text{and}\\*
\mathcal{G}&=\{j^1_{(o,o)}(\{f,g\})\mid\{f,g\}\in\mathcal{T}(\mathbb{R}^n),\ f(o)=g(o)=o\},
\end{align*}
where $T\mathbb{R}^n$ is naturally trivialized.
\end{definition}
\begin{lemma}
\label{lem6.13}
\begin{enumerate}
\item
The set $\mathcal{G}$ is a group of which the product is the composition, and is isomorphic to $T\mathrm{GL}_n(\mathbb{R})$.
\item
The bundle $\mathcal{P}'(TM)$ is isomorphic to $\mathcal{P}^1(TM)$ as $T\mathrm{GL}_n(\mathbb{R}^n)$-bundles.
\end{enumerate}
\end{lemma}
\begin{proof}
We have $D_{(o,o)}(\{f,g\})=\begin{pmatrix}
Df(o)\\
Dg(o) & Df(o)
\end{pmatrix}$.
On the other hand, we have
\stepcounter{theorem}
\begin{align*}
\tag{\thetheorem}
\label{eq6.14}
\{f,g\}.\{f',g'\}(\xi,\eta)&=\{f,g\}(f'(\xi),g'(\xi)+Df'(\xi)\eta)\\*
&=(f\circ f'(\xi),g\circ f'(\xi)+Df(f'(\xi))(g'(\xi)+Df'(\xi)\eta))\\*
&=(f\circ f'(\xi),h(\xi)+D(f\circ f')(\xi)\eta),
\end{align*}
where $h(\xi)=g\circ f'(\xi)+Df(f'(\xi))g'(\xi)$.
Hence we have
\begin{align*}
&\hphantom{{}={}}
D_{(o,o)}(\{f,g\}.\{f',g'\})\\*
&=\begin{pmatrix}
Df(f'(o))Df'(o)\\
Dg(f'(o))Df'(o)+Hf(f'(o))Df'(o)g'(o)+Df(f'(o))Dg'(o) & Df(f'(o))Df'(o)
\end{pmatrix}
\end{align*}
If $f'(o)=o$ and if $g'(o)=o$, then we have
\begin{align*}
&\hphantom{{}={}}
D_{(o,o)}(\{f,g\}.\{f',g'\})\\*
&=\begin{pmatrix}
Df(o)Df'(o)\\
Dg(o)Df'(o)+Df(o)Dg'(o) & Df(o)Df'(o)
\end{pmatrix}\\*
&=\begin{pmatrix}
Df(o)\\
Dg(o) & Df(o)
\end{pmatrix}\begin{pmatrix}
Df'(o)\\
Dg'(o) & Df'(o)
\end{pmatrix}.
\end{align*}
Similarly, if $(U,\varphi)$, $(\widehat{U},\widehat{\varphi})$ be charts and $\phi$ the transition function, then we have
\[
D\phi\circ\{f,g\}(\xi,\eta)=(\phi\circ f(\xi),D\phi(f(\xi))(g(\xi)+Df(\xi)\eta)).
\]
Hence we have
\[
D\phi(j^1_{(o,o)}(\{f,g\}))=\begin{pmatrix}
D\phi(f(o)) \\
H\phi(f(o))g(o) & D\phi(f(o))
\end{pmatrix}\begin{pmatrix}
Df(o)\\
Dg(o) & Df(o)
\end{pmatrix}.
\]
Therefore $\mathcal{P}'(TM)$ and $\mathcal{P}^1(TM)$ are isomorphic as $T\mathrm{GL}_n(\mathbb{R})$-bundles.
\end{proof}
We set
\begin{align*}
\mathcal{P}^1_0(TM)&=\{j^1_{(o,o)}(\{f,g\})\mid\{f,g\}\in\mathcal{T}_0\}.
\end{align*}
The bundle $\mathcal{P}^1_0(TM)$ is a principal $T\mathrm{GL}_n(\mathbb{R})$-bundle which is the restriction of $\mathcal{P}^1(TM)$ to the zero section of $TM\to M$.
Recall that we have an isomorphism between $T\mathrm{GL}_n(\mathbb{R})$ and $\mathrm{GL}_n(\mathbb{R})\ltimes\mathfrak{gl}_n(\mathbb{R})$.
We have the following
\begin{lemma}
\label{lem6.18}
The bundles $\mathcal{P}^1_0(TM)$ and $P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})$ are isomorphic as principal $\mathrm{GL}_n(\mathbb{R})\ltimes\mathfrak{gl}_n(\mathbb{R})$-bundles.
\end{lemma}
\begin{proof}
By the proof of Lemma~\ref{lem6.13}, we see that the transition function on $\mathcal{P}^1_0(TM)$ is given by $\begin{pmatrix}
D\phi(f(o))\\
& D\phi(f(o))
\end{pmatrix}$ because we have $g(o)=o$.
If we associate $j^1_{(o,o)}(\{f,g\})$ with $((f(o),Df(o)),Df(o)^{-1}Dg(o))$, then obtain the desired isomorphism.
\end{proof}
Let now consider a pair of a connection on $TM$ and its infinitesimal deformation.
This induces a connection on $\mathcal{P}^1(TM)$.
By restricting this latter connection to $\mathcal{P}^1_0(TM)$, we obtain a connection on $P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})$ by Lemma~\ref{lem6.18}.
By Lemma~\ref{lem6.3}, we see that this is the connection given by Theorem~\ref{thm6.16}
\begin{remark}
There is a description of $\widetilde{P}^2(M)\ltimes\underline{\widetilde{\mathfrak{g}}^2_n}$ similar to that of $P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})$.
We consider mappings from $T^2\mathbb{R}^n$ to $T^2M$ locally defined by
\begin{align*}
&\hphantom{{}={}}
\{f,g;F,G\}(\xi,\eta;\dot{\xi},\dot{\eta})\\*
&=(f(\xi),Df(\xi)\eta+g(\xi);F(\xi)\dot{\xi},(DF(\xi)^i{}_{\alpha j}\eta^\alpha+G(\xi))\dot{\xi}+F(\xi)\dot{\eta}),
\end{align*}
where we assume that $Df(o)=F(o)$, $Dg(o)=G(o)$ and that $Df(o)$ is regular as a matrix.
We set
\begin{align*}
\mathcal{T}^2(M)&=\{\text{$\{f,g;F,G\}$ as above}\},\\*
\mathcal{T}^2_0(M)&=\{\{f,g;F,G\}\in\mathcal{T}^2(M)\mid g(o)=o\},\\*
\mathcal{P}^2(TM)&=\{j^1_{(o,o)}(\{f,g;F,G\})\mid\{f,g;F,G\}\in\mathcal{T}^2(M)\},\\*
\mathcal{P}^2_0(TM)&=\{j^1_{(o,o)}(\{f,g;F,G\})\mid\{f,g;F,G\}\in\mathcal{T}^2_0(M)\},\\*
\mathcal{G}^2&=\{j^1_{(o,o)}(\{f,g;F,G\})\mid\{f,g;F,G\}\in\mathcal{T}^2(M),\ f(o)=g(o)=o\}
\end{align*}
We can show that $\mathcal{G}^2$ is a group isomorphic to $\widetilde{G}_n^2\ltimes\widetilde{\mathfrak{g}}^2_n$ of which $\mathrm{GL}_n(\mathbb{R})\ltimes\mathfrak{gl}_n(\mathbb{R})$ is a subgroup.
We can also show the following
\begin{lemma}
The bundle $\mathcal{P}^2_0(TM)$ is isomorphic to $\widetilde{P}^2(M)\ltimes\underline{\widetilde{\mathfrak{g}}^2_n}$ as a $\widetilde{G}^2_n\ltimes\widetilde{\mathfrak{g}^2}$-bundle.
\end{lemma}
We omit the proof because they are parallel to the case of $\mathcal{P}^1_0(TM)$.
\end{remark}
\section{Application to deformations of foliations}
\label{sec6}
We consider regular (non-singular) foliations.
Associated with such foliations, there are natural connections called \textit{Bott connections}.
If foliations are deformed, Bott connections are also deformed according to deformations.
We will discuss infinitesimal deformations of Bott connections associated with infinitesimal deformations of foliations.
Let $\mathcal{F}$ be a foliation of $M$, of codimension $q$.
Let $\{(U_\lambda,\varphi_\lambda)\}$ be a foliation atlas, that is, we have homeomorphisms $U_\lambda\cong V_\lambda\times T_\lambda$ such that the restriction of $\mathcal{F}$ to $U_\lambda$ is given by $\{V_\lambda\times\{y\}\}_{y\in T_\lambda}$, where $V_\lambda$ and $T_\lambda$ are balls in $\mathbb{R}^{\dim M-q}$ and $\mathbb{R}^q$, respectively.
Let $p_\lambda$ denote the projection from $U_\lambda$ to $T_\lambda$.
Then, the transition function from $U_\lambda$ to $U_\mu$ is of the form $(x_\lambda,y_\lambda)\mapsto(\psi_{\mu\lambda}(x_\lambda,y_\lambda),\gamma_{\mu\lambda}(y_\lambda))$.
We refer to $\gamma_{\mu\lambda}$ as the \textit{holonomy map}.
Let $T\mathcal{F}$ be the tangent bundle of $\mathcal{F}$, which is the subbundle of $TM$ which consists of vectors tangent to leaves of $\mathcal{F}$.
The bundle $T\mathcal{F}$ locally consists of vectors tangent to $V_\lambda\times\{y\}$.
\begin{definition}
The quotient bundle $Q(\mathcal{F})=TM/T\mathcal{F}$ is called as the \textit{normal bundle} of $\mathcal{F}$.
\end{definition}
\begin{definition}
A connection on $Q(\mathcal{F})$ is called a \textit{Bott connection} if and only if $\nabla_XY=\mathcal{L}_XY$ holds for $X\in T\mathcal{F}$, where $\mathcal{L}_X$ denotes the Lie derivative with respect to $X$.
\end{definition}
It is well-known that Bott connections always exist in the $C^\infty$-category.
Bott connections enjoy the following property.
\begin{lemma}
\label{lem7.3}
Let $\nabla$ be a Bott connection on $Q(\mathcal{F})$.
Let $(U,\varphi)$ be a foliation chart and $(x,y)$ be coordinates for $U\cong V\times T$.
If\/ $\theta^i{}_j$ denote the components of connection matrix with respect to $\pdif{}{y^1},\ldots,\pdif{}{y^q}$, then $\theta^i{}_j$ do not involve $dx^k$'s.
\end{lemma}
\begin{proof}
This is because we have $\nabla_{\pdif{}{x^k}}\pdif{}{y^j}=\mathcal{L}_{\pdif{}{x^k}}\pdif{}{y^j}=0$.
\end{proof}
\begin{remark}
Lemma~\ref{lem7.3} does \textit{not} mean that $\theta^i{}_j$ are independent of $x^k$.
\end{remark}
In the setting of Lemma~\ref{lem7.3}, we can represent $\theta^i{}_j$ as $\theta^i{}_j=\Gamma^i{}_{jk}dy^k$.
The functions $\Gamma^i{}_{jk}$ are referred as the \textit{Christoffel symbols}.
Note that the order of lower indices are always reversed as in the previous sections.
\begin{proposition}
Let $\nabla$ a connection on $Q(\mathcal{F})$.
Let $e=(e_i)$ be a local trivialization of $Q(\mathcal{F})$ which is \textup{foliated} or locally projectable in the sense that each $e_i$ is of the form $f^\alpha{}_i\pdif{}{y^\alpha}$ with $f^\alpha{}_i$ being functions on $y$ independent of $x$, where $(x,y)$ are local coordinates on a foliation chart.
Let $(\theta^i{}_j)$ be the connection matrix of\/ $\nabla$ with respect to $e$.
Then, $\nabla$ is a Bott connection if and only if\/ $\theta^i{}_j|_{T\mathcal{F}}=0$.
\end{proposition}
\begin{proof}
Let $\Gamma^i{}_j$ be the connection matrix of $\nabla$ with respect to $\left(\pdif{}{y^i}\right)$.
If we set $F=(f^i{}_j)$, then we have $\theta^i{}_j=(F^{-1})^i{}_\alpha dF^\alpha{}_j+(F^{-1})^i{}_\alpha\Gamma^\alpha{}_\beta F^\beta{}_j$.
As $e$ is foliated, $\theta^i{}_j$ do not involve $dx^i$ if and only if so do not $\Gamma^i{}_j$.
\end{proof}
By Theorem~\ref{thm4.6} we can form $\widetilde{G}^r_q$-bundles $\widetilde{P}^r(\mathcal{F})$ by pasting $p_\lambda{}^*\widetilde{P}^r(T_\lambda)$.
To be precise, suppose that $U_\lambda\cap U_\mu\neq\varnothing$, and let $u_\lambda\in p_\lambda^*\widetilde{P}^r(T_\lambda)$ and $u_\mu\in p_\mu^*\widetilde{P}^r(T_\mu)$.
We have naturally have $p_\lambda^*\widetilde{P}^r(T_\lambda)\cong V_\lambda\times T_\lambda\times\widetilde{G}^r_q$.
Let we represent $u_\lambda=(x_\lambda,y_\lambda,g_\lambda)$ and $u_\mu=(x_\mu,y_\mu,g_\mu)$.
Then, $\gamma_{\mu\lambda}$ gives a locally defined map from $\widetilde{P}^r(T_\lambda)$ to $\widetilde{P}^r(T_\mu)$, which we represent by $\gamma_{\mu\lambda*}$.
\begin{definition}
We say that $u_\lambda\sim u_\mu$ if and only if $x_\lambda=x_\mu$ and $(y_\mu,g_\mu)=\gamma_{\mu\lambda*}(y_\lambda,g_\lambda)$.
We set $\widetilde{P}^r(\mathcal{F})=\left(\bigsqcup_{\lambda}p_\lambda^*\widetilde{P}^r(T_\lambda)\right)/\sim$.
The natural projection from $\widetilde{P}^r(\mathcal{F})$ to $M$ is represented by $\pi^r$.
\end{definition}
It is easy to see that $\widetilde{P}^r(\mathcal{F})$ is independent of the choice of foliation atlases.
It is clear that $\widetilde{P}^1(\mathcal{F})$ is isomorphic to the frame bundle of $Q(\mathcal{F})$ which is represented by~$P^1(\mathcal{F})$.
There are natural foliations of $\widetilde{P}^r(\mathcal{F})$.
Indeed, if $U\cong V\times T$ is a foliation chart, then $\widetilde{P}^r(\mathcal{F})$ is trivial, namely, we have $\widetilde{P}^r(\mathcal{F})|_U\cong U\times\widetilde{G}_q^r\cong V\times T\times\widetilde{G}_q^r$.
The transition functions are of the form $(x,y,g)\mapsto(\psi(x,y),\gamma(y),\gamma_*(y)g)$.
Therefore, we have a foliation of $\widetilde{P}^r(\mathcal{F})$ locally defined by asking $y$ and $g$ to be constant, to which we refer as $\mathcal{F}^r$.
We always equip $\widetilde{P}^r(\mathcal{F})$ with the foliation~$\mathcal{F}^r$.
\begin{definition}
A connection on $\widetilde{P}^r(\mathcal{F})$ is said to be a \textit{Bott connection} if it is a Bott connection for $\mathcal{F}^r$.
\end{definition}
The following is easy.
\begin{lemma}
A connection on $P^1(\mathcal{F})$ is a Bott connection if and only if it is associated with a Bott connection on $Q(\mathcal{F})$.
\end{lemma}
\begin{definition}[Canonical form]
Let $u\in\widetilde{P}^r(\mathcal{F})$ and $X\in T_u\widetilde{P}^r(\mathcal{F})$.
We choose a foliation chart $U\cong V\times T$ which contains $\pi^r(u)$.
Let $p\colon U\to T$ be the projection, $\theta_T$ the canonical form of $\widetilde{P}^r(T)$ and set
\[
\theta(X)=p^*\theta_T.
\]
We call $\theta$ the \textit{canonical form} on $\widetilde{P}^r(\mathcal{F})$.
\end{definition}
By Theorem~\ref{thm4.6}, the canonical form on $\widetilde{P}^r(\mathcal{F})$ is well-defined.
We have the following.
The proof is just a combination of Lemma~\ref{lem7.3} and Theorem~\ref{thmG11} so that omitted.
\begin{theorem}
There is a one-to-one correspondence between the following objects\/\textup{:}
\begin{enumerate}
\item
Bott connections on $Q(\mathcal{F})$.
\item
Sections of $\widetilde{P}^2(\mathcal{F})\to P^1(\mathcal{F})$ which are equivariant under the $\mathrm{GL}_q(\mathbb{R})$-actions and that preserve foliations.
\item
Sections of $\widetilde{P}^2(\mathcal{F})/\mathrm{GL}_q(\mathbb{R})\to M$ which preserve foliations.
\end{enumerate}
\end{theorem}
Next, we discuss deformations of foliations and Bott connections.
\begin{definition}
A connection $\nabla$ on $P^1(\mathcal{F})\ltimes\underline{\mathfrak{gl}_q}(\mathbb{R})$ is said to be a \textit{Bott connection} if $\iota^*\nabla$ is a Bott connection on $P^1(\mathcal{F})$, where $\iota$ is the inclusion obtained by the natural identification of $P^1(\mathcal{F})$ with $P^1(\mathcal{F})\ltimes\{o\}$.
\end{definition}
Let $\omega=(\omega^i)$ be a local trivialization of $Q^*(\mathcal{F})$.
By the Frobenius theorem, there exists a $\mathfrak{gl}_q(\mathbb{R})$-valued $1$-form, say $\theta$, such that
\[
d\omega+\theta\wedge\omega=0.
\]
The $1$-form $\theta$ is essentially the connection form of a Bott connection, say $\nabla$, with respect to the frame dual to $\omega$.
Assume now that $\mathcal{F}$, $\nabla$, $\omega$, $\theta$ are smooth $1$-parameter families.
Let $t$ be the parameter and assume that $\mathcal{F}_0=\mathcal{F}$ and so on.
If we represent derivatives with respect to $t$ at $t=0$ by adding a dot, then we have
\[
d\dot\omega+\dot\theta\wedge\omega+\theta\wedge\dot\omega=0.
\]
By considering a foliation atlas, we consider $\omega$, etc.~are family defined on $M$.
Then a $\mathfrak{gl}_q(\mathbb{R})$-valued $1$-form $\dot\theta$ gives rise to a global $\mathrm{Hom}(Q(\mathcal{F}),Q(\mathcal{F}))$-valued $1$-form on $M$ independent of the choice of foliation atlases.
In general, we adopt this property as an infinitesimal deformation of connections.
\begin{definition}[\cite{12}, see also~\cite{DuchampKalka}]
Let $\mathcal{F}$ be a foliation and $\nabla$ a Bott connection on $Q(\mathcal{F})$.
We fix a family $\omega$ of local trivializations of $Q(\mathcal{F})$, and let $\theta$ be the family of the connection forms with respect to the dual of $\omega$.
A $Q(\mathcal{F})$-valued global $1$-form $\dot\omega$ and a $\mathrm{Hom}(Q(\mathcal{F}),Q(\mathcal{F}))$-valued $1$-form $\dot\theta$ are said to be \textit{infinitesimal deformations} of $\omega$ and $\theta$, respectively, if we have
\[
d\dot\omega+\theta\wedge\dot\omega+\dot\theta\wedge\omega=0
\]
in the sense that if we choose a local trivialization and if we represent $\omega,\theta,\dot{\omega}$ and $\dot\theta$ by components, then we have
\[
d\dot\omega^i+\theta^i{}_\alpha\wedge\dot{\omega}^\alpha+\dot{\theta}^i{}_\alpha\wedge\omega^\alpha=0.
\]
\end{definition}
By Lemma~\ref{lem7.3} and Theorem~\ref{thm6.16}, we have the following
\begin{theorem}
\label{thm7.12}
There is a one-to-one correspondence between the following\textup{:}
\begin{enumerate}
\item
Pairs of Bott connections on $Q(\mathcal{F})$ and their infinitesimal deformations.
\item
Bott connections on $P^1(\mathcal{F})\ltimes\underline{\mathfrak{gl}_q}(\mathbb{R})$.
\item
Sections from $P^1(\mathcal{F})\ltimes\underline{\mathfrak{gl}_q}(\mathbb{R})$ to $\widetilde{P}^2(\mathcal{F})\ltimes\underline{\widetilde{\mathfrak{g}}^2_q}$ equivariant under the $\mathrm{GL}_q(\mathbb{R})\ltimes\mathfrak{gl}_q(\mathbb{R})$-action and preserving foliations.
\end{enumerate}
\end{theorem}
The normal bundle $Q(\mathcal{F})$ is also equipped with a foliation.
Indeed, if $(x,y,v)$ denote the coordinates for $Q(\mathcal{F})$ on a foliation chart, then we have $(\widehat{x},\widehat{y},\widehat{v})=(\psi(x,y),\gamma(y),D\gamma_y(v))$ so that foliations locally defined by $y=\text{const.}$ and $v=\text{const.}$ give rise to a foliation of $Q(\mathcal{F})$ which is represented by $\mathcal{F}^{(2)}$.
It is easy to see that the normal bundle $Q(\mathcal{F}^{(2)})$ is the $2$-normal bundle $Q^{(2)}(\mathcal{F})$ in~\cite{asuke:2015-2}.
The bundle $Q^{(2)}(\mathcal{F})$ plays a role of $T^2M$.
Namely, we always have the \textit{vertical subbundle} of $Q^{(2)}(\mathcal{F})$ which we represent as $Q(\mathcal{F})^V$.
On the other hand, given a Bott connection on $Q(\mathcal{F})$, we can define the \textit{horizontal subbundle} of $Q^{(2)}(\mathcal{F})$ which we represent by $Q(\mathcal{F})^H$.
If $\varpi$ denotes the projection from $Q(\mathcal{F})$ to $M$, then the both lifts are isomorphic to $\varpi^*(Q(\mathcal{F}))$.
If in addition an infinitesimal deformation of the Bott connection is given, then we can define a connection on $Q^{(2)}(\mathcal{F})$ by considering $\begin{pmatrix}
\theta\\
\dot{\theta} & \theta
\end{pmatrix}$.
This is the construction given in~\cite{asuke:2015-2}, where characteristic classes for infinitesimal deformations of foliations are studied by means of these connections (cf.~\cite{Fuks}, \cite{13}).
They are obtained as differential forms on $Q(\mathcal{F})$ and then shown to project down to $M$.
If we make use of Theorem~\ref{thm7.12}, then we can obtain a Bott connection on $P^1(\mathcal{F})\ltimes\mathfrak{gl}_q(\mathbb{R})$ which is valued in the Lie algebra of $\mathrm{GL}_q(\mathbb{R})\ltimes\mathfrak{gl}_q(\mathbb{R})\cong T\mathrm{GL}_q(\mathbb{R})$, and we can avoid bundles over $Q(\mathcal{F})$ to obtain these classes.
If we denote by $\mathcal{B}$ the space of Bott connections on $P^1(\mathcal{F})\ltimes\underline{\mathfrak{gl}_q}(\mathbb{R})$, then these classes are functionals on $\mathcal{B}$.
For example, we can consider the derivative of the Godbillon--Vey class, $D\mathrm{GV}$ for short, with respect to infinitesimal deformations of foliations.
It is known that if the foliation under consideration admits a transverse projective structure (not necessarily flat), then $D\mathrm{GV}$ vanishes for any infinitesimal deformations of foliations~\cite{asuke:2015}.
This means that if $\mathcal{F}$ admits a transverse projective structure, then $D\mathrm{GV}$ as a functional on $\mathcal{B}$ is identically equal to zero.
Similarly, some characteristic classes are introduced for deformations of flat connections in~\cite{Lue}.
If $\mathcal{C}$ denotes the space of connections on $P^1(M)\ltimes\underline{\mathfrak{gl}_n}(\mathbb{R})$, then these classes can be regarded as functionals on $\mathcal{C}$ and results can be understood as properties of such functionals.
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\begin{document}
\baselineskip=17pt
\title{Littlewood's principles in reverse real analysis}
\author{Rafael Reno S. Cantuba\\
Department of Mathematics and Statistics\\
De La Salle University\\
2401 Taft Ave., Manila\\
1004 Metro Manila, Philippines\\
E-mail: rafael\[email protected]}
\date{}
\maketitle
\renewcommand{\arabic{footnote}}{}
\footnote{2020 \emph{Mathematics Subject Classification}: Primary 28A20; Secondary 26A03, 12J15.}
\footnote{\emph{Key words and phrases}: Littlewood's three principles, Lusin's theorem, Egoroff's theorem, completeness axiom, ordered field.}
\renewcommand{\arabic{footnote}}{\arabic{footnote}}
\setcounter{footnote}{0}
\begin{abstract}
If local forms of Littlewood's three principles are stated as axioms for an ordered field, then each principle is equivalent to the completeness axiom.
\end{abstract}
\section{Introduction}
In the past decade, the papers \cite{dev14,pro13,tei13} gave an engaging account of how some of the traditional theorems from calculus and real analysis are each equivalent to the least upper bound property of, or completeness axiom for, the field $\mathbb{R}$ of all real numbers. In a mathematical theory, theorems are proven from a collection of axioms, but in the aforementioned papers, the opposite thought process was exhibited: in the appropriate universe of discourse (which in this case is the class of all ordered fields) selected traditional theorems were each stated as an axiom (for an arbitrary ordered field) and the statement of the completeness axiom was proven as a consequence. As the author, J. Propp, of \cite{pro13} pointed out, this intellectual exercise has the flavor of \emph{Reverse Mathematics} \cite[p. 392]{pro13}, which is interesting in its own right. See, for instance, \cite[Section 1.1]{sim09} or \cite[Chapter 1]{sti18}. Propp further pointed out that this kind of investigation ``sheds light on the landscape of mathematical theories and structures,'' and that ``arguably the oldest form of mathematics in reverse'' is the quest for a list of equivalences for the parallel postulate in Euclidean geometry \cite[pp. 392--393]{pro13}. An independent work \cite{tei13} gave more technical proofs and initiated a list of `completeness properties' for an ordered field that was expanded in \cite{dev14}, so that, currently, 72 characterizations of the completeness axiom have been identified. An almost full cast of the traditional calculus theorems, ranging from the Intermediate Value Theorem to convergence tests and L'H\^opital's rule, appear in the list, but we also find some forms of the Arzel\`a-Ascoli Theorem, and the Lipschitz property of $C^1$ functions. This paper is inspired by the question, of possibly including in the list, some of the theorems of measure theory. We decided to start with the most fundamental\textemdash Littlewood's three principles, the well-known heuristics for understanding measure theory.
A quote from \emph{the} J. E. Littlewood is now inevitable. From \cite[p. 26]{lit44}, the three principles are:
\begin{enumerate}\item\label{prin1} Every measurable set is nearly a finite sum (meaning union) of intervals.
\item\label{prin2} Every (measurable) function is nearly continuous.
\item\label{prin3} Every convergent sequence of (measurable) functions is nearly uniformly convergent.
\end{enumerate}
The principle \ref{prin2} is also known as Lusin's Theorem \cite[pp. 72, 74]{roy88}, while \ref{prin3} is also referred to as Egoroff's Theorem \cite[p. 40]{lit44}. We shall keep the form of each of the above principles as a `one-directional' implication. For instance, \ref{prin2} may be rephrased as ``If $f$ is a measurable function, then $f$ is nearly continuous.'' In the textbooks, we often find the biconditional form ``$f$ is a measurable function if and only if $f$ is nearly continuous,'' but this is \emph{not} what we shall use. The choice of which statement appears in the hypothesis and which statement appears in the conclusion of the one-directional implication is based on how J. E. Littlewood originally stated the principles: the `nearly' part is always in the conclusion of the conditional statement. Principles \ref{prin1} and \ref{prin3} shall be handled similarly. Also, we shall be using `local' forms of such principles. That is, there is a given closed and bounded interval $I$ (in an arbitrary ordered field) such that we shall consider only the measurable sets contained in $I$ and functions with domain $I$ (or a subset of $I$). This `local' perspective means that, among the many forms of the first principle, we shall indeed be using that form which involves the symmetric difference of two sets, where the second set is the union of a finite number of intervals, which was originally in the aforementioned quote from J. E. Littlewood.
The issue of how to define Lebesgue measure and Lebesgue integrals in an arbitrary ordered field $\mathbb{F}$ was one of the first issues we had to deal with. If the usual outer measure approach is to be used, then we have to define outer measure as the infimum of some set, but then, the \emph{Existence of Infima}, or the assertion that any nonempty subset of $\mathbb{F}$ that has a lower bound has an infimum, is one of the equivalent forms of the completeness axiom \cite[p. 108]{tei13}, yet $\mathbb{F}$ need not be complete. What we found as a suitable approach is the Riesz method \cite[Chapter II]{cha95}, in which step functions form the starting point for establishing the definition of the Lebesgue integral, and the measurability of a set is defined later as the Lebesgue measurability of its characteristic function.
\section{Preliminaries}\label{PrelSec}
Let $\mathbb{F}$ be an ordered field. Thus, $\mathbb{F}$ has characteristic zero, and consequently, contains a subfield isomorphic to the field $\mathbb{Q}$ of all rational numbers. The usual ordering in $\mathbb{Q}$ is consistent with the order relation $<$ on $\mathbb{F}$ defined by $a<b$ if and only if $b-a\in\mathbb{P}$, where $\mathbb{P}$ is the set of all positive elements of $\mathbb{F}$. The relation $<$ on $\mathbb{F}$ obeys a \emph{Trichotomy Law} which states that for any $a,b\in\mathbb{F}$, exactly one of the assertions $a<b$, $b<a$ or $a=b$, is true.
The relation $>$ is defined by $a>b$ if and only if $b<a$, and the relations $<$ and $>$ may be extended in the usual manner to obtain the relations $\leq$ and $\geq$, respectively. Using these order relations, some standard notions of analysis may be defined for the field $\mathbb{F}$.
Given $a,b\in\mathbb{F}$, the open interval with left endpoint $a$ and right endpoint $b$ is $\lpar a,b\rpar:=\{x\in\mathbb{F}\ :\ a<x<b\}$. The intervals $\lbrak a,b\rbrak$, $\left( a,b\right]$ and $\left[ a,b\right)$ may be defined in the obvious manner, by taking the union of $\lpar a,b\rpar$ with one or both of its endpoints. The length of any of the four aforementioned intervals is defined to be $b-a$.
Given functions $f,g:\lbrak a,b\rbrak\longrightarrow\mathbb{F}$, by $f\geq g$, we mean $f(x)\geq g(x)$ for all $x\in\lbrak a,b\rbrak$. A function $f:\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ is continuous at $c\in\lbrak a,b\rbrak$ if, for each $\varepsilon\in\mathbb{P}$, there exists $\delta\in\mathbb{P}$ such that for any $x\in\lbrak a,b\rbrak\cap\left( c-\delta,c+\delta\right)$, $f(x)-f(c)\in\left(-\varepsilon,\varepsilon\right)$. We say that $f:\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ is a continuous function if $f$ is continuous at each element of $\lbrak a,b\rbrak$.
Since $\mathbb{F}$ contains $\mathbb{Q}$ as a subfield, we may view the set of all positive integers $\mathbb{N}\subseteq\mathbb{Q}$ as a subset of $\mathbb{F}$. A sequence in $\mathcal{S}\subseteq\mathbb{F}$ is a function\linebreak $a:\mathbb{N}\longrightarrow\mathcal{S}$. The traditional notation is $a_n:=a(n)$ for any $n\in\mathbb{N}$, and instead of referring to $a$ as a sequence, we say that $\lpar a_n\rpar$ is a sequence. If indeed $\lpar a_n\rpar$ is a sequence in some subset of $\mathbb{F}$, $a_n$ may further be equal to some other expression determined by $n$. A sequence $\lpar a_n\rpar$ converges to $L\in\mathbb{F}$ if, for each $\varepsilon\in\mathbb{P}$, there exists $N\in\mathbb{N}$ such that for any $n\in\mathbb{N}$ with $n\geq N$, we have $a_n-L\in\left(-\varepsilon,\varepsilon\right)$. If indeed $\lpar a_n\rpar$ converges to $L$, then using the Trichotomy Law in $\mathbb{F}$, the element $L$ is unique, and we define $\displaystyle\lim_{n\rightarrow\infty} a_n:=L$. Given sequences $\lpar a_n\rpar$ and $\lpar b_n\rpar$ such that for each $n\in\mathbb{N}$, $b_n=\displaystyle\sum_{k=1}^na_k$, if $\lpar b_n\rpar$ converges, then $\displaystyle\sum_{k=0}^\infty a_k:=\displaystyle\lim_{n\rightarrow\infty} b_n$.
Given $\lbrak a,b\rbrak\subseteq\mathbb{F}$, we shall also be considering sequences of functions $\lbrak a,b\rbrak\longrightarrow\mathcal{S}$ for some $\mathcal{S}\subseteq\mathbb{F}$. By such, we simply mean that each $n\in\mathbb{N}$ is assigned to a unique function $\varphi_n:\lbrak a,b\rbrak\longrightarrow\mathcal{S}$, and we denote the function sequence by $\left(\varphi_n\right)$. We say that $\left(\varphi_n\right)$ is \emph{monotonically decreasing} if $\varphi_{n}\geq \varphi_{n+1}$ for any $n\in\mathbb{N}$. Let $\left(\varphi_n\right)$ be a sequence of functions $\lbrak a,b\rbrak\longrightarrow\mathcal{S}$. We say that the functions in $\left(\varphi_n\right)$ \emph{converge uniformly} to a function $f:\lbrak a,b\rbrak\longrightarrow\mathcal{S}$ if for each $\varepsilon\in\mathbb{P}$, there exists $N\in\mathbb{N}$ such that for any $n\geq N$ and any $x\in\lbrak a,b\rbrak$, $f(x)-\varphi_n(x)\in\left(-\varepsilon,\varepsilon\right)$. If indeed the functions in $\left(\varphi_n\right)$ converge to $f$ and each $\varphi_n$ is a continuous at any element of $\lbrak a,b\rbrak$, then by routine instantiation of quantifiers, $f$ is also continuous at any element of $\lbrak a,b\rbrak$.
By a \emph{partition} of $\lbrak a,b\rbrak\neq\emptyset$, we mean a finite subset of $\lbrak a,b\rbrak$ that contains the endpoints $a$ and $b$. If $a=b$, then the only possible partition of $\lbrak a,b\rbrak$ is the singleton $\{a\}=\{b\}$. For the non-degenerate case, which is when $a<b$, we use the traditional notation, in which, if indeed we have a partition $\Delta$ of $\lbrak a,b\rbrak$ with cardinality $n$, then the elements of $\Delta$ are indexed as\linebreak $a=x_0<x_1<\cdots<x_n=b$. A function $\varphi:\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ is a \emph{step function} if there exists a partition $\{x_0,x_1,\ldots,x_n\}$ of $\lbrak a,b\rbrak$ such that for each $i\in\{1,2,\ldots,n\}$, the restriction $\left.\varphi\right|_{\left( x_{i-1},x_i\right)}$ is a constant function, that is, a function with a one-element range, say $\{M_i\}$ for some $M_i\in\mathbb{F}$. In such a case, the integral of $\varphi$ over $\lbrak a,b\rbrak$ is defined as
\begin{eqnarray}
\displaystyle\int_a^b\varphi:=\sum_{i=1}^nM_i(x_i-x_{i-1}).\label{IntegralStep}
\end{eqnarray}
For the degenerate case $a=b$, the summation in \eqref{IntegralStep} is an empty sum, and so $\displaystyle\int_a^b\varphi=0$. If $a<b$, and if $M_i\in\mathbb{P}\cup\{0\}$ for any $i\in\{1,2,\ldots,n\}$, then $\displaystyle\int_a^b\varphi\geq 0$.
Consider an arbitrary nondegenerate $\lbrak a,b\rbrak\subseteq\mathbb{F}$. We say that $\mathcal{S}\subseteq\lbrak a,b\rbrak$ is a \emph{null set} or \emph{has measure zero} if, for each $\varepsilon\in\mathbb{P}$, there exists a countable collection $\{\left( a_n,b_n\right)\subseteq\lbrak a,b\rbrak\ :\ n\in\mathbb{N}\}$ of open intervals such that
\begin{eqnarray}
\mathcal{S}\subseteq\bigcup_{n=1}^\infty\left( a_n,b_n\right),\qquad\sum_{n=1}^\infty(b_n-a_n)<\varepsilon.\nonumber
\end{eqnarray}
A statement $\mathscr{P}(x)$ is said to hold \emph{almost everywhere in $\lbrak a,b\rbrak$} if\linebreak $\{x\in\lbrak a,b\rbrak\ :\ \neg\mathscr{P}(x)\}$ has measure zero. We say that a sequence $\left(\varphi_n\right)$ of functions $\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ converges to a function $f:\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ almost everywhere in $\lbrak a,b\rbrak$ if $f(x)=\displaystyle\lim_{n\rightarrow\infty}\varphi_n(x)$ almost everywhere in $\lbrak a,b\rbrak$. A function $f:\lbrak a,b\rbrak\longrightarrow\mathbb{P}\cup\{0\}$ is \emph{Lebesgue measurable} if there exists a monotonically decreasing sequence $\left(\varphi_n\right)$ of step functions $\lbrak a,b\rbrak\longrightarrow\mathbb{P}\cup\{0\}$ that converge to $f$ almost everywhere in $\lbrak a,b\rbrak$. A function $f:\lbrak a,b\rbrak\longrightarrow\mathbb{P}\cup\{0\}$ \emph{has a Lebesgue integral} if $f$ is Lebesgue measurable and if there exists $\displaystyle\int_a^bf\in\mathbb{F}$ such that for any sequence $\left(\varphi_n\right)$ of monotonically decreasing step functions $\lbrak a,b\rbrak\longrightarrow\mathbb{P}\cup\{0\}$ that converge to $f$ almost everywhere in $\lbrak a,b\rbrak$, the sequence $\left(\displaystyle\int_a^b\varphi_n\right)$ of integrals converges to $\displaystyle\int_a^bf$. The notion of Lebesgue measurability and of having a Lebesgue integral may then be extended to a function $\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ in the usual manner, which is by considering the nonnegative and negative `parts' of a function.
For any sets $X$ and $Y$, we define $X\backslash Y:=\{x\in X\ :\ x\notin Y\}$,\linebreak and $X\triangle Y:=(X\backslash Y)\cup(Y\backslash X)$.
We say that $E\subseteq\lbrak a,b\rbrak$ is a \emph{measurable set} if its \emph{characteristic function}
\begin{eqnarray}
\chi_E(x):=\begin{cases}1, & \mbox{if }x\in E,\\
0, & \mbox{if }x\in \lbrak a,b\rbrak\backslash E,\end{cases}\nonumber
\end{eqnarray}
is a measurable function. If $E$ has measure zero, or $\chi_E$ has a Lebesgue integral, then $E$ \emph{has Lebesgue measure}. In the former case, $\displaystyle\int_a^b\chi_E=0$.
By a \emph{cut}\footnote{This definition of cut was taken from \cite[p. 60]{mon08}. We chose it because it is apparently more concise. The definition in \cite{dev14,pro13,tei13} is based on the more traditional, which is that a cut is a pair of subsets of $\mathbb{F}$.} of $\mathbb{F}$ we mean a nonempty proper subset $A$ of $\mathbb{F}$ such that for any $a\in A$ and any $b\in\mathbb{F}\backslash A$, $a<b$. We say that $c\in\mathbb{F}$ is a \emph{cut point} of a cut $A\subseteq\mathbb{F}$ if for any $a\in A$ and any $b\in\mathbb{F}\backslash A$, $a\leq c\leq b$. A cut of $\mathbb{F}$ that does not have a cut point is a \emph{gap}. We say that $\mathbb{F}$ is a \emph{complete ordered field} if $\mathbb{F}$ satisfies the axiom:
\begin{enumerate}
\item[\mathbb{C}A] {\it Cut Axiom.} Every cut of $\mathbb{F}$ is not a gap.
\end{enumerate}
Otherwise, $\mathbb{F}$ is said to be \emph{incomplete}.
Let $a\in\mathbb{F}$. If $b\in\mathbb{F}$ such that $a\leq b$, then we define $\max\{a,b\}:=b$, $\min\{a,b\}:=a$, and $|a|:=\max\{-a,a\}$.
\section{Equivalence of Littlewood's Principles to the Cut Axiom}
We now prove the equivalence of Littlewood's three principles to \mathbb{C}A, with a couple more statements added to our list of equivalences.
\begin{theorem} For an arbitrary ordered field $\mathbb{F}$, each of the statements
\begin{enumerate}\item[{\bf(LIP)}] \emph{Lebesgue Integral Property.}\footnote{The names {\bf(LIP)}\ and {\bf(LMP)}\ are not popularly used in standard real analysis. Also, these two statements reduce to trivialities when $\mathbb{F}$ is complete. We chose to name these two statements as such, in analogy to what in \cite{dev14} was called the \emph{Darboux Integral Property}, which states that every Darboux integrable function $f:\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ has a Darboux integral. According to \cite[p. 271]{dev14}, the Darboux Integral Property (in conjunction with some other statement) is one of the equivalent forms of \mathbb{C}A.} Given $\lbrak a,b\rbrak\subseteq\mathbb{F}$, every Lebesgue measurable function $\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ has a Lebesgue integral.
\item[{\bf(LMP)}] \emph{Lebesgue Measure Property.} Given $\lbrak a,b\rbrak\subseteq\mathbb{F}$, every measurable subset of $\lbrak a,b\rbrak$ has Lebesgue measure.
\item[{\bf(LP1)}] \emph{Littlewood's First Principle.} Given $\lbrak a,b\rbrak\subseteq\mathbb{F}$, for each measurable set $E\subseteq\lbrak a,b\rbrak$ and each $\varepsilon\in\mathbb{P}$, there exist intervals $I_1,I_2,\ldots, I_m\subseteq\lbrak a,b\rbrak$ such that if $U=\displaystyle\bigcup_{i=1}^mI_i$, then $\displaystyle\int_a^b\chi_{E\triangle U}<\varepsilon$.
\item[{\bf(LP1)}i] \emph{Littlewood's Second Principle.} Given $\lbrak a,b\rbrak\subseteq\mathbb{F}$, for each Lebesgue measurable function $f:\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ and each $\varepsilon\in\mathbb{P}$, there exists a measurable set $E\subseteq\lbrak a,b\rbrak$ such that $f:\lbrak a,b\rbrak\backslash E\longrightarrow\mathbb{F}$ is a continuous function and that $\displaystyle\int_a^b\chi_{E}<\varepsilon$.
\item[{\bf(LP1)}ii] \emph{Littlewood's Third Principle.} Given $\lbrak a,b\rbrak\subseteq\mathbb{F}$, for each sequence $\left( \varphi_n\right)$ of Lebesgue measurable functions $\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ that converge to\linebreak $f:\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ almost everywhere in $\lbrak a,b\rbrak$, and each $\varepsilon\in\mathbb{P}$, there exists a measurable set $E\subseteq\lbrak a,b\rbrak$ such that the functions $\varphi_n:\lbrak a,b\rbrak\backslash E\longrightarrow\mathbb{F}$ converge uniformly to $f:\lbrak a,b\rbrak\backslash E\longrightarrow\mathbb{F}$, and that $\displaystyle\int_a^b\chi_{E}<\varepsilon$.
\end{enumerate}
is equivalent to \mathbb{C}A.
\end{theorem}
\begin{proof} We shall prove
\begin{eqnarray}
\mbox{\bf(CA)} \Longrightarrow \mbox{\bf(LIP)} \Longrightarrow \mbox{\bf(LMP)} \Longrightarrow \mathbb{C}A,\label{TheCycle1}\\
\mbox{\bf(CA)} \Longrightarrow \mbox{\bf(LP1)}ii \Longrightarrow \mbox{\bf(LP1)}i \Longrightarrow \mbox{\bf(LP1)} \Longrightarrow \mathbb{C}A.\label{TheCycle2}
\end{eqnarray}
Let {\bf($\star$)}\ be one of the statements {\bf(LIP)}, {\bf(LMP)}, {\bf(LP1)}, {\bf(LP1)}i, or {\bf(LP1)}ii. From standard real analysis, {\bf($\star$)}\ is true for the ordered field $\mathbb{R}$. Suppose $\mathbb{F}$ is an ordered field that satisfies $\neg\mbox{\bf($\star$)}$. Then $\mathbb{F}$ cannot be $\mathbb{R}$, but we have a well-known fact\footnote{See, for instance, \cite[pp. 601--605]{spi08}.} that any complete ordered field is isomorphic to $\mathbb{R}$, so $\mathbb{F}$ is incomplete, and thus, $\neg\mbox{\bf(CA)}$ holds in $\mathbb{F}$. We have thus proven\linebreak $\mbox{\bf(CA)}\Longrightarrow\mbox{\bf($\star$)}$ by contraposition. In particular,
\begin{eqnarray}
\mbox{\bf(CA)} &\Longrightarrow& \mbox{\bf(LIP)},\nonumber\\
\mbox{\bf(CA)} & \Longrightarrow & \mbox{\bf(LP1)}ii.\nonumber
\end{eqnarray}
The implication
\begin{eqnarray}
\mbox{\bf(LIP)} &\Longrightarrow& \mbox{\bf(LMP)},\nonumber
\end{eqnarray}
is trivial, while {\bf(LP1)}i\ is well-known as a consequence of {\bf(LP1)}ii. One proof of
\begin{eqnarray}
\mbox{\bf(LP1)}ii\Longrightarrow\mbox{\bf(LP1)}i,\nonumber
\end{eqnarray}
can be found in \cite[p. 110]{cha95}, and in this proof, there is a straightforward use of {\bf(LP1)}ii\ to carry out a `modus ponens' argument to prove {\bf(LP1)}i, and all notions [mainly, continuity and uniform convergence] used in the proof are valid for the arbitrary ordered field $\mathbb{F}$, where such necessary notions have been defined for $\mathbb{F}$ in Section~\ref{PrelSec}. To complete the proof of \eqref{TheCycle1}--\eqref{TheCycle2}, only three implications remain, which we prove in the following.
\newline
\noindent $\neg\mbox{\bf(CA)}\Longrightarrow \neg\mbox{\bf(LMP)}$. Suppose $\mathbb{F}$ is incomplete. We proceed by contradiction, so suppose {\bf(LMP)}\ holds. By \cite[Lemma~B, p. 110]{tei13}, there exist a gap $A\subseteq\mathbb{F}$ and some strictly increasing and non-convergent sequence $\lpar a_n\rpar$ in $A$ such that
\begin{eqnarray}
\forall x\in A\quad \left[\ a_1<x \ \Longrightarrow\ \exists!n\in\mathbb{N}\ x\in\left( a_n,a_{n+1}\right]\ \right].\label{TeiSeq}
\end{eqnarray}
Let $a:=a_1$, let $b\in\mathbb{F}\backslash A$, and let $E:=\lbrak a,b\rbrak\backslash A$. For each $n\in\mathbb{N}$, define $\varphi_n:\lbrak a,b\rbrak\longrightarrow\mathbb{F}$ by $\varphi_n:x\mapsto 1-\chi_{\left[ a,a_{n}\right]}(x)$. Since, for any $n\in\mathbb{N}$,\linebreak $\left[ a,a_n\right]\subseteq\left[ a,a_{n+1}\right]$, we find that $\left(\varphi_n\right)$ is a monotonically decreasing sequence of step functions, with $\displaystyle\int_a^b\varphi_n=b-a_{n}$ for any $n$. Let $x\in\lbrak a,b\rbrak$, and let $\varepsilon\in\mathbb{P}$. If $x\in A$, then $\chi_E(x)=0$, and by \eqref{TeiSeq}, there exists $N\in\mathbb{N}$ such that for any $n\geq N+1$, $\varphi_n(x)=0$, and furthermore, $\chi_E(x)-\varphi_n(x)=0\in\left(-\varepsilon,\varepsilon\right)$. If $x\in\lbrak a,b\rbrak\backslash A$, since $\lpar a_n\rpar$ is a sequence in $A$, by the definition of cut, we have, for any $n\in N$, $x>a_{n}$, so $x\notin\left[ a,a_{n}\right]$. This implies $\varphi_n(x)=1$, but $\chi_E(x)=1$, so we further obtain $\chi_E(x)-\varphi_n(x)=0\in\left(-\varepsilon,\varepsilon\right)$. We have thus shown that the statement $\chi_E(x)=\displaystyle\lim_{n\rightarrow\infty}\varphi_n(x)$ is true for any $x\in\lbrak a,b\rbrak$, and consequently, almost everywhere in $\lbrak a,b\rbrak$. This means that $E$ is measurable, and by the {\bf(LMP)}, there exists $\displaystyle\int_a^b\chi_E\in\mathbb{F}$ such that $\displaystyle\int_a^b\chi_E=\displaystyle\lim_{n\rightarrow\infty}\displaystyle\int_a^b\varphi_n=\displaystyle\lim_{n\rightarrow\infty}(b-a_n)$. That is, the sequence $\left( b-a_n\right)$ converges to $\displaystyle\int_a^b\chi_E$. Routine arguments may be used, to show that, in $\mathbb{F}$, the usual linearity and constant rules for sequence limits hold, so $(a_n-b)$ converges to $-\displaystyle\int_a^b\chi_E$ and taking the sum of $(a_n-b)$ with the constant sequence $(b)$, we find that $\lpar a_n\rpar$ converges to $b-\displaystyle\int_a^b\chi_E$, contradicting the fact that $\lpar a_n\rpar$ is non-convergent. Therefore, $E$ does not have Lebesgue measure.\newline
\noindent $\mbox{\bf(LP1)}i\Longrightarrow\mbox{\bf(LP1)}$. Let $\varepsilon\in\mathbb{P}$. If $E\subseteq\lbrak a,b\rbrak$ is a measurable set, then $\chi_E$ is a measurable function, and by {\bf(LP1)}i, there exists a measurable set $F\subseteq\lbrak a,b\rbrak$ such that $\chi_E:\lbrak a,b\rbrak\backslash F\longrightarrow\mathbb{P}\cup\{0\}$ is a continuous function and that $\displaystyle\int_a^b\chi_{F}<\frac{\varepsilon}{2}$. Since the integral $\displaystyle\int_a^b\chi_{F}$ of the nonnegative characteristic function $\chi_F$, is nonnegative,
\begin{eqnarray}
-\frac{\varepsilon}{2}<\displaystyle\int_a^b\chi_{F}<\frac{\varepsilon}{2},\label{xLebInt}
\end{eqnarray}
which, in particular, means that $F$ has Lebesgue measure. Since $ F$ is measurable, there exists a monotonically decreasing sequence $\left(\varphi_n\right)$ of step functions $\lbrak a,b\rbrak\longrightarrow\mathbb{P}\cup\{0\}$ that converge to $\chi_{ F}$ almost everywhere in $\lbrak a,b\rbrak$, and that
\begin{eqnarray}
\displaystyle\int_a^b\chi_{ F} = \displaystyle\lim_{n\rightarrow\infty}\displaystyle\int_a^b\varphi_n.\nonumber
\end{eqnarray}
Consequently, there exists $N\in\mathbb{N}$ such that for all $n\geq N$,
\begin{eqnarray}
-\frac{\varepsilon}{2}< \displaystyle\int_a^b\varphi_n-\displaystyle\int_a^b\chi_{ F}<\frac{\varepsilon}{2},\nonumber
\end{eqnarray}
which, in conjunction with \eqref{xLebInt}, completes the proof that
\begin{eqnarray}
\displaystyle\lim_{n\rightarrow\infty}\displaystyle\int_a^b\varphi_n=0.\label{xLebInt2}
\end{eqnarray}
For each $m\in\mathbb{N}$, define $P_m=\{x\in\lbrak a,b\rbrak\ :\ \chi_{ F}\geq\frac{1}{m}\}$. Since $\chi_{ F}$ takes on only values of $0$ or $1$,
\begin{eqnarray}
\bigcup_{m\in\mathbb{N}} P_m =\{x\in\lbrak a,b\rbrak\ :\ \chi_{ F}(x)=1\}= F.\label{xLebIntII2}
\end{eqnarray}
Since $\left(\varphi_n\right)$ is monotonically decreasing and converges to $\chi_F$ at $x$, by a routine argument, for any $n\in\mathbb{N}$, $\varphi_n\geq \chi_{ F}$, and so,
\begin{eqnarray}
\varphi_n(x)\geq\frac{1}{m},\label{xLebIntII1}
\end{eqnarray}
for all $x\in P_m$. Since $\varphi_n$ is a step function, if we take the values of $\varphi_n$ that are at least $\frac{1}{m}$, then we have a finite subset of $\mathbb{P}\cup\{0\}$, the inverse image of which, under $\varphi_n$, is the union of a finite number of intervals, and this union contains $P_m$. Let such intervals be collected in the set $\mathscr{C}_n$, and let $\Sigma_n$ be the sum of the lengths of the intervals in $\mathscr{C}_n$. By \eqref{xLebIntII1}, $\displaystyle\int_a^b\varphi_n\geq \frac{\Sigma_n}{m}$, and by \eqref{xLebInt2}, given $\varepsilon\in\mathbb{P}$, there exists $N\in\mathbb{N}$ such that for all $n\geq N$,
$\frac{\displaystyle\Sigma_n}{m}\leq \displaystyle\int_a^b\varphi_n<\frac{\varepsilon}{m}$, and so, $\Sigma_n<\varepsilon$. At this point, we have proven that, for any $m\in\mathbb{N}$, $P_m$ has measure zero, and by \eqref{xLebIntII2}, so does $ F$.
Let $\left( \phi_n\right)$ be a monotonically decreasing sequence of step functions\linebreak $\lbrak a,b\rbrak\longrightarrow\mathbb{P}\cup\{0\}$ such that $\chi_E(x)=\displaystyle\lim_{n\rightarrow\infty}\phi_n(x)$ for all $x\in\lbrak a,b\rbrak\backslash Q$, where $Q$ has measure zero. For each $n\in\mathbb{N}$, define $\psi_n:\lbrak a,b\rbrak\longrightarrow\mathbb{P}\cup\{0\}$ by
\begin{eqnarray}
\psi_n(x):=\begin{cases} 1, & \mbox{if }\phi_n(x)\geq \frac{1}{2},\\
0, & \mbox{if }\phi_n(x)<\frac{1}{2},\end{cases}\nonumber
\end{eqnarray}
which is a step function. By a routine epsilon argument,
\begin{eqnarray}
\displaystyle\lim_{n\rightarrow\infty}\left|\psi_n(x)-\chi_E(x)\right|=0,\label{Lusin1}
\end{eqnarray}
for any $x\in\lbrak a,b\rbrak\backslash Q$. By an argument similar to that done earlier in this proof, we have $\psi_n=\chi_{B_n}$, where $B_n$ is the union of a finite number of intervals in $\lbrak a,b\rbrak$. By a routine argument, $\left|\psi_n-\chi_E\right|=\chi_{E\triangle B_n}$. Let $D_n$ be the set of all discontinuities of $\psi_n$. Since $\psi_n$ is a step function, and $\chi_E:\lbrak a,b\rbrak\backslash F\longrightarrow\mathbb{P}\cup\{0\}$ is a continuous function, the set $D_n\cup F$ of all discontinuities of $\chi_{E\triangle B_n}=\left|\psi_n-\chi_E\right|$ is a set of measure zero, and so is $R:=Q\cup F\cup\displaystyle\bigcup_{n\in\mathbb{N}}D_n$.
Let $x\in\lbrak a,b\rbrak\backslash R$. Thus, for each $n\in\mathbb{N}$, $\chi_{E\triangle B_n}$ is continuous at $x$, and since $\chi_{E\triangle B_n}$ takes on a value of only $0$ or $1$, we find that there exists an interval $Z_x\subseteq\lbrak a,b\rbrak$ that contains $x$ and that $\chi_{E\triangle B_n}$ is zero on all of $Z_x$, or there exists an interval $O_x\subseteq\lbrak a,b\rbrak$ that contains $x$ and that $\chi_{E\triangle B_n}$ has value $1$ on all of $O_x$. Let $\varepsilon_0=\min\left\{1,\varepsilon\right\}$. By \eqref{Lusin1}, there exists $N\in\mathbb{N}$ such that for all $n\geq N$,
\begin{eqnarray}
-1 < & \chi_{E\triangle B_n}(x) & < 1,\nonumber
\end{eqnarray}
and since $\chi_{E\triangle B_n}(x)$ can be only $0$ or $1$, we only have $\chi_{E\triangle B_n}(x)=0$, so $O_x=\emptyset$. Thus, $x\in Z_x$, and we have proven
\begin{eqnarray}
\lbrak a,b\rbrak\backslash R\subseteq\bigcup_{x\in\lbrak a,b\rbrak\backslash R}Z_x\subseteq\{x\in\lbrak a,b\rbrak\ :\ \chi_{E\triangle B_n}=0\},\nonumber
\end{eqnarray}
from which we deduce that $E\triangle B_n=\{x\in\lbrak a,b\rbrak\ :\ \chi_{E\triangle B_n}\neq 0\}\subseteq R$.
At this point, we have proven that for all $n\geq N$, $\chi_{E\triangle B_n}$ has measure zero, and hence has Lebesgue measure. In particular, $\displaystyle\int_a^b\chi_{E\triangle B_n}=0$. Thus, we may state that, for each $\varepsilon\in\mathbb{P}$, there exists a union $U=B_N$ of a finite number of intervals such that $ \displaystyle\int_a^b\chi_{E\triangle U}=0<\varepsilon$. Therefore, $\mathbb{F}$ satisfies {\bf(LP1)}.
\newline
\noindent $\neg\mbox{\bf(CA)}\Longrightarrow\neg\mbox{\bf(LP1)}$. Suppose $\mathbb{F}$ is incomplete, and that, tending towards a contradiction, {\bf(LP1)}\ holds. As shown in the proof of $\neg\mbox{\bf(CA)}\Longrightarrow \neg\mbox{\bf(LMP)}$, there exist a gap $A\subseteq\mathbb{F}$, some $a\in A$ and $b\in F\backslash A$ such that $E:=\lbrak a,b\rbrak\backslash A$ is a measurable set, but does not have Lebesgue measure. Let $\varepsilon\in\mathbb{P}$. By {\bf(LP1)}, there exist intervals $I_1,I_2,\ldots,I_\mu\subseteq\lbrak a,b\rbrak$ such that if $U=\displaystyle\bigcup_{k=1}^\mu I_k$, then $\displaystyle\int_a^b\chi_{E\triangle U}<\frac{\varepsilon}{2}$. By an argument similar to one of those that were done in the proof of $\mbox{\bf(LP1)}i\Longrightarrow\mbox{\bf(LP1)}$, $E\triangle U$ has measure zero.
Without loss of generality, we assume $I_1,I_2,\ldots,I_\mu$ are pairwise disjoint, and that, for some index $K\in\{1,2,\ldots,\mu\}$, the interval $I_K$, if nonempty, intersects both $\lbrak a,b\rbrak\cap A$ and $\lbrak a,b\rbrak\backslash A$, and we further assume that all intervals $I_k$ with $k<K$ are subsets of $\lbrak a,b\rbrak\backslash A$, while all intervals $I_k$ with $k>K$ are subsets of $\lbrak a,b\rbrak\cap A$.
Let $T:=\displaystyle\bigcup_{k=1}^KI_k$. Since $E\triangle U$ has measure zero, the sets $I_K\backslash E\subseteq E\triangle U$ and $E\triangle T\subseteq E\triangle U$ also have measure zero. Furthermore, the equation
\begin{eqnarray}
\chi_E(x)=\chi_T(x)+\chi_{E\triangle T}(x),\label{LebInt4}
\end{eqnarray}
is true for all $x\in\lbrak a,b\rbrak$ except those $x\in I_K\backslash E$. Hence, \eqref{LebInt4} is true almost everywhere in $\lbrak a,b\rbrak$. Since $E\triangle T$ has measure zero, it has Lebesgue measure, so $\chi_{E\triangle T}$ has a Lebesgue integral. The other function in the right-hand side of \eqref{LebInt4}, which is $\chi_T$, is a step function because $T$ is an interval, and thus, $\chi_T$ also has a Lebesgue integral, and, since \eqref{LebInt4} is true almost everywhere in $\lbrak a,b\rbrak$, by a routine argument, we find that $\chi_E$ also has a Lebesgue integral, contradicting the fact that $E$ does not have Lebesgue measure. Therefore, {\bf(LP1)}\ is false in $\mathbb{F}$.
\end{proof}
\end{document} |
\begin{eqnarray}gin{document}
\hoffset -34pt
\title{{\large \bf A note on the asymptotic behavior of conformal metrics with negative
curvatures near isolated singularities}
\thanks{\mbox{Keywords. Singularities, curvatures, metrics.}}}
\author{\normalsize Tanran Zhang }
\date{}
\boldsymbol{a}ketitle \begin{array}selineskip 21pt
\noindent
\begin{eqnarray}gin{minipage}{138mm}
\renewcommand{\begin{array}selinestretch}{1} \normalsize
\begin{eqnarray}gin{abstract}
{The asymptotic behavior of conformal metrics with negative curvatures near an isolated singularity for at most second order derivatives was described by Kraus and Roth in one of their papers in 2008. Our work improves one estimate of theirs and shows the estimate for higher order derivatives near an isolated singularity by means of potential theory. We also give some limits of Minda-type for SK-metrics near the origin. Combining these limits with the Ahlfors' lemma, we provide some observations SK-metrics.}
\end{abstract}
\end{minipage}\\
\\
\\
\renewcommand{\begin{array}selinestretch}{1} \normalsize
\section{Introduction}
\vspace*{4mm}
The research of conformal metrics has a long history, since the time of Liouville and Picard, see \cite{Liouville1, Picard,Picard2}. For a conformal metric $\lambda(z)|dz|$ on a subdomain $G$ of the complex plane $\boldsymbol{a}thbb{C}$, we can define its (generalized) Gaussian curvature $\kappa_{\lambda}(z)$. Let $u(z)= \log \lambda(z)$. If $\kappa_{\lambda}(z)=0$, then $u(z)$ satisfies the Laplace equation $\Delta u=0$, which means $u(z)$ is harmonic on $G$. So that the property of $u(z)$ can be studied by means of potential theory, see, e.g. \cite{PDE}. If $\kappa_{\lambda}(z)=-4$, then $\Delta \log \lambda= 4\lambda^2$ and $u(z)$ is the solution to the Liouville equation
\begin{eqnarray}\label{liouville equation}
\Delta u=4e^{2u}.
\end{eqnarray}
Each solution to equation (\ref{liouville equation}) belongs to a class of subharmonic functions and it is corresponding to a kind of special metric, called the SK-metric, according to Heins, see \cite{Heins}. The existence and the uniqueness of the solutions to equation (\ref{liouville equation}) are subject to some suitable boundary conditions. Through out our study, we are concerned only with the asymptotic behavior near an isolated singularity of the solution to equation (\ref{liouville equation}), so it is sufficient to consider the behavior in the punctured unit disk $\boldsymbol{a}thbb{D} \begin{array}ckslash\{0\}$, where the origin is an isolated singularity of some order $\alpha\leq 1$. Near the singularity, we need some more refined invariants to estimate the asymptotic behavior, like the growth of the density. The assignment of the order of the singularity is such an invariant.
\partialr As for equation (\ref{liouville equation}), Liouville proved that, in any disk $D$ contained in the punctured unit disk $\boldsymbol{a}thbb{D}\begin{array}ckslash\{0\}$ every solution $u$ to \eqref{liouville equation} can be written as $$u(z)=\log\frac{|f'(z)|}{1-|f(z)|^2},$$
where $f$ is a holomorphic function in $D$, see \cite{Roth1}. Based on Liouville's results, Nitsche described the behavior of $u(z)$ with constant curvature $\kappa(z)\equiv -4$ near the isolated singularities on plane domains in \cite{Nitsche}. Subsequently, Kraus and Roth extended Nitsche's results to the solutions of the more general equation
\begin{eqnarray}\label{general equation}
\Delta u=-\kappa(z) e^{2u}
\end{eqnarray}
with strictly negative, H\"{o}lder continuous curvature functions $\kappa(z)$ in \cite{Rothbehaviour}. In fact, equation (1.2) has an exquisite geometric interpretation: Every solution $u$ to \eqref{general equation} induces a conformal metric $e^{u(z)}|dz|$ with Gaussian curvature function $\kappa(z)$ and vice versa (for more details, see \cite{Rothbehaviour}). Our first result is the estimates for some terms of $u(z)$ near the singularity. We improve the estimate of the mixed derivatives when the order of $u$ is $\alpha=1$ and obtain the estimate for higher order derivatives near the origin. We show in \cite{zhang3} that, our result is sharp by use of the generalized hyperbolic metric $\lambda_{\alpha, \begin{eqnarray}ta, \gamma}$ on the thrice-punctured sphere $\boldsymbol{a}thbb{P}\begin{array}ckslash \{z_1, z_2, z_3\}$ with singularities of order $\alpha, \begin{eqnarray}ta, \gamma \leq 1$ at $z_1, z_2, z_3$, which was given by Kraus, Roth and Sugawa for $\alpha+\begin{eqnarray}ta+\gamma >2$, see \cite{Rothhyper}.
\partialr As an extremal case of the SK-metric, the hyperbolic metric, also called the Poincar{\'e} metric, plays an important role on (punctured) disks. Early in 1997, Minda \cite{Mindametric} investigated the behavior of the density of the hyperbolic metric in a neighborhood of a puncture on the plane domain using the uniformization theorem. His method offers us a way to describe the asymptotic behavior on an arbitrary hyperbolic region. Our second result is to extend Minda's work and to give some limits of Minda's type.
\partialr This paper is divided into four sections. In Section 2 the notations and the definitions are introduced. Section 3 is contributed to potential theory. The main results and their proofs are given in Section 4.
\section {Preliminaries }
\subsection{Singularities and orders}
\setcounter{equation}{0}
\vspace*{3mm}
If $G \subseteq \boldsymbol{a}thbb{C}$ is a domain, then every positive, upper semi-continuous, real-valued function $\lambda: G \rightarrow (0, + \infty)$ on $G$ induces a conformal metric $\lambda(z)|dz|$, see \cite{Heins,Roth1}. In our discussion we take the linear notation for a conformal metric $ds=\lambda(z)|dz|$. Let $\boldsymbol{a}thbb{P}$ denote the Riemann sphere $\boldsymbol{a}thbb{C}\cup \{\infty\}$ and let $\Omega\subseteq\boldsymbol{a}thbb{P}$ be a subdomain. For a point $p \in \Omega$, let $z$ be local coordinates such that $z(p)=0$. We say a conformal metric $\lambda(z)|dz|$ on the punctured domain $\Omega^*:=\Omega \begin{array}ckslash \{p\}$ has a singularity of order $\alpha\leq 1$ at the point $p$, if, in local coordinates $z$,
\begin{eqnarray} \label{singularity}
\log\lambda(z)=\left\{
\begin{eqnarray}gin{array}{ll}
-\alpha\log|z|+v(z) & \mbox{if\ }\ \alpha < 1 \\
-\log|z|-\log\log(1/|z|)+w(z)&\mbox{if\ }\ \alpha=1,\end{array} \right.
\end{eqnarray}
where $v(z), w(z) = \textit {O}(1)$ as $z(p)\rightarrow 0$ with $\textit {O}$ and $\textit {o}$ being the Landau symbols throughout our study. Let $M_u(r):=\sup_{|z|=r}u(z)$ for a real-valued function $u(z)$ defined in a punctured neighborhood of $z=0$ and call
\begin{eqnarray}\label{order of u}
\alpha(u):=\lim_{r\rightarrow 0^+}\frac{M_u(r)}{\log(1/r)}
\end{eqnarray}
the order of $u(z)$ if this limit exists. For $u(z):=\log \lambda(z)$, $\alpha(u)$ in (\ref{singularity}) is equal to $\alpha$ in \eqref{order of u}. In fact, if $\alpha(u)\leq 1$ in \eqref{order of u}, then $v(z)$ is continuous at $z=0$ and $w(z)=\textit {O}(1)$ as $z \rightarrow 0$, see Theorem 3.1 in \cite{Rothbehaviour}. We call the point $p$ a conical singularity or corner of order $\alpha$ if $\alpha< 1$ and a cusp if $\alpha=1$. The generalized Gaussian curvature $\kappa_{\lambda}(z)$ of the density function $\lambda(z)$ is defined by
$$\kappa_{\lambda}(z)=-\frac{1}{\lambda(z)^2}{\liminf_{r\rightarrow 0}\frac{4}{r^2}\left(\frac 1 {2\pi}\int_0^{2\pi}\log\lambda(z+re^{it})dt-\log\lambda(z)\right)}.$$
We say a conformal metric $\lambda(z)|dz|$ on a domain $\Omega\subseteq\boldsymbol{a}thbb{C}$ is regular, if its density $\lambda(z)$ is positive and twice continuously differentiable, i.e. $\lambda(z)>0$ and $\lambda(z) \in C^2(\Omega)$. If $\lambda(z)|dz|$ is a regular conformal metric, then $$\kappa_{\lambda}(z)=-\frac{\Delta\log\lambda(z)}{\lambda(z)^2},$$
where $\Delta$ denotes the Laplace operator. It is well known that, if $a<\kappa_{\lambda}(z)<b<0$ with constants $a, b \in \boldsymbol{a}thbb{R}$, the metric $\lambda(z)|dz|$ only has corners or cusps at isolated singularities (see \cite{McOwen1}).
\partialr The Gaussian curvature is a conformal invariant. Let $\lambda(z)|dz|$ be a conformal metric on a domain $G \in \boldsymbol{a}thbb{C}$ and $f: \Omega \rightarrow G$ be a holomorphic mapping of a Riemann surface $\Omega$ into $G$. Then we can define the pullback $f^{*}\lambda(w)|dw|$ of $\lambda(z)|dz|$ by
\begin{eqnarray}
f^{*}\lambda(w)|dw|:=\lambda(f(w))|f'(w)||dw|. \nonumber
\end{eqnarray}
It is easy to see that $f^{*}\lambda(w)|dw|$ is a conformal metric on $\Omega \begin{array}ckslash\{\mbox{critical points of}\ f\}$ with Gaussian curvature
\begin{eqnarray}
\kappa_{f^{*}{\lambda}}(w)=\kappa_{\lambda}(f(w)). \nonumber
\end{eqnarray}
Using this conformal invariance, we can easily build relations between Riemann surfaces with conformal metrics. Here we can see that, on the punctured domain $\Omega \begin{array}ckslash\{\mbox{critical points}$ $\mbox{of}\ f\}$, the critical points of $f$ are the source of the singularities.
\partialr The hyperbolic metric is a complete metric with some constant Gaussian curvature, here we take the constant to be $-4$. We call an upper semi-continuous metric $\lambda(z)|dz|$ on a Riemann surface $\Omega$ an SK-metric if its Gaussian curvature is bounded above by $-4$ at every $z\in \Omega$. The hyperbolic metric on the unit disk $\boldsymbol{a}thbb{D}$ is defined by
\begin{eqnarray} \label{hyperbolic metric}
\lambda_{\boldsymbol{a}thbb{D}}(z)|dz|=\frac {|dz|}{1-{|z|}^2}.
\end{eqnarray}
The following result is a fundamental theorem about SK-metrics by Ahlfors, see \cite{Ahlforslemma}, also \cite{Heins}, which claims that the hyperbolic metric $\lambda_{\boldsymbol{a}thbb{D}}(z)|dz|$ on the unit disk $\boldsymbol{a}thbb{D}$ is the unique maximal SK-metric on $\boldsymbol{a}thbb{D}$.
\begin{eqnarray}gin{theorema} $\boldsymbol{a}thrm{[1]}.$ \label{Ahlfors lemma}
\textsl{Let $ds$ be the hyperbolic metric on $\boldsymbol{a}thbb{D}$ given in (\ref{hyperbolic metric}) and $d\ell$ be the metrics on $\boldsymbol{a}thbb{D}$ induce by an SK-metric on a Riemann surface $\Omega$. If the function $f(z)$ is analytic in $\boldsymbol{a}thbb{D}$, then the inequality
\begin{eqnarray}
d\ell \leq ds \nonumber
\end{eqnarray}
will hold throughout the circle. }
\end{theorema}
On the punctured unit disk $\boldsymbol{a}thbb{D}^*:=\boldsymbol{a}thbb{D} \begin{array}ckslash \{0\}$, the hyperbolic metric is expressed by
$$\lambda_{\boldsymbol{a}thbb{D}^*}(z)|dz|=\frac{|dz|}{2|z|\log(1/|z|)}$$
with the constant curvature $-4$. We denote $\boldsymbol{a}thbb{D}_R:=\{z \in {\boldsymbol{a}thbb{C}}:|z|<R\}$ and ${\boldsymbol{a}thbb{D}_R}^*:=\boldsymbol{a}thbb{D}_R\begin{array}ckslash \{0\}$ for $R>0$. On the punctured disk $\boldsymbol{a}thbb{D}_R^*$, the (generalized) hyperbolic metric with a conical singularity at the origin is given in \cite{Rothhyper}. For its detailed proof, see \cite{zhang2}.
\begin{eqnarray}gin{theorema} $\boldsymbol{a}thrm{[7,14]}.$ \label{maximal}
\textsl{ For $R>0$, let
\begin{eqnarray}n
\lambda_{\alpha,R}(z):=\left\{
\begin{eqnarray}gin{array}{ll}
\displaystyle \frac{(1-\alpha)R^{1-\alpha}|z|^{-\alpha}}{R^{2(1-\alpha)}-|z|^{2(1-\alpha)}} =\frac{1-\alpha}{2|z|\sinh \left((1-\alpha)\log ({R}/{|z|})\right)} & \mbox{if\ }\ \alpha<1, \\
\displaystyle \frac{1}{2|z|\log ({R}/{|z|})} &\mbox{if\ }\ \alpha=1
\end{array} \right.
\end{eqnarray}n
for $z \in \boldsymbol{a}thbb{D}_R^*$. Then given an arbitrary SK-metric $\sigma(z)$ on $\boldsymbol{a}thbb{D}_R^*$ with a singularity at $z=0$ of order $\alpha$, we have $\sigma(z)\leq\lambda_{\alpha,R}(z)$.}
\end{theorema}
\subsection{Regularity and Logarithmic potential}
If equation \eqref{general equation} has a $C^2$-solution $u(z)$, then the higher regularity properties of $u(z)$ only depends on the smoothness of $\kappa(z)$, according to Gilbarg and Trudinger [2, p.\,109]. Here we need the H\"{o}lder spaces $C^{n, \,\nu}(\boldsymbol{a}thbb{D}_R)$, consisting of functions whose $n$-th order partial derivatives are locally H\"{o}lder continuous with exponent $\nu$ in $\boldsymbol{a}thbb{D}_R$, $0<\nu \leq 1$, which are defined as the subspaces of $C^n(\boldsymbol{a}thbb{D}_R)$. The following result can be obtained immediately from the standard regularity theorem, see, e.g. [2,\,Theorem 6.17].
\begin{eqnarray}gin{lemma} $\boldsymbol{a}thrm{(Regularity\ theorem)}$ \label{regularity coro}
\textsl{ Let $u$ be a $C^2$-solution to the equation $\Delta u=-\kappa(z) e^{2u}$ in $\boldsymbol{a}thbb{D}^*$, where $\kappa \in C^{n,\,\nu}(\boldsymbol{a}thbb{D}^*)$. Then $u \in C^{n+2,\,\nu}(\boldsymbol{a}thbb{D}^*)$. If $\kappa$ lies in $C^{\infty}(\boldsymbol{a}thbb{D}^*)$, then $u \in C^{\infty}(\boldsymbol{a}thbb{D}^*)$.}
\end{lemma}
\vspace*{2mm}
We shall use potential theory as employed by Kraus and Roth in \cite{Rothbehaviour}. Here we list some elementary facts without proof. \vspace*{2mm}
For a bounded, integrable function $f(z)$ defined on a domain $\Omega \subseteq \boldsymbol{a}thbb{C}$, the integral
$$\frac {1}{2\pi}\int _{\Omega}L(z-\zeta) f(\zeta) d\sigma_{\zeta}$$
is called the logarithmic potential of $f$, where $L(z-\zeta)=\log|z-\zeta|$ and $d\sigma_{\zeta}$ is the area element on domain $\Omega$. Write $z=x_1+ix_2$, $\zeta=y_1+iy_2$ and set $0<r\leq 1$. The following lemma was mentioned in \cite{Rothbehaviour}. It is a consequence of the famous Riesz decomposition theorem, and can be obtained from Theorem 4.5.1 and Exercise 3.7.3 in \cite{Ransford}.
\begin{eqnarray}gin{lemmaa} $\boldsymbol{a}thrm{[6]}.$ \label{poisson jensen}
\textsl{Let $u$ be a subharmonic function on $\boldsymbol{a}thbb{D}_r$ such that $u\in C^2({\boldsymbol{a}thbb{D}_r}^*)$, $\Delta u$ is integrable in $\boldsymbol{a}thbb{D}_r$ and
$$\lim_{r\rightarrow 0}\frac{\sup_{|z|=r}u(z)}{\log(1/r)}=0.$$
Then $u(z)=h(z)+\omega(z)$ for $z\in \boldsymbol{a}thbb{D}_r$, where $h$ is a harmonic function on $\boldsymbol{a}thbb{D}_r$ and $\omega(z)$ is the logarithmic potential of $\Delta u$.}
\end{lemmaa}
\begin{eqnarray}gin{lemmaa} $\boldsymbol{a}thrm{[2,\,p.\:\!54]},$ \label{newton potential}
\textsl{ Let $f: \boldsymbol{a}thbb{D}_r\rightarrow\boldsymbol{a}thbb{R}$ be a locally bounded, integrable function in $\boldsymbol{a}thbb{D}_r$ and $\omega$ be the logarithmic potential of $f$. Then $\omega \in C^1({\boldsymbol{a}thbb{D}_r})$ and for any $z=x_1+ix_2 \in \boldsymbol{a}thbb{D}_r$,
$$\frac{\partial\:\!{\omega}}{\partial x_j}(z)=\frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_r}\frac {\partial L}{\partial x_j}(z-\zeta) f(\zeta) d\sigma_{\zeta}$$ for $j\ \in \{1,2\}$.\\
If, in addition, $f$ is locally H\"{o}lder continuous with exponent $\nu \leq 1$, then $\omega\in C^2(\boldsymbol{a}thbb{D}_r)$ and for $z \in \boldsymbol{a}thbb{D}_r$,
\begin{eqnarray}
&&\frac{\partial^2\:\!\omega}{\partial x_l\partial x_j}(z)=\frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_R}\frac{\partial^2 L}{\partial x_l\partial x_j}(z-\zeta)\left(f(\zeta)-f(z)\right) d\sigma_{\zeta} \nonumber\\
&& \qquad \qquad \quad \: \ \,-\frac{1}{2\pi}f(z)\int_{\partial \boldsymbol{a}thbb{D}_R}\frac{\partial L}{\partial x_j}(z-\zeta)N_l(\zeta)|d\zeta|, \nonumber
\end{eqnarray}
where $N(\zeta)=(N_1(\zeta),N_2(\zeta))$ is the unit outward normal at the point $\zeta \in\partial \boldsymbol{a}thbb{D}_R$ with $R>r$ and $f$ is extended to vanish outside of $\boldsymbol{a}thbb{D}_r$.}
\end{lemmaa}
\partialr There is a similar proposition for higher order derivatives of the logarithmic potential. Define a multi-index $\boldsymbol{j}=(j_1, j_2)$, $|\boldsymbol{j}|=j_1+j_2$, $j_1, j_2=0,1,2,\ldots \,$, so $(\zeta-z)^{\boldsymbol{j}}=(y_1-x_1)^{j_1}(y_2-x_2)^{j_2}$, $\boldsymbol{j} !=j_1!j_2!$. For $z=x_1+ix_2$, denote $$\frac{\partial}{\partial x_1}=\partial_1,\ \frac{\partial}{\partial x_2}=\partial_2,\ \displaystyle \partial^{\boldsymbol{j}}=\partial^{j_1}_1 \partial^{j_2}_2.$$ For a given multi-index $\boldsymbol{j}=(j_1, j_2)$, we can choose $\boldsymbol{e}_{\tau}=(0,1)$ or $(1,0)$ for $\tau=1,2,\ldots \,$ such that $\boldsymbol{j}=\boldsymbol{e}_1+\boldsymbol{e}_2+\cdots+\boldsymbol{e}_n$ with $n=|\boldsymbol{j}|$. Write $\zeta=y_1+iy_2$, set \begin{eqnarray}n
P_n[f](z,\zeta):=\left\{
\begin{eqnarray}gin{array}{ll}
\vspace*{2mm}
\displaystyle \sum_{|\boldsymbol{a}|\leq n} \frac{(\zeta-z)^{\boldsymbol{a}}{\partial}^{\boldsymbol{a}}}{\boldsymbol{a} !}f(z) & \mbox{if\ }\ n \geq 1 \\
f(z) & \mbox{if\ }\ n=0,
\end{array} \right.
\end{eqnarray}n
where $\boldsymbol{a}$ is a multi-index. For $m=1,2$, it holds that $$\frac{\partial P_n[f]}{\partial y_m}(z,\zeta)=P_{n-1}[\partial_m f](z,\zeta),$$
see \cite{zhang2} for more details. Using this notation, we can present the analogue of Lemma \ref{newton potential} as follows.
\begin{eqnarray}gin{lemmaa} $\boldsymbol{a}thrm{[14]}.$ \label{new higher prop}
\textsl{ Let $0<r<1$, $f: \boldsymbol{a}thbb{D}_r\rightarrow\boldsymbol{a}thbb{R}$ and $f \in C^{n-2,\,\nu}(\boldsymbol{a}thbb{D}_r)$ with $0<\nu \leq 1$, $n\geq 3$, $\omega$ be the logarithmic potential of $f$. Then
$\omega(z)\in C^{n}(\boldsymbol{a}thbb{D}_r)$ and for $n=|\boldsymbol{j}|$,
\begin{eqnarray} \label{new higher}
&& \partial^{\boldsymbol{j}}\omega(z) \nonumber \\
&=& \frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_R}\partial^{\boldsymbol{j}} L(z-\zeta) \cdot \left(f(\zeta)-P_{n-2}[f](z,\zeta)\right)d\sigma_{\zeta} \nonumber \\
&& -\frac{1}{2\pi} \sum^{n-1}_{\tau=1} \int_{\partial \boldsymbol{a}thbb{D}_R}\partial^{\boldsymbol{\theta}_{\tau}} L(z-\zeta) \cdot P_{\tau-1}[\partial^{\boldsymbol{\phi}_{\tau}}f](z,\zeta)\cdot \langle N(\zeta),\boldsymbol{e}_{\tau+1} \rangle |d\zeta|,
\end{eqnarray}
where $\boldsymbol{\theta}_{\tau}:=\boldsymbol{e}_1+ \cdots +\boldsymbol{e}_{\tau}$, $\boldsymbol{\phi}_{\tau}:=\boldsymbol{e}_{\tau+2}+ \cdots +\boldsymbol{e}_n$ for $\tau=1, \ldots, \,n-1$ and $\boldsymbol{\phi}_{n-1}:=(0,0)$, $N(\zeta)=(N_1(\zeta),N_2(\zeta))$ is the unit outward normal at the point $\zeta \in \partial \boldsymbol{a}thbb{D}_R$ with $R>r$, $\langle \ , \ \rangle$ is the inner product and the function $f$ is extended to vanish outside of $\boldsymbol{a}thbb{D}_r$.}
\end{lemmaa}
\section{Main estimates}
\setcounter{equation}{0}
We denote $$\partial^n=\frac{\partial ^n}{\partial z^n},\ \begin{array}r{\partial}^n=\frac {\partial^n}{\partial\begin{array}r{z}^n}$$ for $n\geq1$. The following theorem is given by Kraus and Roth in \cite{Rothbehaviour}.
\begin{eqnarray}gin{theorema} $\boldsymbol{a}thrm{[6]}.$ \label{original}
\textsl{ Let $\kappa:\boldsymbol{a}thbb{D}\rightarrow\boldsymbol{a}thbb{R}$ be a locally H\"{o}lder continuous function with $\kappa(0)<0$. If $u:\boldsymbol{a}thbb{D}^*\rightarrow \boldsymbol{a}thbb{R}$ is a $C^2$-solution to $\Delta u=-\kappa(z) e^{2u}$ in $\boldsymbol{a}thbb{D}^*$, then $u$ has an order $\alpha \in (-\infty,1]$ and
\begin{eqnarray}gin{align}
&u(z)=-\alpha\log|z|+v(z), & \textrm{if\ } \ \alpha<1,\nonumber \\
&u(z)=-\log|z|-\log\log(1/|z|)+w(z), & \textrm{if\ } \ \alpha=1,\nonumber
\end{align}
where the remainder functions $v(z)$ and $w(z)$ are continuous in $\boldsymbol{a}thbb{D}$. Moreover, the first partial derivatives with respect to $z$ and $\begin{array}r{z}$,
\begin{eqnarray}gin{align}
& \partial v(z),\,\begin{array}r{\partial} v(z)\ \mbox{are continuous at}\ z=0& \mbox{if\ }&\ \alpha<1/2;\nonumber
\end{align}
and
\begin{eqnarray}gin{align}
& \partial v(z),\,\begin{array}r{\partial} v(z)=\textit {O}(1) &\mbox{if\ }& \ \alpha=1/2;\nonumber\\
& \partial v(z),\,\begin{array}r{\partial} v(z)=\textit {O}(|z|^{1-2\alpha}) &\mbox{if\ }&\ 1/2<\alpha<1,\nonumber\\
& \partial w(z),\,\begin{array}r{\partial} w(z)=\textit {O}(|z|^{-1}(\log(1/|z|))^{-2}) &\mbox{if\ }& \ \alpha=1,\nonumber
\end{align}
when $z$ approaches $0$. In addition, the second partial derivatives,
\begin{eqnarray}gin{align}
&\partial^2 v(z),\,\partial \begin{array}r{\partial}v(z) \ \textrm{and}\ \begin{array}r{\partial}^2 v(z) \ \textrm{are\ continuous \ at}\ z=0 & \textrm{if\ }& \ \alpha \leq 0;\nonumber\end{align}
and
\begin{eqnarray}gin{align}
& \partial^2 v(z),\,\partial \begin{array}r{\partial}v(z),\,\begin{array}r{\partial}^2 v(z)= \textit {O}(|z|^{-2\alpha}) &\textrm{if\ }&\ 0<\alpha<1,\nonumber\\
& \partial^2 w(z),\,\partial \begin{array}r{\partial} w(z),\,\begin{array}r{\partial}^2 w(z)=\textit {O}(|z|^{-2}(\log(1/|z|))^{-2}) & \textrm{if\ }& \ \alpha=1,
\end{align}
when $z$ tends to $z=0$.}
\end{theorema}
\partialr In the work of Kraus and Roth, the proof of Theorem \ref{original} was based on Lemma \ref{newton potential}. Since we have obtained a similar statement in Lemma \ref{new higher prop}, the estimate for higher order derivatives of the remainder functions $v(z)$, $w(z)$ can be given. We consider $v(z)$ and $w(z)$ separately.
\begin{eqnarray}gin{theorem} \label{estimate v}
\textsl{ Let $\kappa(z)$, $u(z)$, $v(z)$ and $\alpha$ be the same as in Theorem \ref{original}. If $0<\alpha<1$ and if, in addition, $\kappa(z) \in C^{n-2,\,\nu}(\boldsymbol{a}thbb{D}^*)$ for an integer $n \geq 3$, $0<\nu \leq 1$, then for $n_1$, $n_2 \geq 1$, $n_1+n_2 =n$, near the origin, the remainder function $v(z)$ satisfies
$$\partial^n v(z),\,\begin{array}r{\partial}^n v(z),\,\begin{array}r{\partial}^{n_1}\partial^{n_2}v(z)=\textit{O}(|z|^{2-2\alpha-n}).$$}
\end{theorem}
\textbf{Proof.} Lemma \ref{regularity coro} shows that $u(z) \in C^{n,\,\nu}(\boldsymbol{a}thbb{D}^*)$. Due to Kraus and Roth \cite{Rothbehaviour} we have $$v(z)=h(z)+\frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_r}L(z-\zeta)f(\zeta)
d\sigma_{\zeta}$$ for $z\in \boldsymbol{a}thbb{D}_r^*$, $0<r<1$ and a harmonic function $h$ on $\boldsymbol{a}thbb{D}_r$, where $q(z)=-\kappa(z)e^{2v(z)}$, $f(z)=q(z)|z|^{-2\alpha}$. Now fix $0<R<1$, choose $z \in \boldsymbol{a}thbb{D}_{R/2}^*$ and let $r=|z|/2$. Then for a multi-index $\boldsymbol{j}$, $|\boldsymbol{j}|=n \geq 3$, rearranging (\ref{new higher}) leads to
\begin{eqnarray} \label{for v}
&&\partial^{\boldsymbol{j}}v(z) \nonumber \\
&=&\partial^{\boldsymbol{j}}h(z)+\frac{1}{2\pi}\int_{\boldsymbol{a}thbb{D}_{R}\begin{array}ckslash \boldsymbol{a}thbb{D}_{r}}\partial^{\boldsymbol{j}}L(z-\zeta)f(\zeta)d\sigma_{\zeta}+ \frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{j}}L(z-\zeta)
\left( f(\zeta)-f(z)\right)d\sigma_{\zeta} \nonumber\\
&&+\frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{j}}L(z-\zeta)\sum_{1\leq |\boldsymbol{a}|\leq n} \frac{(\zeta-z)^{\boldsymbol{a}}\partial^{\boldsymbol{a}}f(z)}{\boldsymbol{a}!}d\sigma_{\zeta} \nonumber\\
&&-\frac{1}{2\pi} \sum^{n-1}_{\tau=1} \int_{\partial \boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{\theta}_{\tau}} L(z-\zeta) \cdot P_{\tau-1}[\partial^{\boldsymbol{\phi}_{\tau}}f](z,\zeta)\cdot \langle N(\zeta),\boldsymbol{e}_{\tau+1} \rangle |d\zeta|
\end{eqnarray}
for $z=x_1+ix_2$ and a harmonic function $h$ on $\boldsymbol{a}thbb{D}_R$, and the same symbols $\boldsymbol{\theta}_{\tau}$, $\boldsymbol{\phi}_{\tau}$ are used here as in \eqref{new higher}.
It is known that
\begin{eqnarray} \label{logn}
\left| \partial^{\boldsymbol{j}}L(z-\zeta) \right|
\leq \frac{n!}{|z-\zeta|^{n}},
\end{eqnarray}
see [2, p.\,17]. Denote $M=\sup_{\zeta \in \boldsymbol{a}thbb{D}_R}|q(\zeta)|$ and let $C_n>0$, $n \in \boldsymbol{a}thbb{N}$, be some constants. Then
$$\left| \int_{\boldsymbol{a}thbb{D}_{R}\begin{array}ckslash \boldsymbol{a}thbb{D}_{r}}\partial^{\boldsymbol{j}}L(z-\zeta)f(\zeta)d\sigma_{\zeta} \right| \leq M \int_{\boldsymbol{a}thbb{D}_{R}\begin{array}ckslash \boldsymbol{a}thbb{D}_{r}}
\frac{n!}{|z-\zeta|^{n}} \frac{1}{|\zeta|^{2\alpha}} d\sigma_{\zeta} \leq
\frac{C_1}{|z|^{2\alpha+n-2}},$$
and
\begin{eqnarray}
&&\left| \int_{\boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{j}}L(z-\zeta)
\left( f(\zeta)-f(z)\right)d\sigma_{\zeta} \right| \nonumber \\
&\leq& \int_{\boldsymbol{a}thbb{D}_{r}} \frac{n!}{|z-\zeta|^{n}} \frac{|q(\zeta)-q(z)|}{|\zeta|^{2\alpha}} d\sigma_{\zeta}
+ M \int_{\boldsymbol{a}thbb{D}_{r}}\frac{n!}{|z-\zeta|^{n}}
\frac{(|\zeta|^{\alpha}+|z|^{\alpha})||\zeta|^{\alpha}-|z|^{\alpha}|}{|z|^{2\alpha}|\zeta|^{2\alpha}} d\sigma_{\zeta} \nonumber \\
&\leq& \frac{C_2}{|z|^{2\alpha+n-2}}, \nonumber
\end{eqnarray}
see \cite{Rothbehaviour}. When one of $j_1$ and $j_2$ is zero, there is no cancelation. Thus we obtain
$\partial^n v, \begin{array}r{\partial}^n v=\textit{O}(|z|^{2-2\alpha-n}).$ If neither $j_1$ nor $j_2$ is zero, the first three integrals in (\ref{for v}) are canceled, so we have to consider the last term in (\ref{for v}). In the last sum in (\ref{for v}) letting $\tau=1$, we get the integral
$$\partial^{\boldsymbol{\phi}_1}f(z) \cdot \int_{\partial \boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{e}_1} L(z-\zeta) \cdot \langle N(\zeta),\boldsymbol{e}_{2} \rangle |d\zeta|. $$
Writing $\zeta=re^{i\theta}$ and taking $\boldsymbol{e}_1=(1,0)$, $\boldsymbol{e}_2=(0,1)$ without loss of generality, we have
\begin{eqnarray} \label{sincos}
&&\left|\int_{\partial \boldsymbol{a}thbb{D}_r}\partial_1L(z-\zeta)N_2(\zeta)|d\zeta|\right|=\left|\int_{\partial \boldsymbol{a}thbb{D}_r}\frac{x_1-r\cos\theta}{|z-\zeta|^2}\sin\theta|d\zeta|\right|\nonumber\\
&=&\left|\int^{2\pi}_0 \frac{x_1-r\cos\theta}{|z-\zeta|^2} r \sin\theta d\theta\right| \leq 2\pi \frac{|x_1-r\cos\theta|}{|z-\zeta|^2} r |\sin\theta| \leq 6\pi.
\end{eqnarray}
So it is evident that,
$$\left|\int_{\partial \boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{e}_1} L(z-\zeta) \cdot \langle N(\zeta),\boldsymbol{e}_{2} \rangle |d\zeta| \right| \leq 6\pi$$
holds for all kinds of $\boldsymbol{e}_1$ and $\boldsymbol{e}_2$. Now consider $\partial^{\boldsymbol{\phi}_1}f(z)$. Since $f(z)=q(z)|z|^{-2 \alpha}$, then the term $q(z)\cdot \partial^{\boldsymbol{\phi}_1}(|z|^{-2 \alpha})$ appears with some coefficient. Note that
$$\left|\partial^{\boldsymbol{\phi}_1} \left(\frac{1}{|z|^{2\alpha}}\right)\right| \leq \frac{C_3}{|z|^{2\alpha+n-2}},$$
so
$$\left| q(z) \partial^{\boldsymbol{\phi}_1} \left(\frac{1}{|z|^{2\alpha}}\right) \cdot \int_{\partial \boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{e}_1} L(z-\zeta) \cdot \langle N(\zeta),\boldsymbol{e}_{2} \rangle |d\zeta| \right| \leq \frac{6\pi M C_3}{|z|^{2\alpha+n-2}}.$$ Therefore $\begin{array}r{\partial}^{n_1}\partial^{n_2}v=\textit{O}(|z|^{2-2\alpha-n}). $ \hspace*{\fill} $\Box$\\
\partialr The following result is for the higher order derivatives of the remainder functions $w(z)$ when the order is $1$.
\begin{eqnarray}gin{theorem} \label{for w theorem}
\textsl{ Let $\kappa(z)$, $u(z)$, $w(z)$ and $\alpha$ be the same as in Theorem \ref{original}. If $\alpha=1$ and if, in addition, $\kappa(z) \in C^{n-2,\,\nu}(\boldsymbol{a}thbb{D}^*)$ for an integer $n \geq 3$, $0<\nu \leq 1$, then for $n_1$, $n_2 \geq 1$, $n_1+n_2 =n$, near the origin, the remainder functions $w(z)$ satisfies
\begin{eqnarray}
\begin{array}r{\partial}^nw(z),\,\partial^nw(z)=\textit {O}(|z|^{-n}(\log(1/|z|))^{-2}), \nonumber \\
\begin{array}r{\partial}^{n_1}\partial^{n_2}w(z)=\textit {O}(|z|^{-n}(\log(1/|z|))^{-3}). \label{new mixed}
\end{eqnarray}}
\end{theorem}
\partialr The proof is based on the following lemma.
\begin{eqnarray}gin{lemmaa}\label{estimate of kzew-1} $\boldsymbol{a}thrm{[6]}.$ \ \textsl{Let $\kappa:\boldsymbol{a}thbb{D}\rightarrow\boldsymbol{a}thbb{R}$ be a continuous function with $\kappa(0)<0$ and
\begin{eqnarray}
\kappa(z)=\kappa(0)+\frac{s(z)}{(\log(1/|z|))^2},\nonumber\end{eqnarray}
where $s(z)=\textit {O}(1)$ as $z\rightarrow 0$. If $u: \boldsymbol{a}thbb{D}^* \rightarrow\boldsymbol{a}thbb{R}$ is a solution to $\Delta u=-\kappa(z) e^{2u}$ with $u(z)=-\log|z|-\log\log(1/|z|)+w(z)$ where $w(z)=\textit {O}(1)$ for $z\rightarrow0$, then there exists a disk $\boldsymbol{a}thbb{D}_{\rho}$ such that
\begin{eqnarray} \label{lemma}
\left|-\kappa(z) e^{2w(z)}-1\right|\leq\frac C {\log(1/|z|)},\ \ \ z\in \boldsymbol{a}thbb{D}_{\rho},
\end{eqnarray}
for some constant $C>0$.}
\end{lemmaa}
\textbf{Proof of Theorem \ref{for w theorem}.} Lemma \ref{regularity coro} shows that $u(z) \in C^{n,\,\nu}(\boldsymbol{a}thbb{D}^*)$.
For $w(z)$ defined as in Theorem \ref{original}, we first show that
\begin{eqnarray}\label{w(z)}
w(z)=h(z)+\frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_r}L(z-\zeta)\frac{-\kappa(\zeta)e^{2w(\zeta)}-1}{|\zeta|^2(\log(1/|\zeta|))^2}
d\sigma_{\zeta}\end{eqnarray}
for $z\in \boldsymbol{a}thbb{D}_r^*$, $0<r<1$, where $h$ is harmonic on $\boldsymbol{a}thbb{D}_r$. Let
$$t(z):=-\log\log(1/|z|), \quad p(z):=w(z)+t(z)=u(z)+\log|z|$$
for $z\in \boldsymbol{a}thbb{D}_r^*$. Since
$$\Delta p(z)=-\kappa(z) e^{2u}=\frac{-\kappa(z) e^{2w(z)}}{|z|^2(\log(1/|z|))^2}>0,$$
$p(z)$ is subharmonic on $\boldsymbol{a}thbb{D}_r^*$ and $\lim_{z\rightarrow0}p(z)=-\infty$, then $p(z)$ is subharmonic on $\boldsymbol{a}thbb{D}_r$. By Lemma \ref{poisson jensen}, as $z \boldsymbol{a}psto \Delta p(z)$ is integrable over $\boldsymbol{a}thbb{D}_r$,
\begin{eqnarray}
p(z)=h_p(z)+\frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_r}L(z-\zeta)\frac{-\kappa(\zeta)e^{2w(\zeta)}}{|\zeta|^2(\log(1/|\zeta|))^2}
d\sigma_{\zeta},\ z\in \boldsymbol{a}thbb{D}_r,\nonumber\end{eqnarray} where $h_p(z)$ is harmonic on $\boldsymbol{a}thbb{D}_r$.
For $t(z)$, we also have
\begin{eqnarray}
t(z)=h_t(z)+\frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_r}L(z-\zeta)\frac{1}{|\zeta|^2(\log(1/|\zeta|))^2}d\sigma_{\zeta},\ z\in \boldsymbol{a}thbb{D}_r,\nonumber\end{eqnarray} where $h_t(z)$ is harmonic on $\boldsymbol{a}thbb{D}_r$. Setting $w(z)=p(z)-t(z)$ gives \eqref{w(z)} with $h(z)=h_p(z)-h_t(z)$.
Now set $R<1 / e^2$. So there exists a number ${\rho}>0$ such that the inequality (\ref{lemma}) holds in the disk $\boldsymbol{a}thbb{D}_{\rho}$. Let $\widetilde{\rho}=\min\{R/2,\,\rho\}$. We choose $z\in \boldsymbol{a}thbb{D}_{\widetilde{\rho}}$ and set $r=|z|/2$. Let $q(z)=-\kappa(z)e^{2w(z)}-1$, $f(z)=q(z)|z|^{-2\alpha}$. Then from (\ref{new higher}), we have
\begin{eqnarray} \label{for w}
&& \partial^{\boldsymbol{j}}w(z)\nonumber \\
&=&\partial^{\boldsymbol{j}}h(z)+\frac{1}{2\pi}\int_{\boldsymbol{a}thbb{D}_{\widetilde{\rho}}\begin{array}ckslash \boldsymbol{a}thbb{D}_{r}}\partial^{\boldsymbol{j}}L(z-\zeta)f(\zeta)d\sigma_{\zeta}+ \frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{j}}L(z-\zeta)
\left( f(\zeta)-f(z)\right)d\sigma_{\zeta} \nonumber\\
&&+\frac 1 {2\pi}\int_{\boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{j}}L(z-\zeta)\sum_{1\leq |\boldsymbol{a}|\leq n} \frac{(\zeta-z)^{\boldsymbol{a}}\partial^{\boldsymbol{a}}f(z)}{\boldsymbol{a}!}\;d\sigma_{\zeta} \nonumber\\
&&-\frac{1}{2\pi} \sum^{n-1}_{\tau=1} \int_{\partial \boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{\theta}_{\tau}} L(z-\zeta) \cdot P_{\tau-1}[\partial^{\boldsymbol{\phi}_{\tau}}f](z,\zeta)\cdot \langle N(\zeta),\boldsymbol{e}_{\tau+1} \rangle |d\zeta|
\end{eqnarray}
for a harmonic function $h$ on $\boldsymbol{a}thbb{D}_{\widetilde{\rho}}$. We can obtain
$$\left| \int_{\boldsymbol{a}thbb{D}_{\widetilde{\rho}}\begin{array}ckslash \boldsymbol{a}thbb{D}_{r}}\partial^{\boldsymbol{j}}L(z-\zeta)f(\zeta)d\sigma_{\zeta} \right| \leq
\frac{C_4}{|z|^n (\log(1/|z|))^2},$$
$$ \left| \int_{\boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{j}}L(z-\zeta)
\left( f(\zeta)-f(z)\right)d\sigma_{\zeta} \right| \leq \frac{C_5}{|z|^n (\log(1/|z|))^2},$$
by \eqref{logn} and Theorem \ref{logn} in \cite{Rothbehaviour}. So $ \begin{array}r{\partial}^nw(z)$, $\partial^nw(z)=\textit {O}(|z|^{-n}(\log(1/|z|))^{-2})$. For the mixed partial derivatives, since the first three integrals are canceled, we have to estimate the last term in (\ref{for w}). Letting $\tau=1$ in the last sum of (\ref{for w}), the term
$$ \partial^{\boldsymbol{\phi}_{1}}f(z)\int_{\partial \boldsymbol{a}thbb{D}_r}\partial^{\boldsymbol{e}_1} L(z-\zeta) \cdot \langle N(\zeta),\boldsymbol{e}_{2} \rangle |d\zeta|$$
appears. Now consider $\partial^{\boldsymbol{\phi}_{1}}f(z)$. Our aim is $\begin{array}r{\partial}^{n_1}\partial^{n_2}w(z)=\textit {O}(|z|^{-n}(\log(1/|z|))^{-3})$. Since $f(z)=q(z)|z|^{-2}(\log(1/|z|))^{-2}$, then $q(z)\cdot \partial^{\boldsymbol{\phi}_{1}} (|z|^{-2}(\log(1/|z|))^{-2})$ appears in $ \partial^{\boldsymbol{\phi}_{1}} f(z)$ with some coefficient. We can calculate that
$$\left| \partial^{\boldsymbol{\phi}_{1}} \frac{1}{|z|^{2}(\log(1/|z|))^{2}}\right| \leq \frac{C_6}{|z|^{n}(\log(1/|z|))^2},$$
thus
$$\left| q(z) \partial^{\boldsymbol{\phi}_{1}} \frac{1}{|z|^{2}(\log(1/|z|))^{2}} \cdot \partial^{\boldsymbol{e}_1} L(z-\zeta) \cdot \langle N(\zeta),\boldsymbol{e}_{2} \rangle |d\zeta|\right| \leq \frac{6\pi C_7 \cdot C_6}{|z|^{n}(\log(1/|z|))^{3}}$$
provided (\ref{logn}) and (\ref{sincos}). So $ \begin{array}r{\partial}^{n_1}\partial^{n_2}w(z)=\textit {O}(|z|^{-n}(\log(1/|z|))^{-3})$ as desired. \hspace{\fill} $\Box$\\
The second order derivative of $w(z)$ in Theorem \ref{original} is contained in Theorem \ref{for w theorem}. However, for the mixed partial derivative, the estimate (\ref{new mixed}) is more accurate than (4.1). We take it to be a corollary as following.
\begin{eqnarray}gin{corollary} \label{second partial derivative order}
\textsl{ Let $\kappa:\boldsymbol{a}thbb{D}\rightarrow\boldsymbol{a}thbb{R}$ be a locally H\"{o}lder continuous function with $\kappa(0)<0$. If $u:\boldsymbol{a}thbb{D}^* \rightarrow \boldsymbol{a}thbb{R}$ is a $C^2$-solution to $\Delta u=-\kappa(z) e^{2u}$ in $\boldsymbol{a}thbb{D}^*$ with the order $\alpha=1$ at the point $z=0$, then for the remainder function $w(z)$ as in Theorem \ref{original}, the second partial derivatives satisfy
$$w_{z\begin{array}r{z}}(z)=\textit {O}(|z|^{-2}(\log(1/|z|))^{-3}).$$ }
\end{corollary}
As for the sharpness of Theorems \ref{original}, \ref{estimate v} and \ref{for w theorem}, the generalized hyperbolic metric on the thrice-punctured sphere makes a convictive case here. Theorems 3.3 and 4.2 in \cite{zhang3} verify that Theorems \ref{original}, \ref{estimate v} and \ref{for w theorem} are sharp, see \cite{zhang3} for details.
\vspace*{3mm}
\section{Minda-type theorems}
\setcounter{equation}{0}
The following result is Minda's theorem. It is a general estimate for the hyperbolic metric near the singularity.
\begin{eqnarray}gin{theorema} $\boldsymbol{a}thrm{[10]}.$ \label{Minda original}
\textsl{Suppose $\Omega$ is a hyperbolic region in the complex plane and $p \in \boldsymbol{a}thbb{C}$ is an isolated boundary point of $\Omega$. Let the hyperbolic metric on $\Omega$ with the constant Gaussian curvature $-1$ be $\lambda_{\Omega}(\omega)|d\omega|$. Then
\begin{eqnarray}n
\lim_{\omega\rightarrow p}|\omega-p|\log(1/|\omega-p|)\lambda_{\Omega}(\omega)=\frac{1}{2}\,.
\end{eqnarray}n}
\end{theorema}
The following theorem is due to Kraus and Roth.
\begin{eqnarray}gin{theorema} $\boldsymbol{a}thrm{[6]}.$ \label{estimate for cusps original}
\textsl{Let $\lambda(z)|dz|$ be a regular conformal metric on a domain $\Omega\subseteq\boldsymbol{a}thbb{C}$ with an isolated singularity at $z=p$. Suppose that its curvature $\kappa :\Omega\rightarrow \boldsymbol{a}thbb{R}$ has a H\"{o}lder continuous extension to $\Omega\cup\{p\}$ such that $\kappa(p)< 0$. Then $\log\lambda$ has an order $\alpha\leq1$ at $z=p$ and
\begin{eqnarray}n
\lim_{z\rightarrow p}|z-p|\log(1/|z-p|)\lambda(z)=\left\{\begin{eqnarray}gin{array}{ll}
0&\mbox{if\ }\ \alpha< 1\\
\displaystyle \frac 1 {\sqrt{-\kappa(p)}}&\mbox{if\ }\ \alpha=1.\end{array}\right.
\end{eqnarray}n
\vspace*{1mm} }
\end{theorema}
\partialr We obtain the following result in relation to Theorem \ref{estimate for cusps original}.
\begin{eqnarray}gin{theorem}\label{coro}
\textsl{ Let $\lambda(z)|dz|$ be a regular conformal metric on a domain $\Omega\subseteq\boldsymbol{a}thbb{C}$ with an isolated singularity at $z=p$. Suppose that the curvature $\kappa :\Omega\rightarrow \boldsymbol{a}thbb{R}$ has a H\"{o}lder continuous extension to $\Omega\cup\{p\}$ such that $\kappa(p)< 0$ and the order of $\log\lambda$ is $\alpha=1$ at $z=p$. Then \vspace*{2mm} \\
(i)$\ \displaystyle \lim_{z\rightarrow p} (z-p)|z-p|\log(1/|z-p|)\lambda_{z}(z)=-{\frac 1 {2\sqrt{-\kappa(p)}}}, \hspace*{\fill} $ \vspace*{2mm}\\
(ii)$\ \displaystyle \ \lim_{z\rightarrow p}(z-p)^2|z-p|\log(1/|z-p|)\lambda_{zz}(z)=\displaystyle {\frac 3{4\sqrt{-\kappa(p)}}}, \hspace*{\fill} $ \vspace*{2mm}\\
(iii)$\ \displaystyle \ \lim_{z\rightarrow p}|z-p|^3\log(1/|z-p|)\lambda_{z\begin{array}r{z}}(z)=\displaystyle {\frac 1 {4\sqrt{-\kappa(p)}}}. \hspace*{\fill} $ }
\end{theorem}
\textbf{Proof.} Let $\boldsymbol{a}thbb{H}$ be the upper half-plane. For each simply closed curve $\gamma: [0,1] \rightarrow \Omega$ around $p$ with $\gamma(0)=\gamma(1)$, consider the lift $\widetilde{\gamma}$ of $\gamma$ in $\boldsymbol{a}thbb{H}$. Since there exists an automorphism $g$ on $\boldsymbol{a}thbb{H}$ such that $\widetilde{\gamma}(1)=g(\widetilde{\gamma}(0))$, we may assume that $g(z)=z+1$ on $\boldsymbol{a}thbb{H}$. Let $\pi: \boldsymbol{a}thbb{H}\rightarrow \Omega$ be the regular covering projection, we have $\pi \circ g = \pi$. Define $\varphi: \boldsymbol{a}thbb{H} \rightarrow \boldsymbol{a}thbb{D}^*$, $z \boldsymbol{a}psto e^{2 \pi iz}$, then the quotient space $\boldsymbol{a}thbb{H}/ \langle g \rangle$ is conformally equivalent to $\boldsymbol{a}thbb{D}^*$. Hence there exists a conformal mapping $\rho: \boldsymbol{a}thbb{D}^* \rightarrow \Omega$ such that $\rho \circ \varphi=\pi$ and $\rho$ can be extended to $\boldsymbol{a}thbb{D} \rightarrow \Omega$ holomorphically by setting $\rho (0)=p$.
So it is sufficient to consider the case $p=0$ and $\boldsymbol{a}thbb{D}^*=\Omega$.
\partialr Let $u(z):=\log\lambda(z)$, so $\lambda_z(z)=u_z(z)\lambda(z)$. It holds
$$\lim_{z\rightarrow 0}z u_z(z)=-\frac 1 2,\; \;
\lim_{z\rightarrow 0}z^2 u_{zz}(z)=\frac 1 2,\; \; \lim_{z\rightarrow0}|z|^2u_{z\begin{array}r{z}}=0$$
by Theorem \ref{original}. In combination with Theorem \ref{estimate for cusps original}, we have
\begin{eqnarray}n
&&\lim_{z\rightarrow 0}z|z|\log(1/|z|)\lambda_z(z)=\lim_{z\rightarrow 0}z|z|\log(1/|z|) u_z(z)\lambda(z)\\
&=&\lim_{z\rightarrow 0}|z|\log(1/|z|)\lambda(z) \cdot z u_{z}(z)=-\frac 1 {2\sqrt{-\kappa(0)}}
\end{eqnarray}n
for the first case,
\begin{eqnarray}n
&&\lim_{z\rightarrow 0}z^2|z|\log(1/|z|)\lambda_{zz}(z)=\lim_{z\rightarrow 0}z^2|z|\log(1/|z|)(u_{zz} \lambda+u_z \lambda_z)\\
&=&\lim_{z\rightarrow 0}(z^2u_{zz}\cdot|z|\log(1/|z|)\lambda)+\lim_{z\rightarrow 0}(z|z|\log(1/|z|)\lambda_z\cdot zu_z)\\
&=&\frac 1 {2\sqrt{-\kappa(0)}}+(-\frac 1 2)\cdot(-\frac 1 {2\sqrt{-\kappa(0)}})=\frac 3{4\sqrt{-\kappa(0)}}\nonumber
\end{eqnarray}n
for the second case and
\begin{eqnarray}n
&&\lim_{z\rightarrow0}|z|^3\log(1/|z|)\lambda_{z\begin{array}r{z}}(z)=\lim_{z\rightarrow0}|z|^3\log(1/|z|)(u_{z\begin{array}r{z}}\lambda+u_{z}
\lambda_{\begin{array}r{z}})\nonumber\\
&=&\lim_{z\rightarrow0}(|z|^2u_{z\begin{array}r{z}}\cdot|z|\log(1/|z|)\lambda)
+\lim_{z\rightarrow0}(\begin{array}r{z}|z|\log(1/|z|)\lambda_{\begin{array}r{z}}\cdot zu_z)\nonumber\\
&=&-\frac 1 {2\sqrt{-\kappa(0)}}\cdot(-\frac 1 2)=\frac 1 {4\sqrt{-\kappa(0)}}\nonumber
\end{eqnarray}n
for the last case as desired.
$\Box$
\vspace*{2mm}
\partialr Theorem \ref{coro} is given for a regular conformal metric with a (locally) H\"oder continuous Gaussian curvature $\kappa$. Considering Theorems \ref{for v} and \ref{for w theorem}, if we add the assumption that $\kappa$ is $n$-th order (locally) H\"older continuous, we can obtain the higher order version of Theorems \ref{estimate for cusps original} and \ref{coro}.
\begin{eqnarray}gin{theorem}\label{general u}
\textsl{Let $\kappa:\boldsymbol{a}thbb{D}\rightarrow\boldsymbol{a}thbb{R}$ be of class $C^{n-2,\,\nu}(\boldsymbol{a}thbb{D}^*)$ for an integer $n \geq 3$, $0<\nu \leq 1$ and $\kappa(0)<0$. If $u:\boldsymbol{a}thbb{D}^*\rightarrow \boldsymbol{a}thbb{R}$ is a $C^{n,\,\nu}$-solution to $\Delta u=-\kappa(z) e^{2u}$ in $\boldsymbol{a}thbb{D}^*$, then $u$ has order $\alpha \in (-\infty, 1]$ and for $n_1,\,n_2 \geq 1$, $n_1+n_2\leq n$, \vspace*{2mm} \\
(i) $ \displaystyle \ \lim_{z\rightarrow0}z^n\partial^nu(z)=\frac {\alpha}2(-1)^n(n-1)!=\lim_{z\rightarrow0}\begin{array}r{z}^n\begin{array}r{\partial}^nu(z),$ \vspace*{2mm} \\
(ii) $ \displaystyle \ \lim_{z\rightarrow 0}\begin{array}r{z}^{n_1} z^{n_2}\begin{array}r{\partial}^{n_1}\partial^{n_2}u(z)=0.$}
\end{theorem}
\textbf{Proof.} When $0<\alpha<1$,\ $u(z)=-\alpha\log|z|+v(z)$. Theorems 4.1 and 4.2 imply that
$$\lim_{z \rightarrow 0}z^n \partial^n v(z)=0, \ \ \lim_{z \rightarrow 0}\begin{array}r{z}^{n_1} z^{n_2} \begin{array}r{\partial}^{n_1}\partial^{n_2} v(z)=0$$
for $n_1$, $n_2$, $n \geq 1$. Since
\begin{eqnarray} \label{pa logz}
\partial^{n}\log|z|=\frac{(-1)^{n-1}(n-1)!}{2z^n}, \quad \begin{array}r{\partial}^{n_1}\partial^{n_2}\log|z|=0,
\end{eqnarray}
so $$\lim_{z \rightarrow 0}z^n \partial^n u(z)=-\alpha \lim_{z \rightarrow 0}z^n \partial^n \log|z|+\lim_{z \rightarrow 0}z^n \partial^n v(z)=\frac{\alpha}{2z^n}(-1)^{n}(n-1)!,$$
$$\lim_{z \rightarrow 0}\begin{array}r{z}^{n_1} z^{n_2} \begin{array}r{\partial}^{n_1}\partial^{n_2} u(z)=0.$$
When $\alpha=1$,\ $u(z)=-\log|z|-\log\log(1/|z|)+w(z)$. We have
$$\lim_{z \rightarrow 0}z^n \partial^n w(z)=0, \ \ \lim_{z \rightarrow 0}\begin{array}r{z}^{n_1} z^{n_2} \begin{array}r{\partial}^{n_1}\partial^{n_2} w(z)=0$$
for $n_1,\,n_2,\,n \geq 1$, from Theorems 4.1 and 4.3. By induction,
$$\partial^n\log\log(1/|z|)=\sum^n_{j=1}\frac{C^{(n)}_j}{z^n(\log(1/|z|))^j}$$
with constant $C^{(n)}_{j}$ for $1 \leq j \leq n$. If we fix $n_2$, then
$$\begin{array}r{\partial}^{n_1}\partial^{n_2}\log\log(1/|z|)=\sum^{n_2}_{j=1}\frac{C^{(n_1,\, n_2)}_{j}}{\begin{array}r{z}^{n_1}z^{n_2} (\log(1/|z|))^{j+1}}$$
with constant $C^{(n_1,\,n_2)}_{j}$ for $1 \leq j \leq n_2$.
So $$\lim_{z \rightarrow 0}z^n \partial^n \log\log(1/|z|)=0, \ \ \lim_{z \rightarrow 0}\begin{array}r{z}^{n_1}z^{n_2} \begin{array}r{\partial}^{n_1}\partial^{n_2} \log\log(1/|z|)=0$$ for $n_1,\,n_2,\,n \geq 1$. Combining with \eqref{pa logz} leads to
$$\lim_{z \rightarrow 0}z^n \partial^n u(z)=-\alpha \lim_{z \rightarrow 0}z^n \partial^n \log|z|+\lim_{z \rightarrow 0}z^n \partial^n v(z)=\frac{(-1)^{n}(n-1)!}{2z^n},$$
$$\hspace*{51mm}\lim_{z \rightarrow 0}\begin{array}r{z}^{n_1} z^{n_2} \begin{array}r{\partial}^{n_1}\partial^{n_2} u(z)=0.\hspace*{51mm} \Box $$
From the proof above, we can obtain a stronger limit for the mixed derivative of $u(z)$ when the order $\alpha=1$,
$$\displaystyle \ \lim_{z\rightarrow 0}\begin{array}r{z}^{n_1} z^{n_2}(\log(1/|z|))^2\begin{array}r{\partial}^{n_1}\partial^{n_2}u(z)=C^{(n_1,\;n_2)}_1=\frac{(-1)^{n_1+n_2-1}}{4}(n_1-1)!(n_2-1)!,$$
see \cite{zhang3} for more details.
\vspace*{2mm}
\partialr On the basis of Theorem \ref{general u}, we can provide the following result as a higher order estimate for a conformal metric with the negative curvature near the origin when $\alpha=1$.
\begin{eqnarray}gin{theorem} \label{higher lambda for cusps}
\textsl{Let $\kappa$ and $u$ be the same as in Theorem \ref{general u}. If the order of $u$ is $\alpha=1$, then for $n_1,\,n_2 \geq 0$, $n_1+n_2 \leq n$, the limit
$$l_{n_1,n_2}:=\frac{1}{n_1!n_2!}\lim_{z \rightarrow0}|z|\log(1/|z|){\begin{array}r{z}}^{n_1} z^{n_2} {\begin{array}r{\partial}}^{n_1} \partial^{n_2} \lambda(z)$$
exists. Moreover, the numbers $l_{n_1,n_2}$ satisfy the following \vspace*{1mm}\\
(i)$ \ \displaystyle l_{n_1,n_2}={-\frac{1}{2} \choose n_1}{-\frac{1}{2} \choose n_2}\frac 1 {\sqrt{-\kappa(0)}}$, \vspace*{3mm} \\
(ii)$\ l_{n_1,n_2}=l_{n_2,n_1}$,\\
where
$${\tau \choose j}=\frac{\tau(\tau-1)\cdots (\tau-j+1)}{j\,!}$$
is the binomial coefficient. }
\end{theorem}
\textbf{Proof.} We write $\lambda(z)=e^{u(z)}$, then $\partial \lambda (z)=\lambda(z)\, \partial u(z)$, and
$$\partial^{n}\lambda(z)=\sum_{j=0}^{{n}-1}{{n}-1 \choose j}\partial^{{n}-j}u(z)
\,\partial^j\lambda(z)$$
by induction, where $\partial^0\lambda(z)=\begin{array}r{\partial}^0\lambda(z)=\lambda(z),$
so
$$l_{0,{n_2}}=\frac{1}{{n_2}!}\lim_{z\rightarrow0}\sum_{j=0}^{{n_2}-1}{{n_2}-1 \choose j}
z^{{n_2}-j}\partial^{{n_2}-j}u(z)\cdot|z|\log(1/|z|)z^j\partial^j\lambda(z).$$
Theorem \ref{estimate for cusps original} gives that $l_{0,0}=1/\sqrt{-\kappa(0)}$. From the existence of $\lim_{z\rightarrow0}z^{{n_2}-j}\partial^{{n_2}-j}u(z)$ and $l_{0,0}$, we know that $l_{0,\;{n_2}}$ exists. Next, limit (ii) in Theorem \ref{general u} enables us to write $l_{{n_1},{n_2}}$ as a sum of the terms only containing pure derivatives of $u(z)$,
\begin{eqnarray}\label{ind}
l_{{n_1},{n_2}}=\frac{1}{{n_1}!{n_2}!}\lim_{z\rightarrow0}\sum_{j=0}^{{n_2}-1}{{n_2}-1 \choose j}
z^{{n_2}-j}\partial^{{n_2}-j}u(z)\,|z|\log(1/|z|)\begin{array}r{z}^{n_1} z^j\begin{array}r{\partial}^{n_1}\partial^j\lambda(z),
\end{eqnarray}
thus the existence of $l_{0,{n_2}}$ guarantees $l_{{n_1},{n_2}}$ exists.
By Theorem \ref{coro}, it is known that $l_{0,1}$ is a real number, so $l_{1,0}=\overline{l_{0,1}}=l_{0,1}$.
Since
\begin{eqnarray} \label{pa bar n lam}
{\displaystyle\begin{array}r{\partial}^{n_2}\lambda(z)=\sum_{j=0}^{{n_2}-1}{{n_2}-1 \choose j}
\begin{array}r{\partial}^{{n_2}-j}u(z)\,\begin{array}r{\partial}^j\lambda(z)},
\end{eqnarray}
then $l_{{n_2},0}=l_{0,{n_2}}$ by induction. From \eqref{ind}, \eqref{pa bar n lam}, and (i) of Theorem \ref{general u}, we have
\begin{eqnarray}n
&&l_{{n_1},{n_2}}\\
&=&\sum_{j=0}^{{n_2}-1} \lim_{z\rightarrow0} \frac{1}{{n_1}!{n_2}!} \frac{({n_2}-1)!}{j!({n_2}-1-j)!}
z^{{n_2}-j}\partial^{{n_2}-j}u(z)\cdot|z|\log(1/|z|)\begin{array}r{z}^{n_1} z^j\begin{array}r{\partial}^{n_1}\partial^j\lambda(z) \\
&=&\frac{1}{{n_2}}\sum_{j=0}^{{n_2}-1} \frac{1}{{n_1}!} \frac{1}{j!({n_2}-1-j)!}\lim_{z\rightarrow0}
z^{{n_2}-j}\partial^{{n_2}-j}u(z)\cdot\lim_{z\rightarrow0}|z|\log(1/|z|)\begin{array}r{z}^{n_1}z^j\begin{array}r{\partial}^{n_1}\partial^j\lambda(z) \\
&=&\frac{1}{{n_2}}\sum_{j=0}^{{n_2}-1} \frac{(-1)^{{n_2}-j}}{2}\frac{1}{{n_1}!j!} \lim_{z\rightarrow0}|z|\log(1/|z|)\begin{array}r{z}^{n_1} z^j\begin{array}r{\partial}^{n_1}\partial^j\lambda(z)
=\frac{1}{2{n_2}}\sum_{j=1}^{{n_2}-1}(-1)^{{n_2}-j}l_{{n_1},j}.
\end{eqnarray}n
Then
$$ {n_2} \cdot l_{{n_1},{n_2}}=\frac{1}2\sum_{j=0}^{{n_2}-2}(-1)^{{n_2}-j}\,l_{{n_1},j}-\frac{1}{2}\,l_{{n_1},{n_2}-1}
=-({n_2}-1)l_{{n_1},{n_2}-1}-\frac{1}{2}\,l_{{n_1},{n_2}-1}.$$
Since $l_{0,{n_2}}=l_{{n_2},0}$,
\begin{eqnarray}n
l_{{n_1},{n_2}}&=&\frac{-\frac{1}{2}-{n_2}+1}{{n_2}}l_{{n_1},{n_2}-1}=
{-\frac{1}{2} \choose {n_2}}l_{{n_1},0}\\
&=&{-\frac{1}{2} \choose {n_2}}l_{0,{n_1}}={-\frac{1}{2} \choose {n_2}}{-\frac{1}{2} \choose {n_1}}l_{0,0}.
\end{eqnarray}n
Thus (i) is valid and (ii) follows form (i).
$\Box$\\
However, when the order $\alpha<1$, the analogous limit
\begin{eqnarray} \label{lza}
l':=\lim_{z\rightarrow0}|z|^{\alpha}\lambda(z)
\end{eqnarray}
may also exist but cannot be described only in terms of the curvature of $\lambda(z)$. To discuss the limit \eqref{lza} for an SK-metric, we consider the limit
\begin{eqnarray} \label{upper lza}
l:=\limsup_{z\rightarrow0}|z|^{\alpha}\lambda(z)
\end{eqnarray}
instead, since the definition \eqref{singularity} of a corner guarantees that $l<\infty$ for $l$ defined above. Based on Theorem \ref{Ahlfors lemma} and Corollary 4.4 in \cite{Rothhyper}, we have the following result corresponding to Theorem \ref{higher lambda for cusps} in the thrice-punctured sphere.
\begin{eqnarray}gin{theorem} \label{limitfor3}
\textsl{Let $0<\alpha,\ \begin{eqnarray}ta <1$ and $0<\gamma \leq 1$ such that $\alpha+\begin{eqnarray}ta+\gamma>2$ and $\lambda(z)$ be an SK-metric on the thrice-punctured Riemann sphere $\widehat{\boldsymbol{a}thbb{C}}\begin{array}ckslash \{0,\,1,\,\infty\}$ of orders $\alpha,\, \begin{eqnarray}ta,\,\gamma$ at $0$, $1$, $\infty$, respectively, with the curvature $\kappa(z)$. Then the upper limit \eqref{upper lza} satisfies
\begin{eqnarray} \label{3lgene}
l \leq \frac{\deltata}{1-\deltata^2}(1-\alpha),
\end{eqnarray}
where
\begin{eqnarray}n
\deltata=\frac{\Gamma( c)}{\Gamma(2- c)}
\left(\frac{\Gamma(1- a)\Gamma(1- b)\Gamma( a+1- c)
\Gamma( b+1- c)}{\Gamma( a)\Gamma( b)\Gamma( c- a)
\Gamma( c- b)} \right)^{1/2}
\end{eqnarray}n
with
\begin{eqnarray}n
a=\frac{\alpha+\begin{eqnarray}ta-\gamma}{2},\ b=\frac{\alpha+\begin{eqnarray}ta+\gamma-2}{2},\ c=\alpha.
\end{eqnarray}n}
\end{theorem}
Now we consider the upper limit $l$ in the once-punctured unit disk $\boldsymbol{a}thbb{D}^*$. The following result is evident if we combine Theorem \ref{Ahlfors lemma} with Theorem \ref{maximal}.
\begin{eqnarray}gin{theorem}
\textsl{If $\lambda(z)|dz|$ is an an SK-metric on $\boldsymbol{a}thbb{D}^*$ with the order $\alpha \in (0,\,1)$ at the origin, then the upper limit $l$ defined in \eqref{upper lza} satisfies $l \leq 1-\alpha$.}
\end{theorem}
When the SK-metric satisfies some stronger continuity assumption, the upper limit $l$ in \eqref{upper lza} will become the limit $l'$ in \eqref{lza} at the origin, which enables us to consider the derivatives of $l$ locally. For instance, if the $\kappa(z)$ and $u(z)$ satisfy the assumption of Theorem \ref{general u}, then $u(z)$ is of class $C^{\;\!2}$ in a neighborhood of $z=0$ and $l=l'$ locally holds near the origin. Therefore we state the following result in terms of the recurrence relation similar to Theorem \ref{higher lambda for cusps}.
\begin{eqnarray}gin{theorem}
\textsl{Let the functions $\kappa(z)$ and $u(z)$ satisfy the assumption of Theorem \ref{general u} for an integer $n$ and let $\lambda(z):=e^{u(z)}$. If the order of $u(z)$ is $\alpha \in (-\infty, \,1)$, then for $n_1$, $n_2\geq 0$, $n_1+n_2 \leq n$, the limit
$$l_{n_1,n_2}:=\frac{1}{n_1!n_2!}\lim_{z \rightarrow0}|z|{\begin{array}r{z}}^{n_1} z^{n_2} {\begin{array}r{\partial}}^{n_1} \partial^{n_2} \lambda(z)$$
exists and satisfies the following \vspace*{1mm}\\
(i)$ \ \displaystyle l_{n_1,n_2}={-\frac{\alpha}{2} \choose n_1}{-\frac{\alpha}{2} \choose n_2}l'$, \vspace*{3mm} \\
(ii)$\ l_{n_1,n_2}=l_{n_2,n_1}$, \vspace*{1mm} \\
where $l'$ is defined by \eqref{lza}.}
\end{theorem}
\vspace*{5mm}
\hspace*{-17pt}\textbf{Acknowledgement.} I would like to thank Prof. Toshiyoki Sugawa for his helpful comments, suggestion and encouragement. I also want to thank Prof. Toshihiro Nakanishi for his suggestion on the SK-metrics.\\
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\providecommand{\href}[2]{#2}
\begin{eqnarray}gin{thebibliography}{10}
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\end{thebibliography}
\end{document} |
\begin{document}
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\normalsize
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\begin{center}
{\large \bf A Constraint between Noncommutative Parameters of
Quantum Theories in Noncommutative Space}
\end{center}
\begin{center}
Jian-Zu Zhang
\end{center}
\begin{center}
Institute for Theoretical Physics, East China University of
Science and Technology, \\
Box 316, Shanghai 200237, P. R. China
\end{center}
\begin{abstract}
In two-dimensional noncommutive space for the case of both
position - position and momentum - momentum noncommuting, a
constraint between noncommutative parameters is investigated. The
related topic of guaranteeing Bose - Einstein statistics in
noncommutive space in the general case are elucidated: Bose -
Einstein statistics is guaranteed by the deformed Heisenberg -
Weyl algebra itself, independent of dynamics. A special character
of a dynamical system is represented by a constraint between
noncommutative parameters. The general feature of the constraint
for any system is a direct proportionality between noncommutative
parameters with a proportional coefficient depending on
characteristic parameters of the system under study. The
constraint for a harmonic oscillator is illustrated.
\end{abstract}
\begin{flushleft}
${\ast}$ E-mail: [email protected] $\;$ Fax:
+86-21-64251138$\;$ Tel: +86-21-64252613
\end{flushleft}
\section{Introduction}
\setcounter{equation}{0}
Physics in noncommutative space \citer{CDS,DN} has been
extensively investigated in literature. This is motivated by
studies of the low energy effective theory of D-brane with a
nonzero NS - NS $B$ field background. Effects of spatial
noncommutativity are apparent only near the string scale, thus we
need to work at a level of noncommutative quantum field theory.
But based on the incomplete decoupling mechanism one expects that
quantum mechanics in noncommutative space (NCQM) may clarify some
low energy phenomenological consequences, and lead to qualitative
understanding of effects of spatial noncommutativity. In
literature NCQM and its applications \citer{CST,Wu} have been
studied in detail. But an important issue about whether in
noncommutive space the concept of identical particles being still
meaningful and whether Bose-Einstein statistics and Fermi-Dirac
statistics being still maintained has not been resolved.
On the fundamental level of quantum field theory the annihilation
and creation operators appear in the expansion of the (free) field
operator $\Psi(\hat{x})=\int d^3k a_k(t)\Phi_k(\hat{x})+H.c.$ The
consistent multi-particle interpretation requires the usual
(anti)commutation relations among $a_k$ and $a^\dagger_k$.
Introduction of the Moyal type deformation of coordinates may
yield a deformation of the algebra between the creation and
annihilation operators. Various authors \cite{BMPV,BGMPQV} argue
important consequences for Pauli's principle in the case of
Fermi-Dirac statistics. In noncommutative quantum field theory
Poincar\'e invariance is broken and is replaced by a twisted
Poincar\'e symmetry. On the other hand this is only possible if
the statistics is equally twisted \cite{BGMPQV}. Whether the
deformed Heisenberg - Weyl algebra is consistent with Bose -
Einstein statistics is still an open issue at the level of quantum
field theory.
In this paper our study is restricted in the context of
non-relativistic quantum mechanics to elucidate this problem. We
follow the standard procedure of establishing Bose - Einstein
statistics in the ordinary quantum mechanics in commutative space,
and investigate whether Bose - Einstein statistics can be
maintained in noncommutative space. For a two dimensional
isotropic harmonic oscillator the consistency of the deformed
Heisenberg - Weyl algebra with Bose - Einstein statistics is
elucidated \cite{JZZ04a}. But this example is special. We need to
clarify the situation for general cases. We find that in the case
of both noncommutativities of position - position and momentum -
momentum Bose - Einstein statistics is guaranteed by the deformed
Heisenberg - Weyl algebra itself, independent of dynamics. The
special character of a dynamical system is represented by a
relation between noncommutative parameters. The general feature of
such a relation for any system is a direct proportionality between
noncommutative parameters with a proportional coefficient
depending on characteristic parameters of the system under study,
and such a proportional coefficient can be fixed up to a
dimensionless constant. The speciality of a two dimensional
isotropic harmonic oscillator is that the relation between
noncommutative parameters can be completely determined.
In order to demonstrate consistency between the deformed
Heisenberg - Weyl algebra and Bose - Einstein statistics, in the
following we first review the necessary background.
\section{The Deformed Heisenberg - Weyl Algebra}
\setcounter{equation}{0}
The start point is the deformed Heisenberg - Weyl algebra. We
consider the case of both position - position noncommutativity
(space-time noncommutativity is not considered) and momentum -
momentum noncommutativity. In this case the consistent deformed
Heisenberg - Weyl algebra is \cite{JZZ04a}:
\begin{equation}
\label{Eq:xp}
[\hat x_{i},\hat x_{j}]=i\xi^2\theta\epsilon_{ij}, \qquad [\hat
p_{i},\hat p_{j}]=i\xi^2\eta\epsilon_{ij}, \qquad
[\hat x_{i},\hat p_{j}]=i\hbar\delta_{ij},\;(i,j=1,2),
\end{equation}
where $\theta$ and $\eta$ are constant parameters, independent of
the position and momentum. Here we consider the noncommutativity
of the intrinsic canonical momentum. It means that the parameter
$\eta$, like the parameter $\theta$, should be extremely small.
This is guaranteed by a direct proportionality provided by a
constraint between them (See Eq.~(\ref{Eq:cc1}) below). The
$\epsilon_{ij}$ is a two-dimensional antisymmetric unit tensor,
$\epsilon_{12}=-\epsilon_{21}=1,$ $\epsilon_{11}=\epsilon_{22}=0$.
In Eq.~(\ref{Eq:xp}) the scaling factor
$\xi=(1+\theta\eta/4\hbar^2)^{-1/2}$ is a dimensionless constant.
When $\eta=0,$ we have $\xi=1$. The deformed Heisenberg - Weyl
algebra (\ref{Eq:xp}) reduces to the one of only position -
position noncommuting.
In literature there is a tacit confusion about the difference
between the intrinsic noncommutativity of the canonical momenta
discussed here and the noncommutativity of the mechanical momenta
of a particle in an external magnetic field with a vector
potential $A_i(x_j)$ in commutative space. In the later case the
mechanical momentum is
\begin{equation}
\label{Eq:pm}
p_{mech,i}=\mu \dot x_i=p_i-\frac{q}{c}A_i,
\end{equation}
where $p_i=-i\hbar \partial_i$ is the canonical momentum in
commutative space, satisfying $[p_i,p_j]=0$. The commutator
between $p_{mech,i}$ and $p_{mech,j}$ is
\begin{equation}
\label{Eq:pm-pm}
[p_{mech,i},p_{mech,j}]=-\frac{q}{c}
\left([p_i,A_j]+[A_i,p_j]\right)= i\frac{\hbar
q}{c}\left(\partial_i A_j-\partial_j A_i\right)=i\frac{\hbar
q}{c}\epsilon_{ij3}B_3.
\end{equation}
Such a noncommutativity is determined by the external magnetic
field $\vec B$ which, unlike the noncommutative parameter $\eta$,
may not be extremely small. Thus the noncommutativity the
mechanical momenta of a particle in an external magnetic field in
commutative space is essentially different from the intrinsic
noncommutativity, the second equation in Eq.~(\ref{Eq:xp}), of the
canonical momentum in noncommutative space.
The deformed Heisenberg - Weyl algebra (\ref{Eq:xp}) can be
realizations by undeformed variables as follows (henceforth
summation convention is used)
\begin{equation}
\label{Eq:hat-x-p}
\hat x_{i}=\xi(x_{i}-\frac{1}{2\hbar}\theta\epsilon_{ij}p_{j}),
\quad
\hat p_{i}=\xi(p_{i}+\frac{1}{2\hbar}\eta\epsilon_{ij}x_{j}),
\end{equation}
where $x_{i}$ and $p_{i}$ satisfy the undeformed Heisenberg - Weyl
algebra
$[x_{i},x_{j}]=[p_{i},p_{j}]=0,\;
[x_{i},p_{j}]=i\hbar\delta_{ij}.$
It should be emphasized that for the case of both position -
position and momentum - momentum noncommuting the scaling factor
$\xi$ in Eqs.~ (\ref{Eq:xp}) and (\ref{Eq:hat-x-p}) guarantees
consistency of the framework, and plays an essential role in
dynamics. One may argues that only three parameters $\hbar$,
$\theta$ and $\eta$ can appear in three commutators (\ref{Eq:xp}),
thus $\xi$ is an additional spurious parameter and can be set to
$1.$ If one re-scales $\hat x_{i}$ and $\hat p_{i}$ so that
$\xi=1$ in Eqs.~(\ref{Eq:xp}) and (\ref{Eq:hat-x-p}), it is easy
to check that Eq.~(\ref{Eq:hat-x-p}) leads to
$[\hat x_{i},\hat
p_{j}]=i\hbar\left(1+\theta\eta/4\hbar^2\right)\delta_{ij},$
thus the Heisenberg commutation relation cannot be maintained.
\section{Consistency Between The Deformed Heisenberg - Weyl
Algebra and Bose - Einstein Statistics} \setcounter{equation}{0}
In noncommutative space the concept of identical particles being
meaningful and Bose-Einstein statistics being maintained in the
general case are elucidated by the following theorem:
{\bf Theorem} {\it In the case of both position - position and
momentum - momentum noncommuting the deformed Heisenberg - Weyl
algebra is consistent with Bose - Einstein statistics.}
Proving this theorem includes two aspects. The first aspect is to
construct the general representations of the deformed annihilation
and creation operators which satisfy the complete and closed
deformed bosonic algebra \cite{JZZ04a}. The second aspect is, by
generalizing one - particle quantum mechanics, to establish the
Fock space of identical bosons.
In the context of quantum mechanics the general representation of
the deformed annihilation and creation operator $\hat a_i$ and
$\hat a_i^\dagger$ by $\hat x_i$ and $\hat p_i$ is
\begin{equation}
\label{Eq:hat-a}
\hat a_i=c_1(\hat x_i+ic_2\hat p_i),\;
\hat a_i^\dagger=c_1(\hat x_i-ic_2\hat p_i),
\end{equation}
where $c_1$ and $c_2$ are constants and may depend on
characteristic parameters, the mass $\mu$, the frequency $\omega$
etc., of the system under study. $c_1$ and $c_2$ can be fixed as
follows. Operators $\hat a_i$ and $\hat a_i^\dagger$ should
satisfy the bosonic commutation relations $[\hat a_1,\hat
a_1^\dagger]=[\hat a_2,\hat a_2^\dagger]=1$ (to keep the physical
meaning of $\hat a_i$ and $\hat a_i^\dagger$). From this
requirement and the deformed Heisenberg - Weyl algebra
(\ref{Eq:xp}) it follows that
\begin{equation}
\label{Eq:c1-c2}
c_1=\sqrt{1/2\hbar c_2}.
\end{equation}
Following the standard procedure in quantum mechanics, starting
from a system with one particle, the state vector space of a
many-particle system can be constructed by generalizing one -
particle formulism. Then Bose - Einstein statistics for a
identical - boson system can be developed in the standard way.
Bose - Einstein statistics should be maintained at the deformed
level described by $\hat a_i$, thus operators $\hat a_i$ and $\hat
a_j$ should be commuting: $[\hat a_i,\hat a_j]=0$. From this
equation and the deformed Heisenberg - Weyl algebra (\ref{Eq:xp})
it follows that $ic_1^2\xi^2\epsilon_{ij}(\theta-c_2^2\eta)=0$.
Thus {\it the condition of guaranteeing Bose - Einstein
statistics} reads
\begin{equation}
\label{Eq:c-2}
c_2=\sqrt{\frac{\theta}{\eta}}.
\end{equation}
From Eqs.~(\ref{Eq:hat-a}), (\ref{Eq:c1-c2}) and (\ref{Eq:c-2}) we
obtain the following deformed annihilation and creation operators
$\hat a_i$ and $\hat a_i^\dagger$:
\begin{equation}
\label{Eq:aa+1}
\hat a_i=\sqrt{\frac{1}{2\hbar}\sqrt{\frac{\eta}{\theta}}}\left
(\hat x_i +i\sqrt{\frac{\theta}{\eta}}\hat p_i\right),
\hat
a_i^\dagger=\sqrt{\frac{1}{2\hbar}\sqrt{\frac{\eta}{\theta}}}\left
(\hat x_i-i\sqrt{\frac{\theta}{\eta}}\hat p_i\right),
\end{equation}
From Eqs.~(\ref{Eq:xp}) and (\ref{Eq:aa+1}) it follows that the
deformed bosonic algebra of $\hat a_i$ and $\hat a_j^\dagger$
reads \cite{JZZ04a}
\begin{equation}
\label{Eq:[a,a+]1}
[\hat a_i,\hat a_j^\dagger]=\delta_{ij}
+\frac{i}{\hbar}\xi^2\sqrt{\theta\eta}\;\epsilon_{ij},\;
[\hat a_i,\hat a_j]=0,\;(i,j=1,2).
\end{equation}
In Eqs.~(\ref{Eq:[a,a+]1}) the three equations $[\hat a_1,\hat
a_1^\dagger]=[\hat a_2,\hat a_2^\dagger]=1,\;[\hat a_1,\hat
a_2]=0$ are the same as the undeformed bosonic algebra in
commutative space; The equation
\begin{equation}
\label{Eq:[a,a+]2}
[\hat a_1,\hat a_2^\dagger] =\frac{i}{\hbar}\xi^2\sqrt{\theta\eta}
\end{equation}
is a new type. Eqs.~(\ref{Eq:[a,a+]1}) constitute a complete and
closed deformed bosonic algebra. Because of noncommutativity of
space, different degrees of freedom are correlated at the level of
the deformed Heisenberg - Weyl algebra (1); Eq.~(\ref{Eq:[a,a+]2})
represents such correlations at the level of the deformed
annihilation and creation operators.
Now we consider the second aspect. Following the standard
procedure of constructing the Fock space of many - particle
systems in commutative space, we shall take
Eqs.~(\ref{Eq:[a,a+]1}) as the {\it definition relations} for the
complete and closed deformed bosonic algebra without making
further reference to its $\hat x_i$, $\hat p_i$ representations,
generalize it to many - particle systems and find a basis of the
Fock space.
We introduce the following auxiliary operators, the tilde
annihilation and creation operators
\begin{equation}
\label{Eq:tilde-a}
\tilde a_1=\frac{1}{\sqrt{2\alpha_1}} \left(\hat a_1+i\hat
a_2\right),\;
\tilde a_2=\frac{1}{\sqrt{2\alpha_2}} \left(\hat a_1-i\hat
a_2\right),
\end{equation}
where $\alpha_{1,2}=1\pm \xi^2\sqrt{\theta\eta}/\hbar$.
From Eqs.~(\ref{Eq:[a,a+]1}) it follows that the commutation
relations of $\tilde a_i$ and $\tilde a_j^\dagger$ read
\begin{equation}
\label{Eq:tilde[a,a+]}
\left[\tilde a_i,\tilde a_j^\dagger\right]=\delta_{ij},\;
\left[\tilde a_i,\tilde a_j\right]=\left[\tilde a_i^\dagger,\tilde
a_j^\dagger\right]=0,\;(i,j=1,2).
\end{equation}
Thus $\tilde a_i$ and $\tilde a_i^\dagger$ are explained as the
deformed annihilation and creation operators in the tilde system.
The tilde number operators $\tilde N_1=\tilde a_1^\dagger\tilde
a_1$ and $\tilde N_2=\tilde a_2^\dagger\tilde a_2$ commute each
other, $[\tilde N_1,\tilde N_2]= 0.$ A general tilde state
\begin{equation}
\label{Eq:tilde-state}
\widetilde {|m,n\rangle}\equiv (m!n!)^{-1/2}(\tilde
a_1^\dagger)^m(\tilde a_2^\dagger)^n\widetilde {|0,0\rangle},
\end{equation}
where the vacuum state $\widetilde {|0,0\rangle}$ in the tilde
system is defined as $\tilde a_i\widetilde
{|0,0\rangle}=0\;(i=1,2),$ is the common eigenstate of $\tilde
N_1$ and $\tilde N_2$:
$\tilde N_1\widetilde {|m,n\rangle}=m\widetilde {|m,n\rangle}$,
$\tilde N_2\widetilde {|m,n\rangle}=n\widetilde {|m,n\rangle}$,
$(m, n=0, 1, 2,\cdots)$,
and satisfies $\widetilde {\langle m^{\prime},n^{\prime}}
\widetilde {|m,n\rangle}=
\delta_{m^{\prime}m}\delta_{n^{\prime}n}$. Thus $\{\widetilde
{|m,n\rangle}\}$ constitute an orthogonal normalized complete
basis of the tilde Fock space. In the tilde Fock space all
calculations are the same as the case in commutative space, thus
the concept of identical particles is maintained and the formalism
of the deformed Bosonic symmetry which restricts the states under
permutations of identical particles in multi - boson systems can
be similarly developed.
The theorem is proved.
It should be emphasized that in the case of both position -
position and momentum - momentum noncommuting the special feature
is when $[\hat a_{i},\hat a_{j}]=[\hat a_{i}^\dagger,\hat
a_{j}^\dagger]=0$ are satisfied, Bose - Einstein statistics is not
guaranteed. The reason is as follows. Because the new type
(\ref{Eq:[a,a+]2}) of bosonic commutation relations correlates
different degrees of freedom, the number operators $\hat N_1=\hat
a_1^\dagger\hat a_1$ and $\hat N_2=\hat a_2^\dagger\hat a_2$ do
not commute, $[\hat N_1, \hat N_2]\ne 0.$ They have not common
eigenstates. The vacuum state of the hat system is defined as
$\hat a_i|0,0\rangle=0,\;(i=1,2)$. A general hat state $\widehat
{|m,n\rangle}$ is defined as
\begin{equation*}
\widehat {|m,n\rangle}\equiv c(\hat a_1^\dagger)^m(\hat
a_2^\dagger)^n|0,0\rangle
\end{equation*}
where $c$ is the normalization constant, these states $\widehat
{|m,n\rangle}$ are not the eigenstate of $\hat N_1$ and $\hat
N_2$:
\begin{equation*}
\hat N_1\widehat {|m,n\rangle} =m\widehat
{|m,n\rangle}+\frac{i}{\hbar}m\xi^2 \sqrt{\theta\eta}\widehat
{|m+1,n-1\rangle},\; \nonumber
\end{equation*}
\begin{equation*}
\hat N_2\widehat {|m,n\rangle} =n\widehat
{|m,n\rangle}+\frac{i}{\hbar}n\xi^2 \sqrt{\theta\eta}\widehat
{|m-1,n+1\rangle}. \nonumber
\end{equation*}
Because of Eq.~(\ref{Eq:[a,a+]2}), in calculations of the above
equations we should take care of the ordering of $a_i$ and
$a_j^\dagger$ for even $i \ne j$ in the state $\widehat
{|m,n\rangle}$. The states $\widehat {|m,n\rangle}$ are not
orthogonal each other. For example, the inner product between
$\widehat {|1,0\rangle}$ and $\widehat {|0,1\rangle}$ is
\begin{equation*}
\label{Eq:1-2b} \widehat {\langle 1,0|}\widehat {
1,0\rangle}=-\frac{i}{\hbar}\xi^2 \sqrt{\theta\eta}. \nonumber
\end{equation*}
Thus $\{\widehat {|m,n\rangle}\}$ do not constitute an orthogonal
complete basis of the Fock space of a identical - boson system.
Now we investigate two issues related to this theorem: the tilde
phase space and the constraint between noncommutative parameters.
\section{The Tilde Phase Space}
\setcounter{equation}{0}
First we consider tilde phase space variables. Using
Eqs.~(\ref{Eq:aa+1}) and (\ref{Eq:tilde-a}) we rewrite $\tilde
a_i$ as
\begin{eqnarray}
\label{Eq:tilde-aa+1}
\sqrt{\alpha_1}\; \tilde a_1 =\left
(\frac{\eta}{4\theta\hbar^2}\right)^{1/4}\left (\tilde x
+i\sqrt{\frac{\theta}{\eta}}\;\tilde p^\dagger\right), \;
\nonumber\\
\sqrt{\alpha_2}\; \tilde a_2 =\left
(\frac{\eta}{4\theta\hbar^2}\right)^{1/4}\left (\tilde x^\dagger
+i\sqrt{\frac{\theta}{\eta}}\;\tilde p\right).
\end{eqnarray}
Where the tilde coordinate and momentum $(\tilde x, \tilde p)$ are
related to $(\hat x, \hat p)$ by
\begin{equation}
\label{Eq:tilde-x,p}
\tilde x=\frac{1}{\sqrt{2}}\left(\hat x_1 +i\hat x_2\right),\;
\tilde p=\frac{1}{\sqrt{2}}\left (\hat p_1-i\hat p_2\right).
\end{equation}
The tilde phase variables $(\tilde x, \tilde p)$ satisfy the
following commutation relations:
\begin{equation}
\label{Eq:tilde-[x,p]}
[\tilde x,\tilde x^\dagger]=\xi^2\theta, \;
[\tilde p,\tilde p^\dagger]= -\xi^2\eta, \;
[\tilde x,\tilde p]=[\tilde x^\dagger,\tilde p^\dagger]=i\hbar, \;
[\tilde x,\tilde p^\dagger]=[\tilde x^\dagger,\tilde p]=0.
\end{equation}
A Hamiltonian
$\hat H(\hat x,\hat p)=\hat p_i\hat p_i/2\mu +V(\hat x_i)$
with potential $V(\hat x_i)$ in the hat system is rewritten as
\begin{equation}
\label{Eq:tilde-H}
\hat H(\hat x,\hat p)= \tilde H(\tilde x, \tilde x^\dagger, \tilde
p, \tilde p^\dagger) =\left (\tilde p\tilde p^\dagger+ \tilde
p^\dagger\tilde p\right)/2\mu+\tilde V(\tilde x, \tilde x^\dagger)
\end{equation}
in the tilde system.
In some cases calculations in the tilde system are simpler than
ones in the hat system. For example, in the hat system the
Hamiltonian of a two-dimensional isotropic harmonic oscillator is
$\hat H(\hat x,\hat p)= \hat p_i\hat p_i/2\mu + \mu\omega^2 \hat
x_i\hat x_i/2$.
In the tilde system it is rewritten as
\begin{equation}
\label{Eq:tilde-H1}
\tilde H(\tilde x, \tilde x^\dagger, \tilde p,
\tilde p^\dagger)=
\left (\tilde p\tilde p^\dagger+\tilde p^\dagger\tilde
p\right)/2\mu+ \mu\omega^2\left (\tilde x\tilde x^\dagger+ \tilde
x^\dagger\tilde x\right)/2 =\hbar\left (\tilde \omega_i \tilde
N_i+\omega\right),
\end{equation}
where
$\tilde \omega_{1,2}=\alpha_{1,2}\; \omega$
are effective frequencies, the tilde number operators $\tilde N_1$
and $\tilde N_2$ have eigenvalues $n_1, n_2=0, 1, 2,\cdots$. From
Eq.~(\ref{Eq:tilde-H1}) it follows that the energy eigenvalues of
$\tilde H(\tilde x, \tilde x^\dagger, \tilde p, \tilde p^\dagger)$
are
\begin{equation}
\label{Eq:tilde-E1}
\tilde E_{n_1,n_2} =\hbar\left (\tilde
\omega_i n_i+\omega \right)=\hbar\omega \left (n_1+n_2+1
\right)+\hbar\omega\sqrt{\theta\eta}\left (n_1- n_2 \right).
\end{equation}
The last term represents the shift of the energy level originated
from effects of spacial noncommutativity. There is no shift for
zero-point energy $\omega$. It is worth noting that
Eq.~(\ref{Eq:tilde-E1}) gives the {\it exact} (non-perturbational)
eigenvalues.
Ref.~\cite{JLR} also investigated the structure of a
noncommutative Fock space and obtained eigenvectors of several
pairs of commuting hermitian operators which can serve as basis
vectors in the noncommutative Fock space. Calculations in such a
noncommutative Fock space are much complex than the above
(commutative) tilde Fock space.
\section{The Constraint Between Noncommutative Parameters}
\setcounter{equation}{0}
The structure of the deformed annihilation and creation operators
$\hat a_{i}$ and $\hat a_{i}^\dagger$ in Eqs.~(\ref{Eq:aa+1}) are
determined by the deformed Heisenberg - Weyl algebra
(\ref{Eq:xp}), independent of dynamics. The special character of a
dynamical system is encoded in the dependence of the factor
$\sqrt{\theta/\eta}$ on characteristic parameters of the system
under study. This put a constraint between $\theta$ and $\eta$
which can be determined as follows.
The general representation of the undeformed annihilation operator
$a_i$ by $x_i$ and $p_i$ is
$a_i=c_1^{\prime}(x_i +ic_2^{\prime} p_i),$
where the constants $c_1^{\prime}$ can be fixed as follows.
Operators $a_i$ and $a_i^\dagger$ should satisfy bosonic
commutation relations $[a_1,a_1^\dagger]=[a_2,a_2^\dagger]=1$.
From this requirement the undeformed Heisenberg - Weyl algebra
leads to $c_1^{\prime}=\sqrt{1/2\hbar c_2^{\prime}}$. The
undeformed bosonic commutation relation $[a_{i},a_{j}]=0$ is
automatically satisfied, so $c_2^{\prime}$ is a free parameter.
Thus the general representation of the undeformed annihilation
operator reads
\begin{equation}
\label{Eq:aa+2}
a_i=\frac{1}{\sqrt{2\hbar c_2^{\prime}}}(x_i +ic_2^{\prime}p_i),
\end{equation}
operators $a_i$ and $a_i^\dagger$ satisfy the undeformed bosonic
algebra
$[a_{i},a_{j}]=[a_i^\dagger,a_j^\dagger]=0, \;
[a_{i},a^{\dagger}_{j}]=i\delta_{ij}.$
From Eqs.~(\ref{Eq:xp}), (\ref{Eq:hat-x-p}), (\ref{Eq:hat-a}),
(\ref{Eq:c-2}) and (\ref{Eq:aa+2}) it follows that $\hat a_{i}$
can be represented by $a_{i}$ as follows:
\begin{equation}
\label{Eq:aa+3}
\hat
a_{i}=\xi(a_{i}+\frac{i}{2\hbar}\sqrt{\theta\eta}\epsilon_{ij}a_{j}),
\end{equation}
Similar to Eqs.~(\ref{Eq:xp}) and (\ref{Eq:hat-x-p}), it should be
emphasized that for the case of both position - position and
momentum - momentum noncommuting the scaling factor $\xi$ in
Eq.~(\ref{Eq:aa+3}) guarantees consistency of the framework.
Specially, it maintains the bosonic commutation relation $[\hat
a_{i},\hat a^{\dagger}_{j}]=i\delta_{ij}.$
In the limit $\theta,\eta\to 0$, the deformed operators $\hat
x_{i}, \hat p_{i}, \hat a_{i}$ reduce to the undeformed ones
$x_{i}, p_{i}, a_{i}$. Eq.~(\ref{Eq:c-2}) indicates that in this
limit $\theta/\eta$ should keep finite. From Eqs.~(\ref{Eq:xp}),
(\ref{Eq:hat-x-p}), (\ref{Eq:hat-a}), (\ref{Eq:c-2}),
(\ref{Eq:aa+1}), (\ref{Eq:aa+2}) and (\ref{Eq:aa+3}), it follows
that
\begin{equation}
\label{Eq:c-c^prime}
c_1=c_1^{\prime},\;c_2=c_2^{\prime}.
\end{equation}
From Eqs.~(\ref{Eq:c-2}) and (\ref{Eq:c-c^prime}) we obtain the
following constrained condition
\footnote {\; Eq.~(\ref{Eq:aa+1}) is the most general
representation of the physical annihilation operator $\hat a_i$ in
noncommutative space. In literature there is an extensively tacit
understanding about the definition of the physical annihilation
operator such that {\it`` \,it is possible to construct an
infinity of the creation/annihilation operators which satisfy
exactly the bosonic commutation relations, but do not require any
constraint on the parameters such as Eq.~(\ref{Eq:cc1})"}. For
example, similar to the Landau creation and annihilation operators
(acting within or across Landau levels) involve mixing of spatial
directions in an external magnetic field, we may define the
following annihilation operator
$$
\hat {a_i^{\prime}} = \frac{\nu^{-1}}{\sqrt{2 \hbar c_2^{\prime}}}
\left[ \left( \delta_{ij} - \frac{i c_2^{\prime} \eta}{2 \hbar}
\epsilon_{ij} \right) \hat x_j + i \left(c_2^{\prime} \delta_{ij}
- \frac{i \theta}{2 \hbar} \epsilon_{ij} \right) \hat p_j
\right],$$
where $\nu = \xi (1 - \theta\eta/4 \hbar^2)$. These operators
automatically satisfy the bosonic commutation relations
$[\hat {a_i^{\prime}} , \hat {a_j^{\prime}}^{\dagger} ] =
\delta_{ij}, \hspace{0.5 cm}
[\hat {a_i^{\prime}} , \hat {a_j^{\prime}}] = [\hat
{a_i^{\prime}}^{\dagger} , \hat {a_j^{\prime}}^{\dagger}]=0. $
Moreover no constraint on the parameters $\theta$ and $\eta$ is
required apart from the obvious one $\eta \theta \ne 4 \hbar^2$.
The previous construction also indicates that it is not compulsory
to consider both position and momentum noncommutativity. Indeed,
if we take $\eta =0$, $\nu = \xi =1$ in the previous expression
for the creation/annihilation operators, we get:
$$
\hat {a_i^{\prime\prime}} = \frac{1}{\sqrt{2 \hbar c_2'}} \left[
\hat x_i + i \left(c_2' \delta_{ij} - \frac{i \theta}{2 \hbar}
\epsilon_{ij} \right) \hat p_j \right],$$ This is also perfectly
consistent.
In order to clarify the meaning of $\hat {a_i^{\prime}}$ we insert
Eqs.~(\ref{Eq:hat-x-p}) into it. It follows that
$[( \delta_{ij} -i c_2^{\prime} \eta \epsilon_{ij}/2 \hbar) \hat
x_j + i(c_2^{\prime} \delta_{ij} -i \theta \epsilon_{ij}/2 \hbar)
\hat p_j]=\xi (1 - \theta \eta/ 4 \hbar^2)(x_i
+ic_2^{\prime}p_i),$
thus
$\hat {a_i^{\prime}}=(x_i +ic_2^{\prime}p_i)/\sqrt{2 \hbar c_2'},$
which elucidates that $\hat {a_i^{\prime}}$ is just the undeformed
annihilation operator $a_i$ in Eq.~(\ref{Eq:aa+2}), not the
annihilation operator in noncommutative space. This explains that
$\hat {a_i^{\prime}}$ and $\hat {a_i^{\prime}}^{\dagger}$
automatically satisfy the undeformed bosonic commutation
relations, and no constraint on the parameters $\theta$ and $\eta$
is required.
For the case $\eta = 0$, $\nu = \xi =1$, inserting
Eqs.~(\ref{Eq:hat-x-p}) into $\hat {a_i^{\prime\prime}},$ we
obtain
$$
\hat {a_i^{\prime\prime}} = \frac{1}{\sqrt{2 \hbar
c_2^{\prime}}}(x_i +ic_2^{\prime}p_i),$$
which is the annihilation operator in commutative space again.}
\begin{equation}
\label{Eq:cc1}
\eta=K\theta,
\end{equation}
where the coefficient $K={c_2^{\prime}}^{-2}$ is a constant with a
dimension $(mass/time)^2$. Eq.~(\ref{Eq:cc1}) shows that the
general feature of such a constraint for any system is a direct
proportionality between noncommutative parameters $\eta$ and
$\theta$.
In Eq.~(\ref{Eq:cc1}) the proportional coefficient $K$ is not
determined. In the context of quantum mechanics for simple cases
the dimensional analysis can determine $c_2^{\prime}$ up to a
dimensionless constant.
As an example, we consider a harmonic oscillator. The dimension of
$c_2^{\prime}$ in Eqs.~(\ref{Eq:aa+2}) is $time/mass$. The
characteristic parameters in the Hamiltonian of a harmonic
oscillator are the mass $\mu$, frequency $\omega$ and $\hbar$. The
unique product of $\mu^{t_1}$, $\omega^{t_2}$ and $\hbar^{t_3}$
possessing the dimension $time/mass$ is $\mu^{-1}\omega^{-1}$. So
one obtains $c_2^{\prime}=\gamma/\mu\omega$, where $\gamma$ is a
dimensionless constant and can be determined as follows.
The position $x_{i}$ and momentum $p_{i}$ are, respectively,
represented by $a_i$ and $a_i^\dagger$ as
$$x_{i}=\sqrt{\frac{\gamma\hbar}{2\mu\omega}}\left(a_{i}+
a_i^\dagger\right),\;
p_{i}=-i\sqrt{\frac{\hbar\mu\omega}{2\gamma}}\left(a_{i}-
a_i^\dagger\right).$$
In the vacuum state $|0>$ the expectations of the kinetic and the
potential energy, respectively, read
\begin{equation}
\label{Eq:Ek,Ep}
\overline {E_k}=<0|\frac{1}{2\mu}p_{i}^2|0>
=\frac{\hbar\omega}{4\gamma},\;
\overline {E_p}=<0|\frac{1}{2}\mu\omega^2x_{i}^2|0>
=\frac{\gamma\hbar\omega}{4}.
\end{equation}
The special character of a harmonic oscillator is that in any
state the expectation of the kinetic energy equals to the one of
the potential energy. The condition of $\overline {E_k}=\overline
{E_p}$ leads to $\gamma=\pm 1$. Because of $\overline {E_k}\ge 0$,
the only solution is $\gamma=1$. Thus the constraint between
$\theta$ and $\eta$ for a harmonic oscillator reads
\begin{equation}
\label{Eq:cc3}
\eta=\mu^2\omega^2\theta.
\end{equation}
The method of determining such a dimensionless constant for a
harmonic oscillator, $\overline {E_k}=\overline {E_p}$, cannot be
applied to general cases. A complete determination of the
proportional coefficient $K$ in (\ref{Eq:cc1}) based on
fundamental principles for general cases is worth elucidating in
further studies.
\section{Discussions}
\setcounter{equation}{0}
We clarify the following two points to conclude the paper.
(i) Bose - Einstein statistics can be investigated at two levels:
the fundamental level of quantum field theory and the level of
quantum mechanics. Whether the deformed Heisenberg - Weyl algebra
is consistent with Bose - Einstein statistics is still an open
issue at the level of quantum field theory. Following the standard
procedure of investigating Bose - Einstein statistics in quantum
mechanics discussions restricted at the level of quantum mechanics
are allowed and meaningful. At short distances, where spatial
noncommutativity might be relevant, one also expects quantum
mechanics to break down and to be replaced by noncommutative
quantum field theory. But studies at the level of noncommutative
quantum mechanics may explores some qualitative features of
spatial noncommutativity, and some results may survive at the
level of noncommutative quantum field theory. It is therefore
hoped that studies at the level of noncommutative quantum
mechanics may give some clue for further development.
(ii) The constrained condition (\ref{Eq:cc1}) is fixed by the most
fundamental requirement, thus can apply to any dynamical system.
Ordinary quantum mechanics is a most successful theory which has
been fully confirmed by experiments. If NCQM is a realistic
physics, possible modifications from NCQM to ordinary quantum
mechanics should be extremely small. It means that both
noncommutative parameters $\theta$ and $\eta$ should be extremely
small. This is guaranteed by Eq.~(\ref{Eq:cc1}). Furthermore, it
is understood that noncommutativity between positions is
fundamental and the parameter $\theta$ keeps the same for all
systems. Noncommutativity between momenta arises naturally as a
consequence of noncommutativity between coordinates, as momenta
are defined to be the partial derivatives of the action with
respect to the noncommutative coordinates \cite{SGT}. This means
that noncommutativity between momenta depends on dynamics. Thus
$\eta$ and the proportional coefficient $K$ between $\eta$ and
$\theta$ may depend on characteristic parameters of the
Hamiltonian (or the action) of the system under study. In simple
cases when dimensional analysis works, it can determine $K$ up to
a dimensionless constant. In order to completely fix $K$
considerations from dynamics may be necessary.
ACKNOWLEDGMENTS
This work has been supported by the Natural Science Foundation of
China under the grant number 10575037 and by the Shanghai
Education Development Foundation.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Bayesian Regression and Classification Using
Gaussian Process Priors Indexed by Probability Density Functions}
\author[mymainaddress]{A. Fradi}
\author[mymainaddress]{Y. Feunteun}
\author[mymainaddress]{C. Samir}
\author[mymainaddress]{M. Baklouti}
\author[mysecondaryaddress]{F. Bachoc}
\author[mysecondaryaddress]{J-M Loubes}
\address[mymainaddress]{CNRS-LIMOS, UCA, France}
\address[mysecondaryaddress]{Institut de
Mathématiques de Toulouse, France}
\begin{abstract}
In this paper, we introduce the notion of Gaussian processes indexed by probability density functions for extending the Mat\'ern family of covariance functions. We use some tools from information geometry to improve the efficiency and the computational aspects of the Bayesian learning model. We particularly show how a Bayesian inference with a Gaussian process prior (covariance parameters estimation and prediction) can be put into action on the space of probability density functions. Our framework has the capacity of classifiying and infering on data observations that lie on nonlinear subspaces.
Extensive experiments on multiple synthetic, semi-synthetic and real data demonstrate the effectiveness and the efficiency of the proposed methods in comparison with current state-of-the-art methods.
\end{abstract}
\begin{keyword}
\textbf{Information geometry, Learning on nonlinear manifolds, Bayesian regression and classification \sep Gaussian process prior \sep HMC sampling }
\end{keyword}
\end{frontmatter}
\section{Introduction}
In recent years, Gaussian processes on manifolds have become very
popular in various fields including machine learning, data mining, medical imaging, computer
vision, etc. The main purpose consists in inferring the
unknown target value at an observed location on the manifold as a prediction by conditioning on known inputs and targets.
The random field, usually
Gaussian, and the forecast can be seen as the posterior mean, leading to an optimal unbiased predictor~\cite{bachoc2017gaussian,AbtWel1998}. Bayesian regression and classification models
focus on the use of priors for the parameters to define and estimate a conditional predictive expectation. In this work, we consider a very common problem in several contexts of applications in science and technology: learning a Bayesian regression and classification models with Probability Density Functions as inputs.
Probability Density Functions (PDFs) are inherently infinite-dimensional objects. Hence, it is not straightforward to extend traditional machine learning methods from finite vectors to functions. For example, in functional data analysis~\cite{Anuj-book-2016} with applications in medical~\cite{Samir-wacv-16,Ramsay-1991}, it is very common to compare/classify functions. The mathematical formulation leads to a wide range of applications where it is crucial to characterize a population or to build predictive models.
In particular, multiple frameworks exist for comparing PDFs in different representations
including Frobenius, Fisher-Rao, log-Euclidean, Jensen-Shannon
and Wasserstein distances~\cite{Srivastava-2007,Mitsuhiro-Geodesic-2015,Samir-wacv-16,Nguyen-Div-JS,bachoc2017gaussian}. In this work, we extend this formulation to PDFs space $\mathcal{P}$ with the Mat\'ern covariance functions.
\textcolor{black}{There is a rich literature on statistical inference on manifolds among which the Fisher information matrix~\cite{Rao-45} has played a central role. Recently, there has been increasing interest in applying information geometry for machine learning and data mining tasks~\cite{Amari-Rao-87,Nihat-book,Amari-book,Barbaresco-2013}. The Fisher information matrix determines a Riemannian structure on a parametrized space
of probability measures. Study of geometry of $\mathcal{P}$
with the Riemannian structure, which we call information geometry, contributes greatly to statistical inference, refer to~\cite{Nihat-book,Amari-book} for more details. Such methods are based on parametric models that are of great interest in many applications. However, aspects of PDFs other than parametric families may be important in various contexts~\cite{Nguyen-Div-JS,Nielsen2013,Srivastava-2007,shishido2005,Samir-wacv-16,Fukumizu-2010}. In particular,
the consistency of regression and classification with PDFs inputs was established in~\cite{DBLP:journals/corr/SutherlandOPS15,DBLP:conf/aistats/OlivaNPSX14,pmlr-v31-poczos13a} with the help of kernel density estimation~\cite{Botev-2010}.
More recently,~\cite{bachoc2017gaussian} studied the dissimilarity between PDFs with the Wasserstein distance and~\cite{Zhang-2019} used a nonparametric framework to compare spherical populations.}
The main aim of this paper is to learn a Bayesian inference on Gaussian processes. For instance, one can think of a Gaussian process as defining PDFs and inference taking place directly in the function-space. Moreover, the index space is that of PDFs when choosing the underlying metric in order to evaluate the dissimilarity between them~\cite{Atkinson-81-RaoDistance}. The only drawback is that performing Kriging on PDFs space $\mathcal{P}$ is not straightforward due to its geometry. \textcolor{black}{For this end, we exploit an isometric embedding by combinng the square root transform~\cite{Paper-Bhattacharyya43} and the distance induced by the Fisher-Rao metric which make the covariance function non-degenerate and simplify the optimization process.}
Gaussian processes (GPs) have been widely used to provide a probabilistic framework for a large variety of machine learning methods~\cite{rasmussen06gaussian}. Optimization techniques are usually required to fit a GP model $Z$, that is to select a GP covariance function. For $p_i$ and $p_j$ in $\mathcal{P}$, the main issue would be to build a proper covariance between $Z(p_i)$ and $Z(p_j)$. In particular, this covariance can define a notion of stationarity for the process. Another important task is the classification process where we wish to assign an input PDF $p_i$ to one of the given classes~\cite{NIPS2011_4241}.
To search for the covariance function hyperparameter, we use several methods for maximizing the marginal likelihood. Our aim is then to select those optimizing performance criteria for regression and classification: The first method is based on the gradient descent for finding a local maximum of the marginal likelihood. The second method is a special case of MCMC methods, called Hamiltonian Monte-Carlo (HMC)~\cite{Duane1987216}. The objective is to perform sampling from a probability distribution for
which the marginal likelihood and its gradient are known. The latter has the advantage to simulate from a physical system governed by Hamiltonian dynamics.
The remainder of the paper is organized as follows. In Section~\ref{sec:Proposed}, we introduce the problem formulation and we give a background of some Riemannian representations. Section~\ref{sec:GP} extends the usual notion of GPs indexed by finite vectors to those indexed by PDFs with theoretical results for the Mat\'{e}rn covariance function.
We also give details of
the proposed model for predicting and classifying PDFs as well as estimating the covariance function parameters.
In Section~\ref{sec:results},
we present and discuss experimental results with some comparison studies. We conclude the paper in Section~\ref{sec:clc}.
\section{Problem formulation and geometry background}
\label{sec:Proposed}
\noindent Let $p_1,\dots, p_n$ denote a finite set of observed PDFs and $y_1, \dots , y_n$ denote their corresponding outputs with real values (quantitative or qualitative).
In this work, we focus on \textbf{nonparametric PDFs} restricted to be defined on $\Omega=[0,1]$.
Our main goals throughout this paper are: i) Fitting the proposed model's parameters
in order to
better explain the link between $p_i$ and $y_i$, $i=1,\dots,n$, ii)
evaluating the corresponding predictive expectation at
an unobserved PDF $p^* \in \mathcal{P}$ and iii)
studying the properties of the GP with the Mat\'ern covariance function.
In the particular case where $y_i \in \{-1,+1\}$, we will assign each unobserved PDF $p^*$ to its predicted class after learning the model parameters.
To reach such goal, we follow the same idea of nonparametric information geometry that has been discovered by~\cite{Rao-45} and developed later in other works, see for example~\cite{Friedrich91,Srivastava-2007,Mitsuhiro-Geodesic-2015,Nihat-book,Fukumizu-2013}. Thus, the notion of similarity/dissimilarity between any $p_i$ and $p_j$ is measured using the induced Rao distance~\cite{Rao-82,Atkinson-81-RaoDistance} between them on the underlying space.
In this paper, we look at the space of PDFs as a Riemannian manifold, as detailed in the next section, which plays an important role in the proposed methods.
\subsection{Riemannian structure of PDFs space}
\label{sec:background}
For more details about the geometric structure concerning the Fisher information metric, refer to~\cite{Friedrich91,shishido2005,Bauer-2016}. For example,~\cite{Friedrich91} showed that $\mathcal{P}$ with a Riemannian structure has a positive constant curvature. Furthermore, the action of orientation preserving diffeomorphism acts by isometry on $\mathcal{P}$ with respect to the Fisher information metric. We will exploit these nice properties to define an isometric embedding from $\mathcal{P}$ to ${\cal{E}}$ detailed in~(\ref{eq:IsometryEquality}). Then, we use this embedding to construct a Gaussian process on PDFs with a proper covariance function~(\ref{eq:cov:C:K}) and a predictive model~(\ref{predictor}).\\
We first note that
the space of PDFs defined over $\Omega$ with values in $\ensuremath{\mathbb{R}}_{+}$ can be viewed in different manners. The case where $\Omega$ is finite and the statistical model is parametric has been largely studied in the literature~\cite{Amari-book,Nihat-book}. In contrast, if $\Omega$ is infinite which is the case here, different modeling options have been explored~\cite{Friedrich91,Pistone-95,Anuj-book-2016,Bauer-2016}. We start with the ideas developed in~\cite{Friedrich91,shishido2005,Srivastava-2007,Nihat-book} where $\mathcal{P}$ is an infinite dimensional smooth manifold. That is, $\mathcal{P}$ is the space of probability measures that satisfy the normalization constraint. Since we are interested in statistical PDFs analysis on $\mathcal{P}$, we need some geometrical tools~\cite{Helgason1978,Hyperbolic-Lee},\emph{e.g. } geodesic. For the rest of the paper, we view $\mathcal{P}$ as a smooth manifold~(\ref{eq:P}) and we impose a Riemannian structure on it with the Fisher-Rao metric~(\ref{eq:Fisher-Rao}). Let
\begin{eqnarray}
\label{eq:P}
\mathcal{P}=\{p:\Omega \to \mathbb{R} \hspace{0.05cm} | \hspace{0.05cm} p \geq 0 \hspace{0.1cm}\text{and} \int_{\Omega} p(t) dt=1 \}.
\end{eqnarray}
be the space of all PDFs (positive almost everywhere) including nonparametric models. We identify any tangent space of $\mathcal{P}$, locally at each $p$, by
\begin{eqnarray}
\label{eq:TangentP}
T_p(\mathcal{P})=\{f:\Omega \to \mathbb{R} \hspace{0.05cm} | \hspace{0.05cm} \int_{\Omega} f=0 \}
\end{eqnarray}
As detailed in~\cite{Cencov-book-1982,Friedrich91,Bauer-2016}, the tangent space contains functions that are infinitesimally differentiable. But following~\cite{Helgason1978}, we have a constructive method of great importance that allows one to form a local version of any arbitrary $f$ that is continuously differentiable in a small neighborhood and null outside. Now that we have a smooth manifold and its tangent space, we can introduce a Riemannian metric. This choice is very important since it will determine the structure of $\mathcal{P}$ and consequently the covariance function of the Gaussian process. More details about the importance of the metric and the induced Riemannian structure are discussed in~\cite{Mitsuhiro-Geodesic-2015,Samir-FoCom12,bauer-metrics}. We also define and denote by $\mathcal{P}_+$ the interior of $\mathcal{P}$. For the following, we consider without justification that any probability density can be locally perturbed to be smooth enough~\cite{Helgason1978}. This is true in finite dimensional cases but the generalization to infinite dimensional cases is not straightforward. Among several metrics, we are particularly interested in the Fisher-Rao metric defined, for any tangent vectors $f_1,f_2 \in T_p(\mathcal{P})$, by
\begin{eqnarray}
\label{eq:Fisher-Rao}
<f_1,f_2>_p=\int_{\Omega} \frac{f_1(t) f_2(t)}{p(t)} dt.
\end{eqnarray}
Although this metric has nice properties with an increasing interest~\cite{shishido2005,Amari-Rao-87,Cencov-book-1982}, $\mathcal{P}$ equipped with $<.,.>_p$ is still numerically intractable. Therefore, instead of working on $\mathcal{P}$ directly, we consider a mapping from $\mathcal{P}$ to the Hilbert upper-hemisphere (positive part) around the unity $1_{\mathcal{P}}$ such that $1_{\mathcal{P}}(t)=1$ for all $t$ in $\Omega$~\cite{Paper-Bhattacharyya43}. Thus, we exploit the Riemannian isometry between $\mathcal{P}$ and the upper-hemisphere to extend the notion of GPs to the space of PDFs. Indeed, we first define the map
\begin{eqnarray}
\label{eq:SRDF}
\Psi : \cal{P} & \rightarrow & \cal{H} \\
p &\mapsto & \phi=2\sqrt{p}, \quad (\mathtt{p=\Psi^{-1}(\phi)=\frac{1}{4}\phi^2}) \nonumber
\end{eqnarray}
where
\begin{eqnarray}
\label{eq:HilbertSphareH}
\mathcal{H} = \{ \phi: \Omega \to \mathbb{R} \hspace{0.05cm} | \hspace{0.05cm}
\phi \geq 0 \hspace{0.1cm}
\text{and}
\int_{\Omega} \phi(t)^2 dt = 4 \}.
\end{eqnarray}
Note that $\phi$ is well defined since $p$ is nonnegative and
$\Psi$ is a Riemannian isometry from $\cal{P}_+$ to $\cal{H}$ without the boundary~\cite{Hyperbolic-Lee}. On the other hand, any element $\phi \in \mathcal{H}$ can be uniquely projected as $\frac{1}{2} \phi$ to have a unit norm. For simplicity and without loss of generality, we interpret $\cal{H}$ as the elements of unit Hilbert upper-hemisphere $\mathcal{S}_+^{\infty}$ up to a multiplicative factor ($2$ here). From that point of view, we identify $\cal{H}$ with $\mathcal{S}_+^{\infty}$ and we define $\Psi(1_{\mathcal{P}})=1_{\mathcal{H}}$ to be the "unity pole" on $\cal{H}$. Note that $1_{\mathcal{H}}$ as the image of the uniform pdf $1_{\mathcal{P}}$ is a fixed point, \emph{i.e. } $1_{\mathcal{H}}=\sqrt{1_{\mathcal{P}}}=1_{\mathcal{P}}$. In this setup, we have
\begin{eqnarray}
\|\phi \|_2^2 =\int_{\Omega} \phi(t)^2 dt=1,
\end{eqnarray}
for any $\phi$ in $\cal{H}$ which allow us to consider $\cal{H}$, when equipped with the integral inner product $<.,.>_{2}$, as the
unit upper-hemisphere (positive part). Furthermore, for arbitrary directions $f_1, f_2$ in $T_p(\cal{\mathcal{P}})$ the Fisher-Rao metric as defined in~({eq:Fisher-Rao}) becomes $<.,.>_{2}$ as follows:
\begin{eqnarray}
\label{eq:FR2L2}
<f_1,f_2>_p= < D_{f_1} \Psi ,D_{f_1} \Psi>_2.
\end{eqnarray}
with $D_{f_i} \Psi(p)(t) = \frac{f_i(t)}{\sqrt{p(t)}}$ for all $t \in \Omega$ and $i=1,2$.
One of the main advantages of this formulation is to exploit the nice properties of the unit Hilbert sphere such as geodesic, exponential map, log map, and the parallel transport. For the rest of the paper, the geodesic distance $d_{\cal P}(p_1,p_2)$ between two PDFs $p_1$ and $p_2$ in $\cal P$, under the Fisher-Rao metric, is given by the geodesic distance $d_{\cal H}(\phi_1,\phi_2)$ (up to a factor 2) between their corresponding $\phi_1$ and $\phi_2$ on $\mathcal{H}$. We remind that the arc-length (geodesic distance) between distinct and non antipodal $\phi_1$ and $\phi_2$ on $\mathcal{H}$ is the angle
$\beta=\arccos \left( <\phi_1,\phi_2>_2 \right)$. We also remind some geometric tools that will be needed for next sections as a lemma:
\begin{lem}
With $\mathcal{H}$ defined from~(\ref{eq:HilbertSphareH}) with unit norm and $T_{\phi}(\cal{H} )$ its tangent space at $\phi$, we have the following:
\begin{itemize}
\item The exponential map is a bijective isometry from the tangent space $T_{\phi}(\cal{H})$ to
$\cal{H}$. For any
$w \in T_{\phi}(\cal{H})$, we write
\begin{eqnarray}
\label{eq:ExpMap}
\Exp{\phi}{w}=\cos(\|w\|_2)\phi + \sin(\|w\|_2) \frac{w}{\|w\|_2}.
\end{eqnarray}
\item Its inverse, the log map
is defined from $\mathcal{H}$ to $T_{\phi_1}(\cal{H} )$ as
\begin{eqnarray}
\label{eq:LogMap}
\Log{\phi_1}{\phi_2}= \frac{\beta}{\sin(\beta)}(\phi_2-\cos(\beta)\phi_1).
\end{eqnarray}
\item
For any two elements $\phi_1$ and $\phi_2$ on $\mathcal{H}$ the map $\Gamma: T_{\phi_1}(\mathcal{H}) \rightarrow T_{\phi_2}(\mathcal{H})$
parallel transports a vector $w$ from $\phi_1$ to $\phi_2$ and is given by:
\begin{eqnarray}
\label{eq:ParallelTransport}
\Gamma_{\phi_1 \rightarrowtail \phi_2}(w)= w-2\frac{(\phi_1+\phi_2)}{||\phi_1+\phi_2||_2^2}<w,\phi_2>_2
\end{eqnarray}
\end{itemize}
\end{lem}
For more details, we refer to~\cite{Hyperbolic-Lee}. As a special case, we consider the unity pole $\phi=1_{\cal H}$ and we denote ${\cal{E}}=T_{1}(\cal{H})$ the tangent space of $\cal{H}$ at $1_{\cal H}$. For simplicity, we note $\Log{1}{.}$ the log map from $\cal H$ to ${\cal{E}}$ and $\Exp{1}{.}$ its inverse. This choice is motivated by two reasons: The numerical implementation and the fact that $1_{\mathcal{H}}$ is the center of the geodesic disc $[0,\frac{\pi}{2}[$. Indeed and since all elements are on the positive part, the exponential map and its inverse are diffeomorphisms. So, one can consider any point on $\mathcal{H}$ instead of $1_{\mathcal{H}}$ to define the tangent space, \emph{e.g. } the Fr\'echet mean. However, this is without loss for the numerical precision. Furthermore we can use the properties of the log map to show that:
\begin{equation}
\label{eq:IsometryEquality}
||\Log{1}{\phi_i}-\Log{1}{\phi_j}||_2=d_{\cal H}(\phi_i,\phi_j)=\frac{1}{2}d_{\cal P}(p_i,p_j)
\end{equation}
for any two $p_i$, $p_j$ on $\cal P$. Note that the multiplicative factor $\frac{1}{2}$ is important to guarantee the isometry but will not have any impact on the covariance function defined in~(\ref{eq:cov:C:K}) as it is implicit in the hyperparameter.
\section{Gaussian Processes on PDFs}
\label{sec:GP}
In this section, we focus on constructing GPs on $ \mathcal{P}$.
A GP $Z$ on $\mathcal{P}$ is a random field indexed by $ \mathcal{P}$ so that $(Z( p_1),\ldots,Z( p_n))$ is a multivariate Gaussian vector for any $n \in \mathbb{N}\backslash\lbrace{0}\rbrace$ and $p_1, \ldots , p_n \in \mathcal{P}$.
A GP is completely specified by its mean function and its covariance function. We define a mean function $m: \mathcal{P} \to \mathbb{R}$ and the covariance function
$C: \cal{P} \times \cal{P} \to \mathbb{R}$
of a real process $Z$ as
\begin{align}
m(p_i)=&\mathbb{E} \big[Z(p_i)\big]. \\
C(p_i,p_j)=&\mathbb{E} \big[(Z(p_i)-m(p_i)) (Z(p_j)-m(p_j)) \big].
\end{align}
Thus, if a GP is assumed to have zero mean function ($m \equiv 0$), defining the covariance function completely defines the process behavior.
In this paper, we assume that the GPs are centered and we only focus on the issue of constructing a proper covariance function $C$ on $ \cal{P}$.
\subsection{Constructing covariance functions on $ \cal{P}$}
\label{sec:Covariance}
\noindent A covariance function $C$ on $ \cal{P}$ must satisfy the following conditions. For any $n \in \mathbb{N}\backslash\lbrace{0}\rbrace$ and $p_1, \ldots , p_n \in \cal{P}$, the matrix $\mathbf{C}=[C(p_i,p_j)]_{i,j=1}^{n}$ is symmetric nonnegative definite. Furthermore, $C$ is called non-degenerate when the above matrix is invertible whenever $p_1,\ldots,p_n$ are two-by-two distinct~\cite{bachoc2017gaussian}.
\textcolor{black}{The strategy that we adopt to construct covariance functions is to exploit the full isometry map $\Log{1}{.}$ to $\mathcal{E}$ given in~(\ref{eq:IsometryEquality}). That is, we construct covariance functions of the form
\begin{equation}
\label{eq:cov:C:K}
C( p_i , p_j ) =
K( \| \Log{1}{ \phi_i } - \Log{1}{ \phi_j } \|_2 ),
\end{equation}
where $K:\mathbb{R}^+ \to \mathbb{R}$.}
\begin{Prop} \label{prop:preservation:invertible}
Let
$K :\mathbb{R}^{+} \rightarrow \mathbb{R}$ be such that $K (u_i,u_j) = K( \|u_i - u_j\|_2 )$ is a covariance function on $\cal{E}$ and $C$ as defined as in~(\ref{eq:cov:C:K}). Then
\begin{enumerate}
\item $C$ is a covariance function.
\item If $[K(\|u_i-u_j\|_2)]_{i,j=1}^{n}$ is invertible, then $C$ is
non-degenerate.
\end{enumerate}
\end{Prop}
A closely related proof when dealing with Cumulative Density Functions (CDFs)
is given in~\cite{bachoc2017gaussian}.
In practice, we can select the function $K$ from the Mat\'ern family, letting for $t \geq 0$
\begin{eqnarray}
K_{\theta}(t)
=
\frac{\delta^2 }{ \Gamma(\nu)2^{\nu-1}} \Big(\frac{2 \sqrt{\nu t}}{\alpha} \Big)^\nu K_\nu \Big(\frac{2 \sqrt{\nu t}}{\alpha} \Big),
\end{eqnarray}
\textcolor{black}{where $K_\nu$ is a modified Bessel function of the second kind and $\Gamma$ is the gamma function.
We note
$\theta=(\delta^2,\alpha,\nu) \in
\Theta$
where $\delta^2>0$ is the variance parameter,
$\alpha>0$ is the correlation length parameter
and $\nu=\frac{1}{2}+k (k \in \mathbb{N})$ is the smoothness parameter.
The Mat\'{e}rn form~\cite{Ste1999} has the desirable property that GPs have realizations (sample paths) that are $k$ times differentiable~\cite{genton2015}, which prove its smoothness as function of $\nu$.
As $\nu \to \infty$, the Mat\'{e}rn covariance function approaches the squared exponential form, whose realizations are infinitely differentiable. For $\nu = \frac{1}{2}$, the Mat\'{e}rn takes the exponential form.
From Proposition~\ref{prop:preservation:invertible}, the Mat\'ern covariance function defined by $C(p_i , p_j) = K_{\theta}( \| \Log{1}{ \phi_i } - \Log{1}{ \phi_j } \|_2 )$ is indeed non-degenerate.}
\subsection{Regression on $\mathcal{P}$}
\label{sec:Regression}
\noindent
Having set out the conditions on the covariance function, we can define the regression model on $\cal{P}$ by
\begin{eqnarray}
\label{reg-model}
y_i=Z(p_i) + \epsilon_i, \hspace{0.2cm} i=1,\dots,n,
\end{eqnarray} where $Z$ is a zero mean GP indexed by $\mathcal{P}$
with a covariance function in the set $\{C_{\theta}; \theta \in \Theta\}$ and $\epsilon_i
\overset{\text{iid}}{\sim}
\mathcal{N}(0,\gamma^2)$.
Here $\gamma^2$ is the observation noise variance, that we suppose to be known for simplicity.
Moreover,
we note $\mathbf{y}=(y_1,\dots,y_n)^T$,
$\mathbf{p}=(p_1,\dots,p_n)^T$ and $\mathbf{v}=(v_1,\dots,v_n)^T=
(\Log{1}{\Psi(p_1)},\dots,\Log{1}{\Psi(p_n)})^T$.
The likelihood term is $\mathbb{P}(\mathbf{y}|Z(\mathbf{p}))=\mathcal{N}(Z(\mathbf{p}),\gamma^2 I_n)$ where
$I_n$ is the identity matrix. Moreover,
the prior on $Z(\mathbf{p})$ is $\mathbb{P}(Z(\mathbf{p}))=\mathcal{N}(0,\mathbf{C}_{\theta})$ with
$\mathbf{C}_{\theta}=[K_{\theta}(\|v_i-v_j\|_2)]_{i,j=1}^{n}$.
We use the product of likelihood and prior terms to perform the integration yielding the log-marginal likelihood
\begin{eqnarray}
l_{r}(\theta) =
-\mathbf{y}^T (\mathbf{C}_{\theta}+\gamma^2 I_n)^{-1} \mathbf{y}
-\log |\mathbf{C}_{\theta}+\gamma^2 I_n| -\frac{n}{2} \log 2\pi.
\label{likelihood-reg}
\end{eqnarray}
Let $\theta=\{\theta^j\}_{j=1}^{3}=(\delta^2,\alpha,\nu)$ denote the parameters of the Mat\'ern covariance function $K_{\theta}$.
The partial derivatives of $l_{r}(\theta)$ with respect to $\theta^j$ are
\begin{eqnarray}
\label{grad1}
\frac{\partial l_{r}(\theta)}{\partial \theta^j}=\frac{1}{2} \mathbf{y}^T \mathbf{C}_{\theta}^{-1} \frac{\partial \mathbf{C}_{\theta}}{\partial \theta^j} \mathbf{C}_{\theta}^{-1} \mathbf{y} - \text{tr}\big[\mathbf{C}_{\theta}^{-1}\frac{\partial \mathbf{C}_{\theta}}{\partial \theta^j}\big].
\end{eqnarray}
For an unobserved PDF $p^*$ and by deriving the conditional distribution, we arrive at the key predictive equation
\begin{eqnarray}
\mathbb{P}(Z(p^*)|\mathbf{p},\mathbf{y},p^*)= \mathcal{N}(\mu(p^*),\sigma^2(p^*)),
\end{eqnarray}
with
\begin{equation}
\left\lbrace
\begin{aligned}
\mu(p^*)&={\mathbf{C}_{\theta}^*}^T (\mathbf{C}_{\theta}+\gamma^2 I_n)^{-1} \mathbf{y},\\
\sigma^2(p^*)&=C_{\theta}^{**} -
{\mathbf{C}_{\theta}^*}^T(\mathbf{C}_{\theta}+\gamma^2 I_n)^{-1} \mathbf{C}_{\theta}^*,\\
\end{aligned}
\right.
\end{equation}
where $\mathbf{C}^*_{\theta}=K_{\theta}(\mathbf{v},v^*)$ and
$C_{\theta}^{**}=K_{\theta}(v^*,v^*)$ for $v^*=\Log{1}{\Psi(p^*)}$.
As we have introduced GP regression indexed by PDFs, we will present GP classifier in the next section.
\subsection{Classification on $\mathcal{P}$}
\label{sec:Classification}
\noindent For the classification part, we focus on the case of binary outputs, i.e., $y_i \in \{-1,+1\}$.
We first adapt the Laplace approximation to GPc indexed by PDFs in Section~\ref{sec:laplace}. We also give the approximate marginal likelihood and the Gaussian predictive distribution in Section~\ref{sec:marginal}.
\subsubsection{Approximation of the posterior}
\label{sec:laplace}
The likelihood is the product of individual likelihoods
$
\mathbb{P}(\mathbf{y}|Z(\mathbf{p}))=\prod_{i=1}^{n} \mathbb{P}(y_i|Z(p_i))
$
where $\mathbb{P}(y_i|Z(p_i))=\sigma(y_i Z(p_i))$ and $\sigma(.)$ refers to the sigmoid function satisfying $\sigma(t)=\frac{1}{1+\exp(-t)}$. As for regression, the prior law of GPc is
$
\mathbb{P}(Z(\mathbf{p}))=\mathcal{N}(0,\mathbf{C}_\theta)
$.
From the Bayes' rule, the posterior distribution of $Z(\mathbf{p})$ satisfies
\begin{eqnarray}
\label{posterior}
\mathbb{P}(Z(\mathbf{p})|\mathbf{y}) =
\frac{\mathbb{P}(\mathbf{y}|Z(\mathbf{p})) \times \mathbb{P}(Z(\mathbf{p}))}{\mathbb{P}(\mathbf{y}|\mathbf{p},\theta)},
\propto \mathbb{P}(\mathbf{y}|Z(\mathbf{p})) \times \mathbb{P}(Z(\mathbf{p})),
\end{eqnarray}
where $\mathbb{P}(\mathbf{y}|\mathbf{p},\theta)$ is the exact marginal likelihood.
The log-posterior is simply proportional to $
\log \mathbb{P} (\mathbf{y}|Z(\mathbf{p})) - \frac{1}{2} Z(\mathbf{p})^T \mathbf{C}_{\theta}^{-1} Z(\mathbf{p})$.
For the Laplace approximation, we approximate the posterior given in~(\ref{posterior})
by a Gaussian distribution. We can find the maximum a posterior (MAP) estimator denoted by $\hat{Z}(\mathbf{p})$, iteratively, according to
\begin{eqnarray}
Z^{k+1}(\mathbf{p})=(\mathbf{C}_{\theta}+\mathbf{W})^{-1} (\mathbf{W} Z^{k}(\mathbf{p}) + \nabla \mathbb{P} (\mathbf{y}|Z^k(\mathbf{p})) ),
\end{eqnarray}
where $\mathbf{W}$ is a $n \times n$ diagonal matrix with entries
$\mathbf{W}_{ii}=\frac{\exp(-\hat{Z}(p_i))}{(1+\exp(-\hat{Z}(p_i)))^2}$.
Using the MAP estimator, we can specify the Laplace approximation of the posterior by
\begin{eqnarray}
\hat{\mathbb{P}}(Z(\mathbf{p})|\mathbf{p},\mathbf{y})= \mathcal{N}(\hat{Z}(\mathbf{p}),(\mathbf{C}_{\theta}^{-1}+\mathbf{W})^{-1}).
\end{eqnarray}
\subsubsection{Predictive distribution}
\label{sec:marginal}
We evaluate the approximate marginal likelihood denoted by $\hat{\mathbb{P}}(\mathbf{y}|\mathbf{p},\theta)$
instead of the exact marginal likelihood $\mathbb{P}(\mathbf{y}|\mathbf{p},\theta)$ given in the denominator of~(\ref{posterior}).
Integrating out $Z(\mathbf{p})$, the log-marginal likelihood is approximated by
\begin{eqnarray}
\label{log-marginal}
l_{c}( \theta)=-\frac{1}{2} \hat{Z}(\mathbf{p})^T {\mathbf{C}}_{\theta}^{-1} \hat{Z}(\mathbf{p}) + \log p (\mathbf{y}|\hat{Z}(\mathbf{p}))
- \frac{1}{2} \log \big|I_n +\mathbf{W}^{\frac{1}{2}} \mathbf{C}_{\theta} \mathbf{W}^{\frac{1}{2}}\big|.
\end{eqnarray}
The partial derivatives of $l_c(\theta)$
with respect to $\theta^j$
satisfy
\begin{eqnarray}
\label{grad2}
\frac{\partial l_{c}(\theta)}{\partial \theta^j}=\frac{\partial l_{c}(\theta)}{\partial \theta^j}|_{\hat{Z}(\mathbf{p})}+\sum_{i=1}^{n} \frac{\partial l_{c}(\theta)}{\partial \hat{Z}(p_i)} \frac{\partial \hat{Z}(p_i)}{\partial \theta^j}.
\end{eqnarray}
The first term, obtained when we assume that $\hat{Z}(\mathbf{p})$ (as well as $\mathbf{W}$) does not depend on $\theta$, satisfies
\begin{eqnarray}
\frac{\partial l_{c}(\theta)}{\partial \theta^j}|_{\hat{Z}(\mathbf{p})} =
\frac{1}{2} {\hat{Z}(\mathbf{p})}^{T} \mathbf{C}_{\theta}^{-1} \frac{\partial \mathbf{C}_{\theta}}{\partial \theta^j} \mathbf{C}_{\theta}^{-1} \hat{Z}(\mathbf{p})
-\frac{1}{2} \text{tr}\big[(\mathbf{C}_{\theta}+{\mathbf{W}}^{-1})^{-1} \frac{\partial \mathbf{C}_{\theta}}{\partial \theta^j} \big].
\end{eqnarray}
The second term, obtained when we suppose that only $\hat{Z}(\mathbf{p})$ (as well as $\mathbf{W}$) depends on $\theta$, is determined by
\begin{eqnarray}
\frac{\partial l_{c}(\theta)}{\partial \hat{Z}(p_i)}=-\frac{1}{2}\big[(\mathbf{C}_\theta^{-1} + \mathbf{W})^{-1} \big]_{ii}
\frac{\partial^3 \log p(\mathbf{y}|\hat{Z}(\mathbf{p}))}{\partial^3 \hat{Z}(p_i) },
\end{eqnarray}
and
\begin{eqnarray}
\frac{\partial \hat{Z}(\mathbf{p})}{\partial \theta^j}=\big( I_n + \mathbf{C}_{\theta} \mathbf{W} \big)^{-1}
\frac{\partial \mathbf{C}_{\theta}}{\partial \theta^j} \nabla \log p (\mathbf{y}|\hat{Z}(\mathbf{p})).
\end{eqnarray}
Given an unobserved PDF $p^*$, the predictive distribution at $Z(p^*)$ is given by
\begin{eqnarray}
\hat{\mathbb{P}}(Z(p^*)|\mathbf{p},\mathbf{y},p^*)= \mathcal{N}(\mu(p^*),\sigma^2(p^*)),
\end{eqnarray}
with
\begin{equation}
\left\lbrace
\begin{aligned}
\mu(p^*)&={\mathbf{C}_{\theta}^*}^T \mathbf{C}_{\theta}^{-1} \hat{Z}(\mathbf{p}),\\
\sigma^2(p^*)&=C_{\theta}^{**} -
{\mathbf{C}_{\theta}^*}^T(\mathbf{C}_{\theta}+\mathbf{W}^{-1})^{-1} \mathbf{C}_{\theta}^*.\\
\end{aligned}
\right.
\end{equation}
Finally, using the moments of prediction, the predictor
for $y^*=+1$ is
\begin{eqnarray}
\pi(p^*)= \int_{\mathbb{R}}
\sigma(Z^*) \hat{\mathbb{P}}(Z^*|\mathbf{p},\mathbf{y},p^*) d Z^*,
\label{predictor}
\end{eqnarray}
where we note $Z^*=Z(p^*)$ for simplicity.
\subsection{Covariance parameters estimation}
\noindent \textcolor{black}{The marginal likelihoods for both regression and classification depend on the covariance parameters controlling the stationarity of the GP. To show potential applications of this framework,
we explore several optimization methods
in Section~\ref{section:gradient}
and Section~\ref{section:hmc}.}
\subsubsection{log-marginal likelihood gradient}
\label{section:gradient}
\textcolor{black}{In the marginal likelihood estimation, the parameters
are obtained by maximizing the log-marginal likelihood with respect to $\theta$}, i.e., finding
\begin{eqnarray}
\hat{\theta}=\argmax_{\theta} l_{l}(\theta),
\end{eqnarray}
\textcolor{black}{where $l_{l}(\theta)$ is given in~(\ref{likelihood-reg}) by $l_{r}(\theta)$
for regression or
$l_{c}(\theta)$ in(~\ref{log-marginal})
for classification. We summarize the main steps in Algorithm~\ref{algo1}}.
\\[1 cm]
\textcolor{black}{
\begin{algorithm}[H]
\caption{Gradient descent.}
\begin{algorithmic}[1]
\REQUIRE log-marginal likelihood $l_l$ and its gradient $\nabla l_l$
\ENSURE $\hat{\theta}$ \\
\REPEAT
\STATE $\nabla l_{l}(\theta(k))=\{ \frac{\partial l_{l}(\theta(k))}{\partial \theta^j}\}_{j=1}^3$
from~(\ref{grad1})
or~(\ref{grad2})
\STATE Find the step-size $\lambda$
(e.g., by backtracking line search)
\STATE Evaluate $\theta(k+1)=\theta(k)-\lambda \nabla l_{l}(\theta(k))$
\STATE Set $k=k+1$
\UNTIL{$||\nabla l_{l}||_2$ is small enough or a maximum iterations is reached}
\end{algorithmic}
\label{algo1}
\end{algorithm}
}
\subsubsection{HMC sampling}
\label{section:hmc}
\textcolor{black}{Generally, the marginal likelihoods are non-convex functions. Indeed, conventional optimization routines may not find the most probable candidate leading to a lost of
robustness and uncertainty quantification.
To deal with such limitations, we use weak prior distributions for $\delta^2$ and $\alpha$ whereas $\nu$ is simply estimated by cross-validation~\cite{Neal97montecarlo}:
\begin{eqnarray}
\mathbb{P}(\delta^2,\alpha)= \mathbb{P}(\delta^2) \times \mathbb{P}(\alpha),
\end{eqnarray}
with $\delta^2$ and $\alpha$ being independent}. Following~\cite{Gelman06priordistributions},
$\delta^2$ will be assigned a half-Cauchy (positive-only) prior, \emph{i.e. } $\mathbb{P}(\delta^2)=\mathcal{C}(0,b_{\delta^2})$ and $\alpha$ an inverse gamma, \emph{i.e. } $\mathbb{P}(\alpha)=\mathcal{IG}(a_\alpha,b_\alpha)$.
Consequently,
the log-marginal posterior is proportional to
\begin{eqnarray}
l_{p}(\delta^2,\alpha)= l_l(\theta) + \log \mathbb{P}(\delta^2) + \log \mathbb{P}(\alpha).
\end{eqnarray}
When sampling from continuous variables, HMC can prove to be a more powerful tool than usual MCMC sampling. We define the Hamiltonian as the sum of a potential energy
and a kinetic energy:
\begin{eqnarray}
E((\theta^1,\theta^2),(s^1,s^2))&=&E^1(\theta^1,\theta^2)+E^2(s^1,s^2)
-l_{p}(\theta^1,\theta^2) + \frac{1}{2} \sum_{j=1}^{2} {s^j}^2,
\label{Hamiltonian}
\end{eqnarray}
which means that $(s^1,s^2) \sim \mathcal{N}(0,I_2)$.
Instead of sampling from $\exp\big(l_{p}(\theta^1,\theta^2)\big)$ directly, HMC operates by sampling from the distribution $\exp \big(- E((\theta^1,\theta^2), (s^1,s^2))\big)$. The differential equations are given by
\begin{equation}
\frac{d \theta^j}{d t}=\frac{\partial E}{\partial s^j}=s^j \quad \text{and} \quad
\frac{d s^j}{d t}=-\frac{\partial E}{\partial \theta^j} = -\frac{\partial E^1}{\partial \theta^j},
\end{equation}
for $j=1,2$.
In practice, we can not simulate Hamiltonian dynamics exactly because of time discretization. To maintain invariance of the Markov chain, however, care must be taken to preserve the properties of volume conservation and time reversibility. The leap-frog algorithm, summarized in Algorithm~\ref{algo3}, maintains these properties~\cite{1206.1901}.
\begin{algorithm}[H]
\caption{Leap-frog.}
\begin{algorithmic}[1]
\FOR{$k=1,2,\dots$}
\STATE $ s^j (k+\frac{\lambda}{2}) = s^j (k) -\frac{\lambda}{2}
\frac{\partial}{\partial \theta^j} E^1(\theta^1(k),\theta^2(k))$
where $\lambda$ is a finite step-size
\STATE $\theta^j(k+\lambda)=\theta^j(k) + \lambda s^j (k+\frac{\lambda}{2}) $
\STATE $ s^j (k+\lambda)=s^j (k+\frac{\lambda}{2}) -\frac{\lambda}{2}
\frac{\partial}{\partial \theta^j} E^1(\theta^1(k+\lambda),\theta^2(k+\lambda)) $
\ENDFOR
\end{algorithmic}
\label{algo3}
\end{algorithm}
We thus perform a half-step update of the velocity at time $k+\frac{\lambda}{2}$, which is then used to compute $\theta^j(k + \lambda)$ and $s^j(k + \lambda)$. A new state $((\theta^1(N),\theta^2(N)),(s^1(N),s^2(N)))$ is then accepted with the probability
\begin{eqnarray}
\label{acceptance}
\min \Big(1,\frac{\exp\big(-E((\theta^1(N),\theta^2(N),(s^1(N),s^2(N)) \big)}{\exp\big(-E((\theta^1(1),\theta^2(1),(s^1(1),s^2(1)) \big)} \Big).
\end{eqnarray}
We summarize the HMC sampling in Algorithm~\ref{algo4}.
\begin{algorithm}[H]
\caption{HMC sampling.}
\begin{algorithmic}[1]
\REQUIRE log-marginal posteriors $l_p$ and its gradient $\nabla l_p$
\ENSURE $\hat{\theta}$
\STATE Sample a new velocity from a Gaussian distribution $(s^1(1),s^2(1)) \sim \mathcal{N}(0,I_2)$
\STATE Perform $N$ leapfrog steps to obtain the new state $(\theta^1(N),\theta^2(N))$ and velocity $(s^1(N),s^2(N))$ from Algorithm~\ref{algo3}
\STATE Perform accept/reject of $(\theta^1(N),\theta^2(N))$ with acceptance probability defined in~(\ref{acceptance}).
\end{algorithmic}
\label{algo4}
\end{algorithm}
\section{Experimental Results}
\label{sec:results}
\noindent In this section, we test and illustrate the proposed methods using synthetic, semi-synthetic and real data. For all experiments, we study
the empirical results of a Gaussian process indexed by PDFs for both regression and classification.
\noindent \textbf{Baselines.}
We compare results of GP indexed by PDFs (GPP) where the parameters are estimated by gradient descend (G-GPP) and HMC (HMC-GPP) to: Functional Linear Model (\textbf{FLM})~\cite{Ramsay-1991} for regression, Nonparametric Kernel Wasserstein (\textbf{NKW})~\cite{5513626} for regression,
A GPP based on the Wasserstein distance (\textbf{W-GPP})~\cite{NIPS2017_7149,bachoc2017gaussian} for classification, and
a GPP based on the Jensen-Shannon divergence (\textbf{JS-GPP})~\cite{Nguyen-Div-JS} for classification.
\noindent \textbf{Performance metrics.}
For regression, we illustrate the performance of the proposed framework in terms of
root mean square error (RMSE) and negative log-marginal likelihood (NLML). For classification, we consider
accuracy, area under curve (AUC) and NLML.
\subsection{Regression}
\noindent \textbf{Dataset.} We first consider a synthetic dataset where
we observe a finite set of functions simulated according to ~(\ref{reg-model}) as
$Z(p_i)= h(<\sqrt{p_i},\sqrt{\tilde{p}}>_2)=0.5<\sqrt{p_i},\sqrt{\tilde{p}}>_2+0.5$.
In this example, we consider
a truncated Fourier basis (TFB) with random Gaussian coefficients
to form the original functions satisfying
$g_i(t)=\delta_{i,1}\sqrt{2} \sin(2 \pi t) + \delta_{i,2}\sqrt{2} \cos(2 \pi t)$ with $\delta_{i,1},\delta_{i,2} \sim \mathcal{N}(0,1)$.
We also take $\tilde{g}(t)=-0.5\sqrt{2} \sin(2 \pi t) + 0.5 \sqrt{2} \cos(2 \pi t)$.
We suppose that $\tilde{p}$ and $p_i$s refer to the corresponding PDFs of $\tilde{g}$ and $g_i$s
estimated from samples using the nonparametric kernel method (bandwidths were selected using the method given in~\cite{Botev-2010}). Examples of $n=100$ estimates are displayed in Fig.~\ref{fig:DatasetRegression} with colors depending on their output levels.
\begin{figure}
\caption{Examples of PDFs input for regression. The output with continuous value in $[-3,4]$ is illustrated by a colorbar.}
\label{fig:DatasetRegression}
\end{figure}
\noindent \textbf{Regression results.}
Focusing on RMSE, we summarize all results in Table~\ref{tab:RegressionRMSE}.
Accordingly, the
proposed G-GPP gives better precision than FLM.
On the other hand,
HMC-GPP substantially outperforms NKW with a significant margin.
As illustrated in Table~\ref{tab:RegressionAccuracy}, we note that the proposed methods are more efficient than the baseline FLM when maximizing the log-marginal likelihood. Again, this is a very simple explanation on how the quality of GPP strongly depends on parameters estimation method.
In addition, G-GPP stated in Algorithm~\ref{algo1}
is very effective from a computational point of view.
\begin{table}[h!]
\centering
\caption{ Regression: RMSE as a performance metric.}
{\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{G-GPP} & \multicolumn{2}{|c|}{HMC-GPP} & \multicolumn{2}{|c|}{FLM} &
\multicolumn{2}{|c|}{NKW} \\
\hline
mean & std & mean & std & mean & std & mean & std
\\
\hline
$\mathbf{0.07}$ & 0.03 & 0.13 & 0.31 & 0.10 & 0.04 & 0.28 & 0.01 \\
\hline
\end{tabular}}
\label{tab:RegressionRMSE}
\end{table}
\begin{table}[h!]
\centering
\caption{Regression: negative log-marginal likelihood as a performance metric.}
{\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{G-GPP} & \multicolumn{2}{|c|}{HMC-GPP} & \multicolumn{2}{|c|}{FLM} \\
\hline
mean & std & mean & std & mean & std
\\
\hline
73.28 & 1.14 & $\mathbf{21.89}$ & 5.32 & 329.66 & 6.52 \\
\hline
\end{tabular}}
\label{tab:RegressionAccuracy}
\end{table}
\subsection{Classification}
In this section, we perform some extensive experiments to evaluate the proposed methods using a second category of datasets.
\subsubsection{Datasets for classification}
\noindent \textbf{Synthetic datasets}.
We consider a dataset of two synthetic PDFs of beta and inverse gamma distributions. This choice is very crucial for many reasons since beta is defined on $[0, 1]$, parametrized by two positive parameters, and has been widely used to represent a large family of PDFs with finite support in various fields. Increasingly, the inverse gamma plays an important role to characterize random fluctuations affecting wireless channels~\cite{Wireless-Paper}. In both examples, the covariance matrix with $\mathbb{L}^2$ distance and Total Variation TV-distance have a very low rank.
We performed this experiment
by simulating $n=200$ pairs of PDFs slightly different for the two classes. Each observation represents a density when we add a random white noise. We refer to these datasets as Beta and InvGamma, see random examples in Fig.~\ref{fig:DatasetsClassification} (a\&b).
We also illustrate the Fr\'echet mean for each class.
The search of the mean is performed using a gradient approach detailed in~\cite{Srivastava-2007}. \\
\noindent \textbf{Semi-synthetic dataset}. Data represent clinical growth charts for children from $2$ to $12$ years~\cite{Ramsay-1991}. We refer to this dataset as Growth. We simulate the charts from centers for disease control and prevention~\cite{Kuczmarski} through the available quantile values. The main goal is to classify observations by gender.
Each simulation represents the size growth (the increase) of a child according to his age ($120$ months). We represent observations as
nonparametric PDF and we display some examples in Fig.~\ref{fig:DatasetsClassification} (c). For each class: girls (red) and boys (blue) we show the Fr\'echet mean in black.
\begin{figure}
\caption{
Synthetic PDFs for (a) InvGamma and (b) Beta with class 1 (red) and class 2 (blue). Semi-synthetic PDFs for (c) Growth with girls (red) and boys (blue).
Real PDFs for (d) Temp with uninfected (red) and infected (blue).
Real PDFs for (e) Plants with disease (red) and healthy (blue).
The Fr\'echet mean for each class in black.}
\label{fig:DatasetsClassification}
\end{figure}
\noindent \textbf{ Real dataset}. The first public dataset consists of $1500$ images representing maize leaves~\cite{DeChant} with
specific textures whereas the goal is to distinguish healthy and non-healthy plants. We refer to this dataset as Plants.
Motivated by this application, we first represent each image with its wavelet-deconvolved version and form a high-dimensional vector of $262144$ components.
\begin{figure}
\caption{Two examples from maize plants dataset where (top) is a healthy leaf and (bottom) is a leaf with disease. For each class: an original image (left), the extracted features (middle), and the normalized histogram (right).}
\label{fig:ExamplesMaize}
\end{figure}
Fig.~\ref{fig:ExamplesMaize} illustrates an example
of two original images (left): a healthy plant (top) and a plant with disease (bottom), their wavelet-deconvolved versions (middle), and the corresponding histograms (right). We also display PDFs from histograms for each example in Fig.~\ref{fig:ExamplesMaize} (right column in black).
A second real dataset with $1717$ observations gives the body temperature of dogs.
For this dataset, temporal measures of infected and uninfected dogs are stored during $24$ hours. The infection by a parasite is suspected to cause persistent fever despite veterinary medicine~\cite{Kumar}. The main goal is to learn the relationship between the infection and a dominant pattern from temporal temperatures. We display some examples of infected (blue) and uninfected (red) in Fig.~\ref{fig:DatasetsClassification} (c) and we refer to this dataset as Temp. The PDF estimates were obtained using an automatic bandwidth selection method described in~\cite{Botev-2010}. We illustrate some examples of PDFs from real datasets in Fig.~\ref{fig:DatasetsClassification} (d\&e).
\\
We remind that high-dimensional inputs make traditional machine learning techniques fail to solve the problem at hand. However,
the spectral histograms as marginal distributions of the wavelet-deconvolved image can be used to
represent/classify original images~\cite{Liu-2003}. In fact,
instead of comparing the histograms, a better way to compare two images (here a set of repetitive features) would be to compare their corresponding densities.
\subsubsection{Classification results}
We learn the model parameters from $75 \%$ of
the dataset
whereas the rest is kept for test. This subdivision has been performed randomly $100$ times.
The performance is given as a mean and the corresponding standard deviation (std) in order to reduce the bias (class imbalance and sample representativeness) introduced by the random train/test split. \\
\begin{figure}
\caption{
Boxplots of the classification accuracy (left) and AUC (right) on synthetic and semi-synthetic datasets: (a) InvGamma, (b) Beta, and (c) Growth. In each subfigure, the performance is given for different methods:
G-GPP (red), HMC-GPP (light blue), W-GPP (violet), and JS-GPP (dark blue).}
\label{fig:Boxplots-fig1}
\end{figure}
\begin{figure}
\caption{
Boxplots of the classification accuracy (left) and AUC (right) on real datasets: (a) Temp and (b) Plants. In each subfigure, the performance is given for different methods:
G-GPP (red), HMC-GPP (light blue), W-GPP (violet), and JS-GPP (dark blue).}
\label{fig:Boxplots-fig2}
\end{figure}
\noindent \textbf{Results on synthetic datasets.} We summarize all evaluation results on synthetic datasets in Fig.~\ref{fig:Boxplots-fig1} (a\&b).
Accordingly,
one can observe that both HMC-GPP, W-GPP and JS-GPP reach the best accuracy values for InvGamma with a little margin for the proposed HMC-GPP. On the other hand, G-GPP and HMC-GPP heavily outperform W-GPP and JS-GPP for Beta.
Again, this simply shows how each optimization method impacts the quality of the predictive distributions.
\noindent \textbf{Results on semi-synthetic data.}
We summarize all results in Fig.~\ref{fig:Boxplots-fig1} (c) where we show accuracy and AUC values on the Growth dataset as boxplots from $100$ tests.
One can observe that G-GPP
gives the best accuracy with a significant margin. Note that we have used $10^3$ HMC iterations in Algorithm~\ref{algo4}. Furthermore,
we set the ``Burn-in" and ``Thinning" in order to ensure a fast convergence of the Markov chain and to reduce sample autocorrelations.\\
\noindent \textbf{Results on real data.}
We further investigate whether our proposed methods can be used with real data.
Fig.~\ref{fig:Boxplots-fig2} (a\&b) shows the boxplots of accuracy and AUC values for Temp and Plants, respectively. In short,
we highlight that the proposed methods successfully modeled these datasets with improved results in comparison with W-GPP.
Fortunately, the experiments have shown that the problem of big iterations, usually needed to simulate the Markov chains for complex inputs
is partially solved by considering the proposed HMC sampling (Algorithm~\ref{algo4}). In closing, we can state that the
leap-frog algorithm (Algorithm~\ref{algo3}), based on Hamiltonian dynamics, allows us to early search the best directions giving the best minimum of the Hamiltonian defined in~(\ref{Hamiltonian}).
\subsubsection{Summary of all classification results}
\begin{table}[h!]
\centering
\caption{Classification: negative log-marginal likelihood as a performance metric.}
\label{table3}
\begin{adjustbox}{width=\textwidth}
\begin{tabular}{|c||l|c|c|l|c|c|c|c|c|c|}
\hline
\textbf{Datasets} & \multicolumn{4}{c||}{\textbf{Synthetic}} & \multicolumn{2}{c||}{\textbf{Semi-synthetic}} & \multicolumn{4}{c||}{\textbf{Real data}} \\
\hline
&\multicolumn{2}{>{\columncolor{mycolor3}}c|}{\textbf{InvGamma}}
&\multicolumn{2}{>{\columncolor{mycolor3}}c|}{\textbf{Beta}}
&\multicolumn{2}{>{\columncolor{mycolor1}}c|}{\textbf{Growth}}
&\multicolumn{2}{>{\columncolor{mycolor2}}c|}{\textbf{Temp}}
&\multicolumn{2}{>{\columncolor{mycolor2}}c|}{\textbf{Plants}} \\
\cline{2-11}
Method&mean & std& mean & std& mean & std& mean & std& mean & std
\\
\hline
G-GPP & $\boldsymbol{30.50}$ & 2.43 & $\boldsymbol{4.41}$ & 0.06 & 68.03 & 3.43 & $\boldsymbol{98.66}$ & 0.73 & 98.65 & 0.72 \\
\hline
HMC-GPP & 105.35 &0.22 & 105.28 & 0.21& $\boldsymbol{61.65}$ &2.24 & 105.36 & 0.22 & $\boldsymbol{9.33}$ & 0.21 \\
\hline
JS-GPP & 32.2 &2.38 & 42.87 & 2.73& 62.0 &3.02 & 116.65 & 4.13 & 10.26 & 0.12 \\
\hline
\end{tabular}
\end{adjustbox}
\end{table}
We also confirm all previous results from Table~\ref{table3}, which summarizes the mean and the std of NLML values for all datasets.
These clearly show that at least one of the proposed methods (G-GPP or HMC-GPP) better minimizes the NLML than JS-GPP.
This brings more quite accurate estimates, which prove the predictive power of our approaches.
\section{Conclusion}
\label{sec:clc}
\noindent In this paper, we have introduced a novel framework to extend Bayesian learning models and Gaussian processes when the index support is identified with the space of probability density functions (PDFs). We have detailed and applied different numerical methods to learn regression and classification models on PDFs. Furthermore, we showed new theoretical results for the Mat\'ern covariance function defined on the space of PDFs. Extensive experiments on multiple and varied datasets have demonstrated the effectiveness and efficiency of the proposed methods in comparison with current state-of-the-art methods.
\section*{ Acknowledgements }
\noindent This work was partially funded by the French National Centre for Scientific Research.
\end{document} |
\begin{document}
\str{1.05}
\title
{Bayesian modeling of temporal dependence in large sparse contingency tables}
\author{\textsc{Tsuyoshi Kunihama and David B. Dunson} \\
\textit{\small Department of Statistical Science, Duke University, Durham, NC 27708-0251, USA} \\
\texttt{\small [email protected]} \\
\texttt{\small [email protected]}
}
\date{May, 2012}
\maketitle
\begin{abstract}
\noindent
In many applications, it is of interest to study trends over time in relationships among categorical variables, such as age group, ethnicity, religious affiliation, political party and preference for particular policies. At each time point, a sample of individuals provide responses to a set of questions, with different individuals sampled at each time. In such settings, there tends to be abundant missing data and the variables being measured may change over time. At each time point, one obtains a large sparse contingency table, with the number of cells often much larger than the number of individuals being surveyed. To borrow information across time in modeling large sparse contingency tables, we propose a Bayesian autoregressive tensor factorization approach. The proposed model relies on a probabilistic Parafac factorization of the joint pmf characterizing the categorical data distribution at each time point, with autocorrelation included across times. Efficient computational methods are developed relying on MCMC. The methods are evaluated through simulation examples and applied to social survey data.
\\
\noindent
\textit{Key words:} Dynamic model; Multivariate categorical data; Nonparametric Bayes; Panel data; Parafac; Probabilistic tensor factorization; Stick-breaking.
\end{abstract}
\str{1.55}
\section{Introduction}
Time-indexed multivariate categorical data are collected in many areas, with partially-overlapping categorical variables measured for different subjects at the different time points. As a motivating application, we consider social science surveys that are conducted at regular time intervals, containing many categorical questions such as gender, race, age group, ethnicity, religious affiliation, political party and preference for particular policies. For such surveys and other types of time-indexed multivariate categorical data, it is common for the variables measured (questions asked) to vary somewhat over time while a subset of the variables will be measured at all times. In addition, the number of variables measured can be moderate to large leading to a contingency table with an {\em enormous} number of cells, the vast majority of which are empty. Given the fact that social science data often contain complex interactions, it becomes extremely challenging to build realistic and computationally tractable models that allow ultra-sparse data. We define ultra-sparse contingency tables as having exponentially or super-exponentially more cells than the sample size.
Let ${\bf x}_{ti} = (x_{ti1},\ldots,x_{tip})'$ denote the multivariate response for the $i$th subject in the survey at time $t$, with the $j$th categorical question having $d_j$ elements, $x_{tij} \in \{1,\ldots,d_j\}, j=1,\ldots, p$. We accommodate the case in which the specific variables measured can vary across time by introducing missingness indicators, $m_{ti} = (m_{ti1},\ldots, m_{tip})'$, with $m_{tij} = 1$ if variable $j$ is missing for subject $i$ at time $t$; we allow design-based missingness in which certain variables are not measured for any subjects at a particular time and for individual-specific missingness in which certain individuals fail to answer all the questions posed to them. In both cases we assume missing at random.
There is a rich literature on the analysis of contingency tables (\cite{Agresti02}; \cite{FienbergRinaldo07}). Log linear models are perhaps the most commonly used modeling framework. Routine implementations rely on maximum likelihood estimation, though there is also a rich Bayesian literature. For large, sparse contingency tables, maximum likelihood estimates do not exist in many cases except for overly-simplistic log-linear models and richer classes of models become challenging to implement computationally. There is a rich literature on graphical modeling approaches to estimating conditional independence structures in categorical variables, with \cite{DobraLenkoski11} proposing a recent Bayesian approach.
Although their method is computationally efficient, except for very small tables, the number of possible graphical models is so enormous that is becomes infeasible to visit more than a vanishingly small fraction of the models making accurate model selection or averaging difficult. To facilitate scaling to large tables, \cite{DunsonXing09} and \cite{BhattacharyaDunson11} recently proposed Bayesian probabilistic tensor factorizations. These methods express the probability tensor corresponding to the joint probability mass function of the categorical variables as a convex combination of independent components. Such methods have not yet been developed for time-indexed contingency tables.
There is a rich literature on categorical time series and longitudinal data analysis in which the same categorical variable is repeatedly measured for each subject over time. For example, Markov models, state space models and random effects models are routinely applied in such settings. However, these models are not relevant to the problem of incorporating dependence over time in modeling of large sparse contingency tables. As different subjects are measured at different times, we are not faced with the problem of incorporating within-subject dependence in repeated observations; instead our goal is to include dependence in the parameters characterizing the time-dependent joint pmfs for the categorical variables. To our knowledge, this problem has not yet been addressed in the literature. Although one can potentially adapt log-linear or graphical models developed for a contingency tables at one time in a somewhat straightforward manner, the hurdles mentioned above for the static case are compounded in the dynamic setting.
To facilitate routine implementations in ultra sparse cases, we propose to adapt the Dunson and Xing (DX) (2009) probabilistic Parafac factorization to the dynamic setting. The DX model induces a tensor factorization through a Dirichlet process (DP) mixture of product multinomial distributions for the categorical observations. There is an increasingly rich literature proposing nonparametric Bayes dynamic models, which allow time-indexed dependent random probability measures. Perhaps the most common approach relies on a dependent DP (\cite{MacEachern99, MacEachern00}), which incorporates time dependence in the weights and/or atoms in a stick-breaking representation (\cite{GriffinSteel06}; \cite{RodriguezHorst08}; \cite{ChungDunson11}). Most applications of dependent DPs fix the weights and allow the atoms to vary, as varying weights can lead to computational complexities. For dynamic modeling of contingency tables, it is more parsimonious to allow varying weights and varying atoms can lead to a substantial computational burden. An alternative approach, which allows varying weights in a computationally convenient and flexible manner, relies of dynamic mixtures of DPs (\cite{Dunson06}; \cite{RenDunsonLindrothCarin10}). Recently, a class of probit stick-breaking processes was proposed (\cite{ChungDunson09}), which has the appealing feature of allowing one to induce time dependence in random probability measures through Gaussian time series models (\cite{RodriguezDunson11}).
We propose a new nonparametric state space model for time-indexed ultra sparse contingency tables. Relying on a DX-type probabilistic Parafac factorization, we place a dynamic model on the weights, which relies on transformed normal random variables in a similar manner to probit stick-breaking. The model is nonparametric in the sense that the induced prior for each time-indexed joint pmf assigns positive probability in arbitrarily small neighborhoods of any ``true'' data-generating pmf. Hence, our model can allow higher-order interactions and complex dependences, while shrinking towards a low-dimensional structure and borrowing information across time to address the curse of dimensionality. In addition, and crucially for the approach to be useful in the motivating applications, posterior computation can be implemented via a highly efficient Markov chain Monte Carlo (MCMC) algorithm relying on a slice sampler related to \cite{KalliGriffinWalker11}. Finally, the factorization produces a low-dimensional representation of the joint pmf, which is otherwise characterized by a daunting number of parameters in many cases, as the number of cells of the tables can be truly massive.
\str{1.3}
\section{Model specification}
\subsection{Modeling of multivariate categorical data}
We review the nonparametric Bayes approach of \cite{DunsonXing09} for a static large sparse contingency table. Let ${\bf x}_i=(x_{i1},\ldots,x_{ip})'$ be multivariate categorical data for the $i$th subject, with $x_{ij} \in \{1,\ldots, d_j \}$, $j=1,\ldots, p$. Let
\begin{align*}
\bm{\pi} = \left\{ \pi_{c_1\cdots c_p}, \, c_j=1,\ldots,d_j, \, j=1,\ldots,p \right\} \in \Pi_{d_1\cdots d_p}
\end{align*}
be a probability tensor where $\pi_{c_1\cdots c_p}=P(x_{i1}=c_1,\ldots,x_{ip}=c_p)$ is a cell probability and $\Pi_{d_1\cdots d_p}$ is the set of all probability tensors of size $d_1\times \cdots \times d_p$. \cite{DunsonXing09} show that any $\bm{\pi} \in \Pi_{d_1\cdots d_p}$ can be decomposed as
\begin{align}
\bm{\pi} &= \sum_{h=1}^{k} \nu_{h} \Psi_h, \ \ \ \ \Psi_h = \bm{\psi}_h^{(1)} \otimes \cdots \otimes \bm{\psi}_h^{(p)}
\label{eq:mix}
\end{align}
where $\bm{\nu}=(\nu_1,\ldots,\nu_k)'$ is a probability vector, $\Psi_h\in \Pi_{d_1\cdots d_p}$ and $\bm{\psi}_h^{(j)}=(\psi^{(j)}_{h1},\ldots,\psi^{(j)}_{hd_j})'$ is a $d_j\times 1$ probability vector for $h=1,\ldots,k$ and $j=1,\ldots,p$. This expression relies on a Parafac tensor factorization (\cite{Harshman70} and \cite{Kolda01}). It follows that any multivariate categorical data distribution can be expressed as a mixture of product multinomials,
\begin{align*}
P(x_{i1}=c_1,\ldots,x_{ip}=c_p) = \pi_{c_1\cdots c_p} = \sum_{h=1}^{k} \nu_{h} \prod_{j=1}^p \psi_{hc_j}^{(j)}.
\end{align*}
By introducing a latent class index $s_i\in\{1,\ldots,k\}$ for the $i$th subject, the multivariate responses ${\bf x}_i=(x_{i1},\ldots,x_{ip})'$ are conditionally independent given $s_i$. Instead of conditioning on a fixed $k$, \cite{DunsonXing09} developed a nonparametric Bayes approach that lets
\begin{align}
\bm{\pi} &= \sum_{h=1}^{\infty} \nu_{h} \Psi_h, \ \ \ \ \Psi_h = \bm{\psi}_h^{(1)} \otimes \cdots \otimes \bm{\psi}_h^{(p)}, \label{eq:dx1} \\
\bm{\psi}_h^{(j)}&\sim \text{Dirichlet}(a_{j1},\ldots,a_{jd_j}), \ \text{independently for} \ j=1,\ldots, p, \nonumber \\
& \hspace{46mm} h=1,\ldots,\infty, \nonumber \\
\nu_{h} &= V_h \prod_{l<h} (1 - V_l), \nonumber \\
V_h &\sim \text{beta}(1, \alpha), \ \text{independently for} \ h=1,\ldots,\infty, \nonumber
\end{align}
where $a_{jl}>0$ for $l=1,\ldots,d_j$ and $\alpha>0$. Although (2) allows infinitely many components, the number $k_n$ occupied by the $n$ subjects in the sample will tend to be $k_n << n$, so few components will be occupied. The model corresponds to a Dirichlet process mixture of product multinomial distributions relying on a stick-breaking representation (\cite{Sethuraman94}). A prior is induced on the joint pmf which has large support in the sense of assigning positive probability to $L_1$ neighborhoods of any true joint pmf.
\subsection{Modeling of time-indexed multivariate categorical data} \label{sq:extention}
Relying on the DX type probabilistic Parafac factorization, we propose a new nonparametric Bayes approach for time-indexed large sparse contingency tables. In a dynamic setting, we obtain the time-indexed multivariate response ${\bf x}_{ti}=(x_{ti1},\ldots,x_{tip})'$, $x_{tij} \in \{1,\ldots,d_j\}$, for the $i$th subject at time $t$ for $i=1,\ldots, n_t$, $t=1,\ldots, T$ and $j=1,\ldots, p$. At time $t$ we have a probability tensor $\pi_t$ for the multivariate categorical response given by
\begin{align*}
\bm{\pi}_t = \left\{ \pi_{tc_1\cdots c_p}, \, c_j=1,\ldots,d_j, \, j=1,\ldots,p \right\} \in \Pi_{d_1\cdots d_p}
\end{align*}
where $\pi_{tc_1\cdots c_p}=P(x_{ti1}=c_1,\ldots,x_{tip}=c_p)$ is a cell probability at time $t$. Relying on the probabilistic Parafac factorization, each probability tensor $\bm{\pi}_t$ can be expressed as a mixture of product multinomials
\begin{align}
\bm{\pi}_t &= \sum_{h=1}^{k_t} \nu_{th} \Psi_{th}, \ \ \ \ \Psi_{th} = \bm{\psi}_{th}^{(1)} \otimes \cdots \otimes \bm{\psi}_{th}^{(p)} \label{eq:pit}
\end{align}
where $k_t\in \mathbb{N}$, $\bm{\nu}_t=(\nu_{t1},\ldots,\nu_{tk_t})'$ is a probability vector, $\Psi_{th}\in \Pi_{d_1\cdots d_p}$ and $\bm{\psi}_{th}^{(j)}=(\psi^{(j)}_{th1},\ldots,\psi^{(j)}_{thd_j})'$ is a $d_j\times 1$ probability vector for $h=1,\ldots,k_t$. Letting $s_{ti} \in \{1,\ldots,k_t\}$ denote a latent class index for the $i$th subject at time $t$, the observations ${\bf x}_{ti}$ are conditionally independent given $s_{ti}$.
To borrow information across time, we place a dynamic structure on the probability tensor $\bm{\pi}_t$ in (\ref{eq:pit}) assuming time varying weights $\nu_{th}$ and static atoms $\psi_{th}^{(j)} = \psi_h^{(j)}$. Time dependence is induced in the weights through a state space model, which assumes that stick-breaking increments on $\nu_{th}$ arise through transforming Gaussian autoregressive processes using a monotone differentiable link function $g: \Re \to (0,1)$. This characterization is motivated by the probit stick-breaking process (\cite{ChungDunson09}; \cite{RodriguezDunson11}), and leads to a parsimonious but flexible characterization of time-dependence in joint pmfs underlying large, sparse contingence tables.
Similarly to expression (2), we develop a nonparametric Bayes approach that sets the number of components to $k_t = \infty$, though the number of occupied components will tend to be much less than the sample size and can vary across time. The specific model is
\begin{align}
\bm{\pi}_t &= \sum_{h=1}^{\infty} \nu_{th} \Psi_h, \ \ \ \ \Psi_h = \bm{\psi}_h^{(1)} \otimes \cdots \otimes \bm{\psi}_h^{(p)},
\label{eq:new1} \\
\bm{\psi}_h^{(j)}&\sim \text{Dirichlet}(a_{j1},\ldots,a_{jd_j}), \ \text{independently for} \ j=1,\ldots, p, \label{eq:new2} \\
& \hspace{46mm} h=1,\ldots,\infty, \nonumber \\
\nu_{th}&=g(W_{th})\prod_{l<h}\{1-g(W_{tl})\}, \label{eq:new3} \\
W_{th} &= \alpha_{th} + \varepsilon_{th}, \ \ \varepsilon_{th} \sim N(0, \sigma_{\varepsilon}^2), \label{eq:new4} \\
\alpha_{th} &= \mu + \phi \alpha_{t-1h} + \eta_{th}, \ \ \eta_{th} \sim N(0, \sigma^2_{\eta}), \label{eq:new5}
\end{align}
where $|\phi|<1$, $\{\varepsilon_{th}\}$ and $\{\eta_{th}\}$ are sequences of independently normally distributed random variables with mean 0 and variance $\sigma_{\varepsilon}^2$ and $\sigma_{\eta}^2$ respectively. The parameter $\phi$ controls the autocorrelation over time in the weights $\nu_{th}$ on the different components. For sake of parsimony and simplicity in modeling and computation, we include a single time-stationary correlation parameter $\phi$ instead of allowing dependence to be time or element specific. In the limiting case in which $\phi = 0$, the weights $\nu_{th}$ will be modeled as independent. This does not mean that independent priors are placed on the unknown joint pmfs at each time, as the incorporation of common atoms automatically induces some degree of {\em a priori} dependence. However, in applications one typically expects that the joint pmfs will be quite similar over time, and by using varying weights one does not rule out arbitrarily large changes in the pmfs over time. When $\phi$ is close to one, there will be very high time dependence in the weights, leading to effective collapsing on a model that assumes a single time stationary joint pmf. For the initial state variables, we assume the stationary distributions, $\alpha_{1h} \sim N(\mu/(1-\phi), \sigma_{\eta}^2/(1-\phi^2))$ independently for $h=1,\ldots, \infty$. Also, we choose priors $\mu \sim N(\mu_0, \sigma^2_0)$, $\phi \sim U(-1, 1)$, $\sigma^2_{\varepsilon} \sim IG(m_{\varepsilon}/2, S_{\varepsilon}/2)$ and $\sigma^2_{\eta}\sim IG(m_{\eta}/2, S_{\eta}/2)$ respectively.
Expressions (4)-(8) induce a prior on the time-dependent joint pmfs, but it is not immediately obvious how the chosen hyperpriors in the hierarchical specification impact the properties of the prior for $\{ \bm{\pi}_t \}$. In particular, it is important to obtain characterizations of the moments of the induced prior for the cell probabilities, as well as the prior covariance between different elements and across time. Such expressions are provided in Lemma 1, with the proof provided in Appendix A. Lemma 2 shows that the prior is well defined in the sense that $\sum_{h=1}^{\infty} \nu_{th}$ converges to one almost surely.
Lemma 1. The expectation, variance and covariance of the joint prior on the elements of $\{ \bm{\pi}_t \}$ induced through (4)-(8) are
\begin{align*}
&E\{\pi_{tc_1\cdots c_p}\} = \prod_{j=1}^p \frac{a_{jc_j}}{\hat{a}_{j}}, \hspace{5mm}
V\{\pi_{tc_1\cdots c_p}\} = \left( \prod_{j=1}^p \frac{a_{jc_j}(a_{jc_j}+1)}{\hat{a}_j(\hat{a}_j+1)} - \prod_{j=1}^p \frac{a^2_{jc_j}}{\hat{a}^2_{j}} \right)\left( \frac{\beta_2 }{2\beta_1-\beta_2} \right),\\
&Cov\{\pi_{tc_1\cdots c_p}, \pi_{t+k c'_1\cdots c'_p}\} = \left( \prod_{j=1}^p \frac{a_{jc_j} \{ a_{jc'_j} + 1(c_j=c'_j) \}}{\hat{a}_j (\hat{a}_j + 1)} - \prod_{j=1}^p \frac{a_{jc_j} a_{jc'_j} }{\hat{a}^2_j} \right)\left( \frac{\gamma_k}{ 2\beta_1-\gamma_k } \right),
\end{align*}
where $\beta_{1} = E\{g(W_{th})\}$, $\beta_{2} = E\{g^2(W_{th})\}$, $\gamma_k = E \left\{ g(W_{th}) g(W_{t+kh})\right\}$, $\hat{a}_j=\sum_{l=1}^{d_j} a_{jl}$ and $1(\cdot)$ is an indicator function.
The expectation of cell probabilities can be expressed as the product of expectations of Dirichlet priors for atoms. The variance and covariance are expressed as the product of two terms, the first one is related to atoms and the second one comes from time varying weights. As $\mu \rightarrow \infty$, then $\beta_2 / (2\beta_1-\beta_2) \rightarrow 1$ and $\gamma_k / (2\beta_1-\gamma_k) \rightarrow 1$, and the variance and covariance will be influenced only by atoms. In such a case, the measure corresponding to the stick-breaking process will become a point mass at a random atom almost surely. In addition, $\beta_1$, $\beta_2$ and $\gamma_k$ do not depend on time $t$, hence all expectation, variance and covariance are independent of $t$ though the covariance depends on the time difference $k$. Also, the covariance between cell probabilities with $c_j=c'_j$ for all $j$ is always positive and, on the other hand, those with $c_j\neq c'_j$ for all $j$ have negative covariance. In a special case in which the hyperparameters in the Dirichlet prior are $a_{j1}= \cdots = a_{jd_j}=a$ the variance and covariance is zero in the limit as $a \to \infty$. The proof is in Appendix A.
Lemma 2. $\sum_{h=1}^{\infty} \nu_{th} = 1$ almost surely.
Lemma 2 is important in showing that the prior is well defined. The proof is in Appendix B.
Our proposed prior setting is parsimonious but highly flexible in the sense that the induced prior assigns positive probability in arbitrarily small neighborhoods of any true data-generating pmf.
Let $\Pi$ denote the space having elements of the form $\bm{\pi}=\{ \bm{\pi}_t \in \Pi_{d_1\cdots d_p}, \, t\in\{1,\ldots,T\} \}$. We show in Theorem 1 that the proposed prior has large support on $\Pi$.
\noindent
\textit{Theorem 1.}
Let $\mathcal{Q}$ denote the prior on $\Pi$ through the proposed model and $\mathcal{N}_{\epsilon}(\bm{\pi}^0)$ denote an $L_1$ neighborhood around an arbitrary $\bm{\pi}^0 \in \Pi$. Then for any $\bm{\pi}^0\in\Pi$ and $\epsilon > 0$, the prior assigns positive probability in the $\epsilon$-neighborhood, $\mathcal{Q}\left\{ \mathcal{N}_{\epsilon}(\bm{\pi}^0)\right\}>0$.
Since the proposed prior is defined on a space with finitely many components, a straightforward extension of theorem 4.3.1 in \cite{GhoshRamamoorthi03} ensures that the posterior concentrates in arbitrary small neighborhoods of any true data-generating distribution as the sample size increases.
\section{MCMC algorithm for posterior computation}
For posterior computation in DP mixtures, one common approach is marginalizing out the random probability measure with the Polya urn scheme (\cite{BushMacEachern96}). Avoiding marginalization, \cite{IshwaranJames01} developed the blocked Gibbs sampler relying on truncation approximation of the stick-breaking representation. Without truncation, \cite{Walker07} and \cite{PapaspiliopoulosRoberts08} proposed the slice sampler and retrospective MCMC methods respectively. Though the slice sampler is simpler to implement, conditional constraints on sticks can cause slow mixing of the chain. \cite{KalliGriffinWalker11} proposed a more efficient slice sampler avoiding such a mixing problem.
Relying on a slice sampler related to \cite{KalliGriffinWalker11}, we developed a simple and efficient MCMC algorithm for the proposed model. In the motivating application, we have two types of missing data, design-based missingness and
individual-specific missingness. We assume missing at random for both cases and handle the missing data using missingness indicators, $m_{ti} = (m_{ti1},\ldots, m_{tip})'$, with $m_{tij} = 1$ if variable $j$ is missing for subject $i$ at time $t$. In addition, we introduce latent variables $u_{t}=(u_{t1},\ldots,u_{tn_t})'$ for the slice sampler. The likelihood of $\{u_{t}\}$ and $\{{\bf x}_t\}$ given $\{m_{ti}\}$, $\{\bm{\nu}_{t}\}$ and $\left\{\bm{\psi}_{h}^{(j)}\right\}$ can be expressed as
\begin{align*}
\prod_{t=1}^{T} \prod_{i=1}^{n_t} \left\{ \sum_{h=1}^{\infty} 1(u_{ti} < \nu_{th}) \prod_{j:\,m_{tij}=0} \prod_{l=1}^{d_j} \left( \psi_{hl}^{(j)} \right)^{1(x_{tij}=l)} \right\}.
\end{align*}
This representation is consistent with the original model setting if latent variables $\{u_{t}\}$ are marginalized out. In a special case in which $g$ is a probit link function, the data augmentation approach in \cite{AlbertChib01} can improve efficiency of the posterior sampling by introducing independent normal latent variables $\{z_{tih}\}$ with mean $W_{th}$ and variance 1 satisfying
\begin{align*}
P(z_{tih} > 0, \, z_{til} \leq 0, \, l<h) &=\Phi(W_{th}) \prod_{l<h} \{ 1 - \Phi(W_{th}) \} = \nu_{th} = P(s_{ti}=h).
\end{align*}
We propose the following MCMC sampling steps:
\begin{enumerate}
\item For $h=1,\ldots,k^*$, with $k^*=\max\{s_{ti}\}$, update $\bm{\psi}_h^{(j)}$ from the following Dirichlet full conditional posterior distribution,
\begin{align*}
\text{Dirichlet}\left( a_{j1} + \sum_{(t,i) \in A_{jh}} 1(x_{tij}=1),\, \ldots,\, a_{jd_j} + \sum_{(t,i) \in A_{jh}} 1(x_{tij}=d_j) \right).
\end{align*}
where $A_{jh}=\{(t,i): m_{tij}=0,\, s_{ti}=h\}$.
\item Update $z_{tih}$ from the marginal (w.r.t. $u_{ti}$) conditional posterior distribution,
\begin{align*}
z_{tih}\,|\,\cdots \sim \begin{cases}
N_{-}(W_{th}, 1) & h<s_{ti}, \\
N_{+}(W_{th}, 1) & h=s_{ti},
\end{cases}
\end{align*}
where $N_{-}(W_{th}, 1)$ and $N_{+}(W_{th}, 1)$ denote the normal distributions with mean $W_{th}$ and variance 1 truncated on $(-\infty, 0]$ and $(0, \infty)$ respectively.
\item Update $W_{th}$ from the normal marginal (w.r.t. $u_{ti}$) conditional posterior distribution, $N(\hat{W}_{th},\sigma^2_{W_{th}})$ where
\begin{align*}
\hat{W}_{th}=\sigma^2_{W_{th}} \left(\sum_{i:s_{ti} \geq h}^{n_t}z_{tih}+\sigma^{-2}_{\varepsilon}\alpha_{th}\right), \ \
\sigma^2_{W_{th}}= \frac{1}{\sum_{i=1}^{n_t} 1(s_{ti} \geq h) + \sigma^{-2}_{\varepsilon}}.
\end{align*}
\item Update $u_{ti}$ from the full conditional distribution, Uniform$(0, \nu_{ts_{ti}})$.
\item Update $s_{ti}$ from the multinomial full conditional distribution,
\begin{align*}
Pr(s_{ti}=h\,|\, \cdots) = \frac{1(h\in B_{ti}) \prod_{j:m_{tij}=0} \psi_{hx_{tij}}^{(j)} }{ \sum_{l\in B_{ti}} \prod_{j:m_{tij}=0} \psi_{lx_{tij}}^{(j)} },
\end{align*}
where $B_{ti}=\{h:\, \nu_{th} > u_{ti}\}$. To identify the elements in $\{B_{ti}\}$, we first update $\alpha_{th}$ and $W_{th}$ for $t=1,\ldots,T$ and $h=1,\ldots,\tilde{k}$ where $\tilde{k}$ is the smallest number with $\sum_{h=1}^{\tilde{k}}\nu_{th}>1-\min\{s_{ti}\}$ for all $t$.
\item For $h=1,\ldots,k^*$, update $\alpha_{th}$ using the forward filtering backward sampling algorithm by \cite{FruhwirthSchnatter94} and \cite{CarterKohn94}, or Kalman filter and the simulation smoother by \cite{deJongShephard95} and \cite{DurbinKoopman02}.
\item Update $\mu$ from the conditional posterior, $N(\mu_*, \sigma^2_{\mu})$ where $\mu_*=\sigma^2_{\mu}(\hat{\sigma}^{-2}\hat{\mu}+\sigma_{0}^{-2}\mu_{0})$, $\sigma^2_{\mu}=(\hat{\sigma}^{-2}+\sigma_{0}^{-2})^{-1}$ and
\begin{align*}
\hat{\mu} = \frac{\sum_{h=1}^{k^*} \sum_{t=2}^T ( \alpha_{th}- \phi\alpha_{t-1h} ) + (1+\phi) \sum_{h=1}^{k^*} \alpha_{1h} }{k^*\left\{T-1+(1+\phi)/(1-\phi)\right\}}, \ \
\hat{\sigma}^2= \frac{\sigma^2_{\eta} }{k^*\left\{T-1+(1+\phi)/(1-\phi)\right\}}.
\end{align*}
\item Update $\phi$ using the independence MH algorithm in which the proposal distribution is constructed relying on the mode and Hessian of the logarithm of the conditional posterior densities $\pi(\phi|\cdots)$. First, we compute $\hat{\phi}$ which maximizes (or approximately maximizes) the conditional posterior density. Then, we generated a candidate from a truncated normal distribution $TN_{(-1,1)}(\phi_*,\sigma_{\phi}^2)$, where
\begin{eqnarray*}
\phi_*=\hat{\phi}+\sigma_{\phi}^{2}\left.\frac{\partial \log\pi(\phi|\cdots)}{\partial\phi}\right|_{\phi=\hat{\phi}},\hspace{1em}
\sigma_{\phi}^{2}=\left\{ -\left.\frac{\partial \log\pi(\phi|\cdots)}{\partial^2\phi}\right|_{\phi=\hat{\phi}} \right\}^{-1}. \label{eq:phi}
\end{eqnarray*}
\item Update $\sigma^2_{\varepsilon}$ from the conditional distribution, $IG(\hat{m}_{\varepsilon}/2, \hat{S}_{\varepsilon}/2)$ where $\hat{m}_{\varepsilon}=Tk^*+m_{\varepsilon}$ and $\hat{S}_{\varepsilon}=\sum_{t=1}^T\sum_{h=1}^{k^*}( W_{th} - \alpha_{th} )^2 + S_{\varepsilon}$.
\item Update $\sigma^2_{\eta}$ from the conditional distribution, $IG(\hat{m}_{\eta}/2, \hat{S}_{\eta}/2)$ where $\hat{m}_{\eta}=Tk^*+m_{\eta}$ and $\hat{S}_{\eta}=\sum_{h=1}^{k^*} \sum_{t=2}^T( \alpha_{th} - \mu - \phi\alpha_{t-1h} )^2 + (1-\phi^2) \sum_{h=1}^{k^*} \{ \alpha_{1h} - \mu/(1-\phi) \}^2 + S_{\eta}$.
\end{enumerate}
In a case in which $g$ is another link function, we update $W_{th}$ using the independent MH algorithm, instead of step 2 and 3 above. We generate a candidate from a normal distribution relying on the mode and Hessian of the logarithm of the conditional posterior densities of $W_{th}$.
\section{Simulation study}
In this section, we assess the impact of borrowing of information over time by comparing our proposed method to static approaches, such as Dunson and Xing (DX) (2009), applied separately at each time on simulated data. First, we simulate time-indexed contingency tables from the model shown in expressions (\ref{eq:new1})-(\ref{eq:new5}) with $T=10$, $P=20$, $d_j=4$ for all $j$, $\mu=0$, $\phi =0.8$, $\sigma_{\varepsilon}=0.1$ and $\sigma_{\eta}=0.8$. At the respective time points we generated 120, 110, 150, 80, 100, 120, 100, 140, 110 and 150 observations, tiny sample sizes compared with the number of cells. For prior distributions, we assumed $\bm{\psi}_h^{(j)}\sim\text{Dirichlet}(1,\ldots,1)$, $\mu \sim N(0, 1)$, $\phi \sim U(-1, 1)$, $\sigma^2_{\varepsilon} \sim IG(2.5, 0.025)$, $\sigma^2_{\eta}\sim IG(2.5, 0.025)$. We draw 60,000 MCMC samples after the initial 20,000 samples are discarded as a burn-in period and every fifth sample is saved. We observed that the sample paths were stable and the sample autocorrelations dropped smoothly. Therefore, the chains apparently converged and mixed rapidly.
We first assess performance in estimation of cell probabilities. We picked several cells randomly and report true values, posterior means and 95\% credible intervals in Figure \ref{fig:dif1} (the proposed method) and Figure \ref{fig:dif2} (DX method). The proposed approach covers all true values in 95\% intervals and interval widths are much narrower than for the DX approach consistently across time.
We additionally investigate performance in estimating associations among the categorical variables using the following measure of dependence from Dunson and Xing (2009)
\begin{align}
\rho_{tjj'}^2 = \frac{1}{\min\{d_j,d_{j'}\}-1} \sum_{c_j=1}^{d_j} \sum_{c_{j'}=1}^{d_{j'}} \frac{ \left( \pi_{tc_j c_{j'}}-\bar{\psi}_{tc_j}^{(j)} \bar{\psi}_{tc_{j'}}^{(j')} \right)^2 }{\bar{\psi}_{tc_j}^{(j)} \bar{\psi}_{tc_{j'}}^{(j')}},
\label{eq:measure}
\end{align}
where $\bar{\psi}_{tl}^{(j)}\equiv P(x_{tij}=l)\approx \sum_{h=1}^{k^*}\nu_{th} \psi^{(j)}_{hl}$. The first row of Figure \ref{fig:dif3} reports plots of all pairs of true values ($y$-axis) and posterior means ($x$-axis) of $\rho_{tjj'}$ at time $t=2$ and $7$. At each time point, coordinate points by our approach locate closely to the $y=x$ line, compared to widely scattered points by the DX method. In addition, Table \ref{tb:correlation} shows correlations between true values and posterior means of $\rho_{tjj'}$. Although correlations by the DX method are high, the proposed method consistently produces higher correlations.
\begin{table}[H]
\centering
\small
\begin{tabular}{clrrrrrrrrrr}
\hline
& \multicolumn{1}{c}{t=1} & \multicolumn{1}{c}{t=2} &\multicolumn{1}{c}{t=3} &\multicolumn{1}{c}{t=4} &\multicolumn{1}{c}{t=5} &\multicolumn{1}{c}{t=6} &\multicolumn{1}{c}{t=7} &\multicolumn{1}{c}{t=8} &\multicolumn{1}{c}{t=9} &\multicolumn{1}{c}{t=10} &\multicolumn{1}{c}{Total} \\
\hline
Proposed & 0.948 & 0.977 & 0.990 & 0.977 & 0.983 & 0.986 & 0.985 & 0.965 & 0.969 & 0.968 & 0.974 \\
DX & 0.837 & 0.794 & 0.880 & 0.761 & 0.766 & 0.921 & 0.846 & 0.817 & 0.831 & 0.793 & 0.841 \\
\hline
\end{tabular}
\caption{Correlations between true values and posterior means of $\rho_{tjj'}$ using the first simulation data.}
\normalsize
\label{tb:correlation}
\end{table}
Log linear models provide a standard choice for the analysis of contingency tables. However, one issue is that flexible log-linear models that accommodate arbitrary interactions among the variables and allow time dependence cannot be applied directly to large, sparse tables. Certainly, maximum likelihood estimates typically do not exist and Bayesian methods that allow an unknown dependence structure do not scale beyond small tables. \cite{DahindenKalischBuehlmann10} proposed an approach for high-dimensional log-linear models with interactions, which relies on solving several low-dimensional subproblems that are then combined. An earlier approach by \cite{DahindenParmigianiEmerickBuehlmann07} instead relied on L1 penalized log-linear models allowing sparsity of tables. Also, \cite{DahindenParmigianiEmerickBuehlmann07} proposed an efficient estimation algorithm for model selection for two level categorical variables.
As a second alternative to our proposed approach, we implemented the method of Dahinden et al. (DH) (2007) in a second simulation example with $T=8$, $P=13$ and $d_j=2$ for all $j$. Other settings are the same as in the first simulation case. As DH did not consider time-indexed contingency tables, we applied their approach separately at each time point using the logilasso R package, with 5-way cross validation used to choose penalty parameters. The second row of Figure \ref{fig:dif3} and Table 2 summarize the resulting dependence measures $\rho_{tjj'}$ at time $t=2$ and $7$ for each method. For the proposed method, the posterior means are close to true values and correlations between estimates and true values are uniformly high. The DH method has a tendency to underestimate dependence, particularly when true values are low, and has the lowest correlation between the estimates and truth.
\begin{table}[H]
\centering
\small
\begin{tabular}{clrrrrrrrrrr}
\hline
& \multicolumn{1}{c}{t=1} & \multicolumn{1}{c}{t=2} &\multicolumn{1}{c}{t=3} &\multicolumn{1}{c}{t=4} &\multicolumn{1}{c}{t=5} &\multicolumn{1}{c}{t=6} &\multicolumn{1}{c}{t=7} &\multicolumn{1}{c}{t=8} &\multicolumn{1}{c}{Total} \\
\hline
Proposed & 0.951 & 0.978 & 0.979 & 0.984 & 0.986 & 0.969 & 0.981 & 0.944 & 0.965 \\
DX & 0.872 & 0.803 & 0.838 & 0.599 & 0.807 & 0.884 & 0.932 & 0.827 & 0.696 \\
DH & 0.705 & 0.557 & 0.733 & 0.466 & 0.725 & 0.506 & 0.763 & 0.487 & 0.562 \\
\hline
\end{tabular}
\caption{Correlations between true values and posterior means of $\rho_{tjj'}$ using the second simulation data.}
\normalsize
\label{tb:correlation2}
\end{table}
Finally, to gauge robustness we also simulated data from a time-dependent log-linear model in which all the coefficients of the main effects and interactions between two variables independently follow random walk processes with variance 1 and other higher interactions are zero. The third row of Figure \ref{fig:dif3} and Table \ref{tb:correlation3} report the estimation results. Although we find less difference among them in this case, the proposed method still shows the best performance.
\begin{table}[H]
\centering
\small
\begin{tabular}{clrrrrrrrrrr}
\hline
& \multicolumn{1}{c}{t=1} & \multicolumn{1}{c}{t=2} &\multicolumn{1}{c}{t=3} &\multicolumn{1}{c}{t=4} &\multicolumn{1}{c}{t=5} &\multicolumn{1}{c}{t=6} &\multicolumn{1}{c}{t=7} &\multicolumn{1}{c}{t=8} &\multicolumn{1}{c}{Total} \\
\hline
Proposed & 0.725 & 0.827 & 0.768 & 0.798 & 0.818 & 0.916 & 0.791 & 0.807 & 0.817 \\
DX & 0.642 & 0.640 & 0.726 & 0.664 & 0.611 & 0.864 & 0.769 & 0.713 & 0.724 \\
DH & 0.371 & 0.716 & 0.821 & 0.491 & 0.611 & 0.877 & 0.764 & 0.715 & 0.624 \\
\hline
\end{tabular}
\caption{Correlations between true values and posterior means of $\rho_{tjj'}$ using the third simulation data.}
\normalsize
\label{tb:correlation3}
\end{table}
\section{Analysis of social survey data}
In this section, we apply the proposed method to data from the General Social Survey (GSS, http://www3.norc.org/GSS+Website). Our focus is on studying associations among demographic and preference variables over time. We select $p=29$ categorical variables from 1994 to 2010, including gender, ethnicity, preference for particular policies and many more listed in the supplemental materials. The GSS was conducted every two years across this time period. The numbers of observations are 2,992 (1994), 2,904 (1996), 2,832 (1998), 2,817 (2000), 2,765 (2002), 2,812 (2004), 4,510 (2006), 2,023 (2008) and 2,044 (2010) respectively. There are abundant missing data in which only a subset of the variables were recorded for an individual, and compared to the number of cells, the sample size is quite small at each time point.
We first compared our proposed approach to log-linear models. Unfortunately, current methodology for fitting log-linear models that allow flexible dependence structures cannot accommodate these data due to the large sparse structure, time variation and abundant missing data. Hence, in order to provide a comparison, we initially focused on a bivariate subset of the data consisting of religious preference ($i=1,\ldots,5$) and attitude towards abortion ($j=1,2$) from 1994 to 2010. We consider the following log-linear Poisson models.
\begin{align*}
\text{Model 1:} \hspace{10mm} N_{tij} &\sim \text{Poisson}(N_{t}\, \mu_{ij}),
\hspace{5mm} \log \mu_{ij} = \lambda + \lambda^{R}_{i} + \lambda^{A}_{j} + \lambda^{RA}_{ij},
\end{align*}
where $N_{tij}$ is count of the cell $ij$ at time $t$, $N_{t} = \sum_{i}\sum_{j} N_{tij}$, $\lambda^{R}_{i}$ is an effect of the first variable (religious preference), $\lambda^{A}_{j}$ is an effect of the second variable (view of abortion) and $\lambda^{RA}_{ij}$ is an association term. For identifiability, we assume constraints $\lambda^{R}_{5}=\lambda^{A}_{2}=\lambda^{RA}_{5j}=\lambda^{RA}_{i2}=0$. Model 1 assumes no time-dependence in cell probabilities $\mu_{ij}/\sum_{i'}\sum_{j'} \mu_{i'j'}$.
\begin{align*}
\text{Model 2:} \hspace{10mm} N_{tij} &\sim \text{Poisson}(N_{t}\, \mu_{tij}),
\hspace{5mm} \log \mu_{tij} = \lambda_t + \lambda^{R}_{ti} + \lambda^{A}_{tj} + \lambda^{RA}_{tij}, \\
\bm{\beta}_t &= (\lambda_t, \lambda^{R}_{t1},\ldots,\lambda^{R}_{t4},\lambda^{A}_{t1},\lambda^{RA}_{t11},\ldots,\lambda^{RA}_{t41})', \\
\beta_{tl} &= \mu_l + \phi_l \beta_{t-1l} + \varepsilon_{tl}, \ \ \varepsilon_{tl} \sim N(0, \sigma^2_l), \ \ \text{independently for} \ l=1,\ldots,10,
\end{align*}
where $\lambda^{R}_{ti}$, $\lambda^{A}_{tj}$ and $\lambda^{RA}_{tij}$ are effects of the first variable, the second variable and interactions at time $t$ respectively. We assume $\lambda^{R}_{t5}=\lambda^{A}_{t2}=\lambda^{RA}_{t5j}=\lambda^{RA}_{ti2}=0$ at each time point and $\bm{\beta}_0=\bm{0}$ for the initial values. Model 2 is a time dependent hierarchical model where all parameters in the log-linear model follow AR(1) process independently.
We firstly estimate all models using the data from 1994 to 2008. Then, relying on the estimated parameters, we predict the contingency table in 2010 (Table \ref{tb:contingency}). For the proposed model, we used the same MCMC settings as in the simulation study. For log-linear models, we estimated parameters using an MCMC algorithm where missing values are imputed from conditional probabilities given observed data at each iteration. For example, we generate the religious preference $i$ given the view of abortion $j$ with probability $\mu_{tij}/\sum_{i'} \mu_{ti'j}$. For priors, we assumed $\bm{\beta} = (\lambda, \lambda^{R}_{1},\ldots,\lambda^{R}_{4},\lambda^{A}_{1},\lambda^{RA}_{11},\ldots,\lambda^{RA}_{41})'\sim N(\bm{0},I)$ for Model 1, $\mu_l\sim N(0,1)$, $\phi_l\sim U(-1,1)$ and $\sigma_l^2\sim IG(2.5,0.025)$ for all $l$ for Model 2. Using Gibbs sampling, we generated posterior samples of $\mu_l$ and $\sigma^2_j$ from normal and Inverse-Gamma distributions respectively. For $\bm{\beta}$, $\phi_l$, $\bm{\beta}_t$, we used a MH algorithm in which candidates were generated from normal distributions relying on the mode and Hessian of the logarithm of the conditional posterior densities. We generated 10,000 MCMC samples after the 1,000 burn-in for Model 1 and 20,000 MCMC samples after the 2,000 burn-in for Model 2 and, for both cases, every fifth sample was saved.
We generated replications at every fifth MCMC iteration and computed average of the following predictive criteria,
\begin{align*}
\text{Absolute deviation (AD):}&\hspace{5mm} \sum_{i=1}^5 \sum_{j=1}^2 \left| N^{rep}_{ij} - N^{obs}_{ij} \right|, \\
\text{Mean absolute percentage error (MAPE):}&\hspace{5mm} \frac{1}{10} \sum_{i=1}^5 \sum_{j=1}^2 \left| \frac{N^{rep}_{ij} - N^{obs}_{ij}}{N^{obs}_{ij}} \right|,
\end{align*}
where $N^{rep}_{ij}$ and $N^{obs}_{ij}$ are the replication and observation of count of the cell $ij$ respectively. To keep the same total number of replications among all methods, predictions are generated from cell probabilities $\mu_{ij}/\sum_{i'}\sum_{j'} \mu_{i'j'}$ for Model 1 and $\mu_{2010ij}/\sum_{i'}\sum_{j'} \mu_{2010i'j'}$ for Model 2. Table \ref{tb:pred-result} reports the prediction results. Although Model 2 produces better performance than Model 1 by incorporating time-dependence, the proposed method clearly outperforms log-linear models in terms of both predictive criteria.
\begin{table}[H]
\centering
\small
\begin{tabular}{crrrrrr}
\hline
& \multicolumn{1}{c}{Protestant} & \multicolumn{1}{c}{Catholic} &\multicolumn{1}{c}{Jewish} &\multicolumn{1}{c}{None} &\multicolumn{1}{c}{Other} &\multicolumn{1}{c}{Total} \\
\hline
Agree & 216 & 103 & 21 & 137 & 60 & 537 \\
Disagree & 372 & 182 & 7 & 81 & 47 & 689 \\
Total & 588 & 285 & 28 & 218 & 107 & 1226 \\
\hline
\end{tabular}
\caption{Contingency table of the religious preference and view of abortion in 2010.}
\normalsize
\label{tb:contingency}
\end{table}
\begin{table}[H]
\centering
\small
\begin{tabular}{crrrrr}
\hline
& \multicolumn{1}{c}{Proposed} & \multicolumn{1}{c}{Model 1} &\multicolumn{1}{c}{Model 2} \\
\hline
AD & 194.4 & 208.6 & 204.5 \\
MAPE & 0.216 & 0.232 & 0.227 \\
\hline
\end{tabular}
\normalsize
\caption{Prediction results.}
\label{tb:pred-result}
\end{table}
Next, we apply the proposed method to all 29 categorical variables. We generated 30,000 MCMC samples after the initial 10,000 samples are discarded as the burn-in and every fifth sample are saved. We observed the sample paths are stable and the sample autocorrelations are small. Table \ref{tb:real-para} shows the estimation result of parameters in the time dependent stick-breaking processes. Concerning the measure of time dependence $\phi$, the posterior mean is close to 1 and the 95\% credible interval locates near 1, which means the weights of the stick-breaking processes have strong time dependence over time.
\begin{table}[H]
\centering
\small
\begin{tabular}{clrr}
\hline
Parameter & \multicolumn{1}{c}{Mean} & \multicolumn{1}{c}{Stdev.} &\multicolumn{1}{c}{95\% interval} \\
\hline
$\mu$ & -0.012 & 0.004 & [-0.023, -0.005] \hspace{2mm} \\
$\phi$ & 0.988 & 0.004 & [0.978, 0.994] \hspace{2mm} \\
$\sigma_{\varepsilon}$ & 0.062 & 0.009 & [0.046, 0.082] \hspace{2mm} \\
$\sigma_{\eta}$ & 0.126 & 0.011 & [0.104, 0.149] \hspace{2mm} \\
\hline
\end{tabular}
\caption{Estimation result of parameters in the proposed stick-breaking process.}
\normalsize
\label{tb:real-para}
\end{table}
Then, we investigate cross interactions among the variables over time. Figure \ref{fig:table} show the posterior means of $\rho_{tjj'}$ for all pairs in 2002 and 2010. Additional results for other years are included in the supplemental materials. We find the structure of interactions is complex at each time point. Also, though each interaction gradually changes over time, all tables look similar to one another, implying they have close dependence. This is consistent with the result of the strong dependent weights in the stick-breaking processes. Some categorical variables such as Race [$j=3$], Attitude toward abortion [6], Political party affiliation [9] and Think of self as liberal or conservative [14] intricately correlate with many other variables. On the other hand, zodiac [11] shows little interactions with all other variables. Among all pairs of variables, \{Age [1], Marital status [10]\}, \{Attitude toward abortion [6], Attitude toward homosexual [16]\} and \{Attitude toward homosexual [16], Attitude toward Marijuana [19]\} show strong interactions in the whole period. Also, we observed several pairs of variables showing relatively close interactions over time, such as \{Attitude toward abortion [6], Think of self as liberal or conservative [14]\}, \{Race [3], Political party affiliation [9]\} and \{Marital status [10], Having gun [17]\}. In addition, the views of government expense show moderate interactions, especially to the environment [23], nation's health [24], halting the rising crime [25], dealing with drug addiction [26] and education system [27].
Next, we study trends of dependence between categorical variables. Figure \ref{fig:rhoplot1} reports the posterior means and 95\% credible intervals of $\rho_{tjj'}$ for pairs with close interactions. We observed various patterns of time paths. For \{Age, Marital status\}, the interaction increased around 2000 then declined sharply to a lower level. \{Race, Political party affiliation\} and \{Race, Having gun\} have peaks in 2006 and the interactions have steeply decreased after that. In addition, we can see similar trends in \{Attitude toward abortion, Think of self as lib or con\}, \{Attitude toward abortion, Attitude toward homosexual\}, \{Attitude toward homosexual, Attitude toward Marijuana\}, \{Religion, Attitude toward abortion\} and \{Religion, Attitude toward Marijuana\}. The interactions have roughly increased over time, especially in the 2000s. On the other hand, the dependence in \{Race, Death penalty for murder\} decreased at first and kept stable in the middle of the period then declined again. \{Having gun, Family income\} gradually increased over the period but the difference is small. For \{Marital status, Having gun\}, the interaction dropped in the middle of the period but recovered recently at the same level as the beginning.
\section{Discussion}
We have demonstrated that the proposed approach is useful in analyzing time-indexed large sparse contingency tables. One interesting extension is to accommodate joint modeling of mixed scale variables consisting of not only categorical data but also continuous and count variables. In such a case, one can potentially model the observed data vector for the $i$th subject at time $t$, $y_{ti} = (y_{ti1},\ldots,y_{tip})'$, as conditionally independent given latent class variables $x_{ti} = (x_{ti1},\ldots,x_{tip})'$, with $x_{ti}$ modeled exactly as proposed in this article. For example, consider the simple case in which $p=2$ with $y_{ti1} \in \Re$ continuous and $y_{ti2} \in \{1,\ldots,d_2\}$ categorical. Then, one can let $y_{ti1} \sim N( \mu_{x_{ti1}}, \sigma_{x_{ti1}}^2 )$ and $y_{ti2}=x_{ti2}$, with the proposed probabilistic tensor factorization approach flexibly accommodating dependence in $y_{ti1}$ and $y_{ti2}$ through dependence in $x_{ti1}$ and $x_{ti2}$. The induced marginal distribution for the continuous variable $y_{ti1}$ will be a mixture of normals, with the probability weight on each component potentially varying with the categorical variable $y_{ti2}$. This same strategy can be generalized to more complex settings involving many categorical, count, continuous and even functional observations.
Another interesting direction in terms of generalizations is to accommodate dependence in the observations; for example, one may collected multivariate categorical longitudinal data in which the same variables are measured repeatedly on the sample study subjects or the data may have a nested structure. Log linear and logistic regression-type models can be easily generalized to such settings, but clearly encounter computational challenges in large sparse settings. Potentially the simplex factor model of Bhattacharya and Dunson (2012) can be generalized to accommodate such dependence structures through the latent factors, with some challenges arising in terms of developing computationally efficient implementations and models that are both flexible and interpretable.
\section{Proof of Lemma 1} \label{ap:2}
The expectation of cell probability is
\begin{align*}
E\{\pi_{tc_1\cdots c_p}\} &= E\left\{ \sum_{h=1}^{\infty} \nu_{th} \prod_{j=1}^p \psi^{(j)}_{hc_j} \right\} = \sum_{h=1}^{\infty} \left[ E\{\nu_{th}\} \prod_{j=1}^p E\left\{\psi^{(j)}_{hc_j}\right\} \right], \\
&= \prod_{j=1}^p E\left\{\psi^{(j)}_{hc_j}\right\} \sum_{h=1}^{\infty} E\{\nu_{th}\} = \prod_{j=1}^p E\left\{\psi^{(j)}_{hc_j}\right\} = \prod_{j=1}^p \frac{a_{jc_j}}{\hat{a}_j}. \\
\end{align*}
The marginal distribution of $W_{th}$ can be expressed as $N(\mu/(1-\phi), \sigma^2_{\eta}/(1-\phi^2)+\sigma^2_{\varepsilon})$, independent of $t$ and $h$. Hence, we set $\beta_{1} = E\{g(W_{th})\}$ and $\beta_{2} = E\{g^2(W_{th})\}$.
The second moment of cell probability is
\begin{align*}
E\{ \pi_{tc_1\cdots c_p}^2 \} &= E \left[ \left\{ \sum_{h=1}^{\infty} \nu_{th} \prod_{j=1}^p \psi^{(j)}_{hc_j} \right\} \left\{ \sum_{l=1}^{\infty} \nu_{tl} \prod_{j=1}^p \psi^{(j)}_{lc_j} \right\} \right], \\
&= \sum_{h=1}^{\infty} \sum_{l=1}^{\infty} E\{\nu_{th}\nu_{tl}\} E\left\{ \prod_{j=1}^p \psi^{(j)}_{hc_j} \psi^{(j)}_{lc_j} \right\}, \\
&= \left[ \prod_{j=1}^p E \left\{ \left( \psi^{(j)}_{hc_j} \right)^2 \right\} - \prod_{j=1}^p E^2\left\{ \psi^{(j)}_{hc_j} \right\} \right] \sum_{h=1}^{\infty} E\{ \nu_{th}^2 \} + \prod_{j=1}^p E^2\left\{ \psi^{(j)}_{hc_j} \right\} \sum_{h=1}^{\infty} \sum_{l=1}^{\infty} E\{\nu_{th}\nu_{tl}\}, \\
&= \left( \prod_{j=1}^p \frac{a_{jc_j}(a_{jc_j}+1)}{\hat{a}_j(\hat{a}_j+1)} - \prod_{j=1}^p \frac{a^2_{jc_j}}{\hat{a}^2_{j}} \right) \sum_{h=1}^{\infty} E\{ \nu_{th}^2 \} + \prod_{j=1}^p \frac{a^2_{jc_j}}{\hat{a}^2_j},
\end{align*}
where
\begin{align*}
\sum_{h=1}^{\infty} E\{ \nu_{th}^2 \} &= \sum_{h=1}^{\infty} E \left[ g^2(W_{th}) \prod_{l<h} \{1 - g(W_{tl})\}^2 \right], \\
&= \sum_{h=1}^{\infty} \beta_2 \{ 1 -2\beta_1 + \beta_2 \}^{h-1},\\
&= \frac{\beta_2}{2\beta_1-\beta_2}.
\end{align*}
Hence,
\begin{align}
V\{\pi_{tc_1\cdots c_p}\} = \left( \prod_{j=1}^p \frac{a_{jc_j}(a_{jc_j}+1)}{\hat{a}_j(\hat{a}_j+1)} - \prod_{j=1}^p \frac{a^2_{jc_j}}{\hat{a}^2_{j}} \right)\left( \frac{\beta_2 }{2\beta_1-\beta_2} \right).
\label{eq:variance}
\end{align}
Similarly,
\begin{align*}
E&\{\pi_{tc_1\cdots c_p}\pi_{t+kc'_1\cdots c'_p}\} = E \left[ \left\{ \sum_{h=1}^{\infty} \nu_{th} \prod_{j=1}^p \psi^{(j)}_{hc_j} \right\} \left\{ \sum_{l=1}^{\infty} \nu_{t+kl} \prod_{i=1}^p \psi^{(i)}_{lc'_i} \right\} \right], \\
&= \left[ \prod_{j=1}^p E\left\{ \psi^{(j)}_{hc_j} \psi^{(j)}_{hc'_j} \right\} - \prod_{j=1}^p E\left\{ \psi^{(j)}_{hc_j} \right\} E\left\{ \psi^{(j)}_{lc'_j} \right\} \right] \sum_{h=1}^{\infty} E\{ \nu_{th} \nu_{t+kh} \}
+ \prod_{j=1}^p E\left\{ \psi^{(j)}_{hc_j} \right\} E\left\{ \psi^{(j)}_{lc'_j} \right\}, \\
&= \left( \prod_{j=1}^p \frac{a_{jc_j} \{ a_{jc'_j} + 1(c_j=c'_j) \}}{\hat{a}_j (\hat{a}_j + 1)} - \prod_{j=1}^p \frac{a_{jc_j} a_{jc'_j} }{\hat{a}^2_j} \right) \sum_{h=1}^{\infty} E\{ \nu_{th} \nu_{t+kh} \} + \prod_{j=1}^p \frac{a_{jc_j} a_{jc'_j} }{\hat{a}^2_j},
\end{align*}
where
\begin{align*}
E\{ \nu_{th} \nu_{t+kh} \}&= E \left\{ \left[ g(W_{th}) \prod_{l<h} \{1 - g(W_{tl})\} \right] \left[ g(W_{t+kh}) \prod_{l<h} \{1 - g(W_{t+kl})\} \right] \right\}, \\
&= E \left\{ g(W_{th}) g(W_{t+kh}) \right\} \prod_{l<h} E \left[ \{1 - g(W_{tl})\}\{1 - g(W_{t+kl})\} \right], \\
&= E \left\{ g(W_{th}) g(W_{t+kh}) \right\} \prod_{l<h} \left[ 1 - 2\beta_1 + E\{ g(W_{tl}) g(W_{t+kl}) \} \right].
\end{align*}
From (\ref{eq:new4}) and (\ref{eq:new5}), $E \left\{ g(W_{th}) g(W_{t+kh}) \right\}$ can be expressed as
\begin{align*}
E \left\{ g(W_{th}) g(W_{t+kh}) \right\} &= E \left\{ g(\alpha_{th}+\varepsilon_{th}) g(\alpha_{t+kh}+\varepsilon_{t+kh}) \right\}, \\
&= E \left\{ g\left(\alpha_{th} + \varepsilon_{th}\right) g\left(\frac{1-\phi^k}{1-\phi}\mu + \phi^k \alpha_{th} + \sum_{i=0}^{k-1} \phi^i w_{t+k-ih} + \varepsilon_{t+kh}\right) \right\}.
\end{align*}
Since $\alpha_{th}$, $\varepsilon_{th}$, $w_{t+k-ih}$ ($i=0,\ldots,k-1$) and $\varepsilon_{t+kh}$ are independent of one another and their distributions do not depend on $t$ or $h$, hence $\gamma_k\equiv E \left\{ g(W_{th}) g(W_{t+kh}) \right\}$ is dependent on time difference $k$ but independent of time $t$.
In addition,
\begin{align*}
\sum_{h=1}^{\infty} E\{ \nu_{th} \nu_{t+kh} \} &= \sum_{h=1}^{\infty} \gamma_k \prod_{l<h} \left\{ 1 - 2\beta_1 + \gamma_k \right\}, \\
&= \frac{\gamma_k}{2\beta_1 - \gamma_k}.
\end{align*}
Hence,
\begin{align*}
Cov\{\pi_{tc_1\cdots c_p}, \pi_{t+k c'_1\cdots c'_p}\} = \left( \prod_{j=1}^p \frac{a_{jc_j} \{ a_{jc'_j} + 1(c_j=c'_j) \}}{\hat{a}_j (\hat{a}_j + 1)} - \prod_{j=1}^p \frac{a_{jc_j} a_{jc'_j} }{\hat{a}^2_j} \right)\left( \frac{\gamma_k}{ 2\beta_1-\gamma_k } \right).
\end{align*}
Since $\beta_2 /(2\beta_1-\beta_2)>0$, $\gamma_k/(2\beta_1-\gamma_k)>0$ and (\ref{eq:variance}), cell probabilities with $c_j= c'_j$ for all $j$ have positive covariance and, on the other hand, those with $c_j\neq c'_j$ for all $j$ have negative covariance.
In a case where $a_{j1}=\cdots=a_{jc_j}=a$, the variance and covariance are expressed as
\begin{align*}
V\{\pi_{tc_1\cdots c_p}\} &= \left( \prod_{j=1}^p \frac{1+1/a}{d^2_j+d_j/a} - \prod_{j=1}^p \frac{1}{d^2_j} \right)\left( \frac{\beta_2 }{2\beta_1-\beta_2} \right), \\
Cov\{\pi_{tc_1\cdots c_p}, \pi_{t+k c'_1\cdots c'_p}\} &= \left( \prod_{j=1}^p \frac{1+1(c_j=c'_j)/a}{d_j^2 + d_j/a } - \prod_{j=1}^p \frac{1}{d_j^2} \right) \left( \frac{\gamma_k}{ 2\beta_1-\gamma_k } \right).
\end{align*}
Hence, $V\{\pi_{tc_1\cdots c_p}\}\rightarrow 0$ and $Cov\{\pi_{tc_1\cdots c_p}, \pi_{t+k c'_1\cdots c'_p}\}\rightarrow 0$ as $a\rightarrow \infty$.
\section{Proof of Lemma 2} \label{ap:1}
To prove $\sum_{h=1}^{\infty}\nu_{th}=1$ a.s., it is enough to show $\sum_{h=1}^{\infty}E\{\log(1-g(W_{th})\}=-\infty$ (\cite{IshwaranJames01}). $g$ is a non-negative monotone increasing link function: $\Re \to (0,1)$, therefore $0<\beta_1 =E\{g(W_{th})\}<1$. Then, using Jensen's inequality,
\begin{align*}
E[\log\{1-g(W_{th})\}] \leq \log[1 - E\{g(W_{th})\}] = \log(1 - \beta_1) < 0.
\end{align*}
Therefore, $\sum_{h=1}^{\infty}E\{\log(1-g(W_{th})\}=-\infty$ at each time point.
\section{Proof of theorem} \label{ap:3}
The proposed prior probability assigned to $\mathcal{N}_{\epsilon}(\bm{\pi}^0)$ can be expressed as
\begin{align*}
\mathcal{Q} \left\{ \mathcal{N}_{\epsilon}(\bm{\pi}^0) \right\} = \int 1( \| \bm{\pi} - \bm{\pi}^0 \| < \epsilon ) d \mathcal{Q}(\bm{\nu}_t, \bm{\psi}_{h}^{(j)}, t\in\{1,\ldots,T\}, h=1,\ldots,\infty, j=1,\ldots,p).
\end{align*}
where $\bm{\nu}_t$ is a probability vector induced by the proposed stick breaking process and we use the $L_1$ distance
\begin{align*}
\| \bm{\pi} - \bm{\pi}^0 \| = \sum_{t=1}^T p_t \sum_{c_1=1}^{d_1} \cdots \sum_{c_p=1}^{d_p} | \pi_{tc_1\cdots c_p} - \pi_{tc_1\cdots c_p}^0 |,
\end{align*}
where $p_t$ is a probability mass function for time $t\in\{1,\ldots,T\}$.
For any $\bm{\pi}^0 \in \Pi$, each component in $\bm{\pi}^0$ can be expressed as
\begin{align*}
\bm{\pi}^0_t &= \sum_{h=1}^{k_t} \nu^0_{th} \Psi_{th}, \ \ \ \ \Psi_{th} = \bm{\psi}_{th}^{(1)} \otimes \cdots \otimes \bm{\psi}_{th}^{(p)},
\end{align*}
where $k_t\in \mathbb{N}$, $\bm{\nu}^0_t = (\nu^0_{t1},\ldots,\nu^0_{tk_t})'$ is a probability vector, $\Psi_{th}\in \Pi_{d_1\cdots d_p}$ and $\bm{\psi}_{th}^{(j)}=(\psi^{(j)}_{th1},\ldots,\psi^{(j)}_{thd_j})'$ is a $d_j\times 1$ probability vector. We define $k^+_0=0$ and $k^{+}_t=\sum_{i=1}^{t}k_i$ for $t=1,\ldots,T$. Then, we construct $\bm{\pi}=\{\bm{\pi}_t,t\in\{1,\ldots,T\}\} \in \Pi$ induced by the proposed prior such that the component with the index $h$ in $\bm{\pi}^0_t$ is approximated by the component with the index $k^+_{t-1}+h$ in $\bm{\pi}_t$. Let $\tilde{\bm{\nu}}_t=(\tilde{\nu}_{t1},\tilde{\nu}_{t2},\ldots)'$ be a probability vector where $\tilde{\nu}_{tm}=\nu^0_{tm-k^+_{t-1}}$ for $k^+_{t-1} < m\leq k^+_{t}$ and $\tilde{\nu}_{tm}=0$ otherwise, i.e., $\tilde{\nu}_{tk^+_{t-1}+h}=\nu^0_{th}$ for $1 \leq h\leq k_{t}$. For any $\epsilon$, we define a set $D(\bm{\pi}^0,\epsilon)\subset \Pi$
such that for any $\bm{\pi} \in D(\bm{\pi}^0,\epsilon)$, each $\bm{\pi}_t$ can be expressed as (\ref{eq:new1}) satisfying $\bm{\nu}\in\mathcal{N}_{\epsilon'}(\tilde{\bm{\nu}})$, where $\bm{\nu}= \{\bm{\nu}_t,t\in\{1,\ldots,T\}\}$, $\tilde{\bm{\nu}}= \{\tilde{\bm{\nu}}_t,t\in\{1,\ldots,T\}\}$ and $\epsilon'=\epsilon/2\prod_{j=1}^{p}d_j$, and $\bm{\psi}_{k^+_{t-1}+h}^{(j)}\in \mathcal{N}_{\epsilon''}\left(\bm{\psi}_{th}^{(j)}\right)$ for $h=1,\ldots,k_t$ and $t=1,\ldots,T$ where $\epsilon''=\epsilon/2\sum_t p_t k_t p \prod_{j}d_j$.
We consider the intervals $(a_{th}, b_{th})$ in the real line for $W_{th}$ in the proposed prior for $h=1,\ldots,k^+_{t}$ and $t=1,\ldots,T$ where
\begin{equation*}
a_{th} = \begin{cases}
g^{-1}\{\max(\tilde{\nu}_{th}-\tilde{\epsilon}, 0)\}, & (h=1),\\
g^{-1}\left\{ \frac{\max(\tilde{\nu}_{th}-\tilde{\epsilon}, 0)}{\prod_{l<h} \{1-g(W_{tl})\} } \right\}, & (h=2,\ldots, k^+_{t}),
\end{cases}
b_{th} = \begin{cases}
g^{-1}\{\tilde{\nu}_{th} + \tilde{\epsilon}\}, & (h=1),\\
g^{-1}\left\{ \frac{\tilde{\nu}_{th}+\tilde{\epsilon}}{\prod_{l<h} \{1-g(W_{tl})\} } \right\}, & (h=2,\ldots, k^+_{t}),
\end{cases}
\end{equation*}
where $\tilde{\epsilon}=\epsilon'/2\sum_t p_t k_t^+$. In this case, it is straightforward to check $| \nu_{th} - \tilde{\nu}_{th} | < \tilde{\epsilon}$ for $h=1,\ldots,k^+_{t}$ and the proposed prior assigns positive probability to these intervals. Then, the distance between $\bm{\nu}$ and $\tilde{\bm{\nu}}$ is
\begin{align}
\| \bm{\nu} - \tilde{\bm{\nu}} \| &= \sum_{t=1}^T p_t \sum_{h=1}^{\infty} | \nu_{th} - \tilde{\nu}_{th} |, \nonumber \\
&= \sum_{t=1}^T p_t \sum_{h=1}^{k_t^+} | \nu_{th} - \tilde{\nu}_{th} | + \sum_{t=1}^T p_t \sum_{h>k_t^+} \nu_{th}, \label{eq:nu} \\
&< 2 \tilde{\epsilon} \sum_{t=1}^T p_t k_t^+ = \epsilon'. \nonumber
\end{align}
For the second component in (\ref{eq:nu}), $\sum_{h>k_t^+} \nu_{th} < k_t^+ \tilde{\epsilon}$ because $\nu_{th}>\tilde{\nu}_{th}-\tilde{\epsilon}$ for $h=1,\ldots,k_t^+$ and $\sum_{h=1}^{k_t^+} \nu_{th} > 1 -k_t^+ \tilde{\epsilon}$. In addition, it is straightforward to show that the proposed prior assigns positive probability to $\mathcal{N}_{\epsilon''}\left(\bm{\psi}_{th}^{(j)}\right)$. Therefore, since $D(\bm{\pi}^0,\epsilon)$ contains such case, $\mathcal{Q}\{D(\bm{\pi}^0,\epsilon)\}>0$.
For any $\bm{\pi}\in D(\bm{\pi}^0,\epsilon)$,
\begin{align*}
\| \bm{\pi} - \bm{\pi}^0 \| &= \sum_{t=1}^T p_t \sum_{c_1=1}^{d_1} \cdots \sum_{c_p=1}^{d_p} | \pi_{tc_1\cdots c_p} - \pi_{tc_1\cdots c_p}^0 |, \\
&= \sum_{t=1}^T p_t \sum_{c_1=1}^{d_1} \cdots \sum_{c_p=1}^{d_p} \left| \sum_{h=1}^{\infty} \nu_{th}\prod_{j=1}^p \psi_{hc_j}^{(j)} - \sum_{l=1}^{k_t} \nu^0_{tl}\prod_{j=1}^p \psi_{tlc_j}^{(j)} \right|, \\
&= \sum_{t=1}^T p_t \sum_{c_1=1}^{d_1} \cdots \sum_{c_p=1}^{d_p} \left| \sum_{h=1}^{k_t} \left( \nu_{tk^+_{t-1}+h}\prod_{j=1}^p \psi_{k^+_{t-1}+hc_j}^{(j)} - \nu^0_{th}\prod_{j=1}^p \psi_{thc_j}^{(j)} \right) + \sum_{l\leq k^+_{t-1}, k^+_{t}<l} \nu_{tl}\prod_{j=1}^p \psi_{lc_j}^{(j)} \right|, \\
&\leq \sum_{t=1}^T p_t \sum_{c_1=1}^{d_1} \cdots \sum_{c_p=1}^{d_p} \left( \sum_{h=1}^{k_t} \left| \nu_{tk^+_{t-1}+h}\prod_{j=1}^p \psi_{k^+_{t-1}+hc_j}^{(j)} - \nu^0_{th}\prod_{j=1}^p \psi_{thc_j}^{(j)} \right| + \sum_{l\leq k^+_{t-1}, k^+_{t}<l} \nu_{tl} \right), \\
&\leq \sum_{t=1}^T p_t \sum_{c_1=1}^{d_1} \cdots \sum_{c_p=1}^{d_p} \left( \sum_{h=1}^{k_t} \left| \nu_{tk^+_{t-1}+h} - \nu^0_{th} \right| + \sum_{l=1}^{k_t} \sum_{j=1}^p \left| \psi_{k^+_{t-1}+lc_j}^{(j)} - \psi_{tlc_j}^{(j)} \right| + \sum_{l\leq k^+_{t-1}, k^+_{t}<l} \nu_{tl} \right), \\
&= \sum_{c_1=1}^{d_1} \cdots \sum_{c_p=1}^{d_p} \sum_{t=1}^T p_t \sum_{h=1}^{\infty} | \nu_{th} - \tilde{\nu}_{th} | + \sum_{t=1}^T p_t \sum_{l=1}^{k_t} \sum_{j=1}^p \sum_{c_1=1}^{d_1} \cdots \sum_{c_p=1}^{d_p} \left| \psi_{k^+_{t-1}+lc_j}^{(j)} - \psi_{tlc_j}^{(j)} \right|, \\
&< \prod_{j=1}^{p}d_j \epsilon' + \sum_{t=1}^{T} p_t k_t p \prod_{j=1}^p d_j \epsilon'', \\
&= \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.
\end{align*}
Therefore $\bm{\pi}\in \mathcal{N}_{\epsilon}(\bm{\pi}^0)$ and $D(\bm{\pi}^0,\epsilon) \subset \mathcal{N}_{\epsilon}(\bm{\pi}^0)$. Hence, $\mathcal{Q}\{ \mathcal{N}_{\epsilon}(\bm{\pi}^0) \} > 0$.
\section{Figures}
\begin{figure}
\caption{Estimation results of cell probabilities by the proposed method.}
\label{fig:dif1}
\end{figure}
\begin{figure}
\caption{Estimation results of cell probabilities by DX method.}
\label{fig:dif2}
\end{figure}
\begin{figure}
\caption{Plots of true and estimated values of $\rho_{tjj'}
\label{fig:dif3}
\end{figure}
\begin{figure}
\caption{Posterior means of $\rho_{tjj'}
\label{fig:table}
\end{figure}
\begin{figure}
\caption{Estimation results of $\rho_{tjj'}
\label{fig:rhoplot1}
\end{figure}
\section{Supplemental materials}
\begin{table}[H]
\centering
\small
\begin{tabular}{clr}
\hline
\multicolumn{1}{c}{No.} & \multicolumn{1}{l}{Categorical variable (Name in GSS)} & \multicolumn{1}{r}{\scalebox{0.7}[1]{\# of categories}} \\
\hline
1 & Age group* (AGE) & 8 \\
2 & Sex (SEX) & 2\\
3 & Race (RACE) & 3 \\
4 & Religious preference** (RELIG) & 5\\
5 & Region (REGION) & 9 \\
6 & Attitude toward abortion (ABANY) & 2 \\
7 & Should Govetnment help pay for medical care? (HELPSICK) & 5 \\
8 & Highest degree (DEGREE) & 5 \\
9 & Political party affiliation (PARTYID) & 8 \\
10 & Current marital status (MARITAL) & 5 \\
11 & Astrological sign (ZODIAC) & 12 \\
12 & Confidence in banks and financial institutions (CONFINAN) & 3 \\
13 & Confidence in U.S. Supreme Court (CONJUDGE) & 3 \\
14 & Think of self as liberal or conservative (POLVIEWS) & 7 \\
15 & Belief in life after death (POSTLIFE) & 2 \\
16 & Attitude toward homosexual sex relations (HOMOSEX) & 5 \\
17 & Have gun in home (OWNGUN) & 2 \\
18 & Subjective class identification (CLASS) & 4 \\
19 & Should Marijuana be made legal (GRASS) & 2 \\
20 & Total family income (INCOME) & 12 \\
21 & Favor or oppose death penalty for murder (CAPPUN) & 2 \\
22 & \scalebox{0.83}[1]{Attitude toward spending money on space exploration program (NATSPAC)} & 3 \\
23 & \scalebox{0.83}[1]{Attitude toward spending money on improving and protecting environment (NATENVIR)} & 3 \\
24 & \scalebox{0.83}[1]{Attitude toward spending money on improving and protecting the nations's health (NATHEAL)} & 3 \\
25 & \scalebox{0.83}[1]{Attitude toward spending money on halting the rising crime rate (NATCRIME)} & 3 \\
26 & \scalebox{0.83}[1]{Attitude toward spending money on dealing with drug addiction (NATDRUG)} & 3 \\
27 & \scalebox{0.83}[1]{Attitude toward spending money on improving the nation's education system (NATEDUC)} & 3 \\
28 & \scalebox{0.83}[1]{Attitude toward spending money on the military, armaments and defense (NATARMS)} & 3 \\
29 & \scalebox{0.83}[1]{Attitude toward spending money on foreigh aid (NATAID)} & 3 \\
\hline
\end{tabular}
\begin{minipage}{15cm}
{\footnotesize *The category of Age group is different from the original one: 1. 18 or 19 years old, 2. 20s, 3. 30s, 4. 40s, 5. 50s, 6. 60s, 7. 70s, 8. more than 80 years old. \\
**The category of Religious preference is different from the original one: 1. Protestant, 2. Catholic, 3. Jewish, 4. None, 5. Others.}
\end{minipage}
\normalsize
\caption{List of categorical variables.}
\end{table}
\begin{figure}
\caption{Posterior means of $\rho_{tjj'}
\end{figure}
\begin{figure}
\caption{Posterior means of $\rho_{tjj'}
\end{figure}
\begin{figure}
\caption{Posterior means of $\rho_{tjj'}
\end{figure}
\begin{figure}
\caption{Posterior means of $\rho_{tjj'}
\end{figure}
\end{document} |
\begin{document}
\begin{center}
{\bf Some 2-adic conjectures concerning \\
polyomino tilings of Aztec diamonds} \\
\ \\
James Propp, UMass Lowell \\
August 10, 2022 \\
\ \\
{\it Dedicated to Michael Larsen on the occasion of his 60th birthday}
\end{center}
\begin{abstract}
\noindent
For various sets of tiles, we count the ways to tile
an Aztec diamond of order $n$ using tiles from that set.
The resulting function $f(n)$ often has interesting behavior
when one looks at $n$ and $f(n)$ modulo powers of 2.
\end{abstract}
\begin{section}{Introduction}
\label{sec:intro}
I had a great time working on domino tilings of Aztec diamonds
with Noam Elkies, Greg Kuperberg, and Michael Larsen back in the late 1980s,
and the paper we wrote together~\cite{EKLP}
had a huge impact on my career.
So I’d like to honor Michael by proposing
some new problems about tilings of Aztec diamonds
(and other regions), many of which are more challenging
than the one I shared with him thirty-something years ago
and have a number-theoretic slant that I think he will enjoy.
Ideally the solutions to these problems will involve
interesting applications of algebra to combinatorics.
\begin{figure}
\caption{A domino tiling of the Aztec diamond of order 4.}
\label{fig:dominos}
\end{figure}
Here is some general background.
The main result of~\cite{EKLP} was that the number of domino-tilings
of an Aztec diamond of order $n$ is $2^{n(n+1)/2}$
(\href{https://oeis.org/A006125}{A006125}),
where a domino is a rectangle in $\mathds{R}^2$ of the form
$[i,i+1] \times [j,j+2]$ or $[i,i+2] \times [j,j+1]$ (with $i,j \in \mathds{Z}$)
and the Aztec diamond of order $n$
is the union of the squares $[i,i+1] \times [j,j+1]$
contained within the region $\{(x,y): \ |x|+|y| \leq n+1\}$.
Figure~\ref{fig:dominos} shows one of the $2^{(4)(5)/2}$ domino tilings
of the Aztec diamond of order 4.
Mihai Ciucu~\cite{Ci} proved combinatorially
that the number of domino tilings of the $2n$-by-$2n$ square
(\href{https://oeis.org/A004003}{A004003})
can be written in the form $2^n f(n)^2$
where $f(n)$ is the number of domino tilings of the region
exemplified for $n=4$ in Figure~\ref{fig:ciucu}
(\href{https://oeis.org/A065072}{A065072}).
\begin{figure}
\caption{Ciucu's way of halving the 8-by-8 square.}
\label{fig:ciucu}
\end{figure}
Henry Cohn~\cite{Co} proved that the function sending $n$ to $f(n)$
is uniformly continuous under the 2-adic metric
and thus extends to a function defined on all of $\mathds{Z}$
and indeed all of $\mathds{Z}_2$;
moreover, he showed that this extension satisfies
\begin{equation}
f(-1-n) = \left\{\begin{array}{rl}
f(n) & \mbox{when $n$ is congruent to 0 or 3 (mod 4)}, \\
-f(n) & \mbox{when $n$ is congruent to 1 or 2 (mod 4)}.
\end{array} \right.
\end{equation}
Barkley and Liu~\cite{BL} have recently proved results
about 2-divisibility for the number of perfect matchings of a graph,
including as a special case the number of domino tilings of a rectangle,
but there is more refined work still to be done along the lines of Cohn's paper.
For instance, the mod 8 residue of
the number of domino tilings of the $2n$-by-$(2n+2)$ rectangle
appears to depend only on the mod 4 residue of $n$;
the same goes for the number of domino tilings of the $2n$-by-$4n$ rectangle.
In this article we extend the discussion
to other sorts of tiles, specifically, tetrominos.
A {\em tetromino} is a connected subset of the grid
that is a union of four grid-squares,
just as a domino is a union of two grid-squares.
Up to symmetry, there are five kinds of tetrominos: straight tetrominos,
skew tetrominos, L-tetrominos, square tetrominos, and T-tetrominos.
They are shown in Figure~\ref{fig:the-six}, preceded by the domino.
These six tiles can be placed on a square grid in
2, 2, 4, 8, 1, and 4 translationally-inequivalent ways, respectively
(where rotations and reflections are permitted).
These are the sorts of tiles considered in this article.
({\em Trominos} -- unions of three grid-squares --
will be considered elsewhere.)
\begin{figure}
\caption{A domino, a straight tetromino, a skew tetromino,
an L-tetromino, a square tetromino, and a T-tetromino.}
\label{fig:the-six}
\end{figure}
\end{section}
\begin{section}{Skew and straight tetrominos}
\label{sec:skewstraight}
\begin{figure}
\caption{Tiling the Aztec diamond of order 3 with skew and straight tetrominos.}
\label{fig:olympiad}
\end{figure}
I'll start with a warm-up puzzle that's
roughly at the level of a math olympiad:
Prove that an Aztec diamond of order $n$ can be tiled
by skew and straight tetrominos
(as shown in Figure~\ref{fig:olympiad} for $n=3$)
only if $n$ is congruent to 0 or 3 (mod 4).
The puzzle can be solved using a valuation argument
(sometimes called a generalized coloring argument):
one can construct a mapping from the grid-cells to an appropriate abelian group
(a ``weight function'') and show that when $n$ is 1 or 2 (mod 4),
the sum of the weights of the tiles
can't equal the sum of the weights of the region being tiled,
where the weight of a tile or a region being tiled
is the sum of the weights of the constituent cells.
Readers who are already familiar with this technique
might enjoy the challenge of attempting to solve the problem purely mentally.
\end{section}
\begin{section}{Dominos and square tetrominos}
\label{sec:domsquare}
In this section we use dominos and square tetrominos.
Thus an Aztec diamond of order 1
(better known as the 2-by-2 square) can be tiled in 3 ways:
with two horizontal dominos, two vertical dominos, or a single square tetromino.
The Aztec diamond of order 2 can be tiled in $2^{(2)(3)/2} = 8$ ways
using dominos, and can be tiled in an additional 11 ways
if one or more square tetrominos are included,
as shown in Figure~\ref{fig:eleven}.
Thus there are a total of $8+11=19$ ways
to tile an Aztec diamond of order 2
using dominos and square tetrominos.
\begin{figure}
\caption{Tiling the Aztec diamond of order 2
with dominos and at least one square tetromino.}
\label{fig:eleven}
\end{figure}
Define $M(n)$ (with $n \geq 0$) as
the number of tilings of the Aztec diamond of order $n$
using dominos and square tetrominos.
This is \href{https://oeis.org/A356512}{A356512}.
Trivially we have $M(0) = 1$ (since the Aztec diamond of order 0 is empty)
and we have already seen that $M(1) = 3$ and $M(2) = 19$.
Figure~\ref{fig:Mtable} shows the terms of the sequence $M(n)$
for $n$ ranging from 0 to 12,
computed using a program written by David desJardins.
The reader may wish to pause here to consider the problem of showing that $M(n)$ is always odd;
a solution will be given in section~\ref{sec:cong}.
\begin{figure}
\caption{Enumeration of tilings of Aztec diamonds
using dominos and square tetrominos.}
\label{fig:Mtable}
\end{figure}
These numbers grow quadratic-exponentially as a function of $n$,
and I have no conjectural formula for the $n$th term,
nor a conjectural recurrence relation for the sequence,
nor any efficient method of computing terms.
Nonetheless, something systematic is going on.
I have already mentioned that all the terms are odd.
Taking this observation further, one notices that
the numbers’ residues mod 4 are
$$1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1,$$
the residues mod 8 are
$$1, 3, 3, 5, 5, 7, 7, 1, 1, 3, 3, 5, 5,$$
and the residues mod 16 are
$$1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13.$$
{\bf Conjecture 1}: For all $k \geq 1$,
the mod $2^k$ residue of $M(n)$ is periodic with period dividing $2^k$.
That is, $2^k$ divides $M(n+2^k) - M(n)$ for all $k,n$.
I tried to prove this conjecture by reducing it to an assertion
about alternating-sign matrices but I was unsuccessful.
Note that if the conjecture is true
then $M(n) \equiv n + 1 + (1 + (-1)^{n + 1})/2$ (mod 8).
This congruence might also hold mod 16
but it certainly cannot hold mod $2^k$ for all $k$,
since that would require that $M(n)$ actually equals
$n + 1 + (1 + (-1)^{n + 1})/2$, which is clearly not the case for $n \geq 2$.
And indeed $M(2) = 19 \not\equiv 3$ (mod 32).
A deeper consequence of Conjecture 1 is that
the function sending $n$ to $M(n)$ is 2-adically continuous.
Moreover, the function appears to satisfy a kind of symmetry
analogous to the functional equation (1)
mentioned at the end of section~\ref{sec:intro}.
{\bf Conjecture 2}: For all $k \geq 1$,
if $n + n' \equiv -3$ (mod $2^k$) then $M(n) + M(n') \equiv 0$ (mod $2^k$).
That is, if one extends $M: \mathds{N} \rightarrow \mathds{N}$
to the 2-adic function $\widehat{M}: \mathds{Z}_2 \rightarrow \mathds{Z}_2$,
one has $\widehat{M}(-3-n) = - \widehat{M}(n)$.
Although in this article I am limiting myself
to discussion of tilings of Aztec diamonds,
I have also looked at tilings of other regions
using dominos and square tetrominos,
and the same phenomenon of 2-adic continuity arises fairly broadly there.
For instance, for the $2n$-by-$2n$ square, the $2n$-by-$(2n+2)$ rectangle, and the $2n$-by-$4n$ rectangle,
the number of tilings with dominos and square tetrominos
always seems to be congruent to $2n+1$ mod 8.
\end{section}
\begin{section}{Skew tetrominos and square tetrominos}
\label{sec:skewsquare}
In~\cite{Pr} I considered tilings of Aztec diamonds
by skew tetrominos and square tetrominos.
If we require that all skew tetrominos be horizontal,
interesting numerical patterns appear.
(Of course we would get the same result
if we required that all skew tetrominos be vertical.)
In this section we allow horizontal skew tetrominos and square tetrominos
as seen in Figure~\ref{fig:tetra}, which depicts
all six tilings of the Aztec diamond of order 3
using square tetrominos and horizontal skew tetrominos.
\begin{figure}
\caption{Tiling the Aztec diamond of order 3
with horizontal skew tetrominos and square tetrominos.}
\label{fig:tetra}
\end{figure}
\begin{figure}
\caption{Enumeration of tilings of Aztec diamonds
using horizontal skew tetrominos and square tetrominos.}
\label{fig:Ltable}
\end{figure}
Define $L(n)$ (with $n \geq 0$) as
the number of tilings of the Aztec diamond of order $n$
using horizontal skew tetrominos and square tetrominos.
This is \href{https://oeis.org/A356513}{A356513}.
Trivially we have $L(0) = 1$ and $L(1) = 1$.
Figure~\ref{fig:Ltable} shows the terms
of the sequence $L(n)$ for $n$ ranging from 0 to 15,
again computed using the program written by David desJardins.
The sequence grows quadratic-exponentially,
and once again, I have no conjectural formula,
but as before there are patterns that call out for explanation.
Noticing that all but the first two terms are even,
one might naturally think to look at the multiplicity of 2
in the prime factorization of $L(n)$, obtaining the sequence
0,0,1,1,3,2,5,3,7,4,9,5,11,6,13,7,\dots,
which (once we throw out the initial 0)
we recognize as an interspersal of the arithmetic progressions
0,1,2,3,4,5,6,7,\dots and 1,3,5,7,9,11,13,\dots.
{\bf Conjecture 3}:
For $n \geq 1$,
the multiplicity of 2 in the prime factorization of $L(n)$
is $n-1$ if $n$ is even and $(n-1)/2$ if $n$ is odd.
Going further,
let $L_0(m) = L(2m) / 2^{2m-1}$ and $L_1(m) = L(2m-1) / 2^{m-1}$,
so that (if Conjecture 3 holds)
$L_0(m)$ and $L_1(m)$ are odd integers for all $m$. These two new sequences
are shown in Figure~\ref{fig:JKtable}.
\begin{figure}
\caption{Values of $L_0(m)$ and $L_1(m)$.}
\label{fig:JKtable}
\end{figure}
The mod 4 residues of the $L_0$ sequence go $1, 1, 3, 3, 1, 1, 3$
while those of the $L_1$ sequence go $1, 3, 3, 1, 1, 3, 3, 1$.
That's not much evidence to go on,
so perhaps it would be prudent not to make a conjecture,
but I choose to be hopeful.
{\bf Conjecture 4}: For all $k \geq 0$, the mod $2^k$ residue of $L_0(m)$
is periodic with period dividing $2^k$. Likewise for $L_1(m)$.
We do not observe such patterns in the numbers of tilings
when both horizontal and vertical skew tetrominos are allowed
along with square tetrominos as in~\cite{Pr}.
More specifically, if we count tilings of Aztec diamonds
in which we are permitted to use all four kinds of skew tetrominos
as well as square tetrominos, the resulting sequence, taken mod 4, goes
1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, \dots;
if there is a period here, and it is a power of 2,
it must be at least 16.
The prime $p=2$ appears to be special
for the enumerative problems I described above;
looking at the $M$ and $L$ sequences
mod 3 or mod 5 yields no discernible patterns.
\end{section}
\begin{section}{Assorted congruential problems}
\label{sec:cong}
For each of the sixty-three nonempty subsets of
the set of six tiles shown in Figure~\ref{fig:the-six},
we can ask in how many ways
it is possible to tile the Aztec diamond of order $n$
using only tiles from that set,
allowing translations, rotations, and reflections of tiles.
These are the enumerative problems considered in this section.
(One could expand the set of tiling problems
by distinguishing between different orientations of the tiles,
as was done in the preceding section
where we permitted horizontal skew tetrominos
but forbade vertical skew tetrominos;
since there are $2+2+4+8+1+4=21$ different tiles up to translation,
we would obtain over two million different problems,
and even if we mod out the $2^{21}-1$ problems by a dihedral action of order 8,
that is still too many problems to consider exhaustively.
One that appears to be interesting is discussed at the end of this section.)
In each of the sixty-three cases, I used the aforementioned program
to count the tilings of the Aztec diamond of order $n$,
with $n$ going from 1 to 8, using the allowed tiles.
Although no 2-adic continuity phenomena arose from these experiments,
there were definite patterns in the parity,
and in a few cases there were congruence patterns modulo higher powers of 2.
Here I will adopt a six-bit code to represent
the sixty-three tiling problems,
in which the six successive bits (from left to right) equal 1 or 0
according to whether or not dominos, straight tetrominos, skew tetrominos,
L-tetrominos, square tetrominos, and T-tetrominos are allowed.
For instance, the case treated in section~\ref{sec:skewstraight},
in which only straight tetrominos and skew tetrominos are allowed
(see Figure~\ref{fig:olympiad}), would be assigned the code 011000;
the case treated in section~\ref{sec:domsquare},
in which only dominos and square tetrominos are allowed
(see Figure~\ref{fig:eleven}), would be assigned the code 100010;
and the case of unconstrained skew and square tetrominos
(briefly discussed in section~\ref{sec:skewsquare})
would be assigned the code 001010.
In one-third of the 63 cases,
I observed that for all $n$ between 1 and 8,
the number of tilings of the Aztec diamond of order $n$ is even.
These were the cases associated with the six-bit codes
001001, 001100, 001101, 011001, 011100, 011101, 100001,
100100, 100101, 101000, 101001, 101100, 101101, 110000,
110001, 110100, 110101, 111000, 111001, 111100, and 111101.
Presumably some (perhaps all) of these examples can be resolved
by showing that there are no tilings that are invariant
under the full dihedral group, since in that case all orbits
would contain an even number of tilings.
Three of the 21 cases were especially interesting.
In case 011100, all terms were divisible by 8;
in case 100001, all terms after the first were divisible by 8;
and in case 110001, all terms were congruent to 2 (mod 4).
There were also four cases in which I observed that
the number of tilings of the Aztec diamond of order $n$ is even
for all $n$ between 2 and 8
(with the number of tilings being the odd number 1 in the case $n=1$).
These were the cases associated with the six-bit codes
001010, 001110, 011010, and 011110.
In the cases 001101, 100001, 100011, and 111000
it appears that the exponent of 2 in the number of tilings
may be going to infinity with $n$,
though with such scant evidence it would be rash
to place too much faith in this guess.
Additionally, there is one case in which the number of tilings
of the Aztec diamond of order $n$ is always odd,
namely, tilings using only dominos and square tetrominos.
Indeed, if we assign each tiling weight $(-1)^s$ where $s$ is the number of square tetrominos,
I claim that the sum of the weights is 1.
We can prove this using a sign-reversing involution
that scans through the tiling in some fashion in search of a 2-by-2 block
that is tiled either with a square tetromino or with two vertical dominos
and switches between the two possibilities.
The fixed points of this involution are tilings that use only horizontal dominos,
and there is just one of those.
Finally, leaving the small world of the $2^6-1$ problems
and dipping our toe into the big world of the $2^{21}-1$ problems,
we consider tilings of the Aztec diamond of order $n$
by dominos and horizontal straight tetrominos.
This is \href{https://oeis.org/A356523}{A356523}, and begins
1, 2, 11, 209, 12748, 2432209, 1473519065, \dots.
It appears that the number of tilings is
even when $n \equiv 1$ (mod 3) and odd otherwise;
this has been verified for $1 \leq n \leq 16$.
\end{section}
\begin{section}{Reduction to perfect matchings}
\label{sec:matchings}
The $L$ sequence from section~\ref{sec:skewsquare}
has an interpretation in terms of perfect matchings.
To see why, suppose we have a tiling of the Aztec diamond of order $n$
using horizontal skew tetrominos and square tetrominos.
Dividing each tetromino into two horizontal dominos
gives us a tiling of the Aztec diamond by horizontal dominos,
but it is easy to see that there is exactly one such tiling (call it $T$).
Hence each tetromino is obtained by gluing together two dominos in $T$.
That is, the tetromino tilings correspond to perfect matchings in the graph
whose vertices correspond to the dominos in $T$
with an edge joining two vertices if the corresponding dominos
form a horizontal skew tetromino or square tetromino.
It is not hard to see that this graph is similar
to the $n$-by-$n$ square except that the diagonal has been ``doubled'';
for instance, the right panel of Figure~\ref{fig:double-double}
shows the graph for $n=4$.
\begin{figure}
\caption{Deriving the square graph with doubled diagonal.}
\label{fig:double-double}
\end{figure}
A similar analysis can be applied to tilings of Aztec diamonds
using horizontal skew tetrominos and horizontal straight tetrominos.
In this case the Aztec diamond splits
into two non-interacting halves (top half and bottom half),
each of which can be tiled independently of the other,
and the tilings of either half correspond
to perfect matchings of a triangle graph
as shown in Figure~\ref{fig:triangles}.
Thus the number of such tetromino tilings of the Aztec diamond of order $n$
is equal to the square of the $n$th term of sequence
\href{https://oeis.org/A071093}{A071093}.
Studying the first 25 terms, I find that
the sequence seems to have 2-adic properties of its own.
The largest power of 2 dividing the $n$th term of the sequence A071093
appears to be $\lfloor n/2 \rfloor$,
and the 2-free part appears to satisfies 2-adic continuity:
for instance, its value mod 16 seems to be determined by $n$ mod 16.
\begin{figure}
\caption{Deriving the double triangle graph.}
\label{fig:triangles}
\end{figure}
What if we superimpose the two graphs,
obtaining the graph shown at the right half of
Figure~\ref{fig:superimpose}?
This is equivalent to tiling an Aztec diamond using
horizontal skew tetrominos, horizontal straight tetrominos, and square tetrominos.
Then, counting the tilings, we obtain the integer sequence
1, 2, 10, 116, 3212, 209152, 32133552, 11631456480, 9922509270288,
19946786274879008, 94492874103638971552, 1054865198752147761744448,
\dots.
This is \href{https://oeis.org/A356514}{A356514}.
It appears that the number of tilings is divisible
by $2^{\lfloor n/2 \rfloor}$.
\begin{figure}
\caption{Superimposing Figures~\ref{fig:double-double}
\label{fig:superimpose}
\end{figure}
\end{section}
\begin{section}{Some thoughts}
\label{sec:thoughts}
The articles of Lovasz~\cite{Lo}, Ciucu~\cite{Ci},
Pachter~\cite{Pa}, and Barkley and Liu~\cite{BL}
give ways to find the largest power of 2
that divides the number of perfect matchings of a graph.
This should provide traction for Conjecture 3,
since we saw in section~\ref{sec:matchings} that
the $L$ sequence has an interpretation in terms of perfect matchings
of certain graphs.
Graphs of this kind appear in the paper of Ciucu~\cite{Ci};
in particular, his Lemma 1.1 shows that
the number of perfect matchings is divisible by $2^{\lfloor n/2 \rfloor}$.
By bringing ideas from Pachter~\cite{Pa}, one might be able to prove Conjecture 3,
as well as some of the other 2-divisibility conjectures from this article.
The only work I know of that provides detailed 2-adic information
about the 2-free part of numbers that count tilings
is the work of Cohn~\cite{Co}.
Cohn's approach presupposes the existence of an exact formula
(in Cohn's case, an explicit product of algebraic integers);
perhaps something similar can be done for perfect matchings
of the square graph with doubled diagonal,
yielding a proof of Conjecture 4.
Conjectures 1 and 2 seem harder.
The product formula exploited by Cohn
was discovered by Temperley and Fischer~\cite{TF}
and independently by Kasteleyn~\cite{Ka} at about the same time;
those researchers made use of the fact that,
just as determinants and Pfaffians of matrices can be expressed as sums of terms
associated with perfect matchings of the set of rows and columns,
one can conversely express the number of perfect matchings of a planar graph
in terms of the determinant or Pfaffian of an associated matrix.
I know of no way of recast the $M$ sequence
as enumerating perfect matchings of graphs.
However, it is easy to recast the $M$ sequence as enumerating
perfect matchings of certain hypergraphs.
Can any of the existing notions of hyperdeterminants be brought to bear?
Perhaps a reading of~\cite{GKZ} would suggest possible approaches.
Kuperberg's elegant solution~\cite{Ku}
to the alternating sign matrix conjecture
exploits the power of the Yang-Baxter equation in statistical mechanics.
It's possible that tools for analyzing the new problems described in this article
will be found in the existing literature at
the interface between algebra and statistical mechanics.
In any case, inasmuch as Conjectures 1 and 2 are reminiscent of Cohn's work,
and inasmuch as Cohn's argument hinges on an exact product formula,
one might hope that an exact formula of some kind
can be found for the $M$ sequence. Such an exact formula would have other uses.
In~\cite{CEP} and~\cite{CLP}, Henry Cohn, Noam Elkies, Michael Larsen and I
used exact enumeration results to prove
concentration theorems for random tilings.
One might hope that the curious 2-adic phenomena discussed in this article
hint at the existence of algebraic machinery that
could be applied to the task of showing us
what random tilings associated with Conjecture 1
look like in the limit as size goes to infinity.
Preliminary experiments suggest that
there is a ``frozen region'' near the boundary,
but I have no idea how far into the interior it extends.
\end{section}
\noindent
I thank David desJardins for the software that made this research possible,
and Noam Elkies for helpful comments.
Above all I thank Michael Larsen for his many years
of friendship and mathematical camaraderie.
\label{sec:biblio}
\end{document} |
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\title{Learning the Physics of Particle Transport via Transformers}
\begin{abstract}
Particle physics simulations are the cornerstone of nuclear engineering applications. Among them radiotherapy (RT) is crucial for society, with 50\% of cancer patients receiving radiation treatments. For the most precise targeting of tumors, next generation RT treatments aim for real-time correction during radiation delivery, necessitating particle transport algorithms that yield precise dose distributions in sub-second times even in highly heterogeneous patient geometries. This is infeasible with currently available, purely physics based simulations. In this study, we present a data-driven dose calculation algorithm predicting the dose deposited by mono-energetic proton beams for arbitrary energies and patient geometries. Our approach frames particle transport as sequence modeling, where convolutional layers extract important spatial features into tokens and the transformer self-attention mechanism routes information between such tokens in the sequence and a beam energy token. We train our network and evaluate prediction accuracy using computationally expensive but accurate Monte Carlo (MC) simulations, considered the gold standard in particle physics. Our proposed model is 33 times faster than current clinical analytic pencil beam algorithms, improving upon their accuracy in the most heterogeneous and challenging geometries. With a relative error of $0.34\pm0.2$\% and very high gamma pass rate of $99.59\pm0.7$\% (1\%, 3 mm), it also greatly outperforms the only published similar data-driven proton dose algorithm, even at a finer grid resolution. Offering MC precision 400 times faster, our model could overcome a major obstacle that has so far prohibited real-time adaptive proton treatments and significantly increase cancer treatment efficacy. Its potential to model physics interactions of other particles could also boost heavy ion treatment planning procedures limited by the speed of traditional methods.
\end{abstract}
\section{Introduction}
\label{sec:Introduction}
Despite significant research efforts cancer remains a leading cause of death, responsible for more than 10 million deaths in 2020 worldwide \cite{GCO2021,Sung2021}. With more than 50\% of the patients receiving radiation treatments, radiotherapy (RT) is at the forefront of current standard of care, playing a crucially important role in improving societal health. Sophisticated computational methods and particle transport simulations have been key to this success \cite{Bernier2004}, enabling highly personalized treatments. Traditional physics based algorithms improved all steps in the RT workflow (imaging, segmentation, dose calculation, optimization), but so far they proved too slow and inaccurate for real-time adaptive treatments promising ultimate precision with fewest adverse side-effects. Deep learning is key to overcome these limitations and realize the full potential of real-time adaptation.
Our study focuses on learning particle transport physics --- fundamental to all steps of RT from Computed Tomography (CT) image reconstruction to simulating the actually delivered patient dose --- to provide the necessary sub-second speed and high accuracy required for real-time adaptation. We frame the transport problem as sequence modelling, with a particle beam going through varying geometries and materials, using convolutional layers to learn relevant spatial features and the transformer self-attention mechanism to combine information from the feature tokens and a beam energy token. We train our algorithm to specifically learn proton transport in lung cancer patients with highly heterogeneous geometries to predict dose based on CT images alone, but the model could in theory be easily adapted to other particles (photons, electrons, heavy ions) and quantities (e.g., particle flux or secondary particle emission prediction).
\paragraph{Contributions} Our specific contributions are as follows:
\begin{itemize}
\item We frame particle transport physics as a sequence modelling task and propose a novel algorithm using convolutional encoder and decoder layers together with transformer causal self-attention to predict dose distributions.
\item We train our algorithm using highly variable geometries from lung cancer patients and demonstrate that it outperforms both current clinical pencil beam algorithms (PBA), being 33 times faster and more precise in the most complex geometries, and 'gold standard' Monte Carlo (MC) methods, offering MC accuracy 400 times faster. Our model is also more accurate than the only published data driven proton dose calculation algorithm using (Long Short-Term Memory) LSTM cells.
\item We highlight the direct societal impact of the presented algorithm by showcasing how it could improve current radiotherapy practice and enable real-time adaptive treatments. While we train our model to learn proton physics to predict dose distributions, we also detail extensions to make it a general particle transport simulator, accounting for e.g., beam shape or energy spectrum changes.
\end{itemize}
\section{Background}
\label{sec:Background}
Here we describe RT workflow and the critical role of particle transport and dose calculation methods, and frame our work in terms of unsolved challenges and related literature.
\subsubsection{Radiotherapy workflow}
RT treatments usually follow a 4-step procedure. First, high quality anatomical information is acquired --- typically as CT images \cite{Pereira2014} --- on which tumors to irradiate and organs at risk (OARs) to protect are delineated. Second, the irradiation modality is chosen, with most patients receiving photon treatments, but proton therapy spreading quickly due to protons' finite range and significantly better ability to focus dose on tumors \cite{Lundkvist2005}. Third, the beam angles and beamlet intensities to irradiate the patient with are optimized during treatment planning. This is the most complex and computationally expensive task, requiring solving large scale multi-criteria optimization problems and typically several iterations between planners and physicians before an acceptable, clinically 'best' plan is achieved \cite{Hussein2018,Meyer2018}. Last, for quality assurance purposes detailed dose calculations are performed to test plan robustness against anatomical changes or decide to adapt a plan for future irradiations.
\subsubsection{Particle transport \& dose calculation}
Accurate particle transport algorithms are crucial for all these steps. CT image reconstruction relies on simulating photon interactions with tissues and detectors; plan optimization requires the spatial dose distribution (typically in more than 1 million voxels) from each available proton or photon beamlet (in the thousands); while for plan evaluation the dose must be calculated for many different geometries. Ideally these calculations should be quick and precise, but current analytical pencil beam algorithms and stochastic MC dose calculation tools offer a trade-off. PBA yields results without the computational burden of MC engines, but its accuracy is severely compromised in highly heterogeneous or complex geometries, making slow and clinically rarely affordable MC approaches necessary. The problem is most acute for next generation real-time adaptive treatments promising ultimate precision with fewest side effects by correcting treatments during irradiation, e.g., to account for anatomical changes due to breathing, coughs or intestinal movements. To finally become reality, such adaptive treatments require algorithms that deliver MC accuracy in sub-second speed.
\subsubsection{Related work}
Deep learning has achieved significant improvements in all steps of the RT workflow \cite{Meyer2018}, but only imaging, treatment planning and dose calculation are relevant to our work. U-net \cite{Ronneberger2015} and Generative Adversarial Networks \cite{Goodfellow2014} (and their variants) have been widely applied to improve image quality, e.g., to generate synthetic CT images from Magnetic Resonance Images (offering better soft tissue contrast than CT without additional patient dose) or low dose Cone-Beam CT (CBCT) images \cite{Edmund2017,Zhang2021}; to predict stopping power from CBCT \cite{Harms2020}; or correct scatter artifacts in CBCT reconstruction \cite{Lalonde2020}. These works represent image to image transformation, producing more useful images for the RT workflow than their easier/faster to obtain or lower patient dose input.
In treatment planning, deep learning methods aim to predict an optimal 3D patient dose distribution achievable with a given radiotherapy technique based on historical data. The most successful works use ResNet based convolutional networks \cite{Chen2019,Fan2019}, 2/3D U-net \cite{Kearney2018,Nguyen2019, Kajikawa2019} or hierarchically densely connected U-net (HDU-net) \cite{Nguyen2019a,BarraganMontero2019} architectures, with segmented structure masks as input. Some also utilize the CT image \cite{Kearney2018} and manually encoded beam configuration information \cite{Nguyen2019a,BarraganMontero2019}. These works basically mimic 'optimal' plans for new patients that should be achievable based on past ones, only outputting final dose distributions, but not the required beam intensity (i.e., fluence) maps needed to deliver such plans, which must be obtained via additional, costly optimization. Thus, they are mostly used for Quality Assurance (QA) purposes to aid planning, not replace it. Only few papers attempt to jointly predict dose distribution and fluence maps \cite{Lee2019,Wang2020}, and all have been applied to photon treatments.
Practically all applications of deep learning to dose calculations learn how to improve cheaper and faster physics based calculations. Most works try to predict low noise MC photon dose distributions from high noise MC doses \cite{Peng2019, Peng2019a,Bai2021,Neph2021}, or deterministic particle transport based photon distributions from simple analytical calculations \cite{Xing2020a,Dong2020}, using CNNs, U-net or HDU-net architectures with 2/3D patches. A few papers manually encode some physics information as additional input such as fluence maps \cite{Fan2020,Xing2020}, total energy released per unit mass maps \cite{Zhu2020} or beam information \cite{Kontaxis2020,Tsekas2021}. We are only aware of 2 papers \cite{Wu2021,Javaid2021} using deep learning to predict accurate low noise MC proton dose distributions, both using cheap physics models (noisy MC and PBA) as input. While all these works provide significant speed-up compared to pure physics based algorithms, some even reaching sub-second speeds, they all require physics models to produce their input, do not generalize easily (e.g., to different beam energies) and are trained with full plan data, unsuitable for real-time adaptation needing the individual dose distribution from each beamlet alone.
Most related to ours is the work from \cite{Neishabouri2021}, using LSTM networks to sequentially calculate proton pencil beam dose distributions from relative stopping power slices. Although requiring a separate model per beam energy, this LSTM-based dose engine offers excellent inference times and close to PBA accuracy when tested on external patient data. Our approach builds upon the methodology of \cite{Neishabouri2021}, but uses a different architecture, works on finer resolution and --- most crucially --- also learns the physics of energy dependence in particle transport via a single model.
\paragraph{Transformer} The backbone of the presented model is the Transformer, which was first introduced by \cite{Vaswani2017} for machine translation tasks. The Transformer and similar attention-based architectures have completely replaced recurrent neural network variants like LSTM in natural language processing applications since then \cite{Devlin2019, Brown2020}. A main reason behind their large-scale adoption and success is the ability to process long-term dependencies by directly accessing information at any point in the past without needing internal memory, which is essential to introduce beam energy dependence in our model.
Transformer-based architectures have also achieved state-of-the-art performance in computer vision tasks like image classification \cite{Ramachandran2019, Dosovitskiy2020}. Inspired by \cite{Cordonnier2019}, \cite{Dosovitskiy2020} present the Vision Transformer (ViT), circumventing the quadratic cost of computing the attention weight matrix by dividing the input image into a patch sequence. Following the Transformer basis of ViT, our model also adopts a patch-based processing of the inputs.
The price that Transformers pay for their generality in both language and vision is the need for a self-supervised pre-training stage with large amounts of text or image data \cite{Devlin2019, Brown2020, Dosovitskiy2020}. For image classification, several approaches try to achieve state-of-the-art performance without costly pre-training \cite{Touvron2020, DAscoli2021}. As in the concurrent work of \cite{Hassani2021}, our model can be directly trained on a relatively small dataset by using a convolutional encoder first extracting important features from the patched input data.
\subsubsection{Self-attention} The proposed model leverages the self-attention (SA) mechanism \cite{Vaswani2017} allowing dynamic routing of information between the $L$ elements in a sequence $\bm{z}\in\mathbb{R}^{L\times D}$. SA is based on the interaction between a series of queries $\bm{Q}\in\mathbb{R}^{L\times D_h}$, keys $\bm{K}\in\mathbb{R}^{L\times D_h}$, and values $\bm{V}\in\mathbb{R}^{L\times D_h}$ obtained through a learned linear transformation of the input sequence
\begin{equation}
[\bm{Q}, \bm{K}, \bm{V}] = \bm{z}\bm{W}_{QKV},
\end{equation}
\noindent with learned weights $\bm{W}_{QKV}\in\mathbb{R}^{D\times 3D_h}$. Intuitively, every sequence element emits a query and a key vector with the information to gather from and to offer to the rest of the sequence, respectively. Each of the $L$ elements in the output sequence is a weighted sum of the values, where the weights --- referred to as attention matrix $\bm{A}\in\mathbb{R}^{L\times L}$ --- are obtained by matching queries against key vectors via inner products
\begin{equation}
\bm{A} = \text{softmax}\Big(\frac{\bm{Q}\bm{K}^T}{\sqrt{D_h}}\Big),
\end{equation}
\begin{equation}
\text{SA}(\bm{z})=\bm{A}\bm{V}.
\end{equation}
Multi-head self-attention (MSA) runs $N_h$ parallel SA operations to extract different features and inter-dependencies, translating into setting $D_h=D/N_h$. The outputs of the different operations, called \textit{heads}, are concatenated and are linearly projected with learned weights $\bm{W}_h\in\mathbb{R}^{N_hD_h\times D}$ as
\begin{equation}
\text{MSA}({\bm{z}}) = \underset{h\in\{N_h\}}{\text{concat}}[\text{SA}_h(\bm{z})]\bm{W}_h.
\end{equation}
By definition, MSA is invariant to the relative order of elements in the sequence. To account for positional information, fixed \cite{Vaswani2017} or learned \cite{Dosovitskiy2020} positional embedding can be added or concatenated to the input right before the first MSA operation. In addition, in MSA every element in the sequence can retrieve information at any past and future point. For some prediction tasks where the elements cannot or need not attend to future information, a binary mask is used to stop information flow from the future to the present. Such SA mechanism variant is referred to as causal SA and is particularly suited for modeling proton interactions and energy deposition physics that mostly occur sequentially in the forward beam direction.
\section{Methods}
\label{sec:Methods}
Our objective is to implicitly capture the physics of particle transport through a data-driven approach that enables accurate dose calculations in sub-second speed. The presented transformer-based parametric model exploiting the forward sequential nature of proton transport physics is well-suited for this. This section describes the model's building blocks, the dataset and the training and evaluation procedures.
\subsubsection{Proposed model} We introduce a parametric model that computes the output dose distribution $\bm{y}\in\mathbb{R}^{L\times H\times W}$ given the input geometry data $\bm{x}\in\mathbb{R}^{L\times H\times W}$ and particle energy $\epsilon\in E\subset\mathbb{R}^+$, where $L, H$ and $W$ are the depth, height and width of the geometric 3D grid, respectively. The model --- referred to as Dose Transformer Algorithm (DoTA) --- captures the relationship between the inputs and the output dose distribution through a nonlinear mapping $f_{\bm{\theta}}(\bm{x},\epsilon):\mathbb{R}^{L\times H\times W}\times E\rightarrow\mathbb{R}^{L\times H\times W}$, performed by a series of artificial neural networks. \figref{Architecture} shows the architecture of the model, which processes the 3D input geometry $\bm{x}$ as a sequence of $L$ 2D images in the beam's eye view $\{\bm{x}_i|\bm{x}_i\in\mathbb{R}^{1\times H\times W},\forall i=1,...,L \}$.
\begin{figure*}
\caption{Model architecture. We treat the input and output 3D volumes as a sequence of 2D slices. A convolutional encoder reduces the dimension of the input and extracts important geometrical features. The particle energy is added at the beginning of the resulting sequence. A transformer encoder with causal self-attention then routes information between the encoded input slices. Finally, a convolutional decoder transforms the low-dimensional sequence into an output sequence of 2D dose slices.}
\label{fig:Architecture}
\end{figure*}
\subsubsection{Convolutional encoder} First, a convolutional encoder extracts important features such as geometry contrasts and edges from the input CT slices. The convolutional encoder contains two blocks, each with a convolutional, a Group Normalization (GN) \cite{Wu2020} and a pooling layer, followed by a Rectified Linear Unit (ReLU) activation. After the second block, a convolution with $K$ filters results in a sequence of elements of reduced embedding dimension $D=H'\times W'\times K$, where $H'$ and $W'$ are the reduced height and width of the images. The layers in the convolutional encoder share weights and are applied independently to every element $\bm{x}_i$ in the sequence. We refer to the output of the convolutional encoder as tokens $\{\bm{z}_i|\bm{z}_i\in\mathbb{R}^{D},\forall i=1,...,L \}$.
\subsubsection{Transformer encoder} The interaction between tokens $\bm{z}_i$ is modeled in the transformer encoder through causal MSA, with each token routing information from all preceding tokens. To account for the relative positional information of sequence elements we add a learnable embedding to each token. We include an extra energy token $\bm{z}_e=\bm{W}_e\epsilon\in\mathbb{R}^D$ at the beginning of the sequence, where $\bm{W}_e\in\mathbb{R}^{D\times 1}$ is a learned linear projection of the beam energy $\epsilon$. The transformer encoder alternates MSA and Multi-layer Perceptron (MLP) layers, with Layer Normalization (LN) \cite{Ba2016} and residual connections applied before and after every layer, respectively. A stack of $N$ transformer encoder blocks computes the operations
\begin{equation}
\bm{z}_0 = [\bm{z}_e;\bm{z}] + \bm{r}_p,
\end{equation}
\begin{equation}
\bm{s}_n = \bm{z}_{n-1} + \text{MSA}(\text{LN}(\bm{z}_{n-1})), \qquad n=1...\:N
\end{equation}
\begin{equation}
\bm{z}_n = \bm{s}_{n} + \text{MLP}(\text{LN}(\bm{s}_{n})), \quad\qquad n=1...\:N
\end{equation}
\noindent where $\bm{r}_p\in\mathbb{R}^{(L+1)\times D}$ is the learnable positional embedding and MLP is a two layer feed-forward network with Dropout \cite{Srivastava2014} and Gaussian Error Linear Unit (GELU) activations \cite{Hendrycks2016}.
\subsubsection{Convolutional decoder} To produce output dose volume $\bm{y}$ with the same dimension as the input, each token is transformed via a convolutional decoder with shared weights into the output slices $\{\bm{y}_i|\bm{y}_i\in\mathbb{R}^{1\times H\times W},\forall i=1,...,L \}$. Instead of normal convolutional layers, the decoder contains transposed convolutions that increase the dimension of their input. Similarly to the convolutional encoder, two final dimension-preserving convolutions transform the output of the second block into the 2D dose slices.
\subsubsection{Dataset} The models are trained using a dataset with pairs of sliced CT images and dose distributions corresponding to mono-energetic proton beams with different energies. The 3D CT scans from 4 lung cancer patients are highly heterogeneous due to the air, bones and organs present in the thorax, and cover a volume of $512\times 512\times 100 $ $\text{mm}^3$ with a resolution of $1\times 1\times 3$ mm. Since each proton beam has approximately 20 mm diameter and travels up to 250 mm through a small volume only, we crop and extract blocks $\bm{x}\in\mathbb{R}^{256\times 48\times 16}$ maintaining the original CT resolution. Many different blocks can be obtained from the same patient by rotating the CT scan along the Z direction in steps of $5\degree$ and applying shifts in YZ plane with $5$ mm steps.
The output ground-truth dose distributions are calculated using the open source Monte Carlo particle transport software MCsquare \cite{Souris2016}, taking CT slices and calculating output blocks $\bm{y}\in\mathbb{R}^{256\times 48\times 16}$ with the same size and resolution as the input via random sampling of proton trajectories. Dose distributions are estimated using 3 million primary particles ensuring low MC noise levels of 0.6\%. For each input CT block we generate 4 dose distributions corresponding to 4 randomly sampled energies between 80 and 130~MeV, rounded to 1 decimal. We mask MC noise by zeroing out dose values below the noise level.
The entire training dataset consists of 63,048 pairs of input-output blocks, 10\% of which are used as a validation set. We apply data augmentation during training and randomly rotate the volumes $180\degree$ in beam's eye view (YZ plane), doubling the number of samples. A test set of 3,618 input-output pairs from an external patient is used to evaluate generalization to unseen geometries and energies.
\subsubsection{Training details}
The best performing model consists of one transformer encoder block with 16 heads and convolutional layers with a 3$\times$3 kernel. Using size preserving zero-padding results in halving (or doubling, for the decoder) the H and W dimensions after each convolutional block. The token embedding dimension $D=H/4\times W/4\times K$ is constant throughout the transformer encoder layers, with height $H=48$, width $W=16$, $K=10$ kernels and $D=480$ in our particular case. The models are trained with Tensorflow \cite{TF2015} using the LAMB optimizer \cite{You2019} and mini-batches of 8 samples, limited by the maximum internal memory of the Nvidia Tesla T4{\textregistered} Graphics Processing Unit (GPU) used during our experiments. We use the mean squared error (MSE) as loss function and a scheduled learning rate starting at $10^{-3}$ that is halved every 4 epochs. In Appendix A we perform a model hyperparameter search varying the number of transformer layers $N$, convolutional filters $K$ and attention heads $N_h$.
\subsubsection{Gamma analysis}
We compare the predicted and ground-truth 3D dose distributions from the test set using gamma analysis \cite{Low1998}. Intuitively, for a set reference points and their corresponding reference dose values, this method searches for similar dose values within small spheres around each point. The similarity is quantified using a maximum dose difference threshold (usually expressed as a percentage of the reference dose): e.g., dose values are accepted similar if within 1\% of the reference dose. The radius of the sphere is referred to as distance-to-agreement criterion. Mathematically, gamma values are calculated for individual points in the predicted dose grid as
\begin{equation}
\gamma(\bm{p}) = \underset{\bm{\hat{p}}}{\min}\{\Gamma(\bm{p},\bm{\hat{p}})\},
\end{equation}
\begin{equation}
\Gamma_{\delta, \Delta}(\bm{p},\bm{\hat{p}})=\sqrt{\frac{\abs{\bm{p}-\bm{\hat{p}}}^2}{\delta^2}+\frac{\abs{D(\bm{p})-D(\bm{\hat{p}})}^2}{\Delta^2}},
\end{equation}
\noindent where $\bm{p}$ and $\bm{\hat{p}}$ are the coordinates of the points in the predicted and ground truth dose grids, respectively. $D(\bm{p})$ is the dose at any point $\bm{p}$, $\delta$ is the distance-to-agreement and $\Delta$ the dose difference criterion.
We use the publicly available gamma evaluation functions from PyMedPhys\footnote{see \url{https://docs.pymedphys.com}}, with $\delta=3$ mm and $\Delta=1\%$. The 3 mm distance-to-agreement criterion ensures a neighborhood search of at least one voxel, while the dose difference criterion of 1\% disregards uncertainty due to MC noise. Gamma values are calculated for each voxel and a voxel centered at $\bm{p}$ is considered to pass the gamma evaluation if $\gamma(\bm{p})<1$. For the entire grid, the gamma pass rate can be calculated as the fraction of passed voxels over total number of voxels.
\subsubsection{Error analysis} The sample average relative error is used as an additional method to explicitly compare dose differences between two grids. Given the predicted output $\bm{y}$ and the ground truth dose distribution $\bm{\hat{y}}$, the average relative error $\rho$ can be calculated as
\begin{equation}
\rho = \frac{\norm{\bm{y}-\bm{\hat{y}}}_{L_1}}{\max_j{\bm{\hat{y}}_j}}\times 100,
\end{equation}
\noindent where $\max_j{\bm{\hat{y}}_j}$ is the maximum dose value among all voxels in the ground-truth dose grid.
\section{Experiments}
\label{sec:Experiments}
We compare the performance of the presented DoTA model to both state-of-the-art and clinically used methods. The experiments first focus on evaluating the accuracy all models: the gamma evaluation serves as a tool to assess dosimetric differences, while the relative error allows direct comparison of the predicted output and ground truth grids. Last, we report calculation times and evaluate DoTAs' potential to displace other algorithms as a fast dose calculation tool.
\subsubsection{Baselines} Our approach is compared to PBAs, the group of analytical dose calculation methods mostly used in the clinic. In particular, we calculate dose distributions for the entire test set using the PBA included in the open-source treatment planning software matRad\footnote{Available at \url{https://github.com/e0404/matRad}.} \cite{Wieser2017}. The DoTA model is also compared to the only published data-driven approach based on LSTM cells \cite{Neishabouri2021}. Since the LSTM models in \cite{Neishabouri2021} are trained for a single energy, we additionally train a model using 104.25 megaelectronvolt (MeV) proton beams.
\subsubsection{Gamma evaluation} The gamma pass rate is calculated for every test sample using the MC dose distributions as reference with two settings. In the first \textit{unmasked} setting identical to \cite{Neishabouri2021}, voxels with exactly 0 gamma value are excluded from the pass rate calculation. These are typically voxels not receiving any dose in both the predicted and ground-truth grids, having no clinical relevance. However, the outputs of the model's last linear layer are hardly ever exactly 0, taking very small values instead. Thus, in the second, stricter \textit{masked} setting, we mask voxels in the predicted dose grid that are below 0.01\% of the maximum dose.
\tabref{gpr} summarizes the results of the gamma evaluation for both settings. We report mean, standard deviation, maximum and minimum pass rates across the entire test set. Even with energy dependence and a finer grid resolution, DoTA outperforms the LSTM model in all aspects: the average pass rate is higher, the spread lower, and the minimum is almost 2\% higher. The performance of PBA and DoTA is very similar: their average values are very close in both the masked and unmasked setting, and their gamma pass rate distributions (left plot of \figref{gpr}) almost overlap. The minimum pass rate is significantly higher for DoTA, indicating that PBA struggles with the most heterogeneous and complicated samples. To verify this, we divide the dose beamlet into 4 equal sections along depth and score the number of failed voxels per section across the entire test set. The right plot \figreff{gpr} shows the proportion of voxels failing the gamma evaluation per section, out of the total number of failed voxels. The higher proportion in the $4^{\text{th}}$, last section of the beam represents inaccuracies in the high dose, clinically most relevant regions where the effects of heterogeneities are most evident.
\begin{table*}[]
\centering
\caption{Gamma analysis results ($\delta=3 \text{mm}$, $\Delta=1\%$). For PBA and DoTA, gamma pass rates are calculated across the same test set. Pass rates of the LSTM models are directly obtained from \cite{Neishabouri2021}. Mask indicates whether the predicted low dose values below 0.01\% of the maximum dose are masked before gamma evaluation. Mean, standard deviation (Std), minimum (Min) and maximum (Max) values across the test set are shown for each model, mask and energy combination.}
\begin{tabular}{@{}lcccccc@{}}
\toprule
\textbf{Model} & \textbf{Mask} & \textbf{Energy (MeV)} & \textbf{Mean (\%)} & \textbf{Std (\%)} & \textbf{Min (\%)} & \textbf{Max (\%)} \\ \midrule
\multirow{3}{*}{LSTM \cite{Neishabouri2021}} & \ding{55} & 67.85 & 98.56 & 1.30 & 95.35 & 99.79 \\
& \ding{55} & 104.25 & 97.74 & 1.48 & 92.57 & 99.74 \\
& \ding{55} & 134.68 & 94.51 & 2.99 & 85.37 & 99.02 \\ \midrule
\multirow{3}{*}{DoTA (ours)} & \ding{55} & 104.25 & 99.55 & 0.71 & 93.45 & 100 \\
& \ding{55} & [80, 130] & 99.59 & 0.70 & 92.79 & 100 \\
& \ding{51} & [80, 130] & 99.12 & 1.45 & 87.32 & 100 \\ \midrule
\multirow{2}{*}{PBA \cite{Wieser2017}} & \ding{55} & [80, 130] & 99.45 & 1.16 & 89.61 & 100 \\
& \ding{51} & [80, 130] & 99.16 & 1.73 & 83.35 & 100 \\ \bottomrule
\end{tabular}
\label{tab:gpr}
\end{table*}
\begin{figure}
\caption{(Left) Distribution of the gamma pass rates across the test set for the PBA and DoTA. (Right) Distribution of the failed voxels along the beam, where each bin shows the ratio between the number of voxels in the test set that fail the gamma evaluation within a section of the beam and the total number of failed voxels.}
\label{fig:gpr}
\end{figure}
\subsubsection{Error evaluation} To explicitly compare the performance of PBA and DoTA, we calculate the sample average relative error $\rho$ of the test set. \tabref{err} shows the mean, standard deviation, maximum and minimum errors observed across all test samples, and the left plot in \figref{err} displays the distribution of $\rho$ values for both models. Though PBA achieves low errors in the most homogeneous samples, our approach is clearly superior with a lower mean $\rho$ and a twice lower maximum error. The depth profile in the right plot of \figref{err} shows the same trend as the gamma evaluation: DoTA outperforms PBA in the last high dose regions of the beam.
\begin{table}[h!]
\centering
\caption{Average relative error between predicted and reference MC dose distributions. Reported values include mean, standard deviation (Std), minimum (Min) and maximum (Max) values across the test set, for both the PBA and DoTA.}
\begin{tabular}{@{}lllll@{}}
\toprule
\textbf{Model} & \textbf{Mean (\%)} & \textbf{Std (\%)} & \textbf{Min (\%)} & \textbf{Max (\%)} \\ \midrule
\begin{tabular}[c]{@{}l@{}}DoTA\\ (ours)\end{tabular} & 0.3430 & 0.1999 & 0.0780 & 1.4250 \\
PBA & 0.3863 & 0.3154 & 0.0683 & 2.7317 \\ \bottomrule
\end{tabular}
\label{tab:err}
\end{table}
\begin{figure}
\caption{(Left) Distribution of the average relative errors across the test set for the PBA and DoTA. (Right) Comparison of the average relative error per beam section, where each bin shows the mean relative error across the test set for equally sized sections of the beam.}
\label{fig:err}
\end{figure}
\subsubsection{Time evaluation} Besides high prediction accuracy, fast inference times are critical for clinical dose calculation algorithms. \tabref{time} reports run times of the LSTM, DoTA, PBA and MC dose algorithms. Though dependent on hardware, the data-driven approaches are clearly faster than clinically used PBA and MC baselines. While LSTM seems faster than DoTA, this is partially due to the LSTM model having 2.67 times smaller input/output and being run on better hardware. Our proposed approach offers a 33 and 400 times speed-up compared to the PBA and MC methods, respectively.
\begin{table}[h!]
\caption{Mean inference time and standard deviation (Std) across the test set for each model. Reported run times only account for the dose calculation and disregard pre-processing steps. The values for the LSTM model are taken directly from \cite{Neishabouri2021}. The DoTA runtimes include the per-sample runtime obtained using the maximum GPU capacity corresponding to a batches of 8 sample.}
\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
\textbf{Model} & \textbf{Mean (ms)} & \textbf{Std (ms)} \\ \midrule
LSTM$^a$ & 6.0 & - \\
DoTA$^b$ (1 per batch) & 70.5 & 10.2 \\
DoTA$^b$ (8 per batch) & 31.2 & 1.0 \\
PBA$^c$ & 1,030.7 & 108.9 \\
MC$^c$ & 13,295.2 & 3,180.0 \\ \bottomrule
\end{tabular}\\
\end{center}
\label{tab:time}
\footnotesize{$^a$ Nvidia{\small\textregistered} Quadro RTX 6000 64 Gb RAM}\\
\footnotesize{$^b$ Debian20 4 vCPUs 15 Gb RAM - Nvidia{\small\textregistered} Tesla T4 16 Gb RAM}\\
\footnotesize{$^c$ Ubuntu 20.04 intel{\small\textregistered} Core\textsuperscript{\tiny TM}} i7-8550U 1.8 GHz 16Gb RAM
\end{table}
\begin{figure*}
\caption{PBA}
\label{fig:worst_pba}
\caption{DoTA}
\label{fig:worst_dota}
\caption{Worst performing sample in the gamma evaluation across the test set, for (a) PBA and (b) DoTA. Each plot displays the central slice of the 3D input CT grid, the MC ground truth dose distribution, the model prediction and the gamma values.}
\label{fig:worst}
\end{figure*}
\subsubsection{Additional geometries} \figref{worst} displays the worst performing test sample in terms of gamma pass rate for the PBA and DoTA models. Both samples consist of a beam traversing the lungs, where most of the energy is deposited in a highly heterogeneous region, with the PBA sample completely misplacing the high energy peak, while DoTA still yielding reasonable prediction. In Appendix B we evaluate the model in additional geometries unseen during training.
\section{Conclusions}
\label{sec:Conclusions}
After their recent success in natural language processing and computer vision tasks, transformer-based architectures prove to excel in problems that involve sequential image processing too. Framing particle transport as sequence modeling of 2D geometry slices, we use their power to build a fast and accurate dose calculation algorithm that implicitly learns proton transport physics and has the potential for profound social impact by enabling next generation, real-time adaptive radiotherapy cancer treatments.
Our evaluation shows that the presented DoTA model has the right attributes to replace proton dose calculation algorithms currently used in clinical practice. Compared to PBAs, DoTA achieves 33x faster inference times while being better suited for heterogeneous patient geometries. The high gamma pass rate in unseen geometries from an external patient also demonstrate that the model predicts close to high accuracy MC dose distributions in sub-second times.
Such speed and accuracy increase could directly improve current RT practice by allowing comprehensive plan robustness analysis (now performed by checking only few potential geometries), quick dosimetric quality assurance of daily treatments (mostly done by analysing anatomy changes via comparing pre-treatment CT/CBCT images to the planning CT instead of corresponding dose distributions variations) and precise evaluation of needing plan adaptation. Crucially, the sub-second speed for individual pencil beam dose calculation and incorporation of energy dependence make our model well suited for real-time treatment adaptation.
To our knowledge DoTA is the first deep learning method to implicitly learn particle transport physics, predicting dose using only CT and beam energy as input, as opposed to previous works only learning corrections for 'cheap' physics based predictions or predicting dose under fixed conditions. The flexibility to incorporate additional beam characteristics, e.g., changes in beam shape (provided as 0th image slice) or energy spectrum (as the $\epsilon$ 0th token), or to predict additional quantities (e.g., particle flux) holds the potential to be a fast, general particle transport algorithm. Since photons (used in photon therapy and CT/CBCT imaging) and heavy ions (in carbon and helium therapy) share similar, mostly forward scatter physics, training our algorithm for different particles could open door to several further applications: e.g., predicting physical or radiobiological dose (requiring DNA scale simulations) in heavy ion treatments; or real-time CBCT image reconstruction to provide input for real-time adaptation. Attending to future information too could even allow learning large angle scatter physics crucial for electron therapy. Thus, the presented algorithm could significantly contribute to improving cancer treatments, having profound societal impact even on the short term.
\section{Acknowledgments}
The authors would like to thank Kevin Wielinga for his contributions to this project. This work is supported by KWF Kanker Bestrijding [grant number 11711], and is part of the KWF research project PAREL. Zolt\'an Perk\'o would like to thank the support of the NWO VENI grant ALLEGRO (016.Veni.198.055) during the time of this study.
\section*{Code availability}
The code, weights and results are publicly available at \url{https://github.com/} (to be released after the review process).
\section{CRediT authorship contribution statement.}
\textbf{Oscar Pastor-Serrano}: Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Data Curation, Writing – original draft, Visualization.
\noindent \textbf{Zolt\'an Perk\'o}: Conceptualization, Methodology, Formal Analysis, Resources, Writing – original draft, Writing – Review \& editing, Supervision, Project Administration, Funding Acquisition.
\small
\normalsize
\appendix
\section{Model hyperparameters}
\label{app:hyperparameter}
While the dimension of the inputs and outputs is fixed, the different choices of model hyperparameters offer great flexibility in the design DoTA's architecture and have an effect on the final accuracy. To build the best performing algorithm, we experiment with varying the number of transformer blocks $N$, the number of filters $K$ after the last layer in the convolutional encoder, and the number of attention heads $N_h$. Given the internal memory limitations of the GPU used in our experiments, we only report combinations that are compatible with a mini-batch size of 8 samples. \tabref{hs} shows the results of the hyperparameter search, showing the performance of all models on the test set, highlighting that the final combination of hyperparameters chosen based on the lowest validation set error indeed yields the lowest test set error, showcasing good generalization of the model.
\begin{table}[h!]
\centering
\caption{Model hyperparameter tuning experiment. A separate model is trained for each combination of transformer blocks $N$, convolutional filters $K$ and attention heads $N_h$, using the same learning rate and batch size.}
\begin{tabular}{@{}cccc@{}}
\toprule
\textbf{\begin{tabular}[c]{@{}c@{}}Transformer\\ blocks $N$\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Convolution\\ filters $K$\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Attention \\ heads $N_h$\end{tabular}} & \begin{tabular}[c]{@{}c@{}}\textbf{Test MSE} \\ {[}Gy/$10^9$ part.$]^2$\end{tabular} \\ \midrule
\textbf{1} & \textbf{10} & \textbf{16} & \textbf{0.0192} \\
1 & 16 & 16 & 0.0263 \\
2 & 8 & 8 & 0.0560 \\
2 & 8 & 16 & 0.0451 \\
2 & 10 & 8 & 0.0273 \\
2 & 16 & 8 & 0.0466 \\
2 & 16 & 16 & 0.0396 \\
4 & 8 & 8 & 0.0843 \\ \bottomrule
\end{tabular}
\label{tab:hs}
\end{table}
\section{Additional test geometries}
\label{app:geometries}
\figref{median} shows the accuracy that DoTA achieves in the two samples corresponding to the 2 gamma pass rates closest to the median gamma pass rate among the test set samples (99.5915\% and 99.592\% for the left and right of \figref{fig:median}, respectively). The predicted dose distributions closely follow the Monte Carlo ground truth, with only a handful of voxels being inaccurate. Additionally, to verify that the model learns the basic proton physics beyond lung patient geometries, we evaluate DoTA in a series of homogeneous volumes that are unseen during training. \figref{water} displays one such prediction for a 90 MeV mono-energetic pencil beam travelling through a water phantom, clearly demonstrating that the model implicitly captures physics and generalizes very well to simple, but unseen geometries entirely different from the training samples.
\begin{figure}
\caption{DoTA's predicted dose distribution in a uniform volume of water for a 90 MeV mono-energetic proton beam.}
\label{fig:water}
\end{figure}
\begin{figure*}
\caption{Predicted dose distribution and gamma values for the test samples corresponding to the 2 gamma pass rates closest to the median gamma pass rate.}
\label{fig:median}
\end{figure*}
\end{document} |
\begin{document}
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\title{Non-standard modalities\\in paraconsistent G\"{o}del logic\thanks{The research of Marta B\'ilkov\'a was supported by the grant 22-01137S of the Czech Science Foundation. The research of Sabine Frittella and Daniil Kozhemiachenko was funded by the grant ANR JCJC 2019, project PRELAP (ANR-19-CE48-0006). This research is part of the MOSAIC project financed by the European Union's Marie Sk\l{}odowska-Curie grant No.~101007627.}}
\titlerunning{Paraconsistent non-standard modalities}
\author{Marta B\'ilkov\'a\inst{1}\orcidID{0000-0002-3490-2083} \and Sabine Frittella\inst{2}\orcidID{0000-0003-4736-8614}\and Daniil Kozhemiachenko\inst{2}\orcidID{0000-0002-1533-8034}}
\authorrunning{B\'ilkov\'a et al.}
\institute{The Czech Academy of Sciences, Institute of Computer Science, Prague\\
\email{[email protected]}
\and
INSA Centre Val de Loire, Univ.\ Orl\'{e}ans, LIFO EA 4022, France\\
\email{[email protected], [email protected]}}
\maketitle
\begin{abstract}
We introduce a paraconsistent expansion of the G\"{o}del logic with a De Morgan negation $\neg$ and modalities $\blacksquare$ and $\blacklozenge$. We dub the logic $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ and equip it with Kripke semantics on frames with two (possibly fuzzy) relations: $R^+$ and $R^-$ (interpreted as the degree of trust in affirmations and denials by a given source) and valuations $v_1$ and $v_2$ (positive and negative support) ranging over $[0,1]$ and connected via $\neg$.
We motivate the semantics of $\blacksquare\psihi$ (resp., $\blacklozenge\psihi$) as infima (suprema) of both positive and negative supports of $\psihi$ in $R^+$- and $R^-$-accessible states, respectively. We then prove several instructive semantical properties of $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$. Finally, we devise a~tableaux system for $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ over finitely branching frames and establish the complexity of satisfiability and validity.
\keywords{G\"{o}del logic \and modal logic \and non-standard modalities \and constraint tableaux}
\end{abstract}
\section{Introduction\label{sec:introduction}}
When aggregating information from different sources, two of the simplest strategies are as follows: either one is sceptical and cautious regarding the information they provide thus requiring that they agree, or one is credulous and trusts their sources. In the classical setting, these two strategies can be modelled with $\Box$ and $\lozenge$ modalities defined on Kripke frames where states are sources, the accessibility relation represents references between them, and $w\vDash\psihi$ is construed as ‘$w$ says that $\psihi$ is true’. However, the sources can contradict themselves or be silent regarding a given question (as opposed to providing a~clear denial). Furthermore, a~source can provide a degree to their confirmation or denial. In all of these cases, classical logic struggles to formalise reasoning with such information.
\textbf{Paraconsistent reasoning about imperfect data}
In the situation described above, one can use the following setting. A~source $w$ gives a~statement $\psihi$ two valuations over $[0,1]$: $v_1$ standing for the degree with which $w$ \emph{asserts} $\psihi$ (positive support or support of truth) and $v_2$ for the degree of \emph{denial} (negative support or support of falsity). \emph{Classically}, $v_1(\psihi,w)+v_2(\psihi,w)=1$; if a source provides \emph{contradictory information}, then $v_1(\psihi,w)+v_2(\psihi,w)>1$; if the source provides \emph{insufficient information}, then $v_1(\psihi,w)+v_2(\psihi,w)<1$.
Now, if we account for the nonclassical information provided by the sources, the two aggregations described above can be formalised as follows. For the \emph{sceptical} case, the agent considers \emph{infima of positive and negative supports}. For the \emph{credulous aggregation}, one takes \emph{suprema of positive and negative supports}.
These two aggregation strategies were initially proposed and analysed in~\cite{BilkovaFrittellaMajerNazari2020}. There, however, they were described in a two-layered framework\varphiootnote{We refer our readers to~\cite{BaldiCintulaNoguera2020} and~\cite{BilkovaFrittellaKozhemiachenkoMajer2023IJAR} for an exposition of two-layered modal logics.} which prohibits the nesting of modalities. Furthermore, the Belnap--Dunn logic~\cite{Belnap2019} ($\textsf{BD}$) that lacks implication was chosen as the propositional fragment. In this paper, we extend that approach to the Kripke semantics to incorporate possible references between the sources and the sources' ability to give modalised statements. Furthermore, we use a paraconsistent expansion $\mathsf{G}^2$ from~\cite{BilkovaFrittellaKozhemiachenko2021} of G\"{o}del logic $\mathsf{G}$ as the propositional fragment.
\textbf{Formalising beliefs in modal expansions of $\mathsf{G}$}
When information is aggregated, the agent can further reason with it. For example, if one knows the degrees of certainty of two given statements, one can add them up, subtract them from one another, or compare them. In many contexts, however, an ordinary person does not represent their certainty in a given statement numerically and thus cannot conduct arithmetical operations with them. What they can do instead, is to \emph{compare} their certainty in one statement vs the other.
Thus, since G\"{o}del logic expresses order and comparisons but not arithmetic operations, it can be used as a propositional fragment of a modal logic formalising beliefs. For example, $\mathbf{K45}$ and $\mathbf{KD45}$ G\"{o}del logics can be used to formalise possibilistic reasoning since they are complete w.r.t.\ normalised and, respectively, non-normalised possibilistic frames~\cite{RodriguezTuytEstevaGodo2022}.
Furthermore, adding coimplication $\Yleft$ or, equivalently, Baaz' Delta operator $\triangle$ (cf.~\cite{Baaz1996} for details), results in bi-G\"{o}del (‘symmetric G\"{o}del’ in the terminology of~\cite{GrigoliaKiseliovaOdisharia2016}) logic that can additionally express strict order.
Modal expansions of $\mathsf{G}$ are well-studied. In particular, the Hilbert~\cite{CaicedoRodriguez2010} and Gentzen~\cite{MetcalfeOlivetti2009,MetcalfeOlivetti2011} formalisations of both $\Box$ and $\lozenge$ fragments of the modal logic $\mathfrak{GK}$~\varphiootnote{$\Box$ and $\lozenge$ are not interdefinable in $\mathfrak{GK}$.} are known. There are also complete axiomatisations for both fuzzy~\cite{CaicedoRodriguez2015} and crisp~\cite{RodriguezVidal2021} bi-modal G\"{o}del logics. It is known that they and some of their extensions are both decidable and $\psispace$ complete~\cite{CaicedoMetcalfeRodriguezRogger2013,CaicedoMetcalfeRodriguezRogger2017,DieguezFernandez-Duque2023} even though they lack finite model property.
Furthermore, it is known that the addition of $\Yleft$ or $\triangle$ as well as of a paraconsistent negation $\neg$ that swaps the supports of truth and falsity does not increase the complexity. Namely, satisfiability of $\mathbf{K}\mathsf{biG}$ and $\mathsf{GTL}$ (modal and temporal bi-G\"{o}del logics, respectively) (cf.~\cite{BilkovaFrittellaKozhemiachenko2022IJCAR,BilkovaFrittellaKozhemiachenko2022IGPLarxiv} for the former and~\cite{AguileraDieguezFernandez-DuqueMcLean2022} for the latter) as well as that of $\mathbf{K}\mathsf{G}^2$ (expansion of crisp $\mathfrak{GK}$ with $\neg$\varphiootnote{Note that in the presence of $\neg$, $\psihi\Yleft\psihi'$ is definable as $\neg(\neg\psihi'\rightarrow\neg\psihi)$.}) are in $\psispace$.
\textbf{This paper}
In this paper, we consider an expansion of $\mathsf{G}^2$ with modalities $\blacksquare$ and $\blacklozenge$ that stand for the cautious and credulous aggregation strategies. We equip them with Kripke semantics, construct a sound and complete tableaux calculus, and explore their semantical and computational properties. Our inspiration comes from two sources: modal expansions of G\"{o}del logics that we discussed above and modal expansions of Belnap--Dunn logic with Kripke semantics on bi-valued frames as studied by Priest~\cite{Priest2008FromIftoIs,Priest2008}, Odintsov and Wansing~\cite{OdintsovWansing2010,OdintsovWansing2017}, and others (cf.~\cite{Drobyshevich2020} and references therein to related work in the field). In a sense, $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ can be thought of as a hybrid between modal logics over $\textsf{BD}$
The remaining text is organised as follows. In Section~\ref{sec:language}, we define the language and semantics of $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$. Then, in Section~\ref{sec:definability} we show how to define several important frame classes, in particular, finitely branching frames. We also argue for the use of $\varphibinfoGsquare$ ($\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ over finitely branching frames) for the representation of agents' beliefs. In Section~\ref{sec:tableaux} we present a sound and complete tableaux calculus for $\varphibinfoGsquare$ and in Section~\ref{sec:complexity}, we use it to show that $\varphibinfoGsquare$ validity and satisfiability are $\psispace$ complete. Finally, in Section~\ref{sec:conclusion}, we wrap up the paper and provide a roadmap to future work.
\section{Logical preliminaries\label{sec:language}}
In this section, we provide semantics of $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ over both fuzzy and crisp frames. To make the presentation more approachable, we begin with bi-G\"{o}del algebras.
\begin{definition}\label{def:bi-G_algebra}
The bi-G\"{o}del algebra $[0,1]_{\mathsf{G}}=\langle[0,1],0,1,\wedge_\mathsf{G},\vee_\mathsf{G},\rightarrow_{\mathsf{G}},\Yleft\rangle$ is defined as follows: for all $a,b\in[0,1]$, we have $a\wedge_\mathsf{G}b=\min(a,b)$, $a\vee_\mathsf{G}b=\max(a,b)$. The remaining operations are defined below:
\begin{align*}
a\rightarrow_\mathsf{G}b&=
\begin{cases}
1,\text{ if }a\leq b\\
b\text{ else}
\end{cases}
&
a\Yleft_\mathsf{G}b&=
\begin{cases}
0,\text{ if }a\leq b\\
a\text{ else}
\end{cases}
\end{align*}
\end{definition}
We are now ready to define the language and semantics of $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$.
\begin{definition}\label{def:semantics}
We fix a countable set of propositional variables $\mathtt{Prop}$ and define the language via the following grammar.
\[\mathcal{L}^\neg_{\blacksquare,\blacklozenge}\ni\psihi\coloneqq p\in\mathtt{Prop}\mid\neg\psihi\mid(\psihi\!\wedge\!\psihi)\mid(\psihi\!\rightarrow\!\psihi)\mid\blacksquare\psihi\mid\blacklozenge\psihi\]
Constants $\mathbf{0}$ and $\mathbf{1}$, disjunction $\vee$, and coimplication $\Yleft$ as well as G\"{o}del negation ${\sim}$ can be defined as expected:
\begin{align*}
\mathbf{1}&\coloneqq p\!\rightarrow\!p&\mathbf{0}&\coloneqq\neg\mathbf{1}&{\sim}\psihi&\coloneqq\psihi\!\rightarrow\!\mathbf{0}&\psihi\!\vee\!\psihi'&\coloneqq\neg(\neg\psihi\!\wedge\!\neg\psihi')&\psihi\!\Yleft\!\psihi'&\coloneqq\neg(\neg\psihi'\!\!\rightarrow\!\!\neg\psihi)
\end{align*}
A \emph{fuzzy bi-relational frame is a tuple} $\mathfrak{F}=\langle W,R^+,R^-\rangle$ with $W\neq\varnothing$ and $R^+,R^-:W\times W\rightarrow[0,1]$. In a \emph{crisp frame}, $R^+,R^-:W\times W\rightarrow\{0,1\}$. A~\emph{model} is a~tuple $\mathfrak{M}=\langle W,R^+,R^-,v_1,v_2\rangle$ with $\langle W,R^+,R^-\rangle$ being a frame and $v_1,v_2:\mathtt{Prop}\rightarrow[0,1]$ that are extended to the complex formulas as follows.
\begin{center}
\begin{tabular}{rclrcl}
$v_1(\neg\psihi,w)$&$=$&$v_2(\psihi,w)$&$v_2(\neg\psihi,w)$&$=$&$v_1(\psihi,w)$\\
$v_1(\psihi\wedge\psihi',w)$&$=$&$v_1(\psihi,w)\wedge_\mathsf{G}v_1(\psihi',w)$&$v_2(\psihi\wedge\psihi',w)$&$=$&$v_2(\psihi,w)\vee_\mathsf{G}v_2(\psihi',w)$\\
$v_1(\psihi\rightarrow\psihi',w)$&$=$&$v_1(\psihi,w)\!\rightarrow_\mathsf{G}\!v_1(\psihi',w)$&$v_2(\psihi\rightarrow\psihi',w)$&$=$&$v_2(\psihi',w)\Yleft_\mathsf{G}v_2(\psihi,w)$\\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{rclrcl}
$v_1(\blacksquare\psihi,w)$&$=$&$\inf\limits_{w'\!\in\!W}\!\{wR^+w'\!\!\rightarrow_\mathsf{G}\!\!v_1(\psihi,w')\}$
&
$v_2(\blacksquare\psihi,w)$&$=$&$\inf\limits_{w'\!\in\!W}\!\{wR^-w'\!\!\rightarrow_\mathsf{G}\!\!v_2(\psihi,w')\}$\\
$v_1(\blacklozenge\psihi,w)$&$=$&$\sup\limits_{w'\!\in\!W}\!\{wR^+w'\!\wedge_\mathsf{G}\!v_1(\psihi,w')\}$
&
$v_2(\blacklozenge\psihi,w)$&$=$&$\sup\limits_{w'\!\in\!W}\!\{wR^-w'\!\wedge_\mathsf{G}\!v_2(\psihi,w')\}$
\end{tabular}
\end{center}
We will further write $v(\psihi,w)=(x,y)$ to designate that $v_1(\psihi,w)=x$ and $v_2(\psihi,w)=y$. Moreover, we set $S(w)=\{w':wSw'>0\}$.
We say that $\psihi$ is \emph{$v_1$-valid on $\mathfrak{F}$} ($\mathfrak{F}\models^+\psihi$) iff for every model $\mathfrak{M}$ on $\mathfrak{F}$ and every $w\in\mathfrak{M}$, it holds that $v_1(\psihi,w)=1$. $\psihi$ is \emph{$v_2$-valid on $\mathfrak{F}$} ($\mathfrak{F}\models^-\psihi$) iff for every model $\mathfrak{M}$ on $\mathfrak{F}$ and every $w\in\mathfrak{M}$, it holds that $v_2(\psihi,w)=0$. $\psihi$ is \emph{strongly valid on $\mathfrak{F}$} ($\mathfrak{F}\models\psihi$) iff it is $v_1$ and $v_2$-valid.
$\psihi$ is $v_1$ (resp., $v_2$, strongly) \emph{$\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ valid} iff it is $v_1$ (resp., $v_2$, strongly) valid on every frame. We will further use $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ to designate the set of all $\mathcal{L}^\neg_{\blacksquare,\blacklozenge}$ formulas \emph{strongly valid} on every frame.
\end{definition}
Observe in the definition above that the semantical conditions governing the support of truth of $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ connectives (except for $\neg$) coincide with the semantics of $\mathbf{K}\mathsf{biG}$ (cf.~\cite{BilkovaFrittellaKozhemiachenko2022IJCAR} for the detailed semantics of the latter).
\begin{example}\label{example:restaurant}
A tourist ($t$) wants to go to a restaurant and asks their two friends ($f_1$ and $f_2$) to describe their impressions regarding the politeness of the staff ($s$) and the quality of the desserts ($d$). Of course, the friends' opinions are not always internally consistent, nor is it always the case that one or the other even noticed whether the staff was polite or was eating desserts. Furthermore, $t$ trusts their friends to different degrees when it comes to their positive and negative opinions. The situation is depicted in Fig.~\ref{fig:restaurant}.
The first friend says that half of the staff was really nice but the other half is unwelcoming and rude and that the desserts (except for the tiramisu and souffl\'{e}) are tasty. The second friend, unfortunately, did not have the desserts at all. Furthermore, even though, they praised the staff, they also said that the manager was quite obnoxious.
The tourist now makes up their mind. If they are sceptical w.r.t.\ $s$ and $d$, they look for \emph{trusted rejections}\varphiootnote{We differentiate between a~\emph{rejection} which we treat as \emph{lack of support} and a~\emph{denial, disproof, refutation, counterexample}, etc.\ which we interpret as the \emph{negative support}.} of both positive and negative supports of $s$ and $d$. Thus $t$ uses the values of $R^+$ and $R^-$ as thresholds above which the information provided by the source does not count as a trusted enough rejection. In our case, we have $v(\blacksquare s,t)=(0.5,0.5)$ and $v(\blacksquare d,t)=(0,0)$. On the other hand, if $t$ is credulous, they look for \emph{trusted} confirmations \emph{of both positive and negative supports} and use $R^+$ and $R^-$ as thresholds up to which they accept the information provided by the source. Thus, we have $v(\blacklozenge s,t)=(0.7,0.4)$ and $v(\blacklozenge d,t)=(0.7,0.3)$.
\end{example}
\begin{figure}
\caption{$(x,y)$ stands for $wR^+w'=x,wR^-w'=y$. $R^+$ (resp., $R^-$) is interpreted as the tourist's threshold of trust in positive (negative) statements by the friends.}
\label{fig:restaurant}
\end{figure}
\noindent
\begin{minipage}{0.75\linewidth}
~\quad More formally, note that we can combine $v_1$ and $v_2$ into a~single valuation (denoted with $\bullet$) on the following bi-lattice on the right. Now, if we let $\sqcap$ and $\sqcup$ be the meet and join w.r.t.\ the rightward order, it is clear that $\blacksquare$ can be interpreted as an infinitary $\sqcap$ and $\blacklozenge$ as an infinitary $\sqcup$ across the accessible states, respectively.
\end{minipage}
\begin{minipage}{0.25\linewidth}
\resizebox{1\linewidth}{!}{
\begin{tikzpicture}[>=stealth,relative]
\node (U1) at (0,-2) {$(0,1)$};
\node (U2) at (-2,0) {$(0,0)$};
\node (U3) at (2,0) {$(1,1)$};
\node (U4) at (0,2) {$(1,0) $};
\node (U5) at (0.2,0.6) {$\bullet$};
\node (U6) at (0.2,0.4) {$(x,y)$};
\psiath[-,draw] (U1) to (U2);
\psiath[-,draw] (U1) to (U3);
\psiath[-,draw] (U2) to (U4);
\psiath[-,draw] (U3) to (U4);
\end{tikzpicture}}
\end{minipage}
From here, it is expected that $\blacksquare$ and $\blacklozenge$ are not normal in the following sense: $\blacksquare(p\wedge q)\leftrightarrow(\blacksquare p\wedge\blacksquare q)$, $\blacksquare\mathbf{1}$, $\blacklozenge(p\vee q)\leftrightarrow(\blacklozenge p\vee\blacklozenge q)$, and $\blacklozenge\mathbf{0}\leftrightarrow\mathbf{0}$ are not valid.
Finally, we have called $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ ‘paraconsistent’. In this paper, we consider the logic to be a set of valid formulas. It is clear that the explosion principle for $\rightarrow$ --- $(p\wedge\neg p)\rightarrow q$ --- is not valid. Furthermore, in contrast to $\mathbf{K}$, it is possible to believe in a~contradiction without \emph{believing in every statement}: $\blacklozenge(p\wedge\neg p)\rightarrow\blacklozenge q$ and $\blacksquare(p\wedge\neg p)\rightarrow\blacksquare q$ are not valid.
We end the section by proving that $\blacklozenge$ and $\blacksquare$ \emph{are not interdefinable}.
\begin{theorem}\label{theorem:nondefinability}
$\blacksquare$ and $\blacklozenge$ are not interdefinable.
\end{theorem}
\begin{proof}
Denote with $\mathcal{L}_\blacksquare$ and $\mathcal{L}_\blacklozenge$ the $\blacklozenge$- and $\blacksquare$-free fragments of $\mathcal{L}^\neg_{\blacksquare,\blacklozenge}$. We build a pointed model $\langle\mathfrak{M},w\rangle$ s.t.\ there is no $\blacklozenge$-free formula that has the same value at $w$ as $\blacksquare p$ (and vice versa). Consider Fig.~\ref{fig:nondefinability}.
\begin{figure}
\caption{All variables have the same values in all states exemplified by $p$. $R^+=R^-$, $v(\blacksquare p,w_0)=\left(\varphirac{1}
\label{fig:nondefinability}
\end{figure}
One can check by induction that if $\psihi\in\mathcal{L}^\neg_{\blacksquare,\blacklozenge}$, then
\begin{align*}
v(\psihi,w_1)&\in\left\{(0;1),\left(\varphirac{1}{2};\varphirac{2}{3}\right),\left(\varphirac{2}{3};\varphirac{1}{2}\right),(0;0),(1;1),(1;0)\right\}
\end{align*}
\begin{align*}
v(\psihi,w_2)&\in\left\{(0;1),\left(\varphirac{1}{4};\varphirac{1}{3}\right),\left(\varphirac{1}{3};\varphirac{1}{4}\right),(0;0),(1;1),(1;0)\right\}
\end{align*}
Moreover, on the single-point irreflexive frame whose only state is $u$, it holds for every $\psihi(p)\in\mathcal{L}^\neg_{\blacksquare,\blacklozenge}$, $v(\psihi,u)\in\{v(p,u),v(\neg p,u),(1,0),(1,1),(0,0),(0,1)\}$.
Thus, for every $\blacklozenge$-free $\chi$ and every $\blacksquare$-free $\psisi$ it holds that
\begin{align*}
v(\blacksquare\chi,w_0)&\in\left\{(0;1),\left(\varphirac{1}{3};\varphirac{1}{4}\right),\left(\varphirac{1}{4};\varphirac{1}{3}\right),(0;0),(1;1),(1;0)\right\}=X
\end{align*}
\begin{align*}
v(\blacklozenge\psisi,w_0)&\in\left\{(0;1),\left(\varphirac{1}{2};\varphirac{2}{3}\right),\left(\varphirac{2}{3};\varphirac{1}{2}\right),(0;0),(1;1),(1;0)\right\}=Y
\end{align*}
Since $X$ and $Y$ are closed w.r.t.\ propositional operations, it is now easy to check by induction that for every $\chi'\in\mathcal{L}_\blacksquare$ and $\psisi'\in\mathcal{L}_\blacklozenge$, $v(\chi',w_0)\in X$ and $v(\psisi',w_0)\in Y$.
\end{proof}
\section{Frame definability\label{sec:definability}}
In this section, we explore some classes of frames that can be defined in $\mathcal{L}^\neg_{\blacksquare,\blacklozenge}$. However, since $\blacksquare$ and $\blacklozenge$ are non-normal and since we have two independent relations on frames, we expand the traditional notion of modal definability.
\begin{definition}\label{def:+-definability}
\begin{enumerate}[noitemsep,topsep=2pt]
\item[]
\item $\psihi$ \emph{positively defines} a class of frames $\mathbb{F}$ iff for every $\mathfrak{F}$, it holds that \emph{$\mathfrak{F}\models^+\psihi$ iff $\mathfrak{F}\in\mathbb{F}$}.
\item $\psihi$ \emph{negatively defines} a class of frames $\mathbb{F}$ iff for every $\mathfrak{F}$, every $w\in\mathfrak{F}$, it holds that \emph{$\mathfrak{F}\models^-\psihi$ iff $\mathfrak{F}\in\mathbb{F}$}.
\item $\psihi$ \emph{(strongly) defines} a class of frames $\mathbb{F}$ iff for every $\mathfrak{F}$, it holds that $\mathfrak{F}\in\mathbb{F}$ iff $\mathfrak{F}\models\psihi$.
\end{enumerate}
\end{definition}
With the help of the above definition, we can show that every class of frames definable in $\mathbf{K}\mathsf{biG}$ is \emph{positively definable} in $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$.
\begin{definition}\label{def:+-framecounterparts}
Let $\mathfrak{F}=\langle W,S\rangle$ be a (fuzzy or crisp) frame.
\begin{enumerate}[noitemsep,topsep=2pt]
\item An \emph{$R^+$-counterpart of $\mathfrak{F}$} is any bi-relational frame $\mathfrak{F}^+=\langle W,S,R^-\rangle$.
\item An \emph{$R^-$-counterpart of $\mathfrak{F}$} is any bi-relational frame $\mathfrak{F}^+=\langle W,R^+,S\rangle$.
\end{enumerate}
\end{definition}
\begin{convention}\label{conv:blackcounterpart}
Let $\psihi$ be over $\{\wedge,\vee,\rightarrow,\Yleft,\Box,\lozenge\}$.
\begin{enumerate}[noitemsep,topsep=2pt]
\item We denote with $\psihi^{+\bullet}$ the formula obtained from $\psihi$ by replacing all $\Box$'s and $\lozenge$'s with $\blacksquare$'s and $\blacklozenge$'s.
\item We denote with $\psihi^{-\bullet}$ the formula obtained from $\psihi$ by replacing all $\Box$'s and $\lozenge$'s with $\neg\blacksquare\neg$'s and $\neg\blacklozenge\neg$'s.
\end{enumerate}
\end{convention}
\begin{theorem}\label{theorem:blackcounterparts}
Let $\mathfrak{F}=\langle W,S\rangle$ and let $\mathfrak{F}^+$ and $\mathfrak{F}^-$ be its $R^+$ and $R^-$ counterparts. Then, for any $\psihi$ be over $\{\wedge,\vee,\rightarrow,\Yleft,\Box,\lozenge\}$, it holds that
\[\mathfrak{F}\models_{\mathbf{K}\mathsf{biG}}\psihi\quad\text{iff}\quad\mathfrak{F}^+\models^+\psihi^{+\bullet}\quad\text{iff}\quad\mathfrak{F}^-\models^+\psihi^{-\bullet}\]
\end{theorem}
\begin{proof}
Since the semantics of $\mathbf{K}\mathsf{biG}$ connectives is identical to $v_1$ conditions of Definition~\ref{def:semantics}, we only prove that $\mathfrak{F}\models\psihi$ iff $\mathfrak{F}^-\models^+\psihi^{-\bullet}$. It suffices to prove by induction the following statement.
\begin{center}
\emph{Let $\mathbf{v}$ be a $\mathbf{K}\mathsf{biG}$ valuation on $\mathfrak{F}$, $\mathbf{v}(p,w)=v_1(p,w)$ for every $w\in\mathfrak{F}$, and $v_2$ be arbitrary. Then $\mathbf{v}(\psihi,w)=v_1(\psihi^{-\bullet},w)$ for every $\psihi$.}
\end{center}
The case of $\psihi=p$ holds by Convention~\ref{conv:blackcounterpart}, the cases of propositional connectives are straightforward. Consider $\psihi=\Box\chi$. We have that $\psihi^{-\bullet}=\neg\blacksquare\neg(\chi^{-\bullet})$ and thus
\begin{align*}
v_1(\neg\blacksquare\neg(\chi^{-\bullet}),w)&=v_2(\blacksquare\neg(\chi^{-\bullet}),w)\\
&=\inf\limits_{w'\in W}\{wSw'\rightarrow_\mathsf{G}v_2(\neg(\chi^{-\bullet}))\}\\
&=\inf\limits_{w'\in W}\{wSw'\rightarrow_\mathsf{G}v_1(\chi^{-\bullet})\}\\
&=\inf\limits_{w'\in W}\{wSw'\rightarrow_\mathsf{G}\mathbf{v}(\chi)\}\tag{by IH}\\
&=\mathbf{v}(\Box\chi,w)
\end{align*}
\end{proof}
The above theorem allows us to \emph{positively} define in $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ all classes of frames that are definable in $\mathbf{K}\mathsf{biG}$. In particular, all $\mathbf{K}$-definable frames are positively definable. Moreover, it follows that $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ (as $\mathfrak{GK}$ and $\mathbf{K}\mathsf{biG}$) lacks the finite model property: ${\sim}\Box(p\vee{\sim}p)$ is false on every finite frame, and thus, ${\sim}\blacksquare(p\vee{\sim}p)$ is too. On the other hand, there are infinite models satisfying this formula as shown below ($R^+$ and $R^-$ are crisp).
\[\xymatrix{w_1:p=\left(\varphirac{1}{2},0\right)&\ldots&w_n:p=\left(\varphirac{1}{n+1},0\right)&\ldots\\&w_0:p=(0,0)\ar[ul]|{+}\ar[ur]|{+}\ar[urr]|{+}\ar@(d,l)|{-}&&}\]
Furthermore, Theorem~\ref{theorem:blackcounterparts} gives us a degree of flexibility. For example, one can check that $\neg\blacksquare\neg(p\vee q)\rightarrow(\neg\blacksquare\neg p\vee\neg\blacklozenge\neg q)$ positively defines frames with crisp $R^-$ but not necessarily crisp $R^+$. This models a situation when an agent \emph{completely (dis)believes} in denials given by their sources while may have some degree of trust between $0$ and $1$ when the sources assert something. Let us return to Example~\ref{example:restaurant}.
\begin{example}\label{example:illnesscontinued}
Assume that the tourist \emph{completely trusts} the negative (but not positive) opinions of their friends. Thus, instead of Fig.~\ref{fig:restaurant}, we have the following model.
\[\xymatrix{f_1:\txt{$s=(0.5,0.5)$\\$d=(0.7,0.3)$}~&&~t~\ar[rr]^(.3){(0.7,1)}\ar[ll]_(.3){(0.8,1)}&&~f_2:\txt{$s=(1,0.4)$\\$d=(0,0)$}}\]
The new values for the cautious and credulous aggregation are as follows: $v(\blacksquare s,t)=(0.5,0.4)$, $v(\blacksquare d,t)=(0,0)$, $v(\blacklozenge s,t)=(0.7,0.5)$, and $v(\blacklozenge d,t)=(0.7,0.3)$.
\end{example}
Furthermore, the agent can trust the sources to the same degree no matter whether they confirm or deny statements. This can be modelled with \emph{mono-relational} frames where $R^+\!=\!R^-$. We show that they are \emph{strongly definable}.
\begin{theorem}\label{theorem:1relational}
$\mathfrak{F}$ is mono-relational iff $\mathfrak{F}\models\blacksquare\neg p\leftrightarrow\neg\blacksquare p$ and $\mathfrak{F}\models\blacklozenge\neg p\leftrightarrow\neg\blacklozenge p$.
\end{theorem}
\begin{proof}
Let $\mathfrak{F}$ be mono-relational and $R^+=R^-=R$. Now observe that
\begin{align*}
v_i(\blacksquare\neg p,w)&=\inf\limits_{w'\in W}\{wRw'\rightarrow_\mathsf{G}v_i(\neg p,w')\}\tag{$i\in\{1,2\}$}\\
&=\inf\limits_{w'\in W}\{wRw'\rightarrow_\mathsf{G}v_j(p,w')\}\tag{$i\neq j$}\\
&=v_j(\blacksquare p,w)\\
&=v_i(\neg\blacksquare p,w)
\end{align*}
For the converse, let $R^+\!\neq\!R^-$ and, in particular, $wR^+w'\!=\!x$ and $wR^-w'\!=\!y$. Assume w.l.o.g.\ that $x>y$. We set the valuation of $p$: $v(p,w')=(x,y)$ and for every $w''\neq w'$, we have $v(p,w'')=(1,1)$. It is clear that $v(\neg\blacksquare p,w)=(1,1)$. On the other hand, $v(\neg p,w')=(y,x)$, whence $v_1(\blacksquare\neg p)\neq1$.
The case of $\blacklozenge$ can be tackled in a dual manner.
\end{proof}
In the remainder of the paper, we will be concerned with $\varphibinfoGsquare$ --- $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ over finitely branching (both fuzzy and crisp) frames. This is for several reasons. First, in the context of formalising beliefs and reasoning with data acquired from sources, it is reasonable to assume that every source refers to only a finite number of other sources and that agents have access to a finite number of sources as well. This assumption is implicit in many classical epistemic and doxastic logics since they are often complete w.r.t.\ finitely branching models~\cite{FaginHalpernMosesVardi2003}, although cannot \emph{define} them. Second, in the finitely branching models, the values of modal formulas are \emph{witnessed}: if $v_i(\blacksquare\psihi,w)=x<1$, then, $v_i(\psihi,w')=x$ for some $w'$, and if $v_i(\blacklozenge\psihi,w)=x$, then $wRw'=x$ or $v_i(\psihi,w')=x$ for some $w'$. Intuitively, this means that the degree of $w$'s certainty in $\psihi$ is purely based on the information acquired from sources and from its degree of trust in those. Finally, the restriction to finitely branching frames allows for the construction of a simple constraint tableaux calculus that can be used in establishing the complexity valuation.
\section{Tableaux calculus\label{sec:tableaux}}
In this section, we construct a sound and complete constraint tableaux system $\mathcal{T}\left(\varphibinfoGsquare\right)$ for $\varphibinfoGsquare$. The first constraint tableaux were proposed in~\cite{Haehnle1992,Haehnle1994,Haehnle1999} as a decision procedure for the \L{}ukasiewicz logic $\Luk$. A similar approach for the Rational Pawe\l{}ka logic was proposed in~\cite{diLascioGisolfi2005}. In~\cite{BilkovaFrittellaKozhemiachenko2021}, we constructed constraint tableaux for $\Luk^2$ and $\mathsf{G}^2$ --- the paraconsistent expansions of $\Luk$ and $\mathsf{G}$, and in~\cite{BilkovaFrittellaKozhemiachenko2022IJCAR} for modal expansions of the bi-G\"{o}del logic and $\mathsf{G}^2$.
Constraint tableaux are \emph{analytic} in the sense that their rules have subformula property. Moreover, they provide an easy way of the countermodel extraction from complete open branches. Furthermore, while the propositional connectives of $\mathsf{G}^2$ allow for the construction of an analytic proof system, e.g., a~display calculus extending that of $\mathsf{I}_4\mathsf{C}_4$\varphiootnote{This logic was introduced several times: in~\cite{Wansing2008}, then in~\cite{Leitgeb2019}, and further studied in~\cite{OdintsovWansing2021}. It is, in fact, the propositional fragment of Moisil's modal logic~\cite{Moisil1942}. We are grateful to Heinrich Wansing who pointed this out to us.}~\cite{Wansing2008}, the modal ones are not dual to one another w.r.t.\ $\neg$ nor the G\"{o}del negation ${\sim}$. Thus, it is unlikely that an elegant (hyper-)sequent or display calculus for $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ or $\varphibinfoGsquare$ can be constructed.
The next definitions are adapted from~\cite{BilkovaFrittellaKozhemiachenko2022IJCAR}.
\begin{definition}\label{def:TfbinfoGsquare}
We fix a set of state-labels $\mathsf{W}$ and let $\lesssim\in\!\{<,\leqslant\}$ and $\gtrsim\in\!\{>,\geqslant\}$. Let further $w\!\in\!\mathsf{W}$, $\mathbf{x}\!\in\!\{1,2\}$, $\psihi\!\in\!\mathcal{L}^\neg_{\blacksquare,\blacklozenge}$, and $c\!\in\!\{0,1\}$. A~\emph{structure} is either $w\!:\!\mathbf{x}\!:\!\psihi$, $c$, $w\mathsf{R}^+w'$, or $w\mathsf{R}^+w'$. We denote the set of structures with $\mathsf{Str}$. Structures of the form $w\!:\!\mathbf{x}\!:\!p$, $w\mathsf{R}^+w'$, and $w\mathsf{R}^-w'$ are called \emph{atomic} (denoted $\mathsf{AStr}$).
We define a \emph{constraint tableau} as a downward branching tree whose branches are sets containing constraints $\mathfrak{X}\lesssim\mathfrak{X'}$ ($\mathfrak{X},\mathfrak{X'}\in\mathsf{Str}$). Each branch can be extended by an application of a~rule\varphiootnote{If $\mathfrak{X}\!<\!1,\mathfrak{X}\!<\!\mathfrak{X}'\!\in\!\mathcal{B}$ or $0\!<\!\mathfrak{X}',\mathfrak{X}\!<\!\mathfrak{X}'\!\in\!\mathcal{B}$, the rules are applied only to $\mathfrak{X}\!<\!\mathfrak{X}'$.} below (bars denote branching, $i,j\in\{1,2\}$, $i\neq j$).
\[\scriptsize{\begin{array}{cccc}
\neg_i\!\lesssim\!\dfrac{w\!:\!i\!:\!\neg\psihi\!\lesssim\!\mathfrak{X}}{w\!:\!j\!:\!\psihi\!\lesssim\!\mathfrak{X}}
&
\neg_i\!\gtrsim\!\dfrac{w\!:\!i\!:\!\neg\psihi\!\gtrsim\!\mathfrak{X}}{w\!:\!j\!:\!\psihi\!\gtrsim\!\mathfrak{X}}
&
\rightarrow_1\!\leqslant\!\dfrac{w\!:\!1\!:\!\psihi\!\rightarrow\!\psihi'\!\leqslant\!\mathfrak{X}}{\mathfrak{X}\!\geqslant\!{1}\left|\begin{matrix}\mathfrak{X}\!<\!{1}\\w\!:\!1\!:\!\psihi'\!\leqslant\!\mathfrak{X}\\w\!:\!1\!:\!\psihi\!>\!w\!:\!1\!:\!\psihi'\end{matrix}\right.}
&
\rightarrow_2\!\geqslant\!\dfrac{w\!:\!2\!:\!\psihi\rightarrow\psihi'\!\geqslant\!\mathfrak{X}}{\mathfrak{X}\!\leqslant\!{0}\left|\begin{matrix}\mathfrak{X}\!>\!{0}\\w\!:\!2\!:\!\psihi'\!\geqslant\!\mathfrak{X}\\w\!:\!2\!:\!\psihi'\!>\!w\!:\!2\!:\!\psihi\end{matrix}\right.}
\end{array}}\]
\[\scriptsize{\begin{array}{cccc}
\wedge_1\!\gtrsim\!\dfrac{w\!:\!1\!:\!\psihi\!\wedge\!\psihi'\!\gtrsim\!\mathfrak{X}}{\begin{matrix}w\!:\!1\!:\!\psihi\!\gtrsim\!\mathfrak{X}\\w\!:\!1\!:\!\psihi'\!\gtrsim\!\mathfrak{X}\end{matrix}}
&
\wedge_2\!\lesssim\!\dfrac{w\!:\!2\!:\!\psihi\!\wedge\!\psihi'\!\lesssim\!\mathfrak{X}}{\begin{matrix}w\!:\!2\!:\!\psihi\!\lesssim\!\mathfrak{X}\\w\!:\!2\!:\!\psihi'\!\lesssim\!\mathfrak{X}\end{matrix}}
&
\rightarrow_1\!<\!\dfrac{w\!:\!1\!:\!\psihi\rightarrow\psihi'\!<\!\mathfrak{X}}{\begin{matrix}w\!:\!1\!:\!\psihi'\!<\!\mathfrak{X}\\w\!:\!1\!:\!\psihi\!>\!w\!:\!1\!:\!\psihi'\end{matrix}}
&
\rightarrow_2\!>\!\dfrac{w\!:\!2\!:\!\psihi\rightarrow\psihi'\!>\!\mathfrak{X}}{\begin{matrix}w\!:\!2\!:\!\psihi'\!>\!\mathfrak{X}\\w\!:\!2\!:\!\psihi'\!>\!w\!:\!2\!:\!\psihi\end{matrix}}
\end{array}}\]
\[\scriptsize{\begin{array}{cc}
\wedge_1\!\lesssim\!\dfrac{w\!:\!1\!:\!\psihi\wedge\psihi'\!\lesssim\!\mathfrak{X}}{w\!:\!1\!:\!\psihi\!\lesssim\!\mathfrak{X}\mid w\!:\!1\!:\!\psihi'\!\lesssim\!\mathfrak{X}}
&\quad
\wedge_2\!\gtrsim\!\dfrac{w\!:\!2\!:\!\psihi\wedge\psihi'\!\gtrsim\!\mathfrak{X}}{w\!:\!2\!:\!\psihi\!\gtrsim\!\mathfrak{X}\mid w\!:\!2\!:\!\psihi'\!\gtrsim\!\mathfrak{X}}
\end{array}}\]
\[\scriptsize{\begin{array}{cc}
\rightarrow_1\!\gtrsim\!\dfrac{w\!:\!1\!:\!\psihi\!\rightarrow\!\psihi'\!\gtrsim\!\mathfrak{X}}{w\!:\!1\!:\!\psihi\!\leqslant\!w\!:\!1\!:\!\psihi'\mid w\!:\!1\!:\!\psihi'\!\gtrsim\!\mathfrak{X}}&\rightarrow_2\!\lesssim\!\dfrac{w\!:\!2\!:\!\psihi\rightarrow\psihi'\!\lesssim\!\mathfrak{X}}{w\!:\!2\!:\!\psihi'\!\leqslant\!w\!:\!2\!:\!\psihi\mid w\!:\!2\!:\!\psihi'\!\lesssim\!\mathfrak{X}}
\end{array}}\]
\[\scriptsize{\begin{array}{ccc}
\blacksquare_i\!\!\gtrsim\!\dfrac{w\!:\!i\!:\!\blacksquare\psihi\!\gtrsim\!\mathfrak{X}}{w'\!:\!i\!:\!\psihi\gtrsim\mathfrak{X}\mid w\mathsf{S}w'\!\leqslant\!w'\!:\!i\!:\!\psihi}
&\quad
\blacksquare_i\!\!\leqslant\!\dfrac{w\!:\!i\!:\!\blacksquare\psihi\!\leqslant\!\mathfrak{X}}{\mathfrak{X}\geqslant1\left|\begin{matrix}\mathfrak{X}\!<\!1\\w\mathsf{S}w''\!>\!w''\!:\!i\!:\!\psihi\\w''\!:\!:\!i\!:\!\psihi\leqslant\mathfrak{X}\end{matrix}\right.}
&\quad
\blacksquare_i\!\!<\!\dfrac{w\!:\!i\!:\!\blacksquare\psihi\!<\!\mathfrak{X}}{\begin{matrix}w\mathsf{S}w''\!>\!w''\!:\!i\!:\!\psihi\\w''\!:\!:\!i\!:\!\psihi\!<\!\mathfrak{X}\end{matrix}}
\end{array}}\]
\[\scriptsize{\begin{array}{ccc}
\blacklozenge_i\!\!\gtrsim\!\dfrac{w\!:\!i\!:\!\blacklozenge\psihi\!\gtrsim\!\mathfrak{X}}{\begin{matrix}w\mathsf{S}w''\!\gtrsim\!\mathfrak{X}\\w''\!:\!i\!:\!\psihi\!\gtrsim\!\mathfrak{X}\end{matrix}}
&\quad
\blacklozenge_i\!\!\lesssim\!\dfrac{w\!:\!i\!:\!\blacklozenge\psihi\!\lesssim\!\mathfrak{X}}{w'\!:\!i\!:\!\psihi\lesssim\mathfrak{X}\mid w\mathsf{S}w'\!\lesssim\!\mathfrak{X}}&\quad
\left[\begin{matrix}
w''\text{ is fresh on the branch}\\
\text{if }i\!=\!1,\text{ then }\mathsf{S}\!=\!\mathsf{R}^+\\\text{if }i\!=\!2,\text{ then }\mathsf{S}\!=\!\mathsf{R}^-\\\text{in }\blacksquare_i\!\gtrsim,\blacklozenge_i\!\lesssim~w\mathsf{S}w'\text{ occurs on the branch}
\end{matrix}\right]
\end{array}}\]
A tableau's branch $\mathcal{B}$ is \emph{closed} iff one of the following conditions applies:
\begin{itemize}[noitemsep,topsep=2pt]
\item the transitive closure of $\mathcal{B}$ under $\lesssim$ contains $\mathfrak{X}<\mathfrak{X}$;
\item ${0}\geqslant{1}\in\mathcal{B}$, or $\mathfrak{X}>{1}\in\mathcal{B}$, or $\mathfrak{X}<{0}\in\mathcal{B}$.
\end{itemize}
A tableau is \emph{closed} iff all its branches are closed. We say that there is a \emph{tableau proof} of $\psihi$ iff there are closed tableaux starting from $w\!:\!1\!:\!\psihi<1$ and $w\!:\!2\!:\!\psihi>0$.
An open branch $\mathcal{B}$ is \emph{complete} iff the following condition is met.
\begin{itemize}[noitemsep,topsep=2pt]
\item[$*$]\emph{If all premises of a rule occur on $\mathcal{B}$, then its one conclusion\varphiootnote{Note that branching rules have \emph{two} conclusions.} occurs on~$\mathcal{B}$.}
\end{itemize}
\end{definition}
\begin{convention}\label{conv:TG2meaning}
The table below summarises the interpretations of entries.
\begin{center}
\begin{tabular}{c|c}
\textbf{entry}&\textbf{interpretation}\\\hline
$w\!:1\!:\!\psihi\leqslant w'\!:2\!:\!\psihi'$&$v_1(\psihi,w)\leq v_2(\psihi',w')$\\
$w\!:\!2\!:\!\psihi\leqslant c$&$v_2(\psihi,w)\leq c$ with $c\in\{0,1\}$\\
$w\mathsf{R}^-w'\leqslant w'\!:2\!:\!\psihi$&$wR^-w'\leq v_2(\psihi,w')$
\end{tabular}
\end{center}
\end{convention}
\begin{definition}[Branch realisation]\label{G2branchsatisfaction}
A model $\mathfrak{M}=\langle W,R^+,R^-,v_1,v_2\rangle$ with $W=\{w:w\text{ occurs on }\mathcal{B}\}$ \emph{realises a~branch $\mathcal{B}$} of a tableau iff
there is a function $\mathsf{rl}:\mathsf{Str}\rightarrow[0,1]$ s.t.\ for every $\mathfrak{X},\mathfrak{Y},\mathfrak{Y}',\mathfrak{Z},\mathfrak{Z}'\in\mathsf{Str}$ with $\mathfrak{X}=w:\mathbf{x}:\psihi$, $\mathfrak{Y}=w_i\mathsf{R}^+w_j$, and $\mathfrak{Y}'=w'_i\mathsf{R}^-w'_j$ the following holds ($\mathbf{x}\in\{1,2\}$, ${c}\in\{0,1\}$).
\begin{itemize}[noitemsep,topsep=2pt]
\item If $\mathfrak{Z}\lesssim\mathfrak{Z}'\in\mathcal{B}$, then $\mathsf{rl}(\mathfrak{Z})\lesssim\mathsf{rl}(\mathfrak{Z}')$.
\item $\mathsf{rl}(\mathfrak{X})=v_\mathbf{x}(\psihi,w)$, $\mathsf{rl}(c)=c$, $\mathsf{rl}(\mathfrak{Y})=w_iR^+w_j$, $\mathsf{rl}(\mathfrak{Y}')=w'_iR^-w'_j$
\end{itemize}
\end{definition}
To facilitate the understanding of the rules, we give an example of a~failed tableau proof and extract a~coun\-ter-mo\-del. The proof goes as follows: first, we apply all the possible propositional rules, then the modal rules that introduce new states, and then those that use the states already on the branch. We repeat the process until all structures are decomposed into atomic ones.
\begin{center}
\begin{minipage}{.45\linewidth}
\scriptsize{
\begin{forest}
smullyan tableaux
[w_0\!:\!2\!:\!\neg\blacksquare p\!\rightarrow\!\blacksquare\neg p\!>\!0
[w_0\!:\!2\!:\!\neg\blacksquare p\!<\!w_0\!:\!2\!:\!\blacksquare\neg p
[0\!<\!w_0\!:\!2\!:\!\blacksquare\neg p
[w_0\!:\!1\!:\!\blacksquare p\!<\!w_0\!:\!2\!:\!\blacksquare\neg p
[w_0\mathsf{R}^+w_1\!>\!w_1\!:\!1\!:\!p
[w_1\!:\!1\!:\!p\!<\!w_0\!:\!2\!:\!\blacksquare\neg p
[w_1\!:\!2\!:\!\neg p\!>\!w_1\!:\!1\!:\!p[w_1\!:\!1\!:\!p\!>\!w_1\!:\!1\!:\!p,closed]]
[w_0\mathsf{R}^-w_1\leqslant w_1\!:\!2\!:\!\neg p[w_0\mathsf{R}^-w_1\leqslant w_1\!:\!1\!:\!p[\varphirownie]]]
]]]]]]
\end{forest}}
\end{minipage}
\begin{minipage}{.45\linewidth}
\[\xymatrix{w_0\ar@/^1pc/[rr]|{R^+=1}\ar@/_1pc/[rr]|{R^-=\varphirac{1}{2}}&&w_1:p=\left(\varphirac{1}{2},0\right)}\]
\end{minipage}
\end{center}
We can now extract a model from the complete open branch marked with $\varphirownie$ s.t.\ $v_2(\neg\blacksquare p\!\rightarrow\!\blacksquare\neg p,w_0)>0$. We use $w$'s that occur thereon as the carrier and assign the values of variables and relations so that they correspond to $\lesssim$.
\begin{theorem}[$\mathcal{T}\!\left(\fbinfoGsquare\right)$ completeness]\label{theorem:TinfoG2completeness}
$\psihi$ is \emph{strongly} valid in $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ iff there is a tableau proof of $\psihi$.
\end{theorem}
\begin{proof}
The proof is an easy adaptation of~\cite[Theorem~3]{BilkovaFrittellaKozhemiachenko2022IJCAR}, whence we provide only a sketch thereof. The skipped steps can be seen in Section~\ref{subsec:completenessproof}.
To prove soundness, we need to show that if the premise of the rule is realised, then so is at least one of its conclusions. This can be done by a routine check of the rules. Note that since we work with finitely branching frames, infima and suprema from Definition~\ref{def:semantics} become maxima and minima. Since closed branches are not realisable, the result follows.
To prove completeness, we show that every complete open branch $\mathcal{B}$ is realisable. We show how to construct a realising model from the branch. First, we set $W=\{w:w\text{ occurs in }\mathcal{B}\}$. Denote the set of atomic structures appearing on $\mathcal{B}$ with $\mathsf{AStr}(\mathcal{B})$ and let $\mathcal{B}^+$ be the transitive closure of $\mathcal{B}$ under $\lesssim$. Now, we assign values to them. For $i\in\{1,2\}$, if $w\!:\!i\!:\!p\geqslant1\in\mathcal{B}$, we set $v_i(p,w)=1$. If $w\!:\!i\!:\!p\leqslant0\in\mathcal{B}$, we set $v_i(p,w)=0$. If $w\mathsf{S}w'<\mathfrak{X}\notin\mathcal{B}^+$, we set $w\mathsf{S}w'=1$. If $w\!:\!i\!:\!p$ or $w\mathsf{S}w'$ does not occur on $\mathcal{B}$, we set $v_i(p,w)=0$ and $w\mathsf{S}w'=0$.
For each $\mathsf{str}\in\mathsf{AStr}$, we now set
\[[\mathsf{str}]\!=\!\left\{\mathsf{str}'\left| \; \begin{matrix}\mathsf{str}\leqslant\mathsf{str}'\in\mathcal{B}^+\text{ and }\mathsf{str}<\mathsf{str}
\notin\mathcal{B}^+\\
\text{or}\\
\mathsf{str}\geqslant\mathsf{str}'\in\mathcal{B}^+\text{ and }\mathsf{str}>\mathsf{str}'\notin\mathcal{B}^+
\end{matrix}\right.\right\}\]
Denote the number of $[\mathsf{str}]$'s with $\#^\mathsf{str}$. Since the only possible loop in $\mathcal{B}^+$ is $\mathsf{str}\leqslant\mathsf{str}'\leqslant\ldots\leqslant\mathsf{str}$ where all elements belong to $[\mathsf{str}]$, it is clear that $\#^\mathsf{str}\leq2\cdot|\mathsf{AStr}(\mathcal{B})|\cdot|W|$. Put $[\mathsf{str}]\psirec[\mathsf{str}']$ iff there are $\mathsf{str}_i\in[\mathsf{str}]$ and $\mathsf{str}_j\in[\mathsf{str}']$ s.t.\ $\mathsf{str}_i<\mathsf{str}_j\in\mathcal{B}^+$. We now set the valuation of these structures as follows:
\begin{align*}
\mathsf{str}=\dfrac{|\{[\mathsf{str}']\mid[\mathsf{str}']\psirec[\mathsf{str}]\}|}{\#^\mathsf{str}}
\end{align*}
It is clear that constraints containing only atomic structures and constants are now satisfied. To show that all other constraints are satisfied, we prove that if at least one conclusion of the rule is satisfied, then so is the premise. Again, the proof is a slight modification of~\cite[Theorem~3]{BilkovaFrittellaKozhemiachenko2022IJCAR} and can be done by considering the cases of rules (the details are in Section~\ref{subsec:completenessproof}).
\end{proof}
\section{Complexity\label{sec:complexity}}
In this section, we use the tableaux to provide the upper bound on the size of falsifying (satisfying) models and prove that satisfiability and validity\varphiootnote{Satisfiability and falsifiability (non-validity) are reducible to each other: $\psihi$ is satisfiable iff ${\sim\sim}(\psihi\Yleft\mathbf{0})$ is falsifiable; $\psihi$ is falsifiable iff ${\sim\sim}(\mathbf{1}\Yleft\psihi)$ is satisfiable.} of $\varphibinfoGsquare$ are $\psispace$ complete.
The following statement follows immediately from Theorem~\ref{theorem:TinfoG2completeness}.
\begin{corollary}\label{cor:FMP}
Let $\psihi\in\mathcal{L}^\neg_{\blacksquare,\blacklozenge}$ be \emph{not $\varphibinfoGsquare$ valid}, and let $k$ be the number of modalities in it. Then there is a model $\mathfrak{M}$ of the size $\leq k^{k+1}$ and depth $\leq k$ and $w\in\mathfrak{M}$ s.t.\ $v_1(\psihi,w)\neq1$ or $v_2(\psihi,w)\neq0$.
\end{corollary}
\begin{proof}
In Section~\ref{subsec:FMPproof}.
\end{proof}
We can now prove the $\psispace$ completeness result. The proof of $\psispace$ membership adapts the method from~\cite{BilkovaFrittellaKozhemiachenko2022IJCAR} and is inspired by the proof of the $\psispace$ membership of $\mathbf{K}$ from~\cite{BlackburndeRijkeVenema2010}. For the hardness part, we reduce the validity in $\mathbf{K}$ to $v_1$ and $v_2$ validities. We provide a sketch of the proof (the skipped steps are given in Section~\ref{subsec:PSPACEproof}).
\begin{theorem}\label{theorem:infoG2PSPACE}
$\varphibinfoGsquare$ validity and satisfiability are $\psispace$ complete.
\end{theorem}
\begin{proof}
For the membership, observe from the proof of Theorem~\ref{theorem:TinfoG2completeness} that $\psihi$ is satisfiable (falsifiable) on $\mathfrak{M}=\langle W,R^+,R^-,v_1,v_2\rangle$ iff all variables, $w\mathsf{R}^+w'$'s, and $w\mathsf{R}^-w'$'s have values from $\mathsf{V}=\left\{0,\varphirac{1}{\#^\mathsf{str}},\ldots,\varphirac{\#^\mathsf{str}-1}{\#^\mathsf{str}},1\right\}$ under which $\psihi$ is satisfied (falsified).
Since $\#^\mathsf{str}$ is bounded from above, we can now replace constraints with labelled formulas and relational structures of the form $w\!:\!i\!:\!\psihi\!=\!\mathsf{v}$ or $w\mathsf{S}w'\!=\!\mathsf{v}$ ($\mathsf{v}\in\mathsf{V}$) avoiding comparisons of values of formulas in different states. We close the branch if it contains $w\!:\!i\!:\!\psisi\!=\!\mathsf{v}$ and $w\!:\!i\!:\!\psisi\!=\!\mathsf{v}'$ for $\mathsf{v}\!\neq\!\mathsf{v}'$.
Now we replace the rules from Definition~\ref{def:TfbinfoGsquare} with new ones that work with labelled structures. Below, we give as an example the rules\varphiootnote{For a value $\mathsf{v}>0$ of $\blacklozenge\psihi$ at $w$, we add a new state that witnesses $\mathsf{v}$, and for a~state on the branch, we guess a~value smaller than $\mathsf{v}$. Other modal rules can be rewritten similarly.} that replace $\blacklozenge_i\!\!\lesssim$.
\begin{center}
\scriptsize{\begin{align*}
\dfrac{w\!:\!i\!:\!\blacklozenge\psihi\!=\!\varphirac{r}{\#^\mathsf{str}}}{\left.\begin{matrix}w\mathsf{S}w'\!=\!1\\w\!:\!i\!:\!\psihi\!=\!\varphirac{r}{\#^\mathsf{str}}\end{matrix}\right|\left.\begin{matrix}w\mathsf{S}w'\!=\!\varphirac{r}{\#^\mathsf{str}}\\w\!:\!i\!:\!\psihi\!=\!1\end{matrix}\right|\ldots\left|\begin{matrix}w\mathsf{S}w'\!=\!\varphirac{r}{\#^\mathsf{str}}\\w\!:\!i\!:\!\psihi\!=\!\varphirac{r}{\#^\mathsf{str}}\end{matrix}\right.}
&&
\dfrac{w\!:\!i\!:\!\blacklozenge\psihi\!=\!\varphirac{r}{\#^\mathsf{str}};(w\mathsf{S}w'\text{ occurs on the branch})}{w'\!:\!i\!:\!\psihi\!=\!\varphirac{r-1}{\#^\mathsf{str}}\mid w\mathsf{S}w'\!=\!\varphirac{r-1}{\#^\mathsf{str}}\mid\ldots\mid w'\!:\!i\!:\!\psihi\!=\!0}\end{align*}}
\end{center}
Observe that once all rules are rewritten in this manner, we will not need to compare values of formulas \emph{in different states}.
We then proceed as follows: first, we apply the propositional rules, then \emph{one} modal rule requiring a~new state (e.g., $w_0\!:\!i\!:\!\blacklozenge\psihi\!=\!\varphirac{r}{\#^\mathsf{str}}$), then the rules that use that state guessing the tableau branch when needed. By repeating this process, we are building \emph{the model branch by branch}. The model has the depth bounded by the length of $\psihi$ and we work with modal formulas one by one, whence we need to store subformulas of $\psihi$ and $w\mathsf{S}w'$'s with their values $O(|\psihi|)$ times, so, we need only $O(|\psihi|^2)$ space. Once the branch is constructed, we can delete the entries of the tableau and repeat the process with the next formula at $w_0$ that would introduce a new state.
For hardness, we reduce the $\mathbf{K}$ validity of $\{\mathbf{0},\wedge,\vee,\rightarrow,\Box,\lozenge\}$ formulas to $v_1$-validity and $v_2$-validity in $\varphibinfoGsquare$. For the reduction to $v_1$-validity, we use the idea from~\cite[Theorem~21]{CaicedoMetcalfeRodriguezRogger2017}. Namely, given $\psihi$, we denote with $\psihi^\triangledown$ the formula whose every subformula is prenexed with ${\sim\sim}$ and where $\Box$ and $\lozenge$ are replaced with $\blacksquare$ and $\blacklozenge$. Since semantics for the G\"{o}del modal logic and for the positive support ($v_1$ valuations, Definition~\ref{def:semantics}) coincide, the result follows.
For the reduction to $v_2$-validity, we take $\psihi$ and inductively define $\psihi^\psiartial$:
\begin{align*}
p^\psiartial&=\mathbf{1}\Yleft(\mathbf{1}\Yleft p)\\
(\chi\circ\psisi)^\psiartial&=\chi^\psiartial\bullet\psisi^\psiartial\tag{$\circ,\bullet\in\{\wedge,\vee\}$, $\circ\neq\bullet$}\\
(\chi\rightarrow\psisi)^\psiartial&=\psisi^\psiartial\Yleft\chi^\psiartial\\
(\Box\chi)^\psiartial&=\blacksquare(\chi^\psiartial)\\
(\lozenge\chi)^\psiartial&=\blacklozenge(\chi^\psiartial)
\end{align*}
One can check by induction that for every \emph{crisp} finitely branching $\mathfrak{F}$ and every \emph{classical} valuation $\mathbf{v}$ thereon, it holds that $\mathfrak{F},\mathbf{v},w\vDash\psihi$ iff $v_2(\mathbf{1}\Yleft\psihi^\psiartial,w)=0$ and $\mathfrak{F},\mathbf{v},w\nvDash\psihi$ iff $v_2(\mathbf{1}\Yleft\psihi^\psiartial,w)=1$ provided that $v_2=\mathbf{v}$.
For the converse, let $\mathfrak{M}=\langle W,R^+,R^-,v_1,v_2\rangle$ be a $\varphibinfoGsquare$ model. Let $\mathfrak{M}^!=\langle W,R^!,v^!\rangle$ be s.t.\ $wR^!w'$ iff $wR^-w'=1$ and $w\in v^!(p)$ iff $v_2(p,w)=1$. Again, it is easy to verify that for every $\mathfrak{M}$, $v_2(\psihi^\psiartial,w)=1$ iff $\mathfrak{M}^!,w\vDash\psihi$.
It follows that $\psihi$ is $\mathbf{K}$-valid iff $\mathbf{1}\Yleft\psihi^\psiartial$ is $v_2$-valid.
\end{proof}
\section{Conclusions and future work\label{sec:conclusion}}
We presented a modal expansion $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ of $\mathsf{G}^2$ with non-normal modalities and provided it with Kripke semantics on bi-relational frames with two valuations. We established its connection with the bi-G\"{o}del modal logic $\mathbf{K}\mathsf{biG}$ presented in~\cite{BilkovaFrittellaKozhemiachenko2022IJCAR,BilkovaFrittellaKozhemiachenko2022IGPLarxiv} and obtained decidability and complexity results considering $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ over finitely branching frames.
The next steps are as follows. First of all, we plan to explore the decidability of the full $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ logic. We conjecture that it is also $\psispace$ complete. However, the standard way of proving $\psispace$ completeness of G\"{o}del modal logics described in~\cite{CaicedoMetcalfeRodriguezRogger2013,CaicedoMetcalfeRodriguezRogger2017} and used in~\cite{BilkovaFrittellaKozhemiachenko2022IGPLarxiv} to establish $\psispace$ completeness of $\mathbf{K}\mathsf{biG}$ may not be straightforwardly applicable here as the reduction from $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ validity to $\mathbf{K}\mathsf{biG}$ validity can be hard to obtain for it follows immediately from Theorem~\ref{theorem:1relational} that $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ lacks negation normal forms.
Second, it is interesting to design a complete Hilbert-style axiomatisation of $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ and study its correspondence theory w.r.t.\ \emph{strong validity}. This can be non-trivial since $\blacksquare(p\rightarrow q)\rightarrow(\blacksquare p\rightarrow\blacksquare q)$ and $\blacklozenge(p\vee q)\rightarrow\blacklozenge p\vee\blacklozenge q$ are not $\mathsf{G}^{2\pm}_{\blacksquare,\blacklozenge}$ valid, even though, it is easy to check that the following rules are sound.
\begin{align*}
\dfrac{\psihi\rightarrow\chi}{\blacksquare\psihi\rightarrow\blacksquare\chi}&&\dfrac{\psihi\rightarrow\chi}{\blacklozenge\psihi\rightarrow\blacklozenge\chi}
\end{align*}
The other direction of future research is to study global versions of $\blacksquare$ and $\blacklozenge$ as well as description logics based on them. Description G\"{o}del logics are well-known and studied~\cite{BobilloDelgadoGomez-RamiroStraccia2009,BobilloDelgadoGomez-RamiroStraccia2012} and allow for the representation of uncertain data that cannot be represented in the classical ontologies. Furthermore, they are the only decidable family of fuzzy description logics which contrasts them to e.g., \L{}ukasiewicz description (and global) logics which are not even axiomatisable~\cite{Vidal2021}. On the other hand, there are known description logics over $\textsf{BD}$ (cf., e.g.~\cite{MaHitzlerLin2007}), and thus it makes sense to combine the two approaches.
\appendix
\section{Proofs\label{sec:longproofs}}
\subsection{Proof of Theorem~\ref{theorem:TinfoG2completeness}\label{subsec:completenessproof}}
We fill in the gaps in the sketch. First, we prove the soundness result. Since propositional rules are exactly the same as in $\mathcal{T}\left(\varphibKGsquare\right)$~\cite{BilkovaFrittellaKozhemiachenko2022IJCAR}, we consider only the most interesting cases of modal rules. We tackle $\blacksquare_1\!\!\gtrsim$ and $\blacklozenge_2\!\!\gtrsim$ (cf.\ Definition~\ref{def:TfbinfoGsquare}) and show that in each case, if $\mathfrak{M}=\langle W,R^+,R^-,v_1,v_2\rangle$ realises the premise of the rule, it also realises one of its conclusions.
We begin with $\blacksquare_1\!\!\gtrsim$, assume w.l.o.g.\ that $\mathfrak{X}=w''\!:\!2\!:\!\psisi$, and let $\mathfrak{M}$ realise $w\!:\!1\!:\!\blacksquare\psihi\geqslant w''\!:\!2\!:\!\psisi$. Now, since $R^+$ and $R^-$ are finitely branching, we have that $\min\limits_{w'\in W}\{w\mathsf{R}^+w'\rightarrow_\mathsf{G}v_1(\psihi,w')\}\geq v_2(\psisi,w)$, whence at each $w'\in W$ s.t.\ $wR^+w'>0$\varphiootnote{Recall that if $u\mathsf{S}u'\notin\mathcal{B}$, we set $u\mathsf{S}u'=0$.}, either $v_1(\psihi,w')\geq v_2(\psisi,w'')$ or $w\mathsf{R}^+w'\geq v_2(\psisi,w'')$. Thus, at least one conclusion of the rule is satisfied.
For $\blacklozenge_2\!\!\gtrsim$ we proceed similarly. Let $\mathfrak{M}$ realise $w\!:\!1\!:\!\blacklozenge\psihi\geqslant w''\!:\!2\!:\!\psisi$. Again, by the finite branching, we have that $\min\limits_{w'\in W}\{wR^+w'\wedge_\mathsf{G}v_1(\psihi,w')\}$. Hence, there is some fresh $w'\in W$ s.t.\ $wR^+w',v_1(\psihi,w')\geq v_2(\psisi,w'')$. Thus, the conclusion of the rule is satisfied, as desired.
For completeness, we reason by contraposition. We show by induction on formulas that every complete open branch is realised. The case of atomic constraints holds by the construction of the realising model (recall the proof of Theorem~\ref{theorem:TinfoG2completeness}). We show that other constraints are satisfied. For that, we prove that if at least one conclusion of the rule is satisfied, then so is the premise. The propositional cases are straightforward and can be tackled in the same manner as in~\cite[Theorem~2]{BilkovaFrittellaKozhemiachenko2021}. We consider only the cases of $\blacklozenge_2\!\gtrsim$ and $\blacksquare_1\!\!\gtrsim$ and assume w.l.o.g.\ that $\mathfrak{X}=w''\!:\!2\!:\!\psisi$.
For $\blacksquare_1\!\!\gtrsim$, assume that for every $w'$ s.t.\ $w\mathsf{R}^+w'$ is on the branch, either $w'\!:\!1\!:\!\psihi\geqslant w''\!:\!2\!:\!\psisi$ or $w\mathsf{R}^+w'\leqslant w'\!:\!1\!:\!\psihi$ is realisable. Thus, by the inductive hypothesis, for every $w'\in R^+(w)$, it holds that $v_1(\psihi,w')\geq v_2(\psisi,w'')$ or $wR^+w'\leq v_1(\psihi,w')$. Hence, $v_1(\blacksquare\psihi,w)\geq v_2(\psisi,w'')$ and $w\!:\!1\!:\!\blacksquare\psihi\geqslant w''\!:\!2\!:\!\psisi$ is realised.
For $\blacklozenge_2\!\gtrsim$, let $w\mathsf{R}^-w''\geqslant w''\!:\!2\!:\!\psisi$ and $w'\!:\!1\!:\!\psihi\geqslant w''\!:\!2\!:\!\psisi$ be realised for some $w''\in R(w)$. By the induction hypothesis, we have that $w\mathsf{R}^-w'',v_2(\psihi,w')\geq v_2(\psisi,w'')$, whence, $v_2(\blacklozenge\psihi,w)\geq v_2(\psisi,w'')$ and thus, $w\!:\!2\!:\!\blacklozenge\psihi\geqslant w''\!:\!2\!:\!\psisi$.
Other rules can be considered similarly.
\subsection{Proof of Corollary~\ref{cor:FMP}\label{subsec:FMPproof}}
By theorem~\ref{theorem:TinfoG2completeness}, if $\psihi$ is \emph{not $\varphibinfoGsquare$ valid}, we can build a~falsifying model using tableaux. It is also clear from the rules in Definition~\ref{def:TfbinfoGsquare} that the depth of the constructed model is bounded from above by the maximal number of nested modalities in $\psihi$. The width of the model is bounded by the maximal number of modalities on the same level of nesting.
\subsection{Proof of Theorem~\ref{theorem:infoG2PSPACE}\label{subsec:PSPACEproof}}
We provide the decision algorithm that utilises the rewritten rules. The algorithm is essentially the same as in~\cite{BilkovaFrittellaKozhemiachenko2022IJCAR}. Note also that it is possible to use the original calculus as a decision procedure, although it is not optimal.
Let us show how to build a satisfying model for $\psihi$ using polynomial space. We begin with $w_0\!:\!1\!:\psihi\!=\!1$ (the algorithm for $w_0\!:\!1\!:\psihi\!=\!0$ is the same) and start applying propositional rules (first, those that do not require branching). If we implement a branching rule, we pick one branch and work only with it: either until the branch is closed, in which case we pick another one; until no more rules are applicable (then, the model is constructed); or until we need to apply a modal rule to proceed. At this stage, we need to store only the subformulas of $\psihi$ with labels denoting their value at~$w_0$.
Now we guess a~modal formula (say, $w_0\!:\!2\!:\!\blacklozenge\chi\!=\!\varphirac{1}{\#^\mathsf{str}}$) whose decomposition requires an introduction of a~new state ($w_1$) and apply this rule. Then we apply all modal rules whose implementation requires that $w_0\mathsf{R}^-w_1$ occur on the branch (again, if those require branching, we guess only one branch) and start from the beginning with the propositional rules. If we reach a contradiction, the branch is closed. Again, the only new entries to store are subformulas of $\psihi$ (now, with fewer modalities), their values at $w_1$, and a~relational term $w_0\mathsf{R}^-w_1$ with its value. Since the depth of the model is $O(|\psihi|)$ and since we work with modal formulas one by one, we need to store subformulas of $\psihi$ with their values $O(|\psihi|)$ times, so, we need only $O(|\psihi|^2)$ space.
Finally, if no rule is applicable and there is no contradiction, we mark $w_0\!:\!2\!:\!\blacklozenge\chi\!=\!\varphirac{1}{\#^\mathsf{str}}$ as ‘safe’. Now we \emph{delete all entries of the tableau below it} and pick another unmarked modal formula that requires an introduction of a new state. Dealing with these one by one allows us to construct the model branch by branch. But since the length of each branch of the model is bounded by $O(|\psihi|)$ and since we delete \emph{branches of the model} once they are shown to contain no contradictions, we need only polynomial space.
\end{document} |
\begin{document}
\title{Recoil-ion momentum spectroscopy of photoionization of cold rubidium atoms in a strong laser field}
\author{Renyuan Li}
\affiliation{Shanghai Advanced Research Institute, Chinese Academy of Sciences, 201210 Shanghai, China}
\affiliation{University of Chinese Academy of Sciences, 100049 Beijing, China}
\author{Junyang Yuan}
\affiliation{Shanghai Advanced Research Institute, Chinese Academy of Sciences, 201210 Shanghai, China}
\affiliation{University of Chinese Academy of Sciences, 100049 Beijing, China}
\affiliation{School of Physical Science and Technology, ShanghaiTech University, 201210 Shanghai, China}
\author{Xinya Hou}
\affiliation{College of Physics and Materials Science, Henan Normal University, 453007 Xinxiang, China}
\author{Shuai Zhang}
\affiliation{School of Physical Science and Technology, ShanghaiTech University, 201210 Shanghai, China}
\author{Zhiyuan Zhu}
\affiliation{Shanghai Advanced Research Institute, Chinese Academy of Sciences, 201210 Shanghai, China}
\affiliation{School of Physical Science and Technology, ShanghaiTech University, 201210 Shanghai, China}
\author{Yixuan Ma}
\affiliation{Shanghai Advanced Research Institute, Chinese Academy of Sciences, 201210 Shanghai, China}
\affiliation{School of Physical Science and Technology, ShanghaiTech University, 201210 Shanghai, China}
\author{Qi Gao}
\affiliation{Shanghai Advanced Research Institute, Chinese Academy of Sciences, 201210 Shanghai, China}
\author{Zhongyang Wang}
\affiliation{Shanghai Advanced Research Institute, Chinese Academy of Sciences, 201210 Shanghai, China}
\affiliation{School of Physical Science and Technology, ShanghaiTech University, 201210 Shanghai, China}
\author{T.-M. Yan}
\affiliation{Shanghai Advanced Research Institute, Chinese Academy of Sciences, 201210 Shanghai, China}
\author{Chaochao Qin}
\affiliation{College of Physics and Materials Science, Henan Normal University, 453007 Xinxiang, China}
\author{Yizhu Zhang}
\email{[email protected]}
\affiliation{Shanghai Advanced Research Institute, Chinese Academy of Sciences, 201210 Shanghai, China}
\affiliation{Center for Terahertz Waves and College of Precision Instrument and Optoelectronics Engineering, Key Laboratory of Education, Tianjin University, 300072 Tianjin, China}
\author{Xincheng Wang}
\email{[email protected]}
\affiliation{School of Physical Science and Technology, ShanghaiTech University, 201210 Shanghai, China}
\author{Matthias Weidem{\"u}ller}
\affiliation{Hefei National Laboratory for Physical Sciences at the Microscale and Shanghai Branch, University of Science and Technology of China, Shanghai 201315, China}
\affiliation{CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China}
\affiliation{Physikalisches Institut, Universit{\"a}t Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg,Germany}
\author{Y.H. Jiang}
\email{[email protected]}
\affiliation{Shanghai Advanced Research Institute, Chinese Academy of Sciences, 201210 Shanghai, China}
\affiliation{University of Chinese Academy of Sciences, 100049 Beijing, China}
\affiliation{School of Physical Science and Technology, ShanghaiTech University, 201210 Shanghai, China}
\affiliation{CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China}
\date{\today}
\begin{abstract}
We study photoionization of cold rubidium atoms in a strong infrared laser field using a magneto-optical trap (MOT) recoil ion momentum spectrometer. Three types of cold rubidium target are provided, operating in two-dimension (2D) MOT, 2D molasses, and 3D MOT with densities in the orders of $10^7$ atoms/cm$^3$, $10^8$ atoms/cm$^3$, and $10^9$ atoms/cm$^3$, respectively. The density profile and the temperature of 3D MOT are characterized using the absorption imaging and photoionization. The momentum distributions of Rb$^+$ created by absorption of two- or three-photon illuminate a dipole-like double-peak structure, in good agreement with the results in the strong field approximation. The yielding momentum resolution of $0.12 \pm 0.03$ a.u. is achieved in comparison with theoretical calculations, exhibiting the great prospects for the study of electron correlations in alkali metal atoms through interaction with strong laser pulses.
\end{abstract}
\pacs{}
\maketitle
\section{\label{sec:Introduction,level1}Introduction}
In the past few decades, the COLTRIMS (cold target recoil ion momentum spectroscopy) \cite{0034-4885-66-9-203,DORNER200095}, which enables measurement of recoil ion momentum in $4\pi$ solid angle with a high resolution for investigations of dynamical interaction between atoms/molecules and various projectiles, became one of the most fruitful experimental techniques in atomic, molecular and optical physics. In the COLTRIMS, the targets are limited to the species of atoms/molecules in gas phase and volatilizable liquid, normally prepared by a supersonic jet. The usage of COLTRIMS-like apparatus is obstructive for alkali and alkaline-earth atoms due to their solid phase at room temperature. The heating of alkali atoms to gas phase gives rise to the undesired broadening of ionic momentum distributions due to severe thermal motion, leading to a few ten times lower resolution than commonly-used COLTRIMS.
Meanwhile, the hydrogen-like structure and low first ionization energy of alkali atoms open a new dimension to understand the strong-field ionization dynamics. For instance, time-resolved holography of photo-electrons reveals rich intra- and inter-cycle interferences \cite{Huismans61}, where metastable xenon atoms were prepared. Instead of noble gas, alkali atoms are ideal targets for the study of photoelectron interferences, particularly for intermediate-involved states in the far- and mid-infrared laser fields since alkali atoms can be easily prepared on various initial states with lasers. Besides, many amazing physical phenomena in the strong laser field, for instance, high harmonic generation (HHG) \cite{RevModPhys.81.163,KOHLER2012159}, above threshold ionization \cite{BECKER200235,PhysRevLett.110.013001}, tunneling ionization \cite{RN80,Eckle1525,PhysRevLett.118.143203}, and sequential and nonsequential double ionizations \cite{PhysRevLett.48.1814,PhysRevLett.99.263003,RevModPhys.84.1011,PhysRevLett.112.073002,PhysRevLett.113.103001,PhysRevLett.115.123001}, are still hot topics in the noble gas. However, these studies are still lacking due to highly technical challenges for momentum distribution detection of alkali recoil ion. On the other hand, alkali atoms with the extremely low first ionization potential provide new sights for understanding the mechanisms mentioned above. Furthermore, for HHG \cite{PhysRevLett.84.2822,PhysRevLett.94.113906,PhysRevLett.114.143902}, Kramers-Henneberger transformation \cite{Morales16906,WEI2017240}, and nonlinear ionization \cite{RN79}, alkali atom with respective to noble gas has been found to be of unusual features in the strong laser field.
The magneto-optical trap (MOT) technique was introduced to the COLTRIMS, called as magneto-optical trap recoil ion momentum spectroscopy (MOTRIMS) \cite{DEPAOLA2008139}, for investigations of interesting issues mentioned above with unprecedented recoil-ion momentum resolution. Recently, the rubidium (Rb) atom target was cooled down to a temperature of 200 $\mu$K \cite{doi:10.1063/1.4795475}, that is almost five orders lower than supersonic expansion target with similar atomic mass. The photoassociation of ultracold Rb${_2}$ was thus observed and optimized with the pulse-shaping technique of femtosecond laser pulses \cite{PhysRevLett.100.233003,doi:10.1063/1.4738643,doi:10.1063/1.4795475}. In addition, MOTRIMS had been employed for studies of photoionization dynamics of lithium (Li) atom in the free electron laser and infrared strong laser field \cite{schuricke_strong-field_2011,PhysRevLett.103.103008,PhysRevA.83.023413}, and of the few-body dynamics of Li atom in the ion collision \cite{PhysRevLett.109.113202}.
In this article, a MOTRIMS of Rb atoms for strong-field studies is introduced. Three types of targets, i.e. 3D (three-dimensional) MOT, 2D molasses\cite{PhysRevLett.55.48} and 2D MOT, are prepared to provide targets with various densities and different initial states, allowing to adapt to tunable intensities and different wavelengths of strong-field lasers. The 2D MOT target can be further cooled with six laser beams to form 2D molasses. The 2D molasses give higher density and the mixture of Rb(5s) and Rb(5p) as the initial states. The 2D molasses can be further trapped with a magnetic quadrupole field to form 3D MOT, which has the highest density. Similar to that of 2D molasses, the initial states of 3D MOT are also a mixture of Rb(5s) and Rb(5p).The setup of Rb MOTRIMS apparatus is depicted in the following sections, and the temperature, density and momentum resolution of three types are characterized.
\section{\label{sec:Setup,level1}Experimental setup}
As indicated in Fig. \ref{fig:setup}, the setup mainly consists of four parts, dubbed as laser system, 2D MOT preparation region \cite{WEYERS199730,PhysRevA.80.013409,PhysRevA.66.023410,PhysRevLett.77.3331,PhysRevA.58.3891,PhysRevA.73.033415}, target region and recoil ion momentum spectrometer \cite{doi:10.1063/1.1775310,doi:10.1063/1.2994151}. Briefly, Rb atoms are pre-cooled in the 2D MOT preparation region. Then they are pushed into the main chamber, where they can be further cooled to form 2D molasses or 3D MOT target. The target is intersected with a femtosecond laser at the main chamber. The ions generated during the interaction are then extracted and detected by recoil ion momentum spectrometer.
The 2D MOT preparation region sits on top of the main chamber. A Rb dispenser is installed inside a glass cell, which can be heated to evaporate with a tunable current source. Before loading into the main chamber, the Rb atoms are pre-cooled in the glass cell by the cooling laser and the Rb density can be controlled by the heating current. Cooling laser beams with red detuning are split into four equal groups. The 2D MOT preparation part is separated from the main chamber via a differential tube, and the typical vacuum for the 2D MOT and main chamber are about $1 \times 10^{-8}$ mbar and $5 \times 10^{-10}$ mbar, respectively.
The target region is located at the main chamber, where the Rb atoms can be further cooled down by three pairs of orthogonal cooling lasers (2D molasses), with optional magnetic quadrupole field for trapping (3D MOT). A femtosecond laser of 800 nm was used to ionize the target, which offers a tool for characterization of targets and for further scientific investigations. Besides, the rubidium beam from 2D MOT preparation can also be used directly as a target for experiments of high ionization rate, or when the effect of the excited states have to be carefully considered
The recoil ions generated at the target region are extracted and detected by a standard RIMS (recoil ion momentum spectroscopy) method. MCP (micro channel plate) together with delay-line anode detector are used to record the arriving time and positions of each recoil ion, by which momentum vector and kinetic energy of recoil ions can be reconstructed.
\begin{figure}
\caption{\label{fig:setup}
\label{fig:setup}
\end{figure}
\subsection{\label{sec:laser,level2}Laser system}
A laser system is required to provide beams for cooling, pushing, imaging and repumping of Rb atoms. The lasers with the center wavelength of 780 nm for repumping and cooling, pushing, and imaging are locked on two different crossovers. Since the methodologies are quite similar, only one schematic design of the laser is presented in Fig. \ref{fig:laserSystem}.
Laser beam on the left side of Fig. \ref{fig:laserSystem} passing a Rb vapor cell is used for Doppler-free saturation spectroscopy \cite{doi:10.1119/1.18457} to stabilize the laser frequency. Acoustic-optical modulators (AOM) in a double-pass configuration shift the frequencies by $2 \times 80$ MHz for cooling, pushing and imaging beams, and by $2 \times 110$ MHz for repumping beam, where the laser beams are locked on the $F' = 3/4$ crossover (cooling, pushing and imaging) and $F' = 2/3$ crossover (repumping). With this kind of optical design the working frequency of the laser beams on the right side can be stabilized.
The branch on the right side is a typical arrangement for cooling, pushing and imaging lasers with different detuning by AOM with the double-pass configuration. The imaging beam with a power of about 20 $\mu$W is set on the resonant transition of $|5\text{S}_{1/2}, F = 3, m_F = 3 \rangle \to |5\text{P}_{3/2}, F'=4, m_{F'} = 4 \rangle$. It is used in an absorption imaging experiment to measure the temperature and the density of 3D MOT target, described in detail in Section \ref{sec:Characterization,level1}, and whereas the pushing beam with a power of 0.2 mW is red-detuned by $-$2 $\times$ 12.5 MHz. A power of 190 mW in total (130 mW for the four groups of 2D MOT preparation and 60 mW for three pairs of beams of 2D molasses or 3D MOT) with red-detuning of $\delta$ = $-$2$\Gamma$ is needed. $\Gamma$ = 6 MHz is the natural linewidth of 5s5p, which determines the doppler temperature of about 150 $\mu$K.
Besides, the repumping beam with a power of 16 mW in total (12 mW for 2D MOT preparation and 4 mW for 3D moalsses or 3D MOT) is on resonant of the $|5\text{S}_{1/2}, F = 2, m_F = 2 \rangle \to |5\text{P}_{3/2}, F'=3, m_{F'} = 3\rangle$ transition. It can pump the Rb atoms back into the cooling cycle, avoiding accumulation of the dark state \cite{PhysRevA.53.1702,PhysRevLett.70.2253}. The output laser beams are then transported by polarization-maintained single mode optical fibers to the 2D MOT preparation and target region.
\begin{figure}
\caption{\label{fig:laserSystem}
\label{fig:laserSystem}
\end{figure}
\subsection{\label{sec:2D,level2}2D MOT preparation}
The 2D MOT preparation part is used to provide a pre-cooled Rb beam for target region operation. To provide such an atom beam, normally, Zeeman slower and 2D MOT are adopted. However, comparing with 2D MOT, a Zeeman slower has two main disadvantages. One problem is that the Zeeman slower has cooling effect only in one direction, thus inferring a large spread in transverse velocities. On the other hand, for Zeeman slower, either the laser is on resonance or the magnetic field has to be non-zero at the end.
This means that at the target region, the 3D MOT for example, may be disturbed either by the resonant light or by the magnetic field of Zeeman slower \cite{doi:10.1063/1.1820524}. Here, we choose to use 2D MOT for preparation of a pre-cooled Rb beam leading to much more compact in the system geometry structure.
The Rb atoms generated in the glass cell are cooled with cooling lasers in two perpendicular directions associated with a quadrupole magnetic field, thus dubbed as 2D MOT. The cooling directions of $z$ and $y$ are perpendicular to the pushing beam direction as indicated in Fig. \ref{fig:setup}. To increase the target density in the target zone, four identical cooling regions along the pushing beam direction are designed with a total length of 80 mm. The cooling and repumping lasers mentioned above are split into four beams accordingly by PBS (polarization beam splitter) with a 1/e$^2$ diameter of 20 mm. On the other hand, the quadrupole magnetic field can be generated by a pair of anti-aligned permanent magnet. In our setup, six pairs of such magnets are aligned along the pushing beam, covering all the four cooling regions. Each piece of magnet can be adjusted independently to align the four regions. All the optical fiber couplers of the lasers and permanent magnets are fixed on a metal cage which is mounted around the glass cell.
After the 2D cooling in the glass cell, the Rb atoms will be pushed into the target region through a differential tube with diameter of 800 $\mu$m by a pushing beam with a 1/e$^2$ diameter of 1 mm set at one end of the glass cell. The pushed atom beam can interact directly with ionization laser beam as atom target, dubbed as 2D MOT target.
\subsection{\label{sec:3D,level2}Target region}
The target region is inside the main chamber, where three types of target can be used for various experiments. The space charge effect might occur due to large photoionization cross section of Rb atom, particularly for the creation of Rb$^+$ when the intensity of the femtosecond laser is high. This results in a deterioration of the momentum resolution. In this case, 2D MOT target with low density is expected to be a better option for Rb$^+$ detection. As an additional feature, because of the short lifetime of the excited state, 2D MOT provides a target with pure ground state, and thus these results from 2D MOT can be compared with these from 2D molasses or 3D MOT to study the effect of the excited state via detection of photoelectron in the future. Additional advantage is that the magnetic quadrupole field for trapping atoms in 3D MOT is not needed so that the detection efficiency increases significantly without taking time for switching on and off the magnetic field.
Alternatively, the Rb atoms can be further cooled with cooling and repumping lasers in three orthogonal dimensions to form 2D molasses target. The cooling and repumping lasers are combined in an optical fiber, split into three beams and conduced to the main chamber. Each beam comes from an optical coupler passes through a 1/4 $\lambda$ plate, a view port and then get into the target region to cool the Rb atoms. The remaining beam propagates though another viewport and again a 1/4 $\lambda$ plate and then get reflected by a plano mirror, thus realize cooling in the opposite direction. All laser beams intersect in the target region, where the 2D molasses target is formed.
It should be pointed out that the laser beam in the $x$ direction (see Fig. \ref{fig:setup}) can also be combined with an imaging laser for further absorption imaging experiments. Therefore, a PBS is set in front of the plano mirror, allowing cooling and imaging of the target at the same time.
Furthermore, for trapping cold atoms and increasing the target density, the magnetic quadrupole field generated from a pair of anti-Helmholtz coils shown in Fig. \ref{fig:setup} is adopted to form so-called 3D MOT. A pair of coils are mounted in a distance of 120 mm and each of them has the mean diameter is about 70 mm. The cooling water passing through the coils for maintaining the temperature in constant. 2D MOT, 2D molasses and 3D MOT are proper for different types of experiments and their profiles will be discussed in details later.
\subsection{Recoil ion momentum spectrometer}
At the target region, a femtosecond laser of 800 nm interacts with the Rb targets and creates recoil ions in various charged states with certain momentum. The recoil ion momentum spectrometer is a powerful instrument to detect the momentum of recoil ions with 4$\pi$ solid angle. The spectrometer in our setup consists of electrodes for generation of a uniform extraction field, a field-free drift tube for improvement of the resolution, and a MCP (micro channel plate) plus delay-line anode for momentum detection of individual recoil ion.
There are 34 pieces of ring electrodes made of stainless steel. Every electrode has a same thickness of 1 mm and the distances between adjacent electrodes are always 4 mm. The outer diameter of the electrodes is 100 mm and the inner diameter is 75 mm. At the far end from the detector, there is a repeller, i.e. a disk with a diameter of 100 mm and a thickness of 1 mm. The electrode closest to the detector is connected with a metal tube to provide a field-free drift region. The tube has a length of 670 mm and a inner diameter of 81 mm. By a bias voltage between the drift tube and the repeller, a uniform electric field is generated as all adjacent electrodes are connected by resistors of 1 M$\Omega$. For the photoionization experiment, if not specified, an electric field of 0.5 V/cm is set to balance between a better resolution of recoil ion momentum and a preferable efficiency.
A commercial Z-stack MCP detector with delay-line anode is used for the detection of recoil ions. The active area of the detector is about 80 mm diameter, and a position resolution of 0.1 mm can be realized. The time resolution of the recoil ion momentum spectrometer is about 1 ns, and a time range of 400 $\mu$s for TOF measurement is achieved.
\section{\label{sec:Characterization,level1}Characterization of the setup}
\subsection{Characterization of cold atom targets}
The density profile of 3D MOT is measured as a function of its central position by absorption imaging method and photoionization approach given in Fig. \ref{fig:motSize}. Here, the imaging laser beam is transported from the top of the main chamber in the $x$ direction. A CMOS camera is set after the PBS for imaging 3D MOT in the $y$ and $z$ directions. From the FWHMs (full width at half maxima) of the Gaussian fits, the density distributions of 3D MOT are determined to be about 1.1 mm and 0.7 mm in the $y$ and $z$ directions, respectively. Taking into account imaging of pure background (no imaging laser beams and MOT), imaging with cooling lasers (no MOT beam) \cite{Ketterle99making}, and imaging with lasers and MOT, the number of cold atom trapped is estimated to be around $1.5 \times 10^6$ atoms. By switching off the cooling and repumping lasers together with the magnetic field and taking the absorption images at various times, the expansion of 3D MOT was measured to be 0.11 m/s corresponding to temperature of 130 $\mu$K in the $y$ and $z$ directions.
\begin{figure}
\caption{\label{fig:motSize}
\label{fig:motSize}
\end{figure}
On the other hand, the relative density profiles of 3D MOT can also be obtained by photoionization via scanning the focus spot of the femtosecond laser in the position assuming that the ionization rate is proportional to the target density. Here, the wavelength of 800 nm, the repetition rate of 1 kHz, and the maximum power output of about 1 mJ per pulse were used. The laser intensities are adjusted using a calcite wollaston prism and a half-wave plate before entering the main chamber. Unfocused laser beam first goes through science chamber without ionizing the targets because of its extremely low intensity and then is focused back on the targets by a concave mirror with a focal length of 75 mm mounted on the opposite end of the incident femtosecond laser inside the main chamber. The focus spot in a 1/e$^2$ diameter of 20 $\mu$m and the Rayleigh length of 716 $\mu$m in the $y$ direction are achieved. Rayleigh length may result in relatively worse spatial resolution, which can be realized in the momentum distributions below.
In present setup, the focusing lens is mounted on a multi-axis manipulator with a travel range of 75 mm in the femtosecond laser propagation direction ($y$ direction in Fig. \ref{fig:setup}) and 16 mm in the $x$ and $z$ directions. Unfortunately, the movement of manipulator cannot be controlled in the very precise step. Instead, we use magnetic field from two pairs of coils to move the 3D MOT in the $x$ and $z$ directions. Changing the current of the coils, the 3D MOT target moves almost linearly with scale coefficients of 1.04 mm/A in the $x$ direction and 0.63 mm/A in the $z$ direction. With a step of about 0.08 mm, the 3D MOT density profiles plotted in Fig. \ref{fig:motSize}(b) are scanned by counting ionization rate of Rb$^+$ as a function of position. On the basis of the Gaussian fits, the density distributions of the 3D MOT in the $z$ and $x$ directions are given by 1.22 mm and 0.35 mm in FWHM, respectively. With the number of cold atom and the size of 3D MOT mentioned above, the density is estimated to be about $5 \times 10^9$ atoms/cm$^3$. Furthermore, assuming linear dependence between Rb$^+$ count rate and the atomic density, the atomic densities of 2D MOT and 2D molasses are estimated to be $10^7$ atoms/cm$^3$ and $10^8$ atoms/cm$^3$, respectively.
\subsection{Recoil ions momentum distribution of cold atom targets}
One of the most important applications for Rb MOTRIMS is to study the electron correlation dynamics in the strong laser field by investigations of the momentum distribution for the recoil ions, which represents momentum sum of all outgoing electrons. In the strong laser field, multiphoton ionization and tunnel ionization can occur, where Keldysh parameter \cite{KELDYSH1965Ionization}, $\gamma = \sqrt{I_\text{p}/2U_\text{p}}$, is used to categorize both processes, i.e. $\gamma \gg 1$ for multi-photon ionization and $\gamma \ll 1$ for tunnel ionization. $I_\text{p}$ and $U_\text{p}$ indicate the atomic ionization potential and the ponderomotive potential, respectively.
Density plot of the measured recoil-ion momentum distributions for Rb$^+$ in the $xy$-plane are presented in Fig. \ref{fig:pxpy}. Here, linear polarization of the ionizing femtosecond laser with 800 nm wavelength is set in the $z$ direction. The femtosecond laser intensities are estimated to be $2.0 \times 10^{10}$ W/cm$^2$, 1.0 $\times$ 10$^{11}$ W/cm$^2$ and $9.0 \times 10^{11}$ W/cm$^2$ for 3D MOT, 2D molasses and 2D MOT targets corresponding to Keldysh parameters of about 45, 19 and 6, respectively. This means that multiphoton ionization for creation of Rb$^+$ is dominating in all three cases. Note that, the low intensities are selected so that the space charge effect is negligible.
In the Fig. \ref{fig:pxpy}, one clearly find that momentum distributions of the measured Rb$^+$ recoil-ion in the $p_x=$ 0.40 a.u. (FWHM) and the $p_y=$ 0.60 a.u. for 3D MOT are narrower than $p_x = 0.56$ a.u. and the $p_y = 0.70$ a.u. for 2D molasses and $p_x = 0.69$ a.u. and the $p_y = 0.99$ a.u. for 2D MOT. For the $x$ directions in 2D molasses and 2D MOT, the momentum distributions are essentially determined by the target temperature, whereas these in the $y$ direction, worse than the $x$ direction, are affected by the focusing Rayleigh length of the femtosecond laser. For a direct comparison, the normalized $p_x$ momentum components are plotted in Fig. \ref{fig:pxpy}(d).
\begin{figure}
\caption{\label{fig:pxpy}
\label{fig:pxpy}
\end{figure}
\begin{figure}
\caption{\label{fig:pxpz}
\label{fig:pxpz}
\end{figure}
Density plot of Rb$^+$ momentum distribution together with the calculations in the $z$ and $x$ directions for 3D MOT target are displayed in Fig. \ref{fig:pxpz}, where $z$ represents the polarization direction. Present calculations are under the single-active electron approximation. The momentum distribution is evaluated by $w(\boldsymbol{p}) = |p| |M_{\boldsymbol{p}}|^2$ using the strong field approximation (SFA), where the ionization amplitude $M_{\boldsymbol{p}} = \int \mathrm{d}t \langle \boldsymbol{p} + \boldsymbol{A}(t') | \boldsymbol{r}\cdot \boldsymbol{E}(t') | \Psi_0 \rangle \exp[\mathrm{i} S(t')]$ and $S(t') = \int^{t'} \mathrm{d} t''[(\boldsymbol{p} + \boldsymbol{A}(t''))^2 / 2 + I_{\text{p}}]$ with $\boldsymbol{E}(t')$ and $\boldsymbol{A}(t')$ the electric field and vector potential of the laser field, respectively. The vector potential $\boldsymbol{A}(t')$ takes the $\sin^2$-envelope of 20 cycles (duration $\sim$ 26 fs), and the intensity of the laser field is $10^{10}$ W/cm$^2$.
In Fig. \ref{fig:pxpz}(a), experimental momentum distribution shows a dipole-like double-peak structure reproduced by our calculation shown in Fig. \ref{fig:pxpz}(b) and a nice agreement between experiments and calculations is found. The 3D targets with cooling laser constitute atoms mixed in the ground state and in the first excited state and therefore the recoil ions of Rb$^+$ are created by absorption of three photons and two photons, respectively:
\begin{eqnarray*}
\text{Rb}(^2\text{S}_{1/2}) + 3{h}\nu & = & \text{Rb}^+(^1\text{S}_0) + \text{e}^- \\
\text{Rb}(^2\text{P}_{3/2}) + 2{h}\nu & = & \text{Rb}^+(^1\text{S}_0) + \text{e}^- .
\end{eqnarray*}
According to the ionization energies of 4.18 eV and 2.58 eV from the ground state and the first excited state, respectively, both release an almost same excess-energy of $E_e = 0.47$ eV for the outgoing electron. This results in a recoil momentum of 0.19 a.u., which is good agreement with the observed and calculated results of $\pm 0.18$ a.u. displayed in Fig. \ref{fig:pxpz}. Here, two ionization progresses from the ground state and the first excited state cannot be distinguished only by kinetic energy of recoil-ion since the first excited energy of Rb pumped by cooling laser is almost same with single-photon energy of femtosecond laser. In future work, contributions of the ground state and the first excited state in 3D MOT target will be investigated by Rb$^+$ intensity dependence for the angular distribution and the ionization rate. Limited to present experimental momentum resolution, the substructure from the inner ring indicated by theoretical results cannot be distinguished clearly.
Taking a region of $\sqrt{p_x^2 +p_y^2} < 0.1$ a.u., the experimental momentum distribution of $p_z$ as well as theoretical result indicated with blue curve were plotted in Fig. \ref{fig:pxpz} (c). Again, a double-peak structure is illuminated more clearly and the positions of two peaks are in good agreement between experiment and theory.
For a direct comparison, theoretical result in red line is convoluted with a Gaussian function yielding an experimental momentum resolution of 0.12 a.u.. We also note that 2D molasses and 2D MOT targets show the same momentum resolution with 3D MOT in the $z$ (TOF) direction. Here, the momentum distributions in the $z$ direction is reconstructed with TOF of each event by considering the geometry and voltage settings of the spectrometer. Present momentum resolution is mainly limited to the uncertainties of extraction electric field and other stray fields. By changing the polarization of femtosecond laser to the $x$ direction, i.e., along the 2D beam movement direction shown in Fig. 1, a similar double-peak structure is also observed in three targets. By this way, the momentum resolutions extracted along the $x$ direction are less than 0.2 a.u. for 3D MOT but more worse for 2D molasses and 2D MOT where the velocity distributions make the temperatures up to a few mK high.
It should also be pointed out that it is more convenient to use 2D molasses due to its accepted resolution mentioned above since quadrupole magnetic field for trapping atoms is not applied so that electrons can be detected with the high detection efficiency. Imaging electron momentum distributions is being proposed. We also note that very recently a MOTRIMS-like design for lithium was introduced in Ref. \cite{PhysRevA.97.043427}.
\section{\label{sec:Conclusion,level1}Conclusions}
New magneto-optical trap (MOT) recoil ion momentum spectroscopy for studies of Rb atom in the strong infrared laser field was developed, in which three types of cold Rb target are provided, i.e., 2D MOT, 2D molasses, and 3D MOT. In the present design, three targets can provide densities in the orders of $10^7$ atoms/cm$^3$, $10^8$ atoms/cm$^3$, and $10^9$ atoms/cm$^3$, respectively. The lowest temperature for 3D MOT target up to 130 $\mu$K is achieved. The cold targets with various densities are accessible for different investigations considering the effects of space charge and excited states involved. The momentum distributions of Rb$^+$ created by the multi-photon absorption for three types of atom targets are measured and a dipole-like double-peak feature is well reproduced with the strong field approximation. With the help of calculations in the strong field approximation, the momentum resolution of $0.12 \pm 0.03$ a.u. along the time-of-flight is achieved. Present first observation for momentum distributions demonstrates that MOTRIMS apparatus combined with the femtosecond laser pulses provides a powerful and new tool for investigations of the strong-ionization dynamics in alkali atoms.
In the near future, electron detection will be developed, allowing for ion and electron detections in coincidence. Considering the multi-functions for imaging ions, electrons, and photons, present MOTRIMS for alkali atoms are of great potentials for the study of Rydberg multi-body interactions as well as the strong-correlations in the cold plasma.
\section{Acknowledgment}
We acknowledge financial supports by the Instrument Developing Project of the Chinese Academy of Sciences (YZ201537), National Natural Science Foundation of China (NSFC) (91636105, 11420101003, 61675213, 11604347), and Shanghai Sailing Program (16YF1412600). We are grateful for numerous discussions with Alexander Dorn, Bastian H\"oltkemeier, Xueguang Ren, Jiefei Chen and Peng Chen.
\end{document} |
\begin{document}
\title{
Partial regularity result of elliptic systems with Dini continuous coefficients and $q$-growth}
\begin{center}
\begin{minipage}[t]{10cm}
\small{
\noindent \textbf{Abstract.}
We establish partial regularity result for vector-valued solutions $u:\Omega\to\mathbb{R}^N$ to second order elliptic systems
of the type:
\[
-\mathrm{div}(A(x,u,Du))=f(x,u,Du) \qquad \mathrm{in}\>\Omega ,
\]
where the coefficients $A:\Omega\times\mathbb{R}^N\times\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)\to\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$ satisfies Dini condition respect to
$(x,u)$ with growth order $q\geq 2$. We prove $C^1$-regularity of the solutions outside of singular sets.
\noindent \textbf{Keywords.} Nonlinear elliptic systems, Partial regularity, Dini condition, $\mathcal{A}$-harmonic approximation.
\noindent \textbf{Mathematics~Subject~Classification~(2010):}
35J60, 35B65.
}
\end{minipage}
\end{center}
\section{Introduction}
In this paper, we consider the second order nonlinear elliptic systems in divergence form of the following type:
\begin{equation}
-\mathrm{div}(A(x,u,Du))=f(x,u,Du) \qquad \text{in}\>\Omega .
\label{system}
\end{equation}
Here, $\Omega$ is a bounded domain in $\mathbb{R}^n$, $u$ takes values in $\mathbb{R}^N$ with coefficients
$A:\Omega\times\mathbb{R}^N\times\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)\to\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$.
The regularity theory with the growth of $A(x,\xi,p)$ with respect to $p$ has been proved by Giaquinta and Modica \cite{GMo2}.
They proved that weak solutions of \eqref{system} has H\"{o}lder continuous first derivatives outside of a singular set of
Lebesgue measure zero if $(1+\lvert p \rvert)^{-1}A(x,\xi,p)$ is H\"{o}lder continuous in variables $(x,\xi)$ uniformly with respect to $p$.
In \cite{DG}, Duzaar and Grotowski gave a simplified proof of their result without
$L^q$-$L^2$ estimates for $Du$. The method of proof also gives the optimal result in one step,
i.e. if $(1+\lvert p \rvert)^{-1}A(x,\xi,p)$ is in $C^{0,\alpha}$ for some $0<\alpha<1$ in $(x,\xi)$ then $u$ is in
$C^{1,\alpha}$ outside of the singular set. The essential feature is the use of the $\mathcal{A}$-harmonic approximation lemma
(cf. \cite[Lemma 2.1]{DG}; see also Lemma \ref{A-harm}).
Duzaar and Gastel \cite{DGa} prove under weaker assumptions on $A(x,\xi,p)$ with respect to continuity in the variables
$(x,\xi)$. More precisely, they assume for the continuity of $A(x,\xi,p)$ with respect to the variables $(x,\xi)$ that
\begin{equation}
\lvert A(x,\xi,p)-A(\widetilde{x},\widetilde{\xi},p)\rvert
\leq \kappa (\lvert\xi\rvert)\mu\left( \lvert x-\widetilde{x}\rvert +\lvert\xi-\widetilde{\xi}\rvert\right)(1+\lvert p \rvert),
\label{DGaassumption}
\end{equation}
for all $x,\widetilde{x}\in\Omega$, $\xi,\widetilde{\xi}\in\mathbb{R}^N$, $p\in\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$, where
$\kappa :[0,\infty)\to[1,\infty)$ is nondecreasing, and $\mu:(0,\infty)\to[0,\infty)$ is nondecreasing and
concave with $\mu(0+)=0$. They also have to require that $r\mapsto r^{-\alpha}\mu(r)$ is nonincreasing for
some $0<\alpha<1$, and that
\begin{equation}
F(r)=\int_0^r\frac{\mu(\rho)}{\rho}d\rho<\infty \quad \text{for some }r>0. \label{originDini}
\end{equation}
They conclude that a bounded weak solution of elliptic system \eqref{system} satisfying \eqref{DGaassumption} and
(\ref{originDini}) is in $C^1$ outside a closed singular set with Lebesgue measure zero.
The condition \eqref{originDini} is called Dini condition in the literature, although Dini himself \cite{Di} used
a slightly weaker conditions a century ago. It had some significance for the theory of linear elliptic partial
differential equations in the first half of the century, cf. \cite{HW}.
Qiu \cite{Qiu} extend the result in \cite{DGa}, which is the result under quadratic growth condition,
to the subquadratic case. In this case, the assumptions \eqref{DGaassumption} and \eqref{originDini} are modified as
\[
\lvert A(x,\xi,p)-A(\widetilde{x},\widetilde{\xi},p)\rvert
\leq \kappa (\lvert\xi\rvert) \mu\left( \lvert x-\widetilde{x}\rvert^q +\lvert\xi-\widetilde{\xi}\rvert^q\right)
(1+\lvert p \rvert)^{-2/q},
\]
and
\[
F(r)=\int_0^r\frac{\sqrt{\mu(\rho)}}{\rho}d\rho<\infty \quad \text{for some }r>0,
\]
where $1<q<2$.
In this paper, we consider the regularity theory in case of superquadratic, i.e. $q\geq 2$. Thus, we assume the continuity of
$A(x,\xi,p)$ with respect to the variables $(x,\xi)$ that
\[
\lvert A(x,\xi,p)-A(\widetilde{x},\widetilde{\xi},p)\rvert
\leq \kappa (\lvert\xi\rvert) \mu\left( \lvert x-\widetilde{x}\rvert +\lvert\xi-\widetilde{\xi}\rvert\right)
(1+\lvert p \rvert)^{q-1},
\]
and to obtain the regularity result, we assume the modified Dini condition such that
\[
F(r)=\int_0^r\frac{\mu^\beta(\rho)}{\rho}d\rho<\infty \quad \text{for some}\ r>0\ \text{and}\ \beta\in \left( 0,1\right].
\]
Under these assumptions and $q$-growth condition for
inhomogeneous term, we obtain that a bounded weak solution of \eqref{system} is $C^1$(see Theorem \ref{pr}).
Our result is different from the result of Qiu \cite{Qiu2}. The main difference is the version of $\mathcal{A}$-harmonic
approximation lemma which we used. Lemma 2.1 in \cite{Qiu2} (see also \cite[Lemma 2.1]{DG}) only guarantee the existence of
$\mathcal{A}$-harmonic function $h$ which approximate the rescaled solution $w$ in $L^2$. This restricts the growth order $q<n$ to estimate
$\Xint{-}_{B_\rho(x_0)}\lvert w-h \rvert^qdx$ by the Sobolev-Poincar\'{e} inequality. In contrast our $\mathcal{A}$-harmonic
approximation lemma guarantees the approximation in $L^q$ as well as in $L^2$, and this allows us to obtain the regularity
result at any growth order.
We close this section by briefly summarizing the notation used in this paper.
As note above, we consider a bounded domain $\Omega\subset\mathbb{R}^n$, and maps from $\Omega$ to $\mathbb{R}^N$,
where we take $n\geq 2$, $N\geq 1$. For a given set $X$ we denote by $\mathscr{L}^n(X)$ as $n$-dimensional
Lebesgue measure. We write $B_\rho(x_0):=\{ x\in\mathbb{R}^n\> :\> \lvert x-x_0 \rvert<\rho\}$. For bounded set
$X\subset\mathbb{R}^n$ with $\mathscr{L}^n(X)>0$, we denote the average of a given function $g\in L^1(X,\mathbb{R}^N)$ by
$\Xint{-}_Xgdx$ i.e., $\Xint{-}_Xgdx =\frac{1}{\mathscr{L}^n(X)}\int_Xgdx$. In particular, we write
$g_{x_0,\rho}=\Xint{-}_{B_\rho(x_0)\cap\Omega}gdx$. We write $\mathrm{Bil}(\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N))$
for the space of bilinear forms on the space $\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$ of linear maps from $\mathbb{R}^n$ to
$\mathbb{R}^N$. We denote $c$ a positive constant, possibly varying from line by line. Special occurrences will be denoted by
capital letters $K$, $C_1$, $C_2$ or the like.
\section{Hypothesis and Statement of Results}
\begin{dfn}\label{wsol}
We define $u\in W^{1,q}(\Omega,\mathbb{R}^N),q\geq 2$ is a weak solution of \eqref{system} if $u$ satisfies
\begin{equation}
\int_\Omega \langle A(x,u,Du),D\varphi\rangle dx
=\int_\Omega \langle f,\varphi\rangle dx \label{ws}
\end{equation}
for all $\varphi\in C^{\infty}_0(\Omega,\mathbb{R}^N)$, where $\langle\cdot,\cdot\rangle$ is the standard Euclidean
inner product on $\mathbb{R}^N$ or $\mathbb{R}^{nN}$.
\end{dfn}
We assume following structure condition.
\begin{enumerate}
\item[\bf{(H1)}]
$A(x,\xi,p)$ is differentiable in $p$ with continuous derivatives. Moreover, there exists $L\geq 1$ such that
\[
\left\lvert D_pA(x,\xi,p)\right\rvert\leq L(1+\lvert p \rvert)^{q-2}
\quad \text{for all }(x,\xi,p)\in\Omega\times\mathbb{R}^N\times\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N);
\]
this infers the existence of a modulus of continuity $\omega:[0,\infty)\times [0,\infty)\to [0,1]$ with
$\omega(t,0)=0$ for all $t$ such that $t\mapsto\omega(s,t)$ is nondecreasing for fixed $s$,
$s\mapsto\omega(s,t)$ is concave and nondecreasing for fixed $t$. $\omega(s,t)$ also satisfies
\[
\left\lvert D_p A(x,\xi,p)-D_p A(\widetilde{x},\widetilde{\xi},\widetilde{p})\right\rvert
\leq L\omega\left(\lvert\xi\rvert+\lvert\nu\rvert,\lvert x-\widetilde{x}\rvert^2
+\lvert\xi-\widetilde{\xi}\rvert^2+\lvert p-\widetilde{p}\rvert^2\right)
(1+\lvert p \rvert+\lvert\widetilde{p}\rvert)^{q-2}.
\]
for all $(x,\xi,p),(\widetilde{x},\widetilde{\xi},\widetilde{p})\in\Omega\times\mathbb{R}^N\times\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$ with
$\lvert\xi\rvert+\lvert p \rvert\leq M$.
\item[\bf{(H2)}]
$A(x,\xi,p)$ is uniformly strongly elliptic i.e., for some $\lambda>0$, $A(x,\xi,p)$ satisfies
\[
\left\langle D_pA(x,\xi,p)\nu,\nu\right\rangle \geq \lambda \lvert\nu\rvert^2(1+\lvert p \rvert)^{q-2}
\quad \text{for all }x\in\Omega, \xi\in\mathbb{R}^N, p,\nu\in\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N);
\]
\item[\bf{(H3)}]
There exists a modulus of continuity $\mu:[0,\infty)\to(0,\infty)$, and a nondecreasing function
$\kappa :[0,\infty)\to[1,\infty)$ such that
\begin{equation}
\lvert A(x,\xi,p)-A(\widetilde{x},\widetilde{\xi},p)\rvert
\leq \kappa (\lvert\xi\rvert)\mu\left( \lvert x-\widetilde{x}\rvert+\lvert\xi-\widetilde{\xi}\rvert\right)
(1+\lvert p \rvert)^{q-1}
\end{equation}
for all $x,\widetilde{x}\in\Omega$, $\xi,\widetilde{\xi}\in\mathbb{R}^N$, $p\in\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$. Without loss of generality
we may assume that
\begin{enumerate}
\item[($\mu$1)] $\mu$ is nondecreasing function with $\mu(+0)=0$.
\item[($\mu$2)] $\mu$ is concave; in the proof of the regularity theorem we have to require that $r \mapsto r^{-\alpha}\mu(r)$
is nonincreasing for some exponent $\alpha\in(0,1)$.
\end{enumerate}
We also require modified Dini's condition:
\begin{enumerate}
\item[($\mu$3)] $\displaystyle F(r) := \int_0^r \frac{\mu^\beta(\rho)}{\rho}d\rho <+\infty$ for some $r>0$ and
$\displaystyle \beta\in\left( 0,1\right]$.
\end{enumerate}
\item[\bf{(H4)}]
There exists constants $a$ and $b$, with $a$ possibly depending on $M>0$, such that
\[
\lvert f(x,\xi,p)\rvert\leq a(M)\lvert p \rvert^q +b
\]
for all $x\in\Omega$, $\xi\in\mathbb{R}^N$ with $\lvert\xi\rvert\leq M$, and $p\in\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$.
\end{enumerate}
Using above structure conditions, we state our main theorem.
\begin{thm}\label{pr}
Let $u\in W^{1,q}(\Omega,\mathbb{R}^N)\cap L^\infty(\Omega,\mathbb{R}^N)$ be a bounded weak solution to \eqref{system} under the structure conditions
{\rm (\textbf{H1}), (\textbf{H2}), (\textbf{H3}), (\textbf{H4}), ($\mu$1), ($\mu$2)} and {\rm ($\mu$3)},
satisfying $\lVert u\rVert_\infty\leq M$ and $2^{(10-9q)/2}\lambda >a(M)M$.
Then there is a relatively closed set $\mathrm{Sing}\, u\subset\Omega$, such that the weak solution $u$ satisfies
$u\in C^1(\Omega\setminus\mathrm{Sing}\, u,\mathbb{R}^N)$.
Further $\mathrm{Sing}\, u \subset \Sigma_1\cup\Sigma_2$, where
\begin{align*}
\Sigma_1:&=\left\{ x_0\in\Omega \>\> :\>\>
\liminf_{\rho\searrow 0}\Xint{-}_{B_\rho(x_0)}\lvert Du-(Du)_{x_0,\rho}\rvert^q dx>0\right\}, \quad \text{and}\\
\Sigma_2:&=\left\{ x_0\in\Omega \>\> :\>\>
\limsup_{\rho\searrow 0}\lvert (Du)_{x_0,\rho}\rvert =\infty\right\}
\end{align*}
and in particular, $\mathscr{L}^n(\mathrm{Sing}\, u)=0$. In addition, for $\sigma\in[\alpha,1)$ and
$x_0\in\Omega\setminus\mathrm{Sing}\, u$ the derivative of $u$ has modulus of continuity $r\mapsto r^\sigma+F(r)$ in a
neighborhood of $x_0$.
\end{thm}
\section{Some preliminaries}
In this section we recall the $\mathcal{A}$-harmonic approximation lemma, and some standard estimates
for the proof of the regularity theorem.
First we state the definition of $\mathcal{A}$-harmonic function and present the following version of an
$\mathcal{A}$-harmonic approximation lemma which can be retrieved from the corresponding parabolic version in \cite[Lemma 3.2]{DMS}.
This lemma allowed us to approximate the weak solution $u$ to the solution of constant coefficients elliptic system in
$L^2$ as well as in $L^q$. For more detail about $\mathcal{A}$-harmonic approximation technique, we refer to the survey paper \cite{DM}.
\begin{dfn}[{\cite[Section 1]{DG}}]
For a given $\mathcal{A}\in\mathrm{Bil}(\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N))$, we say that $h\in W^{1,q}(\Omega,\mathbb{R}^N)$
is an $\mathcal{A}$-harmonic function, if $h$ satisfies
\[
\int_{\Omega}\mathcal{A}(Dh,D\varphi)dx=0
\]
for all $\varphi\in C^\infty_0(\Omega,\mathbb{R}^N)$.
\end{dfn}
\begin{lem}[{\cite[Lemma 2.3]{BDHS}}]\label{A-harm}
Let $0<\lambda\leq L$ and $q\geq 2$ be given. For every $\varepsilon>0$, there exists a constant
$\delta=\delta(n,N,q,\lambda,L,\varepsilon)\in(0,1]$ such that the following holds: assume that $\gamma\in[0,1]$ and that $\mathcal{A}$
is a bilinear form on $\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$ with the properties
\begin{equation}
\mathcal{A} (\nu,\nu)\geq \lambda\lvert\nu\rvert^2, \quad \text{and} \quad
\mathcal{A} (\nu,\widetilde{\nu})\leq L\lvert\nu\rvert\lvert\widetilde{\nu}\rvert,
\end{equation}
for all $\nu,\widetilde{\nu}\in\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$. Furthermore, let $w\in W^{1,q}(B_\rho(x_0),\mathbb{R}^N)$ be an approximately
$\mathcal{A}$-harmonic map in the sense that there holds
\[
\left\lvert\Xint{-}_{B_\rho(x_0)}\mathcal{A} (Dw,D\varphi )dx \right\rvert\leq \delta\sup_{B_\rho(x_0)}\lvert D\varphi\rvert
\]
for all $\varphi\in C^\infty_0(B_\rho(x_0),\mathbb{R}^N)$ and that
\[
\Xint{-}_{B_\rho(x_0)}\{\lvert Dw\rvert^2+\gamma^{q-2}\lvert Dw\rvert^q\} dx\leq 1.
\]
Then there exists an $\mathcal{A}$-harmonic function $h\in C^\infty(B_{\rho/2}(x_0),\mathbb{R}^N)$ that satisfies
\begin{equation}
\Xint{-}_{B_{\rho/2}(x_0)}\left\{\lvert Dh\rvert^2+\gamma^{q-2}\lvert Dh\rvert^q\right\} dx\leq \tilde{C}(n,q)
\end{equation}
and, at the same time,
\begin{equation}
\Xint{-}_{B_{\rho/2}(x_0)}\left\{\left\lvert\frac{w-h}{\rho/2}\right\rvert^2
+\gamma^{q-2}\left\lvert\frac{w-h}{\rho/2}\right\rvert^q\right\} dx\leq\varepsilon.
\end{equation}
\end{lem}
Next is a standard estimates for the solutions to homogeneous second order elliptic systems with
constant coefficients, due originally to Campanato \cite[Teorema 9.2]{Cam}.
For convenience, we state the estimate in a slightly general form than the original one.
\begin{thm}[{\cite[Theorem 2.3]{DG}}]\label{Campanato}
Consider $A$, $\lambda$ and $L$ as in Lemma \ref{A-harm}. Then there exists $C_0\geq 1$ depending only on
$n$, $N$, $\lambda$ and $L$ such that any $\mathcal{A}$-harmonic function $h$ on $B_{\rho/2}(x_0)$ satisfies
\begin{equation}
\left(\frac{\rho}{2}\right)^2\sup_{{B_{\rho/4}}(x_0)}\lvert Dh\rvert^2
+\left(\frac{\rho}{2}\right)^4\sup_{B_{\rho/4}(x_0)}\lvert D^2h\rvert^2
\leq C_0\left(\frac{\rho}{2}\right)^2\Xint{-}_{B_{\rho/2}(x_0)}\lvert Dh\rvert^2dx. \label{campanato}
\end{equation}
\end{thm}
We state the Poincar\'{e} inequality in a convenient form.
\begin{lem}[{\cite[Proposition 3.10]{GM}}]\label{Poincare}
There exists $C_P\geq 1$ depending only on $n$ such that every $u\in
W^{1,q}(B_\rho(x_0),\mathbb{R}^N)$ satisfies
\begin{equation}
\int_{B_\rho(x_0)}\lvert u-u_{x_0,\rho}\rvert^qdx
\leq C_P\rho^q\int_{B_\rho(x_0)}\lvert Du\rvert^qdx. \label{poincare}
\end{equation}
\end{lem}
Using Young's inequality, we obtain the following estimates.
\begin{lem}[{\cite[Lemma 3.7]{Kana}}]\label{Young2}
Consider fixed $a,b\geq 0$, $q\geq 1$. Then for any $\varepsilon>0$, there exists
$K=K(q,\varepsilon)\geq 0$ satisfying
\begin{equation}
(a+b)^q\leq (1+\varepsilon)a^q +K b^q. \label{young2}
\end{equation}
\end{lem}
\begin{lem}[{\cite[Lemma 2.1]{GMo}}]\label{GM}
For $\delta \geq 0$, and for all $a,b\in\mathbb{R}^{nN}$ we have
\begin{equation}
4^{-(1+2\delta)}\leq
\frac{\displaystyle\int_0^1(1+\lvert sa+(1-s)b\rvert^2)^{\delta/2}ds}{(1+\lvert a\rvert^2+\lvert b-a\rvert^2)^{\delta/2}}
\leq 4^\delta . \label{GM2}
\end{equation}
\end{lem}
In the followings, we write the modulus of continuity $\mu$ as
\[
\eta(t):=\mu^2\left(\sqrt{t}\right)
\]
by technical reason (cf. {\bf (H3)}). The conditions ($\mu$1), ($\mu$2) and ($\mu$3) are expressed as
\begin{enumerate}
\item[($\eta$1)] $\eta$ is continuous, nondecreasing, and $\eta(+0)=0$,
\item[($\eta$2)] $\eta$ is concave; and $t \mapsto t^{-\alpha}\eta(t)$ is nonincreasing for the same exponent $\alpha$ as
in ($\mu$2),
\item[($\eta$3)] $\displaystyle
\widetilde{F}(t):= \left[ 2F\left(\sqrt{t}\right)\right]^2
=\left[ \int_0^t\frac{\sqrt{\eta^\beta(\tau)}}{\tau}d\tau\right]^2<+\infty$ for some $t>0$.
\end{enumerate}
Changing $\kappa$ by a constant, but keeping $\kappa\geq 1$, we can also assume that
\begin{enumerate}
\item[($\eta$4)] $\eta(1)=1$, implying $t\leq\eta(t)\leq 1$ \quad for $t\in(0,1]$.
\end{enumerate}
From the fact that $\eta$ is nondecreasing, for $t\leq s$ and $\sigma\leq 1/\alpha$, we deduce
$s\eta^{\sigma}(t)\leq s\eta^{\sigma}(s)$.
For $s\leq t$, we use nonincreasing property of $t^{-\alpha}\eta(t)$ and $\eta(s)\leq 1$, and we obtain
$s\eta^{\sigma}(t)\leq t$. Combining both cases we obtain
\[
s\eta^{\sigma}(t)\leq s\eta^{\sigma}(s)+t \quad \text{for}\> s\in[0,1],\ t>0,\ \sigma\leq\frac{1}{\alpha}.
\]
In particular, we have
\begin{enumerate}
\item[($\eta$5)] $s\eta(t)\leq s\eta(s)+t$ \quad for $s\in[0,1]$, $t>0$,
\item[($\eta$6)] $s\sqrt{\eta(t)}\leq s\sqrt{\eta(s)}+t$ \quad for $s\in[0,1]$, $t>0$.
\end{enumerate}
From ($\eta 2$) we infer for $i\in\mathbb{N}\cup\{ 0\}$, $\theta\in(0,1/8]$, $t>0$
\[
\int_{\theta^{2(i+1)}t}^{\theta^{2i}t}\frac{\sqrt{\eta^\beta(\tau)}}{\tau}d\tau
\geq \sqrt{\frac{\eta^\beta(\theta^{2i}t)}{(\theta^{2i}t)^{\alpha\beta}}}\int_{\theta^{2(i+1)}t}^{\theta^{2i}t}
\tau^{(\alpha\beta-2)/2}d\tau
=\frac{2}{\alpha\beta}\left( 1-\theta^{\alpha\beta}\right)\sqrt{\eta^\beta(\theta^{2i}t)} ,
\]
which implies
\begin{align}
\sum_{i=0}^{k-1}\sqrt{\eta^\beta(\theta^{2i}t)}
&\leq \frac{\alpha\beta}{2(1-\theta^{\alpha\beta})}\sqrt{\widetilde{F}(t)} \label{sumeta}
\end{align}
for $k\in\mathbb{N}$. This yields in particular that
\begin{equation}
\eta(t)\leq \frac{\alpha^2\beta^2}{4(1-\theta^{\alpha\beta})^2}\widetilde{F}(t) \label{etaF}
\end{equation}
for all $t\in [0,1]$. Moreover, for $t\in [0,1]$, $\theta\in(0,1/8]$, we have
\begin{align}
t^{-\alpha}\widetilde{F}(t)
&=t^{-\alpha}\left[\sqrt{\widetilde{F}(\theta t)}
+\int_{\theta t}^t\sqrt{\tau^{-\alpha}\eta(\tau)}\tau^{(\alpha-2)/2}d\tau\right]^2 \notag\\
&\leq t^{-\alpha}\left[\sqrt{\widetilde{F}(\theta t)}
+\frac{2}{\alpha}\sqrt{(\theta t)^{-\alpha}\eta(\theta t)}\left\{\sqrt{t^\alpha}-\sqrt{(\theta t)^\alpha}\right\} \right]^2 \notag\\
&\leq \left[\sqrt{t^{-\alpha}\widetilde{F}(\theta t)}
+\sqrt{(\theta t)^{-\alpha}\widetilde{F}(\theta t)}
\frac{1-\theta^{\alpha/2}}{1-\theta^{\alpha\beta}}\right]^2 \notag\\
&\leq 4(\theta t)^{-\alpha}\widetilde{F}(\theta t). \label{Fnoninc}
\end{align}
\section{Caccioppoli-type inequality}
For $s,t\geq 0$ let
\[
\rho_1(s,t):= (1+t)^{-1}\kappa^{-1}(s+t), \qquad G(s,t):= (1+t)^2\kappa^{2q}(s+t).
\]
Note that $\rho_1\leq 1$ and $G\geq 1$.
\begin{lem}\label{Caccioppoli}
Consider $\nu\in\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$ and $\xi\in\mathbb{R}^N$ with $\lvert\xi\rvert\leq M$ fixed. Let
$u\in W^{1,q}(\Omega,\mathbb{R}^N)\cap L^\infty(\Omega,\mathbb{R}^N)$ be a bounded weak solution to \eqref{system} under
the structure conditions {\rm (\textbf{H1}), (\textbf{H2}), (\textbf{H3}), (\textbf{H4}), ($\eta$1), ($\eta$2), ($\eta$3)}
and {\rm ($\eta$4)} with satisfying $\lVert u\rVert_\infty \leq M$ and $2^{(10-9q)/2}\lambda >a(M)M$.
Then for any $x_0\in\Omega$ and $\rho\leq \rho_1(\lvert\xi\rvert,\lvert\nu\rvert)$ such
that $B_\rho(x_0)\Subset\Omega$, there holds
\begin{align}
&\Xint{-}_{B_{\rho/2}(x_0)}\left\{\frac{\lvert Du-\nu\rvert^2}{(1+\lvert\nu\rvert)^2}
+\frac{\lvert Du-\nu\rvert^q}{(1+\lvert\nu\rvert)^q}\right\} dx \notag\\
\leq C_1&\Bigg[ \Xint{-}_{B_\rho(x_0)}\left\{\left\lvert\frac{u-\xi-\nu(x-x_0)}{\rho(1+\lvert\nu\rvert)}\right\rvert^2
+\left\lvert\frac{u-\xi-\nu(x-x_0)}{\rho(1+\lvert\nu\rvert)}\right\rvert^q\right\} dx
+G(\lvert\xi\rvert,\lvert\nu\rvert)\eta(\rho^2)
+\left( a\lvert\nu\rvert +b\right)^2\rho^2 \Bigg] \label{caccioppoli}
\end{align}
with $C_1\geq 1$ depending only on $\lambda$, $q$, $L$, $a(M)$ and $M$.
\end{lem}
\begin{proof}
Assume $x_0\in\Omega$ and $\rho\leq 1$ satisfy $B_{\rho}(x_0)\Subset\Omega$ and $\rho\leq\rho_1(\lvert\xi\rvert,\lvert\nu\rvert)$.
We denote $\xi+\nu(x-x_0)$ by $\ell(x)$ and take a standard cut-off function $\psi\in C^\infty_0(B_\rho(x_0))$
satisfying $0\leq\psi\leq 1$, $\lvert D\psi\rvert\leq 4/\rho$, $\psi\equiv 1$ on $B_{\rho/2}(x_0)$.
Then $\varphi :=\psi^q(u-\ell )$ is admissible as a test function in \eqref{ws}, and obtain
\begin{align}
\Xint{-}_{B_\rho(x_0)}\psi^q\langle &A(x,u,Du),Du-\nu \rangle dx \notag\\
=-\,&\Xint{-}_{B_\rho(x_0)}\langle A(x,u,Du),q\psi^{q-1}
D\psi\otimes (u-\ell)\rangle dx
+\Xint{-}_{B_\rho(x_0)}\langle f,\varphi\rangle dx, \label{system2}
\end{align}
where $\xi\otimes\zeta:=\xi_i\zeta^\alpha$. We further have
\begin{align}
-\,\Xint{-}_{B_\rho(x_0)}\psi^q\langle &A(x,u,\nu),Du-\nu \rangle dx \notag\\
=&\Xint{-}_{B_\rho(x_0)}\langle A(x,u,\nu),q\psi^{q-1} \psi\otimes
(u-\ell)\rangle dx-\Xint{-}_{B_\rho(x_0)}\langle A(x,u,\nu),D\varphi
\rangle dx,
\end{align}
and
\begin{equation}
\Xint{-}_{B_\rho(x_0)}\langle A(x_0,\xi,\nu),D\varphi \rangle dx=0. \label{constcoeff}
\end{equation}
Adding these equations, from \eqref{system2} to \eqref{constcoeff}, we obtain
\begin{align}
&\Xint{-}_{B_\rho(x_0)}\psi^q \langle A(x,u,Du)-A(x,u,\nu), Du-\nu\rangle dx \notag \\
=&-\Xint{-}_{B_\rho(x_0)}\langle A(x,u,Du)-A(x,u,\nu),q\psi^{q-1} D\psi\otimes (u-\ell)\rangle dx \notag \\
&-\Xint{-}_{B_\rho(x_0)}\langle A(x,u,\nu)-A(x,\ell,\nu),D\varphi\rangle dx \notag \\
&-\Xint{-}_{B_\rho(x_0)}\langle A(x,\ell,\nu)-A(x_0,\xi,\nu) ,D\varphi \rangle dx \notag \\
&+\Xint{-}_{B_\rho(x_0)}\langle f,\varphi\rangle dx \notag \\
=:& \>\> \hbox{I}+\hbox{I}I+\hbox{I}II+\hbox{I}V. \label{caccio-devide}
\end{align}
The terms I, II, III and IV are defined above. Using the ellipticity condition ({\bf H2}) to the left hand side
of \eqref{caccio-devide}, we get
\begin{align*}
&\langle A(x,u,Du)-A(x,u,\nu),Du-\nu\rangle \\
=&\int_0^1\left\langle D_p A(x,u,sDu+(1-s)\nu)(Du-\nu),
Du-\nu\right\rangle ds \\
\geq& \lambda \lvert Du-\nu\rvert^2\int_0^1(1+\lvert sDu+(1-s)\nu\rvert)^{q-2}ds.
\end{align*}
Then we estimate above by \eqref{GM2} in Lemma \ref{GM} and obtain
\begin{align}
&\langle A(x,u,Du)-A(x,u,\nu),Du-\nu\rangle \notag \\
\geq& 2^{(12-9q)/2}\lambda
\left\{(1+\lvert\nu\rvert)^{q-2}\lvert Du-\nu\rvert^2+\lvert Du-\nu\rvert^q\right\} .\label{elliptic}
\end{align}
For $\varepsilon >0$ to be fixed later, using ({\bf H1}) and Young's inequality, we have
\begin{align}
\lvert\,\hbox{I}\,\rvert
\leq &\varepsilon\Xint{-}_{B_\rho(x_0)}\psi^q\left\{(1+\lvert\nu\rvert)^{q-2}
\lvert Du-\nu\rvert^2+\lvert Du-\nu\rvert^q\right\}dx \notag\\
&+c(p,L,\varepsilon) \Xint{-}_{B_\rho(x_0)}\left\{ (1+\lvert\nu\rvert)^{q-2}\left\lvert\frac{u-\ell}{\rho}\right\rvert^2
+\left\lvert\frac{u-\ell}{\rho}\right\rvert^q\right\}dx. \label{caccio-I}
\end{align}
In order to estimate II, we first use ({\bf H3}) and $D\varphi=\psi^q(Du-\nu)+q\psi^{q-1}D\psi\otimes
(u-\ell)$, we get
\begin{align*}
\lvert\,\hbox{I}I\,\rvert
\leq &\Xint{-}_{B_\rho(x_0)}\kappa(\lvert\xi\rvert+\lvert\nu\rvert\rho)
\mu\left(\lvert u-\ell\rvert\right)(1+\lvert\nu\rvert)^{q-1} \psi^q\lvert Du-\nu\rvert dx \\
&+\Xint{-}_{B_\rho(x_0)}\kappa(\lvert\xi\rvert+\lvert\nu\rvert\rho)\mu\left(\lvert u-\ell\rvert\right)(1+\lvert\nu\rvert)^{q-1}
q\psi^{q-1}\lvert D\psi\rvert\lvert u-\ell\rvert dx \\
=:& \hbox{I}I_1+\hbox{I}I_2.
\end{align*}
The terms $\hbox{I}I_1$ and $\hbox{I}I_2$ are defined above. Using Young's inequality we estimate $\hbox{I}I_1$ as
\begin{align*}
\lvert\,\hbox{I}I_1\rvert
\leq &\varepsilon\Xint{-}_{B_\rho(x_0)}\psi^q(1+\lvert\nu\rvert)^{q-2}\lvert Du-\nu\rvert^2 dx
+\frac{1}{\varepsilon}\Xint{-}_{B_\rho(x_0)}(1+\lvert\nu\rvert)^q
\kappa^2(\lvert\xi\rvert+\lvert\nu\rvert)\eta\left(\lvert u-\ell\rvert^2\right) dx.
\end{align*}
Note that our choice $\rho\leq\rho_1(\lvert\xi\rvert,\lvert\nu\rvert)$ allow us to apply ($\eta 5$), so that we get
\begin{align*}
\lvert\,\hbox{I}I_1\rvert
\leq &\varepsilon\Xint{-}_{B_\rho(x_0)}\psi^q(1+\lvert\nu\rvert)^{q-2}\lvert Du-\nu\rvert^2dx
+\frac{1}{\varepsilon}\Xint{-}_{B_\rho(x_0)}(1+\lvert\nu\rvert)^{q-2}\left\lvert \frac{u-\ell}{\rho}\right\rvert^2dx \\
&+\frac{1}{\varepsilon}\Xint{-}_{B_\rho(x_0)}(1+\lvert\nu\rvert)^q \kappa^2(\lvert\xi\rvert+\lvert\nu\rvert)
\eta\Bigl(\rho^2(1+\lvert\nu\rvert)^2 \kappa^2(\lvert\xi\rvert+\lvert\nu\rvert)\Bigr)dx.
\end{align*}
Using the definition of $G(\cdot,\cdot)$ and the fact that $\eta(ct)\leq c\eta(t)$ for $c\geq 1$, we deduce
\begin{align*}
\lvert\,\hbox{I}I_1\rvert
\leq &\varepsilon\Xint{-}_{B_\rho(x_0)}\psi^q(1+\lvert\nu\rvert)^{q-2}\lvert Du-\nu\rvert^2dx
+\frac{1}{\varepsilon}\Xint{-}_{B_\rho(x_0)}(1+\lvert\nu\rvert)^{q-2}\left\lvert \frac{u-\ell}{\rho}\right\rvert^2dx \\
&+\frac{1}{\varepsilon}(1+\lvert\nu\rvert)^qG(\lvert\xi\rvert,\lvert\nu\rvert)\eta\left(\rho^2\right).
\end{align*}
Similarly we see
\begin{align*}
\lvert\,\hbox{I}I_2\rvert
\leq &c(q,\varepsilon)
\Xint{-}_{B_\rho(x_0)}(1+\lvert\nu\rvert)^{q-2}\left\lvert \frac{u-\ell}{\rho}\right\rvert^2dx
+c(q,\varepsilon)(1+\lvert\nu\rvert)^qG(\lvert\xi\rvert,\lvert\nu\rvert)\eta\left(\rho^2\right).
\end{align*}
Combining these two estimates and get
\begin{align}
\lvert\,\hbox{I}I\,\rvert
\leq &\varepsilon\Xint{-}_{B_\rho(x_0)}\psi^q(1+\lvert\nu\rvert)^{q-2}\lvert Du-\nu\rvert^2dx
+c(q,\varepsilon)\Xint{-}_{B_\rho(x_0)}(1+\lvert\nu\rvert)^{q-2}\left\lvert\frac{u-\ell}{\rho}\right\rvert^qdx \notag \\
&+c(q,\varepsilon)(1+\lvert\nu\rvert)^qG(\lvert\xi\rvert,\lvert\nu\rvert)\eta\left(\rho^2\right) . \label{caccio-II}
\end{align}
In the same way we derive
\begin{align}
\lvert\,\hbox{I}II\,\rvert
\leq & \Xint{-}_{B_\rho(x_0)}(1+\lvert\nu\rvert)^{q-1}\kappa(\lvert\xi\rvert+\lvert\nu\rvert)
\mu\left( (1+\lvert\nu\rvert)\rho\right)\psi^q\lvert Du-\nu\rvert dx \notag\\
&+ \Xint{-}_{B_\rho(x_0)}(1+\lvert\nu\rvert)^{q-1}\kappa(\lvert\xi\rvert+\lvert\nu\rvert\rho)
\mu\bigl( (1+\lvert\nu\rvert)\rho\bigr) 4q\left\lvert\frac{u-\ell}{\rho}\right\rvert dx \notag\\
\leq & \varepsilon\Xint{-}_{B_\rho(x_0)}\psi^q(1+\lvert\nu\rvert)^{q-2}\lvert Du-\nu\rvert^2dx
+\varepsilon\Xint{-}_{B_\rho(x_0)}(1+\lvert\nu\rvert)^{q-2}\left\lvert\frac{u-\ell}{\rho}\right\rvert^2dx \notag\\
&+c(q,\varepsilon)(1+\lvert\nu\rvert)^qG(\lvert\xi\rvert,\lvert\nu\rvert)\eta(\rho^2). \label{caccio-III}
\end{align}
For $\varepsilon'>0$ to be fixed later, using ({\bf H4}), Lemma \ref{Young2}, and Young's inequality, we have
\begin{align}
\lvert\,\hbox{I}V\,\rvert
\leq & \Xint{-}_{B_\rho(x_0)}(a\lvert Du\rvert^q+b)\psi^q\lvert u-\ell\rvert dx \notag\\
\leq & a(1+\varepsilon')\Xint{-}_{B_\rho(x_0)}\psi^q\lvert Du-\nu\rvert^qlvert u-\ell\rvert dx
+\varepsilon b^2\rho^2 +\frac{1}{\varepsilon}\Xint{-}_{B_\rho(x_0)}\left\lvert\frac{u-\ell}{\rho}\right\rvert^2dx \notag\\
&+\Xint{-}_{B_\rho(x_0)}\left\{ aK(q,\varepsilon')\rho\lvert\nu\rvert^{(q+2)/2}\right\}
(1+\lvert\nu\rvert)^{(q-2)/2}\left\lvert\frac{u-\ell}{\rho}\right\rvert dx \notag\\
\leq & a(1+\varepsilon^\prime)(2M+\lvert\nu\rvert\rho)
\Xint{-}_{B_\rho(x_0)}\psi^q\lvert Du-\nu\rvert^qdx +\frac{2}{\varepsilon}\Xint{-}_{B_\rho(x_0)}
(1+\lvert\nu\rvert)^{q-2}\left\lvert\frac{u-\ell}{\rho}\right\rvert^2dx \notag\\
&+\varepsilon(1+\lvert\nu\rvert)^q\rho^2\left\{ aK(q,\varepsilon')\lvert\nu\rvert +b\right\}^2. \label{caccio-IV}
\end{align}
Combining above estimates, from \eqref{caccio-devide} to \eqref{caccio-IV}, and set
$\lambda'=2^{(12-9q)/2}\lambda$C
$\Lambda :=\lambda'-3\varepsilon-a(1+\varepsilon')(2M+\lvert\nu\rvert\rho)$, this gives
\begin{align}
&\Lambda\Xint{-}_{B_\rho(x_0)}\psi^q \left\{(1+\lvert\nu\rvert)^{q-2}\lvert Du-\nu\rvert^2+\lvert Du-\nu\rvert^q\right\}dx \notag\\
\leq & c(q,L,\varepsilon)
\left[\Xint{-}_{B_\rho(x_0)} \left\{(1+\lvert\nu\rvert)^{q-2}\left\lvert\frac{u-\ell}{\rho}\right\rvert^2
+\left\lvert\frac{u-\ell}{\rho}\right\rvert^q\right\}dx
+(1+\lvert\nu\rvert)^qG(\lvert\xi\rvert,\lvert\nu\rvert)\eta(\rho^2) \right] \notag\\
&+\varepsilon(1+\lvert\nu\rvert)^q\left\{ aK\lvert\nu\rvert +b\right\}^2\rho^2. \label{roughcaccio}
\end{align}
Now choose $\varepsilon=\varepsilon(\lambda ,p,a(M),M)>0$ and $\varepsilon'=\varepsilon'(\lambda ,p,a(M),M)>0$ in a right way
(for more precise way of choosing $\varepsilon$ and $\varepsilon'$, we refer to \cite[Lemma 4.1]{DG}),
we obtain \eqref{caccioppoli}.
\end{proof}
\section{Approximatively $\mathcal{A}$-harmonic functions}
\begin{lem}\label{A-harm2}
Under the same assumption in Lemma \ref{caccioppoli}, take $\xi=u_{x_0,\rho}$. Then for any $x_0\in\Omega$
and $\rho\leq \rho_1(\lvert\xi\rvert,\lvert\nu\rvert)$ satisfy $B_\rho(x_0)\Subset\Omega$, the inequality
\begin{align}
\Xint{-}_{B_\rho(x_0)}\mathcal{A} (Dv,D\varphi )dx \leq C_2(1+\lvert\nu\rvert)&\Biggl[
\omega^{1/2}\left(lvert\xi\rvert+\lvert\nu\rvert,\Phi(x_0,\rho,\nu)\right)\Phi^{1/2}(x_0,\rho,\nu) \notag\\
&+\Phi(x_0,\rho,\nu) +G(\lvert\xi\rvert,\lvert\nu\rvert)\sqrt{\eta(\rho^2)}
+\rho(a\lvert\nu\rvert +b)\Biggr]\sup_{B_\rho(x_0)}\lvert D\varphi\rvert \label{Ah}
\end{align}
holds for all $\varphi\in C^\infty_0(B_\rho(x_0),\mathbb{R}^N)$. Where
\begin{align*}
v :=& u-\ell = u-\xi-\nu(x-x_0), \\
\mathcal{A} (Dv,D\varphi ):=& \frac{1}{(1+\lvert\nu\rvert)^{q-1}}\left\langle D_pA(x_0,\xi,\nu )Dv,D\varphi\right\rangle, \\
\Phi(x_0,\rho,\nu):=& \Xint{-}_{B_\rho(x_0)}\left\{\frac{\lvert Du-\nu\rvert^2}{(1+\lvert\nu\rvert)^2}
+\frac{\lvert Du-\nu\rvert^q}{(1+\lvert\nu\rvert)^q}\right\} dx
\end{align*}
and $C_2\geq 1$ depending only on $n$, $q$, $L$ and $a(M)$.
\end{lem}
\begin{proof}
Assume $x_0\in\Omega$ and $\rho\leq 1$ which satisfies $B_{\rho}(x_0)\Subset\Omega$ and
$\rho\leq\rho_1(\lvert\xi\rvert,\lvert\nu\rvert)$.
Without loss of generality we may assume $\displaystyle\sup_{B_\rho(x_0)}\lvert D\varphi\rvert\leq 1$.
Note that this implies $\displaystyle\sup_{B_\rho(x_0)}\lvert\varphi\rvert\leq\rho\leq 1$. Using the fact that
$\int_{B_\rho(x_0)}A(x_0,\xi,\nu)D\varphi dx=0$ holds for all $\varphi\in C^\infty_0(B_\rho(x_0),\mathbb{R}^N)$ we deduce
\begin{align}
(1+\lvert\nu\rvert)^{q-1}&\Xint{-}_{B_\rho(x_0)}\mathcal{A}(Dv,D\varphi)dx \notag\\
=&\Xint{-}_{B_\rho(x_0)}\int_0^1\langle\left[ D_p A(x_0,\xi,\nu)- D_pA(x_0,\xi,\nu+s(Du-\nu))
\right] (Du-\nu),D\varphi\rangle dsdx \notag\\
&+\Xint{-}_{B_\rho(x_0)}\langle A(x_0,\xi,Du) -A(x,\ell,Du),D\varphi\rangle dx \notag\\
&+\Xint{-}_{B_\rho(x_0)}\langle A(x,\ell,Du)-A(x,u,Du),D\varphi\rangle dx \notag\\
&+\Xint{-}_{B_\rho(x_0)}\langle f,\varphi\rangle dx \notag\\
=&:\hbox{I}+\hbox{I}I+\hbox{I}II+\hbox{I}V \label{Ah-devide}
\end{align}
where terms I, II, III and IV are define above.
We estimate I using the modulus of continuity $\omega(\cdot,\cdot)$ from ({\bf H1}), the Jensen's inequality
and H\"{o}lder's inequality, and we get
\begin{align}
\lvert\,\hbox{I}\,\rvert
&\leq c(q,L)\Xint{-}_{B_\rho(x_0)}\int_0^1 \omega\left(\lvert\xi\rvert+\lvert\nu\rvert,\lvert Du-\nu\rvert^2\right)
(1+\lvert\nu\rvert+\lvert Du-\nu\rvert)^{q-2}\lvert Du-\nu\rvert dsdx \notag\\
&\leq
c\, (1+\lvert\nu\rvert)^{q-1} \Xint{-}_{B_\rho(x_0)}\omega\left(\lvert\xi\rvert+\lvert\nu\rvert,\lvert Du-D\ell\rvert^2\right)
\left\{\frac{\lvert Du-\nu\rvert}{1+\lvert\nu\rvert} +\frac{\lvert Du-\nu\rvert^{q-1}}{(1+\lvert\nu\rvert)^{q-1}}\right\}dx \notag\\
&\leq c\, (1+\lvert\nu\rvert)^{q-1}\left[ \omega^{1/2}\left(\lvert\xi\rvert+\lvert\nu\rvert,
(1+\lvert\nu\rvert)^2\Phi(x_0,\rho,\nu)\right)\Phi^{1/2}(x_0,\rho,\nu)\right. \notag\\
&\qquad\qquad\qquad\quad \left. +\omega^{1/q}\left(\lvert\xi\rvert+\lvert\nu\rvert,(1+\lvert\nu\rvert)^2\Phi(x_0,\rho,\nu)\right)
\Phi^{1/{q_*}}(x_0,\rho,\ell)\right]\notag\\
&\leq c\, (1+\lvert\nu\rvert)^q\left[ \omega^{1/2}\left(\lvert\xi\rvert+\lvert\nu\rvert,\Phi(x_0,\rho,\nu)\right)
\Phi^{1/2}(x_0,\rho,\nu)+\Phi(x_0,\rho,\nu)\right] , \label{Ah-I}
\end{align}
where $q_*>0$ is the dual exponent of $q\geq 2$, i.e., $q_*=q/(q-1)$.
The last inequality following from the fact that $a^{1/q}b^{1/q_*}=a^{1/q}b^{1/q}b^{(q-2)/q}\leq
a^{1/2}b^{1/2}+b$ holds by Young's inequality and the fact that $\omega(s,ct)\leq c\omega(s,t)$ for $c\geq 1$ which deduce from
the concavity of $t\mapsto \omega(s,t)$.
In the same way, using the modulus of continuity $\eta(\cdot)$ from ({\bf H3}), Young's inequality and, we deduce
\begin{align}
\lvert\,\hbox{I}I\,\rvert
\leq &2^{q-2}\kappa(\lvert\xi\rvert+\lvert\nu\rvert)(1+\lvert\nu\rvert)^q \sqrt{\eta(\rho^2)} \notag\\
&+2^{q-2}\Xint{-}_{B_\rho(x_0)}\kappa(\lvert\xi\rvert+\lvert\nu\rvert)
\sqrt{\eta(\rho^2(1+\lvert\nu\rvert)^2)} \lvert Du-\nu\rvert^{q-1}dx \notag\\
\leq &2^{q-1}(1+\lvert\nu\rvert)^qG(\lvert\xi\rvert,\lvert\nu\rvert)\sqrt{\eta(\rho^2)}
+2^{q-2}(1+\lvert\nu\rvert)^q\Phi(x_0,\rho,\nu) .
\end{align}
Here we have used $\eta^{q/2}(\rho^2(1+\lvert\nu\rvert)^2)\leq \sqrt{\eta(\rho^2(1+\lvert\nu\rvert)^2)}$ which follows from
the nondecreasing property of $t\mapsto \eta(t)$, ($\eta$4) and our assumption $\rho\leq\rho_1\leq 1$.
We derive, using again the modulus of continuity $\eta(\cdot)$ from ({\bf H3}),
\begin{align*}
\lvert\,\hbox{I}II\,\rvert
\leq & c(q)\Xint{-}_{B_\rho(x_0)}\kappa(\lvert\xi\rvert+\lvert\nu\rvert)\sqrt{\eta(\lvert u-\ell\rvert^2)}(1+\lvert\nu\rvert)^{q-1}dx \\
&+c(q)\Xint{-}_{B_\rho(x_0)}\kappa(\lvert\xi\rvert+\lvert\nu\rvert)\sqrt{\eta(\lvert u-\ell\rvert^2)}\lvert Du-\nu\rvert^{q-1}dx \\
=: &\hbox{I}II_1+\hbox{I}II_2,
\end{align*}
where the terms $\hbox{I}II_1$ and $\hbox{I}II_2$ are defined above.
Using H\"{o}lder's inequality, Jensen's inequality, ($\eta$6) and the Poincar\'{e} inequality, we have
\begin{align*}
\hbox{I}II_1
&\leq c(q) (1+\lvert\nu\rvert)^{q-1}\kappa(\lvert\xi\rvert+\lvert\nu\rvert)
\eta^{1/2}\left(\Xint{-}_{B_\rho(x_0)}\lvert u-\ell\rvert^2 dx\right) \\
&\leq c\, \rho^{-2}(1+\lvert\nu\rvert)^{q-2}\left\{\rho^2(1+\lvert\nu\rvert)^2
\kappa^2(\lvert\xi\rvert+\lvert\nu\rvert)\eta^{1/2}\Bigl(\rho^2(1+\lvert\nu\rvert)^2\kappa^2(\lvert\xi\rvert+\lvert\nu\rvert)\Bigr)
+\Xint{-}_{B_\rho(x_0)}\lvert u-\ell\rvert^2 dx\right\} \\
&\leq c(q)(1+\lvert\nu\rvert)^qG(\lvert\xi\rvert,\lvert\nu\rvert)\sqrt{\eta\left(\rho^2\right)}
+c(n,q)(1+\lvert\nu\rvert)^q\Phi(x_0,\rho,\nu) .
\end{align*}
Similarly, we have, using Young's inequality, ($\eta$5) and the Poincar\'{e} inequality,
\begin{align*}
\hbox{I}II_2
\leq &c(q)\Xint{-}_{B_\rho(x_0)}\kappa^q(\lvert\xi\rvert+\lvert\nu\rvert)\eta^{q/2}\left(\lvert u-\ell\rvert^2\right) dx
+c(q)\Xint{-}_{B_\rho(x_0)}\lvert Du-\nu\rvert^qdx \\
\leq &c\, \Xint{-}_{B_\rho(x_0)}\Bigl[\rho^{-2}\left\{
\kappa^2(\lvert\xi\rvert+\lvert\nu\rvert)\rho^2\eta\left(\kappa^2(\lvert\xi\rvert+\lvert\nu\rvert)\rho^2\right)
+\lvert u-\ell\rvert^2\right\} \Bigr]^{q/2} dx
+c\, (1+\lvert\nu\rvert)^q\Phi(x_0,\rho,\nu) \\
\leq &c(1+\lvert\nu\rvert)^qG(\lvert\xi\rvert,\lvert\nu\rvert)\sqrt{\eta\left(\rho^2\right)}
+c(n,q)(1+\lvert\nu\rvert)^q\Phi(x_0,\rho,\nu) .
\end{align*}
Thus we obtain
\begin{align}
\lvert\,\hbox{I}II\,\rvert
\leq &c(q)(1+\lvert\nu\rvert)^qG(\lvert\xi\rvert,\lvert\nu\rvert)\sqrt{\eta\left(\rho^2\right)}
+c(n,q)(1+\lvert\nu\rvert)^q\Phi(x_0,\rho,\nu).
\end{align}
Using ({\bf H4}) and recall that $\displaystyle\sup_{B_\rho(x_0)}\lvert\varphi\rvert\leq\rho$ holds, we have
\begin{align}
\lvert\,\hbox{I}V\,\rvert
&\leq \Xint{-}_{B_\rho(x_0)}\rho a(\lvert Du-\nu\rvert+\lvert\nu\rvert)^qdx +b\rho \notag\\
&\leq 2^{q-1}a(1+\lvert\nu\rvert)^q\Phi(x_0,\rho,\nu)
+2^{q-1}\rho(1+\lvert\nu\rvert)^q(a\lvert\nu\rvert +b). \label{25}
\end{align}
Combining these estimates, from \eqref{Ah-devide} to (\ref{25}), we obtain the conclusion.
\end{proof}
\section{Proof of the Regularity Theorem}
Let write $\Phi(\rho) =\Phi(x_0,\rho,(Du)_{x_0,\rho})$ from now on.
Now we are in the position to establish the excess improvement.
\begin{lem}\label{EI}
Assume the same assumption with Lemma \ref{A-harm2}. Let $\theta\in (0,1/8]$ be arbitrary and impose the
following smallness conditions on the excess:
\begin{enumerate}
\item[{\rm (i)}] $\displaystyle{\omega^{1/2}\left(\lvert u_{x_0,\rho}\rvert+\lvert (Du)_{x_0,\rho}\rvert, \Phi(\rho)\right)
+\sqrt{\Phi(\rho)}\leq\frac{\delta}{2}}$ with the constant
$\delta =\delta(n,N,q,\lambda,L,\theta^{n+q+2})$ from Lemma \ref{A-harm};
\item[{\rm (ii)}] $(1+\lvert(Du)_{x_0,\rho}\rvert)\gamma(\rho)\leq \theta^n\left( 2\sqrt{C_0\tilde{C}}\right)^{-1}$, where \\
$C_0$ and $\tilde{C}$ are constants from Theorem \ref{Campanato} and Lemma \ref{A-harm}, and \\
$\displaystyle{\gamma(\rho):=C_2\left[\sqrt{\Phi(\rho)}+2\delta^{-1}
\left\{ G(\lvert u_{x_0,\rho}\rvert,\lvert(Du)_{x_0,\rho}\rvert)\sqrt{\eta(\rho^2)}+\rho(a(1+\lvert(Du)_{x_0,\rho}\rvert)+b)
\right\}\right]}$.
\item[{\rm (iii)}] $\rho\leq\rho_1(\lvert u_{x_0,\rho}\rvert,\lvert(Du)_{x_0,\rho}\rvert)$.
\end{enumerate}
Then there holds the excess improvement estimate
\begin{equation}
\Phi(\theta\rho)\leq C_3\theta^2\Phi(\rho)+H(\lvert u_{x_0,\rho}\rvert,\lvert(Du)_{x_0,\rho}\rvert)\eta(\rho^2),
\end{equation}
with a constant $C_3$ that depends only on $n$, $N$, $\lambda$, $L$, $q$, $a(M)$ and $M$. Here $H(\cdot,\cdot)$ is defined as
\[
H(s,t):=8\delta^{-2}C_3\left[ G^2(1+s,1+t)+\{a(1+t)+b\}\right].
\]
\end{lem}
\begin{proof}
We consider $B_\rho(x_0)\Subset\Omega$ and set $\xi=u_{x_0,\rho}, \nu=(Du)_{x_0,\rho},\ell(x)=\xi+\nu(x-x_0)$.
Assume (i), (ii) and (iii) are satisfied and we rescale $u$ as
\[
w:=\frac{u-\ell}{(1+\lvert\nu\rvert)\gamma}.
\]
Applying Lemma \ref{A-harm2} on $B_\rho(x_0)$ to $w$ and combining (i), we obtain
\begin{align*}
&\Xint{-}_{B_\rho(x_0)}\mathcal{A}(Dw,D\varphi)dx \\
\leq & \left[\omega^{1/2}\left(\lvert\xi\rvert+\lvert\nu\rvert,\sqrt{\Phi(\rho)}\right)
+\sqrt{\Phi(\rho)}+\frac{\delta}{2} \right]
\sup_{B_{\rho}(x_0)}\lvert D\varphi\rvert \\
\leq& \delta\sup_{B_{\rho}(x_0)}\lvert D\varphi\rvert
\end{align*}
for all $\varphi\in C_0^\infty(B_\rho(x_0),\mathbb{R}^N)$.
Moreover, we have, note that $\gamma\geq C_2\sqrt{\Phi(\rho)}$ holds from the definition of $\gamma$,
\begin{align*}
\Xint{-}_{B_{\rho}(x_0)}\left\{\lvert Dw\rvert^2+\gamma^{q-2}\lvert Dw\rvert^p\right\}dx
=&\Xint{-}_{B_{\rho}(x_0)}\left\{ \frac{\lvert Du-\nu\rvert^2}{\gamma^2(1+\lvert\nu\rvert)^2}
+\gamma^{q-2}\frac{\lvert Du-\nu\rvert^q}{\gamma^q(1+\lvert\nu\rvert)^q} \right\} dx \\
\leq & \frac{\Phi(\rho)}{\gamma^2}
\leq \frac{1}{{C_2}^2}
\leq 1.
\end{align*}
Thus, these two inequalities allow us to apply the $\mathcal{A}$-harmonic approximation lemma (Lemma \ref{A-harm}), to
conclude the existence of an $\mathcal{A}$-harmonic function $h$ satisfying
\begin{align}
\Xint{-}_{B_{\rho/2}(x_0)}&\left\{\left\lvert\frac{w-h}{\rho/2}\right\rvert^2
+\gamma^{q-2}\left\lvert\frac{w-h}{\rho/2}\right\rvert^q\right\} dx
\leq\theta^{n+q+2}, \quad \text{and} \\
\Xint{-}_{B_{\rho/2}(x_0)}&\left\{\lvert Dh\rvert^2
+\gamma^{q-2}\lvert Dh\rvert^q\right\}dx\leq \tilde{C}, \label{energybound}
\end{align}
where we taken $\varepsilon=\theta^{n+q+2}$. From Theorem \ref{Campanato} and \eqref{energybound} we have
\[
\sup_{B_{\rho/4}(x_0)}\lvert D^2h\rvert^2\leq 4C_0\tilde{C}\rho^{-2}.
\]
From this we infer the following estimate for $s=2$ as well as for $s=q$,
\[
\sup_{B_{\rho/4}(x_0)}\lvert D^2h\rvert^s
\leq c(n,N,\lambda, L, q,s)\rho^{-s}.
\]
For $\theta\in(0,1/8]$, Taylor's theorem applied to $h$ at $x_0$ yields
\[
\sup_{x\in B_{2\theta\rho}(x_0)}\lvert h(x)-h(x_0)-Dh(x_0)(x-x_0)\rvert^s
\leq c(n,N,\lambda, L, q,s)\theta^{2s}\rho^s.
\]
We have then
\begin{align*}
&\gamma^{s-2}(2\theta\rho)^{-s}\Xint{-}_{B_{2\theta\rho}(x_0)}
\lvert w-h(x_0)-Dh(x_0)(x-x_0)\rvert^sdx \\
\leq & c(s)\gamma^{s-2}(2\theta\rho)^{-s}\left[\Xint{-}_{B_{2\theta\rho}(x_0)}\lvert w-h\rvert^sdx
+\Xint{-}_{B_{2\theta\rho}(x_0)}\lvert h(x)-h(x_0)-Dh(x_0)(x-x_0)\rvert^sdx\right]\\
\leq & c(n,N,\lambda,L,q,s)\theta^2.
\end{align*}
Recall that the mean-value of $u-(\nu+\gamma(1+\lvert\nu\rvert)Dh(x_0))(x-x_0)$ on $B_{2\theta\rho}(x_0)$ is $u_{x_0,2\theta\rho}$,
we have
\begin{align}
&(2\theta\rho)^{-s}\Xint{-}_{B_{2\theta\rho}(x_0)}\lvert u-u_{x_0,2\theta\rho}
-\left(\nu+\gamma(1+\lvert\nu\rvert)Dh(x_0)\right)(x-x_0)\rvert^sdx \notag\\
\leq&
c(s)(2\theta\rho)^{-s}\gamma^s(1+\lvert\nu\rvert)^s
\Xint{-}_{B_{2\theta\rho}(x_0)}\lvert w-h(x_0)-Dh(x_0)(x-x_0)\rvert^sdx \notag\\
\leq & c(n,N,\lambda,L,q,s)(1+\lvert\nu\rvert)^s\theta^2\gamma^2. \label{scaling}
\end{align}
By assumption (ii), we infer $\sqrt{\Phi(\rho)}\leq \theta^n/2$. This yields
\begin{equation}
\lvert(Du)_{x_0,\theta\rho}-\nu\rvert
\leq\theta^{-n}\Xint{-}_{B_{\rho}(x_0)}\lvert Du-\nu\rvert dx
\leq \theta^{-n}(1+\lvert\nu\rvert)\sqrt{\Phi(\rho)}
\leq \frac{1}{2}(1+\lvert\nu\rvert).
\end{equation}
Thus, combining with the estimate $1+\lvert\nu\rvert\leq1+\lvert(Du)_{x_0,\theta\rho}\rvert
+\lvert(Du)_{x_0,\theta\rho}-\nu\rvert$, we obtain
\begin{equation}
1+\lvert\nu\rvert\leq 2(1+\lvert(Du)_{x_0,\theta\rho}\rvert). \label{nuesti}
\end{equation}
Set $P_0=\nu+\gamma(1+\lvert\nu\rvert)Dh(x_0)$. Then Theorem \ref{Campanato}, \eqref{energybound} and assumption (ii) imply
\begin{align}
\lvert P_0\rvert\leq \lvert\nu\rvert+\lvert\gamma(1+\lvert\nu\rvert)Dh(x_0)\rvert
\leq \lvert\nu\rvert+\gamma(1+\lvert\nu\rvert)\sqrt{C_0c(n,q)}
\leq \frac{1}{2}+\lvert\nu\rvert. \label{P-esti}
\end{align}
Therefore, combining with \eqref{nuesti}, we have
\begin{equation}
1+\lvert P_0\rvert\leq 3(1+\lvert(Du)_{x_0,\theta\rho}\rvert).
\end{equation}
Applying Caccioppoli-type inequality (Lemma \ref{caccioppoli}) on $B_{2\theta\rho}(x_0)$ with $\xi=u_{x_0,2\theta\rho}$
and $\nu=P_0$ yields
\begin{align}
\Phi(\theta\rho) \leq &6^q\Phi(x_0,\theta\rho,P_0) \notag\\
\leq & 6^qC_1\Bigg[ \Xint{-}_{B_{2\theta\rho}(x_0)}\left\{
\left\lvert\frac{u-u_{x_0,2\theta\rho}-P_0(x-x_0)}{2\theta\rho(1+\lvert P_0\rvert)}\right\rvert^2+
\left\lvert\frac{u-u_{x_0,2\theta\rho}-P_0(x-x_0)}{2\theta\rho(1+\lvert P_0\rvert)}\right\rvert^q\right\} dx \notag\\
&\qquad +G(\lvert u_{x_0,2\theta\rho}\rvert,\lvert P_0\rvert)\eta((2\theta\rho)^2)
+\left( a\lvert P_0\rvert+b\right)^2 (2\theta\rho)^2 \Bigg]. \label{appliedcaccioppoli}
\end{align}
Using H\"{o}lder's inequality, the Poincar\'{e} inequality and assumption (ii) we have
\begin{align}
\lvert u_{x_0,2\theta\rho}\rvert \leq &\lvert u_{x_0,\rho}\rvert+\left\lvert\Xint{-}_{B_{2\theta\rho}(x_0)}
(u-u_{x_0,\rho}-\nu(x-x_0))dx\right\rvert \notag\\
\leq &\lvert u_{x_0,\rho}\rvert+\left(\Xint{-}_{B_{2\theta\rho}(x_0)}
\lvert u-u_{x_0,\rho}-\nu(x-x_0)\rvert^2dx\right)^{1/2} \notag\\
\leq &\lvert u_{x_0,\rho}\rvert+(2\theta)^{-n/2}\left(\Xint{-}_{B_\rho(x_0)}
\lvert u-u_{x_0,\rho}-\nu(x-x_0)\rvert^2dx\right)^{1/2} \notag\\
\leq &\lvert u_{x_0,\rho}\rvert+\theta^{-n/2}\sqrt{C_P}(1+\lvert\nu\rvert)\sqrt{\Phi(\rho)} \notag\\
\leq &\lvert u_{x_0,\rho}\rvert+\theta^{-n/2}\frac{\sqrt{C_P}}{C_2}(1+\lvert\nu\rvert)\gamma \notag\\
\leq &\lvert u_{x_0,\rho}\rvert+1.
\end{align}
Set $H_0(s,t):=G^2(1+s,1+t)+\{ a(1+t)+b\}^{q_*}$ and using \eqref{P-esti} we obtain
\begin{equation}
G(\lvert u_{x_0,2\theta\rho}\rvert,\lvert P_0\rvert)\eta((2\theta\rho)^2)
+\left( a\lvert P_0\rvert+b\right)^2 (2\theta\rho)^2
\leq H_0(\lvert\xi\rvert,\lvert\nu\rvert)\eta(\rho^2). \label{Hesti}
\end{equation}
The definition of $\gamma$ and $H_0$ imply
\begin{align}
\gamma^2 \leq &2{C_2}^2\left[\Phi(\rho)+4\delta^{-2}
\left\{ G(\lvert\xi\rvert,\lvert\nu\rvert)\sqrt{\eta(\rho^2)}+\rho(a(1+\lvert\nu\rvert)+b)\right\}^2\right] \notag\\
\leq &2{C_2}^2\left[\Phi(\rho)+ 8\delta^{-2}H_0(\lvert\xi\rvert,\lvert\nu\rvert)\eta(\rho^2)\right] . \label{gammaesti}
\end{align}
Plugging \eqref{scaling}, \eqref{Hesti}, \eqref{gammaesti} into \eqref{appliedcaccioppoli}, we deduce
\begin{align*}
\Phi(\theta\rho) \leq &6^qC_1\bigg[ c(n,N,\lambda,L,q)\theta^2\gamma^2
+G(\lvert u_{x_0,2\theta\rho}\rvert,\lvert P_0\rvert)\eta((2\theta\rho)^2)
+\left( a\lvert P_0\rvert+b\right)^2 (2\theta\rho)^2 \bigg] \\
\leq &6^qC_1\bigg[ c\,\theta^2{C_2}^2\{\Phi(\rho)+\delta^{-2}H_0(\lvert\xi\rvert,\lvert\nu\rvert)\eta(\rho^2)\}
+H_0(\lvert\xi\rvert,\lvert\nu\rvert)\eta(\rho^{q_*}) \bigg] \notag\\
\leq & C_3\bigg[ \theta^2\Phi(\rho)+8\delta^{-2}H_0(\lvert\xi\rvert,\lvert\nu\rvert)\eta(\rho^2)\bigg],
\end{align*}
and this complete the proof.
\end{proof}
For $\sigma\in[\alpha,1)$ we find $\theta\in(0,1/8]$ such that $C_3\theta^2\leq \theta^{2\sigma}/2$.
For $T_0 \geq 1$ there exists $\Phi_0>0$ such that
\begin{align}
&\omega^{1/2}\left( 2T_0,\sqrt{2\Phi_0}\right)+\sqrt{2\Phi_0}\leq \frac{\delta}{2}, \label{sc1}\\
&2C_4(1+2T_0)\sqrt{2\Phi_0}\leq \theta^n, \label{sc2}
\end{align}
where $C_4:=C_3\left( 1+\sqrt{C_P}\>\right)$. Note that $\Phi_0<1$. Then choose $0<\rho_0\leq 1$ such that
\begin{align}
&C_5\sqrt{\eta(\rho_0)}\leq\Phi_0, \label{sc3} \\
&\frac{(1+2T_0)(1+\sqrt{C_P})}{\theta^{n/2}}
\sqrt{\frac{C_5\alpha^2\beta^2 \widetilde{F}({\rho_0}^2)}{4(1-\theta^{\alpha\beta})^2}}
\leq \frac{1}{2}T_0, \label{sc4}
\end{align}
where
\[
C_5=C_5(n,N,\lambda,L,q,a,M,\alpha,\sigma,T_0):=\frac{2H(2T_0,2T_0)}{2\theta^{2\alpha}-\theta^{2\sigma}}.
\]
\begin{lem}\label{Iteration}
Assume that for some $T_0\geq 1$ and $B_\rho(x_0)\Subset\Omega$ we have
\begin{enumerate}
\item[(a)] $\lvert u_{x_0,\rho}\rvert+\lvert(Du)_{x_0,\rho}\rvert\leq T_0$,
\item[(b)] $\Phi(\rho)\leq \Phi_0$,
\item[(c)] $\rho\leq \rho_0$.
\end{enumerate}
Then the smallness conditions {\rm (i), (ii)} and {\rm (iii)} are satisfied on $B_{\theta^k\rho}(x_0)$ for $k\in\mathbb{N}\cup\{ 0\}$ in
Lemma \ref{EI}. Moreover, the limit
\[
\Lambda_{x_0}:= \lim_{k\to\infty}(Du)_{x_0,\theta^k\rho}
\]
exists, and the inequality
\begin{equation}
\Xint{-}_{B_r(x_0)}\lvert Du-\Lambda_{x_0}\rvert^2dx \leq C_6
\left[ \left(\frac{r}{\rho}\right)^{2\sigma}\Phi(\rho)+\widetilde{F}(r^2)\right] \label{campanatosemi}
\end{equation}
is valid for $0<r\leq \rho$ with a constant $C_6=C_6(n,N,\lambda,L,q,a(M),M,\alpha,\beta,\sigma,T_0)$.
\end{lem}
\begin{proof}
Inductively we shall derive for $k\in\mathbb{N}\cup\{ 0\}$ the following three assertions:
\begin{enumerate}
\item[($\hbox{I}_k$)] $\Phi(\theta^k\rho)\leq 2\Phi_0$,
\item[($\hbox{I}I_k$)] $\lvert u_{x_0,\theta^k\rho}\rvert+\lvert(Du)_{x_0,\theta^k\rho}\rvert\leq 2T_0$,
\item[($\hbox{I}II_k$)] $\theta^k\rho\leq \rho_1(\lvert u_{x_0,\theta^k\rho}\rvert,\lvert(Du)_{x_0,\theta^k\rho}\rvert)$.
\end{enumerate}
We first note that ($\hbox{I}_k$), ($\hbox{I}I_k$) and \eqref{sc1} imply the smallness condition $(\text{i}_k)$, i.e. (i) with
$\theta^k\rho$ instead of $\rho$. Next we observe that ($\hbox{I}_k$), ($\hbox{I}I_k$), \eqref{sc2} and \eqref{sc3} yield
\begin{align*}
&(1+\lvert(Du)_{x_0,\theta^k\rho}\rvert)\left(2\sqrt{C_0\tilde{C}}\right)\gamma(\theta^k\rho) \\
\leq &(1+\lvert(Du)_{x_0,\theta^k\rho}\rvert)\left[C_3\sqrt{2\Phi_0}
+H(\lvert u_{x_0,\theta^k\rho}\rvert,\lvert(Du)_{x_0,\theta^k\rho}\rvert)\sqrt{\eta({\rho_0}^2)}\right] \\
\leq &(1+2T_0)\left[C_3\sqrt{2\Phi_0}+H(2T_0,2T_0)\sqrt{\eta({\rho_0}^2)}\right] \\
\leq &(1+2T_0)\left[C_3\sqrt{2\Phi_0}+\frac{2\theta^{2\alpha}-\theta^{2\sigma}}{2}\Phi_0\right] \\
\leq &2C_3(1+2T_0)\sqrt{2\Phi_0} \\
\leq &1.
\end{align*}
Thus we have $(\text{ii}_k)$. Note that $C_2\left(2\sqrt{C_0\tilde{C}}\right)\leq C_3$ and $\Phi_0> 1$ are hold
from there definition. Finally $(\text{iii}_k)$ is just ($\hbox{I}II_k$).
By (a), (b) and (c), there holds ($\hbox{I}_0$),($\hbox{I}I_0$) and ($\hbox{I}II_0$).
Now suppose that we have ($\hbox{I}_\ell$),($\hbox{I}I_\ell$) and ($\hbox{I}II_\ell$) for $\ell=0,1,\cdots,k-1$ with some $k\in\mathbb{N}$.
Then we can use Lemma \ref{EI} with $\rho,\theta\rho,\cdots,\theta^{k-1}\rho$, and yields
\begin{align*}
\Phi(\theta^k\rho)
\leq &\left(\frac{1}{2}\theta^{2\sigma}\right)^k\Phi(\rho)+\sum_{\ell=0}^{k-1}\left(\frac{1}{2}\theta^{2\sigma}\right)^\ell
H(\lvert u_{x_0,\theta^{k-1-\ell}\rho}\rvert,\lvert(Du)_{x_0,\theta^{k-1-\ell}}\rvert)\eta((\theta^{k-1-\ell}\rho)^2) \\
\leq &\left(\frac{1}{2}\theta^{2\sigma}\right)^k\Phi(\rho)+H(2T_0,2T_0)\sum_{\ell=0}^{k-1}
\left(\frac{1}{2}\theta^{2\sigma}\right)^\ell\eta((\theta^{k-1-\ell}\rho)^2) .
\end{align*}
The nondecreasing property of $t\mapsto t^{-\alpha}\eta(t)$ and the choice of $\sigma$ implies
\begin{align*}
\sum_{\ell=0}^{k-1}\left(\frac{1}{2}\theta^{2\sigma}\right)^\ell\eta\left((\theta^{k-1-\ell}\rho)^2\right)
\leq &\theta^{-2\alpha}\eta\left((\theta^k\rho)^2\right)\sum_{\ell =0}^{k-1}
\left(\frac{1}{2}\theta^{2\sigma-2\alpha}\right)^\ell \notag\\
\leq &\frac{2\eta\left((\theta^k\rho)^2\right)}{2\theta^{2\alpha}-\theta^{2\sigma}}.
\end{align*}
Therefore we have
\begin{equation}
\Phi(\theta^k\rho)\leq \left(\frac{1}{2}\theta^{2\sigma}\right)^k\Phi(\rho)
+C_5\eta\left((\theta^k\rho)^2\right). \label{iteration}
\end{equation}
Keeping in mind of (b), (c) and the choice of $\rho$, we prove ($\hbox{I}_k$). We next want to show ($\hbox{I}I_k$).
Using the fact that $\Xint{-}_{B_\rho(x_0)}\nu(x-x_0)dx=0$ holds for all $\nu\in\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$,
H\"{o}lder's inequality and the Poincar\'{e} inequality, we obtain
\begin{align*}
\lvert u_{x_0,\theta^k\rho}\rvert\leq &\lvert u_{x_0,\theta^{k-1}\rho}\rvert+
\left\lvert\Xint{-}_{B_{\theta^k\rho}(x_0)}(u-u_{x_0,\theta^{k-1}\rho}-(Du)_{x_0,\theta^{k-1}\rho}(x-x_0))dx\right\rvert \\
\leq &\lvert u_{x_0,\theta^{k-1}\rho}\rvert+
\theta^{-n/2}\sqrt{C_P}(1+\lvert(Du)_{x_0,\theta^{k-1}\rho}\rvert)\Phi^{1/2}(\theta^{k-1}\rho) \\
\leq &\lvert u_{x_0,\rho}\rvert+
\theta^{-n/2}\sqrt{C_P}\sum_{\ell =0}^{k-1}(1+\lvert(Du)_{x_0,\theta^\ell\rho}\rvert)\Phi^{1/2}(\theta^\ell\rho) .
\end{align*}
Similarly we see
\begin{align*}
\lvert(Du)_{x_0,\theta^k\rho}\rvert \leq &\lvert(Du)_{x_0,\theta^{k-1}\rho}\rvert+
\left\lvert\Xint{-}_{B_{\theta^k\rho}(x_0)}(Du-(Du)_{x_0,\theta^{k-1}\rho})dx\right\rvert \\
\leq &\lvert(Du)_{x_0,\rho}\rvert+
\theta^{-n/2}\sum_{\ell =0}^{k-1}(1+\lvert(Du)_{x_0,\theta^\ell\rho}\rvert)\Phi^{1/2}(\theta^\ell\rho) .
\end{align*}
Combining two estimates and using (\ref{iteration}) and (\ref{sumeta}) we infer
\begin{align*}
&\lvert u_{x_0,\theta^k\rho}\rvert+\lvert(Du)_{x_0,\theta^k\rho}\rvert \\
\leq &\lvert u_{x_0,\rho}\rvert+\lvert(Du)_{x_0,\rho}\rvert
+\frac{(1+\sqrt{C_P})(1+2T_0)}{\theta^{n/2}}\sum_{\ell =0}^{k-1}\Phi^{1/2}(\theta^\ell\rho) \\
\leq &\lvert u_{x_0,\rho}\rvert+\lvert(Du)_{x_0,\rho}\rvert
+\frac{(1+\sqrt{C_P})(1+2T_0)}{\theta^{n/2}}\sum_{\ell=0}^{k-1}\left\{\left(\frac{1}{\sqrt{2}}\theta^\sigma\right)^\ell\sqrt{\Phi(\rho)}
+\sqrt{C_5\eta(\theta^{2\ell}\rho^2)}\right\} \\
\leq &\lvert u_{x_0,\rho}\rvert+\lvert(Du)_{x_0,\rho}\rvert
+\frac{(1+\sqrt{C_P})(1+2T_0)}{\theta^{n/2}}\left\{\frac{\sqrt{2\Phi(\rho)}}{\sqrt{2}-\theta^\sigma}
+\sqrt{\frac{C_5\alpha^2\beta^2 \widetilde{F}(\rho^2)}{4(1-\theta^{\alpha\beta})^2}}\right\} \\
\leq & T_0+\frac{(1+\sqrt{C_P})(1+2T_0)}{\theta^{n/2}}\frac{\sqrt{2\Phi_0}}{\sqrt{2}-\theta^\sigma}
+\frac{(1+\sqrt{C_P})(1+2T_0)}{\theta^{n/2}}
\sqrt{\frac{C_5\alpha^2\beta^2 \widetilde{F}(\rho^2)}{4(1-\theta^{\alpha\beta})^2}} \\
\leq & T_0+ \frac{1}{\sqrt{2}-\theta^\sigma}\frac{\theta^{n/2}}{2}
+\frac{1}{2}T_0 \\
\leq & 2T_0.
\end{align*}
This proves ($\hbox{I}I_k$).
By hypothesis (c), ($\hbox{I}I_k$), ($\eta$4), the definition of $H$ and \eqref{sc3}, we easily derive
\begin{align*}
&(1+\lvert(Du)_{x_0,\theta^k\rho}\rvert)\kappa (\lvert u_{x_0,\theta^k\rho}\rvert+\lvert(Du)_{x_0,\theta^k\rho}\rvert)\theta^k\rho \\
\leq &H(2T_0,2T_0)\sqrt{\eta(\rho_0)} \\
\leq &1.
\end{align*}
Thus, we prove ($\hbox{I}II_k$).
We next want to prove that $(Du)_{x_0,\theta^k\rho}$ converges to some limit $\Lambda_{x_0}$ in
$\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$. Arguing as in the proof of ($\hbox{I}I_k$) we deduce for $k>j$
\begin{align}
\lvert(Du)_{x_0,\theta^k\rho}-(Du)_{x_0,\theta^j\rho}\rvert
\leq &\sum_{\ell =j+1}^k\lvert(Du)_{x_0,\theta^\ell\rho}-(Du)_{x_0,\theta^{\ell -1}\rho}\rvert \notag\\
\leq &\sum_{\ell =j+1}^k\theta^{-n/2}(1+\lvert(Du)_{x_0,\theta^{\ell -1}\rho}\rvert)\sqrt{\Phi(\theta^{\ell -1}\rho)} \notag\\
\leq &\frac{(1+2T_0)\sqrt{\theta^{2\sigma j}\Phi(\rho)}}{\theta^{n/2}(\sqrt{2}-\theta^\sigma)}
+\frac{(1+2T_0)}{\theta^{n/2}}
\sqrt{\frac{C_5\alpha^2\beta^2 \widetilde{F}(\theta^{2j}\rho^2)}{4(1-\theta^{\alpha\beta})^2}}. \label{cauchy}
\end{align}
Taking into account our assumption ($\eta$3) we see that
$\{ (Du)_{x_0,\theta^k\rho}\}_k$ is a Cauchy sequence in $\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^N)$. Therefore the limit
\[
\Lambda_{x_0}:=\lim_{k\to\infty}(Du)_{x_0,\theta^k\rho}
\]
exists and from \eqref{cauchy} we infer for $j\in\mathbb{N}\cup\{ 0\}$
\begin{align*}
\lvert(Du)_{x_0,\theta^j\rho}-\Lambda_{x_0}\rvert
\leq &\lvert(Du)_{x_0,\theta^k\rho}-(Du)_{x_0,\theta^j\rho}\rvert+\lvert(Du)_{x_0,\theta^k\rho}-\Lambda_{x_0}\rvert \\
\to & C_7\sqrt{\theta^{2\sigma j}\Phi(\rho)+\widetilde{F}(\theta^{2j}\rho^2)} \quad (\text{as}\ k\to\infty)
\end{align*}
where
\[
C_7:=\frac{\sqrt{2}(1+2T_0)}{\theta^{n/2}}\sqrt{\frac{1}{(\sqrt{2}-\theta^\sigma)^2}
+\frac{C_5\alpha^2\beta^2}{4(1-\theta^{\alpha\beta})^2}} .
\]
Combining this with \eqref{iteration}, and recalling the estimate \eqref{etaF} we arrive at
\begin{align*}
\Xint{-}_{B_{\theta^j\rho}(x_0)}\lvert Du-\Lambda_{x_0}\rvert^2dx
\leq &2(1+2T_0)\Phi(\theta^j\rho)+2\lvert(Du)_{x_0,\theta^j\rho}-\Lambda_{x_0}\rvert^2 \\
\leq & C_8 \left\{\theta^{2\sigma j}\Phi(\rho)+\widetilde{F}(\theta^{2j}\rho^2)\right\}
\end{align*}
with
\[
C_8:=2\left\{1+2T_0+{C_7}^2
+\frac{C_5\alpha^2\beta^2 (1+2T_0)}{4(1-\theta^{\alpha\beta})^2}\right\} .
\]
For $0<r\leq \rho$ we find $j\in\mathbb{N}\cup\{ 0\}$ such that $\theta^{j+1}\rho\leq r \leq \theta^j\rho$. Then using the
above estimate with \eqref{Fnoninc} implies
\begin{align*}
\Xint{-}_{B_r(x_0)}\lvert Du-\Lambda_{x_0}\rvert^2dx
&\leq \theta^{-n}\Xint{-}_{B_{\theta^j\rho}(x_0)}\lvert Du-\Lambda_{x_0}\rvert^2dx \\
&\leq C_8\theta^{-n}\{\theta^{2\sigma j}\Phi(\rho)+\widetilde{F}(\theta^{2j}\rho^2)\} \\
&\leq 4C_8\theta^{-n-2\sigma}\left\{\left(\frac{r}{\rho}\right)^{2\sigma}\Phi(\rho)+\widetilde{F}(r^2)\right\} .
\end{align*}
This proves \eqref{campanatosemi} with $C_6:=4C_8\theta^{-n-2\sigma}$.
\end{proof}
The main theorem (Theorem \ref{pr}) is obtained from Lemma \ref{Iteration} by using standard arguments. \\
\mbox{}\\
{\bf Acknowledgments}\\
The author thanks Professor Hisashi Naito for helpful discussions.
\def$'${$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\mbox{}\\
Taku Kanazawa \\
Graduate School of Mathematics \\
Nagoya University \\
Chikusa-ku, Nagoya, 464-8602, JAPAN \\
E-mail:[email protected]
\end{document} |
\begin{document}
\title{A variational approach to the alternating projections method}
\author{Carlo Alberto De Bernardi
}
\address{Dipartimento di Matematica per le Scienze economiche, finanziarie ed attuariali, Universit\`{a} Cattolica del Sacro Cuore, Via Necchi 9, 20123 Milano, Italy}
\email{[email protected], [email protected]}
\author{Enrico Miglierina
}
\address{Dipartimento di Matematica per le Scienze economiche, finanziarie ed attuariali, Universit\`{a} Cattolica del Sacro Cuore, Via Necchi 9, 20123 Milano, Italy}
\email{[email protected]}
\subjclass[2010]{Primary: 47J25; secondary: 90C25, 90C48}
\keywords{convex feasibility problem, stability, set-convergence, alternating projections method}
\thanks{
}
\begin{abstract}
The 2-sets convex feasibility problem aims at finding a point in
the nonempty intersection of two closed convex sets $A$ and $B$ in a Hilbert space $X$. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann.
In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of sets,
each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to $A$ and $B$. Given a starting point $a_0$, we consider the sequences of points obtained by projecting on the ``perturbed'' sets, i.e., the sequences $\{a_n\}$ and $\{b_n\}$ given by $b_n=P_{B_n}(a_{n-1})$ and $a_n=P_{A_n}(b_n)$.
Under appropriate geometrical and topological
assumptions on the intersection of the limit sets, we ensure that the sequences $\{a_n\}$ and $\{b_n\}$
converge in norm to a point in the intersection of $A$ and $B$. In particular, we consider both when the intersection $A\cap B$ reduces to a singleton and when the interior of $A \cap B$ is nonempty. Finally we consider the case in which the limit sets $A$ and $B$ are subspaces.
\end{abstract}
\mathrm{m}aketitle
\section{Introduction}
The 2-sets convex feasibility problem is the classical problem of finding
a point in the nonempty intersection of two closed and
convex sets $A$ and $B$ in a Hilbert space $X$ (see \cite[Section~4.5]{BorweinZhu} for some basic
results on this subject). Many efforts have been devoted to the study of algorithmic
procedures to solve convex feasibility problems, both from a
theoretical and from a computational point of view (see, e.g.,
\cite{BauschkeBorwein,BCOMB,BorweinSimsTam,Censor,Hundal} and the
references therein).
The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann \cite{vonNeumann}: let us denote by $P_A$ and $P_B$ the projections on the sets $A$ and $B$, respectively, and, given a starting point $c_0\in X$, consider the {\em alternating projections sequences} $\{c_n\}$ and $\{d_n\}$ given by $$d_n=P_{B}(c_{n-1})\ \ \text{and}\ \ c_n=P_{A}(d_n)\ \ \ \ \ (n\in\mathrm{m}athbb{N}).$$
In the case the sequences $\{c_n\}$ and $\{d_n\}$ converge in norm to a point in the intersection of $A$ and $B$, we say that the method of alternating projections converges.
Many concrete problems in applications
can be formulated as a convex feasibility problem. As typical
examples, we mention solution of convex inequalities, partial
differential equations, minimization of convex nonsmooth
functions, medical imaging, computerized tomography and image
reconstruction. For some details and other applications see, e.g.,
\cite{BauschkeBorwein} and the references therein.
Often in concrete applications data are affected by some
uncertainties. Hence stability of solutions of a convex feasibility problem with respect to data
perturbations is a desirable property, both from theoretical and computational point of view. In the present paper we investigate some
``stability'' properties of the alternating projections method in the following sense. Let us suppose that
$\{A_n\}$ and $\{B_n\}$
are two sequences of closed convex sets such that $A_n\rightarrow
A$ and $B_n\rightarrow B$ for the Attouch-Wets {variational} convergence (see Definition~\ref{def:AW}) and let us introduce the definition of {\em perturbed alternating projections sequences}.
\begin{definition}\label{def:perturbedseq} Given $a_0\in X$, the {\em perturbed alternating projections sequences} $\{a_n\}$ and $\{b_n\}$, w.r.t. $\{A_n\}$ and $\{B_n\}$ and with starting point $a_0$, are defined inductively by
$$b_n=P_{B_n}(a_{n-1})\ \ \ \text{and}\ \ \ a_n=P_{A_n}(b_n) \ \ \ \ \ \ \ \ \ (n\in\mathrm{m}athbb{N})$$
\end{definition}
\noindent Our aim is to find some conditions on the limit sets $A$ and $B$ that guarantee, for each choice of the sequences $\{A_n\}$ and $\{B_n\}$ and for each choice of the starting point $a_0$, the convergence in norm
of the corresponding perturbed alternating projections sequences $\{a_n\}$ and $\{b_n\}$. If this is the case, we say that the couple $(A,B)$ is {\em stable}.
The results reported in this paper can be seen as a continuation of the research considered in \cite{DebeMiglMol}. However, compared with the notion of stability studied in that paper, the approach developed here seems to be more interesting also from a computational point of view since it does not require to find an exact solution of the ``perturbed problems'' (i.e. the problems given by the sets $A_n$ and $B_n$) but only to consider projections on the ``perturbed'' sets $A_n$ and $B_n$. Moreover, the techniques used in the proofs are completely different from those of \cite{DebeMiglMol}.
Clearly, in order that the couple $(A,B)$ is stable, it is necessary that the alternating projections sequences $\{c_n\}$ and $\{d_n\}$ converge in norm (indeed, we can consider the particular case in which the sequences of sets $\{A_n\}$ and $\{B_n\}$ are given by $A_n=A$ and $B_n=B$, whenever $n\in\mathrm{m}athbb{N}$). Since, in general, this is not the case (see \cite{Hundal,MatouReich}), we shall restrict our attention to those situations in which the method of alternating projections converges. After some preliminaries, contained in Section~\ref{Notations and preliminaries}, we consider, in Sections~\ref{section:Infinite-dimensional}, \ref{Section:nonempty interior} and \ref{section:subspaces}, respectively, the following three cases:
\begin{enumerate}
\item $A$ and $B$ are separated by a strongly exposing functional $f$ for the set $A$, i.e., there exist $x_0\in A\cap B$ and a linear continuous functional $f$ such that $\inf f(B)=f(x_0)=\sup f(A)$ and such that $f$ strongly exposes $A$ at $x_0$ (see Definition~\ref{def:strexp});
\item the intersection between $A$ and $B$ has nonempty interior;
\item $A$ and $B$ are closed subspaces.
\end{enumerate}
Observe that if (i) is satisfied then the method of alternating projections converges. Indeed, by \cite[Lemma~4.5.11]{BorweinZhu} or by \cite[Theorem~1.4]{KopReichHilbert}, the alternating projections sequences $\{c_n\}$ and $\{d_n\}$ satisfy $\|c_n-d_n\|\to0$. Then it is easy to verify that $f(c_n),f(d_n)\to f(x_0)$ and hence, since $f$ strongly exposes $A$ at $x_0$, we have that $c_n,d_n\to x_0$ in norm.
Similar assumption on the limit sets has been considered by the authors and E.~Molho in the recent paper \cite{DebeMiglMol}, in which they proved, among other things, that if (i) is satisfied and if $x_n\in A_n,\,y_n\in B_n$ are such that
$\|x_n-y_n\|$ coincides with the distance between $A_n$ and $B_n$ then $x_n,y_n\to x_0$ in norm (see the proof of \cite[Theorem~4.5]{DebeMiglMol}).
In Section~\ref{section:Infinite-dimensional} of the present paper, we prove that if $A$ and $B$ are separated by a strongly exposing functional $f$ for the set $A$ then, for each choice of sequences $\{A_n\},\,\{B_n\}$ and starting point $a_0$, the corresponding perturbed alternating projections sequences $\{a_n\}$ and $\{b_n\}$ converge in norm to $x_0$ (cf. Theorem~\ref{puntolur} below). In this case, our approach is essentially based on suitable approximations of the sets $A_n$ and $B_n$ by convex and non-convex cones, respectively.
This result shed a new light also on the celebrated example of Hundal (see \cite{Hundal}) of a convex feasibility problem in a Hilbert space whose corresponding alternating projections sequences do not norm converge. There, $A$ is a convex cone and $B$ is a hyperplane touching the vertex of the cone $A$; this hyperplane is defined by a functional that does not strongly expose the vertex of the cone. Our result prove that, if we consider a hyperplane defined by a functional strongly exposing the vertex of the cone, we obtain not only the norm convergence of the alternating projections, but also the convergence of the perturbed alternating projections, i.e., the couple $(A,B)$ is stable.
{In
Section~\ref{Section:nonempty interior}, we investigate to what extent it is possible to guarantee convergence of the perturbed alternating projections in the case $A\cap B$ is nonempty but does not reduce to a singleton.
Example~\ref{ex: notconverge} show that, in general, even in the finite-dimensional setting and even if $A\cap B$ is bounded, the couple $(A,B)$ may be not stable. On the other hand, Theorem~\ref{theorem:corpilur} ensures that the couple $(A,B)$ is stable whenever $\mathrm{m}athrm{int}\,(A\cap B)\neq \emptyset$. We point out that boundedness of $A \cap B$ is not required. Moreover, we apply the results of this section to investigate the convergence of perturbed alternating projections for the inequality constraints problem.
}
Finally the last section of the paper is devoted to the case (iii) where $A$ and $B$ are closed subspaces. The convex feasibility problem where $A$ and $B$ are subspaces is the original problem studied by von~Neumann. In his, now classical, theorem (see \cite{vonNeumann}), he proved that the alternating projections sequences $\{c_n\}$ and $\{d_n\}$ converge in norm to $P_{A\cap B}(a_0)$. This theorem was rediscovered by several authors and many alternative proofs were provided (see, e.g., \cite{KopReich,KopReichHilbert} and the references therein).
In Section~\ref{section:subspaces}, we study the problem of convergence of perturbed alternating projections sequences in the case in which $A$ and $B$ are subspaces.
Example~\ref{ex:duedimensionale} below shows that even in the finite-dimensional setting it is conceivable that the perturbed projections sequences are unbounded in the case $A\cap B\neq\{0\}$. For this, in Section~\ref{section:subspaces}, we focus on the situation in which $A$ and $B$ are closed subspaces such that $A\cap B=\{0\}$. It turns out that if $A+B$ is a closed subspace then the couple $(A,B)$ is stable (Theorem~\ref{prop:sottospazisommachiusa}). On the other hand, in Theorem~\ref{teo:sommaNONchiusa}, we provide a couple $(A,B)$ of closed subspaces such that $A\cap B=\{0\}$ and such that there exist sequences of sets $\{A_n\},\,\{B_n\}$ and starting point $a_0$ such that the corresponding perturbed projections sequences are unbounded. Our construction is based on the example, contained in \cite{FranchettiLight86}, of two subspaces of a Hilbert space with non-closed sum such that the convergence of the corresponding alternating projection method is not geometric (for the definition of geometric convergence see \cite{FranchettiLight86}, see also \cite{PusReichZas} for some results concerning the convergence rate of the alternating projection algorithm for the case of $n$ subspaces).
\section{Notations and preliminaries}\label{Notations and preliminaries}
Throughout all this paper, if not differently stated, $X$ denotes a real normed space with
the topological dual $X^*$. We
denote by $B_X$ and $S_X$ the closed unit ball and the unit sphere of $X$, respectively.
For $x,y\in X$, $[x,y]$ denotes the closed segment in $X$ with
endpoints $x$ and $y$.
For a subset $K$ of $X$,
$\alpha>0$, and a functional $f\in S_{X^*}$ bounded on $K$, let
$$S(f,\alpha,K)=\{x\in
K;\, f(x)\geq\sup f(K)-\alpha\}$$ be the closed slice of $K$
given by $\alpha$ and $f$.
For $f\inS_{X^*}$ and $\alpha\in(0,1)$, we denote
$$C(f,\alpha)=\{x\in X; f(x)\geq \alpha\|x\|\},\
V(f,\alpha)=\{x\in X; f(x)\leq\alpha\|x\|\}.$$
It is easy to see that $C(f,\alpha)$ and
$V(f,\alpha)$ are nonempty closed cones and that $C(f,\alpha)$ is convex.
For a subset $A$ of $X$, we denote by $\mathrm{int}\,(A)$, $\partial A$, ${\mathrm{conv}}\,(A)$ and
$\subset\subsetonv(A)$ the interior, the boundary, the convex hull and the closed convex
hull of $A$, respectively.
We
denote by $$\textstyle \mathrm{m}athrm{diam}(A)=\sup_{x,y\in A}\|x-y\|,$$
the (possibly infinite) diameter of $A$. For $x\in X$, let
$$\mathrm{m}athrm{dist}(x,A) =\inf_{a\in A} \|a-x\|.$$ Moreover, given $A,B$
nonempty subsets of $X$, we denote by $\mathrm{m}athrm{dist}(A,B)$ the usual
``distance'' between $A$ and $B$, that is,
$$ \mathrm{m}athrm{dist}(A,B)=\inf_{a\in A} \mathrm{m}athrm{dist}(a,B).$$
\mathrm{m}edskip
Let us now introduce some definitions and basic properties concerning convergence of sets. By $\mathrm{m}athrm{c}(X)$ we denote the family of all nonempty closed subsets of
$X$.
Let us introduce the (extended) Hausdorff metric $h$ on
$\mathrm{m}athrm{c}(X)$. For $A,B\in\mathrm{m}athrm{c}(X)$, we define the excess of $A$ over $B$
as
$$e(A,B) = \sup_{a\in A} \mathrm{m}athrm{dist}(a,B).$$
\noindent Moreover, if $A\neq\emptyset$ and $B=\emptyset$ we put
$e(A,B)=\infty$, if $A=\emptyset$ we put $e(A,B)=0$. For $A,B\in \mathrm{m}athrm{c}(X)$, we define
$$h(A,B)=\mathrm{m}ax \bigl\{ e(A,B),e(B,A) \bigr\}.$$
\begin{definition} A sequence $\{A_j\}$ in $\mathrm{m}athrm{c}(X)$ is said to
Hausdorff converge to $A\in\mathrm{m}athrm{c}(X)$ if $$\textstyle \lim_j h(A_j,A)
= 0.$$
\end{definition}
Next we recall the definition of the so called Attouch-Wets convergence (see,
e.g., \cite[Definition~8.2.13]{LUCC}), which can be seen as a
localization of the Hausdorff convergence. If $N\in\mathrm{m}athbb{N}$ and
$A,C\in\mathrm{m}athrm{c}(X)$, define
\begin{eqnarray*}
e_N(A,C) &=& e(A\cap N B_X, C)\in[0,\infty),\\
h_N(A,C) &=& \mathrm{m}ax\{e_N(A,C), e_N(C,A)\}.
\end{eqnarray*}
\begin{definition}\label{def:AW} A sequence $\{A_j\}$ in $\mathrm{m}athrm{c}(X)$ is said to
Attouch-Wets converge to $A\in\mathrm{m}athrm{c}(X)$ if, for each $N\in\mathrm{m}athbb{N}$,
$$\textstyle \lim_j h_N(A_j,A)= 0.$$
\end{definition}
Several times without mentioning it, we shall use
the following two results.
\begin{theorem}[{see, e.g., \cite[Theorem~8.2.14]{LUCC}}] The sequence of sets $\{A_n\}$ Attouch-Wets converges to $A$ if{f}
$$\textstyle \sup_{\|x\|\leq N}{|\mathrm{m}athrm{dist}(x,A_n)-\mathrm{m}athrm{dist}(x,A)|}\to 0
\ \ \ (n\to \infty),$$
whenever $N\in\mathrm{m}athbb{N}$.
\end{theorem}
\begin{fact}\label{fact:AW} Let $A$ be a nonempty closed convex set in a Banach space $X$. Suppose that
$\{A_n\}$ is a sequence of closed convex sets such that $A_n\rightarrow
A$ for the Attouch-Wets convergence. Then, if $\{a_n\}$ is a bounded sequence in $X$ such that $a_n\in A_n$ ($n\in\mathrm{m}athbb{N}$), we have that $\mathrm{m}athrm{dist}(a_n,A)\to 0$.
\end{fact}
\mathrm{m}edskip
\begin{definition}[{see, e.g., \cite[Definition~7.10]{FHHMZ}}]\label{def:strexp} Let $A$ be a nonempty subset of a normed space $X$. A point $a\in A$
is called a strongly exposed point of $A$ if there exists a
support functional $f\in X^*\setminus\{0\}$ for $A$ in $a$ $\bigl($i.e.,
$f (a) = \sup f(A)$$\bigr)$, such that $x_n\to a$ for all sequences
$\{x_n\}$ in $A$ such that $\lim_n f(x_n) = \sup f(A)$. In this
case, we say that $f$ strongly exposes $A$ at $a$.
\end{definition}
\noindent Let us observe that $f\in S_{X^*}$ strongly exposes $A$ at $a$
if{f} $f(a)=\sup f(A)$ and
$$\mathrm{m}athrm{diam}\bigl(S(f,\alpha,A)\bigr)\to0 \text{ as } \alpha\to 0^+.$$
\noindent Let us
recall that a {\em body} in $X$ is a closed convex set in $X$ with
nonempty interior.
\begin{definition}[{see, e.g., \cite[Definition~1.3]{KVZ}}]
Let $A\subset X$ be a body. We say that $x\in\partial A$ is an
{\em LUR (locally uniformly rotund) point} of $A$ if for each
$\mathrm{m}athrm{var}epsilonsilon>0$ there exists $\delta>0$ such that if $y\in A$
and $\mathrm{m}athrm{dist}(\partial A,(x+y)/2)<\delta$ then $\|x-y\|<\mathrm{m}athrm{var}epsilonsilon$.
\end{definition}
If $A=B_X$, the previous definition coincides with the standard definition of
local uniform rotundity of the norm at $x$.
We say that $A$ is an {\em LUR body} if each point in
$\partial A$ is an LUR point of $A$.
\begin{lemma}\label{slicelimitatoselur} Let $A$ be a body in $X$
and suppose that $a\in\partial A$ is an LUR point of $A$. Then, if
$f\in S_{X^*}$ is a support functional for $A$ in $a$, $f$
strongly exposes $A$ at $a$.
\end{lemma}
The lemma is well-known in the case the body is
a ball (see, e.g., \cite[Exercise~8.27]{FHHMZ}) and in the general
case the proof is similar (see, e.g., \cite[Lemma~4.3]{DebeMiglMol}).
The next lemma gives a characterization of those functionals $f$ that strongly expose a set $A$ in terms of containment of $A$ in translations of cones of the form $C(f,\alpha)$.
\begin{lemma}\label{lemma:stronglyexposedVSopiccolo}
Let $A$ be a convex set in $X$ such that $0\in A$. Let $f\in S_{X^*}$ be such that
$f (0) = \inf f(A)$ and let $x_0\in S_X$ be such that $f(x_0)=1$ . Let us consider $\mathrm{m}athrm{var}epsilonsilon:(0,1)\to[0,\infty]$ defined by
$$\mathrm{m}athrm{var}epsilonsilon(\alpha)=\inf\{\lambda>0;\, A\subset C(f,\alpha)-\lambda x_0\}\ \ \ \ (0<\alpha<1).$$
Then $\mathrm{m}athrm{var}epsilonsilon(\alpha)$ is $o(\alpha)$ as $\alpha\to 0^+$ if{f} $({-f})$ strongly exposes $A$ at $0$.
\end{lemma}
\begin{remark}\label{InfMin} Observe that if $\alpha\in(0,1)$ is such that $\mathrm{m}athrm{var}epsilonsilon(\alpha)$ is finite then, in the definition of the function $\mathrm{m}athrm{var}epsilonsilon$, the infimum is actually a minimum. Hence, in this case, we have that
$A\subset C(f,\alpha)-\mathrm{m}athrm{var}epsilonsilon(\alpha) x_0.$
\end{remark}
\begin{proof}[Proof of Lemma~\ref{lemma:stronglyexposedVSopiccolo}] On the contrary, suppose that $\mathrm{m}athrm{var}epsilonsilon(\alpha)$ is not $o(\alpha)$ as $\alpha\to 0^+$, then there exist $M>0$ and $\alpha_n\to0^+$ such that $\mathrm{m}athrm{var}epsilonsilon(\alpha_n)>M\alpha_n$.
Let $z_n\in A\setminus[C(f,\alpha_n)-M\alpha_n x_0]$ and observe that
$$\textstyle f(z_n)+M{\alpha_n}=f(z_n+M\alpha_n x_0)<\alpha_n\|z_n+M\alpha_n x_0\|.$$
Hence it holds
$$\textstyle 0\leq f(z_n)<\alpha_n\|z_n+M\alpha_n x_0\|-M{\alpha_n}=\alpha_n(\|z_n+M\alpha_n x_0\|-{M}).$$
Then $\|z_n+M\alpha_n x_0\|>{M}$ and hence eventually $\|z_n\|>\frac M2$.
So, eventually we have
$$\textstyle 0\leq f(\frac{z_n}{\|z_n\|})<\alpha_n\frac{\|z_n+M\alpha_n x_0\|-M}{\|z_n\|}\leq \alpha_n\frac{\|z_n\|+M\alpha_n-M}{\|z_n\|}\leq \alpha_n.$$
In particular, we have $f(\frac{Mz_n}{2\|z_n\|})\to0$ as $n\to\infty$. Since $A$ is convex and $0\in A$, we have that eventually $\frac{Mz_n}{2\|z_n\|}\in A$, and hence that ${-f}$ does not strongly expose $A$ at $0$.
For the other implication, suppose that $\mathrm{m}athrm{var}epsilonsilon(\alpha)$ is $o(\alpha)$ as $\alpha\to 0^+$. By Remark~\ref{InfMin}, we have that eventually (for $\alpha\to 0^+$) $\mathrm{m}athrm{var}epsilonsilon(\alpha)$ is finite and $$A\subset C(f,\alpha)-\mathrm{m}athrm{var}epsilonsilon(\alpha)x_0.$$ Let $x\in A\cap\{x\in X;\, f(x)\leq\alpha^2\}$, then eventually
$$\alpha\|x+\mathrm{m}athrm{var}epsilonsilon(\alpha)x_0\|\leq f(x+\mathrm{m}athrm{var}epsilonsilon(\alpha)x_0)= f(x)+\mathrm{m}athrm{var}epsilonsilon(\alpha)f(x_0)\leq \alpha^2+\mathrm{m}athrm{var}epsilonsilon(\alpha)$$
and hence $\|x\|\leq\frac{\mathrm{m}athrm{var}epsilonsilon(\alpha)}{\alpha}+\mathrm{m}athrm{var}epsilonsilon(\alpha)+\alpha$. This proves that $({-f})$ strongly exposes $A$ at $0$.
\end{proof}
In the following two lemmas we analyse some relations between the Attouch-Wets convergence of a sequence of sets and the containment of the sets of the sequence in a cone of the form $V(f,\alpha)$ or $C(f,\alpha)$.
\begin{lemma}\label{lemma:definitivamentenellaconca} Let $B, B_n$ ($n\in\mathrm{m}athbb{N}$) be closed convex sets in $X$ such that $B_n\rightarrow
B$ for the Attouch-Wets convergence, and $f\in S_{X^*}$. Suppose that $x_0\in S_X$ is such that $f(x_0)=1$ and suppose that $0\in B\subset \{x\in X;\, f(x)\leq 0\}$.
Then, for each $\alpha\in(0,1)$ and $\mathrm{m}athrm{var}epsilonsilon>0$, there exists $n_0\in \mathrm{m}athbb{N}$ such that $B_n\subset V(f,\alpha)+\mathrm{m}athrm{var}epsilonsilon x_0$, whenever $n\geq n_0$.
\end{lemma}
\begin{proof}
On the contrary, suppose that there exists a sequence of integers $\{n_k\}$ such that, for each $k\in\mathrm{m}athbb{N}$, there exists $$b_{n_k}\in B_{n_k}\setminus[V(f,\alpha)+\mathrm{m}athrm{var}epsilonsilon x_0].$$
Since
$$\mathrm{m}athrm{dist}(B, C(f,\alpha)+\mathrm{m}athrm{var}epsilonsilon x_0)>0,$$
by Fact~\ref{fact:AW}, we can suppose without any loss of generality that $\|b_{n_k}\|\geq1$ ($k\in\mathrm{m}athbb{N}$).
Since $b_{n_k}\not\in V(f,\alpha)+\mathrm{m}athrm{var}epsilonsilon x_0$, we have
$$f(b_{n_k})>\alpha\|b_{n_k}-\mathrm{m}athrm{var}epsilonsilon x_0\|+\mathrm{m}athrm{var}epsilonsilon\geq\alpha\|b_{n_k}\|.$$
Let $\delta=\mathrm{m}in\{\mathrm{m}athrm{var}epsilonsilon,\alpha/2\}$, since $0\in B$ and $B_n\rightarrow
B$ for the Attouch-Wets convergence, we can suppose without any loss of generality that, for each $k\in\mathrm{m}athbb{N}$, there exists $d_k\in (\delta B_X)\cap B_{n_k}$. Let
$$\textstyle w_k=\frac1{\|b_{n_k}\|}b_{n_k}+\frac{\|b_{n_k}\|-1}{\|b_{n_k}\|}d_k\in B_{n_k},$$
and observe that $\|w_k\|\leq1+\mathrm{m}athrm{var}epsilonsilon$. Moreover, we have
$$\textstyle f(w_k)\geq f(b_{n_k})\frac1{\|b_{n_k}\|}-\|d_k\|\geq \alpha -\|d_k\|\geq \frac\alpha2.$$
Since $\{w_k\}$ is a bounded sequence, by Fact~\ref{fact:AW}, $\mathrm{m}athrm{dist}(w_k,B)\to0$.
Hence we get a contradiction since $\{w_k\}\subset \{x\in X;\, f(x)\geq\alpha/2\}$ and
$$\mathrm{m}athrm{dist}(B, \{x\in X;\, f(x)\geq\alpha/2\})>0.$$
\end{proof}
\begin{lemma}\label{lemma:definitivamentenelcono} Let $A, A_n$ ($n\in\mathrm{m}athbb{N}$) be closed convex sets in $X$ such that $A_n\rightarrow
A$ for the Attouch-Wets convergence, $f\in S_{X^*}$, $\alpha\in(0,1)$, and $\mathrm{m}athrm{var}epsilonsilon>0$. Suppose that $x_0\in S_X$ is such that $f(x_0)=1$ and suppose that $0\in A\subset C(f,\alpha)-\mathrm{m}athrm{var}epsilonsilon x_0$.
Then, for each $\beta\in(0,\alpha)$ and $\mathrm{m}athrm{var}epsilonsilon'>\mathrm{m}athrm{var}epsilonsilon$, there exists $n_0\in \mathrm{m}athbb{N}$ such that $A_n\subset C(f,\beta)-\mathrm{m}athrm{var}epsilonsilon' x_0$, whenever $n\geq n_0$.
\end{lemma}
\begin{proof}
Suppose on the contrary that there exists a sequence of integers $\{n_k\}$ such that, for each $k\in\mathrm{m}athbb{N}$, there exists $$a_{n_k}\in A_{n_k}\setminus[C(f,\beta)-\mathrm{m}athrm{var}epsilonsilon' x_0].$$
Since $a_{n_k}+\mathrm{m}athrm{var}epsilonsilon' x_0\not\in C(f,\beta)$, we have
\begin{equation}\label{eq:fuoridalcono} f(a_{n_k}+\mathrm{m}athrm{var}epsilonsilon' x_0)=f(a_{n_k})+\mathrm{m}athrm{var}epsilonsilon'<\beta\|a_{n_k}+\mathrm{m}athrm{var}epsilonsilon' x_0\|.
\end{equation}
Fix any $\gamma\in(\beta,\alpha)$ and let $M\geq1$ be such that $M>\frac{2\mathrm{m}athrm{var}epsilonsilon'}{\alpha-\gamma}$. Finally, let $\theta\in (0,1)$ be such that
\begin{enumerate}
\item[(a)] $M-\theta>\frac{2\mathrm{m}athrm{var}epsilonsilon'}{\alpha-\gamma} $;
\item[(b)] $\frac{\beta M+\theta}{M-\theta}\leq\gamma$.
\end{enumerate}
Since
$$\mathrm{m}athrm{dist}\bigl(C(f,\alpha)-\mathrm{m}athrm{var}epsilonsilon x_0,V(f,\beta)-\mathrm{m}athrm{var}epsilonsilon' x_0\bigr)>0,$$
by Fact~\ref{fact:AW}, we can suppose without any loss of generality that $\|a_{n_k}\|\geq M$ ($k\in\mathrm{m}athbb{N}$).
Moreover, since $0\in A$ and $A_n\rightarrow
A$ for the Attouch-Wets convergence, we can suppose without any loss of generality that, for each $k\in\mathrm{m}athbb{N}$, there exists $c_k\in A_{n_k}\cap\theta B_X$.
Put, for each $k\in\mathrm{m}athbb{N}$,
$$\textstyle b_k=\frac M{\|a_{n_k}\|}a_{n_k}+\frac{\|a_{n_k}\|-M}{\|a_{n_k}\|}c_k\in A_{n_k},$$
and observe that $M-\theta\leq\|b_k\|\leq M+\theta$.
Now, by (\ref{eq:fuoridalcono}), we have $f(a_{n_k})<\beta\|a_{n_k}\|$ and hence
$$f(b_k)\leq M\beta+\theta\leq\|b_k\|\frac{M\beta+\theta}{\|b_k\|}\leq\frac{M\beta+\theta}{M-\theta}\|b_k\|\leq\gamma\|b_k\|.$$
Moreover, since $\{b_k\}$ is bounded and $A\subset C(f,\alpha)-\mathrm{m}athrm{var}epsilonsilon x_0$, by Fact~\ref{fact:AW}, we have that eventually
$f(b_k)\geq\alpha\|b_k\|-2\mathrm{m}athrm{var}epsilonsilon'$ and hence that
$$\alpha\|b_k\|-2\mathrm{m}athrm{var}epsilonsilon'\leq f(b_k)\leq\gamma \|b_k\|.$$
In particular, we have that eventually $\|b_k\|\leq\frac{2\mathrm{m}athrm{var}epsilonsilon'}{\alpha-\gamma}<M-\theta$, a contradiction since $\|b_k\|\geq M-\theta$.
\end{proof}
\section{The case where the intersection of limits sets is a singleton}\label{section:Infinite-dimensional}
In the sequel of the paper, we suppose that $X$ is a real Hilbert space. If $u,v\in X\setminus\{0\}$, we denote as usual
$$\textstyle\cos(u,v)=\frac{\langle u,v\rangle}{\|u\|\|v\|},$$
where $\langle u,v\rangle$ denotes the inner product between $u$ and $v$.
If $K$ is a nonempty closed convex subset of $X$, let us denote by $P_K$ the projection onto the set $K$.
Several times without mentioning it, we shall use the variational characterization of best approximations from convex sets in Hilbert spaces: let $K$ be as above, $x\in X$ and $y_0\in K$, then
$y_0=P_K(x)$ if and only if
\begin{equation}\label{eq:proiezionesuconvesso}
\langle x-y_0,y-y_0\rangle\leq0 \ \ \ \ \text{whenever} \ y\in K.
\end{equation}
It is easy to see that, if $x\not\in K$, (\ref{eq:proiezionesuconvesso}) is equivalent to the following condition:
\begin{equation}\label{eq:proiezionecoseno}
\|y-y_0\|\leq \|x-y\|\cos(y_0-y,x-y) \ \ \ \ \text{whenever} \ y\in K\setminus\{y_0\}.
\end{equation}
Moreover, if $K$ is a subspace of $X$ then (\ref{eq:proiezionesuconvesso}) becomes
\begin{equation}\label{eq:proiezionesusottospazio}
\langle x-y_0,y-y_0\rangle=0 \ \ \ \ \text{whenever} \ y\in K.
\end{equation}
Let us recall the definition of stability for a couple $(A,B)$ of subsets of $X$.
\begin{definition}
Let $A$ and $B$ be closed convex subsets of $X$ such that $A\cap B$ is nonempty. We say that the that
the couple $(A,B)$ is {\em stable} if for each choice of sequences $\{A_n\},\{B_n\}\subset\mathrm{m}athrm{c}(X)$ converging for the Attouch-Wets convergence to $A$ and $B$, respectively, and for each choice of the starting point $a_0$, the corresponding perturbed alternating projections sequences $\{a_n\}$ and $\{b_n\}$ converge in norm.
\end{definition}
\begin{remark}
We remark that in the above definition we can equivalently require that there exists $c\in A\cap B$ such that $a_n,b_n\to c$ in norm.
\end{remark}
\begin{proof}
It suffices to prove that if the perturbed alternating projections sequences $\{a_n\}$ and $\{b_n\}$ converge in norm then they both converge to a point in $A\cap B$.
Let us start by proving that if $a_n\to a$ then $a\in A\cap B$. It is not difficult to prove that, since $$a_{n+1}= P_{A_n}P_{B_n}a_n=P_{A}P_{B}a_n+(P_{A_n}P_{B_n}-P_{A}P_{B})a_n$$ and since $A_n\to A,B_n\to B$ for the Attouch-Wets convergence, we have $a=P_AP_B a$. By \cite[Facts~1.1, (ii)]{BauschkeBorwein93}, we have that $a\in A\cap B$. Similarly, it is easy to see that
$$b_{n+1}=P_{B_n}a_n=P_{B}a_n+(P_{B_n}-P_{B})a_n\to P_Ba=a,$$
and the proof is concluded.
\end{proof}
The main aim of this section is to prove that under the assumption that the sets
$A$ and $B$ are separated by a strongly exposing functional $f$ for the set $A$ (i.e. condition (i) in the introduction) the couple $(A,B)$ is stable.
The following theorem is the main result of this section.
\begin{theorem}\label{puntolur} Let $X$ be a Hilbert space and $A,B$
nonempty closed convex subsets of $X$. Let $\{A_n\}$ and $\{B_n\}$
be two sequences of closed convex sets such that $A_n\rightarrow
A$ and $B_n\rightarrow B$ for the Attouch-Wets convergence. Suppose that there exist $y\in A\cap B$ and a linear continuous functional $f\in S_{X^*}$ such that $\inf f(B)=f(y)=\sup f(A)$ and such that $f$ strongly exposes $A$ at $y$.
Then, for each $a_0\in X$, the corresponding perturbed alternating projections sequences $\{a_n\}$ and $\{b_n\}$ (with starting point $a_0$), converge to $y$ in norm.
\end{theorem}
Before starting with the proof of the theorem we need some preliminary work. First of all, let us observe that without any loss of generality we can suppose that $y=0$ and hence that $$\inf f(A)=f(0)=\sup f(B).$$
Suppose that $x_0\in S_X$ is such that $f(x_0)=1$, i.e., $f$ is represented by $x_0$, in the sense that $f(\cdot)=\langle x_0,\cdot\rangle$. Then it is straightforward to give the following representation of the cones $C(f,\alpha)$ and $V(f,\alpha)$, introduced at the beginning of Section~\ref{Notations and preliminaries}: if we define
$$\textstyle C(\theta):=\{x\in X\setminus\{0\};\, \cos(x,x_0)\geq\sin(\theta)\}\cup\{0\}\ \ \ (\theta\in(0,\frac\pi2)),$$
then the set $C(\theta)$ coincides with $C(f,\sin\theta)$.
Similarly, if we define
$$\textstyle V(\theta):=\{x\in X\setminus\{0\};\, \cos(x,x_0)\leq\sin(\theta)\}\cup\{0\}\ \ \ (\theta\in(0,\frac\pi2)),$$
then the set $V(\theta)$ coincides with $V(f,\sin\theta)$. We shall need the following simple fact.
\begin{fact}\label{fact:coseno} Suppose that $\theta_1,\theta_2\in(0,\frac\pi2)$ are such that $\theta_1<\theta_2$. If $x\in C(\theta_2)\setminus\{0\}$ and $y\in V(\theta_1)\setminus\{0\}$ then $\cos(x,y)\leq\cos(\theta_2-\theta_1)$.
\end{fact}
\begin{proof}
For $z\in X\setminus\{0\}$ let us denote $\theta_z=\frac\pi2-\arccos \cos(z,x_0)$ and observe that
$$z\in C(\theta_2)\Leftrightarrow \theta_z\geq\theta_2\ \ \ \text{and}\ \ \ z\in V(\theta_1)\Leftrightarrow \theta_z\leq\theta_1.$$
Let us define $x_1=x-f(x)x_0$ and $y_1=y-f(y)x_0$, then
$$\textstyle \cos(x,y)\leq\frac{f(x)f(y)}{\|x\|\|y\|}+\frac{\|x_1\|\|y_1\|}{\|x\|\|y\|}=\cos(\theta_x-\theta_y)\leq\cos(\theta_2-\theta_1).$$
\end{proof}
\begin{proof}[Proof of Theorem~\ref{puntolur}]
Fix $M>0$, it suffices to prove that the sequences $\{a_n\}$ and $\{b_n\}$ are eventually contained in $2M B_X$.
Let $f\in S_{X^*}$ and $x_0\in X$ be as above. Let $\alpha\in (0,1)$ and let
$$\mathrm{m}athrm{var}epsilonsilon(\alpha)=\inf\{\lambda>0;\, A\subset C(f,\alpha)-\lambda x_0\}\in[0,\infty],$$
by Lemma~\ref{lemma:stronglyexposedVSopiccolo}, $\mathrm{m}athrm{var}epsilonsilon(\alpha)$ is $o(\alpha)$ as $\alpha\to 0^+$. In particular, we can fix $\beta\in(0,1/3)$ such that if $\theta=\frac12\arcsin (2\beta)$ then $\mathrm{m}athrm{var}epsilonsilon':=2\mathrm{m}athrm{var}epsilonsilon(3\beta)\in\mathrm{m}athbb{R}$ and
\begin{enumerate}
\item[(a)] $\mathrm{m}athrm{var}epsilonsilon'\leq M/2$;
\item[(b)] $\sin\theta+\frac8M\mathrm{m}athrm{var}epsilonsilon'\leq\sin(\frac43\theta)$;
\item[(c)] $\sin(2\theta)-\frac8M{\mathrm{m}athrm{var}epsilonsilon'}\geq\sin(\frac53\theta)$;
\item[(d)] $\cos(\frac13\theta)+\frac2M{\mathrm{m}athrm{var}epsilonsilon'}\leq\cos(\frac16\theta)$.
\end{enumerate}
Since, by Remark~\ref{InfMin}, $0\in A\subset C(f,3\beta)-\mathrm{m}athrm{var}epsilonsilon(3\beta) x_0$, by Lemma~\ref{lemma:definitivamentenelcono}, we have that eventually $$A_n\subset C(f,2\beta)-2\mathrm{m}athrm{var}epsilonsilon(3\beta) x_0=C(2\theta)-\mathrm{m}athrm{var}epsilonsilon' x_0.$$
Since, $0\in B\subset\{x\in X;\, f(x)\leq0\}$, by Lemma~\ref{lemma:definitivamentenellaconca}, we have that eventually $$B_n\subset V(\theta)+\mathrm{m}athrm{var}epsilonsilon' x_0.$$
Since $0\in A\cap B$, $A_n\rightarrow
A$ and $B_n\rightarrow B$ for the Attouch-Wets convergence, eventually there exist $x_n\in A_n\cap \mathrm{m}athrm{var}epsilonsilon'B_X$ and $y_n\in B_n\cap \mathrm{m}athrm{var}epsilonsilon'B_X$.
\begin{claim} Eventually, if $a_n,b_n,b_{n+1}\not\in M B_X$, the following conditions hold:
\begin{enumerate}
\item $a_n-x_n\in C(\frac53\theta)$;
\item $b_n-x_n\in V(\frac43\theta)$;
\item $a_n-y_{n+1}\in C(\frac53\theta)$;
\item $b_{n+1}-y_{n+1}\in V(\frac43\theta)$.
\end{enumerate}
\end{claim}
\begin{proof}[Proof of the claim]
Let us prove (i) and (ii), the proof of (iii) and (iv) is similar. To prove (i), observe that, since $a_n\in A_n\subset C(2\theta)-\mathrm{m}athrm{var}epsilonsilon' x_0$, we have
\begin{eqnarray*}
f(a_n-x_n)&\geq& f(a_n+\mathrm{m}athrm{var}epsilonsilon'x_0)-2\mathrm{m}athrm{var}epsilonsilon'\\
&\geq&\sin(2\theta)(\|a_n+\mathrm{m}athrm{var}epsilonsilon'x_0\|)-2\mathrm{m}athrm{var}epsilonsilon'\\
&\geq&\sin(2\theta)(\|a_n-x_n\|-2\mathrm{m}athrm{var}epsilonsilon')-2\mathrm{m}athrm{var}epsilonsilon'\\
\textstyle&=&\textstyle\|a_n-x_n\|(\sin(2\theta)-\frac{2\mathrm{m}athrm{var}epsilonsilon'\sin(2\theta)+2\mathrm{m}athrm{var}epsilonsilon'}{\|a_n-x_n\|})\\
\textstyle&\geq&\textstyle\|a_n-x_n\|(\sin(2\theta)-\frac8M{\mathrm{m}athrm{var}epsilonsilon'}) \\
&\geq&\textstyle\|a_n-x_n\|\sin(\frac53\theta),
\end{eqnarray*}
where the last inequality holds by (c).
To prove (ii), we proceed similarly: observe that, since $b_n\in B_n\subset V(\theta)+\mathrm{m}athrm{var}epsilonsilon' x_0$, we have
\begin{eqnarray*}
f(b_n-x_n)&\leq& f(b_n-\mathrm{m}athrm{var}epsilonsilon'x_0)+2\mathrm{m}athrm{var}epsilonsilon'\\
&\leq&\sin(\theta)(\|b_n-\mathrm{m}athrm{var}epsilonsilon'x_0\|)+2\mathrm{m}athrm{var}epsilonsilon'\\
&\leq&\sin(\theta)(\|b_n-x_n\|+2\mathrm{m}athrm{var}epsilonsilon')+2\mathrm{m}athrm{var}epsilonsilon'\\
\textstyle&=&\textstyle\|b_n-x_n\|(\sin\theta+\frac{2\mathrm{m}athrm{var}epsilonsilon'\sin\theta+2\mathrm{m}athrm{var}epsilonsilon'}{\|b_n-x_n\|})\\
\textstyle&\leq&\textstyle\|b_n-x_n\|(\sin\theta+\frac8M{\mathrm{m}athrm{var}epsilonsilon'}) \\
&\leq&\textstyle\|b_n-x_n\|\sin(\frac43\theta),
\end{eqnarray*}
where the last inequality holds by (b). The claim is proved.
\end{proof}
Now, since $a_n=P_{A_n}b_n$ and $x_n\in A_n$, by (\ref{eq:proiezionecoseno}), it holds
\begin{equation}\label{eq:normadecresce}
\|a_n-x_n\|\leq\|b_n-x_n\|\cos(a_n-x_n,b_n-x_n).
\end{equation}
Then we can observe that, by (i) and (ii) in our claim and by Fact~\ref{fact:coseno}, we have that eventually, if $a_n,b_n\not\in M B_X$, it holds $\|a_n-x_n\|\leq\|b_n-x_n\|\cos(\frac13\theta)$ and hence
$$\textstyle \|a_n\|\leq\|a_n-x_n\|+\mathrm{m}athrm{var}epsilonsilon'\leq(\|b_n\|+\mathrm{m}athrm{var}epsilonsilon')\cos(\frac13\theta)+\mathrm{m}athrm{var}epsilonsilon'\leq\|b_n\|(\cos(\frac13\theta)+\frac2M\mathrm{m}athrm{var}epsilonsilon')\leq\|b_n\|\cos(\frac16\theta),$$
where the last inequality holds by (d).
Similarly, since $b_{n+1}=P_{B_n}a_n$ and $y_{n+1}\in B_n$, it holds $\|b_{n+1}-y_{n+1}\|\leq\|a_n-y_{n+1}\|\cos(b_{n+1}-y_{n+1},a_n-y_{n+1})$. By (iii) and (iv) in our claim and by Fact~\ref{fact:coseno}, we have that eventually, if $a_n,b_{n+1}\not\in M B_X$, it holds $\|b_{n+1}-y_{n+1}\|\leq\|a_n-y_{n+1}\|\cos(\frac13\theta)$ and hence
$$\textstyle \|b_{n+1}\|\leq(\|a_n\|+\mathrm{m}athrm{var}epsilonsilon')\cos(\frac13\theta)+\mathrm{m}athrm{var}epsilonsilon'\leq\|a_n\|(\cos(\frac13\theta)+\frac2M\mathrm{m}athrm{var}epsilonsilon')\leq\|a_n\|\cos(\frac16\theta),$$
where the last inequality holds by (d).
By (\ref{eq:normadecresce}) and by the observations above, there exists $n_0\in\mathrm{m}athbb{N}$ such that if $n\geq n_0$ then the following conditions hold:
\begin{enumerate}
\item[($\alpha)$] if $a_n,b_n\not\in M B_X$ then $ \|a_n\|\leq\|b_n\|\cos(\frac16\theta)$, and if $a_n,b_{n+1}\not\in M B_X$ then $\|b_{n+1}\|\leq\|a_n\|\cos(\frac16\theta)$;
\item[($\beta)$] if $b_n\in M B_X$ then $\|a_n\|\leq\|b_n\|+2\mathrm{m}athrm{var}epsilonsilon'\leq 2M$, and if $a_n\in M B_X$ then $\|b_{n+1}\|\leq\|a_n\|+2\mathrm{m}athrm{var}epsilonsilon'\leq 2M$.
\end{enumerate}
Now, it is easy to see that there exists $n_1\geq n_0$ such that $a_{n_1}\in MB_X$ or $b_{n_1}\in MB_X$. Indeed, since $\cos(\frac16\theta)<1$, the fact that, for each $n\geq n_0$, $a_n,b_n\not\in M B_X$ contradicts ($\alpha$). By ($\beta$) and taking into account also ($\alpha$), we obtain that $a_n,b_n\in2M B_X$, whenever $n>n_1$.
\end{proof}
\begin{corollary}\label{corollary:puntolur} Let $X$ be a Hilbert space, $B$
a nonempty closed convex subset of $X$, $A$ a body in $X$ and
$y\in\partial A$ an LUR point of $A$. Let $\{A_n\}$ and $\{B_n\}$
be two sequences of closed convex sets such that $A_n\rightarrow
A$ and $B_n\rightarrow B$ for the Attouch-Wets convergence.
Suppose that $A\cap B=\{y\}$.
Then, for each $a_0\in X$, the corresponding perturbed alternating projections sequences $\{a_n\}$ and $\{b_n\}$ (with starting point $a_0$), converge to $y$ in norm.
\end{corollary}
\begin{proof}
Since $(\mathrm{int}\, A)\cap B=\emptyset$, by the Hahn-Banach separation theorem, there exists $f\in S_{X^*}$ such that $$\inf f(A)=f(y)=\sup f(B).$$
Since $y$ is an LUR point of $A$, by Lemma~\ref{slicelimitatoselur}, $f$ strongly exposes $A$ at $y$. The thesis follows by Theorem~\ref{puntolur}.
\end{proof}
It is worth noting that, in the recent paper \cite{GrKaKuReich}, a result concerning the convergence of iterates of nonexpansive mapping has been obtained under a geometrical condition involving LUR points.
\section{The case where the interior of the intersection of limits sets is nonempty} \label{Section:nonempty interior}
The main aim of this section is to prove that, under the assumption that the interior of $A\cap B$ is nonempty, the couple $(A,B)$ is stable.
We start by the following two dimensional fact. Even if the argument used is elementary we include a sketch of a possible proof for the sake of completeness.
\begin{fact}\label{fact:norms}
Let $X$ be a Hilbert space and $\mathrm{m}athrm{var}epsilonsilon,K>0$. Then there exists a constant $\mathrm{m}u>0$ such that, whenever $C$ is a closed convex subset of $X$ containing $\mathrm{m}athrm{var}epsilonsilon B_X$ and $x\in KB_X$, we have
\begin{equation}\label{eq:angleboundedawayfrom0}
\|x-P_Cx\|\leq\mathrm{m}u(\|x\|-\|P_C x\|).
\end{equation}
\end{fact}
\begin{proof} We claim that $\mathrm{m}u=K/\mathrm{m}athrm{var}epsilonsilon$ works. Let us denote by $\theta(u,v)$ the angle between two not null vectors $u$ and $v$.
Let us denote $y=P_C x$.
We can (and do) assume that $y$ and $x$ are not proportional (if else (\ref{eq:angleboundedawayfrom0}) trivially holds). Hence, since $\mathrm{m}athrm{var}epsilonsilon B_X\subset C$, we have that $\mathrm{m}athrm{var}epsilonsilon<\|y\|<\|x\|$. Let $Y=\mathrm{m}athrm{span}\{x,y\}$ and let $w\in\mathrm{m}athrm{var}epsilonsilon S_Y$ be such that:
\begin{enumerate}
\item the line containing $\{y,w\}$ is tangent to $\mathrm{m}athrm{var}epsilonsilon B_Y$;
\item the segment $[y,w]$ intersects the segment $[0,x]$.
\end{enumerate}
Observe that existence of such an element $w$ is guaranteed by the fact that $\|x-y\|\leq \|x\|-\mathrm{m}athrm{var}epsilonsilon$.
Since the vectors $w$ and $w-y$ are orthogonal, we clearly have $\sin\theta(-y,w-y)\geq \mathrm{m}athrm{var}epsilonsilon/K $. Let us denote $z=\frac{\|y\|}{\|x\|}x$, by the variational characterization of best approximations from convex sets in Hilbert spaces and by the fact that $\|z\|=\|y\|$, we have:
\begin{enumerate}
\item $\theta(x-y,w-y)\geq \pi/2$;
\item $\theta(-y,z-y)\leq \pi/2$.
\end{enumerate}
It follows that $\theta(x-y,z-y)\geq \theta(-y,w-y)$ and hence that $$\|x-y\|\leq\frac{K}{\mathrm{m}athrm{var}epsilonsilon}\|x-z\|=\frac{K}{\mathrm{m}athrm{var}epsilonsilon}(\|x\|-\|y\|)$$
\end{proof}
The following theorem is the main result of this section and it is an application of the previous argument.
\begin{theorem}\label{theorem:corpilur} Let $X$ be a Hilbert space and $A,B$
nonempty closed convex subsets of $X$.
Suppose that $\mathrm{int}\,(A\cap B)\neq\emptyset$, then the couple $(A,B)$ is stable.
\end{theorem}
\begin{proof}
Without any loss of generality, we can suppose that $0\in\mathrm{int}\, (A\cap B)$.
Let $\{A_n\}$ and $\{B_n\}$
be two sequences of closed convex sets such that $A_n\rightarrow
A$ and $B_n\rightarrow B$ for the Attouch-Wets convergence. Suppose that $\{a_n\}$ and $\{b_n\}$ are
the corresponding perturbed alternating projections sequences with respect to a given starting point $a_0$.
By Proposition 27 in \cite{PenotZalinescu} we have that $A_n\cap B_n \rightarrow A\cap B$ for the Attouch-Wets convergence. Hence, by Theorem 7.4.2 in \cite{Beer}, we can suppose without any loss of generality that there exists $\mathrm{m}athrm{var}epsilonsilon>0$ such that $\mathrm{m}athrm{var}epsilonsilon B_X\subset A_n\cap B_n $, whenever $n\in \mathrm{m}athbb{N}$. Since $0\in A_n\cap B_n$, we have that $\|a_n\|\leq\|b_n\|, \|b_n\|\leq\|a_{n-1}\|$ and hence there exists $K>0$ such that $\{a_n\},\{b_n\}\subset K B_X$. By Fact~\ref{fact:norms}, we have that there exists $\mathrm{m}u>0$ such that $\|a_n-b_n\|\leq\mathrm{m}u(\|b_n\|-\|a_n\|)$ and
$\|b_n-a_{n-1}\|\leq\mathrm{m}u(\|a_{n-1}\|-\|b_n\|)$. Hence
$$\textstyle \sum_{n=1}^N(\|a_n-b_n\|+\|b_n-a_{n-1}\|)\leq\sum_{n=1}^N\mathrm{m}u(\|a_{n-1}\|-\|a_n\|)=\mathrm{m}u(\|a_{0}\|-\|a_N\|).$$
This proves that the series $\sum_{n\in\mathrm{m}athbb{N}}(a_n-a_{n-1})$ is absolutely convergent and hence convergent, i.e., the sequence $\{a_n\}$ is convergent. Similarly, we have that also the sequence $\{b_n\}$ is convergent and the proof is complete.
\end{proof}
By combining the results contained in Section~\ref{section:Infinite-dimensional} and the previous theorem we have the following corollary. This corollary describes the stability property for the couple $(A,B)$ where $A$ and $B$ are bodies.
\begin{corollary}\label{corollary:corpilur} Let $X$ be a Hilbert space, suppose that at least one of the following conditions holds.
\begin{enumerate}
\item $A$ is a closed convex set with nonempty interior, $f\in X^*\setminus\{0\}$ is such that $f$ strongly exposes $A$ at the origin, and $B=\{x\in X;\, f(x)\geq \alpha\}$, where $\alpha\leq 0$.
\item $A,B$ are bodies in $X$ such that $A$ is LUR and $A\cap B\neq\emptyset$.
\end{enumerate} Then the couple $(A,B)$ is stable.
\end{corollary}
\begin{proof}
(i) If $\alpha<0$ then $\mathrm{int}\,(A\cap B)\neq\emptyset$ and we can apply Theorem~\ref{theorem:corpilur}. If $\alpha=0$ apply Theorem~\ref{puntolur}.
\noindent (ii) If $\mathrm{int}\,(A\cap B)\neq\emptyset$ we can apply Theorem~\ref{theorem:corpilur}. If $\mathrm{int}\,(A\cap B)=\emptyset$, since $A$ and $B$ are bodies, we have that
$\mathrm{int}\,(A)\cap B=\emptyset$. Since $A$ is an LUR body, there exists $y\in\partial A$ such that $A\cap B=\{y\}$. Apply Corollary~\ref{corollary:puntolur}.
\end{proof}
It is worth to remark that the assumptions (i) and (ii) in Corollary \ref{corollary:corpilur} cannot be avoided if we ask for a stable couple of bodies. Indeed, when we consider two bodies with nonempty intersection, the typical situation in which (i) and (ii) fail is the following: there exists a functional $f\in X^*\setminus\{0\}$ separating the bodies $A$ and $B$ but $f$ strongly exposes neither $A$ nor $B$. The following simple 2-dimensional example shows that, in general, in this case we cannot guarantee that the couple $(A,B)$ is stable.
\begin{example}\label{ex: notconverge}
Let $X=\mathrm{m}athbb{R}^2$ and let us consider, for each $h\in\mathrm{m}athbb{N}$, the following subsets of $X$:
\begin{eqnarray*}
A&=&\textstyle {\mathrm{conv}}\,\{(1,1),(-1,1),(1,0),(-1,0)\};\\
C_{2h}&=&\textstyle{\mathrm{conv}}\,\{(1,1),(-1,1),(1,\frac1h),(-1,0)\};\\
C_{2h-1}&=&\textstyle{\mathrm{conv}}\,\{(1,1),(-1,1),(1,0),(-1,\frac1h)\};\\
B&=&\textstyle{\mathrm{conv}}\,\{(1,-1),(-1,-1),(1,0),(-1,0)\};\\
D_{2h}&=&\textstyle{\mathrm{conv}}\,\{(1,-1),(-1,-1),(1,-\frac1h),(-1,0)\};\\
D_{2h-1}&=&\textstyle{\mathrm{conv}}\,\{(1,-1),(-1,-1),(1,0),(-1,-\frac1h)\}.
\end{eqnarray*}
We claim that the couple $(A,B)$ is not stable. To prove this, let us consider the starting point $z_0=(0,0)$ and observe that, if we consider the points $a_k=(P_{C_1} P_{D_1})^k z_0$, it is clear that there exists $N_1\in\mathrm{m}athbb{N}$ such that $$\textstyle \|a_{N_1}-(1,0)\|<\frac12.$$
Define $A_n=C_1$ and $B_n=D_1$ whenever $1\leq n\leq N_1$. Similarly, if we consider the points $a_{N_1+k}=(P_{C_2} P_{D_2})^{k} a_{N_1}$ then there exists $N_2\in\mathrm{m}athbb{N}$ such that $$\textstyle \|a_{N_1+N_2}-(-1,0)\|<\frac12.$$ Define $A_n=C_2$ and $B_n=D_2$ whenever $N_1+1\leq n\leq N_1+N_2$. Then, proceeding inductively, it is easy to construct sequences $\{A_n\}$ and $\{B_n\}$ converging respectively to $A$ and $B$ for the Attouch-Wets convergence and such that the perturbed alternating projections sequences $\{a_n\}$ and $\{b_n\}$, w.r.t. $\{A_n\}$ and $\{B_n\}$ and with starting point $z_0$, do not converge.
\end{example}
\subsection*{Inequality constraints}
Inequality constraints are a typical example of problem that can be solved by projections and reflections methods (see, e.g., \cite[Remark~3.17]{BorweinSimsTam}). It appears very in often in mathematical programming theory. This problem reveals to be a stable problem under mild assumptions. Indeed, in the rest of this section we will show that under suitable additional hypotheses also the method of perturbed alternating projections sequences can be applied to deal with such a problem.
Given a closed convex cone $K$ in a Hilbert space $X$ (recall that a subset $K$ of $X$ is called cone if $\lambda k\in K$, whenever $\lambda\in [0,\infty)$ and $k\in K$), we denote by $K^-$ its {\em negative polar cone}, i.e., the closed convex cone defined by
$$K^-=\{x\in X;\, \langle x,k \rangle\leq 0, \ \text{whenever}\ k\in K\}.$$
Let us suppose that $a\in X\setminus\{0\}$, $b\in \mathrm{m}athbb{R}$, and define $A=\{x\in X;\, \langle a, x\rangle\leq b\}$. Then it is easy to observe that the following assertions hold true.
\begin{itemize}
\item If $\mathrm{int}\, K\neq\emptyset$, $a_1,\ldots,a_n\in X$, $b_1,\ldots,b_n>0$ and $$B:=\{x\in X;\, \langle a_i, x\rangle\leq b_i,\ i=1,\ldots,n\}$$ then $\mathrm{int}\,(B\cap K)\neq \emptyset$.
\item If $\mathrm{int}\, K\neq\emptyset$ and $a\not\in K^-$ then $\mathrm{int}\,(A\cap K)\neq \emptyset$.
\item If $a\in\mathrm{int}\, (K^-)$ and $b=0$ then $A$ and $K$ are separated by a strongly exposing functional for the set $K$.
\end{itemize}
\noindent Hence, by combining the previous observation, Theorem~\ref{theorem:corpilur}, and Theorem~\ref{puntolur}, we obtain the following result about the convergence of perturbed projections for the inequality constraints problem.
\begin{theorem}
Let $K$ be a closed convex cone in a Hilbert space $X$. Suppose that at least one of the following conditions holds true.
\begin{enumerate}
\item $\mathrm{int}\, K\neq\emptyset$, $a_1,\ldots,a_n\in X$, $b_1,\ldots,b_n>0$, and $$B:=\{x\in X;\, \langle a_i, x\rangle\leq b_i,\ i=1,\ldots,n\}.$$
\item $\mathrm{int}\, K\neq\emptyset$, $a\not\in K^-$, $b\in\mathrm{m}athbb{R}$, and $$B:=\{x\in X;\, \langle a, x\rangle\leq b\}.$$
\item $a\in\mathrm{int}\, (K^-)$ and $$B:=\{x\in X;\, \langle a, x\rangle\leq 0\}.$$
\end{enumerate}
Then the couple $(K,B)$ is stable.
\end{theorem}
\noindent As a corollary, we obtain the following finite-dimensional result, where the cone is the standard nonnegative lattice cone in $\mathrm{m}athbb{R}^N$.
\begin{corollary}
Let $X=\mathrm{m}athbb{R}^N$ and $K=\{(x_k)_1^N\in\mathrm{m}athbb{R}^N;\, x_k\geq0, k=1,\ldots,N \}$. Suppose that at least one of the following conditions holds true.
\begin{enumerate}
\item $a_1,\ldots,a_n\in X$, $b_1,\ldots,b_n>0$, and $$B:=\{x\in X;\, \langle a_i, x\rangle\leq b_i,\ i=1,\ldots,n\}.$$
\item $a\not\in K^-$, $b\in\mathrm{m}athbb{R}$, and $$B:=\{x\in X;\, \langle a, x\rangle\leq b\}.$$
\item $a\in\mathrm{int}\, (K^-)$ and $$B:=\{x\in X;\, \langle a, x\rangle\leq 0\}.$$
\end{enumerate}
Then the couple $(K,B)$ is stable.
\end{corollary}
\section{Perturbed alternating projections sequences for subspaces}\label{section:subspaces}
In this section, we study the convergence of the perturbed alternating projections sequences in the case where the limit sets are subspaces. The following elementary example shows that if the intersection of the subspaces is non-trivial, in general, convergence does not hold.
\begin{example}\label{ex:duedimensionale} Let $Z=\mathrm{m}athbb{R}^2$ and let us consider $A_n=A=B=\{(x,0)\in Z;\, x\in\mathrm{m}athbb{R}\}$ ($n\in\mathrm{m}athbb{N}$). For each $h\in\mathrm{m}athbb{N}$, let us consider the line $C_h=\{(x,\frac1h-\frac1{h^2}x);\,x\in\mathrm{m}athbb{R}\}$ passing through the points $(0,\frac1h)$ and $(h,0)$. Let us consider the starting point $z_0=(0,0)$ and observe that, if we consider the points $a_k=(P_A P_{C_1})^k z_0$, it is clear that there exists $N_1\in\mathrm{m}athbb{N}$ such that $\|a_{N_1}\|>\frac12$. Define $B_n=C_1$ whenever $1\leq n\leq N_1$. Similarly, if we consider the points $a_{N_1+k}=(P_A P_{C_2})^{k} a_{N_1}$ then there exists $N_2\in\mathrm{m}athbb{N}$ such that $\|a_{N_1+N_2}\|>1$. Define $B_n=C_2$ whenever $N_1+1\leq n\leq N_1+N_2$. Then, proceeding inductively, it is easy to construct a sequence $\{B_n\}$ such that the perturbed alternating projections sequences $\{a_n\}$ and $\{b_n\}$, w.r.t. $\{A_n\}$ and $\{B_n\}$ and with starting point $z_0$, are unbounded.
\end{example}
In order to avoid such a situation we consider the case in which the intersection of the subspaces reduces to the origin. We have the following theorem.
\begin{theorem}\label{prop:sottospazisommachiusa}
Let $X$ be a Hilbert space and suppose that $U,V\subset X$ are closed subspaces such that $U\cap V=\{0\}$ and $U+V$ is closed. Let $\{A_n\}$ and $\{B_n\}$ be two sequences of closed convex sets such that $A_n\rightarrow
U$ and $B_n\rightarrow V$ for the Attouch-Wets convergence.
Then, for each $a_0\in X$, the corresponding perturbed alternating projections sequences $\{a_n\}$ and $\{b_n\}$, with starting point $a_0$, converge to $0$ in norm.
\end{theorem}
\noindent If $W$ is a subspace of $X$ and $\mathrm{m}athrm{var}epsilonsilon\in(0,1)$, let $W(\mathrm{m}athrm{var}epsilonsilon)\subset X$ be the set defined by
$$W(\mathrm{m}athrm{var}epsilonsilon)=\{w\in X\setminus\{0\};\,\exists u\in W\setminus\{0\}\ \text{such that}\ \cos(u,w)\geq1-\mathrm{m}athrm{var}epsilonsilon\}\cup\mathrm{m}athrm{var}epsilonsilon B_X.$$
An easy computation shows that:
\begin{equation}\label{eq:Uepsilon}
W(\mathrm{m}athrm{var}epsilonsilon)=\{w\in X\setminus\{0\};\,\exists u\in W\cap\|w\|S_X\ \text{such that}\ \|u-w\|^2\leq2\mathrm{m}athrm{var}epsilonsilon\|w\|^2\}\cup\mathrm{m}athrm{var}epsilonsilon B_X.
\end{equation}
Before starting with the proof of the theorem we need the following two lemmas.
\begin{lemma}\label{lemma:defnelconoSottospazio}
Let $X$ be a Hilbert space and $U$ a subspace of $X$. Let $\{A_n\}$ be a sequence of closed convex sets such that $A_n\rightarrow
U$ for the Attouch-Wets convergence. Then, for each $\mathrm{m}athrm{var}epsilonsilon\in(0,1)$, it eventually holds that $A_n\subset U(\mathrm{m}athrm{var}epsilonsilon)$.
\end{lemma}
\begin{proof} On the contrary, suppose that there exist $\mathrm{m}athrm{var}epsilonsilon\in(0,1)$ and a sequence $\{n_k\}$ of integers such that, for each $k\in\mathrm{m}athbb{N}$, there exists $x_{n_k}\in A_{n_k}\setminus U(\mathrm{m}athrm{var}epsilonsilon)$. Since $A_n\rightarrow
U$ for the Attouch-Wets convergence, we can suppose, without any loss of generality, that $\|x_{n_k}\|>1$ (indeed, we can observe that $\mathrm{m}athrm{dist}\bigl(U,X\setminus U(\mathrm{m}athrm{var}epsilonsilon)\bigr)>0$ and use Fact~\ref{fact:AW}). Let $\gamma\in(0,1)$ be such that $\frac{(1-\mathrm{m}athrm{var}epsilonsilon)(1+\frac\gamma{1-\mathrm{m}athrm{var}epsilonsilon})}{(1-\frac\mathrm{m}athrm{var}epsilonsilon2)(1-\gamma)}\leq1$ and let $k\in\mathrm{m}athbb{N}$ be such that there exists $z_k\in A_{n_k}\cap\gamma B_X$. Consider $$\textstyle w_k=\lambda x_{n_k}+(1-\lambda)z_k\in A_{n_k},$$
where $\lambda=\frac1{\|x_{n_k}\|}$, and observe that $1-\gamma\leq\|w_k\|\leq1+\gamma$ and that, for each $u\in U$, we have
\begin{eqnarray*}\textstyle \langle w_k,u\rangle&=&\lambda\langle x_{n_k},u\rangle+(1-\lambda)\langle z_{k},u\rangle\leq \|u\|(1-\mathrm{m}athrm{var}epsilonsilon)\|\lambda x_{n_k}\|+\gamma\|u\|\\
&=& \textstyle \|u\|(1-\mathrm{m}athrm{var}epsilonsilon)(1+\frac\gamma{1-\mathrm{m}athrm{var}epsilonsilon})\\
&=& \textstyle
[(1-\frac\mathrm{m}athrm{var}epsilonsilon2)\|u\|\|w_k\|]\frac{(1-\mathrm{m}athrm{var}epsilonsilon)(1+\frac\gamma{1-\mathrm{m}athrm{var}epsilonsilon})}{(1-\frac\mathrm{m}athrm{var}epsilonsilon2)\|w_k\|}\\
&\leq& \textstyle
[(1-\frac\mathrm{m}athrm{var}epsilonsilon2)\|u\|\|w_k\|]\frac{(1-\mathrm{m}athrm{var}epsilonsilon)(1+\frac\gamma{1-\mathrm{m}athrm{var}epsilonsilon})}{(1-\frac\mathrm{m}athrm{var}epsilonsilon2)(1-\gamma)}\leq (1-\frac\mathrm{m}athrm{var}epsilonsilon2)\|u\|\|w_k\|.
\end{eqnarray*}
Hence, $w_k\in A_{n_k}\setminus U(\frac\mathrm{m}athrm{var}epsilonsilon2)$. Since $\{w_k\}$ is a bounded sequence, by Fact~\ref{fact:AW}, $\mathrm{m}athrm{dist}(w_k,U)\to0$.
We get a contradiction since
$$\textstyle \mathrm{m}athrm{dist}\bigl(U, X\setminus
U(\frac\mathrm{m}athrm{var}epsilonsilon2)\bigr)>0.$$
\end{proof}
\begin{lemma}\label{lemma:approssimazioneprodotto}
Let $U,V$ be closed subspace of a Hilbert space $X$ such that $U\cap V=\{0\}$ and $U+V$ is closed. Let $M\in(0,1)$, then there exist $\mathrm{m}athrm{var}epsilonsilon\in(0,M)$ and $\eta\in(0,1)$ such that, for each $x\in U(\mathrm{m}athrm{var}epsilonsilon)\setminus MB_X$, $y\in V(\mathrm{m}athrm{var}epsilonsilon)\setminus MB_X$ and $z\in\mathrm{m}athrm{var}epsilonsilon B_X$, we have $\cos(x-z,y-z)\leq\eta$.
\end{lemma}
\begin{proof}
By \cite[Lemma~3.5]{FranchettiLight86}, we have that
$$\textstyle \Omega:=\sup\{{<a,b>};\,a\in V\cap S_X,b\in U\cap S_X\}<1.$$
Fix any $\eta\in(\Omega,1)$ and take $\mathrm{m}athrm{var}epsilonsilon\in(0,M)$ such that
$$\textstyle\bigl(\frac{M}{M-\mathrm{m}athrm{var}epsilonsilon}\bigr)^2\bigl(\Omega+ \frac{15\sqrt\mathrm{m}athrm{var}epsilonsilon}{M^2}\bigr)\leq\eta.$$ Suppose that
$x\in U(\mathrm{m}athrm{var}epsilonsilon)\setminus MB_X$, $y\in V(\mathrm{m}athrm{var}epsilonsilon)\setminus MB_X$ and $z\in\mathrm{m}athrm{var}epsilonsilon B_X$. By (\ref{eq:Uepsilon}), there exist $u\in U\cap \|x\|S_X$ and $v\in V\cap \|y\|S_X$ such that $\|x-u\|\leq\sqrt{2\mathrm{m}athrm{var}epsilonsilon}\|x\|$ and $\|y-v\|\leq\sqrt{2\mathrm{m}athrm{var}epsilonsilon}\|y\|$. Hence, $x':=x-u-z\in 3\sqrt\mathrm{m}athrm{var}epsilonsilon B_X$ and $y':=y-v-z\in 3\sqrt\mathrm{m}athrm{var}epsilonsilon B_X$. Then we have:
\begin{eqnarray*}
\textstyle \langle x-z,y-z \rangle &=& \langle u+x',v+y'\rangle \\
&\leq&\textstyle \langle u,v\rangle+\langle
u,y'\rangle+\langle x',v\rangle+\langle x',y'\rangle\\
&\leq&\textstyle \Omega\|x\|\|y\|+ 3\sqrt\mathrm{m}athrm{var}epsilonsilon\|x\|+3\sqrt\mathrm{m}athrm{var}epsilonsilon\|y\|+9\mathrm{m}athrm{var}epsilonsilon\\
&\leq&\textstyle \|x\|\|y\|(\Omega+ \frac{3\sqrt\mathrm{m}athrm{var}epsilonsilon}{\|x\|}+\frac{3\sqrt\mathrm{m}athrm{var}epsilonsilon}{\|y\|}+\frac{9\mathrm{m}athrm{var}epsilonsilon}{\|x\|\|y\|})
\\
&\leq&\textstyle \|x\|\|y\|(\Omega+ \frac{6\sqrt\mathrm{m}athrm{var}epsilonsilon}{M}+\frac{9\mathrm{m}athrm{var}epsilonsilon}{M^2})\\
&\leq&\textstyle \|x\|\|y\|(\Omega+ \frac{15\sqrt\mathrm{m}athrm{var}epsilonsilon}{M^2})\\
&\leq&\textstyle \|x-z\|\|y-z\|\frac{\|x\|}{\|x\|-\mathrm{m}athrm{var}epsilonsilon}\frac{\|y\|}{\|y\|-\mathrm{m}athrm{var}epsilonsilon}(\Omega+ \frac{15\sqrt\mathrm{m}athrm{var}epsilonsilon}{M^2})\\
&\leq&\textstyle \|x-z\|\|y-z\|\bigl(\frac{M}{M-\mathrm{m}athrm{var}epsilonsilon}\bigr)^2(\Omega+ \frac{15\sqrt\mathrm{m}athrm{var}epsilonsilon}{M^2})\\
&\leq&\textstyle\eta \|x-z\|\|y-z\|.
\end{eqnarray*}
\end{proof}
\noindent We are now ready to prove our theorem.
\begin{proof}[Proof of Theorem~\ref{prop:sottospazisommachiusa}]
Fix $M\in(0,1)$, it suffices to prove that eventually $a_n, b_n\in 3 M B_X$ (recall that $\{a_n\}$ and $\{b_n\}$ are defined as in Definition~\ref{def:perturbedseq}). Let $\mathrm{m}athrm{var}epsilonsilon\in(0,M)$ and $\eta\in(0,1)$ be given by Lemma~\ref{lemma:approssimazioneprodotto}. Let us consider the sets $U(\mathrm{m}athrm{var}epsilonsilon),V(\mathrm{m}athrm{var}epsilonsilon)$ and observe that, by Lemma~\ref{lemma:defnelconoSottospazio}, there exists $n_0\in\mathrm{m}athbb{N}$ such that if $n\geq n_0$ then $A_n\subset U(\mathrm{m}athrm{var}epsilonsilon)$ and $B_n\subset V(\mathrm{m}athrm{var}epsilonsilon)$. Let us fix $\mathrm{m}athrm{var}epsilonsilon'\in(0,\mathrm{m}athrm{var}epsilonsilon)$ such that $\eta+\frac{2\mathrm{m}athrm{var}epsilonsilon'}M\leq\frac{\eta+1}2$, then there exists an integer $n_1\geq n_0$ such that, for each $n\geq n_1$, there exist $x_n\in A_n\cap\mathrm{m}athrm{var}epsilonsilon' B_X$ and $y_n\in B_n\cap\mathrm{m}athrm{var}epsilonsilon' B_X$.
Suppose that $n\geq n_1$, we can observe that:
\begin{itemize}
\item by (\ref{eq:proiezionecoseno}) and Lemma~\ref{lemma:approssimazioneprodotto}, if $a_n,b_n\not\in M B_X$, it holds $\|a_n-x_n\|\leq\|b_n-x_n\|\eta$ and hence
$$\textstyle \|a_n\|\leq\|a_n-x_n\|+\mathrm{m}athrm{var}epsilonsilon'\leq\eta(\|b_n\|+\mathrm{m}athrm{var}epsilonsilon')+\mathrm{m}athrm{var}epsilonsilon'\leq\|b_n\|(\eta+\frac{2\mathrm{m}athrm{var}epsilonsilon'}M)\leq\frac{\eta+1}2\|b_n\|;$$
\item similarly, if $a_n,b_{n+1}\not\in M B_X$, it holds
$$\textstyle \|b_{n+1}\|\leq\frac{\eta+1}2\|a_n\|;$$
\item by (\ref{eq:proiezionecoseno}), if $b_n\in M B_X$ then $\|a_n\|\leq\|b_n\|+2\mathrm{m}athrm{var}epsilonsilon'\leq 3M$ and, similarly, if $a_n\in M B_X$ then $\|b_{n+1}\|\leq3M$.
\end{itemize}
By the observations above and since $\frac{\eta+1}2<1$, proceeding as at the end of the proof of Theorem~\ref{puntolur}, it easily follows that eventually $a_n,b_n\in 3M B_X$.
\end{proof}
The remaining part of this section is devoted to proving that the assumption on the closedness of the sum of the subspaces, in Proposition~\ref{prop:sottospazisommachiusa}, cannot be removed. This result is contained in Theorem~\ref{teo:sommaNONchiusa} below and is inspired by the construction contained in \cite[Section~4]{FranchettiLight86}.
Let $X=\ell_2$. For the sake of clearness, we point out that, in the sequel, we sometimes use the following notation: if, for each $h\in \mathrm{m}athbb{N}$, $x^h$ is an element of $X$, we denote by $\{x^h\}$ the corresponding sequence in $X$. Moreover, if $h\in\mathrm{m}athbb{N}$ is fixed, we can consider $x^h$ as a sequence of real numbers and we write $x^h=\{x^h_n\}_n$.
Now, suppose that $\{\theta_n\}\subset\mathrm{m}athbb{R}$ is a bounded sequence and let us consider the linear continuous operator $D:X\to X$ given by $Dx=D\{x_n\}=\{\theta_n x_n\}$ ($x=\{x_n\}\in X$).
Suppose that $b=\{b_n\}\in X$ and consider the closed convex subsets of $Z=X\oplus_2X$ defined as follows:
$$
A=\{(x,0)\in Z;\,x\in X\}\ \ \ \ \text{and}\ \ \ \ V=\{(x,b+Dx)\in Z;\,x\in X\}.
$$
Observe that $A$ is a subspace of $Z$ and $V$ is an affine set in $Z$.
\begin{remark} \label{remark calcolo proiezioni}
If $(\alpha,\beta)\in Z$
then we obtain immediately that $P_A(\alpha,\beta)=(\alpha,0)$.
Now, let us suppose that $(\alpha,0)\in A$ and let us compute $P_V(\alpha,0)$. If we denote $P_V(\alpha,0)=(\{x_n\},\{b_n+\theta_n x_n\})$, by the characterization of best approximation in Hilbert space, we have, for each $\{y_n\}\in X$,
$$\textstyle \bigl\langle(\{x_n-\alpha_n\},\{b_n+\theta_n x_n\}),(\{y_n\},\{\theta_n y_n\})\bigr\rangle=0.$$
Hence, we must have $x_n-\alpha_n+b_n\theta_n+x_n\theta^2_n=0$, whenever $n\in\mathrm{m}athbb{N}$. That is, for each $n\in\mathrm{m}athbb{N}$, it holds
\begin{equation}\label{eq:proiez}
\textstyle x_n=\frac{\alpha_n-\theta_n b_n}{1+\theta_n^2}.
\end{equation}
\end{remark}
\begin{lemma}\label{lemma:convAWpersottospazi}
Let $Z$ be defined as above. Let $\{b^n\}\subset X$ be a norm null sequence. Let $D,D^n:X\to X$ ($n\in\mathrm{m}athbb{N}$) be linear bounded operators such that $D^n\to D$ in the operator norm. Then if we define
$$W=\{(x,Dx)\in Z;\,x\in X\}\ \ \ \text{and}\ \ \ W_n=\{(x,b^n+D^n x)\in Z;\,x\in X\}\ \ \ (n\in\mathrm{m}athbb{N})$$
we have that $W_n\rightarrow W$ for the Attouch-Wets convergence.
\end{lemma}
\begin{proof}
Let us fix $N\in\mathrm{m}athbb{N}$. If $z=(x,Dx)\in W\cap N B_Z$ then we can consider $z'=(x,b^n+D^nx)\in W_n$ and observe that $$\|z-z'\|_Z=\|Dx-D^nx-b^n\|_X\leq N\|D-D^n\|+\|b^n\|_X.$$
Similarly, if $w=(y,b^n+D^ny)\in W_n\cap N B_Z$ then we can consider $w'=(y,Dy)\in W$ and observe that $$\|w-w'\|_Z=\|Dy-D^ny-b^n\|_X\leq N\|D-D^n\|+\|b^n\|_X.$$
Hence, $h_N(W,W_n)\leq N\|D-D^n\|+\|b^n\|\to0$ ($n\to\infty$), and the proof is concluded.
\end{proof}
\begin{theorem}\label{teo:sommaNONchiusa} Let $Z$ be defined as above and $A=\{(x,0)\in Z;\,x\in X\}$, then there exist \begin{enumerate}
\item[(a)] $B$ a closed subspace of $Z$,
\item[(b)] $z_0\in Z$,
\item[(c)] $\{A_n\},\{B_n\}\subset c(Z)$
two sequences of sets converging to $A$ and $B$, respectively, for the Attouch-Wets convergence,
\end{enumerate}
such that the perturbed alternating projections sequences (w.r.t. $\{A_n\}$ and $\{B_n\}$ and with starting point $z_0$), are unbounded.
\end{theorem}
\begin{proof}
Let us consider the sequence $\{a_n\}\subset\mathrm{m}athbb{R}$, given by $a_n=4^{-n}$, and let us consider the operator $D:X\to X$, given by $D\{x_n\}=\{a_n x_n\}$. Then define $B=\{(x,Dx)\in Z;\,x\in X\}$ and, for each $n\in\mathrm{m}athbb{N}$, put $A_n=A$. Now, consider any $z_0=(\{\alpha_n\},0)\in A$ such that $\alpha_n>0$ ($n\in\mathrm{m}athbb{N}$) and $\|z_0\|<1$.
Let us put, $N_0=1$ and, for each $n\in\mathrm{m}athbb{N}$, $\alpha_n^{0,1}=\alpha_n$. We shall define inductively (with respect to $h\in\mathrm{m}athbb{N}$) positive integers $N_h$, countable families of elements of $X$ $$\textstyle \{\alpha^{h,1}_n\}_n,\{\alpha^{h,2}_n\}_n,\{\alpha^{h,3}_n\}_n\ldots,$$ positive real numbers $M_h$, and sets $C_h\subset Z$ such that:
\begin{enumerate}
\item \label{induzione M} $2^h+h>(1+M_h)^2\sum_{n=h+1}^\infty(\alpha_n^{h-1,N_{h-1}})^2> 2^h$
\item \label{induzione C} $C_h=\{(x,b^h+D^hx)\in Z;\, x\in X\}$, where $D^h:X\to X$ is given by $D^h\{x_n\}=\{\theta^h_n x_n\}$ and where $b^h=\{b^h_n\}_n\in X$ and $\theta^h_n\in\mathrm{m}athbb{R}$ are given by
$$\textstyle b^h_n=\begin{cases}
0\ \ \ &\text{if}\ n\leq h\\
\alpha_n^{h-1,N_{h-1}} a_n\frac{1+M_h}{M_h} \ &\text{if}\ n> h
\end{cases}\ \ \ \ \text{and}\ \ \ \ \
\theta^h_n=\begin{cases}
a_n\ &\text{if}\ n\leq h\\
-\frac{1}{M_h}a_n\ &\text{if}\ n> h
\end{cases}
;$$
\item \label{induzione P(h,1)} $(\{\alpha^{h,1}_n\}_n,0)=P_A P_{C_h}(\{\alpha^{h-1,N_{h-1}}_n\}_n,0)$;
\item \label{induzione P(h,t+1)} $(\{\alpha^{h,t+1}_n\}_n,0)=P_A P_{C_h}(\{\alpha^{h,t}_n\}_n,0)$, $t\in\mathrm{m}athbb{N}$;
\item \label{induzione disuguaglianze} $2^h+h>\sum_{n=1}^\infty(\alpha_n^{h,N_{h}})^2\geq\sum_{n=h+1}^\infty(\alpha_n^{h,N_{h}})^2> 2^h$;
\item \label{induzione alfa} $\alpha^{h,t}_n>0$, whenever $n,t\in\mathrm{m}athbb{N}$.
\end{enumerate}
Let us show that this is possible.
Let $h\in\mathrm{m}athbb{N}$ and suppose we already
have $N_{h-1}\in\mathrm{m}athbb{N}$ and sequences $$\{\alpha^{h-1,1}_n\}_n,\ldots,\{\alpha^{h-1,N_{h-1}}_n\}_n\subset X$$ such that the following conditions hold:
\begin{itemize}
\item $\textstyle 2^{h-1}+h-1>\sum_{n=1}^\infty(\alpha_n^{h-1,N_{h-1}})^2$;
\item $\alpha^{h-1,N_{h-1}}_n>0$, whenever $n\in\mathrm{m}athbb{N}$.
\end{itemize}
(Observe that for $h=1$ the two conditions above are trivially satisfied since $\alpha_n^{0,N_0}=\alpha_n>0$ and $\sum_{n=1}^\infty(\alpha_n^{0,N_0})^2=\|z_0\|^2<1$.)
By combining these two relations, we obtain that
$$
2^{h}+h>\sum_{n=1}^\infty(\alpha_n^{h-1,N_{h-1}})^2>\sum_{n=h+1}^\infty(\alpha_n^{h-1,N_{h-1}})^2>0.
$$
Hence there exists a positive real number $M_h$ such that (\ref{induzione M}) holds true. Now, let us consider $C_h$ defined as in (\ref{induzione C}). Then, by the relations in (\ref{induzione P(h,1)}) and (\ref{induzione P(h,t+1)}), we define $\{\alpha^{h,t}_n\}_n$ ($t\in\mathrm{m}athbb{N}$). We just have to prove that there exists $N_h\in\mathrm{m}athbb{N}$ such that (\ref{induzione disuguaglianze}) is satisfied and that (\ref{induzione alfa}) holds true. By taking into account Remark \ref{remark calcolo proiezioni} and the fact that $(\{\alpha^{h,1}_n\}_n,0)=P_A P_{C_h}(\{\alpha^{h-1,N_{h-1}}_n\}_n,0)$, an easy computation shows that, for each $n>h$,
$$\textstyle \alpha^{h,1}_n=\alpha^{h-1,N_{h-1}}_n\frac{1+\frac{1+M_h}{M_h^2}a_n^2}{1+\frac1{M_h^2}a_n^2}.$$
Repeating $N$ times the same argument yields:
$$\textstyle \alpha^{h,N}_n=\alpha^{h-1,N_{h-1}}_n\frac{1+\frac{1+M_h}{M_h^2}a_n^2\sum_{l=0}^{N-1}(1+\frac1{M_h^2}a_n^2)^l}{(1+\frac1{M_h^2}a_n^2)^N}.$$
Moreover, for each $n\leq h$,
$$\textstyle \alpha^{h,1}_n=\alpha^{h-1,N_{h-1}}_n\frac{1}{1+a_n^2}.$$
Repeating $N$ times the same argument yields:
$$\textstyle \alpha^{h,N}_n=\alpha^{h-1,N_{h-1}}_n\frac{1}{(1+a_n^2)^N}.$$
Since $$\textstyle \frac{1+\frac{1+M_h}{M_h^2}a_n^2\sum_{l=0}^{N-1}(1+\frac1{M_h^2}a_n^2)^l}{(1+\frac1{M_h^2}a_n^2)^N}=\frac{-M_h+(1+M_h)(1+\frac1{M_h^2}a_n^2)^N}{(1+\frac1{M_h^2}a_n^2)^N}\to 1+M_h \ \ (N\to \infty)$$
and
$$
\textstyle \frac{1}{(1+a_n^2)^N}\to 0\ \ (N\to \infty),
$$
by (\ref{induzione M}) we obtain that there exists $N_h\in\mathrm{m}athbb{N}$ such that
$$\textstyle 2^h+h>\sum_{n=1}^\infty(\alpha_n^{h,N_{h}})^2\geq\sum_{n=h+1}^\infty(\alpha_n^{h,N_{h}})^2> 2^h.$$
Moreover, it follows immediately that condition (\ref{induzione alfa}) is satisfied.
Now, if $\sum_{k=0}^{h-1} N_k\leq n<\sum_{k=0}^{h} N_k$, put $B_n=C_h$. By our construction, it holds that $a_N=(\{\alpha_n^{h,N_h}\},0)$ where $N=\sum_{k=1}^{h} N_k$. In particular, $$\|b_N\|^2\geq\|P_{A} b_N\|^2=\|P_{A_N} b_N\|^2=\|a_N\|^2\geq \sum_{n=h+1}^\infty(\alpha_n^{h,N})^2> 2^h $$
and hence the the sequences $\{a_n\}$ and $\{b_n\}$ are unbounded.
It remains to prove that $B_n\rightarrow B$ for the Attouch-Wets convergence or, equivalently, that $C_h\rightarrow B$ for the Attouch-Wets convergence. In view of Lemma~\ref{lemma:convAWpersottospazi}, it suffices to prove that the sequence $\{b^h\}$ is norm null and that $D^h\to D$ in the operator norm.
By the inequalities in (\ref{induzione M}) and (\ref{induzione disuguaglianze}), we have
$$\textstyle (1+M_h)^2(2^{h-1}+h-1)\geq(1+M_h)^2\sum_{n=h+1}^\infty(\alpha_n^{h-1,N_{h-1}})^2> 2^h,$$
and hence
$$\textstyle (1+M_h)^2>\frac{2^h}{2^{h-1}+h-1}.$$
Therefore the sequence $\{M_h\}$ is bounded away from $0$. Hence, the sequences $\{\frac1{M_h}\}$ and $\{\frac{1+M_h}{M_h}\}$ are bounded above by a positive constant $K$. Then, by the definition of $b^h$ in (\ref{induzione C}), we have
$$\textstyle \|b^h\|\leq Ka_h\|\{\alpha_n^{h-1,N_{h-1}}\}\|_X\leq \frac K{4^h}\|\{\alpha_n^{h-1,N_{h-1}}\}\|_X\leq\frac K{4^h}\sqrt{2^{h-1}+h-1},$$
where the last inequality holds by (\ref{induzione disuguaglianze}). Moreover, by the definition of $\theta_n^h$ in (\ref{induzione C}), we have that
$$\textstyle \|(D-D^h)x\|^2\leq \sum_{n=h+1}^\infty (a_n-\frac1{M_h}a_n)^2x_n^2\leq(1+K)^2a^2_{h+1}\|x\|^2\ \ \ \ \ (x=\{x_n\}\in X). $$
Therefore, finally we obtain that
$$\textstyle \|D-D^h\|\leq (1+K)a_{h+1}.$$
\end{proof}
\section*{Acknowledgments.}
The research of the authors is partially
supported by GNAMPA-INdAM, Project GNAMPA 2018. The research of the second author is partially
supported by the Ministerio de Ciencia, Innovaci\'on y Universidades (MCIU), Agencia Estatal de Investigaci\'on (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under project PGC2018-096899-B-I00 (MCIU/AEI/FEDER, UE).
The authors thank S.~Reich and E.~Molho for useful remarks that helped them in preparing this paper.
\end{document} |
\begin{document}
\title{Exploiting non-quantum entanglement to widen applicability of limited-entanglement classical simulations of quantum systems.}
\author{N. Ratanje and S. Virmani}
\affiliation{Department of Physics SUPA, University of Strathclyde, John Anderson Building,
107 Rottenrow, Glasgow, G4 0NG, United Kingdom}
\date{\today}
\begin{abstract}
It is known that if the quantum gates in a proposed quantum computer are so noisy that they are incapable of generating
entanglement, then the device can be efficiently simulated classically. If the measurements and single particle
operations are restricted, then the same statement can be true for generalised non-quantum notions of entanglement.
Here we show that this can improve the applicability of limited-entanglement simulation algorithms. In particular, we show that a classical simulation algorithm of Harrow \& Nielsen can efficiently simulate magic state quantum computers that are ideal other than noise on the CNOTs for joint depolarising strengths of $272/489 \sim 56\%$, in contrast to noise levels of $2/3 \sim 66\%$ required if the algorithm uses quantum notions of separability. This suggests that quantum entanglement may not be the most appropriate notion of entanglement to use when discussing the power of stabilizer based quantum computers.
\end{abstract}
\maketitle
\section{Introduction}
Entanglement is often cited as the main reason why quantum devices cannot be efficiently simulated on classical computers. However, while it is the case that devices with limited entanglement can often be efficiently classically simulated \cite{liment,HN,ABIN,VHP}, there can also be highly entangled systems that can be efficiently simulated classically \cite{GK,ENT}. The most prominent example of this arises in fault tolerant quantum computation. In such architectures the stabilizer components generate the quantum entanglement, but acting alone they can be efficiently simulated classically \cite{GK}. Universal quantum computation is usually achieved by injecting a single qubit state, gate, or measurement into the circuit. So fault tolerant quantum computation appears to provide a regime where entanglement is not the source of the computational power.
Motivated by this example, in this paper we will present arguments that ordinary {\it quantum} entanglement may not be the most appropriate notion of entanglement to use when discussing the computational power of stabilizer based computations. We will do this by showing that using a {\it generalised} notion of entanglement, as discussed in \cite{Barnum}, we may significantly enhance the range of applicability of classical simulation algorithms that rely on limited entanglement. In particular we will consider applying the classical simulation algorithm of Harrow \& Nielsen \cite{HN} to quantum computers built using stabilizer components and non-stabilizer single qubit resources (the so-called `magic-state' architectures \cite{BK}). If quantum entanglement is used, the Harrow-Nielsen algorithm requires that the CNOT undergo $2/3\sim$66\% of joint depolarising noise in order to provide an efficient classical simulation. However, we will show that in the case of magic-state architectures, an alternative notion of entanglement can improve the range of applicability of the algorithm to less than $56$\% noise on the CNOT. In the same spirit as the research programme proposed by \cite{Barnum}, this suggests that in the context of fault tolerant quantum computation there may be different notions of entanglement that are more appropriate when discussing computational power.
Note that our work is distinct from the usual approach to analysing noise induced classicality in magic-state computers,
which treats the non-stabiliser resource as the source of non-classicality, and aims to to understand the noise levels required
before this resource loses its power \cite{BK,magicmore}. Here we are treating the single qubit non-stabilizer resources as classical, but assessing the power
brought by the generalised-entangling power of the CNOT.
\section{Generalised entanglement and classical simulation}
A quantum state of two particles $A$ and $B$ is said to be quantum separable if it can be written as a probabilistic mixture of
products of single particle quantum states $\rho_A \otimes \rho_B$. In the context of qubits, $\rho_A$ and $\rho_B$
must be quantum states drawn from the Bloch sphere, which we denote by the symbol $Q$. This conventional definition of
entanglement can be generalised \cite{Barnum}: for any convex set $C$ of single particle Bloch vectors (perhaps including ones outside the Bloch sphere), we define an operator to be $C$-separable if it can be decomposed as a probabilistic mixture of product `states' $\rho_A \otimes \rho_B$ such that $\rho_A,\rho_B \in C$.
The notion of quantum separability is utilised in the algorithm developed by Harrow \& Nielsen \cite{HN} to classically simulate devices with non-entangling gates. The algorithm samples the separable decomposition of the system after each gate has been applied, and stores the product state that results from each such sampling step. The final result of the computation is then obtained by sampling the probabilities of measurement outcomes on each individual particle. The algorithm is efficient as it only ever stores and manipulates product operators, which is computationally inexpensive.
If we allow all single particle measurements, then to be able to perform the final sampling step it is essential that the variables stored correspond to single particle quantum states - otherwise the final sampling will involve negative probability distributions. However, if the measurements in the device are {\it restricted}, then the HN algorithm will work even if the gates are non-entangling w.r.t. any set $C$ that is contained within the normalised dual $M^*$ of the set of available measurement operators $M$ (see e.g. \cite{RV}):
\begin{equation}
C \subset M^* := \{\rho| \mbox{tr}\{\rho\}=1, \mbox{tr}\{\rho B\} \geq 0, \forall B \in M \},
\end{equation}
In the case where $M$ is the set of all quantum measurements, then $M^*$ is the set of quantum states. If $M$ is
a restricted, then $M^*$ will be correspondingly larger. The notion of entanglement changes accordingly \cite{Barnum,Boxworld}. For example,
if $M$ is the set of Pauli measurements, then Bell pairs become separable (see e.g. \cite{RV}).
[Aside: It is important to note one subtlety: for the classical simulation to function with such a modified notion of entanglement we also require that the {\it single particle} operations cannot produce single particle
states outside $C$, i.e. that the single particle states can only be initialised within $C$, and $C$ must remain invariant
under the action of any single particle gates.]
In \cite{RV} such considerations were applied to understand the effects of noise on computation in (among other situations) magic state quantum computation, in which single particle unitaries are drawn from the Clifford group and the measurements $M$ are in Pauli directions only. While this did provide new regimes that could be efficiently simulated classically, it did not lead to significant improvements in the applicability of the HN algorithm, in that the levels of noise required to remove the entangling power of the CNOT were typically no lower than those required to remove quantum entanglement.
However in \cite{RV} the set $C$ was taken to be the {\it full} dual, i.e. $C=M^*$, and this is not required as we can only ever initialise the qubits in genuine quantum states. So in fact one can consider choices of $C$ such that $Q \subset C \subset M^*$. In this work we show that in the context of magic state architectures there exist choices of $C$ such that $Q \subset C \subset M^*$ that can lead to quite large differences in the entangling power of the CNOT (CX) or CSIGN (CZ) gate. In particular we find that the CZ gate requires only 56\% depolarising noise to remove its entangling power, in contrast to 2/3 = 66\% if we choose $C=Q$.
\section{Notation}
We will represent two qubit operators in one of two ways, both relying upon the Pauli expansion:
\begin{equation}
{1 \over 4} \sum_{i,j=0,..,3} \rho_{ij} \sigma_{i} \otimes \sigma_{j}
\end{equation}
where $\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3}$ represent the 2x2 Identity
and the Pauli X,Y,Z operators respectively. In fact as we will only consider trace
normalised operators, we will always have $\rho_{00}=1$.
We will display the
coefficients $\{\rho_{ij}\}$ as a 4 x 4 matrix, with rows/columns numbered
from 0,..,3:
\begin{eqnarray}\left(\begin{array}{cccc}
\rho_{00}=1 & \rho_{01} & \rho_{02} & \rho_{03} \\
\rho_{10} & \rho_{11} & \rho_{12} & \rho_{13} \\
\rho_{20} & \rho_{21} & \rho_{22} & \rho_{23} \\
\rho_{30} & \rho_{31} & \rho_{32} & \rho_{33}
\end{array}\right) \label{matrix}
\end{eqnarray}
or as a column vector formed by sequentially concatenating the columns of this matrix, i.e.
\begin{equation}
(\rho_{00},\rho_{10},\rho_{20},\rho_{30},\rho_{01},..,\rho_{31},\rho_{02},..,\rho_{32},\rho_{03},..,\rho_{33})^T. \label{column}
\end{equation}
Please note that we do not represent 2-qubit operators as density matrices
in the computational basis - we always use this Pauli expansion form.
Although the results can be modified straightforwardly to any 2-qubit
Clifford gate, the only gate that we will consider in this work is the CSIGN or CZ gate, which is
the unitary $\ket{0}\bra{0}\otimes I + \ket{1}\bra{1}\otimes Z$.
We will use the symbol ${\cal C}_\lambda$
to denote an ideal CZ gate followed by joint depolarising noise, where with probability $\lambda$ the output of the gate is replaced with a maximally mixed state represented by a normalized identity.
The sets $C$ that we consider consider here will be truncated cubes of Bloch vectors. We have investigated
a number of sets that are invariant under the single-qubit Clifford group, and truncated cubes are the best of these
in the parameter ranges of interest. The truncated cubes (see fig. (\ref{shape})) are parameterised
by a variable $r \in (0,1]$ and are defined in the following way: $TRUN(r)$ is the truncated cube given by the convex hull of Bloch vectors $(x,y,z)$ of the form
$(|x|,|y|,|z|)=(1,1,r),(1,r,1),(r,1,1)$, i.e. the convex hull of all possible sign choices and choices of either $|x|,|y|$ or $|z|$ equal to $r$. There are hence 24 extremal points of $TRUN(r)$. For two particles $A$ and $B$ will need to consider products of
states selected from $TRUN(r)$. We will use the shorthand $(x_A,y_A,z_A)\otimes(x_B,y_B,z_B)$ to denote to such states,
where $(x_A,y_A,z_A)$ and $(x_B,y_B,z_B)$ are the individual Bloch vectors. In particular the maximally
mixed state can be represented as $(0,0,0)\otimes(0,0,0)$.
\begin{figure}
\caption{A truncated cube.}
\label{shape}
\end{figure}
\section{Problem and basic method}
Our goal is to calculate, for various choices of $C=TRUN(r)$, the level of joint depolarising noise
required to make all output states separable w.r.t. $TRUN(r)$. In particular we must go through all
possible pairs of input extremal points, act upon them with ${\cal C}_\lambda$ and find the minimal $\lambda$
required to make sure that all outputs are separable. There are 24$\times$24 possible pairs of extremal input states,
but in fact it is sufficient to consider only four of them. The reason for this is that if a given operator
$\rho$ is separable w.r.t. $TRUN(r)$, then so is any operator obtained from its matrix
representation (\ref{matrix}) by (a) permuting
any of the last three columns, (b) permuting any of the last three rows, (c) transposition of the matrix, (d)
multiplying any of the last three rows or columns by minus signs. This is because all these operations
are linear, and leave $TRUN(r)$ invariant. Up to these transformations the outputs obtained from the full set of pairs of extremal input states are equivalent to the outputs of only four pairs, which we refer to as Cases 1 to 4:
\begin{eqnarray}
\rho_1:= (1,1,r) \otimes (1,1,r) \,\,
\rho_2:= (r,1,1) \otimes (1,1,r) \nonumber\\
\rho_3:= (1,1,r) \otimes (1,r,1) \,\,
\rho_4:= (r,1,1) \otimes (1,r,1) \nonumber
\end{eqnarray}
Hence our goal is to compute the minimal $\lambda$ required to make sure that the outputs ${\cal C}_\lambda(\rho_i)$ are $TRUN(r)$ separable
for $i=1,..,4$. Consider for instance Case 1. We must compute:
\begin{eqnarray}
\min && \lambda \nonumber \\
{\rm{s.t.}}\,\,\,&& (1-\lambda) CZ(\rho_1) + \lambda ((0,0,0) \otimes (0,0,0)) \nonumber \\
&& \in TRUN(r)-sep \nonumber
\end{eqnarray}
This is a linear programming problem, and hence
can be readily solved using standard numerical tools. In fact, as we shall see it turns out that the structure of
the problem enables us to use computational algebra packages to obtain almost completely
analytic solutions. We can reexpress the linear programme as:
\begin{eqnarray}
\min && p_0 \nonumber \\
{\rm{s.t.}}\,\,\,
&& CZ(\rho_1) = \sum_{n=0,..,576} p_n E_n , \label{sepcond} \\
&& p_n \geq 0 \label{probcond}
\end{eqnarray}
where
\begin{eqnarray}
E_0 := (CZ(\rho_1) - (0,0,0) \otimes (0,0,0)) \,\,\, , \,\,\,p_0 := \lambda \nonumber
\end{eqnarray}
and for $n=1,..,576$ the $E_n$ denote pairs of extremal product states from $TRUN(r)$, and the $p_n$ represent
the probabilities with which they occur in the separable decomposition. We do not need to impose normalisation
$\sum_{n>0} p_n=1$, as this is implicitly required by the other conditions.
A numerical linear programming routine will usually return the minimal value of $p_0$ (i.e. $\lambda$) as well
as the values of $p_n$ (for $n>0$) that achieve it. For our purposes it is important to note
that the values of $p_n$ that achieve the optimum can always be chosen to
be non-zero only on a {\it linearly independent} subset of vectors from $E_n$.
The reason for this that the optimum can always be achieved on an extremal
point of the convex set of $p_n$ that satisfy the constraints,
and it is well known in linear programming theory that the extremal points must be non-zero only on linearly independent subsets of
$E_n$.
One can find such a linearly independent solution by first solving the the programme
numerically, and then one by one removing the vectors $E_n$, each time checking whether the optimal
solution can still be attained. If at any step the removal of a particular $E_n$ prevents us
from reaching the optimal solution, we put that vector back into the problem and carry on removing
others. Eventually we terminate at an optimal solution involving only a linearly independent subset
of $E_n$.
This numerical solution can then be turned into an analytic proof that the value of $\lambda$ is
{\it achievable}. One simply takes the symbolic version of the matrix formed from the optimal linearly
independent subset of $E_n$, and then inverts the condition (\ref{sepcond})
to obtain an analytic expression for the optimal $p_n$. The solutions obtained in this way are
not usually simple, as the optimal separable decompositions are usually not unique, and so the computer
does not necessarily identify the neatest solution. However, for completeness in the appendix we present example
decompositions for the region around $r=1/2$.
While this demonstrates achievability analytically, it does not demonstrate necessity analytically - i.e. that
the solution is optimal. One approach to attempt to fill this gap is to obtain a set of inequalities
(the analogue of entanglement witnesses) that are necessary conditions for any operator to be $TRUN(r)$-separable,
and then turn these into inequalities on $\lambda$ that could match the achievable values of $\lambda$.
In the next section we will see that in the regime around $r=1/2$ (which appears to be the best region) we
have been able to find suitable inequalities for Cases 2-4 above, and hence the optimal $\lambda$ for these cases analytically. In case
1 we have not been able to identify suitable inequalities. However, putting all four cases together for $r=1/2$ we find
that $\lambda=272/489 \sim 0.56$ is analytically achievable, whereas $\lambda=5/9 \sim 0.55$ is necessary. So our analytic solutions
are essentially tight.
\section{Results and Necessary conditions}
As the truncated cubes that we consider are within the dual of ideal Pauli measurements, it is a necessary condition
for separability that the output operators return positive values for the probabilities of measurement
outcomes for the various Pauli operators. It turns out that for values of $r$ or most interest such inequalities
are For instance, consider Case 4. Under the action of ${\cal C}_\lambda$ the output operator is:
\begin{eqnarray}\left(\begin{array}{cccc}
1 & (1-\lambda) r & 1-\lambda & 1-\lambda \\
1-\lambda & (1-\lambda) r & -(1-\lambda)r^2 & 1-\lambda\\
(1-\lambda) r & -(1-\lambda) & (1-\lambda) r & (1-\lambda) r \\
1-\lambda & (1-\lambda) r & 1-\lambda & 1-\lambda
\end{array}\right)
\end{eqnarray}
On this operator the requirement of having a positive probability of getting down, down in a measurement of
the $X$ and $Y$ operators leads to:
\begin{equation}
\lambda \geq 1- (1 /(2+r^2)). \label{4lower}
\end{equation}
Performing identical computations for Cases 1-3 gives:
\begin{eqnarray}
{\rm{Cases 1 \& 3}}: \,\,\,\lambda \geq 1- (1 / (1+2r)) \label{3lower} \\
{\rm{Case 2}}: \,\,\,\lambda \geq 1- (1 / (2+r^2)) \label{2lower}
\end{eqnarray}
We will now see that in cases 2-4 these inequalities are tight. The numerical results for
Cases 1-4 are plotted in figure \ref{lambdaplot}. For all values of $r$, Case 4 matches its lower bound in equation (\ref{4lower}). For values of $r$ decreasing from $r=1$ the other three cases also match the lower bounds (\ref{3lower},\ref{2lower}), but for each of these cases there is a value of $r$ below which these bounds become unattainable.
We see from these results that $r=1/2$ appears to require the least noise. This value of $r$ is also large
enough to include the whole Bloch sphere, and hence all magic-states. At the point $r=1/2$, Cases 2,3 and 4
attain the lower bounds (\ref{2lower}),(\ref{3lower}),(\ref{4lower}), hence showing that the higher of these values $
1-1/(2+r^2)= 5/9 = 0.5555$
is necessary and sufficient for these cases. With Case 1 we have not been able to identify matching lower bounds,
however the method of the previous section can be used to show that in the vicinity of $r=1/2$ we may achieve values of
\begin{equation}
\lambda = (4\, r^2 + 8\, r + 12)/({r^4 + 2\, r^3 + 9\, r^2 + 16\, r + 20}). \nonumber
\end{equation}
At $r=1/2$ this evaluates to $272/489=0.5562$.
\begin{figure}
\caption{Plot of the minimal noise level $\lambda$ required to make the output of a noisy CZ $TRUN(r)$ separable,
for each of the four cases. For all values of $r$, Case 4 matches its lower bound in equation (\ref{4lower}
\label{lambdaplot}
\end{figure}
\section{Summary and Open problems}
We have shown that using a non-quantum notion of entanglement can increase the range of application
of Harrow \& Nielsen's limited entanglement simulation method. However, we do not know whether this persists for more general noise
models, or in the context of multiparticle entanglement.
We also do not know whether the truncated cubes are the optimal set for our problem. In future work \cite{RVprep} we will present lower bounds on the $\lambda$ required for any set $C$, where for any $Q \subset C \subset M^*$ we can show that $\lambda$ must be greater than 0.42. However, truncated
cubes are currently the best among the choices of $C$ that we have tried. In \cite{RVprep} we will also present an analysis of a generalised problem where the sets $Q$ and $M^*$ are reduced and expanded respectively
by noise. In very noisy regimes one can identify the optimal sets that require the least amount of depolarising noise to make the CZ separable (they turn out to be either spheres or octahedra), but for
the most interesting case of noise-free $Q$ and $M^*$ it open whether truncated cubes are the best choice.
\section{Acknowledgments}
This work is supported by EU STREP `Corner', a University of Strathclyde Starter grant, and an EPSRC PhD studentship.
We thank Jon Barrett, Steve Barnett, Vedran Dunjko, and Mark Howard for discussions.
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\begin{widetext}
\section{Appendix: Computer generated expressions for achievable values of $\lambda$ near $r=1/2$.}
In this appendix, for $r=1/2$, we present $TRUN(r)$ separable decompositions that show that the numerically obtained values
of $\lambda$ are achievable. The decompositions have been obtained using the computational methods described in the main text.
They are extremely messy objects - this may be a reflection of the fact that the optimal separable decompositions are usually
non-unique. It is possible that other methods may lead to more concise and illustrative expressions. Indeed, the solution
for Case 4 was guessed by hand, and the expressions are much simpler than the remaining cases.
\section{Case 1}
For Case 1, and a noise level given by
\begin{equation}
\lambda = 1 - \frac{r^4 + 2\, r^3 + 5\, r^2 + 8\, r + 8}{r^4 + 2\, r^3 + 9\, r^2 + 16\, r + 20}
\end{equation}
the output state of ${\cal C}_\lambda$ is given by (using the column vector notation of equation (\ref{column})):
\begin{eqnarray}
\frac{r^4 + 2\, r^3 + 5\, r^2 + 8\, r + 8}{r^4 + 2\, r^3 + 9\, r^2 + 16\, r + 20} \left(\begin{array}{c} \frac{r^4 + 2\, r^3 + 9\, r^2 + 16\, r + 20}{r^4 + 2\, r^3 + 5\, r^2 + 8\, r + 8} \\
r
\\ r
\\ r
\\ r
\\ 1
\\ -1
\\ 1
\\ r
\\ -1
\\ 1
\\ 1
\\ r
\\ 1
\\ 1
\\ r^2 \end{array}\right)
\end{eqnarray}
The separable decomposition of this state is given by the following expression, where the columns of the left matrix correspond
to product operator extrema (in the notation of equation (\ref{column})), and the elements of the right column vector
are the associated probabilities. In the vicinity of $r=1/2$ all the probabilities are positive, and so the decomposition
works for a range of $r$.
\begin{eqnarray}
\left(\begin{array}{ccccccccccccccc} 1 & -1 & 1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & -1 & r & - r & - r\\ r & - r & r & r & - r & - r & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1\\ -1 & -1 & 1 & 1 & 1 & 1 & - r & - r & r & - r & r & - r & 1 & -1 & 1\\ - r & - r & 1 & -1 & -1 & 1 & -1 & 1 & 1 & 1 & -1 & -1 & r & - r & 1\\ - r & r & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & r^2 & r^2 & - r\\ - r^2 & r^2 & r & - r & r & - r & -1 & 1 & -1 & -1 & -1 & -1 & r & r & -1\\ r & r & 1 & -1 & -1 & 1 & r & - r & r & - r & - r & r & r & r & 1\\ -1 & -1 & r & r & r & 1 & - r & - r & -1 & -1 & 1 & 1 & 1 & -1 & r\\ -1 & 1 & r & r & r & -1 & - r & r & -1 & -1 & -1 & -1 & r & r & - r^2\\ - r & r & r^2 & r^2 & - r^2 & - r & - r & - r & 1 & 1 & 1 & 1 & 1 & 1 & - r\\ 1 & 1 & r & r & r & 1 & r^2 & r^2 & - r & r & r & - r & 1 & 1 & r\\ 1 & -1 & 1 & 1 & -1 & - r & 1 & 1 & r & - r & r & - r & 1 & -1 & -1\\ 1 & 1 & 1 & 1 & -1 & r & 1 & -1 & r & - r & - r & r & r & r & r\\ r & r & r & r & r & r^2 & 1 & 1 & - r & r & r & - r & 1 & 1 & 1\\ -1 & 1 & 1 & 1 & -1 & - r & - r & - r & r^2 & r^2 & r^2 & r^2 & 1 & 1 & -1 \end{array}\right)
\left(\begin{array}{c} \frac{2\, \left(r^2 - 2\, r + 4\right)}{ - r^5 - 5\, r^3 + 2\, r^2 + 12\, r + 40}\\ 0\\ \frac{4\, \left(r^2 + r + 1\right)}{\left(r + 1\right)\, \left(r^4 + 2\, r^3 + 9\, r^2 + 16\, r + 20\right)}\\ 0\\ 0\\ -\frac{2\, r^2\, \left(r - 1\right)}{ - r^5 - 5\, r^3 + 2\, r^2 + 12\, r + 40}\\ -\frac{2\, r^2\, \left(r - 1\right)}{ - r^5 - 5\, r^3 + 2\, r^2 + 12\, r + 40}\\ 0\\ \frac{ - r^6 + r^5 + 12\, r^3 + 21\, r^2 + 27\, r - 12}{4\, \left(r^5 + 3\, r^4 + 11\, r^3 + 25\, r^2 + 36\, r + 20\right)}\\ -\frac{r^7 - r^6 + 2\, r^5 - 8\, r^4 - 9\, r^3 + r^2 + 54\, r + 8}{4\, \left(r^6 + r^5 + 5\, r^4 + 3\, r^3 - 14\, r^2 - 52\, r - 40\right)}\\ \frac{ - r^6 - 2\, r^4 + 6\, r^3 - r^2 + 14\, r + 8}{4\, \left( - r^5 - 5\, r^3 + 2\, r^2 + 12\, r + 40\right)}\\ -\frac{ - r^7 + r^6 - 6\, r^5 + 12\, r^4 - 11\, r^3 + 11\, r^2 + 2\, r + 40}{4\, \left(r^6 + r^5 + 5\, r^4 + 3\, r^3 - 14\, r^2 - 52\, r - 40\right)}\\ -\frac{r\, \left( - r^5 - 2\, r^3 + 6\, r^2 + 5\, r + 16\right)}{r^6 + r^5 + 5\, r^4 + 3\, r^3 - 14\, r^2 - 52\, r - 40}\\ \frac{ - r^6 - 2\, r^4 + 6\, r^3 + 5\, r^2 + 12\, r - 8}{\left(r + 1\right)\, \left(r - 2\right)\, \left(r^4 + 2\, r^3 + 9\, r^2 + 16\, r + 20\right)}\\ \frac{2\, \left(r^2 - 2\, r + 4\right)}{ - r^5 - 5\, r^3 + 2\, r^2 + 12\, r + 40} \end{array}\right) \nonumber
\end{eqnarray}
\section{Case 2}
For Case 2, in the vicinity of $r=1/2$, and for a noise level given by
\begin{equation}
\lambda = 1 - {1 \over 2 +r^2}
\end{equation}
the output state of ${\cal C}_\lambda$ is given by (using the column vector notation of equation (\ref{column})):
\begin{eqnarray}
{1 \over 2 +r^2} \left(\begin{array}{c} 2 +r^2 \\ r^2\\ r\\ 1\\ 1\\ 1\\ - r\\ 1\\ 1\\ -1\\ r\\ 1\\ r\\ r\\ 1\\ r \end{array}\right)
\end{eqnarray}
The probabilities $p_m$ appearing in the decomposition are given by:
\begin{eqnarray}
\left(\begin{array}{c} -\frac{3\, r^{17} + 9\, r^{16} - 44\, r^{15} - 121\, r^{14} - 92\, r^{13} - 49\, r^{12} + 692\, r^{11} + 807\, r^{10} + 1458\, r^9 + 305\, r^8 + 1020\, r^7 - 419\, r^6 + 772\, r^5 - 371\, r^4 + 252\, r^3 - 139\, r^2 + 35\, r - 22}{4\, r\, \left(r^2 + 2\right)\, \left(r^{15} + 5\, r^{14} + 2\, r^{13} + 12\, r^{12} + 47\, r^{11} - 129\, r^{10} - 88\, r^9 - 122\, r^8 + 15\, r^7 + 99\, r^6 + 26\, r^5 + 96\, r^4 + r^3 + 25\, r^2 - 4\, r + 14\right)}\\ \frac{r^{17} + 5\, r^{16} + 22\, r^{15} + 73\, r^{14} + 98\, r^{13} + 447\, r^{12} - 154\, r^{11} + 805\, r^{10} - 476\, r^9 + 1245\, r^8 - 766\, r^7 + 1131\, r^6 - 642\, r^5 + 357\, r^4 - 126\, r^3 + 39\, r^2 - 5\, r - 6}{4\, r\, \left(r^2 + 2\right)\, \left(r^{14} + 4\, r^{13} - 2\, r^{12} + 14\, r^{11} + 33\, r^{10} - 162\, r^9 + 74\, r^8 - 196\, r^7 + 211\, r^6 - 112\, r^5 + 138\, r^4 - 42\, r^3 + 43\, r^2 - 18\, r + 14\right)}\\ -\frac{ - r^{18} - 4\, r^{17} + 17\, r^{16} + 87\, r^{15} + 159\, r^{14} + 297\, r^{13} + 355\, r^{12} + 419\, r^{11} + 1041\, r^{10} + 405\, r^9 + 1201\, r^8 - 331\, r^7 + 917\, r^6 - 661\, r^5 + 353\, r^4 - 175\, r^3 + 60\, r^2 - 37\, r - 6}{4\, r\, \left(r^2 + 2\right)\, \left(r^{15} + 5\, r^{14} + 2\, r^{13} + 12\, r^{12} + 47\, r^{11} - 129\, r^{10} - 88\, r^9 - 122\, r^8 + 15\, r^7 + 99\, r^6 + 26\, r^5 + 96\, r^4 + r^3 + 25\, r^2 - 4\, r + 14\right)}\\ -\frac{ - 2\, r^{15} - r^{14} + 3\, r^{13} + 60\, r^{12} + 72\, r^{11} + 233\, r^{10} + 293\, r^9 + 406\, r^8 + 38\, r^7 + 289\, r^6 - 259\, r^5 + 48\, r^4 - 108\, r^3 - 9\, r^2 - 37\, r - 2}{2\, \left(r^2 + 2\right)\, \left( - r^{14} - 6\, r^{13} - 8\, r^{12} - 20\, r^{11} - 67\, r^{10} + 62\, r^9 + 150\, r^8 + 272\, r^7 + 257\, r^6 + 158\, r^5 + 132\, r^4 + 36\, r^3 + 35\, r^2 + 10\, r + 14\right)}\\ \frac{r^{17} + 11\, r^{16} + 35\, r^{15} + 55\, r^{14} + 125\, r^{13} + 17\, r^{12} + 175\, r^{11} + 219\, r^{10} + 579\, r^9 + 195\, r^8 + 585\, r^7 - 235\, r^6 + 431\, r^5 - 213\, r^4 + 101\, r^3 - 39\, r^2 + 16\, r - 10}{2\, \left(r^2 + 2\right)\, \left(r^{15} + 5\, r^{14} + 2\, r^{13} + 12\, r^{12} + 47\, r^{11} - 129\, r^{10} - 88\, r^9 - 122\, r^8 + 15\, r^7 + 99\, r^6 + 26\, r^5 + 96\, r^4 + r^3 + 25\, r^2 - 4\, r + 14\right)}\\ \frac{ - r^{16} + 3\, r^{15} + 58\, r^{14} + 131\, r^{13} + 214\, r^{12} + 487\, r^{11} + 122\, r^{10} + 551\, r^9 - 156\, r^8 + 489\, r^7 - 178\, r^6 + 361\, r^5 - 54\, r^4 + 45\, r^3 - 2\, r^2 - 19\, r - 3}{2\, \left(r^2 + 2\right)\, \left(r^{15} + 5\, r^{14} + 2\, r^{13} + 12\, r^{12} + 47\, r^{11} - 129\, r^{10} - 88\, r^9 - 122\, r^8 + 15\, r^7 + 99\, r^6 + 26\, r^5 + 96\, r^4 + r^3 + 25\, r^2 - 4\, r + 14\right)}\\ \frac{\left(r + 1\right)\, \left(r^{15} + 3\, r^{14} + 15\, r^{13} + 52\, r^{12} + 117\, r^{11} + 49\, r^{10} + 467\, r^9 - 278\, r^8 + 675\, r^7 - 427\, r^6 + 573\, r^5 - 336\, r^4 + 167\, r^3 - 73\, r^2 + 33\, r - 14\right)}{4\, \left(r^2 + 2\right)\, \left(r - 1\right)\, \left(r^{13} + 5\, r^{12} + 3\, r^{11} + 17\, r^{10} + 50\, r^9 - 112\, r^8 - 38\, r^7 - 234\, r^6 - 23\, r^5 - 135\, r^4 + 3\, r^3 - 39\, r^2 + 4\, r - 14\right)}\\ -\frac{2\, \left(r^2 + 1\right)\, \left(2\, r^{12} + 11\, r^{11} + 14\, r^{10} + 31\, r^9 + 68\, r^8 + 41\, r^7 + 73\, r^6 - 23\, r^5 + 31\, r^4 + 5\, r^2 + 4\, r - 1\right)}{\left(r^2 + 2\right)\, \left(r + 1\right)\, \left(r^{13} + 5\, r^{12} + 3\, r^{11} + 17\, r^{10} + 50\, r^9 - 112\, r^8 - 38\, r^7 - 234\, r^6 - 23\, r^5 - 135\, r^4 + 3\, r^3 - 39\, r^2 + 4\, r - 14\right)}\\ \frac{{\left(r - 1\right)}^2\, \left( - r^{13} + 17\, r^{11} + 55\, r^{10} + 84\, r^9 + 89\, r^8 + 106\, r^7 + 146\, r^6 + 63\, r^5 + 142\, r^4 - 27\, r^3 + 55\, r^2 + 14\, r + 25\right)}{2\, \left(r^2 + 2\right)\, \left( - r^{14} - 6\, r^{13} - 8\, r^{12} - 20\, r^{11} - 67\, r^{10} + 62\, r^9 + 150\, r^8 + 272\, r^7 + 257\, r^6 + 158\, r^5 + 132\, r^4 + 36\, r^3 + 35\, r^2 + 10\, r + 14\right)}\\ \frac{r^{16} + 6\, r^{15} + 28\, r^{14} + 45\, r^{13} + 101\, r^{12} - 108\, r^{11} + 826\, r^{10} - 317\, r^9 + 1397\, r^8 - 666\, r^7 + 1032\, r^6 - 685\, r^5 + 531\, r^4 - 256\, r^3 + 162\, r^2 - 67\, r + 18}{4\, \left(r^2 + 2\right)\, \left(r^{14} + 4\, r^{13} - 2\, r^{12} + 14\, r^{11} + 33\, r^{10} - 162\, r^9 + 74\, r^8 - 196\, r^7 + 211\, r^6 - 112\, r^5 + 138\, r^4 - 42\, r^3 + 43\, r^2 - 18\, r + 14\right)}\\ \frac{2\, r^{15} - 5\, r^{14} + 9\, r^{13} - 84\, r^{12} + 2\, r^{11} - 429\, r^{10} - 23\, r^9 - 444\, r^8 + 94\, r^7 - 143\, r^6 - 53\, r^5 + 76\, r^4 - 34\, r^3 + r^2 + 3\, r + 4}{2\, \left(r^2 + 2\right)\, \left(r^{14} + 4\, r^{13} - 2\, r^{12} + 14\, r^{11} + 33\, r^{10} - 162\, r^9 + 74\, r^8 - 196\, r^7 + 211\, r^6 - 112\, r^5 + 138\, r^4 - 42\, r^3 + 43\, r^2 - 18\, r + 14\right)}\\ \frac{r^{16} + 10\, r^{15} + 35\, r^{14} + 78\, r^{13} + 181\, r^{12} + 200\, r^{11} + 191\, r^{10} - 8\, r^9 + 135\, r^8 - 110\, r^7 + 241\, r^6 - 122\, r^5 + 183\, r^4 - 36\, r^3 + 45\, r^2 - 12\, r + 12}{2\, \left(r^2 + 2\right)\, \left( - r^{14} - 6\, r^{13} - 8\, r^{12} - 20\, r^{11} - 67\, r^{10} + 62\, r^9 + 150\, r^8 + 272\, r^7 + 257\, r^6 + 158\, r^5 + 132\, r^4 + 36\, r^3 + 35\, r^2 + 10\, r + 14\right)}\\ -\frac{ - r^{15} + 6\, r^{14} + 31\, r^{13} + 100\, r^{12} + 102\, r^{11} + 291\, r^{10} + 94\, r^9 + 305\, r^8 + 35\, r^7 + 156\, r^6 - 149\, r^5 + 142\, r^4 - 104\, r^3 + 27\, r^2 - 8\, r - 3}{2\, \left(r^2 + 2\right)\, \left(r^{14} + 4\, r^{13} - 2\, r^{12} + 14\, r^{11} + 33\, r^{10} - 162\, r^9 + 74\, r^8 - 196\, r^7 + 211\, r^6 - 112\, r^5 + 138\, r^4 - 42\, r^3 + 43\, r^2 - 18\, r + 14\right)}\\ -\frac{\left(r^2 + 1\right)\, \left(r^{14} + 9\, r^{13} + 29\, r^{12} + 67\, r^{11} + 140\, r^{10} + 86\, r^9 + 180\, r^8 - 6\, r^7 + 77\, r^6 - 119\, r^5 + 65\, r^4 - 29\, r^3 + 22\, r^2 - 8\, r - 2\right)}{\left(r^2 + 2\right)\, \left(r - 1\right)\, \left(r + 1\right)\, \left(r^{13} + 5\, r^{12} + 3\, r^{11} + 17\, r^{10} + 50\, r^9 - 112\, r^8 - 38\, r^7 - 234\, r^6 - 23\, r^5 - 135\, r^4 + 3\, r^3 - 39\, r^2 + 4\, r - 14\right)}\\ \frac{r^{16} - 4\, r^{15} - 20\, r^{14} - 55\, r^{13} + 89\, r^{12} + 574\, r^{11} + 1102\, r^{10} + 1467\, r^9 + 1029\, r^8 + 1248\, r^7 + 144\, r^6 + 607\, r^5 - 201\, r^4 + 230\, r^3 - 74\, r^2 + 29\, r - 22}{4\, r\, \left(r^2 + 2\right)\, \left( - r^{14} - 6\, r^{13} - 8\, r^{12} - 20\, r^{11} - 67\, r^{10} + 62\, r^9 + 150\, r^8 + 272\, r^7 + 257\, r^6 + 158\, r^5 + 132\, r^4 + 36\, r^3 + 35\, r^2 + 10\, r + 14\right)} \end{array}\right) \nonumber
\end{eqnarray}
corresponding to the product extrema given by the columns of the following matrix:
\begin{eqnarray}
\left(\begin{array}{ccccccccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & -1 & -1 & -1 & 1 & - r & 1 & - r & r & 1 & 1 & 1 & 1 & -1 & 1\\ r & 1 & 1 & - r & 1 & 1 & -1 & 1 & -1 & 1 & - r & -1 & 1 & r & - r\\ 1 & - r & - r & 1 & - r & 1 & r & 1 & 1 & r & -1 & - r & r & 1 & 1\\ 1 & -1 & -1 & 1 & r & 1 & 1 & - r & 1 & 1 & 1 & r & 1 & r & 1\\ 1 & 1 & 1 & -1 & r & - r & 1 & r^2 & r & 1 & 1 & r & 1 & - r & 1\\ r & -1 & -1 & - r & r & 1 & -1 & - r & -1 & 1 & - r & - r & 1 & r^2 & - r\\ 1 & r & r & 1 & - r^2 & 1 & r & - r & 1 & r & -1 & - r^2 & r & r & 1\\ r & 1 & 1 & 1 & -1 & 1 & - r & 1 & 1 & r & -1 & -1 & -1 & 1 & r\\ r & -1 & -1 & -1 & -1 & - r & - r & - r & r & r & -1 & -1 & -1 & -1 & r\\ r^2 & 1 & 1 & - r & -1 & 1 & r & 1 & -1 & r & r & 1 & -1 & r & - r^2\\ r & - r & - r & 1 & r & 1 & - r^2 & 1 & 1 & r^2 & 1 & r & - r & 1 & r\\ 1 & - r & r & - r & 1 & r & -1 & 1 & - r & 1 & - r & -1 & r & -1 & 1\\ 1 & r & - r & r & 1 & - r^2 & -1 & - r & - r^2 & 1 & - r & -1 & r & 1 & 1\\ r & - r & r & r^2 & 1 & r & 1 & 1 & r & 1 & r^2 & 1 & r & - r & - r\\ 1 & r^2 & - r^2 & - r & - r & r & - r & 1 & - r & r & r & r & r^2 & -1 & 1 \end{array}\right) \nonumber
\end{eqnarray}
\section{Case 3}
For Case 3, in the vicinity of $r=1/2$, and for a noise level given by
\begin{equation}
\lambda = 1 - {1 \over 1 + 2r}
\end{equation}
the output state of ${\cal C}_\lambda$ is given by (using the column vector notation of equation (\ref{column})):
\begin{eqnarray}
1/(1+2r) \left(\begin{array}{c} 1+2r \\ r \\ 1 \\ 1 \\ r \\ 1 \\ - r \\ r \\ 1 \\ - r \\ r^2 \\ 1 \\ 1 \\ r \\ 1 \\ 1 \end{array}\right) \nonumber
\end{eqnarray}
The probabilities $p_m$ appearing in the separable decomposition are given by:
\begin{eqnarray}
\left(\begin{array}{c} \frac{{\left(r - 1\right)}^2\, \left( - r^6 + r^5 + 10\, r^3 + 7\, r^2 + 13\, r - 6\right)}{4\, \left(2\, r + 1\right)\, {\left(r + 1\right)}^2\, \left( - r^5 + r^4 + 2\, r^2 + r + 1\right)}\\ -\frac{ - 2\, r^9 + r^8 + 8\, r^7 - 4\, r^6 + 12\, r^5 - 18\, r^4 + 32\, r^3 - 60\, r^2 + 46\, r - 15}{4\, \left(2\, r + 1\right)\, {\left(r + 1\right)}^3\, \left( - r^5 + r^4 + 2\, r^2 + r + 1\right)}\\ 0\\ \frac{{\left(r - 1\right)}^2\, \left( - r^4 + 4\, r^2 + 8\, r + 1\right)}{4\, \left(2\, r + 1\right)\, \left( - r^5 + r^4 + 2\, r^2 + r + 1\right)}\\ \frac{{\left(r - 1\right)}^2\, \left( - 3\, r^6 + r^4 + 8\, r^3 + r^2 + 1\right)}{2\, r\, \left(2\, r + 1\right)\, {\left(r + 1\right)}^2\, \left( - r^5 + r^4 + 2\, r^2 + r + 1\right)}\\ -\frac{\left(r - 1\right)\, \left( - 2\, r^5 + r^4 + 4\, r^3 + 10\, r^2 + 2\, r + 1\right)}{\left(2\, r + 1\right)\, \left(r^2 + 1\right)\, {\left(r + 1\right)}^2\, \left( - r^3 + r^2 + r + 1\right)}\\ -\frac{ - r^7 + r^5 + 10\, r^4 + r^3 - 4\, r^2 - 9\, r + 2}{4\, r\, \left(2\, r + 1\right)\, {\left(r + 1\right)}^2\, \left( - r^3 + r^2 + r + 1\right)}\\ \frac{\frac{r^8}{2} - 3\, r^7 + 2\, r^6 + 6\, r^4 - 7\, r^3 + 2\, r - \frac{1}{2}}{r\, \left(2\, r + 1\right)\, {\left(r + 1\right)}^2\, \left( - r^5 + r^4 + 2\, r^2 + r + 1\right)}\\ \frac{\frac{r}{4} + \frac{\frac{r^2}{2} + r - \frac{3}{2}}{ - r^4 + 2\, r^2 + 2\, r + 1} + \frac{1}{4}}{2\, r + 1}\\ \frac{r\, \left( - r^5 - r^4 + 2\, r^3 + 6\, r^2 + 7\, r - 5\right)}{2\, \left(2\, r + 1\right)\, \left(r^2 + 1\right)\, \left(r + 1\right)\, \left( - r^3 + r^2 + r + 1\right)}\\ -\frac{2\, r^8 - 5\, r^7 + 7\, r^6 - 9\, r^5 + 7\, r^4 - 19\, r^3 + 9\, r^2 + r - 1}{4\, r^2\, \left(2\, r + 1\right)\, \left( - r^5 + r^4 + 2\, r^2 + r + 1\right)}\\ \frac{ - 2\, r^9 + 3\, r^8 - 8\, r^7 + 16\, r^6 - 8\, r^5 + 30\, r^4 - 24\, r^3 + 8\, r^2 + 2\, r - 1}{4\, r^2\, \left(2\, r + 1\right)\, \left(r^2 + 1\right)\, \left(r + 1\right)\, \left( - r^3 + r^2 + r + 1\right)}\\ \frac{ - 2\, r^{10} + 2\, r^9 - 3\, r^8 + 6\, r^7 + 6\, r^6 + 14\, r^5 + 20\, r^4 - 30\, r^3 + 28\, r^2 - 8\, r - 1}{4\, r\, \left(2\, r + 1\right)\, {\left(r + 1\right)}^2\, \left( - r^5 + r^4 + 2\, r^2 + r + 1\right)}\\ -\frac{ - 2\, r^{11} + 2\, r^{10} + 3\, r^9 + 3\, r^8 + 16\, r^7 - 16\, r^6 + 14\, r^5 - 58\, r^4 + 58\, r^3 - 26\, r^2 + 7\, r - 1}{4\, r\, \left(2\, r + 1\right)\, {\left(r + 1\right)}^3\, \left( - r^5 + r^4 + 2\, r^2 + r + 1\right)}\\ \frac{1}{r + 1} - \frac{7}{4\, \left(2\, r + 1\right)} + \frac{1}{4} \end{array}\right)
\end{eqnarray}
corresponding to the product extrema given by the columns of the following matrix:
\begin{eqnarray}
= \left(\begin{array}{ccccccccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & -1 & -1 & -1 & - r & 1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & -1 & 1 & - r & r & - r & 1 & 1 & 1 & 1 & 1 & -1\\ - r & - r & r & - r & r & 1 & 1 & -1 & 1 & r & r & r & - r & - r & - r\\ 1 & -1 & 1 & -1 & 1 & - r & 1 & 1 & 1 & - r & 1 & 1 & -1 & -1 & 1\\ 1 & -1 & -1 & 1 & -1 & r^2 & 1 & -1 & -1 & r & 1 & 1 & -1 & -1 & 1\\ 1 & -1 & 1 & -1 & -1 & - r & - r & r & - r & - r & 1 & 1 & -1 & -1 & -1\\ - r & r & r & r & r & - r & 1 & -1 & 1 & - r^2 & r & r & r & r & - r\\ -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & r & - r & r & r & - r\\ -1 & -1 & 1 & -1 & -1 & - r & 1 & -1 & -1 & -1 & r & - r & r & r & - r\\ -1 & -1 & -1 & 1 & -1 & 1 & - r & r & - r & 1 & r & - r & r & r & r\\ r & r & - r & - r & r & 1 & 1 & -1 & 1 & r & r^2 & - r^2 & - r^2 & - r^2 & r^2\\ - r & r & r & - r & - r & 1 & r & - r & r & -1 & 1 & 1 & 1 & -1 & -1\\ - r & r & - r & r & r & - r & r & r & - r & 1 & 1 & 1 & 1 & -1 & -1\\ - r & r & r & - r & r & 1 & - r^2 & - r^2 & - r^2 & -1 & 1 & 1 & 1 & -1 & 1\\ r^2 & - r^2 & r^2 & r^2 & - r^2 & 1 & r & r & r & - r & r & r & - r & r & r \end{array}\right) \nonumber
\end{eqnarray}
\section{Case 4}
The separable decomposition in Case 4 was guessed by hand, and hence is simpler that the expressions derived above using numerical and computational
algebra techniques. For Case 4 and a noise level given by:
\begin{equation}
\lambda = 1 - {1 \over 2 +r^2} \nonumber
\end{equation}
the output state of ${\cal C}_\lambda$ is given by (using the column vector notation of equation (\ref{matrix})):
\begin{eqnarray}{1 \over 2 +r^2} \left(\begin{array}{cccc}
2 +r^2 & r & 1 & 1 \\
1 & r & -r^2 & 1 \\
r & -1 & r & r \\
1 & r & 1 & 1
\end{array}\right)
\end{eqnarray}
The separable decomposition of this state is given by the following expression, which works for all values of $r$ (as the probabilities
are always positive):
\begin{eqnarray}
{1 \over 2 +r^2}\left(\begin{array}{cccc}
1 & r & 1 & 1 \\
1 & r & 1 & 1 \\
r & r^2 & r & r \\
1 & r & 1 & 1
\end{array}\right)
+
{1 +r^2 \over 2 +r^2} \left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right)
\end{eqnarray}
where the rightmost matrix is simply a uniform mixture of all product vectors $(x_A,y_A,z_A)\otimes(x_B,y_B,z_B)$ that satisfy
$|x_A|=|x_B|=|y_A|=|y_B|=1$, $|z_A|=|z_B|=r$, and the anticorrelation condition $x_A=-y_B$ and $x_B=-y_A$ (which ensures the presence
of the central $-1$ elements).
\end{widetext}
\end{document} |
\begin{document}
\markboth{Roy}{convergence diagnostics for MCMC}
\title{Convergence diagnostics for Markov chain Monte Carlo}
\author{Vivekananda Roy
\affil{ Department of statistics, Iowa state University\\
Ames, Iowa, USA; email: [email protected]}
}
\begin{abstract}
Markov chain Monte Carlo (MCMC) is one of the most useful approaches
to scientific computing because of its flexible construction, ease
of use and generality. Indeed, MCMC is indispensable for performing
Bayesian analysis. Two critical
questions that MCMC practitioners need to address are where to start
and when to stop the simulation. Although a great amount of research
has gone into establishing convergence criteria and stopping rules
with sound theoretical foundation, in practice, MCMC users often
decide convergence by applying empirical diagnostic tools. This
review article discusses the most widely used MCMC convergence
diagnostic tools. Some recently proposed stopping rules with
firm theoretical footing are also presented. The convergence diagnostics
and stopping rules are illustrated using three detailed examples.
\end{abstract}
\begin{keywords}
autocorrelation, empirical diagnostics, Gibbs
sampler, Metropolis algorithm, MCMC, stopping rules
\end{keywords}
\maketitle
\section{INTRODUCTION}
\label{sec:int}
Markov chain Monte Carlo (MCMC) methods are now routinely used to fit
complex models in diverse disciplines. A Google search for
``Markov chain Monte Carlo'' returns more than 11.5 million hits. The popularity of
MCMC is mainly due to its widespread usage in computational physics and
Bayesian statistics, although it is also used in frequentist inference
\cite[see e.g.][]{geye:thomp:1995, chri:2004}.
The fundamental idea of MCMC is that if simulating from a target
density $\pi$ is difficult so that the ordinary Monte Carlo method
based on independent and identically distributed (iid) samples cannot
be used for making inference on $\pi$, it may be possible to construct
a Markov chain $\{X_n\}_{n \ge 0}$ with stationary density $\pi$ for
forming Monte Carlo estimators. An introduction to construction of
such Markov chains, including the Gibbs sampler and the
Metropolis-Hasting (MH) sampler, is provided by \cite{geye:2011}
\cite[see also][]{robe:case:2004}. General purpose MH algorithms are
available in the R packages mcmc \citep{r:mcmc} and MCMCpack
\citep{R:mcmcp}. There are several R \citep{r} packages implementing
specific MCMC algorithms for a number of statistical models [see
e.g. MCMCpack \citep{R:mcmcp}, MCMCglmm \citep{r:MCMCglmm}, geoBayes
\citep{r:geoBayes}]. Here, we do not discuss development of MCMC
algorithms, but rather focus on analyzing the Markov chain obtained
from running such an algorithm for determining its convergence.
Two important issues that must be addressed while implementing MCMC
are where to start and when to stop the algorithm. As we discuss now,
these two tasks are related to determining convergence of the
underlying Markov chain to stationarity and convergence of Monte Carlo
estimators to population quantities, respectively. It is known that
under some standard conditions on the Markov chain, for any initial
value, the distribution of $X_n$ converges to the stationary
distribution as $n \rightarrow \infty$ (see
e.g. \citet[][chap. 13]{meyn:twee:1993}, \cite{robe:rose:2004}). Since
$X_0 \not\sim \pi$ and MCMC algorithms produce (serially) correlated
samples, the further the initial distribution from $\pi$, the longer
it takes for $X_n$ to approximate $\pi$. In particular, if the
initial value is not in a high-density ($\pi$) region, the
samples at the earlier iterations may not be close to the target
distribution. In such cases, a common practice is to discard early
realizations in the chain and start collecting samples only after the
effect of the initial value has (practically) worn off. The main
idea behind this method, known as {\it burn-in}, is to use samples
only after the Markov chain gets sufficiently close to the stationary
distribution, although its usefulness for Monte Carlo estimation has
been questioned in the MCMC community \citep{geye:2011}. Thus,
ideally, MCMC algorithms should be initialized at a high-density region,
but if finding such areas is difficult, collection of Monte Carlo
samples can be started only after a certain iteration $n'$ when
approximately $X_{n'} \sim \pi$.
Once the starting value is determined, one needs to decide when to
stop the simulation. (Note that the starting value here refers to the
beginning of collection of samples as opposed to the initial value of
$X_0$ of the Markov chain, although these two values can be the same.)
Often the quantities of interest regarding the target density $\pi$
can be expressed as means of certain functions, say
$E_\pi g \equiv \int_{\mathcal{X}} g(x) \pi(x) dx$ where $g$ is a real
valued function. For example, appropriate choices of $g$ make
$E_\pi g$ different measures of location, spread, and other summary
features of $\pi$. Here, the support of the target density $\pi$ is
denoted by $\mathcal{X}$, which is generally $\mathbb{R}^d$ for some
$d \ge 1$, although it can be non-Euclidean as well. We later in
Section~\ref{sec:diag} consider vector valued functions $g$ as
well. The MCMC estimator of the population mean $E_\pi g$ is the
sample average
$\bar{g}_{n', n} \equiv \sum_{i=n'+1}^{n} g(X_i)/(n-n')$. If no
burn-in is used then $n'=0$. It is known that usually
$\bar{g}_{n', n} \rightarrow E_\pi g $ as $n \rightarrow \infty$ (see
Section~\ref{sec:diag} for details). In practice, however, MCMC users
run the Markov chain for a finite $n^*$ number of iterations, thus
MCMC simulation should be stopped only when $\bar{g}_{n', n^*}$ has
sufficiently converged to $E_\pi g$. The accuracy of the time average
estimator $\bar{g}_{n', n}$ obviously depends on the quality of the
samples. Thus, when implementing MCMC methods, it is necessary to
wisely conclude Markov chain convergence, and subsequently determine
when to stop the simulation. In particular, while premature
termination of the simulation will most likely lead to inaccurate
inference, unnecessarily running longer chains is not desirable either
as it eats up resources.
By performing theoretical analysis on the underlying Markov chain, an
analytical upper bound on its distance to stationarity may be obtained
\citep{rose:1995}, which in turn can provide a rigorous method for
deciding MCMC convergence and thus finding $n'$
\citep{jone:hobe:2001}. Similarly, using a sample size calculation
based on an asymptotic distribution of the (appropriately scaled)
Monte Carlo error $\bar{g}_{n', n^*} - E_\pi g$, an {\it honest}
stopping value $n^*$ can be found. In the absence of such theoretical
analysis, often empirical diagnostic tools are used to check
convergence of MCMC samplers and estimators, although, as shown
through examples in Section~\ref{sec:exam}, these tools cannot
determine convergence with certainty. Since early 1990s with the
increasing use of MCMC, a great deal of research effort has gone into
developing convergence diagnostic tools. These diagnostic methods can
be classified into several categories. For example, corresponding to
the two types of convergence mentioned before, some of these
diagnostic tools are designed to assess convergence of the Markov
chain to the stationary distribution, whereas others check for
convergence of the summary statistics like sample means and sample
quantiles to the corresponding population quantities. The available
MCMC diagnostic methods can be categorized according to other criteria
as well, for example, their level of theoretical foundation, if they
are suitable for checking joint convergence of multiple variables,
whether they are based on multiple (parallel) chains or a single chain
or both, if they are complemented by a visualization tool or not, if
they are based on moments and quantiles or the kernel density of the
observed chain, and so on. Several review articles on MCMC convergence
diagnostics are available in the literature \cite[see
e.g.][]{cowl:carl:1996, broo:robe:1998,
meng:robe:guih:1999}. \cite{cowl:carl:1996} provide a description of
13 convergence diagnostics and summarize these according to the
different criteria mentioned above. While some of these methods are
widely used in practice, several new approaches have been proposed
since then. In this article, we review some of these tools that are
commonly used by MCMC practitioners or that we find promising.
\section{MCMC diagnostics}
\label{sec:diag}
As mentioned in the introduction, MCMC diagnostic tools are needed for
deciding convergence of Markov chains to the stationarity. Also,
although in general the longer the chain is run the better Monte Carlo
estimates it produces, in practice, it is desirable to use some
stopping rules for prudent use of resources. In this section, we
describe some MCMC diagnostics that may be used for deciding Markov
chain convergence or stopping MCMC sampling. In the context of each
method, we also report if it is designed particularly for one of these
two objectives.
\subsection{Honest MCMC}
\label{sec:hone}
In this section, we describe some rigorous methods for finding $n'$
and $n^*$ mentioned in the introduction. Let $f_n$ be the density of
$X_n$. It is known that under some standard conditions \citep[see
e.g.][chap. 13]{meyn:twee:1993},
$\frac{1}{2} \int_{\mathcal{X}} |f_n(x) - \pi (x)| dx \downarrow 0$ as
$n \rightarrow \infty$, that is, $X_n$ converges in the total
variation (TV) norm to a random variable following
$\pi$. \cite{jone:hobe:2001} mention that a rigorous way of deciding
the convergence of the Markov chain to $\pi$ is by finding an iteration
number $n'$ such that
\begin{equation}
\label{eq:totvar}
\frac{1}{2} \int_{\mathcal{X}} |f_{n'}(x) - \pi (x)| dx < 0.01.
\end{equation}
(The cutoff value 0.01 is arbitrary and any predetermined precision
level can be used.) \cite{jone:hobe:2001} propose to use the
smallest $n'$ for which (\ref{eq:totvar}) holds as the honest value
for burn-in.
The above-mentioned burn-in hinges on the TV norm in (\ref{eq:totvar})
which is generally not available. Constructing a quantitative bound to
the TV norm is also often difficult, although significant progress has
been made in this direction \citep{rose:1995, rose:2002, baxe:2005,
andr:fort:viho:2015}. In particular, a key tool for constructing a
quantitative bound to the TV norm is using the {\it drift} and {\it
minorization} (d\&m) technique \citep{rose:1995}. The d\&m technique
has been successfully used to analyze a variety of MCMC algorithms
\cite[see e.g.][]{ fort:moul:robe:rose:2003, jone:hobe:2004,
roy:hobe:2010, vats:2017}. The d\&m conditions, as we explain later
in this section, are also crucial to provide an honest way to check
convergence of MCMC estimators of popular summary measures like
moments and quantiles of the target distributions. Although we
consider the TV norm here, over the last few years, other metrics like
the Wasserstein distance have also been used to study Markov chain
convergence \cite[see e.g.][]{durm:moul:2015,
qin:hobe:2019}. On the other hand, using Stein's method,
\cite{gorh:mack:2015} propose a computable discrepancy measure that
seems promising as it depends on the target only through the
derivative of $\log \pi$, and hence is appropriate in Bayesian
settings where the target is generally known up to the intractable
normalizing constant.
As in the Introduction, let a particular feature of the target density
be expressed as $E_\pi g$ where $g$ is a real valued function.
By the strong law of large numbers for Markov chains, it is known that
if $\{X_n\}_{n \ge 0}$ is appropriately {\it irreducible}, then
$\bar{g}_{n',n} \equiv \sum_{i=n'+1}^{n} g(X_i)/(n-n')$ is a strongly
consistent estimator of $E_\pi g$, that is,
$\bar{g}_{n',n} \rightarrow E_\pi g$ almost surely as
$n \rightarrow \infty$ for any fixed $n'$
\citep{asmu:glyn:2011}. Without loss of generality, we let $n' =0$
when discussing stopping rules, and for the ease of notation, we
simply write $\bar{g}_{n}$ for $\bar{g}_{0,n}$. The law of large numbers
justifies estimating $E_\pi g$ by the sample (time) average estimator
$\bar{g}_n$, as in the ordinary Monte Carlo. If a central limit
theorem (CLT) is available for $\bar{g}_n$ (that is, for the error
$\bar{g}_n - E_\pi g$) then a `sample size calculation' based on the
width of an interval estimator for $E_\pi g$ can be performed for
choosing an appropriate value for $n^*$. Indeed, under some regularity
conditions,
\begin{equation}
\label{eq:clt}
\sqrt{n} (\bar{g}_n - E_\pi g) \stackrel{d}{\rightarrow} N(0,
\sigma^2_g) \;\;\mbox{as}\; n \rightarrow \infty,
\end{equation}
where
$\sigma^2_g \equiv \mbox{Var}_\pi (g(X_0)) + 2 \sum_{i=1}^{\infty}
\mbox{Cov}_\pi (g(X_0), g(X_i)) < \infty$; the subscript $\pi$
indicates that the expectations are calculated assuming
$X_0 \sim \pi$. (Note that, due to the autocorrelations present in a
Markov chain, $\sigma^2_g \neq \mbox{Var}_\pi (g(X_0)) = \lambda^2_g$,
say.) If $\widehat{\sigma}_{g, n}$ is a consistent estimator of
$\sigma_g$, then an estimator of the standard error of $\bar{g}_n$,
based on the sample size $n$ is $\widehat{\sigma}_{g, n}/\sqrt{n}$. Since the
standard error $\widehat{\sigma}_{g, n}/\sqrt{n}$ allows one to judge the
reliability of the MCMC estimate, it should always be reported along
with the point estimate $\bar{g}_n$. The standard error also leads to
a $100(1-\alpha)\%$ confidence interval for $E_\pi g$, namely
$\bar{g}_n \mp z_{\alpha/2} \widehat{\sigma}_{g, n}/\sqrt{n}$. Here
$z_{\alpha/2}$ is the $(1 -\alpha/2)$ quantile of the standard
normal distribution. The MCMC simulation can be stopped if the
half-width of the $100(1-\alpha)\%$ confidence interval
falls below a prespecified threshold, say $\epsilon$. \cite{jone:hobe:2001} refer to
this method as the honest way to stop the chain. Indeed, the fixed-width stopping rule (FWSR)
\citep{fleg:hara:jone:2008, jone:hara:caff:neat:2006} terminates
the simulation the first time after some user-specified $\tilde{n}$ iterations that
\begin{equation}
\label{eq:fwsr}
t_{*} \frac{\widehat{\sigma}_{g, n}}{\sqrt{n}} + \frac{1}{n} \le \epsilon.
\end{equation}
Here, $t_{*}$ is an appropriate quantile. The role of $\tilde{n}$ is
to make sure that the simulation is not stopped prematurely due to
poor estimate of $\widehat{\sigma}_{g, n}$. The value of $\tilde{n}$
should depend on the complexity of the problem. \cite{gong:fleg:2016}
suggest that using $\tilde{n} = 10^4$ works well in
practice.
For validity of the honest stopping rule, a CLT (\ref{eq:clt})
for $\bar{g}_n$ needs to exist, and one would need a consistent
estimator $\widehat{\sigma}_{g, n}$ of $\sigma_g$. For the CLT to
hold, the TV norm in (\ref{eq:totvar}) needs to converge to zero at
certain rate \cite[see][for different conditions guaranteeing a Markov
chain CLT]{jone:2004}. The most common method of establishing a CLT
(\ref{eq:clt}) as well as providing a consistent
estimator of $\sigma_g$ has been by showing the Markov chain
$\{X_n\}_{n \ge 0}$ is {\it geometrically ergodic}, that is, the TV
norm (\ref{eq:totvar}) converges at an exponential rate
\citep{jone:hobe:2001, robe:rose:2004}. Generally, geometric
ergodicity of a Markov chain is proven by constructing an appropriate
d\&m condition \citep{rose:1995, roy:hobe:2010}. For estimation of
$\sigma^2_g$, while \cite{mykl:tier:yu:1995}, and \cite{hobe:jone:pres:rose:2002}
discuss regenerative submission method,
\cite{jone:hara:caff:neat:2006} and \cite{fleg:jone:2010} provide
consistent batch means and spectral variance methods. Availability of
a Markov chain CLT has been demonstrated for myriad MCMC algorithms
for common statistical models. Here we provide an incomplete
list: linear models \citep{roma:hobe:2012, roma:hobe:2015},
generalized linear models including the probit model
\citep{roy:hobe:2007, chak:khar:2017}, the popular logistic model
\citep{choi:hobe:2013, wang:roy:2018b} and the robit model
\citep{roy:2012b}, generalized linear mixed models including the
probit mixed model \citep{wang:roy:2018}, and the logistic mixed model
\citep{wang:roy:2018a}, quantile regression models
\citep{khar:hobe:2012}, multivariate regression models
\citep{roy:hobe:2010, hobe:jung:khar:qin:2018}, penalized regression
and variable selection models \citep{khar:hobe:2013,
roy:chak:2017, vats:2017}.
So far we have described the honest MCMC in the context of estimating
means of univariate functions. The method is applicable to estimation
of vector valued functions as well. In particular, if $g$ is a
$\mathbb{R}^p$ valued function, and if a CLT holds for $\bar{g}_n$,
that is, if
$ \sqrt{n} (\bar{g}_n - E_\pi g) \stackrel{d}{\rightarrow} N(0,
\Sigma_g) \;\;\mbox{as}\; n \rightarrow \infty,$ for some $p \times p$
covariance matrix $\Sigma_g$, then using a consistent estimator
$\widehat{\Sigma}_{g, n}$ of $\Sigma_g$, a $100(1-\alpha)\%$
asymptotic confidence region $C_\alpha(n)$ for $E_\pi g$ can be formed
\cite[for
details see][]{vats:fleg:jone:2019}. \cite{vats:fleg:jone:2019} propose
a fixed-volume stopping rule which terminates the simulation the first
time after $\tilde{n}$ iterations that
\[
(\mbox{Vol}\{C_\alpha(n)\})^{1/p} + \frac{1}{n} \le \varepsilon,
\]
where as in (\ref{eq:fwsr}), $\varepsilon$ is the user's desired level
of accuracy. Note that when
$p=1$, except the $1/n$ terms, the expression above is same as (\ref{eq:fwsr}) with
$\varepsilon = 2\epsilon$. Honest MCMC can also be implemented for
estimation of the quantiles \citep{doss:fleg:jone:neat:2014}. In order
to reduce computational burden, the sequential stopping rules should
be checked only at every $l$ iterations where $l$ is appropriately
chosen. Finally, even if theoretical d\&m analysis is not carried out
establishing a Markov chain CLT, in practice, FWSR can be implemented
using the batch means and spectral variance estimators of
$\sigma_g (\Sigma_g) $ available in the R package mcmcse
\citep{R:mcmcse}.
\subsection{Relative fixed-width stopping rules}
\label{sec:relstop}
FWSR (described in Section~\ref{sec:hone}) explicitly address how well
the estimator $\bar{g}_n$ approximates $E_\pi g$.
\cite{fleg:gong:2015} and \cite{gong:fleg:2016} discuss relative
FWSR in the MCMC setting. \cite{fleg:gong:2015} consider a relative
magnitude rule that terminates the simulation when after $\tilde{n}$
iterations
$t_{*} \widehat{\sigma}_{g, n} n^{-1/2} + n^{-1} \le \epsilon
\bar{g}_n$. \cite{fleg:gong:2015} also consider a relative standard
deviation FWSR (SDFWSR) that terminates the simulation when after
$\tilde{n}$ iterations
$t_{*} \widehat{\sigma}_{g, n} n^{-1/2} + n^{-1} \le \epsilon
\widehat{\lambda}_{g, n}$, where $\widehat{\lambda}_{g, n}$ is a
strongly consistent estimator of the population standard deviation $\lambda_g$.
Asymptotic validity of the relative magnitude and relative standard deviation stopping
rules is established by \cite{glyn:whit:1992} and
\cite{fleg:gong:2015} respectively. This ensures that the simulation
will terminate in a finite time with probability $1$.
In Bayesian statistics applications, \cite{fleg:gong:2015} advocate
the use of relative SDFWSR. In the high-dimensional settings, that is,
where $g$ is a $\mathbb{R}^p$ valued function and $p$ is large,
without a priori knowledge of the magnitude of $E_\pi g$,
\cite{gong:fleg:2016} prefer relative SDFWSR over FWSR based on the marginal chains. In the
multivariate settings, \cite{vats:fleg:jone:2019} argue that stopping
rules based on $p$ marginal chains may not be appropriate as these
ignore cross-correlations between components and may be dictated by
the slowest mixing marginal chain. \cite{vats:fleg:jone:2019} propose
a multivariate relative standard deviation stopping rule involving
volume of the $100(1-\alpha)\%$ asymptotic confidence region, that is,
$\mbox{Vol}\{C_\alpha(n)\}$. Let $\widehat{\Lambda}_{g, n}$ be the sample
covariance matrix. \cite{vats:fleg:jone:2019} propose to stop the
simulation, the first time after $\tilde{n}$ iterations that
\begin{equation}
\label{eq:mulrelsd}
(\mbox{Vol}\{C_\alpha(n)\})^{1/p} + \frac{1}{n} \le \varepsilon (|\widehat{\Lambda}_{g, n}|)^{1/2p},
\end{equation}
where $| \cdot |$ denotes the determinant.
\subsection{Effective sample size}
\label{sec:ess}
For an MCMC-based estimator, effective sample size (ESS) is the number
of independent samples equivalent to (that is, having the same
standard error as) a set of correlated Markov chain samples. Although
ESS (based on $n$ correlated samples) is not uniquely defined, the
most common definition \citep{robe:case:2004} is
\[
\mbox{ESS} = \frac{n}{1 + 2\sum_{i=1}^{\infty}\mbox{Corr}_\pi (g(X_0), g(X_i))}.
\]
\cite{gong:fleg:2016} rewrite the above definition as ESS
$= n \lambda_g^2/\sigma^2_g$. In the multivariate setting, that is,
when $g$ is $\mathbb{R}^p$ valued for some $p\ge 1$,
\cite{vats:fleg:jone:2019} define multivariate ESS (mESS) as
\begin{equation}
\label{eq:mess}
\mbox{mESS} = n\bigg(\frac{|\Lambda_g|}{|\Sigma_g|}\bigg)^{1/p},
\end{equation}
where $\Lambda_g$ is the population covariance matrix. An approach to
terminate MCMC simulation is when
$\widehat{\mbox{ESS}} \; (\widehat{\mbox{mESS}}) $ takes a value
larger than a prespecified number, where
$\widehat{\mbox{ESS}} \;(\widehat{\mbox{mESS}})$ is a consistent
estimator of $\mbox{ESS}\; (\mbox{mESS})$. Indeed,
\cite{vats:fleg:jone:2019} mention that simulation can be terminated
the first time that
\begin{equation}
\label{eq:messcut}
\widehat{\mbox{mESS}} = n\bigg(\frac{|\widehat{\Lambda_{g, n|}}}{|\widehat{\Sigma_{g, n}}|}\bigg)^{1/p} \ge \frac{2^{2/p}\pi}{(p \Gamma(p/2))^{2/p}} \frac{\chi^2_{1-\alpha, p}}{\varepsilon^2},
\end{equation}
where $\varepsilon$ is the desired level of precision for the volume
of the $100(1-\alpha)\%$ asymptotic confidence region, and
$\chi^2_{1-\alpha, p}$ is the $(1- \alpha)$ quantile of
$\chi^2_{p}$. This ESS stopping rule is (approximately) equivalent
to the multivariate relative standard deviation stopping rule given
in (\ref{eq:mulrelsd}) \cite[for details,
see][]{vats:fleg:jone:2019}. Note that
$\widehat{\mbox{ESS}} \; (\widehat{\mbox{mESS}})$ per unit time can
be used to compare different MCMC algorithms (with the same
stationary distribution) in terms of both computational and
statistical efficiency. ESS is implemented in several R packages
including coda \citep{R:coda} and mcmcse \citep{R:mcmcse}. In the
mcmcse package, estimates of ESS both in univariate and multivariate
settings are available. While \cite{gong:fleg:2016} and
\cite{vats:fleg:jone:2019} provide a connection between ESS and
relative SDFWSR stopping rules, \cite{vats:knud:2018} draw
correspondence between ESS and a version of the widely used
Gelman-Rubin (GR) diagnostic presented in the next section.
\subsection{Gelman-Rubin diagnostic}
\label{sec:gelm}
The GR diagnostic appears to be the most popular method for
assessing samples obtained from running MCMC algorithms. The GR
diagnostic relies on multiple chains
$\{X_{i 0}, X_{i 1},\dots,X_{i n-1}\}, i=1,\dots,m$ starting at
initial points that are drawn from a density that is over-dispersed
with respect to the target density $\pi$. \cite{gelm:rubi:1992}
describe methods of creating an initial distribution, although in
practice, these initial points are usually chosen in some ad hoc
way. Using parallel chains, \cite{gelm:rubi:1992} construct two
estimators of the variance of $X$ where $X \sim \pi$, namely, the
within-chain variance estimate,
$W = \sum\limits_{i=1}^m\sum\limits_{j=0}^{n-1}(X_{ij} -
\bar{X_{i\cdot}})^2/(m(n-1))$, and the pooled variance estimate
$\hat{V} = ((n-1)/n)W + B/n$ where
$B/n = \sum\limits_{i=1}^m(\bar{X_{i\cdot}} -
\bar{X_{\cdot\cdot}})^2/(m-1)$ is the between-chain variance estimate,
and $\bar{X_{i\cdot}}$ and $\bar{X_{\cdot\cdot}}$ are the $i^{th}$
chain mean and the overall mean respectively, $i = 1, 2,\dots,
m$. Finally, \cite{gelm:rubi:1992} compare the ratio of these two
estimators to one. In particular, they calculate the potential scale
reduction factor (PSRF) defined by
\begin{equation}\label{eq:psrf}
\hat{R}=\frac{\hat{V}}{W},
\end{equation}
and compare it to one.
\cite{gelm:rubi:1992} argue that since the chains are started from an
over-dispersed initial distribution, in finite samples, the numerator in
(\ref{eq:psrf}) overestimates the target variance whereas the
denominator underestimates it, making $\hat{R}$ larger than
$1$. Simulation is stopped when $\hat{R}$ is sufficiently close to
one. The cutoff value 1.1 is generally used by MCMC practitioners, as
recommended by \cite{gelm:carl:ster:duns:veht:rubi:2014}. Recently,
\cite{vats:knud:2018} propose a modified GR statistic where the
between-chain variance $(B/n)$ is replaced with a particular batch
means estimator of the asymptotic variance for the Monte Carlo
averages $\bar{X}_n$. This modified definition allows for a connection
with ESS and, more importantly, computation of the GR diagnostic based
on a single
chain.
We would like to point out that the expression of $\hat{R}$ given in
(\ref{eq:psrf}), although widely used in practice, differs slightly
from the original definition given by \cite{gelm:rubi:1992}.
\cite{broo:gelm:1998} propose the multivariate PSRF
(MPSRF) to diagnose convergence in the multivariate case. It is denoted by $\hat{R}_{p}$ and
is given by,
\begin{equation}\label{eq:mpsrf}
\hat{R}_{p} = \max_{a} \frac{a^{T}\widehat{V^\ast}a}{a^{T}W^\ast a} = \frac{n-1}{n} +
\bigg(1+\frac{1}{m}\bigg)~\lambda_{1},
\end{equation}
where $\widehat{V^\ast}$ is the pooled covariance matrix, $W^\ast$ is
the within-chain covariance matrix, $B^\ast$ is the between chain
covariance matrix and $\lambda_{1}$ is the largest eigenvalue of the
matrix $({W^\ast}^{-1}B^\ast)/n$. As in the univariate case,
simulation is stopped when $\hat{R}_{p} \approx
1$. \cite{pelt:venn:kask:2009} have proposed a visualization tool
based on linear discriminant analysis and discriminant component
analysis which can be used to complement the diagnostic tools proposed
by \cite{gelm:rubi:1992} and \cite{broo:gelm:1998}. The GR diagnostic
can be easily calculated, and is available in different statistical
packages including the CODA package \citep{R:coda} in R. To conclude
our discussion on the GR diagnostic, note that originally
\cite{gelm:rubi:1992} suggested running $m$ parallel chains, each of
length $2n$. Then discarding the first $n$ simulations, $\hat{R}$ is
computed based on the last $n$ iterations. This leads to the waste of
too many samples, and we do not recommend it.
\subsection{Two spectral density-based methods}
\label{sec:specd}
In this section, we discuss
two diagnostic tools based on asymptotic variance estimates of certain
statistics to check for convergence to
stationarity. \cite{gewe:1992} proposes a diagnostic tool based on the
assumption of existence of the spectral density of a related time
series. Indeed, for the estimation of $E_\pi g$, the asymptotic
variance of $\bar{g}_n$ is $S_g(0)$, the spectral density of
$\{g(X_n), n \ge 0 \}$ (treated as a time series) evaluated at
zero. After $n$ iterations of the Markov chain, let $\bar{g}_{n_A}$
and $\bar{g}_{n_B}$ be the time averages based on the first $n_A$ and
the last $n_B$ observations. \cite{gewe:1992}'s statistic is the
difference $\bar{g}_{nA} - \bar{g}_{nB}$, normalized by its standard
error calculated using a nonparametric estimate of $S_g(0)$ for the
two parts of the Markov chain. Thus, \cite{gewe:1992}'s statistic is
\[ Z_n = \Big(\bar{g}_{n_A} - \bar{g}_{n_B}\Big) \Big/
\sqrt{\widehat{S_g(0)}/n_A + \widehat{S_g(0)}/n_B}.
\]
\cite{gewe:1992}
suggests using $n_A = 0.1n$ and $n_B = 0.5n$. The $Z$ score is
calculated under the assumption of independence of the two parts of
the chain. Thus \cite{gewe:1992}'s convergence diagnostic is a $Z$
test of equality of means where autocorrelation in the samples is
taken into account while calculating the standard error.
\cite{heid:welc:1983} propose another method based on spectral
density estimates. \cite{heid:welc:1983}'s diagnostic is based on
\[B_n(t) = \Big(\sum_{i=0}^{\lfloor nt\rfloor} g(X_i) - \lfloor
nt\rfloor \bar{g}_n\Big) \Big/ \sqrt{n \widehat{S_g(0)}}.
\]
Assuming
that $\{B_n(t), 0 \le t \le 1\}$ is distributed
asymptotically as a Brownian bridge, the Cramer-von Mises statistic
$\int_0^1 B_n(t)^2 dt$ may be used to test the stationarity of the
Markov chain. The stationarity test is successively applied, first on
the whole chain, and then rejecting the first $10 \%, 20 \%, \dots $
and so forth of the samples until the test is passed or $50 \%$ of the samples have been
rejected. Both of these two spectral density-based tools presented
here are implemented in the CODA package \citep{R:coda}. These are
univariate diagnostics although \cite{cowl:carl:1996} mention that for
\cite{gewe:1992}'s statistic, $g$ may be taken to be $-2$ times the log
of the target density when $\mathcal{X} = \mathbb{R}^d$ for some
$d >1$. Finally, we would like to mention that the two spectral
density based methods mentioned here, just like the ESS and the GR
diagnostic, assume the existence of a Markov chain CLT (\ref{eq:clt}),
emphasizing the importance of the theoretical analysis discussed in
Section~\ref{sec:hone}.
\subsection{Raftery-Lewis diagnostic}
\label{sec:raftlew}
Suppose the goal is to estimate a quantile of $g(X)$, that is, to
estimate $u$ such that $P_\pi(g(X) \le u) =q$ for some prespecified
$q$. \cite{raft:lewi:1992} propose a method for calculating an
appropriate burn-in. They also discuss choosing a run length so that
the resulting probability estimate lies in
$[q- \epsilon, q + \epsilon]$ with probability $(1- \alpha)$. Thus the
required accuracy $\epsilon$ is achieved with probability
$(1- \alpha)$. \cite{raft:lewi:1992} consider the binary process
$W_n \equiv I(g(X_n) \le u), n \ge 0$. Although, in general,
$\{W_n\}_{n \ge 0}$ itself is not a Markov chain,
\cite{raft:lewi:1992} assume that for sufficiently large $k$, the
subsequence $\{W_{nk}\}_{n \ge 0}$ is approximately a Markov
chain. They discuss a method for choosing $k$ using model selection
techniques. The transition probability $P(W_{nk} = j | W_{(n-1)k} =i)$
is estimated by the usual estimator
\[
\frac{\sum_{l = 1}^{n} I(W_{lk} = j, W_{(l-1)k} =i)}{\sum_{l = 1}^{n} I(W_{lk} =i)},
\]
for $i, j =0,1$. Here, $I(\cdot)$ is the indicator
function. Using a straightforward eigenvalue analysis of the
two-state empirical transition matrix of $\{W_{nk}\}_{n \ge 0}$,
\cite{raft:lewi:1992} provide an estimate of the burn-in. Using a CLT
for $ \sum_{j=0}^{n-1} W_{jk}/n$, they also give a stopping rule to
achieve the desired level of accuracy.
To implement this univariate method an initial number $n_{\text{min}}$
of iterations is used, and then it is determined if any additional runs are
required using the above techniques. The value
$n_{\text{min}} = \{\Phi^{-1}(1 -\alpha/2)\}^2 q(1-q)/\epsilon^2$ is
based on the standard asymptotic sample size calculation for Bernoulli
$(q)$ population. Since the diagnostic depends on the $q$ values, the
method should be repeated for different quantiles and the largest
among these burn-in estimates can be used. \cite{raft:lewi:1992}'s
diagnostic is available in the CODA package \citep{R:coda}.
\subsection{Kernel density-based methods}
\label{sec:kern}
There are MCMC diagnostics which compute distance between the kernel
density estimates of two chains or two parts of a single chain and
conclude convergence when the distance is close to zero. Unlike the
widely used GR diagnostic \citep{gelm:rubi:1992} which is based on
comparison of some summary moments of MCMC chains, these tools are
intended to assess the convergence of the whole
distributions. \cite{yu:1994} and \cite{boon:merr:krac:2014} estimate
the $L^1$ distance and Hellinger distance between the kernel density
estimates respectively. More recently, \cite{dixi:roy:2017} use the
symmetric Kullback Leibler (KL) divergence to produce two diagnostic
tools based on kernel density estimates of the chains. Below, we
briefly describe the method of \cite{dixi:roy:2017}.
Let $\{X_{ij}: i=1,2; j=1,2,\dots, n\}$ be the $n$ observations
obtained from each of the two Markov chains initialized from two points
well separated with respect to the target density $\pi$. The adaptive
kernel density estimates of observations obtained from the two chains
are denoted by $p_{1n}$ and $p_{2n}$ respectively. The KL divergence
between $p_{in}$ and $p_{jn}$ is denoted by
$KL(p_{in}|p_{jn}), i\neq j,\; i,j=1, 2$, that is,
\[
KL(p_{in}|p_{jn}) = \int_{\mathcal{X}} p_{in}(x) \log\frac{p_{in}(x)}{p_{jn}(x)} dx.
\]
\cite{dixi:roy:2017} find the Monte Carlo estimates of
$KL(p_{in}|p_{jn})$ using samples simulated from $p_{in}$ using the
technique proposed by \citet[][Sec 6.4.1]{silv:1986}. They use the
estimated symmetric KL divergence
($[KL(p_{1n}|p_{2n}) + KL(p_{2n}|p_{1n})]/2$) between $p_{1n}$ and
$p_{2n}$ to assess convergence where a testing of hypothesis framework
is used to determine the cutoff points. The hypotheses are chosen
such that the type 1 error is concluding that the Markov chains have
converged when in fact they have not. The cutoff points for the
symmetric KL divergence are selected to ensure that the probability of
type 1 error is below some level say, 0.05. In case of multiple
($m > 2$) chains, the maximum among ${m \choose 2}$ estimated
symmetric KL divergences (referred to as Tool 1) is used to diagnose
MCMC convergence. Finally, for multivariate examples---that is, when
$\mathcal{X} = \mathbb{R}^d$ for some $d >1$---although multivariate
Tool 1 can be used, in higher dimensions when kernel density
estimation is not reliable, \cite{dixi:roy:2017} recommend assessing
convergence marginally, i.e. one variable at a time, where appropriate
cutoff points are found by adjusting the level of significance using
Bonferroni's correction for multiple comparison.
For multimodal target distributions, if all chains are stuck at the
same mode, then empirical convergence diagnostics based solely on MCMC
samples may falsely treat the target density as unimodal and are prone
to failure. In such situations, \cite{dixi:roy:2017} propose another
tool (Tool 2) that makes use of the KL divergence between the kernel
density estimate of MCMC samples and the target density (generally
known up to the unknown normalizing constant) to detect
divergence. In particular, let $\pi(x) = f(x)/c$, where
$c = \int_{\mathcal{X}} f(x) dx$ is the unknown normalizing
constant. \cite{dixi:roy:2017}'s Tool 2 is given by
\begin{equation}
\label{eq:tool2}
T_2^\ast = \dfrac{\mid \hat{c} - c^\ast \mid}{c^\ast},
\end{equation}
where $\hat{c}$ is a Monte Carlo estimate, as described in section 3.3 of
\cite{dixi:roy:2017}, of the unknown normalizing
constant ($c$), based on the KL divergence between the adaptive kernel
density estimate of the chain and $\pi$, and $c^\ast$ is an estimate of $c$ obtained by
numerical integration. \cite{dixi:roy:2017} discuss that $T_2^\ast$
can be interpreted as the percentage of the target distribution not
yet captured by the Markov chain. Using this interpretation, they
advocate that if $T_2^\ast > 0.05$, then the Markov chain has not yet
captured the target distribution adequately. Since (\ref{eq:tool2}) involves
numerical integration, it cannot be used in high-dimensional examples.
{{\bf A visualization tool:}} \cite{dixi:roy:2017} propose a simple visualization tool to complement
their KL divergence diagnostic tool. This tool can be used for any
diagnostic method (including the GR diagnostic) based on multiple
chains started at distinct initial values, to investigate reasons
behind their divergence. Suppose $m (\ge 3)$ chains are run, and a
diagnostic tool has revealed that the $m$ chains have not mixed
adequately and thus the chains have not yet converged. This
indication of divergence could be due to a variety of reasons. A
common reason for divergence is formation of clusters among multiple
chains. \cite{dixi:roy:2017}'s visualization tool utilizes the tile
plot to display these clusters. As mentioned in
Section~\ref{sec:kern}, for $m$ chains, the KL divergence tool finds the estimated
symmetric KL divergence between each of the ${m \choose 2}$
combinations of chains and reports the maximum among them. In the visualization
tool, if the estimated symmetric KL divergence for a
particular combination is less than or equal to the cutoff value,
then the tool utilizes a gray tile to represent that the two chains
belong to the same cluster, or else it uses a black tile to
represent that the two chains belong to different clusters.
This visualization tool can also be used for
multivariate chains. In cases where the diagnostic tool for $d$
variate chains indicates divergence, for further investigation, the
user can choose a chain from each cluster and implement the
visualization tool marginally i.e. one variable at a time. This will
help the user identify which among the $d$ variables are responsible
for inadequate mixing among the $m$ multivariate chains.
\subsection{Graphical methods}
\label{sec:graph}
In addition to the visualization tool mentioned in
Section~\ref{sec:kern}, we now discuss some of the widely used
graphical methods for MCMC convergence diagnosis. The most common
graphical convergence diagnostic method is the trace plot. The trace
plot is a time series plot that shows the realizations of the Markov
chain at each iteration against the iteration numbers. This graphical
method is used to visualize how the Markov chain is moving around the
state space, that is, how well it is mixing. If the MCMC chain is
stuck in some part of the state space, the trace plots shows flat bits
indicating slow convergence. Such a trace plot is observed for an MH
chain if too many proposals are rejected consecutively. In contrast,
for an MH chain if too many proposals are accepted consecutively, then
trace plots may move slowly not exploring the rest of the state
space. Visible trends or changes in spread of the trace plot imply that
the stationarity has not been reached yet. It is often said that a
good trace plot should look like a hairy caterpillar. For an efficient
MCMC algorithm if the initial value is not in the high-density region,
the beginning of the trace plots shows back-to-back steps in one
direction. On the other hand, if the trace plot shows similar pattern
throughout, then there is no use in throwing burn-in samples.
Unlike iid sampling, MCMC algorithms result in correlated samples. The
lag-$k$ (sample) autocorrelation is defined to be the correlation
between the samples $k$ steps apart. The autocorrelation plot shows
values of the lag-$k$ autocorrelation function (ACF) against
increasing $k$ values. For fast-mixing Markov chains, lag-$k$
autocorrelation values drop down to (practically) zero quickly as $k$
increases. On the other hand, high lag-$k$ autocorrelation values for
larger $k$ indicate the presence of a high degree of correlation and slow
mixing of the Markov chain. Generally, in order to get precise Monte Carlo
estimates, Markov chains need to be run a large multiple of the amount of time
it takes the ACF to be practically zero.
Another graphical method used in practice is the running mean plot
although its use has faced criticism \citep{geye:2011}. The running
mean plot shows the Monte Carlo (time average) estimates against the
iterations. This line plot should stabilize to a fixed number as
iteration increases, but non-convergence of the plot
indicates that the simulation cannot be stopped yet. While the trace
plot is used to diagnose a Markov chain's convergence to stationarity,
the running mean plot is used to decide stopping times.
In the multivariate case, individual trace, autocorrelation and
running mean plots are generally made based on realizations of each
marginal chain. Thus the correlations that may be present among
different components are not visualized through these plots. In
multivariate settings, investigating correlation across different
variables is required to check for the presence of high
cross-correlation \citep{cowl:carl:1996}.
\section{Examples}
\label{sec:exam}
In this section, we use three detailed examples to illustrate the
convergence diagnostics presented in Section~\ref{sec:diag}. Using
these examples, we also demonstrate that empirical convergence
diagnostic tools may give false indication of convergence to
stationarity as well as convergence of Monte Carlo estimates.
\subsection{An exponential target distribution}
Let the target distribution be Exp(1), that is,
$\pi(x) = \exp(-x), \; x>0$. We consider an independence Metropolis
sampler with Exp($\theta$) proposal, that is, the proposal density is
$q(x) = \theta \exp(-\theta x), \; x>0$. We study the independence
chain corresponding to two values of $\theta$, namely, $\theta =0.5$
and $\theta =5$. Using this example, we illustrate the honest
choices of burn-in and stopping time described in
Section~\ref{sec:hone} as well as several other diagnostic tools. It
turns out that, even in this unimodal example, some empirical
diagnostics may lead to premature termination of simulation. We first
consider some graphical diagnostics for Markov chain
convergence. Since the target density is a strictly decreasing
function on $(0, \infty)$, a small value may serve as a reasonable
starting value. We run the Markov chains for 1,000 iterations
initialized at $X_0 =0.1$. Figure~\ref{fig:exp:tr.acf} shows the trace
plots and autocorrelation plots of the Markov chain samples. From the
trace plots we see that while the first chain ($\theta =0.5$) mixes
well, the second chain exhibits several flat bits and suffers from
slow mixing. Thus from the trace plots, we see that there is no need
for burn-in for $\theta =0.5$, that is, $X_0 = 0.1$ seems to be a
reasonable starting value. On the other hand, for $\theta =5$, the
chain can be run longer to find an appropriate burn-in. This is also
corroborated by the autocorrelation plots. When $\theta =0.5$,
autocorrelation is almost negligible after lag 4. On the other hand,
for $\theta =5$, there is significant autocorrelation even after lag
50. Next, using the CODA package \citep{R:coda}, we compute
\cite{gewe:1992}'s and \cite{heid:welc:1983}'s convergence diagnostics
for the identity function $g(x)=x$. Using the default $n_A = 0.1n$ and
$n_B = 0.5n$, \cite{gewe:1992}'s Z scores for the $\theta =0.5$ and
$\theta =5$ chains are 0.733 and 0.605 respectively, failing to reject
the hypothesis of the equality of means from the beginning and end
parts of the chains. Similarly, both the chains pass
\cite{heid:welc:1983}'s test for stationarity. Next, we consider
the \cite{raft:lewi:1992} diagnostic. When the two samplers are run for
38,415 ($\lceil n_{min} \rceil$ corresponding to
$\epsilon= 0.005, \alpha=0.05$, and $q=0.5$) iterations, and
Raftery-Lewis diagnostic is applied for different $q$ values (0.1,
$\dots$, 0.9), the burn-in estimates for the $\theta =5$ chain are
larger than those for the $\theta =0.5$ chain, although the overall
maximum burn-in (981) is less than 1,000. Finally, we consider the
choice of honest burn-in. Since for $\theta < 1$,
$\pi(x)/q(x) = \theta^{-1} \exp(x(\theta -1)) \le \theta^{-1}$ for all
$x >0$, according to \cite{meng:twee:1996}, we know that
\[
\frac{1}{2} \int_{\mathcal{X}} |f_n(x) - \pi (x)| dx \le (1 - \theta)^n,
\]
that is, an analytical upper bound to the TV norm can be obtained. Thus
for $\theta = 0.5$, if $n' = \lceil \log(0.01)/\log(0.5)\rceil = 7$,
then (\ref{eq:totvar}) holds. Thus $n' =7$ can be an honest burn-in
for the independence Metropolis chain with $\theta =0.5$.
Note that, for $\theta < 1$, the independence chain is geometrically
ergodic; for $\theta = 1$, the chain produces iid draws from the
target; and for $\theta > 1$, by \cite{meng:twee:1996}, the independence
chain is {\it subgeometric}. As mentioned by
\cite{jone:hobe:2001}, when $\theta > 1$, the tail of the proposal
density is much lighter than that of the target, making it difficult
for the chain to move to larger values, and when it does move there,
it tends to get stuck.
\begin{figure*}
\caption{Trace (left panels) and autocorrelation function (right
panels) plots of the independence Metropolis chains (top row,
$\theta =0.5$; bottom row, $\theta =5$) for the exponential target
example. The presence of frequent flat bits in the trace plot and high autocorrelation values indicate
slow mixing of the Markov chain with $\theta =5$.}
\label{fig:exp:tr.acf}
\end{figure*}
Next, we consider stopping rules for estimation of the mean of the
stationary distribution, that is, $E_\pi X = 1$. Based on a single
chain, we apply the FWSR (\ref{eq:fwsr}) to determine the sample
size for $\epsilon = 0.005$ and $\alpha =0.05$ (that is,
$t_* = 1.96$). For the independence Metropolis chain with
$\theta =0.5$ starting at $X_{8} = 0.1545$, it takes $n^* = 323,693$
iterations to achieve the cutoff $0.005$. The running estimates of
the mean along with confidence intervals are given in the left panel
of Figure~\ref{fig:exp.run}. We next run the independence Metropolis
chain with $\theta =5$ for 323,700 iterations starting at
$X_0 =0.1$. The corresponding running estimates are given in the
right panel of Figure~\ref{fig:exp.run}. Since a Markov chain CLT is
not available for the independence chain with $\theta >1$, we cannot
compute asymptotic confidence intervals in this case. From the plot
we see that the final estimate (0.778) is far off from the truth
($E_\pi X = 1$). Next, we consider ESS. The cut off value for ESS
mentioned in (\ref{eq:messcut}) with $\varepsilon = 2*0.005=0.01$ is
153,658. The ESS for the two chains are 163,955 and 1,166,
respectively which again shows the presence of large correlation
among the MCMC samples for $\theta =5$. We use the R package mcmcse
\citep{R:mcmcse} for computing ESS. Finally, we consider the GR
diagnostic. We run four parallel chains for 2,000 iterations
starting at 0.1, 1, 2, and 3, respectively each with both
$\theta =0.5$ and $\theta =5$. We calculate \cite{gelm:rubi:1992}'s
PSRF (\ref{eq:psrf}) based on these chains. The plots of iterative
$\hat{R}$ at the increment of every 100 iterations are given in
Figure~\ref{fig:exp.gr}. We see that $\hat{R}$ for
the chain with $\theta =0.5$ reaches below 1.1 in 100 iterations. On
the other hand, the Monte Carlo estimate for $E_\pi X$ and its
standard error based on first 100 iterations for the chain started
at 0.1 are 1.109 and 0.111, respectively. Thus, GR diagnostic leads
to premature termination of simulation and the inference drawn from
the resulting samples can be unreliable. Finally, we note that $\hat{R}$ for the
chains with $\theta =5$ takes large ($>16$) values even after 2,000
iterations showing that simulation cannot be stopped yet in this
case.
\begin{figure*}
\caption{The left plot shows the running estimates of the mean with
confidence interval for $\theta =0.5$. Running mean plot for
$\theta =5$ is given in the right panel. The horizontal line
denotes the truth. The plot in the right panel reveals that even
after 300,000 iterations, the Monte Carlo estimate for the chain
with $\theta =5$ is far off from the truth.}
\label{fig:exp.run}
\end{figure*}
\begin{figure*}
\caption{Iterative $\hat{R}
\label{fig:exp.gr}
\end{figure*}
\subsection{A sixmodal target distribution}
\label{sec:sixmo}
This example is proposed by \cite{lem:chen:lavi:2009} where the
target density is as follows
\begin{equation}\label{eq:six_tar}
\pi(x, y)\propto\exp\bigg(\frac{-x^2}{2}\bigg)\exp\bigg(\frac{((\csc~ y)^5 - x)^2}{2}\bigg),\; -10 \le x, y \le 10.
\end{equation}
The contour plot of the target distribution (known up to the
normalizing constant) is given in Figure~\ref{fig:six_joint} and
marginal densities are plotted in Figure \ref{fig:six_x}. The plots of
the joint and marginal distributions clearly show that the target
distribution is multimodal in nature.
\begin{figure}
\caption{Contour plot of the target distribution in the sixmodal example.}
\label{fig:six_joint}
\end{figure}
\begin{figure}
\caption{Marginal densities of $X$ and $Y$ in the sixmodal example.}
\label{fig:six_x}
\end{figure}
To draw MCMC samples from the target density (\ref{eq:six_tar}), we
use a Metropolis within Gibbs sampler in which $X$ is drawn first and
then $Y$. In this example, we consider only convergence to stationarity,
that is, we do not discuss stopping rules here. Through this example,
we illustrate that when an MCMC sampler is stuck in a local mode, the
empirical convergence diagnostic tools may give false indication of
convergence. [Empirical diagnostics may fail even when modes are not
well defined \citep{geye:thomp:1995}.] In order to illustrate the
diagnostic tools, as in \cite{dixi:roy:2017}, we consider two
cases.
\noindent \textbf{Case 1.} In this case, we run four chains wherein
two chains (chains 1 and 2) are started at a particular mode
while the remaining two chains (chains 3 and 4) are started at
some other mode. Each of the four chains is run for 30,000
iterations. Trace plots of the last one thousand iterations of the
four parallel $X$ and $Y$ marginal chains are given in the left panel
of Figures~\ref{fig:tr.acf.x} and \ref{fig:tr.acf.y}
respectively. Trace plots show the divergence of the Markov
chains. High ACF values can also be seen from the autocorrelation
plots of the marginal chains in Figures~\ref{fig:tr.acf.x} and
\ref{fig:tr.acf.y}.
\begin{figure*}
\caption{Trace (left panel) and autocorrelation function (right panel) plots of the $X$ marginal of the four chains for the sixmodal example in Case 1.
}
\label{fig:tr.acf.x}
\end{figure*}
\begin{figure*}
\caption{Trace (left panel) and autocorrelation function (right panel) plots of the $Y$ marginal of the four chains for the sixmodal example in case 1.}
\label{fig:tr.acf.y}
\end{figure*}
Next, we apply \cite{dixi:roy:2017}'s bivariate KL divergence Tool 1
on the joint chain. The maximum symmetric KL divergence among the six
pairs is 104.89 significantly larger than the cutoff value 0.06.
Finally, we use \cite{dixi:roy:2017}'s visualization tool to identify
clusters among the four chains. The result is given in Figure
\ref{fig:c1_viz} which shows that there are two clusters among the
four chains wherein chain 1 and chain 2 form one cluster, while chain
3 and chain 4 form another cluster.
\begin{figure}
\caption{\cite{dixi:roy:2017}
\label{fig:c1_viz}
\end{figure}
\noindent \textbf{Case 2:} In this case also we run four chains but all the chains are started at
the same local mode. As in Case 1, all four chains are run for
30,000 iterations. The trace and autocorrelation plots of the
marginal chains are given in Figures~\ref{fig:tr.acf.x2} and
\ref{fig:tr.acf.y2}. From these plots one may conclude
mixing of the Markov chains, although the large autocorrelations
result in low ESS for the chains. The minimum and maximum mESS
(\ref{eq:mess}) computed using the R package mcmcse for the four
chains are 412 and 469, respectively.
\begin{figure*}
\caption{Trace (left panel) and autocorrelation function (right panel) plots of the $X$ marginal of the four chains for the sixmodal example in case 2.
}
\label{fig:tr.acf.x2}
\end{figure*}
\begin{figure*}
\caption{Trace (left panel) and autocorrelation function (right panel) plots of the $Y$ marginal of the four chains for the sixmodal example in case 2.}
\label{fig:tr.acf.y2}
\end{figure*}
The adaptive kernel density estimates of the four chains are
visualized in Figure \ref{fig:c2_chains}. This bivariate density plot
does not reveal non-convergence of the chains to the stationary
distribution.
\begin{figure}
\caption{Visualizations of the adaptive kernel density estimates of
the four chains in Case 2 of the sixmodal example. Since the
bivariate density plots look similar, it fails to provide indication
of non-convergence of the chains to the stationary distribution.}
\label{fig:c2_chains}
\end{figure}
Next, we compute the \cite{gewe:1992}'s and \cite{heid:welc:1983}'s
convergence diagnostics for the identity function $g(x)=x$ for all
four individual chains. At level 0.05, the \cite{gewe:1992} diagnostic
fails to reject the hypothesis of the equality of means from the
beginning and end parts of each chain. Similarly, all chains pass
the \cite{heid:welc:1983} test for stationarity. Thus, both
\cite{gewe:1992}'s and \cite{heid:welc:1983}'s diagnostics fail to
detect the non-convergence of the chains to the target distribution.
Also, the Raftery-Lewis diagnostic fails to distinguish between the chains
in Case 1 and Case 2 as it results in similar burn-in estimates in
both cases.
We also calculate the PSRF for the marginal chains as well as the
MPSRF for the joint chain based on the four parallel chains as the GR
diagnostic is often used by practitioners for determining burn-in
\citep[][p. 256]{fleg:hara:jone:2008}. The plots of iterative
$\hat{R}$ at increments of 200 iterations are given in
Figure~\ref{fig:sixmo:gr.run}. PSRFs for the marginal chains reach
below 1.1 before 3,000 iterations. The MPSRF (not shown in the plot)
also reaches below 1.1 before 6,000 iterations. Both the PSRF
and MPSRF values are close to one, which is often used as sign of
convergence to stationarity.
\begin{figure*}
\caption{Iterative $\hat{R}
\label{fig:sixmo:gr.run}
\end{figure*}
Since all four chains are stuck at the same local mode, that is, these
are not run long enough to move between the modes, the convergence
diagnostics, including PSRF, MPSRF get fooled into thinking that the
target distribution is unimodal and hence falsely detect
convergence. \cite{laha:dutt:roy:2017} demonstrate failures of trace
plots, autocorrelation plots and PSRF in diagnosing non-convergence of
MCMC samplers in the context of a statistical model used for analyzing
rank data [See \cite{hobe:roy:robe:2011} for examples of multimodal
targets arising from the popular Bayesian finite mixture models where
empirical convergence diagnostic tools face similar issues.] Since
these diagnostic tools make use of (only) the samples obtained from
the MCMC algorithm, and all observations lie around the same mode,
they fail to diagnose non-convergence. In contrast,
\cite{dixi:roy:2017}'s Tool 2 (\ref{eq:tool2}) uses both MCMC samples
and the target density. Since \cite{dixi:roy:2017}'s Tool 2 requires
only one chain and since the PSRF suggest that the four chains are
similar, we simply choose one of the four chains. Now,
$T_2^\ast = 0.88$ is significantly greater than zero and thus
indicates that the chains are stuck at the same mode. Furthermore, it
also indicates that 88\% of the target distribution is not yet
captured by the Markov chain. Thus, \cite{dixi:roy:2017}'s Tool 2 is
successful in detecting the divergence of the chains.
\subsection{A Bayesian logistic model}
\label{sec:logi}
In this section, we illustrate MCMC convergence diagnostics in the
context of a real data analysis using a popular statistical model. In
particular, we fit a Bayesian logistic model on the {\it Anguilla
australis} distribution dataset provided in the R package dismo
\citep{R:dism}. Data are available on a number of sites with presence
or absence of the short-finned eel ({\it Anguilla australis}) in New
Zealand, and some environmental variables at these sites. In
particular, we fit the Anguilla\_train data available in the dismo
package. Here, the response variable is the presence or absence of
short-finned eel, and six other variables are included as
covariates. The six covariates are: summer air temperature (SeqSumT),
distance to coast (DSDist), area with indigenous forest (USNative),
average slope in the upstream catchment (USSlope), maximum downstream
slope (DSMaxSlope) and fishing method (categorical variable with five
classes: electric, mixture, net, spot and trap). Thus the data
set consists of $(y_i, x_i), i=1,\dots,1,000$, where $y_i$ is the $i$th
observation of the response variable taking value 1 (presence) or zero
(absence), and $x_i = (1, \tilde{x}_i)$ is the ten-dimensional
covariate vector, 1 for the intercept and $\tilde{x}_i$ for the other
nine covariates (with four components for the categorical variable
fishing method). This example was also used by \cite{dixi:roy:2017}
and \cite{boon:merr:krac:2014} to illustrate their MCMC convergence
diagnostic tools.
Denote $\beta =(\beta_0, \beta_1, \dots, \beta_9)$ where $\beta_0$ is
the intercept and $(\beta_1, \dots, \beta_9)$ is the $9 \times 1$ vector of unknown regression
coefficients. We consider the logistic regression model
\[
Y_i|\beta \stackrel{ind}{\sim} \mbox{Bernoulli}(F(x_i^T \beta)), i=1,\dots,1,000,
\]
where $F(\cdot)$ is the cdf of the logistic distribution, that is,
\[
F(x_i^T \beta) = \frac{\exp(x_i^T \beta)}{1+\exp(x_i^T \beta)},i=1,\dots,1,000.
\]
We consider a Bayesian analysis with a diffuse normal prior on $\beta$. Thus,
the posterior density is
\begin{equation}
\label{eq:logipost}
\pi(\beta | y) \propto \ell(\beta | y) \phi_{10}(\beta) = \prod_{i=1}^n F(x_i^T \beta)^{y_i} \{1 - F(x_i^T \beta)\}^{1 - y_i} \phi_{10}(\beta),
\end{equation}
where $\ell(\beta | y)$ is the likelihood function and $\phi_{10}(\beta)$ is the density of
$N(\textbf{0}, 100~I_{10})$. The posterior density (\ref{eq:logipost})
is intractable in the sense that means with respect to this density,
which are required for Bayesian inference, are not available in closed
form.
As in \cite{dixi:roy:2017} and \cite{boon:merr:krac:2014}, we use the
MCMClogit function in the R package MCMCpack \citep{R:mcmcp} to draw
MCMC samples from the target density $\pi(\beta | y)$. The maximum
likelihood estimate (MLE) of $\beta$ is the value of the parameter
where the likelihood function $\ell(\beta | y)$ is maximized. Exact
MLE is not available for the logistic likelihood function, neither is
the mode of the posterior density (\ref{eq:logipost}). But, numerical
optimization methods can be used to find an approximate MLE or
posterior mode, which may then be used as starting values. In order to
assess convergence to stationarity, we run three parallel chains with
the default tuning values for 5,000 iterations, one initialized at the
MLE and the other two initialized at points away from the MLE. Trace
plots of the three chains for the last 1,000 iterations for the
regression coefficients of summer air temperature (left panel) and
distance to coast (right panel) are given in
Figure~\ref{fig:logit:tracedefault}. Trace plots of the other
variables look similar. From these plots we see that, there is not
much overlap between the three parallel chains. From the frequent flat
bits, it follows that the Markov chains move tardily and suffer from
slow mixing. Indeed, the default tuning parameters in the MCMClogit
function result in low (0.11) acceptance rate. We next set the tuning
parameters to achieve around $40\%$ acceptance rate and all analysis
in the remaining section is based on these new tuning values. We run
the three chains longer (30,000 iterations) to obtain reliable ACF
plots. Trace plots of the last 1,000 iterations for each of the three
chains for the nine regression coefficient variables are given in
Figure~\ref{fig:logit:trace}. From the trace plots we see that
convergence of the chains can be further improved. Autocorrelations
for all ten variables for one of the chains based on all 30,000 draws
are given in Figure~\ref{fig:logit:acf}. Autocorrelations for the
other two chains look similar (not included here). Like the trace
plots, the autocorrelation plots also reveal that the Markov chains
suffer from high autocorrelations. It is further corroborated by the
mESS values, which are less than 1,000 for all the three
chains. To sample from (\ref{eq:logipost}) one may use an alternative
MCMC sampler, e.g., the P\'olya-Gamma Gibbs sampler
\citep{pols:scot:wind:2013}, which is known to be geometrically
ergodic \citep{wang:roy:2018b, choi:hobe:2013}. Here we do not use the
P\'olya-Gamma Gibbs sampler as our goal is to illustrate the
convergence diagnostic methods. The MPSRF reaches close to one before
30,000 iterations. Since the Markov chains are 10-dimensional, to
maintain an overall type 1 error rate of $\alpha=0.05$, using
Bonferroni's correction, \cite{dixi:roy:2017} advocate the cutoff
point $0.01$ for the KL Tool 1 for marginal chains. For each of the
ten variables, the maximum symmetric KL divergence among the three
pairs of chains is computed. It turns out that the marginal chains do
not pass the KL Tool 1 test as the maximum symmetric KL divergence
takes the value 7.26 for the variable USSlope. After 30,000
iterations, all marginal chains pass the \cite{heid:welc:1983}
stationarity test. On the other hand, for each of the three parallel
chains, for some of the variables, the \cite{gewe:1992} $Z$ test turns
out to be significant at $0.05$ level. Next, we run the chains for
another 40,000 iterations. For the last 40,000 iterations, all
marginal chains pass the \cite{gewe:1992} $Z$ test, as well as the
KL Tool 1 test. Also, based on these 40,000 iterations, the maximum
burn-in estimate from the Raftery-Lewis diagnostic (with
$\epsilon=0.005, \alpha=0.05$) over different quantiles
$(q=0.1,\dots,0.9)$ is less than 100 for all 10 variables. We thus
use $n' =$70,000 as the burn-in value.
\begin{figure*}
\caption{Trace plots of the three chains with default tuning for the
regression coefficients of summer air temperature (left panel) and
distance to coast (right panel) for the Bayesian logistic model
example. The presence of frequent flat bits indicates slow mixing of the Markov chains.}
\label{fig:logit:tracedefault}
\end{figure*}
\begin{figure*}
\caption{Trace plots of the three chains for the nine regression
coefficients variables for the Bayesian logistic model
example. The plots show improved mixing from tuning the
acceptance rate of the Markov chains.}
\label{fig:logit:trace}
\end{figure*}
\begin{figure*}
\caption{Autocorrelation plots of the ten marginal chains for the Bayesian logistic model example.}
\label{fig:logit:acf}
\end{figure*}
After removing the first 70,000 iterations as initial burn-in, each of
the three chains is run for an additional 15,000 iterations. Table
\ref{tab:logit_tab} presents the PSRF and the maximum symmetric KL
divergence [\cite{dixi:roy:2017}'s KL Tool 1] values based on three
parallel chains for all 10 variables. The half-widths of the 95\%
confidence intervals based on the first chain (started at the MLE) are
also tabulated in Table \ref{tab:logit_tab}. All values are given up
to three decimal places. MPSRF takes the value 1.004. For the three
chains mESS takes values 515, 520 and 502, respectively.
High cross-correlation between the Intercept and SeqSumT regression
coefficient parameters (-0.984) and between USNative and USSlope
(-0.558) suggest that mixing of the Markov chain can improve if it is run on
an appropriate lower dimensional space (that is, after dropping some
variables) or a reparameterization is used. From Table
\ref{tab:logit_tab}, we see that all marginal chains pass the KL Tool
1 diagnostic. Also, all PSRF values as well as the MPSRF value reach
below the cutoff 1.1. On the other hand, the maximum half-width
among the 10 regression parameters is $0.112$, much larger than the
cutoff 0.01. Doing a simple sample size calculation, based on the
pilot sample size 15,000, we find that we need
15,000$\times(0.112/0.01)^2=$ 1,881,600 samples for obtaining
confidence intervals with half-widths below 0.01.
\begin{table}[h]
\caption{Application of various MCMC convergence diagnostic tools to the Bayesian logistic model.}
\centering
\begin{tabular}{|l|lll|lll|}
\hline
\textbf{Variable} &$\mathbf{\hat{R}}$&\textbf{half-width}&\textbf{Tool 1}\\
\hline
Intercept & 1.000 & 0.112 & 0.008 \\
SeqSumT & 1.000 & 0.006 & 0.007 \\
DSDist & 1.001 & 0.000 & 0.005\\
USNative & 1.001 & 0.025 & 0.004 \\
M - mix & 1.000 & 0.031 & 0.005 \\
M - net & 1.001 & 0.031 & 0.004 \\
M - spot & 1.000 & 0.048 & 0.004 \\
M - trap & 1.002 & 0.051 & 0.004 \\
DSMaxSlope & 1.000 & 0.005 & 0.007 \\
USSlope & 1.001 & 0.002 & 0.004 \\
\hline
\end{tabular}
\label{tab:logit_tab}
\end{table}
Finally, we run one of the chains (the chain started at the MLE) for
1,881,600 iterations after a burn-in of $n'=$70,000 iterations. Thus
the chain is stopped after $n^*=$1,951,600 iterations. In this case,
as expected, the maximum half-width of the $95\%$ confidence interval
is below $0.01$. An estimate of mESS calculated using the mcmcse
package is 55,775 which is larger than the cutoff value 55,191 given
in (\ref{eq:messcut}) for $p=10$, $\alpha =0.05$ and
$\varepsilon =0.02$. On the other hand, the chain needs to be run
longer to achieve the cutoff value 220,766 (\ref{eq:messcut})
corresponding to $\varepsilon =0.01$. Table~\ref{tab:logit_est} gives
the estimates of posterior means of all regression coefficients
and their corresponding Monte Carlo standard errors (SE).
\begin{table}[h]
\caption{Estimates of posterior means and standard errors of regression coefficients for the Bayesian logistic model.}
\centering
\begin{tabular}{|l|llllllllll|}
\hline
\textbf{Variable} &
$\beta_0$ &
$\beta_1$ &
$\beta_2$&
$\beta_3$&
$\beta_4$&
$\beta_5$&
$\beta_6$&
$\beta_7$&
$\beta_8$&
$\beta_9$\\
\hline
\textbf{Estimate} &
-10.46 &
0.66&
-0.00&
-1.17&
-0.47&
-1.53&
-1.83&
-2.59&
-0.17&
-0.05\\
\hline
\textbf{SE $\times 10^3$} &
5.73&
0.32&
0.00&
1.52&
1.46&
1.65&
2.76&
2.35&
0.24&
0.08\\
\hline
\end{tabular}
\label{tab:logit_est}
\end{table}
\section{Conclusions and discussion}
In this article, we discuss several measures for diagnosing
convergence of Monte Carlo Markov chains to stationarity as well as
convergence of the sample averages based on these chains. Detection of
the first is often used to decide a suitable burn-in period, while the
second leads to termination of the MCMC simulation. Analytical upper
bounds to the TV norm required to obtain an honest burn-in maybe
difficult to find in practice or may lead to very conservative burn-in
values. On the other hand, empirical diagnostics can falsely detect
convergence when the chains are not run long enough to move between
the modes. For the chains initialized at high density density regions,
there is no need for burn-in. If the global mode of the target
density can be (approximately) found by optimization then it can be
used as the starting value.
Some of the empirical diagnostics for convergence of sample averages may
prematurely terminate the simulation and the resulting inference can
be far from the truth. Thus, use of fixed-width and ESS---based
stopping rules is recommended. Most of the quantitative convergence
diagnostics assumes a Markov chain CLT. While demonstrating the
existence of a Markov chain CLT requires some rigorous theoretical
analysis of the Markov chain, given the great amount of work done in
this direction, validating honest stopping rules does not present as
much of an obstacle as in the past.
None of the three examples discussed here use thinning. Thinning,
that is, discarding all but every $k$th observation, is often used by
MCMC practitioners to reduce high autocorrelations present in the
Markov chain samples. Since it wastes too many samples, it should be
used only if computer storage of the samples is an issue or evaluating
the functions of interest ($g$) is more expensive than sampling the
Markov chain. If thinning is used, convergence diagnostics can be used
on the thinned samples.
Some convergence diagnostic tools use parallel chains initialized at
different points, or two parts of a single chain. In the presence of
multiple modes, if the initial points of the parallel chains are not
in distinct high-density regions, or the chain is not run long enough
to move between the modes, the diagnostics fail to detect the
non-convergence. Thus, single long runs should be used to make final
inference. Running the chain longer may also result in discovering new
parts of the support of the target distribution. In contrast,
recently, \cite{jaco:olea:atch:2017} propose a method for
parallelizing MCMC computations using couplings of Markov chains.
Practitioners should be careful while depending purely on empirical
convergence diagnostic tools, especially if the presence of multiple
modes is suspected. Empirical diagnostics cannot detect convergence
with certainty. Also, if the target is incorrectly assumed to be a
proper density, the empirical diagnostic tools may not provide a red
flag indicating its impropriety \citep{athr:roy:2014b,
hobe:case:1996}. Over the past two decades, much research has been
done to provide honest Monte Carlo sample size calculation for myriad
MCMC algorithms for common statistical models. However, theoretical
analysis of MCMC algorithms is an ongoing area of research and further
important work needs to be done. A potential future study involves
theoretically verifying the convergence (to zero) of
\cite{dixi:roy:2017}'s statistics based on the KL divergence. Another
possible research problem is to construct theoretically valid and
computationally efficient MCMC convergence diagnostics in
ultrahigh-dimensional settings.
\section*{ACKNOWLEDGMENTS}
The author thanks one anonymous editor for careful and detailed
comments on an earlier version of the manuscript. The author thanks
Evangelos Evangelou, Mark Kaiser and Dootika Vats for helpful
comments. These valuable suggestions have substantially improved the
article. The author also thanks Chris Oats for suggesting two references
and Anand Dixit for providing some R codes used in the second and
third examples.
\end{document} |
{\bf e}gin{document}
\markboth{A.~Buffa \& C.~Giannelli}{Adaptive isogeometric methods with hierarchical splines: error estimator and convergence}
\title{\normalsize \bf ADAPTIVE ISOGEOMETRIC METHODS WITH HIERARCHICAL SPLINES:\\ ERROR ESTIMATOR AND CONVERGENCE
}
\author{\footnotesize ANNALISA BUFFA\footnote{Istituto di Matematica Applicata e Tecnologie Informatiche `E.~Magenes' del CNR, via Ferrata 1, 27100 Pavia Italy. E-mail address: [email protected]} \, and CARLOTTA GIANNELLI\footnote{Istituto Nazionale di Alta Matematica,
Unit\`a di Ricerca di Firenze c/o DiMaI `U.~Dini', Universit\`a di Firenze, viale Morgagni 67a, 50134 Firenze, Italy. E-mail address: [email protected]}}
\date{}
\maketitle
{\bf e}gin{abstract}\noindent
The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The adaptivity analysis holds in any space dimensions.
We consider a simple residual-type error estimator for which we provide a posteriori upper and lower bound in terms of local error indicators, taking also into account the critical role of oscillations as in a standard adaptive finite element setting. The error estimates are properly combined with a simple marking strategy to define a sequence of admissible locally refined meshes and corresponding approximate solutions. The design of a refine module that preserves the admissibility of the hierarchical mesh configuration between two consectutive steps of the adaptive loop is presented. The contraction property of the quasi-error, given by the sum of the energy error and the scaled error estimator, leads to the convergence proof of the AIGM.
\end{abstract}
\section{Introduction}
\label{sec:aloop}
The definition of adaptive schemes that provide local mesh refinement is an active area of research in the context of isogeometric analysis \cite{cottrell2009,hughes2005}, an emerging paradigm for the solution of partial differential equations which combines and extends finite
element techniques with computer aided design (CAD) methods related to spline models. Since the CAD standard for spline representation in a multivariate setting relies on tensor-product B-splines, e.g.~see \cite{deboor2001,schumaker2007}, an adaptive isogeometric model necessarily requires suitable extensions of the B-spline model that give the possibility to relax \color{black}the rigidity of the tensor-product structure by allowing hanging nodes.
There are a few different frameworks for the definition of splines on rectangular tiling with hanging nodes. We mention here T-splines \cite{sederberg2004,sederberg2003} that have been used in the context of isogeometric analysis in the pioneering papers
\cite{bazilevs2010,doerfel2010},
and their analysis-suitable \cite{scott2011b} or dual-compatible
\cite{daveiga2012,daveiga2013} versions. Other possibilities are offered by polynomial splines over ({\color{black} hierarchical}) T-meshes
\cite{deng2006,deng2008} or LR-splines \cite{dokken2013,bressan2013}, that have been tested within an isogeometric framework in \cite{nguyen-thanh2011} and \cite{johannessen2014}, respectively.
Finally, hierarchical splines based on the construction presented in \cite{kraft1997} is one of the most promising approach. This is also due to the fact that their construction and properties are closely related to the ones of hierarchical finite elements. Hierarchical B-spline constructions and their use, both as an adaptive modeling tool, as well as a framework for isogeometric analysis that provides local refinement possibilities, has been recently investigated in a number of papers, see e.g.~\cite{vuong2011,giannelli2012,giannelli2014,kiss2014b}.
In the present paper we aim at defining and studying an \emph{adaptive isogeometric method (AIGM) based on hierarchical splines}. The choice, among the adaptive spline models mentioned above, of the hierarchical setting have a twofold motivation. On the one hand, it is a natural extension of the B-spline model that is able to preserve many key properties directly by construction, and the refinement rules are simple and straightforward. In addition, \color{black}although the type of refinement they allow is more restrictive than other solutions\B, the locally structured hierarchical approach allows to defines an effective automatically-driven refinement strategy that, in turns, can be used to design a fully adaptive method.
We consider the simple elliptic model problem:
{\bf e}gin{equation}
\label{eq:mp}
-\textrm{div} (\textbf{A} \nabla u) = f \quad\text{in}\; \Omega,
\qquad
u\myvert{\partial\Omega} = 0,
\end{equation}
where $\Omega\subset\mathbb{R}^{d}$, $d\ge 1$, is a bounded domain with Lipschitz boundary $\partial\Omega$, and $f$ is any square integrable function and
{\bf e}gin{equation}
\label{eq:5}
\forall {\bf x}\in \Omega,\ {\bf x}i\in \mathbb{R}^d \;
\eta_1 |{\bf x}i|^2 \leq \textbf{A}(x){\bf x}i\cdot{\bf x}i \; \text{ and } \; |\textbf{A}({\bf x}){\bf x}i|\leq \eta_2 |{\bf x}i|
\end{equation}
with $0 < \eta_1\leq \eta_2$.
By closely following the framework of adaptive finite elements --- see e.g., the recent reviews in \cite{nsv2009,NochettoCIME} and references therein --- for elliptic partial differential equations, we aim at designing and analyse the four blocks in the following flowchart associated to an AIGM.
{\bf e}gin{center}
\fbox{SOLVE} $\rightarrow$ \fbox{ESTIMATE} $\rightarrow$ \fbox{MARK} $\rightarrow$ \fbox{REFINE}
\end{center}
At our best knowledge, all previous works on error estimators in isogeometric analysis were mainly devoted to numerical experiments with some goal--oriented error estimators based on auxiliary global refinement steps \cite{zv2011,ds2012,kvvb2014}.
Our choices for the different steps of the adaptive loop may be detailed as follows.
{\bf e}gin{itemize}
\item[\fbox{SOLVE}] We want to solve problem \eqref{eq:mp} with hierarchical spline spaces. To this aim, we define a family of \emph{admissible} hierarchical meshes, which uses the concept of truncated basis \cite{giannelli2012}, and we consider the Galerkin method on these spaces. Admissibility is related to the number of levels which are present (with non zero basis functions) on an element, and it is a fundamental assumption in our theory.
\item[\fbox{ESTIMATE}] We define residual based error estimator for our problem. Thanks to the regularity of splines, such an estimator reduces to the $L^2$-norm of the element-by-element residual suitably weighted with the mesh size. We prove that this estimator is \emph{reliable}, i.e., it is an upper bound for the error, and \emph{efficient}, i.e., it is a lower bound of the error (up to oscillations).
\item[\fbox{MARK}] We adopt the D\"orfler marking strategy
\cite{dorfler1996}, namely we mark for refinement all elements with largest error indicator until a certain fixed percentage of the total error indicator is taken into account by the set of marked elements.
\item[\fbox{REFINE}] A refinement procedure constructs the refined mesh starting from the set of marked elements, by following the structure of the recursive refine module generally considered in adaptive finite elements, see e.g.~\cite{morin2001,morin2002}. We construct this routine so that the admissibility of the refined mesh is preserved between two consecutive iterations of the adaptive loop.
\end{itemize}
In general, the refinement procedure identifies the mesh with an increased level of resolution for the next iteration by refinining not only the marked elements, but also a suitable set of elements in their neighbourhood, analogously to the concept of \emph{refinement patches} in an adaptive finite element method. This allows to construct a mesh that preserves a certain class of admissibility. The refinement mechanism is similar to the strategy adopted to bound the number of hanging nodes per side in the refinement of quadrilateral meshes for finite elements \cite{bonito2010}, and is also related to the properties of the domain partitions created by the bisection rule that are needed to prove quasi-optimality of adaptive finite element methods
\cite{binev2004a,cascon2008,stevenson2007,stevenson2008}.
In the present paper we start the numerical analysis of our AIGM method and we provide a convergence result together with the contraction of the quasi-error (i.e., the sum of the error and the error indicator), while the complexity of the refine routine, together with quasi-interpolation operators and optimality of the AIGM, is left to the companion paper \cite{buffa2015b}.
The paper is organized as follows. Some preliminary aspects of hierarchial tensor-product B-spline constructions are reviewed in Section~\ref{sec:hspaces} together with the definition of {\color{black} truncated hierarchical B-splines (THB-splines)} and related properties, before introducing the notion of (strictly) admissible meshes.
The module SOLVE and {\color{black} ESTIMATE} of the adaptive isogeometric method are discussed in Sections~\ref{sec:solve} and \ref{sec:estimate} including {\color{black} an a} posteriori error analysis in terms of both upper and lower bound for the energy error. Section~\ref{sec:mark&refine} recalls a well-known marking strategy and introduces a refinement strategy that preserves the class of admissibility during the iterative loop --- module MARK and REFINE. Finally, Section~\ref{sec:closure} concludes the paper by summarizing the key results of the present study, and outlines the spirit of our companion paper \cite{buffa2015b}.
\section{Hierarchical spline spaces}
\label{sec:hspaces}
We start by considering the hierarchical approach to adaptive mesh refinement, as natural extension of the standard tensor-product B-spline model in a general multivariate setting. In particular, we focus on the truncated hierachical B-spline basis, since it allows us to identify a certain class of admissible mesh configurations.
\subsection{Preliminaries: B-spline hierarchies}
\label{sec:pre}
Hierarchical B-spline spaces are constructed by considering a hierarchy of $N$ tensor-product $d$-variate spline spaces $V^0\subset V^1\subset \ldots...\subset V^{N-1}$ defined on a {\color{black} closed hyper-rectangle} $D$ in $\mathbb{R}^d$ together with a hierarchy of domains $\mathfrak{h}at{\Omega}^0\supseteq\mathfrak{h}at{\Omega}^1\supseteq\ldots\supseteq\mathfrak{h}at{\Omega}^{N-1}$, that are {\color{black} closed} subsets of $D$. The \emph{depth} of the subdomain hierarchy is represented by the integer $N$, and we assume $\mathfrak{h}at{\Omega}^N=\emptyset$.
For each level $\ell$, with $\ell=0,1,\dots,N-1$, the multivariate spline space $V^\ell$ is spanned by the tensor-product B-spline basis $\mathfrak{h}at{{\cal B}}^\ell$ of degree $\mathbf{p}=(p_1,\ldots,p_d)$ defined on {\color{black} a given tensor-product} grid $\mathfrak{h}at{G}^\ell$. The (non-empty) quadrilateral elements (or cells) $\mathfrak{h}at{Q}$ of $\mathfrak{h}at{G}^\ell$ are the Cartesian product of $d$ open intervals between adjacent grid values. For any coordinate direction $i$, for $i=1,\ldots,d$ the knot sequences associated to the grids at the different levels contain non-decreasing real numbers so that each grid value appears in the knot vector as many times as specified by a certain multiplicity. At any level $\ell$, i.e., for the case of standard tensor-product B-splines, the multiplicity of each knot may vary between one (single knots) and $p_i$ or $p_{i+1}$ for the case of continuous and discontinuos functions, respectively. In order to guarantee the nested nature of the spline spaces $V^\ell\subset V^{\ell+1}$, we require that every knot of level $\ell-1$ is also present at level $\ell$ at least with the same multiplicity in the corresponding coordinate direction.
From the classical spline theory, it is known that B-splines are locally linear independent, they are non-negative, they have local support, and form a partition of unity \cite{deboor2001,schumaker2007}. Moreover, there exists a two-scale relation between adjacent bases in the hierarchy so that any function $s\in V^\ell\subset V^{\ell+1}$ can be expressed as
{\bf e}gin{equation}\label{eq:2scale}
s=\sum_{\mathfrak{h}at{{\bf e}ta}\in \mathfrak{h}at{{\cal B}}^{\ell+1}} c_{\mathfrak{h}at{{\bf e}ta}}^{\ell+1}(s) \mathfrak{h}at{{\bf e}ta},
\end{equation}
in terms of {\color{black} the} coefficients $c_{\mathfrak{h}at{{\bf e}ta}}^{\ell+1}$.
{\color{black} The domain ${\Omega}^\ell$ is defined as the union of the closure of elements of ${\mathfrak{h}at G}^{\ell-1}$, namely}
\[
{\color{black} {\mathfrak{h}at{\Omega}}}^\ell=
{\bf i}gcup\left\{
\overline{\mathfrak{h}at{Q}} {\color{black} \,: \mathfrak{h}at{Q}\in \mathfrak{h}at{G}^{\ell-1}}
\right\}.
\]
{\color{black} An element $\mathfrak{h}at{Q}$ of level $\ell$ is \emph{active} if $\mathfrak{h}at{Q}\subset\mathfrak{h}at{\Omega}^\ell$ and any $\mathfrak{h}at{Q}^*$ of level $\ell^*>\ell$ which belongs to any $\mathfrak{h}at{\Omega}^{\ell+1}, \ldots,\mathfrak{h}at{\Omega}^{N-1}$ is not a subset of $\mathfrak{h}at{Q}$.} We denote the collection of active elements of level $\ell$ as
{\bf e}gin{equation}\label{eq:active}
\mathfrak{h}at{{\cal G}}^\ell {\color{black}\, :=\,}
\left\{\mathfrak{h}at{Q}\in \mathfrak{h}at{G}^\ell : \mathfrak{h}at{Q} {\color{black} \,\subset\,} \mathfrak{h}at{\Omega}^\ell \wedge
\nexists\; \mathfrak{h}at{Q}^* {\color{black} \in {\mathfrak{h}at G}^{\ell^*},\, \ell^*>\ell : Q^*} \subset{\mathfrak{h}at{\Omega}}^{\ell^*} {\color{black} \,\wedge\,} \mathfrak{h}at{Q}^* \subset \mathfrak{h}at{Q}\right\}.
\end{equation}
Let {\color{black} $\mathfrak{h}at{\cal Q}$} be the
\emph{mesh} composed by taking the active elements $Q$ at any hierarchical level, namely
{\bf e}gin{equation}\label{eq:mesh}
\mathfrak{h}at{{\cal Q}} {\color{black}\,:=\,} \left\{
\mathfrak{h}at{Q}\in \mathfrak{h}at{{\cal G}}^\ell, \, {\color{black} \ell } =0,\ldots,N-1
\right\}.
\end{equation}
For any $\mathfrak{h}at{Q}\in\mathfrak{h}at{{\cal Q}}$, we define $h_{\mathfrak{h}at{Q}} {\color{black}\, :=\,} |\mathfrak{h}at{Q}|^{1/d}$.
A mesh $\mathfrak{h}at{{\cal Q}}^*$ is a refinement of $\mathfrak{h}at{{\cal Q}}$ if each element $\mathfrak{h}at{Q}^*\in\mathfrak{h}at{{\cal Q}}^*$ either also belongs to $\mathfrak{h}at{{\cal Q}}$ or is obtained by splitting $\mathfrak{h}at{Q}\in\mathfrak{h}at{{\cal Q}}$ in $q^d$ elements via ``$q$-adic'' refinement, for some integer $q\ge 2$. The refinement relation between $\mathfrak{h}at{\mathcal{Q}}$ and $\mathfrak{h}at{\mathcal{Q}}^*$ will be indicated as $\mathfrak{h}at{\mathcal{Q}}^*\succeq\mathfrak{h}at{\mathcal{Q}}$.
{
In particular, we will consider the case of standard dyadic refinement with $q=2$.}
A basis for the hierarchical B-spline space can be constructed by a suitable selection of \emph{active} basis functions at different level of details according to the following definition, see also \cite{kraft1997,vuong2011}.
{\bf e}gin{dfn}\label{dfn:hb}
The hierarchical {\color{black} B-spline} (HB-spline) basis $\mathfrak{h}at{{\cal H}}$ with respect to the mesh $\mathfrak{h}at{\cal Q}$ is defined as
{\bf e}gin{equation*}
\mathfrak{h}at{{\cal H}}(\mathfrak{h}at{{\cal Q}}) {\color{black}\,:=\,}
\left\{
\mathfrak{h}at{{\bf e}ta}\in\mathfrak{h}at{{\cal B}}^\ell :
\mathop{\mathrm{supp}} \mathfrak{h}at{{\bf e}ta} \subseteq\mathfrak{h}at{\Omega}^\ell \wedge
\mathop{\mathrm{supp}} \mathfrak{h}at{{\bf e}ta}\not\subseteq \mathfrak{h}at{\Omega}^{\ell+1},
\, {\color{black} \ell} =0,\ldots,N-1
\right\},
\end{equation*}
where $\mathop{\mathrm{supp}} \mathfrak{h}at{{\bf e}ta}$ denotes the intersection of the support of ${\bf e}ta$ with $\mathfrak{h}at{\Omega}^0$.
\end{dfn}
{\bf e}gin{rmk}\label{rmk:pkref}
Note that the hierarchical approach is not confined to dyadic or $q$-adic (uniform) refinement, but it can also handle different kind of mesh refinements,
including non-uniform configurations.
In addition, by assuming that the degrees may increase (but not decrease) moving from one level to the subsequent in the hierarchy, nested sequence of tensor-product spline spaces can be also considered in the context of $p$- (and $k$-) refinement.
\end{rmk}
\subsection{The truncated basis}
\label{sec:thb}
We define the truncation of a function $\mathfrak{h}at{s} \in V^\ell$ with respect to $\mathfrak{h}at{\cal B}^{\ell+1}$ as the contributions in \eqref{eq:2scale} of only basis functions in $\mathfrak{h}at{\cal B}^{\ell+1}$ that are \emph{passive}, i.e., not included in the hierarchical B-spline basis $\mathfrak{h}at{{\cal H}}(\mathfrak{h}at{\mathcal{Q}})$. More precisely,
{\bf e}gin{equation}\label{eq:trunc}
{\mathop{\mathrm{trunc}}}^{\ell+1} \mathfrak{h}at{s} {\color{black}\,:=\,} \sum_{\mathfrak{h}at{{\bf e}ta}\in \mathfrak{h}at{\mathcal{B}}^{\ell+1}, \, \mathop{\mathrm{supp}}\mathfrak{h}at{{\bf e}ta}\not\subseteq\mathfrak{h}at{\Omega}^{\ell+1}} c_{\mathfrak{h}at{{\bf e}ta}}^{\ell+1}(s) \mathfrak{h}at{{\bf e}ta},
\end{equation}
where $c_{\mathfrak{h}at{{\bf e}ta}}^{\ell+1}(s)$ is the coefficient of the function $s$ with respect to the basis element $\mathfrak{h}at{{\bf e}ta}$
at level $\ell+1$ of the B-spline refinement rule \eqref{eq:2scale}. By recursively applying the truncation to the HB-splines introduced in Definition~\ref{dfn:hb}, we can construct a different hierarchical basis
\cite{giannelli2012}.
{\bf e}gin{dfn}\label{dfn:thb}
The truncated hierarchical {\color{black} B-spline} (THB-spline) basis $\mathfrak{h}at{{\cal T}}$ with respect to the mesh $\mathfrak{h}at{{\cal Q}}$ is defined as
{\bf e}gin{equation*}
\mathfrak{h}at{{\cal T}}(\mathfrak{h}at{{\cal Q}}) {\color{black}\,:=\,}
\left\{
{\color{black} {\mathop{\mathrm{Trunc}}}^{\ell+1}}\,\mathfrak{h}at{{\bf e}ta}:\mathfrak{h}at{{\bf e}ta}\in\mathfrak{h}at{{\cal B}}^\ell
\cap\mathfrak{h}at{{\cal H}}(\mathfrak{h}at{{\cal Q}}),\, {\color{black} \ell} =0,\ldots,N-1\right\},
\end{equation*}
where $ {\color{black} {\mathop{\mathrm{Trunc}}}^{\ell+1}}\,\mathfrak{h}at{{\bf e}ta} {\color{black}\,:=\,} {\mathop{\mathrm{trunc}}}^{N-1}({\mathop{\mathrm{trunc}}}^{N-2}(\ldots ({\mathop{\mathrm{trunc}}}^{\ell+1}(\mathfrak{h}at{{\bf e}ta}))\dots))$, for any $\mathfrak{h}at{{\bf e}ta}\in\mathfrak{h}at{{\cal B}}^\ell\cap\mathfrak{h}at{{\cal H}}(\mathfrak{h}at{{\cal Q}})$.
\end{dfn}
The \emph{level} of a truncated B-spline $\mathfrak{h}at{\tau}\in\mathfrak{h}at{{\cal T}}(\mathfrak{h}at{{\cal Q}})$ is the level of the B-spline from which $\mathfrak{h}at{\tau}$ is derived according to the iterative truncation mechanism introduced in Definition~\ref{dfn:thb}.
For simplicity, we will denote $\mathfrak{h}at{{\cal H}}=\mathfrak{h}at{{\cal H}}(\mathfrak{h}at{{\cal Q}})$, $\mathfrak{h}at{{\cal T}} = \mathfrak{h}at{{\cal T}}(\mathfrak{h}at{{\cal Q}})$ when there will be no ambiguity in the text.
\subsection{Properties of THB-splines}
\label{sec:pro}
The truncated basis $\mathfrak{h}at{{\cal T}}$ not only spans the same hierarchical space of classical HB-splines, namely
{\bf e}gin{itemize}
\item[(i)] $\mathop{\mathrm{span}}{\mathfrak{h}at{\cal T}}=\mathop{\mathrm{span}}{\mathfrak{h}at{\cal H}}$,
\end{itemize}
but it also inherits from the hierarchical B-spline basis $\mathfrak{h}at{{\cal H}}$ the following properties:
{\bf e}gin{itemize}
\item[(ii)] non-negativity:
$\mathfrak{h}at{\tau}\ge 0, \,\forall\,\mathfrak{h}at{\tau}\in\mathfrak{h}at{{\cal T}}$;
\item[(iii)] linear independence:
$\sum_{\mathfrak{h}at{\tau}\in\mathfrak{h}at{{\cal T}}}c_{\mathfrak{h}at{\tau}} \mathfrak{h}at{\tau} = 0 \Leftrightarrow c_{\mathfrak{h}at{\tau}} = 0$, $\forall\,\mathfrak{h}at{\tau}\in\mathfrak{h}at{{\cal T}}$;
\item[(iv)] nested nature of the {\color{black} hierarchical} spline spaces {\color{black} for consecutive levels;}
\item[(v)] the span of a THB-spline basis defined over a sequence of subdomains is contained in the span of a truncated basis defined over a second sequence that is the nested enlargment of the original subdomain hierarchy;
\item[(vi)] completeness of the basis: for a certain class of admissible configurations of the hierarchical mesh, ${\mathop{\mathrm{span}} \mathfrak{h}at{{\cal T}}}$ contains all piecewise polynomial functions defined over the underlying grid.
\end{itemize}
In addition, the truncation mechanism enriches the THB-spline basis functions so that
{\bf e}gin{itemize}
\item[(vii)] they preserve the coefficients of the underlying sequence of B-splines;
\item[(iix)] they form a partition of {\color{black} unity};
\item[(ix)] they are strongly stable with respect to the supremum norm, under reasonable assumptions on the given knot configuration.\footnote{Strong stability of a basis means that the associated stability constants do not depend on the number of hierarchical levels.}
\end{itemize}
Due the two-scale relation \eqref{eq:2scale} between adjacent (non-negative) B-spline bases, the non-negativity of truncated basis functions (ii) is preserved by construction. Properties (i), (iii)-(v), and (vii)-(ix) are detailed in \cite{giannelli2012,giannelli2014}. For the analysis of hierarchical spline space in (vi), we refer to \cite{giannelli2013} for the bivariate case with single knots, and to \cite{mokris2014a} for the general multivariate setting with arbitrary knot multiplicities. As a consequence of property (vii), quasi-interpolants in hierarchical spline spaces can be easily constructed \cite{sm2015}.
Properties (ii) and (iix) imply the convex hull property, a key attribute for geometric modeling applications.
\subsection{Admissible meshes}
\label{sec:ameshes}
The truncation mechanism that characterizes the THB-spline basis can be properly exploited to design suitable refinement strategies that define different \emph{classes of admissible meshes}. A mesh of this kind allows to guarantee that the number of basis functions acting on any mesh point is bounded. In addition, the support of any basis function acting on a single element of an admissible mesh can be compared with the size of the element itself in terms of two constants that do not depend on the {\color{black} overall} number of hierarchical levels. These two properties are the key ingredients for the subsequent analysis --- see e.g., Theorem~\ref{eq:ub} related to the a posteriori upper bound, and the error indicator reduction provided by Lemma~\ref{lma:erred}. We postpone the presentation of the refinement procedure to Section~\ref{sec:mark&refine}, by simply focusing here on the desired mesh configuration.
{\bf e}gin{dfn}\label{dfn:amesh}
A mesh $\mathfrak{h}at{{\cal Q}}$ is admissible of class $m$ if the truncated basis functions in $\mathfrak{h}at{{\cal T}}(\mathfrak{h}at{{\cal Q}})$ which take non-zero values over any element $\mathfrak{h}at{Q}\in\mathfrak{h}at{{\cal Q}}$ belong to at most $m$ {\color{black} successive} levels.
\end{dfn}
For this class of admissible meshes, the number of basis functions acting on a single mesh element does not depend on the number of {\color{black} levels in the hierarchy} but only on $m$, that represents the class of admissibility of the mesh.
{\bf e}gin{crl}\label{crl:thbcor1}
For multivariate tensor-product B-splines of degree {\color{black} $\mathbf{p}=(p_1,$ $\ldots,$ $p_d)$}, the number of truncated basis functions which are non-zero on each element of an admissible mesh is then at most $m\prod_{i=1}^d(p_i+1)$.
\end{crl}
Another important fact that holds for admissible meshes is the following.
{\bf e}gin{crl}\label{crl:thbcor2} If $\mathfrak{h}at{\mathcal{Q}}$ is an admissible mesh of class $m$, given a truncated basis function $\mathfrak{h}at{\tau}\in \mathfrak{h}at{{\cal T}}(\mathfrak{h}at{{\cal Q}})$,
{\bf e}gin{equation}
\label{eq:supp-basis}
|\mathfrak{h}at{Q}| \lesssim | \mathop{\mathrm{supp}} \mathfrak{h}at{\tau} | \lesssim |\mathfrak{h}at{Q}| \qquad \forall \mathfrak{h}at{Q}\in\mathfrak{h}at{{\cal Q}} \ :\ \mathfrak{h}at{Q}\cap\mathop{\mathrm{supp}} \mathfrak{h}at{\tau} \neq \emptyset,
\end{equation}
where the hidden constants in the above inequalities depend on $m$ but not on $\mathfrak{h}at{\tau}$, neither on $\mathfrak{h}at{\mathcal{Q}}$ or $N$.
\end{crl}
In what follows, we will always indicate any inequality which does not depend on the depth $N$ of the spline hierarchy with $\lesssim$.
Since the interplay between the truncated basis funcions that are non-zero on a certain mesh element and the overall mesh configuration is strictly related to the \emph{locality} of the basis functions, we naturally focus on the \emph{support extension} of an element $\mathfrak{h}at{Q}\in\mathfrak{h}at{\mathcal{G}}^\ell$ .
For any fixed level $\ell$, the support extension collects the elements intersected by the set of B-splines in $\mathfrak{h}at{{\cal B}}^\ell$ whose support overlaps $\mathfrak{h}at{Q}$.
We extend this definition to the hierarchical setting as follow.
{\bf e}gin{dfn}\label{dfn:hse}
The support extension $S(\mathfrak{h}at{Q},k) $ of an element {\color{black} $\mathfrak{h}at{Q}\in\mathfrak{h}at{G}^\ell$} with respect to level $k$, with $0\le k\le \ell$,
is defined as
\[
S(\mathfrak{h}at{Q},k) {\color{black}\,:=\,} \left\{
\mathfrak{h}at{Q}'\in \mathfrak{h}at{G}^k:
{\color{black} \exists\, \mathfrak{h}at{{\bf e}ta}\in \mathfrak{h}at{\mathcal{B}}^k,\,}
\mathop{\mathrm{supp}}\mathfrak{h}at{{\bf e}ta}\cap \mathfrak{h}at{Q}'\ne\emptyset \wedge \mathop{\mathrm{supp}}\mathfrak{h}at{{\bf e}ta}\cap \mathfrak{h}at{Q}\ne\emptyset
\right\}.
\]
\end{dfn}
{\color{black} By a slight abuse of notation, we will also denote by $S(\mathfrak{h}at{Q},k) $ the region occupied by the closure of elements in $S(\mathfrak{h}at{Q},k)$.}
In order to identify a specific set of admissible meshes, we also consider the auxiliary subdomains
\[
{\color{black} {\mathfrak{h}at{\omega}}^{\ell} \,:=\,} {\bf i}gcup\left\{
\overline{\mathfrak{h}at{Q}} {\color{black} \,:\, \mathfrak{h}at{Q}}
\in \mathfrak{h}at{G}^{\ell} {\color{black} \,\wedge\,}
S(\mathfrak{h}at{Q},{\ell})\subseteq \mathfrak{h}at{\Omega}^{\ell}
\right\},
\]
for $\ell=0,\dots,N-1$. Any $\mathfrak{h}at{\omega}^\ell$ represent the biggest subset of $\mathfrak{h}at{\Omega}^\ell$ so that the set of B-splines in $\mathfrak{h}at{\cal B}^\ell$ whose support is contained in $\mathfrak{h}at{\Omega}^\ell$ spans the restriction of $V^\ell$ to $\mathfrak{h}at{\omega}^\ell$.
{\bf e}gin{exm}\label{exm:01}
A set of admissible meshes of class $m=2$ corresponds to the restricted hierarchies presented in Appendix A of \cite{giannelli2014} and relies on the following result. If $\mathfrak{h}at{\Omega}^{\ell} \subseteq \mathfrak{h}at{\omega}^{\ell-1}$ for $\ell=1,\ldots,N-1$, then for any element $\mathfrak{h}at{Q}\in {\color{black} \mathfrak{h}at{{\cal G}}^{\ell}}$ the THB-splines whose support overlaps $\mathfrak{h}at{Q}$ belong to at most two different levels: $\ell-1$ and $\ell$. Figure~\ref{fig:exm01} shows three examples of this class of admissible meshes related to the bivariate case of degree $(p_1,p_2)=(p,p)$ for $p=2,3,4$.
{\bf e}gin{figure}[ht!]{\bf e}gin{center}\mathfrak{h}space*{-.75cm}
\subfigure[$(p_1,p_2)=(2,2)$ ]{
\includegraphics[scale=0.25]{exm02p2a}}\mathfrak{h}space*{-.75cm}
\subfigure[$(p_1,p_2)=(3,3)$ ]{
\includegraphics[scale=0.25]{exm02p3a}}\mathfrak{h}space*{-.75cm}
\subfigure[$(p_1,p_2)=(4,4)$ ]{
\includegraphics[scale=0.25]{exm02p4a}}
\caption{Admissible meshes of class 2 considered in Example~\ref{exm:01} with respect to different degrees.}
\label{fig:exm01}
\end{center}\end{figure}
\end{exm}
The following proposition generalizes the class of admissible meshes considered in the previous example to the case of an arbitrary $m\ge 2$.
{\bf e}gin{prn}\label{prop:adm}
Let $\mathfrak{h}at{Q}$ be the mesh of active elements defined according to \eqref{eq:active} and \eqref{eq:mesh} with respect to the domain hierarchy $\mathfrak{h}at{\Omega}^0\supseteq\mathfrak{h}at{\Omega}^1\supseteq\ldots\supseteq\mathfrak{h}at{\Omega}^{N-1}$. If
{\bf e}gin{equation}\label{eq:sameshes}
\mathfrak{h}at{\Omega}^\ell\subseteq \mathfrak{h}at{\omega}^{\ell-m+1},
\end{equation}
for $\ell=m,m+1,\ldots,N-1$, then the mesh $\mathfrak{h}at{{\cal Q}}$ is admissible of class $m$.
\end{prn}
{\bf e}gin{proof}
For any $\mathfrak{h}at{\tau}\in\mathfrak{h}at{{\cal T}}(\mathfrak{h}at{\cal Q})$ introduced at level $\ell-m$, the function $\mathop{\mathrm{trunc}}^{\ell-m+1}\mathfrak{h}at{\tau}$ defined by equation \eqref{eq:trunc} is a linear combination of basis functions $\mathfrak{h}at{{\bf e}ta}\in\mathfrak{h}at{{\cal B}}^{\ell-m+1}$ so that $\mathfrak{h}at{{\bf e}ta}\vert_{\mathfrak{h}at{\omega}^{\ell-m+1}}=0$. Since
\[
\mathfrak{h}at{\omega}^{\ell-m+1} = {\color{black} {\bf i}gcup}
\left\{
{\color{black} \overline{\mathfrak{h}at{Q}}:\,}
\mathfrak{h}at{Q}\in \mathfrak{h}at{G}^{\ell-m+1} {\color{black} \,\wedge\,}
S(\mathfrak{h}at{Q},{\ell-m+1})\subseteq
\mathfrak{h}at{\Omega}^{\ell-m+1}\right\},
\]
if condition \eqref{eq:sameshes} holds for $\ell=m,m+1,\ldots,N-1$, then also $\mathop{\mathrm{trunc}}^{\ell-m+1}\mathfrak{h}at{\tau}\vert_{\mathfrak{h}at{\Omega}^\ell}=0$. Consequently, the truncation of a B-spline introduced at level $\ell-m$ will be non-zero on $\mathfrak{h}at{\Omega}^{\ell-m}\setminus\mathfrak{h}at{\Omega}^{\ell}$. This means that any element $\mathfrak{h}at{Q}\in\mathfrak{h}at{{\cal G}}^\ell$ belongs to the support of THB-splines of only $m$ different levels: {\color{black} $\ell-m+1, \ldots, \ell$}.
\end{proof}
As we will detail later, a relevant set of admissible meshes is the one verifying condition \eqref{eq:sameshes} for $\ell=m,m+1,\ldots,N-1$, where different values of $m\ge2$ can be considered.
{\bf e}gin{dfn}\label{dfn:samesh}
A mesh $\mathfrak{h}at{{\cal Q}}$ is strictly admissible of class $m$ if it verifies the assumptions of Proposition~\ref{prop:adm}.
\end{dfn}
The meshes considered in Example~\ref{exm:01} are strictly admissible of class $2$.
\section{The module SOLVE: the Galerkin method}
\label{sec:solve}
In this section we describe our model problem and introduce its discretization by means of hierarchical splines. Indeed, we have no aim of generality, we consider the most simple elliptic problem.
As a first step, we give a precise definition of the domain $\Omega$ in which our problem is posed.
Given a strictly admissible mesh $\mathfrak{h}at{\cal Q}_0$ and the corresponding set of truncated basis function $\mathfrak{h}at{\cal T}_0$, we suppose that the computational domain $\Omega$ is provided as a linear combination of functions in $\mathfrak{h}at{\cal T}_0$ and control points:
{\bf e}gin{equation}
\label{eq:1}
{\bf x}\in {\color{black}\overline{\Omega}}\,, \quad {\bf x} = {\bf F}(\mathfrak{h}at{{\bf x}}) = \sum_{\mathfrak{h}at\tau\in \mathfrak{h}at{{\cal T}}_0} \mathbf{C}_{\mathfrak{h}at{\tau} } \mathfrak{h}at{\tau}(\mathfrak{h}at{{\bf x}})\qquad \mathfrak{h}at{{\bf x}} \in
{\color{black} \mathfrak{h}at{\Omega}^0}
\end{equation}
where $\mathbf{C}_{\mathfrak{h}at{\tau}}\in \mathbb{R}^d$.
In all what follows, we suppose that the mapping {\color{black} ${\bf F}: \mathfrak{h}at{\Omega}^0\to \overline{\Omega}$} is a bi-Lipschitz homeomorphism:
{\bf e}gin{equation}
\label{eq:Fbound}
\| D^\alpha{\bf F}\|_{L^{\infty}({\color{black} \mathfrak{h}at{\Omega}^0})} \le C_{{\bf F}},
\quad
\| D^\alpha{\bf F}^{-1}\color{black}\|_{L^{\infty}( \Omega)}
\le c_{{\bf F}}^{-1}, \qquad |\alpha|\leq 1
\end{equation}
where $c_{\bf F}$ and and $C_{\bf F}$ are independent constants bounded away from infinity.
We consider then the following problem:
{\bf e}gin{equation}
\label{eq:mp_1}
-\textrm{div} (\textbf{A} \nabla u) = f \quad\text{in}\; \Omega,
\qquad
u\myvert{\partial\Omega} = 0,
\end{equation}
where $\textbf{A}\in C^\infty (\bar{\Omega})$ is the diffusion matrix which verifies \eqref{eq:5}.
In order to define the variational formulation of the problem,
we consider the space of functions in $H^1(\Omega)$ with vanishing trace on $\partial\Omega$
\[
\mathbb{V}:=H_0^1(\Omega):=\left\{
v\in H^1(\Omega):v\myvert{\partial\Omega}=0
\right\},
\]
endowed with the norm $\| u \|^2_{\mathbb{V}} = \| \nabla v\|^2_{L^2(\Omega)^d} + \| v \|^2_{L^2(\Omega)}$.
A weak solution of \eqref{eq:mp} is a function $u\in \mathbb{V}$ satisfying
{\bf e}gin{equation}\label{eq:weak}
u\in\mathbb{V}:\quad
a(u,v) = \langle f,v \rangle, \quad \forall\, v\in\mathbb{V},
\end{equation}
where $a:\mathbb{V}\times\mathbb{V}\rightarrow\mathbb{R}$ is the bilinear form
\[
a(u,v):=\int_{\Omega} \mathbf{A} \nabla u \nabla v, \qquad
\forall\, u,v\in\mathbb{V},
\]
and $\langle \cdot,\cdot \rangle$ stands for the $L^2(\Omega)$ scalar product. We assume that $f\in\mathbb{V}^*$.
The bilinear form $a(u,v)$ is coercive and continuos with constant $\alpha_1$ and $\alpha_2$, respectively:
{\bf e}gin{align}
\label{eq:coer}
& a(u,u) \geq \alpha_1 \| u\|^2_{\mathbb{V}} \qquad & u \in {\color{black} \mathbb{V} ,}\\
\label{eq:cont}
& a(u,v) \leq \alpha_2 \| u \|_\mathbb{V} \|v \|_\mathbb{V}\qquad & u\,,\ v \in {\color{black} \mathbb{V} .}
\end{align}
Moreover, it induces the \emph{energy norm}:
$|||v|||_\Omega:=a(v,v)^{1/2}$, $\forall v\in\mathbb{V}$.
The coercivity and continuity properties of $a(u,v)$ implies the equivalence between the energy and the $H^1(\Omega)$ norms on $\mathbb{V}$. In addition, the Lax-Milgram theorem ensures the existence and uniqueness of the weak solution \eqref{eq:weak}.
{We construct now our module SOLVE as the Galerkin discretization of \eqref{eq:weak} by means of hierarchical splines on $\Omega$. To this aim, we first need to introduce a suitable notation for hierarchical meshes and spaces on $\Omega$. }
We consider an admissible mesh $\mathfrak{h}at{\cal Q}$, such that $\mathfrak{h}at{{\cal Q}} \succeq \mathfrak{h}at{{\cal Q}}_0$ and we denote by $\mathfrak{h}at{{\cal T}}$ the corresponding basis truncated basis functions. Moreover, we construct the corresponding mesh and functions of the physical domain via pullback:
\[ {\cal Q} = \{ Q = {\bf F}(\mathfrak{h}at{Q}): \color{black}\mathfrak{h}at{Q}\in \mathfrak{h}at{{\cal Q}}\}.\]
For all $\mathfrak{h}at \tau\in \mathfrak{h}at{{\cal T}}$, we construct:
{\bf e}gin{equation}
\label{eq:4}
\tau ({\bf x}) = \mathfrak{h}at{\tau}(\mathfrak{h}at{{\bf x}}), \qquad {\bf x} = {\bf F}(\mathfrak{h}at{{\bf x}}).
\end{equation}
and we denote by ${\cal T}$ the collection of all mapped basis functions, and by $\mathbb{S}(\mathcal{Q})$ the space they generate, $ \mathbb{S}(\mathcal{Q}) =\mathop{\mathrm{span}}{\cal T} ({\cal Q})$.
Clearly, ${\cal Q}$ is a hierarchical mesh on the domain $\Omega$ and for it, we will make use of all the nomenclature introduced in Section \ref{sec:hspaces} by simply \color{black}removing the $\mathfrak{h}at{\cdot}$. First, for all elements $Q$, we denote by $\mathfrak{h}at{Q}$ its preimage through ${\bf F}$, i.e., $Q= {\bf F}(\mathfrak{h}at{Q})$, and $h_Q = |Q|^{1/d}$, where $|Q|$ represents the volume of $Q$. Moreover, we set:
{\bf e}gin{itemize}
\item $\Omega^\ell = {\bf F}(\mathfrak{h}at\Omega^\ell)$ and $\omega^\ell = {\bf F}(\mathfrak{h}at\omega^\ell)$;
\item ${\cal G}^\ell =\{Q \in {\cal Q} \ :\ \mathfrak{h}at{Q}\in \mathfrak{h}at{\cal G}^\ell\}$ and $G^\ell = \{ Q\subset \Omega \ :\ \mathfrak{h}at{Q}\in \mathfrak{h}at{G}^\ell\}$;
\item for all ${Q}\in {\cal G}^\ell$, its support extension with respect to level $k$ is
\[S(Q, k) = \{ Q'\in G^k \ :\ \mathfrak{h}at{Q}'\in S(\mathfrak{h}at{Q},k)\}.\]
\end{itemize}
Finally, when ${\cal Q}^\star$ is a refinement of ${\cal Q}$, we will write ${\cal Q}^\star \succeq {\cal Q}$, when their pre-images $\mathfrak{h}at{{\cal Q}}^\star$ and $\mathfrak{h}at{{\cal Q}}$ verifies $\mathfrak{h}at{{\cal Q}}^\star \succeq \mathfrak{h}at{{\cal Q}}$.
We are finally in the position to describe the discrete problem we want to solve adaptively. The Galerkin approximation of \eqref{eq:weak} consists in solving:
{\bf e}gin{equation}\label{eq:gal}
\text{find } \ U\in\mathbb{S}_D(\mathcal{Q}): \quad
a(U,V) = \langle f,V \rangle,\quad \forall\, V\in\mathbb{S}_D(\mathcal{Q}),
\end{equation}
where
\[
\mathbb{S}_D(\mathcal{Q}) = \left\{V\in \mathbb{S}({\cal Q}): V\myvert{\partial\Omega} = 0\right\}.
\]
In the subsequent analysis we assume for simplicity $\mathbb{S}_D({\cal Q})\subset C^1(\Omega)$. This assumption is of course not needed for the development of an adaptive strategy, but it allows us to simplify the analysis, by also showing the specific changes with respect to $C^0$ finite elements. The general case could be treated in a similar way following the classical theory of adaptive finite element methods.
\section{The module ESTIMATE: the residual based error indicator}
\label{sec:estimate}
The residual associated to $U\in\mathbb{S}$ is the functional in $\mathbb{V}^*$ defined by
{\bf e}gin{equation*}
\langle R,v\rangle := \langle f,v\rangle -a(U,v),
\end{equation*}
that satisfies
{\bf e}gin{align*}
&\langle r,v\rangle = a(u-U,v), \quad\forall\, v\in\mathbb{V},\\
&a(u-U,V) = \langle r,V\rangle = 0, \quad \forall\, V\in\mathbb{S}.
\end{align*}
By recalling that all discrete functions are continuous with continuous derivatives, we can
integrate by parts and obtain
{\bf e}gin{equation*}
\langle r,v\rangle = \int_{\Omega} fv - \textbf{A} \nabla U\nabla v
= \int_\Omega fv - \textrm{div} (\textbf{A} \nabla U) v
,
\end{equation*}
where, thanks to our assumption that $\mathbb{S} \subset C^1(\Omega)$, the quantity $r= f - \textrm{div} (\textbf{A} \nabla U) $ belongs to $L^2(\Omega)$. In particular, as we expect, this means that the residual does not contain any edge contribution as in typical finite element indicators
\cite{verfurth2013}.
One of the fundamental ingredient in the module ESTIMATE is the equivalence between the primal norm of the error and the dual norm of the residual:
{\bf e}gin{equation}\label{eq:ee1}
||u-U||_{\mathbb{V}} \le
\frac{1}{\alpha_1} ||r||_{\mathbb{V}^*}\le
\frac{\alpha_2}{\alpha_1}||u-U||_{\mathbb{V}}.
\end{equation}
As it is standard, in order to use the residual as error indicator, we would like to replace the norm $\|\cdot\|_{\mathbb{V}*}$ with the following error indicator
{\bf e}gin{equation}\label{eq:ind}
\varepsilon^2_{\cal Q}(U, {\cal Q})
= \sum_{Q\in{\cal Q}} \varepsilon_{\cal Q}^2(U,Q)
\qquad\text{with}\qquad
\varepsilon^2_{\cal Q}(U,Q)
= h_Q^2 ||r||_{L^2(Q)}^2.
\end{equation}
When no confusion is possible, we may also abbreviate the above notation with $\varepsilon^2_{\cal Q}(U)$.
Following \cite{morin2001} (see also \cite{NochettoCIME}), we will show that the following holds:
{\bf e}gin{equation}
\label{eq:apost-1}
||u-U||_{\mathbb{V}} \lesssim
\varepsilon_{\cal Q}(U,{\cal Q}) \lesssim
||u-U||_{\mathbb{V}}
+
\mathrm{osc}_{\cal Q}(U, {\cal Q}),
\end{equation}
where
{\bf e}gin{equation*}
\label{eq:oscillaz}
\mathrm{osc}^2_{\cal Q}(U, {\cal Q}) = \sum_{Q\in {\cal Q}} \mathrm{osc}^2(U,Q) \quad \text{with} \quad \mathrm{osc}(U,Q) = h_Q \| r-\Pi_{\bf n} r\|_{L^2(Q)}
\end{equation*}
and $\Pi_{\bf n} : L^2(Q) \to \mathbb{Q}_{\bf n}$, ${\bf n}=(n_1,n_2,n_3)$, denotes the $L^2$ projector onto the space of polynomials of degree $n_j$ in the space direction $j$. The degrees $n_j$, $j=1,\ldots,d$ can be fixed large enough so that the oscillation are ``smaller'' than the error
\cite{bonito2010}.
Indeed Theorem \ref{thm:lb} below, will also provide a local version of the lower bound in \eqref{eq:apost-1} that reads:
{\bf e}gin{equation*}
\label{eq:lb-local-1}
\varepsilon_{\cal Q}(U,{Q}) \lesssim
||u-U||_{\mathbb{V}(Q)}
+
\mathrm{osc}_{\cal Q}(U, {Q}).
\end{equation*}
\subsection{A posteriori upper bound}
In this section we prove that the residual based error indicator defined in \eqref{eq:ind} is \emph{reliable}, i.e., it is an upper bound for the Galerkin error.
{\bf e}gin{thm}\label{thm:ub}
Let $u$ be the exact weak solution of the model problem \eqref{eq:weak}. The error of the Galerkin approximation $U\in\mathbb{S}({\cal Q})$ in \eqref{eq:gal} is bounded in terms of the error indicator $\varepsilon_{\cal Q}(U)$ introduced in \eqref{eq:ind} as follows:
{\bf e}gin{equation}
\label{eq:ub}
||u-U||_{\mathbb{V}} \leq C_{\mathrm{up}} \varepsilon_{\cal Q}(U),
\end{equation}
where the constant $C_{\mathrm{up}}$ is independent on the mesh size and on the level of hierarchy.
\end{thm}
{\bf e}gin{proof}
This proof follows exactly the lines of the classical proof of upper bound in residual based error estimators. For completeness we repeat here the steps that can be found in, e.g., Theorem 6 in \cite{NochettoCIME}.
Using (\ref{eq:ee1}), we have $\| u-U\|_{\mathbb{V}} \lesssim \displaystyle \frac{1}{\alpha_1} \| r \|_{\mathbb{V}^\star}$, and we will prove that $ \| r \|_{\mathbb{V}^\star} \les \varepsilon_\mathcal{Q}(U)$.
Since the basis functions in ${\cal T}$ form a partition of unity and the residual is orthogonal to all basis functions $\tau$ in ${\cal T}$, it holds:
\[
\langle r, v\rangle = \sum_{\tau \in {\cal T}}\langle r, \tau \, v \rangle = \sum_{\tau\in {\cal T}} \inf_{c_\tau \in \mathbb{R} } \langle r, \tau \, (v-c_\tau) \rangle.
\]
By standard Cauchy-Schwarz inequality, we estimate the terms in the right hand side as follows:
\[\langle R, \tau\, (v-c_\tau) \rangle = \int_\Omega
r\, \tau (v-c_\tau) \leq \| r\, \tau^{1/2} \|_{L^2(\Omega)} \| \tau^{1/2} (v-c_\tau) \|_{L^2(\Omega)}. \]
We denote by $\omega_\tau =\mathop{\mathrm{supp}}\tau$ and by $h_{\omega_\tau} = |\mathop{\mathrm{supp}} \tau|^{1/d}$, i.e., its size. We can deduce by Poincar\'e inequality that:
\[ \| \tau^{1/2} (v-c_\tau )\|_{L^2(\omega_\tau)} \les h_{\omega_\tau} \| \nabla v\|_{L^2(\omega_\tau)^d}.\]
By taking into account Corollaries \ref{crl:thbcor1} and \ref{crl:thbcor2}, we have
{\bf e}gin{enumerate}
\item $\sum_{\tau\in {\cal T}} \| \nabla v\|^2_{L^2(\omega_\tau)^d} \les \| \nabla v\|^2_{L^2(\Omega)^d} $;
\item let $h$ be the piecewise constant function which takes values $h({\bf x}) = |Q|^{1/d}$, ${\bf x}\in Q$ for all $Q\in \mathcal{Q}$. It holds:
\[ \sum_{\tau\in {\cal T}} h^2_{\omega_\tau} \| r\, \tau^{1/2} \|^2_{L^2(\omega_\tau)} \les \sum_{\tau\in {\cal T}} \int_{\omega_\tau} h^2\, r^2 \tau = \int_\Omega h^2\, r^2 = \varepsilon_\mathcal{Q}(U). \]
\end{enumerate}
The estimate (\ref{eq:ub}) follows.
\end{proof}
\subsection{A posteriori lower bound}
In this section we prove that the residual based error indicator defined in \eqref{eq:ind} is \emph{efficient}, i.e., it is a lower bound of the Galerkin error up to oscillations. \B
{\bf e}gin{thm}
\label{thm:lb}
Let $u$ be the exact weak solution of the model problem \eqref{eq:weak}. The error of the Galerkin approximation $U\in\mathbb{S}({\cal Q})$ in \eqref{eq:gal} bounds the error indicator $\varepsilon_{\cal Q}(U)$ introduced in \eqref{eq:ind} up to oscillations:
{\bf e}gin{equation}
\label{eq:lb-local-2}
\varepsilon_{\cal Q}(U,{Q}) \leq C_{\mathrm{lb}} {\bf i}g(
||u-U||_{\mathbb{V}(Q)}
+
\mathrm{osc}_{\cal Q}(U, {Q}){\bf i}g),
\end{equation}
where the constant $C_{\mathrm{lb}} $ does not depend on $Q$.
\end{thm}
{\bf e}gin{proof}
Again, this proof is classical, and we repeat the steps of the proof in Theorem 7 of \cite{NochettoCIME}.
First, it is easy to see that
\[ \| r\|_{\mathbb{V}^*(Q)} \les \| \nabla (u-U)\|_{L^2(Q) }\]
and that the following Poincar\'e estimate is true:
\[ \| r \| _{\mathbb{V}^*(Q)} \les h_Q \| r\|_{L^2(Q)}. \]
Moreover, let $\mathbb{Q}_{\bf n}$, ${\bf n}=(n_1,n_2,n_3)$ be the space of polynomials of degree $n_j$ in the space direction $j$, then we know that the inverse inequality holds:
\[ \| \bar{r} \|_{\mathbb{V}^*} \gtrsim h_Q \| \bar{r} \|_{L^2(Q)} \qquad \forall \bar{r} \in \mathbb{Q}_{\bf n},\]
where the hidden constant does not depend on $Q$ but it deteriorates with ${\bf n}$.
Finally, if we choose $\bar{r} = \Pi_{\bf n} r$, it holds that $\| \bar{r}\|_{\mathbb{V}^*(Q)} \leq \| r \|_{\mathbb{V}^*(Q)} $.
Now, all these ingredients can be used together in the following estimate, with $\bar{r} = \Pi_{\bf n} r$:
{\bf e}gin{equation}
\label{eq:2}
{\bf e}gin{aligned}
h_Q \|r\|_{L^2(Q)} & \leq h_Q \| \bar{r} \|_{L^2(Q)} + \|r-\bar{r}\|_{L^2(Q)} \\
& \les \|\bar{r}\|_{\mathbb{V}^*(Q)} + h_Q \| r-\bar{r}\|_{L^2(Q)} \\
& \les \|{r}\|_{\mathbb{V}^*(Q)} + h_Q \| r-\bar{r}\|_{L^2(Q)}
\end{aligned}
\end{equation}
The proof is completed by setting $\mathrm{osc}_{\cal Q} (U,Q) = h_Q \| r-\bar{r}\|_{L^2(Q)}$.
\end{proof}
{\bf e}gin{rmk}
It should be noted that the lower bound we have proved here will not be used in the sequel of the present paper. In fact, contraction of the error can be proved without using explicitly the lower bound. We have reported this simple proof here in order to collect the main properties of the estimator we are using. On the other hand, this lower bound will be needed in the companion paper \cite{buffa2015b} where optimality will be addressed.
\end{rmk}
\section{The modules MARK and REFINE}
\label{sec:mark&refine}
We now briefly describe the considered marking strategy, before introducing a refine module that preserves the class of admissibility of a given strictly admissible mesh --- see Section~\ref{sec:ameshes}
--- and its properties. Finally, we conclude this section by discussing the contraction property of our AIGM and its convergence. \B
\subsection{MARK: the marking strategy}
Given an admissible mesh ${\cal Q}$, the Galerkin solution $U\in\mathbb{V({\cal Q})}$, the module
\[
{\cal M} = \text{MARK}\left(
\left\{\varepsilon_{\cal Q}(U,Q)\right\}_{Q\in{\cal Q}},
{\cal Q}\right),
\]
selects and marks a set of elements ${\cal M}\subset{\cal Q}$ according to the so-called \emph{D\"orfler marking} \cite{dorfler1996}, i.e., by considering a fixed parameter $\theta\in(0,1]$ so that
{\bf e}gin{equation}\label{eq:dm}
\varepsilon_{\cal Q}(U,{\cal M}) \ge \theta\,
\varepsilon_{\cal Q}(U,{\cal Q}).
\end{equation}
This marking strategy simply guarantees that the set ${\cal M}$ of marked elements gives a substantial contribution to the total error indicator.
\subsection{REFINE: the refinement strategy}
The support extension $S(\widehat{Q},k)$ of an element $\widehat{Q}\in\widehat{\mathcal{G}}^\ell$ with respect to level $k$, with $0\le k\le \ell$
introduced in Definition~\ref{dfn:hse};
\color{black} analogously, we denote by $S(Q,k)$ the support extension of the physical element $Q$.
In order to guarantee that a mesh after refinement is admissible, we aim at imposing that each active element at level $\ell$, $Q\in \mathcal{G}^\ell$, belongs to the support of basis functions of at most levels $\ell-m+1$, \ldots, $\ell$. To achieve this, given an element $Q\in \mathcal{G}^\ell$ that we want to refine, we select the elements that are active, that are of level $\ell-m+1$ and such that their intersection with $S(Q,\ell-m+2)$ is not empty. We collect all these elements in a neighborhood of $Q$, denoted by ${\cal N}({\color{black} {\mathcal{Q}},\,}Q,m) $ and defined here below in Definition \ref{dfn:neigh}. Clearly, when $Q$ is refined, all elements in ${\cal N}({\color{black} {\mathcal{Q}},\,}Q,m) $ have to be refined as well and this procedure has to be applied recursively in order to guarantee that the final mesh is strictly admissible. \color{black}
{\bf e}gin{dfn}\label{dfn:neigh}
The neighborood of $Q\in {\color{black} {\mathcal{Q}} \,\cap\,}\mathcal{G}^\ell$ with respect to $m$ is defined as
\[
{\cal N}({\color{black} {\mathcal{Q}},\,}Q,m) {\color{black} \,:=\,} \left\{Q'\in{\cal G}^{\ell-m+1}:
{\color{black} \exists\, Q'' }\in S(Q,\ell-m+2)
{\color{black} , Q''\subseteq Q' } \right\},
\]
when ${\color{black} \ell-m+1 > 0}$, and ${\cal N}({\color{black} {\mathcal{Q}},\,} Q,m) = \emptyset$ for ${\color{black} \ell-m+1 \le 0}$.
\end{dfn}
Figure~\ref{fig:exm02} shows the the neighborood of an element $Q$ with respect to $m=2$ when, for simplicity, the identity map is considered.
{\bf e}gin{figure}[ht!]{\bf e}gin{center}
\mathfrak{h}space*{-.75cm}
\subfigure[$(p_1,p_2)=(2,2)$ ]{
\includegraphics[scale=0.195]{exm01p2a}\mathfrak{h}space*{-.75cm}
\includegraphics[scale=0.195]{exm01p2b}\mathfrak{h}space*{-.75cm}
\includegraphics[scale=0.195]{exm01p2c}\mathfrak{h}space*{-.75cm}
\includegraphics[scale=0.195]{exm01p2d}}\\
\mathfrak{h}space*{-.75cm}
\subfigure[$(p_1,p_2)=(3,3)$ ]{
\includegraphics[scale=0.195]{exm01p3a}\mathfrak{h}space*{-.75cm}
\includegraphics[scale=0.195]{exm01p3b}\mathfrak{h}space*{-.75cm}
\includegraphics[scale=0.195]{exm01p3c}\mathfrak{h}space*{-.75cm}
\includegraphics[scale=0.195]{exm01p3d}}\\
\mathfrak{h}space*{-.75cm}
\subfigure[$(p_1,p_2)=(4,4)$ ]{
\includegraphics[scale=0.195]{exm01p4a}\mathfrak{h}space*{-.75cm}
\includegraphics[scale=0.195]{exm01p4b}\mathfrak{h}space*{-.75cm}
\includegraphics[scale=0.195]{exm01p4c}\mathfrak{h}space*{-.75cm}
\includegraphics[scale=0.195]{exm01p4d}}
\caption{Neighborood ${\cal N}({\color{black} \mathcal{Q},\,}Q,2)$ (light gray) of an element $Q$ ({represented by any of the four cells in dark grey}) when dyadic refinement is considered for some low degree cases. Note that the neighborood is always aligned with the grid lines of a previous hierarchical level.}
\label{fig:exm02}
\end{center}\end{figure}
An automatic REFINE module which allows to define \emph{strictly} admissible meshes is presented in Figure~\ref{fig:refine}. The core of the refinement strategy relies on the internal recursive module REFINE$\_$RECURSIVE. {Any element $Q$ on which the recursive procedure is called will be subdivided into its children.} Lemma~\ref{lma:rr} and Proposition~\ref{prn:rr} below shows the distinguishing properties of this procedure.
{\bf e}gin{figure}[ht!]{\bf e}gin{center}{\bf e}gin{footnotesize}
{\bf e}gin{tabular}{l}
\mathfrak{h}line \vspace*{-.35cm}\\
${\cal Q}^\star = $ REFINE(${\cal Q},{\cal M},m$)\\%
\\
\mathfrak{h}line \vspace*{-.25cm}\\
for all $Q\in{\cal Q}\cap{\cal M}$
\\
\mathfrak{h}space*{.5cm} ${\cal Q} = $ REFINE$\_$RECURSIVE(${\cal Q},Q,m$)
\\
end\\
$\mathcal{Q}^\star = \mathcal{Q}$
\\
\mathfrak{h}line
\end{tabular}\mathfrak{h}space*{.5cm}
{\bf e}gin{tabular}{l}
\mathfrak{h}line \vspace*{-.35cm}\\
${\cal Q} = $ REFINE$\_$RECURSIVE($\mathcal{Q},Q,m$)\\%
\\
\mathfrak{h}line \vspace*{-.25cm}\\
for all $Q' \in{\cal N}({\color{black} \mathcal{Q},\,}Q,m)$
\\
\mathfrak{h}space*{.5cm}${\cal Q} = $ REFINE$\_$RECURSIVE(${\cal Q},Q',m$)
\\
end
\\
if {\color{black}$Q$} has not been subdivided
\color{black}\\
\mathfrak{h}space*{.25cm} subdivide {\color{black}$Q$} and
\\
\mathfrak{h}space*{.25cm} update $\mathcal{Q}$ by replacing {\color{black}$Q$} with its children
\\
end\\
\mathfrak{h}line
\end{tabular}
\end{footnotesize}\end{center}
\caption{The REFINE and REFINE$\_$RECURSIVE modules.}\label{fig:refine}
\end{figure}
{\bf e}gin{lma}\label{lma:rr}
(Recursive refinement)
Let ${\cal Q}$ be a {strictly} admissible mesh of class $m$. The call to $\mathcal{Q}^* =$ REFINE$\_$RECURSIVE(${\cal Q},Q,m$) terminates and returns a refined mesh $\mathcal{Q}^*$ with elements that either were already active in ${\cal Q}$ or are obtained by single refinement of an element of $\mathcal{Q}$.
\end{lma}
{\bf e}gin{proof}
For every marked element $Q\in\mathcal{G}^\ell\cap\mathcal{M}$ the REFINE$\_$RECURSIVE routine is recursively called on any element of level $\ell'={\ell-m+1}$ that belongs to the neighborood of $Q$ with respect to $m$ while $\ell'$ is greater or equal than zero. Since at each recursive call the level $\ell'$ of interest is strictly decreasing, the termination condition will be satisfied after a finite number of steps.
In addition, any element $Q$ touched by a call to REFINE$\_$RECURSIVE is subdived in its children only the first time it is reached in the return phase after the set of recursive calls. Every element of ${\cal Q}$ is then refined at most once in the refinement process that generates ${\cal Q}^*$ from ${\cal Q}$.
\end{proof}
By exploiting the truncation mechanism in the context of strictly admissible meshes --- see Definition~\ref{dfn:samesh} ---
it is possible to show that only the supports of truncated basis functions of level $\ell-m+1,\ell-m+2,\ldots,\ell$ will contain an element $Q\in{\cal G}^\ell$, for every refined mesh generated by the REFINE$\_$RECURSIVE module.
{\bf e}gin{prn}\label{prn:rr}
Let $\mathcal{Q}$ be a strictly admissible mesh of class $m\ge 2$ and let $Q_\mathcal{M}$ be an active element of level $\ell$, for some $0\le\ell\le N-1$. The call to $\mathcal{Q}^* =$ REFINE$\_$RECURSIVE $(\mathcal{Q},Q_\mathcal{M},m)$ returns a strictly admissible mesh $\mathcal{Q}^*\succeq\mathcal{Q}$ of class $m$.
\end{prn}
{\bf e}gin{proof}
Let $\Omega^{0}\supseteq\ldots\supseteq\Omega^{N-1}\supseteq\Omega^{N}$, with $\Omega^N=\emptyset$, be the domain hierarchy associated to mesh ${\cal Q}$.
The refined mesh $\mathcal{Q}^*$ = REFINE$\_$RECURSIVE $(\mathcal{Q},Q_\mathcal{M},m)$ contains active elements $Q^*\in\mathcal{G}^{\ell,*}$ with respect to the domain hierarchy $\Omega^{0,*}\supseteq\ldots\supseteq\Omega^{N-1,*}\supseteq\Omega^{N,*}$ where
{\bf e}gin{equation}\label{eq:nested}
\Omega^{0,*}\equiv\Omega^{0}
\quad\text{and}\quad
\Omega^{\ell,*}\supseteq\Omega^{\ell},
\end{equation}
for $\ell=1,\ldots,N$. Note that the maximum level of refinement in $\mathcal{Q}^*$ is necessarily $N$ according to Lemma~\ref{lma:rr}.\\
Let $Q^*\in\mathcal{G}^{\ell,*}$ be an active element of $\mathcal{Q}^*$, then $Q^*\subseteq\Omega^{\ell,*}\setminus\Omega^{\ell+1,*}$, for some $0\le\ell\le N$. We have two possibilities: either $Q^*$ belongs also to $\Omega^\ell$ or not.
{\bf e}gin{itemize}
\item If $Q^*\subseteq\Omega^\ell$ then $0\le\ell\le N-1$. Since the initial mesh $\mathcal{Q}$ is strictly admissible of class $m$, we have: $\Omega^\ell\subseteq\omega^{\ell-m+1}$, namely $Q^*\subseteq\omega^{\ell-m+1}$. Now, the refined subdomain hierarchy is a nested enlargement of the original one according to \eqref{eq:nested}, and, consequently, $\omega^{\ell-m+1}\subseteq\omega^{\ell-m+1,*}$, which implies $Q^*\subseteq\omega^{\ell-m+1,*}$.
\item If $Q^*\subseteq\Omega^{\ell,*}\setminus\Omega^\ell$, then Lemma~\ref{lma:rr} guarantees that $Q^*$ has been obtained by applying a single refinement to an element of $\mathcal{Q}$. Hence, there exists $Q_{\mathcal{M}}^{\#}\in\mathcal{G}^{\ell-1}$ so that $Q_{\mathcal{M}}^{\#}\supseteq Q^*$. Condition \ref{eq:sameshes} on $\mathcal{Q}$ implies
\[
Q_{\mathcal{M}}^{\#}
{\color{black} \,\subset\, {\omega}^{\ell-m}} = {\color{black} {\bf i}gcup}
\left\{
{\color{black} \overline{Q}\,:\,}
Q\in G^{\ell-m}
{\color{black} \,\wedge\,}
S({\color{black} Q},\ell-m)\subseteq
{\Omega}^{\ell-m}\right\}
\]
and, consequently,
{\bf e}gin{equation}\label{eq:intermediate}
S({Q}_{\mathcal{M}}^{\#},\ell-m)\subseteq
\Omega^{\ell-m}.
\end{equation}
Since $Q_{\mathcal{M}}^{\#}$ is an active element of $\mathcal{Q}$ that has been subdivided in the refinement process from $\mathcal{Q}$ to $\mathcal{Q}^*$, the REFINE$\_$RECURSIVE module has been called over this element. More precisely, the call REFINE$\_$RECURSIVE $(\mathcal{Q}^{\#},Q_{\mathcal{M}}^{\#},m)$ belongs to the chain of recursive calls activated by REFINE$\_$RECURSIVE $(\mathcal{Q},Q_{\mathcal{M}},m)$ for some intermediate mesh $\mathcal{Q}^{\#}$ so that $\mathcal{Q}^*\succeq\mathcal{Q}^{\#}\succeq\mathcal{Q}$. This mean that {\color{black} the} recursive routine has been called on any $Q'\in\mathcal{N}(\mathcal{Q}^{\#},Q_{\mathcal{M}}^{\#},m)$ with
\[
\mathcal{N}(\mathcal{Q}^{\#},Q_{\mathcal{M}}^{\#},m)
= \left\{Q'\in{\mathcal{G}}^{\ell-m,\#}:
{\color{black} \exists\, Q'' } \in
S({Q}_{\mathcal{M}}^{\#},\ell-m+1)
{\color{black} , Q''\subseteq Q'}\right\}.
\]
By combinining \eqref{eq:intermediate} with $S({Q}_{\mathcal{M}}^{\#},\ell-m+1)\subseteq S({Q}_{\mathcal{M}}^{\#},\ell-m)$, we obtain $S({Q}_{\mathcal{M}}^{\#},\ell-m+1)\subseteq
{\Omega}^{\ell-m}$. Hence, the coarsest elements in $S({Q}_{\mathcal{M}}^{\#},\ell-m+1)$ are exactly the ones of level $\ell-m$. All these $Q'$ elements of level $\ell-m$ have been subdivided into their children of level $\ell-m+1$ in the refinement step from $\mathcal{Q}^{\#}$ to $\mathcal{Q}^*$ in order to guarantee that
\[
S({Q}_{\mathcal{M}}^{\#},\ell-m+1)\subseteq
\Omega^{\ell-m+1,*}.
\]
Then $Q^*\subseteq\omega^{\ell-m+1,*}$.
\end{itemize}
{In both cases, $Q^*\subseteq\omega^{\ell-m+1,*}$ implies $\widehat{Q}^*\subseteq\widehat{\omega}^{\ell-m+1,*}$. Condition \eqref{eq:sameshes} is then satisfied.}
\end{proof}
The previous results guarantees that the strict class of admissibility of the mesh is preserved by the REFINE$\_$RECURSIVE module. This result extends to the REFINE procedure.
{\bf e}gin{crl}\label{crl:refine}
Let $\mathcal{Q}$ be a strictly admissible mesh of class $m\ge 2$ and $\mathcal{M}$ the set of elements of $\mathcal{Q}$ marked for refinement. The call to $\mathcal{Q}^* =$ REFINE $(\mathcal{Q},\mathcal{M},m)$ terminates and returns a strictly admissible mesh $\mathcal{Q}^*\succeq\mathcal{Q}$ of class $m$.
\end{crl}
{\bf e}gin{proof}
The termination of the REFINE module is directly implied by Lemma ~\ref{lma:rr}.
Since every marked element $Q$ activates a call to REFINE$\_$RECURSIVE (${\cal Q},Q,m$), in order to prove that the final refined mesh ${\cal Q}^*$ preserves the satisfation of \eqref{eq:sameshes} and, consequently, the class $m$ of admissibility of ${\cal Q}$, it is sufficient to prove that this property holds after every recursive call. This is guaranteed by Proposition~\ref{prn:rr}.
\end{proof}
In view of the above corollary, we know that the refine mesh $\mathcal{Q}^*$ preserves the class of admissibility of the initial mesh $\mathcal{Q}$ and, consequently, Corollaries~\ref{crl:thbcor1} and \ref{crl:thbcor2} hold.
{\bf e}gin{exm}\label{exm02}
An example for the case $m=2$ {and the identity map} is shown Figure~\ref{fig:exm03}.
\end{exm}
{\bf e}gin{figure}[ht!]{\bf e}gin{center}
\mathfrak{h}space*{-.75cm}
\subfigure[initial mesh]{
\includegraphics[scale=0.25]{exm02mesh}}
\mathfrak{h}space*{-.75cm}
\subfigure[marked elements]{
\includegraphics[scale=0.25]{exm02marked}}
\caption{The admissible meshes in Fig.~\ref{fig:exm01} are generated by the call to $\mathcal{Q}^* =$ REFINE $(\mathcal{Q},\mathcal{M},2)$ to the mesh $\mathcal{Q}$ depicted in (a) with the marked set $\mathcal{M}$ of elements shown in (b).}
\label{fig:exm03}
\end{center}\end{figure}
\subsection{Contraction of the quasi-error and convergence}
Following the approach by \cite{cascon2008} (see also \cite{NochettoCIME}), we can prove
the contraction of the \emph{quasi-error}, defined as the contribution given by the energy error together with the estimator scaled by a positve factor $\gamma$:
\[
|||u-U|||_{\Omega}^2
+
\gamma\,\varepsilon^2_{\mathcal{Q}}(U,\mathcal{Q})
\]
where, we remind the energy norm is just $||| \cdot |||_\Omega = a(\cdot,\cdot)^{1/2}$ as defined in Section \ref{sec:solve}.
Note that neither the energy error $|||u-U|||_{\Omega}$, nor the estimator $\varepsilon_{\mathcal{Q}}(U,\mathcal{Q})$ considered alone may satisfy a similar contraction property between two consecutive steps of the adaptive procedure in the general setting \cite{NochettoCIME}.
In the case of adaptive finite elements, monotonicity of the error is proved only under additional assumptions, see e.g., \cite{morin2001} and \cite{morin2002}, and indeed, we will not study this property in the present paper.
{We present here only the statement of the contraction theorem. Since its proof follows the analogous one for finite elements with only minor changes, we postpone it to the Appendix.}
{\bf e}gin{thm}\label{thm:contraction}
Let $\theta\in (0,1]$ be the D\"orfler marking parameter introduced in
\eqref{eq:dm}, and let $\{\mathcal{Q}_k,\mathbb{S}(\mathcal{Q}_k),U_k\}_{k\ge0}$ be the sequence
of strictly admissible meshes, hierarchical spline spaces, and discrete solution computed
by the adaptive procedure for the model problem \eqref{eq:mp}.
Then, there exist $\gamma>0$ and $0<\alpha<1$, independent of $k$ such that for all $k>0$ it holds:
{\bf e}gin{equation}\label{eq:contraction}
|||u-U_{k+1}|||_{\Omega}^2+\gamma\,\varepsilon^2_{\mathcal{Q}_{k+1}}(U_{k+1},\mathcal{Q}_{k+1})\le
\alpha^2\left[|||u-U_k|||_{\Omega}^2
+\gamma\,\varepsilon^2_{\mathcal{Q}_k}(U_{k},\mathcal{Q}_k)\right].
\end{equation}
\end{thm}
An immediate consequence of this theorem, it the convergence of the error and of the estimator:
{\bf e}gin{crl}
\label{crl:conv}
Under the same assumption of Theorem \ref{thm:contraction}, both the error and the estimator converge geometrically to $0$. I.e.,
there exists { $\gamma>0$, $0<\alpha<1$} and a constant $M$ such that
\[ |||u-U_{k+1}|||_{\Omega}+\gamma\,\varepsilon_{\mathcal{Q}_{k+1}}(U_{k+1},\mathcal{Q}_{k+1}) \leq M \alpha^{k}, \]
where $M$ depends on the bounds \eqref{eq:5} and \eqref{eq:Fbound}, but not on $k$.
\end{crl}
{\bf e}gin{rmk}\label{rmk:mark}
The questions related to convergence for other marking strategies remain open and may require additional assumptions on the refinement module which should be further investigated. On the other hand, it seems very plausible that any other error estimator verifying the upper bound provided by Theorem ~\ref{thm:ub} could be used to replace the simple residual based error indicator proposed in this paper. As a side remark, we also note that the proof of Theorem~\ref{thm:contraction} does not require the lower bound presented in Theorem~\ref{thm:lb}.
\end{rmk}
\section{Closure}
\label{sec:closure}
A posteriori residual-type estimators for the error associated to the Galerkin approximation of a simple model problem have been presented, based on the truncated basis for hierarchical splines with respect to some class of admissible meshes and a certain multilevel refinement. In the case of the upper bound, two key properties of the basis are exploited together with standard inequalities of (adaptive) finite element methods. First, the partition of unity property and, second, the bound for the number of basis functions that assume non-zero value on any mesh element. In the case of the lower bound, classical arguments of finite element estimates can be directly applied. By taking into account the a posteriori upper bound previously computed (ESTIMATE) and a classical marking strategy (MARK), we introduce a specific refinement procedure (REFINE) to proof the contraction of the quasi-error and, consequently, the convergence of the adaptive isogeometric methods.
Corollary \ref{crl:conv} states convergence, but complexity is not analyzed at this stage. In other words, we have not proven any connection between the error and the number of degrees of freedom that are needed to compute the iterate $U^k$. As it is known from the AFEM theory, this can be studied by analyzing the complexity of REFINE. In order to do this, we need to understand how the refinement module controls the interplay between the number of refined elements $\# \mathcal{Q}_k-\# \mathcal{Q}_0$ introduced up to step $k$ (that influences the degrees of freedom added during the refinement) and the total number of marked elements. Among other things, an estimate of the type: there exists a certain constant $\Lambda_0>0$ such that
\[ \# \mathcal{Q}_k - \# \mathcal{Q}_0 \leq \Lambda_0 \sum_{j=0}^{k-1} \# \mathcal{M}_j\]
is in need. This kind of complexity estimate has been derived for adaptive finite elements in \cite{binev2004a,stevenson2007} for two- and three-dimensional problems, respectively. We will prove an analogous estimate for the adaptive isogeometric method here introduced, together with optimal convergence rates, in the companion paper \cite{buffa2015b}.
Suitable extensions of our adaptive framework may be investigated in order to consider less restrictive mesh configurations. For example, the use of analysis-suitable T-splines combined with semi--structured hierarchical construction has been recently investigated \cite{scott2014a,evans2015}. The possibility of extending the adaptivity theory here presented to this case is a challenging issue, but the wide modeling capabilities of T-splines encapsulated into the hierarchical model would provide a powerful refine module.\B
\section*{Acknowledgment}
This work was supported by the Gruppo Nazionale per il Calcolo Scientifico
(GNCS) of the Istituto Nazionale di Alta Matematica (INdAM)
and by the project DREAMS (MIUR ``Futuro in Ricerca'' RBFR13FBI3).
\appendix
\section*{Appendix}
\label{sec:appendix}
This appendix is devoted to the proof of Theorem \ref{thm:contraction}. Here we basically reproduce the proof of the statement as it was first proved for finite elements. In our presentation, we closely follow Chapter 5 of \cite{NochettoCIME}. We are not adding something new, and the value of this appendix is only show that the same arguments used for finite elements apply also to our case with very minor changes. Indeed, some of the proofs are made easier by the fact that our error indicator does not contain jump terms.
Before proving Theorem \ref{thm:contraction}, we need a few preparatory lemmas.
This first Lemma is nothing else then the Pytaghoras theorem. See Lemma 12 in \cite{NochettoCIME}.
{\bf e}gin{lma}\label{lma:pyt}
Let $\mathcal{Q}$ be an admissible mesh and $\mathcal{Q}^*$ be a refinement of $\mathcal{Q}$, i.e., $ {\cal Q^*}\succeq{\cal Q}$. Let $U$ and $U^*$ be the Galerkin solution of problem \eqref{eq:gal} on $\mathbb{S}_D(\mathcal{Q})$ and $\mathbb{S}_D(\mathcal{Q}^*)$, respectively. It holds:
{\bf e}gin{equation}\label{eq:pyt}
|||u-U^*|||_{\Omega}^2 = |||u-U|||_{\Omega}^2-|||U^*-U|||_{\Omega}^2.
\end{equation}
\end{lma}
{\bf e}gin{proof}
This is an immediate consequence of the Galerkin orthogonality.
\end{proof}
The next Lemma provides a measure of the reduction of the error indicator $\varepsilon_\mathcal{Q}(U)$ with respect to the mesh $\mathcal{Q}$,
see Lemma 13 in \cite{NochettoCIME}.
{\bf e}gin{lma}\label{lma:erred}
Let $\mathcal{Q}$ be a stricty admissible mesh \B, $\mathcal{M}$ be a set of marked elements and $\mathcal{Q}^*$ the corresponding refined mesh. i.e., $\mathcal{Q}^*= \mathrm{REFINE}(\mathcal{Q},\mathcal{M})$.
Then, for all $V\in \mathbb{S}_D(\mathcal{Q})$ it holds
$\forall V\in\mathbb{V}({\cal Q})$,
{\bf e}gin{equation}\label{eq:erred}
\varepsilon_{{\cal Q}^*}^2(U,{\cal Q}^*)\le
\varepsilon_{{\cal Q}}^2(U,{\cal Q})
-\lambda\varepsilon_{{\cal Q}}^2(U,\mathcal{M})
\end{equation}
where $0< \lambda <1$. \B
\end{lma}
{\bf e}gin{proof}
For each $Q\in \mathcal{M}$, we denote by $\mathcal{Q}^*(Q)$ the collection of elements of $\mathcal{Q}^*$ that are created by splitting $Q$. We know that, by construction, for all $Q^\star\in \mathcal{Q}^*(Q)$, it holds $h_{\mathfrak{h}at{Q}^*} \leq \displaystyle \frac{1}{2} \; h_{\mathfrak{h}at{Q}}$. Due to \eqref{eq:Fbound}, there exists then a constant $c({\bf F})$, $c({\bf F}) < 1$, independent of $Q$ such that
$h_{Q^\star} \leq c({\bf F}) h_Q$. \color{black}
If we adopt the notation \[\varepsilon^2_{\mathcal{Q}^*} (U,Q)= \sum_{Q^*\in \mathcal{Q}^*(Q)} h_{Q^*}^2 \| r(U) \|^2_{L^2(Q^*)},\]
it clearly holds:
\[ \forall Q\in \mathcal{M} \qquad \varepsilon_{\mathcal{Q}^*} (U,Q) \leq c({\bf F}) \color{black}\; \varepsilon_{\mathcal{Q}} (U,Q). \]
Moreover, since the mesh size does not increase for all elements in $\mathcal{Q}\setminus \mathcal{M}$, we have:
\[ \forall Q\in \mathcal{Q}\setminus \mathcal{M} \qquad \varepsilon_{\mathcal{Q}^*} (U,Q) \leq \varepsilon_{\mathcal{Q}} (U,Q).\]
Summing up for all $Q\in \mathcal{Q}$, we obtain:
\[ \varepsilon^2_{\mathcal{Q}^*} (U,Q^*) \leq \varepsilon^2_{\mathcal{Q}} (U,\mathcal{Q}\setminus\mathcal{M}) + c^2({\bf F}) \color{black}\; \varepsilon_{\mathcal{Q}} (U,\mathcal{M})\]
which implies then \eqref{eq:erred} with $\lambda = 1 - c^2({\bf F})$.\color{black}
\end{proof}
We turn now to the Lipschitz property of the error indicator $\varepsilon_\mathcal{Q}(U,Q)$, for any $Q$, with respect to the trial function $U$, see Lemma 14 in \cite{NochettoCIME}.
{\bf e}gin{lma}\label{lma:lip}
Let $\mathcal{Q}$ be an admissible mesh and $V\,,\ W\in \mathbb{S}_D(\mathcal{Q})$. There exists a $\Lambda>0$ such that the following holds for all $ Q\in{\cal Q}$
{\bf e}gin{equation}\label{eq:lip}
|\varepsilon_{\cal Q}(V,Q)-\varepsilon_{\cal Q}(W,Q)|\leq \Lambda
\eta_{\cal Q}(\textbf{A},Q)||\nabla(V-W)||_{L^2(Q)},
\end{equation}
where $\eta_{\cal Q}(\textbf{A},Q)= h_Q \| \mathrm{div}(\textbf{A})\|_{L^\infty(Q)} + \| \textbf{A}\|_{L^\infty(Q)}.$
\end{lma}
{\bf e}gin{proof}
By definition, we have:
\[ r(V) - r(W)= \mathrm{\textrm{div}}(\textbf{A}\nabla (V-W )) = \mathrm{\textrm{div}}(\textbf{A}) \cdot \nabla(V-W) + \textbf{A}: D^2(U-W), \]
where $D^2\cdot$ stands for the hessian matrix.
Using the inverse inequality $\| D^2(U-W) \|_{L^2(Q)} \les h_Q^{-1} \| \nabla(V-W)\|_{L^2(Q)}$, applying Cauchy-Schwarz and triangle inequality, we obtain:
{\bf e}gin{equation*}
{\bf e}gin{aligned}
|\varepsilon_{\cal Q}(V,Q)-\varepsilon_{\cal Q}(W,Q)| & \les h_Q \| r(V) - r(W) \|_{L^2(Q)} \\
& \les (h_Q \| \mathrm{div}(\textbf{A})\|_{L^\infty(Q)} + \| \textbf{A}\|_{L^\infty(Q)}) \| \nabla(V-W)\|_{L^2(Q)}
\end{aligned}
\end{equation*}
which ends the proof.
\end{proof}
We can combine the previous results to obtain the last preparatory Lemma. See Proposition 3 in \cite{NochettoCIME}.
{\bf e}gin{lma}\label{lma:esred}
Let $\mathcal{Q}$ be a strictly \color{black} admissible mesh, $\mathcal{M}$ be a set of marked elements and $\mathcal{Q}^*$ the corresponding refined mesh. i.e., $\mathcal{Q}^*= \mathrm{REFINE}(\mathcal{Q},\mathcal{M})$. There exists $\Lambda>0$ so that, $\forall V\in\mathbb{S}_D({\cal Q}), V^*\in\mathbb{S}_D^*({\cal Q}^*)$ and any $\delta>0$,
{\bf e}gin{equation}\label{eq:esred}
\varepsilon^2_{{\cal Q}^*}(V^*,{\cal Q}^*)\le
(1+\delta)\left[\varepsilon^2_{\cal Q}(V,{\cal Q})
-\lambda\varepsilon^2_{\cal Q}(V,{\cal M})\right]
+(1+\delta^{-1})\Lambda^2\eta^2_{\cal Q}(A,{\cal Q})
|||V^*-V|||_{\Omega}^2
\end{equation}
with $\eta^2_{\cal Q^*} = \sup_{Q^*\in{\cal Q}^*}\eta^2_{\cal Q^*}(\textbf{A},Q^*)$. \color{black}
\end{lma}
{\bf e}gin{proof}
Applying triangle inequality and Lemma \ref{lma:lip}, we have:
{\bf e}gin{equation*}
{\bf e}gin{aligned}
\varepsilon^2_{{\cal Q}^*}(V^*,{Q}^*) &\leq (1+\delta) \varepsilon^2_{\mathcal{Q}^*}(V,{Q^*}) + (1+\delta^{-1}) \, | \varepsilon_{{\cal Q}^*}(V^*,{Q}^*) - \varepsilon_{\mathcal{Q}^*}(V,{Q^*}) |^2 \\
& \leq (1+\delta) \varepsilon^2_{\mathcal{Q}^*}(V,{Q^*}) + \eta^2_{\cal Q^*}(A,Q^*) \Lambda \| \nabla(V-V^*)\|^2_{L^2(Q^*)}.
\end{aligned}
\end{equation*}
Summing over the elements, we obtain:
\[ \varepsilon^2_{{\cal Q}^*}(V^*,{\cal Q}^*) \leq (1+\delta) \varepsilon^2_{\mathcal{Q}^*}(V,{\mathcal{Q}^*}) + \eta^2_{\cal Q^*} \|(V-V^*)\|^2_{\mathbb{V}}. \]
The statement follows by applying Lemma \ref{lma:erred}.
\end{proof}
Finally we are now ready to prove Theorem \ref{thm:contraction}.
{\bf e}gin{proofof}{Theorem \ref{thm:contraction}}
By summing up the error orthogonality \eqref{eq:pyt} with the estimator reduction
\eqref{eq:esred} scaled by a constant $\gamma>0$, we obtain
{\bf e}gin{align*}
|||u-U_{k+1}|||_{\Omega}^2+\gamma\,\varepsilon^2_{\mathcal{Q}_{k+1}}(U_{k+1},\mathcal{Q}_{k+1})
\,\le\, &
|||u-U_{k}|||_{\Omega}^2 \\
\,+\, &
\left[\gamma\,(1+\delta^{-1})\Lambda_0-1\right] |||U_{k+1}-U_k|||_{\Omega}^2\\
\,+\, &
\gamma\,(1+\delta)\left[\varepsilon^2_{\mathcal{Q}_k}(U_k,\mathcal{Q}_k)
-\lambda\,\varepsilon^2_{\mathcal{Q}_k}(U_k,\mathcal{M}_k)\right],
\end{align*}
where we have used \eqref{eq:erred} with $\mathcal{Q}=\mathcal{Q}_k, \mathcal{Q}^*=\mathcal{Q}_{k+1}, V=U_k, V^*=U_{k+1}$, and we have set $\Lambda_0=
\Lambda \eta_{\mathcal{Q}_0}^2(A,\mathcal{Q}_0)\ge \Lambda \eta_{\mathcal{Q}_k}^2(A,\mathcal{Q}_k)$.
The choice $\gamma=1/[(1+\delta^{-1})\,\Lambda_0]$ together with the D\"orfler
marking property \eqref{eq:dm} leads to
{\bf e}gin{align*}
|||u-U_{k+1}|||_{\Omega}^2+\gamma\,\varepsilon^2_{\mathcal{Q}_{k+1}}(U_{k+1},\mathcal{Q}_{k+1})
\,\le\, &
|||u-U_{k}|||_{\Omega}^2 \\
\,+\, &
\gamma\,(1+\delta)\left[\varepsilon^2_{\mathcal{Q}_k}(U_k,\mathcal{Q}_k)
-\lambda\,\theta^2\,\varepsilon^2_{\mathcal{Q}_k}(U_k,\mathcal{Q}_k)\right]\\
\,=\, &
|||u-U_{k}|||_{\Omega}^2 +
\gamma\,(1+\delta)(1-\lambda\,\theta^2)\,\varepsilon^2_{\mathcal{Q}_k}(U_k,\mathcal{Q}_k).
\end{align*}
By choosing the parameter $\delta$ so that $(1+\delta)(1-\lambda\,\theta^2)=1-\lambda\,\theta^2/2$, the above inequality reduces to
\[
|||u-U_{k+1}|||_{\Omega}^2+\gamma\,\varepsilon^2_{\mathcal{Q}_{k+1}}(U_{k+1},\mathcal{Q}_{k+1})
\,\le\,
|||u-U_{k}|||_{\Omega}^2 +
\gamma\,\left(1-\frac{\lambda\,\theta^2}{2}\right)\,\varepsilon^2_{\mathcal{Q}_k}(U_k,\mathcal{Q}_k).
\]
The second term on the right-hand side may be written as
\[
-\gamma\,\frac{\lambda\,\theta^2}{4}\,\varepsilon^2_{\mathcal{Q}_k}(U_k,\mathcal{Q}_k)
+
\gamma\,\left(1-\frac{\lambda\,\theta^2}{4}\right)\,\varepsilon^2_{\mathcal{Q}_k}(U_k,\mathcal{Q}_k),
\]
so that taking into account the a posteriori upper bound \eqref{eq:ub} and the associated
constant $C_{\mathrm{up}}$, we obtain the inequality \eqref{eq:contraction}
with $\alpha=\max\left\{1-\gamma\frac{\lambda\,\theta^2}{4\,C_{\mathrm{up}}},1-\frac{\lambda\,\theta^2}{4}\right\}<1$.
\end{proofof}
{\bf i}bliographystyle{plain}
{\bf i}bliography{biblio}
\end{document} |
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